LMFDB - Embedded newform 6084.2.b.a.4393.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6084,2,Mod(4393,6084)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6084, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6084.4393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6084.b (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:	$48.5809845897$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4393.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	6084.4393
Dual form			6084.2.b.a.4393.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{7} -6.00000 q^{17} -2.00000i q^{19} +5.00000 q^{25} +6.00000 q^{29} -2.00000i q^{31} +2.00000i q^{37} -12.0000i q^{41} +4.00000 q^{43} +3.00000 q^{49} -6.00000 q^{53} -12.0000i q^{59} +2.00000 q^{61} +10.0000i q^{67} +12.0000i q^{71} +14.0000i q^{73} +8.00000 q^{79} +12.0000i q^{83} +10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 12 q^{17} + 10 q^{25} + 12 q^{29} + 8 q^{43} + 6 q^{49} - 12 q^{53} + 4 q^{61} + 16 q^{79}+O(q^{100}) $ 2 * q - 12 * q^17 + 10 * q^25 + 12 * q^29 + 8 * q^43 + 6 * q^49 - 12 * q^53 + 4 * q^61 + 16 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6084\mathbb{Z}\right)^\times$.

$n$	$677$	$3043$	$3889$
$\chi(n)$	$1$	$1$	$-1$

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	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	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
$31$	− 2.00000i	− 0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$	− 2.00000i	− 0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.0000i	− 1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	− 12.0000i	− 1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 12.0000i	− 1.10004i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	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
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000i	0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$	12.0000i	0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$	0	0
$151$	− 10.0000i	− 0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$	− 10.0000i	− 0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	10.0000i	0.755929i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 12.0000i	− 0.776215i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$	− 12.0000i	− 0.776215i	0.921614	−	0.388108i	$-0.126871\pi$
$240$	0	0
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	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
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	26.0000i	1.57939i	0.613501	+	0.789694i	$0.289761\pi$
$271$	26.0000i	1.57939i	−0.613501	+	0.789694i	$0.710239\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.0000i	0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$	12.0000i	0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.0000i	0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$	10.0000i	0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	− 22.0000i	− 1.17763i	−0.808267	−	0.588817i	$-0.799594\pi$
$349$	− 22.0000i	− 1.17763i	0.808267	−	0.588817i	$-0.200406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 26.0000i	− 1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$	− 26.0000i	− 1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	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
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.0000i	− 1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$	− 24.0000i	− 1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000i	0.494468i	0.968956	+	0.247234i	$0.0795217\pi$
$409$	10.0000i	0.494468i	−0.968956	+	0.247234i	$0.920478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	24.0000	1.18096
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	34.0000i	1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$	34.0000i	1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 10.0000i	− 0.458831i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000i	0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$	12.0000i	0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	− 14.0000i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	− 14.0000i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 12.0000i	− 0.511217i
$552$	0	0
$553$	16.0000i	0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	0	0
$619$	− 22.0000i	− 0.884255i	−0.896952	−	0.442127i	$-0.854224\pi$
$619$	− 22.0000i	− 0.884255i	0.896952	−	0.442127i	$-0.145776\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	− 10.0000i	− 0.398094i	−0.979990	−	0.199047i	$-0.936215\pi$
$631$	− 10.0000i	− 0.398094i	0.979990	−	0.199047i	$-0.0637846\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 14.0000i	− 0.532585i	−0.963892	−	0.266293i	$-0.914201\pi$
$691$	− 14.0000i	− 0.532585i	0.963892	−	0.266293i	$-0.0857987\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	72.0000i	2.72719i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0000i	1.35392i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	32.0000i	1.19174i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	30.0000	1.11417
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.00000i	0.0735712i	0.999323	+	0.0367856i	$0.0117119\pi$
$739$	2.00000i	0.0735712i	−0.999323	+	0.0367856i	$0.988288\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.0000i	0.876941i
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0000i	0.435000i	0.976060	+	0.217500i	$0.0697902\pi$
$761$	12.0000i	0.435000i	−0.976060	+	0.217500i	$0.930210\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	22.0000i	0.793340i	0.917961	+	0.396670i	$0.129834\pi$
$769$	22.0000i	0.793340i	−0.917961	+	0.396670i	$0.870166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	0	0
$775$	− 10.0000i	− 0.359211i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.0000i	0.926800i	0.886149	+	0.463400i	$0.153371\pi$
$787$	26.0000i	0.926800i	−0.886149	+	0.463400i	$0.846629\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 12.0000i	− 0.426671i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	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
$811$	22.0000i	0.772524i	0.922389	+	0.386262i	$0.126234\pi$
$811$	22.0000i	0.772524i	−0.922389	+	0.386262i	$0.873766\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 12.0000i	− 0.418803i	−0.977830	−	0.209401i	$-0.932848\pi$
$821$	− 12.0000i	− 0.418803i	0.977830	−	0.209401i	$-0.0671515\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 36.0000i	− 1.24286i	−0.783470	−	0.621429i	$-0.786552\pi$
$839$	− 36.0000i	− 1.24286i	0.783470	−	0.621429i	$-0.213448\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	22.0000i	0.755929i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	8.00000i	0.268311i
$890$	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
$899$	− 12.0000i	− 0.400222i
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 24.0000i	− 0.792550i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 36.0000i	− 1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$	− 36.0000i	− 1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$	0	0
$931$	− 6.00000i	− 0.196642i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 12.0000i	− 0.391189i	−0.980685	−	0.195594i	$-0.937336\pi$
$941$	− 12.0000i	− 0.391189i	0.980685	−	0.195594i	$-0.0626636\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	27.0000	0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0000i	0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$	24.0000i	0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 60.0000i	− 1.91370i	−0.290578	−	0.956851i	$-0.593847\pi$
$983$	− 60.0000i	− 1.91370i	0.290578	−	0.956851i	$-0.406153\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.b.a.4393.2		2
3.2	odd	2		2028.2.b.d.337.2		2
13.5	odd	4		6084.2.a.h.1.1		1
13.8	odd	4		468.2.a.c.1.1		1
13.12	even	2	inner	6084.2.b.a.4393.1		2
39.2	even	12		2028.2.i.c.529.1		2
39.5	even	4		2028.2.a.e.1.1		1
39.8	even	4		156.2.a.b.1.1	✓	1
39.11	even	12		2028.2.i.b.529.1		2
39.17	odd	6		2028.2.q.d.361.2		4
39.20	even	12		2028.2.i.b.2005.1		2
39.23	odd	6		2028.2.q.d.1837.1		4
39.29	odd	6		2028.2.q.d.1837.2		4
39.32	even	12		2028.2.i.c.2005.1		2
39.35	odd	6		2028.2.q.d.361.1		4
39.38	odd	2		2028.2.b.d.337.1		2
52.47	even	4		1872.2.a.i.1.1		1
104.21	odd	4		7488.2.a.bf.1.1		1
104.99	even	4		7488.2.a.bb.1.1		1
117.34	odd	12		4212.2.i.g.1405.1		2
117.47	even	12		4212.2.i.f.1405.1		2
117.86	even	12		4212.2.i.f.2809.1		2
117.112	odd	12		4212.2.i.g.2809.1		2
156.47	odd	4		624.2.a.b.1.1		1
156.83	odd	4		8112.2.a.i.1.1		1
195.8	odd	4		3900.2.h.e.1249.2		2
195.47	odd	4		3900.2.h.e.1249.1		2
195.164	even	4		3900.2.a.a.1.1		1
273.125	odd	4		7644.2.a.a.1.1		1
312.125	even	4		2496.2.a.h.1.1		1
312.203	odd	4		2496.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.b.1.1	✓	1	39.8	even	4
468.2.a.c.1.1		1	13.8	odd	4
624.2.a.b.1.1		1	156.47	odd	4
1872.2.a.i.1.1		1	52.47	even	4
2028.2.a.e.1.1		1	39.5	even	4
2028.2.b.d.337.1		2	39.38	odd	2
2028.2.b.d.337.2		2	3.2	odd	2
2028.2.i.b.529.1		2	39.11	even	12
2028.2.i.b.2005.1		2	39.20	even	12
2028.2.i.c.529.1		2	39.2	even	12
2028.2.i.c.2005.1		2	39.32	even	12
2028.2.q.d.361.1		4	39.35	odd	6
2028.2.q.d.361.2		4	39.17	odd	6
2028.2.q.d.1837.1		4	39.23	odd	6
2028.2.q.d.1837.2		4	39.29	odd	6
2496.2.a.h.1.1		1	312.125	even	4
2496.2.a.v.1.1		1	312.203	odd	4
3900.2.a.a.1.1		1	195.164	even	4
3900.2.h.e.1249.1		2	195.47	odd	4
3900.2.h.e.1249.2		2	195.8	odd	4
4212.2.i.f.1405.1		2	117.47	even	12
4212.2.i.f.2809.1		2	117.86	even	12
4212.2.i.g.1405.1		2	117.34	odd	12
4212.2.i.g.2809.1		2	117.112	odd	12
6084.2.a.h.1.1		1	13.5	odd	4
6084.2.b.a.4393.1		2	13.12	even	2	inner
6084.2.b.a.4393.2		2	1.1	even	1	trivial
7488.2.a.bb.1.1		1	104.99	even	4
7488.2.a.bf.1.1		1	104.21	odd	4
7644.2.a.a.1.1		1	273.125	odd	4
8112.2.a.i.1.1		1	156.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{7} -6.00000 q^{17} -2.00000i q^{19} +5.00000 q^{25} +6.00000 q^{29} -2.00000i q^{31} +2.00000i q^{37} -12.0000i q^{41} +4.00000 q^{43} +3.00000 q^{49} -6.00000 q^{53} -12.0000i q^{59} +2.00000 q^{61} +10.0000i q^{67} +12.0000i q^{71} +14.0000i q^{73} +8.00000 q^{79} +12.0000i q^{83} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 12 q^{17} + 10 q^{25} + 12 q^{29} + 8 q^{43} + 6 q^{49} - 12 q^{53} + 4 q^{61} + 16 q^{79}+O(q^{100}) \) 2 * q - 12 * q^17 + 10 * q^25 + 12 * q^29 + 8 * q^43 + 6 * q^49 - 12 * q^53 + 4 * q^61 + 16 * q^79

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