LMFDB - Embedded newform 7200.2.f.bi.6049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(6049,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.6049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.f (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.4
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	7200.6049
Dual form			7200.2.f.bi.6049.2

$q$-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 $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$7$	2.23607i	0.845154i	0.906327	+	0.422577i	$0.138874\pi$
$7$	2.23607i	0.845154i	−0.906327	+	0.422577i	$0.861126\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.47214i	0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$	4.47214i	0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.23607	−0.401610	−0.200805	−	0.979631i	$-0.564356\pi$
$31$	−2.23607	−0.401610	−0.200805	+	0.979631i	$0.564356\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	6.70820i	1.02299i	0.859286	+	0.511496i	$0.170908\pi$
$43$	6.70820i	1.02299i	−0.859286	+	0.511496i	$0.829092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.94427i	− 1.30466i	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	− 8.94427i	− 1.30466i	0.757937	−	0.652328i	$-0.226208\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.23607i	0.273179i	0.990628	+	0.136590i	$0.0436142\pi$
$67$	2.23607i	0.273179i	−0.990628	+	0.136590i	$0.956386\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.0000i	1.13961i
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.94427i	− 0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	− 8.94427i	− 0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	−2.23607	−0.234404
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 8.94427i	− 0.881305i	−0.897678	−	0.440653i	$-0.854747\pi$
$103$	− 8.94427i	− 0.881305i	0.897678	−	0.440653i	$-0.145253\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.47214i	− 0.432338i	−0.976356	−	0.216169i	$-0.930644\pi$
$107$	− 4.47214i	− 0.432338i	0.976356	−	0.216169i	$-0.0693562\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.0000i	1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$	16.0000i	1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.47214	−0.409960
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.8885i	− 1.58735i	−0.608341	−	0.793676i	$-0.708165\pi$
$127$	− 17.8885i	− 1.58735i	0.608341	−	0.793676i	$-0.291835\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	15.0000i	1.30066i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.47214i	0.373979i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	20.1246	1.63772	0.818859	−	0.573995i	$-0.194607\pi$
$151$	20.1246	1.63772	0.818859	+	0.573995i	$0.194607\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.00000i	− 0.239426i	−0.992809	−	0.119713i	$-0.961803\pi$
$157$	− 3.00000i	− 0.239426i	0.992809	−	0.119713i	$-0.0381975\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−10.0000	−0.788110
$162$	0	0
$163$	− 15.6525i	− 1.22600i	−0.790084	−	0.612998i	$-0.789963\pi$
$163$	− 15.6525i	− 1.22600i	0.790084	−	0.612998i	$-0.210037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.3607i	1.73032i	0.501495	+	0.865161i	$0.332784\pi$
$167$	22.3607i	1.73032i	−0.501495	+	0.865161i	$0.667216\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.8885	−1.33705	−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885	−1.33705	−0.668526	+	0.743689i	$0.733075\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.94427i	0.654070i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.4164	0.970777	0.485389	−	0.874299i	$-0.338678\pi$
$191$	13.4164	0.970777	0.485389	+	0.874299i	$0.338678\pi$
$192$	0	0
$193$	− 21.0000i	− 1.51161i	−0.654795	−	0.755807i	$-0.727245\pi$
$193$	− 21.0000i	− 1.51161i	0.654795	−	0.755807i	$-0.272755\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−2.23607	−0.158511	−0.0792553	−	0.996854i	$-0.525254\pi$
$199$	−2.23607	−0.158511	−0.0792553	+	0.996854i	$0.525254\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.4164i	0.941647i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	30.0000	2.07514
$210$	0	0
$211$	−6.70820	−0.461812	−0.230906	−	0.972976i	$-0.574169\pi$
$211$	−6.70820	−0.461812	−0.230906	+	0.972976i	$0.574169\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.339422i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	6.70820i	0.449215i	0.974449	+	0.224607i	$0.0721099\pi$
$223$	6.70820i	0.449215i	−0.974449	+	0.224607i	$0.927890\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.3607i	1.48413i	0.670328	+	0.742065i	$0.266154\pi$
$227$	22.3607i	1.48413i	−0.670328	+	0.742065i	$0.733846\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.47214	0.289278	0.144639	−	0.989484i	$-0.453798\pi$
$239$	4.47214	0.289278	0.144639	+	0.989484i	$0.453798\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.70820i	0.426833i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.94427	−0.564557	−0.282279	−	0.959332i	$-0.591090\pi$
$251$	−8.94427	−0.564557	−0.282279	+	0.959332i	$0.591090\pi$
$252$	0	0
$253$	20.0000i	1.25739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.0000i	1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$	28.0000i	1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$	0	0
$259$	4.47214	0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 31.3050i	− 1.93035i	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	− 31.3050i	− 1.93035i	0.261612	−	0.965173i	$-0.415746\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−26.8328	−1.62998	−0.814989	−	0.579477i	$-0.803257\pi$
$271$	−26.8328	−1.62998	−0.814989	+	0.579477i	$0.803257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 17.0000i	− 1.02143i	−0.859750	−	0.510716i	$-0.829381\pi$
$277$	− 17.0000i	− 1.02143i	0.859750	−	0.510716i	$-0.170619\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	20.1246i	1.19628i	0.801390	+	0.598142i	$0.204094\pi$
$283$	20.1246i	1.19628i	−0.801390	+	0.598142i	$0.795906\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.47214i	− 0.263982i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.0000i	0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$	16.0000i	0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	−15.0000	−0.864586
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 11.1803i	− 0.638096i	−0.947739	−	0.319048i	$-0.896637\pi$
$307$	− 11.1803i	− 0.638096i	0.947739	−	0.319048i	$-0.103363\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.8885	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$311$	−17.8885	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$312$	0	0
$313$	− 21.0000i	− 1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$	− 21.0000i	− 1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	26.8328	1.50235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.4164i	0.746509i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.0000	1.10264
$330$	0	0
$331$	17.8885	0.983243	0.491622	−	0.870809i	$-0.336404\pi$
$331$	17.8885	0.983243	0.491622	+	0.870809i	$0.336404\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.0000i	1.47078i	0.677642	+	0.735392i	$0.263002\pi$
$337$	27.0000i	1.47078i	−0.677642	+	0.735392i	$0.736998\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	20.1246i	1.08663i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.8328i	− 1.44046i	−0.693735	−	0.720231i	$-0.744036\pi$
$347$	− 26.8328i	− 1.44046i	0.693735	−	0.720231i	$-0.255964\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.5410i	1.75083i	0.483375	+	0.875413i	$0.339411\pi$
$367$	33.5410i	1.75083i	−0.483375	+	0.875413i	$0.660589\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.4164	−0.696545
$372$	0	0
$373$	31.0000i	1.60512i	0.596572	+	0.802560i	$0.296529\pi$
$373$	31.0000i	1.60512i	−0.596572	+	0.802560i	$0.703471\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000i	0.309016i
$378$	0	0
$379$	11.1803	0.574295	0.287148	−	0.957886i	$-0.407293\pi$
$379$	11.1803	0.574295	0.287148	+	0.957886i	$0.407293\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.94427i	− 0.457031i	−0.973540	−	0.228515i	$-0.926613\pi$
$383$	− 8.94427i	− 0.457031i	0.973540	−	0.228515i	$-0.0733872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−8.94427	−0.452331
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.0000i	0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$	13.0000i	0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	− 2.23607i	− 0.111386i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.94427i	− 0.443351i
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20.0000i	0.984136i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.47214	0.218478	0.109239	−	0.994016i	$-0.465159\pi$
$419$	4.47214	0.218478	0.109239	+	0.994016i	$0.465159\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11.1803i	− 0.541055i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.47214	0.215415	0.107708	−	0.994183i	$-0.465649\pi$
$431$	4.47214	0.215415	0.107708	+	0.994183i	$0.465649\pi$
$432$	0	0
$433$	− 29.0000i	− 1.39365i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	− 29.0000i	− 1.39365i	0.717241	−	0.696826i	$-0.245405\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.0000i	1.43509i
$438$	0	0
$439$	2.23607	0.106722	0.0533609	−	0.998575i	$-0.483007\pi$
$439$	2.23607	0.106722	0.0533609	+	0.998575i	$0.483007\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.94427i	− 0.424955i	−0.977166	−	0.212478i	$-0.931847\pi$
$443$	− 8.94427i	− 0.424955i	0.977166	−	0.212478i	$-0.0681533\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−8.94427	−0.421169
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 26.8328i	− 1.24703i	−0.781813	−	0.623513i	$-0.785705\pi$
$463$	− 26.8328i	− 1.24703i	0.781813	−	0.623513i	$-0.214295\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.47214i	− 0.206946i	−0.994632	−	0.103473i	$-0.967004\pi$
$467$	− 4.47214i	− 0.206946i	0.994632	−	0.103473i	$-0.0329955\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	30.0000i	1.37940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.8328	1.22602	0.613011	−	0.790074i	$-0.289958\pi$
$479$	26.8328	1.22602	0.613011	+	0.790074i	$0.289958\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.6525i	0.709281i	0.935003	+	0.354641i	$0.115397\pi$
$487$	15.6525i	0.709281i	−0.935003	+	0.354641i	$0.884603\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.0000i	− 0.448561i
$498$	0	0
$499$	−20.1246	−0.900901	−0.450451	−	0.892801i	$-0.648737\pi$
$499$	−20.1246	−0.900901	−0.450451	+	0.892801i	$0.648737\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	40.2492i	1.79462i	0.441397	+	0.897312i	$0.354483\pi$
$503$	40.2492i	1.79462i	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.00000	0.177297	0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000	0.177297	0.0886484	+	0.996063i	$0.471745\pi$
$510$	0	0
$511$	13.4164	0.593507
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 40.0000i	− 1.75920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 33.5410i	− 1.46665i	−0.679880	−	0.733323i	$-0.737968\pi$
$523$	− 33.5410i	− 1.46665i	0.679880	−	0.733323i	$-0.262032\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.47214i	− 0.194809i
$528$	0	0
$529$	3.00000	0.130435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00000i	− 0.0866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.94427	0.385257
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.8328i	− 1.14729i	−0.819105	−	0.573644i	$-0.805529\pi$
$547$	− 26.8328i	− 1.14729i	0.819105	−	0.573644i	$-0.194471\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	40.2492	1.71467
$552$	0	0
$553$	− 20.0000i	− 0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.00000i	0.338971i	0.985533	+	0.169485i	$0.0542106\pi$
$557$	8.00000i	0.338971i	−0.985533	+	0.169485i	$0.945789\pi$
$558$	0	0
$559$	−6.70820	−0.283727
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.47214i	− 0.188478i	−0.995550	−	0.0942390i	$-0.969958\pi$
$563$	− 4.47214i	− 0.188478i	0.995550	−	0.0942390i	$-0.0300418\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	29.0689	1.21649	0.608247	−	0.793747i	$-0.291873\pi$
$571$	29.0689	1.21649	0.608247	+	0.793747i	$0.291873\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.0000i	0.707719i	0.935299	+	0.353860i	$0.115131\pi$
$577$	17.0000i	0.707719i	−0.935299	+	0.353860i	$0.884869\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.0000	0.829740
$582$	0	0
$583$	26.8328i	1.11130i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.4164i	0.553754i	0.960905	+	0.276877i	$0.0892995\pi$
$587$	13.4164i	0.553754i	−0.960905	+	0.276877i	$0.910700\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−44.7214	−1.82727	−0.913633	−	0.406541i	$-0.866735\pi$
$599$	−44.7214	−1.82727	−0.913633	+	0.406541i	$0.866735\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 44.7214i	− 1.81518i	−0.419853	−	0.907592i	$-0.637918\pi$
$607$	− 44.7214i	− 1.81518i	0.419853	−	0.907592i	$-0.362082\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.94427	0.361847
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	0	0
$619$	−38.0132	−1.52788	−0.763939	−	0.645289i	$-0.776737\pi$
$619$	−38.0132	−1.52788	−0.763939	+	0.645289i	$0.776737\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.7771i	1.43338i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	6.70820	0.267049	0.133525	−	0.991045i	$-0.457370\pi$
$631$	6.70820	0.267049	0.133525	+	0.991045i	$0.457370\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28.0000	−1.10593	−0.552967	−	0.833203i	$-0.686504\pi$
$641$	−28.0000	−1.10593	−0.552967	+	0.833203i	$0.686504\pi$
$642$	0	0
$643$	8.94427i	0.352728i	0.984325	+	0.176364i	$0.0564335\pi$
$643$	8.94427i	0.352728i	−0.984325	+	0.176364i	$0.943566\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 17.8885i	− 0.703271i	−0.936137	−	0.351636i	$-0.885626\pi$
$647$	− 17.8885i	− 0.703271i	0.936137	−	0.351636i	$-0.114374\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0000i	0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$	16.0000i	0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.2492	1.56789	0.783944	−	0.620832i	$-0.213205\pi$
$659$	40.2492	1.56789	0.783944	+	0.620832i	$0.213205\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26.8328i	1.03897i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.3607	−0.863224
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	0	0
$679$	−15.6525	−0.600687
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 17.8885i	− 0.684486i	−0.939611	−	0.342243i	$-0.888813\pi$
$683$	− 17.8885i	− 0.684486i	0.939611	−	0.342243i	$-0.111187\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−35.7771	−1.36102	−0.680512	−	0.732737i	$-0.738243\pi$
$691$	−35.7771	−1.36102	−0.680512	+	0.732737i	$0.738243\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	− 13.4164i	− 0.506009i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 22.3607i	− 0.840960i
$708$	0	0
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 10.0000i	− 0.374503i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.3607	−0.833913	−0.416956	−	0.908927i	$-0.636903\pi$
$719$	−22.3607	−0.833913	−0.416956	+	0.908927i	$0.636903\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 33.5410i	− 1.24397i	−0.783030	−	0.621984i	$-0.786327\pi$
$727$	− 33.5410i	− 1.24397i	0.783030	−	0.621984i	$-0.213673\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.4164	−0.496224
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0000i	0.368355i
$738$	0	0
$739$	−17.8885	−0.658041	−0.329020	−	0.944323i	$-0.606718\pi$
$739$	−17.8885	−0.658041	−0.329020	+	0.944323i	$0.606718\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 17.8885i	− 0.656267i	−0.944631	−	0.328134i	$-0.893580\pi$
$743$	− 17.8885i	− 0.656267i	0.944631	−	0.328134i	$-0.106420\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.0000i	− 0.617876i	−0.951082	−	0.308938i	$-0.900027\pi$
$757$	− 17.0000i	− 0.617876i	0.951082	−	0.308938i	$-0.0999735\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	0	0
$763$	− 20.1246i	− 0.728560i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.94427i	0.322959i
$768$	0	0
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.4164	−0.480693
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.5410i	1.19561i	0.801642	+	0.597804i	$0.203960\pi$
$787$	33.5410i	1.19561i	−0.801642	+	0.597804i	$0.796040\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−35.7771	−1.27209
$792$	0	0
$793$	− 5.00000i	− 0.177555i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	17.8885	0.632851
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 26.8328i	− 0.946910i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	2.23607	0.0785190	0.0392595	−	0.999229i	$-0.487500\pi$
$811$	2.23607	0.0785190	0.0392595	+	0.999229i	$0.487500\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	45.0000i	1.57435i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	0	0
$823$	6.70820i	0.233833i	0.993142	+	0.116917i	$0.0373010\pi$
$823$	6.70820i	0.233833i	−0.993142	+	0.116917i	$0.962699\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.3050i	1.08858i	0.838897	+	0.544290i	$0.183201\pi$
$827$	31.3050i	1.08858i	−0.838897	+	0.544290i	$0.816799\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.00000i	0.138592i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.1935	1.69835	0.849174	−	0.528113i	$-0.177100\pi$
$839$	49.1935	1.69835	0.849174	+	0.528113i	$0.177100\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.1246i	0.691490i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.94427	0.306606
$852$	0	0
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	0	0
$859$	53.6656	1.83105	0.915524	−	0.402264i	$-0.131776\pi$
$859$	53.6656	1.83105	0.915524	+	0.402264i	$0.131776\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 26.8328i	− 0.913400i	−0.889621	−	0.456700i	$-0.849031\pi$
$863$	− 26.8328i	− 0.913400i	0.889621	−	0.456700i	$-0.150969\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−2.23607	−0.0757663
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.0000i	0.574049i	0.957923	+	0.287025i	$0.0926662\pi$
$877$	17.0000i	0.574049i	−0.957923	+	0.287025i	$0.907334\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	42.4853i	1.42974i	0.699255	+	0.714872i	$0.253515\pi$
$883$	42.4853i	1.42974i	−0.699255	+	0.714872i	$0.746485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.94427i	− 0.300319i	−0.988662	−	0.150160i	$-0.952021\pi$
$887$	− 8.94427i	− 0.300319i	0.988662	−	0.150160i	$-0.0479788\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 60.0000i	− 2.00782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.4164	−0.447462
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.7214	−1.48168	−0.740842	−	0.671679i	$-0.765573\pi$
$911$	−44.7214	−1.48168	−0.740842	+	0.671679i	$0.765573\pi$
$912$	0	0
$913$	− 40.0000i	− 1.32381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.0000i	− 0.330229i
$918$	0	0
$919$	6.70820	0.221283	0.110642	−	0.993860i	$-0.464709\pi$
$919$	6.70820	0.221283	0.110642	+	0.993860i	$0.464709\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.47214i	− 0.147202i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	13.4164	0.439705
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 43.0000i	− 1.40475i	−0.711808	−	0.702374i	$-0.752123\pi$
$937$	− 43.0000i	− 1.40475i	0.711808	−	0.702374i	$-0.247877\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	0	0
$943$	− 8.94427i	− 0.291266i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.4164i	0.435975i	0.975952	+	0.217987i	$0.0699492\pi$
$947$	13.4164i	0.435975i	−0.975952	+	0.217987i	$0.930051\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.8328	−0.866477
$960$	0	0
$961$	−26.0000	−0.838710
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.8328i	0.862885i	0.902140	+	0.431443i	$0.141995\pi$
$967$	26.8328i	0.862885i	−0.902140	+	0.431443i	$0.858005\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.7771	−1.14814	−0.574071	−	0.818806i	$-0.694637\pi$
$971$	−35.7771	−1.14814	−0.574071	+	0.818806i	$0.694637\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	71.5542	2.28688
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.8885i	0.570556i	0.958445	+	0.285278i	$0.0920859\pi$
$983$	17.8885i	0.570556i	−0.958445	+	0.285278i	$0.907914\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.0000	−0.953945
$990$	0	0
$991$	33.5410	1.06547	0.532733	−	0.846283i	$-0.321165\pi$
$991$	33.5410	1.06547	0.532733	+	0.846283i	$0.321165\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.f.bi.6049.4		4
3.2	odd	2		7200.2.f.bf.6049.3		4
4.3	odd	2	inner	7200.2.f.bi.6049.1		4
5.2	odd	4		7200.2.a.ck.1.1	yes	2
5.3	odd	4		7200.2.a.cj.1.2	yes	2
5.4	even	2	inner	7200.2.f.bi.6049.2		4
12.11	even	2		7200.2.f.bf.6049.2		4
15.2	even	4		7200.2.a.cl.1.1	yes	2
15.8	even	4		7200.2.a.ci.1.2	yes	2
15.14	odd	2		7200.2.f.bf.6049.1		4
20.3	even	4		7200.2.a.cj.1.1	yes	2
20.7	even	4		7200.2.a.ck.1.2	yes	2
20.19	odd	2	inner	7200.2.f.bi.6049.3		4
60.23	odd	4		7200.2.a.ci.1.1	✓	2
60.47	odd	4		7200.2.a.cl.1.2	yes	2
60.59	even	2		7200.2.f.bf.6049.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7200.2.a.ci.1.1	✓	2	60.23	odd	4
7200.2.a.ci.1.2	yes	2	15.8	even	4
7200.2.a.cj.1.1	yes	2	20.3	even	4
7200.2.a.cj.1.2	yes	2	5.3	odd	4
7200.2.a.ck.1.1	yes	2	5.2	odd	4
7200.2.a.ck.1.2	yes	2	20.7	even	4
7200.2.a.cl.1.1	yes	2	15.2	even	4
7200.2.a.cl.1.2	yes	2	60.47	odd	4
7200.2.f.bf.6049.1		4	15.14	odd	2
7200.2.f.bf.6049.2		4	12.11	even	2
7200.2.f.bf.6049.3		4	3.2	odd	2
7200.2.f.bf.6049.4		4	60.59	even	2
7200.2.f.bi.6049.1		4	4.3	odd	2	inner
7200.2.f.bi.6049.2		4	5.4	even	2	inner
7200.2.f.bi.6049.3		4	20.19	odd	2	inner
7200.2.f.bi.6049.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{7} +4.47214 q^{11} +1.00000i q^{13} +2.00000i q^{17} +6.70820 q^{19} +4.47214i q^{23} +6.00000 q^{29} -2.23607 q^{31} -2.00000i q^{37} -2.00000 q^{41} +6.70820i q^{43} -8.94427i q^{47} +2.00000 q^{49} +6.00000i q^{53} +8.94427 q^{59} -5.00000 q^{61} +2.23607i q^{67} -4.47214 q^{71} -6.00000i q^{73} +10.0000i q^{77} -8.94427 q^{79} -8.94427i q^{83} +16.0000 q^{89} -2.23607 q^{91} +7.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{29} - 8 q^{41} + 8 q^{49} - 20 q^{61} + 64 q^{89}+O(q^{100}) \) 4 * q + 24 * q^29 - 8 * q^41 + 8 * q^49 - 20 * q^61 + 64 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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