LMFDB - Embedded newform 7200.2.f.be.6049.1

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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	7200.6049
Dual form			7200.2.f.be.6049.3

$q$-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
$7$	− 4.47214i	− 1.69031i	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	− 4.47214i	− 1.69031i	0.534522	−	0.845154i	$-0.320491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.94427i	− 1.86501i	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	− 8.94427i	− 1.86501i	0.361158	−	0.932505i	$-0.382382\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.94427	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	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
$49$	−13.0000	−1.85714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.94427i	1.09272i	0.837552	+	0.546358i	$0.183986\pi$
$67$	8.94427i	1.09272i	−0.837552	+	0.546358i	$0.816014\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	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
$77$	20.0000i	2.27921i
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	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
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−17.8885	−1.87523
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	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
$103$	4.47214i	0.440653i	0.975426	+	0.220326i	$0.0707122\pi$
$103$	4.47214i	0.440653i	−0.975426	+	0.220326i	$0.929288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.94427i	0.864675i	0.901712	+	0.432338i	$0.142311\pi$
$107$	8.94427i	0.864675i	−0.901712	+	0.432338i	$0.857689\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.94427	−0.819920
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.47214i	− 0.396838i	−0.980117	−	0.198419i	$-0.936419\pi$
$127$	− 4.47214i	− 0.396838i	0.980117	−	0.198419i	$-0.0635807\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.47214	0.390732	0.195366	−	0.980730i	$-0.437410\pi$
$131$	4.47214	0.390732	0.195366	+	0.980730i	$0.437410\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−17.8885	−1.51729	−0.758643	−	0.651506i	$-0.774137\pi$
$139$	−17.8885	−1.51729	−0.758643	+	0.651506i	$0.774137\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.8885i	1.49592i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0000i	0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$	12.0000i	0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−40.0000	−3.15244
$162$	0	0
$163$	− 8.94427i	− 0.700569i	−0.936643	−	0.350285i	$-0.886085\pi$
$163$	− 8.94427i	− 0.700569i	0.936643	−	0.350285i	$-0.113915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.94427i	0.692129i	0.938211	+	0.346064i	$0.112482\pi$
$167$	8.94427i	0.692129i	−0.938211	+	0.346064i	$0.887518\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.47214	0.334263	0.167132	−	0.985935i	$-0.446550\pi$
$179$	4.47214	0.334263	0.167132	+	0.985935i	$0.446550\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	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
$191$	26.8328	1.94155	0.970777	−	0.239983i	$-0.0771417\pi$
$191$	26.8328	1.94155	0.970777	+	0.239983i	$0.0771417\pi$
$192$	0	0
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−17.8885	−1.26809	−0.634043	−	0.773298i	$-0.718606\pi$
$199$	−17.8885	−1.26809	−0.634043	+	0.773298i	$0.718606\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.8328i	1.88329i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 40.0000i	− 2.71538i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	− 13.4164i	− 0.898429i	−0.893424	−	0.449215i	$-0.851704\pi$
$223$	− 13.4164i	− 0.898429i	0.893424	−	0.449215i	$-0.148296\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.8885i	− 1.18730i	−0.804722	−	0.593652i	$-0.797686\pi$
$227$	− 17.8885i	− 1.18730i	0.804722	−	0.593652i	$-0.202314\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.3607	1.41139	0.705697	−	0.708514i	$-0.250634\pi$
$251$	22.3607	1.41139	0.705697	+	0.708514i	$0.250634\pi$
$252$	0	0
$253$	40.0000i	2.51478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	35.7771	2.22308
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.94427i	0.551527i	0.961225	+	0.275764i	$0.0889307\pi$
$263$	8.94427i	0.551527i	−0.961225	+	0.275764i	$0.911069\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	26.8328i	1.59505i	0.603289	+	0.797523i	$0.293857\pi$
$283$	26.8328i	1.59505i	−0.603289	+	0.797523i	$0.706143\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	35.7771i	2.11185i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−35.7771	−2.06904
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.8885i	− 1.02095i	−0.859892	−	0.510477i	$-0.829469\pi$
$307$	− 17.8885i	− 1.02095i	0.859892	−	0.510477i	$-0.170531\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.94427	−0.507183	−0.253592	−	0.967311i	$-0.581612\pi$
$311$	−8.94427	−0.507183	−0.253592	+	0.967311i	$0.581612\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	26.8328	1.50235
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−40.0000	−2.20527
$330$	0	0
$331$	35.7771	1.96649	0.983243	−	0.182298i	$-0.0583536\pi$
$331$	35.7771	1.96649	0.983243	+	0.182298i	$0.0583536\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−40.0000	−2.16612
$342$	0	0
$343$	26.8328i	1.44884i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.7771	1.88824	0.944121	−	0.329598i	$-0.106913\pi$
$359$	35.7771	1.88824	0.944121	+	0.329598i	$0.106913\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.4164i	0.700331i	0.936688	+	0.350165i	$0.113875\pi$
$367$	13.4164i	0.700331i	−0.936688	+	0.350165i	$0.886125\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26.8328	−1.39309
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−17.8885	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$380$	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
$391$	−17.8885	−0.904663
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	− 35.7771i	− 1.78218i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 35.7771i	− 1.77340i
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	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	$-0.534841\pi$
$419$	−4.47214	−0.218478	−0.109239	+	0.994016i	$0.534841\pi$
$420$	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
$427$	− 44.7214i	− 2.16422i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−35.7771	−1.70755	−0.853774	−	0.520644i	$-0.825692\pi$
$439$	−35.7771	−1.70755	−0.853774	+	0.520644i	$0.825692\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 35.7771i	− 1.69982i	−0.526927	−	0.849910i	$-0.676656\pi$
$443$	− 35.7771i	− 1.69982i	0.526927	−	0.849910i	$-0.323344\pi$
$444$	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
$451$	35.7771	1.68468
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	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
$463$	− 40.2492i	− 1.87054i	−0.353935	−	0.935270i	$-0.615157\pi$
$463$	− 40.2492i	− 1.87054i	0.353935	−	0.935270i	$-0.384843\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.94427i	0.413892i	0.978352	+	0.206946i	$0.0663524\pi$
$467$	8.94427i	0.413892i	−0.978352	+	0.206946i	$0.933648\pi$
$468$	0	0
$469$	40.0000	1.84703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.8328	−1.22602	−0.613011	−	0.790074i	$-0.710042\pi$
$479$	−26.8328	−1.22602	−0.613011	+	0.790074i	$0.710042\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.3607i	1.01326i	0.862164	+	0.506630i	$0.169109\pi$
$487$	22.3607i	1.01326i	−0.862164	+	0.506630i	$0.830891\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.47214	0.201825	0.100912	−	0.994895i	$-0.467824\pi$
$491$	4.47214	0.201825	0.100912	+	0.994895i	$0.467824\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000i	1.79425i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 26.8328i	− 1.19642i	−0.801341	−	0.598208i	$-0.795880\pi$
$503$	− 26.8328i	− 1.19642i	0.801341	−	0.598208i	$-0.204120\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−26.8328	−1.18701
$512$	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
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	26.8328i	1.17332i	0.809834	+	0.586659i	$0.199557\pi$
$523$	26.8328i	1.17332i	−0.809834	+	0.586659i	$0.800443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 17.8885i	− 0.779237i
$528$	0	0
$529$	−57.0000	−2.47826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.0000i	1.38607i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	58.1378	2.50417
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 40.0000i	− 1.70097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 17.8885i	− 0.753912i	−0.926231	−	0.376956i	$-0.876971\pi$
$563$	− 17.8885i	− 0.753912i	0.926231	−	0.376956i	$-0.123029\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−35.7771	−1.49722	−0.748612	−	0.663008i	$-0.769280\pi$
$571$	−35.7771	−1.49722	−0.748612	+	0.663008i	$0.769280\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−40.0000	−1.65948
$582$	0	0
$583$	26.8328i	1.11130i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.8328i	− 1.10751i	−0.832680	−	0.553754i	$-0.813195\pi$
$587$	− 26.8328i	− 1.10751i	0.832680	−	0.553754i	$-0.186805\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.7771	−1.46181	−0.730906	−	0.682478i	$-0.760902\pi$
$599$	−35.7771	−1.46181	−0.730906	+	0.682478i	$0.760902\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.3607i	0.907592i	0.891106	+	0.453796i	$0.149931\pi$
$607$	22.3607i	0.907592i	−0.891106	+	0.453796i	$0.850069\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.7771	−1.44739
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	0	0
$619$	17.8885	0.719001	0.359501	−	0.933145i	$-0.382947\pi$
$619$	17.8885	0.719001	0.359501	+	0.933145i	$0.382947\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 17.8885i	− 0.716689i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	−26.8328	−1.06820	−0.534099	−	0.845422i	$-0.679349\pi$
$631$	−26.8328	−1.06820	−0.534099	+	0.845422i	$0.679349\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	52.0000i	2.06032i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	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
$647$	8.94427i	0.351636i	0.984423	+	0.175818i	$0.0562570\pi$
$647$	8.94427i	0.351636i	−0.984423	+	0.175818i	$0.943743\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.4164	0.522629	0.261315	−	0.965254i	$-0.415844\pi$
$659$	13.4164	0.522629	0.261315	+	0.965254i	$0.415844\pi$
$660$	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
$667$	53.6656i	2.07794i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−44.7214	−1.72645
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.94427i	0.342243i	0.985250	+	0.171122i	$0.0547391\pi$
$683$	8.94427i	0.342243i	−0.985250	+	0.171122i	$0.945261\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−17.8885	−0.680512	−0.340256	−	0.940333i	$-0.610514\pi$
$691$	−17.8885	−0.680512	−0.340256	+	0.940333i	$0.610514\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	44.7214i	1.68192i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 80.0000i	− 2.99602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.8885	−0.667130	−0.333565	−	0.942727i	$-0.608252\pi$
$719$	−17.8885	−0.667130	−0.333565	+	0.942727i	$0.608252\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.4164i	− 0.497587i	−0.968557	−	0.248794i	$-0.919966\pi$
$727$	− 13.4164i	− 0.497587i	0.968557	−	0.248794i	$-0.0800341\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 16.0000i	− 0.590973i	−0.955347	−	0.295487i	$-0.904518\pi$
$733$	− 16.0000i	− 0.590973i	0.955347	−	0.295487i	$-0.0954818\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.0000i	− 1.47342i
$738$	0	0
$739$	17.8885	0.658041	0.329020	−	0.944323i	$-0.393282\pi$
$739$	17.8885	0.658041	0.329020	+	0.944323i	$0.393282\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.94427i	0.328134i	0.986449	+	0.164067i	$0.0524612\pi$
$743$	8.94427i	0.328134i	−0.986449	+	0.164067i	$0.947539\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	− 26.8328i	− 0.971413i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 17.8885i	− 0.645918i
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.8328i	0.956487i	0.878227	+	0.478243i	$0.158726\pi$
$787$	26.8328i	0.956487i	−0.878227	+	0.478243i	$0.841274\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	62.6099	2.22615
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	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
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	17.8885	0.628152	0.314076	−	0.949398i	$-0.398305\pi$
$811$	17.8885	0.628152	0.314076	+	0.949398i	$0.398305\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	− 40.2492i	− 1.40300i	−0.712670	−	0.701500i	$-0.752514\pi$
$823$	− 40.2492i	− 1.40300i	0.712670	−	0.701500i	$-0.247486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17.8885i	0.622046i	0.950402	+	0.311023i	$0.100672\pi$
$827$	17.8885i	0.622046i	−0.950402	+	0.311023i	$0.899328\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.0000i	0.900847i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 40.2492i	− 1.38298i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	71.5542	2.45285
$852$	0	0
$853$	− 16.0000i	− 0.547830i	−0.961754	−	0.273915i	$-0.911681\pi$
$853$	− 16.0000i	− 0.547830i	0.961754	−	0.273915i	$-0.0883186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−53.6656	−1.83105	−0.915524	−	0.402264i	$-0.868224\pi$
$859$	−53.6656	−1.83105	−0.915524	+	0.402264i	$0.868224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.8328i	0.913400i	0.889621	+	0.456700i	$0.150969\pi$
$863$	26.8328i	0.913400i	−0.889621	+	0.456700i	$0.849031\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	35.7771	1.21226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	52.0000i	1.75592i	0.478738	+	0.877958i	$0.341094\pi$
$877$	52.0000i	1.75592i	−0.478738	+	0.877958i	$0.658906\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	8.94427i	0.300999i	0.988610	+	0.150499i	$0.0480881\pi$
$883$	8.94427i	0.300999i	−0.988610	+	0.150499i	$0.951912\pi$
$884$	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
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−53.6656	−1.78985
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 53.6656i	− 1.78194i	−0.454064	−	0.890969i	$-0.650026\pi$
$907$	− 53.6656i	− 1.78194i	0.454064	−	0.890969i	$-0.349974\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17.8885	0.592674	0.296337	−	0.955083i	$-0.404235\pi$
$911$	17.8885	0.592674	0.296337	+	0.955083i	$0.404235\pi$
$912$	0	0
$913$	40.0000i	1.32381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.0000i	− 0.660458i
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.7771i	1.17762i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	71.5542i	2.33012i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	80.4984	2.59943
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 40.2492i	− 1.29433i	−0.762351	−	0.647164i	$-0.775955\pi$
$967$	− 40.2492i	− 1.29433i	0.762351	−	0.647164i	$-0.224045\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.3050	−1.00462	−0.502312	−	0.864687i	$-0.667517\pi$
$971$	−31.3050	−1.00462	−0.502312	+	0.864687i	$0.667517\pi$
$972$	0	0
$973$	80.0000i	2.56468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	−17.8885	−0.571720
$980$	0	0
$981$	0	0
$982$	0	0
$983$	44.7214i	1.42639i	0.700966	+	0.713195i	$0.252753\pi$
$983$	44.7214i	1.42639i	−0.700966	+	0.713195i	$0.747247\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	53.6656	1.70474	0.852372	−	0.522935i	$-0.175163\pi$
$991$	53.6656	1.70474	0.852372	+	0.522935i	$0.175163\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 28.0000i	− 0.886769i	−0.896332	−	0.443384i	$-0.853778\pi$
$997$	− 28.0000i	− 0.886769i	0.896332	−	0.443384i	$-0.146222\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.f.be.6049.1		4
3.2	odd	2		7200.2.f.bj.6049.2		4
4.3	odd	2	inner	7200.2.f.be.6049.4		4
5.2	odd	4		7200.2.a.ch.1.2		2
5.3	odd	4		1440.2.a.p.1.1	✓	2
5.4	even	2	inner	7200.2.f.be.6049.3		4
12.11	even	2		7200.2.f.bj.6049.3		4
15.2	even	4		7200.2.a.cg.1.2		2
15.8	even	4		1440.2.a.q.1.1	yes	2
15.14	odd	2		7200.2.f.bj.6049.4		4
20.3	even	4		1440.2.a.p.1.2	yes	2
20.7	even	4		7200.2.a.ch.1.1		2
20.19	odd	2	inner	7200.2.f.be.6049.2		4
40.3	even	4		2880.2.a.bj.1.2		2
40.13	odd	4		2880.2.a.bj.1.1		2
60.23	odd	4		1440.2.a.q.1.2	yes	2
60.47	odd	4		7200.2.a.cg.1.1		2
60.59	even	2		7200.2.f.bj.6049.1		4
120.53	even	4		2880.2.a.bi.1.1		2
120.83	odd	4		2880.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.a.p.1.1	✓	2	5.3	odd	4
1440.2.a.p.1.2	yes	2	20.3	even	4
1440.2.a.q.1.1	yes	2	15.8	even	4
1440.2.a.q.1.2	yes	2	60.23	odd	4
2880.2.a.bi.1.1		2	120.53	even	4
2880.2.a.bi.1.2		2	120.83	odd	4
2880.2.a.bj.1.1		2	40.13	odd	4
2880.2.a.bj.1.2		2	40.3	even	4
7200.2.a.cg.1.1		2	60.47	odd	4
7200.2.a.cg.1.2		2	15.2	even	4
7200.2.a.ch.1.1		2	20.7	even	4
7200.2.a.ch.1.2		2	5.2	odd	4
7200.2.f.be.6049.1		4	1.1	even	1	trivial
7200.2.f.be.6049.2		4	20.19	odd	2	inner
7200.2.f.be.6049.3		4	5.4	even	2	inner
7200.2.f.be.6049.4		4	4.3	odd	2	inner
7200.2.f.bj.6049.1		4	60.59	even	2
7200.2.f.bj.6049.2		4	3.2	odd	2
7200.2.f.bj.6049.3		4	12.11	even	2
7200.2.f.bj.6049.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.47214i q^{7} -4.47214 q^{11} -4.00000i q^{13} -2.00000i q^{17} -8.94427i q^{23} -6.00000 q^{29} +8.94427 q^{31} +8.00000i q^{37} -8.00000 q^{41} -8.94427i q^{47} -13.0000 q^{49} -6.00000i q^{53} +4.47214 q^{59} +10.0000 q^{61} +8.94427i q^{67} -8.94427 q^{71} -6.00000i q^{73} +20.0000i q^{77} +8.94427 q^{79} -8.94427i q^{83} +4.00000 q^{89} -17.8885 q^{91} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{29} - 32 q^{41} - 52 q^{49} + 40 q^{61} + 16 q^{89}+O(q^{100}) \) 4 * q - 24 * q^29 - 32 * q^41 - 52 * q^49 + 40 * q^61 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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