LMFDB - Embedded newform 325.6.b.g.274.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,6,Mod(274,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [12,0,0,-268,0,636] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 $ x^12 + 326x^10 + 38809x^8 + 2034064x^6 + 43897824x^4 + 281822976*x^2 + 377913600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.6
Root			$-1.34530i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.g.274.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.34530i q^{2} -19.6439i q^{3} +30.1902 q^{4} -26.4269 q^{6} -48.6530i q^{7} -83.6645i q^{8} -142.882 q^{9} +283.440 q^{11} -593.052i q^{12} +169.000i q^{13} -65.4530 q^{14} +853.531 q^{16} -789.522i q^{17} +192.219i q^{18} -83.3795 q^{19} -955.734 q^{21} -381.312i q^{22} -1935.13i q^{23} -1643.49 q^{24} +227.356 q^{26} -1966.71i q^{27} -1468.84i q^{28} -222.752 q^{29} -2789.48 q^{31} -3825.52i q^{32} -5567.86i q^{33} -1062.15 q^{34} -4313.62 q^{36} -8369.56i q^{37} +112.170i q^{38} +3319.81 q^{39} +1218.51 q^{41} +1285.75i q^{42} +5638.89i q^{43} +8557.10 q^{44} -2603.34 q^{46} -17776.2i q^{47} -16766.7i q^{48} +14439.9 q^{49} -15509.3 q^{51} +5102.14i q^{52} +10834.4i q^{53} -2645.82 q^{54} -4070.53 q^{56} +1637.90i q^{57} +299.668i q^{58} +5363.36 q^{59} -18670.7 q^{61} +3752.69i q^{62} +6951.63i q^{63} +22166.5 q^{64} -7490.44 q^{66} -13985.1i q^{67} -23835.8i q^{68} -38013.5 q^{69} +50969.5 q^{71} +11954.1i q^{72} +42394.9i q^{73} -11259.6 q^{74} -2517.24 q^{76} -13790.2i q^{77} -4466.15i q^{78} -106279. q^{79} -73354.1 q^{81} -1639.26i q^{82} +75513.6i q^{83} -28853.8 q^{84} +7586.01 q^{86} +4375.71i q^{87} -23713.8i q^{88} -77017.8 q^{89} +8222.36 q^{91} -58422.0i q^{92} +54796.2i q^{93} -23914.4 q^{94} -75148.0 q^{96} -126702. i q^{97} -19426.0i q^{98} -40498.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) $ 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

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 $

$2$	− 1.34530i	− 0.237818i	−0.992905	−	0.118909i	$-0.962060\pi$
$2$	− 1.34530i	− 0.237818i	0.992905	−	0.118909i	$-0.0379396\pi$
$3$	− 19.6439i	− 1.26015i	−0.776532	−	0.630077i	$-0.783023\pi$
$3$	− 19.6439i	− 1.26015i	0.776532	−	0.630077i	$-0.216977\pi$
$4$	30.1902	0.943443
$5$	0	0
$5$	0	0
$6$	−26.4269	−0.299687
$7$	− 48.6530i	− 0.375288i	−0.982237	−	0.187644i	$-0.939915\pi$
$7$	− 48.6530i	− 0.375288i	0.982237	−	0.187644i	$-0.0600851\pi$
$8$	− 83.6645i	− 0.462185i
$9$	−142.882	−0.587990
$10$	0	0
$11$	283.440	0.706284	0.353142	−	0.935570i	$-0.385113\pi$
$11$	283.440	0.706284	0.353142	+	0.935570i	$0.385113\pi$
$12$	− 593.052i	− 1.18888i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−65.4530	−0.0892502
$15$	0	0
$16$	853.531	0.833527
$17$	− 789.522i	− 0.662586i	−0.943528	−	0.331293i	$-0.892515\pi$
$17$	− 789.522i	− 0.662586i	0.943528	−	0.331293i	$-0.107485\pi$
$18$	192.219i	0.139835i
$19$	−83.3795	−0.0529877	−0.0264939	−	0.999649i	$-0.508434\pi$
$19$	−83.3795	−0.0529877	−0.0264939	+	0.999649i	$0.508434\pi$
$20$	0	0
$21$	−955.734	−0.472921
$22$	− 381.312i	− 0.167967i
$23$	− 1935.13i	− 0.762767i	−0.924417	−	0.381383i	$-0.875448\pi$
$23$	− 1935.13i	− 0.762767i	0.924417	−	0.381383i	$-0.124552\pi$
$24$	−1643.49	−0.582425
$25$	0	0
$26$	227.356	0.0659588
$27$	− 1966.71i	− 0.519196i
$28$	− 1468.84i	− 0.354063i
$29$	−222.752	−0.0491843	−0.0245922	−	0.999698i	$-0.507829\pi$
$29$	−222.752	−0.0491843	−0.0245922	+	0.999698i	$0.507829\pi$
$30$	0	0
$31$	−2789.48	−0.521338	−0.260669	−	0.965428i	$-0.583943\pi$
$31$	−2789.48	−0.521338	−0.260669	+	0.965428i	$0.583943\pi$
$32$	− 3825.52i	− 0.660413i
$33$	− 5567.86i	− 0.890027i
$34$	−1062.15	−0.157575
$35$	0	0
$36$	−4313.62	−0.554735
$37$	− 8369.56i	− 1.00507i	−0.864555	−	0.502537i	$-0.832400\pi$
$37$	− 8369.56i	− 1.00507i	0.864555	−	0.502537i	$-0.167600\pi$
$38$	112.170i	0.0126014i
$39$	3319.81	0.349504
$40$	0	0
$41$	1218.51	0.113206	0.0566029	−	0.998397i	$-0.481973\pi$
$41$	1218.51	0.113206	0.0566029	+	0.998397i	$0.481973\pi$
$42$	1285.75i	0.112469i
$43$	5638.89i	0.465075i	0.972587	+	0.232537i	$0.0747028\pi$
$43$	5638.89i	0.465075i	−0.972587	+	0.232537i	$0.925297\pi$
$44$	8557.10	0.666338
$45$	0	0
$46$	−2603.34	−0.181399
$47$	− 17776.2i	− 1.17380i	−0.809658	−	0.586902i	$-0.800348\pi$
$47$	− 17776.2i	− 1.17380i	0.809658	−	0.586902i	$-0.199652\pi$
$48$	− 16766.7i	− 1.05037i
$49$	14439.9	0.859159
$50$	0	0
$51$	−15509.3	−0.834961
$52$	5102.14i	0.261664i
$53$	10834.4i	0.529803i	0.964275	+	0.264902i	$0.0853395\pi$
$53$	10834.4i	0.529803i	−0.964275	+	0.264902i	$0.914661\pi$
$54$	−2645.82	−0.123474
$55$	0	0
$56$	−4070.53	−0.173453
$57$	1637.90i	0.0667727i
$58$	299.668i	0.0116969i
$59$	5363.36	0.200589	0.100295	−	0.994958i	$-0.468021\pi$
$59$	5363.36	0.200589	0.100295	+	0.994958i	$0.468021\pi$
$60$	0	0
$61$	−18670.7	−0.642445	−0.321222	−	0.947004i	$-0.604094\pi$
$61$	−18670.7	−0.642445	−0.321222	+	0.947004i	$0.604094\pi$
$62$	3752.69i	0.123983i
$63$	6951.63i	0.220666i
$64$	22166.5	0.676469
$65$	0	0
$66$	−7490.44	−0.211664
$67$	− 13985.1i	− 0.380607i	−0.981725	−	0.190304i	$-0.939053\pi$
$67$	− 13985.1i	− 0.380607i	0.981725	−	0.190304i	$-0.0609473\pi$
$68$	− 23835.8i	− 0.625112i
$69$	−38013.5	−0.961204
$70$	0	0
$71$	50969.5	1.19995	0.599976	−	0.800018i	$-0.295177\pi$
$71$	50969.5	1.19995	0.599976	+	0.800018i	$0.295177\pi$
$72$	11954.1i	0.271761i
$73$	42394.9i	0.931122i	0.885016	+	0.465561i	$0.154147\pi$
$73$	42394.9i	0.931122i	−0.885016	+	0.465561i	$0.845853\pi$
$74$	−11259.6	−0.239025
$75$	0	0
$76$	−2517.24	−0.0499909
$77$	− 13790.2i	− 0.265060i
$78$	− 4466.15i	− 0.0831183i
$79$	−106279.	−1.91593	−0.957964	−	0.286888i	$-0.907379\pi$
$79$	−106279.	−1.91593	−0.957964	+	0.286888i	$0.907379\pi$
$80$	0	0
$81$	−73354.1	−1.24226
$82$	− 1639.26i	− 0.0269224i
$83$	75513.6i	1.20318i	0.798806	+	0.601589i	$0.205465\pi$
$83$	75513.6i	1.20318i	−0.798806	+	0.601589i	$0.794535\pi$
$84$	−28853.8	−0.446174
$85$	0	0
$86$	7586.01	0.110603
$87$	4375.71i	0.0619799i
$88$	− 23713.8i	− 0.326434i
$89$	−77017.8	−1.03066	−0.515331	−	0.856991i	$-0.672331\pi$
$89$	−77017.8	−1.03066	−0.515331	+	0.856991i	$0.672331\pi$
$90$	0	0
$91$	8222.36	0.104086
$92$	− 58422.0i	− 0.719627i
$93$	54796.2i	0.656966i
$94$	−23914.4	−0.279151
$95$	0	0
$96$	−75148.0	−0.832222
$97$	− 126702.i	− 1.36727i	−0.729822	−	0.683637i	$-0.760397\pi$
$97$	− 126702.i	− 1.36727i	0.729822	−	0.683637i	$-0.239603\pi$
$98$	− 19426.0i	− 0.204323i
$99$	−40498.4	−0.415288
$100$	0	0
$101$	−177513.	−1.73151	−0.865757	−	0.500464i	$-0.833163\pi$
$101$	−177513.	−1.73151	−0.865757	+	0.500464i	$0.833163\pi$
$102$	20864.6i	0.198569i
$103$	149599.i	1.38943i	0.719286	+	0.694714i	$0.244469\pi$
$103$	149599.i	1.38943i	−0.719286	+	0.694714i	$0.755531\pi$
$104$	14139.3	0.128187
$105$	0	0
$106$	14575.5	0.125997
$107$	111466.i	0.941199i	0.882347	+	0.470600i	$0.155962\pi$
$107$	111466.i	0.941199i	−0.882347	+	0.470600i	$0.844038\pi$
$108$	− 59375.3i	− 0.489832i
$109$	85200.4	0.686872	0.343436	−	0.939176i	$-0.388409\pi$
$109$	85200.4	0.686872	0.343436	+	0.939176i	$0.388409\pi$
$110$	0	0
$111$	−164411.	−1.26655
$112$	− 41526.9i	− 0.312813i
$113$	− 204137.i	− 1.50392i	−0.659208	−	0.751961i	$-0.729108\pi$
$113$	− 204137.i	− 1.50392i	0.659208	−	0.751961i	$-0.270892\pi$
$114$	2203.46	0.0158797
$115$	0	0
$116$	−6724.92	−0.0464026
$117$	− 24147.0i	− 0.163079i
$118$	− 7215.34i	− 0.0477037i
$119$	−38412.7	−0.248661
$120$	0	0
$121$	−80712.8	−0.501163
$122$	25117.7i	0.152785i
$123$	− 23936.2i	− 0.142657i
$124$	−84214.9	−0.491852
$125$	0	0
$126$	9352.03	0.0524783
$127$	− 272404.i	− 1.49866i	−0.662196	−	0.749331i	$-0.730375\pi$
$127$	− 272404.i	− 1.49866i	0.662196	−	0.749331i	$-0.269625\pi$
$128$	− 152237.i	− 0.821289i
$129$	110770.	0.586066
$130$	0	0
$131$	42029.3	0.213980	0.106990	−	0.994260i	$-0.465879\pi$
$131$	42029.3	0.213980	0.106990	+	0.994260i	$0.465879\pi$
$132$	− 168095.i	− 0.839689i
$133$	4056.66i	0.0198857i
$134$	−18814.1	−0.0905152
$135$	0	0
$136$	−66055.0	−0.306237
$137$	56817.3i	0.258630i	0.991604	+	0.129315i	$0.0412779\pi$
$137$	56817.3i	0.258630i	−0.991604	+	0.129315i	$0.958722\pi$
$138$	51139.6i	0.228591i
$139$	136297.	0.598342	0.299171	−	0.954200i	$-0.403290\pi$
$139$	136297.	0.598342	0.299171	+	0.954200i	$0.403290\pi$
$140$	0	0
$141$	−349194.	−1.47917
$142$	− 68569.3i	− 0.285370i
$143$	47901.3i	0.195888i
$144$	−121954.	−0.490106
$145$	0	0
$146$	57033.9	0.221437
$147$	− 283655.i	− 1.08267i
$148$	− 252678.i	− 0.948230i
$149$	398843.	1.47176	0.735879	−	0.677113i	$-0.236769\pi$
$149$	398843.	1.47176	0.735879	+	0.677113i	$0.236769\pi$
$150$	0	0
$151$	−16131.0	−0.0575730	−0.0287865	−	0.999586i	$-0.509164\pi$
$151$	−16131.0	−0.0575730	−0.0287865	+	0.999586i	$0.509164\pi$
$152$	6975.90i	0.0244901i
$153$	112808.i	0.389594i
$154$	−18552.0	−0.0630360
$155$	0	0
$156$	100226.	0.329737
$157$	− 9495.80i	− 0.0307456i	−0.999882	−	0.0153728i	$-0.995107\pi$
$157$	− 9495.80i	− 0.0307456i	0.999882	−	0.0153728i	$-0.00489350\pi$
$158$	142977.i	0.455642i
$159$	212829.	0.667634
$160$	0	0
$161$	−94150.2	−0.286257
$162$	98683.3i	0.295431i
$163$	439257.i	1.29494i	0.762091	+	0.647469i	$0.224173\pi$
$163$	439257.i	1.29494i	−0.762091	+	0.647469i	$0.775827\pi$
$164$	36787.0	0.106803
$165$	0	0
$166$	101588.	0.286137
$167$	233417.i	0.647651i	0.946117	+	0.323825i	$0.104969\pi$
$167$	233417.i	0.647651i	−0.946117	+	0.323825i	$0.895031\pi$
$168$	79961.0i	0.218577i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	11913.4	0.0311563
$172$	170239.i	0.438771i
$173$	121696.i	0.309144i	0.987982	+	0.154572i	$0.0493999\pi$
$173$	121696.i	0.309144i	−0.987982	+	0.154572i	$0.950600\pi$
$174$	5886.65	0.0147399
$175$	0	0
$176$	241925.	0.588706
$177$	− 105357.i	− 0.252773i
$178$	103612.i	0.245110i
$179$	280901.	0.655271	0.327636	−	0.944804i	$-0.393748\pi$
$179$	280901.	0.655271	0.327636	+	0.944804i	$0.393748\pi$
$180$	0	0
$181$	324458.	0.736142	0.368071	−	0.929798i	$-0.380018\pi$
$181$	324458.	0.736142	0.368071	+	0.929798i	$0.380018\pi$
$182$	− 11061.6i	− 0.0247536i
$183$	366765.i	0.809580i
$184$	−161902.	−0.352540
$185$	0	0
$186$	73717.4	0.156238
$187$	− 223782.i	− 0.467974i
$188$	− 536668.i	− 1.10742i
$189$	−95686.5	−0.194848
$190$	0	0
$191$	803448.	1.59358	0.796791	−	0.604256i	$-0.206529\pi$
$191$	803448.	1.59358	0.796791	+	0.604256i	$0.206529\pi$
$192$	− 435436.i	− 0.852456i
$193$	− 62706.7i	− 0.121177i	−0.998163	−	0.0605886i	$-0.980702\pi$
$193$	− 62706.7i	− 0.121177i	0.998163	−	0.0605886i	$-0.0192978\pi$
$194$	−170453.	−0.325162
$195$	0	0
$196$	435942.	0.810567
$197$	367605.i	0.674864i	0.941350	+	0.337432i	$0.109558\pi$
$197$	367605.i	0.674864i	−0.941350	+	0.337432i	$0.890442\pi$
$198$	54482.5i	0.0987629i
$199$	−860523.	−1.54039	−0.770193	−	0.637810i	$-0.779840\pi$
$199$	−860523.	−1.54039	−0.770193	+	0.637810i	$0.779840\pi$
$200$	0	0
$201$	−274721.	−0.479624
$202$	238808.i	0.411785i
$203$	10837.6i	0.0184583i
$204$	−468228.	−0.787738
$205$	0	0
$206$	201256.	0.330431
$207$	276495.i	0.448499i
$208$	144247.i	0.231179i
$209$	−23633.1	−0.0374244
$210$	0	0
$211$	−457688.	−0.707723	−0.353861	−	0.935298i	$-0.615132\pi$
$211$	−457688.	−0.707723	−0.353861	+	0.935298i	$0.615132\pi$
$212$	327092.i	0.499839i
$213$	− 1.00124e6i	− 1.51213i
$214$	149955.	0.223834
$215$	0	0
$216$	−164544.	−0.239965
$217$	135717.i	0.195652i
$218$	− 114620.i	− 0.163350i
$219$	832800.	1.17336
$220$	0	0
$221$	133429.	0.183768
$222$	221182.i	0.301208i
$223$	− 925567.i	− 1.24637i	−0.782076	−	0.623183i	$-0.785839\pi$
$223$	− 925567.i	− 1.24637i	0.782076	−	0.623183i	$-0.214161\pi$
$224$	−186123.	−0.247845
$225$	0	0
$226$	−274625.	−0.357659
$227$	1.54248e6i	1.98681i	0.114679	+	0.993403i	$0.463416\pi$
$227$	1.54248e6i	1.98681i	−0.114679	+	0.993403i	$0.536584\pi$
$228$	49448.3i	0.0629962i
$229$	−1.09184e6	−1.37584	−0.687922	−	0.725784i	$-0.741477\pi$
$229$	−1.09184e6	−1.37584	−0.687922	+	0.725784i	$0.741477\pi$
$230$	0	0
$231$	−270893.	−0.334017
$232$	18636.4i	0.0227323i
$233$	141596.i	0.170868i	0.996344	+	0.0854341i	$0.0272277\pi$
$233$	141596.i	0.170868i	−0.996344	+	0.0854341i	$0.972772\pi$
$234$	−32485.0	−0.0387831
$235$	0	0
$236$	161921.	0.189244
$237$	2.08773e6i	2.41437i
$238$	51676.6i	0.0591359i
$239$	791678.	0.896507	0.448253	−	0.893906i	$-0.352046\pi$
$239$	791678.	0.896507	0.448253	+	0.893906i	$0.352046\pi$
$240$	0	0
$241$	1.39653e6	1.54884	0.774421	−	0.632671i	$-0.218041\pi$
$241$	1.39653e6	1.54884	0.774421	+	0.632671i	$0.218041\pi$
$242$	108583.i	0.119186i
$243$	963047.i	1.04624i
$244$	−563671.	−0.606110
$245$	0	0
$246$	−32201.4	−0.0339264
$247$	− 14091.1i	− 0.0146961i
$248$	233380.i	0.240955i
$249$	1.48338e6	1.51619
$250$	0	0
$251$	927565.	0.929309	0.464654	−	0.885492i	$-0.346179\pi$
$251$	927565.	0.929309	0.464654	+	0.885492i	$0.346179\pi$
$252$	209871.i	0.208186i
$253$	− 548494.i	− 0.538730i
$254$	−366465.	−0.356408
$255$	0	0
$256$	504524.	0.481152
$257$	− 1.21697e6i	− 1.14934i	−0.818385	−	0.574670i	$-0.805130\pi$
$257$	− 1.21697e6i	− 1.14934i	0.818385	−	0.574670i	$-0.194870\pi$
$258$	− 149019.i	− 0.139377i
$259$	−407205.	−0.377193
$260$	0	0
$261$	31827.2	0.0289199
$262$	− 56542.0i	− 0.0508883i
$263$	− 188972.i	− 0.168464i	−0.996446	−	0.0842321i	$-0.973156\pi$
$263$	− 188972.i	− 0.168464i	0.996446	−	0.0842321i	$-0.0268437\pi$
$264$	−465832.	−0.411357
$265$	0	0
$266$	5457.43	0.00472916
$267$	1.51293e6i	1.29879i
$268$	− 422211.i	− 0.359081i
$269$	1.68629e6	1.42086	0.710432	−	0.703766i	$-0.248500\pi$
$269$	1.68629e6	1.42086	0.710432	+	0.703766i	$0.248500\pi$
$270$	0	0
$271$	1.71354e6	1.41733	0.708664	−	0.705546i	$-0.249298\pi$
$271$	1.71354e6	1.41733	0.708664	+	0.705546i	$0.249298\pi$
$272$	− 673882.i	− 0.552283i
$273$	− 161519.i	− 0.131165i
$274$	76436.4	0.0615069
$275$	0	0
$276$	−1.14764e6	−0.906841
$277$	− 1.51635e6i	− 1.18741i	−0.804682	−	0.593706i	$-0.797664\pi$
$277$	− 1.51635e6i	− 1.18741i	0.804682	−	0.593706i	$-0.202336\pi$
$278$	− 183360.i	− 0.142296i
$279$	398566.	0.306542
$280$	0	0
$281$	2.01362e6	1.52129	0.760644	−	0.649169i	$-0.224883\pi$
$281$	2.01362e6	1.52129	0.760644	+	0.649169i	$0.224883\pi$
$282$	469771.i	0.351774i
$283$	451062.i	0.334788i	0.985890	+	0.167394i	$0.0535352\pi$
$283$	451062.i	0.334788i	−0.985890	+	0.167394i	$0.946465\pi$
$284$	1.53878e6	1.13209
$285$	0	0
$286$	64441.7	0.0465856
$287$	− 59284.1i	− 0.0424848i
$288$	546597.i	0.388316i
$289$	796511.	0.560980
$290$	0	0
$291$	−2.48893e6	−1.72298
$292$	1.27991e6i	0.878460i
$293$	− 1.60140e6i	− 1.08976i	−0.838514	−	0.544879i	$-0.816575\pi$
$293$	− 1.60140e6i	− 1.08976i	0.838514	−	0.544879i	$-0.183425\pi$
$294$	−381602.	−0.257479
$295$	0	0
$296$	−700235.	−0.464531
$297$	− 557444.i	− 0.366700i
$298$	− 536564.i	− 0.350010i
$299$	327038.	0.211553
$300$	0	0
$301$	274349.	0.174537
$302$	21701.0i	0.0136919i
$303$	3.48704e6i	2.18198i
$304$	−71167.0	−0.0441667
$305$	0	0
$306$	151761.	0.0926524
$307$	− 3.03553e6i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$307$	− 3.03553e6i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$308$	− 416329.i	− 0.250069i
$309$	2.93871e6	1.75089
$310$	0	0
$311$	1.48427e6	0.870184	0.435092	−	0.900386i	$-0.356716\pi$
$311$	1.48427e6	0.870184	0.435092	+	0.900386i	$0.356716\pi$
$312$	− 277751.i	− 0.161536i
$313$	1.57466e6i	0.908504i	0.890873	+	0.454252i	$0.150094\pi$
$313$	1.57466e6i	0.908504i	−0.890873	+	0.454252i	$0.849906\pi$
$314$	−12774.7	−0.00731184
$315$	0	0
$316$	−3.20858e6	−1.80757
$317$	− 2.88960e6i	− 1.61507i	−0.589822	−	0.807533i	$-0.700802\pi$
$317$	− 2.88960e6i	− 1.61507i	0.589822	−	0.807533i	$-0.299198\pi$
$318$	− 286319.i	− 0.158775i
$319$	−63136.8	−0.0347381
$320$	0	0
$321$	2.18962e6	1.18606
$322$	126660.i	0.0680771i
$323$	65830.0i	0.0351089i
$324$	−2.21457e6	−1.17200
$325$	0	0
$326$	590932.	0.307960
$327$	− 1.67367e6i	− 0.865565i
$328$	− 101946.i	− 0.0523221i
$329$	−864868.	−0.440514
$330$	0	0
$331$	2.93760e6	1.47374	0.736872	−	0.676032i	$-0.236302\pi$
$331$	2.93760e6	1.47374	0.736872	+	0.676032i	$0.236302\pi$
$332$	2.27977e6i	1.13513i
$333$	1.19586e6i	0.590974i
$334$	314016.	0.154023
$335$	0	0
$336$	−815749.	−0.394192
$337$	842611.i	0.404159i	0.979369	+	0.202079i	$0.0647699\pi$
$337$	842611.i	0.404159i	−0.979369	+	0.202079i	$0.935230\pi$
$338$	38423.1i	0.0182937i
$339$	−4.01004e6	−1.89517
$340$	0	0
$341$	−790650.	−0.368212
$342$	− 16027.1i	− 0.00740951i
$343$	− 1.52026e6i	− 0.697720i
$344$	471775.	0.214951
$345$	0	0
$346$	163718.	0.0735200
$347$	719797.i	0.320912i	0.987043	+	0.160456i	$0.0512965\pi$
$347$	719797.i	0.320912i	−0.987043	+	0.160456i	$0.948703\pi$
$348$	132103.i	0.0584744i
$349$	2.28139e6	1.00262	0.501309	−	0.865269i	$-0.332852\pi$
$349$	2.28139e6	1.00262	0.501309	+	0.865269i	$0.332852\pi$
$350$	0	0
$351$	332374.	0.143999
$352$	− 1.08430e6i	− 0.466439i
$353$	7769.01i	0.00331840i	0.999999	+	0.00165920i	$0.000528140\pi$
$353$	7769.01i	0.00331840i	−0.999999	+	0.00165920i	$0.999472\pi$
$354$	−141737.	−0.0601140
$355$	0	0
$356$	−2.32518e6	−0.972371
$357$	754573.i	0.313351i
$358$	− 377897.i	− 0.155835i
$359$	−309145.	−0.126598	−0.0632989	−	0.997995i	$-0.520162\pi$
$359$	−309145.	−0.126598	−0.0632989	+	0.997995i	$0.520162\pi$
$360$	0	0
$361$	−2.46915e6	−0.997192
$362$	− 436493.i	− 0.175068i
$363$	1.58551e6i	0.631543i
$364$	248234.	0.0981994
$365$	0	0
$366$	493409.	0.192533
$367$	2.32781e6i	0.902156i	0.892484	+	0.451078i	$0.148960\pi$
$367$	2.32781e6i	0.902156i	−0.892484	+	0.451078i	$0.851040\pi$
$368$	− 1.65170e6i	− 0.635786i
$369$	−174103.	−0.0665640
$370$	0	0
$371$	527126.	0.198829
$372$	1.65431e6i	0.619810i
$373$	− 1.60840e6i	− 0.598579i	−0.954162	−	0.299289i	$-0.903250\pi$
$373$	− 1.60840e6i	− 0.598579i	0.954162	−	0.299289i	$-0.0967496\pi$
$374$	−301054.	−0.111292
$375$	0	0
$376$	−1.48724e6	−0.542515
$377$	− 37645.1i	− 0.0136413i
$378$	128727.i	0.0463383i
$379$	473484.	0.169320	0.0846599	−	0.996410i	$-0.473020\pi$
$379$	473484.	0.169320	0.0846599	+	0.996410i	$0.473020\pi$
$380$	0	0
$381$	−5.35106e6	−1.88855
$382$	− 1.08088e6i	− 0.378982i
$383$	363362.i	0.126574i	0.997995	+	0.0632868i	$0.0201583\pi$
$383$	363362.i	0.126574i	−0.997995	+	0.0632868i	$0.979842\pi$
$384$	−2.99053e6	−1.03495
$385$	0	0
$386$	−84359.4	−0.0288181
$387$	− 805695.i	− 0.273460i
$388$	− 3.82517e6i	− 1.28994i
$389$	901091.	0.301922	0.150961	−	0.988540i	$-0.451763\pi$
$389$	901091.	0.301922	0.150961	+	0.988540i	$0.451763\pi$
$390$	0	0
$391$	−1.52783e6	−0.505398
$392$	− 1.20811e6i	− 0.397091i
$393$	− 825618.i	− 0.269648i
$394$	494539.	0.160495
$395$	0	0
$396$	−1.22265e6	−0.391801
$397$	3.36685e6i	1.07213i	0.844177	+	0.536064i	$0.180090\pi$
$397$	3.36685e6i	1.07213i	−0.844177	+	0.536064i	$0.819910\pi$
$398$	1.15766e6i	0.366331i
$399$	79688.6	0.0250590
$400$	0	0
$401$	1.49612e6	0.464629	0.232315	−	0.972641i	$-0.425370\pi$
$401$	1.49612e6	0.464629	0.232315	+	0.972641i	$0.425370\pi$
$402$	369582.i	0.114063i
$403$	− 471422.i	− 0.144593i
$404$	−5.35914e6	−1.63358
$405$	0	0
$406$	14579.8	0.00438971
$407$	− 2.37227e6i	− 0.709868i
$408$	1.29758e6i	0.385907i
$409$	−1.78884e6	−0.528765	−0.264382	−	0.964418i	$-0.585168\pi$
$409$	−1.78884e6	−0.528765	−0.264382	+	0.964418i	$0.585168\pi$
$410$	0	0
$411$	1.11611e6	0.325914
$412$	4.51642e6i	1.31085i
$413$	− 260944.i	− 0.0752787i
$414$	371969.	0.106661
$415$	0	0
$416$	646513.	0.183166
$417$	− 2.67740e6i	− 0.754003i
$418$	31793.6i	0.00890018i
$419$	−3.06163e6	−0.851956	−0.425978	−	0.904733i	$-0.640070\pi$
$419$	−3.06163e6	−0.851956	−0.425978	+	0.904733i	$0.640070\pi$
$420$	0	0
$421$	−6.32711e6	−1.73980	−0.869901	−	0.493226i	$-0.835817\pi$
$421$	−6.32711e6	−1.73980	−0.869901	+	0.493226i	$0.835817\pi$
$422$	615728.i	0.168309i
$423$	2.53990e6i	0.690185i
$424$	906453.	0.244867
$425$	0	0
$426$	−1.34697e6	−0.359611
$427$	908386.i	0.241102i
$428$	3.36517e6i	0.887967i
$429$	940968.	0.246849
$430$	0	0
$431$	−2.68796e6	−0.696995	−0.348498	−	0.937310i	$-0.613308\pi$
$431$	−2.68796e6	−0.696995	−0.348498	+	0.937310i	$0.613308\pi$
$432$	− 1.67865e6i	− 0.432764i
$433$	1.62239e6i	0.415850i	0.978145	+	0.207925i	$0.0666709\pi$
$433$	1.62239e6i	0.415850i	−0.978145	+	0.207925i	$0.933329\pi$
$434$	182580.	0.0465295
$435$	0	0
$436$	2.57222e6	0.648024
$437$	161351.i	0.0404172i
$438$	− 1.12037e6i	− 0.279045i
$439$	−6.21478e6	−1.53909	−0.769546	−	0.638592i	$-0.779517\pi$
$439$	−6.21478e6	−1.53909	−0.769546	+	0.638592i	$0.779517\pi$
$440$	0	0
$441$	−2.06319e6	−0.505177
$442$	− 179503.i	− 0.0437034i
$443$	− 517107.i	− 0.125191i	−0.998039	−	0.0625953i	$-0.980062\pi$
$443$	− 517107.i	− 0.125191i	0.998039	−	0.0625953i	$-0.0199377\pi$
$444$	−4.96358e6	−1.19492
$445$	0	0
$446$	−1.24517e6	−0.296408
$447$	− 7.83482e6i	− 1.85464i
$448$	− 1.07847e6i	− 0.253871i
$449$	7.08559e6	1.65867	0.829336	−	0.558750i	$-0.188719\pi$
$449$	7.08559e6	1.65867	0.829336	+	0.558750i	$0.188719\pi$
$450$	0	0
$451$	345374.	0.0799555
$452$	− 6.16292e6i	− 1.41886i
$453$	316875.i	0.0725509i
$454$	2.07510e6	0.472498
$455$	0	0
$456$	137034.	0.0308614
$457$	929335.i	0.208153i	0.994569	+	0.104076i	$0.0331886\pi$
$457$	929335.i	0.208153i	−0.994569	+	0.104076i	$0.966811\pi$
$458$	1.46885e6i	0.327200i
$459$	−1.55276e6	−0.344012
$460$	0	0
$461$	7.91310e6	1.73418	0.867091	−	0.498150i	$-0.165987\pi$
$461$	7.91310e6	1.73418	0.867091	+	0.498150i	$0.165987\pi$
$462$	364433.i	0.0794351i
$463$	− 1.60654e6i	− 0.348289i	−0.984720	−	0.174144i	$-0.944284\pi$
$463$	− 1.60654e6i	− 0.348289i	0.984720	−	0.174144i	$-0.0557159\pi$
$464$	−190126.	−0.0409964
$465$	0	0
$466$	190489.	0.0406355
$467$	− 2.57909e6i	− 0.547236i	−0.961838	−	0.273618i	$-0.911780\pi$
$467$	− 2.57909e6i	− 0.547236i	0.961838	−	0.273618i	$-0.0882204\pi$
$468$	− 729002.i	− 0.153856i
$469$	−680415.	−0.142837
$470$	0	0
$471$	−186534.	−0.0387442
$472$	− 448723.i	− 0.0927093i
$473$	1.59829e6i	0.328475i
$474$	2.80862e6	0.574179
$475$	0	0
$476$	−1.15968e6	−0.234597
$477$	− 1.54803e6i	− 0.311519i
$478$	− 1.06504e6i	− 0.213205i
$479$	−1.57546e6	−0.313739	−0.156869	−	0.987619i	$-0.550140\pi$
$479$	−1.57546e6	−0.313739	−0.156869	+	0.987619i	$0.550140\pi$
$480$	0	0
$481$	1.41446e6	0.278758
$482$	− 1.87875e6i	− 0.368342i
$483$	1.84947e6i	0.360728i
$484$	−2.43673e6	−0.472819
$485$	0	0
$486$	1.29559e6	0.248815
$487$	3.52518e6i	0.673534i	0.941588	+	0.336767i	$0.109333\pi$
$487$	3.52518e6i	0.673534i	−0.941588	+	0.336767i	$0.890667\pi$
$488$	1.56207e6i	0.296929i
$489$	8.62870e6	1.63182
$490$	0	0
$491$	−3.53510e6	−0.661756	−0.330878	−	0.943674i	$-0.607345\pi$
$491$	−3.53510e6	−0.661756	−0.330878	+	0.943674i	$0.607345\pi$
$492$	− 722639.i	− 0.134589i
$493$	175868.i	0.0325888i
$494$	−18956.8	−0.00349501
$495$	0	0
$496$	−2.38091e6	−0.434549
$497$	− 2.47982e6i	− 0.450328i
$498$	− 1.99559e6i	− 0.360577i
$499$	−311968.	−0.0560866	−0.0280433	−	0.999607i	$-0.508928\pi$
$499$	−311968.	−0.0560866	−0.0280433	+	0.999607i	$0.508928\pi$
$500$	0	0
$501$	4.58521e6	0.816140
$502$	− 1.24785e6i	− 0.221006i
$503$	6.52751e6i	1.15034i	0.818032	+	0.575172i	$0.195065\pi$
$503$	6.52751e6i	1.15034i	−0.818032	+	0.575172i	$0.804935\pi$
$504$	581604.	0.101989
$505$	0	0
$506$	−737890.	−0.128120
$507$	561049.i	0.0969350i
$508$	− 8.22391e6i	− 1.41390i
$509$	6.28149e6	1.07465	0.537327	−	0.843374i	$-0.319434\pi$
$509$	6.28149e6	1.07465	0.537327	+	0.843374i	$0.319434\pi$
$510$	0	0
$511$	2.06264e6	0.349439
$512$	− 5.55033e6i	− 0.935716i
$513$	163983.i	0.0275110i
$514$	−1.63720e6	−0.273334
$515$	0	0
$516$	3.34416e6	0.552920
$517$	− 5.03850e6i	− 0.829038i
$518$	547813.i	0.0897031i
$519$	2.39058e6	0.389569
$520$	0	0
$521$	−7.39635e6	−1.19378	−0.596888	−	0.802324i	$-0.703596\pi$
$521$	−7.39635e6	−1.19378	−0.596888	+	0.802324i	$0.703596\pi$
$522$	− 42817.1i	− 0.00687767i
$523$	6.26869e6i	1.00213i	0.865411	+	0.501063i	$0.167057\pi$
$523$	6.26869e6i	1.00213i	−0.865411	+	0.501063i	$0.832943\pi$
$524$	1.26887e6	0.201878
$525$	0	0
$526$	−254224.	−0.0400638
$527$	2.20236e6i	0.345431i
$528$	− 4.75234e6i	− 0.741861i
$529$	2.69160e6	0.418187
$530$	0	0
$531$	−766327.	−0.117944
$532$	122471.i	0.0187610i
$533$	205928.i	0.0313977i
$534$	2.03534e6	0.308876
$535$	0	0
$536$	−1.17005e6	−0.175911
$537$	− 5.51799e6i	− 0.825743i
$538$	− 2.26857e6i	− 0.337907i
$539$	4.09284e6	0.606810
$540$	0	0
$541$	−2.81963e6	−0.414189	−0.207095	−	0.978321i	$-0.566401\pi$
$541$	−2.81963e6	−0.414189	−0.207095	+	0.978321i	$0.566401\pi$
$542$	− 2.30522e6i	− 0.337066i
$543$	− 6.37360e6i	− 0.927653i
$544$	−3.02033e6	−0.437580
$545$	0	0
$546$	−217292.	−0.0311933
$547$	9.17262e6i	1.31077i	0.755297	+	0.655383i	$0.227493\pi$
$547$	9.17262e6i	1.31077i	−0.755297	+	0.655383i	$0.772507\pi$
$548$	1.71532e6i	0.244003i
$549$	2.66770e6	0.377751
$550$	0	0
$551$	18572.9	0.00260616
$552$	3.18038e6i	0.444254i
$553$	5.17079e6i	0.719025i
$554$	−2.03995e6	−0.282388
$555$	0	0
$556$	4.11483e6	0.564501
$557$	4.27403e6i	0.583713i	0.956462	+	0.291856i	$0.0942730\pi$
$557$	4.27403e6i	0.583713i	−0.956462	+	0.291856i	$0.905727\pi$
$558$	− 536191.i	− 0.0729010i
$559$	−952973.	−0.128989
$560$	0	0
$561$	−4.39595e6	−0.589719
$562$	− 2.70892e6i	− 0.361789i
$563$	9.29935e6i	1.23646i	0.785995	+	0.618232i	$0.212151\pi$
$563$	9.29935e6i	1.23646i	−0.785995	+	0.618232i	$0.787849\pi$
$564$	−1.05422e7	−1.39552
$565$	0	0
$566$	606813.	0.0796185
$567$	3.56890e6i	0.466205i
$568$	− 4.26433e6i	− 0.554601i
$569$	4.10069e6	0.530977	0.265489	−	0.964114i	$-0.414467\pi$
$569$	4.10069e6	0.530977	0.265489	+	0.964114i	$0.414467\pi$
$570$	0	0
$571$	−1.03048e7	−1.32267	−0.661334	−	0.750092i	$-0.730009\pi$
$571$	−1.03048e7	−1.32267	−0.661334	+	0.750092i	$0.730009\pi$
$572$	1.44615e6i	0.184809i
$573$	− 1.57828e7i	− 2.00816i
$574$	−79755.0	−0.0101036
$575$	0	0
$576$	−3.16719e6	−0.397757
$577$	− 2.17391e6i	− 0.271833i	−0.990720	−	0.135916i	$-0.956602\pi$
$577$	− 2.17391e6i	− 0.271833i	0.990720	−	0.135916i	$-0.0433978\pi$
$578$	− 1.07155e6i	− 0.133411i
$579$	−1.23180e6	−0.152702
$580$	0	0
$581$	3.67396e6	0.451538
$582$	3.34835e6i	0.409755i
$583$	3.07090e6i	0.374191i
$584$	3.54695e6	0.430351
$585$	0	0
$586$	−2.15436e6	−0.259164
$587$	1.56597e7i	1.87581i	0.346892	+	0.937905i	$0.387237\pi$
$587$	1.56597e7i	1.87581i	−0.346892	+	0.937905i	$0.612763\pi$
$588$	− 8.56360e6i	− 1.02144i
$589$	232585.	0.0276245
$590$	0	0
$591$	7.22119e6	0.850433
$592$	− 7.14368e6i	− 0.837757i
$593$	− 5.51737e6i	− 0.644310i	−0.946687	−	0.322155i	$-0.895593\pi$
$593$	− 5.51737e6i	− 0.644310i	0.946687	−	0.322155i	$-0.104407\pi$
$594$	−749930.	−0.0872077
$595$	0	0
$596$	1.20411e7	1.38852
$597$	1.69040e7i	1.94113i
$598$	− 439964.i	− 0.0503112i
$599$	8.76025e6	0.997584	0.498792	−	0.866722i	$-0.333777\pi$
$599$	8.76025e6	0.997584	0.498792	+	0.866722i	$0.333777\pi$
$600$	0	0
$601$	1.21014e7	1.36662	0.683310	−	0.730128i	$-0.260540\pi$
$601$	1.21014e7	1.36662	0.683310	+	0.730128i	$0.260540\pi$
$602$	− 369082.i	− 0.0415080i
$603$	1.99821e6i	0.223794i
$604$	−486997.	−0.0543168
$605$	0	0
$606$	4.69111e6	0.518913
$607$	1.29407e7i	1.42556i	0.701388	+	0.712779i	$0.252564\pi$
$607$	1.29407e7i	1.42556i	−0.701388	+	0.712779i	$0.747436\pi$
$608$	318970.i	0.0349938i
$609$	212892.	0.0232603
$610$	0	0
$611$	3.00418e6	0.325554
$612$	3.40570e6i	0.367560i
$613$	2.31350e6i	0.248667i	0.992240	+	0.124333i	$0.0396792\pi$
$613$	2.31350e6i	0.248667i	−0.992240	+	0.124333i	$0.960321\pi$
$614$	−4.08370e6	−0.437152
$615$	0	0
$616$	−1.15375e6	−0.122507
$617$	− 8.53024e6i	− 0.902086i	−0.892502	−	0.451043i	$-0.851052\pi$
$617$	− 8.53024e6i	− 0.902086i	0.892502	−	0.451043i	$-0.148948\pi$
$618$	− 3.95344e6i	− 0.416394i
$619$	−8.89321e6	−0.932893	−0.466447	−	0.884549i	$-0.654466\pi$
$619$	−8.89321e6	−0.932893	−0.466447	+	0.884549i	$0.654466\pi$
$620$	0	0
$621$	−3.80585e6	−0.396025
$622$	− 1.99679e6i	− 0.206945i
$623$	3.74715e6i	0.386795i
$624$	2.83357e6	0.291321
$625$	0	0
$626$	2.11840e6	0.216059
$627$	464245.i	0.0471605i
$628$	− 286680.i	− 0.0290067i
$629$	−6.60796e6	−0.665948
$630$	0	0
$631$	1.01842e7	1.01824	0.509122	−	0.860695i	$-0.329970\pi$
$631$	1.01842e7	1.01824	0.509122	+	0.860695i	$0.329970\pi$
$632$	8.89177e6i	0.885514i
$633$	8.99076e6i	0.891840i
$634$	−3.88739e6	−0.384091
$635$	0	0
$636$	6.42535e6	0.629874
$637$	2.44034e6i	0.238288i
$638$	84938.0i	0.00826133i
$639$	−7.28260e6	−0.705561
$640$	0	0
$641$	−1.30240e7	−1.25198	−0.625991	−	0.779830i	$-0.715306\pi$
$641$	−1.30240e7	−1.25198	−0.625991	+	0.779830i	$0.715306\pi$
$642$	− 2.94569e6i	− 0.282065i
$643$	− 4.52103e6i	− 0.431231i	−0.976478	−	0.215616i	$-0.930824\pi$
$643$	− 4.52103e6i	− 0.431231i	0.976478	−	0.215616i	$-0.0691758\pi$
$644$	−2.84241e6	−0.270067
$645$	0	0
$646$	88561.1	0.00834952
$647$	− 2.72009e6i	− 0.255459i	−0.991809	−	0.127730i	$-0.959231\pi$
$647$	− 2.72009e6i	− 0.255459i	0.991809	−	0.127730i	$-0.0407690\pi$
$648$	6.13713e6i	0.574153i
$649$	1.52019e6	0.141673
$650$	0	0
$651$	2.66600e6	0.246552
$652$	1.32612e7i	1.22170i
$653$	1.83786e7i	1.68667i	0.537389	+	0.843334i	$0.319411\pi$
$653$	1.83786e7i	1.68667i	−0.537389	+	0.843334i	$0.680589\pi$
$654$	−2.25158e6	−0.205847
$655$	0	0
$656$	1.04004e6	0.0943601
$657$	− 6.05745e6i	− 0.547491i
$658$	1.16351e6i	0.104762i
$659$	1.39753e7	1.25357	0.626786	−	0.779192i	$-0.284370\pi$
$659$	1.39753e7	1.25357	0.626786	+	0.779192i	$0.284370\pi$
$660$	0	0
$661$	9.05787e6	0.806348	0.403174	−	0.915123i	$-0.367907\pi$
$661$	9.05787e6	0.806348	0.403174	+	0.915123i	$0.367907\pi$
$662$	− 3.95195e6i	− 0.350483i
$663$	− 2.62107e6i	− 0.231576i
$664$	6.31780e6	0.556091
$665$	0	0
$666$	1.60879e6	0.140544
$667$	431055.i	0.0375161i
$668$	7.04689e6i	0.611021i
$669$	−1.81817e7	−1.57061
$670$	0	0
$671$	−5.29202e6	−0.453748
$672$	3.65618e6i	0.312323i
$673$	1.68758e7i	1.43624i	0.695919	+	0.718121i	$0.254997\pi$
$673$	1.68758e7i	1.43624i	−0.695919	+	0.718121i	$0.745003\pi$
$674$	1.13356e6	0.0961162
$675$	0	0
$676$	−862261.	−0.0725725
$677$	− 7.96769e6i	− 0.668129i	−0.942550	−	0.334065i	$-0.891580\pi$
$677$	− 7.96769e6i	− 0.668129i	0.942550	−	0.334065i	$-0.108420\pi$
$678$	5.39470e6i	0.450706i
$679$	−6.16446e6	−0.513122
$680$	0	0
$681$	3.03003e7	2.50368
$682$	1.06366e6i	0.0875674i
$683$	− 1.29108e7i	− 1.05901i	−0.848306	−	0.529507i	$-0.822377\pi$
$683$	− 1.29108e7i	− 1.05901i	0.848306	−	0.529507i	$-0.177623\pi$
$684$	359667.	0.0293941
$685$	0	0
$686$	−2.04520e6	−0.165930
$687$	2.14479e7i	1.73378i
$688$	4.81297e6i	0.387652i
$689$	−1.83101e6	−0.146941
$690$	0	0
$691$	−1.37987e7	−1.09936	−0.549682	−	0.835374i	$-0.685251\pi$
$691$	−1.37987e7	−1.09936	−0.549682	+	0.835374i	$0.685251\pi$
$692$	3.67402e6i	0.291660i
$693$	1.97037e6i	0.155853i
$694$	968344.	0.0763187
$695$	0	0
$696$	366092.	0.0286462
$697$	− 962040.i	− 0.0750086i
$698$	− 3.06915e6i	− 0.238440i
$699$	2.78149e6	0.215320
$700$	0	0
$701$	−1.54955e6	−0.119099	−0.0595497	−	0.998225i	$-0.518966\pi$
$701$	−1.54955e6	−0.119099	−0.0595497	+	0.998225i	$0.518966\pi$
$702$	− 447143.i	− 0.0342455i
$703$	697849.i	0.0532566i
$704$	6.28288e6	0.477779
$705$	0	0
$706$	10451.7	0.000789175 0
$707$	8.63653e6i	0.649817i
$708$	− 3.18075e6i	− 0.238477i
$709$	1.43383e7	1.07123	0.535614	−	0.844463i	$-0.320080\pi$
$709$	1.43383e7	1.07123	0.535614	+	0.844463i	$0.320080\pi$
$710$	0	0
$711$	1.51853e7	1.12655
$712$	6.44366e6i	0.476357i
$713$	5.39802e6i	0.397659i
$714$	1.01513e6	0.0745204
$715$	0	0
$716$	8.48046e6	0.618211
$717$	− 1.55516e7i	− 1.12974i
$718$	415893.i	0.0301072i
$719$	1.37368e7	0.990980	0.495490	−	0.868614i	$-0.334989\pi$
$719$	1.37368e7	0.990980	0.495490	+	0.868614i	$0.334989\pi$
$720$	0	0
$721$	7.27845e6	0.521436
$722$	3.32175e6i	0.237150i
$723$	− 2.74332e7i	− 1.95178i
$724$	9.79543e6	0.694508
$725$	0	0
$726$	2.13299e6	0.150192
$727$	1.61824e6i	0.113555i	0.998387	+	0.0567776i	$0.0180826\pi$
$727$	1.61824e6i	0.113555i	−0.998387	+	0.0567776i	$0.981917\pi$
$728$	− 687920.i	− 0.0481071i
$729$	1.09293e6	0.0761684
$730$	0	0
$731$	4.45203e6	0.308152
$732$	1.10727e7i	0.763792i
$733$	− 1.57329e7i	− 1.08156i	−0.841165	−	0.540779i	$-0.818130\pi$
$733$	− 1.57329e7i	− 1.08156i	0.841165	−	0.540779i	$-0.181870\pi$
$734$	3.13160e6	0.214549
$735$	0	0
$736$	−7.40290e6	−0.503741
$737$	− 3.96392e6i	− 0.268817i
$738$	234220.i	0.0158301i
$739$	1.62586e6	0.109515	0.0547574	−	0.998500i	$-0.482561\pi$
$739$	1.62586e6	0.109515	0.0547574	+	0.998500i	$0.482561\pi$
$740$	0	0
$741$	−276804.	−0.0185194
$742$	− 709142.i	− 0.0472850i
$743$	1.87162e7i	1.24379i	0.783101	+	0.621894i	$0.213637\pi$
$743$	1.87162e7i	1.24379i	−0.783101	+	0.621894i	$0.786363\pi$
$744$	4.58449e6	0.303640
$745$	0	0
$746$	−2.16378e6	−0.142353
$747$	− 1.07895e7i	− 0.707457i
$748$	− 6.75602e6i	− 0.441506i
$749$	5.42314e6	0.353221
$750$	0	0
$751$	7.16867e6	0.463809	0.231904	−	0.972739i	$-0.425504\pi$
$751$	7.16867e6	0.463809	0.231904	+	0.972739i	$0.425504\pi$
$752$	− 1.51726e7i	− 0.978396i
$753$	− 1.82210e7i	− 1.17107i
$754$	−50644.0	−0.00324414
$755$	0	0
$756$	−2.88879e6	−0.183828
$757$	− 2.67712e7i	− 1.69796i	−0.528422	−	0.848982i	$-0.677216\pi$
$757$	− 2.67712e7i	− 1.69796i	0.528422	−	0.848982i	$-0.322784\pi$
$758$	− 636979.i	− 0.0402673i
$759$	−1.07746e7	−0.678883
$760$	0	0
$761$	−708017.	−0.0443182	−0.0221591	−	0.999754i	$-0.507054\pi$
$761$	−708017.	−0.0443182	−0.0221591	+	0.999754i	$0.507054\pi$
$762$	7.19879e6i	0.449130i
$763$	− 4.14526e6i	− 0.257775i
$764$	2.42562e7	1.50345
$765$	0	0
$766$	488832.	0.0301014
$767$	906409.i	0.0556334i
$768$	− 9.91081e6i	− 0.606326i
$769$	−1.72011e7	−1.04891	−0.524456	−	0.851437i	$-0.675731\pi$
$769$	−1.72011e7	−1.04891	−0.524456	+	0.851437i	$0.675731\pi$
$770$	0	0
$771$	−2.39061e7	−1.44835
$772$	− 1.89313e6i	− 0.114324i
$773$	7.85325e6i	0.472716i	0.971666	+	0.236358i	$0.0759539\pi$
$773$	7.85325e6i	0.472716i	−0.971666	+	0.236358i	$0.924046\pi$
$774$	−1.08390e6	−0.0650336
$775$	0	0
$776$	−1.06005e7	−0.631934
$777$	7.99907e6i	0.475321i
$778$	− 1.21224e6i	− 0.0718024i
$779$	−101599.	−0.00599852
$780$	0	0
$781$	1.44468e7	0.847507
$782$	2.05539e6i	0.120193i
$783$	438089.i	0.0255363i
$784$	1.23249e7	0.716132
$785$	0	0
$786$	−1.11070e6	−0.0641272
$787$	− 2.05200e7i	− 1.18097i	−0.807047	−	0.590487i	$-0.798936\pi$
$787$	− 2.05200e7i	− 1.18097i	0.807047	−	0.590487i	$-0.201064\pi$
$788$	1.10981e7i	0.636695i
$789$	−3.71214e6	−0.212291
$790$	0	0
$791$	−9.93187e6	−0.564404
$792$	3.38827e6i	0.191940i
$793$	− 3.15535e6i	− 0.178182i
$794$	4.52942e6	0.254971
$795$	0	0
$796$	−2.59793e7	−1.45327
$797$	2.57349e7i	1.43508i	0.696517	+	0.717540i	$0.254732\pi$
$797$	2.57349e7i	1.43508i	−0.696517	+	0.717540i	$0.745268\pi$
$798$	− 107205.i	− 0.00595948i
$799$	−1.40347e7	−0.777745
$800$	0	0
$801$	1.10044e7	0.606019
$802$	− 2.01274e6i	− 0.110497i
$803$	1.20164e7i	0.657636i
$804$	−8.29386e6	−0.452498
$805$	0	0
$806$	−634205.	−0.0343868
$807$	− 3.31253e7i	− 1.79051i
$808$	1.48515e7i	0.800280i
$809$	1.45330e7	0.780699	0.390350	−	0.920667i	$-0.372354\pi$
$809$	1.45330e7	0.780699	0.390350	+	0.920667i	$0.372354\pi$
$810$	0	0
$811$	1.60281e6	0.0855717	0.0427858	−	0.999084i	$-0.486377\pi$
$811$	1.60281e6	0.0855717	0.0427858	+	0.999084i	$0.486377\pi$
$812$	327188.i	0.0174143i
$813$	− 3.36605e7i	− 1.78605i
$814$	−3.19141e6	−0.168819
$815$	0	0
$816$	−1.32377e7	−0.695962
$817$	− 470168.i	− 0.0246432i
$818$	2.40652e6i	0.125750i
$819$	−1.17482e6	−0.0612017
$820$	0	0
$821$	1.22945e6	0.0636579	0.0318289	−	0.999493i	$-0.489867\pi$
$821$	1.22945e6	0.0636579	0.0318289	+	0.999493i	$0.489867\pi$
$822$	− 1.50151e6i	− 0.0775082i
$823$	− 260323.i	− 0.0133972i	−0.999978	−	0.00669859i	$-0.997868\pi$
$823$	− 260323.i	− 0.0133972i	0.999978	−	0.00669859i	$-0.00213224\pi$
$824$	1.25161e7	0.642173
$825$	0	0
$826$	−351048.	−0.0179026
$827$	− 1.33246e7i	− 0.677471i	−0.940882	−	0.338736i	$-0.890001\pi$
$827$	− 1.33246e7i	− 0.677471i	0.940882	−	0.338736i	$-0.109999\pi$
$828$	8.34744e6i	0.423134i
$829$	−3.52026e7	−1.77905	−0.889526	−	0.456885i	$-0.848965\pi$
$829$	−3.52026e7	−1.77905	−0.889526	+	0.456885i	$0.848965\pi$
$830$	0	0
$831$	−2.97871e7	−1.49632
$832$	3.74614e6i	0.187619i
$833$	− 1.14006e7i	− 0.569267i
$834$	−3.60191e6	−0.179315
$835$	0	0
$836$	−713486.	−0.0353077
$837$	5.48610e6i	0.270676i
$838$	4.11881e6i	0.202610i
$839$	3.45100e7	1.69255	0.846273	−	0.532750i	$-0.178841\pi$
$839$	3.45100e7	1.69255	0.846273	+	0.532750i	$0.178841\pi$
$840$	0	0
$841$	−2.04615e7	−0.997581
$842$	8.51186e6i	0.413756i
$843$	− 3.95553e7i	− 1.91706i
$844$	−1.38177e7	−0.667696
$845$	0	0
$846$	3.41693e6	0.164138
$847$	3.92692e6i	0.188081i
$848$	9.24749e6i	0.441605i
$849$	8.86059e6	0.421885
$850$	0	0
$851$	−1.61962e7	−0.766637
$852$	− 3.02275e7i	− 1.42660i
$853$	− 1.67507e7i	− 0.788245i	−0.919058	−	0.394122i	$-0.871049\pi$
$853$	− 1.67507e7i	− 0.788245i	0.919058	−	0.394122i	$-0.128951\pi$
$854$	1.22205e6	0.0573383
$855$	0	0
$856$	9.32571e6	0.435008
$857$	2.15120e6i	0.100053i	0.998748	+	0.0500264i	$0.0159306\pi$
$857$	2.15120e6i	0.100053i	−0.998748	+	0.0500264i	$0.984069\pi$
$858$	− 1.26588e6i	− 0.0587051i
$859$	−1.66088e7	−0.767991	−0.383996	−	0.923335i	$-0.625452\pi$
$859$	−1.66088e7	−0.767991	−0.383996	+	0.923335i	$0.625452\pi$
$860$	0	0
$861$	−1.16457e6	−0.0535374
$862$	3.61612e6i	0.165758i
$863$	− 2.71816e7i	− 1.24236i	−0.783667	−	0.621181i	$-0.786653\pi$
$863$	− 2.71816e7i	− 1.24236i	0.783667	−	0.621181i	$-0.213347\pi$
$864$	−7.52369e6	−0.342884
$865$	0	0
$866$	2.18261e6	0.0988965
$867$	− 1.56466e7i	− 0.706922i
$868$	4.09731e6i	0.184586i
$869$	−3.01237e7	−1.35319
$870$	0	0
$871$	2.36347e6	0.105562
$872$	− 7.12825e6i	− 0.317462i
$873$	1.81035e7i	0.803944i
$874$	217065.	0.00961194
$875$	0	0
$876$	2.51424e7	1.10700
$877$	− 1.52455e6i	− 0.0669333i	−0.999440	−	0.0334667i	$-0.989345\pi$
$877$	− 1.52455e6i	− 0.0669333i	0.999440	−	0.0334667i	$-0.0106548\pi$
$878$	8.36075e6i	0.366023i
$879$	−3.14577e7	−1.37326
$880$	0	0
$881$	2.77728e7	1.20554	0.602769	−	0.797916i	$-0.294064\pi$
$881$	2.77728e7	1.20554	0.602769	+	0.797916i	$0.294064\pi$
$882$	2.77562e6i	0.120140i
$883$	3.15964e7i	1.36376i	0.731466	+	0.681878i	$0.238836\pi$
$883$	3.15964e7i	1.36376i	−0.731466	+	0.681878i	$0.761164\pi$
$884$	4.02825e6	0.173375
$885$	0	0
$886$	−695665.	−0.0297725
$887$	− 9.17500e6i	− 0.391559i	−0.980648	−	0.195780i	$-0.937276\pi$
$887$	− 9.17500e6i	− 0.391559i	0.980648	−	0.195780i	$-0.0627237\pi$
$888$	1.37553e7i	0.585381i
$889$	−1.32533e7	−0.562430
$890$	0	0
$891$	−2.07915e7	−0.877386
$892$	− 2.79430e7i	− 1.17588i
$893$	1.48217e6i	0.0621971i
$894$	−1.05402e7	−0.441067
$895$	0	0
$896$	−7.40681e6	−0.308220
$897$	− 6.42429e6i	− 0.266590i
$898$	− 9.53226e6i	− 0.394462i
$899$	621362.	0.0256416
$900$	0	0
$901$	8.55399e6	0.351040
$902$	− 464632.i	− 0.0190148i
$903$	− 5.38928e6i	− 0.219944i
$904$	−1.70790e7	−0.695090
$905$	0	0
$906$	426292.	0.0172539
$907$	1.42674e7i	0.575874i	0.957649	+	0.287937i	$0.0929694\pi$
$907$	1.42674e7i	0.575874i	−0.957649	+	0.287937i	$0.907031\pi$
$908$	4.65678e7i	1.87444i
$909$	2.53633e7	1.01811
$910$	0	0
$911$	3.34037e7	1.33352	0.666759	−	0.745273i	$-0.267681\pi$
$911$	3.34037e7	1.33352	0.666759	+	0.745273i	$0.267681\pi$
$912$	1.39800e6i	0.0556568i
$913$	2.14036e7i	0.849785i
$914$	1.25024e6	0.0495024
$915$	0	0
$916$	−3.29628e7	−1.29803
$917$	− 2.04485e6i	− 0.0803042i
$918$	2.08893e6i	0.0818121i
$919$	−4.52117e7	−1.76588	−0.882942	−	0.469482i	$-0.844441\pi$
$919$	−4.52117e7	−1.76588	−0.882942	+	0.469482i	$0.844441\pi$
$920$	0	0
$921$	−5.96295e7	−2.31639
$922$	− 1.06455e7i	− 0.412419i
$923$	8.61384e6i	0.332807i
$924$	−8.17831e6	−0.315125
$925$	0	0
$926$	−2.16128e6	−0.0828292
$927$	− 2.13750e7i	− 0.816970i
$928$	852142.i	0.0324819i
$929$	517735.	0.0196820	0.00984098	−	0.999952i	$-0.496867\pi$
$929$	517735.	0.0196820	0.00984098	+	0.999952i	$0.496867\pi$
$930$	0	0
$931$	−1.20399e6	−0.0455249
$932$	4.27481e6i	0.161204i
$933$	− 2.91567e7i	− 1.09657i
$934$	−3.46965e6	−0.130142
$935$	0	0
$936$	−2.02025e6	−0.0753728
$937$	2.58406e7i	0.961508i	0.876856	+	0.480754i	$0.159637\pi$
$937$	2.58406e7i	0.961508i	−0.876856	+	0.480754i	$0.840363\pi$
$938$	915363.i	0.0339693i
$939$	3.09325e7	1.14486
$940$	0	0
$941$	4.58537e7	1.68811	0.844053	−	0.536259i	$-0.180163\pi$
$941$	4.58537e7	1.68811	0.844053	+	0.536259i	$0.180163\pi$
$942$	250945.i	0.00921405i
$943$	− 2.35798e6i	− 0.0863496i
$944$	4.57780e6	0.167196
$945$	0	0
$946$	2.15018e6	0.0781172
$947$	− 604846.i	− 0.0219164i	−0.999940	−	0.0109582i	$-0.996512\pi$
$947$	− 604846.i	− 0.0219164i	0.999940	−	0.0109582i	$-0.00348818\pi$
$948$	6.30289e7i	2.27782i
$949$	−7.16474e6	−0.258247
$950$	0	0
$951$	−5.67630e7	−2.03523
$952$	3.21378e6i	0.114927i
$953$	3.20607e7i	1.14351i	0.820424	+	0.571755i	$0.193737\pi$
$953$	3.20607e7i	1.14351i	−0.820424	+	0.571755i	$0.806263\pi$
$954$	−2.08257e6	−0.0740848
$955$	0	0
$956$	2.39009e7	0.845803
$957$	1.24025e6i	0.0437754i
$958$	2.11947e6i	0.0746127i
$959$	2.76434e6	0.0970609
$960$	0	0
$961$	−2.08480e7	−0.728207
$962$	− 1.90287e6i	− 0.0662935i
$963$	− 1.59264e7i	− 0.553416i
$964$	4.21614e7	1.46124
$965$	0	0
$966$	2.48810e6	0.0857877
$967$	3.19660e7i	1.09931i	0.835391	+	0.549657i	$0.185242\pi$
$967$	3.19660e7i	1.09931i	−0.835391	+	0.549657i	$0.814758\pi$
$968$	6.75280e6i	0.231630i
$969$	1.29316e6	0.0442427
$970$	0	0
$971$	−5.47040e7	−1.86196	−0.930982	−	0.365066i	$-0.881046\pi$
$971$	−5.47040e7	−1.86196	−0.930982	+	0.365066i	$0.881046\pi$
$972$	2.90746e7i	0.987069i
$973$	− 6.63126e6i	− 0.224551i
$974$	4.74243e6	0.160178
$975$	0	0
$976$	−1.59360e7	−0.535495
$977$	3.12938e7i	1.04887i	0.851450	+	0.524436i	$0.175724\pi$
$977$	3.12938e7i	1.04887i	−0.851450	+	0.524436i	$0.824276\pi$
$978$	− 1.16082e7i	− 0.388077i
$979$	−2.18299e7	−0.727940
$980$	0	0
$981$	−1.21736e7	−0.403874
$982$	4.75577e6i	0.157377i
$983$	1.21142e7i	0.399863i	0.979810	+	0.199932i	$0.0640720\pi$
$983$	1.21142e7i	0.399863i	−0.979810	+	0.199932i	$0.935928\pi$
$984$	−2.00261e6	−0.0659339
$985$	0	0
$986$	236595.	0.00775020
$987$	1.69894e7i	0.555116i
$988$	− 425414.i	− 0.0138650i
$989$	1.09120e7	0.354744
$990$	0	0
$991$	−2.58041e7	−0.834650	−0.417325	−	0.908757i	$-0.637032\pi$
$991$	−2.58041e7	−0.834650	−0.417325	+	0.908757i	$0.637032\pi$
$992$	1.06712e7i	0.344298i
$993$	− 5.77058e7i	− 1.85715i
$994$	−3.33610e6	−0.107096
$995$	0	0
$996$	4.47835e7	1.43044
$997$	− 4.21820e7i	− 1.34397i	−0.740566	−	0.671984i	$-0.765442\pi$
$997$	− 4.21820e7i	− 1.34397i	0.740566	−	0.671984i	$-0.234558\pi$
$998$	419691.i	0.0133384i
$999$	−1.64605e7	−0.521831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.g.274.6		12
5.2	odd	4		65.6.a.d.1.4	✓	6
5.3	odd	4		325.6.a.g.1.3		6
5.4	even	2	inner	325.6.b.g.274.7		12
15.2	even	4		585.6.a.m.1.3		6
20.7	even	4		1040.6.a.q.1.6		6
65.12	odd	4		845.6.a.h.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.4	✓	6	5.2	odd	4
325.6.a.g.1.3		6	5.3	odd	4
325.6.b.g.274.6		12	1.1	even	1	trivial
325.6.b.g.274.7		12	5.4	even	2	inner
585.6.a.m.1.3		6	15.2	even	4
845.6.a.h.1.3		6	65.12	odd	4
1040.6.a.q.1.6		6	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.34530i q^{2} -19.6439i q^{3} +30.1902 q^{4} -26.4269 q^{6} -48.6530i q^{7} -83.6645i q^{8} -142.882 q^{9} +283.440 q^{11} -593.052i q^{12} +169.000i q^{13} -65.4530 q^{14} +853.531 q^{16} -789.522i q^{17} +192.219i q^{18} -83.3795 q^{19} -955.734 q^{21} -381.312i q^{22} -1935.13i q^{23} -1643.49 q^{24} +227.356 q^{26} -1966.71i q^{27} -1468.84i q^{28} -222.752 q^{29} -2789.48 q^{31} -3825.52i q^{32} -5567.86i q^{33} -1062.15 q^{34} -4313.62 q^{36} -8369.56i q^{37} +112.170i q^{38} +3319.81 q^{39} +1218.51 q^{41} +1285.75i q^{42} +5638.89i q^{43} +8557.10 q^{44} -2603.34 q^{46} -17776.2i q^{47} -16766.7i q^{48} +14439.9 q^{49} -15509.3 q^{51} +5102.14i q^{52} +10834.4i q^{53} -2645.82 q^{54} -4070.53 q^{56} +1637.90i q^{57} +299.668i q^{58} +5363.36 q^{59} -18670.7 q^{61} +3752.69i q^{62} +6951.63i q^{63} +22166.5 q^{64} -7490.44 q^{66} -13985.1i q^{67} -23835.8i q^{68} -38013.5 q^{69} +50969.5 q^{71} +11954.1i q^{72} +42394.9i q^{73} -11259.6 q^{74} -2517.24 q^{76} -13790.2i q^{77} -4466.15i q^{78} -106279. q^{79} -73354.1 q^{81} -1639.26i q^{82} +75513.6i q^{83} -28853.8 q^{84} +7586.01 q^{86} +4375.71i q^{87} -23713.8i q^{88} -77017.8 q^{89} +8222.36 q^{91} -58422.0i q^{92} +54796.2i q^{93} -23914.4 q^{94} -75148.0 q^{96} -126702. i q^{97} -19426.0i q^{98} -40498.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) \) 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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