LMFDB - Embedded newform 325.6.b.i.274.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.5
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.07859i q^{2} +25.6203i q^{3} -4.94922 q^{4} +155.735 q^{6} -137.301i q^{7} -164.431i q^{8} -413.398 q^{9} +O(q^{10})$	\(q-6.07859i q^{2} +25.6203i q^{3} -4.94922 q^{4} +155.735 q^{6} -137.301i q^{7} -164.431i q^{8} -413.398 q^{9} +169.031 q^{11} -126.800i q^{12} +169.000i q^{13} -834.595 q^{14} -1157.88 q^{16} +487.128i q^{17} +2512.87i q^{18} -2385.49 q^{19} +3517.68 q^{21} -1027.47i q^{22} +4144.47i q^{23} +4212.75 q^{24} +1027.28 q^{26} -4365.63i q^{27} +679.532i q^{28} +4570.45 q^{29} -2957.68 q^{31} +1776.50i q^{32} +4330.61i q^{33} +2961.05 q^{34} +2046.00 q^{36} -5733.02i q^{37} +14500.4i q^{38} -4329.82 q^{39} -1705.07 q^{41} -21382.5i q^{42} +10922.2i q^{43} -836.570 q^{44} +25192.5 q^{46} +7912.12i q^{47} -29665.2i q^{48} -2044.52 q^{49} -12480.4 q^{51} -836.418i q^{52} +26431.1i q^{53} -26536.9 q^{54} -22576.4 q^{56} -61116.8i q^{57} -27781.9i q^{58} -11248.5 q^{59} -40298.6 q^{61} +17978.5i q^{62} +56759.8i q^{63} -26253.6 q^{64} +26324.0 q^{66} +64887.2i q^{67} -2410.91i q^{68} -106182. q^{69} -47076.1 q^{71} +67975.2i q^{72} +42771.8i q^{73} -34848.7 q^{74} +11806.3 q^{76} -23208.1i q^{77} +26319.2i q^{78} -90057.0 q^{79} +11392.9 q^{81} +10364.4i q^{82} -16427.0i q^{83} -17409.8 q^{84} +66391.5 q^{86} +117096. i q^{87} -27793.8i q^{88} +35632.7 q^{89} +23203.8 q^{91} -20511.9i q^{92} -75776.6i q^{93} +48094.5 q^{94} -45514.4 q^{96} +121030. i q^{97} +12427.8i q^{98} -69876.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 6.07859i	− 1.07455i	−0.843406	−	0.537276i	$-0.819453\pi$
$2$	− 6.07859i	− 1.07455i	0.843406	−	0.537276i	$-0.180547\pi$
$3$	25.6203i	1.64354i	0.569819	+	0.821770i	$0.307013\pi$
$3$	25.6203i	1.64354i	−0.569819	+	0.821770i	$0.692987\pi$
$4$	−4.94922	−0.154663
$5$	0	0
$5$	0	0
$6$	155.735	1.76607
$7$	− 137.301i	− 1.05908i	−0.848286	−	0.529539i	$-0.822365\pi$
$7$	− 137.301i	− 1.05908i	0.848286	−	0.529539i	$-0.177635\pi$
$8$	− 164.431i	− 0.908359i
$9$	−413.398	−1.70122
$10$	0	0
$11$	169.031	0.421196	0.210598	−	0.977573i	$-0.432459\pi$
$11$	169.031	0.421196	0.210598	+	0.977573i	$0.432459\pi$
$12$	− 126.800i	− 0.254195i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−834.595	−1.13804
$15$	0	0
$16$	−1157.88	−1.13074
$17$	487.128i	0.408810i	0.978886	+	0.204405i	$0.0655259\pi$
$17$	487.128i	0.408810i	−0.978886	+	0.204405i	$0.934474\pi$
$18$	2512.87i	1.82806i
$19$	−2385.49	−1.51598	−0.757990	−	0.652267i	$-0.773818\pi$
$19$	−2385.49	−1.51598	−0.757990	+	0.652267i	$0.773818\pi$
$20$	0	0
$21$	3517.68	1.74064
$22$	− 1027.47i	− 0.452597i
$23$	4144.47i	1.63361i	0.576911	+	0.816807i	$0.304258\pi$
$23$	4144.47i	1.63361i	−0.576911	+	0.816807i	$0.695742\pi$
$24$	4212.75	1.49292
$25$	0	0
$26$	1027.28	0.298027
$27$	− 4365.63i	− 1.15249i
$28$	679.532i	0.163800i
$29$	4570.45	1.00917	0.504585	−	0.863362i	$-0.331646\pi$
$29$	4570.45	1.00917	0.504585	+	0.863362i	$0.331646\pi$
$30$	0	0
$31$	−2957.68	−0.552774	−0.276387	−	0.961046i	$-0.589137\pi$
$31$	−2957.68	−0.552774	−0.276387	+	0.961046i	$0.589137\pi$
$32$	1776.50i	0.306683i
$33$	4330.61i	0.692252i
$34$	2961.05	0.439287
$35$	0	0
$36$	2046.00	0.263117
$37$	− 5733.02i	− 0.688461i	−0.938885	−	0.344230i	$-0.888140\pi$
$37$	− 5733.02i	− 0.688461i	0.938885	−	0.344230i	$-0.111860\pi$
$38$	14500.4i	1.62900i
$39$	−4329.82	−0.455836
$40$	0	0
$41$	−1705.07	−0.158410	−0.0792051	−	0.996858i	$-0.525238\pi$
$41$	−1705.07	−0.158410	−0.0792051	+	0.996858i	$0.525238\pi$
$42$	− 21382.5i	− 1.87041i
$43$	10922.2i	0.900821i	0.892822	+	0.450410i	$0.148722\pi$
$43$	10922.2i	0.900821i	−0.892822	+	0.450410i	$0.851278\pi$
$44$	−836.570	−0.0651434
$45$	0	0
$46$	25192.5	1.75540
$47$	7912.12i	0.522454i	0.965277	+	0.261227i	$0.0841272\pi$
$47$	7912.12i	0.522454i	−0.965277	+	0.261227i	$0.915873\pi$
$48$	− 29665.2i	− 1.85842i
$49$	−2044.52	−0.121647
$50$	0	0
$51$	−12480.4	−0.671895
$52$	− 836.418i	− 0.0428958i
$53$	26431.1i	1.29249i	0.763131	+	0.646244i	$0.223661\pi$
$53$	26431.1i	1.29249i	−0.763131	+	0.646244i	$0.776339\pi$
$54$	−26536.9	−1.23841
$55$	0	0
$56$	−22576.4	−0.962023
$57$	− 61116.8i	− 2.49157i
$58$	− 27781.9i	− 1.08441i
$59$	−11248.5	−0.420692	−0.210346	−	0.977627i	$-0.567459\pi$
$59$	−11248.5	−0.420692	−0.210346	+	0.977627i	$0.567459\pi$
$60$	0	0
$61$	−40298.6	−1.38665	−0.693323	−	0.720627i	$-0.743854\pi$
$61$	−40298.6	−1.38665	−0.693323	+	0.720627i	$0.743854\pi$
$62$	17978.5i	0.593984i
$63$	56759.8i	1.80173i
$64$	−26253.6	−0.801195
$65$	0	0
$66$	26324.0	0.743861
$67$	64887.2i	1.76592i	0.469444	+	0.882962i	$0.344454\pi$
$67$	64887.2i	1.76592i	−0.469444	+	0.882962i	$0.655546\pi$
$68$	− 2410.91i	− 0.0632278i
$69$	−106182.	−2.68491
$70$	0	0
$71$	−47076.1	−1.10829	−0.554147	−	0.832419i	$-0.686955\pi$
$71$	−47076.1	−1.10829	−0.554147	+	0.832419i	$0.686955\pi$
$72$	67975.2i	1.54532i
$73$	42771.8i	0.939400i	0.882826	+	0.469700i	$0.155638\pi$
$73$	42771.8i	0.939400i	−0.882826	+	0.469700i	$0.844362\pi$
$74$	−34848.7	−0.739787
$75$	0	0
$76$	11806.3	0.234466
$77$	− 23208.1i	− 0.446079i
$78$	26319.2i	0.489820i
$79$	−90057.0	−1.62349	−0.811745	−	0.584012i	$-0.801482\pi$
$79$	−90057.0	−1.62349	−0.811745	+	0.584012i	$0.801482\pi$
$80$	0	0
$81$	11392.9	0.192941
$82$	10364.4i	0.170220i
$83$	− 16427.0i	− 0.261735i	−0.991400	−	0.130868i	$-0.958224\pi$
$83$	− 16427.0i	− 0.261735i	0.991400	−	0.130868i	$-0.0417763\pi$
$84$	−17409.8	−0.269212
$85$	0	0
$86$	66391.5	0.967979
$87$	117096.i	1.65861i
$88$	− 27793.8i	− 0.382597i
$89$	35632.7	0.476841	0.238421	−	0.971162i	$-0.423370\pi$
$89$	35632.7	0.476841	0.238421	+	0.971162i	$0.423370\pi$
$90$	0	0
$91$	23203.8	0.293735
$92$	− 20511.9i	− 0.252660i
$93$	− 75776.6i	− 0.908506i
$94$	48094.5	0.561405
$95$	0	0
$96$	−45514.4	−0.504046
$97$	121030.i	1.30606i	0.757333	+	0.653029i	$0.226502\pi$
$97$	121030.i	1.30606i	−0.757333	+	0.653029i	$0.773498\pi$
$98$	12427.8i	0.130716i
$99$	−69876.9	−0.716549
$100$	0	0
$101$	−33043.6	−0.322318	−0.161159	−	0.986928i	$-0.551523\pi$
$101$	−33043.6	−0.322318	−0.161159	+	0.986928i	$0.551523\pi$
$102$	75862.9i	0.721987i
$103$	− 198815.i	− 1.84653i	−0.384164	−	0.923265i	$-0.625510\pi$
$103$	− 198815.i	− 1.84653i	0.384164	−	0.923265i	$-0.374490\pi$
$104$	27788.8	0.251933
$105$	0	0
$106$	160664.	1.38885
$107$	− 183355.i	− 1.54822i	−0.633049	−	0.774112i	$-0.718197\pi$
$107$	− 183355.i	− 1.54822i	0.633049	−	0.774112i	$-0.281803\pi$
$108$	21606.5i	0.178248i
$109$	178328.	1.43765	0.718827	−	0.695189i	$-0.244679\pi$
$109$	178328.	1.43765	0.718827	+	0.695189i	$0.244679\pi$
$110$	0	0
$111$	146881.	1.13151
$112$	158978.i	1.19754i
$113$	− 22426.7i	− 0.165223i	−0.996582	−	0.0826114i	$-0.973674\pi$
$113$	− 22426.7i	− 0.165223i	0.996582	−	0.0826114i	$-0.0263260\pi$
$114$	−371504.	−2.67733
$115$	0	0
$116$	−22620.2	−0.156081
$117$	− 69864.2i	− 0.471835i
$118$	68374.9i	0.452056i
$119$	66883.1	0.432961
$120$	0	0
$121$	−132480.	−0.822594
$122$	244959.i	1.49002i
$123$	− 43684.4i	− 0.260354i
$124$	14638.2	0.0854937
$125$	0	0
$126$	345020.	1.93605
$127$	− 90854.6i	− 0.499847i	−0.968266	−	0.249924i	$-0.919594\pi$
$127$	− 90854.6i	− 0.499847i	0.968266	−	0.249924i	$-0.0804055\pi$
$128$	216433.i	1.16761i
$129$	−279829.	−1.48054
$130$	0	0
$131$	−78399.9	−0.399151	−0.199575	−	0.979882i	$-0.563956\pi$
$131$	−78399.9	−0.399151	−0.199575	+	0.979882i	$0.563956\pi$
$132$	− 21433.1i	− 0.107066i
$133$	327530.i	1.60554i
$134$	394423.	1.89758
$135$	0	0
$136$	80098.8	0.371346
$137$	111261.i	0.506456i	0.967407	+	0.253228i	$0.0814923\pi$
$137$	111261.i	0.506456i	−0.967407	+	0.253228i	$0.918508\pi$
$138$	645439.i	2.88508i
$139$	226363.	0.993728	0.496864	−	0.867828i	$-0.334485\pi$
$139$	226363.	0.993728	0.496864	+	0.867828i	$0.334485\pi$
$140$	0	0
$141$	−202711.	−0.858675
$142$	286156.i	1.19092i
$143$	28566.2i	0.116819i
$144$	478665.	1.92365
$145$	0	0
$146$	259992.	1.00943
$147$	− 52381.1i	− 0.199931i
$148$	28374.0i	0.106479i
$149$	−443036.	−1.63483	−0.817416	−	0.576047i	$-0.804594\pi$
$149$	−443036.	−1.63483	−0.817416	+	0.576047i	$0.804594\pi$
$150$	0	0
$151$	−1334.00	−0.00476118	−0.00238059	−	0.999997i	$-0.500758\pi$
$151$	−1334.00	−0.00476118	−0.00238059	+	0.999997i	$0.500758\pi$
$152$	392247.i	1.37705i
$153$	− 201378.i	− 0.695477i
$154$	−141072.	−0.479336
$155$	0	0
$156$	21429.2	0.0705010
$157$	− 66009.4i	− 0.213726i	−0.994274	−	0.106863i	$-0.965919\pi$
$157$	− 66009.4i	− 0.213726i	0.994274	−	0.106863i	$-0.0340806\pi$
$158$	547419.i	1.74453i
$159$	−677173.	−2.12425
$160$	0	0
$161$	569039.	1.73012
$162$	− 69253.0i	− 0.207325i
$163$	57419.5i	0.169274i	0.996412	+	0.0846371i	$0.0269731\pi$
$163$	57419.5i	0.169274i	−0.996412	+	0.0846371i	$0.973027\pi$
$164$	8438.78	0.0245002
$165$	0	0
$166$	−99852.7	−0.281248
$167$	− 298875.i	− 0.829275i	−0.909987	−	0.414637i	$-0.863908\pi$
$167$	− 298875.i	− 0.829275i	0.909987	−	0.414637i	$-0.136092\pi$
$168$	− 578414.i	− 1.58112i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	986155.	2.57902
$172$	− 54056.3i	− 0.139324i
$173$	16389.5i	0.0416342i	0.999783	+	0.0208171i	$0.00662677\pi$
$173$	16389.5i	0.0416342i	−0.999783	+	0.0208171i	$0.993373\pi$
$174$	711779.	1.78226
$175$	0	0
$176$	−195717.	−0.476264
$177$	− 288189.i	− 0.691424i
$178$	− 216596.i	− 0.512391i
$179$	−420541.	−0.981015	−0.490507	−	0.871437i	$-0.663189\pi$
$179$	−420541.	−0.981015	−0.490507	+	0.871437i	$0.663189\pi$
$180$	0	0
$181$	−598369.	−1.35760	−0.678801	−	0.734322i	$-0.737500\pi$
$181$	−598369.	−1.35760	−0.678801	+	0.734322i	$0.737500\pi$
$182$	− 141047.i	− 0.315634i
$183$	− 1.03246e6i	− 2.27901i
$184$	681477.	1.48391
$185$	0	0
$186$	−460615.	−0.976237
$187$	82339.7i	0.172189i
$188$	− 39158.8i	− 0.0808044i
$189$	−599405.	−1.22058
$190$	0	0
$191$	207706.	0.411971	0.205985	−	0.978555i	$-0.433960\pi$
$191$	207706.	0.411971	0.205985	+	0.978555i	$0.433960\pi$
$192$	− 672623.i	− 1.31680i
$193$	545285.i	1.05373i	0.849948	+	0.526867i	$0.176633\pi$
$193$	545285.i	1.05373i	−0.849948	+	0.526867i	$0.823367\pi$
$194$	735690.	1.40343
$195$	0	0
$196$	10118.8	0.0188143
$197$	559003.i	1.02624i	0.858317	+	0.513119i	$0.171510\pi$
$197$	559003.i	1.02624i	−0.858317	+	0.513119i	$0.828490\pi$
$198$	424753.i	0.769969i
$199$	800588.	1.43310	0.716549	−	0.697536i	$-0.245720\pi$
$199$	800588.	1.43310	0.716549	+	0.697536i	$0.245720\pi$
$200$	0	0
$201$	−1.66243e6	−2.90237
$202$	200858.i	0.346347i
$203$	− 627527.i	− 1.06879i
$204$	61768.0	0.103917
$205$	0	0
$206$	−1.20851e6	−1.98419
$207$	− 1.71331e6i	− 2.77914i
$208$	− 195682.i	− 0.313612i
$209$	−403221.	−0.638524
$210$	0	0
$211$	−971904.	−1.50285	−0.751427	−	0.659816i	$-0.770634\pi$
$211$	−971904.	−1.50285	−0.751427	+	0.659816i	$0.770634\pi$
$212$	− 130814.i	− 0.199900i
$213$	− 1.20610e6i	− 1.82153i
$214$	−1.11454e6	−1.66365
$215$	0	0
$216$	−717843.	−1.04688
$217$	406092.i	0.585431i
$218$	− 1.08398e6i	− 1.54483i
$219$	−1.09582e6	−1.54394
$220$	0	0
$221$	−82324.7	−0.113383
$222$	− 892831.i	− 1.21587i
$223$	− 401286.i	− 0.540371i	−0.962808	−	0.270186i	$-0.912915\pi$
$223$	− 401286.i	− 0.540371i	0.962808	−	0.270186i	$-0.0870851\pi$
$224$	243915.	0.324802
$225$	0	0
$226$	−136323.	−0.177541
$227$	1.30062e6i	1.67527i	0.546227	+	0.837637i	$0.316064\pi$
$227$	1.30062e6i	1.67527i	−0.546227	+	0.837637i	$0.683936\pi$
$228$	302481.i	0.385354i
$229$	864830.	1.08979	0.544894	−	0.838505i	$-0.316570\pi$
$229$	864830.	1.08979	0.544894	+	0.838505i	$0.316570\pi$
$230$	0	0
$231$	594596.	0.733149
$232$	− 751521.i	− 0.916688i
$233$	1.19230e6i	1.43878i	0.694606	+	0.719390i	$0.255579\pi$
$233$	1.19230e6i	1.43878i	−0.694606	+	0.719390i	$0.744421\pi$
$234$	−424676.	−0.507011
$235$	0	0
$236$	55671.3	0.0650655
$237$	− 2.30728e6i	− 2.66827i
$238$	− 406555.i	− 0.465240i
$239$	1.23057e6	1.39352	0.696759	−	0.717305i	$-0.254625\pi$
$239$	1.23057e6	1.39352	0.696759	+	0.717305i	$0.254625\pi$
$240$	0	0
$241$	1.64306e6	1.82226	0.911129	−	0.412122i	$-0.135212\pi$
$241$	1.64306e6	1.82226	0.911129	+	0.412122i	$0.135212\pi$
$242$	805289.i	0.883921i
$243$	− 768958.i	− 0.835385i
$244$	199447.	0.214463
$245$	0	0
$246$	−265539.	−0.279764
$247$	− 403148.i	− 0.420457i
$248$	486333.i	0.502117i
$249$	420863.	0.430172
$250$	0	0
$251$	−447259.	−0.448100	−0.224050	−	0.974578i	$-0.571928\pi$
$251$	−447259.	−0.448100	−0.224050	+	0.974578i	$0.571928\pi$
$252$	− 280917.i	− 0.278661i
$253$	700543.i	0.688071i
$254$	−552267.	−0.537112
$255$	0	0
$256$	475490.	0.453463
$257$	− 465085.i	− 0.439237i	−0.975586	−	0.219619i	$-0.929519\pi$
$257$	− 465085.i	− 0.439237i	0.975586	−	0.219619i	$-0.0704813\pi$
$258$	1.70097e6i	1.59091i
$259$	−787148.	−0.729134
$260$	0	0
$261$	−1.88941e6	−1.71682
$262$	476560.i	0.428909i
$263$	− 1.88192e6i	− 1.67769i	−0.544369	−	0.838846i	$-0.683231\pi$
$263$	− 1.88192e6i	− 1.67769i	0.544369	−	0.838846i	$-0.316769\pi$
$264$	712085.	0.628813
$265$	0	0
$266$	1.99092e6	1.72524
$267$	912919.i	0.783707i
$268$	− 321141.i	− 0.273123i
$269$	−39508.5	−0.0332897	−0.0166448	−	0.999861i	$-0.505298\pi$
$269$	−39508.5	−0.0332897	−0.0166448	+	0.999861i	$0.505298\pi$
$270$	0	0
$271$	2.01156e6	1.66383	0.831916	−	0.554902i	$-0.187244\pi$
$271$	2.01156e6	1.66383	0.831916	+	0.554902i	$0.187244\pi$
$272$	− 564036.i	− 0.462258i
$273$	594488.i	0.482766i
$274$	676310.	0.544214
$275$	0	0
$276$	525520.	0.415256
$277$	750989.i	0.588077i	0.955794	+	0.294039i	$0.0949994\pi$
$277$	750989.i	0.588077i	−0.955794	+	0.294039i	$0.905001\pi$
$278$	− 1.37596e6i	− 1.06781i
$279$	1.22270e6	0.940392
$280$	0	0
$281$	−116949.	−0.0883550	−0.0441775	−	0.999024i	$-0.514067\pi$
$281$	−116949.	−0.0883550	−0.0441775	+	0.999024i	$0.514067\pi$
$282$	1.23219e6i	0.922691i
$283$	1.08953e6i	0.808671i	0.914611	+	0.404335i	$0.132497\pi$
$283$	1.08953e6i	0.808671i	−0.914611	+	0.404335i	$0.867503\pi$
$284$	232990.	0.171412
$285$	0	0
$286$	173642.	0.125528
$287$	234108.i	0.167769i
$288$	− 734400.i	− 0.521737i
$289$	1.18256e6	0.832875
$290$	0	0
$291$	−3.10081e6	−2.14656
$292$	− 211687.i	− 0.145290i
$293$	− 2.35419e6i	− 1.60204i	−0.598638	−	0.801020i	$-0.704291\pi$
$293$	− 2.35419e6i	− 1.60204i	0.598638	−	0.801020i	$-0.295709\pi$
$294$	−318403.	−0.214837
$295$	0	0
$296$	−942683.	−0.625369
$297$	− 737926.i	− 0.485424i
$298$	2.69303e6i	1.75671i
$299$	−700415.	−0.453083
$300$	0	0
$301$	1.49963e6	0.954040
$302$	8108.85i	0.00511613i
$303$	− 846585.i	− 0.529742i
$304$	2.76211e6	1.71418
$305$	0	0
$306$	−1.22409e6	−0.747327
$307$	1.40614e6i	0.851495i	0.904842	+	0.425748i	$0.139989\pi$
$307$	1.40614e6i	0.851495i	−0.904842	+	0.425748i	$0.860011\pi$
$308$	114862.i	0.0689920i
$309$	5.09369e6	3.03485
$310$	0	0
$311$	1.46661e6	0.859832	0.429916	−	0.902869i	$-0.358543\pi$
$311$	1.46661e6	0.859832	0.429916	+	0.902869i	$0.358543\pi$
$312$	711955.i	0.414063i
$313$	310576.i	0.179187i	0.995978	+	0.0895935i	$0.0285568\pi$
$313$	310576.i	0.179187i	−0.995978	+	0.0895935i	$0.971443\pi$
$314$	−401244.	−0.229659
$315$	0	0
$316$	445712.	0.251094
$317$	− 2.21756e6i	− 1.23944i	−0.784822	−	0.619721i	$-0.787246\pi$
$317$	− 2.21756e6i	− 1.23944i	0.784822	−	0.619721i	$-0.212754\pi$
$318$	4.11625e6i	2.28262i
$319$	772546.	0.425058
$320$	0	0
$321$	4.69761e6	2.54457
$322$	− 3.45895e6i	− 1.85911i
$323$	− 1.16204e6i	− 0.619747i
$324$	−56386.2	−0.0298408
$325$	0	0
$326$	349030.	0.181894
$327$	4.56882e6i	2.36284i
$328$	280366.i	0.143893i
$329$	1.08634e6	0.553320
$330$	0	0
$331$	439684.	0.220582	0.110291	−	0.993899i	$-0.464822\pi$
$331$	439684.	0.220582	0.110291	+	0.993899i	$0.464822\pi$
$332$	81300.7i	0.0404808i
$333$	2.37002e6i	1.17123i
$334$	−1.81674e6	−0.891099
$335$	0	0
$336$	−4.07306e6	−1.96821
$337$	− 65125.6i	− 0.0312376i	−0.999878	−	0.0156188i	$-0.995028\pi$
$337$	− 65125.6i	− 0.0312376i	0.999878	−	0.0156188i	$-0.00497181\pi$
$338$	173611.i	0.0826579i
$339$	574579.	0.271550
$340$	0	0
$341$	−499939.	−0.232826
$342$	− 5.99443e6i	− 2.77129i
$343$	− 2.02690e6i	− 0.930245i
$344$	1.79594e6	0.818269
$345$	0	0
$346$	99625.0	0.0447382
$347$	3.63163e6i	1.61912i	0.587039	+	0.809558i	$0.300293\pi$
$347$	3.63163e6i	1.61912i	−0.587039	+	0.809558i	$0.699707\pi$
$348$	− 579534.i	− 0.256526i
$349$	−755061.	−0.331832	−0.165916	−	0.986140i	$-0.553058\pi$
$349$	−755061.	−0.331832	−0.165916	+	0.986140i	$0.553058\pi$
$350$	0	0
$351$	737791.	0.319643
$352$	300283.i	0.129174i
$353$	− 3.06050e6i	− 1.30724i	−0.756823	−	0.653620i	$-0.773249\pi$
$353$	− 3.06050e6i	− 1.30724i	0.756823	−	0.653620i	$-0.226751\pi$
$354$	−1.75178e6	−0.742972
$355$	0	0
$356$	−176354.	−0.0737497
$357$	1.71356e6i	0.711590i
$358$	2.55629e6i	1.05415i
$359$	−3.68034e6	−1.50713	−0.753567	−	0.657371i	$-0.771668\pi$
$359$	−3.68034e6	−1.50713	−0.753567	+	0.657371i	$0.771668\pi$
$360$	0	0
$361$	3.21446e6	1.29819
$362$	3.63724e6i	1.45882i
$363$	− 3.39416e6i	− 1.35197i
$364$	−114841.	−0.0454300
$365$	0	0
$366$	−6.27591e6	−2.44891
$367$	2.22414e6i	0.861978i	0.902357	+	0.430989i	$0.141835\pi$
$367$	2.22414e6i	0.861978i	−0.902357	+	0.430989i	$0.858165\pi$
$368$	− 4.79880e6i	− 1.84720i
$369$	704873.	0.269491
$370$	0	0
$371$	3.62902e6	1.36885
$372$	375035.i	0.140512i
$373$	3.73922e6i	1.39158i	0.718244	+	0.695791i	$0.244946\pi$
$373$	3.73922e6i	1.39158i	−0.718244	+	0.695791i	$0.755054\pi$
$374$	500509.	0.185026
$375$	0	0
$376$	1.30099e6	0.474576
$377$	772406.i	0.279893i
$378$	3.64353e6i	1.31158i
$379$	−656559.	−0.234788	−0.117394	−	0.993085i	$-0.537454\pi$
$379$	−656559.	−0.234788	−0.117394	+	0.993085i	$0.537454\pi$
$380$	0	0
$381$	2.32772e6	0.821519
$382$	− 1.26256e6i	− 0.442684i
$383$	3.33584e6i	1.16200i	0.813902	+	0.581002i	$0.197339\pi$
$383$	3.33584e6i	1.16200i	−0.813902	+	0.581002i	$0.802661\pi$
$384$	−5.54506e6	−1.91901
$385$	0	0
$386$	3.31456e6	1.13229
$387$	− 4.51521e6i	− 1.53250i
$388$	− 599003.i	− 0.201999i
$389$	−4.44083e6	−1.48796	−0.743978	−	0.668204i	$-0.767063\pi$
$389$	−4.44083e6	−1.48796	−0.743978	+	0.668204i	$0.767063\pi$
$390$	0	0
$391$	−2.01889e6	−0.667837
$392$	336181.i	0.110499i
$393$	− 2.00863e6i	− 0.656021i
$394$	3.39795e6	1.10275
$395$	0	0
$396$	345836.	0.110824
$397$	− 2.72643e6i	− 0.868196i	−0.900866	−	0.434098i	$-0.857067\pi$
$397$	− 2.72643e6i	− 0.868196i	0.900866	−	0.434098i	$-0.142933\pi$
$398$	− 4.86644e6i	− 1.53994i
$399$	−8.39139e6	−2.63877
$400$	0	0
$401$	5.27026e6	1.63671	0.818353	−	0.574715i	$-0.194887\pi$
$401$	5.27026e6	1.63671	0.818353	+	0.574715i	$0.194887\pi$
$402$	1.01052e7i	3.11875i
$403$	− 499848.i	− 0.153312i
$404$	163540.	0.0498506
$405$	0	0
$406$	−3.81447e6	−1.14847
$407$	− 969056.i	− 0.289977i
$408$	2.05215e6i	0.610322i
$409$	−4.58065e6	−1.35400	−0.677000	−	0.735983i	$-0.736720\pi$
$409$	−4.58065e6	−1.35400	−0.677000	+	0.735983i	$0.736720\pi$
$410$	0	0
$411$	−2.85054e6	−0.832381
$412$	983979.i	0.285590i
$413$	1.54443e6i	0.445546i
$414$	−1.04145e7	−2.98634
$415$	0	0
$416$	−300228.	−0.0850586
$417$	5.79947e6i	1.63323i
$418$	2.45101e6i	0.686128i
$419$	−3.94543e6	−1.09789	−0.548945	−	0.835858i	$-0.684970\pi$
$419$	−3.94543e6	−1.09789	−0.548945	+	0.835858i	$0.684970\pi$
$420$	0	0
$421$	4.17743e6	1.14869	0.574347	−	0.818612i	$-0.305256\pi$
$421$	4.17743e6	1.14869	0.574347	+	0.818612i	$0.305256\pi$
$422$	5.90780e6i	1.61490i
$423$	− 3.27085e6i	− 0.888812i
$424$	4.34609e6	1.17404
$425$	0	0
$426$	−7.33140e6	−1.95732
$427$	5.53304e6i	1.46857i
$428$	907465.i	0.239453i
$429$	−731873.	−0.191996
$430$	0	0
$431$	−7.66563e6	−1.98772	−0.993859	−	0.110652i	$-0.964706\pi$
$431$	−7.66563e6	−1.98772	−0.993859	+	0.110652i	$0.964706\pi$
$432$	5.05488e6i	1.30317i
$433$	− 5.49209e6i	− 1.40773i	−0.710336	−	0.703863i	$-0.751457\pi$
$433$	− 5.49209e6i	− 1.40773i	0.710336	−	0.703863i	$-0.248543\pi$
$434$	2.46847e6	0.629076
$435$	0	0
$436$	−882586.	−0.222352
$437$	− 9.88658e6i	− 2.47652i
$438$	6.66107e6i	1.65905i
$439$	2.56331e6	0.634805	0.317403	−	0.948291i	$-0.397189\pi$
$439$	2.56331e6	0.634805	0.317403	+	0.948291i	$0.397189\pi$
$440$	0	0
$441$	845199.	0.206949
$442$	500418.i	0.121836i
$443$	1.41896e6i	0.343526i	0.985138	+	0.171763i	$0.0549463\pi$
$443$	1.41896e6i	0.343526i	−0.985138	+	0.171763i	$0.945054\pi$
$444$	−726948.	−0.175003
$445$	0	0
$446$	−2.43925e6	−0.580657
$447$	− 1.13507e7i	− 2.68691i
$448$	3.60464e6i	0.848528i
$449$	−3.56619e6	−0.834813	−0.417406	−	0.908720i	$-0.637061\pi$
$449$	−3.56619e6	−0.834813	−0.417406	+	0.908720i	$0.637061\pi$
$450$	0	0
$451$	−288210.	−0.0667217
$452$	110995.i	0.0255539i
$453$	− 34177.5i	− 0.00782519i
$454$	7.90593e6	1.80017
$455$	0	0
$456$	−1.00495e7	−2.26324
$457$	− 3.71538e6i	− 0.832172i	−0.909325	−	0.416086i	$-0.863402\pi$
$457$	− 3.71538e6i	− 0.832172i	0.909325	−	0.416086i	$-0.136598\pi$
$458$	− 5.25694e6i	− 1.17103i
$459$	2.12662e6	0.471149
$460$	0	0
$461$	3.74184e6	0.820035	0.410018	−	0.912078i	$-0.365523\pi$
$461$	3.74184e6	0.820035	0.410018	+	0.912078i	$0.365523\pi$
$462$	− 3.61431e6i	− 0.787807i
$463$	3.53774e6i	0.766962i	0.923549	+	0.383481i	$0.125275\pi$
$463$	3.53774e6i	0.766962i	−0.923549	+	0.383481i	$0.874725\pi$
$464$	−5.29203e6	−1.14111
$465$	0	0
$466$	7.24748e6	1.54605
$467$	− 1.97789e6i	− 0.419671i	−0.977737	−	0.209836i	$-0.932707\pi$
$467$	− 1.97789e6i	− 0.419671i	0.977737	−	0.209836i	$-0.0672929\pi$
$468$	345773.i	0.0729754i
$469$	8.90907e6	1.87025
$470$	0	0
$471$	1.69118e6	0.351267
$472$	1.84960e6i	0.382139i
$473$	1.84619e6i	0.379422i
$474$	−1.40250e7	−2.86720
$475$	0	0
$476$	−331019.	−0.0669632
$477$	− 1.09266e7i	− 2.19881i
$478$	− 7.48014e6i	− 1.49741i
$479$	2.88196e6	0.573918	0.286959	−	0.957943i	$-0.407356\pi$
$479$	2.88196e6	0.573918	0.286959	+	0.957943i	$0.407356\pi$
$480$	0	0
$481$	968880.	0.190945
$482$	− 9.98746e6i	− 1.95811i
$483$	1.45789e7i	2.84353i
$484$	655671.	0.127225
$485$	0	0
$486$	−4.67418e6	−0.897665
$487$	− 2.88578e6i	− 0.551366i	−0.961249	−	0.275683i	$-0.911096\pi$
$487$	− 2.88578e6i	− 0.551366i	0.961249	−	0.275683i	$-0.0889040\pi$
$488$	6.62632e6i	1.25957i
$489$	−1.47110e6	−0.278209
$490$	0	0
$491$	5.27189e6	0.986876	0.493438	−	0.869781i	$-0.335740\pi$
$491$	5.27189e6	0.986876	0.493438	+	0.869781i	$0.335740\pi$
$492$	216204.i	0.0402671i
$493$	2.22640e6i	0.412558i
$494$	−2.45057e6	−0.451803
$495$	0	0
$496$	3.42464e6	0.625045
$497$	6.46359e6i	1.17377i
$498$	− 2.55825e6i	− 0.462243i
$499$	308093.	0.0553898	0.0276949	−	0.999616i	$-0.491183\pi$
$499$	308093.	0.0553898	0.0276949	+	0.999616i	$0.491183\pi$
$500$	0	0
$501$	7.65725e6	1.36295
$502$	2.71870e6i	0.481507i
$503$	− 4.27447e6i	− 0.753291i	−0.926358	−	0.376645i	$-0.877078\pi$
$503$	− 4.27447e6i	− 0.753291i	0.926358	−	0.376645i	$-0.122922\pi$
$504$	9.33305e6	1.63662
$505$	0	0
$506$	4.25831e6	0.739368
$507$	− 731740.i	− 0.126426i
$508$	449659.i	0.0773080i
$509$	405253.	0.0693317	0.0346658	−	0.999399i	$-0.488963\pi$
$509$	405253.	0.0693317	0.0346658	+	0.999399i	$0.488963\pi$
$510$	0	0
$511$	5.87260e6	0.994898
$512$	4.03553e6i	0.680340i
$513$	1.04142e7i	1.74715i
$514$	−2.82706e6	−0.471984
$515$	0	0
$516$	1.38494e6	0.228984
$517$	1.33739e6i	0.220056i
$518$	4.78475e6i	0.783492i
$519$	−419903.	−0.0684275
$520$	0	0
$521$	−7.00100e6	−1.12997	−0.564983	−	0.825102i	$-0.691117\pi$
$521$	−7.00100e6	−1.12997	−0.564983	+	0.825102i	$0.691117\pi$
$522$	1.14850e7i	1.84482i
$523$	7.78764e6i	1.24495i	0.782640	+	0.622475i	$0.213873\pi$
$523$	7.78764e6i	1.24495i	−0.782640	+	0.622475i	$0.786127\pi$
$524$	388018.	0.0617339
$525$	0	0
$526$	−1.14394e7	−1.80277
$527$	− 1.44077e6i	− 0.225979i
$528$	− 5.01433e6i	− 0.782759i
$529$	−1.07403e7	−1.66869
$530$	0	0
$531$	4.65010e6	0.715692
$532$	− 1.62102e6i	− 0.248318i
$533$	− 288157.i	− 0.0439351i
$534$	5.54926e6	0.842135
$535$	0	0
$536$	1.06694e7	1.60409
$537$	− 1.07744e7i	− 1.61234i
$538$	240156.i	0.0357715i
$539$	−345586.	−0.0512371
$540$	0	0
$541$	3.36678e6	0.494562	0.247281	−	0.968944i	$-0.420463\pi$
$541$	3.36678e6	0.494562	0.247281	+	0.968944i	$0.420463\pi$
$542$	− 1.22274e7i	− 1.78787i
$543$	− 1.53304e7i	− 2.23128i
$544$	−865383.	−0.125375
$545$	0	0
$546$	3.61365e6	0.518757
$547$	1.32125e7i	1.88806i	0.329860	+	0.944030i	$0.392999\pi$
$547$	1.32125e7i	1.88806i	−0.329860	+	0.944030i	$0.607001\pi$
$548$	− 550656.i	− 0.0783301i
$549$	1.66594e7	2.35900
$550$	0	0
$551$	−1.09028e7	−1.52988
$552$	1.74596e7i	2.43886i
$553$	1.23649e7i	1.71940i
$554$	4.56495e6	0.631920
$555$	0	0
$556$	−1.12032e6	−0.153693
$557$	7.47598e6i	1.02101i	0.859875	+	0.510505i	$0.170541\pi$
$557$	7.47598e6i	1.02101i	−0.859875	+	0.510505i	$0.829459\pi$
$558$	− 7.43228e6i	− 1.01050i
$559$	−1.84585e6	−0.249843
$560$	0	0
$561$	−2.10956e6	−0.282999
$562$	710886.i	0.0949421i
$563$	− 1.32316e7i	− 1.75931i	−0.475617	−	0.879653i	$-0.657775\pi$
$563$	− 1.32316e7i	− 1.75931i	0.475617	−	0.879653i	$-0.342225\pi$
$564$	1.00326e6	0.132805
$565$	0	0
$566$	6.62278e6	0.868959
$567$	− 1.56426e6i	− 0.204339i
$568$	7.74075e6i	1.00673i
$569$	9.80441e6	1.26952	0.634762	−	0.772708i	$-0.281098\pi$
$569$	9.80441e6	1.26952	0.634762	+	0.772708i	$0.281098\pi$
$570$	0	0
$571$	2.98442e6	0.383063	0.191532	−	0.981486i	$-0.438655\pi$
$571$	2.98442e6	0.383063	0.191532	+	0.981486i	$0.438655\pi$
$572$	− 141380.i	− 0.0180675i
$573$	5.32149e6i	0.677091i
$574$	1.42305e6	0.180276
$575$	0	0
$576$	1.08532e7	1.36301
$577$	− 1.47218e7i	− 1.84086i	−0.390911	−	0.920429i	$-0.627840\pi$
$577$	− 1.47218e7i	− 1.84086i	0.390911	−	0.920429i	$-0.372160\pi$
$578$	− 7.18831e6i	− 0.894968i
$579$	−1.39704e7	−1.73185
$580$	0	0
$581$	−2.25544e6	−0.277198
$582$	1.88486e7i	2.30659i
$583$	4.46768e6i	0.544390i
$584$	7.03299e6	0.853312
$585$	0	0
$586$	−1.43102e7	−1.72148
$587$	− 9.66987e6i	− 1.15831i	−0.815217	−	0.579156i	$-0.803382\pi$
$587$	− 9.66987e6i	− 1.15831i	0.815217	−	0.579156i	$-0.196618\pi$
$588$	259245.i	0.0309220i
$589$	7.05552e6	0.837993
$590$	0	0
$591$	−1.43218e7	−1.68666
$592$	6.63815e6i	0.778472i
$593$	1.52788e7i	1.78424i	0.451797	+	0.892121i	$0.350783\pi$
$593$	1.52788e7i	1.78424i	−0.451797	+	0.892121i	$0.649217\pi$
$594$	−4.48554e6	−0.521614
$595$	0	0
$596$	2.19268e6	0.252848
$597$	2.05113e7i	2.35536i
$598$	4.25754e6i	0.486861i
$599$	−7.95473e6	−0.905854	−0.452927	−	0.891548i	$-0.649620\pi$
$599$	−7.95473e6	−0.905854	−0.452927	+	0.891548i	$0.649620\pi$
$600$	0	0
$601$	5.48231e6	0.619123	0.309562	−	0.950879i	$-0.399818\pi$
$601$	5.48231e6	0.619123	0.309562	+	0.950879i	$0.399818\pi$
$602$	− 9.11560e6i	− 1.02517i
$603$	− 2.68242e7i	− 3.00423i
$604$	6602.27	0.000736378 0
$605$	0	0
$606$	−5.14604e6	−0.569235
$607$	− 1.49346e6i	− 0.164521i	−0.996611	−	0.0822605i	$-0.973786\pi$
$607$	− 1.49346e6i	− 0.164521i	0.996611	−	0.0822605i	$-0.0262139\pi$
$608$	− 4.23782e6i	− 0.464926i
$609$	1.60774e7	1.75660
$610$	0	0
$611$	−1.33715e6	−0.144903
$612$	996663.i	0.107565i
$613$	9.31063e6i	1.00076i	0.865807	+	0.500378i	$0.166805\pi$
$613$	9.31063e6i	1.00076i	−0.865807	+	0.500378i	$0.833195\pi$
$614$	8.54733e6	0.914976
$615$	0	0
$616$	−3.81611e6	−0.405200
$617$	245403.i	0.0259517i	0.999916	+	0.0129759i	$0.00413046\pi$
$617$	245403.i	0.0259517i	−0.999916	+	0.0129759i	$0.995870\pi$
$618$	− 3.09625e7i	− 3.26110i
$619$	−2.82756e6	−0.296610	−0.148305	−	0.988942i	$-0.547382\pi$
$619$	−2.82756e6	−0.296610	−0.148305	+	0.988942i	$0.547382\pi$
$620$	0	0
$621$	1.80932e7	1.88272
$622$	− 8.91491e6i	− 0.923934i
$623$	− 4.89240e6i	− 0.505012i
$624$	5.01342e6	0.515433
$625$	0	0
$626$	1.88786e6	0.192546
$627$	− 1.03306e7i	− 1.04944i
$628$	326695.i	0.0330555i
$629$	2.79272e6	0.281449
$630$	0	0
$631$	−1.53391e7	−1.53366	−0.766828	−	0.641853i	$-0.778166\pi$
$631$	−1.53391e7	−1.53366	−0.766828	+	0.641853i	$0.778166\pi$
$632$	1.48081e7i	1.47471i
$633$	− 2.49004e7i	− 2.47000i
$634$	−1.34796e7	−1.33185
$635$	0	0
$636$	3.35148e6	0.328544
$637$	− 345523.i	− 0.0337388i
$638$	− 4.69599e6i	− 0.456747i
$639$	1.94612e7	1.88546
$640$	0	0
$641$	1.02617e7	0.986444	0.493222	−	0.869904i	$-0.335819\pi$
$641$	1.02617e7	0.986444	0.493222	+	0.869904i	$0.335819\pi$
$642$	− 2.85548e7i	− 2.73427i
$643$	− 5.26807e6i	− 0.502486i	−0.967924	−	0.251243i	$-0.919161\pi$
$643$	− 5.26807e6i	− 0.502486i	0.967924	−	0.251243i	$-0.0808394\pi$
$644$	−2.81630e6	−0.267586
$645$	0	0
$646$	−7.06356e6	−0.665951
$647$	− 2.03948e7i	− 1.91540i	−0.287775	−	0.957698i	$-0.592915\pi$
$647$	− 2.03948e7i	− 1.91540i	0.287775	−	0.957698i	$-0.407085\pi$
$648$	− 1.87335e6i	− 0.175259i
$649$	−1.90134e6	−0.177194
$650$	0	0
$651$	−1.04042e7	−0.962179
$652$	− 284182.i	− 0.0261805i
$653$	− 8.69527e6i	− 0.797994i	−0.916952	−	0.398997i	$-0.869358\pi$
$653$	− 8.69527e6i	− 0.797994i	0.916952	−	0.398997i	$-0.130642\pi$
$654$	2.77720e7	2.53900
$655$	0	0
$656$	1.97427e6	0.179121
$657$	− 1.76818e7i	− 1.59813i
$658$	− 6.60342e6i	− 0.594571i
$659$	−1.85493e7	−1.66385	−0.831926	−	0.554886i	$-0.812762\pi$
$659$	−1.85493e7	−1.66385	−0.831926	+	0.554886i	$0.812762\pi$
$660$	0	0
$661$	1.00321e6	0.0893078	0.0446539	−	0.999003i	$-0.485782\pi$
$661$	1.00321e6	0.0893078	0.0446539	+	0.999003i	$0.485782\pi$
$662$	− 2.67266e6i	− 0.237027i
$663$	− 2.10918e6i	− 0.186350i
$664$	−2.70109e6	−0.237750
$665$	0	0
$666$	1.44063e7	1.25854
$667$	1.89421e7i	1.64859i
$668$	1.47920e6i	0.128258i
$669$	1.02811e7	0.888122
$670$	0	0
$671$	−6.81171e6	−0.584049
$672$	6.24916e6i	0.533824i
$673$	− 653810.i	− 0.0556434i	−0.999613	−	0.0278217i	$-0.991143\pi$
$673$	− 653810.i	− 0.0556434i	0.999613	−	0.0278217i	$-0.00885706\pi$
$674$	−395872.	−0.0335664
$675$	0	0
$676$	141355.	0.0118972
$677$	− 831237.i	− 0.0697033i	−0.999392	−	0.0348516i	$-0.988904\pi$
$677$	− 831237.i	− 0.0697033i	0.999392	−	0.0348516i	$-0.0110959\pi$
$678$	− 3.49263e6i	− 0.291795i
$679$	1.66175e7	1.38322
$680$	0	0
$681$	−3.33222e7	−2.75338
$682$	3.03892e6i	0.250184i
$683$	− 4.86860e6i	− 0.399349i	−0.979862	−	0.199675i	$-0.936012\pi$
$683$	− 4.86860e6i	− 0.399349i	0.979862	−	0.199675i	$-0.0639885\pi$
$684$	−4.88070e6	−0.398879
$685$	0	0
$686$	−1.23207e7	−0.999597
$687$	2.21572e7i	1.79111i
$688$	− 1.26466e7i	− 1.01860i
$689$	−4.46686e6	−0.358471
$690$	0	0
$691$	−7.17359e6	−0.571533	−0.285767	−	0.958299i	$-0.592248\pi$
$691$	−7.17359e6	−0.571533	−0.285767	+	0.958299i	$0.592248\pi$
$692$	− 81115.2i	− 0.00643928i
$693$	9.59416e6i	0.758881i
$694$	2.20752e7	1.73983
$695$	0	0
$696$	1.92542e7	1.50661
$697$	− 830589.i	− 0.0647596i
$698$	4.58970e6i	0.356571i
$699$	−3.05470e7	−2.36469
$700$	0	0
$701$	−9.45026e6	−0.726355	−0.363177	−	0.931720i	$-0.618308\pi$
$701$	−9.45026e6	−0.726355	−0.363177	+	0.931720i	$0.618308\pi$
$702$	− 4.48473e6i	− 0.343474i
$703$	1.36761e7i	1.04369i
$704$	−4.43766e6	−0.337460
$705$	0	0
$706$	−1.86035e7	−1.40470
$707$	4.53691e6i	0.341359i
$708$	1.42631e6i	0.106938i
$709$	2.98225e6	0.222807	0.111404	−	0.993775i	$-0.464465\pi$
$709$	2.98225e6	0.222807	0.111404	+	0.993775i	$0.464465\pi$
$710$	0	0
$711$	3.72293e7	2.76192
$712$	− 5.85910e6i	− 0.433143i
$713$	− 1.22580e7i	− 0.903019i
$714$	1.04160e7	0.764640
$715$	0	0
$716$	2.08135e6	0.151727
$717$	3.15276e7i	2.29030i
$718$	2.23713e7i	1.61949i
$719$	−1.65959e7	−1.19723	−0.598617	−	0.801036i	$-0.704283\pi$
$719$	−1.65959e7	−1.19723	−0.598617	+	0.801036i	$0.704283\pi$
$720$	0	0
$721$	−2.72975e7	−1.95562
$722$	− 1.95394e7i	− 1.39498i
$723$	4.20955e7i	2.99495i
$724$	2.96146e6	0.209971
$725$	0	0
$726$	−2.06317e7	−1.45276
$727$	− 47580.6i	− 0.00333882i	−0.999999	−	0.00166941i	$-0.999469\pi$
$727$	− 47580.6i	− 0.00333882i	0.999999	−	0.00166941i	$-0.000531391\pi$
$728$	− 3.81542e6i	− 0.266817i
$729$	2.24694e7	1.56593
$730$	0	0
$731$	−5.32051e6	−0.368264
$732$	5.10988e6i	0.352479i
$733$	6.02222e6i	0.413996i	0.978341	+	0.206998i	$0.0663694\pi$
$733$	6.02222e6i	0.413996i	−0.978341	+	0.206998i	$0.933631\pi$
$734$	1.35196e7	0.926241
$735$	0	0
$736$	−7.36265e6	−0.501002
$737$	1.09679e7i	0.743800i
$738$	− 4.28463e6i	− 0.289583i
$739$	−8.15662e6	−0.549413	−0.274707	−	0.961528i	$-0.588581\pi$
$739$	−8.15662e6	−0.549413	−0.274707	+	0.961528i	$0.588581\pi$
$740$	0	0
$741$	1.03287e7	0.691038
$742$	− 2.20593e7i	− 1.47090i
$743$	473118.i	0.0314411i	0.999876	+	0.0157205i	$0.00500421\pi$
$743$	473118.i	0.0314411i	−0.999876	+	0.0157205i	$0.994996\pi$
$744$	−1.24600e7	−0.825249
$745$	0	0
$746$	2.27292e7	1.49533
$747$	6.79087e6i	0.445270i
$748$	− 407517.i	− 0.0266313i
$749$	−2.51748e7	−1.63969
$750$	0	0
$751$	5.49506e6	0.355527	0.177764	−	0.984073i	$-0.443114\pi$
$751$	5.49506e6	0.355527	0.177764	+	0.984073i	$0.443114\pi$
$752$	− 9.16129e6i	− 0.590761i
$753$	− 1.14589e7i	− 0.736470i
$754$	4.69514e6	0.300760
$755$	0	0
$756$	2.96659e6	0.188778
$757$	− 6.82041e6i	− 0.432585i	−0.976329	−	0.216292i	$-0.930604\pi$
$757$	− 6.82041e6i	− 0.432585i	0.976329	−	0.216292i	$-0.0693964\pi$
$758$	3.99095e6i	0.252292i
$759$	−1.79481e7	−1.13087
$760$	0	0
$761$	1.14435e7	0.716304	0.358152	−	0.933663i	$-0.383407\pi$
$761$	1.14435e7	0.716304	0.358152	+	0.933663i	$0.383407\pi$
$762$	− 1.41492e7i	− 0.882766i
$763$	− 2.44846e7i	− 1.52259i
$764$	−1.02798e6	−0.0637167
$765$	0	0
$766$	2.02772e7	1.24863
$767$	− 1.90100e6i	− 0.116679i
$768$	1.21822e7i	0.745284i
$769$	5.05628e6	0.308330	0.154165	−	0.988045i	$-0.450731\pi$
$769$	5.05628e6	0.308330	0.154165	+	0.988045i	$0.450731\pi$
$770$	0	0
$771$	1.19156e7	0.721904
$772$	− 2.69874e6i	− 0.162974i
$773$	3.54919e6i	0.213639i	0.994278	+	0.106820i	$0.0340667\pi$
$773$	3.54919e6i	0.213639i	−0.994278	+	0.106820i	$0.965933\pi$
$774$	−2.74461e7	−1.64675
$775$	0	0
$776$	1.99010e7	1.18637
$777$	− 2.01669e7i	− 1.19836i
$778$	2.69940e7i	1.59889i
$779$	4.06743e6	0.240147
$780$	0	0
$781$	−7.95731e6	−0.466809
$782$	1.22720e7i	0.717626i
$783$	− 1.99529e7i	− 1.16306i
$784$	2.36731e6	0.137551
$785$	0	0
$786$	−1.22096e7	−0.704929
$787$	− 2.14355e7i	− 1.23367i	−0.787094	−	0.616833i	$-0.788415\pi$
$787$	− 2.14355e7i	− 1.23367i	0.787094	−	0.616833i	$-0.211585\pi$
$788$	− 2.76663e6i	− 0.158721i
$789$	4.82153e7	2.75735
$790$	0	0
$791$	−3.07921e6	−0.174984
$792$	1.14899e7i	0.650883i
$793$	− 6.81047e6i	− 0.384586i
$794$	−1.65728e7	−0.932923
$795$	0	0
$796$	−3.96228e6	−0.221648
$797$	− 1.94471e7i	− 1.08445i	−0.840235	−	0.542223i	$-0.817583\pi$
$797$	− 1.94471e7i	− 1.08445i	0.840235	−	0.542223i	$-0.182417\pi$
$798$	5.10078e7i	2.83550i
$799$	−3.85422e6	−0.213584
$800$	0	0
$801$	−1.47305e7	−0.811214
$802$	− 3.20357e7i	− 1.75873i
$803$	7.22975e6i	0.395671i
$804$	8.22772e6	0.448889
$805$	0	0
$806$	−3.03837e6	−0.164742
$807$	− 1.01222e6i	− 0.0547129i
$808$	5.43338e6i	0.292780i
$809$	−1.45681e7	−0.782585	−0.391292	−	0.920266i	$-0.627972\pi$
$809$	−1.45681e7	−0.782585	−0.391292	+	0.920266i	$0.627972\pi$
$810$	0	0
$811$	1.86252e7	0.994371	0.497185	−	0.867644i	$-0.334367\pi$
$811$	1.86252e7	0.994371	0.497185	+	0.867644i	$0.334367\pi$
$812$	3.10577e6i	0.165302i
$813$	5.15366e7i	2.73457i
$814$	−5.89049e6	−0.311595
$815$	0	0
$816$	1.44508e7	0.759740
$817$	− 2.60548e7i	− 1.36563i
$818$	2.78439e7i	1.45494i
$819$	−9.59241e6	−0.499710
$820$	0	0
$821$	1.61122e7	0.834252	0.417126	−	0.908849i	$-0.363037\pi$
$821$	1.61122e7	0.834252	0.417126	+	0.908849i	$0.363037\pi$
$822$	1.73272e7i	0.894438i
$823$	− 1.15862e7i	− 0.596269i	−0.954524	−	0.298134i	$-0.903636\pi$
$823$	− 1.15862e7i	− 0.596269i	0.954524	−	0.298134i	$-0.0963643\pi$
$824$	−3.26913e7	−1.67731
$825$	0	0
$826$	9.38794e6	0.478762
$827$	− 8.40345e6i	− 0.427262i	−0.976914	−	0.213631i	$-0.931471\pi$
$827$	− 8.40345e6i	− 0.427262i	0.976914	−	0.213631i	$-0.0685289\pi$
$828$	8.47957e6i	0.429831i
$829$	4.77193e6	0.241161	0.120581	−	0.992704i	$-0.461524\pi$
$829$	4.77193e6	0.241161	0.120581	+	0.992704i	$0.461524\pi$
$830$	0	0
$831$	−1.92405e7	−0.966528
$832$	− 4.43685e6i	− 0.222212i
$833$	− 995943.i	− 0.0497304i
$834$	3.52526e7	1.75499
$835$	0	0
$836$	1.99563e6	0.0987561
$837$	1.29121e7i	0.637067i
$838$	2.39826e7i	1.17974i
$839$	2.32624e7	1.14091	0.570453	−	0.821330i	$-0.306768\pi$
$839$	2.32624e7	1.14091	0.570453	+	0.821330i	$0.306768\pi$
$840$	0	0
$841$	377863.	0.0184223
$842$	− 2.53929e7i	− 1.23433i
$843$	− 2.99627e6i	− 0.145215i
$844$	4.81016e6	0.232436
$845$	0	0
$846$	−1.98822e7	−0.955075
$847$	1.81896e7i	0.871192i
$848$	− 3.06041e7i	− 1.46147i
$849$	−2.79140e7	−1.32908
$850$	0	0
$851$	2.37603e7	1.12468
$852$	5.96927e6i	0.281723i
$853$	− 6.88858e6i	− 0.324158i	−0.986778	−	0.162079i	$-0.948180\pi$
$853$	− 6.88858e6i	− 0.324158i	0.986778	−	0.162079i	$-0.0518200\pi$
$854$	3.36330e7	1.57805
$855$	0	0
$856$	−3.01492e7	−1.40634
$857$	− 782343.i	− 0.0363869i	−0.999834	−	0.0181935i	$-0.994209\pi$
$857$	− 782343.i	− 0.0363869i	0.999834	−	0.0181935i	$-0.00579148\pi$
$858$	4.44876e6i	0.206310i
$859$	−1.23216e7	−0.569751	−0.284875	−	0.958565i	$-0.591952\pi$
$859$	−1.23216e7	−0.569751	−0.284875	+	0.958565i	$0.591952\pi$
$860$	0	0
$861$	−5.99790e6	−0.275735
$862$	4.65962e7i	2.13591i
$863$	− 7.42942e6i	− 0.339569i	−0.985481	−	0.169785i	$-0.945693\pi$
$863$	− 7.42942e6i	− 0.339569i	0.985481	−	0.169785i	$-0.0543072\pi$
$864$	7.75554e6	0.353450
$865$	0	0
$866$	−3.33842e7	−1.51268
$867$	3.02976e7i	1.36886i
$868$	− 2.00984e6i	− 0.0905445i
$869$	−1.52224e7	−0.683807
$870$	0	0
$871$	−1.09659e7	−0.489779
$872$	− 2.93226e7i	− 1.30591i
$873$	− 5.00334e7i	− 2.22190i
$874$	−6.00965e7	−2.66116
$875$	0	0
$876$	5.42348e6	0.238791
$877$	7.95778e6i	0.349376i	0.984624	+	0.174688i	$0.0558917\pi$
$877$	7.95778e6i	0.349376i	−0.984624	+	0.174688i	$0.944108\pi$
$878$	− 1.55813e7i	− 0.682132i
$879$	6.03151e7	2.63302
$880$	0	0
$881$	3.11775e7	1.35333	0.676663	−	0.736293i	$-0.263426\pi$
$881$	3.11775e7	1.35333	0.676663	+	0.736293i	$0.263426\pi$
$882$	− 5.13761e6i	− 0.222377i
$883$	− 2.20671e7i	− 0.952452i	−0.879323	−	0.476226i	$-0.842004\pi$
$883$	− 2.20671e7i	− 0.952452i	0.879323	−	0.476226i	$-0.157996\pi$
$884$	407443.	0.0175362
$885$	0	0
$886$	8.62524e6	0.369137
$887$	1.08566e6i	0.0463323i	0.999732	+	0.0231662i	$0.00737468\pi$
$887$	1.08566e6i	0.0463323i	−0.999732	+	0.0231662i	$0.992625\pi$
$888$	− 2.41518e7i	− 1.02782i
$889$	−1.24744e7	−0.529378
$890$	0	0
$891$	1.92576e6	0.0812657
$892$	1.98605e6i	0.0835755i
$893$	− 1.88743e7i	− 0.792030i
$894$	−6.89962e7	−2.88723
$895$	0	0
$896$	2.97164e7	1.23659
$897$	− 1.79448e7i	− 0.744660i
$898$	2.16774e7i	0.897050i
$899$	−1.35179e7	−0.557842
$900$	0	0
$901$	−1.28754e7	−0.528381
$902$	1.75191e6i	0.0716960i
$903$	3.84208e7i	1.56800i
$904$	−3.68764e6	−0.150082
$905$	0	0
$906$	−207751.	−0.00840857
$907$	3.82701e7i	1.54469i	0.635203	+	0.772345i	$0.280916\pi$
$907$	3.82701e7i	1.54469i	−0.635203	+	0.772345i	$0.719084\pi$
$908$	− 6.43706e6i	− 0.259103i
$909$	1.36601e7	0.548334
$910$	0	0
$911$	−4.01079e7	−1.60116	−0.800579	−	0.599227i	$-0.795475\pi$
$911$	−4.01079e7	−1.60116	−0.800579	+	0.599227i	$0.795475\pi$
$912$	7.07660e7i	2.81733i
$913$	− 2.77666e6i	− 0.110242i
$914$	−2.25843e7	−0.894212
$915$	0	0
$916$	−4.28023e6	−0.168550
$917$	1.07644e7i	0.422732i
$918$	− 1.29269e7i	− 0.506275i
$919$	−1.38759e6	−0.0541967	−0.0270984	−	0.999633i	$-0.508627\pi$
$919$	−1.38759e6	−0.0541967	−0.0270984	+	0.999633i	$0.508627\pi$
$920$	0	0
$921$	−3.60256e7	−1.39947
$922$	− 2.27451e7i	− 0.881171i
$923$	− 7.95587e6i	− 0.307385i
$924$	−2.94279e6	−0.113391
$925$	0	0
$926$	2.15045e7	0.824141
$927$	8.21897e7i	3.14136i
$928$	8.11940e6i	0.309495i
$929$	7.82980e6	0.297654	0.148827	−	0.988863i	$-0.452450\pi$
$929$	7.82980e6	0.297654	0.148827	+	0.988863i	$0.452450\pi$
$930$	0	0
$931$	4.87717e6	0.184414
$932$	− 5.90094e6i	− 0.222526i
$933$	3.75749e7i	1.41317i
$934$	−1.20228e7	−0.450959
$935$	0	0
$936$	−1.14878e7	−0.428595
$937$	5.19939e7i	1.93465i	0.253532	+	0.967327i	$0.418408\pi$
$937$	5.19939e7i	1.93465i	−0.253532	+	0.967327i	$0.581592\pi$
$938$	− 5.41546e7i	− 2.00968i
$939$	−7.95703e6	−0.294501
$940$	0	0
$941$	2.54802e7	0.938055	0.469028	−	0.883184i	$-0.344605\pi$
$941$	2.54802e7	0.938055	0.469028	+	0.883184i	$0.344605\pi$
$942$	− 1.02800e7i	− 0.377455i
$943$	− 7.06662e6i	− 0.258781i
$944$	1.30244e7	0.475694
$945$	0	0
$946$	1.12222e7	0.407709
$947$	6.45145e6i	0.233767i	0.993146	+	0.116883i	$0.0372903\pi$
$947$	6.45145e6i	0.233767i	−0.993146	+	0.116883i	$0.962710\pi$
$948$	1.14193e7i	0.412683i
$949$	−7.22843e6	−0.260543
$950$	0	0
$951$	5.68143e7	2.03707
$952$	− 1.09976e7i	− 0.393284i
$953$	2.82313e7i	1.00693i	0.864016	+	0.503465i	$0.167942\pi$
$953$	2.82313e7i	1.00693i	−0.864016	+	0.503465i	$0.832058\pi$
$954$	−6.64181e7	−2.36274
$955$	0	0
$956$	−6.09038e6	−0.215526
$957$	1.97928e7i	0.698599i
$958$	− 1.75183e7i	− 0.616705i
$959$	1.52762e7	0.536377
$960$	0	0
$961$	−1.98813e7	−0.694441
$962$	− 5.88942e6i	− 0.205180i
$963$	7.57986e7i	2.63388i
$964$	−8.13185e6	−0.281836
$965$	0	0
$966$	8.86193e7	3.05552
$967$	5.33839e7i	1.83588i	0.396722	+	0.917939i	$0.370148\pi$
$967$	5.33839e7i	1.83588i	−0.396722	+	0.917939i	$0.629852\pi$
$968$	2.17837e7i	0.747211i
$969$	2.97717e7	1.01858
$970$	0	0
$971$	3.29212e7	1.12054	0.560270	−	0.828310i	$-0.310697\pi$
$971$	3.29212e7	1.12054	0.560270	+	0.828310i	$0.310697\pi$
$972$	3.80574e6i	0.129203i
$973$	− 3.10798e7i	− 1.05244i
$974$	−1.75414e7	−0.592472
$975$	0	0
$976$	4.66610e7	1.56794
$977$	4.57380e7i	1.53300i	0.642247	+	0.766498i	$0.278003\pi$
$977$	4.57380e7i	1.53300i	−0.642247	+	0.766498i	$0.721997\pi$
$978$	8.94223e6i	0.298950i
$979$	6.02302e6	0.200843
$980$	0	0
$981$	−7.37205e7	−2.44577
$982$	− 3.20457e7i	− 1.06045i
$983$	− 5.78671e7i	− 1.91006i	−0.296505	−	0.955031i	$-0.595821\pi$
$983$	− 5.78671e7i	− 1.91006i	0.296505	−	0.955031i	$-0.404179\pi$
$984$	−7.18305e6	−0.236495
$985$	0	0
$986$	1.35333e7	0.443315
$987$	2.78323e7i	0.909404i
$988$	1.99527e6i	0.0650292i
$989$	−4.52667e7	−1.47159
$990$	0	0
$991$	−4.51653e7	−1.46090	−0.730451	−	0.682965i	$-0.760690\pi$
$991$	−4.51653e7	−1.46090	−0.730451	+	0.682965i	$0.760690\pi$
$992$	− 5.25432e6i	− 0.169526i
$993$	1.12648e7i	0.362536i
$994$	3.92895e7	1.26128
$995$	0	0
$996$	−2.08294e6	−0.0665318
$997$	1.89237e7i	0.602933i	0.953477	+	0.301467i	$0.0974761\pi$
$997$	1.89237e7i	0.602933i	−0.953477	+	0.301467i	$0.902524\pi$
$998$	− 1.87277e6i	− 0.0595193i
$999$	−2.50282e7	−0.793445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.5		22
5.2	odd	4		325.6.a.k.1.9	yes	11
5.3	odd	4		325.6.a.j.1.3	✓	11
5.4	even	2	inner	325.6.b.i.274.18		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.3	✓	11	5.3	odd	4
325.6.a.k.1.9	yes	11	5.2	odd	4
325.6.b.i.274.5		22	1.1	even	1	trivial
325.6.b.i.274.18		22	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.07859i q^{2} +25.6203i q^{3} -4.94922 q^{4} +155.735 q^{6} -137.301i q^{7} -164.431i q^{8} -413.398 q^{9} +O(q^{10})\)	\(q-6.07859i q^{2} +25.6203i q^{3} -4.94922 q^{4} +155.735 q^{6} -137.301i q^{7} -164.431i q^{8} -413.398 q^{9} +169.031 q^{11} -126.800i q^{12} +169.000i q^{13} -834.595 q^{14} -1157.88 q^{16} +487.128i q^{17} +2512.87i q^{18} -2385.49 q^{19} +3517.68 q^{21} -1027.47i q^{22} +4144.47i q^{23} +4212.75 q^{24} +1027.28 q^{26} -4365.63i q^{27} +679.532i q^{28} +4570.45 q^{29} -2957.68 q^{31} +1776.50i q^{32} +4330.61i q^{33} +2961.05 q^{34} +2046.00 q^{36} -5733.02i q^{37} +14500.4i q^{38} -4329.82 q^{39} -1705.07 q^{41} -21382.5i q^{42} +10922.2i q^{43} -836.570 q^{44} +25192.5 q^{46} +7912.12i q^{47} -29665.2i q^{48} -2044.52 q^{49} -12480.4 q^{51} -836.418i q^{52} +26431.1i q^{53} -26536.9 q^{54} -22576.4 q^{56} -61116.8i q^{57} -27781.9i q^{58} -11248.5 q^{59} -40298.6 q^{61} +17978.5i q^{62} +56759.8i q^{63} -26253.6 q^{64} +26324.0 q^{66} +64887.2i q^{67} -2410.91i q^{68} -106182. q^{69} -47076.1 q^{71} +67975.2i q^{72} +42771.8i q^{73} -34848.7 q^{74} +11806.3 q^{76} -23208.1i q^{77} +26319.2i q^{78} -90057.0 q^{79} +11392.9 q^{81} +10364.4i q^{82} -16427.0i q^{83} -17409.8 q^{84} +66391.5 q^{86} +117096. i q^{87} -27793.8i q^{88} +35632.7 q^{89} +23203.8 q^{91} -20511.9i q^{92} -75776.6i q^{93} +48094.5 q^{94} -45514.4 q^{96} +121030. i q^{97} +12427.8i q^{98} -69876.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more