LMFDB - Embedded newform 4.36.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,36,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	4.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:	$31.0380522535$
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} - x^{2} - 1597028177x + 23572260890640 $ x^3 - x^2 - 1597028177*x + 23572260890640
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19173.3$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.83743e8 q^{3} +4.90658e11 q^{5} -2.46684e14 q^{7} +3.04783e16 q^{9} +O(q^{10})$	$q-2.83743e8 q^{3} +4.90658e11 q^{5} -2.46684e14 q^{7} +3.04783e16 q^{9} +2.33450e18 q^{11} +3.81916e19 q^{13} -1.39220e20 q^{15} -4.45028e21 q^{17} +2.08695e22 q^{19} +6.99948e22 q^{21} -4.20235e23 q^{23} -2.66964e24 q^{25} +5.54809e24 q^{27} -7.50471e24 q^{29} -1.91757e26 q^{31} -6.62396e26 q^{33} -1.21037e26 q^{35} -3.21127e27 q^{37} -1.08366e28 q^{39} -2.18713e28 q^{41} +3.51246e28 q^{43} +1.49544e28 q^{45} +2.90248e29 q^{47} -3.17966e29 q^{49} +1.26273e30 q^{51} +1.28287e30 q^{53} +1.14544e30 q^{55} -5.92157e30 q^{57} -1.49203e31 q^{59} -4.52662e30 q^{61} -7.51852e30 q^{63} +1.87390e31 q^{65} -1.01659e32 q^{67} +1.19239e32 q^{69} +1.79692e32 q^{71} +4.54379e32 q^{73} +7.57490e32 q^{75} -5.75883e32 q^{77} -1.10706e33 q^{79} -3.09911e33 q^{81} +6.18064e32 q^{83} -2.18356e33 q^{85} +2.12941e33 q^{87} -1.41163e34 q^{89} -9.42127e33 q^{91} +5.44097e34 q^{93} +1.02398e34 q^{95} +6.64379e34 q^{97} +7.11515e34 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * 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
$3$	−2.83743e8	−1.26854	−0.634268	−	0.773114i	$-0.718698\pi$
$3$	−2.83743e8	−1.26854	−0.634268	+	0.773114i	$0.718698\pi$
$4$	0	0
$5$	4.90658e11	0.287610	0.143805	−	0.989606i	$-0.454066\pi$
$5$	4.90658e11	0.287610	0.143805	+	0.989606i	$0.454066\pi$
$6$	0	0
$7$	−2.46684e14	−0.400798	−0.200399	−	0.979714i	$-0.564224\pi$
$7$	−2.46684e14	−0.400798	−0.200399	+	0.979714i	$0.564224\pi$
$8$	0	0
$9$	3.04783e16	0.609182
$10$	0	0
$11$	2.33450e18	1.39258	0.696291	−	0.717759i	$-0.254832\pi$
$11$	2.33450e18	1.39258	0.696291	+	0.717759i	$0.254832\pi$
$12$	0	0
$13$	3.81916e19	1.22450	0.612251	−	0.790663i	$-0.290264\pi$
$13$	3.81916e19	1.22450	0.612251	+	0.790663i	$0.290264\pi$
$14$	0	0
$15$	−1.39220e20	−0.364843
$16$	0	0
$17$	−4.45028e21	−1.30476	−0.652381	−	0.757891i	$-0.726230\pi$
$17$	−4.45028e21	−1.30476	−0.652381	+	0.757891i	$0.726230\pi$
$18$	0	0
$19$	2.08695e22	0.873623	0.436812	−	0.899553i	$-0.356108\pi$
$19$	2.08695e22	0.873623	0.436812	+	0.899553i	$0.356108\pi$
$20$	0	0
$21$	6.99948e22	0.508427
$22$	0	0
$23$	−4.20235e23	−0.621234	−0.310617	−	0.950535i	$-0.600536\pi$
$23$	−4.20235e23	−0.621234	−0.310617	+	0.950535i	$0.600536\pi$
$24$	0	0
$25$	−2.66964e24	−0.917281
$26$	0	0
$27$	5.54809e24	0.495767
$28$	0	0
$29$	−7.50471e24	−0.192030	−0.0960150	−	0.995380i	$-0.530610\pi$
$29$	−7.50471e24	−0.192030	−0.0960150	+	0.995380i	$0.530610\pi$
$30$	0	0
$31$	−1.91757e26	−1.52729	−0.763646	−	0.645635i	$-0.776593\pi$
$31$	−1.91757e26	−1.52729	−0.763646	+	0.645635i	$0.776593\pi$
$32$	0	0
$33$	−6.62396e26	−1.76654
$34$	0	0
$35$	−1.21037e26	−0.115273
$36$	0	0
$37$	−3.21127e27	−1.15650	−0.578252	−	0.815858i	$-0.696265\pi$
$37$	−3.21127e27	−1.15650	−0.578252	+	0.815858i	$0.696265\pi$
$38$	0	0
$39$	−1.08366e28	−1.55332
$40$	0	0
$41$	−2.18713e28	−1.30664	−0.653322	−	0.757080i	$-0.726625\pi$
$41$	−2.18713e28	−1.30664	−0.653322	+	0.757080i	$0.726625\pi$
$42$	0	0
$43$	3.51246e28	0.911829	0.455914	−	0.890024i	$-0.349312\pi$
$43$	3.51246e28	0.911829	0.455914	+	0.890024i	$0.349312\pi$
$44$	0	0
$45$	1.49544e28	0.175207
$46$	0	0
$47$	2.90248e29	1.58875	0.794377	−	0.607426i	$-0.207798\pi$
$47$	2.90248e29	1.58875	0.794377	+	0.607426i	$0.207798\pi$
$48$	0	0
$49$	−3.17966e29	−0.839361
$50$	0	0
$51$	1.26273e30	1.65514
$52$	0	0
$53$	1.28287e30	0.857739	0.428869	−	0.903367i	$-0.358912\pi$
$53$	1.28287e30	0.857739	0.428869	+	0.903367i	$0.358912\pi$
$54$	0	0
$55$	1.14544e30	0.400520
$56$	0	0
$57$	−5.92157e30	−1.10822
$58$	0	0
$59$	−1.49203e31	−1.52710	−0.763549	−	0.645750i	$-0.776545\pi$
$59$	−1.49203e31	−1.52710	−0.763549	+	0.645750i	$0.776545\pi$
$60$	0	0
$61$	−4.52662e30	−0.258525	−0.129262	−	0.991610i	$-0.541261\pi$
$61$	−4.52662e30	−0.258525	−0.129262	+	0.991610i	$0.541261\pi$
$62$	0	0
$63$	−7.51852e30	−0.244159
$64$	0	0
$65$	1.87390e31	0.352179
$66$	0	0
$67$	−1.01659e32	−1.12418	−0.562090	−	0.827076i	$-0.690002\pi$
$67$	−1.01659e32	−1.12418	−0.562090	+	0.827076i	$0.690002\pi$
$68$	0	0
$69$	1.19239e32	0.788058
$70$	0	0
$71$	1.79692e32	0.720290	0.360145	−	0.932896i	$-0.382727\pi$
$71$	1.79692e32	0.720290	0.360145	+	0.932896i	$0.382727\pi$
$72$	0	0
$73$	4.54379e32	1.12013	0.560064	−	0.828449i	$-0.310776\pi$
$73$	4.54379e32	1.12013	0.560064	+	0.828449i	$0.310776\pi$
$74$	0	0
$75$	7.57490e32	1.16360
$76$	0	0
$77$	−5.75883e32	−0.558145
$78$	0	0
$79$	−1.10706e33	−0.685009	−0.342505	−	0.939516i	$-0.611275\pi$
$79$	−1.10706e33	−0.685009	−0.342505	+	0.939516i	$0.611275\pi$
$80$	0	0
$81$	−3.09911e33	−1.23808
$82$	0	0
$83$	6.18064e32	0.161126	0.0805632	−	0.996750i	$-0.474328\pi$
$83$	6.18064e32	0.161126	0.0805632	+	0.996750i	$0.474328\pi$
$84$	0	0
$85$	−2.18356e33	−0.375262
$86$	0	0
$87$	2.12941e33	0.243597
$88$	0	0
$89$	−1.41163e34	−1.08492	−0.542459	−	0.840082i	$-0.682507\pi$
$89$	−1.41163e34	−1.08492	−0.542459	+	0.840082i	$0.682507\pi$
$90$	0	0
$91$	−9.42127e33	−0.490778
$92$	0	0
$93$	5.44097e34	1.93742
$94$	0	0
$95$	1.02398e34	0.251262
$96$	0	0
$97$	6.64379e34	1.13217	0.566083	−	0.824348i	$-0.308458\pi$
$97$	6.64379e34	1.13217	0.566083	+	0.824348i	$0.308458\pi$
$98$	0	0
$99$	7.11515e34	0.848336
$100$	0	0
$101$	−1.14029e35	−0.958056	−0.479028	−	0.877800i	$-0.659011\pi$
$101$	−1.14029e35	−0.958056	−0.479028	+	0.877800i	$0.659011\pi$
$102$	0	0
$103$	−2.30990e35	−1.37702	−0.688512	−	0.725225i	$-0.741736\pi$
$103$	−2.30990e35	−1.37702	−0.688512	+	0.725225i	$0.741736\pi$
$104$	0	0
$105$	3.43435e34	0.146228
$106$	0	0
$107$	3.27204e34	0.100139	0.0500693	−	0.998746i	$-0.484056\pi$
$107$	3.27204e34	0.100139	0.0500693	+	0.998746i	$0.484056\pi$
$108$	0	0
$109$	−7.12314e35	−1.57655	−0.788275	−	0.615323i	$-0.789026\pi$
$109$	−7.12314e35	−1.57655	−0.788275	+	0.615323i	$0.789026\pi$
$110$	0	0
$111$	9.11175e35	1.46707
$112$	0	0
$113$	−1.11897e36	−1.31810	−0.659048	−	0.752101i	$-0.729040\pi$
$113$	−1.11897e36	−1.31810	−0.659048	+	0.752101i	$0.729040\pi$
$114$	0	0
$115$	−2.06192e35	−0.178673
$116$	0	0
$117$	1.16402e36	0.745944
$118$	0	0
$119$	1.09781e36	0.522946
$120$	0	0
$121$	2.63963e36	0.939287
$122$	0	0
$123$	6.20582e36	1.65752
$124$	0	0
$125$	−2.73788e36	−0.551428
$126$	0	0
$127$	−5.48154e36	−0.836253	−0.418126	−	0.908389i	$-0.637313\pi$
$127$	−5.48154e36	−0.836253	−0.418126	+	0.908389i	$0.637313\pi$
$128$	0	0
$129$	−9.96634e36	−1.15669
$130$	0	0
$131$	−2.03882e36	−0.180772	−0.0903859	−	0.995907i	$-0.528810\pi$
$131$	−2.03882e36	−0.180772	−0.0903859	+	0.995907i	$0.528810\pi$
$132$	0	0
$133$	−5.14818e36	−0.350147
$134$	0	0
$135$	2.72221e36	0.142587
$136$	0	0
$137$	−1.05428e37	−0.426921	−0.213461	−	0.976952i	$-0.568474\pi$
$137$	−1.05428e37	−0.426921	−0.213461	+	0.976952i	$0.568474\pi$
$138$	0	0
$139$	2.83397e37	0.890504	0.445252	−	0.895405i	$-0.353114\pi$
$139$	2.83397e37	0.890504	0.445252	+	0.895405i	$0.353114\pi$
$140$	0	0
$141$	−8.23557e37	−2.01539
$142$	0	0
$143$	8.91582e37	1.70522
$144$	0	0
$145$	−3.68224e36	−0.0552296
$146$	0	0
$147$	9.02204e37	1.06476
$148$	0	0
$149$	2.84255e37	0.264818	0.132409	−	0.991195i	$-0.457729\pi$
$149$	2.84255e37	0.264818	0.132409	+	0.991195i	$0.457729\pi$
$150$	0	0
$151$	2.11776e38	1.56235	0.781177	−	0.624310i	$-0.214620\pi$
$151$	2.11776e38	1.56235	0.781177	+	0.624310i	$0.214620\pi$
$152$	0	0
$153$	−1.35637e38	−0.794837
$154$	0	0
$155$	−9.40872e37	−0.439264
$156$	0	0
$157$	−3.38275e38	−1.26190	−0.630951	−	0.775823i	$-0.717335\pi$
$157$	−3.38275e38	−1.26190	−0.630951	+	0.775823i	$0.717335\pi$
$158$	0	0
$159$	−3.64005e38	−1.08807
$160$	0	0
$161$	1.03665e38	0.248990
$162$	0	0
$163$	5.02228e37	0.0971891	0.0485946	−	0.998819i	$-0.484526\pi$
$163$	5.02228e37	0.0971891	0.0485946	+	0.998819i	$0.484526\pi$
$164$	0	0
$165$	−3.25009e38	−0.508074
$166$	0	0
$167$	4.10664e38	0.519934	0.259967	−	0.965618i	$-0.416288\pi$
$167$	4.10664e38	0.519934	0.259967	+	0.965618i	$0.416288\pi$
$168$	0	0
$169$	4.85814e38	0.499405
$170$	0	0
$171$	6.36067e38	0.532195
$172$	0	0
$173$	−1.96129e39	−1.33886	−0.669431	−	0.742874i	$-0.733462\pi$
$173$	−1.96129e39	−1.33886	−0.669431	+	0.742874i	$0.733462\pi$
$174$	0	0
$175$	6.58558e38	0.367644
$176$	0	0
$177$	4.23352e39	1.93718
$178$	0	0
$179$	2.30177e39	0.865235	0.432618	−	0.901577i	$-0.357590\pi$
$179$	2.30177e39	0.865235	0.432618	+	0.901577i	$0.357590\pi$
$180$	0	0
$181$	−2.13992e39	−0.662251	−0.331126	−	0.943587i	$-0.607428\pi$
$181$	−2.13992e39	−0.662251	−0.331126	+	0.943587i	$0.607428\pi$
$182$	0	0
$183$	1.28440e39	0.327948
$184$	0	0
$185$	−1.57564e39	−0.332622
$186$	0	0
$187$	−1.03892e40	−1.81699
$188$	0	0
$189$	−1.36863e39	−0.198702
$190$	0	0
$191$	−1.38915e40	−1.67751	−0.838757	−	0.544506i	$-0.816717\pi$
$191$	−1.38915e40	−1.67751	−0.838757	+	0.544506i	$0.816717\pi$
$192$	0	0
$193$	1.41314e40	1.42211	0.711055	−	0.703137i	$-0.248218\pi$
$193$	1.41314e40	1.42211	0.711055	+	0.703137i	$0.248218\pi$
$194$	0	0
$195$	−5.31706e39	−0.446751
$196$	0	0
$197$	8.36956e39	0.588225	0.294113	−	0.955771i	$-0.404976\pi$
$197$	8.36956e39	0.588225	0.294113	+	0.955771i	$0.404976\pi$
$198$	0	0
$199$	−4.43229e39	−0.261035	−0.130518	−	0.991446i	$-0.541664\pi$
$199$	−4.43229e39	−0.261035	−0.130518	+	0.991446i	$0.541664\pi$
$200$	0	0
$201$	2.88449e40	1.42606
$202$	0	0
$203$	1.85129e39	0.0769652
$204$	0	0
$205$	−1.07313e40	−0.375803
$206$	0	0
$207$	−1.28081e40	−0.378445
$208$	0	0
$209$	4.87198e40	1.21659
$210$	0	0
$211$	3.98494e40	0.842323	0.421161	−	0.906986i	$-0.361623\pi$
$211$	3.98494e40	0.842323	0.421161	+	0.906986i	$0.361623\pi$
$212$	0	0
$213$	−5.09862e40	−0.913713
$214$	0	0
$215$	1.72341e40	0.262251
$216$	0	0
$217$	4.73035e40	0.612136
$218$	0	0
$219$	−1.28927e41	−1.42092
$220$	0	0
$221$	−1.69963e41	−1.59768
$222$	0	0
$223$	−3.82214e40	−0.306882	−0.153441	−	0.988158i	$-0.549035\pi$
$223$	−3.82214e40	−0.306882	−0.153441	+	0.988158i	$0.549035\pi$
$224$	0	0
$225$	−8.13661e40	−0.558791
$226$	0	0
$227$	4.02193e40	0.236582	0.118291	−	0.992979i	$-0.462258\pi$
$227$	4.02193e40	0.236582	0.118291	+	0.992979i	$0.462258\pi$
$228$	0	0
$229$	1.36117e41	0.686738	0.343369	−	0.939201i	$-0.388432\pi$
$229$	1.36117e41	0.686738	0.343369	+	0.939201i	$0.388432\pi$
$230$	0	0
$231$	1.63403e41	0.708026
$232$	0	0
$233$	1.52213e41	0.567184	0.283592	−	0.958945i	$-0.408474\pi$
$233$	1.52213e41	0.567184	0.283592	+	0.958945i	$0.408474\pi$
$234$	0	0
$235$	1.42412e41	0.456941
$236$	0	0
$237$	3.14120e41	0.868959
$238$	0	0
$239$	−3.23009e40	−0.0771349	−0.0385675	−	0.999256i	$-0.512279\pi$
$239$	−3.23009e40	−0.0771349	−0.0385675	+	0.999256i	$0.512279\pi$
$240$	0	0
$241$	−8.01250e41	−1.65375	−0.826875	−	0.562386i	$-0.809884\pi$
$241$	−8.01250e41	−1.65375	−0.826875	+	0.562386i	$0.809884\pi$
$242$	0	0
$243$	6.01769e41	1.07478
$244$	0	0
$245$	−1.56012e41	−0.241408
$246$	0	0
$247$	7.97040e41	1.06975
$248$	0	0
$249$	−1.75371e41	−0.204394
$250$	0	0
$251$	−9.02698e41	−0.914644	−0.457322	−	0.889301i	$-0.651191\pi$
$251$	−9.02698e41	−0.914644	−0.457322	+	0.889301i	$0.651191\pi$
$252$	0	0
$253$	−9.81038e41	−0.865120
$254$	0	0
$255$	6.19570e41	0.476033
$256$	0	0
$257$	−2.85219e42	−1.91139	−0.955693	−	0.294367i	$-0.904891\pi$
$257$	−2.85219e42	−1.91139	−0.955693	+	0.294367i	$0.904891\pi$
$258$	0	0
$259$	7.92171e41	0.463525
$260$	0	0
$261$	−2.28731e41	−0.116981
$262$	0	0
$263$	−1.38534e42	−0.619911	−0.309956	−	0.950751i	$-0.600314\pi$
$263$	−1.38534e42	−0.619911	−0.309956	+	0.950751i	$0.600314\pi$
$264$	0	0
$265$	6.29450e41	0.246694
$266$	0	0
$267$	4.00541e42	1.37626
$268$	0	0
$269$	6.48754e42	1.95621	0.978105	−	0.208112i	$-0.0667319\pi$
$269$	6.48754e42	1.95621	0.978105	+	0.208112i	$0.0667319\pi$
$270$	0	0
$271$	−1.20669e42	−0.319619	−0.159810	−	0.987148i	$-0.551088\pi$
$271$	−1.20669e42	−0.319619	−0.159810	+	0.987148i	$0.551088\pi$
$272$	0	0
$273$	2.67322e42	0.622569
$274$	0	0
$275$	−6.23226e42	−1.27739
$276$	0	0
$277$	9.45964e42	1.70796	0.853982	−	0.520303i	$-0.174181\pi$
$277$	9.45964e42	1.70796	0.853982	+	0.520303i	$0.174181\pi$
$278$	0	0
$279$	−5.84444e42	−0.930399
$280$	0	0
$281$	−1.02911e43	−1.44577	−0.722887	−	0.690966i	$-0.757185\pi$
$281$	−1.02911e43	−1.44577	−0.722887	+	0.690966i	$0.757185\pi$
$282$	0	0
$283$	1.37988e42	0.171230	0.0856149	−	0.996328i	$-0.472715\pi$
$283$	1.37988e42	0.171230	0.0856149	+	0.996328i	$0.472715\pi$
$284$	0	0
$285$	−2.90546e42	−0.318735
$286$	0	0
$287$	5.39530e42	0.523700
$288$	0	0
$289$	8.17144e42	0.702403
$290$	0	0
$291$	−1.88513e43	−1.43619
$292$	0	0
$293$	−9.01953e42	−0.609538	−0.304769	−	0.952426i	$-0.598579\pi$
$293$	−9.01953e42	−0.609538	−0.304769	+	0.952426i	$0.598579\pi$
$294$	0	0
$295$	−7.32075e42	−0.439208
$296$	0	0
$297$	1.29520e43	0.690396
$298$	0	0
$299$	−1.60495e43	−0.760703
$300$	0	0
$301$	−8.66468e42	−0.365459
$302$	0	0
$303$	3.23549e43	1.21533
$304$	0	0
$305$	−2.22102e42	−0.0743542
$306$	0	0
$307$	6.42879e43	1.91959	0.959793	−	0.280708i	$-0.0905694\pi$
$307$	6.42879e43	1.91959	0.959793	+	0.280708i	$0.0905694\pi$
$308$	0	0
$309$	6.55416e43	1.74680
$310$	0	0
$311$	−7.61324e43	−1.81244	−0.906218	−	0.422810i	$-0.861044\pi$
$311$	−7.61324e43	−1.81244	−0.906218	+	0.422810i	$0.861044\pi$
$312$	0	0
$313$	5.53058e43	1.17691	0.588457	−	0.808529i	$-0.299736\pi$
$313$	5.53058e43	1.17691	0.588457	+	0.808529i	$0.299736\pi$
$314$	0	0
$315$	−3.68902e42	−0.0702225
$316$	0	0
$317$	−7.88573e43	−1.34371	−0.671854	−	0.740684i	$-0.734502\pi$
$317$	−7.88573e43	−1.34371	−0.671854	+	0.740684i	$0.734502\pi$
$318$	0	0
$319$	−1.75197e43	−0.267418
$320$	0	0
$321$	−9.28416e42	−0.127029
$322$	0	0
$323$	−9.28751e43	−1.13987
$324$	0	0
$325$	−1.01958e44	−1.12321
$326$	0	0
$327$	2.02114e44	1.99991
$328$	0	0
$329$	−7.15995e43	−0.636769
$330$	0	0
$331$	−7.05747e42	−0.0564493	−0.0282247	−	0.999602i	$-0.508985\pi$
$331$	−7.05747e42	−0.0564493	−0.0282247	+	0.999602i	$0.508985\pi$
$332$	0	0
$333$	−9.78742e43	−0.704522
$334$	0	0
$335$	−4.98796e43	−0.323325
$336$	0	0
$337$	1.23271e44	0.720007	0.360003	−	0.932951i	$-0.382776\pi$
$337$	1.23271e44	0.720007	0.360003	+	0.932951i	$0.382776\pi$
$338$	0	0
$339$	3.17500e44	1.67205
$340$	0	0
$341$	−4.47657e44	−2.12688
$342$	0	0
$343$	1.71886e44	0.737212
$344$	0	0
$345$	5.85053e43	0.226653
$346$	0	0
$347$	9.42984e43	0.330171	0.165085	−	0.986279i	$-0.447210\pi$
$347$	9.42984e43	0.330171	0.165085	+	0.986279i	$0.447210\pi$
$348$	0	0
$349$	5.06575e44	1.60398	0.801989	−	0.597339i	$-0.203775\pi$
$349$	5.06575e44	1.60398	0.801989	+	0.597339i	$0.203775\pi$
$350$	0	0
$351$	2.11890e44	0.607067
$352$	0	0
$353$	7.03174e43	0.182392	0.0911960	−	0.995833i	$-0.470931\pi$
$353$	7.03174e43	0.182392	0.0911960	+	0.995833i	$0.470931\pi$
$354$	0	0
$355$	8.81672e43	0.207162
$356$	0	0
$357$	−3.11497e44	−0.663375
$358$	0	0
$359$	−8.55482e44	−1.65218	−0.826091	−	0.563536i	$-0.809440\pi$
$359$	−8.55482e44	−1.65218	−0.826091	+	0.563536i	$0.809440\pi$
$360$	0	0
$361$	−1.35122e44	−0.236783
$362$	0	0
$363$	−7.48974e44	−1.19152
$364$	0	0
$365$	2.22944e44	0.322160
$366$	0	0
$367$	−2.84210e44	−0.373236	−0.186618	−	0.982433i	$-0.559753\pi$
$367$	−2.84210e44	−0.373236	−0.186618	+	0.982433i	$0.559753\pi$
$368$	0	0
$369$	−6.66600e44	−0.795983
$370$	0	0
$371$	−3.16464e44	−0.343780
$372$	0	0
$373$	1.75086e45	1.73120	0.865600	−	0.500735i	$-0.166937\pi$
$373$	1.75086e45	1.73120	0.865600	+	0.500735i	$0.166937\pi$
$374$	0	0
$375$	7.76853e44	0.699506
$376$	0	0
$377$	−2.86617e44	−0.235141
$378$	0	0
$379$	−6.12102e44	−0.457760	−0.228880	−	0.973455i	$-0.573506\pi$
$379$	−6.12102e44	−0.457760	−0.228880	+	0.973455i	$0.573506\pi$
$380$	0	0
$381$	1.55535e45	1.06082
$382$	0	0
$383$	−7.71048e44	−0.479847	−0.239923	−	0.970792i	$-0.577122\pi$
$383$	−7.71048e44	−0.479847	−0.239923	+	0.970792i	$0.577122\pi$
$384$	0	0
$385$	−2.82561e44	−0.160528
$386$	0	0
$387$	1.07054e45	0.555469
$388$	0	0
$389$	4.82937e44	0.228967	0.114483	−	0.993425i	$-0.463479\pi$
$389$	4.82937e44	0.228967	0.114483	+	0.993425i	$0.463479\pi$
$390$	0	0
$391$	1.87016e45	0.810563
$392$	0	0
$393$	5.78499e44	0.229315
$394$	0	0
$395$	−5.43187e44	−0.197015
$396$	0	0
$397$	−8.75288e44	−0.290615	−0.145307	−	0.989387i	$-0.546417\pi$
$397$	−8.75288e44	−0.290615	−0.145307	+	0.989387i	$0.546417\pi$
$398$	0	0
$399$	1.46076e45	0.444173
$400$	0	0
$401$	2.22212e45	0.619071	0.309535	−	0.950888i	$-0.399827\pi$
$401$	2.22212e45	0.619071	0.309535	+	0.950888i	$0.399827\pi$
$402$	0	0
$403$	−7.32352e45	−1.87017
$404$	0	0
$405$	−1.52060e45	−0.356084
$406$	0	0
$407$	−7.49671e45	−1.61053
$408$	0	0
$409$	−4.00647e45	−0.789958	−0.394979	−	0.918690i	$-0.629248\pi$
$409$	−4.00647e45	−0.789958	−0.394979	+	0.918690i	$0.629248\pi$
$410$	0	0
$411$	2.99145e45	0.541565
$412$	0	0
$413$	3.68060e45	0.612058
$414$	0	0
$415$	3.03258e44	0.0463415
$416$	0	0
$417$	−8.04117e45	−1.12964
$418$	0	0
$419$	7.97513e45	1.03037	0.515184	−	0.857080i	$-0.327724\pi$
$419$	7.97513e45	1.03037	0.515184	+	0.857080i	$0.327724\pi$
$420$	0	0
$421$	8.79872e45	1.04588	0.522941	−	0.852369i	$-0.324835\pi$
$421$	8.79872e45	1.04588	0.522941	+	0.852369i	$0.324835\pi$
$422$	0	0
$423$	8.84626e45	0.967840
$424$	0	0
$425$	1.18806e46	1.19683
$426$	0	0
$427$	1.11665e45	0.103616
$428$	0	0
$429$	−2.52980e46	−2.16313
$430$	0	0
$431$	2.04764e46	1.61399	0.806994	−	0.590559i	$-0.201093\pi$
$431$	2.04764e46	1.61399	0.806994	+	0.590559i	$0.201093\pi$
$432$	0	0
$433$	−6.27229e45	−0.455920	−0.227960	−	0.973671i	$-0.573205\pi$
$433$	−6.27229e45	−0.455920	−0.227960	+	0.973671i	$0.573205\pi$
$434$	0	0
$435$	1.04481e45	0.0700608
$436$	0	0
$437$	−8.77010e45	−0.542725
$438$	0	0
$439$	3.80675e45	0.217483	0.108742	−	0.994070i	$-0.465318\pi$
$439$	3.80675e45	0.217483	0.108742	+	0.994070i	$0.465318\pi$
$440$	0	0
$441$	−9.69105e45	−0.511323
$442$	0	0
$443$	1.94774e46	0.949436	0.474718	−	0.880138i	$-0.342550\pi$
$443$	1.94774e46	0.949436	0.474718	+	0.880138i	$0.342550\pi$
$444$	0	0
$445$	−6.92629e45	−0.312033
$446$	0	0
$447$	−8.06551e45	−0.335931
$448$	0	0
$449$	2.66259e46	1.02564	0.512819	−	0.858497i	$-0.328601\pi$
$449$	2.66259e46	1.02564	0.512819	+	0.858497i	$0.328601\pi$
$450$	0	0
$451$	−5.10585e46	−1.81961
$452$	0	0
$453$	−6.00898e46	−1.98190
$454$	0	0
$455$	−4.62262e45	−0.141152
$456$	0	0
$457$	2.37152e46	0.670647	0.335323	−	0.942103i	$-0.391154\pi$
$457$	2.37152e46	0.670647	0.335323	+	0.942103i	$0.391154\pi$
$458$	0	0
$459$	−2.46905e46	−0.646857
$460$	0	0
$461$	4.97091e46	1.20690	0.603448	−	0.797402i	$-0.293793\pi$
$461$	4.97091e46	1.20690	0.603448	+	0.797402i	$0.293793\pi$
$462$	0	0
$463$	−5.14163e45	−0.115727	−0.0578633	−	0.998325i	$-0.518429\pi$
$463$	−5.14163e45	−0.115727	−0.0578633	+	0.998325i	$0.518429\pi$
$464$	0	0
$465$	2.66965e46	0.557222
$466$	0	0
$467$	6.98000e45	0.135148	0.0675740	−	0.997714i	$-0.478474\pi$
$467$	6.98000e45	0.135148	0.0675740	+	0.997714i	$0.478474\pi$
$468$	0	0
$469$	2.50776e46	0.450569
$470$	0	0
$471$	9.59829e46	1.60077
$472$	0	0
$473$	8.19982e46	1.26980
$474$	0	0
$475$	−5.57140e46	−0.801358
$476$	0	0
$477$	3.90997e46	0.522519
$478$	0	0
$479$	−4.34782e46	−0.540007	−0.270004	−	0.962859i	$-0.587025\pi$
$479$	−4.34782e46	−0.540007	−0.270004	+	0.962859i	$0.587025\pi$
$480$	0	0
$481$	−1.22644e47	−1.41614
$482$	0	0
$483$	−2.94143e46	−0.315852
$484$	0	0
$485$	3.25982e46	0.325622
$486$	0	0
$487$	1.39946e46	0.130078	0.0650391	−	0.997883i	$-0.479283\pi$
$487$	1.39946e46	0.130078	0.0650391	+	0.997883i	$0.479283\pi$
$488$	0	0
$489$	−1.42503e46	−0.123288
$490$	0	0
$491$	−1.70380e47	−1.37244	−0.686218	−	0.727396i	$-0.740730\pi$
$491$	−1.70380e47	−1.37244	−0.686218	+	0.727396i	$0.740730\pi$
$492$	0	0
$493$	3.33981e46	0.250553
$494$	0	0
$495$	3.49110e46	0.243990
$496$	0	0
$497$	−4.43272e46	−0.288691
$498$	0	0
$499$	5.00476e46	0.303825	0.151913	−	0.988394i	$-0.451457\pi$
$499$	5.00476e46	0.303825	0.151913	+	0.988394i	$0.451457\pi$
$500$	0	0
$501$	−1.16523e47	−0.659554
$502$	0	0
$503$	3.79753e46	0.200476	0.100238	−	0.994964i	$-0.468040\pi$
$503$	3.79753e46	0.200476	0.100238	+	0.994964i	$0.468040\pi$
$504$	0	0
$505$	−5.59491e46	−0.275546
$506$	0	0
$507$	−1.37846e47	−0.633513
$508$	0	0
$509$	1.61088e47	0.691039	0.345520	−	0.938412i	$-0.387703\pi$
$509$	1.61088e47	0.691039	0.345520	+	0.938412i	$0.387703\pi$
$510$	0	0
$511$	−1.12088e47	−0.448945
$512$	0	0
$513$	1.15786e47	0.433113
$514$	0	0
$515$	−1.13337e47	−0.396045
$516$	0	0
$517$	6.77582e47	2.21247
$518$	0	0
$519$	5.56502e47	1.69839
$520$	0	0
$521$	−3.24063e47	−0.924634	−0.462317	−	0.886715i	$-0.652982\pi$
$521$	−3.24063e47	−0.924634	−0.462317	+	0.886715i	$0.652982\pi$
$522$	0	0
$523$	9.80294e46	0.261563	0.130782	−	0.991411i	$-0.458251\pi$
$523$	9.80294e46	0.261563	0.130782	+	0.991411i	$0.458251\pi$
$524$	0	0
$525$	−1.86861e47	−0.466370
$526$	0	0
$527$	8.53374e47	1.99275
$528$	0	0
$529$	−2.80990e47	−0.614068
$530$	0	0
$531$	−4.54745e47	−0.930280
$532$	0	0
$533$	−8.35301e47	−1.59999
$534$	0	0
$535$	1.60545e46	0.0288008
$536$	0	0
$537$	−6.53111e47	−1.09758
$538$	0	0
$539$	−7.42289e47	−1.16888
$540$	0	0
$541$	2.29578e47	0.338827	0.169413	−	0.985545i	$-0.445813\pi$
$541$	2.29578e47	0.338827	0.169413	+	0.985545i	$0.445813\pi$
$542$	0	0
$543$	6.07187e47	0.840089
$544$	0	0
$545$	−3.49502e47	−0.453431
$546$	0	0
$547$	6.20054e47	0.754486	0.377243	−	0.926114i	$-0.376872\pi$
$547$	6.20054e47	0.754486	0.377243	+	0.926114i	$0.376872\pi$
$548$	0	0
$549$	−1.37964e47	−0.157489
$550$	0	0
$551$	−1.56620e47	−0.167762
$552$	0	0
$553$	2.73094e47	0.274550
$554$	0	0
$555$	4.47075e47	0.421943
$556$	0	0
$557$	−2.00082e48	−1.77313	−0.886567	−	0.462601i	$-0.846916\pi$
$557$	−2.00082e48	−1.77313	−0.886567	+	0.462601i	$0.846916\pi$
$558$	0	0
$559$	1.34147e48	1.11654
$560$	0	0
$561$	2.94785e48	2.30491
$562$	0	0
$563$	−2.18076e48	−1.60218	−0.801091	−	0.598543i	$-0.795747\pi$
$563$	−2.18076e48	−1.60218	−0.801091	+	0.598543i	$0.795747\pi$
$564$	0	0
$565$	−5.49033e47	−0.379097
$566$	0	0
$567$	7.64500e47	0.496220
$568$	0	0
$569$	1.95181e48	1.19116	0.595582	−	0.803294i	$-0.296921\pi$
$569$	1.95181e48	1.19116	0.595582	+	0.803294i	$0.296921\pi$
$570$	0	0
$571$	−1.74335e48	−1.00058	−0.500292	−	0.865857i	$-0.666774\pi$
$571$	−1.74335e48	−1.00058	−0.500292	+	0.865857i	$0.666774\pi$
$572$	0	0
$573$	3.94162e48	2.12798
$574$	0	0
$575$	1.12188e48	0.569846
$576$	0	0
$577$	−5.93777e47	−0.283822	−0.141911	−	0.989879i	$-0.545325\pi$
$577$	−5.93777e47	−0.283822	−0.141911	+	0.989879i	$0.545325\pi$
$578$	0	0
$579$	−4.00969e48	−1.80400
$580$	0	0
$581$	−1.52467e47	−0.0645791
$582$	0	0
$583$	2.99485e48	1.19447
$584$	0	0
$585$	5.71133e47	0.214541
$586$	0	0
$587$	−4.95184e48	−1.75227	−0.876133	−	0.482069i	$-0.839886\pi$
$587$	−4.95184e48	−1.75227	−0.876133	+	0.482069i	$0.839886\pi$
$588$	0	0
$589$	−4.00188e48	−1.33428
$590$	0	0
$591$	−2.37480e48	−0.746184
$592$	0	0
$593$	−3.92884e48	−1.16361	−0.581804	−	0.813329i	$-0.697653\pi$
$593$	−3.92884e48	−1.16361	−0.581804	+	0.813329i	$0.697653\pi$
$594$	0	0
$595$	5.38651e47	0.150404
$596$	0	0
$597$	1.25763e48	0.331132
$598$	0	0
$599$	3.36641e48	0.835982	0.417991	−	0.908451i	$-0.362734\pi$
$599$	3.36641e48	0.835982	0.417991	+	0.908451i	$0.362734\pi$
$600$	0	0
$601$	5.17376e48	1.21200	0.606001	−	0.795464i	$-0.292773\pi$
$601$	5.17376e48	1.21200	0.606001	+	0.795464i	$0.292773\pi$
$602$	0	0
$603$	−3.09839e48	−0.684830
$604$	0	0
$605$	1.29515e48	0.270148
$606$	0	0
$607$	9.08661e47	0.178896	0.0894478	−	0.995992i	$-0.471490\pi$
$607$	9.08661e47	0.178896	0.0894478	+	0.995992i	$0.471490\pi$
$608$	0	0
$609$	−5.25291e47	−0.0976331
$610$	0	0
$611$	1.10850e49	1.94543
$612$	0	0
$613$	−3.25008e47	−0.0538686	−0.0269343	−	0.999637i	$-0.508574\pi$
$613$	−3.25008e47	−0.0538686	−0.0269343	+	0.999637i	$0.508574\pi$
$614$	0	0
$615$	3.04493e48	0.476720
$616$	0	0
$617$	8.17292e47	0.120889	0.0604445	−	0.998172i	$-0.480748\pi$
$617$	8.17292e47	0.120889	0.0604445	+	0.998172i	$0.480748\pi$
$618$	0	0
$619$	−1.02467e49	−1.43217	−0.716087	−	0.698011i	$-0.754069\pi$
$619$	−1.02467e49	−1.43217	−0.716087	+	0.698011i	$0.754069\pi$
$620$	0	0
$621$	−2.33150e48	−0.307987
$622$	0	0
$623$	3.48228e48	0.434833
$624$	0	0
$625$	6.42631e48	0.758685
$626$	0	0
$627$	−1.38239e49	−1.54329
$628$	0	0
$629$	1.42911e49	1.50896
$630$	0	0
$631$	1.46353e49	1.46180	0.730898	−	0.682487i	$-0.239101\pi$
$631$	1.46353e49	1.46180	0.730898	+	0.682487i	$0.239101\pi$
$632$	0	0
$633$	−1.13070e49	−1.06852
$634$	0	0
$635$	−2.68956e48	−0.240514
$636$	0	0
$637$	−1.21436e49	−1.02780
$638$	0	0
$639$	5.47670e48	0.438788
$640$	0	0
$641$	−1.87013e49	−1.41859	−0.709294	−	0.704913i	$-0.750986\pi$
$641$	−1.87013e49	−1.41859	−0.709294	+	0.704913i	$0.750986\pi$
$642$	0	0
$643$	−4.63269e48	−0.332768	−0.166384	−	0.986061i	$-0.553209\pi$
$643$	−4.63269e48	−0.332768	−0.166384	+	0.986061i	$0.553209\pi$
$644$	0	0
$645$	−4.89006e48	−0.332674
$646$	0	0
$647$	1.11401e49	0.717898	0.358949	−	0.933357i	$-0.383135\pi$
$647$	1.11401e49	0.717898	0.358949	+	0.933357i	$0.383135\pi$
$648$	0	0
$649$	−3.48313e49	−2.12661
$650$	0	0
$651$	−1.34220e49	−0.776516
$652$	0	0
$653$	2.45552e49	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$653$	2.45552e49	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$654$	0	0
$655$	−1.00036e48	−0.0519917
$656$	0	0
$657$	1.38487e49	0.682362
$658$	0	0
$659$	2.88794e49	1.34925	0.674627	−	0.738159i	$-0.264304\pi$
$659$	2.88794e49	1.34925	0.674627	+	0.738159i	$0.264304\pi$
$660$	0	0
$661$	−1.16569e49	−0.516487	−0.258243	−	0.966080i	$-0.583144\pi$
$661$	−1.16569e49	−0.516487	−0.258243	+	0.966080i	$0.583144\pi$
$662$	0	0
$663$	4.82259e49	2.02672
$664$	0	0
$665$	−2.52599e48	−0.100705
$666$	0	0
$667$	3.15374e48	0.119296
$668$	0	0
$669$	1.08450e49	0.389290
$670$	0	0
$671$	−1.05674e49	−0.360017
$672$	0	0
$673$	1.78400e49	0.576940	0.288470	−	0.957489i	$-0.406853\pi$
$673$	1.78400e49	0.576940	0.288470	+	0.957489i	$0.406853\pi$
$674$	0	0
$675$	−1.48114e49	−0.454757
$676$	0	0
$677$	1.09002e49	0.317786	0.158893	−	0.987296i	$-0.449208\pi$
$677$	1.09002e49	0.317786	0.158893	+	0.987296i	$0.449208\pi$
$678$	0	0
$679$	−1.63892e49	−0.453770
$680$	0	0
$681$	−1.14119e49	−0.300113
$682$	0	0
$683$	−2.96392e49	−0.740465	−0.370233	−	0.928939i	$-0.620722\pi$
$683$	−2.96392e49	−0.740465	−0.370233	+	0.928939i	$0.620722\pi$
$684$	0	0
$685$	−5.17292e48	−0.122787
$686$	0	0
$687$	−3.86221e49	−0.871151
$688$	0	0
$689$	4.89949e49	1.05030
$690$	0	0
$691$	−4.67650e49	−0.952916	−0.476458	−	0.879197i	$-0.658080\pi$
$691$	−4.67650e49	−0.952916	−0.476458	+	0.879197i	$0.658080\pi$
$692$	0	0
$693$	−1.75519e49	−0.340012
$694$	0	0
$695$	1.39051e49	0.256117
$696$	0	0
$697$	9.73334e49	1.70486
$698$	0	0
$699$	−4.31894e49	−0.719494
$700$	0	0
$701$	−6.43229e49	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$701$	−6.43229e49	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$702$	0	0
$703$	−6.70177e49	−1.01035
$704$	0	0
$705$	−4.04084e49	−0.579645
$706$	0	0
$707$	2.81291e49	0.383987
$708$	0	0
$709$	1.40300e49	0.182284	0.0911419	−	0.995838i	$-0.470948\pi$
$709$	1.40300e49	0.182284	0.0911419	+	0.995838i	$0.470948\pi$
$710$	0	0
$711$	−3.37413e49	−0.417295
$712$	0	0
$713$	8.05832e49	0.948807
$714$	0	0
$715$	4.37461e49	0.490438
$716$	0	0
$717$	9.16513e48	0.0978484
$718$	0	0
$719$	9.96213e47	0.0101297	0.00506485	−	0.999987i	$-0.498388\pi$
$719$	9.96213e47	0.0101297	0.00506485	+	0.999987i	$0.498388\pi$
$720$	0	0
$721$	5.69815e49	0.551908
$722$	0	0
$723$	2.27349e50	2.09784
$724$	0	0
$725$	2.00349e49	0.176145
$726$	0	0
$727$	9.68047e49	0.811043	0.405522	−	0.914085i	$-0.367090\pi$
$727$	9.68047e49	0.811043	0.405522	+	0.914085i	$0.367090\pi$
$728$	0	0
$729$	−1.56944e49	−0.125318
$730$	0	0
$731$	−1.56314e50	−1.18972
$732$	0	0
$733$	5.44337e49	0.394956	0.197478	−	0.980307i	$-0.436725\pi$
$733$	5.44337e49	0.394956	0.197478	+	0.980307i	$0.436725\pi$
$734$	0	0
$735$	4.42673e49	0.306235
$736$	0	0
$737$	−2.37322e50	−1.56551
$738$	0	0
$739$	5.56111e49	0.349852	0.174926	−	0.984582i	$-0.444031\pi$
$739$	5.56111e49	0.349852	0.174926	+	0.984582i	$0.444031\pi$
$740$	0	0
$741$	−2.26154e50	−1.35702
$742$	0	0
$743$	−1.67574e50	−0.959185	−0.479593	−	0.877491i	$-0.659216\pi$
$743$	−1.67574e50	−0.959185	−0.479593	+	0.877491i	$0.659216\pi$
$744$	0	0
$745$	1.39472e49	0.0761642
$746$	0	0
$747$	1.88375e49	0.0981553
$748$	0	0
$749$	−8.07160e48	−0.0401354
$750$	0	0
$751$	−3.20459e50	−1.52081	−0.760403	−	0.649451i	$-0.774999\pi$
$751$	−3.20459e50	−1.52081	−0.760403	+	0.649451i	$0.774999\pi$
$752$	0	0
$753$	2.56134e50	1.16026
$754$	0	0
$755$	1.03909e50	0.449348
$756$	0	0
$757$	−3.86478e49	−0.159568	−0.0797842	−	0.996812i	$-0.525423\pi$
$757$	−3.86478e49	−0.159568	−0.0797842	+	0.996812i	$0.525423\pi$
$758$	0	0
$759$	2.78362e50	1.09744
$760$	0	0
$761$	−1.82364e50	−0.686607	−0.343303	−	0.939225i	$-0.611546\pi$
$761$	−1.82364e50	−0.686607	−0.343303	+	0.939225i	$0.611546\pi$
$762$	0	0
$763$	1.75717e50	0.631878
$764$	0	0
$765$	−6.65513e49	−0.228603
$766$	0	0
$767$	−5.69830e50	−1.86993
$768$	0	0
$769$	−3.28112e50	−1.02875	−0.514376	−	0.857565i	$-0.671976\pi$
$769$	−3.28112e50	−1.02875	−0.514376	+	0.857565i	$0.671976\pi$
$770$	0	0
$771$	8.09287e50	2.42466
$772$	0	0
$773$	2.84696e50	0.815156	0.407578	−	0.913170i	$-0.366373\pi$
$773$	2.84696e50	0.815156	0.407578	+	0.913170i	$0.366373\pi$
$774$	0	0
$775$	5.11923e50	1.40096
$776$	0	0
$777$	−2.24773e50	−0.587998
$778$	0	0
$779$	−4.56443e50	−1.14151
$780$	0	0
$781$	4.19490e50	1.00306
$782$	0	0
$783$	−4.16368e49	−0.0952020
$784$	0	0
$785$	−1.65977e50	−0.362935
$786$	0	0
$787$	7.44639e50	1.55736	0.778678	−	0.627423i	$-0.215890\pi$
$787$	7.44639e50	1.55736	0.778678	+	0.627423i	$0.215890\pi$
$788$	0	0
$789$	3.93079e50	0.786379
$790$	0	0
$791$	2.76033e50	0.528290
$792$	0	0
$793$	−1.72879e50	−0.316564
$794$	0	0
$795$	−1.78602e50	−0.312940
$796$	0	0
$797$	−6.39179e50	−1.07177	−0.535885	−	0.844291i	$-0.680022\pi$
$797$	−6.39179e50	−1.07177	−0.535885	+	0.844291i	$0.680022\pi$
$798$	0	0
$799$	−1.29168e51	−2.07294
$800$	0	0
$801$	−4.30242e50	−0.660912
$802$	0	0
$803$	1.06075e51	1.55987
$804$	0	0
$805$	5.08642e49	0.0716118
$806$	0	0
$807$	−1.84079e51	−2.48152
$808$	0	0
$809$	1.71253e50	0.221074	0.110537	−	0.993872i	$-0.464743\pi$
$809$	1.71253e50	0.221074	0.110537	+	0.993872i	$0.464743\pi$
$810$	0	0
$811$	−8.82188e50	−1.09068	−0.545338	−	0.838216i	$-0.683599\pi$
$811$	−8.82188e50	−1.09068	−0.545338	+	0.838216i	$0.683599\pi$
$812$	0	0
$813$	3.42389e50	0.405449
$814$	0	0
$815$	2.46422e49	0.0279525
$816$	0	0
$817$	7.33033e50	0.796595
$818$	0	0
$819$	−2.87144e50	−0.298973
$820$	0	0
$821$	5.26416e50	0.525199	0.262600	−	0.964905i	$-0.415420\pi$
$821$	5.26416e50	0.525199	0.262600	+	0.964905i	$0.415420\pi$
$822$	0	0
$823$	9.49543e50	0.907857	0.453929	−	0.891038i	$-0.350022\pi$
$823$	9.49543e50	0.907857	0.453929	+	0.891038i	$0.350022\pi$
$824$	0	0
$825$	1.76836e51	1.62041
$826$	0	0
$827$	−4.29171e48	−0.00376950	−0.00188475	−	0.999998i	$-0.500600\pi$
$827$	−4.29171e48	−0.00376950	−0.00188475	+	0.999998i	$0.500600\pi$
$828$	0	0
$829$	8.30923e50	0.699611	0.349805	−	0.936822i	$-0.386248\pi$
$829$	8.30923e50	0.699611	0.349805	+	0.936822i	$0.386248\pi$
$830$	0	0
$831$	−2.68410e51	−2.16661
$832$	0	0
$833$	1.41504e51	1.09517
$834$	0	0
$835$	2.01495e50	0.149538
$836$	0	0
$837$	−1.06389e51	−0.757181
$838$	0	0
$839$	8.65560e50	0.590831	0.295415	−	0.955369i	$-0.404542\pi$
$839$	8.65560e50	0.590831	0.295415	+	0.955369i	$0.404542\pi$
$840$	0	0
$841$	−1.47100e51	−0.963125
$842$	0	0
$843$	2.92002e51	1.83402
$844$	0	0
$845$	2.38369e50	0.143634
$846$	0	0
$847$	−6.51154e50	−0.376465
$848$	0	0
$849$	−3.91531e50	−0.217211
$850$	0	0
$851$	1.34949e51	0.718460
$852$	0	0
$853$	−1.25142e51	−0.639433	−0.319716	−	0.947513i	$-0.603588\pi$
$853$	−1.25142e51	−0.639433	−0.319716	+	0.947513i	$0.603588\pi$
$854$	0	0
$855$	3.12091e50	0.153065
$856$	0	0
$857$	1.39052e50	0.0654656	0.0327328	−	0.999464i	$-0.489579\pi$
$857$	1.39052e50	0.0654656	0.0327328	+	0.999464i	$0.489579\pi$
$858$	0	0
$859$	−5.12695e50	−0.231728	−0.115864	−	0.993265i	$-0.536964\pi$
$859$	−5.12695e50	−0.231728	−0.115864	+	0.993265i	$0.536964\pi$
$860$	0	0
$861$	−1.53088e51	−0.664332
$862$	0	0
$863$	3.92641e50	0.163609	0.0818045	−	0.996648i	$-0.473932\pi$
$863$	3.92641e50	0.163609	0.0818045	+	0.996648i	$0.473932\pi$
$864$	0	0
$865$	−9.62323e50	−0.385069
$866$	0	0
$867$	−2.31859e51	−0.891023
$868$	0	0
$869$	−2.58442e51	−0.953932
$870$	0	0
$871$	−3.88251e51	−1.37656
$872$	0	0
$873$	2.02491e51	0.689695
$874$	0	0
$875$	6.75392e50	0.221011
$876$	0	0
$877$	−3.47786e51	−1.09350	−0.546750	−	0.837296i	$-0.684135\pi$
$877$	−3.47786e51	−1.09350	−0.546750	+	0.837296i	$0.684135\pi$
$878$	0	0
$879$	2.55922e51	0.773220
$880$	0	0
$881$	−2.02605e51	−0.588264	−0.294132	−	0.955765i	$-0.595031\pi$
$881$	−2.02605e51	−0.588264	−0.294132	+	0.955765i	$0.595031\pi$
$882$	0	0
$883$	5.52091e51	1.54063	0.770315	−	0.637663i	$-0.220099\pi$
$883$	5.52091e51	1.54063	0.770315	+	0.637663i	$0.220099\pi$
$884$	0	0
$885$	2.07721e51	0.557151
$886$	0	0
$887$	7.31525e51	1.88611	0.943053	−	0.332643i	$-0.107940\pi$
$887$	7.31525e51	1.88611	0.943053	+	0.332643i	$0.107940\pi$
$888$	0	0
$889$	1.35221e51	0.335169
$890$	0	0
$891$	−7.23485e51	−1.72413
$892$	0	0
$893$	6.05733e51	1.38797
$894$	0	0
$895$	1.12938e51	0.248850
$896$	0	0
$897$	4.55392e51	0.964978
$898$	0	0
$899$	1.43908e51	0.293286
$900$	0	0
$901$	−5.70913e51	−1.11914
$902$	0	0
$903$	2.45854e51	0.463598
$904$	0	0
$905$	−1.04997e51	−0.190470
$906$	0	0
$907$	1.23219e50	0.0215055	0.0107528	−	0.999942i	$-0.496577\pi$
$907$	1.23219e50	0.0215055	0.0107528	+	0.999942i	$0.496577\pi$
$908$	0	0
$909$	−3.47541e51	−0.583630
$910$	0	0
$911$	−6.49346e51	−1.04931	−0.524656	−	0.851314i	$-0.675806\pi$
$911$	−6.49346e51	−1.04931	−0.524656	+	0.851314i	$0.675806\pi$
$912$	0	0
$913$	1.44287e51	0.224382
$914$	0	0
$915$	6.30199e50	0.0943210
$916$	0	0
$917$	5.02944e50	0.0724530
$918$	0	0
$919$	4.52733e51	0.627799	0.313899	−	0.949456i	$-0.398365\pi$
$919$	4.52733e51	0.627799	0.313899	+	0.949456i	$0.398365\pi$
$920$	0	0
$921$	−1.82412e52	−2.43506
$922$	0	0
$923$	6.86273e51	0.881997
$924$	0	0
$925$	8.57294e51	1.06084
$926$	0	0
$927$	−7.04018e51	−0.838857
$928$	0	0
$929$	−6.76706e51	−0.776471	−0.388235	−	0.921560i	$-0.626915\pi$
$929$	−6.76706e51	−0.776471	−0.388235	+	0.921560i	$0.626915\pi$
$930$	0	0
$931$	−6.63578e51	−0.733285
$932$	0	0
$933$	2.16020e52	2.29914
$934$	0	0
$935$	−5.09752e51	−0.522583
$936$	0	0
$937$	−4.49743e51	−0.444142	−0.222071	−	0.975030i	$-0.571282\pi$
$937$	−4.49743e51	−0.444142	−0.222071	+	0.975030i	$0.571282\pi$
$938$	0	0
$939$	−1.56926e52	−1.49296
$940$	0	0
$941$	−1.33504e52	−1.22370	−0.611850	−	0.790974i	$-0.709574\pi$
$941$	−1.33504e52	−1.22370	−0.611850	+	0.790974i	$0.709574\pi$
$942$	0	0
$943$	9.19109e51	0.811732
$944$	0	0
$945$	−6.71526e50	−0.0571487
$946$	0	0
$947$	−9.72665e51	−0.797698	−0.398849	−	0.917017i	$-0.630590\pi$
$947$	−9.72665e51	−0.797698	−0.398849	+	0.917017i	$0.630590\pi$
$948$	0	0
$949$	1.73535e52	1.37160
$950$	0	0
$951$	2.23752e52	1.70454
$952$	0	0
$953$	1.57100e52	1.15358	0.576792	−	0.816891i	$-0.304304\pi$
$953$	1.57100e52	1.15358	0.576792	+	0.816891i	$0.304304\pi$
$954$	0	0
$955$	−6.81598e51	−0.482469
$956$	0	0
$957$	4.97109e51	0.339229
$958$	0	0
$959$	2.60075e51	0.171109
$960$	0	0
$961$	2.10071e52	1.33262
$962$	0	0
$963$	9.97261e50	0.0610027
$964$	0	0
$965$	6.93369e51	0.409012
$966$	0	0
$967$	−1.31887e52	−0.750307	−0.375154	−	0.926963i	$-0.622410\pi$
$967$	−1.31887e52	−0.750307	−0.375154	+	0.926963i	$0.622410\pi$
$968$	0	0
$969$	2.63526e52	1.44597
$970$	0	0
$971$	−3.45878e52	−1.83057	−0.915285	−	0.402806i	$-0.868035\pi$
$971$	−3.45878e52	−1.83057	−0.915285	+	0.402806i	$0.868035\pi$
$972$	0	0
$973$	−6.99095e51	−0.356912
$974$	0	0
$975$	2.89298e52	1.42483
$976$	0	0
$977$	−3.79825e52	−1.80480	−0.902400	−	0.430900i	$-0.858196\pi$
$977$	−3.79825e52	−1.80480	−0.902400	+	0.430900i	$0.858196\pi$
$978$	0	0
$979$	−3.29545e52	−1.51084
$980$	0	0
$981$	−2.17101e52	−0.960406
$982$	0	0
$983$	−5.22067e51	−0.222864	−0.111432	−	0.993772i	$-0.535544\pi$
$983$	−5.22067e51	−0.222864	−0.111432	+	0.993772i	$0.535544\pi$
$984$	0	0
$985$	4.10659e51	0.169179
$986$	0	0
$987$	2.03158e52	0.807764
$988$	0	0
$989$	−1.47606e52	−0.566459
$990$	0	0
$991$	2.70299e52	1.00128	0.500639	−	0.865656i	$-0.333098\pi$
$991$	2.70299e52	1.00128	0.500639	+	0.865656i	$0.333098\pi$
$992$	0	0
$993$	2.00250e51	0.0716080
$994$	0	0
$995$	−2.17474e51	−0.0750762
$996$	0	0
$997$	−4.14317e52	−1.38092	−0.690459	−	0.723372i	$-0.742591\pi$
$997$	−4.14317e52	−1.38092	−0.690459	+	0.723372i	$0.742591\pi$
$998$	0	0
$999$	−1.78164e52	−0.573356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.36.a.a.1.1	✓	3
4.3	odd	2		16.36.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.36.a.a.1.1	✓	3	1.1	even	1	trivial
16.36.a.c.1.3		3	4.3	odd	2

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

\(f(q)\)	\(=\)	\(q-2.83743e8 q^{3} +4.90658e11 q^{5} -2.46684e14 q^{7} +3.04783e16 q^{9} +O(q^{10})\)	\(q-2.83743e8 q^{3} +4.90658e11 q^{5} -2.46684e14 q^{7} +3.04783e16 q^{9} +2.33450e18 q^{11} +3.81916e19 q^{13} -1.39220e20 q^{15} -4.45028e21 q^{17} +2.08695e22 q^{19} +6.99948e22 q^{21} -4.20235e23 q^{23} -2.66964e24 q^{25} +5.54809e24 q^{27} -7.50471e24 q^{29} -1.91757e26 q^{31} -6.62396e26 q^{33} -1.21037e26 q^{35} -3.21127e27 q^{37} -1.08366e28 q^{39} -2.18713e28 q^{41} +3.51246e28 q^{43} +1.49544e28 q^{45} +2.90248e29 q^{47} -3.17966e29 q^{49} +1.26273e30 q^{51} +1.28287e30 q^{53} +1.14544e30 q^{55} -5.92157e30 q^{57} -1.49203e31 q^{59} -4.52662e30 q^{61} -7.51852e30 q^{63} +1.87390e31 q^{65} -1.01659e32 q^{67} +1.19239e32 q^{69} +1.79692e32 q^{71} +4.54379e32 q^{73} +7.57490e32 q^{75} -5.75883e32 q^{77} -1.10706e33 q^{79} -3.09911e33 q^{81} +6.18064e32 q^{83} -2.18356e33 q^{85} +2.12941e33 q^{87} -1.41163e34 q^{89} -9.42127e33 q^{91} +5.44097e34 q^{93} +1.02398e34 q^{95} +6.64379e34 q^{97} +7.11515e34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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