LMFDB - Embedded newform 2376.2.a.j.1.1

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:	$0$
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			1.1
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)

$f(q)$	$=$	$q-2.64575 q^{7} +1.00000 q^{11} -0.645751 q^{17} +5.29150 q^{19} -2.29150 q^{23} -5.00000 q^{25} +1.35425 q^{29} +9.29150 q^{31} -8.29150 q^{37} +9.93725 q^{41} -0.645751 q^{43} +6.29150 q^{47} +4.00000 q^{53} +7.00000 q^{59} -10.5830 q^{61} +2.70850 q^{67} +14.5830 q^{71} +6.58301 q^{73} -2.64575 q^{77} +3.93725 q^{79} +8.00000 q^{83} +13.2915 q^{89} +7.58301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.666667\pi$
$7$	−2.64575	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$8$	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
$17$	−0.645751	−0.156618	−0.0783088	−	0.996929i	$-0.524952\pi$
$17$	−0.645751	−0.156618	−0.0783088	+	0.996929i	$0.524952\pi$
$18$	0	0
$19$	5.29150	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.29150	−0.477811	−0.238906	−	0.971043i	$-0.576789\pi$
$23$	−2.29150	−0.477811	−0.238906	+	0.971043i	$0.576789\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.35425	0.251478	0.125739	−	0.992063i	$-0.459870\pi$
$29$	1.35425	0.251478	0.125739	+	0.992063i	$0.459870\pi$
$30$	0	0
$31$	9.29150	1.66880	0.834402	−	0.551157i	$-0.185813\pi$
$31$	9.29150	1.66880	0.834402	+	0.551157i	$0.185813\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.29150	−1.36311	−0.681557	−	0.731765i	$-0.738697\pi$
$37$	−8.29150	−1.36311	−0.681557	+	0.731765i	$0.738697\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.93725	1.55194	0.775969	−	0.630771i	$-0.217261\pi$
$41$	9.93725	1.55194	0.775969	+	0.630771i	$0.217261\pi$
$42$	0	0
$43$	−0.645751	−0.0984762	−0.0492381	−	0.998787i	$-0.515679\pi$
$43$	−0.645751	−0.0984762	−0.0492381	+	0.998787i	$0.515679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.29150	0.917710	0.458855	−	0.888511i	$-0.348260\pi$
$47$	6.29150	0.917710	0.458855	+	0.888511i	$0.348260\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−10.5830	−1.35501	−0.677507	−	0.735516i	$-0.736940\pi$
$61$	−10.5830	−1.35501	−0.677507	+	0.735516i	$0.736940\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.70850	0.330896	0.165448	−	0.986219i	$-0.447093\pi$
$67$	2.70850	0.330896	0.165448	+	0.986219i	$0.447093\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.5830	1.73068	0.865342	−	0.501182i	$-0.167101\pi$
$71$	14.5830	1.73068	0.865342	+	0.501182i	$0.167101\pi$
$72$	0	0
$73$	6.58301	0.770482	0.385241	−	0.922816i	$-0.374118\pi$
$73$	6.58301	0.770482	0.385241	+	0.922816i	$0.374118\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.64575	−0.301511
$78$	0	0
$79$	3.93725	0.442976	0.221488	−	0.975163i	$-0.428909\pi$
$79$	3.93725	0.442976	0.221488	+	0.975163i	$0.428909\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.2915	1.40890	0.704448	−	0.709755i	$-0.251195\pi$
$89$	13.2915	1.40890	0.704448	+	0.709755i	$0.251195\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.58301	0.769938	0.384969	−	0.922930i	$-0.374212\pi$
$97$	7.58301	0.769938	0.384969	+	0.922930i	$0.374212\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.6458	1.05929	0.529646	−	0.848219i	$-0.322325\pi$
$101$	10.6458	1.05929	0.529646	+	0.848219i	$0.322325\pi$
$102$	0	0
$103$	−6.58301	−0.648643	−0.324321	−	0.945947i	$-0.605136\pi$
$103$	−6.58301	−0.648643	−0.324321	+	0.945947i	$0.605136\pi$
$104$	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
$113$	2.70850	0.254794	0.127397	−	0.991852i	$-0.459338\pi$
$113$	2.70850	0.254794	0.127397	+	0.991852i	$0.459338\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.70850	0.156618
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.70850	0.240340	0.120170	−	0.992753i	$-0.461656\pi$
$127$	2.70850	0.240340	0.120170	+	0.992753i	$0.461656\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.5830	1.44886	0.724432	−	0.689346i	$-0.242102\pi$
$131$	16.5830	1.44886	0.724432	+	0.689346i	$0.242102\pi$
$132$	0	0
$133$	−14.0000	−1.21395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.5830	−1.24591	−0.622955	−	0.782258i	$-0.714068\pi$
$137$	−14.5830	−1.24591	−0.622955	+	0.782258i	$0.714068\pi$
$138$	0	0
$139$	20.6458	1.75115	0.875575	−	0.483082i	$-0.160483\pi$
$139$	20.6458	1.75115	0.875575	+	0.483082i	$0.160483\pi$
$140$	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
$149$	2.58301	0.211608	0.105804	−	0.994387i	$-0.466258\pi$
$149$	2.58301	0.211608	0.105804	+	0.994387i	$0.466258\pi$
$150$	0	0
$151$	−13.2915	−1.08165	−0.540824	−	0.841136i	$-0.681887\pi$
$151$	−13.2915	−1.08165	−0.540824	+	0.841136i	$0.681887\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.29150	0.182882	0.0914409	−	0.995811i	$-0.470853\pi$
$157$	2.29150	0.182882	0.0914409	+	0.995811i	$0.470853\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.06275	0.477811
$162$	0	0
$163$	6.70850	0.525450	0.262725	−	0.964871i	$-0.415379\pi$
$163$	6.70850	0.525450	0.262725	+	0.964871i	$0.415379\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.58301	−0.509408	−0.254704	−	0.967019i	$-0.581978\pi$
$167$	−6.58301	−0.509408	−0.254704	+	0.967019i	$0.581978\pi$
$168$	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
$179$	−9.58301	−0.716267	−0.358134	−	0.933670i	$-0.616587\pi$
$179$	−9.58301	−0.716267	−0.358134	+	0.933670i	$0.616587\pi$
$180$	0	0
$181$	−9.70850	−0.721627	−0.360813	−	0.932638i	$-0.617501\pi$
$181$	−9.70850	−0.721627	−0.360813	+	0.932638i	$0.617501\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.645751	−0.0472220
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.29150	0.310522	0.155261	−	0.987873i	$-0.450378\pi$
$191$	4.29150	0.310522	0.155261	+	0.987873i	$0.450378\pi$
$192$	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
$197$	5.35425	0.381474	0.190737	−	0.981641i	$-0.438912\pi$
$197$	5.35425	0.381474	0.190737	+	0.981641i	$0.438912\pi$
$198$	0	0
$199$	−19.8745	−1.40887	−0.704433	−	0.709770i	$-0.748799\pi$
$199$	−19.8745	−1.40887	−0.704433	+	0.709770i	$0.748799\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.58301	−0.251478
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.29150	0.366021
$210$	0	0
$211$	20.6458	1.42131	0.710656	−	0.703540i	$-0.248398\pi$
$211$	20.6458	1.42131	0.710656	+	0.703540i	$0.248398\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.5830	−1.66880
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.70850	−0.181374	−0.0906872	−	0.995879i	$-0.528906\pi$
$223$	−2.70850	−0.181374	−0.0906872	+	0.995879i	$0.528906\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.1660	1.13935	0.569674	−	0.821871i	$-0.307069\pi$
$227$	17.1660	1.13935	0.569674	+	0.821871i	$0.307069\pi$
$228$	0	0
$229$	1.70850	0.112901	0.0564503	−	0.998405i	$-0.482022\pi$
$229$	1.70850	0.112901	0.0564503	+	0.998405i	$0.482022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.58301	0.431267	0.215634	−	0.976474i	$-0.430818\pi$
$233$	6.58301	0.431267	0.215634	+	0.976474i	$0.430818\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−3.41699	−0.220108	−0.110054	−	0.993926i	$-0.535102\pi$
$241$	−3.41699	−0.220108	−0.110054	+	0.993926i	$0.535102\pi$
$242$	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
$251$	−2.41699	−0.152559	−0.0762797	−	0.997086i	$-0.524304\pi$
$251$	−2.41699	−0.152559	−0.0762797	+	0.997086i	$0.524304\pi$
$252$	0	0
$253$	−2.29150	−0.144066
$254$	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
$259$	21.9373	1.36311
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.1660	−1.18183	−0.590913	−	0.806735i	$-0.701233\pi$
$263$	−19.1660	−1.18183	−0.590913	+	0.806735i	$0.701233\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.2915	1.05428	0.527141	−	0.849778i	$-0.323264\pi$
$269$	17.2915	1.05428	0.527141	+	0.849778i	$0.323264\pi$
$270$	0	0
$271$	9.22876	0.560607	0.280304	−	0.959911i	$-0.409565\pi$
$271$	9.22876	0.560607	0.280304	+	0.959911i	$0.409565\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	6.58301	0.395534	0.197767	−	0.980249i	$-0.436631\pi$
$277$	6.58301	0.395534	0.197767	+	0.980249i	$0.436631\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.9373	−0.831427	−0.415713	−	0.909496i	$-0.636468\pi$
$281$	−13.9373	−0.831427	−0.415713	+	0.909496i	$0.636468\pi$
$282$	0	0
$283$	2.70850	0.161003	0.0805017	−	0.996754i	$-0.474348\pi$
$283$	2.70850	0.161003	0.0805017	+	0.996754i	$0.474348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.2915	−1.55194
$288$	0	0
$289$	−16.5830	−0.975471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.5830	−1.55300	−0.776498	−	0.630120i	$-0.783006\pi$
$293$	−26.5830	−1.55300	−0.776498	+	0.630120i	$0.783006\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.70850	0.0984762
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.64575	0.493439	0.246720	−	0.969087i	$-0.420647\pi$
$307$	8.64575	0.493439	0.246720	+	0.969087i	$0.420647\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.8745	0.730046	0.365023	−	0.930998i	$-0.381061\pi$
$311$	12.8745	0.730046	0.365023	+	0.930998i	$0.381061\pi$
$312$	0	0
$313$	−35.1660	−1.98770	−0.993850	−	0.110733i	$-0.964680\pi$
$313$	−35.1660	−1.98770	−0.993850	+	0.110733i	$0.964680\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.29150	0.0725380	0.0362690	−	0.999342i	$-0.488453\pi$
$317$	1.29150	0.0725380	0.0362690	+	0.999342i	$0.488453\pi$
$318$	0	0
$319$	1.35425	0.0758234
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.41699	−0.190127
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.6458	−0.917710
$330$	0	0
$331$	19.8745	1.09240	0.546201	−	0.837654i	$-0.316074\pi$
$331$	19.8745	1.09240	0.546201	+	0.837654i	$0.316074\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.58301	−0.358599	−0.179300	−	0.983795i	$-0.557383\pi$
$337$	−6.58301	−0.358599	−0.179300	+	0.983795i	$0.557383\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.29150	0.503163
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.1660	1.35098	0.675491	−	0.737368i	$-0.263932\pi$
$347$	25.1660	1.35098	0.675491	+	0.737368i	$0.263932\pi$
$348$	0	0
$349$	−5.41699	−0.289965	−0.144983	−	0.989434i	$-0.546313\pi$
$349$	−5.41699	−0.289965	−0.144983	+	0.989434i	$0.546313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.4575	1.62109	0.810545	−	0.585676i	$-0.199171\pi$
$353$	30.4575	1.62109	0.810545	+	0.585676i	$0.199171\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.58301	0.136326	0.0681629	−	0.997674i	$-0.478286\pi$
$359$	2.58301	0.136326	0.0681629	+	0.997674i	$0.478286\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−26.5830	−1.38762	−0.693811	−	0.720157i	$-0.744070\pi$
$367$	−26.5830	−1.38762	−0.693811	+	0.720157i	$0.744070\pi$
$368$	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	$-0.587621\pi$
$379$	−10.5830	−0.543612	−0.271806	+	0.962352i	$0.587621\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.5830	1.15394	0.576969	−	0.816766i	$-0.304235\pi$
$383$	22.5830	1.15394	0.576969	+	0.816766i	$0.304235\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.87451	−0.399253	−0.199627	−	0.979872i	$-0.563973\pi$
$389$	−7.87451	−0.399253	−0.199627	+	0.979872i	$0.563973\pi$
$390$	0	0
$391$	1.47974	0.0748337
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.1660	1.36342	0.681711	−	0.731621i	$-0.261236\pi$
$397$	27.1660	1.36342	0.681711	+	0.731621i	$0.261236\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.12549	−0.405768	−0.202884	−	0.979203i	$-0.565031\pi$
$401$	−8.12549	−0.405768	−0.202884	+	0.979203i	$0.565031\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.29150	−0.410995
$408$	0	0
$409$	18.5830	0.918870	0.459435	−	0.888211i	$-0.348052\pi$
$409$	18.5830	0.918870	0.459435	+	0.888211i	$0.348052\pi$
$410$	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
$419$	−25.1660	−1.22944	−0.614720	−	0.788745i	$-0.710731\pi$
$419$	−25.1660	−1.22944	−0.614720	+	0.788745i	$0.710731\pi$
$420$	0	0
$421$	−34.2915	−1.67127	−0.835633	−	0.549288i	$-0.814899\pi$
$421$	−34.2915	−1.67127	−0.835633	+	0.549288i	$0.814899\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.22876	0.156618
$426$	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	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	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
$437$	−12.1255	−0.580041
$438$	0	0
$439$	−15.9373	−0.760644	−0.380322	−	0.924854i	$-0.624187\pi$
$439$	−15.9373	−0.760644	−0.380322	+	0.924854i	$0.624187\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.8745	−1.31548	−0.657740	−	0.753245i	$-0.728487\pi$
$449$	−27.8745	−1.31548	−0.657740	+	0.753245i	$0.728487\pi$
$450$	0	0
$451$	9.93725	0.467927
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.5830	0.682164	0.341082	−	0.940034i	$-0.389207\pi$
$457$	14.5830	0.682164	0.341082	+	0.940034i	$0.389207\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.41699	−0.252295	−0.126147	−	0.992012i	$-0.540261\pi$
$461$	−5.41699	−0.252295	−0.126147	+	0.992012i	$0.540261\pi$
$462$	0	0
$463$	29.1660	1.35546	0.677730	−	0.735311i	$-0.262964\pi$
$463$	29.1660	1.35546	0.677730	+	0.735311i	$0.262964\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−40.7490	−1.88564	−0.942820	−	0.333303i	$-0.891837\pi$
$467$	−40.7490	−1.88564	−0.942820	+	0.333303i	$0.891837\pi$
$468$	0	0
$469$	−7.16601	−0.330896
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.645751	−0.0296917
$474$	0	0
$475$	−26.4575	−1.21395
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.7490	−1.26788	−0.633942	−	0.773380i	$-0.718564\pi$
$479$	−27.7490	−1.26788	−0.633942	+	0.773380i	$0.718564\pi$
$480$	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
$491$	−31.1660	−1.40650	−0.703251	−	0.710941i	$-0.748269\pi$
$491$	−31.1660	−1.40650	−0.703251	+	0.710941i	$0.748269\pi$
$492$	0	0
$493$	−0.874508	−0.0393859
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−38.5830	−1.73068
$498$	0	0
$499$	9.16601	0.410327	0.205163	−	0.978728i	$-0.434227\pi$
$499$	9.16601	0.410327	0.205163	+	0.978728i	$0.434227\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−37.1660	−1.64735	−0.823677	−	0.567059i	$-0.808081\pi$
$509$	−37.1660	−1.64735	−0.823677	+	0.567059i	$0.808081\pi$
$510$	0	0
$511$	−17.4170	−0.770482
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.29150	0.276700
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.00000	0.350486	0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000	0.350486	0.175243	+	0.984525i	$0.443929\pi$
$522$	0	0
$523$	33.9373	1.48397	0.741986	−	0.670415i	$-0.233884\pi$
$523$	33.9373	1.48397	0.741986	+	0.670415i	$0.233884\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−17.7490	−0.771696
$530$	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
$541$	−20.5830	−0.884933	−0.442466	−	0.896785i	$-0.645896\pi$
$541$	−20.5830	−0.884933	−0.442466	+	0.896785i	$0.645896\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.6458	−0.540693	−0.270347	−	0.962763i	$-0.587138\pi$
$547$	−12.6458	−0.540693	−0.270347	+	0.962763i	$0.587138\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.16601	0.305282
$552$	0	0
$553$	−10.4170	−0.442976
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.9373	1.35322	0.676612	−	0.736339i	$-0.263447\pi$
$557$	31.9373	1.35322	0.676612	+	0.736339i	$0.263447\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.5830	0.867470	0.433735	−	0.901040i	$-0.357195\pi$
$563$	20.5830	0.867470	0.433735	+	0.901040i	$0.357195\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.5830	−0.611351	−0.305676	−	0.952136i	$-0.598882\pi$
$569$	−14.5830	−0.611351	−0.305676	+	0.952136i	$0.598882\pi$
$570$	0	0
$571$	4.77124	0.199670	0.0998352	−	0.995004i	$-0.468168\pi$
$571$	4.77124	0.199670	0.0998352	+	0.995004i	$0.468168\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.4575	0.477811
$576$	0	0
$577$	32.7490	1.36336	0.681680	−	0.731651i	$-0.261250\pi$
$577$	32.7490	1.36336	0.681680	+	0.731651i	$0.261250\pi$
$578$	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
$587$	8.16601	0.337047	0.168524	−	0.985698i	$-0.446100\pi$
$587$	8.16601	0.337047	0.168524	+	0.985698i	$0.446100\pi$
$588$	0	0
$589$	49.1660	2.02585
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.22876	0.296849	0.148425	−	0.988924i	$-0.452580\pi$
$593$	7.22876	0.296849	0.148425	+	0.988924i	$0.452580\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0405	1.22742	0.613711	−	0.789531i	$-0.289676\pi$
$599$	30.0405	1.22742	0.613711	+	0.789531i	$0.289676\pi$
$600$	0	0
$601$	28.5830	1.16593	0.582963	−	0.812499i	$-0.301893\pi$
$601$	28.5830	1.16593	0.582963	+	0.812499i	$0.301893\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−42.4575	−1.72330	−0.861649	−	0.507505i	$-0.830568\pi$
$607$	−42.4575	−1.72330	−0.861649	+	0.507505i	$0.830568\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.16601	0.0470947	0.0235474	−	0.999723i	$-0.492504\pi$
$613$	1.16601	0.0470947	0.0235474	+	0.999723i	$0.492504\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.45751	−0.259970	−0.129985	−	0.991516i	$-0.541493\pi$
$617$	−6.45751	−0.259970	−0.129985	+	0.991516i	$0.541493\pi$
$618$	0	0
$619$	−13.2915	−0.534231	−0.267115	−	0.963665i	$-0.586070\pi$
$619$	−13.2915	−0.534231	−0.267115	+	0.963665i	$0.586070\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−35.1660	−1.40890
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.35425	0.213488
$630$	0	0
$631$	5.16601	0.205656	0.102828	−	0.994699i	$-0.467211\pi$
$631$	5.16601	0.205656	0.102828	+	0.994699i	$0.467211\pi$
$632$	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
$641$	14.5830	0.575994	0.287997	−	0.957631i	$-0.407011\pi$
$641$	14.5830	0.575994	0.287997	+	0.957631i	$0.407011\pi$
$642$	0	0
$643$	5.29150	0.208676	0.104338	−	0.994542i	$-0.466728\pi$
$643$	5.29150	0.208676	0.104338	+	0.994542i	$0.466728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.5830	−1.51685	−0.758427	−	0.651758i	$-0.774032\pi$
$647$	−38.5830	−1.51685	−0.758427	+	0.651758i	$0.774032\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.41699	−0.211983	−0.105992	−	0.994367i	$-0.533802\pi$
$653$	−5.41699	−0.211983	−0.105992	+	0.994367i	$0.533802\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.1660	1.68151	0.840755	−	0.541417i	$-0.182112\pi$
$659$	43.1660	1.68151	0.840755	+	0.541417i	$0.182112\pi$
$660$	0	0
$661$	−25.4575	−0.990182	−0.495091	−	0.868841i	$-0.664865\pi$
$661$	−25.4575	−0.990182	−0.495091	+	0.868841i	$0.664865\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.10326	−0.120159
$668$	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	$-0.633754\pi$
$673$	−21.1660	−0.815890	−0.407945	+	0.913007i	$0.633754\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.0627	−1.07854	−0.539269	−	0.842133i	$-0.681300\pi$
$677$	−28.0627	−1.07854	−0.539269	+	0.842133i	$0.681300\pi$
$678$	0	0
$679$	−20.0627	−0.769938
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26.4170	−1.01082	−0.505409	−	0.862880i	$-0.668658\pi$
$683$	−26.4170	−1.01082	−0.505409	+	0.862880i	$0.668658\pi$
$684$	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
$697$	−6.41699	−0.243061
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.5203	0.548423	0.274211	−	0.961669i	$-0.411583\pi$
$701$	14.5203	0.548423	0.274211	+	0.961669i	$0.411583\pi$
$702$	0	0
$703$	−43.8745	−1.65476
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.1660	−1.05929
$708$	0	0
$709$	13.4575	0.505408	0.252704	−	0.967544i	$-0.418680\pi$
$709$	13.4575	0.505408	0.252704	+	0.967544i	$0.418680\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.2915	−0.797373
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.5830	−1.14055	−0.570277	−	0.821453i	$-0.693164\pi$
$719$	−30.5830	−1.14055	−0.570277	+	0.821453i	$0.693164\pi$
$720$	0	0
$721$	17.4170	0.648643
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.77124	−0.251478
$726$	0	0
$727$	−21.2915	−0.789658	−0.394829	−	0.918755i	$-0.629196\pi$
$727$	−21.2915	−0.789658	−0.394829	+	0.918755i	$0.629196\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.416995	0.0154231
$732$	0	0
$733$	39.1660	1.44663	0.723315	−	0.690518i	$-0.242618\pi$
$733$	39.1660	1.44663	0.723315	+	0.690518i	$0.242618\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.70850	0.0997688
$738$	0	0
$739$	15.3542	0.564815	0.282408	−	0.959295i	$-0.408867\pi$
$739$	15.3542	0.564815	0.282408	+	0.959295i	$0.408867\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.5830	1.41547	0.707737	−	0.706476i	$-0.249716\pi$
$743$	38.5830	1.41547	0.707737	+	0.706476i	$0.249716\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	28.0000	1.02310
$750$	0	0
$751$	−45.1660	−1.64813	−0.824066	−	0.566494i	$-0.808299\pi$
$751$	−45.1660	−1.64813	−0.824066	+	0.566494i	$0.808299\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	49.4575	1.79756	0.898782	−	0.438396i	$-0.144453\pi$
$757$	49.4575	1.79756	0.898782	+	0.438396i	$0.144453\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.5830	0.528634	0.264317	−	0.964436i	$-0.414854\pi$
$761$	14.5830	0.528634	0.264317	+	0.964436i	$0.414854\pi$
$762$	0	0
$763$	5.29150	0.191565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.7490	−0.784290	−0.392145	−	0.919904i	$-0.628267\pi$
$769$	−21.7490	−0.784290	−0.392145	+	0.919904i	$0.628267\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.8745	1.43419	0.717093	−	0.696977i	$-0.245472\pi$
$773$	39.8745	1.43419	0.717093	+	0.696977i	$0.245472\pi$
$774$	0	0
$775$	−46.4575	−1.66880
$776$	0	0
$777$	0	0
$778$	0	0
$779$	52.5830	1.88398
$780$	0	0
$781$	14.5830	0.521821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	37.2915	1.32930	0.664649	−	0.747156i	$-0.268581\pi$
$787$	37.2915	1.32930	0.664649	+	0.747156i	$0.268581\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.16601	−0.254794
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.7085	−1.37113	−0.685563	−	0.728014i	$-0.740444\pi$
$797$	−38.7085	−1.37113	−0.685563	+	0.728014i	$0.740444\pi$
$798$	0	0
$799$	−4.06275	−0.143730
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.58301	0.232309
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.3948	1.13894	0.569470	−	0.822012i	$-0.307149\pi$
$809$	32.3948	1.13894	0.569470	+	0.822012i	$0.307149\pi$
$810$	0	0
$811$	−21.9373	−0.770321	−0.385161	−	0.922850i	$-0.625854\pi$
$811$	−21.9373	−0.770321	−0.385161	+	0.922850i	$0.625854\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.41699	−0.119546
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.6458	1.20915	0.604573	−	0.796550i	$-0.293344\pi$
$821$	34.6458	1.20915	0.604573	+	0.796550i	$0.293344\pi$
$822$	0	0
$823$	−1.41699	−0.0493933	−0.0246967	−	0.999695i	$-0.507862\pi$
$823$	−1.41699	−0.0493933	−0.0246967	+	0.999695i	$0.507862\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.5830	−0.993928	−0.496964	−	0.867771i	$-0.665552\pi$
$827$	−28.5830	−0.993928	−0.496964	+	0.867771i	$0.665552\pi$
$828$	0	0
$829$	2.29150	0.0795872	0.0397936	−	0.999208i	$-0.487330\pi$
$829$	2.29150	0.0795872	0.0397936	+	0.999208i	$0.487330\pi$
$830$	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
$839$	10.2915	0.355302	0.177651	−	0.984094i	$-0.443150\pi$
$839$	10.2915	0.355302	0.177651	+	0.984094i	$0.443150\pi$
$840$	0	0
$841$	−27.1660	−0.936759
$842$	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
$853$	42.5830	1.45801	0.729007	−	0.684506i	$-0.239982\pi$
$853$	42.5830	1.45801	0.729007	+	0.684506i	$0.239982\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−57.6863	−1.97053	−0.985263	−	0.171048i	$-0.945285\pi$
$857$	−57.6863	−1.97053	−0.985263	+	0.171048i	$0.945285\pi$
$858$	0	0
$859$	−12.1255	−0.413716	−0.206858	−	0.978371i	$-0.566324\pi$
$859$	−12.1255	−0.413716	−0.206858	+	0.978371i	$0.566324\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.2915	1.71194	0.855971	−	0.517023i	$-0.172960\pi$
$863$	50.2915	1.71194	0.855971	+	0.517023i	$0.172960\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.93725	0.133562
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.5830	−0.830109	−0.415055	−	0.909797i	$-0.636238\pi$
$877$	−24.5830	−0.830109	−0.415055	+	0.909797i	$0.636238\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.16601	−0.0392839	−0.0196419	−	0.999807i	$-0.506253\pi$
$881$	−1.16601	−0.0392839	−0.0196419	+	0.999807i	$0.506253\pi$
$882$	0	0
$883$	14.4575	0.486534	0.243267	−	0.969959i	$-0.421781\pi$
$883$	14.4575	0.486534	0.243267	+	0.969959i	$0.421781\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−55.1660	−1.85229	−0.926147	−	0.377164i	$-0.876899\pi$
$887$	−55.1660	−1.85229	−0.926147	+	0.377164i	$0.876899\pi$
$888$	0	0
$889$	−7.16601	−0.240340
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.2915	1.11406
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.5830	0.419667
$900$	0	0
$901$	−2.58301	−0.0860524
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.7490	0.788573	0.394287	−	0.918988i	$-0.370992\pi$
$907$	23.7490	0.788573	0.394287	+	0.918988i	$0.370992\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.8745	0.824129	0.412065	−	0.911155i	$-0.364808\pi$
$911$	24.8745	0.824129	0.412065	+	0.911155i	$0.364808\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.8745	−1.44886
$918$	0	0
$919$	27.8118	0.917425	0.458713	−	0.888585i	$-0.348311\pi$
$919$	27.8118	0.917425	0.458713	+	0.888585i	$0.348311\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	41.4575	1.36311
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−5.41699	−0.177726	−0.0888629	−	0.996044i	$-0.528323\pi$
$929$	−5.41699	−0.177726	−0.0888629	+	0.996044i	$0.528323\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.3320	−0.990904	−0.495452	−	0.868635i	$-0.664998\pi$
$937$	−30.3320	−0.990904	−0.495452	+	0.868635i	$0.664998\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.6458	−1.52061	−0.760304	−	0.649567i	$-0.774950\pi$
$941$	−46.6458	−1.52061	−0.760304	+	0.649567i	$0.774950\pi$
$942$	0	0
$943$	−22.7712	−0.741534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.1660	−0.817785	−0.408893	−	0.912582i	$-0.634085\pi$
$947$	−25.1660	−0.817785	−0.408893	+	0.912582i	$0.634085\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51.3542	1.66353	0.831764	−	0.555130i	$-0.187331\pi$
$953$	51.3542	1.66353	0.831764	+	0.555130i	$0.187331\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	38.5830	1.24591
$960$	0	0
$961$	55.3320	1.78490
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.0627	−1.15970	−0.579850	−	0.814723i	$-0.696889\pi$
$967$	−36.0627	−1.15970	−0.579850	+	0.814723i	$0.696889\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.1660	−0.807616	−0.403808	−	0.914844i	$-0.632314\pi$
$971$	−25.1660	−0.807616	−0.403808	+	0.914844i	$0.632314\pi$
$972$	0	0
$973$	−54.6235	−1.75115
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.4575	1.61428	0.807139	−	0.590361i	$-0.201015\pi$
$977$	50.4575	1.61428	0.807139	+	0.590361i	$0.201015\pi$
$978$	0	0
$979$	13.2915	0.424798
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.4170	0.555516	0.277758	−	0.960651i	$-0.410409\pi$
$983$	17.4170	0.555516	0.277758	+	0.960651i	$0.410409\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.47974	0.0470530
$990$	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
$997$	−9.16601	−0.290290	−0.145145	−	0.989410i	$-0.546365\pi$
$997$	−9.16601	−0.290290	−0.145145	+	0.989410i	$0.546365\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.i.1.1	✓	2	3.2	odd	2
2376.2.a.j.1.1	yes	2	1.1	even	1	trivial
4752.2.a.x.1.2		2	4.3	odd	2
4752.2.a.y.1.2		2	12.11	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.64575 q^{7} +1.00000 q^{11} -0.645751 q^{17} +5.29150 q^{19} -2.29150 q^{23} -5.00000 q^{25} +1.35425 q^{29} +9.29150 q^{31} -8.29150 q^{37} +9.93725 q^{41} -0.645751 q^{43} +6.29150 q^{47} +4.00000 q^{53} +7.00000 q^{59} -10.5830 q^{61} +2.70850 q^{67} +14.5830 q^{71} +6.58301 q^{73} -2.64575 q^{77} +3.93725 q^{79} +8.00000 q^{83} +13.2915 q^{89} +7.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{11} + 4 q^{17} + 6 q^{23} - 10 q^{25} + 8 q^{29} + 8 q^{31} - 6 q^{37} + 4 q^{41} + 4 q^{43} + 2 q^{47} + 8 q^{53} + 14 q^{59} + 16 q^{67} + 8 q^{71} - 8 q^{73} - 8 q^{79} + 16 q^{83} + 16 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 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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