LMFDB - Embedded newform 750.6.c.a.499.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1810x^{4} + 801025 $ x^8 + 1810*x^4 + 801025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.3
Root			$4.01473 - 4.01473i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.a.499.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +73.7120i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +73.7120i q^{7} +64.0000i q^{8} -81.0000 q^{9} -417.950 q^{11} +144.000i q^{12} +221.666i q^{13} +294.848 q^{14} +256.000 q^{16} -1488.97i q^{17} +324.000i q^{18} -2035.50 q^{19} +663.408 q^{21} +1671.80i q^{22} -4739.58i q^{23} +576.000 q^{24} +886.663 q^{26} +729.000i q^{27} -1179.39i q^{28} -2866.06 q^{29} -8552.25 q^{31} -1024.00i q^{32} +3761.55i q^{33} -5955.89 q^{34} +1296.00 q^{36} -15572.8i q^{37} +8142.01i q^{38} +1994.99 q^{39} +12169.9 q^{41} -2653.63i q^{42} +16143.8i q^{43} +6687.21 q^{44} -18958.3 q^{46} +21918.5i q^{47} -2304.00i q^{48} +11373.5 q^{49} -13400.8 q^{51} -3546.65i q^{52} -18607.2i q^{53} +2916.00 q^{54} -4717.57 q^{56} +18319.5i q^{57} +11464.2i q^{58} +30991.7 q^{59} +5542.80 q^{61} +34209.0i q^{62} -5970.67i q^{63} -4096.00 q^{64} +15046.2 q^{66} +40500.2i q^{67} +23823.6i q^{68} -42656.2 q^{69} -39170.8 q^{71} -5184.00i q^{72} +18427.6i q^{73} -62291.2 q^{74} +32568.1 q^{76} -30808.0i q^{77} -7979.97i q^{78} +101136. q^{79} +6561.00 q^{81} -48679.5i q^{82} +77014.4i q^{83} -10614.5 q^{84} +64575.1 q^{86} +25794.5i q^{87} -26748.8i q^{88} +39906.6 q^{89} -16339.4 q^{91} +75833.3i q^{92} +76970.2i q^{93} +87673.9 q^{94} -9216.00 q^{96} +75390.8i q^{97} -45494.2i q^{98} +33854.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	73.7120i	0.568582i	0.958738	+	0.284291i	$0.0917581\pi$
$7$	73.7120i	0.568582i	−0.958738	+	0.284291i	$0.908242\pi$
$8$	64.0000i	0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−417.950	−1.04146	−0.520731	−	0.853721i	$-0.674340\pi$
$11$	−417.950	−1.04146	−0.520731	+	0.853721i	$0.674340\pi$
$12$	144.000i	0.288675i
$13$	221.666i	0.363781i	0.983319	+	0.181891i	$0.0582217\pi$
$13$	221.666i	0.363781i	−0.983319	+	0.181891i	$0.941778\pi$
$14$	294.848	0.402048
$15$	0	0
$16$	256.000	0.250000
$17$	− 1488.97i	− 1.24958i	−0.780792	−	0.624791i	$-0.785184\pi$
$17$	− 1488.97i	− 1.24958i	0.780792	−	0.624791i	$-0.214816\pi$
$18$	324.000i	0.235702i
$19$	−2035.50	−1.29356	−0.646782	−	0.762675i	$-0.723886\pi$
$19$	−2035.50	−1.29356	−0.646782	+	0.762675i	$0.723886\pi$
$20$	0	0
$21$	663.408	0.328271
$22$	1671.80i	0.736424i
$23$	− 4739.58i	− 1.86819i	−0.357027	−	0.934094i	$-0.616210\pi$
$23$	− 4739.58i	− 1.86819i	0.357027	−	0.934094i	$-0.383790\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	886.663	0.257232
$27$	729.000i	0.192450i
$28$	− 1179.39i	− 0.284291i
$29$	−2866.06	−0.632834	−0.316417	−	0.948620i	$-0.602480\pi$
$29$	−2866.06	−0.632834	−0.316417	+	0.948620i	$0.602480\pi$
$30$	0	0
$31$	−8552.25	−1.59836	−0.799182	−	0.601089i	$-0.794734\pi$
$31$	−8552.25	−1.59836	−0.799182	+	0.601089i	$0.794734\pi$
$32$	− 1024.00i	− 0.176777i
$33$	3761.55i	0.601288i
$34$	−5955.89	−0.883587
$35$	0	0
$36$	1296.00	0.166667
$37$	− 15572.8i	− 1.87009i	−0.354531	−	0.935044i	$-0.615359\pi$
$37$	− 15572.8i	− 1.87009i	0.354531	−	0.935044i	$-0.384641\pi$
$38$	8142.01i	0.914688i
$39$	1994.99	0.210029
$40$	0	0
$41$	12169.9	1.13065	0.565323	−	0.824869i	$-0.308751\pi$
$41$	12169.9	1.13065	0.565323	+	0.824869i	$0.308751\pi$
$42$	− 2653.63i	− 0.232123i
$43$	16143.8i	1.33148i	0.746185	+	0.665739i	$0.231884\pi$
$43$	16143.8i	1.33148i	−0.746185	+	0.665739i	$0.768116\pi$
$44$	6687.21	0.520731
$45$	0	0
$46$	−18958.3	−1.32101
$47$	21918.5i	1.44732i	0.690155	+	0.723662i	$0.257542\pi$
$47$	21918.5i	1.44732i	−0.690155	+	0.723662i	$0.742458\pi$
$48$	− 2304.00i	− 0.144338i
$49$	11373.5	0.676715
$50$	0	0
$51$	−13400.8	−0.721446
$52$	− 3546.65i	− 0.181891i
$53$	− 18607.2i	− 0.909895i	−0.890518	−	0.454948i	$-0.849658\pi$
$53$	− 18607.2i	− 0.909895i	0.890518	−	0.454948i	$-0.150342\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	−4717.57	−0.201024
$57$	18319.5i	0.746839i
$58$	11464.2i	0.447481i
$59$	30991.7	1.15909	0.579543	−	0.814942i	$-0.303231\pi$
$59$	30991.7	1.15909	0.579543	+	0.814942i	$0.303231\pi$
$60$	0	0
$61$	5542.80	0.190724	0.0953618	−	0.995443i	$-0.469599\pi$
$61$	5542.80	0.190724	0.0953618	+	0.995443i	$0.469599\pi$
$62$	34209.0i	1.13021i
$63$	− 5970.67i	− 0.189527i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	15046.2	0.425175
$67$	40500.2i	1.10222i	0.834431	+	0.551112i	$0.185796\pi$
$67$	40500.2i	1.10222i	−0.834431	+	0.551112i	$0.814204\pi$
$68$	23823.6i	0.624791i
$69$	−42656.2	−1.07860
$70$	0	0
$71$	−39170.8	−0.922182	−0.461091	−	0.887353i	$-0.652542\pi$
$71$	−39170.8	−0.922182	−0.461091	+	0.887353i	$0.652542\pi$
$72$	− 5184.00i	− 0.117851i
$73$	18427.6i	0.404727i	0.979310	+	0.202364i	$0.0648622\pi$
$73$	18427.6i	0.404727i	−0.979310	+	0.202364i	$0.935138\pi$
$74$	−62291.2	−1.32235
$75$	0	0
$76$	32568.1	0.646782
$77$	− 30808.0i	− 0.592156i
$78$	− 7979.97i	− 0.148513i
$79$	101136.	1.82322	0.911609	−	0.411059i	$-0.134841\pi$
$79$	101136.	1.82322	0.911609	+	0.411059i	$0.134841\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 48679.5i	− 0.799488i
$83$	77014.4i	1.22709i	0.789659	+	0.613546i	$0.210257\pi$
$83$	77014.4i	1.22709i	−0.789659	+	0.613546i	$0.789743\pi$
$84$	−10614.5	−0.164135
$85$	0	0
$86$	64575.1	0.941497
$87$	25794.5i	0.365367i
$88$	− 26748.8i	− 0.368212i
$89$	39906.6	0.534035	0.267018	−	0.963692i	$-0.413962\pi$
$89$	39906.6	0.534035	0.267018	+	0.963692i	$0.413962\pi$
$90$	0	0
$91$	−16339.4	−0.206839
$92$	75833.3i	0.934094i
$93$	76970.2i	0.922816i
$94$	87673.9	1.02341
$95$	0	0
$96$	−9216.00	−0.102062
$97$	75390.8i	0.813559i	0.913526	+	0.406779i	$0.133348\pi$
$97$	75390.8i	0.813559i	−0.913526	+	0.406779i	$0.866652\pi$
$98$	− 45494.2i	− 0.478510i
$99$	33854.0	0.347154
$100$	0	0
$101$	−58217.0	−0.567866	−0.283933	−	0.958844i	$-0.591639\pi$
$101$	−58217.0	−0.567866	−0.283933	+	0.958844i	$0.591639\pi$
$102$	53603.0i	0.510139i
$103$	103230.i	0.958767i	0.877606	+	0.479383i	$0.159140\pi$
$103$	103230.i	0.958767i	−0.877606	+	0.479383i	$0.840860\pi$
$104$	−14186.6	−0.128616
$105$	0	0
$106$	−74428.8	−0.643393
$107$	26271.9i	0.221836i	0.993830	+	0.110918i	$0.0353791\pi$
$107$	26271.9i	0.221836i	−0.993830	+	0.110918i	$0.964621\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	−188146.	−1.51680	−0.758402	−	0.651787i	$-0.774020\pi$
$109$	−188146.	−1.51680	−0.758402	+	0.651787i	$0.774020\pi$
$110$	0	0
$111$	−140155.	−1.07970
$112$	18870.3i	0.142145i
$113$	− 16140.7i	− 0.118912i	−0.998231	−	0.0594561i	$-0.981063\pi$
$113$	− 16140.7i	− 0.118912i	0.998231	−	0.0594561i	$-0.0189366\pi$
$114$	73278.1	0.528095
$115$	0	0
$116$	45856.9	0.316417
$117$	− 17954.9i	− 0.121260i
$118$	− 123967.i	− 0.819598i
$119$	109755.	0.710489
$120$	0	0
$121$	13631.6	0.0846415
$122$	− 22171.2i	− 0.134862i
$123$	− 109529.i	− 0.652779i
$124$	136836.	0.799182
$125$	0	0
$126$	−23882.7	−0.134016
$127$	251834.i	1.38549i	0.721181	+	0.692747i	$0.243600\pi$
$127$	251834.i	1.38549i	−0.721181	+	0.692747i	$0.756400\pi$
$128$	16384.0i	0.0883883i
$129$	145294.	0.768729
$130$	0	0
$131$	272191.	1.38579	0.692893	−	0.721041i	$-0.256336\pi$
$131$	272191.	1.38579	0.692893	+	0.721041i	$0.256336\pi$
$132$	− 60184.9i	− 0.300644i
$133$	− 150041.i	− 0.735497i
$134$	162001.	0.779391
$135$	0	0
$136$	95294.3	0.441794
$137$	251532.i	1.14497i	0.819917	+	0.572483i	$0.194020\pi$
$137$	251532.i	1.14497i	−0.819917	+	0.572483i	$0.805980\pi$
$138$	170625.i	0.762684i
$139$	246129.	1.08050	0.540252	−	0.841503i	$-0.318329\pi$
$139$	246129.	1.08050	0.540252	+	0.841503i	$0.318329\pi$
$140$	0	0
$141$	197266.	0.835613
$142$	156683.i	0.652081i
$143$	− 92645.3i	− 0.378864i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	73710.5	0.286185
$147$	− 102362.i	− 0.390701i
$148$	249165.i	0.935044i
$149$	469642.	1.73301	0.866505	−	0.499168i	$-0.166361\pi$
$149$	469642.	1.73301	0.866505	+	0.499168i	$0.166361\pi$
$150$	0	0
$151$	−221579.	−0.790836	−0.395418	−	0.918501i	$-0.629400\pi$
$151$	−221579.	−0.790836	−0.395418	+	0.918501i	$0.629400\pi$
$152$	− 130272.i	− 0.457344i
$153$	120607.i	0.416527i
$154$	−123232.	−0.418717
$155$	0	0
$156$	−31919.9	−0.105015
$157$	− 217308.i	− 0.703601i	−0.936075	−	0.351800i	$-0.885570\pi$
$157$	− 217308.i	− 0.703601i	0.936075	−	0.351800i	$-0.114430\pi$
$158$	− 404544.i	− 1.28921i
$159$	−167465.	−0.525328
$160$	0	0
$161$	349364.	1.06222
$162$	− 26244.0i	− 0.0785674i
$163$	286709.i	0.845226i	0.906310	+	0.422613i	$0.138887\pi$
$163$	286709.i	0.845226i	−0.906310	+	0.422613i	$0.861113\pi$
$164$	−194718.	−0.565323
$165$	0	0
$166$	308058.	0.867685
$167$	− 130025.i	− 0.360773i	−0.983596	−	0.180387i	$-0.942265\pi$
$167$	− 130025.i	− 0.360773i	0.983596	−	0.180387i	$-0.0577349\pi$
$168$	42458.1i	0.116061i
$169$	322157.	0.867663
$170$	0	0
$171$	164876.	0.431188
$172$	− 258300.i	− 0.665739i
$173$	− 375852.i	− 0.954777i	−0.878692	−	0.477389i	$-0.841583\pi$
$173$	− 375852.i	− 0.954777i	0.878692	−	0.477389i	$-0.158417\pi$
$174$	103178.	0.258354
$175$	0	0
$176$	−106995.	−0.260365
$177$	− 278925.i	− 0.669199i
$178$	− 159627.i	− 0.377620i
$179$	−335950.	−0.783687	−0.391843	−	0.920032i	$-0.628162\pi$
$179$	−335950.	−0.783687	−0.391843	+	0.920032i	$0.628162\pi$
$180$	0	0
$181$	−744534.	−1.68923	−0.844613	−	0.535377i	$-0.820169\pi$
$181$	−744534.	−1.68923	−0.844613	+	0.535377i	$0.820169\pi$
$182$	65357.7i	0.146257i
$183$	− 49885.2i	− 0.110114i
$184$	303333.	0.660504
$185$	0	0
$186$	307881.	0.652530
$187$	622317.i	1.30139i
$188$	− 350696.i	− 0.723662i
$189$	−53736.0	−0.109424
$190$	0	0
$191$	−954678.	−1.89354	−0.946768	−	0.321917i	$-0.895673\pi$
$191$	−954678.	−1.89354	−0.946768	+	0.321917i	$0.895673\pi$
$192$	36864.0i	0.0721688i
$193$	37252.7i	0.0719888i	0.999352	+	0.0359944i	$0.0114598\pi$
$193$	37252.7i	0.0719888i	−0.999352	+	0.0359944i	$0.988540\pi$
$194$	301563.	0.575273
$195$	0	0
$196$	−181977.	−0.338357
$197$	410453.i	0.753526i	0.926310	+	0.376763i	$0.122963\pi$
$197$	410453.i	0.753526i	−0.926310	+	0.376763i	$0.877037\pi$
$198$	− 135416.i	− 0.245475i
$199$	504617.	0.903294	0.451647	−	0.892197i	$-0.350837\pi$
$199$	504617.	0.903294	0.451647	+	0.892197i	$0.350837\pi$
$200$	0	0
$201$	364502.	0.636370
$202$	232868.i	0.401542i
$203$	− 211263.i	− 0.359818i
$204$	214412.	0.360723
$205$	0	0
$206$	412920.	0.677951
$207$	383906.i	0.622729i
$208$	56746.4i	0.0909453i
$209$	850740.	1.34720
$210$	0	0
$211$	−256267.	−0.396266	−0.198133	−	0.980175i	$-0.563488\pi$
$211$	−256267.	−0.396266	−0.198133	+	0.980175i	$0.563488\pi$
$212$	297715.i	0.454948i
$213$	352537.i	0.532422i
$214$	105088.	0.156862
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	− 630403.i	− 0.908801i
$218$	752585.i	1.07254i
$219$	165849.	0.233669
$220$	0	0
$221$	330054.	0.454574
$222$	560621.i	0.763461i
$223$	− 953626.i	− 1.28415i	−0.766641	−	0.642075i	$-0.778074\pi$
$223$	− 953626.i	− 1.28415i	0.766641	−	0.642075i	$-0.221926\pi$
$224$	75481.1	0.100512
$225$	0	0
$226$	−64562.8	−0.0840836
$227$	− 383348.i	− 0.493774i	−0.969044	−	0.246887i	$-0.920592\pi$
$227$	− 383348.i	− 0.493774i	0.969044	−	0.246887i	$-0.0794077\pi$
$228$	− 293112.i	− 0.373420i
$229$	428221.	0.539609	0.269805	−	0.962915i	$-0.413041\pi$
$229$	428221.	0.539609	0.269805	+	0.962915i	$0.413041\pi$
$230$	0	0
$231$	−277272.	−0.341881
$232$	− 183428.i	− 0.223741i
$233$	856062.i	1.03304i	0.856276	+	0.516518i	$0.172772\pi$
$233$	856062.i	1.03304i	−0.856276	+	0.516518i	$0.827228\pi$
$234$	−71819.7	−0.0857440
$235$	0	0
$236$	−495868.	−0.579543
$237$	− 910225.i	− 1.05263i
$238$	− 439021.i	− 0.502392i
$239$	1.71269e6	1.93947	0.969736	−	0.244154i	$-0.0785103\pi$
$239$	1.71269e6	1.93947	0.969736	+	0.244154i	$0.0785103\pi$
$240$	0	0
$241$	225564.	0.250166	0.125083	−	0.992146i	$-0.460080\pi$
$241$	225564.	0.250166	0.125083	+	0.992146i	$0.460080\pi$
$242$	− 54526.4i	− 0.0598506i
$243$	− 59049.0i	− 0.0641500i
$244$	−88684.8	−0.0953618
$245$	0	0
$246$	−438116.	−0.461584
$247$	− 451201.i	− 0.470574i
$248$	− 547344.i	− 0.565107i
$249$	693130.	0.708462
$250$	0	0
$251$	817259.	0.818795	0.409398	−	0.912356i	$-0.365739\pi$
$251$	817259.	0.818795	0.409398	+	0.912356i	$0.365739\pi$
$252$	95530.7i	0.0947636i
$253$	1.98091e6i	1.94565i
$254$	1.00734e6	0.979692
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 77381.2i	− 0.0730807i	−0.999332	−	0.0365404i	$-0.988366\pi$
$257$	− 77381.2i	− 0.0730807i	0.999332	−	0.0365404i	$-0.0116337\pi$
$258$	− 581176.i	− 0.543574i
$259$	1.14790e6	1.06330
$260$	0	0
$261$	232151.	0.210945
$262$	− 1.08877e6i	− 0.979898i
$263$	− 163971.i	− 0.146177i	−0.997325	−	0.0730883i	$-0.976715\pi$
$263$	− 163971.i	− 0.146177i	0.997325	−	0.0730883i	$-0.0232855\pi$
$264$	−240739.	−0.212587
$265$	0	0
$266$	−600164.	−0.520075
$267$	− 359160.i	− 0.308326i
$268$	− 648003.i	− 0.551112i
$269$	336935.	0.283900	0.141950	−	0.989874i	$-0.454663\pi$
$269$	336935.	0.283900	0.141950	+	0.989874i	$0.454663\pi$
$270$	0	0
$271$	−342041.	−0.282914	−0.141457	−	0.989944i	$-0.545179\pi$
$271$	−342041.	−0.282914	−0.141457	+	0.989944i	$0.545179\pi$
$272$	− 381177.i	− 0.312395i
$273$	147055.i	0.119419i
$274$	1.00613e6	0.809613
$275$	0	0
$276$	682500.	0.539299
$277$	− 1.24632e6i	− 0.975953i	−0.872857	−	0.487977i	$-0.837735\pi$
$277$	− 1.24632e6i	− 0.975953i	0.872857	−	0.487977i	$-0.162265\pi$
$278$	− 984517.i	− 0.764032i
$279$	692732.	0.532788
$280$	0	0
$281$	−2.52356e6	−1.90655	−0.953274	−	0.302106i	$-0.902310\pi$
$281$	−2.52356e6	−1.90655	−0.953274	+	0.302106i	$0.902310\pi$
$282$	− 789065.i	− 0.590867i
$283$	512155.i	0.380133i	0.981771	+	0.190066i	$0.0608703\pi$
$283$	512155.i	0.380133i	−0.981771	+	0.190066i	$0.939130\pi$
$284$	626733.	0.461091
$285$	0	0
$286$	−370581.	−0.267897
$287$	897066.i	0.642865i
$288$	82944.0i	0.0589256i
$289$	−797184.	−0.561453
$290$	0	0
$291$	678517.	0.469708
$292$	− 294842.i	− 0.202364i
$293$	− 420048.i	− 0.285845i	−0.989734	−	0.142922i	$-0.954350\pi$
$293$	− 420048.i	− 0.285845i	0.989734	−	0.142922i	$-0.0456499\pi$
$294$	−409448.	−0.276268
$295$	0	0
$296$	996659.	0.661176
$297$	− 304686.i	− 0.200429i
$298$	− 1.87857e6i	− 1.22542i
$299$	1.05060e6	0.679611
$300$	0	0
$301$	−1.18999e6	−0.757054
$302$	886316.i	0.559205i
$303$	523953.i	0.327858i
$304$	−521089.	−0.323391
$305$	0	0
$306$	482427.	0.294529
$307$	− 2.54087e6i	− 1.53864i	−0.638863	−	0.769320i	$-0.720595\pi$
$307$	− 2.54087e6i	− 1.53864i	0.638863	−	0.769320i	$-0.279405\pi$
$308$	492927.i	0.296078i
$309$	929070.	0.553544
$310$	0	0
$311$	1.32604e6	0.777423	0.388711	−	0.921360i	$-0.372920\pi$
$311$	1.32604e6	0.777423	0.388711	+	0.921360i	$0.372920\pi$
$312$	127679.i	0.0742565i
$313$	1.08656e6i	0.626895i	0.949606	+	0.313447i	$0.101484\pi$
$313$	1.08656e6i	0.626895i	−0.949606	+	0.313447i	$0.898516\pi$
$314$	−869232.	−0.497521
$315$	0	0
$316$	−1.61818e6	−0.911609
$317$	1.18880e6i	0.664449i	0.943200	+	0.332224i	$0.107799\pi$
$317$	1.18880e6i	0.664449i	−0.943200	+	0.332224i	$0.892201\pi$
$318$	669859.i	0.371463i
$319$	1.19787e6	0.659072
$320$	0	0
$321$	236447.	0.128077
$322$	− 1.39746e6i	− 0.751101i
$323$	3.03081e6i	1.61641i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.14684e6	0.597665
$327$	1.69332e6i	0.875727i
$328$	778873.i	0.399744i
$329$	−1.61565e6	−0.822922
$330$	0	0
$331$	999290.	0.501328	0.250664	−	0.968074i	$-0.419351\pi$
$331$	999290.	0.501328	0.250664	+	0.968074i	$0.419351\pi$
$332$	− 1.23223e6i	− 0.613546i
$333$	1.26140e6i	0.623363i
$334$	−520098.	−0.255105
$335$	0	0
$336$	169832.	0.0820677
$337$	2.12986e6i	1.02159i	0.859703	+	0.510795i	$0.170649\pi$
$337$	2.12986e6i	1.02159i	−0.859703	+	0.510795i	$0.829351\pi$
$338$	− 1.28863e6i	− 0.613531i
$339$	−145266.	−0.0686540
$340$	0	0
$341$	3.57441e6	1.66464
$342$	− 659503.i	− 0.304896i
$343$	2.07724e6i	0.953349i
$344$	−1.03320e6	−0.470749
$345$	0	0
$346$	−1.50341e6	−0.675130
$347$	− 3.20299e6i	− 1.42801i	−0.700140	−	0.714005i	$-0.746879\pi$
$347$	− 3.20299e6i	− 1.42801i	0.700140	−	0.714005i	$-0.253121\pi$
$348$	− 412712.i	− 0.182684i
$349$	1.96821e6	0.864984	0.432492	−	0.901638i	$-0.357634\pi$
$349$	1.96821e6	0.864984	0.432492	+	0.901638i	$0.357634\pi$
$350$	0	0
$351$	−161594.	−0.0700097
$352$	427981.i	0.184106i
$353$	1.09121e6i	0.466093i	0.972466	+	0.233046i	$0.0748694\pi$
$353$	1.09121e6i	0.466093i	−0.972466	+	0.233046i	$0.925131\pi$
$354$	−1.11570e6	−0.473195
$355$	0	0
$356$	−638506.	−0.267018
$357$	− 987796.i	− 0.410201i
$358$	1.34380e6i	0.554150i
$359$	−2.51666e6	−1.03060	−0.515298	−	0.857011i	$-0.672319\pi$
$359$	−2.51666e6	−1.03060	−0.515298	+	0.857011i	$0.672319\pi$
$360$	0	0
$361$	1.66717e6	0.673307
$362$	2.97813e6i	1.19446i
$363$	− 122684.i	− 0.0488678i
$364$	261431.	0.103420
$365$	0	0
$366$	−199541.	−0.0778626
$367$	− 821269.i	− 0.318288i	−0.987255	−	0.159144i	$-0.949127\pi$
$367$	− 821269.i	− 0.318288i	0.987255	−	0.159144i	$-0.0508734\pi$
$368$	− 1.21333e6i	− 0.467047i
$369$	−985761.	−0.376882
$370$	0	0
$371$	1.37157e6	0.517350
$372$	− 1.23152e6i	− 0.461408i
$373$	− 1.66082e6i	− 0.618087i	−0.951048	−	0.309044i	$-0.899991\pi$
$373$	− 1.66082e6i	− 0.618087i	0.951048	−	0.309044i	$-0.100009\pi$
$374$	2.48927e6	0.920222
$375$	0	0
$376$	−1.40278e6	−0.511706
$377$	− 635307.i	− 0.230213i
$378$	214944.i	0.0773742i
$379$	4.12550e6	1.47529	0.737646	−	0.675187i	$-0.235937\pi$
$379$	4.12550e6	1.47529	0.737646	+	0.675187i	$0.235937\pi$
$380$	0	0
$381$	2.26650e6	0.799916
$382$	3.81871e6i	1.33893i
$383$	3.69265e6i	1.28630i	0.765742	+	0.643148i	$0.222372\pi$
$383$	3.69265e6i	1.28630i	−0.765742	+	0.643148i	$0.777628\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	149011.	0.0509037
$387$	− 1.30765e6i	− 0.443826i
$388$	− 1.20625e6i	− 0.406779i
$389$	1.85751e6	0.622381	0.311191	−	0.950348i	$-0.399272\pi$
$389$	1.85751e6	0.622381	0.311191	+	0.950348i	$0.399272\pi$
$390$	0	0
$391$	−7.05711e6	−2.33445
$392$	727907.i	0.239255i
$393$	− 2.44972e6i	− 0.800084i
$394$	1.64181e6	0.532823
$395$	0	0
$396$	−541664.	−0.173577
$397$	27173.2i	0.00865295i	0.999991	+	0.00432648i	$0.00137716\pi$
$397$	27173.2i	0.00865295i	−0.999991	+	0.00432648i	$0.998623\pi$
$398$	− 2.01847e6i	− 0.638725i
$399$	−1.35037e6	−0.424639
$400$	0	0
$401$	1.17930e6	0.366239	0.183120	−	0.983091i	$-0.441380\pi$
$401$	1.17930e6	0.366239	0.183120	+	0.983091i	$0.441380\pi$
$402$	− 1.45801e6i	− 0.449981i
$403$	− 1.89574e6i	− 0.581455i
$404$	931471.	0.283933
$405$	0	0
$406$	−845051.	−0.254430
$407$	6.50866e6i	1.94762i
$408$	− 857648.i	− 0.255070i
$409$	−5.94634e6	−1.75769	−0.878843	−	0.477112i	$-0.841684\pi$
$409$	−5.94634e6	−1.75769	−0.878843	+	0.477112i	$0.841684\pi$
$410$	0	0
$411$	2.26379e6	0.661046
$412$	− 1.65168e6i	− 0.479383i
$413$	2.28446e6i	0.659035i
$414$	1.53562e6	0.440336
$415$	0	0
$416$	226986.	0.0643080
$417$	− 2.21516e6i	− 0.623829i
$418$	− 3.40296e6i	− 0.952612i
$419$	−1.57622e6	−0.438613	−0.219306	−	0.975656i	$-0.570379\pi$
$419$	−1.57622e6	−0.438613	−0.219306	+	0.975656i	$0.570379\pi$
$420$	0	0
$421$	217307.	0.0597542	0.0298771	−	0.999554i	$-0.490488\pi$
$421$	217307.	0.0597542	0.0298771	+	0.999554i	$0.490488\pi$
$422$	1.02507e6i	0.280203i
$423$	− 1.77540e6i	− 0.482441i
$424$	1.19086e6	0.321697
$425$	0	0
$426$	1.41015e6	0.376479
$427$	408571.i	0.108442i
$428$	− 420350.i	− 0.110918i
$429$	−833808.	−0.218737
$430$	0	0
$431$	3.10284e6	0.804575	0.402288	−	0.915513i	$-0.368215\pi$
$431$	3.10284e6	0.804575	0.402288	+	0.915513i	$0.368215\pi$
$432$	186624.i	0.0481125i
$433$	403653.i	0.103464i	0.998661	+	0.0517319i	$0.0164741\pi$
$433$	403653.i	0.103464i	−0.998661	+	0.0517319i	$0.983526\pi$
$434$	−2.52161e6	−0.642619
$435$	0	0
$436$	3.01034e6	0.758402
$437$	9.64744e6i	2.41662i
$438$	− 663394.i	− 0.165229i
$439$	−3.16275e6	−0.783255	−0.391628	−	0.920124i	$-0.628088\pi$
$439$	−3.16275e6	−0.783255	−0.391628	+	0.920124i	$0.628088\pi$
$440$	0	0
$441$	−921257.	−0.225572
$442$	− 1.32022e6i	− 0.321432i
$443$	5.54566e6i	1.34259i	0.741190	+	0.671295i	$0.234262\pi$
$443$	5.54566e6i	1.34259i	−0.741190	+	0.671295i	$0.765738\pi$
$444$	2.24248e6	0.539848
$445$	0	0
$446$	−3.81451e6	−0.908032
$447$	− 4.22678e6i	− 1.00055i
$448$	− 301924.i	− 0.0710727i
$449$	−3.25484e6	−0.761928	−0.380964	−	0.924590i	$-0.624408\pi$
$449$	−3.25484e6	−0.761928	−0.380964	+	0.924590i	$0.624408\pi$
$450$	0	0
$451$	−5.08641e6	−1.17752
$452$	258251.i	0.0594561i
$453$	1.99421e6i	0.456589i
$454$	−1.53339e6	−0.349151
$455$	0	0
$456$	−1.17245e6	−0.264048
$457$	3.73798e6i	0.837233i	0.908163	+	0.418617i	$0.137485\pi$
$457$	3.73798e6i	0.837233i	−0.908163	+	0.418617i	$0.862515\pi$
$458$	− 1.71288e6i	− 0.381561i
$459$	1.08546e6	0.240482
$460$	0	0
$461$	941344.	0.206298	0.103149	−	0.994666i	$-0.467108\pi$
$461$	941344.	0.206298	0.103149	+	0.994666i	$0.467108\pi$
$462$	1.10909e6i	0.241747i
$463$	5.53498e6i	1.19995i	0.800019	+	0.599975i	$0.204823\pi$
$463$	5.53498e6i	1.19995i	−0.800019	+	0.599975i	$0.795177\pi$
$464$	−733711.	−0.158209
$465$	0	0
$466$	3.42425e6	0.730467
$467$	3.35878e6i	0.712670i	0.934358	+	0.356335i	$0.115974\pi$
$467$	3.35878e6i	0.712670i	−0.934358	+	0.356335i	$0.884026\pi$
$468$	287279.i	0.0606302i
$469$	−2.98535e6	−0.626705
$470$	0	0
$471$	−1.95577e6	−0.406224
$472$	1.98347e6i	0.409799i
$473$	− 6.74730e6i	− 1.38668i
$474$	−3.64090e6	−0.744325
$475$	0	0
$476$	−1.75608e6	−0.355245
$477$	1.50718e6i	0.303298i
$478$	− 6.85076e6i	− 1.37141i
$479$	4.23998e6	0.844355	0.422178	−	0.906513i	$-0.361266\pi$
$479$	4.23998e6	0.844355	0.422178	+	0.906513i	$0.361266\pi$
$480$	0	0
$481$	3.45195e6	0.680303
$482$	− 902257.i	− 0.176894i
$483$	− 3.14428e6i	− 0.613272i
$484$	−218106.	−0.0423208
$485$	0	0
$486$	−236196.	−0.0453609
$487$	− 3.80025e6i	− 0.726089i	−0.931772	−	0.363044i	$-0.881737\pi$
$487$	− 3.80025e6i	− 0.726089i	0.931772	−	0.363044i	$-0.118263\pi$
$488$	354739.i	0.0674310i
$489$	2.58038e6	0.487991
$490$	0	0
$491$	2.50831e6	0.469545	0.234772	−	0.972050i	$-0.424566\pi$
$491$	2.50831e6	0.469545	0.234772	+	0.972050i	$0.424566\pi$
$492$	1.75246e6i	0.326390i
$493$	4.26748e6i	0.790778i
$494$	−1.80481e6	−0.332746
$495$	0	0
$496$	−2.18937e6	−0.399591
$497$	− 2.88736e6i	− 0.524336i
$498$	− 2.77252e6i	− 0.500958i
$499$	−6.19059e6	−1.11296	−0.556482	−	0.830860i	$-0.687849\pi$
$499$	−6.19059e6	−1.11296	−0.556482	+	0.830860i	$0.687849\pi$
$500$	0	0
$501$	−1.17022e6	−0.208293
$502$	− 3.26903e6i	− 0.578976i
$503$	4.31768e6i	0.760905i	0.924800	+	0.380453i	$0.124232\pi$
$503$	4.31768e6i	0.760905i	−0.924800	+	0.380453i	$0.875768\pi$
$504$	382123.	0.0670080
$505$	0	0
$506$	7.92364e6	1.37578
$507$	− 2.89942e6i	− 0.500946i
$508$	− 4.02934e6i	− 0.692747i
$509$	1.88467e6	0.322435	0.161217	−	0.986919i	$-0.448458\pi$
$509$	1.88467e6	0.322435	0.161217	+	0.986919i	$0.448458\pi$
$510$	0	0
$511$	−1.35834e6	−0.230120
$512$	− 262144.i	− 0.0441942i
$513$	− 1.48388e6i	− 0.248946i
$514$	−309525.	−0.0516759
$515$	0	0
$516$	−2.32470e6	−0.384365
$517$	− 9.16084e6i	− 1.50733i
$518$	− 4.59161e6i	− 0.751865i
$519$	−3.38267e6	−0.551241
$520$	0	0
$521$	−8.80918e6	−1.42181	−0.710904	−	0.703289i	$-0.751714\pi$
$521$	−8.80918e6	−1.42181	−0.710904	+	0.703289i	$0.751714\pi$
$522$	− 928603.i	− 0.149160i
$523$	− 3.52271e6i	− 0.563148i	−0.959539	−	0.281574i	$-0.909143\pi$
$523$	− 3.52271e6i	− 0.563148i	0.959539	−	0.281574i	$-0.0908565\pi$
$524$	−4.35506e6	−0.692893
$525$	0	0
$526$	−655884.	−0.103362
$527$	1.27341e7i	1.99729i
$528$	962958.i	0.150322i
$529$	−1.60273e7	−2.49013
$530$	0	0
$531$	−2.51033e6	−0.386362
$532$	2.40066e6i	0.367748i
$533$	2.69765e6i	0.411308i
$534$	−1.43664e6	−0.218019
$535$	0	0
$536$	−2.59201e6	−0.389695
$537$	3.02355e6i	0.452462i
$538$	− 1.34774e6i	− 0.200747i
$539$	−4.75358e6	−0.704772
$540$	0	0
$541$	−1.35551e6	−0.199117	−0.0995587	−	0.995032i	$-0.531743\pi$
$541$	−1.35551e6	−0.199117	−0.0995587	+	0.995032i	$0.531743\pi$
$542$	1.36816e6i	0.200050i
$543$	6.70080e6i	0.975275i
$544$	−1.52471e6	−0.220897
$545$	0	0
$546$	588219.	0.0844418
$547$	640874.i	0.0915808i	0.998951	+	0.0457904i	$0.0145806\pi$
$547$	640874.i	0.0915808i	−0.998951	+	0.0457904i	$0.985419\pi$
$548$	− 4.02452e6i	− 0.572483i
$549$	−448967.	−0.0635745
$550$	0	0
$551$	5.83387e6	0.818612
$552$	− 2.73000e6i	− 0.381342i
$553$	7.45494e6i	1.03665i
$554$	−4.98527e6	−0.690103
$555$	0	0
$556$	−3.93807e6	−0.540252
$557$	4.52709e6i	0.618274i	0.951017	+	0.309137i	$0.100040\pi$
$557$	4.52709e6i	0.618274i	−0.951017	+	0.309137i	$0.899960\pi$
$558$	− 2.77093e6i	− 0.376738i
$559$	−3.57852e6	−0.484367
$560$	0	0
$561$	5.60085e6	0.751358
$562$	1.00942e7i	1.34813i
$563$	− 9.65324e6i	− 1.28352i	−0.766906	−	0.641759i	$-0.778205\pi$
$563$	− 9.65324e6i	− 1.28352i	0.766906	−	0.641759i	$-0.221795\pi$
$564$	−3.15626e6	−0.417806
$565$	0	0
$566$	2.04862e6	0.268794
$567$	483624.i	0.0631757i
$568$	− 2.50693e6i	− 0.326041i
$569$	−8.04851e6	−1.04216	−0.521080	−	0.853508i	$-0.674471\pi$
$569$	−8.04851e6	−1.04216	−0.521080	+	0.853508i	$0.674471\pi$
$570$	0	0
$571$	−1.39693e6	−0.179302	−0.0896511	−	0.995973i	$-0.528575\pi$
$571$	−1.39693e6	−0.179302	−0.0896511	+	0.995973i	$0.528575\pi$
$572$	1.48232e6i	0.189432i
$573$	8.59210e6i	1.09323i
$574$	3.58826e6	0.454574
$575$	0	0
$576$	331776.	0.0416667
$577$	− 8.77056e6i	− 1.09670i	−0.836249	−	0.548350i	$-0.815256\pi$
$577$	− 8.77056e6i	− 1.09670i	0.836249	−	0.548350i	$-0.184744\pi$
$578$	3.18873e6i	0.397008i
$579$	335275.	0.0415627
$580$	0	0
$581$	−5.67689e6	−0.697702
$582$	− 2.71407e6i	− 0.332134i
$583$	7.77689e6i	0.947621i
$584$	−1.17937e6	−0.143093
$585$	0	0
$586$	−1.68019e6	−0.202123
$587$	1.43463e7i	1.71848i	0.511574	+	0.859239i	$0.329063\pi$
$587$	1.43463e7i	1.71848i	−0.511574	+	0.859239i	$0.670937\pi$
$588$	1.63779e6i	0.195351i
$589$	1.74081e7	2.06759
$590$	0	0
$591$	3.69408e6	0.435049
$592$	− 3.98664e6i	− 0.467522i
$593$	8.71036e6i	1.01718i	0.861008	+	0.508592i	$0.169834\pi$
$593$	8.71036e6i	1.01718i	−0.861008	+	0.508592i	$0.830166\pi$
$594$	−1.21874e6	−0.141725
$595$	0	0
$596$	−7.51427e6	−0.866505
$597$	− 4.54155e6i	− 0.521517i
$598$	− 4.20241e6i	− 0.480558i
$599$	9.91333e6	1.12889	0.564446	−	0.825470i	$-0.309090\pi$
$599$	9.91333e6	1.12889	0.564446	+	0.825470i	$0.309090\pi$
$600$	0	0
$601$	−4.13333e6	−0.466782	−0.233391	−	0.972383i	$-0.574982\pi$
$601$	−4.13333e6	−0.466782	−0.233391	+	0.972383i	$0.574982\pi$
$602$	4.75996e6i	0.535318i
$603$	− 3.28052e6i	− 0.367408i
$604$	3.54526e6	0.395418
$605$	0	0
$606$	2.09581e6	0.231830
$607$	− 1.55701e7i	− 1.71522i	−0.514301	−	0.857610i	$-0.671948\pi$
$607$	− 1.55701e7i	− 1.71522i	0.514301	−	0.857610i	$-0.328052\pi$
$608$	2.08436e6i	0.228672i
$609$	−1.90137e6	−0.207741
$610$	0	0
$611$	−4.85857e6	−0.526509
$612$	− 1.92971e6i	− 0.208264i
$613$	− 1.30239e7i	− 1.39987i	−0.714205	−	0.699936i	$-0.753212\pi$
$613$	− 1.30239e7i	− 1.39987i	0.714205	−	0.699936i	$-0.246788\pi$
$614$	−1.01635e7	−1.08798
$615$	0	0
$616$	1.97171e6	0.209359
$617$	7.57484e6i	0.801052i	0.916285	+	0.400526i	$0.131173\pi$
$617$	7.57484e6i	0.801052i	−0.916285	+	0.400526i	$0.868827\pi$
$618$	− 3.71628e6i	− 0.391415i
$619$	−1.41402e7	−1.48330	−0.741651	−	0.670786i	$-0.765957\pi$
$619$	−1.41402e7	−1.48330	−0.741651	+	0.670786i	$0.765957\pi$
$620$	0	0
$621$	3.45516e6	0.359533
$622$	− 5.30418e6i	− 0.549721i
$623$	2.94160e6i	0.303643i
$624$	510718.	0.0525073
$625$	0	0
$626$	4.34626e6	0.443281
$627$	− 7.65666e6i	− 0.777804i
$628$	3.47693e6i	0.351800i
$629$	−2.31875e7	−2.33683
$630$	0	0
$631$	1.50753e7	1.50728	0.753638	−	0.657290i	$-0.228297\pi$
$631$	1.50753e7	1.50728	0.753638	+	0.657290i	$0.228297\pi$
$632$	6.47271e6i	0.644605i
$633$	2.30641e6i	0.228784i
$634$	4.75521e6	0.469836
$635$	0	0
$636$	2.67944e6	0.262664
$637$	2.52113e6i	0.246176i
$638$	− 4.79148e6i	− 0.466035i
$639$	3.17284e6	0.307394
$640$	0	0
$641$	−1.32678e7	−1.27542	−0.637709	−	0.770278i	$-0.720118\pi$
$641$	−1.32678e7	−1.27542	−0.637709	+	0.770278i	$0.720118\pi$
$642$	− 945788.i	− 0.0905641i
$643$	3.67781e6i	0.350802i	0.984497	+	0.175401i	$0.0561222\pi$
$643$	3.67781e6i	0.350802i	−0.984497	+	0.175401i	$0.943878\pi$
$644$	−5.58982e6	−0.531109
$645$	0	0
$646$	1.21232e7	1.14298
$647$	1.83529e7i	1.72363i	0.507226	+	0.861813i	$0.330671\pi$
$647$	1.83529e7i	1.72363i	−0.507226	+	0.861813i	$0.669329\pi$
$648$	419904.i	0.0392837i
$649$	−1.29530e7	−1.20714
$650$	0	0
$651$	−5.67363e6	−0.524697
$652$	− 4.58735e6i	− 0.422613i
$653$	− 34491.3i	− 0.00316539i	−0.999999	−	0.00158269i	$-0.999496\pi$
$653$	− 34491.3i	− 0.00316539i	0.999999	−	0.00158269i	$-0.000503787\pi$
$654$	6.77327e6	0.619233
$655$	0	0
$656$	3.11549e6	0.282662
$657$	− 1.49264e6i	− 0.134909i
$658$	6.46262e6i	0.581894i
$659$	3.22246e6	0.289051	0.144526	−	0.989501i	$-0.453834\pi$
$659$	3.22246e6	0.289051	0.144526	+	0.989501i	$0.453834\pi$
$660$	0	0
$661$	4.19667e6	0.373595	0.186797	−	0.982398i	$-0.440189\pi$
$661$	4.19667e6	0.373595	0.186797	+	0.982398i	$0.440189\pi$
$662$	− 3.99716e6i	− 0.354492i
$663$	− 2.97049e6i	− 0.262448i
$664$	−4.92892e6	−0.433842
$665$	0	0
$666$	5.04558e6	0.440784
$667$	1.35839e7i	1.18225i
$668$	2.08039e6i	0.180387i
$669$	−8.58264e6	−0.741405
$670$	0	0
$671$	−2.31661e6	−0.198631
$672$	− 679330.i	− 0.0580306i
$673$	− 9.87553e6i	− 0.840471i	−0.907415	−	0.420236i	$-0.861947\pi$
$673$	− 9.87553e6i	− 0.840471i	0.907415	−	0.420236i	$-0.138053\pi$
$674$	8.51944e6	0.722373
$675$	0	0
$676$	−5.15452e6	−0.433832
$677$	− 3.78771e6i	− 0.317618i	−0.987309	−	0.158809i	$-0.949235\pi$
$677$	− 3.78771e6i	− 0.317618i	0.987309	−	0.158809i	$-0.0507654\pi$
$678$	581065.i	0.0485457i
$679$	−5.55720e6	−0.462575
$680$	0	0
$681$	−3.45013e6	−0.285081
$682$	− 1.42977e7i	− 1.17707i
$683$	− 2.21820e7i	− 1.81948i	−0.415175	−	0.909741i	$-0.636280\pi$
$683$	− 2.21820e7i	− 1.81948i	0.415175	−	0.909741i	$-0.363720\pi$
$684$	−2.63801e6	−0.215594
$685$	0	0
$686$	8.30897e6	0.674120
$687$	− 3.85399e6i	− 0.311544i
$688$	4.13281e6i	0.332870i
$689$	4.12458e6	0.331003
$690$	0	0
$691$	−4.56418e6	−0.363636	−0.181818	−	0.983332i	$-0.558198\pi$
$691$	−4.56418e6	−0.363636	−0.181818	+	0.983332i	$0.558198\pi$
$692$	6.01364e6i	0.477389i
$693$	2.49544e6i	0.197385i
$694$	−1.28119e7	−1.00976
$695$	0	0
$696$	−1.65085e6	−0.129177
$697$	− 1.81206e7i	− 1.41283i
$698$	− 7.87284e6i	− 0.611636i
$699$	7.70456e6	0.596423
$700$	0	0
$701$	−7.75134e6	−0.595774	−0.297887	−	0.954601i	$-0.596282\pi$
$701$	−7.75134e6	−0.595774	−0.297887	+	0.954601i	$0.596282\pi$
$702$	646377.i	0.0495043i
$703$	3.16985e7i	2.41908i
$704$	1.71193e6	0.130183
$705$	0	0
$706$	4.36485e6	0.329577
$707$	− 4.29129e6i	− 0.322878i
$708$	4.46281e6i	0.334599i
$709$	−1.33033e7	−0.993900	−0.496950	−	0.867779i	$-0.665547\pi$
$709$	−1.33033e7	−0.993900	−0.496950	+	0.867779i	$0.665547\pi$
$710$	0	0
$711$	−8.19202e6	−0.607739
$712$	2.55402e6i	0.188810i
$713$	4.05341e7i	2.98605i
$714$	−3.95118e6	−0.290056
$715$	0	0
$716$	5.37521e6	0.391843
$717$	− 1.54142e7i	− 1.11976i
$718$	1.00666e7i	0.728741i
$719$	9.95457e6	0.718126	0.359063	−	0.933313i	$-0.383096\pi$
$719$	9.95457e6	0.718126	0.359063	+	0.933313i	$0.383096\pi$
$720$	0	0
$721$	−7.60929e6	−0.545137
$722$	− 6.66870e6i	− 0.476100i
$723$	− 2.03008e6i	− 0.144433i
$724$	1.19125e7	0.844613
$725$	0	0
$726$	−490738.	−0.0345547
$727$	− 1.07599e7i	− 0.755046i	−0.926000	−	0.377523i	$-0.876776\pi$
$727$	− 1.07599e7i	− 0.755046i	0.926000	−	0.377523i	$-0.123224\pi$
$728$	− 1.04572e6i	− 0.0731287i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	2.40377e7	1.66379
$732$	798163.i	0.0550572i
$733$	− 2.60981e6i	− 0.179411i	−0.995968	−	0.0897055i	$-0.971407\pi$
$733$	− 2.60981e6i	− 0.179411i	0.995968	−	0.0897055i	$-0.0285926\pi$
$734$	−3.28507e6	−0.225064
$735$	0	0
$736$	−4.85333e6	−0.330252
$737$	− 1.69271e7i	− 1.14792i
$738$	3.94304e6i	0.266496i
$739$	−2.44216e6	−0.164499	−0.0822494	−	0.996612i	$-0.526210\pi$
$739$	−2.44216e6	−0.164499	−0.0822494	+	0.996612i	$0.526210\pi$
$740$	0	0
$741$	−4.06081e6	−0.271686
$742$	− 5.48629e6i	− 0.365822i
$743$	5.99980e6i	0.398717i	0.979927	+	0.199358i	$0.0638858\pi$
$743$	5.99980e6i	0.398717i	−0.979927	+	0.199358i	$0.936114\pi$
$744$	−4.92609e6	−0.326265
$745$	0	0
$746$	−6.64327e6	−0.437054
$747$	− 6.23817e6i	− 0.409030i
$748$	− 9.95707e6i	− 0.650695i
$749$	−1.93655e6	−0.126132
$750$	0	0
$751$	2.02079e7	1.30744	0.653720	−	0.756736i	$-0.273207\pi$
$751$	2.02079e7	1.30744	0.653720	+	0.756736i	$0.273207\pi$
$752$	5.61113e6i	0.361831i
$753$	− 7.35533e6i	− 0.472732i
$754$	−2.54123e6	−0.162785
$755$	0	0
$756$	859776.	0.0547118
$757$	2.25145e7i	1.42798i	0.700157	+	0.713989i	$0.253114\pi$
$757$	2.25145e7i	1.42798i	−0.700157	+	0.713989i	$0.746886\pi$
$758$	− 1.65020e7i	− 1.04319i
$759$	1.78282e7	1.12332
$760$	0	0
$761$	8.06122e6	0.504591	0.252295	−	0.967650i	$-0.418815\pi$
$761$	8.06122e6	0.504591	0.252295	+	0.967650i	$0.418815\pi$
$762$	− 9.06602e6i	− 0.565626i
$763$	− 1.38686e7i	− 0.862427i
$764$	1.52748e7	0.946768
$765$	0	0
$766$	1.47706e7	0.909549
$767$	6.86980e6i	0.421654i
$768$	− 589824.i	− 0.0360844i
$769$	6.17324e6	0.376441	0.188221	−	0.982127i	$-0.439728\pi$
$769$	6.17324e6	0.376441	0.188221	+	0.982127i	$0.439728\pi$
$770$	0	0
$771$	−696431.	−0.0421932
$772$	− 596044.i	− 0.0359944i
$773$	834496.i	0.0502314i	0.999685	+	0.0251157i	$0.00799542\pi$
$773$	834496.i	0.0502314i	−0.999685	+	0.0251157i	$0.992005\pi$
$774$	−5.23058e6	−0.313832
$775$	0	0
$776$	−4.82501e6	−0.287637
$777$	− 1.03311e7i	− 0.613896i
$778$	− 7.43003e6i	− 0.440090i
$779$	−2.47718e7	−1.46256
$780$	0	0
$781$	1.63715e7	0.960417
$782$	2.82284e7i	1.65071i
$783$	− 2.08936e6i	− 0.121789i
$784$	2.91163e6	0.169179
$785$	0	0
$786$	−9.79889e6	−0.565745
$787$	4.11684e6i	0.236934i	0.992958	+	0.118467i	$0.0377980\pi$
$787$	4.11684e6i	0.236934i	−0.992958	+	0.118467i	$0.962202\pi$
$788$	− 6.56726e6i	− 0.376763i
$789$	−1.47574e6	−0.0843951
$790$	0	0
$791$	1.18976e6	0.0676113
$792$	2.16666e6i	0.122737i
$793$	1.22865e6i	0.0693816i
$794$	108693.	0.00611856
$795$	0	0
$796$	−8.07387e6	−0.451647
$797$	− 1.77929e6i	− 0.0992202i	−0.998769	−	0.0496101i	$-0.984202\pi$
$797$	− 1.77929e6i	− 0.0992202i	0.998769	−	0.0496101i	$-0.0157979\pi$
$798$	5.40147e6i	0.300265i
$799$	3.26360e7	1.80855
$800$	0	0
$801$	−3.23244e6	−0.178012
$802$	− 4.71722e6i	− 0.258970i
$803$	− 7.70183e6i	− 0.421508i
$804$	−5.83203e6	−0.318185
$805$	0	0
$806$	−7.58296e6	−0.411151
$807$	− 3.03241e6i	− 0.163910i
$808$	− 3.72589e6i	− 0.200771i
$809$	2.72658e7	1.46470	0.732348	−	0.680930i	$-0.238424\pi$
$809$	2.72658e7	1.46470	0.732348	+	0.680930i	$0.238424\pi$
$810$	0	0
$811$	3.20647e7	1.71189	0.855943	−	0.517071i	$-0.172978\pi$
$811$	3.20647e7	1.71189	0.855943	+	0.517071i	$0.172978\pi$
$812$	3.38021e6i	0.179909i
$813$	3.07837e6i	0.163340i
$814$	2.60346e7	1.37718
$815$	0	0
$816$	−3.43059e6	−0.180362
$817$	− 3.28607e7i	− 1.72235i
$818$	2.37853e7i	1.24287i
$819$	1.32349e6	0.0689464
$820$	0	0
$821$	−1.45463e7	−0.753175	−0.376588	−	0.926381i	$-0.622903\pi$
$821$	−1.45463e7	−0.753175	−0.376588	+	0.926381i	$0.622903\pi$
$822$	− 9.05516e6i	− 0.467430i
$823$	9.03309e6i	0.464875i	0.972611	+	0.232438i	$0.0746702\pi$
$823$	9.03309e6i	0.464875i	−0.972611	+	0.232438i	$0.925330\pi$
$824$	−6.60672e6	−0.338975
$825$	0	0
$826$	9.13784e6	0.466008
$827$	− 5.07698e6i	− 0.258132i	−0.991636	−	0.129066i	$-0.958802\pi$
$827$	− 5.07698e6i	− 0.258132i	0.991636	−	0.129066i	$-0.0411979\pi$
$828$	− 6.14250e6i	− 0.311365i
$829$	9.07399e6	0.458577	0.229288	−	0.973359i	$-0.426360\pi$
$829$	9.07399e6	0.458577	0.229288	+	0.973359i	$0.426360\pi$
$830$	0	0
$831$	−1.12169e7	−0.563467
$832$	− 907943.i	− 0.0454726i
$833$	− 1.69349e7i	− 0.845610i
$834$	−8.86065e6	−0.441114
$835$	0	0
$836$	−1.36118e7	−0.673598
$837$	− 6.23459e6i	− 0.307605i
$838$	6.30487e6i	0.310146i
$839$	2.72446e7	1.33621	0.668105	−	0.744067i	$-0.267106\pi$
$839$	2.72446e7	1.33621	0.668105	+	0.744067i	$0.267106\pi$
$840$	0	0
$841$	−1.22969e7	−0.599521
$842$	− 869228.i	− 0.0422526i
$843$	2.27120e7i	1.10075i
$844$	4.10028e6	0.198133
$845$	0	0
$846$	−7.10159e6	−0.341137
$847$	1.00481e6i	0.0481256i
$848$	− 4.76344e6i	− 0.227474i
$849$	4.60940e6	0.219470
$850$	0	0
$851$	−7.38085e7	−3.49368
$852$	− 5.64060e6i	− 0.266211i
$853$	− 4.23439e6i	− 0.199259i	−0.995025	−	0.0996296i	$-0.968234\pi$
$853$	− 4.23439e6i	− 0.199259i	0.995025	−	0.0996296i	$-0.0317658\pi$
$854$	1.63428e6	0.0766800
$855$	0	0
$856$	−1.68140e6	−0.0784308
$857$	1.69178e7i	0.786849i	0.919357	+	0.393424i	$0.128710\pi$
$857$	1.69178e7i	0.786849i	−0.919357	+	0.393424i	$0.871290\pi$
$858$	3.33523e6i	0.154671i
$859$	1.13932e7	0.526820	0.263410	−	0.964684i	$-0.415153\pi$
$859$	1.13932e7	0.526820	0.263410	+	0.964684i	$0.415153\pi$
$860$	0	0
$861$	8.07360e6	0.371158
$862$	− 1.24114e7i	− 0.568921i
$863$	− 2.67123e7i	− 1.22091i	−0.792050	−	0.610457i	$-0.790986\pi$
$863$	− 2.67123e7i	− 1.22091i	0.792050	−	0.610457i	$-0.209014\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	1.61461e6	0.0731600
$867$	7.17465e6i	0.324155i
$868$	1.00864e7i	0.454401i
$869$	−4.22699e7	−1.89881
$870$	0	0
$871$	−8.97751e6	−0.400969
$872$	− 1.20414e7i	− 0.536271i
$873$	− 6.10665e6i	− 0.271186i
$874$	3.85897e7	1.70881
$875$	0	0
$876$	−2.65358e6	−0.116835
$877$	− 3.94392e7i	− 1.73153i	−0.500453	−	0.865764i	$-0.666833\pi$
$877$	− 3.94392e7i	− 1.73153i	0.500453	−	0.865764i	$-0.333167\pi$
$878$	1.26510e7i	0.553845i
$879$	−3.78043e6	−0.165032
$880$	0	0
$881$	2.85506e7	1.23930	0.619649	−	0.784879i	$-0.287275\pi$
$881$	2.85506e7	1.23930	0.619649	+	0.784879i	$0.287275\pi$
$882$	3.68503e6i	0.159503i
$883$	3.45567e7i	1.49152i	0.666212	+	0.745762i	$0.267915\pi$
$883$	3.45567e7i	1.49152i	−0.666212	+	0.745762i	$0.732085\pi$
$884$	−5.28087e6	−0.227287
$885$	0	0
$886$	2.21826e7	0.949355
$887$	3.31738e7i	1.41575i	0.706339	+	0.707873i	$0.250345\pi$
$887$	3.31738e7i	1.41575i	−0.706339	+	0.707873i	$0.749655\pi$
$888$	− 8.96993e6i	− 0.381730i
$889$	−1.85632e7	−0.787767
$890$	0	0
$891$	−2.74217e6	−0.115718
$892$	1.52580e7i	0.642075i
$893$	− 4.46151e7i	− 1.87221i
$894$	−1.69071e7	−0.707498
$895$	0	0
$896$	−1.20770e6	−0.0502560
$897$	− 9.45543e6i	− 0.392374i
$898$	1.30194e7i	0.538765i
$899$	2.45112e7	1.01150
$900$	0	0
$901$	−2.77056e7	−1.13699
$902$	2.03456e7i	0.832635i
$903$	1.07099e7i	0.437085i
$904$	1.03301e6	0.0420418
$905$	0	0
$906$	7.97684e6	0.322857
$907$	1.14736e7i	0.463105i	0.972822	+	0.231553i	$0.0743806\pi$
$907$	1.14736e7i	0.463105i	−0.972822	+	0.231553i	$0.925619\pi$
$908$	6.13357e6i	0.246887i
$909$	4.71557e6	0.189289
$910$	0	0
$911$	2.51503e7	1.00403	0.502016	−	0.864859i	$-0.332592\pi$
$911$	2.51503e7	1.00403	0.502016	+	0.864859i	$0.332592\pi$
$912$	4.68980e6i	0.186710i
$913$	− 3.21882e7i	− 1.27797i
$914$	1.49519e7	0.592013
$915$	0	0
$916$	−6.85154e6	−0.269805
$917$	2.00638e7i	0.787932i
$918$	− 4.34185e6i	− 0.170046i
$919$	−4.78910e7	−1.87053	−0.935265	−	0.353948i	$-0.884839\pi$
$919$	−4.78910e7	−1.87053	−0.935265	+	0.353948i	$0.884839\pi$
$920$	0	0
$921$	−2.28679e7	−0.888335
$922$	− 3.76537e6i	− 0.145875i
$923$	− 8.68283e6i	− 0.335472i
$924$	4.43635e6	0.170941
$925$	0	0
$926$	2.21399e7	0.848493
$927$	− 8.36163e6i	− 0.319589i
$928$	2.93484e6i	0.111870i
$929$	4.23921e7	1.61156	0.805778	−	0.592218i	$-0.201748\pi$
$929$	4.23921e7	1.61156	0.805778	+	0.592218i	$0.201748\pi$
$930$	0	0
$931$	−2.31509e7	−0.875374
$932$	− 1.36970e7i	− 0.516518i
$933$	− 1.19344e7i	− 0.448845i
$934$	1.34351e7	0.503934
$935$	0	0
$936$	1.14912e6	0.0428720
$937$	4.39897e6i	0.163682i	0.996645	+	0.0818411i	$0.0260800\pi$
$937$	4.39897e6i	0.163682i	−0.996645	+	0.0818411i	$0.973920\pi$
$938$	1.19414e7i	0.443147i
$939$	9.77908e6	0.361938
$940$	0	0
$941$	4.13696e7	1.52303	0.761513	−	0.648149i	$-0.224457\pi$
$941$	4.13696e7	1.52303	0.761513	+	0.648149i	$0.224457\pi$
$942$	7.82308e6i	0.287244i
$943$	− 5.76802e7i	− 2.11226i
$944$	7.93388e6	0.289772
$945$	0	0
$946$	−2.69892e7	−0.980533
$947$	3.08782e7i	1.11886i	0.828876	+	0.559432i	$0.188981\pi$
$947$	3.08782e7i	1.11886i	−0.828876	+	0.559432i	$0.811019\pi$
$948$	1.45636e7i	0.526317i
$949$	−4.08477e6	−0.147232
$950$	0	0
$951$	1.06992e7	0.383620
$952$	7.02433e6i	0.251196i
$953$	2.48636e7i	0.886811i	0.896321	+	0.443406i	$0.146230\pi$
$953$	2.48636e7i	0.886811i	−0.896321	+	0.443406i	$0.853770\pi$
$954$	6.02873e6	0.214464
$955$	0	0
$956$	−2.74030e7	−0.969736
$957$	− 1.07808e7i	− 0.380516i
$958$	− 1.69599e7i	− 0.597049i
$959$	−1.85409e7	−0.651006
$960$	0	0
$961$	4.45117e7	1.55477
$962$	− 1.38078e7i	− 0.481047i
$963$	− 2.12802e6i	− 0.0739453i
$964$	−3.60903e6	−0.125083
$965$	0	0
$966$	−1.25771e7	−0.433648
$967$	1.39639e7i	0.480220i	0.970746	+	0.240110i	$0.0771835\pi$
$967$	1.39639e7i	0.480220i	−0.970746	+	0.240110i	$0.922817\pi$
$968$	872422.i	0.0299253i
$969$	2.72773e7	0.933237
$970$	0	0
$971$	−4.43443e6	−0.150935	−0.0754675	−	0.997148i	$-0.524045\pi$
$971$	−4.43443e6	−0.150935	−0.0754675	+	0.997148i	$0.524045\pi$
$972$	944784.i	0.0320750i
$973$	1.81427e7i	0.614355i
$974$	−1.52010e7	−0.513422
$975$	0	0
$976$	1.41896e6	0.0476809
$977$	4.65135e7i	1.55899i	0.626411	+	0.779493i	$0.284523\pi$
$977$	4.65135e7i	1.55899i	−0.626411	+	0.779493i	$0.715477\pi$
$978$	− 1.03215e7i	− 0.345062i
$979$	−1.66790e7	−0.556177
$980$	0	0
$981$	1.52399e7	0.505601
$982$	− 1.00332e7i	− 0.332018i
$983$	− 1.74833e7i	− 0.577085i	−0.957467	−	0.288543i	$-0.906829\pi$
$983$	− 1.74833e7i	− 0.577085i	0.957467	−	0.288543i	$-0.0931707\pi$
$984$	7.00985e6	0.230792
$985$	0	0
$986$	1.70699e7	0.559164
$987$	1.45409e7i	0.475114i
$988$	7.21922e6i	0.235287i
$989$	7.65148e7	2.48745
$990$	0	0
$991$	1.40979e7	0.456006	0.228003	−	0.973660i	$-0.426780\pi$
$991$	1.40979e7	0.456006	0.228003	+	0.973660i	$0.426780\pi$
$992$	8.75750e6i	0.282554i
$993$	− 8.99361e6i	− 0.289442i
$994$	−1.15494e7	−0.370762
$995$	0	0
$996$	−1.10901e7	−0.354231
$997$	1.47805e7i	0.470924i	0.971884	+	0.235462i	$0.0756603\pi$
$997$	1.47805e7i	0.470924i	−0.971884	+	0.235462i	$0.924340\pi$
$998$	2.47624e7i	0.786984i
$999$	1.13526e7	0.359899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.a.499.3		8
5.2	odd	4		750.6.a.e.1.2	yes	4
5.3	odd	4		750.6.a.d.1.3	✓	4
5.4	even	2	inner	750.6.c.a.499.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.d.1.3	✓	4	5.3	odd	4
750.6.a.e.1.2	yes	4	5.2	odd	4
750.6.c.a.499.3		8	1.1	even	1	trivial
750.6.c.a.499.6		8	5.4	even	2	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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +73.7120i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +73.7120i q^{7} +64.0000i q^{8} -81.0000 q^{9} -417.950 q^{11} +144.000i q^{12} +221.666i q^{13} +294.848 q^{14} +256.000 q^{16} -1488.97i q^{17} +324.000i q^{18} -2035.50 q^{19} +663.408 q^{21} +1671.80i q^{22} -4739.58i q^{23} +576.000 q^{24} +886.663 q^{26} +729.000i q^{27} -1179.39i q^{28} -2866.06 q^{29} -8552.25 q^{31} -1024.00i q^{32} +3761.55i q^{33} -5955.89 q^{34} +1296.00 q^{36} -15572.8i q^{37} +8142.01i q^{38} +1994.99 q^{39} +12169.9 q^{41} -2653.63i q^{42} +16143.8i q^{43} +6687.21 q^{44} -18958.3 q^{46} +21918.5i q^{47} -2304.00i q^{48} +11373.5 q^{49} -13400.8 q^{51} -3546.65i q^{52} -18607.2i q^{53} +2916.00 q^{54} -4717.57 q^{56} +18319.5i q^{57} +11464.2i q^{58} +30991.7 q^{59} +5542.80 q^{61} +34209.0i q^{62} -5970.67i q^{63} -4096.00 q^{64} +15046.2 q^{66} +40500.2i q^{67} +23823.6i q^{68} -42656.2 q^{69} -39170.8 q^{71} -5184.00i q^{72} +18427.6i q^{73} -62291.2 q^{74} +32568.1 q^{76} -30808.0i q^{77} -7979.97i q^{78} +101136. q^{79} +6561.00 q^{81} -48679.5i q^{82} +77014.4i q^{83} -10614.5 q^{84} +64575.1 q^{86} +25794.5i q^{87} -26748.8i q^{88} +39906.6 q^{89} -16339.4 q^{91} +75833.3i q^{92} +76970.2i q^{93} +87673.9 q^{94} -9216.00 q^{96} +75390.8i q^{97} -45494.2i q^{98} +33854.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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