LMFDB - Embedded newform 720.3.l.a.161.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,3,Mod(161,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.161"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.161
Dual form			720.3.l.a.161.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -0.837722 q^{7} +14.3716i q^{11} -21.8114 q^{13} -23.5454i q^{17} +6.32456 q^{19} -38.8723i q^{23} -5.00000 q^{25} +0.266737i q^{29} -30.2719 q^{31} -1.87320i q^{35} +9.53950 q^{37} +19.3028i q^{41} -19.6228 q^{43} +22.1684i q^{47} -48.2982 q^{49} -49.0012i q^{53} -32.1359 q^{55} -73.2351i q^{59} -48.3246 q^{61} -48.7717i q^{65} -77.2982 q^{67} -104.044i q^{71} +47.6754 q^{73} -12.0394i q^{77} -68.2192 q^{79} +28.2098i q^{83} +52.6491 q^{85} +53.7774i q^{89} +18.2719 q^{91} +14.1421i q^{95} -114.921 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7} - 24 q^{13} - 20 q^{25} + 56 q^{31} + 152 q^{37} + 48 q^{43} - 92 q^{49} - 40 q^{55} - 168 q^{61} - 208 q^{67} + 216 q^{73} + 56 q^{79} + 160 q^{85} - 104 q^{91} - 232 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 - 24 * q^13 - 20 * q^25 + 56 * q^31 + 152 * q^37 + 48 * q^43 - 92 * q^49 - 40 * q^55 - 168 * q^61 - 208 * q^67 + 216 * q^73 + 56 * q^79 + 160 * q^85 - 104 * q^91 - 232 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	−0.837722	−0.119675	−0.0598373	−	0.998208i	$-0.519058\pi$
$7$	−0.837722	−0.119675	−0.0598373	+	0.998208i	$0.519058\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.3716i	1.30651i	0.757137	+	0.653256i	$0.226597\pi$
$11$	14.3716i	1.30651i	−0.757137	+	0.653256i	$0.773403\pi$
$12$	0	0
$13$	−21.8114	−1.67780	−0.838900	−	0.544286i	$-0.816801\pi$
$13$	−21.8114	−1.67780	−0.838900	+	0.544286i	$0.816801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 23.5454i	− 1.38502i	−0.721407	−	0.692512i	$-0.756504\pi$
$17$	− 23.5454i	− 1.38502i	0.721407	−	0.692512i	$-0.243496\pi$
$18$	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
$23$	− 38.8723i	− 1.69010i	−0.534689	−	0.845049i	$-0.679571\pi$
$23$	− 38.8723i	− 1.69010i	0.534689	−	0.845049i	$-0.320429\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.266737i	0.00919784i	0.999989	+	0.00459892i	$0.00146389\pi$
$29$	0.266737i	0.00919784i	−0.999989	+	0.00459892i	$0.998536\pi$
$30$	0	0
$31$	−30.2719	−0.976512	−0.488256	−	0.872700i	$-0.662367\pi$
$31$	−30.2719	−0.976512	−0.488256	+	0.872700i	$0.662367\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.87320i	− 0.0535201i
$36$	0	0
$37$	9.53950	0.257824	0.128912	−	0.991656i	$-0.458851\pi$
$37$	9.53950	0.257824	0.128912	+	0.991656i	$0.458851\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	19.3028i	0.470799i	0.971899	+	0.235399i	$0.0756398\pi$
$41$	19.3028i	0.470799i	−0.971899	+	0.235399i	$0.924360\pi$
$42$	0	0
$43$	−19.6228	−0.456344	−0.228172	−	0.973621i	$-0.573275\pi$
$43$	−19.6228	−0.456344	−0.228172	+	0.973621i	$0.573275\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22.1684i	0.471669i	0.971793	+	0.235834i	$0.0757823\pi$
$47$	22.1684i	0.471669i	−0.971793	+	0.235834i	$0.924218\pi$
$48$	0	0
$49$	−48.2982	−0.985678
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 49.0012i	− 0.924552i	−0.886736	−	0.462276i	$-0.847033\pi$
$53$	− 49.0012i	− 0.924552i	0.886736	−	0.462276i	$-0.152967\pi$
$54$	0	0
$55$	−32.1359	−0.584290
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 73.2351i	− 1.24127i	−0.784098	−	0.620637i	$-0.786874\pi$
$59$	− 73.2351i	− 1.24127i	0.784098	−	0.620637i	$-0.213126\pi$
$60$	0	0
$61$	−48.3246	−0.792206	−0.396103	−	0.918206i	$-0.629638\pi$
$61$	−48.3246	−0.792206	−0.396103	+	0.918206i	$0.629638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 48.7717i	− 0.750335i
$66$	0	0
$67$	−77.2982	−1.15370	−0.576852	−	0.816848i	$-0.695719\pi$
$67$	−77.2982	−1.15370	−0.576852	+	0.816848i	$0.695719\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 104.044i	− 1.46541i	−0.680548	−	0.732703i	$-0.738258\pi$
$71$	− 104.044i	− 1.46541i	0.680548	−	0.732703i	$-0.261742\pi$
$72$	0	0
$73$	47.6754	0.653088	0.326544	−	0.945182i	$-0.394116\pi$
$73$	47.6754	0.653088	0.326544	+	0.945182i	$0.394116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.0394i	− 0.156356i
$78$	0	0
$79$	−68.2192	−0.863534	−0.431767	−	0.901985i	$-0.642110\pi$
$79$	−68.2192	−0.863534	−0.431767	+	0.901985i	$0.642110\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	28.2098i	0.339877i	0.985455	+	0.169938i	$0.0543569\pi$
$83$	28.2098i	0.339877i	−0.985455	+	0.169938i	$0.945643\pi$
$84$	0	0
$85$	52.6491	0.619401
$86$	0	0
$87$	0	0
$88$	0	0
$89$	53.7774i	0.604240i	0.953270	+	0.302120i	$0.0976943\pi$
$89$	53.7774i	0.604240i	−0.953270	+	0.302120i	$0.902306\pi$
$90$	0	0
$91$	18.2719	0.200790
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.1421i	0.148865i
$96$	0	0
$97$	−114.921	−1.18475	−0.592376	−	0.805661i	$-0.701810\pi$
$97$	−114.921	−1.18475	−0.592376	+	0.805661i	$0.701810\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 17.5473i	− 0.173736i	−0.996220	−	0.0868679i	$-0.972314\pi$
$101$	− 17.5473i	− 0.173736i	0.996220	−	0.0868679i	$-0.0276858\pi$
$102$	0	0
$103$	71.5395	0.694558	0.347279	−	0.937762i	$-0.387106\pi$
$103$	71.5395	0.694558	0.347279	+	0.937762i	$0.387106\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	76.3675i	0.713715i	0.934159	+	0.356858i	$0.116152\pi$
$107$	76.3675i	0.713715i	−0.934159	+	0.356858i	$0.883848\pi$
$108$	0	0
$109$	−126.921	−1.16441	−0.582206	−	0.813041i	$-0.697810\pi$
$109$	−126.921	−1.16441	−0.582206	+	0.813041i	$0.697810\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 15.0601i	− 0.133275i	−0.997777	−	0.0666377i	$-0.978773\pi$
$113$	− 15.0601i	− 0.133275i	0.997777	−	0.0666377i	$-0.0212272\pi$
$114$	0	0
$115$	86.9210	0.755835
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.7245i	0.165752i
$120$	0	0
$121$	−85.5438	−0.706973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−158.031	−1.24434	−0.622168	−	0.782884i	$-0.713748\pi$
$127$	−158.031	−1.24434	−0.622168	+	0.782884i	$0.713748\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	211.220i	1.61237i	0.591665	+	0.806184i	$0.298471\pi$
$131$	211.220i	1.61237i	−0.591665	+	0.806184i	$0.701529\pi$
$132$	0	0
$133$	−5.29822	−0.0398363
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 69.2592i	− 0.505542i	−0.967526	−	0.252771i	$-0.918658\pi$
$137$	− 69.2592i	− 0.505542i	0.967526	−	0.252771i	$-0.0813419\pi$
$138$	0	0
$139$	159.842	1.14994	0.574971	−	0.818174i	$-0.305013\pi$
$139$	159.842	1.14994	0.574971	+	0.818174i	$0.305013\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 313.465i	− 2.19206i
$144$	0	0
$145$	−0.596443	−0.00411340
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 9.81897i	− 0.0658991i	−0.999457	−	0.0329496i	$-0.989510\pi$
$149$	− 9.81897i	− 0.0658991i	0.999457	−	0.0329496i	$-0.0104901\pi$
$150$	0	0
$151$	−210.649	−1.39503	−0.697514	−	0.716572i	$-0.745710\pi$
$151$	−210.649	−1.39503	−0.697514	+	0.716572i	$0.745710\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 67.6900i	− 0.436710i
$156$	0	0
$157$	211.276	1.34571	0.672854	−	0.739775i	$-0.265068\pi$
$157$	211.276	1.34571	0.672854	+	0.739775i	$0.265068\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	32.5642i	0.202262i
$162$	0	0
$163$	222.763	1.36664	0.683322	−	0.730117i	$-0.260535\pi$
$163$	222.763	1.36664	0.683322	+	0.730117i	$0.260535\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	33.3644i	0.199787i	0.994998	+	0.0998933i	$0.0318501\pi$
$167$	33.3644i	0.199787i	−0.994998	+	0.0998933i	$0.968150\pi$
$168$	0	0
$169$	306.737	1.81501
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 29.8102i	− 0.172313i	−0.996282	−	0.0861567i	$-0.972541\pi$
$173$	− 29.8102i	− 0.172313i	0.996282	−	0.0861567i	$-0.0274586\pi$
$174$	0	0
$175$	4.18861	0.0239349
$176$	0	0
$177$	0	0
$178$	0	0
$179$	111.841i	0.624808i	0.949949	+	0.312404i	$0.101134\pi$
$179$	111.841i	0.624808i	−0.949949	+	0.312404i	$0.898866\pi$
$180$	0	0
$181$	−49.0790	−0.271155	−0.135577	−	0.990767i	$-0.543289\pi$
$181$	−49.0790	−0.271155	−0.135577	+	0.990767i	$0.543289\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.3310i	0.115303i
$186$	0	0
$187$	338.386	1.80955
$188$	0	0
$189$	0	0
$190$	0	0
$191$	278.947i	1.46046i	0.683203	+	0.730229i	$0.260586\pi$
$191$	278.947i	1.46046i	−0.683203	+	0.730229i	$0.739414\pi$
$192$	0	0
$193$	89.8947	0.465775	0.232888	−	0.972504i	$-0.425183\pi$
$193$	89.8947	0.465775	0.232888	+	0.972504i	$0.425183\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	212.709i	1.07974i	0.841748	+	0.539870i	$0.181527\pi$
$197$	212.709i	1.07974i	−0.841748	+	0.539870i	$0.818473\pi$
$198$	0	0
$199$	−96.4911	−0.484880	−0.242440	−	0.970166i	$-0.577948\pi$
$199$	−96.4911	−0.484880	−0.242440	+	0.970166i	$0.577948\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.223452i	− 0.00110075i
$204$	0	0
$205$	−43.1623	−0.210548
$206$	0	0
$207$	0	0
$208$	0	0
$209$	90.8942i	0.434900i
$210$	0	0
$211$	65.7893	0.311798	0.155899	−	0.987773i	$-0.450173\pi$
$211$	65.7893	0.311798	0.155899	+	0.987773i	$0.450173\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 43.8779i	− 0.204083i
$216$	0	0
$217$	25.3594	0.116864
$218$	0	0
$219$	0	0
$220$	0	0
$221$	513.558i	2.32379i
$222$	0	0
$223$	−102.302	−0.458756	−0.229378	−	0.973337i	$-0.573669\pi$
$223$	−102.302	−0.458756	−0.229378	+	0.973337i	$0.573669\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.5296i	− 0.0551966i	−0.999619	−	0.0275983i	$-0.991214\pi$
$227$	− 12.5296i	− 0.0551966i	0.999619	−	0.0275983i	$-0.00878593\pi$
$228$	0	0
$229$	23.2982	0.101739	0.0508695	−	0.998705i	$-0.483801\pi$
$229$	23.2982	0.101739	0.0508695	+	0.998705i	$0.483801\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 356.382i	− 1.52954i	−0.644306	−	0.764768i	$-0.722854\pi$
$233$	− 356.382i	− 1.52954i	0.644306	−	0.764768i	$-0.277146\pi$
$234$	0	0
$235$	−49.5701	−0.210937
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 175.524i	− 0.734408i	−0.930140	−	0.367204i	$-0.880315\pi$
$239$	− 175.524i	− 0.734408i	0.930140	−	0.367204i	$-0.119685\pi$
$240$	0	0
$241$	104.438	0.433355	0.216677	−	0.976243i	$-0.430478\pi$
$241$	104.438	0.433355	0.216677	+	0.976243i	$0.430478\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 107.998i	− 0.440809i
$246$	0	0
$247$	−137.947	−0.558491
$248$	0	0
$249$	0	0
$250$	0	0
$251$	130.945i	0.521694i	0.965380	+	0.260847i	$0.0840018\pi$
$251$	130.945i	0.521694i	−0.965380	+	0.260847i	$0.915998\pi$
$252$	0	0
$253$	558.658	2.20813
$254$	0	0
$255$	0	0
$256$	0	0
$257$	425.641i	1.65619i	0.560587	+	0.828095i	$0.310575\pi$
$257$	425.641i	1.65619i	−0.560587	+	0.828095i	$0.689425\pi$
$258$	0	0
$259$	−7.99145	−0.0308550
$260$	0	0
$261$	0	0
$262$	0	0
$263$	74.5004i	0.283271i	0.989919	+	0.141636i	$0.0452362\pi$
$263$	74.5004i	0.283271i	−0.989919	+	0.141636i	$0.954764\pi$
$264$	0	0
$265$	109.570	0.413472
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 205.067i	− 0.762331i	−0.924507	−	0.381165i	$-0.875523\pi$
$269$	− 205.067i	− 0.762331i	0.924507	−	0.381165i	$-0.124477\pi$
$270$	0	0
$271$	233.351	0.861073	0.430537	−	0.902573i	$-0.358324\pi$
$271$	233.351	0.861073	0.430537	+	0.902573i	$0.358324\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 71.8582i	− 0.261302i
$276$	0	0
$277$	−423.715	−1.52966	−0.764828	−	0.644235i	$-0.777176\pi$
$277$	−423.715	−1.52966	−0.764828	+	0.644235i	$0.777176\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	402.604i	1.43275i	0.697713	+	0.716377i	$0.254201\pi$
$281$	402.604i	1.43275i	−0.697713	+	0.716377i	$0.745799\pi$
$282$	0	0
$283$	272.333	0.962308	0.481154	−	0.876636i	$-0.340218\pi$
$283$	272.333	0.962308	0.481154	+	0.876636i	$0.340218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16.1704i	− 0.0563427i
$288$	0	0
$289$	−265.386	−0.918290
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 443.188i	− 1.51259i	−0.654232	−	0.756294i	$-0.727008\pi$
$293$	− 443.188i	− 1.51259i	0.654232	−	0.756294i	$-0.272992\pi$
$294$	0	0
$295$	163.759	0.555114
$296$	0	0
$297$	0	0
$298$	0	0
$299$	847.858i	2.83564i
$300$	0	0
$301$	16.4384	0.0546128
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 108.057i	− 0.354285i
$306$	0	0
$307$	390.824	1.27304	0.636522	−	0.771259i	$-0.280373\pi$
$307$	390.824	1.27304	0.636522	+	0.771259i	$0.280373\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.97739i	0.00957362i	0.999989	+	0.00478681i	$0.00152369\pi$
$311$	2.97739i	0.00957362i	−0.999989	+	0.00478681i	$0.998476\pi$
$312$	0	0
$313$	130.105	0.415672	0.207836	−	0.978164i	$-0.433358\pi$
$313$	130.105	0.415672	0.207836	+	0.978164i	$0.433358\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	131.677i	0.415384i	0.978194	+	0.207692i	$0.0665953\pi$
$317$	131.677i	0.415384i	−0.978194	+	0.207692i	$0.933405\pi$
$318$	0	0
$319$	−3.83345	−0.0120171
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 148.914i	− 0.461035i
$324$	0	0
$325$	109.057	0.335560
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 18.5710i	− 0.0564468i
$330$	0	0
$331$	−160.483	−0.484842	−0.242421	−	0.970171i	$-0.577941\pi$
$331$	−160.483	−0.484842	−0.242421	+	0.970171i	$0.577941\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 172.844i	− 0.515952i
$336$	0	0
$337$	−128.114	−0.380160	−0.190080	−	0.981769i	$-0.560875\pi$
$337$	−128.114	−0.380160	−0.190080	+	0.981769i	$0.560875\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 435.056i	− 1.27583i
$342$	0	0
$343$	81.5089	0.237635
$344$	0	0
$345$	0	0
$346$	0	0
$347$	219.637i	0.632959i	0.948599	+	0.316480i	$0.102501\pi$
$347$	219.637i	0.632959i	−0.948599	+	0.316480i	$0.897499\pi$
$348$	0	0
$349$	403.465	1.15606	0.578030	−	0.816016i	$-0.303822\pi$
$349$	403.465	1.15606	0.578030	+	0.816016i	$0.303822\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	54.8192i	0.155295i	0.996981	+	0.0776475i	$0.0247409\pi$
$353$	54.8192i	0.155295i	−0.996981	+	0.0776475i	$0.975259\pi$
$354$	0	0
$355$	232.649	0.655350
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 480.460i	− 1.33833i	−0.743114	−	0.669165i	$-0.766652\pi$
$359$	− 480.460i	− 1.33833i	0.743114	−	0.669165i	$-0.233348\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	0	0
$363$	0	0
$364$	0	0
$365$	106.606i	0.292070i
$366$	0	0
$367$	−522.364	−1.42333	−0.711667	−	0.702517i	$-0.752060\pi$
$367$	−522.364	−1.42333	−0.711667	+	0.702517i	$0.752060\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	41.0494i	0.110645i
$372$	0	0
$373$	−233.285	−0.625428	−0.312714	−	0.949847i	$-0.601238\pi$
$373$	−233.285	−0.625428	−0.312714	+	0.949847i	$0.601238\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.81791i	− 0.0154321i
$378$	0	0
$379$	−248.596	−0.655927	−0.327964	−	0.944690i	$-0.606362\pi$
$379$	−248.596	−0.655927	−0.327964	+	0.944690i	$0.606362\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 468.291i	− 1.22269i	−0.791364	−	0.611346i	$-0.790628\pi$
$383$	− 468.291i	− 1.22269i	0.791364	−	0.611346i	$-0.209372\pi$
$384$	0	0
$385$	26.9210	0.0699247
$386$	0	0
$387$	0	0
$388$	0	0
$389$	484.238i	1.24483i	0.782688	+	0.622414i	$0.213848\pi$
$389$	484.238i	1.24483i	−0.782688	+	0.622414i	$0.786152\pi$
$390$	0	0
$391$	−915.263	−2.34083
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 152.543i	− 0.386184i
$396$	0	0
$397$	−298.943	−0.753005	−0.376503	−	0.926416i	$-0.622873\pi$
$397$	−298.943	−0.753005	−0.376503	+	0.926416i	$0.622873\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 467.509i	− 1.16586i	−0.812523	−	0.582929i	$-0.801907\pi$
$401$	− 467.509i	− 1.16586i	0.812523	−	0.582929i	$-0.198093\pi$
$402$	0	0
$403$	660.272	1.63839
$404$	0	0
$405$	0	0
$406$	0	0
$407$	137.098i	0.336851i
$408$	0	0
$409$	−184.158	−0.450264	−0.225132	−	0.974328i	$-0.572281\pi$
$409$	−184.158	−0.450264	−0.225132	+	0.974328i	$0.572281\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	61.3507i	0.148549i
$414$	0	0
$415$	−63.0790	−0.151998
$416$	0	0
$417$	0	0
$418$	0	0
$419$	429.840i	1.02587i	0.858427	+	0.512936i	$0.171442\pi$
$419$	429.840i	1.02587i	−0.858427	+	0.512936i	$0.828558\pi$
$420$	0	0
$421$	−305.035	−0.724548	−0.362274	−	0.932072i	$-0.618000\pi$
$421$	−305.035	−0.724548	−0.362274	+	0.932072i	$0.618000\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	117.727i	0.277005i
$426$	0	0
$427$	40.4826	0.0948069
$428$	0	0
$429$	0	0
$430$	0	0
$431$	128.880i	0.299025i	0.988760	+	0.149512i	$0.0477704\pi$
$431$	128.880i	0.299025i	−0.988760	+	0.149512i	$0.952230\pi$
$432$	0	0
$433$	243.886	0.563247	0.281624	−	0.959525i	$-0.409127\pi$
$433$	243.886	0.563247	0.281624	+	0.959525i	$0.409127\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 245.850i	− 0.562585i
$438$	0	0
$439$	259.614	0.591376	0.295688	−	0.955285i	$-0.404451\pi$
$439$	259.614	0.591376	0.295688	+	0.955285i	$0.404451\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 541.011i	− 1.22124i	−0.791923	−	0.610622i	$-0.790920\pi$
$443$	− 541.011i	− 1.22124i	0.791923	−	0.610622i	$-0.209080\pi$
$444$	0	0
$445$	−120.250	−0.270224
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 791.947i	− 1.76380i	−0.471436	−	0.881900i	$-0.656264\pi$
$449$	− 791.947i	− 1.76380i	0.471436	−	0.881900i	$-0.343736\pi$
$450$	0	0
$451$	−277.412	−0.615104
$452$	0	0
$453$	0	0
$454$	0	0
$455$	40.8572i	0.0897960i
$456$	0	0
$457$	611.359	1.33777	0.668883	−	0.743367i	$-0.266773\pi$
$457$	611.359	1.33777	0.668883	+	0.743367i	$0.266773\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	586.991i	1.27330i	0.771153	+	0.636650i	$0.219680\pi$
$461$	586.991i	1.27330i	−0.771153	+	0.636650i	$0.780320\pi$
$462$	0	0
$463$	−195.285	−0.421781	−0.210891	−	0.977510i	$-0.567636\pi$
$463$	−195.285	−0.421781	−0.210891	+	0.977510i	$0.567636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 753.763i	− 1.61405i	−0.590515	−	0.807027i	$-0.701075\pi$
$467$	− 753.763i	− 1.61405i	0.590515	−	0.807027i	$-0.298925\pi$
$468$	0	0
$469$	64.7544	0.138069
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 282.011i	− 0.596218i
$474$	0	0
$475$	−31.6228	−0.0665743
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 614.848i	− 1.28361i	−0.766869	−	0.641803i	$-0.778186\pi$
$479$	− 614.848i	− 1.28361i	0.766869	−	0.641803i	$-0.221814\pi$
$480$	0	0
$481$	−208.070	−0.432577
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 256.971i	− 0.529837i
$486$	0	0
$487$	−478.197	−0.981924	−0.490962	−	0.871181i	$-0.663355\pi$
$487$	−478.197	−0.981924	−0.490962	+	0.871181i	$0.663355\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	617.223i	1.25707i	0.777780	+	0.628537i	$0.216346\pi$
$491$	617.223i	1.25707i	−0.777780	+	0.628537i	$0.783654\pi$
$492$	0	0
$493$	6.28043	0.0127392
$494$	0	0
$495$	0	0
$496$	0	0
$497$	87.1599i	0.175372i
$498$	0	0
$499$	−783.096	−1.56933	−0.784665	−	0.619919i	$-0.787165\pi$
$499$	−783.096	−1.56933	−0.784665	+	0.619919i	$0.787165\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 369.395i	− 0.734383i	−0.930145	−	0.367191i	$-0.880319\pi$
$503$	− 369.395i	− 0.734383i	0.930145	−	0.367191i	$-0.119681\pi$
$504$	0	0
$505$	39.2370	0.0776970
$506$	0	0
$507$	0	0
$508$	0	0
$509$	225.635i	0.443291i	0.975127	+	0.221645i	$0.0711427\pi$
$509$	225.635i	0.443291i	−0.975127	+	0.221645i	$0.928857\pi$
$510$	0	0
$511$	−39.9388	−0.0781581
$512$	0	0
$513$	0	0
$514$	0	0
$515$	159.967i	0.310616i
$516$	0	0
$517$	−318.596	−0.616241
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1007.73i	1.93422i	0.254367	+	0.967108i	$0.418133\pi$
$521$	1007.73i	1.93422i	−0.254367	+	0.967108i	$0.581867\pi$
$522$	0	0
$523$	−935.517	−1.78875	−0.894376	−	0.447316i	$-0.852380\pi$
$523$	−935.517	−1.78875	−0.894376	+	0.447316i	$0.852380\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	712.764i	1.35249i
$528$	0	0
$529$	−982.052	−1.85643
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 421.020i	− 0.789906i
$534$	0	0
$535$	−170.763	−0.319183
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 694.124i	− 1.28780i
$540$	0	0
$541$	−399.149	−0.737798	−0.368899	−	0.929469i	$-0.620265\pi$
$541$	−399.149	−0.737798	−0.368899	+	0.929469i	$0.620265\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 283.804i	− 0.520741i
$546$	0	0
$547$	−480.833	−0.879036	−0.439518	−	0.898234i	$-0.644851\pi$
$547$	−480.833	−0.879036	−0.439518	+	0.898234i	$0.644851\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.68699i	0.00306170i
$552$	0	0
$553$	57.1488	0.103343
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 751.542i	− 1.34927i	−0.738152	−	0.674634i	$-0.764301\pi$
$557$	− 751.542i	− 1.34927i	0.738152	−	0.674634i	$-0.235699\pi$
$558$	0	0
$559$	428.000	0.765653
$560$	0	0
$561$	0	0
$562$	0	0
$563$	670.820i	1.19151i	0.803166	+	0.595755i	$0.203147\pi$
$563$	670.820i	1.19151i	−0.803166	+	0.595755i	$0.796853\pi$
$564$	0	0
$565$	33.6754	0.0596026
$566$	0	0
$567$	0	0
$568$	0	0
$569$	275.325i	0.483875i	0.970292	+	0.241937i	$0.0777829\pi$
$569$	275.325i	0.483875i	−0.970292	+	0.241937i	$0.922217\pi$
$570$	0	0
$571$	900.289	1.57669	0.788344	−	0.615235i	$-0.210939\pi$
$571$	900.289	1.57669	0.788344	+	0.615235i	$0.210939\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	194.361i	0.338020i
$576$	0	0
$577$	−596.236	−1.03334	−0.516669	−	0.856185i	$-0.672828\pi$
$577$	−596.236	−1.03334	−0.516669	+	0.856185i	$0.672828\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 23.6320i	− 0.0406746i
$582$	0	0
$583$	704.228	1.20794
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 497.431i	− 0.847412i	−0.905800	−	0.423706i	$-0.860729\pi$
$587$	− 497.431i	− 0.847412i	0.905800	−	0.423706i	$-0.139271\pi$
$588$	0	0
$589$	−191.456	−0.325053
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 898.856i	− 1.51578i	−0.652384	−	0.757889i	$-0.726231\pi$
$593$	− 898.856i	− 1.51578i	0.652384	−	0.757889i	$-0.273769\pi$
$594$	0	0
$595$	−44.1053	−0.0741266
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 28.5701i	− 0.0476964i	−0.999716	−	0.0238482i	$-0.992408\pi$
$599$	− 28.5701i	− 0.0476964i	0.999716	−	0.0238482i	$-0.00759183\pi$
$600$	0	0
$601$	−20.2107	−0.0336284	−0.0168142	−	0.999859i	$-0.505352\pi$
$601$	−20.2107	−0.0336284	−0.0168142	+	0.999859i	$0.505352\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 191.282i	− 0.316168i
$606$	0	0
$607$	713.626	1.17566	0.587831	−	0.808984i	$-0.299982\pi$
$607$	713.626	1.17566	0.587831	+	0.808984i	$0.299982\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 483.524i	− 0.791365i
$612$	0	0
$613$	−76.1530	−0.124230	−0.0621150	−	0.998069i	$-0.519785\pi$
$613$	−76.1530	−0.124230	−0.0621150	+	0.998069i	$0.519785\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	201.599i	0.326741i	0.986565	+	0.163371i	$0.0522367\pi$
$617$	201.599i	0.326741i	−0.986565	+	0.163371i	$0.947763\pi$
$618$	0	0
$619$	204.263	0.329989	0.164995	−	0.986294i	$-0.447239\pi$
$619$	204.263	0.329989	0.164995	+	0.986294i	$0.447239\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 45.0505i	− 0.0723122i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 224.611i	− 0.357093i
$630$	0	0
$631$	639.903	1.01411	0.507055	−	0.861914i	$-0.330734\pi$
$631$	639.903	1.01411	0.507055	+	0.861914i	$0.330734\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 353.367i	− 0.556484i
$636$	0	0
$637$	1053.45	1.65377
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 570.709i	− 0.890342i	−0.895446	−	0.445171i	$-0.853143\pi$
$641$	− 570.709i	− 0.890342i	0.895446	−	0.445171i	$-0.146857\pi$
$642$	0	0
$643$	−453.693	−0.705587	−0.352794	−	0.935701i	$-0.614768\pi$
$643$	−453.693	−0.705587	−0.352794	+	0.935701i	$0.614768\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	983.536i	1.52015i	0.649837	+	0.760074i	$0.274837\pi$
$647$	983.536i	1.52015i	−0.649837	+	0.760074i	$0.725163\pi$
$648$	0	0
$649$	1052.51	1.62174
$650$	0	0
$651$	0	0
$652$	0	0
$653$	544.478i	0.833811i	0.908950	+	0.416905i	$0.136885\pi$
$653$	544.478i	0.833811i	−0.908950	+	0.416905i	$0.863115\pi$
$654$	0	0
$655$	−472.302	−0.721073
$656$	0	0
$657$	0	0
$658$	0	0
$659$	107.933i	0.163783i	0.996641	+	0.0818916i	$0.0260961\pi$
$659$	107.933i	0.163783i	−0.996641	+	0.0818916i	$0.973904\pi$
$660$	0	0
$661$	345.728	0.523038	0.261519	−	0.965198i	$-0.415777\pi$
$661$	345.728	0.523038	0.261519	+	0.965198i	$0.415777\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.8472i	− 0.0178153i
$666$	0	0
$667$	10.3687	0.0155452
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 694.503i	− 1.03503i
$672$	0	0
$673$	−1204.78	−1.79016	−0.895082	−	0.445902i	$-0.852883\pi$
$673$	−1204.78	−1.79016	−0.895082	+	0.445902i	$0.852883\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	574.598i	0.848742i	0.905488	+	0.424371i	$0.139505\pi$
$677$	574.598i	0.848742i	−0.905488	+	0.424371i	$0.860495\pi$
$678$	0	0
$679$	96.2719	0.141785
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 334.387i	− 0.489585i	−0.969575	−	0.244792i	$-0.921280\pi$
$683$	− 334.387i	− 0.489585i	0.969575	−	0.244792i	$-0.0787198\pi$
$684$	0	0
$685$	154.868	0.226085
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1068.79i	1.55121i
$690$	0	0
$691$	−901.149	−1.30412	−0.652061	−	0.758166i	$-0.726096\pi$
$691$	−901.149	−1.30412	−0.652061	+	0.758166i	$0.726096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	357.418i	0.514270i
$696$	0	0
$697$	454.491	0.652068
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 887.934i	− 1.26667i	−0.773879	−	0.633334i	$-0.781686\pi$
$701$	− 887.934i	− 1.26667i	0.773879	−	0.633334i	$-0.218314\pi$
$702$	0	0
$703$	60.3331	0.0858223
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.6998i	0.0207918i
$708$	0	0
$709$	−1026.35	−1.44760	−0.723801	−	0.690008i	$-0.757607\pi$
$709$	−1026.35	−1.44760	−0.723801	+	0.690008i	$0.757607\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1176.74i	1.65040i
$714$	0	0
$715$	700.930	0.980321
$716$	0	0
$717$	0	0
$718$	0	0
$719$	740.080i	1.02932i	0.857395	+	0.514659i	$0.172081\pi$
$719$	740.080i	1.02932i	−0.857395	+	0.514659i	$0.827919\pi$
$720$	0	0
$721$	−59.9302	−0.0831210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.33369i	− 0.00183957i
$726$	0	0
$727$	−1063.75	−1.46321	−0.731603	−	0.681731i	$-0.761227\pi$
$727$	−1063.75	−1.46321	−0.731603	+	0.681731i	$0.761227\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	462.026i	0.632047i
$732$	0	0
$733$	134.749	0.183833	0.0919164	−	0.995767i	$-0.470701\pi$
$733$	134.749	0.183833	0.0919164	+	0.995767i	$0.470701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1110.90i	− 1.50733i
$738$	0	0
$739$	711.429	0.962692	0.481346	−	0.876531i	$-0.340148\pi$
$739$	711.429	0.962692	0.481346	+	0.876531i	$0.340148\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 466.330i	− 0.627631i	−0.949484	−	0.313816i	$-0.898393\pi$
$743$	− 466.330i	− 0.627631i	0.949484	−	0.313816i	$-0.101607\pi$
$744$	0	0
$745$	21.9559	0.0294710
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 63.9748i	− 0.0854136i
$750$	0	0
$751$	−227.359	−0.302742	−0.151371	−	0.988477i	$-0.548369\pi$
$751$	−227.359	−0.302742	−0.151371	+	0.988477i	$0.548369\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 471.026i	− 0.623875i
$756$	0	0
$757$	−552.258	−0.729535	−0.364768	−	0.931099i	$-0.618852\pi$
$757$	−552.258	−0.729535	−0.364768	+	0.931099i	$0.618852\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 303.051i	− 0.398228i	−0.979976	−	0.199114i	$-0.936194\pi$
$761$	− 303.051i	− 0.398228i	0.979976	−	0.199114i	$-0.0638064\pi$
$762$	0	0
$763$	106.325	0.139351
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1597.36i	2.08261i
$768$	0	0
$769$	739.684	0.961878	0.480939	−	0.876754i	$-0.340296\pi$
$769$	739.684	0.961878	0.480939	+	0.876754i	$0.340296\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	603.479i	0.780697i	0.920667	+	0.390348i	$0.127645\pi$
$773$	603.479i	0.780697i	−0.920667	+	0.390348i	$0.872355\pi$
$774$	0	0
$775$	151.359	0.195302
$776$	0	0
$777$	0	0
$778$	0	0
$779$	122.081i	0.156715i
$780$	0	0
$781$	1495.28	1.91457
$782$	0	0
$783$	0	0
$784$	0	0
$785$	472.428i	0.601819i
$786$	0	0
$787$	660.483	0.839241	0.419620	−	0.907700i	$-0.362163\pi$
$787$	660.483	0.839241	0.419620	+	0.907700i	$0.362163\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.6162i	0.0159497i
$792$	0	0
$793$	1054.03	1.32916
$794$	0	0
$795$	0	0
$796$	0	0
$797$	634.969i	0.796699i	0.917234	+	0.398349i	$0.130417\pi$
$797$	634.969i	0.796699i	−0.917234	+	0.398349i	$0.869583\pi$
$798$	0	0
$799$	521.964	0.653272
$800$	0	0
$801$	0	0
$802$	0	0
$803$	685.174i	0.853268i
$804$	0	0
$805$	−72.8157	−0.0904542
$806$	0	0
$807$	0	0
$808$	0	0
$809$	405.048i	0.500677i	0.968158	+	0.250339i	$0.0805419\pi$
$809$	405.048i	0.500677i	−0.968158	+	0.250339i	$0.919458\pi$
$810$	0	0
$811$	−406.034	−0.500659	−0.250329	−	0.968161i	$-0.580539\pi$
$811$	−406.034	−0.500659	−0.250329	+	0.968161i	$0.580539\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	498.113i	0.611182i
$816$	0	0
$817$	−124.105	−0.151904
$818$	0	0
$819$	0	0
$820$	0	0
$821$	550.073i	0.670003i	0.942218	+	0.335002i	$0.108737\pi$
$821$	550.073i	0.670003i	−0.942218	+	0.335002i	$0.891263\pi$
$822$	0	0
$823$	1472.51	1.78920	0.894600	−	0.446868i	$-0.147460\pi$
$823$	1472.51	1.78920	0.894600	+	0.446868i	$0.147460\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1510.47i	− 1.82644i	−0.407466	−	0.913220i	$-0.633587\pi$
$827$	− 1510.47i	− 1.82644i	0.407466	−	0.913220i	$-0.366413\pi$
$828$	0	0
$829$	712.692	0.859701	0.429850	−	0.902900i	$-0.358566\pi$
$829$	712.692	0.859701	0.429850	+	0.902900i	$0.358566\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1137.20i	1.36519i
$834$	0	0
$835$	−74.6050	−0.0893473
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 530.579i	− 0.632394i	−0.948694	−	0.316197i	$-0.897594\pi$
$839$	− 530.579i	− 0.632394i	0.948694	−	0.316197i	$-0.102406\pi$
$840$	0	0
$841$	840.929	0.999915
$842$	0	0
$843$	0	0
$844$	0	0
$845$	685.884i	0.811697i
$846$	0	0
$847$	71.6619	0.0846068
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 370.822i	− 0.435748i
$852$	0	0
$853$	215.232	0.252324	0.126162	−	0.992010i	$-0.459734\pi$
$853$	215.232	0.252324	0.126162	+	0.992010i	$0.459734\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.97441i	− 0.00580445i	−0.999996	−	0.00290222i	$-0.999076\pi$
$857$	− 4.97441i	− 0.00580445i	0.999996	−	0.00290222i	$-0.000923808\pi$
$858$	0	0
$859$	−1652.28	−1.92349	−0.961746	−	0.273941i	$-0.911673\pi$
$859$	−1652.28	−1.92349	−0.961746	+	0.273941i	$0.911673\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	379.077i	0.439255i	0.975584	+	0.219627i	$0.0704841\pi$
$863$	379.077i	0.439255i	−0.975584	+	0.219627i	$0.929516\pi$
$864$	0	0
$865$	66.6577	0.0770609
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 980.421i	− 1.12822i
$870$	0	0
$871$	1685.98	1.93568
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.36602i	0.0107040i
$876$	0	0
$877$	1101.60	1.25610	0.628051	−	0.778173i	$-0.283853\pi$
$877$	1101.60	1.25610	0.628051	+	0.778173i	$0.283853\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 184.877i	− 0.209850i	−0.994480	−	0.104925i	$-0.966540\pi$
$881$	− 184.877i	− 0.209850i	0.994480	−	0.104925i	$-0.0334602\pi$
$882$	0	0
$883$	−978.236	−1.10786	−0.553928	−	0.832565i	$-0.686872\pi$
$883$	−978.236	−1.10786	−0.553928	+	0.832565i	$0.686872\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 667.937i	− 0.753029i	−0.926411	−	0.376514i	$-0.877123\pi$
$887$	− 667.937i	− 0.753029i	0.926411	−	0.376514i	$-0.122877\pi$
$888$	0	0
$889$	132.386	0.148915
$890$	0	0
$891$	0	0
$892$	0	0
$893$	140.205i	0.157005i
$894$	0	0
$895$	−250.083	−0.279423
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 8.07464i	− 0.00898180i
$900$	0	0
$901$	−1153.75	−1.28053
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 109.744i	− 0.121264i
$906$	0	0
$907$	832.622	0.917996	0.458998	−	0.888437i	$-0.348209\pi$
$907$	832.622	0.917996	0.458998	+	0.888437i	$0.348209\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	565.263i	0.620486i	0.950657	+	0.310243i	$0.100410\pi$
$911$	565.263i	0.620486i	−0.950657	+	0.310243i	$0.899590\pi$
$912$	0	0
$913$	−405.421	−0.444053
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 176.944i	− 0.192959i
$918$	0	0
$919$	1185.75	1.29026	0.645128	−	0.764075i	$-0.276804\pi$
$919$	1185.75	1.29026	0.645128	+	0.764075i	$0.276804\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2269.34i	2.45866i
$924$	0	0
$925$	−47.6975	−0.0515649
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 942.995i	− 1.01506i	−0.861633	−	0.507532i	$-0.830558\pi$
$929$	− 942.995i	− 1.01506i	0.861633	−	0.507532i	$-0.169442\pi$
$930$	0	0
$931$	−305.465	−0.328104
$932$	0	0
$933$	0	0
$934$	0	0
$935$	756.654i	0.809255i
$936$	0	0
$937$	−531.281	−0.567002	−0.283501	−	0.958972i	$-0.591496\pi$
$937$	−531.281	−0.567002	−0.283501	+	0.958972i	$0.591496\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	595.650i	0.632996i	0.948593	+	0.316498i	$0.102507\pi$
$941$	595.650i	0.632996i	−0.948593	+	0.316498i	$0.897493\pi$
$942$	0	0
$943$	750.342	0.795696
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 782.246i	− 0.826026i	−0.910725	−	0.413013i	$-0.864476\pi$
$947$	− 782.246i	− 0.826026i	0.910725	−	0.413013i	$-0.135524\pi$
$948$	0	0
$949$	−1039.87	−1.09575
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 426.708i	− 0.447752i	−0.974618	−	0.223876i	$-0.928129\pi$
$953$	− 426.708i	− 0.447752i	0.974618	−	0.223876i	$-0.0718711\pi$
$954$	0	0
$955$	−623.745	−0.653136
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.0200i	0.0605005i
$960$	0	0
$961$	−44.6128	−0.0464234
$962$	0	0
$963$	0	0
$964$	0	0
$965$	201.011i	0.208301i
$966$	0	0
$967$	1210.91	1.25223	0.626116	−	0.779730i	$-0.284644\pi$
$967$	1210.91	1.25223	0.626116	+	0.779730i	$0.284644\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1193.69i	− 1.22934i	−0.788784	−	0.614670i	$-0.789289\pi$
$971$	− 1193.69i	− 1.22934i	0.788784	−	0.614670i	$-0.210711\pi$
$972$	0	0
$973$	−133.903	−0.137619
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 909.475i	− 0.930886i	−0.885078	−	0.465443i	$-0.845895\pi$
$977$	− 909.475i	− 0.930886i	0.885078	−	0.465443i	$-0.154105\pi$
$978$	0	0
$979$	−772.868	−0.789447
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1560.88i	1.58788i	0.607999	+	0.793938i	$0.291972\pi$
$983$	1560.88i	1.58788i	−0.607999	+	0.793938i	$0.708028\pi$
$984$	0	0
$985$	−475.631	−0.482874
$986$	0	0
$987$	0	0
$988$	0	0
$989$	762.782i	0.771265i
$990$	0	0
$991$	1692.63	1.70800	0.854001	−	0.520271i	$-0.174169\pi$
$991$	1692.63	1.70800	0.854001	+	0.520271i	$0.174169\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 215.761i	− 0.216845i
$996$	0	0
$997$	16.5744	0.0166243	0.00831213	−	0.999965i	$-0.497354\pi$
$997$	16.5744	0.0166243	0.00831213	+	0.999965i	$0.497354\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.l.a.161.4		4
3.2	odd	2	inner	720.3.l.a.161.2		4
4.3	odd	2		45.3.c.a.26.3	yes	4
5.2	odd	4		3600.3.c.i.449.4		8
5.3	odd	4		3600.3.c.i.449.6		8
5.4	even	2		3600.3.l.v.1601.2		4
8.3	odd	2		2880.3.l.g.1601.1		4
8.5	even	2		2880.3.l.c.1601.2		4
12.11	even	2		45.3.c.a.26.2	✓	4
15.2	even	4		3600.3.c.i.449.3		8
15.8	even	4		3600.3.c.i.449.5		8
15.14	odd	2		3600.3.l.v.1601.1		4
20.3	even	4		225.3.d.b.224.5		8
20.7	even	4		225.3.d.b.224.4		8
20.19	odd	2		225.3.c.c.26.2		4
24.5	odd	2		2880.3.l.c.1601.4		4
24.11	even	2		2880.3.l.g.1601.3		4
36.7	odd	6		405.3.i.d.296.3		8
36.11	even	6		405.3.i.d.296.2		8
36.23	even	6		405.3.i.d.26.3		8
36.31	odd	6		405.3.i.d.26.2		8
60.23	odd	4		225.3.d.b.224.3		8
60.47	odd	4		225.3.d.b.224.6		8
60.59	even	2		225.3.c.c.26.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.2	✓	4	12.11	even	2
45.3.c.a.26.3	yes	4	4.3	odd	2
225.3.c.c.26.2		4	20.19	odd	2
225.3.c.c.26.3		4	60.59	even	2
225.3.d.b.224.3		8	60.23	odd	4
225.3.d.b.224.4		8	20.7	even	4
225.3.d.b.224.5		8	20.3	even	4
225.3.d.b.224.6		8	60.47	odd	4
405.3.i.d.26.2		8	36.31	odd	6
405.3.i.d.26.3		8	36.23	even	6
405.3.i.d.296.2		8	36.11	even	6
405.3.i.d.296.3		8	36.7	odd	6
720.3.l.a.161.2		4	3.2	odd	2	inner
720.3.l.a.161.4		4	1.1	even	1	trivial
2880.3.l.c.1601.2		4	8.5	even	2
2880.3.l.c.1601.4		4	24.5	odd	2
2880.3.l.g.1601.1		4	8.3	odd	2
2880.3.l.g.1601.3		4	24.11	even	2
3600.3.c.i.449.3		8	15.2	even	4
3600.3.c.i.449.4		8	5.2	odd	4
3600.3.c.i.449.5		8	15.8	even	4
3600.3.c.i.449.6		8	5.3	odd	4
3600.3.l.v.1601.1		4	15.14	odd	2
3600.3.l.v.1601.2		4	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -0.837722 q^{7} +14.3716i q^{11} -21.8114 q^{13} -23.5454i q^{17} +6.32456 q^{19} -38.8723i q^{23} -5.00000 q^{25} +0.266737i q^{29} -30.2719 q^{31} -1.87320i q^{35} +9.53950 q^{37} +19.3028i q^{41} -19.6228 q^{43} +22.1684i q^{47} -48.2982 q^{49} -49.0012i q^{53} -32.1359 q^{55} -73.2351i q^{59} -48.3246 q^{61} -48.7717i q^{65} -77.2982 q^{67} -104.044i q^{71} +47.6754 q^{73} -12.0394i q^{77} -68.2192 q^{79} +28.2098i q^{83} +52.6491 q^{85} +53.7774i q^{89} +18.2719 q^{91} +14.1421i q^{95} -114.921 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7} - 24 q^{13} - 20 q^{25} + 56 q^{31} + 152 q^{37} + 48 q^{43} - 92 q^{49} - 40 q^{55} - 168 q^{61} - 208 q^{67} + 216 q^{73} + 56 q^{79} + 160 q^{85} - 104 q^{91} - 232 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 - 24 * q^13 - 20 * q^25 + 56 * q^31 + 152 * q^37 + 48 * q^43 - 92 * q^49 - 40 * q^55 - 168 * q^61 - 208 * q^67 + 216 * q^73 + 56 * q^79 + 160 * q^85 - 104 * q^91 - 232 * q^97

Introduction

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