LMFDB - Embedded newform 75.9.d.c.74.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 943 x^{18} + 318815 x^{16} + 48938090 x^{14} + 3842259173 x^{12} + 159675554657 x^{10} + \cdots + 336685801 $ x^20 + 943x^18 + 318815x^16 + 48938090x^14 + 3842259173x^12 + 159675554657x^10 + 3390679484573x^8 + 32981662033730x^6 + 124592584990175x^4 + 133865237486743*x^2 + 336685801
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{22}\cdot 5^{32} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.1
Root			$17.2930i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.c.74.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.5009 q^{2} +(69.5124 - 41.5815i) q^{3} +614.301 q^{4} +(-2050.68 + 1226.69i) q^{6} +3174.27i q^{7} -10570.2 q^{8} +(3102.96 - 5780.86i) q^{9} +O(q^{10})$	\(q-29.5009 q^{2} +(69.5124 - 41.5815i) q^{3} +614.301 q^{4} +(-2050.68 + 1226.69i) q^{6} +3174.27i q^{7} -10570.2 q^{8} +(3102.96 - 5780.86i) q^{9} -12996.3i q^{11} +(42701.6 - 25543.6i) q^{12} -8759.90i q^{13} -93643.7i q^{14} +154569. q^{16} -108656. q^{17} +(-91540.0 + 170540. i) q^{18} +78696.7 q^{19} +(131991. + 220651. i) q^{21} +383403. i q^{22} +43589.4 q^{23} +(-734760. + 439524. i) q^{24} +258425. i q^{26} +(-24682.6 - 530868. i) q^{27} +1.94996e6i q^{28} -183211. i q^{29} +780664. q^{31} -1.85394e6 q^{32} +(-540407. - 903407. i) q^{33} +3.20544e6 q^{34} +(1.90615e6 - 3.55119e6i) q^{36} +2.20452e6i q^{37} -2.32162e6 q^{38} +(-364250. - 608922. i) q^{39} -3.04685e6i q^{41} +(-3.89384e6 - 6.50940e6i) q^{42} -4.84516e6i q^{43} -7.98367e6i q^{44} -1.28593e6 q^{46} +3.51057e6 q^{47} +(1.07444e7 - 6.42720e6i) q^{48} -4.31118e6 q^{49} +(-7.55293e6 + 4.51807e6i) q^{51} -5.38121e6i q^{52} +8.64762e6 q^{53} +(728158. + 1.56611e7i) q^{54} -3.35526e7i q^{56} +(5.47040e6 - 3.27232e6i) q^{57} +5.40489e6i q^{58} -5.16824e6i q^{59} +5.78123e6 q^{61} -2.30303e7 q^{62} +(1.83500e7 + 9.84963e6i) q^{63} +1.51233e7 q^{64} +(1.59425e7 + 2.66513e7i) q^{66} -3.67916e7i q^{67} -6.67474e7 q^{68} +(3.03001e6 - 1.81251e6i) q^{69} -971113. i q^{71} +(-3.27989e7 + 6.11048e7i) q^{72} -9.46941e6i q^{73} -6.50352e7i q^{74} +4.83434e7 q^{76} +4.12539e7 q^{77} +(1.07457e7 + 1.79637e7i) q^{78} -5.63493e7 q^{79} +(-2.37900e7 - 3.58756e7i) q^{81} +8.98849e7i q^{82} -5.88128e7 q^{83} +(8.10821e7 + 1.35546e8i) q^{84} +1.42936e8i q^{86} +(-7.61819e6 - 1.27355e7i) q^{87} +1.37374e8i q^{88} -1.92128e7i q^{89} +2.78063e7 q^{91} +2.67770e7 q^{92} +(5.42658e7 - 3.24612e7i) q^{93} -1.03565e8 q^{94} +(-1.28872e8 + 7.70897e7i) q^{96} -1.42760e8i q^{97} +1.27184e8 q^{98} +(-7.51301e7 - 4.03271e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) $ 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9	$ 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) $ 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9 + 560772 * q^16 + 463032 * q^19 + 579144 * q^21 - 2272668 * q^24 + 1763240 * q^31 + 2222552 * q^34 - 1337324 * q^36 - 3653584 * q^39 - 50849208 * q^46 - 18708428 * q^49 - 55465384 * q^51 + 15959596 * q^54 + 44834040 * q^61 + 45870004 * q^64 - 54839600 * q^66 - 67125264 * q^69 + 397844872 * q^76 - 324621848 * q^79 - 187150780 * q^81 + 394693536 * q^84 + 888576928 * q^91 + 184100072 * q^94 - 721614812 * q^96 + 67930400 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−29.5009			−1.84380			−0.921902	−	0.387423i	$-0.873365\pi$
$2$	−29.5009			−1.84380			−0.921902	+	0.387423i	$0.873365\pi$
$3$	69.5124	−	41.5815i	0.858178	−	0.513352i
$3$	69.5124	−	41.5815i	0.858178	−	0.513352i
$4$	614.301			2.39961
$5$		0			0
$5$		0			0
$6$	−2050.68	+	1226.69i	−1.58231	+	0.946520i
$7$			3174.27i			1.32206i	0.750359	+	0.661031i	$0.229881\pi$
$7$			3174.27i			1.32206i	−0.750359	+	0.661031i	$0.770119\pi$
$8$	−10570.2			−2.58061
$9$	3102.96	−	5780.86i	0.472940	−	0.881095i
$10$		0			0
$11$		−	12996.3i		−	0.887668i	−0.896109	−	0.443834i	$-0.853618\pi$
$11$		−	12996.3i		−	0.887668i	0.896109	−	0.443834i	$-0.146382\pi$
$12$	42701.6	−	25543.6i	2.05930	−	1.23185i
$13$		−	8759.90i		−	0.306708i	−0.988171	−	0.153354i	$-0.950992\pi$
$13$		−	8759.90i		−	0.306708i	0.988171	−	0.153354i	$-0.0490075\pi$
$14$		−	93643.7i		−	2.43762i
$15$		0			0
$16$	154569.			2.35853
$17$	−108656.			−1.30094			−0.650470	−	0.759532i	$-0.725428\pi$
$17$	−108656.			−1.30094			−0.650470	+	0.759532i	$0.725428\pi$
$18$	−91540.0	+	170540.i	−0.872009	+	1.62457i
$19$	78696.7			0.603868			0.301934	−	0.953329i	$-0.402368\pi$
$19$	78696.7			0.603868			0.301934	+	0.953329i	$0.402368\pi$
$20$		0			0
$21$	131991.	+	220651.i	0.678683	+	1.13456i
$22$			383403.i			1.63669i
$23$	43589.4			0.155765			0.0778825	−	0.996963i	$-0.475184\pi$
$23$	43589.4			0.155765			0.0778825	+	0.996963i	$0.475184\pi$
$24$	−734760.	+	439524.i	−2.21463	+	1.32476i
$25$		0			0
$26$			258425.i			0.565510i
$27$	−24682.6	−	530868.i	−0.0464447	−	0.998921i
$28$			1.94996e6i			3.17244i
$29$		−	183211.i		−	0.259036i	−0.991577	−	0.129518i	$-0.958657\pi$
$29$		−	183211.i		−	0.259036i	0.991577	−	0.129518i	$-0.0413430\pi$
$30$		0			0
$31$	780664.			0.845312			0.422656	−	0.906290i	$-0.361098\pi$
$31$	780664.			0.845312			0.422656	+	0.906290i	$0.361098\pi$
$32$	−1.85394e6			−1.76806
$33$	−540407.	−	903407.i	−0.455686	−	0.761777i
$34$	3.20544e6			2.39868
$35$		0			0
$36$	1.90615e6	−	3.55119e6i	1.13487	−	2.11429i
$37$			2.20452e6i			1.17627i	0.808763	+	0.588134i	$0.200137\pi$
$37$			2.20452e6i			1.17627i	−0.808763	+	0.588134i	$0.799863\pi$
$38$	−2.32162e6			−1.11341
$39$	−364250.	−	608922.i	−0.157449	−	0.263210i
$40$		0			0
$41$		−	3.04685e6i		−	1.07824i	−0.842228	−	0.539121i	$-0.818757\pi$
$41$		−	3.04685e6i		−	1.07824i	0.842228	−	0.539121i	$-0.181243\pi$
$42$	−3.89384e6	−	6.50940e6i	−1.25136	−	2.09191i
$43$		−	4.84516e6i		−	1.41721i	−0.705606	−	0.708605i	$-0.749325\pi$
$43$		−	4.84516e6i		−	1.41721i	0.705606	−	0.708605i	$-0.250675\pi$
$44$		−	7.98367e6i		−	2.13006i
$45$		0			0
$46$	−1.28593e6			−0.287200
$47$	3.51057e6			0.719426			0.359713	−	0.933063i	$-0.382875\pi$
$47$	3.51057e6			0.719426			0.359713	+	0.933063i	$0.382875\pi$
$48$	1.07444e7	−	6.42720e6i	2.02404	−	1.21076i
$49$	−4.31118e6			−0.747846
$50$		0			0
$51$	−7.55293e6	+	4.51807e6i	−1.11644	+	0.667840i
$52$		−	5.38121e6i		−	0.735982i
$53$	8.64762e6			1.09596			0.547978	−	0.836493i	$-0.315398\pi$
$53$	8.64762e6			1.09596			0.547978	+	0.836493i	$0.315398\pi$
$54$	728158.	+	1.56611e7i	0.0856349	+	1.84181i
$55$		0			0
$56$		−	3.35526e7i		−	3.41173i
$57$	5.47040e6	−	3.27232e6i	0.518226	−	0.309997i
$58$			5.40489e6i			0.477611i
$59$		−	5.16824e6i		−	0.426515i	−0.976996	−	0.213257i	$-0.931593\pi$
$59$		−	5.16824e6i		−	0.426515i	0.976996	−	0.213257i	$-0.0684073\pi$
$60$		0			0
$61$	5.78123e6			0.417543			0.208771	−	0.977964i	$-0.433054\pi$
$61$	5.78123e6			0.417543			0.208771	+	0.977964i	$0.433054\pi$
$62$	−2.30303e7			−1.55859
$63$	1.83500e7	+	9.84963e6i	1.16486	+	0.625256i
$64$	1.51233e7			0.901419
$65$		0			0
$66$	1.59425e7	+	2.66513e7i	0.840195	+	1.40457i
$67$		−	3.67916e7i		−	1.82578i	−0.408202	−	0.912891i	$-0.633844\pi$
$67$		−	3.67916e7i		−	1.82578i	0.408202	−	0.912891i	$-0.366156\pi$
$68$	−6.67474e7			−3.12175
$69$	3.03001e6	−	1.81251e6i	0.133674	−	0.0799622i
$70$		0			0
$71$		−	971113.i		−	0.0382152i	−0.999817	−	0.0191076i	$-0.993917\pi$
$71$		−	971113.i		−	0.0382152i	0.999817	−	0.0191076i	$-0.00608251\pi$
$72$	−3.27989e7	+	6.11048e7i	−1.22048	+	2.27376i
$73$		−	9.46941e6i		−	0.333451i	−0.986003	−	0.166725i	$-0.946681\pi$
$73$		−	9.46941e6i		−	0.333451i	0.986003	−	0.166725i	$-0.0533193\pi$
$74$		−	6.50352e7i		−	2.16881i
$75$		0			0
$76$	4.83434e7			1.44905
$77$	4.12539e7			1.17355
$78$	1.07457e7	+	1.79637e7i	0.290306	+	0.485309i
$79$	−5.63493e7			−1.44671			−0.723353	−	0.690479i	$-0.757400\pi$
$79$	−5.63493e7			−1.44671			−0.723353	+	0.690479i	$0.757400\pi$
$80$		0			0
$81$	−2.37900e7	−	3.58756e7i	−0.552656	−	0.833410i
$82$			8.98849e7i			1.98807i
$83$	−5.88128e7			−1.23925			−0.619626	−	0.784897i	$-0.712716\pi$
$83$	−5.88128e7			−1.23925			−0.619626	+	0.784897i	$0.712716\pi$
$84$	8.10821e7	+	1.35546e8i	1.62858	+	2.72252i
$85$		0			0
$86$			1.42936e8i			2.61306i
$87$	−7.61819e6	−	1.27355e7i	−0.132977	−	0.222299i
$88$			1.37374e8i			2.29073i
$89$		−	1.92128e7i		−	0.306218i	−0.988209	−	0.153109i	$-0.951071\pi$
$89$		−	1.92128e7i		−	0.306218i	0.988209	−	0.153109i	$-0.0489286\pi$
$90$		0			0
$91$	2.78063e7			0.405487
$92$	2.67770e7			0.373776
$93$	5.42658e7	−	3.24612e7i	0.725429	−	0.433943i
$94$	−1.03565e8			−1.32648
$95$		0			0
$96$	−1.28872e8	+	7.70897e7i	−1.51731	+	0.907635i
$97$		−	1.42760e8i		−	1.61257i	−0.591526	−	0.806286i	$-0.701474\pi$
$97$		−	1.42760e8i		−	1.61257i	0.591526	−	0.806286i	$-0.298526\pi$
$98$	1.27184e8			1.37888
$99$	−7.51301e7	−	4.03271e7i	−0.782119	−	0.419814i
$100$		0			0
$101$		−	8.34775e7i		−	0.802202i	−0.916034	−	0.401101i	$-0.868628\pi$
$101$		−	8.34775e7i		−	0.802202i	0.916034	−	0.401101i	$-0.131372\pi$
$102$	2.22818e8	−	1.33287e8i	2.05849	−	1.23137i
$103$		−	1.29545e8i		−	1.15099i	−0.817806	−	0.575494i	$-0.804810\pi$
$103$		−	1.29545e8i		−	1.15099i	0.817806	−	0.575494i	$-0.195190\pi$
$104$			9.25938e7i			0.791496i
$105$		0			0
$106$	−2.55112e8			−2.02073
$107$	1.02862e8			0.784731			0.392365	−	0.919809i	$-0.371657\pi$
$107$	1.02862e8			0.784731			0.392365	+	0.919809i	$0.371657\pi$
$108$	−1.51626e7	−	3.26112e8i	−0.111449	−	2.39702i
$109$	1.82090e8			1.28997			0.644984	−	0.764196i	$-0.276864\pi$
$109$	1.82090e8			1.28997			0.644984	+	0.764196i	$0.276864\pi$
$110$		0			0
$111$	9.16671e7	+	1.53241e8i	0.603840	+	1.00945i
$112$			4.90643e8i			3.11812i
$113$	−8.75507e6			−0.0536965			−0.0268483	−	0.999640i	$-0.508547\pi$
$113$	−8.75507e6			−0.0536965			−0.0268483	+	0.999640i	$0.508547\pi$
$114$	−1.61381e8	+	9.65364e7i	−0.955508	+	0.571573i
$115$		0			0
$116$		−	1.12547e8i		−	0.621586i
$117$	−5.06398e7	−	2.71816e7i	−0.270239	−	0.145055i
$118$			1.52467e8i			0.786410i
$119$		−	3.44903e8i		−	1.71992i
$120$		0			0
$121$	4.54540e7			0.212046
$122$	−1.70551e8			−0.769867
$123$	−1.26693e8	−	2.11794e8i	−0.553518	−	0.925324i
$124$	4.79563e8			2.02842
$125$		0			0
$126$	−5.41341e8	−	2.90573e8i	−2.14778	−	1.15285i
$127$		−	2.17231e7i		−	0.0835040i	−0.999128	−	0.0417520i	$-0.986706\pi$
$127$		−	2.17231e7i		−	0.0835040i	0.999128	−	0.0417520i	$-0.0132939\pi$
$128$	2.84587e7			0.106017
$129$	−2.01469e8	−	3.36799e8i	−0.727527	−	1.21622i
$130$		0			0
$131$			1.72425e8i			0.585485i	0.956191	+	0.292743i	$0.0945679\pi$
$131$			1.72425e8i			0.585485i	−0.956191	+	0.292743i	$0.905432\pi$
$132$	−3.31973e8	−	5.54964e8i	−1.09347	−	1.82797i
$133$			2.49804e8i			0.798350i
$134$			1.08538e9i			3.36639i
$135$		0			0
$136$	1.14851e9			3.35722
$137$	2.76036e8			0.783580			0.391790	−	0.920055i	$-0.371856\pi$
$137$	2.76036e8			0.783580			0.391790	+	0.920055i	$0.371856\pi$
$138$	−8.93879e7	+	5.34707e7i	−0.246469	+	0.147435i
$139$	−4.04596e8			−1.08383			−0.541916	−	0.840433i	$-0.682301\pi$
$139$	−4.04596e8			−1.08383			−0.541916	+	0.840433i	$0.682301\pi$
$140$		0			0
$141$	2.44028e8	−	1.45975e8i	0.617396	−	0.369319i
$142$			2.86487e7i			0.0704614i
$143$	−1.13847e8			−0.272255
$144$	4.79620e8	−	8.93540e8i	1.11544	−	2.07809i
$145$		0			0
$146$			2.79356e8i			0.614818i
$147$	−2.99681e8	+	1.79265e8i	−0.641785	+	0.383908i
$148$			1.35424e9i			2.82259i
$149$			8.41277e8i			1.70685i	0.521219	+	0.853423i	$0.325477\pi$
$149$			8.41277e8i			1.70685i	−0.521219	+	0.853423i	$0.674523\pi$
$150$		0			0
$151$	−1.91529e8			−0.368405			−0.184203	−	0.982888i	$-0.558970\pi$
$151$	−1.91529e8			−0.368405			−0.184203	+	0.982888i	$0.558970\pi$
$152$	−8.31839e8			−1.55835
$153$	−3.37154e8	+	6.28124e8i	−0.615266	+	1.14625i
$154$	−1.21703e9			−2.16380
$155$		0			0
$156$	−2.23759e8	−	3.74061e8i	−0.377817	−	0.631603i
$157$			2.37178e8i			0.390370i	0.980766	+	0.195185i	$0.0625307\pi$
$157$			2.37178e8i			0.390370i	−0.980766	+	0.195185i	$0.937469\pi$
$158$	1.66235e9			2.66744
$159$	6.01117e8	−	3.59581e8i	0.940525	−	0.562611i
$160$		0			0
$161$			1.38365e8i			0.205931i
$162$	7.01826e8	+	1.05836e9i	1.01899	+	1.53664i
$163$		−	6.29506e8i		−	0.891763i	−0.895092	−	0.445881i	$-0.852890\pi$
$163$		−	6.29506e8i		−	0.891763i	0.895092	−	0.445881i	$-0.147110\pi$
$164$		−	1.87169e9i		−	2.58736i
$165$		0			0
$166$	1.73503e9			2.28494
$167$	3.79259e7			0.0487607			0.0243803	−	0.999703i	$-0.492239\pi$
$167$	3.79259e7			0.0487607			0.0243803	+	0.999703i	$0.492239\pi$
$168$	−1.39517e9	−	2.33233e9i	−1.75142	−	2.92787i
$169$	7.38995e8			0.905930
$170$		0			0
$171$	2.44193e8	−	4.54935e8i	0.285593	−	0.532065i
$172$		−	2.97638e9i		−	3.40075i
$173$	−1.49961e9			−1.67415			−0.837077	−	0.547085i	$-0.815737\pi$
$173$	−1.49961e9			−1.67415			−0.837077	+	0.547085i	$0.815737\pi$
$174$	2.24743e8	+	3.75707e8i	0.245183	+	0.409876i
$175$		0			0
$176$		−	2.00883e9i		−	2.09359i
$177$	−2.14903e8	−	3.59257e8i	−0.218952	−	0.366026i
$178$			5.66795e8i			0.564606i
$179$		−	7.72267e8i		−	0.752238i	−0.926571	−	0.376119i	$-0.877258\pi$
$179$		−	7.72267e8i		−	0.752238i	0.926571	−	0.376119i	$-0.122742\pi$
$180$		0			0
$181$	−7.49920e8			−0.698716			−0.349358	−	0.936989i	$-0.613600\pi$
$181$	−7.49920e8			−0.698716			−0.349358	+	0.936989i	$0.613600\pi$
$182$	−8.20309e8			−0.747639
$183$	4.01867e8	−	2.40392e8i	0.358326	−	0.214346i
$184$	−4.60749e8			−0.401969
$185$		0			0
$186$	−1.60089e9	+	9.57632e8i	−1.33755	+	0.800105i
$187$			1.41213e9i			1.15480i
$188$	2.15655e9			1.72635
$189$	1.68512e9	−	7.83492e7i	1.32063	−	0.0614027i
$190$		0			0
$191$		−	8.93822e7i		−	0.0671611i	−0.999436	−	0.0335805i	$-0.989309\pi$
$191$		−	8.93822e7i		−	0.0671611i	0.999436	−	0.0335805i	$-0.0106910\pi$
$192$	1.05126e9	−	6.28849e8i	0.773578	−	0.462745i
$193$		−	8.16090e8i		−	0.588178i	−0.955778	−	0.294089i	$-0.904984\pi$
$193$		−	8.16090e8i		−	0.588178i	0.955778	−	0.294089i	$-0.0950163\pi$
$194$			4.21154e9i			2.97327i
$195$		0			0
$196$	−2.64837e9			−1.79454
$197$	−1.32278e9			−0.878256			−0.439128	−	0.898424i	$-0.644713\pi$
$197$	−1.32278e9			−0.878256			−0.439128	+	0.898424i	$0.644713\pi$
$198$	2.21640e9	+	1.18968e9i	1.44207	+	0.774054i
$199$	1.92328e9			1.22639			0.613197	−	0.789930i	$-0.289883\pi$
$199$	1.92328e9			1.22639			0.613197	+	0.789930i	$0.289883\pi$
$200$		0			0
$201$	−1.52985e9	−	2.55747e9i	−0.937269	−	1.56685i
$202$			2.46266e9i			1.47910i
$203$	5.81562e8			0.342461
$204$	−4.63977e9	+	2.77546e9i	−2.67902	+	1.60256i
$205$		0			0
$206$			3.82168e9i			2.12220i
$207$	1.35256e8	−	2.51985e8i	0.0736675	−	0.137244i
$208$		−	1.35401e9i		−	0.723381i
$209$		−	1.02277e9i		−	0.536034i
$210$		0			0
$211$	2.03655e9			1.02746			0.513729	−	0.857952i	$-0.328264\pi$
$211$	2.03655e9			1.02746			0.513729	+	0.857952i	$0.328264\pi$
$212$	5.31224e9			2.62987
$213$	−4.03803e7	−	6.75045e7i	−0.0196179	−	0.0327955i
$214$	−3.03452e9			−1.44689
$215$		0			0
$216$	2.60900e8	+	5.61137e9i	0.119856	+	2.57783i
$217$			2.47804e9i			1.11755i
$218$	−5.37180e9			−2.37845
$219$	−3.93752e8	−	6.58242e8i	−0.171177	−	0.286160i
$220$		0			0
$221$			9.51814e8i			0.399009i
$222$	−2.70426e9	−	4.52075e9i	−1.11336	−	1.86122i
$223$			4.63258e8i			0.187328i	0.995604	+	0.0936642i	$0.0298580\pi$
$223$			4.63258e8i			0.187328i	−0.995604	+	0.0936642i	$0.970142\pi$
$224$		−	5.88491e9i		−	2.33748i
$225$		0			0
$226$	2.58282e8			0.0990059
$227$	4.20915e9			1.58522			0.792612	−	0.609726i	$-0.208721\pi$
$227$	4.20915e9			1.58522			0.792612	+	0.609726i	$0.208721\pi$
$228$	3.36047e9	−	2.01019e9i	1.24354	−	0.743872i
$229$	4.11215e9			1.49530			0.747648	−	0.664095i	$-0.231183\pi$
$229$	4.11215e9			1.49530			0.747648	+	0.664095i	$0.231183\pi$
$230$		0			0
$231$	2.86766e9	−	1.71540e9i	1.00712	−	0.602445i
$232$			1.93658e9i			0.668471i
$233$	−2.87004e9			−0.973788			−0.486894	−	0.873461i	$-0.661870\pi$
$233$	−2.87004e9			−0.973788			−0.486894	+	0.873461i	$0.661870\pi$
$234$	1.49392e9	+	8.01881e8i	0.498268	+	0.267452i
$235$		0			0
$236$		−	3.17485e9i		−	1.02347i
$237$	−3.91698e9	+	2.34309e9i	−1.24153	+	0.742669i
$238$			1.01749e10i			3.17120i
$239$		−	2.06189e9i		−	0.631937i	−0.948770	−	0.315968i	$-0.897671\pi$
$239$		−	2.06189e9i		−	0.631937i	0.948770	−	0.315968i	$-0.102329\pi$
$240$		0			0
$241$	1.87727e8			0.0556491			0.0278246	−	0.999613i	$-0.491142\pi$
$241$	1.87727e8			0.0556491			0.0278246	+	0.999613i	$0.491142\pi$
$242$	−1.34093e9			−0.390971
$243$	−3.14546e9	−	1.50457e9i	−0.902109	−	0.431507i
$244$	3.55142e9			1.00194
$245$		0			0
$246$	3.73755e9	+	6.24812e9i	1.02058	+	1.70612i
$247$		−	6.89375e8i		−	0.185211i
$248$	−8.25177e9			−2.18142
$249$	−4.08822e9	+	2.44552e9i	−1.06350	+	0.636172i
$250$		0			0
$251$			1.87286e9i			0.471856i	0.971771	+	0.235928i	$0.0758130\pi$
$251$			1.87286e9i			0.471856i	−0.971771	+	0.235928i	$0.924187\pi$
$252$	1.12724e10	+	6.05064e9i	2.79522	+	1.50037i
$253$		−	5.66503e8i		−	0.138268i
$254$			6.40851e8i			0.153965i
$255$		0			0
$256$	−4.71112e9			−1.09689
$257$	−2.62531e9			−0.601795			−0.300898	−	0.953656i	$-0.597286\pi$
$257$	−2.62531e9			−0.601795			−0.300898	+	0.953656i	$0.597286\pi$
$258$	5.94351e9	+	9.93585e9i	1.34142	+	2.24247i
$259$	−6.99773e9			−1.55510
$260$		0			0
$261$	−1.05912e9	−	5.68497e8i	−0.228235	−	0.122508i
$262$		−	5.08670e9i		−	1.07952i
$263$	7.30592e9			1.52705			0.763523	−	0.645781i	$-0.223468\pi$
$263$	7.30592e9			1.52705			0.763523	+	0.645781i	$0.223468\pi$
$264$	5.71221e9	+	9.54919e9i	1.17595	+	1.96585i
$265$		0			0
$266$		−	7.36945e9i		−	1.47200i
$267$	−7.98898e8	−	1.33553e9i	−0.157198	−	0.262790i
$268$		−	2.26011e10i		−	4.38117i
$269$			1.05642e9i			0.201757i	0.994899	+	0.100878i	$0.0321653\pi$
$269$			1.05642e9i			0.201757i	−0.994899	+	0.100878i	$0.967835\pi$
$270$		0			0
$271$	−5.55499e9			−1.02993			−0.514963	−	0.857212i	$-0.672194\pi$
$271$	−5.55499e9			−1.02993			−0.514963	+	0.857212i	$0.672194\pi$
$272$	−1.67948e10			−3.06831
$273$	1.93288e9	−	1.15623e9i	0.347980	−	0.208158i
$274$	−8.14330e9			−1.44477
$275$		0			0
$276$	1.86134e9	−	1.11343e9i	0.320766	−	0.191878i
$277$		−	5.81689e9i		−	0.988034i	−0.869452	−	0.494017i	$-0.835528\pi$
$277$		−	5.81689e9i		−	0.988034i	0.869452	−	0.494017i	$-0.164472\pi$
$278$	1.19359e10			1.99837
$279$	2.42237e9	−	4.51291e9i	0.399782	−	0.744800i
$280$		0			0
$281$			7.93059e9i			1.27198i	0.771698	+	0.635990i	$0.219408\pi$
$281$			7.93059e9i			1.27198i	−0.771698	+	0.635990i	$0.780592\pi$
$282$	−7.19905e9	+	4.30638e9i	−1.13836	+	0.680951i
$283$			8.19763e9i			1.27803i	0.769193	+	0.639017i	$0.220659\pi$
$283$			8.19763e9i			1.27803i	−0.769193	+	0.639017i	$0.779341\pi$
$284$		−	5.96556e8i		−	0.0917018i
$285$		0			0
$286$	3.35857e9			0.501985
$287$	9.67154e9			1.42550
$288$	−5.75271e9	+	1.07174e10i	−0.836185	+	1.55783i
$289$	4.83032e9			0.692444
$290$		0			0
$291$	−5.93617e9	−	9.92359e9i	−0.827817	−	1.38387i
$292$		−	5.81707e9i		−	0.800153i
$293$	−1.02001e10			−1.38399			−0.691995	−	0.721902i	$-0.743268\pi$
$293$	−1.02001e10			−1.38399			−0.691995	+	0.721902i	$0.743268\pi$
$294$	8.84085e9	−	5.28849e9i	1.18333	−	0.707851i
$295$		0			0
$296$		−	2.33022e10i		−	3.03549i
$297$	−6.89934e9	+	3.20784e8i	−0.886710	+	0.0412274i
$298$		−	2.48184e10i		−	3.14709i
$299$		−	3.81839e8i		−	0.0477744i
$300$		0			0
$301$	1.53798e10			1.87364
$302$	5.65026e9			0.679267
$303$	−3.47112e9	−	5.80273e9i	−0.411812	−	0.688433i
$304$	1.21640e10			1.42424
$305$		0			0
$306$	9.94635e9	−	1.85302e10i	1.13443	−	2.11346i
$307$		−	1.59327e10i		−	1.79365i	−0.442389	−	0.896823i	$-0.645869\pi$
$307$		−	1.59327e10i		−	1.79365i	0.442389	−	0.896823i	$-0.354131\pi$
$308$	2.53423e10			2.81607
$309$	−5.38667e9	−	9.00498e9i	−0.590862	−	0.987754i
$310$		0			0
$311$			1.47120e10i			1.57265i	0.617816	+	0.786323i	$0.288018\pi$
$311$			1.47120e10i			1.57265i	−0.617816	+	0.786323i	$0.711982\pi$
$312$	3.85019e9	+	6.43642e9i	0.406316	+	0.679244i
$313$		−	1.09163e10i		−	1.13736i	−0.822560	−	0.568678i	$-0.807455\pi$
$313$		−	1.09163e10i		−	1.13736i	0.822560	−	0.568678i	$-0.192545\pi$
$314$		−	6.99696e9i		−	0.719765i
$315$		0			0
$316$	−3.46154e10			−3.47153
$317$	1.31319e10			1.30044			0.650220	−	0.759746i	$-0.274677\pi$
$317$	1.31319e10			1.30044			0.650220	+	0.759746i	$0.274677\pi$
$318$	−1.77335e10	+	1.06079e10i	−1.73414	+	1.03734i
$319$	−2.38108e9			−0.229938
$320$		0			0
$321$	7.15020e9	−	4.27716e9i	0.673439	−	0.402843i
$322$		−	4.08188e9i		−	0.379696i
$323$	−8.55085e9			−0.785596
$324$	−1.46142e10	−	2.20384e10i	−1.32616	−	1.99986i
$325$		0			0
$326$			1.85710e10i			1.64424i
$327$	1.26575e10	−	7.57156e9i	1.10702	−	0.662208i
$328$			3.22058e10i			2.78253i
$329$			1.11435e10i			0.951126i
$330$		0			0
$331$	−5.11274e9			−0.425934			−0.212967	−	0.977059i	$-0.568313\pi$
$331$	−5.11274e9			−0.425934			−0.212967	+	0.977059i	$0.568313\pi$
$332$	−3.61288e10			−2.97372
$333$	1.27440e10	+	6.84053e9i	1.03640	+	0.556305i
$334$	−1.11885e9			−0.0899052
$335$		0			0
$336$	2.04017e10	+	3.41058e10i	1.60069	+	2.67591i
$337$			2.37856e10i			1.84414i	0.387024	+	0.922070i	$0.373503\pi$
$337$			2.37856e10i			1.84414i	−0.387024	+	0.922070i	$0.626497\pi$
$338$	−2.18010e10			−1.67036
$339$	−6.08587e8	+	3.64049e8i	−0.0460812	+	0.0275652i
$340$		0			0
$341$		−	1.01458e10i		−	0.750356i
$342$	−7.20389e9	+	1.34210e10i	−0.526578	+	0.981023i
$343$			4.61417e9i			0.333363i
$344$			5.12142e10i			3.65727i
$345$		0			0
$346$	4.42399e10			3.08681
$347$	−3.67752e9			−0.253652			−0.126826	−	0.991925i	$-0.540479\pi$
$347$	−3.67752e9			−0.253652			−0.126826	+	0.991925i	$0.540479\pi$
$348$	−4.67986e9	−	7.82340e9i	−0.319092	−	0.533432i
$349$	−8.39415e9			−0.565816			−0.282908	−	0.959147i	$-0.591299\pi$
$349$	−8.39415e9			−0.565816			−0.282908	+	0.959147i	$0.591299\pi$
$350$		0			0
$351$	−4.65034e9	+	2.16217e8i	−0.306377	+	0.0142450i
$352$			2.40945e10i			1.56945i
$353$	2.53478e10			1.63246			0.816229	−	0.577729i	$-0.196061\pi$
$353$	2.53478e10			1.63246			0.816229	+	0.577729i	$0.196061\pi$
$354$	6.33982e9	+	1.05984e10i	0.403705	+	0.674880i
$355$		0			0
$356$		−	1.18025e10i		−	0.734805i
$357$	−1.43416e10	−	2.39750e10i	−0.882925	−	1.47600i
$358$			2.27825e10i			1.38698i
$359$			3.29745e9i			0.198518i	0.995062	+	0.0992590i	$0.0316472\pi$
$359$			3.29745e9i			0.198518i	−0.995062	+	0.0992590i	$0.968353\pi$
$360$		0			0
$361$	−1.07904e10			−0.635344
$362$	2.21233e10			1.28829
$363$	3.15962e9	−	1.89004e9i	0.181973	−	0.108854i
$364$	1.70814e10			0.973013
$365$		0			0
$366$	−1.18554e10	+	7.09178e9i	−0.660683	+	0.395213i
$367$			1.99945e10i			1.10216i	0.834451	+	0.551082i	$0.185785\pi$
$367$			1.99945e10i			1.10216i	−0.834451	+	0.551082i	$0.814215\pi$
$368$	6.73756e9			0.367377
$369$	−1.76134e10	−	9.45427e9i	−0.950033	−	0.509944i
$370$		0			0
$371$			2.74499e10i			1.44892i
$372$	3.33356e10	−	1.99409e10i	1.74075	−	1.04129i
$373$			8.36869e9i			0.432337i	0.976356	+	0.216168i	$0.0693561\pi$
$373$			8.36869e9i			0.432337i	−0.976356	+	0.216168i	$0.930644\pi$
$374$		−	4.16590e10i		−	2.12923i
$375$		0			0
$376$	−3.71074e10			−1.85656
$377$	−1.60491e9			−0.0794485
$378$	−4.97124e10	+	2.31137e9i	−2.43499	+	0.113215i
$379$	−1.46377e10			−0.709440			−0.354720	−	0.934973i	$-0.615424\pi$
$379$	−1.46377e10			−0.709440			−0.354720	+	0.934973i	$0.615424\pi$
$380$		0			0
$381$	−9.03280e8	−	1.51003e9i	−0.0428669	−	0.0716613i
$382$			2.63685e9i			0.123832i
$383$	9.54053e9			0.443381			0.221691	−	0.975117i	$-0.428842\pi$
$383$	9.54053e9			0.443381			0.221691	+	0.975117i	$0.428842\pi$
$384$	1.97823e9	−	1.18335e9i	0.0909814	−	0.0544239i
$385$		0			0
$386$			2.40754e10i			1.08449i
$387$	−2.80092e10	−	1.50343e10i	−1.24870	−	0.670255i
$388$		−	8.76975e10i		−	3.86955i
$389$			2.67145e10i			1.16667i	0.812230	+	0.583337i	$0.198253\pi$
$389$			2.67145e10i			1.16667i	−0.812230	+	0.583337i	$0.801747\pi$
$390$		0			0
$391$	−4.73624e9			−0.202641
$392$	4.55700e10			1.92990
$393$	7.16970e9	+	1.19857e10i	0.300560	+	0.502451i
$394$	3.90230e10			1.61933
$395$		0			0
$396$	−4.61525e10	−	2.47730e10i	−1.87678	−	1.00739i
$397$		−	7.88247e9i		−	0.317322i	−0.987333	−	0.158661i	$-0.949282\pi$
$397$		−	7.88247e9i		−	0.317322i	0.987333	−	0.158661i	$-0.0507177\pi$
$398$	−5.67384e10			−2.26123
$399$	1.03872e10	+	1.73645e10i	0.409835	+	0.685127i
$400$		0			0
$401$		−	1.90264e10i		−	0.735834i	−0.929859	−	0.367917i	$-0.880071\pi$
$401$		−	1.90264e10i		−	0.735834i	0.929859	−	0.367917i	$-0.119929\pi$
$402$	4.51319e10	+	7.54476e10i	1.72814	+	2.88896i
$403$		−	6.83853e9i		−	0.259264i
$404$		−	5.12803e10i		−	1.92498i
$405$		0			0
$406$	−1.71566e10			−0.631432
$407$	2.86507e10			1.04414
$408$	7.98359e10	−	4.77569e10i	2.88110	−	1.72344i
$409$	4.82473e9			0.172417			0.0862084	−	0.996277i	$-0.472525\pi$
$409$	4.82473e9			0.172417			0.0862084	+	0.996277i	$0.472525\pi$
$410$		0			0
$411$	1.91879e10	−	1.14780e10i	0.672451	−	0.402252i
$412$		−	7.95795e10i		−	2.76193i
$413$	1.64054e10			0.563879
$414$	−3.99018e9	+	7.43376e9i	−0.135828	+	0.253051i
$415$		0			0
$416$			1.62403e10i			0.542278i
$417$	−2.81244e10	+	1.68237e10i	−0.930121	+	0.556387i
$418$			3.01726e10i			0.988342i
$419$			3.79749e10i			1.23208i	0.787713	+	0.616042i	$0.211265\pi$
$419$			3.79749e10i			1.23208i	−0.787713	+	0.616042i	$0.788735\pi$
$420$		0			0
$421$	−5.86781e9			−0.186787			−0.0933937	−	0.995629i	$-0.529772\pi$
$421$	−5.86781e9			−0.186787			−0.0933937	+	0.995629i	$0.529772\pi$
$422$	−6.00799e10			−1.89443
$423$	1.08932e10	−	2.02941e10i	0.340245	−	0.633883i
$424$	−9.14070e10			−2.82824
$425$		0			0
$426$	1.19125e9	+	1.99144e9i	0.0361715	+	0.0604684i
$427$			1.83512e10i			0.552017i
$428$	6.31884e10			1.88305
$429$	−7.91376e9	+	4.73391e9i	−0.233643	+	0.139763i
$430$		0			0
$431$		−	1.50537e10i		−	0.436250i	−0.975921	−	0.218125i	$-0.930006\pi$
$431$		−	1.50537e10i		−	0.436250i	0.975921	−	0.218125i	$-0.0699940\pi$
$432$	−3.81516e9	−	8.20555e10i	−0.109541	−	2.35599i
$433$			2.72042e10i			0.773899i	0.922101	+	0.386950i	$0.126471\pi$
$433$			2.72042e10i			0.773899i	−0.922101	+	0.386950i	$0.873529\pi$
$434$		−	7.31042e10i		−	2.06055i
$435$		0			0
$436$	1.11858e11			3.09543
$437$	3.43034e9			0.0940615
$438$	1.16160e10	+	1.94187e10i	0.315618	+	0.527623i
$439$	3.48887e10			0.939348			0.469674	−	0.882840i	$-0.344372\pi$
$439$	3.48887e10			0.939348			0.469674	+	0.882840i	$0.344372\pi$
$440$		0			0
$441$	−1.33774e10	+	2.49224e10i	−0.353686	+	0.658923i
$442$		−	2.80793e10i		−	0.735695i
$443$	−1.32254e10			−0.343395			−0.171698	−	0.985150i	$-0.554925\pi$
$443$	−1.32254e10			−0.343395			−0.171698	+	0.985150i	$0.554925\pi$
$444$	5.63112e10	+	9.41363e10i	1.44898	+	2.42229i
$445$		0			0
$446$		−	1.36665e10i		−	0.345397i
$447$	3.49816e10	+	5.84793e10i	0.876212	+	1.46478i
$448$			4.80054e10i			1.19173i
$449$		−	2.35824e9i		−	0.0580232i	−0.999579	−	0.0290116i	$-0.990764\pi$
$449$		−	2.35824e9i		−	0.0580232i	0.999579	−	0.0290116i	$-0.00923598\pi$
$450$		0			0
$451$	−3.95980e10			−0.957121
$452$	−5.37825e9			−0.128851
$453$	−1.33136e10	+	7.96404e9i	−0.316157	+	0.189121i
$454$	−1.24174e11			−2.92284
$455$		0			0
$456$	−5.78231e10	+	3.45891e10i	−1.33734	+	0.799981i
$457$			2.48874e10i			0.570579i	0.958441	+	0.285289i	$0.0920897\pi$
$457$			2.48874e10i			0.570579i	−0.958441	+	0.285289i	$0.907910\pi$
$458$	−1.21312e11			−2.75703
$459$	2.68191e9	+	5.76818e10i	0.0604217	+	1.29954i
$460$		0			0
$461$		−	2.32674e10i		−	0.515163i	−0.966257	−	0.257581i	$-0.917074\pi$
$461$		−	2.32674e10i		−	0.515163i	0.966257	−	0.257581i	$-0.0829255\pi$
$462$	−8.45984e10	+	5.06057e10i	−1.85692	+	1.11079i
$463$			3.52388e10i			0.766828i	0.923577	+	0.383414i	$0.125252\pi$
$463$			3.52388e10i			0.766828i	−0.923577	+	0.383414i	$0.874748\pi$
$464$		−	2.83187e10i		−	0.610944i
$465$		0			0
$466$	8.46687e10			1.79547
$467$	−8.78873e10			−1.84781			−0.923907	−	0.382616i	$-0.875023\pi$
$467$	−8.78873e10			−1.84781			−0.923907	+	0.382616i	$0.875023\pi$
$468$	−3.11081e10	−	1.66977e10i	−0.648469	−	0.348075i
$469$	1.16786e11			2.41380
$470$		0			0
$471$	9.86222e9	+	1.64868e10i	0.200397	+	0.335007i
$472$			5.46292e10i			1.10067i
$473$	−6.29693e10			−1.25801
$474$	1.15554e11	−	6.91231e10i	2.28914	−	1.36934i
$475$		0			0
$476$		−	2.11874e11i		−	4.12715i
$477$	2.68332e10	−	4.99907e10i	0.518321	−	0.965641i
$478$			6.08275e10i			1.16517i
$479$		−	7.91032e10i		−	1.50263i	−0.659944	−	0.751315i	$-0.729420\pi$
$479$		−	7.91032e10i		−	1.50263i	0.659944	−	0.751315i	$-0.270580\pi$
$480$		0			0
$481$	1.93113e10			0.360771
$482$	−5.53811e9			−0.102606
$483$	5.75341e9	+	9.61806e9i	0.105715	+	0.176725i
$484$	2.79224e10			0.508829
$485$		0			0
$486$	9.27938e10	+	4.43862e10i	1.66331	+	0.795615i
$487$			5.78470e10i			1.02841i	0.857668	+	0.514203i	$0.171912\pi$
$487$			5.78470e10i			1.02841i	−0.857668	+	0.514203i	$0.828088\pi$
$488$	−6.11087e10			−1.07752
$489$	−2.61758e10	−	4.37585e10i	−0.457788	−	0.765292i
$490$		0			0
$491$			1.07166e11i			1.84387i	0.387339	+	0.921937i	$0.373394\pi$
$491$			1.07166e11i			1.84387i	−0.387339	+	0.921937i	$0.626606\pi$
$492$	−7.78275e10	−	1.30105e11i	−1.32823	−	2.22042i
$493$			1.99070e10i			0.336990i
$494$			2.03371e10i			0.341493i
$495$		0			0
$496$	1.20666e11			1.99370
$497$	3.08257e9			0.0505229
$498$	1.20606e11	−	7.21451e10i	1.96088	−	1.17298i
$499$	−2.45077e9			−0.0395276			−0.0197638	−	0.999805i	$-0.506291\pi$
$499$	−2.45077e9			−0.0395276			−0.0197638	+	0.999805i	$0.506291\pi$
$500$		0			0
$501$	2.63632e9	−	1.57701e9i	0.0418454	−	0.0250314i
$502$		−	5.52509e10i		−	0.870010i
$503$	4.84386e10			0.756694			0.378347	−	0.925664i	$-0.376493\pi$
$503$	4.84386e10			0.756694			0.378347	+	0.925664i	$0.376493\pi$
$504$	−1.93963e11	−	1.04112e11i	−3.00606	−	1.61354i
$505$		0			0
$506$			1.67123e10i			0.254938i
$507$	5.13693e10	−	3.07285e10i	0.777449	−	0.465061i
$508$		−	1.33445e10i		−	0.200377i
$509$			3.20903e10i			0.478082i	0.971009	+	0.239041i	$0.0768330\pi$
$509$			3.20903e10i			0.478082i	−0.971009	+	0.239041i	$0.923167\pi$
$510$		0			0
$511$	3.00585e10			0.440842
$512$	1.31697e11			1.91644
$513$	−1.94244e9	−	4.17775e10i	−0.0280464	−	0.603216i
$514$	7.74490e10			1.10959
$515$		0			0
$516$	−1.23763e11	−	2.06896e11i	−1.74578	−	2.91845i
$517$		−	4.56246e10i		−	0.638611i
$518$	2.06439e11			2.86730
$519$	−1.04242e11	+	6.23562e10i	−1.43672	+	0.859430i
$520$		0			0
$521$		−	6.76727e10i		−	0.918465i	−0.888316	−	0.459232i	$-0.848125\pi$
$521$		−	6.76727e10i		−	0.918465i	0.888316	−	0.459232i	$-0.151875\pi$
$522$	3.12449e10	+	1.67711e10i	0.420821	+	0.225882i
$523$			6.05025e10i			0.808661i	0.914613	+	0.404330i	$0.132495\pi$
$523$			6.05025e10i			0.808661i	−0.914613	+	0.404330i	$0.867505\pi$
$524$			1.05921e11i			1.40494i
$525$		0			0
$526$	−2.15531e11			−2.81557
$527$	−8.48236e10			−1.09970
$528$	−8.35301e10	−	1.39639e11i	−1.07475	−	1.79668i
$529$	−7.64109e10			−0.975737
$530$		0			0
$531$	−2.98769e10	−	1.60368e10i	−0.375800	−	0.201716i
$532$			1.53455e11i			1.91573i
$533$	−2.66901e10			−0.330706
$534$	2.35682e10	+	3.93993e10i	0.289842	+	0.484533i
$535$		0			0
$536$			3.88894e11i			4.71164i
$537$	−3.21120e10	−	5.36822e10i	−0.386163	−	0.645555i
$538$		−	3.11654e10i		−	0.372000i
$539$			5.60296e10i			0.663839i
$540$		0			0
$541$	−2.66748e10			−0.311396			−0.155698	−	0.987805i	$-0.549763\pi$
$541$	−2.66748e10			−0.311396			−0.155698	+	0.987805i	$0.549763\pi$
$542$	1.63877e11			1.89898
$543$	−5.21287e10	+	3.11828e10i	−0.599622	+	0.358687i
$544$	2.01442e11			2.30014
$545$		0			0
$546$	−5.70217e10	+	3.41097e10i	−0.641608	+	0.383802i
$547$			1.39645e11i			1.55983i	0.625885	+	0.779915i	$0.284738\pi$
$547$			1.39645e11i			1.55983i	−0.625885	+	0.779915i	$0.715262\pi$
$548$	1.69569e11			1.88029
$549$	1.79389e10	−	3.34205e10i	0.197473	−	0.367895i
$550$		0			0
$551$		−	1.44181e10i		−	0.156423i
$552$	−3.20278e10	+	1.91586e10i	−0.344961	+	0.206352i
$553$		−	1.78868e11i		−	1.91263i
$554$			1.71603e11i			1.82174i
$555$		0			0
$556$	−2.48544e11			−2.60078
$557$	−1.88635e10			−0.195975			−0.0979876	−	0.995188i	$-0.531241\pi$
$557$	−1.88635e10			−0.195975			−0.0979876	+	0.995188i	$0.531241\pi$
$558$	−7.14619e10	+	1.33135e11i	−0.737120	+	1.37327i
$559$	−4.24431e10			−0.434670
$560$		0			0
$561$	5.87184e10	+	9.81605e10i	0.592820	+	0.991026i
$562$		−	2.33959e11i		−	2.34528i
$563$	3.06642e10			0.305210			0.152605	−	0.988287i	$-0.451234\pi$
$563$	3.06642e10			0.305210			0.152605	+	0.988287i	$0.451234\pi$
$564$	1.49907e11	−	8.96725e10i	1.48151	−	0.886222i
$565$		0			0
$566$		−	2.41837e11i		−	2.35644i
$567$	1.13879e11	−	7.55159e10i	1.10182	−	0.730645i
$568$			1.02649e10i			0.0986187i
$569$		−	1.85504e11i		−	1.76972i	−0.465861	−	0.884858i	$-0.654255\pi$
$569$		−	1.85504e11i		−	1.76972i	0.465861	−	0.884858i	$-0.345745\pi$
$570$		0			0
$571$	8.50195e10			0.799786			0.399893	−	0.916562i	$-0.369047\pi$
$571$	8.50195e10			0.799786			0.399893	+	0.916562i	$0.369047\pi$
$572$	−6.99361e10			−0.653307
$573$	−3.71664e9	−	6.21317e9i	−0.0344772	−	0.0576362i
$574$	−2.85319e11			−2.62835
$575$		0			0
$576$	4.69270e10	−	8.74257e10i	0.426317	−	0.794235i
$577$		−	1.80485e11i		−	1.62831i	−0.580648	−	0.814155i	$-0.697201\pi$
$577$		−	1.80485e11i		−	1.62831i	0.580648	−	0.814155i	$-0.302799\pi$
$578$	−1.42499e11			−1.27673
$579$	−3.39343e10	−	5.67284e10i	−0.301942	−	0.504762i
$580$		0			0
$581$		−	1.86688e11i		−	1.63837i
$582$	1.75122e11	+	2.92754e11i	1.52633	+	2.55159i
$583$		−	1.12387e11i		−	0.972845i
$584$			1.00093e11i			0.860507i
$585$		0			0
$586$	3.00911e11			2.55181
$587$	3.34184e10			0.281471			0.140736	−	0.990047i	$-0.455053\pi$
$587$	3.34184e10			0.281471			0.140736	+	0.990047i	$0.455053\pi$
$588$	−1.84094e11	+	1.10123e11i	−1.54004	+	0.921231i
$589$	6.14356e10			0.510457
$590$		0			0
$591$	−9.19494e10	+	5.50030e10i	−0.753701	+	0.450854i
$592$			3.40749e11i			2.77427i
$593$	1.21264e11			0.980647			0.490323	−	0.871541i	$-0.336879\pi$
$593$	1.21264e11			0.980647			0.490323	+	0.871541i	$0.336879\pi$
$594$	2.03536e11	−	9.46339e9i	1.63492	−	0.0760153i
$595$		0			0
$596$			5.16798e11i			4.09577i
$597$	1.33692e11	−	7.99728e10i	1.05246	−	0.629571i
$598$			1.12646e10i			0.0880867i
$599$		−	1.08145e11i		−	0.840041i	−0.907515	−	0.420020i	$-0.862023\pi$
$599$		−	1.08145e11i		−	0.840041i	0.907515	−	0.420020i	$-0.137977\pi$
$600$		0			0
$601$	1.99467e11			1.52888			0.764440	−	0.644695i	$-0.223016\pi$
$601$	1.99467e11			1.52888			0.764440	+	0.644695i	$0.223016\pi$
$602$	−4.53718e11			−3.45462
$603$	−2.12687e11	−	1.14163e11i	−1.60869	−	0.863486i
$604$	−1.17656e11			−0.884030
$605$		0			0
$606$	1.02401e11	+	1.71185e11i	0.759301	+	1.26934i
$607$			7.47882e10i			0.550907i	0.961314	+	0.275453i	$0.0888280\pi$
$607$			7.47882e10i			0.550907i	−0.961314	+	0.275453i	$0.911172\pi$
$608$	−1.45899e11			−1.06767
$609$	4.04258e10	−	2.41822e10i	0.293893	−	0.175803i
$610$		0			0
$611$		−	3.07522e10i		−	0.220654i
$612$	−2.07114e11	+	3.85857e11i	−1.47640	+	2.75056i
$613$			2.70342e11i			1.91457i	0.289148	+	0.957284i	$0.406628\pi$
$613$			2.70342e11i			1.91457i	−0.289148	+	0.957284i	$0.593372\pi$
$614$			4.70030e11i			3.30713i
$615$		0			0
$616$	−4.36062e11			−3.02848
$617$	7.48975e10			0.516805			0.258403	−	0.966037i	$-0.416804\pi$
$617$	7.48975e10			0.516805			0.258403	+	0.966037i	$0.416804\pi$
$618$	1.58911e11	+	2.65655e11i	1.08943	+	1.82122i
$619$	9.53847e10			0.649705			0.324852	−	0.945765i	$-0.394685\pi$
$619$	9.53847e10			0.649705			0.324852	+	0.945765i	$0.394685\pi$
$620$		0			0
$621$	−1.07590e9	−	2.31402e10i	−0.00723446	−	0.155597i
$622$		−	4.34018e11i		−	2.89965i
$623$	6.09867e10			0.404839
$624$	−5.63016e10	−	9.41203e10i	−0.371349	−	0.620790i
$625$		0			0
$626$			3.22039e11i			2.09706i
$627$	−4.25282e10	−	7.10951e10i	−0.275174	−	0.460013i
$628$			1.45699e11i			0.936736i
$629$		−	2.39534e11i		−	1.53025i
$630$		0			0
$631$	−2.63469e11			−1.66193			−0.830965	−	0.556325i	$-0.812211\pi$
$631$	−2.63469e11			−1.66193			−0.830965	+	0.556325i	$0.812211\pi$
$632$	5.95623e11			3.73339
$633$	1.41565e11	−	8.46826e10i	0.881743	−	0.527448i
$634$	−3.87402e11			−2.39776
$635$		0			0
$636$	3.69267e11	−	2.20891e11i	2.25690	−	1.35005i
$637$			3.77655e10i			0.229371i
$638$	7.02438e10			0.423960
$639$	−5.61387e9	−	3.01332e9i	−0.0336712	−	0.0180735i
$640$		0			0
$641$		−	1.27087e11i		−	0.752780i	−0.926461	−	0.376390i	$-0.877165\pi$
$641$		−	1.27087e11i		−	0.752780i	0.926461	−	0.376390i	$-0.122835\pi$
$642$	−2.10937e11	+	1.26180e11i	−1.24169	+	0.742764i
$643$		−	2.47669e11i		−	1.44887i	−0.689345	−	0.724433i	$-0.742101\pi$
$643$		−	2.47669e11i		−	1.44887i	0.689345	−	0.724433i	$-0.257899\pi$
$644$			8.49975e10i			0.494155i
$645$		0			0
$646$	2.52257e11			1.44848
$647$	1.34645e11			0.768374			0.384187	−	0.923255i	$-0.374482\pi$
$647$	1.34645e11			0.768374			0.384187	+	0.923255i	$0.374482\pi$
$648$	2.51465e11	+	3.79211e11i	1.42619	+	2.15071i
$649$	−6.71682e10			−0.378603
$650$		0			0
$651$	1.03040e11	+	1.72254e11i	0.573699	+	0.959061i
$652$		−	3.86706e11i		−	2.13989i
$653$	−1.95881e11			−1.07731			−0.538655	−	0.842527i	$-0.681067\pi$
$653$	−1.95881e11			−1.07731			−0.538655	+	0.842527i	$0.681067\pi$
$654$	−3.73407e11	+	2.23368e11i	−2.04113	+	1.22098i
$655$		0			0
$656$		−	4.70948e11i		−	2.54307i
$657$	−5.47414e10	−	2.93832e10i	−0.293802	−	0.157702i
$658$		−	3.28743e11i		−	1.75369i
$659$			1.11563e11i			0.591531i	0.955261	+	0.295765i	$0.0955747\pi$
$659$			1.11563e11i			0.591531i	−0.955261	+	0.295765i	$0.904425\pi$
$660$		0			0
$661$	−2.12263e11			−1.11191			−0.555953	−	0.831214i	$-0.687647\pi$
$661$	−2.12263e11			−1.11191			−0.555953	+	0.831214i	$0.687647\pi$
$662$	1.50830e11			0.785338
$663$	3.95778e10	+	6.61629e10i	0.204832	+	0.342421i
$664$	6.21663e11			3.19803
$665$		0			0
$666$	−3.75959e11	−	2.01801e11i	−1.91093	−	1.02572i
$667$		−	7.98607e9i		−	0.0403487i
$668$	2.32979e10			0.117007
$669$	1.92630e10	+	3.22022e10i	0.0961654	+	0.160761i
$670$		0			0
$671$		−	7.51349e10i		−	0.370639i
$672$	−2.44703e11	−	4.09075e11i	−1.19995	−	2.00597i
$673$			5.44544e10i			0.265444i	0.991153	+	0.132722i	$0.0423717\pi$
$673$			5.44544e10i			0.265444i	−0.991153	+	0.132722i	$0.957628\pi$
$674$		−	7.01695e11i		−	3.40023i
$675$		0			0
$676$	4.53965e11			2.17388
$677$	−3.73397e11			−1.77753			−0.888763	−	0.458367i	$-0.848435\pi$
$677$	−3.73397e11			−1.77753			−0.888763	+	0.458367i	$0.848435\pi$
$678$	1.79538e10	−	1.07398e10i	0.0849647	−	0.0508248i
$679$	4.53158e11			2.13192
$680$		0			0
$681$	2.92588e11	−	1.75023e11i	1.36041	−	0.813778i
$682$			2.99309e11i			1.38351i
$683$	−3.36543e11			−1.54653			−0.773263	−	0.634085i	$-0.781377\pi$
$683$	−3.36543e11			−1.54653			−0.773263	+	0.634085i	$0.781377\pi$
$684$	1.50008e11	−	2.79467e11i	0.685313	−	1.27675i
$685$		0			0
$686$		−	1.36122e11i		−	0.614656i
$687$	2.85846e11	−	1.70989e11i	1.28323	−	0.767613i
$688$		−	7.48910e11i		−	3.34253i
$689$		−	7.57523e10i		−	0.336139i
$690$		0			0
$691$	−1.37252e11			−0.602013			−0.301007	−	0.953622i	$-0.597323\pi$
$691$	−1.37252e11			−0.602013			−0.301007	+	0.953622i	$0.597323\pi$
$692$	−9.21215e11			−4.01732
$693$	1.28009e11	−	2.38483e11i	0.555019	−	1.03401i
$694$	1.08490e11			0.467684
$695$		0			0
$696$	8.05258e10	+	1.34616e11i	0.343161	+	0.573668i
$697$			3.31058e11i			1.40273i
$698$	2.47635e11			1.04325
$699$	−1.99504e11	+	1.19341e11i	−0.835684	+	0.499896i
$700$		0			0
$701$		−	4.12721e11i		−	1.70917i	−0.519316	−	0.854583i	$-0.673813\pi$
$701$		−	4.12721e11i		−	1.70917i	0.519316	−	0.854583i	$-0.326187\pi$
$702$	1.37189e11	−	6.37859e9i	0.564900	−	0.0262649i
$703$			1.73488e11i			0.710311i
$704$		−	1.96548e11i		−	0.800160i
$705$		0			0
$706$	−7.47783e11			−3.00993
$707$	2.64980e11			1.06056
$708$	−1.32015e11	−	2.20692e11i	−0.525401	−	0.878321i
$709$	−1.59025e11			−0.629333			−0.314667	−	0.949202i	$-0.601893\pi$
$709$	−1.59025e11			−0.629333			−0.314667	+	0.949202i	$0.601893\pi$
$710$		0			0
$711$	−1.74850e11	+	3.25747e11i	−0.684205	+	1.27468i
$712$			2.03083e11i			0.790231i
$713$	3.40287e10			0.131670
$714$	4.23089e11	+	7.07284e11i	1.62794	+	2.72145i
$715$		0			0
$716$		−	4.74405e11i		−	1.80508i
$717$	−8.57364e10	−	1.43327e11i	−0.324406	−	0.542314i
$718$		−	9.72775e10i		−	0.366028i
$719$			1.83117e11i			0.685195i	0.939482	+	0.342597i	$0.111307\pi$
$719$			1.83117e11i			0.685195i	−0.939482	+	0.342597i	$0.888693\pi$
$720$		0			0
$721$	4.11210e11			1.52168
$722$	3.18326e11			1.17145
$723$	1.30494e10	−	7.80596e9i	0.0477569	−	0.0285676i
$724$	−4.60676e11			−1.67665
$725$		0			0
$726$	−9.32114e10	+	5.57579e10i	−0.335523	+	0.200706i
$727$			2.59798e10i			0.0930032i	0.998918	+	0.0465016i	$0.0148073\pi$
$727$			2.59798e10i			0.0930032i	−0.998918	+	0.0465016i	$0.985193\pi$
$728$	−2.93918e11			−1.04641
$729$	−2.81211e11	+	2.62064e10i	−0.995686	+	0.0927891i
$730$		0			0
$731$			5.26454e11i			1.84370i
$732$	2.46868e11	−	1.47673e11i	0.859844	−	0.514348i
$733$			6.63035e10i			0.229678i	0.993384	+	0.114839i	$0.0366353\pi$
$733$			6.63035e10i			0.229678i	−0.993384	+	0.114839i	$0.963365\pi$
$734$		−	5.89855e11i		−	2.03218i
$735$		0			0
$736$	−8.08123e10			−0.275401
$737$	−4.78156e11			−1.62069
$738$	5.19612e11	+	2.78909e11i	1.75168	+	0.940237i
$739$	1.53919e11			0.516078			0.258039	−	0.966135i	$-0.416924\pi$
$739$	1.53919e11			0.516078			0.258039	+	0.966135i	$0.416924\pi$
$740$		0			0
$741$	−2.86652e10	−	4.79201e10i	−0.0950785	−	0.158944i
$742$		−	8.09795e11i		−	2.67153i
$743$	3.56107e9			0.0116849			0.00584246	−	0.999983i	$-0.498140\pi$
$743$	3.56107e9			0.0116849			0.00584246	+	0.999983i	$0.498140\pi$
$744$	−5.73600e11	+	3.43121e11i	−1.87205	+	1.11984i
$745$		0			0
$746$		−	2.46884e11i		−	0.797145i
$747$	−1.82494e11	+	3.39989e11i	−0.586092	+	1.09190i
$748$			8.67472e11i			2.77108i
$749$			3.26512e11i			1.03746i
$750$		0			0
$751$	−3.54316e10			−0.111386			−0.0556931	−	0.998448i	$-0.517737\pi$
$751$	−3.54316e10			−0.111386			−0.0556931	+	0.998448i	$0.517737\pi$
$752$	5.42624e11			1.69679
$753$	7.78761e10	+	1.30187e11i	0.242228	+	0.404937i
$754$	4.73463e10			0.146487
$755$		0			0
$756$	1.03517e12	−	4.81300e10i	3.16901	−	0.147343i
$757$			3.97220e11i			1.20962i	0.796371	+	0.604808i	$0.206750\pi$
$757$			3.97220e11i			1.20962i	−0.796371	+	0.604808i	$0.793250\pi$
$758$	4.31824e11			1.30807
$759$	−2.35560e10	−	3.93790e10i	−0.0709799	−	0.118658i
$760$		0			0
$761$			8.97779e10i			0.267689i	0.991002	+	0.133845i	$0.0427323\pi$
$761$			8.97779e10i			0.267689i	−0.991002	+	0.133845i	$0.957268\pi$
$762$	2.66475e10	+	4.45471e10i	0.0790382	+	0.132129i
$763$			5.78002e11i			1.70542i
$764$		−	5.49076e10i		−	0.161161i
$765$		0			0
$766$	−2.81454e11			−0.817508
$767$	−4.52732e10			−0.130816
$768$	−3.27482e11	+	1.95895e11i	−0.941330	+	0.563092i
$769$	4.29746e11			1.22887			0.614436	−	0.788967i	$-0.289384\pi$
$769$	4.29746e11			1.22887			0.614436	+	0.788967i	$0.289384\pi$
$770$		0			0
$771$	−1.82492e11	+	1.09164e11i	−0.516448	+	0.308933i
$772$		−	5.01325e11i		−	1.41140i
$773$	−1.11151e11			−0.311312			−0.155656	−	0.987811i	$-0.549749\pi$
$773$	−1.11151e11			−0.311312			−0.155656	+	0.987811i	$0.549749\pi$
$774$	8.26295e11	+	4.43526e11i	2.30235	+	1.23582i
$775$		0			0
$776$			1.50900e12i			4.16142i
$777$	−4.86429e11	+	2.90976e11i	−1.33455	+	0.798313i
$778$		−	7.88102e11i		−	2.15112i
$779$		−	2.39777e11i		−	0.651116i
$780$		0			0
$781$	−1.26209e10			−0.0339224
$782$	1.39723e11			0.373630
$783$	−9.72609e10	+	4.52213e9i	−0.258756	+	0.0120308i
$784$	−6.66374e11			−1.76382
$785$		0			0
$786$	−2.11512e11	−	3.53589e11i	−0.554173	−	0.926420i
$787$			3.22875e11i			0.841658i	0.907140	+	0.420829i	$0.138261\pi$
$787$			3.22875e11i			0.841658i	−0.907140	+	0.420829i	$0.861739\pi$
$788$	−8.12583e11			−2.10748
$789$	5.07852e11	−	3.03791e11i	1.31048	−	0.783912i
$790$		0			0
$791$		−	2.77910e10i		−	0.0709901i
$792$	7.94139e11	+	4.26265e11i	2.01835	+	1.08338i
$793$		−	5.06430e10i		−	0.128064i
$794$			2.32540e11i			0.585080i
$795$		0			0
$796$	1.18147e12			2.94287
$797$	4.82426e10			0.119563			0.0597815	−	0.998211i	$-0.480960\pi$
$797$	4.82426e10			0.119563			0.0597815	+	0.998211i	$0.480960\pi$
$798$	−3.06433e11	−	5.12268e11i	−0.755655	−	1.26324i
$799$	−3.81444e11			−0.935930
$800$		0			0
$801$	−1.11067e11	−	5.96166e10i	−0.269807	−	0.144823i
$802$			5.61296e11i			1.35673i
$803$	−1.23068e11			−0.295993
$804$	−9.39787e11	−	1.57106e12i	−2.24908	−	3.75983i
$805$		0			0
$806$			2.01743e11i			0.478033i
$807$	4.39276e10	+	7.34345e10i	0.103572	+	0.173143i
$808$			8.82373e11i			2.07017i
$809$		−	4.74471e10i		−	0.110768i	−0.998465	−	0.0553842i	$-0.982362\pi$
$809$		−	4.74471e10i		−	0.110768i	0.998465	−	0.0553842i	$-0.0176384\pi$
$810$		0			0
$811$	1.83507e11			0.424199			0.212100	−	0.977248i	$-0.431970\pi$
$811$	1.83507e11			0.424199			0.212100	+	0.977248i	$0.431970\pi$
$812$	3.57254e11			0.821775
$813$	−3.86141e11	+	2.30985e11i	−0.883861	+	0.528715i
$814$	−8.45219e11			−1.92518
$815$		0			0
$816$	−1.16745e12	+	6.98352e11i	−2.63315	+	1.57512i
$817$		−	3.81298e11i		−	0.855807i
$818$	−1.42334e11			−0.317903
$819$	8.62817e10	−	1.60744e11i	0.191771	−	0.357273i
$820$		0			0
$821$			8.94845e10i			0.196959i	0.995139	+	0.0984794i	$0.0313978\pi$
$821$			8.94845e10i			0.196959i	−0.995139	+	0.0984794i	$0.968602\pi$
$822$	−5.66060e11	+	3.38610e11i	−1.23987	+	0.741674i
$823$		−	8.24619e11i		−	1.79744i	−0.438524	−	0.898719i	$-0.644499\pi$
$823$		−	8.24619e11i		−	1.79744i	0.438524	−	0.898719i	$-0.355501\pi$
$824$			1.36931e12i			2.97026i
$825$		0			0
$826$	−4.83973e11			−1.03968
$827$	8.26757e11			1.76749			0.883743	−	0.467973i	$-0.155015\pi$
$827$	8.26757e11			1.76749			0.883743	+	0.467973i	$0.155015\pi$
$828$	8.30880e10	−	1.54794e11i	0.176774	−	0.329332i
$829$	2.54789e11			0.539465			0.269732	−	0.962935i	$-0.413065\pi$
$829$	2.54789e11			0.539465			0.269732	+	0.962935i	$0.413065\pi$
$830$		0			0
$831$	−2.41875e11	−	4.04346e11i	−0.507209	−	0.847909i
$832$		−	1.32479e11i		−	0.276473i
$833$	4.68435e11			0.972903
$834$	8.29695e11	−	4.96314e11i	1.71496	−	1.02587i
$835$		0			0
$836$		−	6.28288e11i		−	1.28627i
$837$	−1.92688e10	−	4.14429e11i	−0.0392603	−	0.844400i
$838$		−	1.12029e12i		−	2.27172i
$839$			6.07397e11i			1.22581i	0.790155	+	0.612907i	$0.210000\pi$
$839$			6.07397e11i			1.22581i	−0.790155	+	0.612907i	$0.790000\pi$
$840$		0			0
$841$	4.66680e11			0.932900
$842$	1.73106e11			0.344400
$843$	3.29766e11	+	5.51274e11i	0.652973	+	1.09158i
$844$	1.25105e12			2.46550
$845$		0			0
$846$	−3.21358e11	+	5.98694e11i	−0.627346	+	1.16876i
$847$			1.44283e11i			0.280338i
$848$	1.33665e12			2.58485
$849$	3.40870e11	+	5.69837e11i	0.656081	+	1.09678i
$850$		0			0
$851$			9.60937e10i			0.183222i
$852$	−2.48057e10	−	4.14681e10i	−0.0470753	−	0.0786965i
$853$		−	1.34856e11i		−	0.254726i	−0.991856	−	0.127363i	$-0.959349\pi$
$853$		−	1.34856e11i		−	0.254726i	0.991856	−	0.127363i	$-0.0406514\pi$
$854$		−	5.41376e11i		−	1.01781i
$855$		0			0
$856$	−1.08727e12			−2.02509
$857$	−7.33386e11			−1.35959			−0.679797	−	0.733400i	$-0.737932\pi$
$857$	−7.33386e11			−1.35959			−0.679797	+	0.733400i	$0.737932\pi$
$858$	2.33463e11	−	1.39655e11i	0.430793	−	0.257695i
$859$	−4.51837e11			−0.829868			−0.414934	−	0.909852i	$-0.636195\pi$
$859$	−4.51837e11			−0.829868			−0.414934	+	0.909852i	$0.636195\pi$
$860$		0			0
$861$	6.72292e11	−	4.02157e11i	1.22334	−	0.731784i
$862$			4.44098e11i			0.804359i
$863$	6.73082e11			1.21346			0.606729	−	0.794909i	$-0.292481\pi$
$863$	6.73082e11			1.21346			0.606729	+	0.794909i	$0.292481\pi$
$864$	4.57601e10	+	9.84198e11i	0.0821168	+	1.76615i
$865$		0			0
$866$		−	8.02548e11i		−	1.42692i
$867$	3.35768e11	−	2.00852e11i	0.594241	−	0.355467i
$868$			1.52226e12i			2.68170i
$869$			7.32335e11i			1.28419i
$870$		0			0
$871$	−3.22290e11			−0.559983
$872$	−1.92472e12			−3.32891
$873$	−8.25275e11	−	4.42978e11i	−1.42083	−	0.762650i
$874$	−1.01198e11			−0.173431
$875$		0			0
$876$	−2.41882e11	−	4.04359e11i	−0.410760	−	0.686674i
$877$		−	8.29492e11i		−	1.40221i	−0.713057	−	0.701106i	$-0.752690\pi$
$877$		−	8.29492e11i		−	1.40221i	0.713057	−	0.701106i	$-0.247310\pi$
$878$	−1.02925e12			−1.73197
$879$	−7.09033e11	+	4.24135e11i	−1.18771	+	0.710474i
$880$		0			0
$881$			1.16546e11i			0.193461i	0.995311	+	0.0967305i	$0.0308385\pi$
$881$			1.16546e11i			0.193461i	−0.995311	+	0.0967305i	$0.969161\pi$
$882$	3.94646e11	−	7.35231e11i	0.652128	−	1.21493i
$883$			4.57775e11i			0.753025i	0.926412	+	0.376512i	$0.122877\pi$
$883$			4.57775e11i			0.753025i	−0.926412	+	0.376512i	$0.877123\pi$
$884$			5.84700e11i			0.957468i
$885$		0			0
$886$	3.90161e11			0.633153
$887$	8.01818e11			1.29533			0.647667	−	0.761924i	$-0.275745\pi$
$887$	8.01818e11			1.29533			0.647667	+	0.761924i	$0.275745\pi$
$888$	−9.68939e11	−	1.61979e12i	−1.55828	−	2.60500i
$889$	6.89550e10			0.110397
$890$		0			0
$891$	−4.66251e11	+	3.09183e11i	−0.739791	+	0.490575i
$892$			2.84580e11i			0.449516i
$893$	2.76270e11			0.434438
$894$	−1.03199e12	−	1.72519e12i	−1.61556	−	2.70076i
$895$		0			0
$896$			9.03355e10i			0.140161i
$897$	−1.58774e10	−	2.65426e10i	−0.0245251	−	0.0409990i
$898$			6.95700e10i			0.106983i
$899$		−	1.43026e11i		−	0.218966i
$900$		0			0
$901$	−9.39614e11			−1.42577
$902$	1.16817e12			1.76474
$903$	1.06909e12	−	6.39516e11i	1.60792	−	0.961835i
$904$	9.25428e10			0.138570
$905$		0			0
$906$	3.92763e11	−	2.34946e11i	0.582932	−	0.348703i
$907$		−	4.83758e10i		−	0.0714824i	−0.999361	−	0.0357412i	$-0.988621\pi$
$907$		−	4.83758e10i		−	0.0714824i	0.999361	−	0.0357412i	$-0.0113792\pi$
$908$	2.58568e12			3.80393
$909$	−4.82572e11	−	2.59027e11i	−0.706816	−	0.379394i
$910$		0			0
$911$			9.65899e11i			1.40236i	0.712986	+	0.701178i	$0.247342\pi$
$911$			9.65899e11i			1.40236i	−0.712986	+	0.701178i	$0.752658\pi$
$912$	8.45552e11	−	5.05799e11i	1.22225	−	0.731137i
$913$			7.64351e11i			1.10004i
$914$		−	7.34201e11i		−	1.05204i
$915$		0			0
$916$	2.52610e12			3.58813
$917$	−5.47324e11			−0.774047
$918$	−7.91186e10	−	1.70166e12i	−0.111406	−	2.39609i
$919$	−1.14405e12			−1.60392			−0.801962	−	0.597375i	$-0.796211\pi$
$919$	−1.14405e12			−1.60392			−0.801962	+	0.597375i	$0.796211\pi$
$920$		0			0
$921$	−6.62507e11	−	1.10752e12i	−0.920772	−	1.53927i
$922$			6.86409e11i			0.949859i
$923$	−8.50685e9			−0.0117209
$924$	1.76161e12	−	1.05377e12i	2.41669	−	1.44563i
$925$		0			0
$926$		−	1.03958e12i		−	1.41388i
$927$	−7.48881e11	−	4.01972e11i	−1.01413	−	0.544349i
$928$			3.39663e11i			0.457990i
$929$		−	7.33571e11i		−	0.984870i	−0.870349	−	0.492435i	$-0.836107\pi$
$929$		−	7.33571e11i		−	0.984870i	0.870349	−	0.492435i	$-0.163893\pi$
$930$		0			0
$931$	−3.39276e11			−0.451600
$932$	−1.76307e12			−2.33671
$933$	6.11748e11	+	1.02267e12i	0.807321	+	1.34961i
$934$	2.59275e12			3.40701
$935$		0			0
$936$	5.35272e11	+	2.87315e11i	0.697383	+	0.374330i
$937$		−	3.55464e11i		−	0.461145i	−0.973055	−	0.230573i	$-0.925940\pi$
$937$		−	3.55464e11i		−	0.461145i	0.973055	−	0.230573i	$-0.0740599\pi$
$938$	−3.44530e12			−4.45057
$939$	−4.53914e11	−	7.58816e11i	−0.583864	−	0.976054i
$940$		0			0
$941$			6.76442e11i			0.862724i	0.902179	+	0.431362i	$0.141967\pi$
$941$			6.76442e11i			0.862724i	−0.902179	+	0.431362i	$0.858033\pi$
$942$	−2.90944e11	−	4.86376e11i	−0.369493	−	0.617687i
$943$		−	1.32811e11i		−	0.167952i
$944$		−	7.98847e11i		−	1.00595i
$945$		0			0
$946$	1.85765e12			2.31953
$947$	1.02862e12			1.27896			0.639479	−	0.768809i	$-0.279150\pi$
$947$	1.02862e12			1.27896			0.639479	+	0.768809i	$0.279150\pi$
$948$	−2.40620e12	+	1.43936e12i	−2.97919	+	1.78212i
$949$	−8.29511e10			−0.102272
$950$		0			0
$951$	9.12830e11	−	5.46044e11i	1.11601	−	0.667583i
$952$			3.64569e12i			4.43845i
$953$	1.23011e11			0.149132			0.0745661	−	0.997216i	$-0.476243\pi$
$953$	1.23011e11			0.149132			0.0745661	+	0.997216i	$0.476243\pi$
$954$	−7.91603e11	+	1.47477e12i	−0.955683	+	1.78045i
$955$		0			0
$956$		−	1.26662e12i		−	1.51640i
$957$	−1.65514e11	+	9.90087e10i	−0.197328	+	0.118039i
$958$			2.33361e12i			2.77056i
$959$			8.76212e11i			1.03594i
$960$		0			0
$961$	−2.43455e11			−0.285447
$962$	−5.69701e11			−0.665192
$963$	3.19177e11	−	5.94632e11i	0.371131	−	0.691422i
$964$	1.15321e11			0.133536
$965$		0			0
$966$	−1.69730e11	−	2.83741e11i	−0.194918	−	0.325847i
$967$		−	2.66857e11i		−	0.305191i	−0.988289	−	0.152596i	$-0.951237\pi$
$967$		−	2.66857e11i		−	0.305191i	0.988289	−	0.152596i	$-0.0487632\pi$
$968$	−4.80457e11			−0.547209
$969$	−5.94390e11	+	3.55557e11i	−0.674181	+	0.403287i
$970$		0			0
$971$			1.52105e12i			1.71106i	0.517750	+	0.855532i	$0.326770\pi$
$971$			1.52105e12i			1.71106i	−0.517750	+	0.855532i	$0.673230\pi$
$972$	−1.93226e12	−	9.24261e11i	−2.16471	−	1.03545i
$973$		−	1.28430e12i		−	1.43289i
$974$		−	1.70654e12i		−	1.89618i
$975$		0			0
$976$	8.93597e11			0.984788
$977$	1.09651e12			1.20347			0.601735	−	0.798696i	$-0.294476\pi$
$977$	1.09651e12			1.20347			0.601735	+	0.798696i	$0.294476\pi$
$978$	7.72209e11	+	1.29091e12i	0.844072	+	1.41105i
$979$	−2.49696e11			−0.271820
$980$		0			0
$981$	5.65017e11	−	1.05264e12i	0.610078	−	1.13658i
$982$		−	3.16149e12i		−	3.39974i
$983$	4.45708e11			0.477349			0.238675	−	0.971100i	$-0.423287\pi$
$983$	4.45708e11			0.477349			0.238675	+	0.971100i	$0.423287\pi$
$984$	1.33917e12	+	2.23871e12i	1.42841	+	2.38790i
$985$		0			0
$986$		−	5.87272e11i		−	0.621344i
$987$	4.63363e11	+	7.74612e11i	0.488262	+	0.816235i
$988$		−	4.23484e11i		−	0.444436i
$989$		−	2.11198e11i		−	0.220752i
$990$		0			0
$991$	5.99867e11			0.621957			0.310979	−	0.950417i	$-0.399343\pi$
$991$	5.99867e11			0.621957			0.310979	+	0.950417i	$0.399343\pi$
$992$	−1.44731e12			−1.49456
$993$	−3.55399e11	+	2.12595e11i	−0.365527	+	0.218654i
$994$	−9.09386e10			−0.0931543
$995$		0			0
$996$	−2.51140e12	+	1.50229e12i	−2.55199	+	1.52657i
$997$		−	6.40433e11i		−	0.648176i	−0.946027	−	0.324088i	$-0.894943\pi$
$997$		−	6.40433e11i		−	0.648176i	0.946027	−	0.324088i	$-0.105057\pi$
$998$	7.22998e10			0.0728811
$999$	1.17031e12	−	5.44132e10i	1.17500	−	0.0546314i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.d.c.74.1		20
3.2	odd	2	inner	75.9.d.c.74.19		20
5.2	odd	4		75.9.c.g.26.1		10
5.3	odd	4		15.9.c.a.11.10	yes	10
5.4	even	2	inner	75.9.d.c.74.20		20
15.2	even	4		75.9.c.g.26.10		10
15.8	even	4		15.9.c.a.11.1	✓	10
15.14	odd	2	inner	75.9.d.c.74.2		20
20.3	even	4		240.9.l.b.161.3		10
60.23	odd	4		240.9.l.b.161.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.9.c.a.11.1	✓	10	15.8	even	4
15.9.c.a.11.10	yes	10	5.3	odd	4
75.9.c.g.26.1		10	5.2	odd	4
75.9.c.g.26.10		10	15.2	even	4
75.9.d.c.74.1		20	1.1	even	1	trivial
75.9.d.c.74.2		20	15.14	odd	2	inner
75.9.d.c.74.19		20	3.2	odd	2	inner
75.9.d.c.74.20		20	5.4	even	2	inner
240.9.l.b.161.3		10	20.3	even	4
240.9.l.b.161.4		10	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-29.5009 q^{2} +(69.5124 - 41.5815i) q^{3} +614.301 q^{4} +(-2050.68 + 1226.69i) q^{6} +3174.27i q^{7} -10570.2 q^{8} +(3102.96 - 5780.86i) q^{9} +O(q^{10})\)	\(q-29.5009 q^{2} +(69.5124 - 41.5815i) q^{3} +614.301 q^{4} +(-2050.68 + 1226.69i) q^{6} +3174.27i q^{7} -10570.2 q^{8} +(3102.96 - 5780.86i) q^{9} -12996.3i q^{11} +(42701.6 - 25543.6i) q^{12} -8759.90i q^{13} -93643.7i q^{14} +154569. q^{16} -108656. q^{17} +(-91540.0 + 170540. i) q^{18} +78696.7 q^{19} +(131991. + 220651. i) q^{21} +383403. i q^{22} +43589.4 q^{23} +(-734760. + 439524. i) q^{24} +258425. i q^{26} +(-24682.6 - 530868. i) q^{27} +1.94996e6i q^{28} -183211. i q^{29} +780664. q^{31} -1.85394e6 q^{32} +(-540407. - 903407. i) q^{33} +3.20544e6 q^{34} +(1.90615e6 - 3.55119e6i) q^{36} +2.20452e6i q^{37} -2.32162e6 q^{38} +(-364250. - 608922. i) q^{39} -3.04685e6i q^{41} +(-3.89384e6 - 6.50940e6i) q^{42} -4.84516e6i q^{43} -7.98367e6i q^{44} -1.28593e6 q^{46} +3.51057e6 q^{47} +(1.07444e7 - 6.42720e6i) q^{48} -4.31118e6 q^{49} +(-7.55293e6 + 4.51807e6i) q^{51} -5.38121e6i q^{52} +8.64762e6 q^{53} +(728158. + 1.56611e7i) q^{54} -3.35526e7i q^{56} +(5.47040e6 - 3.27232e6i) q^{57} +5.40489e6i q^{58} -5.16824e6i q^{59} +5.78123e6 q^{61} -2.30303e7 q^{62} +(1.83500e7 + 9.84963e6i) q^{63} +1.51233e7 q^{64} +(1.59425e7 + 2.66513e7i) q^{66} -3.67916e7i q^{67} -6.67474e7 q^{68} +(3.03001e6 - 1.81251e6i) q^{69} -971113. i q^{71} +(-3.27989e7 + 6.11048e7i) q^{72} -9.46941e6i q^{73} -6.50352e7i q^{74} +4.83434e7 q^{76} +4.12539e7 q^{77} +(1.07457e7 + 1.79637e7i) q^{78} -5.63493e7 q^{79} +(-2.37900e7 - 3.58756e7i) q^{81} +8.98849e7i q^{82} -5.88128e7 q^{83} +(8.10821e7 + 1.35546e8i) q^{84} +1.42936e8i q^{86} +(-7.61819e6 - 1.27355e7i) q^{87} +1.37374e8i q^{88} -1.92128e7i q^{89} +2.78063e7 q^{91} +2.67770e7 q^{92} +(5.42658e7 - 3.24612e7i) q^{93} -1.03565e8 q^{94} +(-1.28872e8 + 7.70897e7i) q^{96} -1.42760e8i q^{97} +1.27184e8 q^{98} +(-7.51301e7 - 4.03271e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9}+O(q^{10}) \) 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9	\( 20 q + 1572 q^{4} - 10564 q^{6} - 7844 q^{9} + 560772 q^{16} + 463032 q^{19} + 579144 q^{21} - 2272668 q^{24} + 1763240 q^{31} + 2222552 q^{34} - 1337324 q^{36} - 3653584 q^{39} - 50849208 q^{46} - 18708428 q^{49} - 55465384 q^{51} + 15959596 q^{54} + 44834040 q^{61} + 45870004 q^{64} - 54839600 q^{66} - 67125264 q^{69} + 397844872 q^{76} - 324621848 q^{79} - 187150780 q^{81} + 394693536 q^{84} + 888576928 q^{91} + 184100072 q^{94} - 721614812 q^{96} + 67930400 q^{99}+O(q^{100}) \) 20 * q + 1572 * q^4 - 10564 * q^6 - 7844 * q^9 + 560772 * q^16 + 463032 * q^19 + 579144 * q^21 - 2272668 * q^24 + 1763240 * q^31 + 2222552 * q^34 - 1337324 * q^36 - 3653584 * q^39 - 50849208 * q^46 - 18708428 * q^49 - 55465384 * q^51 + 15959596 * q^54 + 44834040 * q^61 + 45870004 * q^64 - 54839600 * q^66 - 67125264 * q^69 + 397844872 * q^76 - 324621848 * q^79 - 187150780 * q^81 + 394693536 * q^84 + 888576928 * q^91 + 184100072 * q^94 - 721614812 * q^96 + 67930400 * q^99

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