LMFDB - Embedded newform 16.36.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,36,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 36, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	16.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:	$124.152209014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 12422194x - 2645665785 $ x^3 - 12422194*x - 2645665785
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{3}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3626.53$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.54691e8 q^{3} +2.12139e12 q^{5} -3.96640e14 q^{7} -2.61023e16 q^{9} +O(q^{10})$	$q+1.54691e8 q^{3} +2.12139e12 q^{5} -3.96640e14 q^{7} -2.61023e16 q^{9} +6.82681e17 q^{11} -1.17949e19 q^{13} +3.28159e20 q^{15} -1.33083e21 q^{17} +3.58945e22 q^{19} -6.13565e22 q^{21} -6.92195e23 q^{23} +1.58990e24 q^{25} -1.17772e25 q^{27} -5.67568e25 q^{29} -1.02960e26 q^{31} +1.05604e26 q^{33} -8.41427e26 q^{35} +4.50412e27 q^{37} -1.82456e27 q^{39} -6.23104e27 q^{41} +2.75822e27 q^{43} -5.53731e28 q^{45} -1.41103e29 q^{47} -2.21495e29 q^{49} -2.05867e29 q^{51} -2.48503e30 q^{53} +1.44823e30 q^{55} +5.55255e30 q^{57} -5.47495e30 q^{59} +2.30979e31 q^{61} +1.03532e31 q^{63} -2.50216e31 q^{65} -1.55814e31 q^{67} -1.07076e32 q^{69} +1.13262e32 q^{71} +3.23524e32 q^{73} +2.45942e32 q^{75} -2.70778e32 q^{77} +1.44166e32 q^{79} -5.15885e32 q^{81} -3.50421e33 q^{83} -2.82320e33 q^{85} -8.77975e33 q^{87} +9.37543e33 q^{89} +4.67833e33 q^{91} -1.59269e34 q^{93} +7.61461e34 q^{95} +3.59603e33 q^{97} -1.78195e34 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.54691e8	0.691580	0.345790	−	0.938312i	$-0.387611\pi$
$3$	1.54691e8	0.691580	0.345790	+	0.938312i	$0.387611\pi$
$4$	0	0
$5$	2.12139e12	1.24350	0.621748	−	0.783217i	$-0.286423\pi$
$5$	2.12139e12	1.24350	0.621748	+	0.783217i	$0.286423\pi$
$6$	0	0
$7$	−3.96640e14	−0.644438	−0.322219	−	0.946665i	$-0.604429\pi$
$7$	−3.96640e14	−0.644438	−0.322219	+	0.946665i	$0.604429\pi$
$8$	0	0
$9$	−2.61023e16	−0.521717
$10$	0	0
$11$	6.82681e17	0.407236	0.203618	−	0.979050i	$-0.434730\pi$
$11$	6.82681e17	0.407236	0.203618	+	0.979050i	$0.434730\pi$
$12$	0	0
$13$	−1.17949e19	−0.378169	−0.189085	−	0.981961i	$-0.560552\pi$
$13$	−1.17949e19	−0.378169	−0.189085	+	0.981961i	$0.560552\pi$
$14$	0	0
$15$	3.28159e20	0.859977
$16$	0	0
$17$	−1.33083e21	−0.390181	−0.195090	−	0.980785i	$-0.562500\pi$
$17$	−1.33083e21	−0.390181	−0.195090	+	0.980785i	$0.562500\pi$
$18$	0	0
$19$	3.58945e22	1.50259	0.751294	−	0.659967i	$-0.229430\pi$
$19$	3.58945e22	1.50259	0.751294	+	0.659967i	$0.229430\pi$
$20$	0	0
$21$	−6.13565e22	−0.445680
$22$	0	0
$23$	−6.92195e23	−1.02327	−0.511636	−	0.859202i	$-0.670961\pi$
$23$	−6.92195e23	−1.02327	−0.511636	+	0.859202i	$0.670961\pi$
$24$	0	0
$25$	1.58990e24	0.546284
$26$	0	0
$27$	−1.17772e25	−1.05239
$28$	0	0
$29$	−5.67568e25	−1.45229	−0.726144	−	0.687542i	$-0.758690\pi$
$29$	−5.67568e25	−1.45229	−0.726144	+	0.687542i	$0.758690\pi$
$30$	0	0
$31$	−1.02960e26	−0.820043	−0.410022	−	0.912076i	$-0.634479\pi$
$31$	−1.02960e26	−0.820043	−0.410022	+	0.912076i	$0.634479\pi$
$32$	0	0
$33$	1.05604e26	0.281636
$34$	0	0
$35$	−8.41427e26	−0.801356
$36$	0	0
$37$	4.50412e27	1.62211	0.811054	−	0.584971i	$-0.198894\pi$
$37$	4.50412e27	1.62211	0.811054	+	0.584971i	$0.198894\pi$
$38$	0	0
$39$	−1.82456e27	−0.261534
$40$	0	0
$41$	−6.23104e27	−0.372257	−0.186129	−	0.982525i	$-0.559594\pi$
$41$	−6.23104e27	−0.372257	−0.186129	+	0.982525i	$0.559594\pi$
$42$	0	0
$43$	2.75822e27	0.0716029	0.0358014	−	0.999359i	$-0.488602\pi$
$43$	2.75822e27	0.0716029	0.0358014	+	0.999359i	$0.488602\pi$
$44$	0	0
$45$	−5.53731e28	−0.648754
$46$	0	0
$47$	−1.41103e29	−0.772365	−0.386182	−	0.922422i	$-0.626207\pi$
$47$	−1.41103e29	−0.772365	−0.386182	+	0.922422i	$0.626207\pi$
$48$	0	0
$49$	−2.21495e29	−0.584700
$50$	0	0
$51$	−2.05867e29	−0.269841
$52$	0	0
$53$	−2.48503e30	−1.66151	−0.830755	−	0.556638i	$-0.812091\pi$
$53$	−2.48503e30	−1.66151	−0.830755	+	0.556638i	$0.812091\pi$
$54$	0	0
$55$	1.44823e30	0.506396
$56$	0	0
$57$	5.55255e30	1.03916
$58$	0	0
$59$	−5.47495e30	−0.560364	−0.280182	−	0.959947i	$-0.590395\pi$
$59$	−5.47495e30	−0.560364	−0.280182	+	0.959947i	$0.590395\pi$
$60$	0	0
$61$	2.30979e31	1.31917	0.659585	−	0.751630i	$-0.270732\pi$
$61$	2.30979e31	1.31917	0.659585	+	0.751630i	$0.270732\pi$
$62$	0	0
$63$	1.03532e31	0.336214
$64$	0	0
$65$	−2.50216e31	−0.470252
$66$	0	0
$67$	−1.55814e31	−0.172304	−0.0861522	−	0.996282i	$-0.527457\pi$
$67$	−1.55814e31	−0.172304	−0.0861522	+	0.996282i	$0.527457\pi$
$68$	0	0
$69$	−1.07076e32	−0.707675
$70$	0	0
$71$	1.13262e32	0.454008	0.227004	−	0.973894i	$-0.427107\pi$
$71$	1.13262e32	0.454008	0.227004	+	0.973894i	$0.427107\pi$
$72$	0	0
$73$	3.23524e32	0.797548	0.398774	−	0.917049i	$-0.369436\pi$
$73$	3.23524e32	0.797548	0.398774	+	0.917049i	$0.369436\pi$
$74$	0	0
$75$	2.45942e32	0.377799
$76$	0	0
$77$	−2.70778e32	−0.262438
$78$	0	0
$79$	1.44166e32	0.0892050	0.0446025	−	0.999005i	$-0.485798\pi$
$79$	1.44166e32	0.0892050	0.0446025	+	0.999005i	$0.485798\pi$
$80$	0	0
$81$	−5.15885e32	−0.206094
$82$	0	0
$83$	−3.50421e33	−0.913532	−0.456766	−	0.889587i	$-0.650992\pi$
$83$	−3.50421e33	−0.913532	−0.456766	+	0.889587i	$0.650992\pi$
$84$	0	0
$85$	−2.82320e33	−0.485188
$86$	0	0
$87$	−8.77975e33	−1.00437
$88$	0	0
$89$	9.37543e33	0.720553	0.360277	−	0.932846i	$-0.382682\pi$
$89$	9.37543e33	0.720553	0.360277	+	0.932846i	$0.382682\pi$
$90$	0	0
$91$	4.67833e33	0.243706
$92$	0	0
$93$	−1.59269e34	−0.567126
$94$	0	0
$95$	7.61461e34	1.86846
$96$	0	0
$97$	3.59603e33	0.0612799	0.0306399	−	0.999530i	$-0.490245\pi$
$97$	3.59603e33	0.0612799	0.0306399	+	0.999530i	$0.490245\pi$
$98$	0	0
$99$	−1.78195e34	−0.212462
$100$	0	0
$101$	7.27571e34	0.611296	0.305648	−	0.952145i	$-0.401127\pi$
$101$	7.27571e34	0.611296	0.305648	+	0.952145i	$0.401127\pi$
$102$	0	0
$103$	−9.48038e34	−0.565164	−0.282582	−	0.959243i	$-0.591191\pi$
$103$	−9.48038e34	−0.565164	−0.282582	+	0.959243i	$0.591191\pi$
$104$	0	0
$105$	−1.30161e35	−0.554202
$106$	0	0
$107$	−5.43957e35	−1.66475	−0.832374	−	0.554214i	$-0.813019\pi$
$107$	−5.43957e35	−1.66475	−0.832374	+	0.554214i	$0.813019\pi$
$108$	0	0
$109$	−2.19201e35	−0.485153	−0.242577	−	0.970132i	$-0.577993\pi$
$109$	−2.19201e35	−0.485153	−0.242577	+	0.970132i	$0.577993\pi$
$110$	0	0
$111$	6.96746e35	1.12182
$112$	0	0
$113$	−3.57083e35	−0.420627	−0.210313	−	0.977634i	$-0.567448\pi$
$113$	−3.57083e35	−0.420627	−0.210313	+	0.977634i	$0.567448\pi$
$114$	0	0
$115$	−1.46841e36	−1.27244
$116$	0	0
$117$	3.07875e35	0.197297
$118$	0	0
$119$	5.27859e35	0.251447
$120$	0	0
$121$	−2.34419e36	−0.834159
$122$	0	0
$123$	−9.63884e35	−0.257446
$124$	0	0
$125$	−2.80126e36	−0.564194
$126$	0	0
$127$	2.98205e36	0.454935	0.227467	−	0.973786i	$-0.426955\pi$
$127$	2.98205e36	0.454935	0.227467	+	0.973786i	$0.426955\pi$
$128$	0	0
$129$	4.26671e35	0.0495191
$130$	0	0
$131$	−1.67513e37	−1.48525	−0.742627	−	0.669705i	$-0.766421\pi$
$131$	−1.67513e37	−1.48525	−0.742627	+	0.669705i	$0.766421\pi$
$132$	0	0
$133$	−1.42372e37	−0.968325
$134$	0	0
$135$	−2.49840e37	−1.30864
$136$	0	0
$137$	7.05739e36	0.285782	0.142891	−	0.989738i	$-0.454360\pi$
$137$	7.05739e36	0.285782	0.142891	+	0.989738i	$0.454360\pi$
$138$	0	0
$139$	−5.54070e36	−0.174103	−0.0870514	−	0.996204i	$-0.527744\pi$
$139$	−5.54070e36	−0.174103	−0.0870514	+	0.996204i	$0.527744\pi$
$140$	0	0
$141$	−2.18273e37	−0.534152
$142$	0	0
$143$	−8.05216e36	−0.154004
$144$	0	0
$145$	−1.20403e38	−1.80592
$146$	0	0
$147$	−3.42633e37	−0.404367
$148$	0	0
$149$	−1.26594e38	−1.17938	−0.589689	−	0.807631i	$-0.700750\pi$
$149$	−1.26594e38	−1.17938	−0.589689	+	0.807631i	$0.700750\pi$
$150$	0	0
$151$	−1.46030e38	−1.07732	−0.538661	−	0.842523i	$-0.681070\pi$
$151$	−1.46030e38	−1.07732	−0.538661	+	0.842523i	$0.681070\pi$
$152$	0	0
$153$	3.47377e37	0.203564
$154$	0	0
$155$	−2.18417e38	−1.01972
$156$	0	0
$157$	−2.57702e38	−0.961332	−0.480666	−	0.876904i	$-0.659605\pi$
$157$	−2.57702e38	−0.961332	−0.480666	+	0.876904i	$0.659605\pi$
$158$	0	0
$159$	−3.84410e38	−1.14907
$160$	0	0
$161$	2.74552e38	0.659436
$162$	0	0
$163$	1.00523e39	1.94529	0.972643	−	0.232306i	$-0.0746270\pi$
$163$	1.00523e39	1.94529	0.972643	+	0.232306i	$0.0746270\pi$
$164$	0	0
$165$	2.24028e38	0.350213
$166$	0	0
$167$	−4.87583e38	−0.617320	−0.308660	−	0.951172i	$-0.599881\pi$
$167$	−4.87583e38	−0.617320	−0.308660	+	0.951172i	$0.599881\pi$
$168$	0	0
$169$	−8.33666e38	−0.856988
$170$	0	0
$171$	−9.36930e38	−0.783926
$172$	0	0
$173$	−3.15688e37	−0.0215502	−0.0107751	−	0.999942i	$-0.503430\pi$
$173$	−3.15688e37	−0.0215502	−0.0107751	+	0.999942i	$0.503430\pi$
$174$	0	0
$175$	−6.30617e38	−0.352046
$176$	0	0
$177$	−8.46924e38	−0.387536
$178$	0	0
$179$	4.86041e38	0.182702	0.0913512	−	0.995819i	$-0.470881\pi$
$179$	4.86041e38	0.182702	0.0913512	+	0.995819i	$0.470881\pi$
$180$	0	0
$181$	−4.89892e39	−1.51609	−0.758045	−	0.652202i	$-0.773845\pi$
$181$	−4.89892e39	−1.51609	−0.758045	+	0.652202i	$0.773845\pi$
$182$	0	0
$183$	3.57303e39	0.912311
$184$	0	0
$185$	9.55498e39	2.01709
$186$	0	0
$187$	−9.08530e38	−0.158895
$188$	0	0
$189$	4.67131e39	0.678199
$190$	0	0
$191$	8.79853e39	1.06249	0.531246	−	0.847217i	$-0.321724\pi$
$191$	8.79853e39	1.06249	0.531246	+	0.847217i	$0.321724\pi$
$192$	0	0
$193$	1.65403e40	1.66452	0.832262	−	0.554383i	$-0.187046\pi$
$193$	1.65403e40	1.66452	0.832262	+	0.554383i	$0.187046\pi$
$194$	0	0
$195$	−3.87060e39	−0.325217
$196$	0	0
$197$	3.96581e36	0.000278723 0	0.000139362	−	1.00000i	$-0.499956\pi$
$197$	3.96581e36	0.000278723 0	0.000139362		1.00000i	$0.499956\pi$
$198$	0	0
$199$	1.61546e40	0.951409	0.475705	−	0.879605i	$-0.342193\pi$
$199$	1.61546e40	0.951409	0.475705	+	0.879605i	$0.342193\pi$
$200$	0	0
$201$	−2.41029e39	−0.119162
$202$	0	0
$203$	2.25120e40	0.935909
$204$	0	0
$205$	−1.32184e40	−0.462901
$206$	0	0
$207$	1.80679e40	0.533859
$208$	0	0
$209$	2.45045e40	0.611908
$210$	0	0
$211$	−2.70344e40	−0.571444	−0.285722	−	0.958313i	$-0.592233\pi$
$211$	−2.70344e40	−0.571444	−0.285722	+	0.958313i	$0.592233\pi$
$212$	0	0
$213$	1.75206e40	0.313983
$214$	0	0
$215$	5.85124e39	0.0890379
$216$	0	0
$217$	4.08379e40	0.528467
$218$	0	0
$219$	5.00462e40	0.551569
$220$	0	0
$221$	1.56970e40	0.147554
$222$	0	0
$223$	−1.60448e41	−1.28825	−0.644124	−	0.764921i	$-0.722778\pi$
$223$	−1.60448e41	−1.28825	−0.644124	+	0.764921i	$0.722778\pi$
$224$	0	0
$225$	−4.15000e40	−0.285006
$226$	0	0
$227$	1.32023e41	0.776597	0.388299	−	0.921534i	$-0.373063\pi$
$227$	1.32023e41	0.776597	0.388299	+	0.921534i	$0.373063\pi$
$228$	0	0
$229$	−1.27675e41	−0.644147	−0.322073	−	0.946715i	$-0.604380\pi$
$229$	−1.27675e41	−0.644147	−0.322073	+	0.946715i	$0.604380\pi$
$230$	0	0
$231$	−4.18869e40	−0.181497
$232$	0	0
$233$	−1.15249e41	−0.429445	−0.214722	−	0.976675i	$-0.568885\pi$
$233$	−1.15249e41	−0.429445	−0.214722	+	0.976675i	$0.568885\pi$
$234$	0	0
$235$	−2.99333e41	−0.960433
$236$	0	0
$237$	2.23012e40	0.0616924
$238$	0	0
$239$	−1.42881e41	−0.341201	−0.170601	−	0.985340i	$-0.554571\pi$
$239$	−1.42881e41	−0.341201	−0.170601	+	0.985340i	$0.554571\pi$
$240$	0	0
$241$	−1.47367e41	−0.304160	−0.152080	−	0.988368i	$-0.548597\pi$
$241$	−1.47367e41	−0.304160	−0.152080	+	0.988368i	$0.548597\pi$
$242$	0	0
$243$	5.09429e41	0.909859
$244$	0	0
$245$	−4.69877e41	−0.727073
$246$	0	0
$247$	−4.23373e41	−0.568233
$248$	0	0
$249$	−5.42069e41	−0.631781
$250$	0	0
$251$	−1.16925e42	−1.18473	−0.592363	−	0.805671i	$-0.701805\pi$
$251$	−1.16925e42	−1.18473	−0.592363	+	0.805671i	$0.701805\pi$
$252$	0	0
$253$	−4.72548e41	−0.416713
$254$	0	0
$255$	−4.36723e41	−0.335546
$256$	0	0
$257$	6.14627e41	0.411890	0.205945	−	0.978564i	$-0.433973\pi$
$257$	6.14627e41	0.411890	0.205945	+	0.978564i	$0.433973\pi$
$258$	0	0
$259$	−1.78651e42	−1.04535
$260$	0	0
$261$	1.48148e42	0.757684
$262$	0	0
$263$	−5.29428e41	−0.236909	−0.118454	−	0.992959i	$-0.537794\pi$
$263$	−5.29428e41	−0.236909	−0.118454	+	0.992959i	$0.537794\pi$
$264$	0	0
$265$	−5.27170e42	−2.06608
$266$	0	0
$267$	1.45029e42	0.498320
$268$	0	0
$269$	2.61222e42	0.787670	0.393835	−	0.919181i	$-0.371148\pi$
$269$	2.61222e42	0.787670	0.393835	+	0.919181i	$0.371148\pi$
$270$	0	0
$271$	6.35646e42	1.68365	0.841827	−	0.539748i	$-0.181480\pi$
$271$	6.35646e42	1.68365	0.841827	+	0.539748i	$0.181480\pi$
$272$	0	0
$273$	7.23695e41	0.168542
$274$	0	0
$275$	1.08539e42	0.222466
$276$	0	0
$277$	1.32088e41	0.0238488	0.0119244	−	0.999929i	$-0.496204\pi$
$277$	1.32088e41	0.0238488	0.0119244	+	0.999929i	$0.496204\pi$
$278$	0	0
$279$	2.68748e42	0.427831
$280$	0	0
$281$	1.21053e43	1.70065	0.850324	−	0.526260i	$-0.176406\pi$
$281$	1.21053e43	1.70065	0.850324	+	0.526260i	$0.176406\pi$
$282$	0	0
$283$	−1.08291e41	−0.0134378	−0.00671891	−	0.999977i	$-0.502139\pi$
$283$	−1.08291e41	−0.0134378	−0.00671891	+	0.999977i	$0.502139\pi$
$284$	0	0
$285$	1.17791e43	1.29219
$286$	0	0
$287$	2.47148e42	0.239896
$288$	0	0
$289$	−9.86245e42	−0.847759
$290$	0	0
$291$	5.56273e41	0.0423799
$292$	0	0
$293$	−2.38381e42	−0.161097	−0.0805486	−	0.996751i	$-0.525667\pi$
$293$	−2.38381e42	−0.161097	−0.0805486	+	0.996751i	$0.525667\pi$
$294$	0	0
$295$	−1.16145e43	−0.696811
$296$	0	0
$297$	−8.04007e42	−0.428570
$298$	0	0
$299$	8.16438e42	0.386970
$300$	0	0
$301$	−1.09402e42	−0.0461436
$302$	0	0
$303$	1.12548e43	0.422760
$304$	0	0
$305$	4.89996e43	1.64038
$306$	0	0
$307$	4.84424e43	1.44645	0.723226	−	0.690611i	$-0.242658\pi$
$307$	4.84424e43	1.44645	0.723226	+	0.690611i	$0.242658\pi$
$308$	0	0
$309$	−1.46653e43	−0.390856
$310$	0	0
$311$	−1.85715e43	−0.442119	−0.221060	−	0.975260i	$-0.570952\pi$
$311$	−1.85715e43	−0.442119	−0.221060	+	0.975260i	$0.570952\pi$
$312$	0	0
$313$	2.67734e43	0.569740	0.284870	−	0.958566i	$-0.408049\pi$
$313$	2.67734e43	0.569740	0.284870	+	0.958566i	$0.408049\pi$
$314$	0	0
$315$	2.19632e43	0.418081
$316$	0	0
$317$	−2.42612e43	−0.413405	−0.206703	−	0.978404i	$-0.566273\pi$
$317$	−2.42612e43	−0.413405	−0.206703	+	0.978404i	$0.566273\pi$
$318$	0	0
$319$	−3.87468e43	−0.591423
$320$	0	0
$321$	−8.41452e43	−1.15131
$322$	0	0
$323$	−4.77694e43	−0.586281
$324$	0	0
$325$	−1.87527e43	−0.206588
$326$	0	0
$327$	−3.39084e43	−0.335522
$328$	0	0
$329$	5.59669e43	0.497741
$330$	0	0
$331$	−1.83689e44	−1.46924	−0.734620	−	0.678479i	$-0.762639\pi$
$331$	−1.83689e44	−1.46924	−0.734620	+	0.678479i	$0.762639\pi$
$332$	0	0
$333$	−1.17568e44	−0.846282
$334$	0	0
$335$	−3.30541e43	−0.214260
$336$	0	0
$337$	9.63203e42	0.0562593	0.0281297	−	0.999604i	$-0.491045\pi$
$337$	9.63203e42	0.0562593	0.0281297	+	0.999604i	$0.491045\pi$
$338$	0	0
$339$	−5.52375e43	−0.290897
$340$	0	0
$341$	−7.02885e43	−0.333951
$342$	0	0
$343$	2.38109e44	1.02124
$344$	0	0
$345$	−2.27150e44	−0.879992
$346$	0	0
$347$	−2.05198e44	−0.718468	−0.359234	−	0.933247i	$-0.616962\pi$
$347$	−2.05198e44	−0.718468	−0.359234	+	0.933247i	$0.616962\pi$
$348$	0	0
$349$	−1.00117e44	−0.317003	−0.158501	−	0.987359i	$-0.550666\pi$
$349$	−1.00117e44	−0.317003	−0.158501	+	0.987359i	$0.550666\pi$
$350$	0	0
$351$	1.38911e44	0.397981
$352$	0	0
$353$	−2.71539e44	−0.704328	−0.352164	−	0.935938i	$-0.614554\pi$
$353$	−2.71539e44	−0.704328	−0.352164	+	0.935938i	$0.614554\pi$
$354$	0	0
$355$	2.40272e44	0.564557
$356$	0	0
$357$	8.16550e43	0.173896
$358$	0	0
$359$	−8.81356e44	−1.70215	−0.851076	−	0.525043i	$-0.824049\pi$
$359$	−8.81356e44	−1.70215	−0.851076	+	0.525043i	$0.824049\pi$
$360$	0	0
$361$	7.17758e44	1.25777
$362$	0	0
$363$	−3.62625e44	−0.576888
$364$	0	0
$365$	6.86320e44	0.991749
$366$	0	0
$367$	1.11298e45	1.46161	0.730806	−	0.682585i	$-0.239144\pi$
$367$	1.11298e45	1.46161	0.730806	+	0.682585i	$0.239144\pi$
$368$	0	0
$369$	1.62645e44	0.194213
$370$	0	0
$371$	9.85661e44	1.07074
$372$	0	0
$373$	−5.50495e44	−0.544313	−0.272157	−	0.962253i	$-0.587737\pi$
$373$	−5.50495e44	−0.544313	−0.272157	+	0.962253i	$0.587737\pi$
$374$	0	0
$375$	−4.33329e44	−0.390185
$376$	0	0
$377$	6.69441e44	0.549211
$378$	0	0
$379$	1.81534e45	1.35760	0.678799	−	0.734325i	$-0.262501\pi$
$379$	1.81534e45	1.35760	0.678799	+	0.734325i	$0.262501\pi$
$380$	0	0
$381$	4.61295e44	0.314624
$382$	0	0
$383$	−5.42819e44	−0.337813	−0.168907	−	0.985632i	$-0.554024\pi$
$383$	−5.42819e44	−0.337813	−0.168907	+	0.985632i	$0.554024\pi$
$384$	0	0
$385$	−5.74426e44	−0.326341
$386$	0	0
$387$	−7.19958e43	−0.0373564
$388$	0	0
$389$	2.27596e45	1.07906	0.539532	−	0.841965i	$-0.318601\pi$
$389$	2.27596e45	1.07906	0.539532	+	0.841965i	$0.318601\pi$
$390$	0	0
$391$	9.21192e44	0.399261
$392$	0	0
$393$	−2.59127e45	−1.02717
$394$	0	0
$395$	3.05832e44	0.110926
$396$	0	0
$397$	2.97575e45	0.988013	0.494006	−	0.869458i	$-0.335532\pi$
$397$	2.97575e45	0.988013	0.494006	+	0.869458i	$0.335532\pi$
$398$	0	0
$399$	−2.20236e45	−0.669674
$400$	0	0
$401$	−3.35408e45	−0.934429	−0.467215	−	0.884144i	$-0.654742\pi$
$401$	−3.35408e45	−0.934429	−0.467215	+	0.884144i	$0.654742\pi$
$402$	0	0
$403$	1.21440e45	0.310115
$404$	0	0
$405$	−1.09439e45	−0.256277
$406$	0	0
$407$	3.07488e45	0.660580
$408$	0	0
$409$	8.75180e45	1.72560	0.862799	−	0.505547i	$-0.168709\pi$
$409$	8.75180e45	1.72560	0.862799	+	0.505547i	$0.168709\pi$
$410$	0	0
$411$	1.09171e45	0.197641
$412$	0	0
$413$	2.17158e45	0.361119
$414$	0	0
$415$	−7.43379e45	−1.13597
$416$	0	0
$417$	−8.57095e44	−0.120406
$418$	0	0
$419$	−1.99625e45	−0.257911	−0.128955	−	0.991650i	$-0.541162\pi$
$419$	−1.99625e45	−0.257911	−0.128955	+	0.991650i	$0.541162\pi$
$420$	0	0
$421$	−1.23389e46	−1.46670	−0.733348	−	0.679854i	$-0.762043\pi$
$421$	−1.23389e46	−1.46670	−0.733348	+	0.679854i	$0.762043\pi$
$422$	0	0
$423$	3.68311e45	0.402956
$424$	0	0
$425$	−2.11588e45	−0.213150
$426$	0	0
$427$	−9.16156e45	−0.850122
$428$	0	0
$429$	−1.24559e45	−0.106506
$430$	0	0
$431$	1.20357e46	0.948677	0.474339	−	0.880343i	$-0.342687\pi$
$431$	1.20357e46	0.948677	0.474339	+	0.880343i	$0.342687\pi$
$432$	0	0
$433$	−2.06332e46	−1.49979	−0.749893	−	0.661559i	$-0.769895\pi$
$433$	−2.06332e46	−1.49979	−0.749893	+	0.661559i	$0.769895\pi$
$434$	0	0
$435$	−1.86252e46	−1.24894
$436$	0	0
$437$	−2.48460e46	−1.53756
$438$	0	0
$439$	−3.14998e46	−1.79961	−0.899805	−	0.436293i	$-0.856291\pi$
$439$	−3.14998e46	−1.79961	−0.899805	+	0.436293i	$0.856291\pi$
$440$	0	0
$441$	5.78154e45	0.305048
$442$	0	0
$443$	2.49595e46	1.21667	0.608333	−	0.793682i	$-0.291839\pi$
$443$	2.49595e46	1.21667	0.608333	+	0.793682i	$0.291839\pi$
$444$	0	0
$445$	1.98889e46	0.896006
$446$	0	0
$447$	−1.95829e46	−0.815634
$448$	0	0
$449$	3.14792e46	1.21259	0.606294	−	0.795241i	$-0.292656\pi$
$449$	3.14792e46	1.21259	0.606294	+	0.795241i	$0.292656\pi$
$450$	0	0
$451$	−4.25381e45	−0.151596
$452$	0	0
$453$	−2.25895e46	−0.745054
$454$	0	0
$455$	9.92455e45	0.303048
$456$	0	0
$457$	3.21994e46	0.910572	0.455286	−	0.890345i	$-0.349537\pi$
$457$	3.21994e46	0.910572	0.455286	+	0.890345i	$0.349537\pi$
$458$	0	0
$459$	1.56734e46	0.410622
$460$	0	0
$461$	8.96567e45	0.217679	0.108840	−	0.994059i	$-0.465287\pi$
$461$	8.96567e45	0.217679	0.108840	+	0.994059i	$0.465287\pi$
$462$	0	0
$463$	2.74379e46	0.617566	0.308783	−	0.951133i	$-0.400078\pi$
$463$	2.74379e46	0.617566	0.308783	+	0.951133i	$0.400078\pi$
$464$	0	0
$465$	−3.37871e46	−0.705219
$466$	0	0
$467$	8.40024e46	1.62647	0.813235	−	0.581935i	$-0.197704\pi$
$467$	8.40024e46	1.62647	0.813235	+	0.581935i	$0.197704\pi$
$468$	0	0
$469$	6.18019e45	0.111039
$470$	0	0
$471$	−3.98641e46	−0.664838
$472$	0	0
$473$	1.88298e45	0.0291592
$474$	0	0
$475$	5.70686e46	0.820841
$476$	0	0
$477$	6.48649e46	0.866839
$478$	0	0
$479$	−1.51913e47	−1.88679	−0.943395	−	0.331671i	$-0.892388\pi$
$479$	−1.51913e47	−1.88679	−0.943395	+	0.331671i	$0.892388\pi$
$480$	0	0
$481$	−5.31257e46	−0.613431
$482$	0	0
$483$	4.24707e46	0.456052
$484$	0	0
$485$	7.62857e45	0.0762014
$486$	0	0
$487$	9.57009e46	0.889527	0.444764	−	0.895648i	$-0.353288\pi$
$487$	9.57009e46	0.889527	0.444764	+	0.895648i	$0.353288\pi$
$488$	0	0
$489$	1.55500e47	1.34532
$490$	0	0
$491$	9.00218e46	0.725139	0.362570	−	0.931957i	$-0.381900\pi$
$491$	9.00218e46	0.725139	0.362570	+	0.931957i	$0.381900\pi$
$492$	0	0
$493$	7.55335e46	0.566655
$494$	0	0
$495$	−3.78021e46	−0.264196
$496$	0	0
$497$	−4.49242e46	−0.292580
$498$	0	0
$499$	1.31845e47	0.800394	0.400197	−	0.916429i	$-0.368942\pi$
$499$	1.31845e47	0.800394	0.400197	+	0.916429i	$0.368942\pi$
$500$	0	0
$501$	−7.54246e46	−0.426926
$502$	0	0
$503$	−3.29261e46	−0.173820	−0.0869101	−	0.996216i	$-0.527699\pi$
$503$	−3.29261e46	−0.173820	−0.0869101	+	0.996216i	$0.527699\pi$
$504$	0	0
$505$	1.54346e47	0.760144
$506$	0	0
$507$	−1.28960e47	−0.592676
$508$	0	0
$509$	1.43449e47	0.615370	0.307685	−	0.951488i	$-0.400446\pi$
$509$	1.43449e47	0.615370	0.307685	+	0.951488i	$0.400446\pi$
$510$	0	0
$511$	−1.28323e47	−0.513970
$512$	0	0
$513$	−4.22737e47	−1.58131
$514$	0	0
$515$	−2.01115e47	−0.702779
$516$	0	0
$517$	−9.63280e46	−0.314534
$518$	0	0
$519$	−4.88340e45	−0.0149037
$520$	0	0
$521$	−2.33055e47	−0.664966	−0.332483	−	0.943109i	$-0.607886\pi$
$521$	−2.33055e47	−0.664966	−0.332483	+	0.943109i	$0.607886\pi$
$522$	0	0
$523$	4.26380e47	1.13767	0.568836	−	0.822451i	$-0.307394\pi$
$523$	4.26380e47	1.13767	0.568836	+	0.822451i	$0.307394\pi$
$524$	0	0
$525$	−9.75506e46	−0.243468
$526$	0	0
$527$	1.37021e47	0.319965
$528$	0	0
$529$	2.15467e46	0.0470875
$530$	0	0
$531$	1.42909e47	0.292351
$532$	0	0
$533$	7.34945e46	0.140776
$534$	0	0
$535$	−1.15394e48	−2.07011
$536$	0	0
$537$	7.51860e46	0.126353
$538$	0	0
$539$	−1.51211e47	−0.238111
$540$	0	0
$541$	−5.43835e45	−0.00802629	−0.00401314	−	0.999992i	$-0.501277\pi$
$541$	−5.43835e45	−0.00802629	−0.00401314	+	0.999992i	$0.501277\pi$
$542$	0	0
$543$	−7.57817e47	−1.04850
$544$	0	0
$545$	−4.65010e47	−0.603286
$546$	0	0
$547$	4.15694e47	0.505820	0.252910	−	0.967490i	$-0.418612\pi$
$547$	4.15694e47	0.505820	0.252910	+	0.967490i	$0.418612\pi$
$548$	0	0
$549$	−6.02909e47	−0.688233
$550$	0	0
$551$	−2.03726e48	−2.18219
$552$	0	0
$553$	−5.71821e46	−0.0574871
$554$	0	0
$555$	1.47807e48	1.39498
$556$	0	0
$557$	1.81613e45	0.00160947	0.000804733	−	1.00000i	$-0.499744\pi$
$557$	1.81613e45	0.00160947	0.000804733		1.00000i	$0.499744\pi$
$558$	0	0
$559$	−3.25329e46	−0.0270780
$560$	0	0
$561$	−1.40541e47	−0.109889
$562$	0	0
$563$	2.19502e48	1.61266	0.806329	−	0.591468i	$-0.201451\pi$
$563$	2.19502e48	1.61266	0.806329	+	0.591468i	$0.201451\pi$
$564$	0	0
$565$	−7.57512e47	−0.523048
$566$	0	0
$567$	2.04621e47	0.132815
$568$	0	0
$569$	1.19614e48	0.729993	0.364996	−	0.931009i	$-0.381070\pi$
$569$	1.19614e48	0.729993	0.364996	+	0.931009i	$0.381070\pi$
$570$	0	0
$571$	1.26426e48	0.725610	0.362805	−	0.931865i	$-0.381819\pi$
$571$	1.26426e48	0.725610	0.362805	+	0.931865i	$0.381819\pi$
$572$	0	0
$573$	1.36105e48	0.734799
$574$	0	0
$575$	−1.10052e48	−0.558998
$576$	0	0
$577$	2.42022e48	1.15686	0.578428	−	0.815734i	$-0.303667\pi$
$577$	2.42022e48	1.15686	0.578428	+	0.815734i	$0.303667\pi$
$578$	0	0
$579$	2.55863e48	1.15115
$580$	0	0
$581$	1.38991e48	0.588715
$582$	0	0
$583$	−1.69648e48	−0.676626
$584$	0	0
$585$	6.53121e47	0.245339
$586$	0	0
$587$	1.04500e48	0.369787	0.184893	−	0.982759i	$-0.440806\pi$
$587$	1.04500e48	0.369787	0.184893	+	0.982759i	$0.440806\pi$
$588$	0	0
$589$	−3.69568e48	−1.23219
$590$	0	0
$591$	6.13475e44	0.000192760 0
$592$	0	0
$593$	4.87793e48	1.44470	0.722351	−	0.691526i	$-0.243061\pi$
$593$	4.87793e48	1.44470	0.722351	+	0.691526i	$0.243061\pi$
$594$	0	0
$595$	1.11979e48	0.312674
$596$	0	0
$597$	2.49897e48	0.657976
$598$	0	0
$599$	−8.48111e47	−0.210612	−0.105306	−	0.994440i	$-0.533582\pi$
$599$	−8.48111e47	−0.210612	−0.105306	+	0.994440i	$0.533582\pi$
$600$	0	0
$601$	−6.23099e48	−1.45967	−0.729834	−	0.683624i	$-0.760403\pi$
$601$	−6.23099e48	−1.45967	−0.729834	+	0.683624i	$0.760403\pi$
$602$	0	0
$603$	4.06710e47	0.0898942
$604$	0	0
$605$	−4.97293e48	−1.03727
$606$	0	0
$607$	−2.65746e48	−0.523197	−0.261598	−	0.965177i	$-0.584250\pi$
$607$	−2.65746e48	−0.523197	−0.261598	+	0.965177i	$0.584250\pi$
$608$	0	0
$609$	3.48240e48	0.647256
$610$	0	0
$611$	1.66429e48	0.292085
$612$	0	0
$613$	5.89105e48	0.976415	0.488208	−	0.872727i	$-0.337651\pi$
$613$	5.89105e48	0.976415	0.488208	+	0.872727i	$0.337651\pi$
$614$	0	0
$615$	−2.04477e48	−0.320133
$616$	0	0
$617$	−2.86204e48	−0.423337	−0.211668	−	0.977342i	$-0.567890\pi$
$617$	−2.86204e48	−0.423337	−0.211668	+	0.977342i	$0.567890\pi$
$618$	0	0
$619$	9.42149e48	1.31684	0.658420	−	0.752651i	$-0.271225\pi$
$619$	9.42149e48	1.31684	0.658420	+	0.752651i	$0.271225\pi$
$620$	0	0
$621$	8.15212e48	1.07688
$622$	0	0
$623$	−3.71867e48	−0.464352
$624$	0	0
$625$	−1.05698e49	−1.24786
$626$	0	0
$627$	3.79062e48	0.423183
$628$	0	0
$629$	−5.99421e48	−0.632915
$630$	0	0
$631$	−8.74950e48	−0.873915	−0.436958	−	0.899482i	$-0.643944\pi$
$631$	−8.74950e48	−0.873915	−0.436958	+	0.899482i	$0.643944\pi$
$632$	0	0
$633$	−4.18197e48	−0.395199
$634$	0	0
$635$	6.32607e48	0.565710
$636$	0	0
$637$	2.61252e48	0.221116
$638$	0	0
$639$	−2.95640e48	−0.236864
$640$	0	0
$641$	5.35983e48	0.406571	0.203285	−	0.979120i	$-0.434838\pi$
$641$	5.35983e48	0.406571	0.203285	+	0.979120i	$0.434838\pi$
$642$	0	0
$643$	−2.04799e49	−1.47108	−0.735538	−	0.677483i	$-0.763071\pi$
$643$	−2.04799e49	−1.47108	−0.735538	+	0.677483i	$0.763071\pi$
$644$	0	0
$645$	9.05133e47	0.0615768
$646$	0	0
$647$	−2.13324e49	−1.37472	−0.687360	−	0.726317i	$-0.741231\pi$
$647$	−2.13324e49	−1.37472	−0.687360	+	0.726317i	$0.741231\pi$
$648$	0	0
$649$	−3.73764e48	−0.228200
$650$	0	0
$651$	6.31724e48	0.365477
$652$	0	0
$653$	−2.78018e49	−1.52438	−0.762189	−	0.647354i	$-0.775876\pi$
$653$	−2.78018e49	−1.52438	−0.762189	+	0.647354i	$0.775876\pi$
$654$	0	0
$655$	−3.55360e49	−1.84691
$656$	0	0
$657$	−8.44474e48	−0.416095
$658$	0	0
$659$	2.64746e49	1.23690	0.618450	−	0.785824i	$-0.287761\pi$
$659$	2.64746e49	1.23690	0.618450	+	0.785824i	$0.287761\pi$
$660$	0	0
$661$	2.31405e49	1.02529	0.512646	−	0.858600i	$-0.328665\pi$
$661$	2.31405e49	1.02529	0.512646	+	0.858600i	$0.328665\pi$
$662$	0	0
$663$	2.42818e48	0.102046
$664$	0	0
$665$	−3.02026e49	−1.20411
$666$	0	0
$667$	3.92868e49	1.48609
$668$	0	0
$669$	−2.48198e49	−0.890927
$670$	0	0
$671$	1.57685e49	0.537213
$672$	0	0
$673$	3.41502e49	1.10441	0.552204	−	0.833709i	$-0.313787\pi$
$673$	3.41502e49	1.10441	0.552204	+	0.833709i	$0.313787\pi$
$674$	0	0
$675$	−1.87245e49	−0.574904
$676$	0	0
$677$	1.81760e49	0.529903	0.264951	−	0.964262i	$-0.414644\pi$
$677$	1.81760e49	0.529903	0.264951	+	0.964262i	$0.414644\pi$
$678$	0	0
$679$	−1.42633e48	−0.0394911
$680$	0	0
$681$	2.04227e49	0.537079
$682$	0	0
$683$	1.52341e49	0.380587	0.190294	−	0.981727i	$-0.439056\pi$
$683$	1.52341e49	0.380587	0.190294	+	0.981727i	$0.439056\pi$
$684$	0	0
$685$	1.49715e49	0.355368
$686$	0	0
$687$	−1.97501e49	−0.445479
$688$	0	0
$689$	2.93107e49	0.628332
$690$	0	0
$691$	1.25243e49	0.255203	0.127602	−	0.991825i	$-0.459272\pi$
$691$	1.25243e49	0.255203	0.127602	+	0.991825i	$0.459272\pi$
$692$	0	0
$693$	7.06795e48	0.136918
$694$	0	0
$695$	−1.17540e49	−0.216496
$696$	0	0
$697$	8.29244e48	0.145247
$698$	0	0
$699$	−1.78279e49	−0.296995
$700$	0	0
$701$	4.20207e49	0.665883	0.332942	−	0.942947i	$-0.391959\pi$
$701$	4.20207e49	0.665883	0.332942	+	0.942947i	$0.391959\pi$
$702$	0	0
$703$	1.61673e50	2.43736
$704$	0	0
$705$	−4.63041e49	−0.664216
$706$	0	0
$707$	−2.88584e49	−0.393942
$708$	0	0
$709$	9.39998e48	0.122129	0.0610644	−	0.998134i	$-0.480551\pi$
$709$	9.39998e48	0.122129	0.0610644	+	0.998134i	$0.480551\pi$
$710$	0	0
$711$	−3.76307e48	−0.0465398
$712$	0	0
$713$	7.12681e49	0.839128
$714$	0	0
$715$	−1.70817e49	−0.191503
$716$	0	0
$717$	−2.21023e49	−0.235968
$718$	0	0
$719$	7.29315e49	0.741583	0.370792	−	0.928716i	$-0.379086\pi$
$719$	7.29315e49	0.741583	0.370792	+	0.928716i	$0.379086\pi$
$720$	0	0
$721$	3.76030e49	0.364213
$722$	0	0
$723$	−2.27963e49	−0.210351
$724$	0	0
$725$	−9.02375e49	−0.793363
$726$	0	0
$727$	4.92620e49	0.412724	0.206362	−	0.978476i	$-0.433838\pi$
$727$	4.92620e49	0.412724	0.206362	+	0.978476i	$0.433838\pi$
$728$	0	0
$729$	1.04614e50	0.835334
$730$	0	0
$731$	−3.67071e48	−0.0279380
$732$	0	0
$733$	−1.28080e50	−0.929310	−0.464655	−	0.885492i	$-0.653822\pi$
$733$	−1.28080e50	−0.929310	−0.464655	+	0.885492i	$0.653822\pi$
$734$	0	0
$735$	−7.26857e49	−0.502829
$736$	0	0
$737$	−1.06371e49	−0.0701685
$738$	0	0
$739$	−1.99479e50	−1.25493	−0.627466	−	0.778644i	$-0.715908\pi$
$739$	−1.99479e50	−1.25493	−0.627466	+	0.778644i	$0.715908\pi$
$740$	0	0
$741$	−6.54918e49	−0.392978
$742$	0	0
$743$	−1.05855e50	−0.605907	−0.302954	−	0.953005i	$-0.597973\pi$
$743$	−1.05855e50	−0.605907	−0.302954	+	0.953005i	$0.597973\pi$
$744$	0	0
$745$	−2.68554e50	−1.46655
$746$	0	0
$747$	9.14681e49	0.476606
$748$	0	0
$749$	2.15755e50	1.07283
$750$	0	0
$751$	1.20549e50	0.572090	0.286045	−	0.958216i	$-0.407659\pi$
$751$	1.20549e50	0.572090	0.286045	+	0.958216i	$0.407659\pi$
$752$	0	0
$753$	−1.80872e50	−0.819333
$754$	0	0
$755$	−3.09786e50	−1.33965
$756$	0	0
$757$	1.18495e50	0.489240	0.244620	−	0.969619i	$-0.421337\pi$
$757$	1.18495e50	0.489240	0.244620	+	0.969619i	$0.421337\pi$
$758$	0	0
$759$	−7.30989e49	−0.288190
$760$	0	0
$761$	2.89900e50	1.09148	0.545741	−	0.837954i	$-0.316248\pi$
$761$	2.89900e50	1.09148	0.545741	+	0.837954i	$0.316248\pi$
$762$	0	0
$763$	8.69439e49	0.312651
$764$	0	0
$765$	7.36920e49	0.253131
$766$	0	0
$767$	6.45766e49	0.211912
$768$	0	0
$769$	6.18664e49	0.193974	0.0969870	−	0.995286i	$-0.469079\pi$
$769$	6.18664e49	0.193974	0.0969870	+	0.995286i	$0.469079\pi$
$770$	0	0
$771$	9.50771e49	0.284855
$772$	0	0
$773$	−6.37753e50	−1.82604	−0.913022	−	0.407911i	$-0.866257\pi$
$773$	−6.37753e50	−1.82604	−0.913022	+	0.407911i	$0.866257\pi$
$774$	0	0
$775$	−1.63695e50	−0.447977
$776$	0	0
$777$	−2.76357e50	−0.722941
$778$	0	0
$779$	−2.23660e50	−0.559349
$780$	0	0
$781$	7.73218e49	0.184888
$782$	0	0
$783$	6.68436e50	1.52837
$784$	0	0
$785$	−5.46685e50	−1.19541
$786$	0	0
$787$	−6.23653e50	−1.30432	−0.652161	−	0.758080i	$-0.726138\pi$
$787$	−6.23653e50	−1.30432	−0.652161	+	0.758080i	$0.726138\pi$
$788$	0	0
$789$	−8.18976e49	−0.163841
$790$	0	0
$791$	1.41634e50	0.271068
$792$	0	0
$793$	−2.72438e50	−0.498869
$794$	0	0
$795$	−8.15483e50	−1.42886
$796$	0	0
$797$	9.07097e50	1.52101	0.760507	−	0.649329i	$-0.224950\pi$
$797$	9.07097e50	1.52101	0.760507	+	0.649329i	$0.224950\pi$
$798$	0	0
$799$	1.87783e50	0.301362
$800$	0	0
$801$	−2.44720e50	−0.375925
$802$	0	0
$803$	2.20864e50	0.324790
$804$	0	0
$805$	5.82432e50	0.820006
$806$	0	0
$807$	4.04086e50	0.544736
$808$	0	0
$809$	1.43609e51	1.85389	0.926943	−	0.375202i	$-0.122427\pi$
$809$	1.43609e51	1.85389	0.926943	+	0.375202i	$0.122427\pi$
$810$	0	0
$811$	−6.18627e50	−0.764828	−0.382414	−	0.923991i	$-0.624907\pi$
$811$	−6.18627e50	−0.764828	−0.382414	+	0.923991i	$0.624907\pi$
$812$	0	0
$813$	9.83286e50	1.16438
$814$	0	0
$815$	2.13249e51	2.41896
$816$	0	0
$817$	9.90049e49	0.107590
$818$	0	0
$819$	−1.22115e50	−0.127146
$820$	0	0
$821$	−9.87204e50	−0.984922	−0.492461	−	0.870335i	$-0.663903\pi$
$821$	−9.87204e50	−0.984922	−0.492461	+	0.870335i	$0.663903\pi$
$822$	0	0
$823$	1.56871e51	1.49985	0.749923	−	0.661525i	$-0.230091\pi$
$823$	1.56871e51	1.49985	0.749923	+	0.661525i	$0.230091\pi$
$824$	0	0
$825$	1.67900e50	0.153853
$826$	0	0
$827$	3.34712e49	0.0293985	0.0146992	−	0.999892i	$-0.495321\pi$
$827$	3.34712e49	0.0293985	0.0146992	+	0.999892i	$0.495321\pi$
$828$	0	0
$829$	−1.24277e51	−1.04638	−0.523188	−	0.852217i	$-0.675257\pi$
$829$	−1.24277e51	−1.04638	−0.523188	+	0.852217i	$0.675257\pi$
$830$	0	0
$831$	2.04327e49	0.0164933
$832$	0	0
$833$	2.94772e50	0.228139
$834$	0	0
$835$	−1.03435e51	−0.767635
$836$	0	0
$837$	1.21258e51	0.863005
$838$	0	0
$839$	−7.06560e50	−0.482297	−0.241149	−	0.970488i	$-0.577524\pi$
$839$	−7.06560e50	−0.482297	−0.241149	+	0.970488i	$0.577524\pi$
$840$	0	0
$841$	1.69401e51	1.10914
$842$	0	0
$843$	1.87258e51	1.17613
$844$	0	0
$845$	−1.76853e51	−1.06566
$846$	0	0
$847$	9.29800e50	0.537564
$848$	0	0
$849$	−1.67516e49	−0.00929333
$850$	0	0
$851$	−3.11773e51	−1.65986
$852$	0	0
$853$	−4.40952e50	−0.225311	−0.112656	−	0.993634i	$-0.535936\pi$
$853$	−4.40952e50	−0.225311	−0.112656	+	0.993634i	$0.535936\pi$
$854$	0	0
$855$	−1.98759e51	−0.974810
$856$	0	0
$857$	−2.01998e51	−0.951004	−0.475502	−	0.879715i	$-0.657733\pi$
$857$	−2.01998e51	−0.951004	−0.475502	+	0.879715i	$0.657733\pi$
$858$	0	0
$859$	2.47558e51	1.11891	0.559456	−	0.828860i	$-0.311010\pi$
$859$	2.47558e51	1.11891	0.559456	+	0.828860i	$0.311010\pi$
$860$	0	0
$861$	3.82315e50	0.165908
$862$	0	0
$863$	−1.15182e51	−0.479948	−0.239974	−	0.970779i	$-0.577139\pi$
$863$	−1.15182e51	−0.479948	−0.239974	+	0.970779i	$0.577139\pi$
$864$	0	0
$865$	−6.69696e49	−0.0267976
$866$	0	0
$867$	−1.52563e51	−0.586293
$868$	0	0
$869$	9.84195e49	0.0363274
$870$	0	0
$871$	1.83781e50	0.0651602
$872$	0	0
$873$	−9.38647e49	−0.0319708
$874$	0	0
$875$	1.11109e51	0.363588
$876$	0	0
$877$	1.21793e50	0.0382938	0.0191469	−	0.999817i	$-0.493905\pi$
$877$	1.21793e50	0.0382938	0.0191469	+	0.999817i	$0.493905\pi$
$878$	0	0
$879$	−3.68753e50	−0.111412
$880$	0	0
$881$	1.43159e51	0.415663	0.207832	−	0.978165i	$-0.433359\pi$
$881$	1.43159e51	0.415663	0.207832	+	0.978165i	$0.433359\pi$
$882$	0	0
$883$	−4.21625e51	−1.17656	−0.588280	−	0.808658i	$-0.700195\pi$
$883$	−4.21625e51	−1.17656	−0.588280	+	0.808658i	$0.700195\pi$
$884$	0	0
$885$	−1.79665e51	−0.481900
$886$	0	0
$887$	−1.06484e51	−0.274549	−0.137274	−	0.990533i	$-0.543834\pi$
$887$	−1.06484e51	−0.274549	−0.137274	+	0.990533i	$0.543834\pi$
$888$	0	0
$889$	−1.18280e51	−0.293177
$890$	0	0
$891$	−3.52185e50	−0.0839288
$892$	0	0
$893$	−5.06481e51	−1.16055
$894$	0	0
$895$	1.03108e51	0.227190
$896$	0	0
$897$	1.26295e51	0.267621
$898$	0	0
$899$	5.84365e51	1.19094
$900$	0	0
$901$	3.30714e51	0.648289
$902$	0	0
$903$	−1.69235e50	−0.0319120
$904$	0	0
$905$	−1.03925e52	−1.88525
$906$	0	0
$907$	−5.18332e51	−0.904649	−0.452325	−	0.891853i	$-0.649405\pi$
$907$	−5.18332e51	−0.904649	−0.452325	+	0.891853i	$0.649405\pi$
$908$	0	0
$909$	−1.89913e51	−0.318923
$910$	0	0
$911$	−9.24676e51	−1.49423	−0.747116	−	0.664694i	$-0.768562\pi$
$911$	−9.24676e51	−1.49423	−0.747116	+	0.664694i	$0.768562\pi$
$912$	0	0
$913$	−2.39226e51	−0.372023
$914$	0	0
$915$	7.57978e51	1.13446
$916$	0	0
$917$	6.64424e51	0.957154
$918$	0	0
$919$	4.29279e51	0.595275	0.297638	−	0.954679i	$-0.403801\pi$
$919$	4.29279e51	0.595275	0.297638	+	0.954679i	$0.403801\pi$
$920$	0	0
$921$	7.49359e51	1.00034
$922$	0	0
$923$	−1.33592e51	−0.171692
$924$	0	0
$925$	7.16109e51	0.886132
$926$	0	0
$927$	2.47460e51	0.294856
$928$	0	0
$929$	4.47733e51	0.513740	0.256870	−	0.966446i	$-0.417309\pi$
$929$	4.47733e51	0.513740	0.256870	+	0.966446i	$0.417309\pi$
$930$	0	0
$931$	−7.95047e51	−0.878564
$932$	0	0
$933$	−2.87284e51	−0.305761
$934$	0	0
$935$	−1.92734e51	−0.197586
$936$	0	0
$937$	−1.28073e50	−0.0126478	−0.00632388	−	0.999980i	$-0.502013\pi$
$937$	−1.28073e50	−0.0126478	−0.00632388	+	0.999980i	$0.502013\pi$
$938$	0	0
$939$	4.14159e51	0.394021
$940$	0	0
$941$	3.26378e50	0.0299160	0.0149580	−	0.999888i	$-0.495239\pi$
$941$	3.26378e50	0.0299160	0.0149580	+	0.999888i	$0.495239\pi$
$942$	0	0
$943$	4.31310e51	0.380921
$944$	0	0
$945$	9.90965e51	0.843338
$946$	0	0
$947$	1.57573e52	1.29228	0.646140	−	0.763219i	$-0.276382\pi$
$947$	1.57573e52	1.29228	0.646140	+	0.763219i	$0.276382\pi$
$948$	0	0
$949$	−3.81594e51	−0.301608
$950$	0	0
$951$	−3.75299e51	−0.285903
$952$	0	0
$953$	−9.47096e51	−0.695454	−0.347727	−	0.937596i	$-0.613046\pi$
$953$	−9.47096e51	−0.695454	−0.347727	+	0.937596i	$0.613046\pi$
$954$	0	0
$955$	1.86651e52	1.32121
$956$	0	0
$957$	−5.99377e51	−0.409017
$958$	0	0
$959$	−2.79924e51	−0.184168
$960$	0	0
$961$	−5.16308e51	−0.327529
$962$	0	0
$963$	1.41985e52	0.868528
$964$	0	0
$965$	3.50883e52	2.06983
$966$	0	0
$967$	−1.42806e52	−0.812424	−0.406212	−	0.913779i	$-0.633151\pi$
$967$	−1.42806e52	−0.812424	−0.406212	+	0.913779i	$0.633151\pi$
$968$	0	0
$969$	−7.38949e51	−0.405460
$970$	0	0
$971$	4.08920e51	0.216422	0.108211	−	0.994128i	$-0.465488\pi$
$971$	4.08920e51	0.216422	0.108211	+	0.994128i	$0.465488\pi$
$972$	0	0
$973$	2.19766e51	0.112198
$974$	0	0
$975$	−2.90087e51	−0.142872
$976$	0	0
$977$	2.55940e52	1.21614	0.608071	−	0.793883i	$-0.291944\pi$
$977$	2.55940e52	1.21614	0.608071	+	0.793883i	$0.291944\pi$
$978$	0	0
$979$	6.40042e51	0.293435
$980$	0	0
$981$	5.72166e51	0.253113
$982$	0	0
$983$	−1.82270e52	−0.778087	−0.389044	−	0.921219i	$-0.627194\pi$
$983$	−1.82270e52	−0.778087	−0.389044	+	0.921219i	$0.627194\pi$
$984$	0	0
$985$	8.41302e48	0.000346592 0
$986$	0	0
$987$	8.65757e51	0.344228
$988$	0	0
$989$	−1.90922e51	−0.0732693
$990$	0	0
$991$	−8.47738e51	−0.314031	−0.157015	−	0.987596i	$-0.550187\pi$
$991$	−8.47738e51	−0.314031	−0.157015	+	0.987596i	$0.550187\pi$
$992$	0	0
$993$	−2.84149e52	−1.01610
$994$	0	0
$995$	3.42702e52	1.18307
$996$	0	0
$997$	−4.09490e52	−1.36483	−0.682414	−	0.730966i	$-0.739070\pi$
$997$	−4.09490e52	−1.36483	−0.682414	+	0.730966i	$0.739070\pi$
$998$	0	0
$999$	−5.30459e52	−1.70709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.36.a.d.1.2		3
4.3	odd	2		1.36.a.a.1.3	✓	3
12.11	even	2		9.36.a.b.1.1		3
20.3	even	4		25.36.b.a.24.1		6
20.7	even	4		25.36.b.a.24.6		6
20.19	odd	2		25.36.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.36.a.a.1.3	✓	3	4.3	odd	2
9.36.a.b.1.1		3	12.11	even	2
16.36.a.d.1.2		3	1.1	even	1	trivial
25.36.a.a.1.1		3	20.19	odd	2
25.36.b.a.24.1		6	20.3	even	4
25.36.b.a.24.6		6	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1.54691e8 q^{3} +2.12139e12 q^{5} -3.96640e14 q^{7} -2.61023e16 q^{9} +O(q^{10})\)	\(q+1.54691e8 q^{3} +2.12139e12 q^{5} -3.96640e14 q^{7} -2.61023e16 q^{9} +6.82681e17 q^{11} -1.17949e19 q^{13} +3.28159e20 q^{15} -1.33083e21 q^{17} +3.58945e22 q^{19} -6.13565e22 q^{21} -6.92195e23 q^{23} +1.58990e24 q^{25} -1.17772e25 q^{27} -5.67568e25 q^{29} -1.02960e26 q^{31} +1.05604e26 q^{33} -8.41427e26 q^{35} +4.50412e27 q^{37} -1.82456e27 q^{39} -6.23104e27 q^{41} +2.75822e27 q^{43} -5.53731e28 q^{45} -1.41103e29 q^{47} -2.21495e29 q^{49} -2.05867e29 q^{51} -2.48503e30 q^{53} +1.44823e30 q^{55} +5.55255e30 q^{57} -5.47495e30 q^{59} +2.30979e31 q^{61} +1.03532e31 q^{63} -2.50216e31 q^{65} -1.55814e31 q^{67} -1.07076e32 q^{69} +1.13262e32 q^{71} +3.23524e32 q^{73} +2.45942e32 q^{75} -2.70778e32 q^{77} +1.44166e32 q^{79} -5.15885e32 q^{81} -3.50421e33 q^{83} -2.82320e33 q^{85} -8.77975e33 q^{87} +9.37543e33 q^{89} +4.67833e33 q^{91} -1.59269e34 q^{93} +7.61461e34 q^{95} +3.59603e33 q^{97} -1.78195e34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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