LMFDB - Embedded newform 1872.2.d.d.287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(287,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1872.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.d (of order $2$, degree $1$, 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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 $ x^8 + 4x^6 + 7x^4 + 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.4
Root			$-1.40294 + 1.01575i$ of defining polynomial
Character	$\chi$	$=$	1872.287
Dual form			1872.2.d.d.287.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} +3.96812i q^{7} -3.16228 q^{11} +1.00000 q^{13} -5.47723i q^{17} -4.97615i q^{19} +5.61177 q^{23} +3.00000 q^{25} -4.24264i q^{29} +7.43222i q^{31} +5.61177 q^{35} +9.74597 q^{37} +12.3687i q^{41} +3.46410i q^{43} +6.63568 q^{47} -8.74597 q^{49} +2.64880i q^{53} +4.47214i q^{55} +7.34847 q^{59} -7.74597 q^{61} -1.41421i q^{65} +8.44025i q^{67} +13.6730 q^{71} +5.74597 q^{73} -12.5483i q^{77} -1.00803i q^{79} +7.34847 q^{83} -7.74597 q^{85} -12.3687i q^{89} +3.96812i q^{91} -7.03734 q^{95} +13.7460 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} + 24 q^{25} + 16 q^{37} - 8 q^{49} - 16 q^{73} + 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 + 24 * q^25 + 16 * q^37 - 8 * q^49 - 16 * q^73 + 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	3.96812i	1.49981i	0.661547	+	0.749904i	$0.269900\pi$
$7$	3.96812i	1.49981i	−0.661547	+	0.749904i	$0.730100\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.16228	−0.953463	−0.476731	−	0.879049i	$-0.658179\pi$
$11$	−3.16228	−0.953463	−0.476731	+	0.879049i	$0.658179\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.47723i	− 1.32842i	−0.747545	−	0.664211i	$-0.768768\pi$
$17$	− 5.47723i	− 1.32842i	0.747545	−	0.664211i	$-0.231232\pi$
$18$	0	0
$19$	− 4.97615i	− 1.14161i	−0.821086	−	0.570804i	$-0.806632\pi$
$19$	− 4.97615i	− 1.14161i	0.821086	−	0.570804i	$-0.193368\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.61177	1.17013	0.585067	−	0.810985i	$-0.301068\pi$
$23$	5.61177	1.17013	0.585067	+	0.810985i	$0.301068\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.24264i	− 0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	− 4.24264i	− 0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	0	0
$31$	7.43222i	1.33487i	0.744670	+	0.667433i	$0.232607\pi$
$31$	7.43222i	1.33487i	−0.744670	+	0.667433i	$0.767393\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.61177	0.948562
$36$	0	0
$37$	9.74597	1.60223	0.801114	−	0.598512i	$-0.204241\pi$
$37$	9.74597	1.60223	0.801114	+	0.598512i	$0.204241\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.3687i	1.93166i	0.259175	+	0.965830i	$0.416549\pi$
$41$	12.3687i	1.93166i	−0.259175	+	0.965830i	$0.583451\pi$
$42$	0	0
$43$	3.46410i	0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$	3.46410i	0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.63568	0.967914	0.483957	−	0.875092i	$-0.339199\pi$
$47$	6.63568	0.967914	0.483957	+	0.875092i	$0.339199\pi$
$48$	0	0
$49$	−8.74597	−1.24942
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.64880i	0.363840i	0.983313	+	0.181920i	$0.0582313\pi$
$53$	2.64880i	0.363840i	−0.983313	+	0.181920i	$0.941769\pi$
$54$	0	0
$55$	4.47214i	0.603023i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.34847	0.956689	0.478345	−	0.878172i	$-0.341237\pi$
$59$	7.34847	0.956689	0.478345	+	0.878172i	$0.341237\pi$
$60$	0	0
$61$	−7.74597	−0.991769	−0.495885	−	0.868388i	$-0.665156\pi$
$61$	−7.74597	−0.991769	−0.495885	+	0.868388i	$0.665156\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.41421i	− 0.175412i
$66$	0	0
$67$	8.44025i	1.03114i	0.856847	+	0.515571i	$0.172420\pi$
$67$	8.44025i	1.03114i	−0.856847	+	0.515571i	$0.827580\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6730	1.62269	0.811345	−	0.584568i	$-0.198736\pi$
$71$	13.6730	1.62269	0.811345	+	0.584568i	$0.198736\pi$
$72$	0	0
$73$	5.74597	0.672515	0.336257	−	0.941770i	$-0.390839\pi$
$73$	5.74597	0.672515	0.336257	+	0.941770i	$0.390839\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.5483i	− 1.43001i
$78$	0	0
$79$	− 1.00803i	− 0.113413i	−0.998391	−	0.0567064i	$-0.981940\pi$
$79$	− 1.00803i	− 0.113413i	0.998391	−	0.0567064i	$-0.0180599\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.34847	0.806599	0.403300	−	0.915068i	$-0.367863\pi$
$83$	7.34847	0.806599	0.403300	+	0.915068i	$0.367863\pi$
$84$	0	0
$85$	−7.74597	−0.840168
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.3687i	− 1.31108i	−0.755162	−	0.655538i	$-0.772442\pi$
$89$	− 12.3687i	− 1.31108i	0.755162	−	0.655538i	$-0.227558\pi$
$90$	0	0
$91$	3.96812i	0.415972i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.03734	−0.722016
$96$	0	0
$97$	13.7460	1.39569	0.697846	−	0.716248i	$-0.254142\pi$
$97$	13.7460	1.39569	0.697846	+	0.716248i	$0.254142\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.7909i	− 1.67076i	−0.549673	−	0.835380i	$-0.685248\pi$
$101$	− 16.7909i	− 1.67076i	0.549673	−	0.835380i	$-0.314752\pi$
$102$	0	0
$103$	1.44803i	0.142679i	0.997452	+	0.0713395i	$0.0227274\pi$
$103$	1.44803i	0.142679i	−0.997452	+	0.0713395i	$0.977273\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.03734	−0.680326	−0.340163	−	0.940367i	$-0.610482\pi$
$107$	−7.03734	−0.680326	−0.340163	+	0.940367i	$0.610482\pi$
$108$	0	0
$109$	−3.74597	−0.358799	−0.179399	−	0.983776i	$-0.557415\pi$
$109$	−3.74597	−0.358799	−0.179399	+	0.983776i	$0.557415\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.7909i	1.57956i	0.613391	+	0.789779i	$0.289805\pi$
$113$	16.7909i	1.57956i	−0.613391	+	0.789779i	$0.710195\pi$
$114$	0	0
$115$	− 7.93624i	− 0.740058i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.7343	1.99238
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	− 14.8644i	− 1.31901i	−0.751702	−	0.659503i	$-0.770767\pi$
$127$	− 14.8644i	− 1.31901i	0.751702	−	0.659503i	$-0.229233\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.13836	0.186830	0.0934149	−	0.995627i	$-0.470222\pi$
$131$	2.13836	0.186830	0.0934149	+	0.995627i	$0.470222\pi$
$132$	0	0
$133$	19.7460	1.71219
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.24264i	− 0.362473i	−0.983440	−	0.181237i	$-0.941990\pi$
$137$	− 4.24264i	− 0.362473i	0.983440	−	0.181237i	$-0.0580100\pi$
$138$	0	0
$139$	19.3366i	1.64011i	0.572287	+	0.820054i	$0.306056\pi$
$139$	19.3366i	1.64011i	−0.572287	+	0.820054i	$0.693944\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.16228	−0.264443
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 9.89949i	− 0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$	− 9.89949i	− 0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$	0	0
$151$	7.43222i	0.604826i	0.953177	+	0.302413i	$0.0977921\pi$
$151$	7.43222i	0.604826i	−0.953177	+	0.302413i	$0.902208\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.5107	0.844244
$156$	0	0
$157$	15.4919	1.23639	0.618195	−	0.786024i	$-0.287864\pi$
$157$	15.4919	1.23639	0.618195	+	0.786024i	$0.287864\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.2682i	1.75498i
$162$	0	0
$163$	− 10.8963i	− 0.853466i	−0.904378	−	0.426733i	$-0.859664\pi$
$163$	− 10.8963i	− 0.853466i	0.904378	−	0.426733i	$-0.140336\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.8592	−1.38199	−0.690994	−	0.722861i	$-0.742827\pi$
$167$	−17.8592	−1.38199	−0.690994	+	0.722861i	$0.742827\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.64880i	0.201384i	0.994918	+	0.100692i	$0.0321057\pi$
$173$	2.64880i	0.201384i	−0.994918	+	0.100692i	$0.967894\pi$
$174$	0	0
$175$	11.9044i	0.899885i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.2609	−1.36488	−0.682441	−	0.730941i	$-0.739082\pi$
$179$	−18.2609	−1.36488	−0.682441	+	0.730941i	$0.739082\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 13.7829i	− 1.01334i
$186$	0	0
$187$	17.3205i	1.26660i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.4097	−1.11501	−0.557504	−	0.830174i	$-0.688241\pi$
$191$	−15.4097	−1.11501	−0.557504	+	0.830174i	$0.688241\pi$
$192$	0	0
$193$	−11.7460	−0.845493	−0.422747	−	0.906248i	$-0.638934\pi$
$193$	−11.7460	−0.845493	−0.422747	+	0.906248i	$0.638934\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.41421i	− 0.100759i	−0.998730	−	0.0503793i	$-0.983957\pi$
$197$	− 1.41421i	− 0.100759i	0.998730	−	0.0503793i	$-0.0160430\pi$
$198$	0	0
$199$	1.00803i	0.0714577i	0.999362	+	0.0357288i	$0.0113753\pi$
$199$	1.00803i	0.0714577i	−0.999362	+	0.0357288i	$0.988625\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.8353	1.18161
$204$	0	0
$205$	17.4919	1.22169
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.7360i	1.08848i
$210$	0	0
$211$	− 9.95231i	− 0.685145i	−0.939491	−	0.342573i	$-0.888702\pi$
$211$	− 9.95231i	− 0.685145i	0.939491	−	0.342573i	$-0.111298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.89898	0.334108
$216$	0	0
$217$	−29.4919	−2.00204
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.47723i	− 0.368438i
$222$	0	0
$223$	− 9.44829i	− 0.632704i	−0.948642	−	0.316352i	$-0.897542\pi$
$223$	− 9.44829i	− 0.632704i	0.948642	−	0.316352i	$-0.102458\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.58785	−0.304507	−0.152253	−	0.988342i	$-0.548653\pi$
$227$	−4.58785	−0.304507	−0.152253	+	0.988342i	$0.548653\pi$
$228$	0	0
$229$	−5.49193	−0.362917	−0.181459	−	0.983399i	$-0.558082\pi$
$229$	−5.49193	−0.362917	−0.181459	+	0.983399i	$0.558082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.41421i	0.0926482i	0.998926	+	0.0463241i	$0.0147507\pi$
$233$	1.41421i	0.0926482i	−0.998926	+	0.0463241i	$0.985249\pi$
$234$	0	0
$235$	− 9.38427i	− 0.612162i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.1996	0.659759	0.329879	−	0.944023i	$-0.392992\pi$
$239$	10.1996	0.659759	0.329879	+	0.944023i	$0.392992\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.3687i	0.790205i
$246$	0	0
$247$	− 4.97615i	− 0.316625i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2982	1.59681	0.798405	−	0.602121i	$-0.205678\pi$
$251$	25.2982	1.59681	0.798405	+	0.602121i	$0.205678\pi$
$252$	0	0
$253$	−17.7460	−1.11568
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.05496i	− 0.0658064i	−0.999459	−	0.0329032i	$-0.989525\pi$
$257$	− 1.05496i	− 0.0658064i	0.999459	−	0.0329032i	$-0.0104753\pi$
$258$	0	0
$259$	38.6732i	2.40303i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.85115	0.175810	0.0879048	−	0.996129i	$-0.471983\pi$
$263$	2.85115	0.175810	0.0879048	+	0.996129i	$0.471983\pi$
$264$	0	0
$265$	3.74597	0.230113
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.5108i	1.61639i	0.588914	+	0.808196i	$0.299556\pi$
$269$	26.5108i	1.61639i	−0.588914	+	0.808196i	$0.700444\pi$
$270$	0	0
$271$	− 15.8085i	− 0.960295i	−0.877188	−	0.480148i	$-0.840583\pi$
$271$	− 15.8085i	− 0.960295i	0.877188	−	0.480148i	$-0.159417\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.48683	−0.572078
$276$	0	0
$277$	6.25403	0.375768	0.187884	−	0.982191i	$-0.439837\pi$
$277$	6.25403	0.375768	0.187884	+	0.982191i	$0.439837\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.54024i	0.569123i	0.958658	+	0.284561i	$0.0918480\pi$
$281$	9.54024i	0.569123i	−0.958658	+	0.284561i	$0.908152\pi$
$282$	0	0
$283$	− 11.8403i	− 0.703835i	−0.936031	−	0.351918i	$-0.885530\pi$
$283$	− 11.8403i	− 0.703835i	0.936031	−	0.351918i	$-0.114470\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−49.0803	−2.89712
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.41421i	0.0826192i	0.999146	+	0.0413096i	$0.0131530\pi$
$293$	1.41421i	0.0826192i	−0.999146	+	0.0413096i	$0.986847\pi$
$294$	0	0
$295$	− 10.3923i	− 0.605063i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.61177	0.324537
$300$	0	0
$301$	−13.7460	−0.792304
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.9545i	0.627250i
$306$	0	0
$307$	− 23.3047i	− 1.33007i	−0.746813	−	0.665035i	$-0.768417\pi$
$307$	− 23.3047i	− 1.33007i	0.746813	−	0.665035i	$-0.231583\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.0110	−1.47495	−0.737475	−	0.675375i	$-0.763982\pi$
$311$	−26.0110	−1.47495	−0.737475	+	0.675375i	$0.763982\pi$
$312$	0	0
$313$	−23.4919	−1.32784	−0.663921	−	0.747802i	$-0.731109\pi$
$313$	−23.4919	−1.32784	−0.663921	+	0.747802i	$0.731109\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.89949i	− 0.556011i	−0.960579	−	0.278006i	$-0.910327\pi$
$317$	− 9.89949i	− 0.556011i	0.960579	−	0.278006i	$-0.0896734\pi$
$318$	0	0
$319$	13.4164i	0.751175i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−27.2555	−1.51654
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	0	0
$328$	0	0
$329$	26.3312i	1.45168i
$330$	0	0
$331$	8.44025i	0.463918i	0.972726	+	0.231959i	$0.0745136\pi$
$331$	8.44025i	0.463918i	−0.972726	+	0.231959i	$0.925486\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.9363	0.652151
$336$	0	0
$337$	27.2379	1.48374	0.741871	−	0.670542i	$-0.233938\pi$
$337$	27.2379	1.48374	0.741871	+	0.670542i	$0.233938\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 23.5027i	− 1.27274i
$342$	0	0
$343$	− 6.92820i	− 0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.94681	−0.372924	−0.186462	−	0.982462i	$-0.559702\pi$
$347$	−6.94681	−0.372924	−0.186462	+	0.982462i	$0.559702\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.35255i	− 0.338112i	−0.985606	−	0.169056i	$-0.945928\pi$
$353$	− 6.35255i	− 0.338112i	0.985606	−	0.169056i	$-0.0540719\pi$
$354$	0	0
$355$	− 19.3366i	− 1.02628i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.6572	1.45969	0.729845	−	0.683613i	$-0.239592\pi$
$359$	27.6572	1.45969	0.729845	+	0.683613i	$0.239592\pi$
$360$	0	0
$361$	−5.76210	−0.303268
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.12602i	− 0.425336i
$366$	0	0
$367$	− 20.3446i	− 1.06198i	−0.847378	−	0.530990i	$-0.821820\pi$
$367$	− 20.3446i	− 1.06198i	0.847378	−	0.530990i	$-0.178180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.5107	−0.545691
$372$	0	0
$373$	34.9839	1.81140	0.905698	−	0.423924i	$-0.139347\pi$
$373$	34.9839	1.81140	0.905698	+	0.423924i	$0.139347\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.24264i	− 0.218507i
$378$	0	0
$379$	− 18.8326i	− 0.967364i	−0.875244	−	0.483682i	$-0.839299\pi$
$379$	− 18.8326i	− 0.967364i	0.875244	−	0.483682i	$-0.160701\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.48683	−0.484755	−0.242377	−	0.970182i	$-0.577927\pi$
$383$	−9.48683	−0.484755	−0.242377	+	0.970182i	$0.577927\pi$
$384$	0	0
$385$	−17.7460	−0.904418
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.5563i	− 0.788738i	−0.918952	−	0.394369i	$-0.870963\pi$
$389$	− 15.5563i	− 0.788738i	0.918952	−	0.394369i	$-0.129037\pi$
$390$	0	0
$391$	− 30.7369i	− 1.55443i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.42558	−0.0717285
$396$	0	0
$397$	19.4919	0.978272	0.489136	−	0.872208i	$-0.337312\pi$
$397$	19.4919	0.978272	0.489136	+	0.872208i	$0.337312\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.8701i	1.34183i	0.741536	+	0.670913i	$0.234098\pi$
$401$	26.8701i	1.34183i	−0.741536	+	0.670913i	$0.765902\pi$
$402$	0	0
$403$	7.43222i	0.370225i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−30.8195	−1.52766
$408$	0	0
$409$	6.25403	0.309242	0.154621	−	0.987974i	$-0.450584\pi$
$409$	6.25403	0.309242	0.154621	+	0.987974i	$0.450584\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	29.1596i	1.43485i
$414$	0	0
$415$	− 10.3923i	− 0.510138i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.03734	−0.343797	−0.171898	−	0.985115i	$-0.554990\pi$
$419$	−7.03734	−0.343797	−0.171898	+	0.985115i	$0.554990\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 16.4317i	− 0.797053i
$426$	0	0
$427$	− 30.7369i	− 1.48746i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.21011	−0.250962	−0.125481	−	0.992096i	$-0.540047\pi$
$431$	−5.21011	−0.250962	−0.125481	+	0.992096i	$0.540047\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 27.9250i	− 1.33583i
$438$	0	0
$439$	3.46410i	0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$	3.46410i	0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.0056	−1.66317	−0.831584	−	0.555399i	$-0.812566\pi$
$443$	−35.0056	−1.66317	−0.831584	+	0.555399i	$0.812566\pi$
$444$	0	0
$445$	−17.4919	−0.829197
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.1971i	0.717195i	0.933492	+	0.358598i	$0.116745\pi$
$449$	15.1971i	0.717195i	−0.933492	+	0.358598i	$0.883255\pi$
$450$	0	0
$451$	− 39.1132i	− 1.84177i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.61177	0.263084
$456$	0	0
$457$	−1.74597	−0.0816729	−0.0408364	−	0.999166i	$-0.513002\pi$
$457$	−1.74597	−0.0816729	−0.0408364	+	0.999166i	$0.513002\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.3687i	− 0.576066i	−0.957620	−	0.288033i	$-0.906999\pi$
$461$	− 12.3687i	− 0.576066i	0.957620	−	0.288033i	$-0.0930013\pi$
$462$	0	0
$463$	34.7050i	1.61288i	0.591316	+	0.806440i	$0.298609\pi$
$463$	34.7050i	1.61288i	−0.591316	+	0.806440i	$0.701391\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.0962	1.62406	0.812029	−	0.583617i	$-0.198363\pi$
$467$	35.0962	1.62406	0.812029	+	0.583617i	$0.198363\pi$
$468$	0	0
$469$	−33.4919	−1.54651
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.9545i	− 0.503686i
$474$	0	0
$475$	− 14.9285i	− 0.684965i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.7209	−0.718304	−0.359152	−	0.933279i	$-0.616934\pi$
$479$	−15.7209	−0.718304	−0.359152	+	0.933279i	$0.616934\pi$
$480$	0	0
$481$	9.74597	0.444378
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 19.4397i	− 0.882713i
$486$	0	0
$487$	25.3208i	1.14739i	0.819068	+	0.573697i	$0.194491\pi$
$487$	25.3208i	1.14739i	−0.819068	+	0.573697i	$0.805509\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.0430	−1.89737	−0.948687	−	0.316217i	$-0.897587\pi$
$491$	−42.0430	−1.89737	−0.948687	+	0.316217i	$0.897587\pi$
$492$	0	0
$493$	−23.2379	−1.04658
$494$	0	0
$495$	0	0
$496$	0	0
$497$	54.2562i	2.43372i
$498$	0	0
$499$	13.3524i	0.597735i	0.954295	+	0.298868i	$0.0966089\pi$
$499$	13.3524i	0.597735i	−0.954295	+	0.298868i	$0.903391\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.7184	−1.59261	−0.796303	−	0.604898i	$-0.793214\pi$
$503$	−35.7184	−1.59261	−0.796303	+	0.604898i	$0.793214\pi$
$504$	0	0
$505$	−23.7460	−1.05668
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.88338i	0.172128i	0.996290	+	0.0860640i	$0.0274289\pi$
$509$	3.88338i	0.172128i	−0.996290	+	0.0860640i	$0.972571\pi$
$510$	0	0
$511$	22.8007i	1.00864i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.04783	0.0902381
$516$	0	0
$517$	−20.9839	−0.922869
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.1502i	0.751364i	0.926749	+	0.375682i	$0.122591\pi$
$521$	17.1502i	0.751364i	−0.926749	+	0.375682i	$0.877409\pi$
$522$	0	0
$523$	8.50427i	0.371866i	0.982562	+	0.185933i	$0.0595307\pi$
$523$	8.50427i	0.371866i	−0.982562	+	0.185933i	$0.940469\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.7079	1.77327
$528$	0	0
$529$	8.49193	0.369214
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.3687i	0.535746i
$534$	0	0
$535$	9.95231i	0.430276i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.6572	1.19128
$540$	0	0
$541$	9.49193	0.408090	0.204045	−	0.978962i	$-0.434591\pi$
$541$	9.49193	0.408090	0.204045	+	0.978962i	$0.434591\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.29760i	0.226924i
$546$	0	0
$547$	40.2492i	1.72093i	0.509507	+	0.860466i	$0.329828\pi$
$547$	40.2492i	1.72093i	−0.509507	+	0.860466i	$0.670172\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.1120	−0.899403
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.4009i	1.03390i	0.856016	+	0.516949i	$0.172932\pi$
$557$	24.4009i	1.03390i	−0.856016	+	0.516949i	$0.827068\pi$
$558$	0	0
$559$	3.46410i	0.146516i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.3834	−1.44909	−0.724544	−	0.689229i	$-0.757949\pi$
$563$	−34.3834	−1.44909	−0.724544	+	0.689229i	$0.757949\pi$
$564$	0	0
$565$	23.7460	0.999000
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.8084i	1.33348i	0.745292	+	0.666739i	$0.232310\pi$
$569$	31.8084i	1.33348i	−0.745292	+	0.666739i	$0.767690\pi$
$570$	0	0
$571$	2.89607i	0.121197i	0.998162	+	0.0605983i	$0.0193009\pi$
$571$	2.89607i	0.121197i	−0.998162	+	0.0605983i	$0.980699\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.8353	0.702081
$576$	0	0
$577$	−33.4919	−1.39429	−0.697144	−	0.716931i	$-0.745546\pi$
$577$	−33.4919	−1.39429	−0.697144	+	0.716931i	$0.745546\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	29.1596i	1.20974i
$582$	0	0
$583$	− 8.37624i	− 0.346908i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.8966	−1.02759	−0.513795	−	0.857913i	$-0.671761\pi$
$587$	−24.8966	−1.02759	−0.513795	+	0.857913i	$0.671761\pi$
$588$	0	0
$589$	36.9839	1.52389
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.1971i	− 0.624070i	−0.950071	−	0.312035i	$-0.898989\pi$
$593$	− 15.1971i	− 0.624070i	0.950071	−	0.312035i	$-0.101011\pi$
$594$	0	0
$595$	− 30.7369i	− 1.26009i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.70230	0.232990	0.116495	−	0.993191i	$-0.462834\pi$
$599$	5.70230	0.232990	0.116495	+	0.993191i	$0.462834\pi$
$600$	0	0
$601$	27.4919	1.12142	0.560710	−	0.828012i	$-0.310528\pi$
$601$	27.4919	1.12142	0.560710	+	0.828012i	$0.310528\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.41421i	0.0574960i
$606$	0	0
$607$	− 37.2251i	− 1.51092i	−0.655194	−	0.755461i	$-0.727413\pi$
$607$	− 37.2251i	− 1.51092i	0.655194	−	0.755461i	$-0.272587\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.63568	0.268451
$612$	0	0
$613$	5.74597	0.232077	0.116039	−	0.993245i	$-0.462980\pi$
$613$	5.74597	0.232077	0.116039	+	0.993245i	$0.462980\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.9800i	− 1.16669i	−0.812225	−	0.583345i	$-0.801744\pi$
$617$	− 28.9800i	− 1.16669i	0.812225	−	0.583345i	$-0.198256\pi$
$618$	0	0
$619$	26.2008i	1.05310i	0.850145	+	0.526549i	$0.176514\pi$
$619$	26.2008i	1.05310i	−0.850145	+	0.526549i	$0.823486\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	49.0803	1.96636
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 53.3809i	− 2.12843i
$630$	0	0
$631$	− 26.3288i	− 1.04813i	−0.851677	−	0.524066i	$-0.824414\pi$
$631$	− 26.3288i	− 1.04813i	0.851677	−	0.524066i	$-0.175586\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.0215	−0.834213
$636$	0	0
$637$	−8.74597	−0.346528
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.8526i	0.468149i	0.972219	+	0.234075i	$0.0752060\pi$
$641$	11.8526i	0.468149i	−0.972219	+	0.234075i	$0.924794\pi$
$642$	0	0
$643$	3.40008i	0.134086i	0.997750	+	0.0670431i	$0.0213565\pi$
$643$	3.40008i	0.134086i	−0.997750	+	0.0670431i	$0.978644\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0484	1.29926	0.649632	−	0.760248i	$-0.274923\pi$
$647$	33.0484	1.29926	0.649632	+	0.760248i	$0.274923\pi$
$648$	0	0
$649$	−23.2379	−0.912167
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.1060i	1.45207i	0.687658	+	0.726035i	$0.258639\pi$
$653$	37.1060i	1.45207i	−0.687658	+	0.726035i	$0.741361\pi$
$654$	0	0
$655$	− 3.02410i	− 0.118161i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−43.4686	−1.69329	−0.846647	−	0.532154i	$-0.821383\pi$
$659$	−43.4686	−1.69329	−0.846647	+	0.532154i	$0.821383\pi$
$660$	0	0
$661$	−22.2540	−0.865582	−0.432791	−	0.901494i	$-0.642471\pi$
$661$	−22.2540	−0.865582	−0.432791	+	0.901494i	$0.642471\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 27.9250i	− 1.08289i
$666$	0	0
$667$	− 23.8087i	− 0.921877i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.4949	0.945615
$672$	0	0
$673$	−33.2379	−1.28123	−0.640613	−	0.767864i	$-0.721320\pi$
$673$	−33.2379	−1.28123	−0.640613	+	0.767864i	$0.721320\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.7748i	0.414110i	0.978329	+	0.207055i	$0.0663879\pi$
$677$	10.7748i	0.414110i	−0.978329	+	0.207055i	$0.933612\pi$
$678$	0	0
$679$	54.5456i	2.09327i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.0564	−1.83883	−0.919414	−	0.393291i	$-0.871336\pi$
$683$	−48.0564	−1.83883	−0.919414	+	0.393291i	$0.871336\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.64880i	0.100911i
$690$	0	0
$691$	33.2570i	1.26516i	0.774497	+	0.632578i	$0.218003\pi$
$691$	33.2570i	1.26516i	−0.774497	+	0.632578i	$0.781997\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.3460	1.03729
$696$	0	0
$697$	67.7460	2.56606
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.89949i	0.373899i	0.982370	+	0.186949i	$0.0598600\pi$
$701$	9.89949i	0.373899i	−0.982370	+	0.186949i	$0.940140\pi$
$702$	0	0
$703$	− 48.4974i	− 1.82911i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	66.6284	2.50582
$708$	0	0
$709$	41.4919	1.55826	0.779131	−	0.626861i	$-0.215661\pi$
$709$	41.4919	1.55826	0.779131	+	0.626861i	$0.215661\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	41.7079i	1.56197i
$714$	0	0
$715$	4.47214i	0.167248i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.3565	0.833758	0.416879	−	0.908962i	$-0.363124\pi$
$719$	22.3565	0.833758	0.416879	+	0.908962i	$0.363124\pi$
$720$	0	0
$721$	−5.74597	−0.213991
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.7279i	− 0.472703i
$726$	0	0
$727$	18.7685i	0.696087i	0.937478	+	0.348043i	$0.113154\pi$
$727$	18.7685i	0.696087i	−0.937478	+	0.348043i	$0.886846\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.9737	0.701766
$732$	0	0
$733$	19.2379	0.710568	0.355284	−	0.934758i	$-0.384384\pi$
$733$	19.2379	0.710568	0.355284	+	0.934758i	$0.384384\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 26.6904i	− 0.983155i
$738$	0	0
$739$	− 19.2726i	− 0.708953i	−0.935065	−	0.354476i	$-0.884659\pi$
$739$	− 19.2726i	− 0.708953i	0.935065	−	0.354476i	$-0.115341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.5242	−0.606213	−0.303107	−	0.952957i	$-0.598024\pi$
$743$	−16.5242	−0.606213	−0.303107	+	0.952957i	$0.598024\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 27.9250i	− 1.02036i
$750$	0	0
$751$	34.2010i	1.24801i	0.781419	+	0.624006i	$0.214496\pi$
$751$	34.2010i	1.24801i	−0.781419	+	0.624006i	$0.785504\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.5107	0.382525
$756$	0	0
$757$	−39.4919	−1.43536	−0.717679	−	0.696374i	$-0.754796\pi$
$757$	−39.4919	−1.43536	−0.717679	+	0.696374i	$0.754796\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 46.3098i	− 1.67873i	−0.543569	−	0.839364i	$-0.682927\pi$
$761$	− 46.3098i	− 1.67873i	0.543569	−	0.839364i	$-0.317073\pi$
$762$	0	0
$763$	− 14.8644i	− 0.538129i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.34847	0.265338
$768$	0	0
$769$	39.7460	1.43328	0.716638	−	0.697445i	$-0.245680\pi$
$769$	39.7460	1.43328	0.716638	+	0.697445i	$0.245680\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 15.5563i	− 0.559523i	−0.960070	−	0.279761i	$-0.909745\pi$
$773$	− 15.5563i	− 0.559523i	0.960070	−	0.279761i	$-0.0902554\pi$
$774$	0	0
$775$	22.2967i	0.800920i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	61.5484	2.20520
$780$	0	0
$781$	−43.2379	−1.54717
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 21.9089i	− 0.781962i
$786$	0	0
$787$	10.4563i	0.372728i	0.982481	+	0.186364i	$0.0596703\pi$
$787$	10.4563i	0.372728i	−0.982481	+	0.186364i	$0.940330\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−66.6284	−2.36903
$792$	0	0
$793$	−7.74597	−0.275067
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 29.3392i	− 1.03925i	−0.854395	−	0.519624i	$-0.826072\pi$
$797$	− 29.3392i	− 1.03925i	0.854395	−	0.519624i	$-0.173928\pi$
$798$	0	0
$799$	− 36.3451i	− 1.28580i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.1703	−0.641217
$804$	0	0
$805$	31.4919	1.10994
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 19.9786i	− 0.702411i	−0.936298	−	0.351205i	$-0.885772\pi$
$809$	− 19.9786i	− 0.702411i	0.936298	−	0.351205i	$-0.114228\pi$
$810$	0	0
$811$	23.8727i	0.838285i	0.907920	+	0.419142i	$0.137669\pi$
$811$	23.8727i	0.838285i	−0.907920	+	0.419142i	$0.862331\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.4097	−0.539779
$816$	0	0
$817$	17.2379	0.603078
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 51.9666i	− 1.81365i	−0.421509	−	0.906824i	$-0.638499\pi$
$821$	− 51.9666i	− 1.81365i	0.421509	−	0.906824i	$-0.361501\pi$
$822$	0	0
$823$	− 25.2567i	− 0.880395i	−0.897901	−	0.440197i	$-0.854908\pi$
$823$	− 25.2567i	− 0.880395i	0.897901	−	0.440197i	$-0.145092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.9871	−0.868886	−0.434443	−	0.900699i	$-0.643055\pi$
$827$	−24.9871	−0.868886	−0.434443	+	0.900699i	$0.643055\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	47.9036i	1.65976i
$834$	0	0
$835$	25.2567i	0.874046i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.97072	0.275180	0.137590	−	0.990489i	$-0.456064\pi$
$839$	7.97072	0.275180	0.137590	+	0.990489i	$0.456064\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.41421i	− 0.0486504i
$846$	0	0
$847$	− 3.96812i	− 0.136346i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	54.6921	1.87482
$852$	0	0
$853$	21.7460	0.744568	0.372284	−	0.928119i	$-0.378575\pi$
$853$	21.7460	0.744568	0.372284	+	0.928119i	$0.378575\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 21.5725i	− 0.736901i	−0.929647	−	0.368451i	$-0.879888\pi$
$857$	− 21.5725i	− 0.736901i	0.929647	−	0.368451i	$-0.120112\pi$
$858$	0	0
$859$	− 40.6892i	− 1.38830i	−0.719831	−	0.694149i	$-0.755781\pi$
$859$	− 40.6892i	− 1.38830i	0.719831	−	0.694149i	$-0.244219\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.7636	−0.468517	−0.234259	−	0.972174i	$-0.575266\pi$
$863$	−13.7636	−0.468517	−0.234259	+	0.972174i	$0.575266\pi$
$864$	0	0
$865$	3.74597	0.127367
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.18768i	0.108135i
$870$	0	0
$871$	8.44025i	0.285987i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	44.8941	1.51770
$876$	0	0
$877$	−53.2379	−1.79772	−0.898858	−	0.438240i	$-0.855602\pi$
$877$	−53.2379	−1.79772	−0.898858	+	0.438240i	$0.855602\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.43033i	0.250334i	0.992136	+	0.125167i	$0.0399467\pi$
$881$	7.43033i	0.250334i	−0.992136	+	0.125167i	$0.960053\pi$
$882$	0	0
$883$	33.7610i	1.13615i	0.822977	+	0.568074i	$0.192311\pi$
$883$	33.7610i	1.13615i	−0.822977	+	0.568074i	$0.807689\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.41509	−0.215398	−0.107699	−	0.994184i	$-0.534348\pi$
$887$	−6.41509	−0.215398	−0.107699	+	0.994184i	$0.534348\pi$
$888$	0	0
$889$	58.9839	1.97826
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 33.0202i	− 1.10498i
$894$	0	0
$895$	25.8248i	0.863227i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.5322	1.05166
$900$	0	0
$901$	14.5081	0.483334
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.7990i	− 0.658141i
$906$	0	0
$907$	− 5.04017i	− 0.167356i	−0.996493	−	0.0836781i	$-0.973333\pi$
$907$	− 5.04017i	− 0.167356i	0.996493	−	0.0836781i	$-0.0266667\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−51.1282	−1.69395	−0.846976	−	0.531632i	$-0.821579\pi$
$911$	−51.1282	−1.69395	−0.846976	+	0.531632i	$0.821579\pi$
$912$	0	0
$913$	−23.2379	−0.769062
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.48528i	0.280209i
$918$	0	0
$919$	− 25.6967i	− 0.847657i	−0.905742	−	0.423829i	$-0.860686\pi$
$919$	− 25.6967i	− 0.847657i	0.905742	−	0.423829i	$-0.139314\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.6730	0.450053
$924$	0	0
$925$	29.2379	0.961336
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 21.5725i	− 0.707769i	−0.935289	−	0.353885i	$-0.884861\pi$
$929$	− 21.5725i	− 0.707769i	0.935289	−	0.353885i	$-0.115139\pi$
$930$	0	0
$931$	43.5213i	1.42635i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.4949	0.801069
$936$	0	0
$937$	24.9839	0.816187	0.408094	−	0.912940i	$-0.366194\pi$
$937$	24.9839	0.816187	0.408094	+	0.912940i	$0.366194\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 4.96116i	− 0.161729i	−0.996725	−	0.0808645i	$-0.974232\pi$
$941$	− 4.96116i	− 0.161729i	0.996725	−	0.0808645i	$-0.0257681\pi$
$942$	0	0
$943$	69.4101i	2.26030i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.6999	0.835134	0.417567	−	0.908646i	$-0.362883\pi$
$947$	25.6999	0.835134	0.417567	+	0.908646i	$0.362883\pi$
$948$	0	0
$949$	5.74597	0.186522
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.88338i	0.125795i	0.998020	+	0.0628976i	$0.0200341\pi$
$953$	3.88338i	0.125795i	−0.998020	+	0.0628976i	$0.979966\pi$
$954$	0	0
$955$	21.7926i	0.705193i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.8353	0.543640
$960$	0	0
$961$	−24.2379	−0.781868
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.6113i	0.534737i
$966$	0	0
$967$	− 11.4644i	− 0.368669i	−0.982864	−	0.184334i	$-0.940987\pi$
$967$	− 11.4644i	− 0.368669i	0.982864	−	0.184334i	$-0.0590130\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.1972	0.969074	0.484537	−	0.874771i	$-0.338988\pi$
$971$	30.1972	0.969074	0.484537	+	0.874771i	$0.338988\pi$
$972$	0	0
$973$	−76.7298	−2.45985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.7628i	1.36810i	0.729433	+	0.684052i	$0.239784\pi$
$977$	42.7628i	1.36810i	−0.729433	+	0.684052i	$0.760216\pi$
$978$	0	0
$979$	39.1132i	1.25006i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.6682	1.71175	0.855875	−	0.517183i	$-0.173019\pi$
$983$	53.6682	1.71175	0.855875	+	0.517183i	$0.173019\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.4397i	0.618148i
$990$	0	0
$991$	− 54.1056i	− 1.71872i	−0.511369	−	0.859361i	$-0.670862\pi$
$991$	− 54.1056i	− 1.71872i	0.511369	−	0.859361i	$-0.329138\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.42558	0.0451938
$996$	0	0
$997$	−44.9839	−1.42465	−0.712327	−	0.701848i	$-0.752359\pi$
$997$	−44.9839	−1.42465	−0.712327	+	0.701848i	$0.752359\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.d.d.287.4	yes	8
3.2	odd	2	inner	1872.2.d.d.287.8	yes	8
4.3	odd	2	inner	1872.2.d.d.287.1	✓	8
8.3	odd	2		7488.2.d.g.4031.5		8
8.5	even	2		7488.2.d.g.4031.8		8
12.11	even	2	inner	1872.2.d.d.287.5	yes	8
24.5	odd	2		7488.2.d.g.4031.4		8
24.11	even	2		7488.2.d.g.4031.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1872.2.d.d.287.1	✓	8	4.3	odd	2	inner
1872.2.d.d.287.4	yes	8	1.1	even	1	trivial
1872.2.d.d.287.5	yes	8	12.11	even	2	inner
1872.2.d.d.287.8	yes	8	3.2	odd	2	inner
7488.2.d.g.4031.1		8	24.11	even	2
7488.2.d.g.4031.4		8	24.5	odd	2
7488.2.d.g.4031.5		8	8.3	odd	2
7488.2.d.g.4031.8		8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} +3.96812i q^{7} -3.16228 q^{11} +1.00000 q^{13} -5.47723i q^{17} -4.97615i q^{19} +5.61177 q^{23} +3.00000 q^{25} -4.24264i q^{29} +7.43222i q^{31} +5.61177 q^{35} +9.74597 q^{37} +12.3687i q^{41} +3.46410i q^{43} +6.63568 q^{47} -8.74597 q^{49} +2.64880i q^{53} +4.47214i q^{55} +7.34847 q^{59} -7.74597 q^{61} -1.41421i q^{65} +8.44025i q^{67} +13.6730 q^{71} +5.74597 q^{73} -12.5483i q^{77} -1.00803i q^{79} +7.34847 q^{83} -7.74597 q^{85} -12.3687i q^{89} +3.96812i q^{91} -7.03734 q^{95} +13.7460 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} + 24 q^{25} + 16 q^{37} - 8 q^{49} - 16 q^{73} + 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 + 24 * q^25 + 16 * q^37 - 8 * q^49 - 16 * q^73 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more