LMFDB - Embedded newform 3600.3.c.i.449.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.8
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.i.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.16228i	1.02318i	0.859229	+	0.511591i	$0.170944\pi$
$7$	7.16228i	1.02318i	−0.859229	+	0.511591i	$0.829056\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.42736i	0.493396i	0.969092	+	0.246698i	$0.0793456\pi$
$11$	5.42736i	0.493396i	−0.969092	+	0.246698i	$0.920654\pi$
$12$	0	0
$13$	9.81139i	0.754722i	0.926066	+	0.377361i	$0.123168\pi$
$13$	9.81139i	0.754722i	−0.926066	+	0.377361i	$0.876832\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.2317	0.719511	0.359756	−	0.933047i	$-0.382860\pi$
$17$	12.2317	0.719511	0.359756	+	0.933047i	$0.382860\pi$
$18$	0	0
$19$	6.32456	0.332871	0.166436	−	0.986052i	$-0.446774\pi$
$19$	6.32456	0.332871	0.166436	+	0.986052i	$0.446774\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	12.0394	0.523454	0.261727	−	0.965142i	$-0.415708\pi$
$23$	12.0394	0.523454	0.261727	+	0.965142i	$0.415708\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 44.9881i	− 1.55131i	−0.631155	−	0.775657i	$-0.717419\pi$
$29$	− 44.9881i	− 1.55131i	0.631155	−	0.775657i	$-0.282581\pi$
$30$	0	0
$31$	58.2719	1.87974	0.939869	−	0.341535i	$-0.110947\pi$
$31$	58.2719	1.87974	0.939869	+	0.341535i	$0.110947\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 66.4605i	− 1.79623i	−0.439761	−	0.898115i	$-0.644937\pi$
$37$	− 66.4605i	− 1.79623i	0.439761	−	0.898115i	$-0.355063\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 16.4743i	− 0.401813i	−0.979610	−	0.200906i	$-0.935611\pi$
$41$	− 16.4743i	− 0.401813i	0.979610	−	0.200906i	$-0.0643887\pi$
$42$	0	0
$43$	43.6228i	1.01448i	0.861804	+	0.507242i	$0.169335\pi$
$43$	43.6228i	1.01448i	−0.861804	+	0.507242i	$0.830665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	40.0570	0.852276	0.426138	−	0.904658i	$-0.359874\pi$
$47$	40.0570	0.852276	0.426138	+	0.904658i	$0.359874\pi$
$48$	0	0
$49$	−2.29822	−0.0469025
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.2242	0.249512	0.124756	−	0.992187i	$-0.460185\pi$
$53$	13.2242	0.249512	0.124756	+	0.992187i	$0.460185\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 25.1519i	− 0.426303i	−0.977019	−	0.213151i	$-0.931627\pi$
$59$	− 25.1519i	− 0.426303i	0.977019	−	0.213151i	$-0.0683728\pi$
$60$	0	0
$61$	−35.6754	−0.584843	−0.292422	−	0.956289i	$-0.594461\pi$
$61$	−35.6754	−0.584843	−0.292422	+	0.956289i	$0.594461\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	26.7018i	0.398534i	0.979945	+	0.199267i	$0.0638561\pi$
$67$	26.7018i	0.398534i	−0.979945	+	0.199267i	$0.936144\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	92.7301i	1.30606i	0.757333	+	0.653029i	$0.226502\pi$
$71$	92.7301i	1.30606i	−0.757333	+	0.653029i	$0.773498\pi$
$72$	0	0
$73$	60.3246i	0.826364i	0.910649	+	0.413182i	$0.135583\pi$
$73$	60.3246i	0.826364i	−0.910649	+	0.413182i	$0.864417\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−38.8723	−0.504834
$78$	0	0
$79$	−96.2192	−1.21796	−0.608982	−	0.793184i	$-0.708422\pi$
$79$	−96.2192	−1.21796	−0.608982	+	0.793184i	$0.708422\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	79.1215	0.953271	0.476635	−	0.879101i	$-0.341856\pi$
$83$	79.1215	0.953271	0.476635	+	0.879101i	$0.341856\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 107.443i	− 1.20722i	−0.797278	−	0.603612i	$-0.793728\pi$
$89$	− 107.443i	− 1.20722i	0.797278	−	0.603612i	$-0.206272\pi$
$90$	0	0
$91$	−70.2719	−0.772219
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.07900i	0.0111237i	0.999985	+	0.00556187i	$0.00177041\pi$
$97$	1.07900i	0.0111237i	−0.999985	+	0.00556187i	$0.998230\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	170.282i	1.68596i	0.537942	+	0.842982i	$0.319202\pi$
$101$	170.282i	1.68596i	−0.537942	+	0.842982i	$0.680798\pi$
$102$	0	0
$103$	128.460i	1.24719i	0.781748	+	0.623595i	$0.214328\pi$
$103$	128.460i	1.24719i	−0.781748	+	0.623595i	$0.785672\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	76.3675	0.713715	0.356858	−	0.934159i	$-0.383848\pi$
$107$	76.3675	0.713715	0.356858	+	0.934159i	$0.383848\pi$
$108$	0	0
$109$	13.0790	0.119991	0.0599954	−	0.998199i	$-0.480891\pi$
$109$	13.0790	0.119991	0.0599954	+	0.998199i	$0.480891\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.7170	−0.183336	−0.0916680	−	0.995790i	$-0.529220\pi$
$113$	−20.7170	−0.183336	−0.0916680	+	0.995790i	$0.529220\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	87.6068i	0.736191i
$120$	0	0
$121$	91.5438	0.756560
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 38.0306i	− 0.299454i	−0.988727	−	0.149727i	$-0.952161\pi$
$127$	− 38.0306i	− 0.299454i	0.988727	−	0.149727i	$-0.0478394\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 83.9409i	− 0.640770i	−0.947287	−	0.320385i	$-0.896188\pi$
$131$	− 83.9409i	− 0.640770i	0.947287	−	0.320385i	$-0.103812\pi$
$132$	0	0
$133$	45.2982i	0.340588i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.5936	−0.113822	−0.0569109	−	0.998379i	$-0.518125\pi$
$137$	−15.5936	−0.113822	−0.0569109	+	0.998379i	$0.518125\pi$
$138$	0	0
$139$	67.8420	0.488072	0.244036	−	0.969766i	$-0.421529\pi$
$139$	67.8420	0.488072	0.244036	+	0.969766i	$0.421529\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−53.2499	−0.372377
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	233.426i	1.56662i	0.621634	+	0.783308i	$0.286469\pi$
$149$	233.426i	1.56662i	−0.621634	+	0.783308i	$0.713531\pi$
$150$	0	0
$151$	−185.351	−1.22749	−0.613745	−	0.789505i	$-0.710338\pi$
$151$	−185.351	−1.22749	−0.613745	+	0.789505i	$0.710338\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	111.276i	0.708765i	0.935100	+	0.354383i	$0.115309\pi$
$157$	111.276i	0.708765i	−0.935100	+	0.354383i	$0.884691\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	86.2298i	0.535589i
$162$	0	0
$163$	− 118.763i	− 0.728607i	−0.931280	−	0.364304i	$-0.881307\pi$
$163$	− 118.763i	− 0.728607i	0.931280	−	0.364304i	$-0.118693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	221.194	1.32452	0.662258	−	0.749276i	$-0.269598\pi$
$167$	221.194	1.32452	0.662258	+	0.749276i	$0.269598\pi$
$168$	0	0
$169$	72.7367	0.430394
$170$	0	0
$171$	0	0
$172$	0	0
$173$	190.807	1.10293	0.551466	−	0.834198i	$-0.314069\pi$
$173$	190.807	1.10293	0.551466	+	0.834198i	$0.314069\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	58.1005i	0.324584i	0.986743	+	0.162292i	$0.0518886\pi$
$179$	58.1005i	0.324584i	−0.986743	+	0.162292i	$0.948111\pi$
$180$	0	0
$181$	−162.921	−0.900116	−0.450058	−	0.892999i	$-0.648597\pi$
$181$	−162.921	−0.900116	−0.450058	+	0.892999i	$0.648597\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	66.3858i	0.355004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	100.062i	0.523884i	0.965084	+	0.261942i	$0.0843630\pi$
$191$	100.062i	0.523884i	−0.965084	+	0.261942i	$0.915637\pi$
$192$	0	0
$193$	− 61.8947i	− 0.320698i	−0.987060	−	0.160349i	$-0.948738\pi$
$193$	− 61.8947i	− 0.320698i	0.987060	−	0.160349i	$-0.0512619\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.8791	0.126290	0.0631449	−	0.998004i	$-0.479887\pi$
$197$	24.8791	0.126290	0.0631449	+	0.998004i	$0.479887\pi$
$198$	0	0
$199$	−156.491	−0.786387	−0.393194	−	0.919456i	$-0.628630\pi$
$199$	−156.491	−0.786387	−0.393194	+	0.919456i	$0.628630\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	322.217	1.58728
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	34.3256i	0.164237i
$210$	0	0
$211$	−237.789	−1.12696	−0.563482	−	0.826128i	$-0.690539\pi$
$211$	−237.789	−1.12696	−0.563482	+	0.826128i	$0.690539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	417.359i	1.92332i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	120.010i	0.543031i
$222$	0	0
$223$	182.302i	0.817500i	0.912646	+	0.408750i	$0.134035\pi$
$223$	182.302i	0.817500i	−0.912646	+	0.408750i	$0.865965\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−406.078	−1.78889	−0.894444	−	0.447180i	$-0.852428\pi$
$227$	−406.078	−1.78889	−0.894444	+	0.447180i	$0.852428\pi$
$228$	0	0
$229$	27.2982	0.119206	0.0596031	−	0.998222i	$-0.481016\pi$
$229$	27.2982	0.119206	0.0596031	+	0.998222i	$0.481016\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	356.382	1.52954	0.764768	−	0.644306i	$-0.222854\pi$
$233$	356.382	1.52954	0.764768	+	0.644306i	$0.222854\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 271.690i	− 1.13678i	−0.822760	−	0.568389i	$-0.807567\pi$
$239$	− 271.690i	− 1.13678i	0.822760	−	0.568389i	$-0.192433\pi$
$240$	0	0
$241$	−224.438	−0.931280	−0.465640	−	0.884974i	$-0.654176\pi$
$241$	−224.438	−0.931280	−0.465640	+	0.884974i	$0.654176\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	62.0527i	0.251225i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	318.775i	1.27002i	0.772504	+	0.635010i	$0.219004\pi$
$251$	318.775i	1.27002i	−0.772504	+	0.635010i	$0.780996\pi$
$252$	0	0
$253$	65.3423i	0.258270i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	371.975	1.44738	0.723688	−	0.690128i	$-0.242446\pi$
$257$	371.975	1.44738	0.723688	+	0.690128i	$0.242446\pi$
$258$	0	0
$259$	476.009	1.83787
$260$	0	0
$261$	0	0
$262$	0	0
$263$	238.549	0.907031	0.453515	−	0.891248i	$-0.350170\pi$
$263$	238.549	0.907031	0.453515	+	0.891248i	$0.350170\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 125.871i	− 0.467922i	−0.972246	−	0.233961i	$-0.924831\pi$
$269$	− 125.871i	− 0.467922i	0.972246	−	0.233961i	$-0.0751688\pi$
$270$	0	0
$271$	258.649	0.954425	0.477212	−	0.878788i	$-0.341647\pi$
$271$	258.649	0.954425	0.477212	+	0.878788i	$0.341647\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 227.715i	− 0.822074i	−0.911619	−	0.411037i	$-0.865167\pi$
$277$	− 227.715i	− 0.822074i	0.911619	−	0.411037i	$-0.134833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 241.384i	− 0.859016i	−0.903063	−	0.429508i	$-0.858687\pi$
$281$	− 241.384i	− 0.859016i	0.903063	−	0.429508i	$-0.141313\pi$
$282$	0	0
$283$	− 208.333i	− 0.736159i	−0.929794	−	0.368080i	$-0.880015\pi$
$283$	− 208.333i	− 0.736159i	0.929794	−	0.368080i	$-0.119985\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	117.994	0.411128
$288$	0	0
$289$	−139.386	−0.482304
$290$	0	0
$291$	0	0
$292$	0	0
$293$	201.693	0.688372	0.344186	−	0.938901i	$-0.388155\pi$
$293$	201.693	0.688372	0.344186	+	0.938901i	$0.388155\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	118.124i	0.395062i
$300$	0	0
$301$	−312.438	−1.03800
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	342.824i	1.11669i	0.829608	+	0.558346i	$0.188564\pi$
$307$	342.824i	1.11669i	−0.829608	+	0.558346i	$0.811436\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	217.640i	0.699807i	0.936786	+	0.349903i	$0.113786\pi$
$311$	217.640i	0.699807i	−0.936786	+	0.349903i	$0.886214\pi$
$312$	0	0
$313$	281.895i	0.900622i	0.892872	+	0.450311i	$0.148687\pi$
$313$	281.895i	0.900622i	−0.892872	+	0.450311i	$0.851313\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4013	0.0485847	0.0242923	−	0.999705i	$-0.492267\pi$
$317$	15.4013	0.0485847	0.0242923	+	0.999705i	$0.492267\pi$
$318$	0	0
$319$	244.167	0.765412
$320$	0	0
$321$	0	0
$322$	0	0
$323$	77.3600	0.239505
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	286.899i	0.872034i
$330$	0	0
$331$	−375.517	−1.13449	−0.567247	−	0.823548i	$-0.691991\pi$
$331$	−375.517	−1.13449	−0.567247	+	0.823548i	$0.691991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 188.114i	− 0.558201i	−0.960262	−	0.279101i	$-0.909964\pi$
$337$	− 188.114i	− 0.558201i	0.960262	−	0.279101i	$-0.0900363\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	316.262i	0.927456i
$342$	0	0
$343$	334.491i	0.975193i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−513.793	−1.48067	−0.740336	−	0.672237i	$-0.765334\pi$
$347$	−513.793	−1.48067	−0.740336	+	0.672237i	$0.765334\pi$
$348$	0	0
$349$	−112.535	−0.322451	−0.161225	−	0.986918i	$-0.551545\pi$
$349$	−112.535	−0.322451	−0.161225	+	0.986918i	$0.551545\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	428.172	1.21295	0.606475	−	0.795102i	$-0.292583\pi$
$353$	428.172	1.21295	0.606475	+	0.795102i	$0.292583\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 56.1961i	− 0.156535i	−0.996932	−	0.0782676i	$-0.975061\pi$
$359$	− 56.1961i	− 0.156535i	0.996932	−	0.0782676i	$-0.0249389\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 154.364i	− 0.420610i	−0.977636	−	0.210305i	$-0.932554\pi$
$367$	− 154.364i	− 0.420610i	0.977636	−	0.210305i	$-0.0674456\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	94.7151i	0.255297i
$372$	0	0
$373$	557.285i	1.49406i	0.664790	+	0.747030i	$0.268521\pi$
$373$	557.285i	1.49406i	−0.664790	+	0.747030i	$0.731479\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	441.396	1.17081
$378$	0	0
$379$	147.404	0.388928	0.194464	−	0.980910i	$-0.437703\pi$
$379$	147.404	0.388928	0.194464	+	0.980910i	$0.437703\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	736.619	1.92329	0.961644	−	0.274302i	$-0.0884467\pi$
$383$	736.619	1.92329	0.961644	+	0.274302i	$0.0884467\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 296.408i	− 0.761975i	−0.924580	−	0.380987i	$-0.875584\pi$
$389$	− 296.408i	− 0.761975i	0.924580	−	0.380987i	$-0.124416\pi$
$390$	0	0
$391$	147.263	0.376631
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	457.057i	1.15128i	0.817704	+	0.575638i	$0.195246\pi$
$397$	457.057i	1.15128i	−0.817704	+	0.575638i	$0.804754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	391.141i	0.975415i	0.873007	+	0.487707i	$0.162167\pi$
$401$	391.141i	0.975415i	−0.873007	+	0.487707i	$0.837833\pi$
$402$	0	0
$403$	571.728i	1.41868i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	360.705	0.886253
$408$	0	0
$409$	411.842	1.00695	0.503474	−	0.864010i	$-0.332055\pi$
$409$	411.842	1.00695	0.503474	+	0.864010i	$0.332055\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	180.145	0.436186
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 653.447i	− 1.55954i	−0.626066	−	0.779770i	$-0.715336\pi$
$419$	− 653.447i	− 1.55954i	0.626066	−	0.779770i	$-0.284664\pi$
$420$	0	0
$421$	125.035	0.296995	0.148497	−	0.988913i	$-0.452556\pi$
$421$	125.035	0.296995	0.148497	+	0.988913i	$0.452556\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 255.517i	− 0.598401i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	397.208i	0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$431$	397.208i	0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$432$	0	0
$433$	560.114i	1.29357i	0.762674	+	0.646783i	$0.223886\pi$
$433$	560.114i	1.29357i	−0.762674	+	0.646783i	$0.776114\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	76.1441	0.174243
$438$	0	0
$439$	−664.386	−1.51341	−0.756704	−	0.653758i	$-0.773191\pi$
$439$	−664.386	−1.51341	−0.756704	+	0.653758i	$0.773191\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−371.305	−0.838160	−0.419080	−	0.907949i	$-0.637647\pi$
$443$	−371.305	−0.838160	−0.419080	+	0.907949i	$0.637647\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 585.471i	− 1.30395i	−0.758243	−	0.651973i	$-0.773942\pi$
$449$	− 585.471i	− 1.30395i	0.758243	−	0.651973i	$-0.226058\pi$
$450$	0	0
$451$	89.4121	0.198253
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 168.641i	− 0.369017i	−0.982831	−	0.184508i	$-0.940931\pi$
$457$	− 168.641i	− 0.369017i	0.982831	−	0.184508i	$-0.0590693\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 298.492i	− 0.647487i	−0.946145	−	0.323744i	$-0.895058\pi$
$461$	− 298.492i	− 0.647487i	0.946145	−	0.323744i	$-0.104942\pi$
$462$	0	0
$463$	595.285i	1.28571i	0.765987	+	0.642856i	$0.222251\pi$
$463$	595.285i	1.28571i	−0.765987	+	0.642856i	$0.777749\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	623.655	1.33545	0.667725	−	0.744408i	$-0.267268\pi$
$467$	623.655	1.33545	0.667725	+	0.744408i	$0.267268\pi$
$468$	0	0
$469$	−191.246	−0.407773
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−236.756	−0.500542
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	131.857i	0.275276i	0.990483	+	0.137638i	$0.0439510\pi$
$479$	131.857i	0.275276i	−0.990483	+	0.137638i	$0.956049\pi$
$480$	0	0
$481$	652.070	1.35565
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	41.8028i	0.0858375i	0.999079	+	0.0429187i	$0.0136657\pi$
$487$	41.8028i	0.0858375i	−0.999079	+	0.0429187i	$0.986334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 178.817i	− 0.364189i	−0.983281	−	0.182095i	$-0.941712\pi$
$491$	− 178.817i	− 0.364189i	0.983281	−	0.182095i	$-0.0582877\pi$
$492$	0	0
$493$	− 550.280i	− 1.11619i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−664.159	−1.33634
$498$	0	0
$499$	−39.0961	−0.0783489	−0.0391744	−	0.999232i	$-0.512473\pi$
$499$	−39.0961	−0.0783489	−0.0391744	+	0.999232i	$0.512473\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−578.698	−1.15049	−0.575247	−	0.817980i	$-0.695094\pi$
$503$	−578.698	−1.15049	−0.575247	+	0.817980i	$0.695094\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	355.743i	0.698905i	0.936954	+	0.349453i	$0.113632\pi$
$509$	355.743i	0.698905i	−0.936954	+	0.349453i	$0.886368\pi$
$510$	0	0
$511$	−432.061	−0.845521
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	217.404i	0.420510i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	810.952i	1.55653i	0.627936	+	0.778265i	$0.283900\pi$
$521$	810.952i	1.55653i	−0.627936	+	0.778265i	$0.716100\pi$
$522$	0	0
$523$	− 720.483i	− 1.37760i	−0.724953	−	0.688798i	$-0.758139\pi$
$523$	− 720.483i	− 1.37760i	0.724953	−	0.688798i	$-0.241861\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	712.764	1.35249
$528$	0	0
$529$	−384.052	−0.725996
$530$	0	0
$531$	0	0
$532$	0	0
$533$	161.636	0.303257
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12.4733i	− 0.0231415i
$540$	0	0
$541$	347.149	0.641680	0.320840	−	0.947133i	$-0.396035\pi$
$541$	347.149	0.641680	0.320840	+	0.947133i	$0.396035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 720.833i	− 1.31779i	−0.752234	−	0.658896i	$-0.771024\pi$
$547$	− 720.833i	− 1.31779i	0.752234	−	0.658896i	$-0.228976\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 284.530i	− 0.516388i
$552$	0	0
$553$	− 689.149i	− 1.24620i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	429.102	0.770380	0.385190	−	0.922837i	$-0.374136\pi$
$557$	429.102	0.770380	0.385190	+	0.922837i	$0.374136\pi$
$558$	0	0
$559$	−428.000	−0.765653
$560$	0	0
$561$	0	0
$562$	0	0
$563$	670.820	1.19151	0.595755	−	0.803166i	$-0.296853\pi$
$563$	670.820	1.19151	0.595755	+	0.803166i	$0.296853\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	368.663i	0.647914i	0.946072	+	0.323957i	$0.105013\pi$
$569$	368.663i	0.647914i	−0.946072	+	0.323957i	$0.894987\pi$
$570$	0	0
$571$	−124.289	−0.217669	−0.108834	−	0.994060i	$-0.534712\pi$
$571$	−124.289	−0.217669	−0.108834	+	0.994060i	$0.534712\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 504.236i	− 0.873893i	−0.899487	−	0.436947i	$-0.856060\pi$
$577$	− 504.236i	− 0.873893i	0.899487	−	0.436947i	$-0.143940\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	566.690i	0.975370i
$582$	0	0
$583$	71.7722i	0.123108i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.2256	0.0668238	0.0334119	−	0.999442i	$-0.489363\pi$
$587$	39.2256	0.0668238	0.0334119	+	0.999442i	$0.489363\pi$
$588$	0	0
$589$	368.544	0.625711
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−621.670	−1.04835	−0.524174	−	0.851611i	$-0.675626\pi$
$593$	−621.670	−1.04835	−0.524174	+	0.851611i	$0.675626\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1119.77i	1.86940i	0.355436	+	0.934701i	$0.384332\pi$
$599$	1119.77i	1.86940i	−0.355436	+	0.934701i	$0.615668\pi$
$600$	0	0
$601$	−323.789	−0.538751	−0.269375	−	0.963035i	$-0.586817\pi$
$601$	−323.789	−0.538751	−0.269375	+	0.963035i	$0.586817\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1025.63i	1.68966i	0.535031	+	0.844832i	$0.320300\pi$
$607$	1025.63i	1.68966i	−0.535031	+	0.844832i	$0.679700\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	393.014i	0.643232i
$612$	0	0
$613$	904.153i	1.47496i	0.675367	+	0.737482i	$0.263985\pi$
$613$	904.153i	1.47496i	−0.675367	+	0.737482i	$0.736015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−710.716	−1.15189	−0.575945	−	0.817488i	$-0.695366\pi$
$617$	−710.716	−1.15189	−0.575945	+	0.817488i	$0.695366\pi$
$618$	0	0
$619$	−583.737	−0.943032	−0.471516	−	0.881858i	$-0.656293\pi$
$619$	−583.737	−0.943032	−0.471516	+	0.881858i	$0.656293\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	769.537	1.23521
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 812.924i	− 1.29241i
$630$	0	0
$631$	20.0968	0.0318491	0.0159246	−	0.999873i	$-0.494931\pi$
$631$	20.0968	0.0318491	0.0159246	+	0.999873i	$0.494931\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 22.5487i	− 0.0353983i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	341.607i	0.532928i	0.963845	+	0.266464i	$0.0858553\pi$
$641$	341.607i	0.532928i	−0.963845	+	0.266464i	$0.914145\pi$
$642$	0	0
$643$	469.693i	0.730471i	0.930915	+	0.365235i	$0.119011\pi$
$643$	469.693i	0.730471i	−0.930915	+	0.365235i	$0.880989\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	572.099	0.884234	0.442117	−	0.896957i	$-0.354228\pi$
$647$	572.099	0.884234	0.442117	+	0.896957i	$0.354228\pi$
$648$	0	0
$649$	136.508	0.210336
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−213.540	−0.327014	−0.163507	−	0.986542i	$-0.552281\pi$
$653$	−213.540	−0.327014	−0.163507	+	0.986542i	$0.552281\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 420.983i	− 0.638820i	−0.947617	−	0.319410i	$-0.896515\pi$
$659$	− 420.983i	− 0.638820i	0.947617	−	0.319410i	$-0.103485\pi$
$660$	0	0
$661$	434.272	0.656992	0.328496	−	0.944505i	$-0.393458\pi$
$661$	434.272	0.656992	0.328496	+	0.944505i	$0.393458\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 541.631i	− 0.812041i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 193.623i	− 0.288560i
$672$	0	0
$673$	72.7801i	0.108143i	0.998537	+	0.0540714i	$0.0172199\pi$
$673$	72.7801i	0.108143i	−0.998537	+	0.0540714i	$0.982780\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	172.106	0.254219	0.127109	−	0.991889i	$-0.459430\pi$
$677$	172.106	0.254219	0.127109	+	0.991889i	$0.459430\pi$
$678$	0	0
$679$	−7.72811	−0.0113816
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−792.592	−1.16046	−0.580228	−	0.814454i	$-0.697037\pi$
$683$	−792.592	−1.16046	−0.580228	+	0.814454i	$0.697037\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	129.747i	0.188313i
$690$	0	0
$691$	−154.851	−0.224097	−0.112049	−	0.993703i	$-0.535741\pi$
$691$	−154.851	−0.224097	−0.112049	+	0.993703i	$0.535741\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 201.509i	− 0.289109i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 950.544i	− 1.35598i	−0.735070	−	0.677991i	$-0.762851\pi$
$701$	− 950.544i	− 1.35598i	0.735070	−	0.677991i	$-0.237149\pi$
$702$	0	0
$703$	− 420.333i	− 0.597913i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1219.61	−1.72505
$708$	0	0
$709$	−390.350	−0.550564	−0.275282	−	0.961363i	$-0.588771\pi$
$709$	−390.350	−0.550564	−0.275282	+	0.961363i	$0.588771\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	701.561	0.983956
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	655.227i	0.911303i	0.890158	+	0.455651i	$0.150594\pi$
$719$	655.227i	0.911303i	−0.890158	+	0.455651i	$0.849406\pi$
$720$	0	0
$721$	−920.070	−1.27610
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1424.25i	1.95908i	0.201254	+	0.979539i	$0.435498\pi$
$727$	1424.25i	1.95908i	−0.201254	+	0.979539i	$0.564502\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	533.580i	0.729932i
$732$	0	0
$733$	− 946.749i	− 1.29161i	−0.763503	−	0.645805i	$-0.776522\pi$
$733$	− 946.749i	− 1.29161i	0.763503	−	0.645805i	$-0.223478\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−144.920	−0.196635
$738$	0	0
$739$	591.429	0.800310	0.400155	−	0.916447i	$-0.368956\pi$
$739$	591.429	0.800310	0.400155	+	0.916447i	$0.368956\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−732.202	−0.985467	−0.492734	−	0.870180i	$-0.664002\pi$
$743$	−732.202	−0.985467	−0.492734	+	0.870180i	$0.664002\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	546.965i	0.730261i
$750$	0	0
$751$	215.359	0.286764	0.143382	−	0.989667i	$-0.454202\pi$
$751$	215.359	0.286764	0.143382	+	0.989667i	$0.454202\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 276.258i	− 0.364938i	−0.983212	−	0.182469i	$-0.941591\pi$
$757$	− 276.258i	− 0.364938i	0.983212	−	0.182469i	$-0.0584090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 893.373i	− 1.17395i	−0.809606	−	0.586973i	$-0.800319\pi$
$761$	− 893.373i	− 1.17395i	0.809606	−	0.586973i	$-0.199681\pi$
$762$	0	0
$763$	93.6754i	0.122773i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	246.775	0.321740
$768$	0	0
$769$	−284.316	−0.369722	−0.184861	−	0.982765i	$-0.559183\pi$
$769$	−284.316	−0.369722	−0.184861	+	0.982765i	$0.559183\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1059.64	−1.37081	−0.685405	−	0.728162i	$-0.740375\pi$
$773$	−1059.64	−1.37081	−0.685405	+	0.728162i	$0.740375\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 104.193i	− 0.133752i
$780$	0	0
$781$	−503.280	−0.644404
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 875.517i	− 1.11247i	−0.831024	−	0.556237i	$-0.812245\pi$
$787$	− 875.517i	− 1.11247i	0.831024	−	0.556237i	$-0.187755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 148.381i	− 0.187586i
$792$	0	0
$793$	− 350.026i	− 0.441394i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−742.449	−0.931555	−0.465777	−	0.884902i	$-0.654225\pi$
$797$	−742.449	−0.931555	−0.465777	+	0.884902i	$0.654225\pi$
$798$	0	0
$799$	489.964	0.613222
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−327.403	−0.407725
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	113.720i	0.140568i	0.997527	+	0.0702842i	$0.0223906\pi$
$809$	113.720i	0.140568i	−0.997527	+	0.0702842i	$0.977609\pi$
$810$	0	0
$811$	1466.03	1.80769	0.903844	−	0.427863i	$-0.140733\pi$
$811$	1466.03	1.80769	0.903844	+	0.427863i	$0.140733\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	275.895i	0.337692i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 550.073i	− 0.670003i	−0.942218	−	0.335002i	$-0.891263\pi$
$821$	− 550.073i	− 0.670003i	0.942218	−	0.335002i	$-0.108737\pi$
$822$	0	0
$823$	− 1392.51i	− 1.69199i	−0.533187	−	0.845997i	$-0.679006\pi$
$823$	− 1392.51i	− 1.69199i	0.533187	−	0.845997i	$-0.320994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−955.922	−1.15589	−0.577945	−	0.816075i	$-0.696145\pi$
$827$	−955.922	−1.15589	−0.577945	+	0.816075i	$0.696145\pi$
$828$	0	0
$829$	1652.69	1.99360	0.996798	−	0.0799552i	$-0.0254777\pi$
$829$	1652.69	1.99360	0.996798	+	0.0799552i	$0.0254777\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−28.1111	−0.0337469
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1568.11i	1.86903i	0.355927	+	0.934514i	$0.384165\pi$
$839$	1568.11i	1.86903i	−0.355927	+	0.934514i	$0.615835\pi$
$840$	0	0
$841$	−1182.93	−1.40657
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	655.662i	0.774099i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 800.147i	− 0.940243i
$852$	0	0
$853$	− 651.232i	− 0.763461i	−0.924274	−	0.381730i	$-0.875328\pi$
$853$	− 651.232i	− 0.763461i	0.924274	−	0.381730i	$-0.124672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	299.131	0.349044	0.174522	−	0.984653i	$-0.444162\pi$
$857$	299.131	0.349044	0.174522	+	0.984653i	$0.444162\pi$
$858$	0	0
$859$	1095.72	1.27558	0.637788	−	0.770212i	$-0.279850\pi$
$859$	1095.72	1.27558	0.637788	+	0.770212i	$0.279850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1221.95	1.41593	0.707965	−	0.706247i	$-0.249613\pi$
$863$	1221.95	1.41593	0.707965	+	0.706247i	$0.249613\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 522.216i	− 0.600939i
$870$	0	0
$871$	−261.982	−0.300782
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 766.399i	− 0.873887i	−0.899489	−	0.436944i	$-0.856061\pi$
$877$	− 766.399i	− 0.873887i	0.899489	−	0.436944i	$-0.143939\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 310.097i	− 0.351983i	−0.984392	−	0.175992i	$-0.943687\pi$
$881$	− 310.097i	− 0.351983i	0.984392	−	0.175992i	$-0.0563132\pi$
$882$	0	0
$883$	122.236i	0.138433i	0.997602	+	0.0692165i	$0.0220499\pi$
$883$	122.236i	0.138433i	−0.997602	+	0.0692165i	$0.977950\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−265.444	−0.299261	−0.149630	−	0.988742i	$-0.547808\pi$
$887$	−265.444	−0.299261	−0.149630	+	0.988742i	$0.547808\pi$
$888$	0	0
$889$	272.386	0.306396
$890$	0	0
$891$	0	0
$892$	0	0
$893$	253.343	0.283698
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2621.54i	− 2.91606i
$900$	0	0
$901$	161.754	0.179527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	672.622i	0.741590i	0.928715	+	0.370795i	$0.120915\pi$
$907$	672.622i	0.741590i	−0.928715	+	0.370795i	$0.879085\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1402.48i	− 1.53949i	−0.638350	−	0.769746i	$-0.720383\pi$
$911$	− 1402.48i	− 1.53949i	0.638350	−	0.769746i	$-0.279617\pi$
$912$	0	0
$913$	429.421i	0.470340i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	601.208	0.655625
$918$	0	0
$919$	−338.255	−0.368068	−0.184034	−	0.982920i	$-0.558916\pi$
$919$	−338.255	−0.368068	−0.184034	+	0.982920i	$0.558916\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−909.811	−0.985711
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 148.207i	− 0.159533i	−0.996814	−	0.0797667i	$-0.974582\pi$
$929$	− 148.207i	− 0.159533i	0.996814	−	0.0797667i	$-0.0254175\pi$
$930$	0	0
$931$	−14.5352	−0.0156125
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1416.72i	1.51197i	0.654587	+	0.755987i	$0.272843\pi$
$937$	1416.72i	1.51197i	−0.654587	+	0.755987i	$0.727157\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1398.92i	− 1.48663i	−0.668939	−	0.743317i	$-0.733251\pi$
$941$	− 1398.92i	− 1.48663i	0.668939	−	0.743317i	$-0.266749\pi$
$942$	0	0
$943$	− 198.342i	− 0.210330i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1050.57	−1.10937	−0.554686	−	0.832060i	$-0.687161\pi$
$947$	−1050.57	−1.10937	−0.554686	+	0.832060i	$0.687161\pi$
$948$	0	0
$949$	−591.868	−0.623675
$950$	0	0
$951$	0	0
$952$	0	0
$953$	551.928	0.579148	0.289574	−	0.957156i	$-0.406486\pi$
$953$	551.928	0.579148	0.289574	+	0.957156i	$0.406486\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 111.686i	− 0.116461i
$960$	0	0
$961$	2434.61	2.53342
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 357.093i	− 0.369279i	−0.982806	−	0.184639i	$-0.940888\pi$
$967$	− 357.093i	− 0.369279i	0.982806	−	0.184639i	$-0.0591117\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 308.206i	− 0.317411i	−0.987326	−	0.158705i	$-0.949268\pi$
$971$	− 308.206i	− 0.317411i	0.987326	−	0.158705i	$-0.0507320\pi$
$972$	0	0
$973$	485.903i	0.499387i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	253.280	0.259243	0.129621	−	0.991564i	$-0.458624\pi$
$977$	253.280	0.259243	0.129621	+	0.991564i	$0.458624\pi$
$978$	0	0
$979$	583.132	0.595640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1068.73	1.08722	0.543609	−	0.839339i	$-0.317058\pi$
$983$	1068.73	1.08722	0.543609	+	0.839339i	$0.317058\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	525.194i	0.531035i
$990$	0	0
$991$	−280.631	−0.283179	−0.141590	−	0.989925i	$-0.545221\pi$
$991$	−280.631	−0.283179	−0.141590	+	0.989925i	$0.545221\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	356.574i	0.357647i	0.983881	+	0.178824i	$0.0572292\pi$
$997$	356.574i	0.357647i	−0.983881	+	0.178824i	$0.942771\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.i.449.8		8
3.2	odd	2	inner	3600.3.c.i.449.7		8
4.3	odd	2		225.3.d.b.224.1		8
5.2	odd	4		720.3.l.a.161.1		4
5.3	odd	4		3600.3.l.v.1601.4		4
5.4	even	2	inner	3600.3.c.i.449.2		8
12.11	even	2		225.3.d.b.224.7		8
15.2	even	4		720.3.l.a.161.3		4
15.8	even	4		3600.3.l.v.1601.3		4
15.14	odd	2	inner	3600.3.c.i.449.1		8
20.3	even	4		225.3.c.c.26.4		4
20.7	even	4		45.3.c.a.26.1	✓	4
20.19	odd	2		225.3.d.b.224.8		8
40.27	even	4		2880.3.l.g.1601.4		4
40.37	odd	4		2880.3.l.c.1601.3		4
60.23	odd	4		225.3.c.c.26.1		4
60.47	odd	4		45.3.c.a.26.4	yes	4
60.59	even	2		225.3.d.b.224.2		8
120.77	even	4		2880.3.l.c.1601.1		4
120.107	odd	4		2880.3.l.g.1601.2		4
180.7	even	12		405.3.i.d.296.1		8
180.47	odd	12		405.3.i.d.296.4		8
180.67	even	12		405.3.i.d.26.4		8
180.167	odd	12		405.3.i.d.26.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.1	✓	4	20.7	even	4
45.3.c.a.26.4	yes	4	60.47	odd	4
225.3.c.c.26.1		4	60.23	odd	4
225.3.c.c.26.4		4	20.3	even	4
225.3.d.b.224.1		8	4.3	odd	2
225.3.d.b.224.2		8	60.59	even	2
225.3.d.b.224.7		8	12.11	even	2
225.3.d.b.224.8		8	20.19	odd	2
405.3.i.d.26.1		8	180.167	odd	12
405.3.i.d.26.4		8	180.67	even	12
405.3.i.d.296.1		8	180.7	even	12
405.3.i.d.296.4		8	180.47	odd	12
720.3.l.a.161.1		4	5.2	odd	4
720.3.l.a.161.3		4	15.2	even	4
2880.3.l.c.1601.1		4	120.77	even	4
2880.3.l.c.1601.3		4	40.37	odd	4
2880.3.l.g.1601.2		4	120.107	odd	4
2880.3.l.g.1601.4		4	40.27	even	4
3600.3.c.i.449.1		8	15.14	odd	2	inner
3600.3.c.i.449.2		8	5.4	even	2	inner
3600.3.c.i.449.7		8	3.2	odd	2	inner
3600.3.c.i.449.8		8	1.1	even	1	trivial
3600.3.l.v.1601.3		4	15.8	even	4
3600.3.l.v.1601.4		4	5.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.16228i q^{7} +5.42736i q^{11} +9.81139i q^{13} +12.2317 q^{17} +6.32456 q^{19} +12.0394 q^{23} -44.9881i q^{29} +58.2719 q^{31} -66.4605i q^{37} -16.4743i q^{41} +43.6228i q^{43} +40.0570 q^{47} -2.29822 q^{49} +13.2242 q^{53} -25.1519i q^{59} -35.6754 q^{61} +26.7018i q^{67} +92.7301i q^{71} +60.3246i q^{73} -38.8723 q^{77} -96.2192 q^{79} +79.1215 q^{83} -107.443i q^{89} -70.2719 q^{91} +1.07900i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 112 q^{31} + 184 q^{49} - 336 q^{61} - 112 q^{79} - 208 q^{91}+O(q^{100}) \) 8 * q + 112 * q^31 + 184 * q^49 - 336 * q^61 - 112 * q^79 - 208 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more