LMFDB - Embedded newform 256.12.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,12,Mod(1,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,21040,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$196.695854223$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9138x^{2} + 12641832 $ x^4 - 9138*x^2 + 12641832
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$41.2252$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+329.802 q^{3} +12216.3 q^{5} -87187.3 q^{7} -68377.9 q^{9} -343802. q^{11} +774529. q^{13} +4.02896e6 q^{15} -2.60742e6 q^{17} +1.33083e7 q^{19} -2.87545e7 q^{21} -4.56234e7 q^{23} +1.00410e8 q^{25} -8.09745e7 q^{27} +1.15036e8 q^{29} -3.59943e7 q^{31} -1.13386e8 q^{33} -1.06511e9 q^{35} -3.36790e8 q^{37} +2.55441e8 q^{39} +5.13921e8 q^{41} -4.70170e8 q^{43} -8.35326e8 q^{45} +2.11278e9 q^{47} +5.62429e9 q^{49} -8.59932e8 q^{51} +4.25150e9 q^{53} -4.20000e9 q^{55} +4.38910e9 q^{57} +3.15524e8 q^{59} +4.85644e8 q^{61} +5.96168e9 q^{63} +9.46189e9 q^{65} +4.92012e9 q^{67} -1.50467e10 q^{69} +2.43357e10 q^{71} +1.87316e10 q^{73} +3.31155e10 q^{75} +2.99751e10 q^{77} -2.54101e10 q^{79} -1.45926e10 q^{81} +4.47849e10 q^{83} -3.18531e10 q^{85} +3.79392e10 q^{87} -7.21926e10 q^{89} -6.75290e10 q^{91} -1.18710e10 q^{93} +1.62579e11 q^{95} -3.03613e9 q^{97} +2.35084e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 21040 q^{5} + 461076 q^{9} - 981072 q^{13} - 15482760 q^{17} - 14846976 q^{21} + 108919500 q^{25} + 597997232 q^{29} - 41842944 q^{33} + 353870704 q^{37} + 1861173400 q^{41} - 2684766480 q^{45} + 9205489060 q^{49}+ \cdots + 273341582264 q^{97}+O(q^{100}) $ 4 * q + 21040 * q^5 + 461076 * q^9 - 981072 * q^13 - 15482760 * q^17 - 14846976 * q^21 + 108919500 * q^25 + 597997232 * q^29 - 41842944 * q^33 + 353870704 * q^37 + 1861173400 * q^41 - 2684766480 * q^45 + 9205489060 * q^49 - 793307344 * q^53 - 1077596928 * q^57 - 970979856 * q^61 + 23215691840 * q^65 - 25513915392 * q^69 + 49895617320 * q^73 + 68236479488 * q^77 - 19150006428 * q^81 - 46288531040 * q^85 - 243583370776 * q^89 + 251490557952 * q^93 + 273341582264 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	329.802	0.783585	0.391792	−	0.920054i	$-0.371855\pi$
$3$	329.802	0.783585	0.391792	+	0.920054i	$0.371855\pi$
$4$	0	0
$5$	12216.3	1.74826	0.874129	−	0.485694i	$-0.161433\pi$
$5$	12216.3	1.74826	0.874129	+	0.485694i	$0.161433\pi$
$6$	0	0
$7$	−87187.3	−1.96071	−0.980356	−	0.197237i	$-0.936803\pi$
$7$	−87187.3	−1.96071	−0.980356	+	0.197237i	$0.936803\pi$
$8$	0	0
$9$	−68377.9	−0.385995
$10$	0	0
$11$	−343802.	−0.643648	−0.321824	−	0.946799i	$-0.604296\pi$
$11$	−343802.	−0.643648	−0.321824	+	0.946799i	$0.604296\pi$
$12$	0	0
$13$	774529.	0.578561	0.289280	−	0.957244i	$-0.406584\pi$
$13$	774529.	0.578561	0.289280	+	0.957244i	$0.406584\pi$
$14$	0	0
$15$	4.02896e6	1.36991
$16$	0	0
$17$	−2.60742e6	−0.445392	−0.222696	−	0.974888i	$-0.571486\pi$
$17$	−2.60742e6	−0.445392	−0.222696	+	0.974888i	$0.571486\pi$
$18$	0	0
$19$	1.33083e7	1.23304	0.616522	−	0.787338i	$-0.288541\pi$
$19$	1.33083e7	1.23304	0.616522	+	0.787338i	$0.288541\pi$
$20$	0	0
$21$	−2.87545e7	−1.53638
$22$	0	0
$23$	−4.56234e7	−1.47803	−0.739017	−	0.673686i	$-0.764710\pi$
$23$	−4.56234e7	−1.47803	−0.739017	+	0.673686i	$0.764710\pi$
$24$	0	0
$25$	1.00410e8	2.05640
$26$	0	0
$27$	−8.09745e7	−1.08604
$28$	0	0
$29$	1.15036e8	1.04147	0.520734	−	0.853719i	$-0.325658\pi$
$29$	1.15036e8	1.04147	0.520734	+	0.853719i	$0.325658\pi$
$30$	0	0
$31$	−3.59943e7	−0.225810	−0.112905	−	0.993606i	$-0.536016\pi$
$31$	−3.59943e7	−0.225810	−0.112905	+	0.993606i	$0.536016\pi$
$32$	0	0
$33$	−1.13386e8	−0.504353
$34$	0	0
$35$	−1.06511e9	−3.42783
$36$	0	0
$37$	−3.36790e8	−0.798454	−0.399227	−	0.916852i	$-0.630722\pi$
$37$	−3.36790e8	−0.798454	−0.399227	+	0.916852i	$0.630722\pi$
$38$	0	0
$39$	2.55441e8	0.453351
$40$	0	0
$41$	5.13921e8	0.692763	0.346382	−	0.938094i	$-0.387410\pi$
$41$	5.13921e8	0.692763	0.346382	+	0.938094i	$0.387410\pi$
$42$	0	0
$43$	−4.70170e8	−0.487729	−0.243865	−	0.969809i	$-0.578415\pi$
$43$	−4.70170e8	−0.487729	−0.243865	+	0.969809i	$0.578415\pi$
$44$	0	0
$45$	−8.35326e8	−0.674819
$46$	0	0
$47$	2.11278e9	1.34374	0.671871	−	0.740668i	$-0.265491\pi$
$47$	2.11278e9	1.34374	0.671871	+	0.740668i	$0.265491\pi$
$48$	0	0
$49$	5.62429e9	2.84439
$50$	0	0
$51$	−8.59932e8	−0.349002
$52$	0	0
$53$	4.25150e9	1.39645	0.698224	−	0.715879i	$-0.253974\pi$
$53$	4.25150e9	1.39645	0.698224	+	0.715879i	$0.253974\pi$
$54$	0	0
$55$	−4.20000e9	−1.12526
$56$	0	0
$57$	4.38910e9	0.966194
$58$	0	0
$59$	3.15524e8	0.0574574	0.0287287	−	0.999587i	$-0.490854\pi$
$59$	3.15524e8	0.0574574	0.0287287	+	0.999587i	$0.490854\pi$
$60$	0	0
$61$	4.85644e8	0.0736214	0.0368107	−	0.999322i	$-0.488280\pi$
$61$	4.85644e8	0.0736214	0.0368107	+	0.999322i	$0.488280\pi$
$62$	0	0
$63$	5.96168e9	0.756825
$64$	0	0
$65$	9.46189e9	1.01147
$66$	0	0
$67$	4.92012e9	0.445209	0.222605	−	0.974909i	$-0.428544\pi$
$67$	4.92012e9	0.445209	0.222605	+	0.974909i	$0.428544\pi$
$68$	0	0
$69$	−1.50467e10	−1.15817
$70$	0	0
$71$	2.43357e10	1.60075	0.800375	−	0.599500i	$-0.204634\pi$
$71$	2.43357e10	1.60075	0.800375	+	0.599500i	$0.204634\pi$
$72$	0	0
$73$	1.87316e10	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$73$	1.87316e10	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$74$	0	0
$75$	3.31155e10	1.61137
$76$	0	0
$77$	2.99751e10	1.26201
$78$	0	0
$79$	−2.54101e10	−0.929089	−0.464545	−	0.885550i	$-0.653782\pi$
$79$	−2.54101e10	−0.929089	−0.464545	+	0.885550i	$0.653782\pi$
$80$	0	0
$81$	−1.45926e10	−0.465013
$82$	0	0
$83$	4.47849e10	1.24796	0.623982	−	0.781439i	$-0.285514\pi$
$83$	4.47849e10	1.24796	0.623982	+	0.781439i	$0.285514\pi$
$84$	0	0
$85$	−3.18531e10	−0.778660
$86$	0	0
$87$	3.79392e10	0.816078
$88$	0	0
$89$	−7.21926e10	−1.37040	−0.685200	−	0.728355i	$-0.740285\pi$
$89$	−7.21926e10	−1.37040	−0.685200	+	0.728355i	$0.740285\pi$
$90$	0	0
$91$	−6.75290e10	−1.13439
$92$	0	0
$93$	−1.18710e10	−0.176942
$94$	0	0
$95$	1.62579e11	2.15568
$96$	0	0
$97$	−3.03613e9	−0.0358984	−0.0179492	−	0.999839i	$-0.505714\pi$
$97$	−3.03613e9	−0.0358984	−0.0179492	+	0.999839i	$0.505714\pi$
$98$	0	0
$99$	2.35084e10	0.248445
$100$	0	0
$101$	2.00866e10	0.190168	0.0950842	−	0.995469i	$-0.469688\pi$
$101$	2.00866e10	0.190168	0.0950842	+	0.995469i	$0.469688\pi$
$102$	0	0
$103$	5.41042e10	0.459861	0.229930	−	0.973207i	$-0.426150\pi$
$103$	5.41042e10	0.459861	0.229930	+	0.973207i	$0.426150\pi$
$104$	0	0
$105$	−3.51274e11	−2.68599
$106$	0	0
$107$	−1.56922e11	−1.08162	−0.540808	−	0.841146i	$-0.681881\pi$
$107$	−1.56922e11	−1.08162	−0.540808	+	0.841146i	$0.681881\pi$
$108$	0	0
$109$	9.29760e10	0.578796	0.289398	−	0.957209i	$-0.406545\pi$
$109$	9.29760e10	0.578796	0.289398	+	0.957209i	$0.406545\pi$
$110$	0	0
$111$	−1.11074e11	−0.625657
$112$	0	0
$113$	1.99483e11	1.01853	0.509266	−	0.860609i	$-0.329917\pi$
$113$	1.99483e11	1.01853	0.509266	+	0.860609i	$0.329917\pi$
$114$	0	0
$115$	−5.57350e11	−2.58399
$116$	0	0
$117$	−5.29606e10	−0.223322
$118$	0	0
$119$	2.27334e11	0.873285
$120$	0	0
$121$	−1.67112e11	−0.585717
$122$	0	0
$123$	1.69492e11	0.542839
$124$	0	0
$125$	6.30145e11	1.84687
$126$	0	0
$127$	2.99734e11	0.805036	0.402518	−	0.915412i	$-0.368135\pi$
$127$	2.99734e11	0.805036	0.402518	+	0.915412i	$0.368135\pi$
$128$	0	0
$129$	−1.55063e11	−0.382177
$130$	0	0
$131$	2.06627e11	0.467945	0.233972	−	0.972243i	$-0.424827\pi$
$131$	2.06627e11	0.467945	0.233972	+	0.972243i	$0.424827\pi$
$132$	0	0
$133$	−1.16032e12	−2.41764
$134$	0	0
$135$	−9.89211e11	−1.89869
$136$	0	0
$137$	5.55695e10	0.0983725	0.0491862	−	0.998790i	$-0.484337\pi$
$137$	5.55695e10	0.0983725	0.0491862	+	0.998790i	$0.484337\pi$
$138$	0	0
$139$	8.41080e11	1.37485	0.687426	−	0.726254i	$-0.258740\pi$
$139$	8.41080e11	1.37485	0.687426	+	0.726254i	$0.258740\pi$
$140$	0	0
$141$	6.96798e11	1.05294
$142$	0	0
$143$	−2.66284e11	−0.372390
$144$	0	0
$145$	1.40532e12	1.82075
$146$	0	0
$147$	1.85490e12	2.22882
$148$	0	0
$149$	5.74018e11	0.640326	0.320163	−	0.947362i	$-0.396262\pi$
$149$	5.74018e11	0.640326	0.320163	+	0.947362i	$0.396262\pi$
$150$	0	0
$151$	8.98713e11	0.931639	0.465820	−	0.884880i	$-0.345760\pi$
$151$	8.98713e11	0.931639	0.465820	+	0.884880i	$0.345760\pi$
$152$	0	0
$153$	1.78290e11	0.171919
$154$	0	0
$155$	−4.39718e11	−0.394775
$156$	0	0
$157$	1.62105e12	1.35627	0.678136	−	0.734936i	$-0.262788\pi$
$157$	1.62105e12	1.35627	0.678136	+	0.734936i	$0.262788\pi$
$158$	0	0
$159$	1.40215e12	1.09424
$160$	0	0
$161$	3.97778e12	2.89800
$162$	0	0
$163$	−1.11004e12	−0.755623	−0.377812	−	0.925882i	$-0.623323\pi$
$163$	−1.11004e12	−0.755623	−0.377812	+	0.925882i	$0.623323\pi$
$164$	0	0
$165$	−1.38517e12	−0.881739
$166$	0	0
$167$	−1.02719e12	−0.611943	−0.305972	−	0.952041i	$-0.598981\pi$
$167$	−1.02719e12	−0.611943	−0.305972	+	0.952041i	$0.598981\pi$
$168$	0	0
$169$	−1.19227e12	−0.665267
$170$	0	0
$171$	−9.09994e11	−0.475949
$172$	0	0
$173$	−1.41195e12	−0.692732	−0.346366	−	0.938100i	$-0.612584\pi$
$173$	−1.41195e12	−0.692732	−0.346366	+	0.938100i	$0.612584\pi$
$174$	0	0
$175$	−8.75450e12	−4.03202
$176$	0	0
$177$	1.04060e11	0.0450227
$178$	0	0
$179$	−2.12297e12	−0.863481	−0.431740	−	0.901998i	$-0.642100\pi$
$179$	−2.12297e12	−0.863481	−0.431740	+	0.901998i	$0.642100\pi$
$180$	0	0
$181$	3.95208e12	1.51214	0.756072	−	0.654488i	$-0.227116\pi$
$181$	3.95208e12	1.51214	0.756072	+	0.654488i	$0.227116\pi$
$182$	0	0
$183$	1.60166e11	0.0576886
$184$	0	0
$185$	−4.11434e12	−1.39590
$186$	0	0
$187$	8.96437e11	0.286676
$188$	0	0
$189$	7.05994e12	2.12942
$190$	0	0
$191$	5.07410e12	1.44436	0.722180	−	0.691705i	$-0.243140\pi$
$191$	5.07410e12	1.44436	0.722180	+	0.691705i	$0.243140\pi$
$192$	0	0
$193$	−6.04183e12	−1.62406	−0.812032	−	0.583613i	$-0.801639\pi$
$193$	−6.04183e12	−1.62406	−0.812032	+	0.583613i	$0.801639\pi$
$194$	0	0
$195$	3.12055e12	0.792575
$196$	0	0
$197$	−2.26540e12	−0.543977	−0.271988	−	0.962301i	$-0.587681\pi$
$197$	−2.26540e12	−0.543977	−0.271988	+	0.962301i	$0.587681\pi$
$198$	0	0
$199$	−1.90553e12	−0.432836	−0.216418	−	0.976301i	$-0.569437\pi$
$199$	−1.90553e12	−0.432836	−0.216418	+	0.976301i	$0.569437\pi$
$200$	0	0
$201$	1.62266e12	0.348859
$202$	0	0
$203$	−1.00297e13	−2.04202
$204$	0	0
$205$	6.27822e12	1.21113
$206$	0	0
$207$	3.11963e12	0.570514
$208$	0	0
$209$	−4.57542e12	−0.793646
$210$	0	0
$211$	−5.51515e12	−0.907828	−0.453914	−	0.891045i	$-0.649973\pi$
$211$	−5.51515e12	−0.907828	−0.453914	+	0.891045i	$0.649973\pi$
$212$	0	0
$213$	8.02596e12	1.25432
$214$	0	0
$215$	−5.74375e12	−0.852676
$216$	0	0
$217$	3.13824e12	0.442749
$218$	0	0
$219$	6.17771e12	0.828677
$220$	0	0
$221$	−2.01952e12	−0.257686
$222$	0	0
$223$	7.92520e11	0.0962352	0.0481176	−	0.998842i	$-0.484678\pi$
$223$	7.92520e11	0.0962352	0.0481176	+	0.998842i	$0.484678\pi$
$224$	0	0
$225$	−6.86585e12	−0.793762
$226$	0	0
$227$	−2.98398e12	−0.328589	−0.164295	−	0.986411i	$-0.552535\pi$
$227$	−2.98398e12	−0.328589	−0.164295	+	0.986411i	$0.552535\pi$
$228$	0	0
$229$	−1.31479e12	−0.137962	−0.0689811	−	0.997618i	$-0.521975\pi$
$229$	−1.31479e12	−0.137962	−0.0689811	+	0.997618i	$0.521975\pi$
$230$	0	0
$231$	9.88585e12	0.988891
$232$	0	0
$233$	−9.18610e12	−0.876342	−0.438171	−	0.898892i	$-0.644374\pi$
$233$	−9.18610e12	−0.876342	−0.438171	+	0.898892i	$0.644374\pi$
$234$	0	0
$235$	2.58104e13	2.34921
$236$	0	0
$237$	−8.38030e12	−0.728020
$238$	0	0
$239$	7.02201e12	0.582469	0.291235	−	0.956652i	$-0.405934\pi$
$239$	7.02201e12	0.582469	0.291235	+	0.956652i	$0.405934\pi$
$240$	0	0
$241$	−3.97192e12	−0.314707	−0.157354	−	0.987542i	$-0.550296\pi$
$241$	−3.97192e12	−0.314707	−0.157354	+	0.987542i	$0.550296\pi$
$242$	0	0
$243$	9.53173e12	0.721668
$244$	0	0
$245$	6.87081e13	4.97273
$246$	0	0
$247$	1.03077e13	0.713390
$248$	0	0
$249$	1.47701e13	0.977886
$250$	0	0
$251$	2.09269e12	0.132586	0.0662931	−	0.997800i	$-0.478883\pi$
$251$	2.09269e12	0.132586	0.0662931	+	0.997800i	$0.478883\pi$
$252$	0	0
$253$	1.56854e13	0.951335
$254$	0	0
$255$	−1.05052e13	−0.610146
$256$	0	0
$257$	2.39680e13	1.33352	0.666761	−	0.745271i	$-0.267680\pi$
$257$	2.39680e13	1.33352	0.666761	+	0.745271i	$0.267680\pi$
$258$	0	0
$259$	2.93638e13	1.56554
$260$	0	0
$261$	−7.86594e12	−0.402002
$262$	0	0
$263$	7.29414e11	0.0357452	0.0178726	−	0.999840i	$-0.494311\pi$
$263$	7.29414e11	0.0357452	0.0178726	+	0.999840i	$0.494311\pi$
$264$	0	0
$265$	5.19377e13	2.44135
$266$	0	0
$267$	−2.38092e13	−1.07382
$268$	0	0
$269$	−6.09880e12	−0.264002	−0.132001	−	0.991250i	$-0.542140\pi$
$269$	−6.09880e12	−0.264002	−0.132001	+	0.991250i	$0.542140\pi$
$270$	0	0
$271$	4.32582e13	1.79779	0.898893	−	0.438169i	$-0.144373\pi$
$271$	4.32582e13	1.79779	0.898893	+	0.438169i	$0.144373\pi$
$272$	0	0
$273$	−2.22712e13	−0.888891
$274$	0	0
$275$	−3.45213e13	−1.32360
$276$	0	0
$277$	3.57553e13	1.31735	0.658675	−	0.752427i	$-0.271117\pi$
$277$	3.57553e13	1.31735	0.658675	+	0.752427i	$0.271117\pi$
$278$	0	0
$279$	2.46121e12	0.0871617
$280$	0	0
$281$	1.76325e13	0.600384	0.300192	−	0.953879i	$-0.402949\pi$
$281$	1.76325e13	0.600384	0.300192	+	0.953879i	$0.402949\pi$
$282$	0	0
$283$	−4.45842e13	−1.46001	−0.730005	−	0.683441i	$-0.760482\pi$
$283$	−4.45842e13	−1.46001	−0.730005	+	0.683441i	$0.760482\pi$
$284$	0	0
$285$	5.36187e13	1.68916
$286$	0	0
$287$	−4.48073e13	−1.35831
$288$	0	0
$289$	−2.74732e13	−0.801626
$290$	0	0
$291$	−1.00132e12	−0.0281295
$292$	0	0
$293$	4.87336e13	1.31843	0.659214	−	0.751955i	$-0.270889\pi$
$293$	4.87336e13	1.31843	0.659214	+	0.751955i	$0.270889\pi$
$294$	0	0
$295$	3.85454e12	0.100450
$296$	0	0
$297$	2.78392e13	0.699031
$298$	0	0
$299$	−3.53366e13	−0.855133
$300$	0	0
$301$	4.09929e13	0.956296
$302$	0	0
$303$	6.62458e12	0.149013
$304$	0	0
$305$	5.93279e12	0.128709
$306$	0	0
$307$	−3.59860e13	−0.753134	−0.376567	−	0.926389i	$-0.622896\pi$
$307$	−3.59860e13	−0.753134	−0.376567	+	0.926389i	$0.622896\pi$
$308$	0	0
$309$	1.78437e13	0.360340
$310$	0	0
$311$	3.80512e13	0.741627	0.370814	−	0.928707i	$-0.379079\pi$
$311$	3.80512e13	0.741627	0.370814	+	0.928707i	$0.379079\pi$
$312$	0	0
$313$	1.18672e13	0.223282	0.111641	−	0.993749i	$-0.464389\pi$
$313$	1.18672e13	0.223282	0.111641	+	0.993749i	$0.464389\pi$
$314$	0	0
$315$	7.28298e13	1.32313
$316$	0	0
$317$	−1.04037e13	−0.182541	−0.0912705	−	0.995826i	$-0.529093\pi$
$317$	−1.04037e13	−0.182541	−0.0912705	+	0.995826i	$0.529093\pi$
$318$	0	0
$319$	−3.95497e13	−0.670339
$320$	0	0
$321$	−5.17531e13	−0.847537
$322$	0	0
$323$	−3.47004e13	−0.549187
$324$	0	0
$325$	7.77707e13	1.18976
$326$	0	0
$327$	3.06636e13	0.453535
$328$	0	0
$329$	−1.84207e14	−2.63469
$330$	0	0
$331$	7.18424e13	0.993864	0.496932	−	0.867789i	$-0.334460\pi$
$331$	7.18424e13	0.993864	0.496932	+	0.867789i	$0.334460\pi$
$332$	0	0
$333$	2.30290e13	0.308199
$334$	0	0
$335$	6.01057e13	0.778340
$336$	0	0
$337$	8.66988e13	1.08655	0.543274	−	0.839556i	$-0.317185\pi$
$337$	8.66988e13	1.08655	0.543274	+	0.839556i	$0.317185\pi$
$338$	0	0
$339$	6.57899e13	0.798106
$340$	0	0
$341$	1.23749e13	0.145343
$342$	0	0
$343$	−3.17969e14	−3.61632
$344$	0	0
$345$	−1.83815e14	−2.02477
$346$	0	0
$347$	3.44960e13	0.368093	0.184046	−	0.982918i	$-0.441080\pi$
$347$	3.44960e13	0.368093	0.184046	+	0.982918i	$0.441080\pi$
$348$	0	0
$349$	1.00311e14	1.03707	0.518533	−	0.855057i	$-0.326478\pi$
$349$	1.00311e14	1.03707	0.518533	+	0.855057i	$0.326478\pi$
$350$	0	0
$351$	−6.27171e13	−0.628343
$352$	0	0
$353$	−6.53687e13	−0.634759	−0.317379	−	0.948299i	$-0.602803\pi$
$353$	−6.53687e13	−0.634759	−0.317379	+	0.948299i	$0.602803\pi$
$354$	0	0
$355$	2.97293e14	2.79852
$356$	0	0
$357$	7.49751e13	0.684293
$358$	0	0
$359$	−4.45057e13	−0.393909	−0.196955	−	0.980413i	$-0.563105\pi$
$359$	−4.45057e13	−0.393909	−0.196955	+	0.980413i	$0.563105\pi$
$360$	0	0
$361$	6.06210e13	0.520395
$362$	0	0
$363$	−5.51138e13	−0.458959
$364$	0	0
$365$	2.28831e14	1.84886
$366$	0	0
$367$	−6.03017e12	−0.0472788	−0.0236394	−	0.999721i	$-0.507525\pi$
$367$	−6.03017e12	−0.0472788	−0.0236394	+	0.999721i	$0.507525\pi$
$368$	0	0
$369$	−3.51408e13	−0.267403
$370$	0	0
$371$	−3.70677e14	−2.73803
$372$	0	0
$373$	−2.17811e14	−1.56200	−0.780999	−	0.624532i	$-0.785290\pi$
$373$	−2.17811e14	−1.56200	−0.780999	+	0.624532i	$0.785290\pi$
$374$	0	0
$375$	2.07823e14	1.44718
$376$	0	0
$377$	8.90989e13	0.602553
$378$	0	0
$379$	1.29514e14	0.850749	0.425375	−	0.905017i	$-0.360142\pi$
$379$	1.29514e14	0.850749	0.425375	+	0.905017i	$0.360142\pi$
$380$	0	0
$381$	9.88527e13	0.630814
$382$	0	0
$383$	1.06806e13	0.0662218	0.0331109	−	0.999452i	$-0.489459\pi$
$383$	1.06806e13	0.0662218	0.0331109	+	0.999452i	$0.489459\pi$
$384$	0	0
$385$	3.66186e14	2.20632
$386$	0	0
$387$	3.21492e13	0.188261
$388$	0	0
$389$	−1.43137e14	−0.814761	−0.407381	−	0.913258i	$-0.633558\pi$
$389$	−1.43137e14	−0.814761	−0.407381	+	0.913258i	$0.633558\pi$
$390$	0	0
$391$	1.18960e14	0.658305
$392$	0	0
$393$	6.81459e13	0.366674
$394$	0	0
$395$	−3.10418e14	−1.62429
$396$	0	0
$397$	1.67555e14	0.852725	0.426362	−	0.904552i	$-0.359795\pi$
$397$	1.67555e14	0.852725	0.426362	+	0.904552i	$0.359795\pi$
$398$	0	0
$399$	−3.82674e14	−1.89443
$400$	0	0
$401$	3.37877e13	0.162729	0.0813644	−	0.996684i	$-0.474072\pi$
$401$	3.37877e13	0.162729	0.0813644	+	0.996684i	$0.474072\pi$
$402$	0	0
$403$	−2.78786e13	−0.130645
$404$	0	0
$405$	−1.78268e14	−0.812962
$406$	0	0
$407$	1.15789e14	0.513924
$408$	0	0
$409$	7.46538e13	0.322533	0.161266	−	0.986911i	$-0.448442\pi$
$409$	7.46538e13	0.322533	0.161266	+	0.986911i	$0.448442\pi$
$410$	0	0
$411$	1.83269e13	0.0770832
$412$	0	0
$413$	−2.75096e13	−0.112657
$414$	0	0
$415$	5.47107e14	2.18176
$416$	0	0
$417$	2.77390e14	1.07731
$418$	0	0
$419$	−1.38266e14	−0.523043	−0.261522	−	0.965198i	$-0.584224\pi$
$419$	−1.38266e14	−0.523043	−0.261522	+	0.965198i	$0.584224\pi$
$420$	0	0
$421$	3.57094e14	1.31593	0.657963	−	0.753050i	$-0.271418\pi$
$421$	3.57094e14	1.31593	0.657963	+	0.753050i	$0.271418\pi$
$422$	0	0
$423$	−1.44467e14	−0.518678
$424$	0	0
$425$	−2.61812e14	−0.915906
$426$	0	0
$427$	−4.23420e13	−0.144350
$428$	0	0
$429$	−8.78211e13	−0.291799
$430$	0	0
$431$	−1.31241e14	−0.425055	−0.212527	−	0.977155i	$-0.568169\pi$
$431$	−1.31241e14	−0.425055	−0.212527	+	0.977155i	$0.568169\pi$
$432$	0	0
$433$	5.07589e14	1.60261	0.801307	−	0.598253i	$-0.204138\pi$
$433$	5.07589e14	1.60261	0.801307	+	0.598253i	$0.204138\pi$
$434$	0	0
$435$	4.63477e14	1.42672
$436$	0	0
$437$	−6.07171e14	−1.82248
$438$	0	0
$439$	4.24766e14	1.24336	0.621678	−	0.783273i	$-0.286451\pi$
$439$	4.24766e14	1.24336	0.621678	+	0.783273i	$0.286451\pi$
$440$	0	0
$441$	−3.84577e14	−1.09792
$442$	0	0
$443$	−5.86227e14	−1.63247	−0.816235	−	0.577720i	$-0.803943\pi$
$443$	−5.86227e14	−1.63247	−0.816235	+	0.577720i	$0.803943\pi$
$444$	0	0
$445$	−8.81927e14	−2.39581
$446$	0	0
$447$	1.89312e14	0.501750
$448$	0	0
$449$	−5.48965e14	−1.41968	−0.709839	−	0.704364i	$-0.751232\pi$
$449$	−5.48965e14	−1.41968	−0.709839	+	0.704364i	$0.751232\pi$
$450$	0	0
$451$	−1.76687e14	−0.445896
$452$	0	0
$453$	2.96397e14	0.730018
$454$	0	0
$455$	−8.24956e14	−1.98321
$456$	0	0
$457$	5.19789e14	1.21980	0.609899	−	0.792479i	$-0.291210\pi$
$457$	5.19789e14	1.21980	0.609899	+	0.792479i	$0.291210\pi$
$458$	0	0
$459$	2.11135e14	0.483715
$460$	0	0
$461$	8.03876e13	0.179818	0.0899092	−	0.995950i	$-0.471342\pi$
$461$	8.03876e13	0.179818	0.0899092	+	0.995950i	$0.471342\pi$
$462$	0	0
$463$	−7.27623e13	−0.158932	−0.0794659	−	0.996838i	$-0.525321\pi$
$463$	−7.27623e13	−0.158932	−0.0794659	+	0.996838i	$0.525321\pi$
$464$	0	0
$465$	−1.45020e14	−0.309339
$466$	0	0
$467$	−6.33095e14	−1.31894	−0.659472	−	0.751729i	$-0.729220\pi$
$467$	−6.33095e14	−1.31894	−0.659472	+	0.751729i	$0.729220\pi$
$468$	0	0
$469$	−4.28972e14	−0.872927
$470$	0	0
$471$	5.34623e14	1.06275
$472$	0	0
$473$	1.61646e14	0.313926
$474$	0	0
$475$	1.33629e15	2.53564
$476$	0	0
$477$	−2.90708e14	−0.539022
$478$	0	0
$479$	7.86403e14	1.42495	0.712475	−	0.701697i	$-0.247574\pi$
$479$	7.86403e14	1.42495	0.712475	+	0.701697i	$0.247574\pi$
$480$	0	0
$481$	−2.60854e14	−0.461954
$482$	0	0
$483$	1.31188e15	2.27083
$484$	0	0
$485$	−3.70903e13	−0.0627597
$486$	0	0
$487$	9.62029e14	1.59140	0.795699	−	0.605693i	$-0.207104\pi$
$487$	9.62029e14	1.59140	0.795699	+	0.605693i	$0.207104\pi$
$488$	0	0
$489$	−3.66092e14	−0.592095
$490$	0	0
$491$	1.11508e15	1.76343	0.881715	−	0.471783i	$-0.156389\pi$
$491$	1.11508e15	1.76343	0.881715	+	0.471783i	$0.156389\pi$
$492$	0	0
$493$	−2.99948e14	−0.463861
$494$	0	0
$495$	2.87187e14	0.434346
$496$	0	0
$497$	−2.12176e15	−3.13861
$498$	0	0
$499$	−9.02155e14	−1.30535	−0.652677	−	0.757636i	$-0.726354\pi$
$499$	−9.02155e14	−1.30535	−0.652677	+	0.757636i	$0.726354\pi$
$500$	0	0
$501$	−3.38770e14	−0.479509
$502$	0	0
$503$	−6.31803e14	−0.874899	−0.437450	−	0.899243i	$-0.644118\pi$
$503$	−6.31803e14	−0.874899	−0.437450	+	0.899243i	$0.644118\pi$
$504$	0	0
$505$	2.45384e14	0.332463
$506$	0	0
$507$	−3.93211e14	−0.521293
$508$	0	0
$509$	3.66995e13	0.0476115	0.0238058	−	0.999717i	$-0.492422\pi$
$509$	3.66995e13	0.0476115	0.0238058	+	0.999717i	$0.492422\pi$
$510$	0	0
$511$	−1.63316e15	−2.07354
$512$	0	0
$513$	−1.07763e15	−1.33914
$514$	0	0
$515$	6.60955e14	0.803955
$516$	0	0
$517$	−7.26377e14	−0.864897
$518$	0	0
$519$	−4.65663e14	−0.542814
$520$	0	0
$521$	6.54478e14	0.746943	0.373471	−	0.927642i	$-0.378167\pi$
$521$	6.54478e14	0.746943	0.373471	+	0.927642i	$0.378167\pi$
$522$	0	0
$523$	−1.55382e15	−1.73637	−0.868184	−	0.496243i	$-0.834713\pi$
$523$	−1.55382e15	−1.73637	−0.868184	+	0.496243i	$0.834713\pi$
$524$	0	0
$525$	−2.88725e15	−3.15943
$526$	0	0
$527$	9.38523e13	0.100574
$528$	0	0
$529$	1.12869e15	1.18459
$530$	0	0
$531$	−2.15748e13	−0.0221783
$532$	0	0
$533$	3.98046e14	0.400806
$534$	0	0
$535$	−1.91701e15	−1.89094
$536$	0	0
$537$	−7.00160e14	−0.676610
$538$	0	0
$539$	−1.93364e15	−1.83079
$540$	0	0
$541$	−2.04067e15	−1.89316	−0.946580	−	0.322469i	$-0.895487\pi$
$541$	−2.04067e15	−1.89316	−0.946580	+	0.322469i	$0.895487\pi$
$542$	0	0
$543$	1.30340e15	1.18489
$544$	0	0
$545$	1.13582e15	1.01188
$546$	0	0
$547$	−1.88570e15	−1.64642	−0.823212	−	0.567734i	$-0.807820\pi$
$547$	−1.88570e15	−1.64642	−0.823212	+	0.567734i	$0.807820\pi$
$548$	0	0
$549$	−3.32073e13	−0.0284175
$550$	0	0
$551$	1.53094e15	1.28417
$552$	0	0
$553$	2.21544e15	1.82168
$554$	0	0
$555$	−1.35692e15	−1.09381
$556$	0	0
$557$	−1.19371e15	−0.943398	−0.471699	−	0.881760i	$-0.656359\pi$
$557$	−1.19371e15	−0.943398	−0.471699	+	0.881760i	$0.656359\pi$
$558$	0	0
$559$	−3.64160e14	−0.282181
$560$	0	0
$561$	2.95646e14	0.224635
$562$	0	0
$563$	1.20186e15	0.895486	0.447743	−	0.894162i	$-0.352228\pi$
$563$	1.20186e15	0.895486	0.447743	+	0.894162i	$0.352228\pi$
$564$	0	0
$565$	2.43695e15	1.78066
$566$	0	0
$567$	1.27229e15	0.911756
$568$	0	0
$569$	7.09317e14	0.498567	0.249283	−	0.968431i	$-0.419805\pi$
$569$	7.09317e14	0.498567	0.249283	+	0.968431i	$0.419805\pi$
$570$	0	0
$571$	1.56094e15	1.07619	0.538093	−	0.842885i	$-0.319145\pi$
$571$	1.56094e15	1.07619	0.538093	+	0.842885i	$0.319145\pi$
$572$	0	0
$573$	1.67345e15	1.13178
$574$	0	0
$575$	−4.58106e15	−3.03944
$576$	0	0
$577$	1.00933e15	0.657003	0.328501	−	0.944504i	$-0.393456\pi$
$577$	1.00933e15	0.657003	0.328501	+	0.944504i	$0.393456\pi$
$578$	0	0
$579$	−1.99260e15	−1.27259
$580$	0	0
$581$	−3.90467e15	−2.44690
$582$	0	0
$583$	−1.46167e15	−0.898821
$584$	0	0
$585$	−6.46984e14	−0.390424
$586$	0	0
$587$	3.08654e15	1.82794	0.913969	−	0.405783i	$-0.133001\pi$
$587$	3.08654e15	1.82794	0.913969	+	0.405783i	$0.133001\pi$
$588$	0	0
$589$	−4.79023e14	−0.278434
$590$	0	0
$591$	−7.47132e14	−0.426252
$592$	0	0
$593$	8.19038e14	0.458673	0.229337	−	0.973347i	$-0.426344\pi$
$593$	8.19038e14	0.458673	0.229337	+	0.973347i	$0.426344\pi$
$594$	0	0
$595$	2.77718e15	1.52673
$596$	0	0
$597$	−6.28446e14	−0.339163
$598$	0	0
$599$	−1.57478e15	−0.834395	−0.417197	−	0.908816i	$-0.636988\pi$
$599$	−1.57478e15	−0.834395	−0.417197	+	0.908816i	$0.636988\pi$
$600$	0	0
$601$	2.82611e15	1.47021	0.735104	−	0.677954i	$-0.237133\pi$
$601$	2.82611e15	1.47021	0.735104	+	0.677954i	$0.237133\pi$
$602$	0	0
$603$	−3.36427e14	−0.171849
$604$	0	0
$605$	−2.04149e15	−1.02398
$606$	0	0
$607$	1.49123e15	0.734525	0.367262	−	0.930117i	$-0.380295\pi$
$607$	1.49123e15	0.734525	0.367262	+	0.930117i	$0.380295\pi$
$608$	0	0
$609$	−3.30781e15	−1.60009
$610$	0	0
$611$	1.63641e15	0.777436
$612$	0	0
$613$	−2.41453e15	−1.12668	−0.563338	−	0.826226i	$-0.690483\pi$
$613$	−2.41453e15	−1.12668	−0.563338	+	0.826226i	$0.690483\pi$
$614$	0	0
$615$	2.07057e15	0.949022
$616$	0	0
$617$	−1.05127e15	−0.473312	−0.236656	−	0.971593i	$-0.576051\pi$
$617$	−1.05127e15	−0.473312	−0.236656	+	0.971593i	$0.576051\pi$
$618$	0	0
$619$	−1.16435e15	−0.514973	−0.257486	−	0.966282i	$-0.582894\pi$
$619$	−1.16435e15	−0.514973	−0.257486	+	0.966282i	$0.582894\pi$
$620$	0	0
$621$	3.69433e15	1.60521
$622$	0	0
$623$	6.29427e15	2.68696
$624$	0	0
$625$	2.79521e15	1.17239
$626$	0	0
$627$	−1.50898e15	−0.621889
$628$	0	0
$629$	8.78154e14	0.355625
$630$	0	0
$631$	−2.63120e14	−0.104711	−0.0523556	−	0.998629i	$-0.516673\pi$
$631$	−2.63120e14	−0.104711	−0.0523556	+	0.998629i	$0.516673\pi$
$632$	0	0
$633$	−1.81891e15	−0.711360
$634$	0	0
$635$	3.66165e15	1.40741
$636$	0	0
$637$	4.35617e15	1.64565
$638$	0	0
$639$	−1.66402e15	−0.617881
$640$	0	0
$641$	−1.52056e14	−0.0554988	−0.0277494	−	0.999615i	$-0.508834\pi$
$641$	−1.52056e14	−0.0554988	−0.0277494	+	0.999615i	$0.508834\pi$
$642$	0	0
$643$	1.44509e15	0.518483	0.259242	−	0.965813i	$-0.416527\pi$
$643$	1.44509e15	0.518483	0.259242	+	0.965813i	$0.416527\pi$
$644$	0	0
$645$	−1.89430e15	−0.668144
$646$	0	0
$647$	−1.89856e15	−0.658339	−0.329170	−	0.944271i	$-0.606769\pi$
$647$	−1.89856e15	−0.658339	−0.329170	+	0.944271i	$0.606769\pi$
$648$	0	0
$649$	−1.08478e14	−0.0369824
$650$	0	0
$651$	1.03500e15	0.346931
$652$	0	0
$653$	−1.32626e15	−0.437127	−0.218563	−	0.975823i	$-0.570137\pi$
$653$	−1.32626e15	−0.437127	−0.218563	+	0.975823i	$0.570137\pi$
$654$	0	0
$655$	2.52422e15	0.818088
$656$	0	0
$657$	−1.28083e15	−0.408208
$658$	0	0
$659$	2.07026e15	0.648867	0.324433	−	0.945909i	$-0.394826\pi$
$659$	2.07026e15	0.648867	0.324433	+	0.945909i	$0.394826\pi$
$660$	0	0
$661$	9.26477e14	0.285579	0.142790	−	0.989753i	$-0.454393\pi$
$661$	9.26477e14	0.285579	0.142790	+	0.989753i	$0.454393\pi$
$662$	0	0
$663$	−6.66042e14	−0.201919
$664$	0	0
$665$	−1.41748e16	−4.22666
$666$	0	0
$667$	−5.24835e15	−1.53933
$668$	0	0
$669$	2.61375e14	0.0754084
$670$	0	0
$671$	−1.66966e14	−0.0473863
$672$	0	0
$673$	−2.59168e14	−0.0723599	−0.0361800	−	0.999345i	$-0.511519\pi$
$673$	−2.59168e14	−0.0723599	−0.0361800	+	0.999345i	$0.511519\pi$
$674$	0	0
$675$	−8.13068e15	−2.23335
$676$	0	0
$677$	3.41525e15	0.922964	0.461482	−	0.887150i	$-0.347318\pi$
$677$	3.41525e15	0.922964	0.461482	+	0.887150i	$0.347318\pi$
$678$	0	0
$679$	2.64712e14	0.0703865
$680$	0	0
$681$	−9.84120e14	−0.257477
$682$	0	0
$683$	2.59376e15	0.667752	0.333876	−	0.942617i	$-0.391643\pi$
$683$	2.59376e15	0.667752	0.333876	+	0.942617i	$0.391643\pi$
$684$	0	0
$685$	6.78855e14	0.171980
$686$	0	0
$687$	−4.33619e14	−0.108105
$688$	0	0
$689$	3.29291e15	0.807930
$690$	0	0
$691$	9.78422e14	0.236264	0.118132	−	0.992998i	$-0.462309\pi$
$691$	9.78422e14	0.236264	0.118132	+	0.992998i	$0.462309\pi$
$692$	0	0
$693$	−2.04964e15	−0.487129
$694$	0	0
$695$	1.02749e16	2.40360
$696$	0	0
$697$	−1.34001e15	−0.308551
$698$	0	0
$699$	−3.02959e15	−0.686688
$700$	0	0
$701$	5.65457e15	1.26168	0.630842	−	0.775911i	$-0.282709\pi$
$701$	5.65457e15	1.26168	0.630842	+	0.775911i	$0.282709\pi$
$702$	0	0
$703$	−4.48211e15	−0.984529
$704$	0	0
$705$	8.51230e15	1.84080
$706$	0	0
$707$	−1.75129e15	−0.372865
$708$	0	0
$709$	−5.15753e15	−1.08115	−0.540577	−	0.841294i	$-0.681794\pi$
$709$	−5.15753e15	−1.08115	−0.540577	+	0.841294i	$0.681794\pi$
$710$	0	0
$711$	1.73749e15	0.358624
$712$	0	0
$713$	1.64218e15	0.333756
$714$	0	0
$715$	−3.25302e15	−0.651033
$716$	0	0
$717$	2.31587e15	0.456414
$718$	0	0
$719$	−8.70637e15	−1.68977	−0.844887	−	0.534945i	$-0.820332\pi$
$719$	−8.70637e15	−1.68977	−0.844887	+	0.534945i	$0.820332\pi$
$720$	0	0
$721$	−4.71720e15	−0.901655
$722$	0	0
$723$	−1.30994e15	−0.246600
$724$	0	0
$725$	1.15508e16	2.14168
$726$	0	0
$727$	−5.72086e15	−1.04477	−0.522386	−	0.852709i	$-0.674958\pi$
$727$	−5.72086e15	−1.04477	−0.522386	+	0.852709i	$0.674958\pi$
$728$	0	0
$729$	5.72861e15	1.03050
$730$	0	0
$731$	1.22593e15	0.217231
$732$	0	0
$733$	6.79839e15	1.18668	0.593341	−	0.804952i	$-0.297809\pi$
$733$	6.79839e15	1.18668	0.593341	+	0.804952i	$0.297809\pi$
$734$	0	0
$735$	2.26601e16	3.89655
$736$	0	0
$737$	−1.69155e15	−0.286558
$738$	0	0
$739$	1.30629e15	0.218019	0.109010	−	0.994041i	$-0.465232\pi$
$739$	1.30629e15	0.218019	0.109010	+	0.994041i	$0.465232\pi$
$740$	0	0
$741$	3.39949e15	0.559002
$742$	0	0
$743$	6.49965e14	0.105306	0.0526528	−	0.998613i	$-0.483232\pi$
$743$	6.49965e14	0.105306	0.0526528	+	0.998613i	$0.483232\pi$
$744$	0	0
$745$	7.01239e15	1.11945
$746$	0	0
$747$	−3.06230e15	−0.481708
$748$	0	0
$749$	1.36816e16	2.12074
$750$	0	0
$751$	−5.74255e15	−0.877173	−0.438587	−	0.898689i	$-0.644521\pi$
$751$	−5.74255e15	−0.877173	−0.438587	+	0.898689i	$0.644521\pi$
$752$	0	0
$753$	6.90171e14	0.103893
$754$	0	0
$755$	1.09790e16	1.62875
$756$	0	0
$757$	−5.29523e15	−0.774208	−0.387104	−	0.922036i	$-0.626525\pi$
$757$	−5.29523e15	−0.774208	−0.387104	+	0.922036i	$0.626525\pi$
$758$	0	0
$759$	5.17308e15	0.745451
$760$	0	0
$761$	−1.07060e15	−0.152059	−0.0760296	−	0.997106i	$-0.524224\pi$
$761$	−1.07060e15	−0.152059	−0.0760296	+	0.997106i	$0.524224\pi$
$762$	0	0
$763$	−8.10632e15	−1.13485
$764$	0	0
$765$	2.17805e15	0.300559
$766$	0	0
$767$	2.44382e14	0.0332426
$768$	0	0
$769$	−9.93161e15	−1.33176	−0.665878	−	0.746061i	$-0.731943\pi$
$769$	−9.93161e15	−1.33176	−0.665878	+	0.746061i	$0.731943\pi$
$770$	0	0
$771$	7.90470e15	1.04493
$772$	0	0
$773$	4.99261e15	0.650640	0.325320	−	0.945604i	$-0.394528\pi$
$773$	4.99261e15	0.650640	0.325320	+	0.945604i	$0.394528\pi$
$774$	0	0
$775$	−3.61420e15	−0.464358
$776$	0	0
$777$	9.68424e15	1.22673
$778$	0	0
$779$	6.83942e15	0.854207
$780$	0	0
$781$	−8.36667e15	−1.03032
$782$	0	0
$783$	−9.31501e15	−1.13108
$784$	0	0
$785$	1.98032e16	2.37111
$786$	0	0
$787$	−7.77761e15	−0.918301	−0.459151	−	0.888358i	$-0.651846\pi$
$787$	−7.77761e15	−0.918301	−0.459151	+	0.888358i	$0.651846\pi$
$788$	0	0
$789$	2.40562e14	0.0280094
$790$	0	0
$791$	−1.73924e16	−1.99705
$792$	0	0
$793$	3.76146e14	0.0425945
$794$	0	0
$795$	1.71291e16	1.91301
$796$	0	0
$797$	−6.99888e15	−0.770917	−0.385458	−	0.922725i	$-0.625957\pi$
$797$	−6.99888e15	−0.770917	−0.385458	+	0.922725i	$0.625957\pi$
$798$	0	0
$799$	−5.50890e15	−0.598491
$800$	0	0
$801$	4.93637e15	0.528968
$802$	0	0
$803$	−6.43996e15	−0.680688
$804$	0	0
$805$	4.85938e16	5.06645
$806$	0	0
$807$	−2.01139e15	−0.206868
$808$	0	0
$809$	1.13614e16	1.15270	0.576349	−	0.817204i	$-0.304477\pi$
$809$	1.13614e16	1.15270	0.576349	+	0.817204i	$0.304477\pi$
$810$	0	0
$811$	1.88346e16	1.88513	0.942565	−	0.334023i	$-0.108406\pi$
$811$	1.88346e16	1.88513	0.942565	+	0.334023i	$0.108406\pi$
$812$	0	0
$813$	1.42666e16	1.40872
$814$	0	0
$815$	−1.35606e16	−1.32102
$816$	0	0
$817$	−6.25718e15	−0.601391
$818$	0	0
$819$	4.61749e15	0.437869
$820$	0	0
$821$	−1.01440e16	−0.949124	−0.474562	−	0.880222i	$-0.657394\pi$
$821$	−1.01440e16	−0.949124	−0.474562	+	0.880222i	$0.657394\pi$
$822$	0	0
$823$	1.69907e16	1.56860	0.784302	−	0.620380i	$-0.213022\pi$
$823$	1.69907e16	1.56860	0.784302	+	0.620380i	$0.213022\pi$
$824$	0	0
$825$	−1.13852e16	−1.03715
$826$	0	0
$827$	−8.28965e15	−0.745171	−0.372585	−	0.927998i	$-0.621529\pi$
$827$	−8.28965e15	−0.745171	−0.372585	+	0.927998i	$0.621529\pi$
$828$	0	0
$829$	−1.12816e16	−1.00074	−0.500371	−	0.865811i	$-0.666803\pi$
$829$	−1.12816e16	−1.00074	−0.500371	+	0.865811i	$0.666803\pi$
$830$	0	0
$831$	1.17921e16	1.03226
$832$	0	0
$833$	−1.46649e16	−1.26687
$834$	0	0
$835$	−1.25485e16	−1.06983
$836$	0	0
$837$	2.91462e15	0.245240
$838$	0	0
$839$	−1.63246e16	−1.35566	−0.677831	−	0.735218i	$-0.737080\pi$
$839$	−1.63246e16	−1.35566	−0.677831	+	0.735218i	$0.737080\pi$
$840$	0	0
$841$	1.03284e15	0.0846556
$842$	0	0
$843$	5.81523e15	0.470452
$844$	0	0
$845$	−1.45651e16	−1.16306
$846$	0	0
$847$	1.45700e16	1.14842
$848$	0	0
$849$	−1.47040e16	−1.14404
$850$	0	0
$851$	1.53655e16	1.18014
$852$	0	0
$853$	3.95556e15	0.299908	0.149954	−	0.988693i	$-0.452087\pi$
$853$	3.95556e15	0.299908	0.149954	+	0.988693i	$0.452087\pi$
$854$	0	0
$855$	−1.11168e16	−0.832081
$856$	0	0
$857$	−1.65618e15	−0.122381	−0.0611905	−	0.998126i	$-0.519490\pi$
$857$	−1.65618e15	−0.122381	−0.0611905	+	0.998126i	$0.519490\pi$
$858$	0	0
$859$	−1.44522e16	−1.05432	−0.527158	−	0.849767i	$-0.676742\pi$
$859$	−1.44522e16	−1.05432	−0.527158	+	0.849767i	$0.676742\pi$
$860$	0	0
$861$	−1.47775e16	−1.06435
$862$	0	0
$863$	−1.30297e16	−0.926563	−0.463282	−	0.886211i	$-0.653328\pi$
$863$	−1.30297e16	−0.926563	−0.463282	+	0.886211i	$0.653328\pi$
$864$	0	0
$865$	−1.72488e16	−1.21107
$866$	0	0
$867$	−9.06072e15	−0.628142
$868$	0	0
$869$	8.73605e15	0.598007
$870$	0	0
$871$	3.81077e15	0.257581
$872$	0	0
$873$	2.07604e14	0.0138566
$874$	0	0
$875$	−5.49406e16	−3.62117
$876$	0	0
$877$	1.96397e16	1.27831	0.639156	−	0.769077i	$-0.279284\pi$
$877$	1.96397e16	1.27831	0.639156	+	0.769077i	$0.279284\pi$
$878$	0	0
$879$	1.60724e16	1.03310
$880$	0	0
$881$	−1.63850e16	−1.04011	−0.520053	−	0.854134i	$-0.674088\pi$
$881$	−1.63850e16	−1.04011	−0.520053	+	0.854134i	$0.674088\pi$
$882$	0	0
$883$	−2.44046e16	−1.52999	−0.764994	−	0.644038i	$-0.777258\pi$
$883$	−2.44046e16	−1.52999	−0.764994	+	0.644038i	$0.777258\pi$
$884$	0	0
$885$	1.27123e15	0.0787113
$886$	0	0
$887$	1.97735e15	0.120921	0.0604607	−	0.998171i	$-0.480743\pi$
$887$	1.97735e15	0.120921	0.0604607	+	0.998171i	$0.480743\pi$
$888$	0	0
$889$	−2.61330e16	−1.57844
$890$	0	0
$891$	5.01696e15	0.299305
$892$	0	0
$893$	2.81175e16	1.65689
$894$	0	0
$895$	−2.59349e16	−1.50959
$896$	0	0
$897$	−1.16541e16	−0.670069
$898$	0	0
$899$	−4.14065e15	−0.235174
$900$	0	0
$901$	−1.10855e16	−0.621967
$902$	0	0
$903$	1.35195e16	0.749339
$904$	0	0
$905$	4.82798e16	2.64362
$906$	0	0
$907$	2.16469e15	0.117099	0.0585497	−	0.998284i	$-0.481352\pi$
$907$	2.16469e15	0.117099	0.0585497	+	0.998284i	$0.481352\pi$
$908$	0	0
$909$	−1.37348e15	−0.0734040
$910$	0	0
$911$	2.47285e16	1.30571	0.652856	−	0.757482i	$-0.273571\pi$
$911$	2.47285e16	1.30571	0.652856	+	0.757482i	$0.273571\pi$
$912$	0	0
$913$	−1.53971e16	−0.803250
$914$	0	0
$915$	1.95664e15	0.100855
$916$	0	0
$917$	−1.80152e16	−0.917505
$918$	0	0
$919$	1.86239e16	0.937208	0.468604	−	0.883408i	$-0.344757\pi$
$919$	1.86239e16	0.937208	0.468604	+	0.883408i	$0.344757\pi$
$920$	0	0
$921$	−1.18682e16	−0.590145
$922$	0	0
$923$	1.88487e16	0.926131
$924$	0	0
$925$	−3.38172e16	−1.64195
$926$	0	0
$927$	−3.69953e15	−0.177504
$928$	0	0
$929$	8.59493e15	0.407527	0.203763	−	0.979020i	$-0.434683\pi$
$929$	8.59493e15	0.407527	0.203763	+	0.979020i	$0.434683\pi$
$930$	0	0
$931$	7.48498e16	3.50726
$932$	0	0
$933$	1.25493e16	0.581128
$934$	0	0
$935$	1.09512e16	0.501183
$936$	0	0
$937$	1.18734e16	0.537041	0.268521	−	0.963274i	$-0.413465\pi$
$937$	1.18734e16	0.537041	0.268521	+	0.963274i	$0.413465\pi$
$938$	0	0
$939$	3.91382e15	0.174961
$940$	0	0
$941$	1.39439e16	0.616088	0.308044	−	0.951372i	$-0.400326\pi$
$941$	1.39439e16	0.616088	0.308044	+	0.951372i	$0.400326\pi$
$942$	0	0
$943$	−2.34468e16	−1.02393
$944$	0	0
$945$	8.62465e16	3.72277
$946$	0	0
$947$	−4.36290e16	−1.86144	−0.930722	−	0.365726i	$-0.880821\pi$
$947$	−4.36290e16	−1.86144	−0.930722	+	0.365726i	$0.880821\pi$
$948$	0	0
$949$	1.45082e16	0.611855
$950$	0	0
$951$	−3.43115e15	−0.143036
$952$	0	0
$953$	−1.16561e16	−0.480333	−0.240167	−	0.970732i	$-0.577202\pi$
$953$	−1.16561e16	−0.480333	−0.240167	+	0.970732i	$0.577202\pi$
$954$	0	0
$955$	6.19869e16	2.52511
$956$	0	0
$957$	−1.30436e16	−0.525267
$958$	0	0
$959$	−4.84495e15	−0.192880
$960$	0	0
$961$	−2.41129e16	−0.949010
$962$	0	0
$963$	1.07300e16	0.417498
$964$	0	0
$965$	−7.38089e16	−2.83928
$966$	0	0
$967$	4.64312e15	0.176589	0.0882946	−	0.996094i	$-0.471858\pi$
$967$	4.64312e15	0.176589	0.0882946	+	0.996094i	$0.471858\pi$
$968$	0	0
$969$	−1.14442e16	−0.430335
$970$	0	0
$971$	1.80931e16	0.672677	0.336339	−	0.941741i	$-0.390811\pi$
$971$	1.80931e16	0.672677	0.336339	+	0.941741i	$0.390811\pi$
$972$	0	0
$973$	−7.33315e16	−2.69569
$974$	0	0
$975$	2.56489e16	0.932274
$976$	0	0
$977$	5.09418e16	1.83086	0.915428	−	0.402482i	$-0.131852\pi$
$977$	5.09418e16	1.83086	0.915428	+	0.402482i	$0.131852\pi$
$978$	0	0
$979$	2.48199e16	0.882056
$980$	0	0
$981$	−6.35750e15	−0.223412
$982$	0	0
$983$	2.37026e16	0.823666	0.411833	−	0.911259i	$-0.364889\pi$
$983$	2.37026e16	0.823666	0.411833	+	0.911259i	$0.364889\pi$
$984$	0	0
$985$	−2.76748e16	−0.951012
$986$	0	0
$987$	−6.07519e16	−2.06450
$988$	0	0
$989$	2.14508e16	0.720881
$990$	0	0
$991$	1.50638e16	0.500643	0.250322	−	0.968163i	$-0.419464\pi$
$991$	1.50638e16	0.500643	0.250322	+	0.968163i	$0.419464\pi$
$992$	0	0
$993$	2.36937e16	0.778777
$994$	0	0
$995$	−2.32785e16	−0.756708
$996$	0	0
$997$	1.55892e16	0.501186	0.250593	−	0.968092i	$-0.419374\pi$
$997$	1.55892e16	0.501186	0.250593	+	0.968092i	$0.419374\pi$
$998$	0	0
$999$	2.72714e16	0.867157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.12.a.i.1.3		4
4.3	odd	2	inner	256.12.a.i.1.2		4
8.3	odd	2		256.12.a.h.1.3		4
8.5	even	2		256.12.a.h.1.2		4
16.3	odd	4		128.12.b.c.65.3	✓	8
16.5	even	4		128.12.b.c.65.4	yes	8
16.11	odd	4		128.12.b.c.65.6	yes	8
16.13	even	4		128.12.b.c.65.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.c.65.3	✓	8	16.3	odd	4
128.12.b.c.65.4	yes	8	16.5	even	4
128.12.b.c.65.5	yes	8	16.13	even	4
128.12.b.c.65.6	yes	8	16.11	odd	4
256.12.a.h.1.2		4	8.5	even	2
256.12.a.h.1.3		4	8.3	odd	2
256.12.a.i.1.2		4	4.3	odd	2	inner
256.12.a.i.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+329.802 q^{3} +12216.3 q^{5} -87187.3 q^{7} -68377.9 q^{9} -343802. q^{11} +774529. q^{13} +4.02896e6 q^{15} -2.60742e6 q^{17} +1.33083e7 q^{19} -2.87545e7 q^{21} -4.56234e7 q^{23} +1.00410e8 q^{25} -8.09745e7 q^{27} +1.15036e8 q^{29} -3.59943e7 q^{31} -1.13386e8 q^{33} -1.06511e9 q^{35} -3.36790e8 q^{37} +2.55441e8 q^{39} +5.13921e8 q^{41} -4.70170e8 q^{43} -8.35326e8 q^{45} +2.11278e9 q^{47} +5.62429e9 q^{49} -8.59932e8 q^{51} +4.25150e9 q^{53} -4.20000e9 q^{55} +4.38910e9 q^{57} +3.15524e8 q^{59} +4.85644e8 q^{61} +5.96168e9 q^{63} +9.46189e9 q^{65} +4.92012e9 q^{67} -1.50467e10 q^{69} +2.43357e10 q^{71} +1.87316e10 q^{73} +3.31155e10 q^{75} +2.99751e10 q^{77} -2.54101e10 q^{79} -1.45926e10 q^{81} +4.47849e10 q^{83} -3.18531e10 q^{85} +3.79392e10 q^{87} -7.21926e10 q^{89} -6.75290e10 q^{91} -1.18710e10 q^{93} +1.62579e11 q^{95} -3.03613e9 q^{97} +2.35084e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 21040 q^{5} + 461076 q^{9} - 981072 q^{13} - 15482760 q^{17} - 14846976 q^{21} + 108919500 q^{25} + 597997232 q^{29} - 41842944 q^{33} + 353870704 q^{37} + 1861173400 q^{41} - 2684766480 q^{45} + 9205489060 q^{49}+ \cdots + 273341582264 q^{97}+O(q^{100}) \) 4 * q + 21040 * q^5 + 461076 * q^9 - 981072 * q^13 - 15482760 * q^17 - 14846976 * q^21 + 108919500 * q^25 + 597997232 * q^29 - 41842944 * q^33 + 353870704 * q^37 + 1861173400 * q^41 - 2684766480 * q^45 + 9205489060 * q^49 - 793307344 * q^53 - 1077596928 * q^57 - 970979856 * q^61 + 23215691840 * q^65 - 25513915392 * q^69 + 49895617320 * q^73 + 68236479488 * q^77 - 19150006428 * q^81 - 46288531040 * q^85 - 243583370776 * q^89 + 251490557952 * q^93 + 273341582264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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