LMFDB - Embedded newform 207.8.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,8,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.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:	$64.6637002752$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 $ x^8 - 757x^6 - 1170x^5 + 170343x^4 + 424132x^3 - 9973075x^2 - 5161010x + 130545120
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-11.6014$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.6014 q^{2} +85.2018 q^{4} -493.797 q^{5} -368.336 q^{7} +624.915 q^{8} +O(q^{10})$	\(q-14.6014 q^{2} +85.2018 q^{4} -493.797 q^{5} -368.336 q^{7} +624.915 q^{8} +7210.15 q^{10} -7537.54 q^{11} -417.322 q^{13} +5378.24 q^{14} -20030.5 q^{16} +24880.5 q^{17} +20861.4 q^{19} -42072.4 q^{20} +110059. q^{22} -12167.0 q^{23} +165711. q^{25} +6093.50 q^{26} -31382.9 q^{28} +5058.15 q^{29} -228761. q^{31} +212485. q^{32} -363291. q^{34} +181883. q^{35} +588441. q^{37} -304606. q^{38} -308581. q^{40} -191846. q^{41} +208779. q^{43} -642212. q^{44} +177656. q^{46} +1.36255e6 q^{47} -687871. q^{49} -2.41962e6 q^{50} -35556.6 q^{52} +1.13105e6 q^{53} +3.72202e6 q^{55} -230179. q^{56} -73856.3 q^{58} +1.59555e6 q^{59} -1.18637e6 q^{61} +3.34024e6 q^{62} -538678. q^{64} +206072. q^{65} +2.64240e6 q^{67} +2.11986e6 q^{68} -2.65576e6 q^{70} +2.80200e6 q^{71} -5.16484e6 q^{73} -8.59208e6 q^{74} +1.77743e6 q^{76} +2.77635e6 q^{77} -3.28919e6 q^{79} +9.89100e6 q^{80} +2.80123e6 q^{82} -590082. q^{83} -1.22859e7 q^{85} -3.04847e6 q^{86} -4.71032e6 q^{88} -2.78595e6 q^{89} +153715. q^{91} -1.03665e6 q^{92} -1.98952e7 q^{94} -1.03013e7 q^{95} -289056. q^{97} +1.00439e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8}+O(q^{10}) $ 8 * q - 24 * q^2 + 562 * q^4 - 378 * q^5 + 126 * q^7 - 4188 * q^8	\( 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8} + 11720 q^{10} - 6932 q^{11} + 12404 q^{13} - 30222 q^{14} + 27058 q^{16} - 24434 q^{17} - 14682 q^{19} + 3760 q^{20} + 36294 q^{22} - 97336 q^{23} + 144644 q^{25} - 325840 q^{26} - 21566 q^{28} - 255356 q^{29} + 450764 q^{31} - 647588 q^{32} + 191822 q^{34} - 1022616 q^{35} + 206240 q^{37} - 737372 q^{38} + 590028 q^{40} - 1053344 q^{41} + 1587806 q^{43} - 589366 q^{44} + 292008 q^{46} - 443336 q^{47} + 1944828 q^{49} + 1556112 q^{50} - 614236 q^{52} + 375530 q^{53} + 407792 q^{55} + 1316922 q^{56} - 1413384 q^{58} - 624008 q^{59} - 2005568 q^{61} + 3908272 q^{62} - 5082310 q^{64} - 646124 q^{65} - 2712286 q^{67} + 2289698 q^{68} - 16499468 q^{70} + 6287176 q^{71} - 10358312 q^{73} + 2000150 q^{74} - 25107464 q^{76} + 2156840 q^{77} - 8800574 q^{79} - 2384344 q^{80} - 31799800 q^{82} - 384948 q^{83} - 17826684 q^{85} + 11563928 q^{86} - 25202782 q^{88} + 3445530 q^{89} - 16316740 q^{91} - 6837854 q^{92} - 24237616 q^{94} - 26164288 q^{95} - 28043764 q^{97} + 9998012 q^{98}+O(q^{100}) \) 8 * q - 24 * q^2 + 562 * q^4 - 378 * q^5 + 126 * q^7 - 4188 * q^8 + 11720 * q^10 - 6932 * q^11 + 12404 * q^13 - 30222 * q^14 + 27058 * q^16 - 24434 * q^17 - 14682 * q^19 + 3760 * q^20 + 36294 * q^22 - 97336 * q^23 + 144644 * q^25 - 325840 * q^26 - 21566 * q^28 - 255356 * q^29 + 450764 * q^31 - 647588 * q^32 + 191822 * q^34 - 1022616 * q^35 + 206240 * q^37 - 737372 * q^38 + 590028 * q^40 - 1053344 * q^41 + 1587806 * q^43 - 589366 * q^44 + 292008 * q^46 - 443336 * q^47 + 1944828 * q^49 + 1556112 * q^50 - 614236 * q^52 + 375530 * q^53 + 407792 * q^55 + 1316922 * q^56 - 1413384 * q^58 - 624008 * q^59 - 2005568 * q^61 + 3908272 * q^62 - 5082310 * q^64 - 646124 * q^65 - 2712286 * q^67 + 2289698 * q^68 - 16499468 * q^70 + 6287176 * q^71 - 10358312 * q^73 + 2000150 * q^74 - 25107464 * q^76 + 2156840 * q^77 - 8800574 * q^79 - 2384344 * q^80 - 31799800 * q^82 - 384948 * q^83 - 17826684 * q^85 + 11563928 * q^86 - 25202782 * q^88 + 3445530 * q^89 - 16316740 * q^91 - 6837854 * q^92 - 24237616 * q^94 - 26164288 * q^95 - 28043764 * q^97 + 9998012 * 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 $

$2$	−14.6014	−1.29060	−0.645298	−	0.763931i	$-0.723267\pi$
$2$	−14.6014	−1.29060	−0.645298	+	0.763931i	$0.723267\pi$
$3$	0	0
$3$	0	0
$4$	85.2018	0.665639
$5$	−493.797	−1.76666	−0.883332	−	0.468749i	$-0.844705\pi$
$5$	−493.797	−1.76666	−0.883332	+	0.468749i	$0.844705\pi$
$6$	0	0
$7$	−368.336	−0.405883	−0.202942	−	0.979191i	$-0.565050\pi$
$7$	−368.336	−0.405883	−0.202942	+	0.979191i	$0.565050\pi$
$8$	624.915	0.431525
$9$	0	0
$10$	7210.15	2.28005
$11$	−7537.54	−1.70748	−0.853739	−	0.520701i	$-0.825670\pi$
$11$	−7537.54	−1.70748	−0.853739	+	0.520701i	$0.825670\pi$
$12$	0	0
$13$	−417.322	−0.0526829	−0.0263414	−	0.999653i	$-0.508386\pi$
$13$	−417.322	−0.0526829	−0.0263414	+	0.999653i	$0.508386\pi$
$14$	5378.24	0.523832
$15$	0	0
$16$	−20030.5	−1.22256
$17$	24880.5	1.22825	0.614127	−	0.789207i	$-0.289508\pi$
$17$	24880.5	1.22825	0.614127	+	0.789207i	$0.289508\pi$
$18$	0	0
$19$	20861.4	0.697759	0.348879	−	0.937168i	$-0.386562\pi$
$19$	20861.4	0.697759	0.348879	+	0.937168i	$0.386562\pi$
$20$	−42072.4	−1.17596
$21$	0	0
$22$	110059.	2.20366
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	0	0
$25$	165711.	2.12110
$26$	6093.50	0.0679923
$27$	0	0
$28$	−31382.9	−0.270172
$29$	5058.15	0.0385122	0.0192561	−	0.999815i	$-0.493870\pi$
$29$	5058.15	0.0385122	0.0192561	+	0.999815i	$0.493870\pi$
$30$	0	0
$31$	−228761.	−1.37917	−0.689583	−	0.724207i	$-0.742206\pi$
$31$	−228761.	−1.37917	−0.689583	+	0.724207i	$0.742206\pi$
$32$	212485.	1.14631
$33$	0	0
$34$	−363291.	−1.58518
$35$	181883.	0.717059
$36$	0	0
$37$	588441.	1.90984	0.954920	−	0.296864i	$-0.0959409\pi$
$37$	588441.	1.90984	0.954920	+	0.296864i	$0.0959409\pi$
$38$	−304606.	−0.900525
$39$	0	0
$40$	−308581.	−0.762359
$41$	−191846.	−0.434720	−0.217360	−	0.976092i	$-0.569745\pi$
$41$	−191846.	−0.434720	−0.217360	+	0.976092i	$0.569745\pi$
$42$	0	0
$43$	208779.	0.400449	0.200225	−	0.979750i	$-0.435833\pi$
$43$	208779.	0.400449	0.200225	+	0.979750i	$0.435833\pi$
$44$	−642212.	−1.13656
$45$	0	0
$46$	177656.	0.269108
$47$	1.36255e6	1.91431	0.957153	−	0.289584i	$-0.0935169\pi$
$47$	1.36255e6	1.91431	0.957153	+	0.289584i	$0.0935169\pi$
$48$	0	0
$49$	−687871.	−0.835259
$50$	−2.41962e6	−2.73748
$51$	0	0
$52$	−35556.6	−0.0350678
$53$	1.13105e6	1.04355	0.521777	−	0.853082i	$-0.325269\pi$
$53$	1.13105e6	1.04355	0.521777	+	0.853082i	$0.325269\pi$
$54$	0	0
$55$	3.72202e6	3.01654
$56$	−230179.	−0.175149
$57$	0	0
$58$	−73856.3	−0.0497038
$59$	1.59555e6	1.01141	0.505707	−	0.862705i	$-0.331232\pi$
$59$	1.59555e6	1.01141	0.505707	+	0.862705i	$0.331232\pi$
$60$	0	0
$61$	−1.18637e6	−0.669217	−0.334608	−	0.942357i	$-0.608604\pi$
$61$	−1.18637e6	−0.669217	−0.334608	+	0.942357i	$0.608604\pi$
$62$	3.34024e6	1.77995
$63$	0	0
$64$	−538678.	−0.256862
$65$	206072.	0.0930729
$66$	0	0
$67$	2.64240e6	1.07334	0.536669	−	0.843793i	$-0.319682\pi$
$67$	2.64240e6	1.07334	0.536669	+	0.843793i	$0.319682\pi$
$68$	2.11986e6	0.817574
$69$	0	0
$70$	−2.65576e6	−0.925434
$71$	2.80200e6	0.929104	0.464552	−	0.885546i	$-0.346215\pi$
$71$	2.80200e6	0.929104	0.464552	+	0.885546i	$0.346215\pi$
$72$	0	0
$73$	−5.16484e6	−1.55391	−0.776956	−	0.629554i	$-0.783237\pi$
$73$	−5.16484e6	−1.55391	−0.776956	+	0.629554i	$0.783237\pi$
$74$	−8.59208e6	−2.46483
$75$	0	0
$76$	1.77743e6	0.464456
$77$	2.77635e6	0.693037
$78$	0	0
$79$	−3.28919e6	−0.750576	−0.375288	−	0.926908i	$-0.622456\pi$
$79$	−3.28919e6	−0.750576	−0.375288	+	0.926908i	$0.622456\pi$
$80$	9.89100e6	2.15986
$81$	0	0
$82$	2.80123e6	0.561048
$83$	−590082.	−0.113276	−0.0566382	−	0.998395i	$-0.518038\pi$
$83$	−590082.	−0.113276	−0.0566382	+	0.998395i	$0.518038\pi$
$84$	0	0
$85$	−1.22859e7	−2.16991
$86$	−3.04847e6	−0.516818
$87$	0	0
$88$	−4.71032e6	−0.736819
$89$	−2.78595e6	−0.418898	−0.209449	−	0.977820i	$-0.567167\pi$
$89$	−2.78595e6	−0.418898	−0.209449	+	0.977820i	$0.567167\pi$
$90$	0	0
$91$	153715.	0.0213831
$92$	−1.03665e6	−0.138795
$93$	0	0
$94$	−1.98952e7	−2.47060
$95$	−1.03013e7	−1.23270
$96$	0	0
$97$	−289056.	−0.0321574	−0.0160787	−	0.999871i	$-0.505118\pi$
$97$	−289056.	−0.0321574	−0.0160787	+	0.999871i	$0.505118\pi$
$98$	1.00439e7	1.07798
$99$	0	0
$100$	1.41189e7	1.41189
$101$	−5.19146e6	−0.501377	−0.250689	−	0.968068i	$-0.580657\pi$
$101$	−5.19146e6	−0.501377	−0.250689	+	0.968068i	$0.580657\pi$
$102$	0	0
$103$	−6.83585e6	−0.616399	−0.308200	−	0.951322i	$-0.599726\pi$
$103$	−6.83585e6	−0.616399	−0.308200	+	0.951322i	$0.599726\pi$
$104$	−260791.	−0.0227340
$105$	0	0
$106$	−1.65149e7	−1.34681
$107$	−5.98559e6	−0.472350	−0.236175	−	0.971711i	$-0.575894\pi$
$107$	−5.98559e6	−0.472350	−0.236175	+	0.971711i	$0.575894\pi$
$108$	0	0
$109$	−1.00111e7	−0.740436	−0.370218	−	0.928945i	$-0.620717\pi$
$109$	−1.00111e7	−0.740436	−0.370218	+	0.928945i	$0.620717\pi$
$110$	−5.43468e7	−3.89313
$111$	0	0
$112$	7.37795e6	0.496218
$113$	−2.09806e6	−0.136787	−0.0683933	−	0.997658i	$-0.521787\pi$
$113$	−2.09806e6	−0.136787	−0.0683933	+	0.997658i	$0.521787\pi$
$114$	0	0
$115$	6.00803e6	0.368375
$116$	430964.	0.0256353
$117$	0	0
$118$	−2.32973e7	−1.30533
$119$	−9.16439e6	−0.498528
$120$	0	0
$121$	3.73273e7	1.91548
$122$	1.73227e7	0.863689
$123$	0	0
$124$	−1.94909e7	−0.918027
$125$	−4.32496e7	−1.98060
$126$	0	0
$127$	−9.76439e6	−0.422992	−0.211496	−	0.977379i	$-0.567834\pi$
$127$	−9.76439e6	−0.422992	−0.211496	+	0.977379i	$0.567834\pi$
$128$	−1.93326e7	−0.814807
$129$	0	0
$130$	−3.00895e6	−0.120120
$131$	−4.72387e7	−1.83590	−0.917948	−	0.396700i	$-0.870155\pi$
$131$	−4.72387e7	−1.83590	−0.917948	+	0.396700i	$0.870155\pi$
$132$	0	0
$133$	−7.68400e6	−0.283209
$134$	−3.85828e7	−1.38525
$135$	0	0
$136$	1.55482e7	0.530022
$137$	2.85915e7	0.949980	0.474990	−	0.879991i	$-0.342452\pi$
$137$	2.85915e7	0.949980	0.474990	+	0.879991i	$0.342452\pi$
$138$	0	0
$139$	1.97562e7	0.623951	0.311976	−	0.950090i	$-0.399009\pi$
$139$	1.97562e7	0.623951	0.311976	+	0.950090i	$0.399009\pi$
$140$	1.54968e7	0.477303
$141$	0	0
$142$	−4.09132e7	−1.19910
$143$	3.14558e6	0.0899548
$144$	0	0
$145$	−2.49770e6	−0.0680382
$146$	7.54140e7	2.00547
$147$	0	0
$148$	5.01362e7	1.27126
$149$	−2.13465e7	−0.528658	−0.264329	−	0.964433i	$-0.585150\pi$
$149$	−2.13465e7	−0.528658	−0.264329	+	0.964433i	$0.585150\pi$
$150$	0	0
$151$	6.56691e7	1.55218	0.776090	−	0.630622i	$-0.217200\pi$
$151$	6.56691e7	1.55218	0.776090	+	0.630622i	$0.217200\pi$
$152$	1.30366e7	0.301100
$153$	0	0
$154$	−4.05386e7	−0.894431
$155$	1.12962e8	2.43652
$156$	0	0
$157$	3.83258e7	0.790392	0.395196	−	0.918597i	$-0.370677\pi$
$157$	3.83258e7	0.790392	0.395196	+	0.918597i	$0.370677\pi$
$158$	4.80269e7	0.968691
$159$	0	0
$160$	−1.04924e8	−2.02515
$161$	4.48155e6	0.0846325
$162$	0	0
$163$	1.29604e7	0.234403	0.117201	−	0.993108i	$-0.462608\pi$
$163$	1.29604e7	0.234403	0.117201	+	0.993108i	$0.462608\pi$
$164$	−1.63456e7	−0.289366
$165$	0	0
$166$	8.61605e6	0.146194
$167$	−9.60453e7	−1.59576	−0.797881	−	0.602815i	$-0.794046\pi$
$167$	−9.60453e7	−1.59576	−0.797881	+	0.602815i	$0.794046\pi$
$168$	0	0
$169$	−6.25744e7	−0.997225
$170$	1.79392e8	2.80048
$171$	0	0
$172$	1.77884e7	0.266555
$173$	−1.54094e7	−0.226268	−0.113134	−	0.993580i	$-0.536089\pi$
$173$	−1.54094e7	−0.226268	−0.113134	+	0.993580i	$0.536089\pi$
$174$	0	0
$175$	−6.10373e7	−0.860918
$176$	1.50980e8	2.08750
$177$	0	0
$178$	4.06789e7	0.540629
$179$	−8.38575e7	−1.09284	−0.546419	−	0.837512i	$-0.684010\pi$
$179$	−8.38575e7	−1.09284	−0.546419	+	0.837512i	$0.684010\pi$
$180$	0	0
$181$	−1.50898e8	−1.89151	−0.945754	−	0.324883i	$-0.894675\pi$
$181$	−1.50898e8	−1.89151	−0.945754	+	0.324883i	$0.894675\pi$
$182$	−2.24445e6	−0.0275969
$183$	0	0
$184$	−7.60334e6	−0.0899792
$185$	−2.90571e8	−3.37404
$186$	0	0
$187$	−1.87538e8	−2.09722
$188$	1.16092e8	1.27424
$189$	0	0
$190$	1.50414e8	1.59092
$191$	−2.94854e7	−0.306189	−0.153095	−	0.988212i	$-0.548924\pi$
$191$	−2.94854e7	−0.306189	−0.153095	+	0.988212i	$0.548924\pi$
$192$	0	0
$193$	1.42403e8	1.42583	0.712917	−	0.701248i	$-0.247373\pi$
$193$	1.42403e8	1.42583	0.712917	+	0.701248i	$0.247373\pi$
$194$	4.22063e6	0.0415022
$195$	0	0
$196$	−5.86079e7	−0.555981
$197$	6.70489e7	0.624827	0.312413	−	0.949946i	$-0.398863\pi$
$197$	6.70489e7	0.624827	0.312413	+	0.949946i	$0.398863\pi$
$198$	0	0
$199$	1.79632e8	1.61584	0.807918	−	0.589295i	$-0.200594\pi$
$199$	1.79632e8	1.61584	0.807918	+	0.589295i	$0.200594\pi$
$200$	1.03555e8	0.915307
$201$	0	0
$202$	7.58028e7	0.647076
$203$	−1.86310e6	−0.0156315
$204$	0	0
$205$	9.47331e7	0.768003
$206$	9.98132e7	0.795523
$207$	0	0
$208$	8.35916e6	0.0644081
$209$	−1.57243e8	−1.19141
$210$	0	0
$211$	1.75301e8	1.28468	0.642342	−	0.766418i	$-0.277963\pi$
$211$	1.75301e8	1.28468	0.642342	+	0.766418i	$0.277963\pi$
$212$	9.63673e7	0.694631
$213$	0	0
$214$	8.73982e7	0.609613
$215$	−1.03095e8	−0.707459
$216$	0	0
$217$	8.42610e7	0.559780
$218$	1.46176e8	0.955605
$219$	0	0
$220$	3.17122e8	2.00793
$221$	−1.03832e7	−0.0647079
$222$	0	0
$223$	−2.10447e7	−0.127080	−0.0635399	−	0.997979i	$-0.520239\pi$
$223$	−2.10447e7	−0.127080	−0.0635399	+	0.997979i	$0.520239\pi$
$224$	−7.82658e7	−0.465269
$225$	0	0
$226$	3.06347e7	0.176536
$227$	3.24371e8	1.84057	0.920284	−	0.391251i	$-0.127957\pi$
$227$	3.24371e8	1.84057	0.920284	+	0.391251i	$0.127957\pi$
$228$	0	0
$229$	−4.97183e6	−0.0273585	−0.0136792	−	0.999906i	$-0.504354\pi$
$229$	−4.97183e6	−0.0273585	−0.0136792	+	0.999906i	$0.504354\pi$
$230$	−8.77259e7	−0.475423
$231$	0	0
$232$	3.16091e6	0.0166190
$233$	5.34348e7	0.276744	0.138372	−	0.990380i	$-0.455813\pi$
$233$	5.34348e7	0.276744	0.138372	+	0.990380i	$0.455813\pi$
$234$	0	0
$235$	−6.72826e8	−3.38193
$236$	1.35944e8	0.673237
$237$	0	0
$238$	1.33813e8	0.643398
$239$	2.65448e8	1.25773	0.628864	−	0.777515i	$-0.283520\pi$
$239$	2.65448e8	1.25773	0.628864	+	0.777515i	$0.283520\pi$
$240$	0	0
$241$	−1.42389e8	−0.655263	−0.327631	−	0.944806i	$-0.606250\pi$
$241$	−1.42389e8	−0.655263	−0.327631	+	0.944806i	$0.606250\pi$
$242$	−5.45032e8	−2.47211
$243$	0	0
$244$	−1.01081e8	−0.445457
$245$	3.39669e8	1.47562
$246$	0	0
$247$	−8.70591e6	−0.0367599
$248$	−1.42956e8	−0.595144
$249$	0	0
$250$	6.31507e8	2.55616
$251$	−2.71119e8	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$251$	−2.71119e8	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$252$	0	0
$253$	9.17092e7	0.356034
$254$	1.42574e8	0.545912
$255$	0	0
$256$	3.51234e8	1.30845
$257$	3.70271e8	1.36067	0.680336	−	0.732900i	$-0.261834\pi$
$257$	3.70271e8	1.36067	0.680336	+	0.732900i	$0.261834\pi$
$258$	0	0
$259$	−2.16744e8	−0.775172
$260$	1.75577e7	0.0619529
$261$	0	0
$262$	6.89752e8	2.36940
$263$	−2.67912e7	−0.0908128	−0.0454064	−	0.998969i	$-0.514458\pi$
$263$	−2.67912e7	−0.0908128	−0.0454064	+	0.998969i	$0.514458\pi$
$264$	0	0
$265$	−5.58508e8	−1.84361
$266$	1.12197e8	0.365508
$267$	0	0
$268$	2.25137e8	0.714456
$269$	4.19430e7	0.131379	0.0656896	−	0.997840i	$-0.479075\pi$
$269$	4.19430e7	0.131379	0.0656896	+	0.997840i	$0.479075\pi$
$270$	0	0
$271$	3.59386e8	1.09690	0.548452	−	0.836182i	$-0.315217\pi$
$271$	3.59386e8	1.09690	0.548452	+	0.836182i	$0.315217\pi$
$272$	−4.98369e8	−1.50162
$273$	0	0
$274$	−4.17476e8	−1.22604
$275$	−1.24905e9	−3.62173
$276$	0	0
$277$	−5.22440e7	−0.147692	−0.0738460	−	0.997270i	$-0.523527\pi$
$277$	−5.22440e7	−0.147692	−0.0738460	+	0.997270i	$0.523527\pi$
$278$	−2.88468e8	−0.805269
$279$	0	0
$280$	1.13662e8	0.309429
$281$	−5.82346e8	−1.56570	−0.782850	−	0.622210i	$-0.786235\pi$
$281$	−5.82346e8	−1.56570	−0.782850	+	0.622210i	$0.786235\pi$
$282$	0	0
$283$	3.59974e8	0.944102	0.472051	−	0.881571i	$-0.343514\pi$
$283$	3.59974e8	0.944102	0.472051	+	0.881571i	$0.343514\pi$
$284$	2.38736e8	0.618448
$285$	0	0
$286$	−4.59299e7	−0.116095
$287$	7.06638e7	0.176445
$288$	0	0
$289$	2.08701e8	0.508607
$290$	3.64700e7	0.0878098
$291$	0	0
$292$	−4.40053e8	−1.03435
$293$	3.37944e8	0.784888	0.392444	−	0.919776i	$-0.371630\pi$
$293$	3.37944e8	0.784888	0.392444	+	0.919776i	$0.371630\pi$
$294$	0	0
$295$	−7.87879e8	−1.78683
$296$	3.67725e8	0.824143
$297$	0	0
$298$	3.11690e8	0.682284
$299$	5.07755e6	0.0109851
$300$	0	0
$301$	−7.69009e7	−0.162536
$302$	−9.58863e8	−2.00324
$303$	0	0
$304$	−4.17863e8	−0.853054
$305$	5.85828e8	1.18228
$306$	0	0
$307$	−3.52294e8	−0.694898	−0.347449	−	0.937699i	$-0.612952\pi$
$307$	−3.52294e8	−0.694898	−0.347449	+	0.937699i	$0.612952\pi$
$308$	2.36550e8	0.461312
$309$	0	0
$310$	−1.64940e9	−3.14457
$311$	3.08514e8	0.581585	0.290792	−	0.956786i	$-0.406081\pi$
$311$	3.08514e8	0.581585	0.290792	+	0.956786i	$0.406081\pi$
$312$	0	0
$313$	2.48147e8	0.457407	0.228704	−	0.973496i	$-0.426551\pi$
$313$	2.48147e8	0.457407	0.228704	+	0.973496i	$0.426551\pi$
$314$	−5.59612e8	−1.02008
$315$	0	0
$316$	−2.80245e8	−0.499613
$317$	3.68196e8	0.649190	0.324595	−	0.945853i	$-0.394772\pi$
$317$	3.68196e8	0.649190	0.324595	+	0.945853i	$0.394772\pi$
$318$	0	0
$319$	−3.81260e7	−0.0657588
$320$	2.65998e8	0.453788
$321$	0	0
$322$	−6.54370e7	−0.109226
$323$	5.19042e8	0.857025
$324$	0	0
$325$	−6.91547e7	−0.111746
$326$	−1.89241e8	−0.302519
$327$	0	0
$328$	−1.19887e8	−0.187592
$329$	−5.01878e8	−0.776985
$330$	0	0
$331$	−1.67853e8	−0.254409	−0.127204	−	0.991877i	$-0.540600\pi$
$331$	−1.67853e8	−0.254409	−0.127204	+	0.991877i	$0.540600\pi$
$332$	−5.02761e7	−0.0754012
$333$	0	0
$334$	1.40240e9	2.05949
$335$	−1.30481e9	−1.89623
$336$	0	0
$337$	−1.72217e8	−0.245116	−0.122558	−	0.992461i	$-0.539110\pi$
$337$	−1.72217e8	−0.245116	−0.122558	+	0.992461i	$0.539110\pi$
$338$	9.13675e8	1.28701
$339$	0	0
$340$	−1.04678e9	−1.44438
$341$	1.72430e9	2.35489
$342$	0	0
$343$	5.56709e8	0.744901
$344$	1.30469e8	0.172804
$345$	0	0
$346$	2.24999e8	0.292021
$347$	−7.96731e8	−1.02367	−0.511834	−	0.859085i	$-0.671034\pi$
$347$	−7.96731e8	−1.02367	−0.511834	+	0.859085i	$0.671034\pi$
$348$	0	0
$349$	1.09341e9	1.37687	0.688436	−	0.725297i	$-0.258298\pi$
$349$	1.09341e9	1.37687	0.688436	+	0.725297i	$0.258298\pi$
$350$	8.91232e8	1.11110
$351$	0	0
$352$	−1.60161e9	−1.95730
$353$	−6.64482e8	−0.804029	−0.402015	−	0.915633i	$-0.631690\pi$
$353$	−6.64482e8	−0.804029	−0.402015	+	0.915633i	$0.631690\pi$
$354$	0	0
$355$	−1.38362e9	−1.64141
$356$	−2.37368e8	−0.278835
$357$	0	0
$358$	1.22444e9	1.41041
$359$	2.55461e8	0.291403	0.145702	−	0.989329i	$-0.453456\pi$
$359$	2.55461e8	0.291403	0.145702	+	0.989329i	$0.453456\pi$
$360$	0	0
$361$	−4.58675e8	−0.513133
$362$	2.20333e9	2.44117
$363$	0	0
$364$	1.30968e7	0.0142334
$365$	2.55038e9	2.74524
$366$	0	0
$367$	7.04463e8	0.743921	0.371961	−	0.928249i	$-0.378686\pi$
$367$	7.04463e8	0.743921	0.371961	+	0.928249i	$0.378686\pi$
$368$	2.43711e8	0.254922
$369$	0	0
$370$	4.24275e9	4.35453
$371$	−4.16606e8	−0.423561
$372$	0	0
$373$	−1.53844e9	−1.53497	−0.767484	−	0.641068i	$-0.778491\pi$
$373$	−1.53844e9	−1.53497	−0.767484	+	0.641068i	$0.778491\pi$
$374$	2.73832e9	2.70666
$375$	0	0
$376$	8.51480e8	0.826070
$377$	−2.11088e6	−0.00202894
$378$	0	0
$379$	−2.20215e7	−0.0207782	−0.0103891	−	0.999946i	$-0.503307\pi$
$379$	−2.20215e7	−0.0207782	−0.0103891	+	0.999946i	$0.503307\pi$
$380$	−8.77689e8	−0.820536
$381$	0	0
$382$	4.30528e8	0.395166
$383$	−1.80851e9	−1.64484	−0.822421	−	0.568879i	$-0.807377\pi$
$383$	−1.80851e9	−1.64484	−0.822421	+	0.568879i	$0.807377\pi$
$384$	0	0
$385$	−1.37095e9	−1.22436
$386$	−2.07929e9	−1.84018
$387$	0	0
$388$	−2.46281e7	−0.0214052
$389$	−1.89367e8	−0.163110	−0.0815549	−	0.996669i	$-0.525989\pi$
$389$	−1.89367e8	−0.163110	−0.0815549	+	0.996669i	$0.525989\pi$
$390$	0	0
$391$	−3.02721e8	−0.256109
$392$	−4.29861e8	−0.360435
$393$	0	0
$394$	−9.79010e8	−0.806399
$395$	1.62420e9	1.32601
$396$	0	0
$397$	1.15077e9	0.923040	0.461520	−	0.887130i	$-0.347304\pi$
$397$	1.15077e9	0.923040	0.461520	+	0.887130i	$0.347304\pi$
$398$	−2.62288e9	−2.08539
$399$	0	0
$400$	−3.31927e9	−2.59318
$401$	−1.47249e9	−1.14037	−0.570186	−	0.821516i	$-0.693129\pi$
$401$	−1.47249e9	−1.14037	−0.570186	+	0.821516i	$0.693129\pi$
$402$	0	0
$403$	9.54670e7	0.0726584
$404$	−4.42322e8	−0.333736
$405$	0	0
$406$	2.72039e7	0.0201739
$407$	−4.43539e9	−3.26101
$408$	0	0
$409$	−9.24969e8	−0.668491	−0.334246	−	0.942486i	$-0.608482\pi$
$409$	−9.24969e8	−0.668491	−0.334246	+	0.942486i	$0.608482\pi$
$410$	−1.38324e9	−0.991182
$411$	0	0
$412$	−5.82427e8	−0.410299
$413$	−5.87699e8	−0.410516
$414$	0	0
$415$	2.91381e8	0.200121
$416$	−8.86745e7	−0.0603910
$417$	0	0
$418$	2.29598e9	1.53763
$419$	7.26576e8	0.482539	0.241269	−	0.970458i	$-0.422436\pi$
$419$	7.26576e8	0.482539	0.241269	+	0.970458i	$0.422436\pi$
$420$	0	0
$421$	−2.41347e8	−0.157636	−0.0788178	−	0.996889i	$-0.525115\pi$
$421$	−2.41347e8	−0.157636	−0.0788178	+	0.996889i	$0.525115\pi$
$422$	−2.55965e9	−1.65801
$423$	0	0
$424$	7.06808e8	0.450320
$425$	4.12297e9	2.60525
$426$	0	0
$427$	4.36984e8	0.271624
$428$	−5.09983e8	−0.314415
$429$	0	0
$430$	1.50533e9	0.913044
$431$	−3.23006e9	−1.94330	−0.971652	−	0.236417i	$-0.924027\pi$
$431$	−3.23006e9	−1.94330	−0.971652	+	0.236417i	$0.924027\pi$
$432$	0	0
$433$	−2.93355e9	−1.73654	−0.868272	−	0.496088i	$-0.834769\pi$
$433$	−2.93355e9	−1.73654	−0.868272	+	0.496088i	$0.834769\pi$
$434$	−1.23033e9	−0.722451
$435$	0	0
$436$	−8.52961e8	−0.492863
$437$	−2.53820e8	−0.145493
$438$	0	0
$439$	−1.39654e9	−0.787819	−0.393910	−	0.919149i	$-0.628878\pi$
$439$	−1.39654e9	−0.787819	−0.393910	+	0.919149i	$0.628878\pi$
$440$	2.32594e9	1.30171
$441$	0	0
$442$	1.51609e8	0.0835118
$443$	−9.43130e8	−0.515417	−0.257708	−	0.966223i	$-0.582967\pi$
$443$	−9.43130e8	−0.515417	−0.257708	+	0.966223i	$0.582967\pi$
$444$	0	0
$445$	1.37570e9	0.740052
$446$	3.07283e8	0.164009
$447$	0	0
$448$	1.98415e8	0.104256
$449$	3.54094e9	1.84610	0.923052	−	0.384676i	$-0.125687\pi$
$449$	3.54094e9	1.84610	0.923052	+	0.384676i	$0.125687\pi$
$450$	0	0
$451$	1.44605e9	0.742274
$452$	−1.78759e8	−0.0910506
$453$	0	0
$454$	−4.73629e9	−2.37543
$455$	−7.59039e7	−0.0377767
$456$	0	0
$457$	−2.03166e9	−0.995738	−0.497869	−	0.867252i	$-0.665884\pi$
$457$	−2.03166e9	−0.995738	−0.497869	+	0.867252i	$0.665884\pi$
$458$	7.25958e7	0.0353087
$459$	0	0
$460$	5.11895e8	0.245205
$461$	4.97341e8	0.236429	0.118215	−	0.992988i	$-0.462283\pi$
$461$	4.97341e8	0.236429	0.118215	+	0.992988i	$0.462283\pi$
$462$	0	0
$463$	1.07915e9	0.505301	0.252650	−	0.967558i	$-0.418698\pi$
$463$	1.07915e9	0.505301	0.252650	+	0.967558i	$0.418698\pi$
$464$	−1.01317e8	−0.0470837
$465$	0	0
$466$	−7.80225e8	−0.357165
$467$	1.05677e9	0.480144	0.240072	−	0.970755i	$-0.422829\pi$
$467$	1.05677e9	0.480144	0.240072	+	0.970755i	$0.422829\pi$
$468$	0	0
$469$	−9.73291e8	−0.435650
$470$	9.82422e9	4.36471
$471$	0	0
$472$	9.97084e8	0.436450
$473$	−1.57368e9	−0.683758
$474$	0	0
$475$	3.45695e9	1.48001
$476$	−7.80823e8	−0.331840
$477$	0	0
$478$	−3.87592e9	−1.62322
$479$	1.21111e8	0.0503512	0.0251756	−	0.999683i	$-0.491986\pi$
$479$	1.21111e8	0.0503512	0.0251756	+	0.999683i	$0.491986\pi$
$480$	0	0
$481$	−2.45569e8	−0.100616
$482$	2.07908e9	0.845680
$483$	0	0
$484$	3.18035e9	1.27502
$485$	1.42735e8	0.0568113
$486$	0	0
$487$	4.13714e9	1.62311	0.811556	−	0.584274i	$-0.198621\pi$
$487$	4.13714e9	1.62311	0.811556	+	0.584274i	$0.198621\pi$
$488$	−7.41382e8	−0.288784
$489$	0	0
$490$	−4.95966e9	−1.90443
$491$	3.63021e8	0.138403	0.0692016	−	0.997603i	$-0.477955\pi$
$491$	3.63021e8	0.138403	0.0692016	+	0.997603i	$0.477955\pi$
$492$	0	0
$493$	1.25849e8	0.0473028
$494$	1.27119e8	0.0474422
$495$	0	0
$496$	4.58220e9	1.68612
$497$	−1.03208e9	−0.377108
$498$	0	0
$499$	−4.68516e9	−1.68800	−0.844001	−	0.536342i	$-0.819806\pi$
$499$	−4.68516e9	−1.68800	−0.844001	+	0.536342i	$0.819806\pi$
$500$	−3.68495e9	−1.31837
$501$	0	0
$502$	3.95873e9	1.39667
$503$	−3.24026e9	−1.13525	−0.567626	−	0.823286i	$-0.692138\pi$
$503$	−3.24026e9	−1.13525	−0.567626	+	0.823286i	$0.692138\pi$
$504$	0	0
$505$	2.56353e9	0.885765
$506$	−1.33909e9	−0.459496
$507$	0	0
$508$	−8.31943e8	−0.281560
$509$	2.41699e9	0.812388	0.406194	−	0.913787i	$-0.366856\pi$
$509$	2.41699e9	0.812388	0.406194	+	0.913787i	$0.366856\pi$
$510$	0	0
$511$	1.90240e9	0.630707
$512$	−2.65395e9	−0.873872
$513$	0	0
$514$	−5.40649e9	−1.75608
$515$	3.37552e9	1.08897
$516$	0	0
$517$	−1.02703e10	−3.26863
$518$	3.16477e9	1.00043
$519$	0	0
$520$	1.28778e8	0.0401633
$521$	−5.28397e9	−1.63692	−0.818461	−	0.574562i	$-0.805173\pi$
$521$	−5.28397e9	−1.63692	−0.818461	+	0.574562i	$0.805173\pi$
$522$	0	0
$523$	−1.32356e9	−0.404566	−0.202283	−	0.979327i	$-0.564836\pi$
$523$	−1.32356e9	−0.404566	−0.202283	+	0.979327i	$0.564836\pi$
$524$	−4.02482e9	−1.22204
$525$	0	0
$526$	3.91190e8	0.117203
$527$	−5.69170e9	−1.69397
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	8.15502e9	2.37936
$531$	0	0
$532$	−6.54690e8	−0.188515
$533$	8.00615e7	0.0229023
$534$	0	0
$535$	2.95567e9	0.834483
$536$	1.65127e9	0.463172
$537$	0	0
$538$	−6.12428e8	−0.169558
$539$	5.18486e9	1.42619
$540$	0	0
$541$	6.76224e9	1.83612	0.918058	−	0.396445i	$-0.129756\pi$
$541$	6.76224e9	1.83612	0.918058	+	0.396445i	$0.129756\pi$
$542$	−5.24755e9	−1.41566
$543$	0	0
$544$	5.28673e9	1.40796
$545$	4.94344e9	1.30810
$546$	0	0
$547$	8.50953e7	0.0222305	0.0111153	−	0.999938i	$-0.496462\pi$
$547$	8.50953e7	0.0222305	0.0111153	+	0.999938i	$0.496462\pi$
$548$	2.43604e9	0.632344
$549$	0	0
$550$	1.82379e10	4.67419
$551$	1.05520e8	0.0268723
$552$	0	0
$553$	1.21153e9	0.304646
$554$	7.62837e8	0.190611
$555$	0	0
$556$	1.68326e9	0.415326
$557$	1.99497e9	0.489150	0.244575	−	0.969630i	$-0.421352\pi$
$557$	1.99497e9	0.489150	0.244575	+	0.969630i	$0.421352\pi$
$558$	0	0
$559$	−8.71281e7	−0.0210968
$560$	−3.64321e9	−0.876650
$561$	0	0
$562$	8.50308e9	2.02069
$563$	3.64076e9	0.859829	0.429915	−	0.902870i	$-0.358544\pi$
$563$	3.64076e9	0.859829	0.429915	+	0.902870i	$0.358544\pi$
$564$	0	0
$565$	1.03602e9	0.241656
$566$	−5.25614e9	−1.21845
$567$	0	0
$568$	1.75101e9	0.400931
$569$	−2.21399e9	−0.503828	−0.251914	−	0.967750i	$-0.581060\pi$
$569$	−2.21399e9	−0.503828	−0.251914	+	0.967750i	$0.581060\pi$
$570$	0	0
$571$	1.63533e9	0.367602	0.183801	−	0.982963i	$-0.441160\pi$
$571$	1.63533e9	0.367602	0.183801	+	0.982963i	$0.441160\pi$
$572$	2.68009e8	0.0598774
$573$	0	0
$574$	−1.03179e9	−0.227720
$575$	−2.01620e9	−0.442280
$576$	0	0
$577$	4.85985e9	1.05319	0.526596	−	0.850115i	$-0.323468\pi$
$577$	4.85985e9	1.05319	0.526596	+	0.850115i	$0.323468\pi$
$578$	−3.04733e9	−0.656406
$579$	0	0
$580$	−2.12809e8	−0.0452889
$581$	2.17349e8	0.0459770
$582$	0	0
$583$	−8.52531e9	−1.78185
$584$	−3.22758e9	−0.670552
$585$	0	0
$586$	−4.93446e9	−1.01297
$587$	6.39733e9	1.30547	0.652733	−	0.757588i	$-0.273623\pi$
$587$	6.39733e9	1.30547	0.652733	+	0.757588i	$0.273623\pi$
$588$	0	0
$589$	−4.77227e9	−0.962325
$590$	1.15042e10	2.30607
$591$	0	0
$592$	−1.17868e10	−2.33490
$593$	2.73477e9	0.538554	0.269277	−	0.963063i	$-0.413215\pi$
$593$	2.73477e9	0.538554	0.269277	+	0.963063i	$0.413215\pi$
$594$	0	0
$595$	4.52535e9	0.880730
$596$	−1.81876e9	−0.351895
$597$	0	0
$598$	−7.41396e7	−0.0141774
$599$	4.75928e8	0.0904790	0.0452395	−	0.998976i	$-0.485595\pi$
$599$	4.75928e8	0.0904790	0.0452395	+	0.998976i	$0.485595\pi$
$600$	0	0
$601$	4.25604e9	0.799733	0.399866	−	0.916573i	$-0.369057\pi$
$601$	4.25604e9	0.799733	0.399866	+	0.916573i	$0.369057\pi$
$602$	1.12286e9	0.209768
$603$	0	0
$604$	5.59513e9	1.03319
$605$	−1.84321e10	−3.38401
$606$	0	0
$607$	−6.01074e9	−1.09086	−0.545428	−	0.838157i	$-0.683633\pi$
$607$	−6.01074e9	−1.09086	−0.545428	+	0.838157i	$0.683633\pi$
$608$	4.43272e9	0.799849
$609$	0	0
$610$	−8.55392e9	−1.52585
$611$	−5.68623e8	−0.100851
$612$	0	0
$613$	5.65645e8	0.0991819	0.0495910	−	0.998770i	$-0.484208\pi$
$613$	5.65645e8	0.0991819	0.0495910	+	0.998770i	$0.484208\pi$
$614$	5.14400e9	0.896833
$615$	0	0
$616$	1.73498e9	0.299063
$617$	−7.85121e9	−1.34567	−0.672834	−	0.739793i	$-0.734923\pi$
$617$	−7.85121e9	−1.34567	−0.672834	+	0.739793i	$0.734923\pi$
$618$	0	0
$619$	3.13703e9	0.531620	0.265810	−	0.964025i	$-0.414361\pi$
$619$	3.13703e9	0.531620	0.265810	+	0.964025i	$0.414361\pi$
$620$	9.62454e9	1.62184
$621$	0	0
$622$	−4.50474e9	−0.750591
$623$	1.02617e9	0.170024
$624$	0	0
$625$	8.41040e9	1.37796
$626$	−3.62329e9	−0.590328
$627$	0	0
$628$	3.26543e9	0.526116
$629$	1.46407e10	2.34577
$630$	0	0
$631$	−2.25037e9	−0.356576	−0.178288	−	0.983978i	$-0.557056\pi$
$631$	−2.25037e9	−0.356576	−0.178288	+	0.983978i	$0.557056\pi$
$632$	−2.05547e9	−0.323892
$633$	0	0
$634$	−5.37619e9	−0.837842
$635$	4.82163e9	0.747284
$636$	0	0
$637$	2.87064e8	0.0440038
$638$	5.56694e8	0.0848681
$639$	0	0
$640$	9.54637e9	1.43949
$641$	2.64631e9	0.396860	0.198430	−	0.980115i	$-0.436416\pi$
$641$	2.64631e9	0.396860	0.198430	+	0.980115i	$0.436416\pi$
$642$	0	0
$643$	−8.30752e9	−1.23235	−0.616173	−	0.787611i	$-0.711318\pi$
$643$	−8.30752e9	−1.23235	−0.616173	+	0.787611i	$0.711318\pi$
$644$	3.81836e8	0.0563347
$645$	0	0
$646$	−7.57875e9	−1.10607
$647$	4.09897e9	0.594990	0.297495	−	0.954723i	$-0.403849\pi$
$647$	4.09897e9	0.594990	0.297495	+	0.954723i	$0.403849\pi$
$648$	0	0
$649$	−1.20265e10	−1.72697
$650$	1.00976e9	0.144218
$651$	0	0
$652$	1.10425e9	0.156027
$653$	5.29390e9	0.744011	0.372006	−	0.928230i	$-0.378670\pi$
$653$	5.29390e9	0.744011	0.372006	+	0.928230i	$0.378670\pi$
$654$	0	0
$655$	2.33263e10	3.24341
$656$	3.84277e9	0.531473
$657$	0	0
$658$	7.32814e9	1.00277
$659$	−9.66411e9	−1.31542	−0.657708	−	0.753273i	$-0.728474\pi$
$659$	−9.66411e9	−1.31542	−0.657708	+	0.753273i	$0.728474\pi$
$660$	0	0
$661$	4.20815e9	0.566743	0.283372	−	0.959010i	$-0.408547\pi$
$661$	4.20815e9	0.566743	0.283372	+	0.959010i	$0.408547\pi$
$662$	2.45090e9	0.328339
$663$	0	0
$664$	−3.68751e8	−0.0488816
$665$	3.79434e9	0.500334
$666$	0	0
$667$	−6.15425e7	−0.00803036
$668$	−8.18323e9	−1.06220
$669$	0	0
$670$	1.90521e10	2.44726
$671$	8.94233e9	1.14267
$672$	0	0
$673$	−5.81522e9	−0.735383	−0.367691	−	0.929948i	$-0.619852\pi$
$673$	−5.81522e9	−0.735383	−0.367691	+	0.929948i	$0.619852\pi$
$674$	2.51462e9	0.316346
$675$	0	0
$676$	−5.33145e9	−0.663792
$677$	−4.45305e9	−0.551565	−0.275783	−	0.961220i	$-0.588937\pi$
$677$	−4.45305e9	−0.551565	−0.275783	+	0.961220i	$0.588937\pi$
$678$	0	0
$679$	1.06470e8	0.0130521
$680$	−7.67766e9	−0.936370
$681$	0	0
$682$	−2.51772e10	−3.03922
$683$	−5.09118e9	−0.611428	−0.305714	−	0.952123i	$-0.598895\pi$
$683$	−5.09118e9	−0.611428	−0.305714	+	0.952123i	$0.598895\pi$
$684$	0	0
$685$	−1.41184e10	−1.67829
$686$	−8.12874e9	−0.961366
$687$	0	0
$688$	−4.18195e9	−0.489575
$689$	−4.72011e8	−0.0549774
$690$	0	0
$691$	3.95531e9	0.456045	0.228022	−	0.973656i	$-0.426774\pi$
$691$	3.95531e9	0.456045	0.228022	+	0.973656i	$0.426774\pi$
$692$	−1.31291e9	−0.150613
$693$	0	0
$694$	1.16334e10	1.32114
$695$	−9.75554e9	−1.10231
$696$	0	0
$697$	−4.77323e9	−0.533946
$698$	−1.59653e10	−1.77699
$699$	0	0
$700$	−5.20049e9	−0.573061
$701$	−1.26904e10	−1.39143	−0.695716	−	0.718317i	$-0.744913\pi$
$701$	−1.26904e10	−1.39143	−0.695716	+	0.718317i	$0.744913\pi$
$702$	0	0
$703$	1.22757e10	1.33261
$704$	4.06031e9	0.438586
$705$	0	0
$706$	9.70239e9	1.03768
$707$	1.91220e9	0.203501
$708$	0	0
$709$	−1.25513e10	−1.32260	−0.661300	−	0.750122i	$-0.729995\pi$
$709$	−1.25513e10	−1.32260	−0.661300	+	0.750122i	$0.729995\pi$
$710$	2.02028e10	2.11840
$711$	0	0
$712$	−1.74098e9	−0.180765
$713$	2.78334e9	0.287576
$714$	0	0
$715$	−1.55328e9	−0.158920
$716$	−7.14481e9	−0.727436
$717$	0	0
$718$	−3.73010e9	−0.376084
$719$	5.65642e9	0.567532	0.283766	−	0.958894i	$-0.408416\pi$
$719$	5.65642e9	0.567532	0.283766	+	0.958894i	$0.408416\pi$
$720$	0	0
$721$	2.51789e9	0.250186
$722$	6.69731e9	0.662247
$723$	0	0
$724$	−1.28568e10	−1.25906
$725$	8.38190e8	0.0816883
$726$	0	0
$727$	−1.59955e10	−1.54393	−0.771963	−	0.635667i	$-0.780725\pi$
$727$	−1.59955e10	−1.54393	−0.771963	+	0.635667i	$0.780725\pi$
$728$	9.60586e7	0.00922734
$729$	0	0
$730$	−3.72392e10	−3.54300
$731$	5.19453e9	0.491853
$732$	0	0
$733$	−1.22763e10	−1.15134	−0.575670	−	0.817682i	$-0.695259\pi$
$733$	−1.22763e10	−1.15134	−0.575670	+	0.817682i	$0.695259\pi$
$734$	−1.02862e10	−0.960102
$735$	0	0
$736$	−2.58530e9	−0.239022
$737$	−1.99172e10	−1.83270
$738$	0	0
$739$	−2.44144e9	−0.222531	−0.111265	−	0.993791i	$-0.535490\pi$
$739$	−2.44144e9	−0.222531	−0.111265	+	0.993791i	$0.535490\pi$
$740$	−2.47571e10	−2.24589
$741$	0	0
$742$	6.08304e9	0.546647
$743$	−2.03182e9	−0.181729	−0.0908646	−	0.995863i	$-0.528963\pi$
$743$	−2.03182e9	−0.181729	−0.0908646	+	0.995863i	$0.528963\pi$
$744$	0	0
$745$	1.05408e10	0.933961
$746$	2.24634e10	1.98102
$747$	0	0
$748$	−1.59786e10	−1.39599
$749$	2.20471e9	0.191719
$750$	0	0
$751$	1.99854e10	1.72176	0.860880	−	0.508809i	$-0.169914\pi$
$751$	1.99854e10	1.72176	0.860880	+	0.508809i	$0.169914\pi$
$752$	−2.72926e10	−2.34036
$753$	0	0
$754$	3.08218e7	0.00261854
$755$	−3.24272e10	−2.74218
$756$	0	0
$757$	−1.68290e10	−1.41001	−0.705006	−	0.709202i	$-0.749056\pi$
$757$	−1.68290e10	−1.41001	−0.705006	+	0.709202i	$0.749056\pi$
$758$	3.21545e8	0.0268163
$759$	0	0
$760$	−6.43743e9	−0.531943
$761$	2.47157e9	0.203295	0.101648	−	0.994820i	$-0.467589\pi$
$761$	2.47157e9	0.203295	0.101648	+	0.994820i	$0.467589\pi$
$762$	0	0
$763$	3.68744e9	0.300531
$764$	−2.51221e9	−0.203811
$765$	0	0
$766$	2.64068e10	2.12283
$767$	−6.65859e8	−0.0532842
$768$	0	0
$769$	−2.24237e10	−1.77813	−0.889067	−	0.457777i	$-0.848646\pi$
$769$	−2.24237e10	−1.77813	−0.889067	+	0.457777i	$0.848646\pi$
$770$	2.00179e10	1.58016
$771$	0	0
$772$	1.21330e10	0.949091
$773$	1.92307e10	1.49750	0.748748	−	0.662854i	$-0.230655\pi$
$773$	1.92307e10	1.49750	0.748748	+	0.662854i	$0.230655\pi$
$774$	0	0
$775$	−3.79082e10	−2.92535
$776$	−1.80635e8	−0.0138767
$777$	0	0
$778$	2.76503e9	0.210509
$779$	−4.00217e9	−0.303329
$780$	0	0
$781$	−2.11202e10	−1.58642
$782$	4.42016e9	0.330533
$783$	0	0
$784$	1.37784e10	1.02116
$785$	−1.89252e10	−1.39636
$786$	0	0
$787$	8.73845e9	0.639032	0.319516	−	0.947581i	$-0.396480\pi$
$787$	8.73845e9	0.639032	0.319516	+	0.947581i	$0.396480\pi$
$788$	5.71269e9	0.415909
$789$	0	0
$790$	−2.37156e10	−1.71135
$791$	7.72792e8	0.0555194
$792$	0	0
$793$	4.95099e8	0.0352563
$794$	−1.68028e10	−1.19127
$795$	0	0
$796$	1.53049e10	1.07556
$797$	1.61543e10	1.13027	0.565137	−	0.824997i	$-0.308823\pi$
$797$	1.61543e10	1.13027	0.565137	+	0.824997i	$0.308823\pi$
$798$	0	0
$799$	3.39010e10	2.35125
$800$	3.52110e10	2.43144
$801$	0	0
$802$	2.15004e10	1.47176
$803$	3.89301e10	2.65327
$804$	0	0
$805$	−2.21298e9	−0.149517
$806$	−1.39396e9	−0.0937727
$807$	0	0
$808$	−3.24422e9	−0.216357
$809$	−2.10541e10	−1.39803	−0.699015	−	0.715107i	$-0.746378\pi$
$809$	−2.10541e10	−1.39803	−0.699015	+	0.715107i	$0.746378\pi$
$810$	0	0
$811$	−6.03214e9	−0.397098	−0.198549	−	0.980091i	$-0.563623\pi$
$811$	−6.03214e9	−0.397098	−0.198549	+	0.980091i	$0.563623\pi$
$812$	−1.58740e8	−0.0104049
$813$	0	0
$814$	6.47631e10	4.20865
$815$	−6.39982e9	−0.414110
$816$	0	0
$817$	4.35542e9	0.279417
$818$	1.35059e10	0.862753
$819$	0	0
$820$	8.07143e9	0.511213
$821$	−2.34470e10	−1.47872	−0.739361	−	0.673309i	$-0.764872\pi$
$821$	−2.34470e10	−1.47872	−0.739361	+	0.673309i	$0.764872\pi$
$822$	0	0
$823$	−2.27370e10	−1.42179	−0.710893	−	0.703300i	$-0.751709\pi$
$823$	−2.27370e10	−1.42179	−0.710893	+	0.703300i	$0.751709\pi$
$824$	−4.27182e9	−0.265992
$825$	0	0
$826$	8.58125e9	0.529810
$827$	−1.91123e10	−1.17501	−0.587507	−	0.809219i	$-0.699890\pi$
$827$	−1.91123e10	−1.17501	−0.587507	+	0.809219i	$0.699890\pi$
$828$	0	0
$829$	−1.83939e8	−0.0112133	−0.00560663	−	0.999984i	$-0.501785\pi$
$829$	−1.83939e8	−0.0112133	−0.00560663	+	0.999984i	$0.501785\pi$
$830$	−4.25458e9	−0.258276
$831$	0	0
$832$	2.24802e8	0.0135322
$833$	−1.71146e10	−1.02591
$834$	0	0
$835$	4.74269e10	2.81917
$836$	−1.33974e10	−0.793047
$837$	0	0
$838$	−1.06091e10	−0.622763
$839$	−9.03520e8	−0.0528166	−0.0264083	−	0.999651i	$-0.508407\pi$
$839$	−9.03520e8	−0.0528166	−0.0264083	+	0.999651i	$0.508407\pi$
$840$	0	0
$841$	−1.72243e10	−0.998517
$842$	3.52401e9	0.203444
$843$	0	0
$844$	1.49360e10	0.855136
$845$	3.08991e10	1.76176
$846$	0	0
$847$	−1.37490e10	−0.777461
$848$	−2.26554e10	−1.27581
$849$	0	0
$850$	−6.02013e10	−3.36232
$851$	−7.15956e9	−0.398229
$852$	0	0
$853$	−1.51293e10	−0.834635	−0.417317	−	0.908761i	$-0.637030\pi$
$853$	−1.51293e10	−0.834635	−0.417317	+	0.908761i	$0.637030\pi$
$854$	−6.38059e9	−0.350557
$855$	0	0
$856$	−3.74049e9	−0.203831
$857$	4.11557e9	0.223356	0.111678	−	0.993744i	$-0.464378\pi$
$857$	4.11557e9	0.223356	0.111678	+	0.993744i	$0.464378\pi$
$858$	0	0
$859$	1.02304e10	0.550703	0.275351	−	0.961344i	$-0.411206\pi$
$859$	1.02304e10	0.550703	0.275351	+	0.961344i	$0.411206\pi$
$860$	−8.78384e9	−0.470912
$861$	0	0
$862$	4.71635e10	2.50802
$863$	1.78056e9	0.0943015	0.0471508	−	0.998888i	$-0.484986\pi$
$863$	1.78056e9	0.0943015	0.0471508	+	0.998888i	$0.484986\pi$
$864$	0	0
$865$	7.60911e9	0.399740
$866$	4.28340e10	2.24118
$867$	0	0
$868$	7.17919e9	0.372612
$869$	2.47924e10	1.28159
$870$	0	0
$871$	−1.10273e9	−0.0565465
$872$	−6.25607e9	−0.319517
$873$	0	0
$874$	3.70614e9	0.187772
$875$	1.59304e10	0.803894
$876$	0	0
$877$	3.47384e10	1.73905	0.869523	−	0.493893i	$-0.164427\pi$
$877$	3.47384e10	1.73905	0.869523	+	0.493893i	$0.164427\pi$
$878$	2.03914e10	1.01676
$879$	0	0
$880$	−7.45538e10	−3.68791
$881$	−2.81250e10	−1.38572	−0.692862	−	0.721070i	$-0.743651\pi$
$881$	−2.81250e10	−1.38572	−0.692862	+	0.721070i	$0.743651\pi$
$882$	0	0
$883$	2.00161e10	0.978399	0.489200	−	0.872172i	$-0.337289\pi$
$883$	2.00161e10	0.978399	0.489200	+	0.872172i	$0.337289\pi$
$884$	−8.84666e8	−0.0430721
$885$	0	0
$886$	1.37711e10	0.665195
$887$	−1.97644e10	−0.950936	−0.475468	−	0.879733i	$-0.657721\pi$
$887$	−1.97644e10	−0.950936	−0.475468	+	0.879733i	$0.657721\pi$
$888$	0	0
$889$	3.59658e9	0.171685
$890$	−2.00871e10	−0.955109
$891$	0	0
$892$	−1.79305e9	−0.0845893
$893$	2.84247e10	1.33572
$894$	0	0
$895$	4.14086e10	1.93068
$896$	7.12088e9	0.330716
$897$	0	0
$898$	−5.17027e10	−2.38257
$899$	−1.15711e9	−0.0531148
$900$	0	0
$901$	2.81410e10	1.28175
$902$	−2.11143e10	−0.957976
$903$	0	0
$904$	−1.31111e9	−0.0590269
$905$	7.45130e10	3.34166
$906$	0	0
$907$	7.73861e8	0.0344380	0.0172190	−	0.999852i	$-0.494519\pi$
$907$	7.73861e8	0.0344380	0.0172190	+	0.999852i	$0.494519\pi$
$908$	2.76370e10	1.22515
$909$	0	0
$910$	1.10831e9	0.0487545
$911$	−4.47493e9	−0.196097	−0.0980487	−	0.995182i	$-0.531260\pi$
$911$	−4.47493e9	−0.196097	−0.0980487	+	0.995182i	$0.531260\pi$
$912$	0	0
$913$	4.44777e9	0.193417
$914$	2.96652e10	1.28510
$915$	0	0
$916$	−4.23608e8	−0.0182109
$917$	1.73997e10	0.745160
$918$	0	0
$919$	−2.87268e10	−1.22091	−0.610454	−	0.792052i	$-0.709013\pi$
$919$	−2.87268e10	−1.22091	−0.610454	+	0.792052i	$0.709013\pi$
$920$	3.75451e9	0.158963
$921$	0	0
$922$	−7.26189e9	−0.305135
$923$	−1.16934e9	−0.0489478
$924$	0	0
$925$	9.75110e10	4.05096
$926$	−1.57572e10	−0.652139
$927$	0	0
$928$	1.07478e9	0.0441470
$929$	−2.38092e10	−0.974294	−0.487147	−	0.873320i	$-0.661962\pi$
$929$	−2.38092e10	−0.974294	−0.487147	+	0.873320i	$0.661962\pi$
$930$	0	0
$931$	−1.43499e10	−0.582809
$932$	4.55274e9	0.184212
$933$	0	0
$934$	−1.54304e10	−0.619672
$935$	9.26056e10	3.70507
$936$	0	0
$937$	−2.55610e10	−1.01505	−0.507527	−	0.861636i	$-0.669440\pi$
$937$	−2.55610e10	−1.01505	−0.507527	+	0.861636i	$0.669440\pi$
$938$	1.42114e10	0.562248
$939$	0	0
$940$	−5.73260e10	−2.25115
$941$	−3.51742e9	−0.137613	−0.0688067	−	0.997630i	$-0.521919\pi$
$941$	−3.51742e9	−0.137613	−0.0688067	+	0.997630i	$0.521919\pi$
$942$	0	0
$943$	2.33419e9	0.0906453
$944$	−3.19597e10	−1.23652
$945$	0	0
$946$	2.29780e10	0.882456
$947$	4.81870e10	1.84376	0.921882	−	0.387471i	$-0.126651\pi$
$947$	4.81870e10	1.84376	0.921882	+	0.387471i	$0.126651\pi$
$948$	0	0
$949$	2.15540e9	0.0818646
$950$	−5.04765e10	−1.91010
$951$	0	0
$952$	−5.72696e9	−0.215127
$953$	4.14619e10	1.55176	0.775879	−	0.630882i	$-0.217307\pi$
$953$	4.14619e10	1.55176	0.775879	+	0.630882i	$0.217307\pi$
$954$	0	0
$955$	1.45598e10	0.540933
$956$	2.26166e10	0.837193
$957$	0	0
$958$	−1.76840e9	−0.0649830
$959$	−1.05313e10	−0.385581
$960$	0	0
$961$	2.48191e10	0.902098
$962$	3.58566e9	0.129854
$963$	0	0
$964$	−1.21318e10	−0.436168
$965$	−7.03183e10	−2.51897
$966$	0	0
$967$	1.23269e10	0.438389	0.219195	−	0.975681i	$-0.429657\pi$
$967$	1.23269e10	0.438389	0.219195	+	0.975681i	$0.429657\pi$
$968$	2.33264e10	0.826577
$969$	0	0
$970$	−2.08414e9	−0.0733204
$971$	6.05848e9	0.212372	0.106186	−	0.994346i	$-0.466136\pi$
$971$	6.05848e9	0.212372	0.106186	+	0.994346i	$0.466136\pi$
$972$	0	0
$973$	−7.27690e9	−0.253251
$974$	−6.04081e10	−2.09478
$975$	0	0
$976$	2.37636e10	0.818160
$977$	−3.09327e10	−1.06118	−0.530588	−	0.847630i	$-0.678029\pi$
$977$	−3.09327e10	−1.06118	−0.530588	+	0.847630i	$0.678029\pi$
$978$	0	0
$979$	2.09992e10	0.715259
$980$	2.89404e10	0.982231
$981$	0	0
$982$	−5.30062e9	−0.178623
$983$	−1.95459e9	−0.0656325	−0.0328162	−	0.999461i	$-0.510448\pi$
$983$	−1.95459e9	−0.0656325	−0.0328162	+	0.999461i	$0.510448\pi$
$984$	0	0
$985$	−3.31086e10	−1.10386
$986$	−1.83758e9	−0.0610488
$987$	0	0
$988$	−7.41759e8	−0.0244688
$989$	−2.54022e9	−0.0834994
$990$	0	0
$991$	−2.87355e10	−0.937911	−0.468956	−	0.883222i	$-0.655370\pi$
$991$	−2.87355e10	−0.937911	−0.468956	+	0.883222i	$0.655370\pi$
$992$	−4.86082e10	−1.58095
$993$	0	0
$994$	1.50698e10	0.486694
$995$	−8.87017e10	−2.85464
$996$	0	0
$997$	−4.10733e9	−0.131258	−0.0656291	−	0.997844i	$-0.520905\pi$
$997$	−4.10733e9	−0.131258	−0.0656291	+	0.997844i	$0.520905\pi$
$998$	6.84101e10	2.17853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.e.1.3		8
3.2	odd	2		69.8.a.d.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.6	✓	8	3.2	odd	2
207.8.a.e.1.3		8	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-14.6014 q^{2} +85.2018 q^{4} -493.797 q^{5} -368.336 q^{7} +624.915 q^{8} +O(q^{10})\)	\(q-14.6014 q^{2} +85.2018 q^{4} -493.797 q^{5} -368.336 q^{7} +624.915 q^{8} +7210.15 q^{10} -7537.54 q^{11} -417.322 q^{13} +5378.24 q^{14} -20030.5 q^{16} +24880.5 q^{17} +20861.4 q^{19} -42072.4 q^{20} +110059. q^{22} -12167.0 q^{23} +165711. q^{25} +6093.50 q^{26} -31382.9 q^{28} +5058.15 q^{29} -228761. q^{31} +212485. q^{32} -363291. q^{34} +181883. q^{35} +588441. q^{37} -304606. q^{38} -308581. q^{40} -191846. q^{41} +208779. q^{43} -642212. q^{44} +177656. q^{46} +1.36255e6 q^{47} -687871. q^{49} -2.41962e6 q^{50} -35556.6 q^{52} +1.13105e6 q^{53} +3.72202e6 q^{55} -230179. q^{56} -73856.3 q^{58} +1.59555e6 q^{59} -1.18637e6 q^{61} +3.34024e6 q^{62} -538678. q^{64} +206072. q^{65} +2.64240e6 q^{67} +2.11986e6 q^{68} -2.65576e6 q^{70} +2.80200e6 q^{71} -5.16484e6 q^{73} -8.59208e6 q^{74} +1.77743e6 q^{76} +2.77635e6 q^{77} -3.28919e6 q^{79} +9.89100e6 q^{80} +2.80123e6 q^{82} -590082. q^{83} -1.22859e7 q^{85} -3.04847e6 q^{86} -4.71032e6 q^{88} -2.78595e6 q^{89} +153715. q^{91} -1.03665e6 q^{92} -1.98952e7 q^{94} -1.03013e7 q^{95} -289056. q^{97} +1.00439e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8}+O(q^{10}) \) 8 * q - 24 * q^2 + 562 * q^4 - 378 * q^5 + 126 * q^7 - 4188 * q^8	\( 8 q - 24 q^{2} + 562 q^{4} - 378 q^{5} + 126 q^{7} - 4188 q^{8} + 11720 q^{10} - 6932 q^{11} + 12404 q^{13} - 30222 q^{14} + 27058 q^{16} - 24434 q^{17} - 14682 q^{19} + 3760 q^{20} + 36294 q^{22} - 97336 q^{23} + 144644 q^{25} - 325840 q^{26} - 21566 q^{28} - 255356 q^{29} + 450764 q^{31} - 647588 q^{32} + 191822 q^{34} - 1022616 q^{35} + 206240 q^{37} - 737372 q^{38} + 590028 q^{40} - 1053344 q^{41} + 1587806 q^{43} - 589366 q^{44} + 292008 q^{46} - 443336 q^{47} + 1944828 q^{49} + 1556112 q^{50} - 614236 q^{52} + 375530 q^{53} + 407792 q^{55} + 1316922 q^{56} - 1413384 q^{58} - 624008 q^{59} - 2005568 q^{61} + 3908272 q^{62} - 5082310 q^{64} - 646124 q^{65} - 2712286 q^{67} + 2289698 q^{68} - 16499468 q^{70} + 6287176 q^{71} - 10358312 q^{73} + 2000150 q^{74} - 25107464 q^{76} + 2156840 q^{77} - 8800574 q^{79} - 2384344 q^{80} - 31799800 q^{82} - 384948 q^{83} - 17826684 q^{85} + 11563928 q^{86} - 25202782 q^{88} + 3445530 q^{89} - 16316740 q^{91} - 6837854 q^{92} - 24237616 q^{94} - 26164288 q^{95} - 28043764 q^{97} + 9998012 q^{98}+O(q^{100}) \) 8 * q - 24 * q^2 + 562 * q^4 - 378 * q^5 + 126 * q^7 - 4188 * q^8 + 11720 * q^10 - 6932 * q^11 + 12404 * q^13 - 30222 * q^14 + 27058 * q^16 - 24434 * q^17 - 14682 * q^19 + 3760 * q^20 + 36294 * q^22 - 97336 * q^23 + 144644 * q^25 - 325840 * q^26 - 21566 * q^28 - 255356 * q^29 + 450764 * q^31 - 647588 * q^32 + 191822 * q^34 - 1022616 * q^35 + 206240 * q^37 - 737372 * q^38 + 590028 * q^40 - 1053344 * q^41 + 1587806 * q^43 - 589366 * q^44 + 292008 * q^46 - 443336 * q^47 + 1944828 * q^49 + 1556112 * q^50 - 614236 * q^52 + 375530 * q^53 + 407792 * q^55 + 1316922 * q^56 - 1413384 * q^58 - 624008 * q^59 - 2005568 * q^61 + 3908272 * q^62 - 5082310 * q^64 - 646124 * q^65 - 2712286 * q^67 + 2289698 * q^68 - 16499468 * q^70 + 6287176 * q^71 - 10358312 * q^73 + 2000150 * q^74 - 25107464 * q^76 + 2156840 * q^77 - 8800574 * q^79 - 2384344 * q^80 - 31799800 * q^82 - 384948 * q^83 - 17826684 * q^85 + 11563928 * q^86 - 25202782 * q^88 + 3445530 * q^89 - 16316740 * q^91 - 6837854 * q^92 - 24237616 * q^94 - 26164288 * q^95 - 28043764 * q^97 + 9998012 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more