LMFDB - Embedded newform 1863.2.c.a.1862.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1863,2,Mod(1862,1863)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1863, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1863.1862");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1863 = 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1863.c (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:	$14.8761298966$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.57352136505929721.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{9} + x^{6} - 24x^{3} + 64 $ x^12 - 3x^9 + x^6 - 24x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 207)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1862.5
Root			$0.229129 - 1.39553i$ of defining polynomial
Character	$\chi$	$=$	1863.1862
Dual form			1863.2.c.a.1862.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.998666i q^{2} +1.00267 q^{4} -2.99866i q^{8} +O(q^{10})$	$q-0.998666i q^{2} +1.00267 q^{4} -2.99866i q^{8} -7.15143 q^{13} -0.989326 q^{16} -4.79583i q^{23} -5.00000 q^{25} +7.14189i q^{26} -5.62946i q^{29} -10.3540 q^{31} -5.00931i q^{32} -6.02711i q^{41} -4.78943 q^{46} +10.7192i q^{47} +7.00000 q^{49} +4.99333i q^{50} -7.17050 q^{52} -5.62195 q^{58} -5.59647i q^{59} +10.3402i q^{62} -6.98128 q^{64} -16.3838i q^{71} +12.3860 q^{73} -6.01907 q^{82} -4.80862i q^{92} +10.7049 q^{94} -6.99066i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 24 q^{4}+O(q^{10}) $ 12 * q - 24 * q^4	$ 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} + 84 q^{49} + 6 q^{52} - 30 q^{58} - 54 q^{64} - 66 q^{82} + 78 q^{94}+O(q^{100}) $ 12 * q - 24 * q^4 + 48 * q^16 - 60 * q^25 + 84 * q^49 + 6 * q^52 - 30 * q^58 - 54 * q^64 - 66 * q^82 + 78 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1863\mathbb{Z}\right)^\times$.

$n$	$649$	$1703$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 0.998666i	− 0.706163i	−0.935593	−	0.353082i	$-0.885134\pi$
$2$	− 0.998666i	− 0.706163i	0.935593	−	0.353082i	$-0.114866\pi$
$3$	0	0
$3$	0	0
$4$	1.00267	0.501333
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.99866i	− 1.06019i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−7.15143	−1.98345	−0.991725	−	0.128379i	$-0.959023\pi$
$13$	−7.15143	−1.98345	−0.991725	+	0.128379i	$0.959023\pi$
$14$	0	0
$15$	0	0
$16$	−0.989326	−0.247332
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.79583i	− 1.00000i
$23$	− 4.79583i	− 1.00000i
$24$	0	0
$25$	−5.00000	−1.00000
$26$	7.14189i	1.40064i
$27$	0	0
$28$	0	0
$29$	− 5.62946i	− 1.04536i	−0.852527	−	0.522682i	$-0.824931\pi$
$29$	− 5.62946i	− 1.04536i	0.852527	−	0.522682i	$-0.175069\pi$
$30$	0	0
$31$	−10.3540	−1.85963	−0.929816	−	0.368025i	$-0.880034\pi$
$31$	−10.3540	−1.85963	−0.929816	+	0.368025i	$0.880034\pi$
$32$	− 5.00931i	− 0.885530i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.02711i	− 0.941277i	−0.882326	−	0.470638i	$-0.844024\pi$
$41$	− 6.02711i	− 0.941277i	0.882326	−	0.470638i	$-0.155976\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−4.78943	−0.706163
$47$	10.7192i	1.56356i	0.623554	+	0.781780i	$0.285688\pi$
$47$	10.7192i	1.56356i	−0.623554	+	0.781780i	$0.714312\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	4.99333i	0.706163i
$51$	0	0
$52$	−7.17050	−0.994370
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−5.62195	−0.738198
$59$	− 5.59647i	− 0.728599i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	− 5.59647i	− 0.728599i	0.931282	−	0.364299i	$-0.118692\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	10.3402i	1.31320i
$63$	0	0
$64$	−6.98128	−0.872660
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 16.3838i	− 1.94440i	−0.234151	−	0.972200i	$-0.575231\pi$
$71$	− 16.3838i	− 1.94440i	0.234151	−	0.972200i	$-0.424769\pi$
$72$	0	0
$73$	12.3860	1.44967	0.724836	−	0.688921i	$-0.241915\pi$
$73$	12.3860	1.44967	0.724836	+	0.688921i	$0.241915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	−6.01907	−0.664695
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	− 4.80862i	− 0.501333i
$93$	0	0
$94$	10.7049	1.10413
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 6.99066i	− 0.706163i
$99$	0	0
$100$	−5.01333	−0.501333
$101$	− 4.39551i	− 0.437370i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	− 4.39551i	− 0.437370i	0.975796	−	0.218685i	$-0.0701767\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	21.4447i	2.10283i
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	− 5.64447i	− 0.524076i
$117$	0	0
$118$	−5.58901	−0.514510
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	−10.3816	−0.932295
$125$	0	0
$126$	0	0
$127$	−19.1377	−1.69819	−0.849096	−	0.528239i	$-0.822853\pi$
$127$	−19.1377	−1.69819	−0.849096	+	0.528239i	$0.822853\pi$
$128$	− 3.04666i	− 0.269289i
$129$	0	0
$130$	0	0
$131$	5.12489i	0.447763i	0.974616	+	0.223882i	$0.0718729\pi$
$131$	5.12489i	0.447763i	−0.974616	+	0.223882i	$0.928127\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	7.12699	0.604503	0.302252	−	0.953228i	$-0.402262\pi$
$139$	7.12699	0.604503	0.302252	+	0.953228i	$0.402262\pi$
$140$	0	0
$141$	0	0
$142$	−16.3619	−1.37306
$143$	0	0
$144$	0	0
$145$	0	0
$146$	− 12.3695i	− 1.02371i
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−2.42743	−0.197542	−0.0987709	−	0.995110i	$-0.531491\pi$
$151$	−2.42743	−0.197542	−0.0987709	+	0.995110i	$0.531491\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.8754	−0.930156	−0.465078	−	0.885270i	$-0.653974\pi$
$163$	−11.8754	−0.930156	−0.465078	+	0.885270i	$0.653974\pi$
$164$	− 6.04318i	− 0.471893i
$165$	0	0
$166$	0	0
$167$	25.5804i	1.97947i	0.142901	+	0.989737i	$0.454357\pi$
$167$	25.5804i	1.97947i	−0.142901	+	0.989737i	$0.545643\pi$
$168$	0	0
$169$	38.1430	2.93408
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.1833i	1.45848i	0.684257	+	0.729241i	$0.260127\pi$
$173$	19.1833i	1.45848i	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 22.7735i	− 1.70217i	−0.525030	−	0.851084i	$-0.675946\pi$
$179$	− 22.7735i	− 1.70217i	0.525030	−	0.851084i	$-0.324054\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−14.3811	−1.06019
$185$	0	0
$186$	0	0
$187$	0	0
$188$	10.7478i	0.783865i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1.12165	−0.0807383	−0.0403692	−	0.999185i	$-0.512853\pi$
$193$	−1.12165	−0.0807383	−0.0403692	+	0.999185i	$0.512853\pi$
$194$	0	0
$195$	0	0
$196$	7.01867	0.501333
$197$	27.4656i	1.95684i	0.206623	+	0.978421i	$0.433752\pi$
$197$	27.4656i	1.95684i	−0.206623	+	0.978421i	$0.566248\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	14.9933i	1.06019i
$201$	0	0
$202$	−4.38965	−0.308854
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	7.07510	0.490570
$209$	0	0
$210$	0	0
$211$	26.9199	1.85324	0.926620	−	0.375999i	$-0.122700\pi$
$211$	26.9199	1.85324	0.926620	+	0.375999i	$0.122700\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.9199	1.40090	0.700449	−	0.713702i	$-0.252983\pi$
$223$	20.9199	1.40090	0.700449	+	0.713702i	$0.252983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−16.8808	−1.10828
$233$	28.3051i	1.85433i	0.374657	+	0.927163i	$0.377760\pi$
$233$	28.3051i	1.85433i	−0.374657	+	0.927163i	$0.622240\pi$
$234$	0	0
$235$	0	0
$236$	− 5.61140i	− 0.365271i
$237$	0	0
$238$	0	0
$239$	− 30.3024i	− 1.96010i	−0.198752	−	0.980050i	$-0.563689\pi$
$239$	− 30.3024i	− 1.96010i	0.198752	−	0.980050i	$-0.436311\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	10.9853i	0.706163i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	31.0481i	1.97156i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	19.1121i	1.19920i
$255$	0	0
$256$	−17.0052	−1.06282
$257$	− 27.1382i	− 1.69283i	−0.532522	−	0.846416i	$-0.678756\pi$
$257$	− 27.1382i	− 1.69283i	0.532522	−	0.846416i	$-0.321244\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	5.11805	0.316194
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 24.3104i	− 1.48223i	−0.671377	−	0.741116i	$-0.734297\pi$
$269$	− 24.3104i	− 1.48223i	0.671377	−	0.741116i	$-0.265703\pi$
$270$	0	0
$271$	32.9199	1.99974	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	32.9199	1.99974	0.999870	+	0.0161307i	$0.00513477\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−33.0940	−1.98843	−0.994213	−	0.107430i	$-0.965738\pi$
$277$	−33.0940	−1.98843	−0.994213	+	0.107430i	$0.965738\pi$
$278$	− 7.11748i	− 0.426878i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	− 16.4275i	− 0.974793i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	12.4190	0.726769
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	34.2971i	1.98345i
$300$	0	0
$301$	0	0
$302$	2.42419i	0.139497i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.9199	0.851522	0.425761	−	0.904836i	$-0.360006\pi$
$307$	14.9199	0.851522	0.425761	+	0.904836i	$0.360006\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.3131i	1.26526i	0.774455	+	0.632629i	$0.218024\pi$
$311$	22.3131i	1.26526i	−0.774455	+	0.632629i	$0.781976\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 35.5724i	− 1.99795i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	− 35.5724i	− 1.99795i	0.0453045	−	0.998973i	$-0.485574\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	35.7572	1.98345
$326$	11.8596i	0.656842i
$327$	0	0
$328$	−18.0733	−0.997929
$329$	0	0
$330$	0	0
$331$	6.28996	0.345728	0.172864	−	0.984946i	$-0.444698\pi$
$331$	6.28996	0.345728	0.172864	+	0.984946i	$0.444698\pi$
$332$	0	0
$333$	0	0
$334$	25.5463	1.39783
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 38.0921i	− 2.07194i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	19.1577	1.02993
$347$	− 9.59166i	− 0.514907i	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	− 9.59166i	− 0.514907i	0.966291	−	0.257454i	$-0.0828835\pi$
$348$	0	0
$349$	−37.1537	−1.98879	−0.994395	−	0.105732i	$-0.966282\pi$
$349$	−37.1537	−1.98879	−0.994395	+	0.105732i	$0.966282\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 36.2944i	− 1.93176i	−0.258998	−	0.965878i	$-0.583392\pi$
$353$	− 36.2944i	− 1.93176i	0.258998	−	0.965878i	$-0.416608\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−22.7431	−1.20201
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	4.74464i	0.247332i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	32.1433	1.65767
$377$	40.2587i	2.07343i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	1.12016i	0.0570144i
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	− 20.9906i	− 1.06019i
$393$	0	0
$394$	27.4289	1.38185
$395$	0	0
$396$	0	0
$397$	−10.8890	−0.546504	−0.273252	−	0.961942i	$-0.588099\pi$
$397$	−10.8890	−0.546504	−0.273252	+	0.961942i	$0.588099\pi$
$398$	0	0
$399$	0	0
$400$	4.94663	0.247332
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	74.0459	3.68849
$404$	− 4.40723i	− 0.219268i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.0300	1.43544	0.717720	−	0.696332i	$-0.245186\pi$
$409$	29.0300	1.43544	0.717720	+	0.696332i	$0.245186\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	35.8238i	1.75641i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 26.8840i	− 1.30869i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	31.0332	1.48113	0.740566	−	0.671984i	$-0.234557\pi$
$439$	31.0332	1.48113	0.740566	+	0.671984i	$0.234557\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.3184i	− 0.870334i	−0.900350	−	0.435167i	$-0.856689\pi$
$443$	− 18.3184i	− 0.870334i	0.900350	−	0.435167i	$-0.143311\pi$
$444$	0	0
$445$	0	0
$446$	− 20.8920i	− 0.989263i
$447$	0	0
$448$	0	0
$449$	− 38.3667i	− 1.81063i	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	− 38.3667i	− 1.81063i	0.424736	−	0.905317i	$-0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 39.5198i	− 1.84062i	−0.391189	−	0.920310i	$-0.627936\pi$
$461$	− 39.5198i	− 1.84062i	0.391189	−	0.920310i	$-0.372064\pi$
$462$	0	0
$463$	8.91987	0.414542	0.207271	−	0.978284i	$-0.433542\pi$
$463$	8.91987	0.414542	0.207271	+	0.978284i	$0.433542\pi$
$464$	5.56937i	0.258552i
$465$	0	0
$466$	28.2673	1.30946
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−16.7819	−0.772451
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−30.2620	−1.38415
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−11.0293	−0.501333
$485$	0	0
$486$	0	0
$487$	43.1590	1.95572	0.977860	−	0.209259i	$-0.0671053\pi$
$487$	43.1590	1.95572	0.977860	+	0.209259i	$0.0671053\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	44.2119i	1.99526i	0.0688370	+	0.997628i	$0.478071\pi$
$491$	44.2119i	1.99526i	−0.0688370	+	0.997628i	$0.521929\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	10.2435	0.459946
$497$	0	0
$498$	0	0
$499$	40.4812	1.81219	0.906093	−	0.423079i	$-0.139051\pi$
$499$	40.4812	1.81219	0.906093	+	0.423079i	$0.139051\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−19.1887	−0.851360
$509$	40.2891i	1.78578i	0.450274	+	0.892891i	$0.351326\pi$
$509$	40.2891i	1.78578i	−0.450274	+	0.892891i	$0.648674\pi$
$510$	0	0
$511$	0	0
$512$	10.8892i	0.481237i
$513$	0	0
$514$	−27.1019	−1.19542
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	5.13855i	0.224479i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	43.1025i	1.86698i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−24.2780	−1.04670
$539$	0	0
$540$	0	0
$541$	−16.4500	−0.707242	−0.353621	−	0.935389i	$-0.615050\pi$
$541$	−16.4500	−0.707242	−0.353621	+	0.935389i	$0.615050\pi$
$542$	− 32.8759i	− 1.41214i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.8943	0.722350	0.361175	−	0.932498i	$-0.382376\pi$
$547$	16.8943	0.722350	0.361175	+	0.932498i	$0.382376\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	33.0498i	1.40415i
$555$	0	0
$556$	7.14599	0.303058
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−49.1295	−2.06143
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	23.9792i	1.00000i
$576$	0	0
$577$	2.29657	0.0956073	0.0478037	−	0.998857i	$-0.484778\pi$
$577$	2.29657	0.0956073	0.0478037	+	0.998857i	$0.484778\pi$
$578$	16.9773i	0.706163i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	− 37.1414i	− 1.53692i
$585$	0	0
$586$	0	0
$587$	− 37.8925i	− 1.56399i	−0.623284	−	0.781996i	$-0.714202\pi$
$587$	− 37.8925i	− 1.56399i	0.623284	−	0.781996i	$-0.285798\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.79744i	− 0.279137i	−0.990212	−	0.139569i	$-0.955428\pi$
$593$	− 6.79744i	− 0.279137i	0.990212	−	0.139569i	$-0.0445716\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	34.2513	1.40064
$599$	− 36.7734i	− 1.50252i	−0.660006	−	0.751260i	$-0.729446\pi$
$599$	− 36.7734i	− 1.50252i	0.660006	−	0.751260i	$-0.270554\pi$
$600$	0	0
$601$	−16.5994	−0.677105	−0.338552	−	0.940948i	$-0.609937\pi$
$601$	−16.5994	−0.677105	−0.338552	+	0.940948i	$0.609937\pi$
$602$	0	0
$603$	0	0
$604$	−2.43391	−0.0990342
$605$	0	0
$606$	0	0
$607$	44.9199	1.82324	0.911621	−	0.411033i	$-0.134832\pi$
$607$	44.9199	1.82324	0.911621	+	0.411033i	$0.134832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 76.6579i	− 3.10124i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	− 14.9000i	− 0.601314i
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	22.2833	0.893479
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−35.5250	−1.41088
$635$	0	0
$636$	0	0
$637$	−50.0600	−1.98345
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.9017i	1.52938i	0.644397	+	0.764691i	$0.277108\pi$
$647$	38.9017i	1.52938i	−0.644397	+	0.764691i	$0.722892\pi$
$648$	0	0
$649$	0	0
$650$	− 35.7095i	− 1.40064i
$651$	0	0
$652$	−11.9071	−0.466318
$653$	− 3.35712i	− 0.131374i	−0.997840	−	0.0656872i	$-0.979076\pi$
$653$	− 3.35712i	− 0.131374i	0.997840	−	0.0656872i	$-0.0209240\pi$
$654$	0	0
$655$	0	0
$656$	5.96278i	0.232807i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 6.28157i	− 0.244140i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−26.9979	−1.04536
$668$	25.6487i	0.992376i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.25795	−0.164132	−0.0820659	−	0.996627i	$-0.526152\pi$
$673$	−4.25795	−0.164132	−0.0820659	+	0.996627i	$0.526152\pi$
$674$	0	0
$675$	0	0
$676$	38.2447	1.47095
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 20.1035i	− 0.769238i	−0.923075	−	0.384619i	$-0.874333\pi$
$683$	− 20.1035i	− 0.769238i	0.923075	−	0.384619i	$-0.125667\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.91987	0.111077	0.0555386	−	0.998457i	$-0.482312\pi$
$691$	2.91987	0.111077	0.0555386	+	0.998457i	$0.482312\pi$
$692$	19.2345i	0.731185i
$693$	0	0
$694$	−9.57887	−0.363609
$695$	0	0
$696$	0	0
$697$	0	0
$698$	37.1041i	1.40441i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−36.2460	−1.36414
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	49.6560i	1.85963i
$714$	0	0
$715$	0	0
$716$	− 22.8342i	− 0.853354i
$717$	0	0
$718$	0	0
$719$	− 3.19455i	− 0.119137i	−0.998224	−	0.0595683i	$-0.981028\pi$
$719$	− 3.19455i	− 0.119137i	0.998224	−	0.0595683i	$-0.0189724\pi$
$720$	0	0
$721$	0	0
$722$	− 18.9746i	− 0.706163i
$723$	0	0
$724$	0	0
$725$	28.1473i	1.04536i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−24.0238	−0.885530
$737$	0	0
$738$	0	0
$739$	−45.3360	−1.66771	−0.833856	−	0.551981i	$-0.813872\pi$
$739$	−45.3360	−1.66771	−0.833856	+	0.551981i	$0.813872\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 10.6048i	− 0.386718i
$753$	0	0
$754$	40.2050	1.46418
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 12.3264i	− 0.446832i	−0.974723	−	0.223416i	$-0.928279\pi$
$761$	− 12.3264i	− 0.446832i	0.974723	−	0.223416i	$-0.0717209\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.0228i	1.44514i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−1.12464	−0.0404768
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	51.7700	1.85963
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.92528	−0.247332
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	27.5388i	0.981030i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	10.8745i	0.385921i
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	25.0466i	0.885530i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	− 73.9471i	− 2.60467i
$807$	0	0
$808$	−13.1806	−0.463693
$809$	55.5564i	1.95326i	0.214930	+	0.976629i	$0.431048\pi$
$809$	55.5564i	1.95326i	−0.214930	+	0.976629i	$0.568952\pi$
$810$	0	0
$811$	−55.8340	−1.96060	−0.980298	−	0.197523i	$-0.936710\pi$
$811$	−55.8340	−1.96060	−0.980298	+	0.197523i	$0.936710\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 28.9912i	− 1.01365i
$819$	0	0
$820$	0	0
$821$	19.1833i	0.669503i	0.942306	+	0.334751i	$0.108652\pi$
$821$	19.1833i	0.669503i	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	−54.7840	−1.90965	−0.954826	−	0.297167i	$-0.903958\pi$
$823$	−54.7840	−1.90965	−0.954826	+	0.297167i	$0.903958\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−50.8397	−1.76574	−0.882869	−	0.469620i	$-0.844391\pi$
$829$	−50.8397	−1.76574	−0.882869	+	0.469620i	$0.844391\pi$
$830$	0	0
$831$	0	0
$832$	49.9262	1.73088
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−2.69084	−0.0927875
$842$	0	0
$843$	0	0
$844$	26.9917	0.929091
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−44.8397	−1.53528	−0.767642	−	0.640879i	$-0.778570\pi$
$853$	−44.8397	−1.53528	−0.767642	+	0.640879i	$0.778570\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 36.8498i	− 1.25877i	−0.777095	−	0.629383i	$-0.783308\pi$
$857$	− 36.8498i	− 1.25877i	0.777095	−	0.629383i	$-0.216692\pi$
$858$	0	0
$859$	−55.1697	−1.88236	−0.941182	−	0.337900i	$-0.890284\pi$
$859$	−55.1697	−1.88236	−0.941182	+	0.337900i	$0.890284\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.2811i	1.57543i	0.616043	+	0.787713i	$0.288735\pi$
$863$	46.2811i	1.57543i	−0.616043	+	0.787713i	$0.711265\pi$
$864$	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
$877$	−56.8397	−1.91934	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−56.8397	−1.91934	−0.959671	+	0.281126i	$0.909292\pi$
$878$	− 30.9918i	− 1.04592i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	50.9199	1.71359	0.856795	−	0.515657i	$-0.172452\pi$
$883$	50.9199	1.71359	0.856795	+	0.515657i	$0.172452\pi$
$884$	0	0
$885$	0	0
$886$	−18.2940	−0.614598
$887$	− 56.2661i	− 1.88923i	−0.328178	−	0.944616i	$-0.606434\pi$
$887$	− 56.2661i	− 1.88923i	0.328178	−	0.944616i	$-0.393566\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	20.9757	0.702317
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−38.3155	−1.27860
$899$	58.2874i	1.94399i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−39.4671	−1.29978
$923$	117.168i	3.85662i
$924$	0	0
$925$	0	0
$926$	− 8.90797i	− 0.292734i
$927$	0	0
$928$	−28.1997	−0.925702
$929$	60.9583i	1.99998i	0.00493945	+	0.999988i	$0.498428\pi$
$929$	60.9583i	1.99998i	−0.00493945	+	0.999988i	$0.501572\pi$
$930$	0	0
$931$	0	0
$932$	28.3806i	0.929636i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−28.9050	−0.941277
$944$	5.53674i	0.180205i
$945$	0	0
$946$	0	0
$947$	17.3930i	0.565195i	0.959239	+	0.282598i	$0.0911961\pi$
$947$	17.3930i	0.565195i	−0.959239	+	0.282598i	$0.908804\pi$
$948$	0	0
$949$	−88.5777	−2.87535
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	− 30.3832i	− 0.982663i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	76.2052	2.45823
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.4820	0.594343	0.297171	−	0.954824i	$-0.403957\pi$
$967$	18.4820	0.594343	0.297171	+	0.954824i	$0.403957\pi$
$968$	32.9853i	1.06019i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	− 43.1014i	− 1.38106i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	44.1529	1.40898
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−3.08013	−0.0978435	−0.0489218	−	0.998803i	$-0.515578\pi$
$991$	−3.08013	−0.0978435	−0.0489218	+	0.998803i	$0.515578\pi$
$992$	51.8664i	1.64676i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−62.8397	−1.99015	−0.995077	−	0.0991016i	$-0.968403\pi$
$997$	−62.8397	−1.99015	−0.995077	+	0.0991016i	$0.968403\pi$
$998$	− 40.4272i	− 1.27970i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1863.2.c.a.1862.5		12
3.2	odd	2	inner	1863.2.c.a.1862.8		12
9.2	odd	6		207.2.g.a.68.4	✓	12
9.4	even	3		207.2.g.a.137.4	yes	12
9.5	odd	6		621.2.g.a.413.3		12
9.7	even	3		621.2.g.a.206.3		12
23.22	odd	2	CM	1863.2.c.a.1862.5		12
69.68	even	2	inner	1863.2.c.a.1862.8		12
207.22	odd	6		207.2.g.a.137.4	yes	12
207.68	even	6		621.2.g.a.413.3		12
207.137	even	6		207.2.g.a.68.4	✓	12
207.160	odd	6		621.2.g.a.206.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.g.a.68.4	✓	12	9.2	odd	6
207.2.g.a.68.4	✓	12	207.137	even	6
207.2.g.a.137.4	yes	12	9.4	even	3
207.2.g.a.137.4	yes	12	207.22	odd	6
621.2.g.a.206.3		12	9.7	even	3
621.2.g.a.206.3		12	207.160	odd	6
621.2.g.a.413.3		12	9.5	odd	6
621.2.g.a.413.3		12	207.68	even	6
1863.2.c.a.1862.5		12	1.1	even	1	trivial
1863.2.c.a.1862.5		12	23.22	odd	2	CM
1863.2.c.a.1862.8		12	3.2	odd	2	inner
1863.2.c.a.1862.8		12	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 1863 = 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1863.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.998666i q^{2} +1.00267 q^{4} -2.99866i q^{8} +O(q^{10})\)	\(q-0.998666i q^{2} +1.00267 q^{4} -2.99866i q^{8} -7.15143 q^{13} -0.989326 q^{16} -4.79583i q^{23} -5.00000 q^{25} +7.14189i q^{26} -5.62946i q^{29} -10.3540 q^{31} -5.00931i q^{32} -6.02711i q^{41} -4.78943 q^{46} +10.7192i q^{47} +7.00000 q^{49} +4.99333i q^{50} -7.17050 q^{52} -5.62195 q^{58} -5.59647i q^{59} +10.3402i q^{62} -6.98128 q^{64} -16.3838i q^{71} +12.3860 q^{73} -6.01907 q^{82} -4.80862i q^{92} +10.7049 q^{94} -6.99066i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{4}+O(q^{10}) \) 12 * q - 24 * q^4	\( 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} + 84 q^{49} + 6 q^{52} - 30 q^{58} - 54 q^{64} - 66 q^{82} + 78 q^{94}+O(q^{100}) \) 12 * q - 24 * q^4 + 48 * q^16 - 60 * q^25 + 84 * q^49 + 6 * q^52 - 30 * q^58 - 54 * q^64 - 66 * q^82 + 78 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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