LMFDB - Embedded newform 207.12.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,12,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, 12, names="a")

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 12 $
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:	$159.047038376$
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} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 $ x^8 - 2672x^6 - 1234x^5 + 2202967x^4 + 2386582x^3 - 543567396x^2 - 1204011928x + 23305583840
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-9.43512$ 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)

$f(q)$	$=$	$q-14.8702 q^{2} -1826.88 q^{4} -2118.93 q^{5} -68580.9 q^{7} +57620.3 q^{8} +O(q^{10})$	\(q-14.8702 q^{2} -1826.88 q^{4} -2118.93 q^{5} -68580.9 q^{7} +57620.3 q^{8} +31509.0 q^{10} +573905. q^{11} -209778. q^{13} +1.01981e6 q^{14} +2.88461e6 q^{16} -7.02586e6 q^{17} -1.37759e7 q^{19} +3.87103e6 q^{20} -8.53411e6 q^{22} -6.43634e6 q^{23} -4.43382e7 q^{25} +3.11945e6 q^{26} +1.25289e8 q^{28} +1.90149e8 q^{29} +1.11283e8 q^{31} -1.60901e8 q^{32} +1.04476e8 q^{34} +1.45318e8 q^{35} +3.70363e8 q^{37} +2.04850e8 q^{38} -1.22094e8 q^{40} +4.78021e8 q^{41} +1.32298e9 q^{43} -1.04845e9 q^{44} +9.57100e7 q^{46} -1.04187e9 q^{47} +2.72601e9 q^{49} +6.59320e8 q^{50} +3.83238e8 q^{52} +3.65404e9 q^{53} -1.21607e9 q^{55} -3.95165e9 q^{56} -2.82757e9 q^{58} -9.91859e8 q^{59} -5.85821e6 q^{61} -1.65481e9 q^{62} -3.51505e9 q^{64} +4.44505e8 q^{65} +1.92877e10 q^{67} +1.28354e10 q^{68} -2.16092e9 q^{70} +5.27575e9 q^{71} -2.46322e10 q^{73} -5.50739e9 q^{74} +2.51668e10 q^{76} -3.93589e10 q^{77} +9.77106e9 q^{79} -6.11230e9 q^{80} -7.10829e9 q^{82} +1.67805e10 q^{83} +1.48873e10 q^{85} -1.96731e10 q^{86} +3.30686e10 q^{88} +3.63782e10 q^{89} +1.43867e10 q^{91} +1.17584e10 q^{92} +1.54929e10 q^{94} +2.91901e10 q^{95} +5.73626e10 q^{97} -4.05364e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{2} + 5120 q^{4} + 11466 q^{5} - 54118 q^{7} + 155568 q^{8}+O(q^{10}) $ 8 * q + 32 * q^2 + 5120 * q^4 + 11466 * q^5 - 54118 * q^7 + 155568 * q^8	\( 8 q + 32 q^{2} + 5120 q^{4} + 11466 q^{5} - 54118 q^{7} + 155568 q^{8} + 517892 q^{10} + 291462 q^{11} - 2211306 q^{13} + 939584 q^{14} - 8561344 q^{16} + 5775330 q^{17} - 21015588 q^{19} + 65503576 q^{20} - 83047784 q^{22} - 51490744 q^{23} - 36491644 q^{25} + 119299562 q^{26} - 392796032 q^{28} + 322285430 q^{29} - 415184840 q^{31} - 31831744 q^{32} - 28224252 q^{34} - 603721008 q^{35} + 176642018 q^{37} - 554685496 q^{38} + 1337904816 q^{40} - 357962218 q^{41} + 2500461376 q^{43} - 5064743472 q^{44} - 205962976 q^{46} - 261795200 q^{47} + 2656605924 q^{49} - 1642758328 q^{50} + 3841657212 q^{52} - 3542935060 q^{53} - 10100187604 q^{55} - 7995463104 q^{56} - 9113565454 q^{58} - 930905396 q^{59} - 25338655048 q^{61} - 4385691666 q^{62} - 34067008768 q^{64} + 25954746658 q^{65} - 3123467482 q^{67} + 37358480280 q^{68} - 35719175696 q^{70} + 52612263236 q^{71} - 67014176274 q^{73} - 10171443276 q^{74} + 17955918576 q^{76} + 44516617816 q^{77} - 27683357604 q^{79} - 74357773216 q^{80} + 73615849126 q^{82} + 12253964262 q^{83} + 58779027600 q^{85} - 90522557252 q^{86} + 33736356800 q^{88} - 10662817760 q^{89} - 28336741418 q^{91} - 32954076160 q^{92} + 285145948346 q^{94} + 64104297380 q^{95} - 124519454530 q^{97} - 215615498272 q^{98}+O(q^{100}) \) 8 * q + 32 * q^2 + 5120 * q^4 + 11466 * q^5 - 54118 * q^7 + 155568 * q^8 + 517892 * q^10 + 291462 * q^11 - 2211306 * q^13 + 939584 * q^14 - 8561344 * q^16 + 5775330 * q^17 - 21015588 * q^19 + 65503576 * q^20 - 83047784 * q^22 - 51490744 * q^23 - 36491644 * q^25 + 119299562 * q^26 - 392796032 * q^28 + 322285430 * q^29 - 415184840 * q^31 - 31831744 * q^32 - 28224252 * q^34 - 603721008 * q^35 + 176642018 * q^37 - 554685496 * q^38 + 1337904816 * q^40 - 357962218 * q^41 + 2500461376 * q^43 - 5064743472 * q^44 - 205962976 * q^46 - 261795200 * q^47 + 2656605924 * q^49 - 1642758328 * q^50 + 3841657212 * q^52 - 3542935060 * q^53 - 10100187604 * q^55 - 7995463104 * q^56 - 9113565454 * q^58 - 930905396 * q^59 - 25338655048 * q^61 - 4385691666 * q^62 - 34067008768 * q^64 + 25954746658 * q^65 - 3123467482 * q^67 + 37358480280 * q^68 - 35719175696 * q^70 + 52612263236 * q^71 - 67014176274 * q^73 - 10171443276 * q^74 + 17955918576 * q^76 + 44516617816 * q^77 - 27683357604 * q^79 - 74357773216 * q^80 + 73615849126 * q^82 + 12253964262 * q^83 + 58779027600 * q^85 - 90522557252 * q^86 + 33736356800 * q^88 - 10662817760 * q^89 - 28336741418 * q^91 - 32954076160 * q^92 + 285145948346 * q^94 + 64104297380 * q^95 - 124519454530 * q^97 - 215615498272 * 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 $

$2$	−14.8702	−0.328589	−0.164295	−	0.986411i	$-0.552535\pi$
$2$	−14.8702	−0.328589	−0.164295	+	0.986411i	$0.552535\pi$
$3$	0	0
$3$	0	0
$4$	−1826.88	−0.892029
$5$	−2118.93	−0.303237	−0.151618	−	0.988439i	$-0.548449\pi$
$5$	−2118.93	−0.303237	−0.151618	+	0.988439i	$0.548449\pi$
$6$	0	0
$7$	−68580.9	−1.54228	−0.771141	−	0.636665i	$-0.780314\pi$
$7$	−68580.9	−1.54228	−0.771141	+	0.636665i	$0.780314\pi$
$8$	57620.3	0.621700
$9$	0	0
$10$	31509.0	0.0996403
$11$	573905.	1.07444	0.537218	−	0.843444i	$-0.319475\pi$
$11$	573905.	1.07444	0.537218	+	0.843444i	$0.319475\pi$
$12$	0	0
$13$	−209778.	−0.156701	−0.0783504	−	0.996926i	$-0.524965\pi$
$13$	−209778.	−0.156701	−0.0783504	+	0.996926i	$0.524965\pi$
$14$	1.01981e6	0.506777
$15$	0	0
$16$	2.88461e6	0.687745
$17$	−7.02586e6	−1.20014	−0.600068	−	0.799949i	$-0.704860\pi$
$17$	−7.02586e6	−1.20014	−0.600068	+	0.799949i	$0.704860\pi$
$18$	0	0
$19$	−1.37759e7	−1.27636	−0.638181	−	0.769886i	$-0.720313\pi$
$19$	−1.37759e7	−1.27636	−0.638181	+	0.769886i	$0.720313\pi$
$20$	3.87103e6	0.270496
$21$	0	0
$22$	−8.53411e6	−0.353048
$23$	−6.43634e6	−0.208514
$23$	−6.43634e6	−0.208514
$24$	0	0
$25$	−4.43382e7	−0.908047
$26$	3.11945e6	0.0514902
$27$	0	0
$28$	1.25289e8	1.37576
$29$	1.90149e8	1.72150	0.860748	−	0.509032i	$-0.169996\pi$
$29$	1.90149e8	1.72150	0.860748	+	0.509032i	$0.169996\pi$
$30$	0	0
$31$	1.11283e8	0.698136	0.349068	−	0.937097i	$-0.386498\pi$
$31$	1.11283e8	0.698136	0.349068	+	0.937097i	$0.386498\pi$
$32$	−1.60901e8	−0.847686
$33$	0	0
$34$	1.04476e8	0.394351
$35$	1.45318e8	0.467677
$36$	0	0
$37$	3.70363e8	0.878047	0.439024	−	0.898475i	$-0.355324\pi$
$37$	3.70363e8	0.878047	0.439024	+	0.898475i	$0.355324\pi$
$38$	2.04850e8	0.419399
$39$	0	0
$40$	−1.22094e8	−0.188522
$41$	4.78021e8	0.644371	0.322185	−	0.946677i	$-0.395583\pi$
$41$	4.78021e8	0.644371	0.322185	+	0.946677i	$0.395583\pi$
$42$	0	0
$43$	1.32298e9	1.37239	0.686196	−	0.727417i	$-0.259279\pi$
$43$	1.32298e9	1.37239	0.686196	+	0.727417i	$0.259279\pi$
$44$	−1.04845e9	−0.958428
$45$	0	0
$46$	9.57100e7	0.0685155
$47$	−1.04187e9	−0.662639	−0.331319	−	0.943519i	$-0.607494\pi$
$47$	−1.04187e9	−0.662639	−0.331319	+	0.943519i	$0.607494\pi$
$48$	0	0
$49$	2.72601e9	1.37863
$50$	6.59320e8	0.298374
$51$	0	0
$52$	3.83238e8	0.139782
$53$	3.65404e9	1.20021	0.600103	−	0.799923i	$-0.295126\pi$
$53$	3.65404e9	1.20021	0.600103	+	0.799923i	$0.295126\pi$
$54$	0	0
$55$	−1.21607e9	−0.325809
$56$	−3.95165e9	−0.958837
$57$	0	0
$58$	−2.82757e9	−0.565665
$59$	−9.91859e8	−0.180619	−0.0903096	−	0.995914i	$-0.528786\pi$
$59$	−9.91859e8	−0.180619	−0.0903096	+	0.995914i	$0.528786\pi$
$60$	0	0
$61$	−5.85821e6	−0.000888077 0	−0.000444039	−	1.00000i	$-0.500141\pi$
$61$	−5.85821e6	−0.000888077 0	−0.000444039		1.00000i	$0.500141\pi$
$62$	−1.65481e9	−0.229400
$63$	0	0
$64$	−3.51505e9	−0.409205
$65$	4.44505e8	0.0475175
$66$	0	0
$67$	1.92877e10	1.74530	0.872650	−	0.488346i	$-0.162400\pi$
$67$	1.92877e10	1.74530	0.872650	+	0.488346i	$0.162400\pi$
$68$	1.28354e10	1.07056
$69$	0	0
$70$	−2.16092e9	−0.153673
$71$	5.27575e9	0.347027	0.173514	−	0.984831i	$-0.444488\pi$
$71$	5.27575e9	0.347027	0.173514	+	0.984831i	$0.444488\pi$
$72$	0	0
$73$	−2.46322e10	−1.39068	−0.695340	−	0.718681i	$-0.744746\pi$
$73$	−2.46322e10	−1.39068	−0.695340	+	0.718681i	$0.744746\pi$
$74$	−5.50739e9	−0.288517
$75$	0	0
$76$	2.51668e10	1.13855
$77$	−3.93589e10	−1.65708
$78$	0	0
$79$	9.77106e9	0.357267	0.178633	−	0.983916i	$-0.442832\pi$
$79$	9.77106e9	0.357267	0.178633	+	0.983916i	$0.442832\pi$
$80$	−6.11230e9	−0.208550
$81$	0	0
$82$	−7.10829e9	−0.211733
$83$	1.67805e10	0.467602	0.233801	−	0.972285i	$-0.424884\pi$
$83$	1.67805e10	0.467602	0.233801	+	0.972285i	$0.424884\pi$
$84$	0	0
$85$	1.48873e10	0.363926
$86$	−1.96731e10	−0.450953
$87$	0	0
$88$	3.30686e10	0.667977
$89$	3.63782e10	0.690551	0.345275	−	0.938501i	$-0.387786\pi$
$89$	3.63782e10	0.690551	0.345275	+	0.938501i	$0.387786\pi$
$90$	0	0
$91$	1.43867e10	0.241677
$92$	1.17584e10	0.186001
$93$	0	0
$94$	1.54929e10	0.217736
$95$	2.91901e10	0.387040
$96$	0	0
$97$	5.73626e10	0.678241	0.339120	−	0.940743i	$-0.389871\pi$
$97$	5.73626e10	0.678241	0.339120	+	0.940743i	$0.389871\pi$
$98$	−4.05364e10	−0.453004
$99$	0	0
$100$	8.10005e10	0.810005
$101$	−1.17247e11	−1.11003	−0.555014	−	0.831841i	$-0.687287\pi$
$101$	−1.17247e11	−1.11003	−0.555014	+	0.831841i	$0.687287\pi$
$102$	0	0
$103$	−8.05838e10	−0.684925	−0.342463	−	0.939531i	$-0.611261\pi$
$103$	−8.05838e10	−0.684925	−0.342463	+	0.939531i	$0.611261\pi$
$104$	−1.20875e10	−0.0974209
$105$	0	0
$106$	−5.43364e10	−0.394374
$107$	−5.11195e10	−0.352351	−0.176176	−	0.984359i	$-0.556373\pi$
$107$	−5.11195e10	−0.352351	−0.176176	+	0.984359i	$0.556373\pi$
$108$	0	0
$109$	−2.34009e11	−1.45675	−0.728377	−	0.685177i	$-0.759725\pi$
$109$	−2.34009e11	−1.45675	−0.728377	+	0.685177i	$0.759725\pi$
$110$	1.80832e10	0.107057
$111$	0	0
$112$	−1.97829e11	−1.06070
$113$	9.72965e10	0.496782	0.248391	−	0.968660i	$-0.420098\pi$
$113$	9.72965e10	0.496782	0.248391	+	0.968660i	$0.420098\pi$
$114$	0	0
$115$	1.36382e10	0.0632293
$116$	−3.47379e11	−1.53562
$117$	0	0
$118$	1.47492e10	0.0593495
$119$	4.81840e11	1.85095
$120$	0	0
$121$	4.40555e10	0.154412
$122$	8.71130e7	0.000291812 0
$123$	0	0
$124$	−2.03300e11	−0.622758
$125$	1.97413e11	0.578591
$126$	0	0
$127$	−3.16122e11	−0.849051	−0.424525	−	0.905416i	$-0.639559\pi$
$127$	−3.16122e11	−0.849051	−0.424525	+	0.905416i	$0.639559\pi$
$128$	3.81796e11	0.982146
$129$	0	0
$130$	−6.60990e9	−0.0156137
$131$	2.48312e11	0.562348	0.281174	−	0.959657i	$-0.409276\pi$
$131$	2.48312e11	0.562348	0.281174	+	0.959657i	$0.409276\pi$
$132$	0	0
$133$	9.44760e11	1.96851
$134$	−2.86813e11	−0.573486
$135$	0	0
$136$	−4.04832e11	−0.746125
$137$	−2.69695e11	−0.477429	−0.238715	−	0.971090i	$-0.576726\pi$
$137$	−2.69695e11	−0.477429	−0.238715	+	0.971090i	$0.576726\pi$
$138$	0	0
$139$	−9.35942e11	−1.52992	−0.764958	−	0.644081i	$-0.777240\pi$
$139$	−9.35942e11	−1.52992	−0.764958	+	0.644081i	$0.777240\pi$
$140$	−2.65478e11	−0.417181
$141$	0	0
$142$	−7.84517e10	−0.114029
$143$	−1.20393e11	−0.168365
$144$	0	0
$145$	−4.02914e11	−0.522021
$146$	3.66287e11	0.456962
$147$	0	0
$148$	−6.76607e11	−0.783244
$149$	1.52904e12	1.70566	0.852831	−	0.522186i	$-0.174883\pi$
$149$	1.52904e12	1.70566	0.852831	+	0.522186i	$0.174883\pi$
$150$	0	0
$151$	4.04164e11	0.418972	0.209486	−	0.977812i	$-0.432821\pi$
$151$	4.04164e11	0.418972	0.209486	+	0.977812i	$0.432821\pi$
$152$	−7.93770e11	−0.793514
$153$	0	0
$154$	5.85276e11	0.544499
$155$	−2.35801e11	−0.211701
$156$	0	0
$157$	1.85179e12	1.54933	0.774665	−	0.632372i	$-0.217919\pi$
$157$	1.85179e12	1.54933	0.774665	+	0.632372i	$0.217919\pi$
$158$	−1.45298e11	−0.117394
$159$	0	0
$160$	3.40939e11	0.257050
$161$	4.41410e11	0.321588
$162$	0	0
$163$	1.39551e12	0.949950	0.474975	−	0.879999i	$-0.342457\pi$
$163$	1.39551e12	0.949950	0.474975	+	0.879999i	$0.342457\pi$
$164$	−8.73285e11	−0.574798
$165$	0	0
$166$	−2.49530e11	−0.153649
$167$	−2.13124e12	−1.26967	−0.634836	−	0.772647i	$-0.718932\pi$
$167$	−2.13124e12	−1.26967	−0.634836	+	0.772647i	$0.718932\pi$
$168$	0	0
$169$	−1.74815e12	−0.975445
$170$	−2.21378e11	−0.119582
$171$	0	0
$172$	−2.41693e12	−1.22421
$173$	−1.26158e12	−0.618960	−0.309480	−	0.950906i	$-0.600155\pi$
$173$	−1.26158e12	−0.618960	−0.309480	+	0.950906i	$0.600155\pi$
$174$	0	0
$175$	3.04076e12	1.40046
$176$	1.65549e12	0.738938
$177$	0	0
$178$	−5.40952e11	−0.226907
$179$	4.34895e12	1.76886	0.884429	−	0.466674i	$-0.154548\pi$
$179$	4.34895e12	1.76886	0.884429	+	0.466674i	$0.154548\pi$
$180$	0	0
$181$	−4.47831e12	−1.71349	−0.856747	−	0.515737i	$-0.827518\pi$
$181$	−4.47831e12	−1.71349	−0.856747	+	0.515737i	$0.827518\pi$
$182$	−2.13934e11	−0.0794123
$183$	0	0
$184$	−3.70864e11	−0.129633
$185$	−7.84774e11	−0.266256
$186$	0	0
$187$	−4.03218e12	−1.28947
$188$	1.90337e12	0.591093
$189$	0	0
$190$	−4.34064e11	−0.127177
$191$	−1.33967e12	−0.381341	−0.190671	−	0.981654i	$-0.561066\pi$
$191$	−1.33967e12	−0.381341	−0.190671	+	0.981654i	$0.561066\pi$
$192$	0	0
$193$	−3.43747e12	−0.924002	−0.462001	−	0.886879i	$-0.652868\pi$
$193$	−3.43747e12	−0.924002	−0.462001	+	0.886879i	$0.652868\pi$
$194$	−8.52995e11	−0.222862
$195$	0	0
$196$	−4.98008e12	−1.22978
$197$	−4.01962e12	−0.965207	−0.482603	−	0.875839i	$-0.660309\pi$
$197$	−4.01962e12	−0.965207	−0.482603	+	0.875839i	$0.660309\pi$
$198$	0	0
$199$	1.99472e12	0.453097	0.226548	−	0.974000i	$-0.427256\pi$
$199$	1.99472e12	0.453097	0.226548	+	0.974000i	$0.427256\pi$
$200$	−2.55478e12	−0.564533
$201$	0	0
$202$	1.74349e12	0.364743
$203$	−1.30406e13	−2.65503
$204$	0	0
$205$	−1.01289e12	−0.195397
$206$	1.19830e12	0.225059
$207$	0	0
$208$	−6.05128e11	−0.107770
$209$	−7.90604e12	−1.37137
$210$	0	0
$211$	−4.55008e12	−0.748972	−0.374486	−	0.927233i	$-0.622181\pi$
$211$	−4.55008e12	−0.748972	−0.374486	+	0.927233i	$0.622181\pi$
$212$	−6.67547e12	−1.07062
$213$	0	0
$214$	7.60160e11	0.115779
$215$	−2.80332e12	−0.416160
$216$	0	0
$217$	−7.63189e12	−1.07672
$218$	3.47976e12	0.478673
$219$	0	0
$220$	2.22160e12	0.290631
$221$	1.47387e12	0.188062
$222$	0	0
$223$	−6.35776e12	−0.772018	−0.386009	−	0.922495i	$-0.626147\pi$
$223$	−6.35776e12	−0.772018	−0.386009	+	0.922495i	$0.626147\pi$
$224$	1.10348e13	1.30737
$225$	0	0
$226$	−1.44682e12	−0.163237
$227$	−1.62023e13	−1.78416	−0.892080	−	0.451878i	$-0.850754\pi$
$227$	−1.62023e13	−1.78416	−0.892080	+	0.451878i	$0.850754\pi$
$228$	0	0
$229$	2.41365e12	0.253267	0.126633	−	0.991950i	$-0.459583\pi$
$229$	2.41365e12	0.253267	0.126633	+	0.991950i	$0.459583\pi$
$230$	−2.02803e11	−0.0207764
$231$	0	0
$232$	1.09565e13	1.07025
$233$	1.51989e13	1.44995	0.724976	−	0.688774i	$-0.241851\pi$
$233$	1.51989e13	1.44995	0.724976	+	0.688774i	$0.241851\pi$
$234$	0	0
$235$	2.20766e12	0.200937
$236$	1.81200e12	0.161118
$237$	0	0
$238$	−7.16507e12	−0.608201
$239$	−1.02197e13	−0.847717	−0.423858	−	0.905728i	$-0.639325\pi$
$239$	−1.02197e13	−0.847717	−0.423858	+	0.905728i	$0.639325\pi$
$240$	0	0
$241$	−6.38405e12	−0.505827	−0.252914	−	0.967489i	$-0.581389\pi$
$241$	−6.38405e12	−0.505827	−0.252914	+	0.967489i	$0.581389\pi$
$242$	−6.55115e11	−0.0507380
$243$	0	0
$244$	1.07022e10	0.000792191 0
$245$	−5.77623e12	−0.418052
$246$	0	0
$247$	2.88987e12	0.200007
$248$	6.41217e12	0.434031
$249$	0	0
$250$	−2.93558e12	−0.190118
$251$	−4.94359e12	−0.313211	−0.156605	−	0.987661i	$-0.550055\pi$
$251$	−4.94359e12	−0.313211	−0.156605	+	0.987661i	$0.550055\pi$
$252$	0	0
$253$	−3.69385e12	−0.224035
$254$	4.70080e12	0.278989
$255$	0	0
$256$	1.52142e12	0.0864829
$257$	2.89848e13	1.61264	0.806320	−	0.591480i	$-0.201456\pi$
$257$	2.89848e13	1.61264	0.806320	+	0.591480i	$0.201456\pi$
$258$	0	0
$259$	−2.53998e13	−1.35420
$260$	−8.12056e11	−0.0423870
$261$	0	0
$262$	−3.69246e12	−0.184782
$263$	3.37198e12	0.165245	0.0826224	−	0.996581i	$-0.473670\pi$
$263$	3.37198e12	0.165245	0.0826224	+	0.996581i	$0.473670\pi$
$264$	0	0
$265$	−7.74266e12	−0.363947
$266$	−1.40488e13	−0.646831
$267$	0	0
$268$	−3.52363e13	−1.55686
$269$	1.90487e13	0.824572	0.412286	−	0.911055i	$-0.364731\pi$
$269$	1.90487e13	0.824572	0.412286	+	0.911055i	$0.364731\pi$
$270$	0	0
$271$	2.01073e13	0.835647	0.417823	−	0.908528i	$-0.362793\pi$
$271$	2.01073e13	0.835647	0.417823	+	0.908528i	$0.362793\pi$
$272$	−2.02669e13	−0.825388
$273$	0	0
$274$	4.01042e12	0.156878
$275$	−2.54459e13	−0.975638
$276$	0	0
$277$	−3.85765e13	−1.42130	−0.710648	−	0.703548i	$-0.751598\pi$
$277$	−3.85765e13	−1.42130	−0.710648	+	0.703548i	$0.751598\pi$
$278$	1.39177e13	0.502713
$279$	0	0
$280$	8.37329e12	0.290755
$281$	1.54736e13	0.526873	0.263436	−	0.964677i	$-0.415144\pi$
$281$	1.54736e13	0.526873	0.263436	+	0.964677i	$0.415144\pi$
$282$	0	0
$283$	−1.52107e13	−0.498110	−0.249055	−	0.968489i	$-0.580120\pi$
$283$	−1.52107e13	−0.498110	−0.249055	+	0.968489i	$0.580120\pi$
$284$	−9.63815e12	−0.309558
$285$	0	0
$286$	1.79027e12	0.0553229
$287$	−3.27831e13	−0.993801
$288$	0	0
$289$	1.50908e13	0.440326
$290$	5.99142e12	0.171530
$291$	0	0
$292$	4.50000e13	1.24053
$293$	−6.67427e13	−1.80564	−0.902822	−	0.430014i	$-0.858509\pi$
$293$	−6.67427e13	−1.80564	−0.902822	+	0.430014i	$0.858509\pi$
$294$	0	0
$295$	2.10168e12	0.0547704
$296$	2.13404e13	0.545882
$297$	0	0
$298$	−2.27371e13	−0.560462
$299$	1.35020e12	0.0326744
$300$	0	0
$301$	−9.07314e13	−2.11662
$302$	−6.01002e12	−0.137670
$303$	0	0
$304$	−3.97380e13	−0.877812
$305$	1.24131e10	0.000269298 0
$306$	0	0
$307$	−9.49792e12	−0.198778	−0.0993888	−	0.995049i	$-0.531689\pi$
$307$	−9.49792e12	−0.198778	−0.0993888	+	0.995049i	$0.531689\pi$
$308$	7.19038e13	1.47817
$309$	0	0
$310$	3.50642e12	0.0695625
$311$	−1.92082e13	−0.374373	−0.187186	−	0.982324i	$-0.559937\pi$
$311$	−1.92082e13	−0.374373	−0.187186	+	0.982324i	$0.559937\pi$
$312$	0	0
$313$	9.70177e13	1.82540	0.912698	−	0.408635i	$-0.133995\pi$
$313$	9.70177e13	1.82540	0.912698	+	0.408635i	$0.133995\pi$
$314$	−2.75366e13	−0.509093
$315$	0	0
$316$	−1.78505e13	−0.318692
$317$	2.10018e13	0.368494	0.184247	−	0.982880i	$-0.441015\pi$
$317$	2.10018e13	0.368494	0.184247	+	0.982880i	$0.441015\pi$
$318$	0	0
$319$	1.09128e14	1.84964
$320$	7.44815e12	0.124086
$321$	0	0
$322$	−6.56387e12	−0.105670
$323$	9.67872e13	1.53181
$324$	0	0
$325$	9.30118e12	0.142292
$326$	−2.07516e13	−0.312143
$327$	0	0
$328$	2.75437e13	0.400605
$329$	7.14526e13	1.02198
$330$	0	0
$331$	−3.65032e12	−0.0504983	−0.0252491	−	0.999681i	$-0.508038\pi$
$331$	−3.65032e12	−0.0504983	−0.0252491	+	0.999681i	$0.508038\pi$
$332$	−3.06559e13	−0.417114
$333$	0	0
$334$	3.16920e13	0.417200
$335$	−4.08694e13	−0.529239
$336$	0	0
$337$	−1.07930e14	−1.35263	−0.676314	−	0.736613i	$-0.736424\pi$
$337$	−1.07930e14	−1.35263	−0.676314	+	0.736613i	$0.736424\pi$
$338$	2.59955e13	0.320520
$339$	0	0
$340$	−2.71973e13	−0.324632
$341$	6.38659e13	0.750102
$342$	0	0
$343$	−5.13452e13	−0.583958
$344$	7.62308e13	0.853216
$345$	0	0
$346$	1.87601e13	0.203384
$347$	2.46710e13	0.263253	0.131627	−	0.991299i	$-0.457980\pi$
$347$	2.46710e13	0.263253	0.131627	+	0.991299i	$0.457980\pi$
$348$	0	0
$349$	1.80182e14	1.86282	0.931409	−	0.363973i	$-0.118580\pi$
$349$	1.80182e14	1.86282	0.931409	+	0.363973i	$0.118580\pi$
$350$	−4.52168e13	−0.460177
$351$	0	0
$352$	−9.23421e13	−0.910784
$353$	−1.75917e14	−1.70823	−0.854117	−	0.520081i	$-0.825902\pi$
$353$	−1.75917e14	−1.70823	−0.854117	+	0.520081i	$0.825902\pi$
$354$	0	0
$355$	−1.11790e13	−0.105231
$356$	−6.64584e13	−0.615992
$357$	0	0
$358$	−6.46700e13	−0.581228
$359$	1.07222e14	0.948993	0.474496	−	0.880257i	$-0.342630\pi$
$359$	1.07222e14	0.948993	0.474496	+	0.880257i	$0.342630\pi$
$360$	0	0
$361$	7.32840e13	0.629100
$362$	6.65936e13	0.563035
$363$	0	0
$364$	−2.62828e13	−0.215583
$365$	5.21940e13	0.421706
$366$	0	0
$367$	3.35932e13	0.263384	0.131692	−	0.991291i	$-0.457959\pi$
$367$	3.35932e13	0.263384	0.131692	+	0.991291i	$0.457959\pi$
$368$	−1.85664e13	−0.143405
$369$	0	0
$370$	1.16698e13	0.0874889
$371$	−2.50597e14	−1.85106
$372$	0	0
$373$	7.35501e13	0.527454	0.263727	−	0.964597i	$-0.415048\pi$
$373$	7.35501e13	0.527454	0.263727	+	0.964597i	$0.415048\pi$
$374$	5.99594e13	0.423705
$375$	0	0
$376$	−6.00331e13	−0.411963
$377$	−3.98891e13	−0.269760
$378$	0	0
$379$	2.26744e14	1.48943	0.744714	−	0.667384i	$-0.232586\pi$
$379$	2.26744e14	1.48943	0.744714	+	0.667384i	$0.232586\pi$
$380$	−5.33267e13	−0.345251
$381$	0	0
$382$	1.99212e13	0.125305
$383$	−1.59732e14	−0.990375	−0.495187	−	0.868786i	$-0.664901\pi$
$383$	−1.59732e14	−0.990375	−0.495187	+	0.868786i	$0.664901\pi$
$384$	0	0
$385$	8.33989e13	0.502489
$386$	5.11159e13	0.303617
$387$	0	0
$388$	−1.04794e14	−0.605011
$389$	−3.08043e14	−1.75343	−0.876714	−	0.481012i	$-0.840269\pi$
$389$	−3.08043e14	−1.75343	−0.876714	+	0.481012i	$0.840269\pi$
$390$	0	0
$391$	4.52208e13	0.250246
$392$	1.57073e14	0.857096
$393$	0	0
$394$	5.97726e13	0.317156
$395$	−2.07042e13	−0.108337
$396$	0	0
$397$	3.32327e14	1.69129	0.845645	−	0.533746i	$-0.179216\pi$
$397$	3.32327e14	1.69129	0.845645	+	0.533746i	$0.179216\pi$
$398$	−2.96620e13	−0.148883
$399$	0	0
$400$	−1.27899e14	−0.624505
$401$	−8.82588e13	−0.425074	−0.212537	−	0.977153i	$-0.568173\pi$
$401$	−8.82588e13	−0.425074	−0.212537	+	0.977153i	$0.568173\pi$
$402$	0	0
$403$	−2.33447e13	−0.109398
$404$	2.14195e14	0.990177
$405$	0	0
$406$	1.93917e14	0.872414
$407$	2.12553e14	0.943405
$408$	0	0
$409$	−1.46893e14	−0.634633	−0.317317	−	0.948320i	$-0.602782\pi$
$409$	−1.46893e14	−0.634633	−0.317317	+	0.948320i	$0.602782\pi$
$410$	1.50620e13	0.0642053
$411$	0	0
$412$	1.47217e14	0.610973
$413$	6.80225e13	0.278566
$414$	0	0
$415$	−3.55568e13	−0.141794
$416$	3.37535e13	0.132833
$417$	0	0
$418$	1.17565e14	0.450617
$419$	1.82043e14	0.688647	0.344323	−	0.938851i	$-0.388108\pi$
$419$	1.82043e14	0.688647	0.344323	+	0.938851i	$0.388108\pi$
$420$	0	0
$421$	−3.53391e14	−1.30228	−0.651139	−	0.758958i	$-0.725709\pi$
$421$	−3.53391e14	−1.30228	−0.651139	+	0.758958i	$0.725709\pi$
$422$	6.76608e13	0.246104
$423$	0	0
$424$	2.10547e14	0.746168
$425$	3.11514e14	1.08978
$426$	0	0
$427$	4.01761e11	0.00136966
$428$	9.33890e13	0.314308
$429$	0	0
$430$	4.16860e13	0.136746
$431$	8.37205e13	0.271148	0.135574	−	0.990767i	$-0.456712\pi$
$431$	8.37205e13	0.271148	0.135574	+	0.990767i	$0.456712\pi$
$432$	0	0
$433$	2.73556e13	0.0863700	0.0431850	−	0.999067i	$-0.486250\pi$
$433$	2.73556e13	0.0863700	0.0431850	+	0.999067i	$0.486250\pi$
$434$	1.13488e14	0.353799
$435$	0	0
$436$	4.27505e14	1.29947
$437$	8.86661e13	0.266140
$438$	0	0
$439$	1.60445e14	0.469646	0.234823	−	0.972038i	$-0.424549\pi$
$439$	1.60445e14	0.469646	0.234823	+	0.972038i	$0.424549\pi$
$440$	−7.00702e13	−0.202555
$441$	0	0
$442$	−2.19168e13	−0.0617952
$443$	−3.29401e14	−0.917284	−0.458642	−	0.888621i	$-0.651664\pi$
$443$	−3.29401e14	−0.917284	−0.458642	+	0.888621i	$0.651664\pi$
$444$	0	0
$445$	−7.70829e13	−0.209401
$446$	9.45414e13	0.253677
$447$	0	0
$448$	2.41065e14	0.631110
$449$	4.42737e14	1.14496	0.572481	−	0.819918i	$-0.305981\pi$
$449$	4.42737e14	1.14496	0.572481	+	0.819918i	$0.305981\pi$
$450$	0	0
$451$	2.74339e14	0.692335
$452$	−1.77749e14	−0.443144
$453$	0	0
$454$	2.40932e14	0.586255
$455$	−3.04845e13	−0.0732853
$456$	0	0
$457$	−1.58449e14	−0.371834	−0.185917	−	0.982565i	$-0.559526\pi$
$457$	−1.58449e14	−0.371834	−0.185917	+	0.982565i	$0.559526\pi$
$458$	−3.58915e13	−0.0832207
$459$	0	0
$460$	−2.49153e13	−0.0564024
$461$	2.05796e14	0.460343	0.230172	−	0.973150i	$-0.426071\pi$
$461$	2.05796e14	0.460343	0.230172	+	0.973150i	$0.426071\pi$
$462$	0	0
$463$	4.02346e14	0.878828	0.439414	−	0.898285i	$-0.355186\pi$
$463$	4.02346e14	0.878828	0.439414	+	0.898285i	$0.355186\pi$
$464$	5.48507e14	1.18395
$465$	0	0
$466$	−2.26011e14	−0.476438
$467$	−5.67176e14	−1.18161	−0.590806	−	0.806814i	$-0.701190\pi$
$467$	−5.67176e14	−1.18161	−0.590806	+	0.806814i	$0.701190\pi$
$468$	0	0
$469$	−1.32277e15	−2.69174
$470$	−3.28284e13	−0.0660255
$471$	0	0
$472$	−5.71513e13	−0.112291
$473$	7.59268e14	1.47455
$474$	0	0
$475$	6.10797e14	1.15900
$476$	−8.80261e14	−1.65110
$477$	0	0
$478$	1.51970e14	0.278550
$479$	3.20416e14	0.580589	0.290294	−	0.956937i	$-0.406247\pi$
$479$	3.20416e14	0.580589	0.290294	+	0.956937i	$0.406247\pi$
$480$	0	0
$481$	−7.76939e13	−0.137591
$482$	9.49323e13	0.166209
$483$	0	0
$484$	−8.04839e13	−0.137740
$485$	−1.21547e14	−0.205668
$486$	0	0
$487$	−4.43098e14	−0.732977	−0.366488	−	0.930423i	$-0.619440\pi$
$487$	−4.43098e14	−0.732977	−0.366488	+	0.930423i	$0.619440\pi$
$488$	−3.37552e11	−0.000552118 0
$489$	0	0
$490$	8.58939e13	0.137367
$491$	1.03321e15	1.63396	0.816979	−	0.576668i	$-0.195647\pi$
$491$	1.03321e15	1.63396	0.816979	+	0.576668i	$0.195647\pi$
$492$	0	0
$493$	−1.33596e15	−2.06603
$494$	−4.29731e13	−0.0657201
$495$	0	0
$496$	3.21009e14	0.480140
$497$	−3.61816e14	−0.535214
$498$	0	0
$499$	2.42908e14	0.351471	0.175735	−	0.984437i	$-0.443770\pi$
$499$	2.42908e14	0.351471	0.175735	+	0.984437i	$0.443770\pi$
$500$	−3.60650e14	−0.516120
$501$	0	0
$502$	7.35123e13	0.102918
$503$	−1.21106e15	−1.67703	−0.838514	−	0.544880i	$-0.816575\pi$
$503$	−1.21106e15	−1.67703	−0.838514	+	0.544880i	$0.816575\pi$
$504$	0	0
$505$	2.48438e14	0.336601
$506$	5.49284e13	0.0736155
$507$	0	0
$508$	5.77515e14	0.757378
$509$	−3.69683e14	−0.479603	−0.239801	−	0.970822i	$-0.577082\pi$
$509$	−3.69683e14	−0.479603	−0.239801	+	0.970822i	$0.577082\pi$
$510$	0	0
$511$	1.68930e15	2.14482
$512$	−8.04541e14	−1.01056
$513$	0	0
$514$	−4.31010e14	−0.529896
$515$	1.70752e14	0.207695
$516$	0	0
$517$	−5.97937e14	−0.711963
$518$	3.77701e14	0.444974
$519$	0	0
$520$	2.56125e13	0.0295416
$521$	1.66121e14	0.189591	0.0947955	−	0.995497i	$-0.469780\pi$
$521$	1.66121e14	0.189591	0.0947955	+	0.995497i	$0.469780\pi$
$522$	0	0
$523$	1.37994e14	0.154206	0.0771030	−	0.997023i	$-0.475433\pi$
$523$	1.37994e14	0.154206	0.0771030	+	0.997023i	$0.475433\pi$
$524$	−4.53635e14	−0.501631
$525$	0	0
$526$	−5.01421e13	−0.0542977
$527$	−7.81860e14	−0.837858
$528$	0	0
$529$	4.14265e13	0.0434783
$530$	1.15135e14	0.119589
$531$	0	0
$532$	−1.72596e15	−1.75597
$533$	−1.00278e14	−0.100973
$534$	0	0
$535$	1.08319e14	0.106846
$536$	1.11137e15	1.08505
$537$	0	0
$538$	−2.83259e14	−0.270945
$539$	1.56447e15	1.48125
$540$	0	0
$541$	−1.11102e15	−1.03071	−0.515357	−	0.856976i	$-0.672341\pi$
$541$	−1.11102e15	−1.03071	−0.515357	+	0.856976i	$0.672341\pi$
$542$	−2.99001e14	−0.274584
$543$	0	0
$544$	1.13047e15	1.01734
$545$	4.95848e14	0.441742
$546$	0	0
$547$	−1.17036e15	−1.02185	−0.510925	−	0.859625i	$-0.670697\pi$
$547$	−1.17036e15	−1.02185	−0.510925	+	0.859625i	$0.670697\pi$
$548$	4.92699e14	0.425881
$549$	0	0
$550$	3.78387e14	0.320584
$551$	−2.61947e15	−2.19725
$552$	0	0
$553$	−6.70108e14	−0.551006
$554$	5.73642e14	0.467022
$555$	0	0
$556$	1.70985e15	1.36473
$557$	−1.83380e15	−1.44927	−0.724634	−	0.689133i	$-0.757991\pi$
$557$	−1.83380e15	−1.44927	−0.724634	+	0.689133i	$0.757991\pi$
$558$	0	0
$559$	−2.77533e14	−0.215055
$560$	4.19187e14	0.321643
$561$	0	0
$562$	−2.30096e14	−0.173125
$563$	8.49508e14	0.632953	0.316477	−	0.948600i	$-0.397500\pi$
$563$	8.49508e14	0.632953	0.316477	+	0.948600i	$0.397500\pi$
$564$	0	0
$565$	−2.06165e14	−0.150643
$566$	2.26187e14	0.163673
$567$	0	0
$568$	3.03991e14	0.215747
$569$	1.97273e15	1.38660	0.693298	−	0.720651i	$-0.256157\pi$
$569$	1.97273e15	1.38660	0.693298	+	0.720651i	$0.256157\pi$
$570$	0	0
$571$	1.62788e15	1.12234	0.561168	−	0.827702i	$-0.310352\pi$
$571$	1.62788e15	1.12234	0.561168	+	0.827702i	$0.310352\pi$
$572$	2.19942e14	0.150186
$573$	0	0
$574$	4.87493e14	0.326552
$575$	2.85376e14	0.189341
$576$	0	0
$577$	1.53888e15	1.00170	0.500851	−	0.865534i	$-0.333021\pi$
$577$	1.53888e15	1.00170	0.500851	+	0.865534i	$0.333021\pi$
$578$	−2.24404e14	−0.144686
$579$	0	0
$580$	7.36073e14	0.465658
$581$	−1.15082e15	−0.721173
$582$	0	0
$583$	2.09707e15	1.28954
$584$	−1.41932e15	−0.864586
$585$	0	0
$586$	9.92481e14	0.593315
$587$	1.62013e15	0.959487	0.479743	−	0.877409i	$-0.340730\pi$
$587$	1.62013e15	0.959487	0.479743	+	0.877409i	$0.340730\pi$
$588$	0	0
$589$	−1.53302e15	−0.891074
$590$	−3.12525e13	−0.0179970
$591$	0	0
$592$	1.06835e15	0.603873
$593$	−1.70158e15	−0.952910	−0.476455	−	0.879199i	$-0.658078\pi$
$593$	−1.70158e15	−0.952910	−0.476455	+	0.879199i	$0.658078\pi$
$594$	0	0
$595$	−1.02099e15	−0.561276
$596$	−2.79336e15	−1.52150
$597$	0	0
$598$	−2.00778e13	−0.0107364
$599$	2.43837e15	1.29197	0.645983	−	0.763351i	$-0.276447\pi$
$599$	2.43837e15	1.29197	0.645983	+	0.763351i	$0.276447\pi$
$600$	0	0
$601$	−3.03781e15	−1.58034	−0.790171	−	0.612887i	$-0.790008\pi$
$601$	−3.03781e15	−1.58034	−0.790171	+	0.612887i	$0.790008\pi$
$602$	1.34920e15	0.695497
$603$	0	0
$604$	−7.38358e14	−0.373735
$605$	−9.33505e13	−0.0468233
$606$	0	0
$607$	−2.46255e15	−1.21296	−0.606482	−	0.795097i	$-0.707420\pi$
$607$	−2.46255e15	−1.21296	−0.606482	+	0.795097i	$0.707420\pi$
$608$	2.21655e15	1.08195
$609$	0	0
$610$	−1.84586e11	−8.84883e−5 0
$611$	2.18562e14	0.103836
$612$	0	0
$613$	8.07779e13	0.0376929	0.0188465	−	0.999822i	$-0.494001\pi$
$613$	8.07779e13	0.0376929	0.0188465	+	0.999822i	$0.494001\pi$
$614$	1.41236e14	0.0653162
$615$	0	0
$616$	−2.26787e15	−1.03021
$617$	−2.12085e14	−0.0954864	−0.0477432	−	0.998860i	$-0.515203\pi$
$617$	−2.12085e14	−0.0954864	−0.0477432	+	0.998860i	$0.515203\pi$
$618$	0	0
$619$	−2.96321e15	−1.31058	−0.655291	−	0.755377i	$-0.727454\pi$
$619$	−2.96321e15	−1.31058	−0.655291	+	0.755377i	$0.727454\pi$
$620$	4.30780e14	0.188843
$621$	0	0
$622$	2.85630e14	0.123015
$623$	−2.49485e15	−1.06502
$624$	0	0
$625$	1.74665e15	0.732597
$626$	−1.44268e15	−0.599805
$627$	0	0
$628$	−3.38299e15	−1.38205
$629$	−2.60212e15	−1.05378
$630$	0	0
$631$	2.88905e15	1.14972	0.574862	−	0.818250i	$-0.305056\pi$
$631$	2.88905e15	1.14972	0.574862	+	0.818250i	$0.305056\pi$
$632$	5.63012e14	0.222113
$633$	0	0
$634$	−3.12302e14	−0.121083
$635$	6.69840e14	0.257464
$636$	0	0
$637$	−5.71856e14	−0.216033
$638$	−1.62276e15	−0.607770
$639$	0	0
$640$	−8.08999e14	−0.297823
$641$	−3.29686e15	−1.20332	−0.601660	−	0.798753i	$-0.705494\pi$
$641$	−3.29686e15	−1.20332	−0.601660	+	0.798753i	$0.705494\pi$
$642$	0	0
$643$	−4.92409e15	−1.76671	−0.883355	−	0.468704i	$-0.844721\pi$
$643$	−4.92409e15	−1.76671	−0.883355	+	0.468704i	$0.844721\pi$
$644$	−8.06401e14	−0.286866
$645$	0	0
$646$	−1.43925e15	−0.503335
$647$	−4.52679e15	−1.56970	−0.784851	−	0.619685i	$-0.787260\pi$
$647$	−4.52679e15	−1.56970	−0.784851	+	0.619685i	$0.787260\pi$
$648$	0	0
$649$	−5.69233e14	−0.194064
$650$	−1.38311e14	−0.0467555
$651$	0	0
$652$	−2.54942e15	−0.847383
$653$	3.02494e15	0.996998	0.498499	−	0.866890i	$-0.333885\pi$
$653$	3.02494e15	0.996998	0.498499	+	0.866890i	$0.333885\pi$
$654$	0	0
$655$	−5.26156e14	−0.170525
$656$	1.37891e15	0.443163
$657$	0	0
$658$	−1.06252e15	−0.335810
$659$	−1.43759e15	−0.450573	−0.225287	−	0.974293i	$-0.572332\pi$
$659$	−1.43759e15	−0.450573	−0.225287	+	0.974293i	$0.572332\pi$
$660$	0	0
$661$	−3.08710e15	−0.951575	−0.475788	−	0.879560i	$-0.657837\pi$
$661$	−3.08710e15	−0.951575	−0.475788	+	0.879560i	$0.657837\pi$
$662$	5.42811e13	0.0165932
$663$	0	0
$664$	9.66899e14	0.290708
$665$	−2.00188e15	−0.596925
$666$	0	0
$667$	−1.22387e15	−0.358957
$668$	3.89351e15	1.13258
$669$	0	0
$670$	6.07738e14	0.173902
$671$	−3.36206e12	−0.000954182 0
$672$	0	0
$673$	4.91496e15	1.37226	0.686131	−	0.727478i	$-0.259308\pi$
$673$	4.91496e15	1.37226	0.686131	+	0.727478i	$0.259308\pi$
$674$	1.60495e15	0.444459
$675$	0	0
$676$	3.19366e15	0.870125
$677$	1.19609e14	0.0323240	0.0161620	−	0.999869i	$-0.494855\pi$
$677$	1.19609e14	0.0323240	0.0161620	+	0.999869i	$0.494855\pi$
$678$	0	0
$679$	−3.93397e15	−1.04604
$680$	8.57813e14	0.226253
$681$	0	0
$682$	−9.49702e14	−0.246475
$683$	6.17337e15	1.58931	0.794655	−	0.607061i	$-0.207652\pi$
$683$	6.17337e15	1.58931	0.794655	+	0.607061i	$0.207652\pi$
$684$	0	0
$685$	5.71465e14	0.144774
$686$	7.63515e14	0.191882
$687$	0	0
$688$	3.81630e15	0.943857
$689$	−7.66536e14	−0.188073
$690$	0	0
$691$	−4.07938e15	−0.985066	−0.492533	−	0.870294i	$-0.663929\pi$
$691$	−4.07938e15	−0.985066	−0.492533	+	0.870294i	$0.663929\pi$
$692$	2.30476e15	0.552131
$693$	0	0
$694$	−3.66863e14	−0.0865021
$695$	1.98320e15	0.463927
$696$	0	0
$697$	−3.35851e15	−0.773332
$698$	−2.67934e15	−0.612102
$699$	0	0
$700$	−5.55508e15	−1.24926
$701$	−7.86951e15	−1.75590	−0.877948	−	0.478756i	$-0.841088\pi$
$701$	−7.86951e15	−1.75590	−0.877948	+	0.478756i	$0.841088\pi$
$702$	0	0
$703$	−5.10207e15	−1.12071
$704$	−2.01730e15	−0.439665
$705$	0	0
$706$	2.61593e15	0.561307
$707$	8.04089e15	1.71197
$708$	0	0
$709$	1.46428e15	0.306951	0.153475	−	0.988152i	$-0.450953\pi$
$709$	1.46428e15	0.306951	0.153475	+	0.988152i	$0.450953\pi$
$710$	1.66234e14	0.0345779
$711$	0	0
$712$	2.09612e15	0.429315
$713$	−7.16256e14	−0.145571
$714$	0	0
$715$	2.55104e14	0.0510545
$716$	−7.94500e15	−1.57787
$717$	0	0
$718$	−1.59441e15	−0.311829
$719$	5.54772e15	1.07673	0.538364	−	0.842713i	$-0.319043\pi$
$719$	5.54772e15	1.07673	0.538364	+	0.842713i	$0.319043\pi$
$720$	0	0
$721$	5.52651e15	1.05635
$722$	−1.08975e15	−0.206715
$723$	0	0
$724$	8.18133e15	1.52849
$725$	−8.43089e15	−1.56320
$726$	0	0
$727$	−3.53006e15	−0.644677	−0.322339	−	0.946624i	$-0.604469\pi$
$727$	−3.53006e15	−0.644677	−0.322339	+	0.946624i	$0.604469\pi$
$728$	8.28969e14	0.150250
$729$	0	0
$730$	−7.76137e14	−0.138568
$731$	−9.29510e15	−1.64706
$732$	0	0
$733$	−3.42088e15	−0.597126	−0.298563	−	0.954390i	$-0.596507\pi$
$733$	−3.42088e15	−0.597126	−0.298563	+	0.954390i	$0.596507\pi$
$734$	−4.99540e14	−0.0865449
$735$	0	0
$736$	1.03562e15	0.176755
$737$	1.10693e16	1.87521
$738$	0	0
$739$	9.78682e15	1.63342	0.816709	−	0.577050i	$-0.195796\pi$
$739$	9.78682e15	1.63342	0.816709	+	0.577050i	$0.195796\pi$
$740$	1.43368e15	0.237509
$741$	0	0
$742$	3.72644e15	0.608237
$743$	−2.89890e15	−0.469672	−0.234836	−	0.972035i	$-0.575455\pi$
$743$	−2.89890e15	−0.469672	−0.234836	+	0.972035i	$0.575455\pi$
$744$	0	0
$745$	−3.23992e15	−0.517220
$746$	−1.09371e15	−0.173316
$747$	0	0
$748$	7.36629e15	1.15024
$749$	3.50582e15	0.543425
$750$	0	0
$751$	−4.79339e15	−0.732188	−0.366094	−	0.930578i	$-0.619305\pi$
$751$	−4.79339e15	−0.732188	−0.366094	+	0.930578i	$0.619305\pi$
$752$	−3.00540e15	−0.455727
$753$	0	0
$754$	5.93161e14	0.0886401
$755$	−8.56397e14	−0.127048
$756$	0	0
$757$	−6.22282e15	−0.909830	−0.454915	−	0.890535i	$-0.650330\pi$
$757$	−6.22282e15	−0.909830	−0.454915	+	0.890535i	$0.650330\pi$
$758$	−3.37173e15	−0.489410
$759$	0	0
$760$	1.68194e15	0.240623
$761$	6.95542e15	0.987887	0.493944	−	0.869494i	$-0.335555\pi$
$761$	6.95542e15	0.987887	0.493944	+	0.869494i	$0.335555\pi$
$762$	0	0
$763$	1.60485e16	2.24672
$764$	2.44741e15	0.340168
$765$	0	0
$766$	2.37526e15	0.325426
$767$	2.08070e14	0.0283032
$768$	0	0
$769$	−2.09625e15	−0.281092	−0.140546	−	0.990074i	$-0.544886\pi$
$769$	−2.09625e15	−0.281092	−0.140546	+	0.990074i	$0.544886\pi$
$770$	−1.24016e15	−0.165112
$771$	0	0
$772$	6.27982e15	0.824237
$773$	6.55473e15	0.854216	0.427108	−	0.904201i	$-0.359532\pi$
$773$	6.55473e15	0.854216	0.427108	+	0.904201i	$0.359532\pi$
$774$	0	0
$775$	−4.93410e15	−0.633940
$776$	3.30525e15	0.421662
$777$	0	0
$778$	4.58067e15	0.576157
$779$	−6.58515e15	−0.822450
$780$	0	0
$781$	3.02778e15	0.372858
$782$	−6.72445e14	−0.0822280
$783$	0	0
$784$	7.86348e15	0.948148
$785$	−3.92382e15	−0.469814
$786$	0	0
$787$	−1.24777e16	−1.47324	−0.736620	−	0.676307i	$-0.763579\pi$
$787$	−1.24777e16	−1.47324	−0.736620	+	0.676307i	$0.763579\pi$
$788$	7.34334e15	0.860992
$789$	0	0
$790$	3.07877e14	0.0355982
$791$	−6.67268e15	−0.766178
$792$	0	0
$793$	1.22892e12	0.000139162 0
$794$	−4.94178e15	−0.555739
$795$	0	0
$796$	−3.64411e15	−0.404176
$797$	−3.38386e15	−0.372728	−0.186364	−	0.982481i	$-0.559670\pi$
$797$	−3.38386e15	−0.372728	−0.186364	+	0.982481i	$0.559670\pi$
$798$	0	0
$799$	7.32006e15	0.795257
$800$	7.13408e15	0.769739
$801$	0	0
$802$	1.31243e15	0.139674
$803$	−1.41365e16	−1.49420
$804$	0	0
$805$	−9.35318e14	−0.0975174
$806$	3.47142e14	0.0359471
$807$	0	0
$808$	−6.75580e15	−0.690104
$809$	1.06483e16	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$809$	1.06483e16	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$810$	0	0
$811$	8.28981e15	0.829716	0.414858	−	0.909886i	$-0.363831\pi$
$811$	8.28981e15	0.829716	0.414858	+	0.909886i	$0.363831\pi$
$812$	2.38236e16	2.36837
$813$	0	0
$814$	−3.16072e15	−0.309993
$815$	−2.95699e15	−0.288060
$816$	0	0
$817$	−1.82252e16	−1.75167
$818$	2.18433e15	0.208534
$819$	0	0
$820$	1.85043e15	0.174300
$821$	2.00661e16	1.87748	0.938741	−	0.344624i	$-0.111994\pi$
$821$	2.00661e16	1.87748	0.938741	+	0.344624i	$0.111994\pi$
$822$	0	0
$823$	1.00366e16	0.926592	0.463296	−	0.886203i	$-0.346667\pi$
$823$	1.00366e16	0.926592	0.463296	+	0.886203i	$0.346667\pi$
$824$	−4.64327e15	−0.425818
$825$	0	0
$826$	−1.01151e15	−0.0915336
$827$	−9.11917e15	−0.819738	−0.409869	−	0.912144i	$-0.634426\pi$
$827$	−9.11917e15	−0.819738	−0.409869	+	0.912144i	$0.634426\pi$
$828$	0	0
$829$	1.06230e16	0.942316	0.471158	−	0.882049i	$-0.343836\pi$
$829$	1.06230e16	0.942316	0.471158	+	0.882049i	$0.343836\pi$
$830$	5.28738e14	0.0465920
$831$	0	0
$832$	7.37379e14	0.0641228
$833$	−1.91525e16	−1.65455
$834$	0	0
$835$	4.51595e15	0.385011
$836$	1.44433e16	1.22330
$837$	0	0
$838$	−2.70702e15	−0.226282
$839$	7.33970e15	0.609520	0.304760	−	0.952429i	$-0.401424\pi$
$839$	7.33970e15	0.609520	0.304760	+	0.952429i	$0.401424\pi$
$840$	0	0
$841$	2.39563e16	1.96355
$842$	5.25501e15	0.427914
$843$	0	0
$844$	8.31243e15	0.668105
$845$	3.70422e15	0.295791
$846$	0	0
$847$	−3.02136e15	−0.238146
$848$	1.05405e16	0.825436
$849$	0	0
$850$	−4.63229e15	−0.358090
$851$	−2.38378e15	−0.183086
$852$	0	0
$853$	2.62255e16	1.98841	0.994203	−	0.107523i	$-0.0342918\pi$
$853$	2.62255e16	1.98841	0.994203	+	0.107523i	$0.0342918\pi$
$854$	−5.97428e12	−0.000450057 0
$855$	0	0
$856$	−2.94552e15	−0.219057
$857$	5.55008e15	0.410114	0.205057	−	0.978750i	$-0.434262\pi$
$857$	5.55008e15	0.410114	0.205057	+	0.978750i	$0.434262\pi$
$858$	0	0
$859$	7.00228e14	0.0510831	0.0255415	−	0.999674i	$-0.491869\pi$
$859$	7.00228e14	0.0510831	0.0255415	+	0.999674i	$0.491869\pi$
$860$	5.12131e15	0.371227
$861$	0	0
$862$	−1.24494e15	−0.0890964
$863$	1.74210e16	1.23884	0.619419	−	0.785061i	$-0.287368\pi$
$863$	1.74210e16	1.23884	0.619419	+	0.785061i	$0.287368\pi$
$864$	0	0
$865$	2.67321e15	0.187692
$866$	−4.06785e14	−0.0283802
$867$	0	0
$868$	1.39425e16	0.960468
$869$	5.60766e15	0.383860
$870$	0	0
$871$	−4.04614e15	−0.273490
$872$	−1.34837e16	−0.905664
$873$	0	0
$874$	−1.31849e15	−0.0874506
$875$	−1.35388e16	−0.892350
$876$	0	0
$877$	−1.23403e16	−0.803205	−0.401602	−	0.915814i	$-0.631547\pi$
$877$	−1.23403e16	−0.803205	−0.401602	+	0.915814i	$0.631547\pi$
$878$	−2.38585e15	−0.154320
$879$	0	0
$880$	−3.50788e15	−0.224073
$881$	−2.58898e16	−1.64347	−0.821735	−	0.569870i	$-0.806993\pi$
$881$	−2.58898e16	−1.64347	−0.821735	+	0.569870i	$0.806993\pi$
$882$	0	0
$883$	−3.56947e14	−0.0223779	−0.0111890	−	0.999937i	$-0.503562\pi$
$883$	−3.56947e14	−0.0223779	−0.0111890	+	0.999937i	$0.503562\pi$
$884$	−2.69258e15	−0.167757
$885$	0	0
$886$	4.89827e15	0.301409
$887$	−4.35822e15	−0.266519	−0.133260	−	0.991081i	$-0.542544\pi$
$887$	−4.35822e15	−0.266519	−0.133260	+	0.991081i	$0.542544\pi$
$888$	0	0
$889$	2.16799e16	1.30948
$890$	1.14624e15	0.0688067
$891$	0	0
$892$	1.16148e16	0.688663
$893$	1.43527e16	0.845767
$894$	0	0
$895$	−9.21514e15	−0.536383
$896$	−2.61839e16	−1.51475
$897$	0	0
$898$	−6.58361e15	−0.376222
$899$	2.11604e16	1.20184
$900$	0	0
$901$	−2.56728e16	−1.44041
$902$	−4.07948e15	−0.227494
$903$	0	0
$904$	5.60626e15	0.308849
$905$	9.48925e15	0.519595
$906$	0	0
$907$	−2.75194e15	−0.148867	−0.0744335	−	0.997226i	$-0.523715\pi$
$907$	−2.75194e15	−0.148867	−0.0744335	+	0.997226i	$0.523715\pi$
$908$	2.95995e16	1.59152
$909$	0	0
$910$	4.53313e14	0.0240808
$911$	1.35179e15	0.0713770	0.0356885	−	0.999363i	$-0.488638\pi$
$911$	1.35179e15	0.0713770	0.0356885	+	0.999363i	$0.488638\pi$
$912$	0	0
$913$	9.63043e15	0.502408
$914$	2.35617e15	0.122181
$915$	0	0
$916$	−4.40943e15	−0.225921
$917$	−1.70294e16	−0.867300
$918$	0	0
$919$	−1.00536e16	−0.505927	−0.252963	−	0.967476i	$-0.581405\pi$
$919$	−1.00536e16	−0.505927	−0.252963	+	0.967476i	$0.581405\pi$
$920$	7.85836e14	0.0393096
$921$	0	0
$922$	−3.06024e15	−0.151264
$923$	−1.10674e15	−0.0543794
$924$	0	0
$925$	−1.64212e16	−0.797309
$926$	−5.98298e15	−0.288773
$927$	0	0
$928$	−3.05953e16	−1.45929
$929$	−1.88670e16	−0.894575	−0.447288	−	0.894390i	$-0.647610\pi$
$929$	−1.88670e16	−0.894575	−0.447288	+	0.894390i	$0.647610\pi$
$930$	0	0
$931$	−3.75531e16	−1.75963
$932$	−2.77664e16	−1.29340
$933$	0	0
$934$	8.43404e15	0.388265
$935$	8.54391e15	0.391015
$936$	0	0
$937$	−2.56131e16	−1.15849	−0.579247	−	0.815152i	$-0.696653\pi$
$937$	−2.56131e16	−1.15849	−0.579247	+	0.815152i	$0.696653\pi$
$938$	1.96699e16	0.884477
$939$	0	0
$940$	−4.03312e15	−0.179241
$941$	6.57238e15	0.290389	0.145194	−	0.989403i	$-0.453619\pi$
$941$	6.57238e15	0.290389	0.145194	+	0.989403i	$0.453619\pi$
$942$	0	0
$943$	−3.07671e15	−0.134361
$944$	−2.86113e15	−0.124220
$945$	0	0
$946$	−1.12905e16	−0.484520
$947$	−3.14635e16	−1.34240	−0.671201	−	0.741276i	$-0.734221\pi$
$947$	−3.14635e16	−1.34240	−0.671201	+	0.741276i	$0.734221\pi$
$948$	0	0
$949$	5.16729e15	0.217921
$950$	−9.08270e15	−0.380834
$951$	0	0
$952$	2.77638e16	1.15073
$953$	2.72372e16	1.12241	0.561205	−	0.827677i	$-0.310338\pi$
$953$	2.72372e16	1.12241	0.561205	+	0.827677i	$0.310338\pi$
$954$	0	0
$955$	2.83867e15	0.115637
$956$	1.86702e16	0.756188
$957$	0	0
$958$	−4.76466e15	−0.190775
$959$	1.84959e16	0.736331
$960$	0	0
$961$	−1.30245e16	−0.512606
$962$	1.15533e15	0.0452108
$963$	0	0
$964$	1.16629e16	0.451213
$965$	7.28376e15	0.280192
$966$	0	0
$967$	2.83130e16	1.07681	0.538407	−	0.842685i	$-0.319026\pi$
$967$	2.83130e16	1.07681	0.538407	+	0.842685i	$0.319026\pi$
$968$	2.53849e15	0.0959977
$969$	0	0
$970$	1.80744e15	0.0675801
$971$	2.55713e16	0.950708	0.475354	−	0.879795i	$-0.342320\pi$
$971$	2.55713e16	0.950708	0.475354	+	0.879795i	$0.342320\pi$
$972$	0	0
$973$	6.41877e16	2.35956
$974$	6.58897e15	0.240848
$975$	0	0
$976$	−1.68987e13	−0.000610771 0
$977$	1.73935e16	0.625125	0.312563	−	0.949897i	$-0.398813\pi$
$977$	1.73935e16	0.625125	0.312563	+	0.949897i	$0.398813\pi$
$978$	0	0
$979$	2.08776e16	0.741952
$980$	1.05524e16	0.372915
$981$	0	0
$982$	−1.53641e16	−0.536900
$983$	−2.03114e16	−0.705822	−0.352911	−	0.935657i	$-0.614808\pi$
$983$	−2.03114e16	−0.705822	−0.352911	+	0.935657i	$0.614808\pi$
$984$	0	0
$985$	8.51729e15	0.292686
$986$	1.98661e16	0.678874
$987$	0	0
$988$	−5.27943e15	−0.178412
$989$	−8.51518e15	−0.286164
$990$	0	0
$991$	−1.72761e16	−0.574172	−0.287086	−	0.957905i	$-0.592686\pi$
$991$	−1.72761e16	−0.574172	−0.287086	+	0.957905i	$0.592686\pi$
$992$	−1.79056e16	−0.591800
$993$	0	0
$994$	5.38029e15	0.175865
$995$	−4.22669e15	−0.137396
$996$	0	0
$997$	−2.58661e16	−0.831588	−0.415794	−	0.909459i	$-0.636496\pi$
$997$	−2.58661e16	−0.831588	−0.415794	+	0.909459i	$0.636496\pi$
$998$	−3.61211e15	−0.115489
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.12.a.a.1.4		8
3.2	odd	2		23.12.a.a.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.5	✓	8	3.2	odd	2
207.12.a.a.1.4		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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-14.8702 q^{2} -1826.88 q^{4} -2118.93 q^{5} -68580.9 q^{7} +57620.3 q^{8} +O(q^{10})\)	\(q-14.8702 q^{2} -1826.88 q^{4} -2118.93 q^{5} -68580.9 q^{7} +57620.3 q^{8} +31509.0 q^{10} +573905. q^{11} -209778. q^{13} +1.01981e6 q^{14} +2.88461e6 q^{16} -7.02586e6 q^{17} -1.37759e7 q^{19} +3.87103e6 q^{20} -8.53411e6 q^{22} -6.43634e6 q^{23} -4.43382e7 q^{25} +3.11945e6 q^{26} +1.25289e8 q^{28} +1.90149e8 q^{29} +1.11283e8 q^{31} -1.60901e8 q^{32} +1.04476e8 q^{34} +1.45318e8 q^{35} +3.70363e8 q^{37} +2.04850e8 q^{38} -1.22094e8 q^{40} +4.78021e8 q^{41} +1.32298e9 q^{43} -1.04845e9 q^{44} +9.57100e7 q^{46} -1.04187e9 q^{47} +2.72601e9 q^{49} +6.59320e8 q^{50} +3.83238e8 q^{52} +3.65404e9 q^{53} -1.21607e9 q^{55} -3.95165e9 q^{56} -2.82757e9 q^{58} -9.91859e8 q^{59} -5.85821e6 q^{61} -1.65481e9 q^{62} -3.51505e9 q^{64} +4.44505e8 q^{65} +1.92877e10 q^{67} +1.28354e10 q^{68} -2.16092e9 q^{70} +5.27575e9 q^{71} -2.46322e10 q^{73} -5.50739e9 q^{74} +2.51668e10 q^{76} -3.93589e10 q^{77} +9.77106e9 q^{79} -6.11230e9 q^{80} -7.10829e9 q^{82} +1.67805e10 q^{83} +1.48873e10 q^{85} -1.96731e10 q^{86} +3.30686e10 q^{88} +3.63782e10 q^{89} +1.43867e10 q^{91} +1.17584e10 q^{92} +1.54929e10 q^{94} +2.91901e10 q^{95} +5.73626e10 q^{97} -4.05364e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{2} + 5120 q^{4} + 11466 q^{5} - 54118 q^{7} + 155568 q^{8}+O(q^{10}) \) 8 * q + 32 * q^2 + 5120 * q^4 + 11466 * q^5 - 54118 * q^7 + 155568 * q^8	\( 8 q + 32 q^{2} + 5120 q^{4} + 11466 q^{5} - 54118 q^{7} + 155568 q^{8} + 517892 q^{10} + 291462 q^{11} - 2211306 q^{13} + 939584 q^{14} - 8561344 q^{16} + 5775330 q^{17} - 21015588 q^{19} + 65503576 q^{20} - 83047784 q^{22} - 51490744 q^{23} - 36491644 q^{25} + 119299562 q^{26} - 392796032 q^{28} + 322285430 q^{29} - 415184840 q^{31} - 31831744 q^{32} - 28224252 q^{34} - 603721008 q^{35} + 176642018 q^{37} - 554685496 q^{38} + 1337904816 q^{40} - 357962218 q^{41} + 2500461376 q^{43} - 5064743472 q^{44} - 205962976 q^{46} - 261795200 q^{47} + 2656605924 q^{49} - 1642758328 q^{50} + 3841657212 q^{52} - 3542935060 q^{53} - 10100187604 q^{55} - 7995463104 q^{56} - 9113565454 q^{58} - 930905396 q^{59} - 25338655048 q^{61} - 4385691666 q^{62} - 34067008768 q^{64} + 25954746658 q^{65} - 3123467482 q^{67} + 37358480280 q^{68} - 35719175696 q^{70} + 52612263236 q^{71} - 67014176274 q^{73} - 10171443276 q^{74} + 17955918576 q^{76} + 44516617816 q^{77} - 27683357604 q^{79} - 74357773216 q^{80} + 73615849126 q^{82} + 12253964262 q^{83} + 58779027600 q^{85} - 90522557252 q^{86} + 33736356800 q^{88} - 10662817760 q^{89} - 28336741418 q^{91} - 32954076160 q^{92} + 285145948346 q^{94} + 64104297380 q^{95} - 124519454530 q^{97} - 215615498272 q^{98}+O(q^{100}) \) 8 * q + 32 * q^2 + 5120 * q^4 + 11466 * q^5 - 54118 * q^7 + 155568 * q^8 + 517892 * q^10 + 291462 * q^11 - 2211306 * q^13 + 939584 * q^14 - 8561344 * q^16 + 5775330 * q^17 - 21015588 * q^19 + 65503576 * q^20 - 83047784 * q^22 - 51490744 * q^23 - 36491644 * q^25 + 119299562 * q^26 - 392796032 * q^28 + 322285430 * q^29 - 415184840 * q^31 - 31831744 * q^32 - 28224252 * q^34 - 603721008 * q^35 + 176642018 * q^37 - 554685496 * q^38 + 1337904816 * q^40 - 357962218 * q^41 + 2500461376 * q^43 - 5064743472 * q^44 - 205962976 * q^46 - 261795200 * q^47 + 2656605924 * q^49 - 1642758328 * q^50 + 3841657212 * q^52 - 3542935060 * q^53 - 10100187604 * q^55 - 7995463104 * q^56 - 9113565454 * q^58 - 930905396 * q^59 - 25338655048 * q^61 - 4385691666 * q^62 - 34067008768 * q^64 + 25954746658 * q^65 - 3123467482 * q^67 + 37358480280 * q^68 - 35719175696 * q^70 + 52612263236 * q^71 - 67014176274 * q^73 - 10171443276 * q^74 + 17955918576 * q^76 + 44516617816 * q^77 - 27683357604 * q^79 - 74357773216 * q^80 + 73615849126 * q^82 + 12253964262 * q^83 + 58779027600 * q^85 - 90522557252 * q^86 + 33736356800 * q^88 - 10662817760 * q^89 - 28336741418 * q^91 - 32954076160 * q^92 + 285145948346 * q^94 + 64104297380 * q^95 - 124519454530 * q^97 - 215615498272 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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