LMFDB - Embedded newform 252.9.bg.a.29.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(29,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("252.29");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bg (of order $6$, degree $2$, 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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.6
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-75.7417 + 28.7087i) q^{3} +(820.829 - 473.906i) q^{5} +(453.746 - 785.912i) q^{7} +(4912.62 - 4348.90i) q^{9} +O(q^{10})$	\(q+(-75.7417 + 28.7087i) q^{3} +(820.829 - 473.906i) q^{5} +(453.746 - 785.912i) q^{7} +(4912.62 - 4348.90i) q^{9} +(13313.8 + 7686.70i) q^{11} +(-9352.38 - 16198.8i) q^{13} +(-48565.7 + 59459.4i) q^{15} +142076. i q^{17} -147550. q^{19} +(-11805.0 + 72552.8i) q^{21} +(-409302. + 236310. i) q^{23} +(253861. - 439699. i) q^{25} +(-247239. + 470428. i) q^{27} +(901911. + 520718. i) q^{29} +(-479658. - 830792. i) q^{31} +(-1.22908e6 - 199983. i) q^{33} -860132. i q^{35} +2.84553e6 q^{37} +(1.17341e6 + 958430. i) q^{39} +(-2.94432e6 + 1.69991e6i) q^{41} +(-1.25049e6 + 2.16591e6i) q^{43} +(1.97145e6 - 5.89782e6i) q^{45} +(-2.42892e6 - 1.40234e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-4.07882e6 - 1.07611e7i) q^{51} +1.51408e7i q^{53} +1.45711e7 q^{55} +(1.11757e7 - 4.23599e6i) q^{57} +(1.02848e7 - 5.93795e6i) q^{59} +(-2.63223e6 + 4.55915e6i) q^{61} +(-1.18877e6 - 5.83418e6i) q^{63} +(-1.53534e7 - 8.86429e6i) q^{65} +(-7.87249e6 - 1.36355e7i) q^{67} +(2.42170e7 - 2.96491e7i) q^{69} -3.46144e7i q^{71} -1.15301e7 q^{73} +(-6.60462e6 + 4.05916e7i) q^{75} +(1.20821e7 - 6.97562e6i) q^{77} +(2.47322e6 - 4.28373e6i) q^{79} +(5.22088e6 - 4.27289e7i) q^{81} +(-1.25698e7 - 7.25717e6i) q^{83} +(6.73306e7 + 1.16620e8i) q^{85} +(-8.32614e7 - 1.35474e7i) q^{87} +1.13712e8i q^{89} -1.69744e7 q^{91} +(6.01811e7 + 4.91552e7i) q^{93} +(-1.21114e8 + 6.99250e7i) q^{95} +(3.29234e7 - 5.70251e7i) q^{97} +(9.88340e7 - 2.01384e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−75.7417	+	28.7087i	−0.935083	+	0.354429i
$3$	−75.7417	+	28.7087i	−0.935083	+	0.354429i
$4$		0			0
$5$	820.829	−	473.906i	1.31333	−	0.758249i	0.330680	−	0.943743i	$-0.392722\pi$
$5$	820.829	−	473.906i	1.31333	−	0.758249i	0.982645	+	0.185494i	$0.0593885\pi$
$6$		0			0
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$8$		0			0
$9$	4912.62	−	4348.90i	0.748760	−	0.662841i
$10$		0			0
$11$	13313.8	+	7686.70i	0.909347	+	0.525012i	0.880221	−	0.474564i	$-0.157394\pi$
$11$	13313.8	+	7686.70i	0.909347	+	0.525012i	0.0291262	+	0.999576i	$0.490728\pi$
$12$		0			0
$13$	−9352.38	−	16198.8i	−0.327453	−	0.567165i	0.654553	−	0.756016i	$-0.272857\pi$
$13$	−9352.38	−	16198.8i	−0.327453	−	0.567165i	−0.982006	+	0.188851i	$0.939524\pi$
$14$		0			0
$15$	−48565.7	+	59459.4i	−0.959323	+	1.17451i
$16$		0			0
$17$			142076.i			1.70108i	0.525909	+	0.850541i	$0.323725\pi$
$17$			142076.i			1.70108i	−0.525909	+	0.850541i	$0.676275\pi$
$18$		0			0
$19$	−147550.			−1.13221			−0.566104	−	0.824334i	$-0.691550\pi$
$19$	−147550.			−1.13221			−0.566104	+	0.824334i	$0.691550\pi$
$20$		0			0
$21$	−11805.0	+	72552.8i	−0.0607000	+	0.373059i
$22$		0			0
$23$	−409302.	+	236310.i	−1.46262	+	0.844445i	−0.999132	−	0.0416567i	$-0.986736\pi$
$23$	−409302.	+	236310.i	−1.46262	+	0.844445i	−0.463490	+	0.886102i	$0.653403\pi$
$24$		0			0
$25$	253861.	−	439699.i	0.649883	−	1.12563i
$26$		0			0
$27$	−247239.	+	470428.i	−0.465223	+	0.885193i
$28$		0			0
$29$	901911.	+	520718.i	1.27518	+	0.736226i	0.975958	−	0.217958i	$-0.0699395\pi$
$29$	901911.	+	520718.i	1.27518	+	0.736226i	0.299222	+	0.954183i	$0.403273\pi$
$30$		0			0
$31$	−479658.	−	830792.i	−0.519379	−	0.899592i	−0.999746	−	0.0225239i	$-0.992830\pi$
$31$	−479658.	−	830792.i	−0.519379	−	0.899592i	0.480367	−	0.877068i	$-0.340504\pi$
$32$		0			0
$33$	−1.22908e6	−	199983.i	−1.03639	−	0.168631i
$34$		0			0
$35$		−	860132.i		−	0.573182i
$36$		0			0
$37$	2.84553e6			1.51830			0.759149	−	0.650917i	$-0.225616\pi$
$37$	2.84553e6			1.51830			0.759149	+	0.650917i	$0.225616\pi$
$38$		0			0
$39$	1.17341e6	+	958430.i	0.507215	+	0.414288i
$40$		0			0
$41$	−2.94432e6	+	1.69991e6i	−1.04196	+	0.601574i	−0.920387	−	0.391008i	$-0.872126\pi$
$41$	−2.94432e6	+	1.69991e6i	−1.04196	+	0.601574i	−0.121570	+	0.992583i	$0.538793\pi$
$42$		0			0
$43$	−1.25049e6	+	2.16591e6i	−0.365768	+	0.633529i	−0.988899	−	0.148588i	$-0.952527\pi$
$43$	−1.25049e6	+	2.16591e6i	−0.365768	+	0.633529i	0.623131	+	0.782118i	$0.285860\pi$
$44$		0			0
$45$	1.97145e6	−	5.89782e6i	0.480768	−	1.43827i
$46$		0			0
$47$	−2.42892e6	−	1.40234e6i	−0.497762	−	0.287383i	0.230027	−	0.973184i	$-0.426119\pi$
$47$	−2.42892e6	−	1.40234e6i	−0.497762	−	0.287383i	−0.727789	+	0.685801i	$0.759452\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	−4.07882e6	−	1.07611e7i	−0.602912	−	1.59065i
$52$		0			0
$53$			1.51408e7i			1.91887i	0.281929	+	0.959435i	$0.409026\pi$
$53$			1.51408e7i			1.91887i	−0.281929	+	0.959435i	$0.590974\pi$
$54$		0			0
$55$	1.45711e7			1.59236
$56$		0			0
$57$	1.11757e7	−	4.23599e6i	1.05871	−	0.401287i
$58$		0			0
$59$	1.02848e7	−	5.93795e6i	0.848768	−	0.490037i	−0.0114669	−	0.999934i	$-0.503650\pi$
$59$	1.02848e7	−	5.93795e6i	0.848768	−	0.490037i	0.860235	+	0.509898i	$0.170317\pi$
$60$		0			0
$61$	−2.63223e6	+	4.55915e6i	−0.190110	+	0.329279i	−0.945286	−	0.326242i	$-0.894218\pi$
$61$	−2.63223e6	+	4.55915e6i	−0.190110	+	0.329279i	0.755177	+	0.655521i	$0.227551\pi$
$62$		0			0
$63$	−1.18877e6	−	5.83418e6i	−0.0754632	−	0.370354i
$64$		0			0
$65$	−1.53534e7	−	8.86429e6i	−0.860105	−	0.496582i
$66$		0			0
$67$	−7.87249e6	−	1.36355e7i	−0.390672	−	0.676664i	0.601866	−	0.798597i	$-0.294424\pi$
$67$	−7.87249e6	−	1.36355e7i	−0.390672	−	0.676664i	−0.992538	+	0.121933i	$0.961091\pi$
$68$		0			0
$69$	2.42170e7	−	2.96491e7i	1.06838	−	1.30802i
$70$		0			0
$71$		−	3.46144e7i		−	1.36215i	−0.732215	−	0.681073i	$-0.761513\pi$
$71$		−	3.46144e7i		−	1.36215i	0.732215	−	0.681073i	$-0.238487\pi$
$72$		0			0
$73$	−1.15301e7			−0.406016			−0.203008	−	0.979177i	$-0.565072\pi$
$73$	−1.15301e7			−0.406016			−0.203008	+	0.979177i	$0.565072\pi$
$74$		0			0
$75$	−6.60462e6	+	4.05916e7i	−0.208739	+	1.28289i
$76$		0			0
$77$	1.20821e7	−	6.97562e6i	0.343701	−	0.198436i
$78$		0			0
$79$	2.47322e6	−	4.28373e6i	0.0634970	−	0.109980i	−0.832529	−	0.553981i	$-0.813108\pi$
$79$	2.47322e6	−	4.28373e6i	0.0634970	−	0.109980i	0.896026	+	0.444001i	$0.146441\pi$
$80$		0			0
$81$	5.22088e6	−	4.27289e7i	0.121284	−	0.992618i
$82$		0			0
$83$	−1.25698e7	−	7.25717e6i	−0.264860	−	0.152917i	0.361690	−	0.932299i	$-0.382200\pi$
$83$	−1.25698e7	−	7.25717e6i	−0.264860	−	0.152917i	−0.626549	+	0.779382i	$0.715533\pi$
$84$		0			0
$85$	6.73306e7	+	1.16620e8i	1.28984	+	2.23407i
$86$		0			0
$87$	−8.32614e7	−	1.35474e7i	−1.45334	−	0.236471i
$88$		0			0
$89$			1.13712e8i			1.81237i	0.422885	+	0.906183i	$0.361017\pi$
$89$			1.13712e8i			1.81237i	−0.422885	+	0.906183i	$0.638983\pi$
$90$		0			0
$91$	−1.69744e7			−0.247531
$92$		0			0
$93$	6.01811e7	+	4.91552e7i	0.804504	+	0.657110i
$94$		0			0
$95$	−1.21114e8	+	6.99250e7i	−1.48696	+	0.858495i
$96$		0			0
$97$	3.29234e7	−	5.70251e7i	0.371893	−	0.644138i	−0.617964	−	0.786207i	$-0.712042\pi$
$97$	3.29234e7	−	5.70251e7i	0.371893	−	0.644138i	0.989857	+	0.142069i	$0.0453754\pi$
$98$		0			0
$99$	9.88340e7	−	2.01384e7i	1.02888	−	0.209644i
$100$		0			0
$101$	−6.54283e7	−	3.77751e7i	−0.628753	−	0.363011i	0.151516	−	0.988455i	$-0.451585\pi$
$101$	−6.54283e7	−	3.77751e7i	−0.628753	−	0.363011i	−0.780269	+	0.625444i	$0.784918\pi$
$102$		0			0
$103$	8.79822e7	+	1.52390e8i	0.781711	+	1.35396i	0.930944	+	0.365161i	$0.118986\pi$
$103$	8.79822e7	+	1.52390e8i	0.781711	+	1.35396i	−0.149234	+	0.988802i	$0.547681\pi$
$104$		0			0
$105$	2.46933e7	+	6.51479e7i	0.203152	+	0.535973i
$106$		0			0
$107$			1.98645e8i			1.51545i	0.652572	+	0.757727i	$0.273690\pi$
$107$			1.98645e8i			1.51545i	−0.652572	+	0.757727i	$0.726310\pi$
$108$		0			0
$109$	5.72278e7			0.405416			0.202708	−	0.979239i	$-0.435026\pi$
$109$	5.72278e7			0.405416			0.202708	+	0.979239i	$0.435026\pi$
$110$		0			0
$111$	−2.15526e8	+	8.16917e7i	−1.41973	+	0.538128i
$112$		0			0
$113$	−1.81274e8	+	1.04659e8i	−1.11179	+	0.641891i	−0.939292	−	0.343119i	$-0.888517\pi$
$113$	−1.81274e8	+	1.04659e8i	−1.11179	+	0.641891i	−0.172496	+	0.985010i	$0.555183\pi$
$114$		0			0
$115$	−2.23978e8	+	3.87941e8i	−1.28060	+	2.21806i
$116$		0			0
$117$	−1.16392e8	−	3.89059e7i	−0.621124	−	0.207622i
$118$		0			0
$119$	1.11659e8	+	6.44665e7i	0.556810	+	0.321474i
$120$		0			0
$121$	1.09912e7	+	1.90374e7i	0.0512750	+	0.0888109i
$122$		0			0
$123$	1.74206e8	−	2.13282e8i	0.761101	−	0.931822i
$124$		0			0
$125$		−	1.10985e8i		−	0.454594i
$126$		0			0
$127$	−2.82789e8			−1.08705			−0.543523	−	0.839394i	$-0.682910\pi$
$127$	−2.82789e8			−1.08705			−0.543523	+	0.839394i	$0.682910\pi$
$128$		0			0
$129$	3.25336e7	−	1.99950e8i	0.117483	−	0.722041i
$130$		0			0
$131$	−3.33033e8	+	1.92277e8i	−1.13084	+	0.652893i	−0.944147	−	0.329525i	$-0.893111\pi$
$131$	−3.33033e8	+	1.92277e8i	−1.13084	+	0.652893i	−0.186697	+	0.982418i	$0.559778\pi$
$132$		0			0
$133$	−6.69505e7	+	1.15962e8i	−0.213967	+	0.370602i
$134$		0			0
$135$	1.99980e7	+	5.03309e8i	0.0602076	+	1.51530i
$136$		0			0
$137$	4.85306e8	+	2.80191e8i	1.37763	+	0.795376i	0.991874	−	0.127224i	$-0.0406068\pi$
$137$	4.85306e8	+	2.80191e8i	1.37763	+	0.795376i	0.385757	+	0.922600i	$0.373940\pi$
$138$		0			0
$139$	2.08791e8	+	3.61636e8i	0.559309	+	0.968752i	0.997554	+	0.0698966i	$0.0222669\pi$
$139$	2.08791e8	+	3.61636e8i	0.559309	+	0.968752i	−0.438245	+	0.898856i	$0.644400\pi$
$140$		0			0
$141$	2.24230e8	+	3.64842e7i	0.567306	+	0.0923057i
$142$		0			0
$143$		−	2.87556e8i		−	0.687667i
$144$		0			0
$145$	9.87085e8			2.23297
$146$		0			0
$147$	5.16636e7	+	4.21982e7i	0.110641	+	0.0903702i
$148$		0			0
$149$	7.50746e6	−	4.33444e6i	0.0152317	−	0.00879402i	−0.492365	−	0.870389i	$-0.663867\pi$
$149$	7.50746e6	−	4.33444e6i	0.0152317	−	0.00879402i	0.507597	+	0.861595i	$0.330534\pi$
$150$		0			0
$151$	2.09136e8	−	3.62235e8i	0.402274	−	0.696759i	−0.591726	−	0.806139i	$-0.701553\pi$
$151$	2.09136e8	−	3.62235e8i	0.402274	−	0.696759i	0.994000	+	0.109380i	$0.0348867\pi$
$152$		0			0
$153$	6.17874e8	+	6.97965e8i	1.12755	+	1.27370i
$154$		0			0
$155$	−7.87434e8	−	4.54625e8i	−1.36423	−	0.787638i
$156$		0			0
$157$	−6.67571e7	−	1.15627e8i	−0.109875	−	0.190309i	0.805845	−	0.592127i	$-0.201712\pi$
$157$	−6.67571e7	−	1.15627e8i	−0.109875	−	0.190309i	−0.915719	+	0.401818i	$0.868378\pi$
$158$		0			0
$159$	−4.34674e8	−	1.14679e9i	−0.680103	−	1.79430i
$160$		0			0
$161$			4.28900e8i			0.638341i
$162$		0			0
$163$	3.75104e8			0.531376			0.265688	−	0.964059i	$-0.414401\pi$
$163$	3.75104e8			0.531376			0.265688	+	0.964059i	$0.414401\pi$
$164$		0			0
$165$	−1.10364e9	+	4.18317e8i	−1.48899	+	0.564378i
$166$		0			0
$167$	−1.50917e8	+	8.71321e7i	−0.194032	+	0.112024i	−0.593869	−	0.804562i	$-0.702400\pi$
$167$	−1.50917e8	+	8.71321e7i	−0.194032	+	0.112024i	0.399837	+	0.916586i	$0.369067\pi$
$168$		0			0
$169$	2.32931e8	−	4.03449e8i	0.285549	−	0.494586i
$170$		0			0
$171$	−7.24858e8	+	6.41682e8i	−0.847752	+	0.750473i
$172$		0			0
$173$	1.32635e9	+	7.65768e8i	1.48072	+	0.854895i	0.999761	−	0.0218465i	$-0.00695450\pi$
$173$	1.32635e9	+	7.65768e8i	1.48072	+	0.854895i	0.480961	+	0.876742i	$0.340288\pi$
$174$		0			0
$175$	−2.30377e8	−	3.99024e8i	−0.245633	−	0.425448i
$176$		0			0
$177$	−6.08520e8	+	7.45015e8i	−0.619986	+	0.759053i
$178$		0			0
$179$			8.15760e8i			0.794603i	0.917688	+	0.397302i	$0.130053\pi$
$179$			8.15760e8i			0.794603i	−0.917688	+	0.397302i	$0.869947\pi$
$180$		0			0
$181$	−1.42982e9			−1.33219			−0.666096	−	0.745866i	$-0.732036\pi$
$181$	−1.42982e9			−1.33219			−0.666096	+	0.745866i	$0.732036\pi$
$182$		0			0
$183$	6.84819e7	−	4.20886e8i	0.0610621	−	0.375284i
$184$		0			0
$185$	2.33570e9	−	1.34851e9i	1.99402	−	1.15125i
$186$		0			0
$187$	−1.09210e9	+	1.89157e9i	−0.893088	+	1.54687i
$188$		0			0
$189$	2.57531e8	+	4.07763e8i	0.201829	+	0.319566i
$190$		0			0
$191$	1.07531e9	+	6.20830e8i	0.807979	+	0.466487i	0.846253	−	0.532781i	$-0.178853\pi$
$191$	1.07531e9	+	6.20830e8i	0.807979	+	0.466487i	−0.0382748	+	0.999267i	$0.512186\pi$
$192$		0			0
$193$	−2.42024e7	−	4.19197e7i	−0.0174433	−	0.0302127i	0.857172	−	0.515030i	$-0.172219\pi$
$193$	−2.42024e7	−	4.19197e7i	−0.0174433	−	0.0302127i	−0.874615	+	0.484818i	$0.838886\pi$
$194$		0			0
$195$	1.41738e9	+	2.30620e8i	0.980272	+	0.159499i
$196$		0			0
$197$			2.32193e8i			0.154165i	0.997025	+	0.0770824i	$0.0245604\pi$
$197$			2.32193e8i			0.154165i	−0.997025	+	0.0770824i	$0.975440\pi$
$198$		0			0
$199$	−1.76971e9			−1.12847			−0.564233	−	0.825615i	$-0.690828\pi$
$199$	−1.76971e9			−1.12847			−0.564233	+	0.825615i	$0.690828\pi$
$200$		0			0
$201$	9.87735e8	+	8.06770e8i	0.605140	+	0.494272i
$202$		0			0
$203$	8.18477e8	−	4.72548e8i	0.481973	−	0.278267i
$204$		0			0
$205$	−1.61119e9	+	2.79066e9i	−0.912286	+	1.58013i
$206$		0			0
$207$	−9.83052e8	+	2.94091e9i	−0.535421	+	1.60177i
$208$		0			0
$209$	−1.96445e9	−	1.13418e9i	−1.02957	−	0.594422i
$210$		0			0
$211$	−1.17650e9	−	2.03776e9i	−0.593557	−	1.02807i	−0.993749	−	0.111640i	$-0.964390\pi$
$211$	−1.17650e9	−	2.03776e9i	−0.593557	−	1.02807i	0.400192	−	0.916431i	$-0.368944\pi$
$212$		0			0
$213$	9.93737e8	+	2.62176e9i	0.482784	+	1.27372i
$214$		0			0
$215$			2.37045e9i			1.10937i
$216$		0			0
$217$	−8.70572e8			−0.392614
$218$		0			0
$219$	8.73312e8	−	3.31015e8i	0.379658	−	0.143904i
$220$		0			0
$221$	2.30146e9	−	1.32875e9i	0.964794	−	0.557024i
$222$		0			0
$223$	4.62906e8	−	8.01777e8i	0.187186	−	0.324216i	−0.757125	−	0.653270i	$-0.773397\pi$
$223$	4.62906e8	−	8.01777e8i	0.187186	−	0.324216i	0.944311	+	0.329054i	$0.106730\pi$
$224$		0			0
$225$	−6.65088e8	−	3.26409e9i	−0.259507	−	1.27360i
$226$		0			0
$227$	1.32245e9	+	7.63518e8i	0.498054	+	0.287552i	0.727909	−	0.685673i	$-0.240492\pi$
$227$	1.32245e9	+	7.63518e8i	0.498054	+	0.287552i	−0.229856	+	0.973225i	$0.573825\pi$
$228$		0			0
$229$	1.60256e9	+	2.77572e9i	0.582738	+	1.00933i	0.995153	+	0.0983362i	$0.0313520\pi$
$229$	1.60256e9	+	2.77572e9i	0.582738	+	1.00933i	−0.412415	+	0.910996i	$0.635315\pi$
$230$		0			0
$231$	−7.14860e8	+	8.75209e8i	−0.251058	+	0.307372i
$232$		0			0
$233$		−	1.59587e9i		−	0.541470i	−0.962654	−	0.270735i	$-0.912733\pi$
$233$		−	1.59587e9i		−	0.541470i	0.962654	−	0.270735i	$-0.0872667\pi$
$234$		0			0
$235$	−2.65830e9			−0.871631
$236$		0			0
$237$	−6.43450e7	+	3.95460e8i	−0.0203949	+	0.125346i
$238$		0			0
$239$	−2.03448e9	+	1.17461e9i	−0.623538	+	0.360000i	−0.778245	−	0.627961i	$-0.783890\pi$
$239$	−2.03448e9	+	1.17461e9i	−0.623538	+	0.360000i	0.154707	+	0.987960i	$0.450557\pi$
$240$		0			0
$241$	3.09146e9	−	5.35457e9i	0.916423	−	1.58729i	0.111618	−	0.993751i	$-0.464397\pi$
$241$	3.09146e9	−	5.35457e9i	0.916423	−	1.58729i	0.804805	−	0.593540i	$-0.202270\pi$
$242$		0			0
$243$	8.31256e8	+	3.38625e9i	0.238402	+	0.971167i
$244$		0			0
$245$	−6.75988e8	−	3.90282e8i	−0.187618	−	0.108321i
$246$		0			0
$247$	1.37995e9	+	2.39014e9i	0.370745	+	0.642149i
$248$		0			0
$249$	1.16040e9	+	1.88808e8i	0.301864	+	0.0491160i
$250$		0			0
$251$			5.80021e9i			1.46133i	0.682736	+	0.730666i	$0.260790\pi$
$251$			5.80021e9i			1.46133i	−0.682736	+	0.730666i	$0.739210\pi$
$252$		0			0
$253$	−7.26579e9			−1.77338
$254$		0			0
$255$	−8.44775e9	−	6.90003e9i	−1.99793	−	1.63189i
$256$		0			0
$257$	−4.82683e9	+	2.78677e9i	−1.10645	+	0.638807i	−0.937906	−	0.346888i	$-0.887238\pi$
$257$	−4.82683e9	+	2.78677e9i	−1.10645	+	0.638807i	−0.168539	+	0.985695i	$0.553905\pi$
$258$		0			0
$259$	1.29115e9	−	2.23634e9i	0.286931	−	0.496979i
$260$		0			0
$261$	6.69529e9	−	1.36423e9i	1.44280	−	0.293985i
$262$		0			0
$263$	6.90312e8	+	3.98552e8i	0.144285	+	0.0833032i	0.570405	−	0.821364i	$-0.306786\pi$
$263$	6.90312e8	+	3.98552e8i	0.144285	+	0.0833032i	−0.426119	+	0.904667i	$0.640120\pi$
$264$		0			0
$265$	7.17531e9	+	1.24280e10i	1.45498	+	2.52010i
$266$		0			0
$267$	−3.26453e9	−	8.61274e9i	−0.642355	−	1.69471i
$268$		0			0
$269$		−	5.43885e9i		−	1.03872i	−0.854556	−	0.519359i	$-0.826171\pi$
$269$		−	5.43885e9i		−	1.03872i	0.854556	−	0.519359i	$-0.173829\pi$
$270$		0			0
$271$	3.08470e9			0.571921			0.285961	−	0.958241i	$-0.407687\pi$
$271$	3.08470e9			0.571921			0.285961	+	0.958241i	$0.407687\pi$
$272$		0			0
$273$	1.28567e9	−	4.87315e8i	0.231462	−	0.0877322i
$274$		0			0
$275$	6.75967e9	−	3.90270e9i	1.18194	−	0.682392i
$276$		0			0
$277$	2.82407e9	−	4.89144e9i	0.479686	−	0.830840i	−0.520043	−	0.854140i	$-0.674084\pi$
$277$	2.82407e9	−	4.89144e9i	0.479686	−	0.830840i	0.999729	+	0.0233001i	$0.00741732\pi$
$278$		0			0
$279$	−5.96940e9	−	1.99538e9i	−0.985177	−	0.329313i
$280$		0			0
$281$	2.15568e9	+	1.24458e9i	0.345748	+	0.199618i	0.662811	−	0.748787i	$-0.269363\pi$
$281$	2.15568e9	+	1.24458e9i	0.345748	+	0.199618i	−0.317063	+	0.948404i	$0.602697\pi$
$282$		0			0
$283$	−1.07393e9	−	1.86009e9i	−0.167428	−	0.289994i	0.770087	−	0.637939i	$-0.220213\pi$
$283$	−1.07393e9	−	1.86009e9i	−0.167428	−	0.289994i	−0.937515	+	0.347945i	$0.886880\pi$
$284$		0			0
$285$	7.16589e9	−	8.77325e9i	1.08615	−	1.32978i
$286$		0			0
$287$			3.08530e9i			0.454748i
$288$		0			0
$289$	−1.32098e10			−1.89368
$290$		0			0
$291$	−8.56560e8	+	5.26437e9i	−0.119450	+	0.734132i
$292$		0			0
$293$	7.88286e9	−	4.55117e9i	1.06958	−	0.617522i	0.141513	−	0.989936i	$-0.454803\pi$
$293$	7.88286e9	−	4.55117e9i	1.06958	−	0.617522i	0.928067	+	0.372414i	$0.121470\pi$
$294$		0			0
$295$	5.62805e9	−	9.74808e9i	0.743139	−	1.28716i
$296$		0			0
$297$	−6.90771e9	+	4.36271e9i	−0.887786	+	0.560701i
$298$		0			0
$299$	7.65589e9	+	4.42013e9i	0.957880	+	0.553032i
$300$		0			0
$301$	1.13481e9	+	1.96555e9i	0.138247	+	0.239451i
$302$		0			0
$303$	6.04013e9	+	9.82783e8i	0.716598	+	0.116597i
$304$		0			0
$305$			4.98971e9i			0.576601i
$306$		0			0
$307$	3.84969e9			0.433384			0.216692	−	0.976240i	$-0.430473\pi$
$307$	3.84969e9			0.433384			0.216692	+	0.976240i	$0.430473\pi$
$308$		0			0
$309$	−1.10388e10	−	9.01640e9i	−1.21085	−	0.989007i
$310$		0			0
$311$	−9.48787e9	+	5.47782e9i	−1.01421	+	0.585553i	−0.912421	−	0.409253i	$-0.865789\pi$
$311$	−9.48787e9	+	5.47782e9i	−1.01421	+	0.585553i	−0.101787	+	0.994806i	$0.532456\pi$
$312$		0			0
$313$	−1.57941e9	+	2.73563e9i	−0.164558	+	0.285023i	−0.936498	−	0.350672i	$-0.885953\pi$
$313$	−1.57941e9	+	2.73563e9i	−0.164558	+	0.285023i	0.771940	+	0.635695i	$0.219286\pi$
$314$		0			0
$315$	−3.74063e9	−	4.22550e9i	−0.379929	−	0.429176i
$316$		0			0
$317$	−7.44048e9	−	4.29576e9i	−0.736824	−	0.425406i	0.0840893	−	0.996458i	$-0.473202\pi$
$317$	−7.44048e9	−	4.29576e9i	−0.736824	−	0.425406i	−0.820913	+	0.571053i	$0.806535\pi$
$318$		0			0
$319$	8.00521e9	+	1.38654e10i	0.773054	+	1.33897i
$320$		0			0
$321$	−5.70285e9	−	1.50457e10i	−0.537120	−	1.41707i
$322$		0			0
$323$		−	2.09634e10i		−	1.92598i
$324$		0			0
$325$	−9.49680e9			−0.851224
$326$		0			0
$327$	−4.33453e9	+	1.64294e9i	−0.379097	+	0.143691i
$328$		0			0
$329$	−2.20423e9	+	1.27261e9i	−0.188136	+	0.108621i
$330$		0			0
$331$	−3.83503e9	+	6.64246e9i	−0.319489	+	0.553372i	−0.980382	−	0.197109i	$-0.936845\pi$
$331$	−3.83503e9	+	6.64246e9i	−0.319489	+	0.553372i	0.660892	+	0.750481i	$0.270178\pi$
$332$		0			0
$333$	1.39790e10	−	1.23749e10i	1.13684	−	1.00639i
$334$		0			0
$335$	−1.29239e10	−	7.46163e9i	−1.02616	−	0.592454i
$336$		0			0
$337$	9.32250e9	+	1.61470e10i	0.722791	+	1.25191i	0.959877	+	0.280422i	$0.0904743\pi$
$337$	9.32250e9	+	1.61470e10i	0.722791	+	1.25191i	−0.237086	+	0.971489i	$0.576192\pi$
$338$		0			0
$339$	1.07254e10	−	1.31312e10i	0.812110	−	0.994272i
$340$		0			0
$341$		−	1.47479e10i		−	1.09072i
$342$		0			0
$343$	−7.47359e8			−0.0539949
$344$		0			0
$345$	5.82716e9	−	3.58134e10i	0.411321	−	2.52795i
$346$		0			0
$347$	1.22273e10	−	7.05941e9i	0.843357	−	0.486912i	−0.0150471	−	0.999887i	$-0.504790\pi$
$347$	1.22273e10	−	7.05941e9i	0.843357	−	0.486912i	0.858404	+	0.512975i	$0.171456\pi$
$348$		0			0
$349$	−1.34784e9	+	2.33452e9i	−0.0908522	+	0.157361i	−0.907870	−	0.419252i	$-0.862292\pi$
$349$	−1.34784e9	+	2.33452e9i	−0.0908522	+	0.157361i	0.817018	+	0.576613i	$0.195626\pi$
$350$		0			0
$351$	9.93264e9	−	3.94654e8i	0.654390	−	0.0260009i
$352$		0			0
$353$	−2.19645e9	−	1.26812e9i	−0.141456	−	0.0816697i	0.427602	−	0.903967i	$-0.359359\pi$
$353$	−2.19645e9	−	1.26812e9i	−0.141456	−	0.0816697i	−0.569058	+	0.822298i	$0.692692\pi$
$354$		0			0
$355$	−1.64040e10	−	2.84125e10i	−1.03285	−	1.78894i
$356$		0			0
$357$	−1.03080e10	−	1.67721e9i	−0.634603	−	0.103256i
$358$		0			0
$359$			1.34238e10i			0.808158i	0.914724	+	0.404079i	$0.132408\pi$
$359$			1.34238e10i			0.808158i	−0.914724	+	0.404079i	$0.867592\pi$
$360$		0			0
$361$	4.78756e9			0.281893
$362$		0			0
$363$	−1.37904e9	−	1.12638e9i	−0.0794235	−	0.0648722i
$364$		0			0
$365$	−9.46426e9	+	5.46419e9i	−0.533231	+	0.307861i
$366$		0			0
$367$	−4.17310e9	+	7.22802e9i	−0.230035	+	0.398433i	−0.957818	−	0.287375i	$-0.907218\pi$
$367$	−4.17310e9	+	7.22802e9i	−0.230035	+	0.398433i	0.727783	+	0.685808i	$0.240551\pi$
$368$		0			0
$369$	−7.07161e9	+	2.11555e10i	−0.381428	+	1.14109i
$370$		0			0
$371$	1.18993e10	+	6.87009e9i	0.628098	+	0.362632i
$372$		0			0
$373$	1.73776e10	+	3.00989e10i	0.897749	+	1.55495i	0.830365	+	0.557220i	$0.188132\pi$
$373$	1.73776e10	+	3.00989e10i	0.897749	+	1.55495i	0.0673844	+	0.997727i	$0.478535\pi$
$374$		0			0
$375$	3.18624e9	+	8.40619e9i	0.161121	+	0.425083i
$376$		0			0
$377$		−	1.94798e10i		−	0.964317i
$378$		0			0
$379$	−6.28689e9			−0.304705			−0.152352	−	0.988326i	$-0.548685\pi$
$379$	−6.28689e9			−0.304705			−0.152352	+	0.988326i	$0.548685\pi$
$380$		0			0
$381$	2.14189e10	−	8.11852e9i	1.01648	−	0.385280i
$382$		0			0
$383$	3.04263e10	−	1.75666e10i	1.41401	−	0.816381i	0.418250	−	0.908332i	$-0.362644\pi$
$383$	3.04263e10	−	1.75666e10i	1.41401	−	0.816381i	0.995764	+	0.0919510i	$0.0293103\pi$
$384$		0			0
$385$	6.61157e9	−	1.14516e10i	0.300928	−	0.521222i
$386$		0			0
$387$	3.27615e9	+	1.60785e10i	0.146056	+	0.716807i
$388$		0			0
$389$	2.78297e10	+	1.60675e10i	1.21537	+	0.701696i	0.963925	−	0.266175i	$-0.0857598\pi$
$389$	2.78297e10	+	1.60675e10i	1.21537	+	0.701696i	0.251448	+	0.967871i	$0.419093\pi$
$390$		0			0
$391$	−3.35740e10	−	5.81520e10i	−1.43647	−	2.48804i
$392$		0			0
$393$	1.97045e10	−	2.41243e10i	0.826028	−	1.01131i
$394$		0			0
$395$		−	4.68828e9i		−	0.192586i
$396$		0			0
$397$	−5.41052e9			−0.217809			−0.108905	−	0.994052i	$-0.534734\pi$
$397$	−5.41052e9			−0.217809			−0.108905	+	0.994052i	$0.534734\pi$
$398$		0			0
$399$	1.74183e9	−	1.07052e10i	0.0687250	−	0.422380i
$400$		0			0
$401$	−1.20767e10	+	6.97247e9i	−0.467057	+	0.269655i	−0.715007	−	0.699117i	$-0.753576\pi$
$401$	−1.20767e10	+	6.97247e9i	−0.467057	+	0.269655i	0.247950	+	0.968773i	$0.420243\pi$
$402$		0			0
$403$	−8.97189e9	+	1.55398e10i	−0.340145	+	0.589148i
$404$		0			0
$405$	−1.59640e10	−	3.75473e10i	−0.593366	−	1.39559i
$406$		0			0
$407$	3.78847e10	+	2.18728e10i	1.38066	+	0.797124i
$408$		0			0
$409$	−2.05695e10	−	3.56275e10i	−0.735074	−	1.27319i	−0.954691	−	0.297600i	$-0.903814\pi$
$409$	−2.05695e10	−	3.56275e10i	−0.735074	−	1.27319i	0.219616	−	0.975586i	$-0.429519\pi$
$410$		0			0
$411$	−4.48018e10	−	7.28966e9i	−1.57010	−	0.255470i
$412$		0			0
$413$		−	1.07773e10i		−	0.370433i
$414$		0			0
$415$	−1.37569e10			−0.463796
$416$		0			0
$417$	−2.61963e10	−	2.13968e10i	−0.866354	−	0.707628i
$418$		0			0
$419$	2.00718e10	−	1.15885e10i	0.651223	−	0.375984i	−0.137701	−	0.990474i	$-0.543971\pi$
$419$	2.00718e10	−	1.15885e10i	0.651223	−	0.375984i	0.788925	+	0.614490i	$0.210638\pi$
$420$		0			0
$421$	−1.29729e10	+	2.24698e10i	−0.412962	+	0.715272i	−0.995212	−	0.0977383i	$-0.968839\pi$
$421$	−1.29729e10	+	2.24698e10i	−0.412962	+	0.715272i	0.582250	+	0.813010i	$0.302173\pi$
$422$		0			0
$423$	−1.80310e10	+	3.67398e9i	−0.563194	+	0.114756i
$424$		0			0
$425$	6.24707e10	+	3.60675e10i	1.91479	+	1.10550i
$426$		0			0
$427$	2.38873e9	+	4.13739e9i	0.0718546	+	0.124456i
$428$		0			0
$429$	8.25537e9	+	2.17800e10i	0.243729	+	0.643026i
$430$		0			0
$431$			5.29304e10i			1.53390i	0.641708	+	0.766949i	$0.278226\pi$
$431$			5.29304e10i			1.53390i	−0.641708	+	0.766949i	$0.721774\pi$
$432$		0			0
$433$	−1.05559e10			−0.300292			−0.150146	−	0.988664i	$-0.547974\pi$
$433$	−1.05559e10			−0.300292			−0.150146	+	0.988664i	$0.547974\pi$
$434$		0			0
$435$	−7.47636e10	+	2.83380e10i	−2.08801	+	0.791429i
$436$		0			0
$437$	6.03926e10	−	3.48677e10i	1.65599	−	0.956087i
$438$		0			0
$439$	−2.88020e10	+	4.98865e10i	−0.775468	+	1.34315i	0.159063	+	0.987268i	$0.449153\pi$
$439$	−2.88020e10	+	4.98865e10i	−0.775468	+	1.34315i	−0.934531	+	0.355882i	$0.884181\pi$
$440$		0			0
$441$	−5.12455e9	−	1.71297e9i	−0.135488	−	0.0452893i
$442$		0			0
$443$	−2.34003e9	−	1.35101e9i	−0.0607583	−	0.0350788i	0.469313	−	0.883032i	$-0.344502\pi$
$443$	−2.34003e9	−	1.35101e9i	−0.0607583	−	0.0350788i	−0.530071	+	0.847953i	$0.677835\pi$
$444$		0			0
$445$	5.38887e10	+	9.33380e10i	1.37423	+	2.38023i
$446$		0			0
$447$	−4.44192e8	+	5.43827e8i	−0.0111260	+	0.0136217i
$448$		0			0
$449$			3.58814e10i			0.882844i	0.897300	+	0.441422i	$0.145526\pi$
$449$			3.58814e10i			0.882844i	−0.897300	+	0.441422i	$0.854474\pi$
$450$		0			0
$451$	−5.22666e10			−1.26333
$452$		0			0
$453$	−5.44104e9	+	3.34403e10i	−0.129208	+	0.794105i
$454$		0			0
$455$	−1.39331e10	+	8.04428e9i	−0.325089	+	0.187690i
$456$		0			0
$457$	−5.52432e9	+	9.56840e9i	−0.126653	+	0.219369i	−0.922378	−	0.386289i	$-0.873757\pi$
$457$	−5.52432e9	+	9.56840e9i	−0.126653	+	0.219369i	0.795725	+	0.605658i	$0.207090\pi$
$458$		0			0
$459$	−6.68366e10	−	3.51267e10i	−1.50579	−	0.791382i
$460$		0			0
$461$	−1.17771e10	−	6.79951e9i	−0.260756	−	0.150548i	0.363923	−	0.931429i	$-0.381437\pi$
$461$	−1.17771e10	−	6.79951e9i	−0.260756	−	0.150548i	−0.624680	+	0.780881i	$0.714770\pi$
$462$		0			0
$463$	1.43189e10	+	2.48011e10i	0.311592	+	0.539693i	0.978707	−	0.205261i	$-0.0658045\pi$
$463$	1.43189e10	+	2.48011e10i	0.311592	+	0.539693i	−0.667115	+	0.744955i	$0.732471\pi$
$464$		0			0
$465$	7.26933e10	+	1.18279e10i	1.55483	+	0.252985i
$466$		0			0
$467$			1.82843e10i			0.384424i	0.981353	+	0.192212i	$0.0615660\pi$
$467$			1.82843e10i			0.384424i	−0.981353	+	0.192212i	$0.938434\pi$
$468$		0			0
$469$	−1.42884e10			−0.295321
$470$		0			0
$471$	8.37579e9	+	6.84125e9i	0.170193	+	0.139012i
$472$		0			0
$473$	−3.32974e10	+	1.92243e10i	−0.665221	+	0.384065i
$474$		0			0
$475$	−3.74572e10	+	6.48778e10i	−0.735802	+	1.27445i
$476$		0			0
$477$	6.58459e10	+	7.43810e10i	1.27191	+	1.43677i
$478$		0			0
$479$	3.76600e10	+	2.17430e10i	0.715382	+	0.413026i	0.813051	−	0.582193i	$-0.197805\pi$
$479$	3.76600e10	+	2.17430e10i	0.715382	+	0.413026i	−0.0976684	+	0.995219i	$0.531138\pi$
$480$		0			0
$481$	−2.66125e10	−	4.60942e10i	−0.497171	−	0.861125i
$482$		0			0
$483$	−1.23132e10	−	3.24856e10i	−0.226246	−	0.596901i
$484$		0			0
$485$		−	6.24104e10i		−	1.12795i
$486$		0			0
$487$	1.25971e10			0.223952			0.111976	−	0.993711i	$-0.464282\pi$
$487$	1.25971e10			0.223952			0.111976	+	0.993711i	$0.464282\pi$
$488$		0			0
$489$	−2.84111e10	+	1.07688e10i	−0.496880	+	0.188335i
$490$		0			0
$491$	1.96375e10	−	1.13377e10i	0.337879	−	0.195074i	−0.321455	−	0.946925i	$-0.604172\pi$
$491$	1.96375e10	−	1.13377e10i	0.337879	−	0.195074i	0.659333	+	0.751851i	$0.270839\pi$
$492$		0			0
$493$	−7.39816e10	+	1.28140e11i	−1.25238	+	2.16919i
$494$		0			0
$495$	7.15821e10	−	6.33681e10i	1.19230	−	1.05548i
$496$		0			0
$497$	−2.72039e10	−	1.57062e10i	−0.445867	−	0.257422i
$498$		0			0
$499$	−2.57425e10	−	4.45872e10i	−0.415191	−	0.719132i	0.580258	−	0.814433i	$-0.302952\pi$
$499$	−2.57425e10	−	4.45872e10i	−0.415191	−	0.719132i	−0.995448	+	0.0953013i	$0.969619\pi$
$500$		0			0
$501$	8.92928e9	−	1.09322e10i	0.141731	−	0.173523i
$502$		0			0
$503$			8.37001e10i			1.30754i	0.756694	+	0.653769i	$0.226813\pi$
$503$			8.37001e10i			1.30754i	−0.756694	+	0.653769i	$0.773187\pi$
$504$		0			0
$505$	−7.16072e10			−1.10101
$506$		0			0
$507$	−6.06011e9	+	3.72451e10i	−0.0917167	+	0.563685i
$508$		0			0
$509$	8.41227e9	−	4.85683e9i	0.125326	−	0.0723571i	−0.436026	−	0.899934i	$-0.643615\pi$
$509$	8.41227e9	−	4.85683e9i	0.125326	−	0.0723571i	0.561353	+	0.827577i	$0.310281\pi$
$510$		0			0
$511$	−5.23175e9	+	9.06166e9i	−0.0767297	+	0.132900i
$512$		0			0
$513$	3.64802e10	−	6.94119e10i	0.526729	−	1.00222i
$514$		0			0
$515$	1.44437e11	+	8.33906e10i	2.05328	+	1.18546i
$516$		0			0
$517$	−2.15587e10	−	3.73408e10i	−0.301759	−	0.522662i
$518$		0			0
$519$	−1.22444e11	−	1.99228e10i	−1.68760	−	0.274587i
$520$		0			0
$521$		−	1.00471e11i		−	1.36361i	−0.731532	−	0.681807i	$-0.761194\pi$
$521$		−	1.00471e11i		−	1.36361i	0.731532	−	0.681807i	$-0.238806\pi$
$522$		0			0
$523$	−1.84736e10			−0.246914			−0.123457	−	0.992350i	$-0.539398\pi$
$523$	−1.84736e10			−0.246914			−0.123457	+	0.992350i	$0.539398\pi$
$524$		0			0
$525$	2.89046e10	+	2.36089e10i	0.380478	+	0.310770i
$526$		0			0
$527$	1.18036e11	−	6.81479e10i	1.53028	−	0.883507i
$528$		0			0
$529$	7.25297e10	−	1.25625e11i	0.926176	−	1.60418i
$530$		0			0
$531$	2.47019e10	−	7.38986e10i	0.310708	−	0.929518i
$532$		0			0
$533$	5.50729e10	+	3.17963e10i	0.682384	+	0.393975i
$534$		0			0
$535$	9.41390e10	+	1.63053e11i	1.14909	+	1.99028i
$536$		0			0
$537$	−2.34194e10	−	6.17871e10i	−0.281630	−	0.743020i
$538$		0			0
$539$		−	1.26607e10i		−	0.150003i
$540$		0			0
$541$	6.70083e10			0.782239			0.391120	−	0.920340i	$-0.372088\pi$
$541$	6.70083e10			0.782239			0.391120	+	0.920340i	$0.372088\pi$
$542$		0			0
$543$	1.08297e11	−	4.10483e10i	1.24571	−	0.472168i
$544$		0			0
$545$	4.69742e10	−	2.71206e10i	0.532443	−	0.307406i
$546$		0			0
$547$	6.07873e10	−	1.05287e11i	0.678990	−	1.17605i	−0.296295	−	0.955096i	$-0.595751\pi$
$547$	6.07873e10	−	1.05287e11i	0.678990	−	1.17605i	0.975285	−	0.220949i	$-0.0709154\pi$
$548$		0			0
$549$	6.89616e9	+	3.38446e10i	0.0759133	+	0.372564i
$550$		0			0
$551$	−1.33077e11	−	7.68322e10i	−1.44377	−	0.833560i
$552$		0			0
$553$	−2.24442e9	−	3.88746e9i	−0.0239996	−	0.0415686i
$554$		0			0
$555$	−1.38195e11	+	1.69194e11i	−1.45654	+	1.78325i
$556$		0			0
$557$			1.05086e11i			1.09175i	0.837867	+	0.545874i	$0.183802\pi$
$557$			1.05086e11i			1.09175i	−0.837867	+	0.545874i	$0.816198\pi$
$558$		0			0
$559$	4.67802e10			0.479087
$560$		0			0
$561$	2.84128e10	−	1.74623e11i	0.286855	−	1.76299i
$562$		0			0
$563$	4.37399e10	−	2.52533e10i	0.435356	−	0.251353i	−0.266270	−	0.963899i	$-0.585791\pi$
$563$	4.37399e10	−	2.52533e10i	0.435356	−	0.251353i	0.701626	+	0.712546i	$0.252458\pi$
$564$		0			0
$565$	−9.91967e10	+	1.71814e11i	−0.973427	+	1.68602i
$566$		0			0
$567$	−3.12122e10	−	2.34913e10i	−0.301990	−	0.227287i
$568$		0			0
$569$	7.67114e10	+	4.42893e10i	0.731831	+	0.422523i	0.819092	−	0.573663i	$-0.194478\pi$
$569$	7.67114e10	+	4.42893e10i	0.731831	+	0.422523i	−0.0872607	+	0.996186i	$0.527811\pi$
$570$		0			0
$571$	2.24428e10	+	3.88721e10i	0.211122	+	0.365674i	0.952066	−	0.305893i	$-0.0989550\pi$
$571$	2.24428e10	+	3.88721e10i	0.211122	+	0.365674i	−0.740944	+	0.671567i	$0.765622\pi$
$572$		0			0
$573$	−9.92690e10	−	1.61520e10i	−0.920863	−	0.149833i
$574$		0			0
$575$			2.39960e11i			2.19516i
$576$		0			0
$577$	−1.86903e11			−1.68622			−0.843110	−	0.537742i	$-0.819278\pi$
$577$	−1.86903e11			−1.68622			−0.843110	+	0.537742i	$0.819278\pi$
$578$		0			0
$579$	3.03659e9	+	2.48025e9i	0.0270192	+	0.0220689i
$580$		0			0
$581$	−1.14070e10	+	6.58583e9i	−0.100108	+	0.0577971i
$582$		0			0
$583$	−1.16383e11	+	2.01581e11i	−1.00743	+	1.74492i
$584$		0			0
$585$	−1.13975e11	+	2.32235e10i	−0.973167	+	0.198292i
$586$		0			0
$587$	1.02663e10	+	5.92723e9i	0.0864690	+	0.0499229i	0.542611	−	0.839984i	$-0.317436\pi$
$587$	1.02663e10	+	5.92723e9i	0.0864690	+	0.0499229i	−0.456142	+	0.889907i	$0.650769\pi$
$588$		0			0
$589$	7.07737e10	+	1.22584e11i	0.588045	+	1.01852i
$590$		0			0
$591$	−6.66598e9	−	1.75867e10i	−0.0546404	−	0.144157i
$592$		0			0
$593$		−	2.34195e11i		−	1.89391i	−0.321367	−	0.946955i	$-0.604142\pi$
$593$		−	2.34195e11i		−	1.89391i	0.321367	−	0.946955i	$-0.395858\pi$
$594$		0			0
$595$	1.22204e11			0.975030
$596$		0			0
$597$	1.34041e11	−	5.08060e10i	1.05521	−	0.399961i
$598$		0			0
$599$	−2.13631e11	+	1.23340e11i	−1.65943	+	0.958070i	−0.686445	+	0.727181i	$0.740830\pi$
$599$	−2.13631e11	+	1.23340e11i	−1.65943	+	0.958070i	−0.972980	+	0.230888i	$0.925837\pi$
$600$		0			0
$601$	−6.41787e10	+	1.11161e11i	−0.491918	+	0.852027i	−0.999957	−	0.00930704i	$-0.997037\pi$
$601$	−6.41787e10	+	1.11161e11i	−0.491918	+	0.852027i	0.508038	+	0.861334i	$0.330371\pi$
$602$		0			0
$603$	−9.79741e10	−	3.27496e10i	−0.741041	−	0.247706i
$604$		0			0
$605$	1.80439e10	+	1.04176e10i	0.134681	+	0.0777584i
$606$		0			0
$607$	8.67626e8	+	1.50277e9i	0.00639113	+	0.0110698i	0.869203	−	0.494455i	$-0.164632\pi$
$607$	8.67626e8	+	1.50277e9i	0.00639113	+	0.0110698i	−0.862812	+	0.505525i	$0.831299\pi$
$608$		0			0
$609$	−4.84266e10	+	5.92891e10i	−0.352059	+	0.431028i
$610$		0			0
$611$			5.24608e10i			0.376418i
$612$		0			0
$613$	−1.82835e10			−0.129484			−0.0647422	−	0.997902i	$-0.520623\pi$
$613$	−1.82835e10			−0.129484			−0.0647422	+	0.997902i	$0.520623\pi$
$614$		0			0
$615$	4.19179e10	−	2.57625e11i	0.293021	−	1.80089i
$616$		0			0
$617$	−4.02578e10	+	2.32429e10i	−0.277785	+	0.160380i	−0.632420	−	0.774625i	$-0.717938\pi$
$617$	−4.02578e10	+	2.32429e10i	−0.277785	+	0.160380i	0.354635	+	0.935005i	$0.384605\pi$
$618$		0			0
$619$	−5.58678e10	+	9.67658e10i	−0.380539	+	0.659112i	−0.991139	−	0.132826i	$-0.957595\pi$
$619$	−5.58678e10	+	9.67658e10i	−0.380539	+	0.659112i	0.610601	+	0.791939i	$0.290928\pi$
$620$		0			0
$621$	−9.97189e9	−	2.50972e11i	−0.0670519	−	1.68756i
$622$		0			0
$623$	8.93676e10	+	5.15964e10i	0.593236	+	0.342505i
$624$		0			0
$625$	4.65679e10	+	8.06580e10i	0.305187	+	0.528600i
$626$		0			0
$627$	1.81352e11	+	2.95075e10i	1.17341	+	0.190925i
$628$		0			0
$629$			4.04282e11i			2.58275i
$630$		0			0
$631$	2.95503e11			1.86400			0.931998	−	0.362464i	$-0.118064\pi$
$631$	2.95503e11			1.86400			0.931998	+	0.362464i	$0.118064\pi$
$632$		0			0
$633$	1.47612e11	+	1.20568e11i	0.919403	+	0.750958i
$634$		0			0
$635$	−2.32121e11	+	1.34015e11i	−1.42764	+	0.824251i
$636$		0			0
$637$	−7.70209e9	+	1.33404e10i	−0.0467790	+	0.0810236i
$638$		0			0
$639$	−1.50535e11	−	1.70047e11i	−0.902886	−	1.01992i
$640$		0			0
$641$	1.27443e11	+	7.35794e10i	0.754892	+	0.435837i	0.827459	−	0.561526i	$-0.189786\pi$
$641$	1.27443e11	+	7.35794e10i	0.754892	+	0.435837i	−0.0725668	+	0.997364i	$0.523119\pi$
$642$		0			0
$643$	−1.95669e10	−	3.38908e10i	−0.114466	−	0.198261i	0.803100	−	0.595844i	$-0.203182\pi$
$643$	−1.95669e10	−	3.38908e10i	−0.114466	−	0.198261i	−0.917566	+	0.397583i	$0.869849\pi$
$644$		0			0
$645$	−6.80527e10	−	1.79542e11i	−0.393194	−	1.03736i
$646$		0			0
$647$		−	3.37649e11i		−	1.92685i	−0.267979	−	0.963425i	$-0.586356\pi$
$647$		−	3.37649e11i		−	1.92685i	0.267979	−	0.963425i	$-0.413644\pi$
$648$		0			0
$649$	1.82573e11			1.02910
$650$		0			0
$651$	6.59386e10	−	2.49930e10i	0.367127	−	0.139154i
$652$		0			0
$653$	1.17371e11	−	6.77644e10i	0.645520	−	0.372691i	−0.141218	−	0.989979i	$-0.545102\pi$
$653$	1.17371e11	−	6.77644e10i	0.645520	−	0.372691i	0.786738	+	0.617288i	$0.211768\pi$
$654$		0			0
$655$	−1.82242e11	+	3.15653e11i	−0.990110	+	1.71492i
$656$		0			0
$657$	−5.66431e10	+	5.01434e10i	−0.304008	+	0.269124i
$658$		0			0
$659$	−2.46475e11	−	1.42302e11i	−1.30686	−	0.754518i	−0.325292	−	0.945613i	$-0.605463\pi$
$659$	−2.46475e11	−	1.42302e11i	−1.30686	−	0.754518i	−0.981572	+	0.191095i	$0.938796\pi$
$660$		0			0
$661$	−5.33932e10	−	9.24798e10i	−0.279692	−	0.484441i	0.691616	−	0.722265i	$-0.256899\pi$
$661$	−5.33932e10	−	9.24798e10i	−0.279692	−	0.484441i	−0.971308	+	0.237824i	$0.923566\pi$
$662$		0			0
$663$	−1.36170e11	+	1.66714e11i	−0.704737	+	0.862815i
$664$		0			0
$665$			1.26913e11i			0.648961i
$666$		0			0
$667$	−4.92205e11			−2.48681
$668$		0			0
$669$	−1.20433e10	+	7.40174e10i	−0.0601230	+	0.369513i
$670$		0			0
$671$	−7.00896e10	+	4.04663e10i	−0.345751	+	0.199619i
$672$		0			0
$673$	9.25133e10	−	1.60238e11i	0.450966	−	0.781096i	−0.547480	−	0.836819i	$-0.684413\pi$
$673$	9.25133e10	−	1.60238e11i	0.450966	−	0.781096i	0.998446	+	0.0557226i	$0.0177463\pi$
$674$		0			0
$675$	1.44083e11	+	2.28134e11i	0.694060	+	1.09894i
$676$		0			0
$677$	−1.84037e10	−	1.06254e10i	−0.0876093	−	0.0505813i	0.455555	−	0.890207i	$-0.349441\pi$
$677$	−1.84037e10	−	1.06254e10i	−0.0876093	−	0.0505813i	−0.543165	+	0.839626i	$0.682774\pi$
$678$		0			0
$679$	−2.98778e10	−	5.17498e10i	−0.140562	−	0.243461i
$680$		0			0
$681$	−1.22084e11	−	1.98642e10i	−0.567638	−	0.0923599i
$682$		0			0
$683$		−	4.06165e11i		−	1.86647i	−0.359272	−	0.933233i	$-0.616975\pi$
$683$		−	4.06165e11i		−	1.86647i	0.359272	−	0.933233i	$-0.383025\pi$
$684$		0			0
$685$	5.31137e11			2.41237
$686$		0			0
$687$	−2.01068e11	−	1.64230e11i	−0.902645	−	0.737270i
$688$		0			0
$689$	2.45263e11	−	1.41603e11i	1.08832	−	0.628340i
$690$		0			0
$691$	−1.28507e11	+	2.22581e11i	−0.563658	+	0.976284i	0.433515	+	0.901146i	$0.357273\pi$
$691$	−1.28507e11	+	2.22581e11i	−0.563658	+	0.976284i	−0.997173	+	0.0751378i	$0.976060\pi$
$692$		0			0
$693$	2.90186e10	−	8.68125e10i	0.125818	−	0.376400i
$694$		0			0
$695$	3.42763e11	+	1.97894e11i	1.46911	+	0.848191i
$696$		0			0
$697$	−2.41516e11	−	4.18318e11i	−1.02333	−	1.77245i
$698$		0			0
$699$	4.58155e10	+	1.20874e11i	0.191913	+	0.506319i
$700$		0			0
$701$			3.86157e10i			0.159916i	0.996798	+	0.0799580i	$0.0254786\pi$
$701$			3.86157e10i			0.159916i	−0.996798	+	0.0799580i	$0.974521\pi$
$702$		0			0
$703$	−4.19860e11			−1.71903
$704$		0			0
$705$	2.01344e11	−	7.63165e10i	0.815048	−	0.308931i
$706$		0			0
$707$	−5.93757e10	+	3.42806e10i	−0.237646	+	0.137205i
$708$		0			0
$709$	2.35065e11	−	4.07145e11i	0.930258	−	1.61125i	0.147378	−	0.989080i	$-0.452917\pi$
$709$	2.35065e11	−	4.07145e11i	0.930258	−	1.61125i	0.782880	−	0.622173i	$-0.213750\pi$
$710$		0			0
$711$	−6.47957e9	−	3.18001e10i	−0.0253552	−	0.124437i
$712$		0			0
$713$	3.92649e11	+	2.26696e11i	1.51931	+	0.877175i
$714$		0			0
$715$	−1.36274e11	−	2.36034e11i	−0.521423	−	0.903131i
$716$		0			0
$717$	1.20374e11	−	1.47374e11i	0.455465	−	0.557629i
$718$		0			0
$719$		−	1.77844e11i		−	0.665462i	−0.943022	−	0.332731i	$-0.892030\pi$
$719$		−	1.77844e11i		−	0.665462i	0.943022	−	0.332731i	$-0.107970\pi$
$720$		0			0
$721$	1.59686e11			0.590918
$722$		0			0
$723$	−8.04297e10	+	4.94316e11i	−0.294350	+	1.80906i
$724$		0			0
$725$	4.57919e11	−	2.64380e11i	1.65744	−	0.956921i
$726$		0			0
$727$	1.36057e10	−	2.35658e10i	0.0487061	−	0.0843615i	−0.840644	−	0.541587i	$-0.817824\pi$
$727$	1.36057e10	−	2.35658e10i	0.0487061	−	0.0843615i	0.889351	+	0.457226i	$0.151157\pi$
$728$		0			0
$729$	−1.60176e11	−	2.32616e11i	−0.567135	−	0.823625i
$730$		0			0
$731$	−3.07724e11	−	1.77664e11i	−1.07768	−	0.622201i
$732$		0			0
$733$	−8.46238e10	−	1.46573e11i	−0.293141	−	0.507735i	0.681410	−	0.731902i	$-0.261367\pi$
$733$	−8.46238e10	−	1.46573e11i	−0.293141	−	0.507735i	−0.974551	+	0.224167i	$0.928034\pi$
$734$		0			0
$735$	6.24050e10	+	1.01538e10i	0.213831	+	0.0347922i
$736$		0			0
$737$		−	2.42054e11i		−	0.820430i
$738$		0			0
$739$	2.59122e11			0.868813			0.434406	−	0.900717i	$-0.356958\pi$
$739$	2.59122e11			0.868813			0.434406	+	0.900717i	$0.356958\pi$
$740$		0			0
$741$	−1.73138e11	−	1.41417e11i	−0.574273	−	0.469060i
$742$		0			0
$743$	−1.24010e11	+	7.15972e10i	−0.406913	+	0.234931i	−0.689462	−	0.724321i	$-0.742153\pi$
$743$	−1.24010e11	+	7.15972e10i	−0.406913	+	0.234931i	0.282550	+	0.959253i	$0.408820\pi$
$744$		0			0
$745$	4.10823e9	−	7.11566e9i	0.0133361	−	0.0230988i
$746$		0			0
$747$	−9.33113e10	+	1.90130e10i	−0.299676	+	0.0610618i
$748$		0			0
$749$	1.56117e11	+	9.01344e10i	0.496049	+	0.286394i
$750$		0			0
$751$	−1.12374e11	−	1.94637e11i	−0.353269	−	0.611881i	0.633551	−	0.773701i	$-0.281597\pi$
$751$	−1.12374e11	−	1.94637e11i	−0.353269	−	0.611881i	−0.986820	+	0.161821i	$0.948263\pi$
$752$		0			0
$753$	−1.66517e11	−	4.39318e11i	−0.517938	−	1.36647i
$754$		0			0
$755$		−	3.96444e11i		−	1.22009i
$756$		0			0
$757$	−5.30630e11			−1.61588			−0.807938	−	0.589267i	$-0.799417\pi$
$757$	−5.30630e11			−1.61588			−0.807938	+	0.589267i	$0.799417\pi$
$758$		0			0
$759$	5.50323e11	−	2.08592e11i	1.65825	−	0.628535i
$760$		0			0
$761$	1.22330e11	−	7.06272e10i	0.364749	−	0.210588i	−0.306413	−	0.951899i	$-0.599129\pi$
$761$	1.22330e11	−	7.06272e10i	0.364749	−	0.210588i	0.671162	+	0.741311i	$0.265796\pi$
$762$		0			0
$763$	2.59669e10	−	4.49760e10i	0.0766164	−	0.132703i
$764$		0			0
$765$	8.37938e11	+	2.80096e11i	2.44662	+	0.817825i
$766$		0			0
$767$	−1.92375e11	−	1.11068e11i	−0.555863	−	0.320928i
$768$		0			0
$769$	−1.68912e11	−	2.92565e11i	−0.483010	−	0.836598i	0.516799	−	0.856106i	$-0.327123\pi$
$769$	−1.68912e11	−	2.92565e11i	−0.483010	−	0.836598i	−0.999810	+	0.0195082i	$0.993790\pi$
$770$		0			0
$771$	2.85588e11	−	3.49647e11i	0.808207	−	0.989493i
$772$		0			0
$773$			5.64980e9i			0.0158240i	0.999969	+	0.00791198i	$0.00251849\pi$
$773$			5.64980e9i			0.0158240i	−0.999969	+	0.00791198i	$0.997482\pi$
$774$		0			0
$775$	−4.87065e11			−1.35014
$776$		0			0
$777$	−3.35915e10	+	2.06451e11i	−0.0921606	+	0.566414i
$778$		0			0
$779$	4.34436e11	−	2.50822e11i	1.17971	−	0.681107i
$780$		0			0
$781$	2.66071e11	−	4.60848e11i	0.715143	−	1.23866i
$782$		0			0
$783$	−4.67948e11	+	2.95542e11i	−1.24495	+	0.786272i
$784$		0			0
$785$	−1.09592e11	−	6.32731e10i	−0.288603	−	0.166625i
$786$		0			0
$787$	6.60320e10	+	1.14371e11i	0.172130	+	0.298137i	0.939164	−	0.343469i	$-0.111602\pi$
$787$	6.60320e10	+	1.14371e11i	0.172130	+	0.298137i	−0.767035	+	0.641606i	$0.778269\pi$
$788$		0			0
$789$	−6.37273e10	−	1.03690e10i	−0.164444	−	0.0267565i
$790$		0			0
$791$			1.89954e11i			0.485224i
$792$		0			0
$793$	9.84704e10			0.249008
$794$		0			0
$795$	−9.00263e11	−	7.35325e11i	−2.25373	−	1.84082i
$796$		0			0
$797$	−2.38395e11	+	1.37638e11i	−0.590832	+	0.341117i	−0.765427	−	0.643523i	$-0.777472\pi$
$797$	−2.38395e11	+	1.37638e11i	−0.590832	+	0.341117i	0.174594	+	0.984640i	$0.444139\pi$
$798$		0			0
$799$	1.99239e11	−	3.45091e11i	0.488862	−	0.846734i
$800$		0			0
$801$	4.94522e11	+	5.58623e11i	1.20131	+	1.35703i
$802$		0			0
$803$	−1.53509e11	−	8.86286e10i	−0.369209	−	0.213163i
$804$		0			0
$805$	2.03258e11	+	3.52053e11i	0.484021	+	0.838349i
$806$		0			0
$807$	1.56143e11	+	4.11948e11i	0.368152	+	0.971288i
$808$		0			0
$809$		−	2.31340e11i		−	0.540078i	−0.962849	−	0.270039i	$-0.912963\pi$
$809$		−	2.31340e11i		−	0.540078i	0.962849	−	0.270039i	$-0.0870365\pi$
$810$		0			0
$811$	1.71306e11			0.395995			0.197997	−	0.980203i	$-0.436556\pi$
$811$	1.71306e11			0.395995			0.197997	+	0.980203i	$0.436556\pi$
$812$		0			0
$813$	−2.33641e11	+	8.85580e10i	−0.534794	+	0.202705i
$814$		0			0
$815$	3.07896e11	−	1.77764e11i	0.697869	−	0.402915i
$816$		0			0
$817$	1.84510e11	−	3.19581e11i	0.414125	−	0.717286i
$818$		0			0
$819$	−8.33889e10	+	7.38201e10i	−0.185342	+	0.164074i
$820$		0			0
$821$	−4.64052e11	−	2.67921e11i	−1.02140	−	0.589703i	−0.106888	−	0.994271i	$-0.534089\pi$
$821$	−4.64052e11	−	2.67921e11i	−1.02140	−	0.589703i	−0.914508	+	0.404568i	$0.867422\pi$
$822$		0			0
$823$	1.61775e10	+	2.80203e10i	0.0352625	+	0.0610764i	0.883118	−	0.469151i	$-0.155440\pi$
$823$	1.61775e10	+	2.80203e10i	0.0352625	+	0.0610764i	−0.847856	+	0.530227i	$0.822107\pi$
$824$		0			0
$825$	−3.99948e11	+	4.89659e11i	−0.863351	+	1.05701i
$826$		0			0
$827$			6.44183e10i			0.137717i	0.997626	+	0.0688585i	$0.0219357\pi$
$827$			6.44183e10i			0.137717i	−0.997626	+	0.0688585i	$0.978064\pi$
$828$		0			0
$829$	−1.01472e11			−0.214846			−0.107423	−	0.994213i	$-0.534260\pi$
$829$	−1.01472e11			−0.214846			−0.107423	+	0.994213i	$0.534260\pi$
$830$		0			0
$831$	−7.34731e10	+	4.51562e11i	−0.154072	+	0.946919i
$832$		0			0
$833$	1.01330e11	−	5.85029e10i	0.210454	−	0.121506i
$834$		0			0
$835$	−8.25848e10	+	1.43041e11i	−0.169885	+	0.294249i
$836$		0			0
$837$	5.09418e11	−	2.02407e10i	1.03794	−	0.0412405i
$838$		0			0
$839$	−3.57056e11	−	2.06146e11i	−0.720590	−	0.416033i	0.0943799	−	0.995536i	$-0.469913\pi$
$839$	−3.57056e11	−	2.06146e11i	−0.720590	−	0.416033i	−0.814970	+	0.579504i	$0.803246\pi$
$840$		0			0
$841$	2.92172e11	+	5.06057e11i	0.584056	+	1.01162i
$842$		0			0
$843$	−1.99005e11	−	3.23800e10i	−0.394053	−	0.0641160i
$844$		0			0
$845$		−	4.41550e11i		−	0.866069i
$846$		0			0
$847$	1.99490e10			0.0387602
$848$		0			0
$849$	1.34742e11	+	1.10056e11i	0.259341	+	0.211827i
$850$		0			0
$851$	−1.16468e12	+	6.72429e11i	−2.22070	+	1.28212i
$852$		0			0
$853$	2.47387e11	−	4.28487e11i	0.467284	−	0.809360i	−0.532017	−	0.846733i	$-0.678566\pi$
$853$	2.47387e11	−	4.28487e11i	0.467284	−	0.809360i	0.999301	+	0.0373738i	$0.0118992\pi$
$854$		0			0
$855$	−2.90888e11	+	8.70225e11i	−0.544329	+	1.62842i
$856$		0			0
$857$	4.78716e11	+	2.76387e11i	0.887472	+	0.512382i	0.873115	−	0.487515i	$-0.162097\pi$
$857$	4.78716e11	+	2.76387e11i	0.887472	+	0.512382i	0.0143569	+	0.999897i	$0.495430\pi$
$858$		0			0
$859$	−3.58374e11	−	6.20721e11i	−0.658208	−	1.14005i	−0.981079	−	0.193607i	$-0.937981\pi$
$859$	−3.58374e11	−	6.20721e11i	−0.658208	−	1.14005i	0.322871	−	0.946443i	$-0.395352\pi$
$860$		0			0
$861$	−8.85752e10	−	2.33686e11i	−0.161176	−	0.425227i
$862$		0			0
$863$			4.98894e11i			0.899426i	0.893173	+	0.449713i	$0.148474\pi$
$863$			4.98894e11i			0.899426i	−0.893173	+	0.449713i	$0.851526\pi$
$864$		0			0
$865$	1.45161e12			2.59289
$866$		0			0
$867$	1.00054e12	−	3.79238e11i	1.77075	−	0.671174i
$868$		0			0
$869$	6.58556e10	−	3.80217e10i	0.115482	−	0.0666734i
$870$		0			0
$871$	−1.47253e11	+	2.55050e11i	−0.255854	+	0.443151i
$872$		0			0
$873$	−8.62560e10	−	4.23323e11i	−0.148502	−	0.728811i
$874$		0			0
$875$	−8.72243e10	−	5.03590e10i	−0.148801	−	0.0859102i
$876$		0			0
$877$	3.78324e11	+	6.55276e11i	0.639537	+	1.10771i	0.985535	+	0.169475i	$0.0542071\pi$
$877$	3.78324e11	+	6.55276e11i	0.639537	+	1.10771i	−0.345998	+	0.938235i	$0.612460\pi$
$878$		0			0
$879$	−4.66403e11	+	5.71020e11i	−0.781278	+	0.956524i
$880$		0			0
$881$		−	1.36437e11i		−	0.226479i	−0.993568	−	0.113239i	$-0.963877\pi$
$881$		−	1.36437e11i		−	0.226479i	0.993568	−	0.113239i	$-0.0361227\pi$
$882$		0			0
$883$	−7.71632e10			−0.126931			−0.0634655	−	0.997984i	$-0.520215\pi$
$883$	−7.71632e10			−0.126931			−0.0634655	+	0.997984i	$0.520215\pi$
$884$		0			0
$885$	−1.46424e11	+	8.99910e11i	−0.238692	+	1.46699i
$886$		0			0
$887$	−4.01721e11	+	2.31933e11i	−0.648977	+	0.374687i	−0.788064	−	0.615593i	$-0.788917\pi$
$887$	−4.01721e11	+	2.31933e11i	−0.648977	+	0.374687i	0.139087	+	0.990280i	$0.455583\pi$
$888$		0			0
$889$	−1.28314e11	+	2.22247e11i	−0.205432	+	0.355819i
$890$		0			0
$891$	3.97954e11	−	5.28751e11i	0.631425	−	0.838959i
$892$		0			0
$893$	3.58388e11	+	2.06915e11i	0.563570	+	0.325377i
$894$		0			0
$895$	3.86593e11	+	6.69599e11i	0.602507	+	1.04357i
$896$		0			0
$897$	−7.06767e11	−	1.14997e11i	−1.09171	−	0.177631i
$898$		0			0
$899$		−	9.99067e11i		−	1.52952i
$900$		0			0
$901$	−2.15115e12			−3.26415
$902$		0			0
$903$	−1.42381e11	−	1.16295e11i	−0.214141	−	0.174908i
$904$		0			0
$905$	−1.17364e12	+	6.77600e11i	−1.74960	+	1.01013i
$906$		0			0
$907$	−8.97633e10	+	1.55475e11i	−0.132639	+	0.229737i	−0.924693	−	0.380714i	$-0.875678\pi$
$907$	−8.97633e10	+	1.55475e11i	−0.132639	+	0.229737i	0.792054	+	0.610451i	$0.209012\pi$
$908$		0			0
$909$	−4.85704e11	+	9.89668e10i	−0.711404	+	0.144955i
$910$		0			0
$911$	4.05664e11	+	2.34210e11i	0.588970	+	0.340042i	0.764690	−	0.644398i	$-0.222892\pi$
$911$	4.05664e11	+	2.34210e11i	0.588970	+	0.340042i	−0.175720	+	0.984440i	$0.556225\pi$
$912$		0			0
$913$	−1.11567e11	−	1.93240e11i	−0.160566	−	0.278109i
$914$		0			0
$915$	−1.43248e11	−	3.77929e11i	−0.204364	−	0.539170i
$916$		0			0
$917$			3.48980e11i			0.493541i
$918$		0			0
$919$	9.56118e11			1.34045			0.670223	−	0.742159i	$-0.266198\pi$
$919$	9.56118e11			1.34045			0.670223	+	0.742159i	$0.266198\pi$
$920$		0			0
$921$	−2.91582e11	+	1.10520e11i	−0.405250	+	0.153604i
$922$		0			0
$923$	−5.60713e11	+	3.23728e11i	−0.772562	+	0.446039i
$924$		0			0
$925$	7.22369e11	−	1.25118e12i	0.986715	−	1.70904i
$926$		0			0
$927$	1.09495e12	+	3.66006e11i	1.48278	+	0.495644i
$928$		0			0
$929$	−9.45855e10	−	5.46090e10i	−0.126988	−	0.0733164i	0.435160	−	0.900353i	$-0.356692\pi$
$929$	−9.45855e10	−	5.46090e10i	−0.126988	−	0.0733164i	−0.562148	+	0.827036i	$0.690025\pi$
$930$		0			0
$931$	6.07570e10	+	1.05234e11i	0.0808720	+	0.140074i
$932$		0			0
$933$	5.61366e11	−	6.87284e11i	0.740832	−	0.907005i
$934$		0			0
$935$			2.07020e12i			2.70873i
$936$		0			0
$937$	2.31192e11			0.299926			0.149963	−	0.988692i	$-0.452085\pi$
$937$	2.31192e11			0.299926			0.149963	+	0.988692i	$0.452085\pi$
$938$		0			0
$939$	4.10912e10	−	2.52544e11i	0.0528550	−	0.324844i
$940$		0			0
$941$	−1.06268e12	+	6.13540e11i	−1.35533	+	0.782501i	−0.988990	−	0.147980i	$-0.952723\pi$
$941$	−1.06268e12	+	6.13540e11i	−1.35533	+	0.782501i	−0.366340	+	0.930481i	$0.619389\pi$
$942$		0			0
$943$	8.03411e11	−	1.39155e12i	1.01599	−	1.75975i
$944$		0			0
$945$	4.04630e11	+	2.12658e11i	0.507377	+	0.266658i
$946$		0			0
$947$	−8.64986e11	−	4.99400e11i	−1.07550	−	0.620938i	−0.145819	−	0.989311i	$-0.546582\pi$
$947$	−8.64986e11	−	4.99400e11i	−1.07550	−	0.620938i	−0.929678	+	0.368373i	$0.879915\pi$
$948$		0			0
$949$	1.07834e11	+	1.86774e11i	0.132951	+	0.230278i
$950$		0			0
$951$	6.86881e11	+	1.11762e11i	0.839768	+	0.136638i
$952$		0			0
$953$		−	3.34249e10i		−	0.0405227i	−0.999795	−	0.0202614i	$-0.993550\pi$
$953$		−	3.34249e10i		−	0.0405227i	0.999795	−	0.0202614i	$-0.00644983\pi$
$954$		0			0
$955$	1.17686e12			1.41485
$956$		0			0
$957$	−1.00439e12	−	8.20372e11i	−1.19744	−	0.978055i
$958$		0			0
$959$	4.40411e11	−	2.54272e11i	0.520696	−	0.300624i
$960$		0			0
$961$	−3.36977e10	+	5.83661e10i	−0.0395099	+	0.0684332i
$962$		0			0
$963$	8.63887e11	+	9.75867e11i	1.00450	+	1.13471i
$964$		0			0
$965$	−3.97320e10	−	2.29393e10i	−0.0458174	−	0.0264527i
$966$		0			0
$967$	−7.29647e10	−	1.26379e11i	−0.0834463	−	0.144533i	0.821282	−	0.570523i	$-0.193259\pi$
$967$	−7.29647e10	−	1.26379e11i	−0.0834463	−	0.144533i	−0.904728	+	0.425990i	$0.859926\pi$
$968$		0			0
$969$	6.01832e11	+	1.58780e12i	0.682622	+	1.80095i
$970$		0			0
$971$		−	8.13411e11i		−	0.915025i	−0.889203	−	0.457513i	$-0.848740\pi$
$971$		−	8.13411e11i		−	0.915025i	0.889203	−	0.457513i	$-0.151260\pi$
$972$		0			0
$973$	3.78952e11			0.422798
$974$		0			0
$975$	7.19304e11	−	2.72641e11i	0.795965	−	0.301699i
$976$		0			0
$977$	−3.78623e11	+	2.18598e11i	−0.415555	+	0.239921i	−0.693174	−	0.720770i	$-0.743788\pi$
$977$	−3.78623e11	+	2.18598e11i	−0.415555	+	0.239921i	0.277619	+	0.960691i	$0.410455\pi$
$978$		0			0
$979$	−8.74070e11	+	1.51393e12i	−0.951514	+	1.64807i
$980$		0			0
$981$	2.81138e11	−	2.48878e11i	0.303559	−	0.268726i
$982$		0			0
$983$	3.85592e11	+	2.22622e11i	0.412966	+	0.238426i	0.692063	−	0.721837i	$-0.256702\pi$
$983$	3.85592e11	+	2.22622e11i	0.412966	+	0.238426i	−0.279098	+	0.960263i	$0.590035\pi$
$984$		0			0
$985$	1.10038e11	+	1.90591e11i	0.116895	+	0.202469i
$986$		0			0
$987$	1.30417e11	−	1.59670e11i	0.137425	−	0.168250i
$988$		0			0
$989$		−	1.18201e12i		−	1.23548i
$990$		0			0
$991$	1.55614e12			1.61345			0.806723	−	0.590929i	$-0.201239\pi$
$991$	1.55614e12			1.61345			0.806723	+	0.590929i	$0.201239\pi$
$992$		0			0
$993$	9.97749e10	−	6.13210e11i	0.102618	−	0.630685i
$994$		0			0
$995$	−1.45262e12	+	8.38673e11i	−1.48204	+	0.855658i
$996$		0			0
$997$	−4.08150e11	+	7.06937e11i	−0.413085	+	0.715484i	−0.995225	−	0.0976042i	$-0.968882\pi$
$997$	−4.08150e11	+	7.06937e11i	−0.413085	+	0.715484i	0.582140	+	0.813088i	$0.302215\pi$
$998$		0			0
$999$	−7.03526e11	+	1.33862e12i	−0.706347	+	1.34399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.6	✓	96
9.5	odd	6	inner	252.9.bg.a.113.6	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.6	✓	96	1.1	even	1	trivial
252.9.bg.a.113.6	yes	96	9.5	odd	6	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-75.7417 + 28.7087i) q^{3} +(820.829 - 473.906i) q^{5} +(453.746 - 785.912i) q^{7} +(4912.62 - 4348.90i) q^{9} +O(q^{10})\)	\(q+(-75.7417 + 28.7087i) q^{3} +(820.829 - 473.906i) q^{5} +(453.746 - 785.912i) q^{7} +(4912.62 - 4348.90i) q^{9} +(13313.8 + 7686.70i) q^{11} +(-9352.38 - 16198.8i) q^{13} +(-48565.7 + 59459.4i) q^{15} +142076. i q^{17} -147550. q^{19} +(-11805.0 + 72552.8i) q^{21} +(-409302. + 236310. i) q^{23} +(253861. - 439699. i) q^{25} +(-247239. + 470428. i) q^{27} +(901911. + 520718. i) q^{29} +(-479658. - 830792. i) q^{31} +(-1.22908e6 - 199983. i) q^{33} -860132. i q^{35} +2.84553e6 q^{37} +(1.17341e6 + 958430. i) q^{39} +(-2.94432e6 + 1.69991e6i) q^{41} +(-1.25049e6 + 2.16591e6i) q^{43} +(1.97145e6 - 5.89782e6i) q^{45} +(-2.42892e6 - 1.40234e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-4.07882e6 - 1.07611e7i) q^{51} +1.51408e7i q^{53} +1.45711e7 q^{55} +(1.11757e7 - 4.23599e6i) q^{57} +(1.02848e7 - 5.93795e6i) q^{59} +(-2.63223e6 + 4.55915e6i) q^{61} +(-1.18877e6 - 5.83418e6i) q^{63} +(-1.53534e7 - 8.86429e6i) q^{65} +(-7.87249e6 - 1.36355e7i) q^{67} +(2.42170e7 - 2.96491e7i) q^{69} -3.46144e7i q^{71} -1.15301e7 q^{73} +(-6.60462e6 + 4.05916e7i) q^{75} +(1.20821e7 - 6.97562e6i) q^{77} +(2.47322e6 - 4.28373e6i) q^{79} +(5.22088e6 - 4.27289e7i) q^{81} +(-1.25698e7 - 7.25717e6i) q^{83} +(6.73306e7 + 1.16620e8i) q^{85} +(-8.32614e7 - 1.35474e7i) q^{87} +1.13712e8i q^{89} -1.69744e7 q^{91} +(6.01811e7 + 4.91552e7i) q^{93} +(-1.21114e8 + 6.99250e7i) q^{95} +(3.29234e7 - 5.70251e7i) q^{97} +(9.88340e7 - 2.01384e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more