LMFDB - Embedded newform 450.6.f.f.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, 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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 $ x^16 - 252x^14 + 27174x^12 - 1635700x^10 + 60061815x^8 - 1376564028x^6 + 19220200150x^4 - 149540021784*x^2 + 498214340649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{8}\cdot 5^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.4
Root			$-7.20885 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.f.143.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})$	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(45.2548 + 45.2548i) q^{8} -28.2771i q^{11} +(283.237 - 283.237i) q^{13} -662.815 q^{14} -256.000 q^{16} +(137.842 - 137.842i) q^{17} +2809.99i q^{19} +(79.9797 + 79.9797i) q^{22} +(902.809 + 902.809i) q^{23} +1602.23i q^{26} +(1874.73 - 1874.73i) q^{28} +827.573 q^{29} +2049.64 q^{31} +(724.077 - 724.077i) q^{32} +779.754i q^{34} +(-1320.83 - 1320.83i) q^{37} +(-7947.85 - 7947.85i) q^{38} -10791.6i q^{41} +(-7831.78 + 7831.78i) q^{43} -452.433 q^{44} -5107.06 q^{46} +(11749.4 - 11749.4i) q^{47} +10650.8i q^{49} +(-4531.80 - 4531.80i) q^{52} +(-22576.3 - 22576.3i) q^{53} +10605.0i q^{56} +(-2340.73 + 2340.73i) q^{58} +33192.4 q^{59} +32011.7 q^{61} +(-5797.27 + 5797.27i) q^{62} +4096.00i q^{64} +(27419.3 + 27419.3i) q^{67} +(-2205.48 - 2205.48i) q^{68} +54641.5i q^{71} +(14250.4 - 14250.4i) q^{73} +7471.75 q^{74} +44959.8 q^{76} +(3313.23 - 3313.23i) q^{77} +79274.9i q^{79} +(30523.2 + 30523.2i) q^{82} +(-36508.8 - 36508.8i) q^{83} -44303.2i q^{86} +(1279.67 - 1279.67i) q^{88} -69924.7 q^{89} +66374.0 q^{91} +(14444.9 - 14444.9i) q^{92} +66464.6i q^{94} +(69381.8 + 69381.8i) q^{97} +(-30124.9 - 30124.9i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 528 q^{7}+O(q^{10}) $ 16 * q - 528 * q^7	$ 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) $ 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$2$	−2.82843	+	2.82843i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	117.170	+	117.170i	0.903800	+	0.903800i	0.995763	−	0.0919622i	$-0.0293139\pi$
$7$	117.170	+	117.170i	0.903800	+	0.903800i	−0.0919622	+	0.995763i	$0.529314\pi$
$8$	45.2548	+	45.2548i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$		−	28.2771i		−	0.0704616i	−0.999379	−	0.0352308i	$-0.988783\pi$
$11$		−	28.2771i		−	0.0704616i	0.999379	−	0.0352308i	$-0.0112166\pi$
$12$		0			0
$13$	283.237	−	283.237i	0.464828	−	0.464828i	−0.435406	−	0.900234i	$-0.643395\pi$
$13$	283.237	−	283.237i	0.464828	−	0.464828i	0.900234	+	0.435406i	$0.143395\pi$
$14$	−662.815			−0.903800
$15$		0			0
$16$	−256.000			−0.250000
$17$	137.842	−	137.842i	0.115681	−	0.115681i	−0.646897	−	0.762577i	$-0.723934\pi$
$17$	137.842	−	137.842i	0.115681	−	0.115681i	0.762577	+	0.646897i	$0.223934\pi$
$18$		0			0
$19$			2809.99i			1.78575i	0.450305	+	0.892875i	$0.351315\pi$
$19$			2809.99i			1.78575i	−0.450305	+	0.892875i	$0.648685\pi$
$20$		0			0
$21$		0			0
$22$	79.9797	+	79.9797i	0.0352308	+	0.0352308i
$23$	902.809	+	902.809i	0.355858	+	0.355858i	0.862284	−	0.506426i	$-0.169034\pi$
$23$	902.809	+	902.809i	0.355858	+	0.355858i	−0.506426	+	0.862284i	$0.669034\pi$
$24$		0			0
$25$		0			0
$26$			1602.23i			0.464828i
$27$		0			0
$28$	1874.73	−	1874.73i	0.451900	−	0.451900i
$29$	827.573			0.182731			0.0913653	−	0.995817i	$-0.470877\pi$
$29$	827.573			0.182731			0.0913653	+	0.995817i	$0.470877\pi$
$30$		0			0
$31$	2049.64			0.383067			0.191533	−	0.981486i	$-0.438654\pi$
$31$	2049.64			0.383067			0.191533	+	0.981486i	$0.438654\pi$
$32$	724.077	−	724.077i	0.125000	−	0.125000i
$33$		0			0
$34$			779.754i			0.115681i
$35$		0			0
$36$		0			0
$37$	−1320.83	−	1320.83i	−0.158615	−	0.158615i	0.623338	−	0.781953i	$-0.285776\pi$
$37$	−1320.83	−	1320.83i	−0.158615	−	0.158615i	−0.781953	+	0.623338i	$0.785776\pi$
$38$	−7947.85	−	7947.85i	−0.892875	−	0.892875i
$39$		0			0
$40$		0			0
$41$		−	10791.6i		−	1.00259i	−0.865275	−	0.501297i	$-0.832856\pi$
$41$		−	10791.6i		−	1.00259i	0.865275	−	0.501297i	$-0.167144\pi$
$42$		0			0
$43$	−7831.78	+	7831.78i	−0.645936	+	0.645936i	−0.952008	−	0.306073i	$-0.900985\pi$
$43$	−7831.78	+	7831.78i	−0.645936	+	0.645936i	0.306073	+	0.952008i	$0.400985\pi$
$44$	−452.433			−0.0352308
$45$		0			0
$46$	−5107.06			−0.355858
$47$	11749.4	−	11749.4i	0.775838	−	0.775838i	−0.203282	−	0.979120i	$-0.565161\pi$
$47$	11749.4	−	11749.4i	0.775838	−	0.775838i	0.979120	+	0.203282i	$0.0651609\pi$
$48$		0			0
$49$			10650.8i			0.633710i
$50$		0			0
$51$		0			0
$52$	−4531.80	−	4531.80i	−0.232414	−	0.232414i
$53$	−22576.3	−	22576.3i	−1.10398	−	1.10398i	−0.993925	−	0.110059i	$-0.964896\pi$
$53$	−22576.3	−	22576.3i	−1.10398	−	1.10398i	−0.110059	−	0.993925i	$-0.535104\pi$
$54$		0			0
$55$		0			0
$56$			10605.0i			0.451900i
$57$		0			0
$58$	−2340.73	+	2340.73i	−0.0913653	+	0.0913653i
$59$	33192.4			1.24139			0.620695	−	0.784052i	$-0.286850\pi$
$59$	33192.4			1.24139			0.620695	+	0.784052i	$0.286850\pi$
$60$		0			0
$61$	32011.7			1.10150			0.550749	−	0.834671i	$-0.314342\pi$
$61$	32011.7			1.10150			0.550749	+	0.834671i	$0.314342\pi$
$62$	−5797.27	+	5797.27i	−0.191533	+	0.191533i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	27419.3	+	27419.3i	0.746223	+	0.746223i	0.973768	−	0.227544i	$-0.0730696\pi$
$67$	27419.3	+	27419.3i	0.746223	+	0.746223i	−0.227544	+	0.973768i	$0.573070\pi$
$68$	−2205.48	−	2205.48i	−0.0578403	−	0.0578403i
$69$		0			0
$70$		0			0
$71$			54641.5i			1.28640i	0.765698	+	0.643200i	$0.222394\pi$
$71$			54641.5i			1.28640i	−0.765698	+	0.643200i	$0.777606\pi$
$72$		0			0
$73$	14250.4	−	14250.4i	0.312982	−	0.312982i	−0.533081	−	0.846064i	$-0.678966\pi$
$73$	14250.4	−	14250.4i	0.312982	−	0.312982i	0.846064	+	0.533081i	$0.178966\pi$
$74$	7471.75			0.158615
$75$		0			0
$76$	44959.8			0.892875
$77$	3313.23	−	3313.23i	0.0636833	−	0.0636833i
$78$		0			0
$79$			79274.9i			1.42912i	0.699576	+	0.714559i	$0.253372\pi$
$79$			79274.9i			1.42912i	−0.699576	+	0.714559i	$0.746628\pi$
$80$		0			0
$81$		0			0
$82$	30523.2	+	30523.2i	0.501297	+	0.501297i
$83$	−36508.8	−	36508.8i	−0.581704	−	0.581704i	0.353667	−	0.935371i	$-0.384935\pi$
$83$	−36508.8	−	36508.8i	−0.581704	−	0.581704i	−0.935371	+	0.353667i	$0.884935\pi$
$84$		0			0
$85$		0			0
$86$		−	44303.2i		−	0.645936i
$87$		0			0
$88$	1279.67	−	1279.67i	0.0176154	−	0.0176154i
$89$	−69924.7			−0.935741			−0.467871	−	0.883797i	$-0.654979\pi$
$89$	−69924.7			−0.935741			−0.467871	+	0.883797i	$0.654979\pi$
$90$		0			0
$91$	66374.0			0.840223
$92$	14444.9	−	14444.9i	0.177929	−	0.177929i
$93$		0			0
$94$			66464.6i			0.775838i
$95$		0			0
$96$		0			0
$97$	69381.8	+	69381.8i	0.748715	+	0.748715i	0.974238	−	0.225523i	$-0.0724091\pi$
$97$	69381.8	+	69381.8i	0.748715	+	0.748715i	−0.225523	+	0.974238i	$0.572409\pi$
$98$	−30124.9	−	30124.9i	−0.316855	−	0.316855i
$99$		0			0
$100$		0			0
$101$			163404.i			1.59389i	0.604049	+	0.796947i	$0.293553\pi$
$101$			163404.i			1.59389i	−0.604049	+	0.796947i	$0.706447\pi$
$102$		0			0
$103$	−78791.9	+	78791.9i	−0.731793	+	0.731793i	−0.970975	−	0.239181i	$-0.923121\pi$
$103$	−78791.9	+	78791.9i	−0.731793	+	0.731793i	0.239181	+	0.970975i	$0.423121\pi$
$104$	25635.7			0.232414
$105$		0			0
$106$	127711.			1.10398
$107$	−59824.6	+	59824.6i	−0.505150	+	0.505150i	−0.913034	−	0.407884i	$-0.866267\pi$
$107$	−59824.6	+	59824.6i	−0.505150	+	0.505150i	0.407884	+	0.913034i	$0.366267\pi$
$108$		0			0
$109$			3640.35i			0.0293479i	0.999892	+	0.0146739i	$0.00467103\pi$
$109$			3640.35i			0.0293479i	−0.999892	+	0.0146739i	$0.995329\pi$
$110$		0			0
$111$		0			0
$112$	−29995.6	−	29995.6i	−0.225950	−	0.225950i
$113$	−141328.	−	141328.i	−1.04120	−	1.04120i	−0.999114	−	0.0420834i	$-0.986600\pi$
$113$	−141328.	−	141328.i	−1.04120	−	1.04120i	−0.0420834	−	0.999114i	$-0.513400\pi$
$114$		0			0
$115$		0			0
$116$		−	13241.2i		−	0.0913653i
$117$		0			0
$118$	−93882.3	+	93882.3i	−0.620695	+	0.620695i
$119$	32302.1			0.209104
$120$		0			0
$121$	160251.			0.995035
$122$	−90542.7	+	90542.7i	−0.550749	+	0.550749i
$123$		0			0
$124$		−	32794.3i		−	0.191533i
$125$		0			0
$126$		0			0
$127$	−99907.0	−	99907.0i	−0.549650	−	0.549650i	0.376689	−	0.926340i	$-0.377062\pi$
$127$	−99907.0	−	99907.0i	−0.549650	−	0.549650i	−0.926340	+	0.376689i	$0.877062\pi$
$128$	−11585.2	−	11585.2i	−0.0625000	−	0.0625000i
$129$		0			0
$130$		0			0
$131$			189944.i			0.967047i	0.875331	+	0.483524i	$0.160643\pi$
$131$			189944.i			0.967047i	−0.875331	+	0.483524i	$0.839357\pi$
$132$		0			0
$133$	−329247.	+	329247.i	−1.61396	+	1.61396i
$134$	−155107.			−0.746223
$135$		0			0
$136$	12476.1			0.0578403
$137$	266688.	−	266688.i	1.21395	−	1.21395i	0.244236	−	0.969716i	$-0.421463\pi$
$137$	266688.	−	266688.i	1.21395	−	1.21395i	0.969716	−	0.244236i	$-0.0785371\pi$
$138$		0			0
$139$			242739.i			1.06562i	0.846234	+	0.532811i	$0.178864\pi$
$139$			242739.i			1.06562i	−0.846234	+	0.532811i	$0.821136\pi$
$140$		0			0
$141$		0			0
$142$	−154549.	−	154549.i	−0.643200	−	0.643200i
$143$	−8009.12	−	8009.12i	−0.0327525	−	0.0327525i
$144$		0			0
$145$		0			0
$146$			80612.4i			0.312982i
$147$		0			0
$148$	−21133.3	+	21133.3i	−0.0793073	+	0.0793073i
$149$	−479131.			−1.76803			−0.884013	−	0.467462i	$-0.845168\pi$
$149$	−479131.			−1.76803			−0.884013	+	0.467462i	$0.845168\pi$
$150$		0			0
$151$	181321.			0.647151			0.323576	−	0.946202i	$-0.395115\pi$
$151$	181321.			0.647151			0.323576	+	0.946202i	$0.395115\pi$
$152$	−127166.	+	127166.i	−0.446437	+	0.446437i
$153$		0			0
$154$			18742.5i			0.0636833i
$155$		0			0
$156$		0			0
$157$	261002.	+	261002.i	0.845075	+	0.845075i	0.989514	−	0.144439i	$-0.0461377\pi$
$157$	261002.	+	261002.i	0.845075	+	0.845075i	−0.144439	+	0.989514i	$0.546138\pi$
$158$	−224223.	−	224223.i	−0.714559	−	0.714559i
$159$		0			0
$160$		0			0
$161$			211565.i			0.643249i
$162$		0			0
$163$	−414824.	+	414824.i	−1.22291	+	1.22291i	−0.256317	+	0.966593i	$0.582509\pi$
$163$	−414824.	+	414824.i	−1.22291	+	1.22291i	−0.966593	+	0.256317i	$0.917491\pi$
$164$	−172665.			−0.501297
$165$		0			0
$166$	206525.			0.581704
$167$	−282143.	+	282143.i	−0.782850	+	0.782850i	−0.980311	−	0.197461i	$-0.936731\pi$
$167$	−282143.	+	282143.i	−0.782850	+	0.782850i	0.197461	+	0.980311i	$0.436731\pi$
$168$		0			0
$169$			210846.i			0.567870i
$170$		0			0
$171$		0			0
$172$	125308.	+	125308.i	0.322968	+	0.322968i
$173$	516050.	+	516050.i	1.31092	+	1.31092i	0.920737	+	0.390183i	$0.127588\pi$
$173$	516050.	+	516050.i	1.31092	+	1.31092i	0.390183	+	0.920737i	$0.372412\pi$
$174$		0			0
$175$		0			0
$176$			7238.93i			0.0176154i
$177$		0			0
$178$	197777.	−	197777.i	0.467871	−	0.467871i
$179$	−183417.			−0.427865			−0.213933	−	0.976848i	$-0.568627\pi$
$179$	−183417.			−0.427865			−0.213933	+	0.976848i	$0.568627\pi$
$180$		0			0
$181$	333015.			0.755557			0.377779	−	0.925896i	$-0.376688\pi$
$181$	333015.			0.755557			0.377779	+	0.925896i	$0.376688\pi$
$182$	−187734.	+	187734.i	−0.420112	+	0.420112i
$183$		0			0
$184$			81712.9i			0.177929i
$185$		0			0
$186$		0			0
$187$	−3897.78	−	3897.78i	−0.00815105	−	0.00815105i
$188$	−187990.	−	187990.i	−0.387919	−	0.387919i
$189$		0			0
$190$		0			0
$191$			872731.i			1.73100i	0.500910	+	0.865500i	$0.332999\pi$
$191$			872731.i			1.73100i	−0.500910	+	0.865500i	$0.667001\pi$
$192$		0			0
$193$	−198341.	+	198341.i	−0.383283	+	0.383283i	−0.872284	−	0.489000i	$-0.837362\pi$
$193$	−198341.	+	198341.i	−0.383283	+	0.383283i	0.489000	+	0.872284i	$0.337362\pi$
$194$	−392483.			−0.748715
$195$		0			0
$196$	170412.			0.316855
$197$	63308.1	−	63308.1i	0.116223	−	0.116223i	−0.646603	−	0.762827i	$-0.723811\pi$
$197$	63308.1	−	63308.1i	0.116223	−	0.116223i	0.762827	+	0.646603i	$0.223811\pi$
$198$		0			0
$199$			354056.i			0.633782i	0.948462	+	0.316891i	$0.102639\pi$
$199$			354056.i			0.633782i	−0.948462	+	0.316891i	$0.897361\pi$
$200$		0			0
$201$		0			0
$202$	−462177.	−	462177.i	−0.796947	−	0.796947i
$203$	96967.0	+	96967.0i	0.165152	+	0.165152i
$204$		0			0
$205$		0			0
$206$		−	445714.i		−	0.731793i
$207$		0			0
$208$	−72508.8	+	72508.8i	−0.116207	+	0.116207i
$209$	79458.3			0.125827
$210$		0			0
$211$	−295633.			−0.457138			−0.228569	−	0.973528i	$-0.573405\pi$
$211$	−295633.			−0.457138			−0.228569	+	0.973528i	$0.573405\pi$
$212$	−361220.	+	361220.i	−0.551992	+	0.551992i
$213$		0			0
$214$		−	338419.i		−	0.505150i
$215$		0			0
$216$		0			0
$217$	240158.	+	240158.i	0.346216	+	0.346216i
$218$	−10296.5	−	10296.5i	−0.0146739	−	0.0146739i
$219$		0			0
$220$		0			0
$221$		−	78084.2i		−	0.107543i
$222$		0			0
$223$	−72165.1	+	72165.1i	−0.0971773	+	0.0971773i	−0.754024	−	0.656847i	$-0.771890\pi$
$223$	−72165.1	+	72165.1i	−0.0971773	+	0.0971773i	0.656847	+	0.754024i	$0.271890\pi$
$224$	169681.			0.225950
$225$		0			0
$226$	799474.			1.04120
$227$	674789.	−	674789.i	0.869168	−	0.869168i	−0.123213	−	0.992380i	$-0.539320\pi$
$227$	674789.	−	674789.i	0.869168	−	0.869168i	0.992380	+	0.123213i	$0.0393197\pi$
$228$		0			0
$229$		−	1.07253e6i		−	1.35152i	−0.737122	−	0.675760i	$-0.763816\pi$
$229$		−	1.07253e6i		−	1.35152i	0.737122	−	0.675760i	$-0.236184\pi$
$230$		0			0
$231$		0			0
$232$	37451.7	+	37451.7i	0.0456827	+	0.0456827i
$233$	−668430.	−	668430.i	−0.806615	−	0.806615i	0.177505	−	0.984120i	$-0.443197\pi$
$233$	−668430.	−	668430.i	−0.806615	−	0.806615i	−0.984120	+	0.177505i	$0.943197\pi$
$234$		0			0
$235$		0			0
$236$		−	531078.i		−	0.620695i
$237$		0			0
$238$	−91364.1	+	91364.1i	−0.104552	+	0.104552i
$239$	1.02490e6			1.16061			0.580306	−	0.814399i	$-0.302933\pi$
$239$	1.02490e6			1.16061			0.580306	+	0.814399i	$0.302933\pi$
$240$		0			0
$241$	−465821.			−0.516626			−0.258313	−	0.966061i	$-0.583167\pi$
$241$	−465821.			−0.516626			−0.258313	+	0.966061i	$0.583167\pi$
$242$	−453259.	+	453259.i	−0.497518	+	0.497518i
$243$		0			0
$244$		−	512187.i		−	0.550749i
$245$		0			0
$246$		0			0
$247$	795894.	+	795894.i	0.830066	+	0.830066i
$248$	92756.3	+	92756.3i	0.0957667	+	0.0957667i
$249$		0			0
$250$		0			0
$251$			1.02118e6i			1.02310i	0.859254	+	0.511550i	$0.170928\pi$
$251$			1.02118e6i			1.02310i	−0.859254	+	0.511550i	$0.829072\pi$
$252$		0			0
$253$	25528.8	−	25528.8i	0.0250743	−	0.0250743i
$254$	565159.			0.549650
$255$		0			0
$256$	65536.0			0.0625000
$257$	510335.	−	510335.i	0.481973	−	0.481973i	−0.423789	−	0.905761i	$-0.639300\pi$
$257$	510335.	−	510335.i	0.481973	−	0.481973i	0.905761	+	0.423789i	$0.139300\pi$
$258$		0			0
$259$		−	309525.i		−	0.286712i
$260$		0			0
$261$		0			0
$262$	−537243.	−	537243.i	−0.483524	−	0.483524i
$263$	320756.	+	320756.i	0.285947	+	0.285947i	0.835475	−	0.549528i	$-0.185193\pi$
$263$	320756.	+	320756.i	0.285947	+	0.285947i	−0.549528	+	0.835475i	$0.685193\pi$
$264$		0			0
$265$		0			0
$266$		−	1.86250e6i		−	1.61396i
$267$		0			0
$268$	438708.	−	438708.i	0.373112	−	0.373112i
$269$	85841.8			0.0723300			0.0361650	−	0.999346i	$-0.488486\pi$
$269$	85841.8			0.0723300			0.0361650	+	0.999346i	$0.488486\pi$
$270$		0			0
$271$	−2.28993e6			−1.89409			−0.947043	−	0.321107i	$-0.895945\pi$
$271$	−2.28993e6			−1.89409			−0.947043	+	0.321107i	$0.895945\pi$
$272$	−35287.7	+	35287.7i	−0.0289202	+	0.0289202i
$273$		0			0
$274$			1.50861e6i			1.21395i
$275$		0			0
$276$		0			0
$277$	−431331.	−	431331.i	−0.337762	−	0.337762i	0.517762	−	0.855525i	$-0.326765\pi$
$277$	−431331.	−	431331.i	−0.337762	−	0.337762i	−0.855525	+	0.517762i	$0.826765\pi$
$278$	−686570.	−	686570.i	−0.532811	−	0.532811i
$279$		0			0
$280$		0			0
$281$			226762.i			0.171319i	0.996324	+	0.0856594i	$0.0272997\pi$
$281$			226762.i			0.171319i	−0.996324	+	0.0856594i	$0.972700\pi$
$282$		0			0
$283$	1.26414e6	−	1.26414e6i	0.938271	−	0.938271i	−0.0599317	−	0.998202i	$-0.519088\pi$
$283$	1.26414e6	−	1.26414e6i	0.938271	−	0.938271i	0.998202	+	0.0599317i	$0.0190883\pi$
$284$	874263.			0.643200
$285$		0			0
$286$	45306.4			0.0327525
$287$	1.26445e6	−	1.26445e6i	0.906145	−	0.906145i
$288$		0			0
$289$			1.38186e6i			0.973236i
$290$		0			0
$291$		0			0
$292$	−228006.	−	228006.i	−0.156491	−	0.156491i
$293$	1.45042e6	+	1.45042e6i	0.987015	+	0.987015i	0.999917	−	0.0129015i	$-0.00410680\pi$
$293$	1.45042e6	+	1.45042e6i	0.987015	+	0.987015i	−0.0129015	+	0.999917i	$0.504107\pi$
$294$		0			0
$295$		0			0
$296$		−	119548.i		−	0.0793073i
$297$		0			0
$298$	1.35519e6	−	1.35519e6i	0.884013	−	0.884013i
$299$	511418.			0.330825
$300$		0			0
$301$	−1.83530e6			−1.16759
$302$	−512853.	+	512853.i	−0.323576	+	0.323576i
$303$		0			0
$304$		−	719357.i		−	0.446437i
$305$		0			0
$306$		0			0
$307$	800295.	+	800295.i	0.484623	+	0.484623i	0.906604	−	0.421982i	$-0.138665\pi$
$307$	800295.	+	800295.i	0.484623	+	0.484623i	−0.421982	+	0.906604i	$0.638665\pi$
$308$	−53011.7	−	53011.7i	−0.0318416	−	0.0318416i
$309$		0			0
$310$		0			0
$311$		−	1.63357e6i		−	0.957714i	−0.877893	−	0.478857i	$-0.841051\pi$
$311$		−	1.63357e6i		−	0.957714i	0.877893	−	0.478857i	$-0.158949\pi$
$312$		0			0
$313$	1.99538e6	−	1.99538e6i	1.15124	−	1.15124i	0.164933	−	0.986305i	$-0.447259\pi$
$313$	1.99538e6	−	1.99538e6i	1.15124	−	1.15124i	0.986305	−	0.164933i	$-0.0527407\pi$
$314$	−1.47645e6			−0.845075
$315$		0			0
$316$	1.26840e6			0.714559
$317$	537671.	−	537671.i	0.300517	−	0.300517i	−0.540699	−	0.841216i	$-0.681840\pi$
$317$	537671.	−	537671.i	0.300517	−	0.300517i	0.841216	+	0.540699i	$0.181840\pi$
$318$		0			0
$319$		−	23401.4i		−	0.0128755i
$320$		0			0
$321$		0			0
$322$	−598396.	−	598396.i	−0.321624	−	0.321624i
$323$	387336.	+	387336.i	0.206577	+	0.206577i
$324$		0			0
$325$		0			0
$326$		−	2.34660e6i		−	1.22291i
$327$		0			0
$328$	488371.	−	488371.i	0.250649	−	0.250649i
$329$	2.75336e6			1.40240
$330$		0			0
$331$	−31025.8			−0.0155651			−0.00778257	−	0.999970i	$-0.502477\pi$
$331$	−31025.8			−0.0155651			−0.00778257	+	0.999970i	$0.502477\pi$
$332$	−584141.	+	584141.i	−0.290852	+	0.290852i
$333$		0			0
$334$		−	1.59604e6i		−	0.782850i
$335$		0			0
$336$		0			0
$337$	702444.	+	702444.i	0.336928	+	0.336928i	0.855210	−	0.518282i	$-0.173428\pi$
$337$	702444.	+	702444.i	0.336928	+	0.336928i	−0.518282	+	0.855210i	$0.673428\pi$
$338$	−596363.	−	596363.i	−0.283935	−	0.283935i
$339$		0			0
$340$		0			0
$341$		−	57958.0i		−	0.0269915i
$342$		0			0
$343$	721328.	−	721328.i	0.331053	−	0.331053i
$344$	−708852.			−0.322968
$345$		0			0
$346$	−2.91922e6			−1.31092
$347$	1.84881e6	−	1.84881e6i	0.824266	−	0.824266i	−0.162451	−	0.986717i	$-0.551940\pi$
$347$	1.84881e6	−	1.84881e6i	0.824266	−	0.824266i	0.986717	+	0.162451i	$0.0519398\pi$
$348$		0			0
$349$			285458.i			0.125452i	0.998031	+	0.0627262i	$0.0199795\pi$
$349$			285458.i			0.125452i	−0.998031	+	0.0627262i	$0.980021\pi$
$350$		0			0
$351$		0			0
$352$	−20474.8	−	20474.8i	−0.00880771	−	0.00880771i
$353$	1.11482e6	+	1.11482e6i	0.476178	+	0.476178i	0.903907	−	0.427729i	$-0.140686\pi$
$353$	1.11482e6	+	1.11482e6i	0.476178	+	0.476178i	−0.427729	+	0.903907i	$0.640686\pi$
$354$		0			0
$355$		0			0
$356$			1.11880e6i			0.467871i
$357$		0			0
$358$	518782.	−	518782.i	0.213933	−	0.213933i
$359$	1.09213e6			0.447237			0.223618	−	0.974677i	$-0.428213\pi$
$359$	1.09213e6			0.447237			0.223618	+	0.974677i	$0.428213\pi$
$360$		0			0
$361$	−5.41994e6			−2.18890
$362$	−941908.	+	941908.i	−0.377779	+	0.377779i
$363$		0			0
$364$		−	1.06198e6i		−	0.420112i
$365$		0			0
$366$		0			0
$367$	−1.38502e6	−	1.38502e6i	−0.536774	−	0.536774i	0.385806	−	0.922580i	$-0.373923\pi$
$367$	−1.38502e6	−	1.38502e6i	−0.536774	−	0.536774i	−0.922580	+	0.385806i	$0.873923\pi$
$368$	−231119.	−	231119.i	−0.0889644	−	0.0889644i
$369$		0			0
$370$		0			0
$371$		−	5.29054e6i		−	1.99556i
$372$		0			0
$373$	−2.32237e6	+	2.32237e6i	−0.864291	+	0.864291i	−0.991833	−	0.127542i	$-0.959291\pi$
$373$	−2.32237e6	+	2.32237e6i	−0.864291	+	0.864291i	0.127542	+	0.991833i	$0.459291\pi$
$374$	22049.2			0.00815105
$375$		0			0
$376$	1.06343e6			0.387919
$377$	234400.	−	234400.i	0.0849383	−	0.0849383i
$378$		0			0
$379$		−	2.91676e6i		−	1.04304i	−0.853238	−	0.521522i	$-0.825364\pi$
$379$		−	2.91676e6i		−	1.04304i	0.853238	−	0.521522i	$-0.174636\pi$
$380$		0			0
$381$		0			0
$382$	−2.46846e6	−	2.46846e6i	−0.865500	−	0.865500i
$383$	1.46470e6	+	1.46470e6i	0.510212	+	0.510212i	0.914591	−	0.404379i	$-0.132512\pi$
$383$	1.46470e6	+	1.46470e6i	0.510212	+	0.510212i	−0.404379	+	0.914591i	$0.632512\pi$
$384$		0			0
$385$		0			0
$386$		−	1.12199e6i		−	0.383283i
$387$		0			0
$388$	1.11011e6	−	1.11011e6i	0.374357	−	0.374357i
$389$	2.43062e6			0.814410			0.407205	−	0.913337i	$-0.366503\pi$
$389$	2.43062e6			0.814410			0.407205	+	0.913337i	$0.366503\pi$
$390$		0			0
$391$	248891.			0.0823317
$392$	−481999.	+	481999.i	−0.158428	+	0.158428i
$393$		0			0
$394$			358125.i			0.116223i
$395$		0			0
$396$		0			0
$397$	−499831.	−	499831.i	−0.159165	−	0.159165i	0.623032	−	0.782197i	$-0.285901\pi$
$397$	−499831.	−	499831.i	−0.159165	−	0.159165i	−0.782197	+	0.623032i	$0.785901\pi$
$398$	−1.00142e6	−	1.00142e6i	−0.316891	−	0.316891i
$399$		0			0
$400$		0			0
$401$			5.80023e6i			1.80129i	0.434554	+	0.900646i	$0.356906\pi$
$401$			5.80023e6i			1.80129i	−0.434554	+	0.900646i	$0.643094\pi$
$402$		0			0
$403$	580536.	−	580536.i	0.178060	−	0.178060i
$404$	2.61447e6			0.796947
$405$		0			0
$406$	−548528.			−0.165152
$407$	−37349.3	+	37349.3i	−0.0111762	+	0.0111762i
$408$		0			0
$409$		−	6.25836e6i		−	1.84992i	−0.380067	−	0.924959i	$-0.624099\pi$
$409$		−	6.25836e6i		−	1.84992i	0.380067	−	0.924959i	$-0.375901\pi$
$410$		0			0
$411$		0			0
$412$	1.26067e6	+	1.26067e6i	0.365897	+	0.365897i
$413$	3.88916e6	+	3.88916e6i	1.12197	+	1.12197i
$414$		0			0
$415$		0			0
$416$		−	410171.i		−	0.116207i
$417$		0			0
$418$	−224742.	+	224742.i	−0.0629134	+	0.0629134i
$419$	−1.90211e6			−0.529298			−0.264649	−	0.964345i	$-0.585256\pi$
$419$	−1.90211e6			−0.529298			−0.264649	+	0.964345i	$0.585256\pi$
$420$		0			0
$421$	396767.			0.109101			0.0545507	−	0.998511i	$-0.482627\pi$
$421$	396767.			0.109101			0.0545507	+	0.998511i	$0.482627\pi$
$422$	836178.	−	836178.i	0.228569	−	0.228569i
$423$		0			0
$424$		−	2.04337e6i		−	0.551992i
$425$		0			0
$426$		0			0
$427$	3.75082e6	+	3.75082e6i	0.995535	+	0.995535i
$428$	957194.	+	957194.i	0.252575	+	0.252575i
$429$		0			0
$430$		0			0
$431$			1.37490e6i			0.356514i	0.983984	+	0.178257i	$0.0570459\pi$
$431$			1.37490e6i			0.356514i	−0.983984	+	0.178257i	$0.942954\pi$
$432$		0			0
$433$	361016.	−	361016.i	0.0925351	−	0.0925351i	−0.659324	−	0.751859i	$-0.729157\pi$
$433$	361016.	−	361016.i	0.0925351	−	0.0925351i	0.751859	+	0.659324i	$0.229157\pi$
$434$	−1.35854e6			−0.346216
$435$		0			0
$436$	58245.6			0.0146739
$437$	−2.53688e6	+	2.53688e6i	−0.635473	+	0.635473i
$438$		0			0
$439$		−	844631.i		−	0.209173i	−0.994516	−	0.104586i	$-0.966648\pi$
$439$		−	844631.i		−	0.209173i	0.994516	−	0.104586i	$-0.0333519\pi$
$440$		0			0
$441$		0			0
$442$	220856.	+	220856.i	0.0537716	+	0.0537716i
$443$	−904722.	−	904722.i	−0.219031	−	0.219031i	0.589059	−	0.808090i	$-0.299499\pi$
$443$	−904722.	−	904722.i	−0.219031	−	0.219031i	−0.808090	+	0.589059i	$0.799499\pi$
$444$		0			0
$445$		0			0
$446$		−	408227.i		−	0.0971773i
$447$		0			0
$448$	−479930.	+	479930.i	−0.112975	+	0.112975i
$449$	−7.62319e6			−1.78452			−0.892259	−	0.451525i	$-0.850880\pi$
$449$	−7.62319e6			−1.78452			−0.892259	+	0.451525i	$0.850880\pi$
$450$		0			0
$451$	−305154.			−0.0706445
$452$	−2.26125e6	+	2.26125e6i	−0.520599	+	0.520599i
$453$		0			0
$454$			3.81718e6i			0.869168i
$455$		0			0
$456$		0			0
$457$	−3.38843e6	−	3.38843e6i	−0.758941	−	0.758941i	0.217188	−	0.976130i	$-0.430311\pi$
$457$	−3.38843e6	−	3.38843e6i	−0.758941	−	0.758941i	−0.976130	+	0.217188i	$0.930311\pi$
$458$	3.03358e6	+	3.03358e6i	0.675760	+	0.675760i
$459$		0			0
$460$		0			0
$461$		−	3.37767e6i		−	0.740226i	−0.928987	−	0.370113i	$-0.879319\pi$
$461$		−	3.37767e6i		−	0.740226i	0.928987	−	0.370113i	$-0.120681\pi$
$462$		0			0
$463$	841979.	−	841979.i	0.182536	−	0.182536i	−0.609924	−	0.792460i	$-0.708800\pi$
$463$	841979.	−	841979.i	0.182536	−	0.182536i	0.792460	+	0.609924i	$0.208800\pi$
$464$	−211859.			−0.0456827
$465$		0			0
$466$	3.78121e6			0.806615
$467$	774792.	−	774792.i	0.164397	−	0.164397i	−0.620115	−	0.784511i	$-0.712914\pi$
$467$	774792.	−	774792.i	0.164397	−	0.164397i	0.784511	+	0.620115i	$0.212914\pi$
$468$		0			0
$469$			6.42545e6i			1.34887i
$470$		0			0
$471$		0			0
$472$	1.50212e6	+	1.50212e6i	0.310348	+	0.310348i
$473$	221460.	+	221460.i	0.0455137	+	0.0455137i
$474$		0			0
$475$		0			0
$476$		−	516833.i		−	0.104552i
$477$		0			0
$478$	−2.89886e6	+	2.89886e6i	−0.580306	+	0.580306i
$479$	−2.64210e6			−0.526150			−0.263075	−	0.964775i	$-0.584737\pi$
$479$	−2.64210e6			−0.526150			−0.263075	+	0.964775i	$0.584737\pi$
$480$		0			0
$481$	−748218.			−0.147457
$482$	1.31754e6	−	1.31754e6i	0.258313	−	0.258313i
$483$		0			0
$484$		−	2.56402e6i		−	0.497518i
$485$		0			0
$486$		0			0
$487$	190754.	+	190754.i	0.0364461	+	0.0364461i	0.725095	−	0.688649i	$-0.241796\pi$
$487$	190754.	+	190754.i	0.0364461	+	0.0364461i	−0.688649	+	0.725095i	$0.741796\pi$
$488$	1.44868e6	+	1.44868e6i	0.275375	+	0.275375i
$489$		0			0
$490$		0			0
$491$			6.69770e6i			1.25378i	0.779107	+	0.626891i	$0.215673\pi$
$491$			6.69770e6i			1.25378i	−0.779107	+	0.626891i	$0.784327\pi$
$492$		0			0
$493$	114075.	−	114075.i	0.0211384	−	0.0211384i
$494$	−4.50226e6			−0.830066
$495$		0			0
$496$	−524709.			−0.0957667
$497$	−6.40236e6	+	6.40236e6i	−1.16265	+	1.16265i
$498$		0			0
$499$			4.04738e6i			0.727650i	0.931467	+	0.363825i	$0.118529\pi$
$499$			4.04738e6i			0.727650i	−0.931467	+	0.363825i	$0.881471\pi$
$500$		0			0
$501$		0			0
$502$	−2.88833e6	−	2.88833e6i	−0.511550	−	0.511550i
$503$	−4.85945e6	−	4.85945e6i	−0.856381	−	0.856381i	0.134528	−	0.990910i	$-0.457048\pi$
$503$	−4.85945e6	−	4.85945e6i	−0.856381	−	0.856381i	−0.990910	+	0.134528i	$0.957048\pi$
$504$		0			0
$505$		0			0
$506$			144413.i			0.0250743i
$507$		0			0
$508$	−1.59851e6	+	1.59851e6i	−0.274825	+	0.274825i
$509$	3.04228e6			0.520481			0.260240	−	0.965544i	$-0.416198\pi$
$509$	3.04228e6			0.520481			0.260240	+	0.965544i	$0.416198\pi$
$510$		0			0
$511$	3.33945e6			0.565747
$512$	−185364.	+	185364.i	−0.0312500	+	0.0312500i
$513$		0			0
$514$			2.88689e6i			0.481973i
$515$		0			0
$516$		0			0
$517$	−332239.	−	332239.i	−0.0546668	−	0.0546668i
$518$	875468.	+	875468.i	0.143356	+	0.143356i
$519$		0			0
$520$		0			0
$521$			1.34709e6i			0.217422i	0.994073	+	0.108711i	$0.0346723\pi$
$521$			1.34709e6i			0.217422i	−0.994073	+	0.108711i	$0.965328\pi$
$522$		0			0
$523$	−3.71364e6	+	3.71364e6i	−0.593671	+	0.593671i	−0.938621	−	0.344950i	$-0.887896\pi$
$523$	−3.71364e6	+	3.71364e6i	−0.593671	+	0.593671i	0.344950	+	0.938621i	$0.387896\pi$
$524$	3.03911e6			0.483524
$525$		0			0
$526$	−1.81447e6			−0.285947
$527$	282528.	−	282528.i	0.0443134	−	0.0443134i
$528$		0			0
$529$		−	4.80621e6i		−	0.746731i
$530$		0			0
$531$		0			0
$532$	5.26796e6	+	5.26796e6i	0.806981	+	0.806981i
$533$	−3.05658e6	−	3.05658e6i	−0.466034	−	0.466034i
$534$		0			0
$535$		0			0
$536$			2.48171e6i			0.373112i
$537$		0			0
$538$	−242797.	+	242797.i	−0.0361650	+	0.0361650i
$539$	301173.			0.0446523
$540$		0			0
$541$	1.31604e7			1.93320			0.966598	−	0.256298i	$-0.0825027\pi$
$541$	1.31604e7			1.93320			0.966598	+	0.256298i	$0.0825027\pi$
$542$	6.47691e6	−	6.47691e6i	0.947043	−	0.947043i
$543$		0			0
$544$		−	199617.i		−	0.0289202i
$545$		0			0
$546$		0			0
$547$	−5.13609e6	−	5.13609e6i	−0.733946	−	0.733946i	0.237453	−	0.971399i	$-0.423687\pi$
$547$	−5.13609e6	−	5.13609e6i	−0.733946	−	0.733946i	−0.971399	+	0.237453i	$0.923687\pi$
$548$	−4.26700e6	−	4.26700e6i	−0.606976	−	0.606976i
$549$		0			0
$550$		0			0
$551$			2.32547e6i			0.326311i
$552$		0			0
$553$	−9.28866e6	+	9.28866e6i	−1.29164	+	1.29164i
$554$	2.43998e6			0.337762
$555$		0			0
$556$	3.88383e6			0.532811
$557$	4.26355e6	−	4.26355e6i	0.582282	−	0.582282i	−0.353248	−	0.935530i	$-0.614923\pi$
$557$	4.26355e6	−	4.26355e6i	0.582282	−	0.582282i	0.935530	+	0.353248i	$0.114923\pi$
$558$		0			0
$559$			4.43650e6i			0.600498i
$560$		0			0
$561$		0			0
$562$	−641381.	−	641381.i	−0.0856594	−	0.0856594i
$563$	−4.98183e6	−	4.98183e6i	−0.662396	−	0.662396i	0.293548	−	0.955944i	$-0.405164\pi$
$563$	−4.98183e6	−	4.98183e6i	−0.662396	−	0.662396i	−0.955944	+	0.293548i	$0.905164\pi$
$564$		0			0
$565$		0			0
$566$			7.15104e6i			0.938271i
$567$		0			0
$568$	−2.47279e6	+	2.47279e6i	−0.321600	+	0.321600i
$569$	−1.53062e7			−1.98193			−0.990963	−	0.134136i	$-0.957174\pi$
$569$	−1.53062e7			−1.98193			−0.990963	+	0.134136i	$0.957174\pi$
$570$		0			0
$571$	834423.			0.107102			0.0535508	−	0.998565i	$-0.482946\pi$
$571$	834423.			0.107102			0.0535508	+	0.998565i	$0.482946\pi$
$572$	−128146.	+	128146.i	−0.0163763	+	0.0163763i
$573$		0			0
$574$			7.15282e6i			0.906145i
$575$		0			0
$576$		0			0
$577$	−1.12011e6	−	1.12011e6i	−0.140062	−	0.140062i	0.633599	−	0.773661i	$-0.281577\pi$
$577$	−1.12011e6	−	1.12011e6i	−0.140062	−	0.140062i	−0.773661	+	0.633599i	$0.781577\pi$
$578$	−3.90848e6	−	3.90848e6i	−0.486618	−	0.486618i
$579$		0			0
$580$		0			0
$581$		−	8.55549e6i		−	1.05149i
$582$		0			0
$583$	−638391.	+	638391.i	−0.0777885	+	0.0777885i
$584$	1.28980e6			0.156491
$585$		0			0
$586$	−8.20480e6			−0.987015
$587$	1.49924e6	−	1.49924e6i	0.179587	−	0.179587i	−0.611589	−	0.791176i	$-0.709469\pi$
$587$	1.49924e6	−	1.49924e6i	0.179587	−	0.179587i	0.791176	+	0.611589i	$0.209469\pi$
$588$		0			0
$589$			5.75948e6i			0.684061i
$590$		0			0
$591$		0			0
$592$	338133.	+	338133.i	0.0396536	+	0.0396536i
$593$	−5.77287e6	−	5.77287e6i	−0.674147	−	0.674147i	0.284522	−	0.958669i	$-0.408165\pi$
$593$	−5.77287e6	−	5.77287e6i	−0.674147	−	0.674147i	−0.958669	+	0.284522i	$0.908165\pi$
$594$		0			0
$595$		0			0
$596$			7.66610e6i			0.884013i
$597$		0			0
$598$	−1.44651e6	+	1.44651e6i	−0.165413	+	0.165413i
$599$	1.36531e7			1.55476			0.777379	−	0.629032i	$-0.216549\pi$
$599$	1.36531e7			1.55476			0.777379	+	0.629032i	$0.216549\pi$
$600$		0			0
$601$	1.24801e7			1.40939			0.704696	−	0.709509i	$-0.251083\pi$
$601$	1.24801e7			1.40939			0.704696	+	0.709509i	$0.251083\pi$
$602$	5.19102e6	−	5.19102e6i	0.583797	−	0.583797i
$603$		0			0
$604$		−	2.90114e6i		−	0.323576i
$605$		0			0
$606$		0			0
$607$	7.28190e6	+	7.28190e6i	0.802182	+	0.802182i	0.983436	−	0.181255i	$-0.0580158\pi$
$607$	7.28190e6	+	7.28190e6i	0.802182	+	0.802182i	−0.181255	+	0.983436i	$0.558016\pi$
$608$	2.03465e6	+	2.03465e6i	0.223219	+	0.223219i
$609$		0			0
$610$		0			0
$611$		−	6.65574e6i		−	0.721262i
$612$		0			0
$613$	−4.08795e6	+	4.08795e6i	−0.439394	+	0.439394i	−0.891808	−	0.452414i	$-0.850563\pi$
$613$	−4.08795e6	+	4.08795e6i	−0.439394	+	0.439394i	0.452414	+	0.891808i	$0.350563\pi$
$614$	−4.52715e6			−0.484623
$615$		0			0
$616$	299880.			0.0318416
$617$	−8.93170e6	+	8.93170e6i	−0.944542	+	0.944542i	−0.998541	−	0.0539990i	$-0.982803\pi$
$617$	−8.93170e6	+	8.93170e6i	−0.944542	+	0.944542i	0.0539990	+	0.998541i	$0.482803\pi$
$618$		0			0
$619$			1.35977e7i			1.42639i	0.700966	+	0.713195i	$0.252752\pi$
$619$			1.35977e7i			1.42639i	−0.700966	+	0.713195i	$0.747248\pi$
$620$		0			0
$621$		0			0
$622$	4.62042e6	+	4.62042e6i	0.478857	+	0.478857i
$623$	−8.19310e6	−	8.19310e6i	−0.845723	−	0.845723i
$624$		0			0
$625$		0			0
$626$			1.12876e7i			1.15124i
$627$		0			0
$628$	4.17604e6	−	4.17604e6i	0.422537	−	0.422537i
$629$	−364133.			−0.0366973
$630$		0			0
$631$	1.43251e7			1.43227			0.716133	−	0.697963i	$-0.245910\pi$
$631$	1.43251e7			1.43227			0.716133	+	0.697963i	$0.245910\pi$
$632$	−3.58757e6	+	3.58757e6i	−0.357279	+	0.357279i
$633$		0			0
$634$			3.04153e6i			0.300517i
$635$		0			0
$636$		0			0
$637$	3.01669e6	+	3.01669e6i	0.294566	+	0.294566i
$638$	66189.0	+	66189.0i	0.00643775	+	0.00643775i
$639$		0			0
$640$		0			0
$641$			8.36174e6i			0.803807i	0.915682	+	0.401903i	$0.131651\pi$
$641$			8.36174e6i			0.803807i	−0.915682	+	0.401903i	$0.868349\pi$
$642$		0			0
$643$	9.10919e6	−	9.10919e6i	0.868865	−	0.868865i	−0.123482	−	0.992347i	$-0.539406\pi$
$643$	9.10919e6	−	9.10919e6i	0.868865	−	0.868865i	0.992347	+	0.123482i	$0.0394061\pi$
$644$	3.38504e6			0.321624
$645$		0			0
$646$	−2.19110e6			−0.206577
$647$	−6.96716e6	+	6.96716e6i	−0.654328	+	0.654328i	−0.954032	−	0.299704i	$-0.903112\pi$
$647$	−6.96716e6	+	6.96716e6i	−0.654328	+	0.654328i	0.299704	+	0.954032i	$0.403112\pi$
$648$		0			0
$649$		−	938584.i		−	0.0874704i
$650$		0			0
$651$		0			0
$652$	6.63718e6	+	6.63718e6i	0.611455	+	0.611455i
$653$	2.40446e6	+	2.40446e6i	0.220666	+	0.220666i	0.808779	−	0.588113i	$-0.200129\pi$
$653$	2.40446e6	+	2.40446e6i	0.220666	+	0.220666i	−0.588113	+	0.808779i	$0.700129\pi$
$654$		0			0
$655$		0			0
$656$			2.76264e6i			0.250649i
$657$		0			0
$658$	−7.78768e6	+	7.78768e6i	−0.701202	+	0.701202i
$659$	1.11541e7			1.00051			0.500257	−	0.865877i	$-0.333239\pi$
$659$	1.11541e7			1.00051			0.500257	+	0.865877i	$0.333239\pi$
$660$		0			0
$661$	8.37133e6			0.745231			0.372615	−	0.927986i	$-0.378461\pi$
$661$	8.37133e6			0.745231			0.372615	+	0.927986i	$0.378461\pi$
$662$	87754.2	−	87754.2i	0.00778257	−	0.00778257i
$663$		0			0
$664$		−	3.30440e6i		−	0.290852i
$665$		0			0
$666$		0			0
$667$	747141.	+	747141.i	0.0650261	+	0.0650261i
$668$	4.51429e6	+	4.51429e6i	0.391425	+	0.391425i
$669$		0			0
$670$		0			0
$671$		−	905197.i		−	0.0776134i
$672$		0			0
$673$	7.04166e6	−	7.04166e6i	0.599291	−	0.599291i	−0.340833	−	0.940124i	$-0.610709\pi$
$673$	7.04166e6	−	7.04166e6i	0.599291	−	0.599291i	0.940124	+	0.340833i	$0.110709\pi$
$674$	−3.97362e6			−0.336928
$675$		0			0
$676$	3.37354e6			0.283935
$677$	7.32224e6	−	7.32224e6i	0.614005	−	0.614005i	−0.329982	−	0.943987i	$-0.607043\pi$
$677$	7.32224e6	−	7.32224e6i	0.614005	−	0.614005i	0.943987	+	0.329982i	$0.107043\pi$
$678$		0			0
$679$			1.62590e7i			1.35338i
$680$		0			0
$681$		0			0
$682$	163930.	+	163930.i	0.0134958	+	0.0134958i
$683$	−9.77498e6	−	9.77498e6i	−0.801796	−	0.801796i	0.181580	−	0.983376i	$-0.441879\pi$
$683$	−9.77498e6	−	9.77498e6i	−0.801796	−	0.801796i	−0.983376	+	0.181580i	$0.941879\pi$
$684$		0			0
$685$		0			0
$686$			4.08045e6i			0.331053i
$687$		0			0
$688$	2.00494e6	−	2.00494e6i	0.161484	−	0.161484i
$689$	−1.27889e7			−1.02632
$690$		0			0
$691$	2.14497e7			1.70894			0.854470	−	0.519501i	$-0.173882\pi$
$691$	2.14497e7			1.70894			0.854470	+	0.519501i	$0.173882\pi$
$692$	8.25680e6	−	8.25680e6i	0.655460	−	0.655460i
$693$		0			0
$694$			1.04584e7i			0.824266i
$695$		0			0
$696$		0			0
$697$	−1.48754e6	−	1.48754e6i	−0.115981	−	0.115981i
$698$	−807398.	−	807398.i	−0.0627262	−	0.0627262i
$699$		0			0
$700$		0			0
$701$		−	1.45950e7i		−	1.12179i	−0.827888	−	0.560893i	$-0.810458\pi$
$701$		−	1.45950e7i		−	1.12179i	0.827888	−	0.560893i	$-0.189542\pi$
$702$		0			0
$703$	3.71152e6	−	3.71152e6i	0.283246	−	0.283246i
$704$	115823.			0.00880771
$705$		0			0
$706$	−6.30639e6			−0.476178
$707$	−1.91461e7	+	1.91461e7i	−1.44056	+	1.44056i
$708$		0			0
$709$		−	1.88701e7i		−	1.40980i	−0.709306	−	0.704900i	$-0.750992\pi$
$709$		−	1.88701e7i		−	1.40980i	0.709306	−	0.704900i	$-0.249008\pi$
$710$		0			0
$711$		0			0
$712$	−3.16443e6	−	3.16443e6i	−0.233935	−	0.233935i
$713$	1.85044e6	+	1.85044e6i	0.136317	+	0.136317i
$714$		0			0
$715$		0			0
$716$			2.93467e6i			0.213933i
$717$		0			0
$718$	−3.08900e6	+	3.08900e6i	−0.223618	+	0.223618i
$719$	−3.25833e6			−0.235057			−0.117528	−	0.993070i	$-0.537497\pi$
$719$	−3.25833e6			−0.235057			−0.117528	+	0.993070i	$0.537497\pi$
$720$		0			0
$721$	−1.84641e7			−1.32279
$722$	1.53299e7	−	1.53299e7i	1.09445	−	1.09445i
$723$		0			0
$724$		−	5.32824e6i		−	0.377779i
$725$		0			0
$726$		0			0
$727$	−1.56667e7	−	1.56667e7i	−1.09936	−	1.09936i	−0.994485	−	0.104879i	$-0.966554\pi$
$727$	−1.56667e7	−	1.56667e7i	−1.09936	−	1.09936i	−0.104879	−	0.994485i	$-0.533446\pi$
$728$	3.00375e6	+	3.00375e6i	0.210056	+	0.210056i
$729$		0			0
$730$		0			0
$731$			2.15910e6i			0.149444i
$732$		0			0
$733$	−1.29191e7	+	1.29191e7i	−0.888121	+	0.888121i	−0.994343	−	0.106221i	$-0.966125\pi$
$733$	−1.29191e7	+	1.29191e7i	−0.888121	+	0.888121i	0.106221	+	0.994343i	$0.466125\pi$
$734$	7.83487e6			0.536774
$735$		0			0
$736$	1.30741e6			0.0889644
$737$	775337.	−	775337.i	0.0525801	−	0.0525801i
$738$		0			0
$739$		−	8.83812e6i		−	0.595318i	−0.954672	−	0.297659i	$-0.903794\pi$
$739$		−	8.83812e6i		−	0.595318i	0.954672	−	0.297659i	$-0.0962058\pi$
$740$		0			0
$741$		0			0
$742$	1.49639e7	+	1.49639e7i	0.997781	+	0.997781i
$743$	−1.16046e7	−	1.16046e7i	−0.771187	−	0.771187i	0.207127	−	0.978314i	$-0.433589\pi$
$743$	−1.16046e7	−	1.16046e7i	−0.771187	−	0.771187i	−0.978314	+	0.207127i	$0.933589\pi$
$744$		0			0
$745$		0			0
$746$		−	1.31373e7i		−	0.864291i
$747$		0			0
$748$	−62364.5	+	62364.5i	−0.00407552	+	0.00407552i
$749$	−1.40193e7			−0.913110
$750$		0			0
$751$	−1.83362e7			−1.18634			−0.593171	−	0.805076i	$-0.702124\pi$
$751$	−1.83362e7			−1.18634			−0.593171	+	0.805076i	$0.702124\pi$
$752$	−3.00785e6	+	3.00785e6i	−0.193959	+	0.193959i
$753$		0			0
$754$			1.32596e6i			0.0849383i
$755$		0			0
$756$		0			0
$757$	2.21679e6	+	2.21679e6i	0.140600	+	0.140600i	0.773904	−	0.633303i	$-0.218301\pi$
$757$	2.21679e6	+	2.21679e6i	0.140600	+	0.140600i	−0.633303	+	0.773904i	$0.718301\pi$
$758$	8.24984e6	+	8.24984e6i	0.521522	+	0.521522i
$759$		0			0
$760$		0			0
$761$			1.85062e7i			1.15839i	0.815188	+	0.579197i	$0.196634\pi$
$761$			1.85062e7i			1.15839i	−0.815188	+	0.579197i	$0.803366\pi$
$762$		0			0
$763$	−426541.	+	426541.i	−0.0265246	+	0.0265246i
$764$	1.39637e7			0.865500
$765$		0			0
$766$	−8.28557e6			−0.510212
$767$	9.40132e6	−	9.40132e6i	0.577033	−	0.577033i
$768$		0			0
$769$		−	2.16451e7i		−	1.31991i	−0.751307	−	0.659953i	$-0.770576\pi$
$769$		−	2.16451e7i		−	1.31991i	0.751307	−	0.659953i	$-0.229424\pi$
$770$		0			0
$771$		0			0
$772$	3.17346e6	+	3.17346e6i	0.191642	+	0.191642i
$773$	1.55356e7	+	1.55356e7i	0.935143	+	0.935143i	0.998021	−	0.0628778i	$-0.0200278\pi$
$773$	1.55356e7	+	1.55356e7i	0.935143	+	0.935143i	−0.0628778	+	0.998021i	$0.520028\pi$
$774$		0			0
$775$		0			0
$776$			6.27973e6i			0.374357i
$777$		0			0
$778$	−6.87484e6	+	6.87484e6i	−0.407205	+	0.407205i
$779$	3.03242e7			1.79038
$780$		0			0
$781$	1.54510e6			0.0906419
$782$	−703969.	+	703969.i	−0.0411658	+	0.0411658i
$783$		0			0
$784$		−	2.72660e6i		−	0.158428i
$785$		0			0
$786$		0			0
$787$	−9.00535e6	−	9.00535e6i	−0.518279	−	0.518279i	0.398771	−	0.917050i	$-0.369437\pi$
$787$	−9.00535e6	−	9.00535e6i	−0.518279	−	0.518279i	−0.917050	+	0.398771i	$0.869437\pi$
$788$	−1.01293e6	−	1.01293e6i	−0.0581117	−	0.0581117i
$789$		0			0
$790$		0			0
$791$		−	3.31190e7i		−	1.88207i
$792$		0			0
$793$	9.06691e6	−	9.06691e6i	0.512007	−	0.512007i
$794$	2.82747e6			0.159165
$795$		0			0
$796$	5.66490e6			0.316891
$797$	−2.44170e7	+	2.44170e7i	−1.36159	+	1.36159i	−0.489697	+	0.871893i	$0.662893\pi$
$797$	−2.44170e7	+	2.44170e7i	−1.36159	+	1.36159i	−0.871893	+	0.489697i	$0.837107\pi$
$798$		0			0
$799$		−	3.23913e6i		−	0.179499i
$800$		0			0
$801$		0			0
$802$	−1.64055e7	−	1.64055e7i	−0.900646	−	0.900646i
$803$	−402960.	−	402960.i	−0.0220533	−	0.0220533i
$804$		0			0
$805$		0			0
$806$			3.28401e6i			0.178060i
$807$		0			0
$808$	−7.39483e6	+	7.39483e6i	−0.398474	+	0.398474i
$809$	−2.80955e7			−1.50926			−0.754632	−	0.656149i	$-0.772184\pi$
$809$	−2.80955e7			−1.50926			−0.754632	+	0.656149i	$0.772184\pi$
$810$		0			0
$811$	−2.35077e7			−1.25504			−0.627522	−	0.778599i	$-0.715931\pi$
$811$	−2.35077e7			−1.25504			−0.627522	+	0.778599i	$0.715931\pi$
$812$	1.55147e6	−	1.55147e6i	0.0825760	−	0.0825760i
$813$		0			0
$814$		−	211279.i		−	0.0111762i
$815$		0			0
$816$		0			0
$817$	−2.20072e7	−	2.20072e7i	−1.15348	−	1.15348i
$818$	1.77013e7	+	1.77013e7i	0.924959	+	0.924959i
$819$		0			0
$820$		0			0
$821$		−	1.08516e7i		−	0.561870i	−0.959727	−	0.280935i	$-0.909355\pi$
$821$		−	1.08516e7i		−	0.561870i	0.959727	−	0.280935i	$-0.0906446\pi$
$822$		0			0
$823$	1.20680e7	−	1.20680e7i	0.621061	−	0.621061i	−0.324742	−	0.945803i	$-0.605277\pi$
$823$	1.20680e7	−	1.20680e7i	0.621061	−	0.621061i	0.945803	+	0.324742i	$0.105277\pi$
$824$	−7.13143e6			−0.365897
$825$		0			0
$826$	−2.20004e7			−1.12197
$827$	2.12279e7	−	2.12279e7i	1.07930	−	1.07930i	0.0827289	−	0.996572i	$-0.473636\pi$
$827$	2.12279e7	−	2.12279e7i	1.07930	−	1.07930i	0.996572	−	0.0827289i	$-0.0263636\pi$
$828$		0			0
$829$		−	6.04829e6i		−	0.305665i	−0.988252	−	0.152833i	$-0.951160\pi$
$829$		−	6.04829e6i		−	0.305665i	0.988252	−	0.152833i	$-0.0488396\pi$
$830$		0			0
$831$		0			0
$832$	1.16014e6	+	1.16014e6i	0.0581035	+	0.0581035i
$833$	1.46813e6	+	1.46813e6i	0.0733080	+	0.0733080i
$834$		0			0
$835$		0			0
$836$		−	1.27133e6i		−	0.0629134i
$837$		0			0
$838$	5.37998e6	−	5.37998e6i	0.264649	−	0.264649i
$839$	−1.15583e7			−0.566879			−0.283440	−	0.958990i	$-0.591476\pi$
$839$	−1.15583e7			−0.566879			−0.283440	+	0.958990i	$0.591476\pi$
$840$		0			0
$841$	−1.98263e7			−0.966610
$842$	−1.12223e6	+	1.12223e6i	−0.0545507	+	0.0545507i
$843$		0			0
$844$			4.73013e6i			0.228569i
$845$		0			0
$846$		0			0
$847$	1.87767e7	+	1.87767e7i	0.899313	+	0.899313i
$848$	5.77953e6	+	5.77953e6i	0.275996	+	0.275996i
$849$		0			0
$850$		0			0
$851$		−	2.38492e6i		−	0.112888i
$852$		0			0
$853$	6.22871e6	−	6.22871e6i	0.293107	−	0.293107i	−0.545200	−	0.838306i	$-0.683546\pi$
$853$	6.22871e6	−	6.22871e6i	0.293107	−	0.293107i	0.838306	+	0.545200i	$0.183546\pi$
$854$	−2.12178e7			−0.995535
$855$		0			0
$856$	−5.41471e6			−0.252575
$857$	−2.14608e6	+	2.14608e6i	−0.0998145	+	0.0998145i	−0.755251	−	0.655436i	$-0.772485\pi$
$857$	−2.14608e6	+	2.14608e6i	−0.0998145	+	0.0998145i	0.655436	+	0.755251i	$0.272485\pi$
$858$		0			0
$859$			1.59281e7i			0.736514i	0.929724	+	0.368257i	$0.120045\pi$
$859$			1.59281e7i			0.736514i	−0.929724	+	0.368257i	$0.879955\pi$
$860$		0			0
$861$		0			0
$862$	−3.88879e6	−	3.88879e6i	−0.178257	−	0.178257i
$863$	1.87460e7	+	1.87460e7i	0.856806	+	0.856806i	0.990960	−	0.134155i	$-0.0428319\pi$
$863$	1.87460e7	+	1.87460e7i	0.856806	+	0.856806i	−0.134155	+	0.990960i	$0.542832\pi$
$864$		0			0
$865$		0			0
$866$			2.04221e6i			0.0925351i
$867$		0			0
$868$	3.84252e6	−	3.84252e6i	0.173108	−	0.173108i
$869$	2.24166e6			0.100698
$870$		0			0
$871$	1.55323e7			0.693731
$872$	−164743.	+	164743.i	−0.00733697	+	0.00733697i
$873$		0			0
$874$		−	1.43508e7i		−	0.635473i
$875$		0			0
$876$		0			0
$877$	−1.91128e7	−	1.91128e7i	−0.839125	−	0.839125i	0.149619	−	0.988744i	$-0.452195\pi$
$877$	−1.91128e7	−	1.91128e7i	−0.839125	−	0.839125i	−0.988744	+	0.149619i	$0.952195\pi$
$878$	2.38898e6	+	2.38898e6i	0.104586	+	0.104586i
$879$		0			0
$880$		0			0
$881$		−	3.02404e7i		−	1.31265i	−0.754479	−	0.656324i	$-0.772110\pi$
$881$		−	3.02404e7i		−	1.31265i	0.754479	−	0.656324i	$-0.227890\pi$
$882$		0			0
$883$	2.28275e7	−	2.28275e7i	0.985272	−	0.985272i	−0.0146211	−	0.999893i	$-0.504654\pi$
$883$	2.28275e7	−	2.28275e7i	0.985272	−	0.985272i	0.999893	+	0.0146211i	$0.00465420\pi$
$884$	−1.24935e6			−0.0537716
$885$		0			0
$886$	5.11788e6			0.219031
$887$	−4.94314e6	+	4.94314e6i	−0.210957	+	0.210957i	−0.804674	−	0.593717i	$-0.797660\pi$
$887$	−4.94314e6	+	4.94314e6i	−0.210957	+	0.210957i	0.593717	+	0.804674i	$0.297660\pi$
$888$		0			0
$889$		−	2.34123e7i		−	0.993548i
$890$		0			0
$891$		0			0
$892$	1.15464e6	+	1.15464e6i	0.0485886	+	0.0485886i
$893$	3.30157e7	+	3.30157e7i	1.38545	+	1.38545i
$894$		0			0
$895$		0			0
$896$		−	2.71489e6i		−	0.112975i
$897$		0			0
$898$	2.15616e7	−	2.15616e7i	0.892259	−	0.892259i
$899$	1.69623e6			0.0699980
$900$		0			0
$901$	−6.22394e6			−0.255419
$902$	863107.	−	863107.i	0.0353222	−	0.0353222i
$903$		0			0
$904$		−	1.27916e7i		−	0.520599i
$905$		0			0
$906$		0			0
$907$	1.85920e7	+	1.85920e7i	0.750428	+	0.750428i	0.974559	−	0.224131i	$-0.0719544\pi$
$907$	1.85920e7	+	1.85920e7i	0.750428	+	0.750428i	−0.224131	+	0.974559i	$0.571954\pi$
$908$	−1.07966e7	−	1.07966e7i	−0.434584	−	0.434584i
$909$		0			0
$910$		0			0
$911$		−	6.12298e6i		−	0.244437i	−0.992503	−	0.122219i	$-0.960999\pi$
$911$		−	6.12298e6i		−	0.244437i	0.992503	−	0.122219i	$-0.0390009\pi$
$912$		0			0
$913$	−1.03236e6	+	1.03236e6i	−0.0409878	+	0.0409878i
$914$	1.91679e7			0.758941
$915$		0			0
$916$	−1.71605e7			−0.675760
$917$	−2.22558e7	+	2.22558e7i	−0.874018	+	0.874018i
$918$		0			0
$919$		−	2.02334e7i		−	0.790279i	−0.918621	−	0.395140i	$-0.870696\pi$
$919$		−	2.02334e7i		−	0.790279i	0.918621	−	0.395140i	$-0.129304\pi$
$920$		0			0
$921$		0			0
$922$	9.55349e6	+	9.55349e6i	0.370113	+	0.370113i
$923$	1.54765e7	+	1.54765e7i	0.597955	+	0.597955i
$924$		0			0
$925$		0			0
$926$			4.76295e6i			0.182536i
$927$		0			0
$928$	599227.	−	599227.i	0.0228413	−	0.0228413i
$929$	1.16624e7			0.443353			0.221676	−	0.975120i	$-0.428847\pi$
$929$	1.16624e7			0.443353			0.221676	+	0.975120i	$0.428847\pi$
$930$		0			0
$931$	−2.99285e7			−1.13165
$932$	−1.06949e7	+	1.06949e7i	−0.403308	+	0.403308i
$933$		0			0
$934$			4.38289e6i			0.164397i
$935$		0			0
$936$		0			0
$937$	3.00134e6	+	3.00134e6i	0.111678	+	0.111678i	0.760737	−	0.649060i	$-0.224837\pi$
$937$	3.00134e6	+	3.00134e6i	0.111678	+	0.111678i	−0.649060	+	0.760737i	$0.724837\pi$
$938$	−1.81739e7	−	1.81739e7i	−0.674437	−	0.674437i
$939$		0			0
$940$		0			0
$941$			3.96549e6i			0.145990i	0.997332	+	0.0729949i	$0.0232557\pi$
$941$			3.96549e6i			0.145990i	−0.997332	+	0.0729949i	$0.976744\pi$
$942$		0			0
$943$	9.74274e6	−	9.74274e6i	0.356781	−	0.356781i
$944$	−8.49725e6			−0.310348
$945$		0			0
$946$	−1.25277e6			−0.0455137
$947$	1.84772e7	−	1.84772e7i	0.669518	−	0.669518i	−0.288087	−	0.957604i	$-0.593019\pi$
$947$	1.84772e7	−	1.84772e7i	0.669518	−	0.669518i	0.957604	+	0.288087i	$0.0930191\pi$
$948$		0			0
$949$		−	8.07249e6i		−	0.290966i
$950$		0			0
$951$		0			0
$952$	1.46183e6	+	1.46183e6i	0.0522761	+	0.0522761i
$953$	1.18785e7	+	1.18785e7i	0.423672	+	0.423672i	0.886466	−	0.462794i	$-0.153153\pi$
$953$	1.18785e7	+	1.18785e7i	0.423672	+	0.423672i	−0.462794	+	0.886466i	$0.653153\pi$
$954$		0			0
$955$		0			0
$956$		−	1.63984e7i		−	0.580306i
$957$		0			0
$958$	7.47297e6	−	7.47297e6i	0.263075	−	0.263075i
$959$	6.24957e7			2.19434
$960$		0			0
$961$	−2.44281e7			−0.853260
$962$	2.11628e6	−	2.11628e6i	0.0737285	−	0.0737285i
$963$		0			0
$964$			7.45313e6i			0.258313i
$965$		0			0
$966$		0			0
$967$	−3.78721e6	−	3.78721e6i	−0.130243	−	0.130243i	0.638980	−	0.769223i	$-0.279356\pi$
$967$	−3.78721e6	−	3.78721e6i	−0.130243	−	0.130243i	−0.769223	+	0.638980i	$0.779356\pi$
$968$	7.25215e6	+	7.25215e6i	0.248759	+	0.248759i
$969$		0			0
$970$		0			0
$971$		−	4.75911e6i		−	0.161986i	−0.996715	−	0.0809930i	$-0.974191\pi$
$971$		−	4.75911e6i		−	0.161986i	0.996715	−	0.0809930i	$-0.0258092\pi$
$972$		0			0
$973$	−2.84418e7	+	2.84418e7i	−0.963109	+	0.963109i
$974$	−1.07907e6			−0.0364461
$975$		0			0
$976$	−8.19499e6			−0.275375
$977$	−1.13234e7	+	1.13234e7i	−0.379525	+	0.379525i	−0.870931	−	0.491406i	$-0.836483\pi$
$977$	−1.13234e7	+	1.13234e7i	−0.379525	+	0.379525i	0.491406	+	0.870931i	$0.336483\pi$
$978$		0			0
$979$			1.97727e6i			0.0659339i
$980$		0			0
$981$		0			0
$982$	−1.89440e7	−	1.89440e7i	−0.626891	−	0.626891i
$983$	−3.71041e7	−	3.71041e7i	−1.22472	−	1.22472i	−0.965933	−	0.258790i	$-0.916676\pi$
$983$	−3.71041e7	−	3.71041e7i	−1.22472	−	1.22472i	−0.258790	−	0.965933i	$-0.583324\pi$
$984$		0			0
$985$		0			0
$986$			645304.i			0.0211384i
$987$		0			0
$988$	1.27343e7	−	1.27343e7i	0.415033	−	0.415033i
$989$	−1.41412e7			−0.459722
$990$		0			0
$991$	−1.18623e6			−0.0383693			−0.0191847	−	0.999816i	$-0.506107\pi$
$991$	−1.18623e6			−0.0383693			−0.0191847	+	0.999816i	$0.506107\pi$
$992$	1.48410e6	−	1.48410e6i	0.0478833	−	0.0478833i
$993$		0			0
$994$		−	3.62172e7i		−	1.16265i
$995$		0			0
$996$		0			0
$997$	−1.48509e7	−	1.48509e7i	−0.473167	−	0.473167i	0.429771	−	0.902938i	$-0.358594\pi$
$997$	−1.48509e7	−	1.48509e7i	−0.473167	−	0.473167i	−0.902938	+	0.429771i	$0.858594\pi$
$998$	−1.14477e7	−	1.14477e7i	−0.363825	−	0.363825i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.f.107.4	✓	16
3.2	odd	2	inner	450.6.f.f.107.8	yes	16
5.2	odd	4		450.6.f.g.143.1	yes	16
5.3	odd	4	inner	450.6.f.f.143.8	yes	16
5.4	even	2		450.6.f.g.107.5	yes	16
15.2	even	4		450.6.f.g.143.5	yes	16
15.8	even	4	inner	450.6.f.f.143.4	yes	16
15.14	odd	2		450.6.f.g.107.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.f.f.107.4	✓	16	1.1	even	1	trivial
450.6.f.f.107.8	yes	16	3.2	odd	2	inner
450.6.f.f.143.4	yes	16	15.8	even	4	inner
450.6.f.f.143.8	yes	16	5.3	odd	4	inner
450.6.f.g.107.1	yes	16	15.14	odd	2
450.6.f.g.107.5	yes	16	5.4	even	2
450.6.f.g.143.1	yes	16	5.2	odd	4
450.6.f.g.143.5	yes	16	15.2	even	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(45.2548 + 45.2548i) q^{8} +O(q^{10})\)	\(q+(-2.82843 + 2.82843i) q^{2} -16.0000i q^{4} +(117.170 + 117.170i) q^{7} +(45.2548 + 45.2548i) q^{8} -28.2771i q^{11} +(283.237 - 283.237i) q^{13} -662.815 q^{14} -256.000 q^{16} +(137.842 - 137.842i) q^{17} +2809.99i q^{19} +(79.9797 + 79.9797i) q^{22} +(902.809 + 902.809i) q^{23} +1602.23i q^{26} +(1874.73 - 1874.73i) q^{28} +827.573 q^{29} +2049.64 q^{31} +(724.077 - 724.077i) q^{32} +779.754i q^{34} +(-1320.83 - 1320.83i) q^{37} +(-7947.85 - 7947.85i) q^{38} -10791.6i q^{41} +(-7831.78 + 7831.78i) q^{43} -452.433 q^{44} -5107.06 q^{46} +(11749.4 - 11749.4i) q^{47} +10650.8i q^{49} +(-4531.80 - 4531.80i) q^{52} +(-22576.3 - 22576.3i) q^{53} +10605.0i q^{56} +(-2340.73 + 2340.73i) q^{58} +33192.4 q^{59} +32011.7 q^{61} +(-5797.27 + 5797.27i) q^{62} +4096.00i q^{64} +(27419.3 + 27419.3i) q^{67} +(-2205.48 - 2205.48i) q^{68} +54641.5i q^{71} +(14250.4 - 14250.4i) q^{73} +7471.75 q^{74} +44959.8 q^{76} +(3313.23 - 3313.23i) q^{77} +79274.9i q^{79} +(30523.2 + 30523.2i) q^{82} +(-36508.8 - 36508.8i) q^{83} -44303.2i q^{86} +(1279.67 - 1279.67i) q^{88} -69924.7 q^{89} +66374.0 q^{91} +(14444.9 - 14444.9i) q^{92} +66464.6i q^{94} +(69381.8 + 69381.8i) q^{97} +(-30124.9 - 30124.9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 528 q^{7}+O(q^{10}) \) 16 * q - 528 * q^7	\( 16 q - 528 q^{7} + 192 q^{13} - 4096 q^{16} - 2688 q^{22} - 8448 q^{28} + 13024 q^{31} - 47328 q^{37} - 55440 q^{43} + 44544 q^{46} - 3072 q^{52} - 101184 q^{58} + 28400 q^{61} + 242256 q^{67} - 430944 q^{73} - 7168 q^{76} + 158208 q^{82} - 43008 q^{88} - 185472 q^{91} + 457152 q^{97}+O(q^{100}) \) 16 * q - 528 * q^7 + 192 * q^13 - 4096 * q^16 - 2688 * q^22 - 8448 * q^28 + 13024 * q^31 - 47328 * q^37 - 55440 * q^43 + 44544 * q^46 - 3072 * q^52 - 101184 * q^58 + 28400 * q^61 + 242256 * q^67 - 430944 * q^73 - 7168 * q^76 + 158208 * q^82 - 43008 * q^88 - 185472 * q^91 + 457152 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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