LMFDB - Embedded newform 414.8.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,8,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("414.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	414.a (trivial)

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:	yes
Analytic conductor:	$129.327400550$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 $ x^4 - 2x^3 - 8367x^2 - 89140*x + 11077220
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$90.4389$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +64.0000 q^{4} +10.0669 q^{5} +971.172 q^{7} -512.000 q^{8} +O(q^{10})$	\(q-8.00000 q^{2} +64.0000 q^{4} +10.0669 q^{5} +971.172 q^{7} -512.000 q^{8} -80.5353 q^{10} +2066.60 q^{11} +9535.57 q^{13} -7769.38 q^{14} +4096.00 q^{16} +20856.4 q^{17} +41075.9 q^{19} +644.282 q^{20} -16532.8 q^{22} +12167.0 q^{23} -78023.7 q^{25} -76284.5 q^{26} +62155.0 q^{28} -24659.9 q^{29} +244895. q^{31} -32768.0 q^{32} -166851. q^{34} +9776.71 q^{35} +520379. q^{37} -328607. q^{38} -5154.26 q^{40} -123395. q^{41} +7429.38 q^{43} +132262. q^{44} -97336.0 q^{46} +1.38129e6 q^{47} +119632. q^{49} +624189. q^{50} +610276. q^{52} -1.01252e6 q^{53} +20804.2 q^{55} -497240. q^{56} +197279. q^{58} -2.93571e6 q^{59} -2.36149e6 q^{61} -1.95916e6 q^{62} +262144. q^{64} +95993.7 q^{65} +3.19614e6 q^{67} +1.33481e6 q^{68} -78213.6 q^{70} +3.49894e6 q^{71} +2.48917e6 q^{73} -4.16304e6 q^{74} +2.62886e6 q^{76} +2.00702e6 q^{77} -4.80101e6 q^{79} +41234.1 q^{80} +987160. q^{82} -7.77498e6 q^{83} +209959. q^{85} -59435.0 q^{86} -1.05810e6 q^{88} -539607. q^{89} +9.26068e6 q^{91} +778688. q^{92} -1.10503e7 q^{94} +413508. q^{95} -1.04621e7 q^{97} -957058. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8}+O(q^{10}) $ 4 * q - 32 * q^2 + 256 * q^4 - 270 * q^5 + 2022 * q^7 - 2048 * q^8	\( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8} + 2160 q^{10} - 4120 q^{11} + 8036 q^{13} - 16176 q^{14} + 16384 q^{16} - 37182 q^{17} + 5702 q^{19} - 17280 q^{20} + 32960 q^{22} + 48668 q^{23} + 121480 q^{25} - 64288 q^{26} + 129408 q^{28} - 217716 q^{29} + 222852 q^{31} - 131072 q^{32} + 297456 q^{34} - 68440 q^{35} + 486428 q^{37} - 45616 q^{38} + 138240 q^{40} - 338336 q^{41} + 730974 q^{43} - 263680 q^{44} - 389344 q^{46} - 338248 q^{47} - 310552 q^{49} - 971840 q^{50} + 514304 q^{52} + 375502 q^{53} + 424840 q^{55} - 1035264 q^{56} + 1741728 q^{58} - 71392 q^{59} + 2101164 q^{61} - 1782816 q^{62} + 1048576 q^{64} - 1578780 q^{65} + 4337162 q^{67} - 2379648 q^{68} + 547520 q^{70} - 2288016 q^{71} - 1107328 q^{73} - 3891424 q^{74} + 364928 q^{76} - 5826200 q^{77} + 60610 q^{79} - 1105920 q^{80} + 2706688 q^{82} - 1485464 q^{83} - 8843820 q^{85} - 5847792 q^{86} + 2109440 q^{88} - 1485090 q^{89} - 2898412 q^{91} + 3114752 q^{92} + 2705984 q^{94} - 8545200 q^{95} + 1935444 q^{97} + 2484416 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 270 * q^5 + 2022 * q^7 - 2048 * q^8 + 2160 * q^10 - 4120 * q^11 + 8036 * q^13 - 16176 * q^14 + 16384 * q^16 - 37182 * q^17 + 5702 * q^19 - 17280 * q^20 + 32960 * q^22 + 48668 * q^23 + 121480 * q^25 - 64288 * q^26 + 129408 * q^28 - 217716 * q^29 + 222852 * q^31 - 131072 * q^32 + 297456 * q^34 - 68440 * q^35 + 486428 * q^37 - 45616 * q^38 + 138240 * q^40 - 338336 * q^41 + 730974 * q^43 - 263680 * q^44 - 389344 * q^46 - 338248 * q^47 - 310552 * q^49 - 971840 * q^50 + 514304 * q^52 + 375502 * q^53 + 424840 * q^55 - 1035264 * q^56 + 1741728 * q^58 - 71392 * q^59 + 2101164 * q^61 - 1782816 * q^62 + 1048576 * q^64 - 1578780 * q^65 + 4337162 * q^67 - 2379648 * q^68 + 547520 * q^70 - 2288016 * q^71 - 1107328 * q^73 - 3891424 * q^74 + 364928 * q^76 - 5826200 * q^77 + 60610 * q^79 - 1105920 * q^80 + 2706688 * q^82 - 1485464 * q^83 - 8843820 * q^85 - 5847792 * q^86 + 2109440 * q^88 - 1485090 * q^89 - 2898412 * q^91 + 3114752 * q^92 + 2705984 * q^94 - 8545200 * q^95 + 1935444 * q^97 + 2484416 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	10.0669	0.0360165	0.0180082	−	0.999838i	$-0.494267\pi$
$5$	10.0669	0.0360165	0.0180082	+	0.999838i	$0.494267\pi$
$6$	0	0
$7$	971.172	1.07017	0.535085	−	0.844798i	$-0.320279\pi$
$7$	971.172	1.07017	0.535085	+	0.844798i	$0.320279\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−80.5353	−0.0254675
$11$	2066.60	0.468146	0.234073	−	0.972219i	$-0.424795\pi$
$11$	2066.60	0.468146	0.234073	+	0.972219i	$0.424795\pi$
$12$	0	0
$13$	9535.57	1.20377	0.601887	−	0.798581i	$-0.294416\pi$
$13$	9535.57	1.20377	0.601887	+	0.798581i	$0.294416\pi$
$14$	−7769.38	−0.756725
$15$	0	0
$16$	4096.00	0.250000
$17$	20856.4	1.02960	0.514799	−	0.857311i	$-0.327867\pi$
$17$	20856.4	1.02960	0.514799	+	0.857311i	$0.327867\pi$
$18$	0	0
$19$	41075.9	1.37388	0.686942	−	0.726713i	$-0.258953\pi$
$19$	41075.9	1.37388	0.686942	+	0.726713i	$0.258953\pi$
$20$	644.282	0.0180082
$21$	0	0
$22$	−16532.8	−0.331029
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	−78023.7	−0.998703
$26$	−76284.5	−0.851196
$27$	0	0
$28$	62155.0	0.535085
$29$	−24659.9	−0.187758	−0.0938789	−	0.995584i	$-0.529927\pi$
$29$	−24659.9	−0.187758	−0.0938789	+	0.995584i	$0.529927\pi$
$30$	0	0
$31$	244895.	1.47643	0.738217	−	0.674563i	$-0.235668\pi$
$31$	244895.	1.47643	0.738217	+	0.674563i	$0.235668\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−166851.	−0.728036
$35$	9776.71	0.0385438
$36$	0	0
$37$	520379.	1.68894	0.844470	−	0.535603i	$-0.179916\pi$
$37$	520379.	1.68894	0.844470	+	0.535603i	$0.179916\pi$
$38$	−328607.	−0.971482
$39$	0	0
$40$	−5154.26	−0.0127338
$41$	−123395.	−0.279611	−0.139805	−	0.990179i	$-0.544648\pi$
$41$	−123395.	−0.279611	−0.139805	+	0.990179i	$0.544648\pi$
$42$	0	0
$43$	7429.38	0.0142499	0.00712497	−	0.999975i	$-0.497732\pi$
$43$	7429.38	0.0142499	0.00712497	+	0.999975i	$0.497732\pi$
$44$	132262.	0.234073
$45$	0	0
$46$	−97336.0	−0.147442
$47$	1.38129e6	1.94063	0.970316	−	0.241839i	$-0.0777507\pi$
$47$	1.38129e6	1.94063	0.970316	+	0.241839i	$0.0777507\pi$
$48$	0	0
$49$	119632.	0.145265
$50$	624189.	0.706190
$51$	0	0
$52$	610276.	0.601887
$53$	−1.01252e6	−0.934192	−0.467096	−	0.884206i	$-0.654700\pi$
$53$	−1.01252e6	−0.934192	−0.467096	+	0.884206i	$0.654700\pi$
$54$	0	0
$55$	20804.2	0.0168610
$56$	−497240.	−0.378362
$57$	0	0
$58$	197279.	0.132765
$59$	−2.93571e6	−1.86094	−0.930468	−	0.366372i	$-0.880600\pi$
$59$	−2.93571e6	−1.86094	−0.930468	+	0.366372i	$0.880600\pi$
$60$	0	0
$61$	−2.36149e6	−1.33208	−0.666041	−	0.745915i	$-0.732012\pi$
$61$	−2.36149e6	−1.33208	−0.666041	+	0.745915i	$0.732012\pi$
$62$	−1.95916e6	−1.04400
$63$	0	0
$64$	262144.	0.125000
$65$	95993.7	0.0433557
$66$	0	0
$67$	3.19614e6	1.29827	0.649133	−	0.760675i	$-0.275132\pi$
$67$	3.19614e6	1.29827	0.649133	+	0.760675i	$0.275132\pi$
$68$	1.33481e6	0.514799
$69$	0	0
$70$	−78213.6	−0.0272546
$71$	3.49894e6	1.16020	0.580100	−	0.814545i	$-0.303013\pi$
$71$	3.49894e6	1.16020	0.580100	+	0.814545i	$0.303013\pi$
$72$	0	0
$73$	2.48917e6	0.748903	0.374451	−	0.927247i	$-0.377831\pi$
$73$	2.48917e6	0.748903	0.374451	+	0.927247i	$0.377831\pi$
$74$	−4.16304e6	−1.19426
$75$	0	0
$76$	2.62886e6	0.686942
$77$	2.00702e6	0.500996
$78$	0	0
$79$	−4.80101e6	−1.09556	−0.547782	−	0.836621i	$-0.684528\pi$
$79$	−4.80101e6	−1.09556	−0.547782	+	0.836621i	$0.684528\pi$
$80$	41234.1	0.00900412
$81$	0	0
$82$	987160.	0.197715
$83$	−7.77498e6	−1.49254	−0.746270	−	0.665644i	$-0.768157\pi$
$83$	−7.77498e6	−1.49254	−0.746270	+	0.665644i	$0.768157\pi$
$84$	0	0
$85$	209959.	0.0370825
$86$	−59435.0	−0.0100762
$87$	0	0
$88$	−1.05810e6	−0.165514
$89$	−539607.	−0.0811358	−0.0405679	−	0.999177i	$-0.512917\pi$
$89$	−539607.	−0.0811358	−0.0405679	+	0.999177i	$0.512917\pi$
$90$	0	0
$91$	9.26068e6	1.28824
$92$	778688.	0.104257
$93$	0	0
$94$	−1.10503e7	−1.37223
$95$	413508.	0.0494824
$96$	0	0
$97$	−1.04621e7	−1.16391	−0.581953	−	0.813222i	$-0.697711\pi$
$97$	−1.04621e7	−1.16391	−0.581953	+	0.813222i	$0.697711\pi$
$98$	−957058.	−0.102718
$99$	0	0
$100$	−4.99351e6	−0.499351
$101$	4.89157e6	0.472415	0.236207	−	0.971703i	$-0.424096\pi$
$101$	4.89157e6	0.472415	0.236207	+	0.971703i	$0.424096\pi$
$102$	0	0
$103$	6.99268e6	0.630541	0.315270	−	0.949002i	$-0.397905\pi$
$103$	6.99268e6	0.630541	0.315270	+	0.949002i	$0.397905\pi$
$104$	−4.88221e6	−0.425598
$105$	0	0
$106$	8.10013e6	0.660574
$107$	−1.26394e7	−0.997434	−0.498717	−	0.866765i	$-0.666195\pi$
$107$	−1.26394e7	−0.997434	−0.498717	+	0.866765i	$0.666195\pi$
$108$	0	0
$109$	−1.88878e7	−1.39697	−0.698486	−	0.715623i	$-0.746143\pi$
$109$	−1.88878e7	−1.39697	−0.698486	+	0.715623i	$0.746143\pi$
$110$	−166434.	−0.0119225
$111$	0	0
$112$	3.97792e6	0.267543
$113$	9.49674e6	0.619156	0.309578	−	0.950874i	$-0.399812\pi$
$113$	9.49674e6	0.619156	0.309578	+	0.950874i	$0.399812\pi$
$114$	0	0
$115$	122484.	0.00750996
$116$	−1.57823e6	−0.0938789
$117$	0	0
$118$	2.34857e7	1.31588
$119$	2.02551e7	1.10185
$120$	0	0
$121$	−1.52164e7	−0.780840
$122$	1.88919e7	0.941924
$123$	0	0
$124$	1.56733e7	0.738217
$125$	−1.57194e6	−0.0719862
$126$	0	0
$127$	−1.65209e7	−0.715684	−0.357842	−	0.933782i	$-0.616487\pi$
$127$	−1.65209e7	−0.715684	−0.357842	+	0.933782i	$0.616487\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	−767950.	−0.0306571
$131$	−1.15691e7	−0.449624	−0.224812	−	0.974402i	$-0.572177\pi$
$131$	−1.15691e7	−0.449624	−0.224812	+	0.974402i	$0.572177\pi$
$132$	0	0
$133$	3.98918e7	1.47029
$134$	−2.55691e7	−0.918013
$135$	0	0
$136$	−1.06785e7	−0.364018
$137$	4.83328e7	1.60590	0.802952	−	0.596043i	$-0.203261\pi$
$137$	4.83328e7	1.60590	0.802952	+	0.596043i	$0.203261\pi$
$138$	0	0
$139$	−3.30182e7	−1.04280	−0.521401	−	0.853312i	$-0.674591\pi$
$139$	−3.30182e7	−1.04280	−0.521401	+	0.853312i	$0.674591\pi$
$140$	625709.	0.0192719
$141$	0	0
$142$	−2.79916e7	−0.820385
$143$	1.97062e7	0.563541
$144$	0	0
$145$	−248249.	−0.00676237
$146$	−1.99134e7	−0.529554
$147$	0	0
$148$	3.33043e7	0.844470
$149$	6.04124e6	0.149615	0.0748074	−	0.997198i	$-0.476166\pi$
$149$	6.04124e6	0.149615	0.0748074	+	0.997198i	$0.476166\pi$
$150$	0	0
$151$	−1.73272e7	−0.409551	−0.204775	−	0.978809i	$-0.565646\pi$
$151$	−1.73272e7	−0.409551	−0.204775	+	0.978809i	$0.565646\pi$
$152$	−2.10309e7	−0.485741
$153$	0	0
$154$	−1.60562e7	−0.354258
$155$	2.46534e6	0.0531760
$156$	0	0
$157$	−3.58063e7	−0.738433	−0.369216	−	0.929343i	$-0.620374\pi$
$157$	−3.58063e7	−0.738433	−0.369216	+	0.929343i	$0.620374\pi$
$158$	3.84081e7	0.774681
$159$	0	0
$160$	−329873.	−0.00636688
$161$	1.18163e7	0.223146
$162$	0	0
$163$	−8.41945e7	−1.52275	−0.761373	−	0.648315i	$-0.775474\pi$
$163$	−8.41945e7	−1.52275	−0.761373	+	0.648315i	$0.775474\pi$
$164$	−7.89728e6	−0.139805
$165$	0	0
$166$	6.21998e7	1.05538
$167$	1.15659e8	1.92163	0.960817	−	0.277182i	$-0.0894006\pi$
$167$	1.15659e8	1.92163	0.960817	+	0.277182i	$0.0894006\pi$
$168$	0	0
$169$	2.81785e7	0.449071
$170$	−1.67967e6	−0.0262213
$171$	0	0
$172$	475480.	0.00712497
$173$	2.95752e7	0.434276	0.217138	−	0.976141i	$-0.430328\pi$
$173$	2.95752e7	0.434276	0.217138	+	0.976141i	$0.430328\pi$
$174$	0	0
$175$	−7.57744e7	−1.06878
$176$	8.46477e6	0.117036
$177$	0	0
$178$	4.31686e6	0.0573717
$179$	7.92741e7	1.03311	0.516554	−	0.856255i	$-0.327214\pi$
$179$	7.92741e7	1.03311	0.516554	+	0.856255i	$0.327214\pi$
$180$	0	0
$181$	9.63504e7	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$181$	9.63504e7	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$182$	−7.40854e7	−0.910926
$183$	0	0
$184$	−6.22950e6	−0.0737210
$185$	5.23862e6	0.0608297
$186$	0	0
$187$	4.31017e7	0.482002
$188$	8.84028e7	0.970316
$189$	0	0
$190$	−3.30806e6	−0.0349894
$191$	1.35606e8	1.40819	0.704094	−	0.710107i	$-0.251353\pi$
$191$	1.35606e8	1.40819	0.704094	+	0.710107i	$0.251353\pi$
$192$	0	0
$193$	−1.75736e8	−1.75959	−0.879795	−	0.475354i	$-0.842320\pi$
$193$	−1.75736e8	−1.75959	−0.879795	+	0.475354i	$0.842320\pi$
$194$	8.36969e7	0.823006
$195$	0	0
$196$	7.65647e6	0.0726327
$197$	1.22530e8	1.14186	0.570929	−	0.821000i	$-0.306583\pi$
$197$	1.22530e8	1.14186	0.570929	+	0.821000i	$0.306583\pi$
$198$	0	0
$199$	2.14948e8	1.93352	0.966760	−	0.255686i	$-0.0823015\pi$
$199$	2.14948e8	1.93352	0.966760	+	0.255686i	$0.0823015\pi$
$200$	3.99481e7	0.353095
$201$	0	0
$202$	−3.91325e7	−0.334048
$203$	−2.39490e7	−0.200933
$204$	0	0
$205$	−1.24221e6	−0.0100706
$206$	−5.59414e7	−0.445860
$207$	0	0
$208$	3.90577e7	0.300943
$209$	8.48873e7	0.643178
$210$	0	0
$211$	2.30223e8	1.68717	0.843586	−	0.536994i	$-0.180440\pi$
$211$	2.30223e8	1.68717	0.843586	+	0.536994i	$0.180440\pi$
$212$	−6.48010e7	−0.467096
$213$	0	0
$214$	1.01115e8	0.705293
$215$	74790.9	0.000513233 0
$216$	0	0
$217$	2.37835e8	1.58004
$218$	1.51102e8	0.987809
$219$	0	0
$220$	1.33147e6	0.00843048
$221$	1.98877e8	1.23940
$222$	0	0
$223$	2.16447e8	1.30703	0.653515	−	0.756914i	$-0.273294\pi$
$223$	2.16447e8	1.30703	0.653515	+	0.756914i	$0.273294\pi$
$224$	−3.18234e7	−0.189181
$225$	0	0
$226$	−7.59739e7	−0.437809
$227$	2.11703e8	1.20126	0.600630	−	0.799527i	$-0.294917\pi$
$227$	2.11703e8	1.20126	0.600630	+	0.799527i	$0.294917\pi$
$228$	0	0
$229$	−1.41898e7	−0.0780825	−0.0390412	−	0.999238i	$-0.512430\pi$
$229$	−1.41898e7	−0.0780825	−0.0390412	+	0.999238i	$0.512430\pi$
$230$	−979873.	−0.00531034
$231$	0	0
$232$	1.26259e7	0.0663824
$233$	−2.97428e8	−1.54041	−0.770204	−	0.637798i	$-0.779846\pi$
$233$	−2.97428e8	−1.54041	−0.770204	+	0.637798i	$0.779846\pi$
$234$	0	0
$235$	1.39054e7	0.0698948
$236$	−1.87886e8	−0.930468
$237$	0	0
$238$	−1.62041e8	−0.779122
$239$	−2.69292e8	−1.27594	−0.637972	−	0.770060i	$-0.720226\pi$
$239$	−2.69292e8	−1.27594	−0.637972	+	0.770060i	$0.720226\pi$
$240$	0	0
$241$	3.06355e7	0.140982	0.0704912	−	0.997512i	$-0.477543\pi$
$241$	3.06355e7	0.140982	0.0704912	+	0.997512i	$0.477543\pi$
$242$	1.21731e8	0.552137
$243$	0	0
$244$	−1.51135e8	−0.666041
$245$	1.20433e6	0.00523195
$246$	0	0
$247$	3.91682e8	1.65384
$248$	−1.25386e8	−0.521998
$249$	0	0
$250$	1.25755e7	0.0509020
$251$	2.34129e8	0.934539	0.467270	−	0.884115i	$-0.345238\pi$
$251$	2.34129e8	0.934539	0.467270	+	0.884115i	$0.345238\pi$
$252$	0	0
$253$	2.51443e7	0.0976151
$254$	1.32167e8	0.506065
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−3.87910e8	−1.42549	−0.712747	−	0.701421i	$-0.752549\pi$
$257$	−3.87910e8	−1.42549	−0.712747	+	0.701421i	$0.752549\pi$
$258$	0	0
$259$	5.05378e8	1.80745
$260$	6.14360e6	0.0216778
$261$	0	0
$262$	9.25527e7	0.317932
$263$	3.05352e8	1.03504	0.517518	−	0.855672i	$-0.326856\pi$
$263$	3.05352e8	1.03504	0.517518	+	0.855672i	$0.326856\pi$
$264$	0	0
$265$	−1.01929e7	−0.0336463
$266$	−3.19134e8	−1.03965
$267$	0	0
$268$	2.04553e8	0.649133
$269$	2.30284e8	0.721324	0.360662	−	0.932697i	$-0.382551\pi$
$269$	2.30284e8	0.721324	0.360662	+	0.932697i	$0.382551\pi$
$270$	0	0
$271$	−3.89654e7	−0.118929	−0.0594643	−	0.998230i	$-0.518939\pi$
$271$	−3.89654e7	−0.118929	−0.0594643	+	0.998230i	$0.518939\pi$
$272$	8.54277e7	0.257399
$273$	0	0
$274$	−3.86662e8	−1.13555
$275$	−1.61243e8	−0.467538
$276$	0	0
$277$	3.37749e8	0.954804	0.477402	−	0.878685i	$-0.341579\pi$
$277$	3.37749e8	0.954804	0.477402	+	0.878685i	$0.341579\pi$
$278$	2.64146e8	0.737373
$279$	0	0
$280$	−5.00567e6	−0.0136273
$281$	−4.32675e8	−1.16329	−0.581647	−	0.813441i	$-0.697592\pi$
$281$	−4.32675e8	−1.16329	−0.581647	+	0.813441i	$0.697592\pi$
$282$	0	0
$283$	2.64590e8	0.693938	0.346969	−	0.937877i	$-0.387211\pi$
$283$	2.64590e8	0.693938	0.346969	+	0.937877i	$0.387211\pi$
$284$	2.23932e8	0.580100
$285$	0	0
$286$	−1.57649e8	−0.398484
$287$	−1.19838e8	−0.299232
$288$	0	0
$289$	2.46498e7	0.0600718
$290$	1.98599e6	0.00478172
$291$	0	0
$292$	1.59307e8	0.374451
$293$	1.03680e8	0.240802	0.120401	−	0.992725i	$-0.461582\pi$
$293$	1.03680e8	0.240802	0.120401	+	0.992725i	$0.461582\pi$
$294$	0	0
$295$	−2.95536e7	−0.0670244
$296$	−2.66434e8	−0.597130
$297$	0	0
$298$	−4.83299e7	−0.105794
$299$	1.16019e8	0.251004
$300$	0	0
$301$	7.21521e6	0.0152499
$302$	1.38617e8	0.289596
$303$	0	0
$304$	1.68247e8	0.343471
$305$	−2.37729e7	−0.0479769
$306$	0	0
$307$	−3.01316e8	−0.594344	−0.297172	−	0.954824i	$-0.596044\pi$
$307$	−3.01316e8	−0.594344	−0.297172	+	0.954824i	$0.596044\pi$
$308$	1.28449e8	0.250498
$309$	0	0
$310$	−1.97227e7	−0.0376011
$311$	−6.35884e8	−1.19872	−0.599358	−	0.800481i	$-0.704578\pi$
$311$	−6.35884e8	−1.19872	−0.599358	+	0.800481i	$0.704578\pi$
$312$	0	0
$313$	−2.65449e8	−0.489302	−0.244651	−	0.969611i	$-0.578673\pi$
$313$	−2.65449e8	−0.489302	−0.244651	+	0.969611i	$0.578673\pi$
$314$	2.86451e8	0.522151
$315$	0	0
$316$	−3.07265e8	−0.547782
$317$	−4.07834e8	−0.719078	−0.359539	−	0.933130i	$-0.617066\pi$
$317$	−4.07834e8	−0.719078	−0.359539	+	0.933130i	$0.617066\pi$
$318$	0	0
$319$	−5.09620e7	−0.0878980
$320$	2.63898e6	0.00450206
$321$	0	0
$322$	−9.45300e7	−0.157788
$323$	8.56695e8	1.41455
$324$	0	0
$325$	−7.44000e8	−1.20221
$326$	6.73556e8	1.07674
$327$	0	0
$328$	6.31783e7	0.0988574
$329$	1.34147e9	2.07681
$330$	0	0
$331$	−2.72660e8	−0.413260	−0.206630	−	0.978419i	$-0.566250\pi$
$331$	−2.72660e8	−0.413260	−0.206630	+	0.978419i	$0.566250\pi$
$332$	−4.97598e8	−0.746270
$333$	0	0
$334$	−9.25270e8	−1.35880
$335$	3.21753e7	0.0467590
$336$	0	0
$337$	5.05642e8	0.719679	0.359839	−	0.933014i	$-0.382832\pi$
$337$	5.05642e8	0.719679	0.359839	+	0.933014i	$0.382832\pi$
$338$	−2.25428e8	−0.317541
$339$	0	0
$340$	1.34374e7	0.0185412
$341$	5.06099e8	0.691186
$342$	0	0
$343$	−6.83618e8	−0.914712
$344$	−3.80384e6	−0.00503811
$345$	0	0
$346$	−2.36601e8	−0.307079
$347$	1.05304e9	1.35297	0.676487	−	0.736454i	$-0.263501\pi$
$347$	1.05304e9	1.35297	0.676487	+	0.736454i	$0.263501\pi$
$348$	0	0
$349$	−1.26100e8	−0.158792	−0.0793958	−	0.996843i	$-0.525299\pi$
$349$	−1.26100e8	−0.158792	−0.0793958	+	0.996843i	$0.525299\pi$
$350$	6.06195e8	0.755743
$351$	0	0
$352$	−6.77182e7	−0.0827572
$353$	3.34219e8	0.404407	0.202204	−	0.979343i	$-0.435190\pi$
$353$	3.34219e8	0.404407	0.202204	+	0.979343i	$0.435190\pi$
$354$	0	0
$355$	3.52236e7	0.0417863
$356$	−3.45349e7	−0.0405679
$357$	0	0
$358$	−6.34193e8	−0.730518
$359$	−5.37104e8	−0.612671	−0.306336	−	0.951924i	$-0.599103\pi$
$359$	−5.37104e8	−0.612671	−0.306336	+	0.951924i	$0.599103\pi$
$360$	0	0
$361$	7.93361e8	0.887555
$362$	−7.70804e8	−0.854011
$363$	0	0
$364$	5.92683e8	0.644122
$365$	2.50583e7	0.0269728
$366$	0	0
$367$	−2.86339e8	−0.302378	−0.151189	−	0.988505i	$-0.548310\pi$
$367$	−2.86339e8	−0.302378	−0.151189	+	0.988505i	$0.548310\pi$
$368$	4.98360e7	0.0521286
$369$	0	0
$370$	−4.19089e7	−0.0430131
$371$	−9.83327e8	−0.999745
$372$	0	0
$373$	1.16695e9	1.16432	0.582159	−	0.813075i	$-0.302208\pi$
$373$	1.16695e9	1.16432	0.582159	+	0.813075i	$0.302208\pi$
$374$	−3.44813e8	−0.340827
$375$	0	0
$376$	−7.07222e8	−0.686117
$377$	−2.35146e8	−0.226018
$378$	0	0
$379$	−7.02743e8	−0.663070	−0.331535	−	0.943443i	$-0.607566\pi$
$379$	−7.02743e8	−0.663070	−0.331535	+	0.943443i	$0.607566\pi$
$380$	2.64645e7	0.0247412
$381$	0	0
$382$	−1.08484e9	−0.995739
$383$	4.77966e8	0.434712	0.217356	−	0.976092i	$-0.430257\pi$
$383$	4.77966e8	0.434712	0.217356	+	0.976092i	$0.430257\pi$
$384$	0	0
$385$	2.02045e7	0.0180441
$386$	1.40589e9	1.24422
$387$	0	0
$388$	−6.69575e8	−0.581953
$389$	−1.64085e9	−1.41334	−0.706668	−	0.707545i	$-0.749803\pi$
$389$	−1.64085e9	−1.41334	−0.706668	+	0.707545i	$0.749803\pi$
$390$	0	0
$391$	2.53760e8	0.214686
$392$	−6.12517e7	−0.0513591
$393$	0	0
$394$	−9.80243e8	−0.807415
$395$	−4.83314e7	−0.0394584
$396$	0	0
$397$	9.42161e8	0.755716	0.377858	−	0.925864i	$-0.376661\pi$
$397$	9.42161e8	0.755716	0.377858	+	0.925864i	$0.376661\pi$
$398$	−1.71959e9	−1.36720
$399$	0	0
$400$	−3.19585e8	−0.249676
$401$	−1.49153e8	−0.115512	−0.0577561	−	0.998331i	$-0.518395\pi$
$401$	−1.49153e8	−0.115512	−0.0577561	+	0.998331i	$0.518395\pi$
$402$	0	0
$403$	2.33521e9	1.77729
$404$	3.13060e8	0.236207
$405$	0	0
$406$	1.91592e8	0.142081
$407$	1.07541e9	0.790670
$408$	0	0
$409$	−1.15135e9	−0.832101	−0.416051	−	0.909341i	$-0.636586\pi$
$409$	−1.15135e9	−0.832101	−0.416051	+	0.909341i	$0.636586\pi$
$410$	9.93766e6	0.00712099
$411$	0	0
$412$	4.47531e8	0.315270
$413$	−2.85108e9	−1.99152
$414$	0	0
$415$	−7.82700e7	−0.0537560
$416$	−3.12461e8	−0.212799
$417$	0	0
$418$	−6.79099e8	−0.454795
$419$	−3.94962e8	−0.262305	−0.131152	−	0.991362i	$-0.541868\pi$
$419$	−3.94962e8	−0.262305	−0.131152	+	0.991362i	$0.541868\pi$
$420$	0	0
$421$	−3.31295e8	−0.216385	−0.108193	−	0.994130i	$-0.534506\pi$
$421$	−3.31295e8	−0.216385	−0.108193	+	0.994130i	$0.534506\pi$
$422$	−1.84178e9	−1.19301
$423$	0	0
$424$	5.18408e8	0.330287
$425$	−1.62729e9	−1.02826
$426$	0	0
$427$	−2.29341e9	−1.42556
$428$	−8.08924e8	−0.498717
$429$	0	0
$430$	−598327.	−0.000362910 0
$431$	6.07719e8	0.365622	0.182811	−	0.983148i	$-0.441480\pi$
$431$	6.07719e8	0.365622	0.182811	+	0.983148i	$0.441480\pi$
$432$	0	0
$433$	1.70910e9	1.01172	0.505860	−	0.862615i	$-0.331175\pi$
$433$	1.70910e9	1.01172	0.505860	+	0.862615i	$0.331175\pi$
$434$	−1.90268e9	−1.11725
$435$	0	0
$436$	−1.20882e9	−0.698486
$437$	4.99771e8	0.286474
$438$	0	0
$439$	−1.60133e9	−0.903349	−0.451674	−	0.892183i	$-0.649173\pi$
$439$	−1.60133e9	−0.903349	−0.451674	+	0.892183i	$0.649173\pi$
$440$	−1.06518e7	−0.00596125
$441$	0	0
$442$	−1.59102e9	−0.876390
$443$	2.60952e9	1.42609	0.713045	−	0.701118i	$-0.247315\pi$
$443$	2.60952e9	1.42609	0.713045	+	0.701118i	$0.247315\pi$
$444$	0	0
$445$	−5.43218e6	−0.00292223
$446$	−1.73158e9	−0.924210
$447$	0	0
$448$	2.54587e8	0.133771
$449$	−3.28207e8	−0.171114	−0.0855571	−	0.996333i	$-0.527267\pi$
$449$	−3.28207e8	−0.171114	−0.0855571	+	0.996333i	$0.527267\pi$
$450$	0	0
$451$	−2.55008e8	−0.130899
$452$	6.07791e8	0.309578
$453$	0	0
$454$	−1.69363e9	−0.849419
$455$	9.32264e7	0.0463980
$456$	0	0
$457$	1.35864e9	0.665884	0.332942	−	0.942947i	$-0.391959\pi$
$457$	1.35864e9	0.665884	0.332942	+	0.942947i	$0.391959\pi$
$458$	1.13519e8	0.0552126
$459$	0	0
$460$	7.83898e6	0.00375498
$461$	−1.49808e9	−0.712168	−0.356084	−	0.934454i	$-0.615888\pi$
$461$	−1.49808e9	−0.712168	−0.356084	+	0.934454i	$0.615888\pi$
$462$	0	0
$463$	−3.81745e9	−1.78748	−0.893738	−	0.448590i	$-0.851927\pi$
$463$	−3.81745e9	−1.78748	−0.893738	+	0.448590i	$0.851927\pi$
$464$	−1.01007e8	−0.0469394
$465$	0	0
$466$	2.37942e9	1.08923
$467$	−2.11226e9	−0.959706	−0.479853	−	0.877349i	$-0.659310\pi$
$467$	−2.11226e9	−0.959706	−0.479853	+	0.877349i	$0.659310\pi$
$468$	0	0
$469$	3.10400e9	1.38937
$470$	−1.11243e8	−0.0494231
$471$	0	0
$472$	1.50309e9	0.657940
$473$	1.53535e7	0.00667105
$474$	0	0
$475$	−3.20489e9	−1.37210
$476$	1.29633e9	0.550923
$477$	0	0
$478$	2.15434e9	0.902228
$479$	−3.66057e9	−1.52186	−0.760929	−	0.648836i	$-0.775256\pi$
$479$	−3.66057e9	−1.52186	−0.760929	+	0.648836i	$0.775256\pi$
$480$	0	0
$481$	4.96211e9	2.03310
$482$	−2.45084e8	−0.0996896
$483$	0	0
$484$	−9.73847e8	−0.390420
$485$	−1.05321e8	−0.0419198
$486$	0	0
$487$	−1.88367e9	−0.739014	−0.369507	−	0.929228i	$-0.620473\pi$
$487$	−1.88367e9	−0.739014	−0.369507	+	0.929228i	$0.620473\pi$
$488$	1.20908e9	0.470962
$489$	0	0
$490$	−9.63463e6	−0.00369955
$491$	1.56036e8	0.0594892	0.0297446	−	0.999558i	$-0.490531\pi$
$491$	1.56036e8	0.0594892	0.0297446	+	0.999558i	$0.490531\pi$
$492$	0	0
$493$	−5.14316e8	−0.193315
$494$	−3.13346e9	−1.16944
$495$	0	0
$496$	1.00309e9	0.369109
$497$	3.39808e9	1.24161
$498$	0	0
$499$	−2.59287e9	−0.934174	−0.467087	−	0.884211i	$-0.654697\pi$
$499$	−2.59287e9	−0.934174	−0.467087	+	0.884211i	$0.654697\pi$
$500$	−1.00604e8	−0.0359931
$501$	0	0
$502$	−1.87303e9	−0.660819
$503$	−1.41030e8	−0.0494109	−0.0247055	−	0.999695i	$-0.507865\pi$
$503$	−1.41030e8	−0.0494109	−0.0247055	+	0.999695i	$0.507865\pi$
$504$	0	0
$505$	4.92430e7	0.0170147
$506$	−2.01154e8	−0.0690243
$507$	0	0
$508$	−1.05734e9	−0.357842
$509$	1.57066e9	0.527924	0.263962	−	0.964533i	$-0.414971\pi$
$509$	1.57066e9	0.527924	0.263962	+	0.964533i	$0.414971\pi$
$510$	0	0
$511$	2.41742e9	0.801454
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	3.10328e9	1.00798
$515$	7.03947e7	0.0227099
$516$	0	0
$517$	2.85457e9	0.908499
$518$	−4.04302e9	−1.27806
$519$	0	0
$520$	−4.91488e7	−0.0153286
$521$	−1.08322e9	−0.335572	−0.167786	−	0.985823i	$-0.553662\pi$
$521$	−1.08322e9	−0.335572	−0.167786	+	0.985823i	$0.553662\pi$
$522$	0	0
$523$	4.46212e9	1.36391	0.681954	−	0.731395i	$-0.261130\pi$
$523$	4.46212e9	1.36391	0.681954	+	0.731395i	$0.261130\pi$
$524$	−7.40421e8	−0.224812
$525$	0	0
$526$	−2.44282e9	−0.731881
$527$	5.10762e9	1.52013
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	8.15433e7	0.0237915
$531$	0	0
$532$	2.55308e9	0.735145
$533$	−1.17664e9	−0.336588
$534$	0	0
$535$	−1.27240e8	−0.0359241
$536$	−1.63642e9	−0.459007
$537$	0	0
$538$	−1.84227e9	−0.510053
$539$	2.47232e8	0.0680054
$540$	0	0
$541$	3.63866e9	0.987985	0.493993	−	0.869466i	$-0.335537\pi$
$541$	3.63866e9	0.987985	0.493993	+	0.869466i	$0.335537\pi$
$542$	3.11723e8	0.0840952
$543$	0	0
$544$	−6.83422e8	−0.182009
$545$	−1.90142e8	−0.0503141
$546$	0	0
$547$	2.28626e9	0.597269	0.298635	−	0.954368i	$-0.403469\pi$
$547$	2.28626e9	0.597269	0.298635	+	0.954368i	$0.403469\pi$
$548$	3.09330e9	0.802952
$549$	0	0
$550$	1.28995e9	0.330600
$551$	−1.01293e9	−0.257957
$552$	0	0
$553$	−4.66261e9	−1.17244
$554$	−2.70199e9	−0.675149
$555$	0	0
$556$	−2.11317e9	−0.521401
$557$	2.29871e9	0.563626	0.281813	−	0.959469i	$-0.409064\pi$
$557$	2.29871e9	0.563626	0.281813	+	0.959469i	$0.409064\pi$
$558$	0	0
$559$	7.08433e7	0.0171537
$560$	4.00454e7	0.00963595
$561$	0	0
$562$	3.46140e9	0.822573
$563$	−3.55963e8	−0.0840669	−0.0420334	−	0.999116i	$-0.513384\pi$
$563$	−3.55963e8	−0.0840669	−0.0420334	+	0.999116i	$0.513384\pi$
$564$	0	0
$565$	9.56029e7	0.0222998
$566$	−2.11672e9	−0.490688
$567$	0	0
$568$	−1.79146e9	−0.410193
$569$	−6.89772e9	−1.56968	−0.784842	−	0.619696i	$-0.787256\pi$
$569$	−6.89772e9	−1.56968	−0.784842	+	0.619696i	$0.787256\pi$
$570$	0	0
$571$	9.04590e8	0.203341	0.101671	−	0.994818i	$-0.467581\pi$
$571$	9.04590e8	0.203341	0.101671	+	0.994818i	$0.467581\pi$
$572$	1.26119e9	0.281771
$573$	0	0
$574$	9.58703e8	0.211589
$575$	−9.49314e8	−0.208244
$576$	0	0
$577$	−3.27571e9	−0.709889	−0.354945	−	0.934887i	$-0.615500\pi$
$577$	−3.27571e9	−0.709889	−0.354945	+	0.934887i	$0.615500\pi$
$578$	−1.97198e8	−0.0424772
$579$	0	0
$580$	−1.58879e7	−0.00338119
$581$	−7.55084e9	−1.59727
$582$	0	0
$583$	−2.09246e9	−0.437338
$584$	−1.27446e9	−0.264777
$585$	0	0
$586$	−8.29442e8	−0.170272
$587$	−7.50283e8	−0.153106	−0.0765529	−	0.997066i	$-0.524391\pi$
$587$	−7.50283e8	−0.153106	−0.0765529	+	0.997066i	$0.524391\pi$
$588$	0	0
$589$	1.00593e10	2.02845
$590$	2.36429e8	0.0473934
$591$	0	0
$592$	2.13147e9	0.422235
$593$	−1.36947e9	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$593$	−1.36947e9	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$594$	0	0
$595$	2.03907e8	0.0396846
$596$	3.86640e8	0.0748074
$597$	0	0
$598$	−9.28154e8	−0.177487
$599$	5.06032e9	0.962020	0.481010	−	0.876715i	$-0.340270\pi$
$599$	5.06032e9	0.962020	0.481010	+	0.876715i	$0.340270\pi$
$600$	0	0
$601$	−9.68350e9	−1.81958	−0.909791	−	0.415067i	$-0.863758\pi$
$601$	−9.68350e9	−1.81958	−0.909791	+	0.415067i	$0.863758\pi$
$602$	−5.77216e7	−0.0107833
$603$	0	0
$604$	−1.10894e9	−0.204775
$605$	−1.53182e8	−0.0281231
$606$	0	0
$607$	−5.82449e9	−1.05705	−0.528527	−	0.848916i	$-0.677256\pi$
$607$	−5.82449e9	−1.05705	−0.528527	+	0.848916i	$0.677256\pi$
$608$	−1.34598e9	−0.242871
$609$	0	0
$610$	1.90183e8	0.0339248
$611$	1.31714e10	2.33608
$612$	0	0
$613$	−2.39156e9	−0.419343	−0.209671	−	0.977772i	$-0.567239\pi$
$613$	−2.39156e9	−0.419343	−0.209671	+	0.977772i	$0.567239\pi$
$614$	2.41053e9	0.420265
$615$	0	0
$616$	−1.02759e9	−0.177129
$617$	6.83517e9	1.17152	0.585762	−	0.810483i	$-0.300795\pi$
$617$	6.83517e9	1.17152	0.585762	+	0.810483i	$0.300795\pi$
$618$	0	0
$619$	−8.54071e9	−1.44736	−0.723680	−	0.690135i	$-0.757551\pi$
$619$	−8.54071e9	−1.44736	−0.723680	+	0.690135i	$0.757551\pi$
$620$	1.57782e8	0.0265880
$621$	0	0
$622$	5.08707e9	0.847621
$623$	−5.24051e8	−0.0868292
$624$	0	0
$625$	6.07977e9	0.996110
$626$	2.12360e9	0.345988
$627$	0	0
$628$	−2.29161e9	−0.369216
$629$	1.08532e10	1.73893
$630$	0	0
$631$	1.16903e10	1.85236	0.926178	−	0.377086i	$-0.123074\pi$
$631$	1.16903e10	1.85236	0.926178	+	0.377086i	$0.123074\pi$
$632$	2.45812e9	0.387341
$633$	0	0
$634$	3.26267e9	0.508465
$635$	−1.66315e8	−0.0257764
$636$	0	0
$637$	1.14076e9	0.174867
$638$	4.07696e8	0.0621533
$639$	0	0
$640$	−2.11118e7	−0.00318344
$641$	4.35893e9	0.653697	0.326849	−	0.945077i	$-0.394013\pi$
$641$	4.35893e9	0.653697	0.326849	+	0.945077i	$0.394013\pi$
$642$	0	0
$643$	3.50189e9	0.519474	0.259737	−	0.965679i	$-0.416364\pi$
$643$	3.50189e9	0.519474	0.259737	+	0.965679i	$0.416364\pi$
$644$	7.56240e8	0.111573
$645$	0	0
$646$	−6.85356e9	−1.00024
$647$	4.71652e9	0.684632	0.342316	−	0.939585i	$-0.388789\pi$
$647$	4.71652e9	0.684632	0.342316	+	0.939585i	$0.388789\pi$
$648$	0	0
$649$	−6.06693e9	−0.871189
$650$	5.95200e9	0.850092
$651$	0	0
$652$	−5.38845e9	−0.761373
$653$	2.98375e9	0.419340	0.209670	−	0.977772i	$-0.432761\pi$
$653$	2.98375e9	0.419340	0.209670	+	0.977772i	$0.432761\pi$
$654$	0	0
$655$	−1.16465e8	−0.0161939
$656$	−5.05426e8	−0.0699027
$657$	0	0
$658$	−1.07318e10	−1.46853
$659$	5.13716e8	0.0699237	0.0349619	−	0.999389i	$-0.488869\pi$
$659$	5.13716e8	0.0699237	0.0349619	+	0.999389i	$0.488869\pi$
$660$	0	0
$661$	−2.64445e9	−0.356148	−0.178074	−	0.984017i	$-0.556987\pi$
$661$	−2.64445e9	−0.356148	−0.178074	+	0.984017i	$0.556987\pi$
$662$	2.18128e9	0.292219
$663$	0	0
$664$	3.98079e9	0.527692
$665$	4.01587e8	0.0529547
$666$	0	0
$667$	−3.00037e8	−0.0391502
$668$	7.40216e9	0.960817
$669$	0	0
$670$	−2.57402e8	−0.0330636
$671$	−4.88024e9	−0.623608
$672$	0	0
$673$	8.08792e9	1.02278	0.511392	−	0.859348i	$-0.329130\pi$
$673$	8.08792e9	1.02278	0.511392	+	0.859348i	$0.329130\pi$
$674$	−4.04514e9	−0.508890
$675$	0	0
$676$	1.80343e9	0.224535
$677$	−9.80601e9	−1.21460	−0.607298	−	0.794474i	$-0.707747\pi$
$677$	−9.80601e9	−1.21460	−0.607298	+	0.794474i	$0.707747\pi$
$678$	0	0
$679$	−1.01605e10	−1.24558
$680$	−1.07499e8	−0.0131106
$681$	0	0
$682$	−4.04879e9	−0.488743
$683$	−1.16162e10	−1.39505	−0.697526	−	0.716559i	$-0.745716\pi$
$683$	−1.16162e10	−1.39505	−0.697526	+	0.716559i	$0.745716\pi$
$684$	0	0
$685$	4.86562e8	0.0578391
$686$	5.46895e9	0.646799
$687$	0	0
$688$	3.04307e7	0.00356248
$689$	−9.65491e9	−1.12456
$690$	0	0
$691$	1.04349e9	0.120313	0.0601566	−	0.998189i	$-0.480840\pi$
$691$	1.04349e9	0.120313	0.0601566	+	0.998189i	$0.480840\pi$
$692$	1.89281e9	0.217138
$693$	0	0
$694$	−8.42428e9	−0.956698
$695$	−3.32392e8	−0.0375581
$696$	0	0
$697$	−2.57357e9	−0.287887
$698$	1.00880e9	0.112283
$699$	0	0
$700$	−4.84956e9	−0.534391
$701$	3.23469e9	0.354666	0.177333	−	0.984151i	$-0.443253\pi$
$701$	3.23469e9	0.354666	0.177333	+	0.984151i	$0.443253\pi$
$702$	0	0
$703$	2.13751e10	2.32041
$704$	5.41746e8	0.0585182
$705$	0	0
$706$	−2.67375e9	−0.285959
$707$	4.75056e9	0.505564
$708$	0	0
$709$	1.27054e10	1.33883	0.669416	−	0.742888i	$-0.266544\pi$
$709$	1.27054e10	1.33883	0.669416	+	0.742888i	$0.266544\pi$
$710$	−2.81789e8	−0.0295474
$711$	0	0
$712$	2.76279e8	0.0286858
$713$	2.97964e9	0.307858
$714$	0	0
$715$	1.98380e8	0.0202968
$716$	5.07354e9	0.516554
$717$	0	0
$718$	4.29683e9	0.433224
$719$	1.09711e10	1.10077	0.550387	−	0.834910i	$-0.314480\pi$
$719$	1.09711e10	1.10077	0.550387	+	0.834910i	$0.314480\pi$
$720$	0	0
$721$	6.79110e9	0.674787
$722$	−6.34689e9	−0.627596
$723$	0	0
$724$	6.16643e9	0.603877
$725$	1.92405e9	0.187514
$726$	0	0
$727$	−1.65278e10	−1.59531	−0.797654	−	0.603116i	$-0.793926\pi$
$727$	−1.65278e10	−1.59531	−0.797654	+	0.603116i	$0.793926\pi$
$728$	−4.74147e9	−0.455463
$729$	0	0
$730$	−2.00466e8	−0.0190727
$731$	1.54950e8	0.0146717
$732$	0	0
$733$	−1.79161e9	−0.168028	−0.0840138	−	0.996465i	$-0.526774\pi$
$733$	−1.79161e9	−0.168028	−0.0840138	+	0.996465i	$0.526774\pi$
$734$	2.29071e9	0.213813
$735$	0	0
$736$	−3.98688e8	−0.0368605
$737$	6.60513e9	0.607778
$738$	0	0
$739$	−1.29335e10	−1.17886	−0.589429	−	0.807820i	$-0.700647\pi$
$739$	−1.29335e10	−1.17886	−0.589429	+	0.807820i	$0.700647\pi$
$740$	3.35271e8	0.0304148
$741$	0	0
$742$	7.86662e9	0.706927
$743$	1.98134e10	1.77214	0.886069	−	0.463554i	$-0.153426\pi$
$743$	1.98134e10	1.77214	0.886069	+	0.463554i	$0.153426\pi$
$744$	0	0
$745$	6.08167e7	0.00538860
$746$	−9.33560e9	−0.823297
$747$	0	0
$748$	2.75851e9	0.241001
$749$	−1.22751e10	−1.06743
$750$	0	0
$751$	1.43776e10	1.23864	0.619322	−	0.785137i	$-0.287408\pi$
$751$	1.43776e10	1.23864	0.619322	+	0.785137i	$0.287408\pi$
$752$	5.65778e9	0.485158
$753$	0	0
$754$	1.88117e9	0.159819
$755$	−1.74431e8	−0.0147506
$756$	0	0
$757$	1.22101e10	1.02302	0.511508	−	0.859278i	$-0.329087\pi$
$757$	1.22101e10	1.02302	0.511508	+	0.859278i	$0.329087\pi$
$758$	5.62194e9	0.468861
$759$	0	0
$760$	−2.11716e8	−0.0174947
$761$	6.00939e8	0.0494293	0.0247146	−	0.999695i	$-0.492132\pi$
$761$	6.00939e8	0.0494293	0.0247146	+	0.999695i	$0.492132\pi$
$762$	0	0
$763$	−1.83433e10	−1.49500
$764$	8.67875e9	0.704094
$765$	0	0
$766$	−3.82373e9	−0.307388
$767$	−2.79937e10	−2.24015
$768$	0	0
$769$	1.27908e10	1.01427	0.507137	−	0.861866i	$-0.330704\pi$
$769$	1.27908e10	1.01427	0.507137	+	0.861866i	$0.330704\pi$
$770$	−1.61636e8	−0.0127591
$771$	0	0
$772$	−1.12471e10	−0.879795
$773$	5.32783e9	0.414880	0.207440	−	0.978248i	$-0.433487\pi$
$773$	5.32783e9	0.414880	0.207440	+	0.978248i	$0.433487\pi$
$774$	0	0
$775$	−1.91076e10	−1.47452
$776$	5.35660e9	0.411503
$777$	0	0
$778$	1.31268e10	0.999380
$779$	−5.06857e9	−0.384153
$780$	0	0
$781$	7.23090e9	0.543143
$782$	−2.03008e9	−0.151806
$783$	0	0
$784$	4.90014e8	0.0363164
$785$	−3.60459e8	−0.0265958
$786$	0	0
$787$	−1.10076e9	−0.0804971	−0.0402485	−	0.999190i	$-0.512815\pi$
$787$	−1.10076e9	−0.0804971	−0.0402485	+	0.999190i	$0.512815\pi$
$788$	7.84194e9	0.570929
$789$	0	0
$790$	3.86651e8	0.0279013
$791$	9.22297e9	0.662603
$792$	0	0
$793$	−2.25181e10	−1.60353
$794$	−7.53729e9	−0.534372
$795$	0	0
$796$	1.37567e10	0.966760
$797$	−1.41736e10	−0.991694	−0.495847	−	0.868410i	$-0.665142\pi$
$797$	−1.41736e10	−0.991694	−0.495847	+	0.868410i	$0.665142\pi$
$798$	0	0
$799$	2.88088e10	1.99807
$800$	2.55668e9	0.176547
$801$	0	0
$802$	1.19323e9	0.0816795
$803$	5.14412e9	0.350595
$804$	0	0
$805$	1.18953e8	0.00803694
$806$	−1.86817e10	−1.25674
$807$	0	0
$808$	−2.50448e9	−0.167024
$809$	−4.00127e9	−0.265692	−0.132846	−	0.991137i	$-0.542412\pi$
$809$	−4.00127e9	−0.265692	−0.132846	+	0.991137i	$0.542412\pi$
$810$	0	0
$811$	−2.05460e10	−1.35255	−0.676275	−	0.736649i	$-0.736407\pi$
$811$	−2.05460e10	−1.35255	−0.676275	+	0.736649i	$0.736407\pi$
$812$	−1.53273e9	−0.100466
$813$	0	0
$814$	−8.60331e9	−0.559088
$815$	−8.47579e8	−0.0548439
$816$	0	0
$817$	3.05169e8	0.0195778
$818$	9.21081e9	0.588385
$819$	0	0
$820$	−7.95013e7	−0.00503530
$821$	−1.60430e10	−1.01177	−0.505887	−	0.862600i	$-0.668835\pi$
$821$	−1.60430e10	−1.01177	−0.505887	+	0.862600i	$0.668835\pi$
$822$	0	0
$823$	2.08789e10	1.30560	0.652798	−	0.757532i	$-0.273595\pi$
$823$	2.08789e10	1.30560	0.652798	+	0.757532i	$0.273595\pi$
$824$	−3.58025e9	−0.222930
$825$	0	0
$826$	2.28087e10	1.40822
$827$	−1.50170e9	−0.0923241	−0.0461621	−	0.998934i	$-0.514699\pi$
$827$	−1.50170e9	−0.0923241	−0.0461621	+	0.998934i	$0.514699\pi$
$828$	0	0
$829$	2.73315e10	1.66618	0.833090	−	0.553137i	$-0.186570\pi$
$829$	2.73315e10	1.66618	0.833090	+	0.553137i	$0.186570\pi$
$830$	6.26160e8	0.0380112
$831$	0	0
$832$	2.49969e9	0.150472
$833$	2.49510e9	0.149565
$834$	0	0
$835$	1.16433e9	0.0692105
$836$	5.43279e9	0.321589
$837$	0	0
$838$	3.15969e9	0.185477
$839$	−2.62962e10	−1.53718	−0.768591	−	0.639740i	$-0.779042\pi$
$839$	−2.62962e10	−1.53718	−0.768591	+	0.639740i	$0.779042\pi$
$840$	0	0
$841$	−1.66418e10	−0.964747
$842$	2.65036e9	0.153008
$843$	0	0
$844$	1.47342e10	0.843586
$845$	2.83671e8	0.0161740
$846$	0	0
$847$	−1.47777e10	−0.835632
$848$	−4.14727e9	−0.233548
$849$	0	0
$850$	1.30183e10	0.727091
$851$	6.33146e9	0.352168
$852$	0	0
$853$	−1.09100e10	−0.601871	−0.300936	−	0.953644i	$-0.597299\pi$
$853$	−1.09100e10	−0.601871	−0.300936	+	0.953644i	$0.597299\pi$
$854$	1.83473e10	1.00802
$855$	0	0
$856$	6.47139e9	0.352646
$857$	−1.11091e10	−0.602900	−0.301450	−	0.953482i	$-0.597471\pi$
$857$	−1.11091e10	−0.602900	−0.301450	+	0.953482i	$0.597471\pi$
$858$	0	0
$859$	−1.92427e10	−1.03584	−0.517918	−	0.855430i	$-0.673293\pi$
$859$	−1.92427e10	−1.03584	−0.517918	+	0.855430i	$0.673293\pi$
$860$	4.78662e6	0.000256616 0
$861$	0	0
$862$	−4.86175e9	−0.258534
$863$	−1.41798e10	−0.750985	−0.375493	−	0.926825i	$-0.622526\pi$
$863$	−1.41798e10	−0.750985	−0.375493	+	0.926825i	$0.622526\pi$
$864$	0	0
$865$	2.97730e8	0.0156411
$866$	−1.36728e10	−0.715395
$867$	0	0
$868$	1.52215e10	0.790018
$869$	−9.92175e9	−0.512884
$870$	0	0
$871$	3.04770e10	1.56282
$872$	9.67054e9	0.493905
$873$	0	0
$874$	−3.99817e9	−0.202568
$875$	−1.52662e9	−0.0770376
$876$	0	0
$877$	2.27875e10	1.14077	0.570385	−	0.821377i	$-0.306794\pi$
$877$	2.27875e10	1.14077	0.570385	+	0.821377i	$0.306794\pi$
$878$	1.28107e10	0.638764
$879$	0	0
$880$	8.52141e7	0.00421524
$881$	3.16523e10	1.55952	0.779758	−	0.626081i	$-0.215342\pi$
$881$	3.16523e10	1.55952	0.779758	+	0.626081i	$0.215342\pi$
$882$	0	0
$883$	−1.28325e10	−0.627263	−0.313631	−	0.949545i	$-0.601546\pi$
$883$	−1.28325e10	−0.627263	−0.313631	+	0.949545i	$0.601546\pi$
$884$	1.27282e10	0.619701
$885$	0	0
$886$	−2.08761e10	−1.00840
$887$	−6.07730e9	−0.292400	−0.146200	−	0.989255i	$-0.546704\pi$
$887$	−6.07730e9	−0.292400	−0.146200	+	0.989255i	$0.546704\pi$
$888$	0	0
$889$	−1.60447e10	−0.765904
$890$	4.34574e7	0.00206633
$891$	0	0
$892$	1.38526e10	0.653515
$893$	5.67379e10	2.66620
$894$	0	0
$895$	7.98046e8	0.0372089
$896$	−2.03670e9	−0.0945906
$897$	0	0
$898$	2.62566e9	0.120996
$899$	−6.03908e9	−0.277212
$900$	0	0
$901$	−2.11174e10	−0.961843
$902$	2.04006e9	0.0925593
$903$	0	0
$904$	−4.86233e9	−0.218905
$905$	9.69952e8	0.0434991
$906$	0	0
$907$	−6.33378e9	−0.281863	−0.140931	−	0.990019i	$-0.545010\pi$
$907$	−6.33378e9	−0.281863	−0.140931	+	0.990019i	$0.545010\pi$
$908$	1.35490e10	0.600630
$909$	0	0
$910$	−7.45811e8	−0.0328083
$911$	4.31202e10	1.88958	0.944792	−	0.327670i	$-0.106264\pi$
$911$	4.31202e10	1.88958	0.944792	+	0.327670i	$0.106264\pi$
$912$	0	0
$913$	−1.60677e10	−0.698726
$914$	−1.08691e10	−0.470851
$915$	0	0
$916$	−9.08150e8	−0.0390412
$917$	−1.12356e10	−0.481174
$918$	0	0
$919$	2.66851e10	1.13414	0.567068	−	0.823671i	$-0.308078\pi$
$919$	2.66851e10	1.13414	0.567068	+	0.823671i	$0.308078\pi$
$920$	−6.27119e7	−0.00265517
$921$	0	0
$922$	1.19847e10	0.503579
$923$	3.33644e10	1.39662
$924$	0	0
$925$	−4.06019e10	−1.68675
$926$	3.05396e10	1.26394
$927$	0	0
$928$	8.08055e8	0.0331912
$929$	−1.03414e10	−0.423178	−0.211589	−	0.977359i	$-0.567864\pi$
$929$	−1.03414e10	−0.423178	−0.211589	+	0.977359i	$0.567864\pi$
$930$	0	0
$931$	4.91401e9	0.199578
$932$	−1.90354e10	−0.770204
$933$	0	0
$934$	1.68981e10	0.678615
$935$	4.33901e8	0.0173600
$936$	0	0
$937$	2.29344e10	0.910748	0.455374	−	0.890300i	$-0.349506\pi$
$937$	2.29344e10	0.910748	0.455374	+	0.890300i	$0.349506\pi$
$938$	−2.48320e10	−0.982431
$939$	0	0
$940$	8.89943e8	0.0349474
$941$	−5.44364e9	−0.212973	−0.106487	−	0.994314i	$-0.533960\pi$
$941$	−5.44364e9	−0.212973	−0.106487	+	0.994314i	$0.533960\pi$
$942$	0	0
$943$	−1.50135e9	−0.0583029
$944$	−1.20247e10	−0.465234
$945$	0	0
$946$	−1.22828e8	−0.00471714
$947$	8.75674e9	0.335056	0.167528	−	0.985867i	$-0.446422\pi$
$947$	8.75674e9	0.335056	0.167528	+	0.985867i	$0.446422\pi$
$948$	0	0
$949$	2.37357e10	0.901509
$950$	2.56392e10	0.970222
$951$	0	0
$952$	−1.03706e10	−0.389561
$953$	3.41231e10	1.27710	0.638549	−	0.769582i	$-0.279535\pi$
$953$	3.41231e10	1.27710	0.638549	+	0.769582i	$0.279535\pi$
$954$	0	0
$955$	1.36513e9	0.0507180
$956$	−1.72347e10	−0.637972
$957$	0	0
$958$	2.92845e10	1.07612
$959$	4.69395e10	1.71859
$960$	0	0
$961$	3.24610e10	1.17986
$962$	−3.96969e10	−1.43762
$963$	0	0
$964$	1.96067e9	0.0704912
$965$	−1.76912e9	−0.0633742
$966$	0	0
$967$	−3.56029e10	−1.26617	−0.633087	−	0.774081i	$-0.718212\pi$
$967$	−3.56029e10	−1.26617	−0.633087	+	0.774081i	$0.718212\pi$
$968$	7.79077e9	0.276069
$969$	0	0
$970$	8.42569e8	0.0296418
$971$	−4.01429e10	−1.40715	−0.703577	−	0.710619i	$-0.748415\pi$
$971$	−4.01429e10	−1.40715	−0.703577	+	0.710619i	$0.748415\pi$
$972$	0	0
$973$	−3.20664e10	−1.11598
$974$	1.50693e10	0.522561
$975$	0	0
$976$	−9.67265e9	−0.333021
$977$	−3.55282e10	−1.21883	−0.609413	−	0.792853i	$-0.708595\pi$
$977$	−3.55282e10	−1.21883	−0.609413	+	0.792853i	$0.708595\pi$
$978$	0	0
$979$	−1.11515e9	−0.0379834
$980$	7.70770e7	0.00261597
$981$	0	0
$982$	−1.24828e9	−0.0420652
$983$	2.12262e10	0.712746	0.356373	−	0.934344i	$-0.384013\pi$
$983$	2.12262e10	0.712746	0.356373	+	0.934344i	$0.384013\pi$
$984$	0	0
$985$	1.23350e9	0.0411257
$986$	4.11453e9	0.136694
$987$	0	0
$988$	2.50677e10	0.826922
$989$	9.03933e7	0.00297132
$990$	0	0
$991$	−2.77326e10	−0.905175	−0.452588	−	0.891720i	$-0.649499\pi$
$991$	−2.77326e10	−0.905175	−0.452588	+	0.891720i	$0.649499\pi$
$992$	−8.02472e9	−0.260999
$993$	0	0
$994$	−2.71846e10	−0.877952
$995$	2.16387e9	0.0696386
$996$	0	0
$997$	−4.52372e10	−1.44565	−0.722823	−	0.691033i	$-0.757156\pi$
$997$	−4.52372e10	−1.44565	−0.722823	+	0.691033i	$0.757156\pi$
$998$	2.07429e10	0.660561
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.8.a.i.1.3		4
3.2	odd	2		138.8.a.h.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.h.1.2	✓	4	3.2	odd	2
414.8.a.i.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} +10.0669 q^{5} +971.172 q^{7} -512.000 q^{8} +O(q^{10})\)	\(q-8.00000 q^{2} +64.0000 q^{4} +10.0669 q^{5} +971.172 q^{7} -512.000 q^{8} -80.5353 q^{10} +2066.60 q^{11} +9535.57 q^{13} -7769.38 q^{14} +4096.00 q^{16} +20856.4 q^{17} +41075.9 q^{19} +644.282 q^{20} -16532.8 q^{22} +12167.0 q^{23} -78023.7 q^{25} -76284.5 q^{26} +62155.0 q^{28} -24659.9 q^{29} +244895. q^{31} -32768.0 q^{32} -166851. q^{34} +9776.71 q^{35} +520379. q^{37} -328607. q^{38} -5154.26 q^{40} -123395. q^{41} +7429.38 q^{43} +132262. q^{44} -97336.0 q^{46} +1.38129e6 q^{47} +119632. q^{49} +624189. q^{50} +610276. q^{52} -1.01252e6 q^{53} +20804.2 q^{55} -497240. q^{56} +197279. q^{58} -2.93571e6 q^{59} -2.36149e6 q^{61} -1.95916e6 q^{62} +262144. q^{64} +95993.7 q^{65} +3.19614e6 q^{67} +1.33481e6 q^{68} -78213.6 q^{70} +3.49894e6 q^{71} +2.48917e6 q^{73} -4.16304e6 q^{74} +2.62886e6 q^{76} +2.00702e6 q^{77} -4.80101e6 q^{79} +41234.1 q^{80} +987160. q^{82} -7.77498e6 q^{83} +209959. q^{85} -59435.0 q^{86} -1.05810e6 q^{88} -539607. q^{89} +9.26068e6 q^{91} +778688. q^{92} -1.10503e7 q^{94} +413508. q^{95} -1.04621e7 q^{97} -957058. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8}+O(q^{10}) \) 4 * q - 32 * q^2 + 256 * q^4 - 270 * q^5 + 2022 * q^7 - 2048 * q^8	\( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8} + 2160 q^{10} - 4120 q^{11} + 8036 q^{13} - 16176 q^{14} + 16384 q^{16} - 37182 q^{17} + 5702 q^{19} - 17280 q^{20} + 32960 q^{22} + 48668 q^{23} + 121480 q^{25} - 64288 q^{26} + 129408 q^{28} - 217716 q^{29} + 222852 q^{31} - 131072 q^{32} + 297456 q^{34} - 68440 q^{35} + 486428 q^{37} - 45616 q^{38} + 138240 q^{40} - 338336 q^{41} + 730974 q^{43} - 263680 q^{44} - 389344 q^{46} - 338248 q^{47} - 310552 q^{49} - 971840 q^{50} + 514304 q^{52} + 375502 q^{53} + 424840 q^{55} - 1035264 q^{56} + 1741728 q^{58} - 71392 q^{59} + 2101164 q^{61} - 1782816 q^{62} + 1048576 q^{64} - 1578780 q^{65} + 4337162 q^{67} - 2379648 q^{68} + 547520 q^{70} - 2288016 q^{71} - 1107328 q^{73} - 3891424 q^{74} + 364928 q^{76} - 5826200 q^{77} + 60610 q^{79} - 1105920 q^{80} + 2706688 q^{82} - 1485464 q^{83} - 8843820 q^{85} - 5847792 q^{86} + 2109440 q^{88} - 1485090 q^{89} - 2898412 q^{91} + 3114752 q^{92} + 2705984 q^{94} - 8545200 q^{95} + 1935444 q^{97} + 2484416 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 270 * q^5 + 2022 * q^7 - 2048 * q^8 + 2160 * q^10 - 4120 * q^11 + 8036 * q^13 - 16176 * q^14 + 16384 * q^16 - 37182 * q^17 + 5702 * q^19 - 17280 * q^20 + 32960 * q^22 + 48668 * q^23 + 121480 * q^25 - 64288 * q^26 + 129408 * q^28 - 217716 * q^29 + 222852 * q^31 - 131072 * q^32 + 297456 * q^34 - 68440 * q^35 + 486428 * q^37 - 45616 * q^38 + 138240 * q^40 - 338336 * q^41 + 730974 * q^43 - 263680 * q^44 - 389344 * q^46 - 338248 * q^47 - 310552 * q^49 - 971840 * q^50 + 514304 * q^52 + 375502 * q^53 + 424840 * q^55 - 1035264 * q^56 + 1741728 * q^58 - 71392 * q^59 + 2101164 * q^61 - 1782816 * q^62 + 1048576 * q^64 - 1578780 * q^65 + 4337162 * q^67 - 2379648 * q^68 + 547520 * q^70 - 2288016 * q^71 - 1107328 * q^73 - 3891424 * q^74 + 364928 * q^76 - 5826200 * q^77 + 60610 * q^79 - 1105920 * q^80 + 2706688 * q^82 - 1485464 * q^83 - 8843820 * q^85 - 5847792 * q^86 + 2109440 * q^88 - 1485090 * q^89 - 2898412 * q^91 + 3114752 * q^92 + 2705984 * q^94 - 8545200 * q^95 + 1935444 * q^97 + 2484416 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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