LMFDB - Embedded newform 2376.2.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(1,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.9724555203$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	2376.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{7} -1.00000 q^{11} -4.64575 q^{17} -5.29150 q^{19} -8.29150 q^{23} -5.00000 q^{25} -6.64575 q^{29} -1.29150 q^{31} +2.29150 q^{37} +5.93725 q^{41} +4.64575 q^{43} +4.29150 q^{47} -4.00000 q^{53} -7.00000 q^{59} +10.5830 q^{61} +13.2915 q^{67} +6.58301 q^{71} -14.5830 q^{73} -2.64575 q^{77} -11.9373 q^{79} -8.00000 q^{83} -2.70850 q^{89} -13.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{11} - 4 q^{17} - 6 q^{23} - 10 q^{25} - 8 q^{29} + 8 q^{31} - 6 q^{37} - 4 q^{41} + 4 q^{43} - 2 q^{47} - 8 q^{53} - 14 q^{59} + 16 q^{67} - 8 q^{71} - 8 q^{73} - 8 q^{79} - 16 q^{83} - 16 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^11 - 4 * q^17 - 6 * q^23 - 10 * q^25 - 8 * q^29 + 8 * q^31 - 6 * q^37 - 4 * q^41 + 4 * q^43 - 2 * q^47 - 8 * q^53 - 14 * q^59 + 16 * q^67 - 8 * q^71 - 8 * q^73 - 8 * q^79 - 16 * q^83 - 16 * q^89 - 6 * q^97

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		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.64575	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$7$	2.64575	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.64575	−1.12676	−0.563380	−	0.826198i	$-0.690499\pi$
$17$	−4.64575	−1.12676	−0.563380	+	0.826198i	$0.690499\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.29150	−1.72890	−0.864449	−	0.502721i	$-0.832332\pi$
$23$	−8.29150	−1.72890	−0.864449	+	0.502721i	$0.832332\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.64575	−1.23409	−0.617043	−	0.786930i	$-0.711669\pi$
$29$	−6.64575	−1.23409	−0.617043	+	0.786930i	$0.711669\pi$
$30$	0	0
$31$	−1.29150	−0.231961	−0.115980	−	0.993252i	$-0.537001\pi$
$31$	−1.29150	−0.231961	−0.115980	+	0.993252i	$0.537001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.29150	0.376721	0.188360	−	0.982100i	$-0.439683\pi$
$37$	2.29150	0.376721	0.188360	+	0.982100i	$0.439683\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.93725	0.927243	0.463622	−	0.886033i	$-0.346550\pi$
$41$	5.93725	0.927243	0.463622	+	0.886033i	$0.346550\pi$
$42$	0	0
$43$	4.64575	0.708470	0.354235	−	0.935156i	$-0.384741\pi$
$43$	4.64575	0.708470	0.354235	+	0.935156i	$0.384741\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.29150	0.625980	0.312990	−	0.949756i	$-0.398669\pi$
$47$	4.29150	0.625980	0.312990	+	0.949756i	$0.398669\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	10.5830	1.35501	0.677507	−	0.735516i	$-0.263060\pi$
$61$	10.5830	1.35501	0.677507	+	0.735516i	$0.263060\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2915	1.62382	0.811908	−	0.583786i	$-0.198429\pi$
$67$	13.2915	1.62382	0.811908	+	0.583786i	$0.198429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.58301	0.781259	0.390629	−	0.920548i	$-0.372257\pi$
$71$	6.58301	0.781259	0.390629	+	0.920548i	$0.372257\pi$
$72$	0	0
$73$	−14.5830	−1.70681	−0.853406	−	0.521247i	$-0.825467\pi$
$73$	−14.5830	−1.70681	−0.853406	+	0.521247i	$0.825467\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.64575	−0.301511
$78$	0	0
$79$	−11.9373	−1.34305	−0.671523	−	0.740984i	$-0.734360\pi$
$79$	−11.9373	−1.34305	−0.671523	+	0.740984i	$0.734360\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.70850	−0.287100	−0.143550	−	0.989643i	$-0.545852\pi$
$89$	−2.70850	−0.287100	−0.143550	+	0.989643i	$0.545852\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.5830	−1.37915	−0.689573	−	0.724217i	$-0.742202\pi$
$97$	−13.5830	−1.37915	−0.689573	+	0.724217i	$0.742202\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.35425	−0.532768	−0.266384	−	0.963867i	$-0.585829\pi$
$101$	−5.35425	−0.532768	−0.266384	+	0.963867i	$0.585829\pi$
$102$	0	0
$103$	14.5830	1.43691	0.718453	−	0.695575i	$-0.244850\pi$
$103$	14.5830	1.43691	0.718453	+	0.695575i	$0.244850\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.5830	−1.02310	−0.511549	−	0.859254i	$-0.670928\pi$
$107$	−10.5830	−1.02310	−0.511549	+	0.859254i	$0.670928\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.2915	−1.25036	−0.625180	−	0.780481i	$-0.714974\pi$
$113$	−13.2915	−1.25036	−0.625180	+	0.780481i	$0.714974\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.2915	−1.12676
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.2915	1.17943	0.589715	−	0.807611i	$-0.299240\pi$
$127$	13.2915	1.17943	0.589715	+	0.807611i	$0.299240\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.58301	0.400419	0.200210	−	0.979753i	$-0.435838\pi$
$131$	4.58301	0.400419	0.200210	+	0.979753i	$0.435838\pi$
$132$	0	0
$133$	−14.0000	−1.21395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.58301	−0.562424	−0.281212	−	0.959646i	$-0.590736\pi$
$137$	−6.58301	−0.562424	−0.281212	+	0.959646i	$0.590736\pi$
$138$	0	0
$139$	15.3542	1.30233	0.651165	−	0.758936i	$-0.274281\pi$
$139$	15.3542	1.30233	0.651165	+	0.758936i	$0.274281\pi$
$140$	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
$149$	18.5830	1.52238	0.761190	−	0.648529i	$-0.224616\pi$
$149$	18.5830	1.52238	0.761190	+	0.648529i	$0.224616\pi$
$150$	0	0
$151$	−2.70850	−0.220414	−0.110207	−	0.993909i	$-0.535151\pi$
$151$	−2.70850	−0.220414	−0.110207	+	0.993909i	$0.535151\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.29150	−0.661734	−0.330867	−	0.943677i	$-0.607341\pi$
$157$	−8.29150	−0.661734	−0.330867	+	0.943677i	$0.607341\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−21.9373	−1.72890
$162$	0	0
$163$	17.2915	1.35438	0.677188	−	0.735810i	$-0.263199\pi$
$163$	17.2915	1.35438	0.677188	+	0.735810i	$0.263199\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.5830	−1.12847	−0.564233	−	0.825615i	$-0.690828\pi$
$167$	−14.5830	−1.12847	−0.564233	+	0.825615i	$0.690828\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−13.2288	−1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5830	−0.865754	−0.432877	−	0.901453i	$-0.642502\pi$
$179$	−11.5830	−0.865754	−0.432877	+	0.901453i	$0.642502\pi$
$180$	0	0
$181$	−20.2915	−1.50826	−0.754128	−	0.656728i	$-0.771940\pi$
$181$	−20.2915	−1.50826	−0.754128	+	0.656728i	$0.771940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.64575	0.339731
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.29150	0.455237	0.227619	−	0.973750i	$-0.426906\pi$
$191$	6.29150	0.455237	0.227619	+	0.973750i	$0.426906\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6458	−0.758478	−0.379239	−	0.925299i	$-0.623814\pi$
$197$	−10.6458	−0.758478	−0.379239	+	0.925299i	$0.623814\pi$
$198$	0	0
$199$	11.8745	0.841762	0.420881	−	0.907116i	$-0.361721\pi$
$199$	11.8745	0.841762	0.420881	+	0.907116i	$0.361721\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−17.5830	−1.23409
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.29150	0.366021
$210$	0	0
$211$	15.3542	1.05703	0.528515	−	0.848924i	$-0.322749\pi$
$211$	15.3542	1.05703	0.528515	+	0.848924i	$0.322749\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.41699	−0.231961
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.2915	−0.890065	−0.445032	−	0.895514i	$-0.646808\pi$
$223$	−13.2915	−0.890065	−0.445032	+	0.895514i	$0.646808\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.1660	1.67033	0.835163	−	0.550002i	$-0.185373\pi$
$227$	25.1660	1.67033	0.835163	+	0.550002i	$0.185373\pi$
$228$	0	0
$229$	12.2915	0.812245	0.406123	−	0.913819i	$-0.366881\pi$
$229$	12.2915	0.812245	0.406123	+	0.913819i	$0.366881\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5830	0.955364	0.477682	−	0.878533i	$-0.341477\pi$
$233$	14.5830	0.955364	0.477682	+	0.878533i	$0.341477\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	0	0
$241$	−24.5830	−1.58353	−0.791765	−	0.610825i	$-0.790838\pi$
$241$	−24.5830	−1.58353	−0.791765	+	0.610825i	$0.790838\pi$
$242$	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
$251$	23.5830	1.48855	0.744273	−	0.667875i	$-0.232796\pi$
$251$	23.5830	1.48855	0.744273	+	0.667875i	$0.232796\pi$
$252$	0	0
$253$	8.29150	0.521282
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.5830	−0.660150	−0.330075	−	0.943955i	$-0.607074\pi$
$257$	−10.5830	−0.660150	−0.330075	+	0.943955i	$0.607074\pi$
$258$	0	0
$259$	6.06275	0.376721
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.1660	−1.42848	−0.714239	−	0.699902i	$-0.753227\pi$
$263$	−23.1660	−1.42848	−0.714239	+	0.699902i	$0.753227\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.70850	−0.409024	−0.204512	−	0.978864i	$-0.565561\pi$
$269$	−6.70850	−0.409024	−0.204512	+	0.978864i	$0.565561\pi$
$270$	0	0
$271$	−17.2288	−1.04657	−0.523286	−	0.852157i	$-0.675294\pi$
$271$	−17.2288	−1.04657	−0.523286	+	0.852157i	$0.675294\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−14.5830	−0.876208	−0.438104	−	0.898924i	$-0.644350\pi$
$277$	−14.5830	−0.876208	−0.438104	+	0.898924i	$0.644350\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.93725	−0.115567	−0.0577834	−	0.998329i	$-0.518403\pi$
$281$	−1.93725	−0.115567	−0.0577834	+	0.998329i	$0.518403\pi$
$282$	0	0
$283$	13.2915	0.790098	0.395049	−	0.918660i	$-0.370728\pi$
$283$	13.2915	0.790098	0.395049	+	0.918660i	$0.370728\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.7085	0.927243
$288$	0	0
$289$	4.58301	0.269589
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.41699	0.316464	0.158232	−	0.987402i	$-0.449421\pi$
$293$	5.41699	0.316464	0.158232	+	0.987402i	$0.449421\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.2915	0.708470
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.35425	0.191437	0.0957185	−	0.995408i	$-0.469485\pi$
$307$	3.35425	0.191437	0.0957185	+	0.995408i	$0.469485\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.8745	1.07028	0.535138	−	0.844765i	$-0.320260\pi$
$311$	18.8745	1.07028	0.535138	+	0.844765i	$0.320260\pi$
$312$	0	0
$313$	7.16601	0.405047	0.202523	−	0.979277i	$-0.435086\pi$
$313$	7.16601	0.405047	0.202523	+	0.979277i	$0.435086\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.29150	0.521863	0.260931	−	0.965357i	$-0.415970\pi$
$317$	9.29150	0.521863	0.260931	+	0.965357i	$0.415970\pi$
$318$	0	0
$319$	6.64575	0.372091
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.5830	1.36784
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.3542	0.625980
$330$	0	0
$331$	−11.8745	−0.652682	−0.326341	−	0.945252i	$-0.605816\pi$
$331$	−11.8745	−0.652682	−0.326341	+	0.945252i	$0.605816\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.5830	0.794387	0.397193	−	0.917735i	$-0.369984\pi$
$337$	14.5830	0.794387	0.397193	+	0.917735i	$0.369984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.29150	0.0699388
$342$	0	0
$343$	−18.5203	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.1660	0.921520	0.460760	−	0.887525i	$-0.347577\pi$
$347$	17.1660	0.921520	0.460760	+	0.887525i	$0.347577\pi$
$348$	0	0
$349$	−26.5830	−1.42296	−0.711478	−	0.702709i	$-0.751974\pi$
$349$	−26.5830	−1.42296	−0.711478	+	0.702709i	$0.751974\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.4575	1.19529	0.597646	−	0.801760i	$-0.296103\pi$
$353$	22.4575	1.19529	0.597646	+	0.801760i	$0.296103\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.5830	0.980774	0.490387	−	0.871505i	$-0.336856\pi$
$359$	18.5830	0.980774	0.490387	+	0.871505i	$0.336856\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.41699	−0.282765	−0.141382	−	0.989955i	$-0.545155\pi$
$367$	−5.41699	−0.282765	−0.141382	+	0.989955i	$0.545155\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.5830	−0.549442
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.5830	0.543612	0.271806	−	0.962352i	$-0.412379\pi$
$379$	10.5830	0.543612	0.271806	+	0.962352i	$0.412379\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.41699	−0.0724050	−0.0362025	−	0.999344i	$-0.511526\pi$
$383$	−1.41699	−0.0724050	−0.0362025	+	0.999344i	$0.511526\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.8745	−1.21049	−0.605243	−	0.796041i	$-0.706924\pi$
$389$	−23.8745	−1.21049	−0.605243	+	0.796041i	$0.706924\pi$
$390$	0	0
$391$	38.5203	1.94805
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.1660	−0.761160	−0.380580	−	0.924748i	$-0.624276\pi$
$397$	−15.1660	−0.761160	−0.380580	+	0.924748i	$0.624276\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.8745	1.99124	0.995619	−	0.0935035i	$-0.0298066\pi$
$401$	39.8745	1.99124	0.995619	+	0.0935035i	$0.0298066\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.29150	−0.113586
$408$	0	0
$409$	−2.58301	−0.127721	−0.0638607	−	0.997959i	$-0.520341\pi$
$409$	−2.58301	−0.127721	−0.0638607	+	0.997959i	$0.520341\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.5203	−0.911322
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.1660	−0.838614	−0.419307	−	0.907844i	$-0.637727\pi$
$419$	−17.1660	−0.838614	−0.419307	+	0.907844i	$0.637727\pi$
$420$	0	0
$421$	−23.7085	−1.15548	−0.577741	−	0.816220i	$-0.696066\pi$
$421$	−23.7085	−1.15548	−0.577741	+	0.816220i	$0.696066\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.2288	1.12676
$426$	0	0
$427$	28.0000	1.35501
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	43.8745	2.09880
$438$	0	0
$439$	−0.0627461	−0.00299471	−0.00149735	−	0.999999i	$-0.500477\pi$
$439$	−0.0627461	−0.00299471	−0.00149735	+	0.999999i	$0.500477\pi$
$440$	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
$449$	−3.87451	−0.182849	−0.0914247	−	0.995812i	$-0.529142\pi$
$449$	−3.87451	−0.182849	−0.0914247	+	0.995812i	$0.529142\pi$
$450$	0	0
$451$	−5.93725	−0.279574
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.58301	−0.307940	−0.153970	−	0.988076i	$-0.549206\pi$
$457$	−6.58301	−0.307940	−0.153970	+	0.988076i	$0.549206\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.5830	1.23809	0.619047	−	0.785354i	$-0.287519\pi$
$461$	26.5830	1.23809	0.619047	+	0.785354i	$0.287519\pi$
$462$	0	0
$463$	−13.1660	−0.611876	−0.305938	−	0.952051i	$-0.598970\pi$
$463$	−13.1660	−0.611876	−0.305938	+	0.952051i	$0.598970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.7490	−1.05270	−0.526349	−	0.850268i	$-0.676440\pi$
$467$	−22.7490	−1.05270	−0.526349	+	0.850268i	$0.676440\pi$
$468$	0	0
$469$	35.1660	1.62382
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.64575	−0.213612
$474$	0	0
$475$	26.4575	1.21395
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.7490	−1.63341	−0.816707	−	0.577052i	$-0.804203\pi$
$479$	−35.7490	−1.63341	−0.816707	+	0.577052i	$0.804203\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.0000	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.1660	−0.503915	−0.251957	−	0.967738i	$-0.581074\pi$
$491$	−11.1660	−0.503915	−0.251957	+	0.967738i	$0.581074\pi$
$492$	0	0
$493$	30.8745	1.39052
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.4170	0.781259
$498$	0	0
$499$	−33.1660	−1.48471	−0.742357	−	0.670004i	$-0.766292\pi$
$499$	−33.1660	−1.48471	−0.742357	+	0.670004i	$0.766292\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.16601	−0.228979	−0.114490	−	0.993424i	$-0.536523\pi$
$509$	−5.16601	−0.228979	−0.114490	+	0.993424i	$0.536523\pi$
$510$	0	0
$511$	−38.5830	−1.70681
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.29150	−0.188740
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.00000	−0.350486	−0.175243	−	0.984525i	$-0.556071\pi$
$521$	−8.00000	−0.350486	−0.175243	+	0.984525i	$0.556071\pi$
$522$	0	0
$523$	18.0627	0.789829	0.394914	−	0.918718i	$-0.370774\pi$
$523$	18.0627	0.789829	0.394914	+	0.918718i	$0.370774\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	45.7490	1.98909
$530$	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
$541$	0.583005	0.0250654	0.0125327	−	0.999921i	$-0.496011\pi$
$541$	0.583005	0.0250654	0.0125327	+	0.999921i	$0.496011\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.35425	−0.314445	−0.157222	−	0.987563i	$-0.550254\pi$
$547$	−7.35425	−0.314445	−0.157222	+	0.987563i	$0.550254\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35.1660	1.49812
$552$	0	0
$553$	−31.5830	−1.34305
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.0627	−0.680600	−0.340300	−	0.940317i	$-0.610529\pi$
$557$	−16.0627	−0.680600	−0.340300	+	0.940317i	$0.610529\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.583005	0.0245707	0.0122854	−	0.999925i	$-0.496089\pi$
$563$	0.583005	0.0245707	0.0122854	+	0.999925i	$0.496089\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.58301	−0.275974	−0.137987	−	0.990434i	$-0.544063\pi$
$569$	−6.58301	−0.275974	−0.137987	+	0.990434i	$0.544063\pi$
$570$	0	0
$571$	31.2288	1.30688	0.653441	−	0.756977i	$-0.273325\pi$
$571$	31.2288	1.30688	0.653441	+	0.756977i	$0.273325\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	41.4575	1.72890
$576$	0	0
$577$	−30.7490	−1.28010	−0.640049	−	0.768334i	$-0.721086\pi$
$577$	−30.7490	−1.28010	−0.640049	+	0.768334i	$0.721086\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−21.1660	−0.878114
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.1660	1.41018	0.705091	−	0.709117i	$-0.250906\pi$
$587$	34.1660	1.41018	0.705091	+	0.709117i	$0.250906\pi$
$588$	0	0
$589$	6.83399	0.281590
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.2288	0.789630	0.394815	−	0.918761i	$-0.370809\pi$
$593$	19.2288	0.789630	0.394815	+	0.918761i	$0.370809\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.0405	1.79945	0.899723	−	0.436461i	$-0.143768\pi$
$599$	44.0405	1.79945	0.899723	+	0.436461i	$0.143768\pi$
$600$	0	0
$601$	7.41699	0.302546	0.151273	−	0.988492i	$-0.451663\pi$
$601$	7.41699	0.302546	0.151273	+	0.988492i	$0.451663\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.4575	0.424457	0.212229	−	0.977220i	$-0.431928\pi$
$607$	10.4575	0.424457	0.212229	+	0.977220i	$0.431928\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−41.1660	−1.66268	−0.831340	−	0.555765i	$-0.812426\pi$
$613$	−41.1660	−1.66268	−0.831340	+	0.555765i	$0.812426\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.4575	−1.87031	−0.935155	−	0.354240i	$-0.884740\pi$
$617$	−46.4575	−1.87031	−0.935155	+	0.354240i	$0.884740\pi$
$618$	0	0
$619$	−2.70850	−0.108864	−0.0544319	−	0.998517i	$-0.517335\pi$
$619$	−2.70850	−0.108864	−0.0544319	+	0.998517i	$0.517335\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.16601	−0.287100
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.6458	−0.424474
$630$	0	0
$631$	−37.1660	−1.47956	−0.739778	−	0.672851i	$-0.765069\pi$
$631$	−37.1660	−1.47956	−0.739778	+	0.672851i	$0.765069\pi$
$632$	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
$641$	6.58301	0.260013	0.130007	−	0.991513i	$-0.458500\pi$
$641$	6.58301	0.260013	0.130007	+	0.991513i	$0.458500\pi$
$642$	0	0
$643$	−5.29150	−0.208676	−0.104338	−	0.994542i	$-0.533272\pi$
$643$	−5.29150	−0.208676	−0.104338	+	0.994542i	$0.533272\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.4170	0.684733	0.342366	−	0.939567i	$-0.388772\pi$
$647$	17.4170	0.684733	0.342366	+	0.939567i	$0.388772\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.5830	1.04027	0.520137	−	0.854083i	$-0.325881\pi$
$653$	26.5830	1.04027	0.520137	+	0.854083i	$0.325881\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.833990	−0.0324876	−0.0162438	−	0.999868i	$-0.505171\pi$
$659$	−0.833990	−0.0324876	−0.0162438	+	0.999868i	$0.505171\pi$
$660$	0	0
$661$	27.4575	1.06797	0.533987	−	0.845493i	$-0.320693\pi$
$661$	27.4575	1.06797	0.533987	+	0.845493i	$0.320693\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	55.1033	2.13361
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.5830	−0.408552
$672$	0	0
$673$	21.1660	0.815890	0.407945	−	0.913007i	$-0.366246\pi$
$673$	21.1660	0.815890	0.407945	+	0.913007i	$0.366246\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.9373	1.68865	0.844323	−	0.535835i	$-0.180003\pi$
$677$	43.9373	1.68865	0.844323	+	0.535835i	$0.180003\pi$
$678$	0	0
$679$	−35.9373	−1.37915
$680$	0	0
$681$	0	0
$682$	0	0
$683$	47.5830	1.82071	0.910357	−	0.413825i	$-0.135807\pi$
$683$	47.5830	1.82071	0.910357	+	0.413825i	$0.135807\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−27.5830	−1.04478
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.5203	0.850578	0.425289	−	0.905057i	$-0.360172\pi$
$701$	22.5203	0.850578	0.425289	+	0.905057i	$0.360172\pi$
$702$	0	0
$703$	−12.1255	−0.457322
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.1660	−0.532768
$708$	0	0
$709$	−39.4575	−1.48186	−0.740929	−	0.671583i	$-0.765615\pi$
$709$	−39.4575	−1.48186	−0.740929	+	0.671583i	$0.765615\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.7085	0.401036
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.41699	0.351195	0.175597	−	0.984462i	$-0.443814\pi$
$719$	9.41699	0.351195	0.175597	+	0.984462i	$0.443814\pi$
$720$	0	0
$721$	38.5830	1.43691
$722$	0	0
$723$	0	0
$724$	0	0
$725$	33.2288	1.23409
$726$	0	0
$727$	−10.7085	−0.397156	−0.198578	−	0.980085i	$-0.563632\pi$
$727$	−10.7085	−0.397156	−0.198578	+	0.980085i	$0.563632\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.5830	−0.798276
$732$	0	0
$733$	−3.16601	−0.116939	−0.0584696	−	0.998289i	$-0.518622\pi$
$733$	−3.16601	−0.116939	−0.0584696	+	0.998289i	$0.518622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.2915	−0.489599
$738$	0	0
$739$	20.6458	0.759466	0.379733	−	0.925096i	$-0.376016\pi$
$739$	20.6458	0.759466	0.379733	+	0.925096i	$0.376016\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.4170	−0.638968	−0.319484	−	0.947592i	$-0.603509\pi$
$743$	−17.4170	−0.638968	−0.319484	+	0.947592i	$0.603509\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−28.0000	−1.02310
$750$	0	0
$751$	−2.83399	−0.103414	−0.0517069	−	0.998662i	$-0.516466\pi$
$751$	−2.83399	−0.103414	−0.0517069	+	0.998662i	$0.516466\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.45751	−0.125665	−0.0628327	−	0.998024i	$-0.520013\pi$
$757$	−3.45751	−0.125665	−0.0628327	+	0.998024i	$0.520013\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.58301	0.238634	0.119317	−	0.992856i	$-0.461930\pi$
$761$	6.58301	0.238634	0.119317	+	0.992856i	$0.461930\pi$
$762$	0	0
$763$	−5.29150	−0.191565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	41.7490	1.50551	0.752754	−	0.658302i	$-0.228725\pi$
$769$	41.7490	1.50551	0.752754	+	0.658302i	$0.228725\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.12549	−0.292254	−0.146127	−	0.989266i	$-0.546681\pi$
$773$	−8.12549	−0.292254	−0.146127	+	0.989266i	$0.546681\pi$
$774$	0	0
$775$	6.45751	0.231961
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.4170	−1.12563
$780$	0	0
$781$	−6.58301	−0.235558
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.7085	0.952055	0.476028	−	0.879430i	$-0.342076\pi$
$787$	26.7085	0.952055	0.476028	+	0.879430i	$0.342076\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−35.1660	−1.25036
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.2915	1.74599	0.872997	−	0.487725i	$-0.162173\pi$
$797$	49.2915	1.74599	0.872997	+	0.487725i	$0.162173\pi$
$798$	0	0
$799$	−19.9373	−0.705329
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.5830	0.514623
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.3948	1.27957	0.639786	−	0.768553i	$-0.279023\pi$
$809$	36.3948	1.27957	0.639786	+	0.768553i	$0.279023\pi$
$810$	0	0
$811$	−6.06275	−0.212892	−0.106446	−	0.994318i	$-0.533947\pi$
$811$	−6.06275	−0.212892	−0.106446	+	0.994318i	$0.533947\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.5830	−0.860050
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.3542	−1.02447	−0.512235	−	0.858845i	$-0.671182\pi$
$821$	−29.3542	−1.02447	−0.512235	+	0.858845i	$0.671182\pi$
$822$	0	0
$823$	−22.5830	−0.787194	−0.393597	−	0.919283i	$-0.628769\pi$
$823$	−22.5830	−0.787194	−0.393597	+	0.919283i	$0.628769\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.41699	0.257914	0.128957	−	0.991650i	$-0.458837\pi$
$827$	7.41699	0.257914	0.128957	+	0.991650i	$0.458837\pi$
$828$	0	0
$829$	−8.29150	−0.287976	−0.143988	−	0.989579i	$-0.545993\pi$
$829$	−8.29150	−0.287976	−0.143988	+	0.989579i	$0.545993\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.291503	0.0100638	0.00503189	−	0.999987i	$-0.498398\pi$
$839$	0.291503	0.0100638	0.00503189	+	0.999987i	$0.498398\pi$
$840$	0	0
$841$	15.1660	0.522966
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.64575	0.0909091
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.0000	−0.651312
$852$	0	0
$853$	21.4170	0.733304	0.366652	−	0.930358i	$-0.380504\pi$
$853$	21.4170	0.733304	0.366652	+	0.930358i	$0.380504\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.6863	−0.740789	−0.370394	−	0.928875i	$-0.620777\pi$
$857$	−21.6863	−0.740789	−0.370394	+	0.928875i	$0.620777\pi$
$858$	0	0
$859$	−43.8745	−1.49698	−0.748489	−	0.663147i	$-0.769221\pi$
$859$	−43.8745	−1.49698	−0.748489	+	0.663147i	$0.769221\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.7085	−1.35169	−0.675846	−	0.737042i	$-0.736222\pi$
$863$	−39.7085	−1.35169	−0.675846	+	0.737042i	$0.736222\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.9373	0.404944
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.41699	−0.115384	−0.0576919	−	0.998334i	$-0.518374\pi$
$877$	−3.41699	−0.115384	−0.0576919	+	0.998334i	$0.518374\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41.1660	−1.38692	−0.693459	−	0.720496i	$-0.743914\pi$
$881$	−41.1660	−1.38692	−0.693459	+	0.720496i	$0.743914\pi$
$882$	0	0
$883$	−38.4575	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$883$	−38.4575	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.8340	0.430923	0.215462	−	0.976512i	$-0.430874\pi$
$887$	12.8340	0.430923	0.215462	+	0.976512i	$0.430874\pi$
$888$	0	0
$889$	35.1660	1.17943
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−22.7085	−0.759911
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.58301	0.286259
$900$	0	0
$901$	18.5830	0.619090
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−39.7490	−1.31984	−0.659922	−	0.751334i	$-0.729411\pi$
$907$	−39.7490	−1.31984	−0.659922	+	0.751334i	$0.729411\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.87451	0.227763	0.113881	−	0.993494i	$-0.463672\pi$
$911$	6.87451	0.227763	0.113881	+	0.993494i	$0.463672\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.1255	0.400419
$918$	0	0
$919$	−19.8118	−0.653530	−0.326765	−	0.945106i	$-0.605959\pi$
$919$	−19.8118	−0.653530	−0.326765	+	0.945106i	$0.605959\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−11.4575	−0.376721
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.5830	0.872160	0.436080	−	0.899908i	$-0.356366\pi$
$929$	26.5830	0.872160	0.436080	+	0.899908i	$0.356366\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	54.3320	1.77495	0.887475	−	0.460856i	$-0.152458\pi$
$937$	54.3320	1.77495	0.887475	+	0.460856i	$0.152458\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.3542	1.34811	0.674055	−	0.738681i	$-0.264551\pi$
$941$	41.3542	1.34811	0.674055	+	0.738681i	$0.264551\pi$
$942$	0	0
$943$	−49.2288	−1.60311
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.1660	−0.557820	−0.278910	−	0.960317i	$-0.589973\pi$
$947$	−17.1660	−0.557820	−0.278910	+	0.960317i	$0.589973\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56.6458	−1.83494	−0.917468	−	0.397810i	$-0.869770\pi$
$953$	−56.6458	−1.83494	−0.917468	+	0.397810i	$0.869770\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.4170	−0.562424
$960$	0	0
$961$	−29.3320	−0.946194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−51.9373	−1.67019	−0.835095	−	0.550106i	$-0.814587\pi$
$967$	−51.9373	−1.67019	−0.835095	+	0.550106i	$0.814587\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.1660	−0.550883	−0.275442	−	0.961318i	$-0.588824\pi$
$971$	−17.1660	−0.550883	−0.275442	+	0.961318i	$0.588824\pi$
$972$	0	0
$973$	40.6235	1.30233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.45751	0.0786228	0.0393114	−	0.999227i	$-0.487484\pi$
$977$	2.45751	0.0786228	0.0393114	+	0.999227i	$0.487484\pi$
$978$	0	0
$979$	2.70850	0.0865640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.5830	−1.23061	−0.615303	−	0.788290i	$-0.710966\pi$
$983$	−38.5830	−1.23061	−0.615303	+	0.788290i	$0.710966\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−38.5203	−1.22487
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.1660	1.05038	0.525189	−	0.850986i	$-0.323995\pi$
$997$	33.1660	1.05038	0.525189	+	0.850986i	$0.323995\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.i.1.2	✓	2
3.2	odd	2		2376.2.a.j.1.2	yes	2
4.3	odd	2		4752.2.a.y.1.1		2
12.11	even	2		4752.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.i.1.2	✓	2	1.1	even	1	trivial
2376.2.a.j.1.2	yes	2	3.2	odd	2
4752.2.a.x.1.1		2	12.11	even	2
4752.2.a.y.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{7} -1.00000 q^{11} -4.64575 q^{17} -5.29150 q^{19} -8.29150 q^{23} -5.00000 q^{25} -6.64575 q^{29} -1.29150 q^{31} +2.29150 q^{37} +5.93725 q^{41} +4.64575 q^{43} +4.29150 q^{47} -4.00000 q^{53} -7.00000 q^{59} +10.5830 q^{61} +13.2915 q^{67} +6.58301 q^{71} -14.5830 q^{73} -2.64575 q^{77} -11.9373 q^{79} -8.00000 q^{83} -2.70850 q^{89} -13.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{11} - 4 q^{17} - 6 q^{23} - 10 q^{25} - 8 q^{29} + 8 q^{31} - 6 q^{37} - 4 q^{41} + 4 q^{43} - 2 q^{47} - 8 q^{53} - 14 q^{59} + 16 q^{67} - 8 q^{71} - 8 q^{73} - 8 q^{79} - 16 q^{83} - 16 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^11 - 4 * q^17 - 6 * q^23 - 10 * q^25 - 8 * q^29 + 8 * q^31 - 6 * q^37 - 4 * q^41 + 4 * q^43 - 2 * q^47 - 8 * q^53 - 14 * q^59 + 16 * q^67 - 8 * q^71 - 8 * q^73 - 8 * q^79 - 16 * q^83 - 16 * q^89 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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