LMFDB - Embedded newform 2376.2.a.l.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 = [3,0,0,0,-2,0,1,0,0,0,-3,0,0] 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:	$3$
Coefficient field:	3.3.788.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
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			$3.35386$ 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)

$f(q)$	$=$	$q-0.459363 q^{5} +3.35386 q^{7} -1.00000 q^{11} -6.24835 q^{13} -4.89449 q^{17} +5.16707 q^{19} -2.54064 q^{23} -4.78899 q^{25} -5.35386 q^{29} -1.54064 q^{31} -1.54064 q^{35} +7.24835 q^{37} +0.435130 q^{41} -10.5209 q^{43} -8.78899 q^{47} +4.24835 q^{49} +0.707712 q^{53} +0.459363 q^{55} -11.0858 q^{59} +6.45936 q^{61} +2.87026 q^{65} -6.45936 q^{67} +6.54516 q^{71} +10.8703 q^{73} -3.35386 q^{77} -0.731945 q^{79} +15.6638 q^{83} +2.24835 q^{85} -17.5780 q^{89} -20.9561 q^{91} -2.37356 q^{95} +8.54064 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{17} - 2 q^{19} - 7 q^{23} + 5 q^{25} - 7 q^{29} - 4 q^{31} - 4 q^{35} + 3 q^{37} - 9 q^{41} - 5 q^{43} - 7 q^{47} - 6 q^{49} - 16 q^{53} + 2 q^{55} - 17 q^{59} + 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 + q^7 - 3 * q^11 - 5 * q^17 - 2 * q^19 - 7 * q^23 + 5 * q^25 - 7 * q^29 - 4 * q^31 - 4 * q^35 + 3 * q^37 - 9 * q^41 - 5 * q^43 - 7 * q^47 - 6 * q^49 - 16 * q^53 + 2 * q^55 - 17 * q^59 + 20 * q^61 - 12 * q^65 - 20 * q^67 + 4 * q^71 + 12 * q^73 - q^77 + 5 * q^79 - 8 * q^83 - 12 * q^85 - 14 * q^89 - 26 * q^91 - 24 * q^95 + 25 * 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
$5$	−0.459363	−0.205433	−0.102717	−	0.994711i	$-0.532754\pi$
$5$	−0.459363	−0.205433	−0.102717	+	0.994711i	$0.532754\pi$
$6$	0	0
$7$	3.35386	1.26764	0.633819	−	0.773481i	$-0.281486\pi$
$7$	3.35386	1.26764	0.633819	+	0.773481i	$0.281486\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.24835	−1.73298	−0.866490	−	0.499194i	$-0.833629\pi$
$13$	−6.24835	−1.73298	−0.866490	+	0.499194i	$0.833629\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.89449	−1.18709	−0.593544	−	0.804801i	$-0.702272\pi$
$17$	−4.89449	−1.18709	−0.593544	+	0.804801i	$0.702272\pi$
$18$	0	0
$19$	5.16707	1.18541	0.592704	−	0.805420i	$-0.298060\pi$
$19$	5.16707	1.18541	0.592704	+	0.805420i	$0.298060\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.54064	−0.529759	−0.264880	−	0.964281i	$-0.585332\pi$
$23$	−2.54064	−0.529759	−0.264880	+	0.964281i	$0.585332\pi$
$24$	0	0
$25$	−4.78899	−0.957797
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.35386	−0.994186	−0.497093	−	0.867697i	$-0.665599\pi$
$29$	−5.35386	−0.994186	−0.497093	+	0.867697i	$0.665599\pi$
$30$	0	0
$31$	−1.54064	−0.276707	−0.138353	−	0.990383i	$-0.544181\pi$
$31$	−1.54064	−0.276707	−0.138353	+	0.990383i	$0.544181\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.54064	−0.260415
$36$	0	0
$37$	7.24835	1.19162	0.595811	−	0.803125i	$-0.296831\pi$
$37$	7.24835	1.19162	0.595811	+	0.803125i	$0.296831\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.435130	0.0679559	0.0339779	−	0.999423i	$-0.489182\pi$
$41$	0.435130	0.0679559	0.0339779	+	0.999423i	$0.489182\pi$
$42$	0	0
$43$	−10.5209	−1.60443	−0.802213	−	0.597037i	$-0.796344\pi$
$43$	−10.5209	−1.60443	−0.802213	+	0.597037i	$0.796344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.78899	−1.28201	−0.641003	−	0.767539i	$-0.721481\pi$
$47$	−8.78899	−1.28201	−0.641003	+	0.767539i	$0.721481\pi$
$48$	0	0
$49$	4.24835	0.606907
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.707712	0.0972117	0.0486058	−	0.998818i	$-0.484522\pi$
$53$	0.707712	0.0972117	0.0486058	+	0.998818i	$0.484522\pi$
$54$	0	0
$55$	0.459363	0.0619405
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.0858	−1.44325	−0.721624	−	0.692285i	$-0.756604\pi$
$59$	−11.0858	−1.44325	−0.721624	+	0.692285i	$0.756604\pi$
$60$	0	0
$61$	6.45936	0.827037	0.413518	−	0.910496i	$-0.364300\pi$
$61$	6.45936	0.827037	0.413518	+	0.910496i	$0.364300\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.87026	0.356012
$66$	0	0
$67$	−6.45936	−0.789137	−0.394568	−	0.918867i	$-0.629106\pi$
$67$	−6.45936	−0.789137	−0.394568	+	0.918867i	$0.629106\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.54516	0.776768	0.388384	−	0.921498i	$-0.373033\pi$
$71$	6.54516	0.776768	0.388384	+	0.921498i	$0.373033\pi$
$72$	0	0
$73$	10.8703	1.27227	0.636134	−	0.771579i	$-0.280533\pi$
$73$	10.8703	1.27227	0.636134	+	0.771579i	$0.280533\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.35386	−0.382207
$78$	0	0
$79$	−0.731945	−0.0823502	−0.0411751	−	0.999152i	$-0.513110\pi$
$79$	−0.731945	−0.0823502	−0.0411751	+	0.999152i	$0.513110\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.6638	1.71932	0.859661	−	0.510865i	$-0.170675\pi$
$83$	15.6638	1.71932	0.859661	+	0.510865i	$0.170675\pi$
$84$	0	0
$85$	2.24835	0.243868
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.5780	−1.86326	−0.931631	−	0.363407i	$-0.881614\pi$
$89$	−17.5780	−1.86326	−0.931631	+	0.363407i	$0.881614\pi$
$90$	0	0
$91$	−20.9561	−2.19679
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.37356	−0.243522
$96$	0	0
$97$	8.54064	0.867170	0.433585	−	0.901113i	$-0.357248\pi$
$97$	8.54064	0.867170	0.433585	+	0.901113i	$0.357248\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.10551	−0.110002	−0.0550010	−	0.998486i	$-0.517516\pi$
$101$	−1.10551	−0.110002	−0.0550010	+	0.998486i	$0.517516\pi$
$102$	0	0
$103$	4.95606	0.488335	0.244168	−	0.969733i	$-0.421485\pi$
$103$	4.95606	0.488335	0.244168	+	0.969733i	$0.421485\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.621911	0.0601224	0.0300612	−	0.999548i	$-0.490430\pi$
$107$	0.621911	0.0601224	0.0300612	+	0.999548i	$0.490430\pi$
$108$	0	0
$109$	−15.2044	−1.45632	−0.728159	−	0.685408i	$-0.759624\pi$
$109$	−15.2044	−1.45632	−0.728159	+	0.685408i	$0.759624\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.6638	1.47352	0.736762	−	0.676152i	$-0.236354\pi$
$113$	15.6638	1.47352	0.736762	+	0.676152i	$0.236354\pi$
$114$	0	0
$115$	1.16707	0.108830
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.4154	−1.50480
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.49670	0.402197
$126$	0	0
$127$	11.3296	1.00534	0.502671	−	0.864478i	$-0.332351\pi$
$127$	11.3296	1.00534	0.502671	+	0.864478i	$0.332351\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.5452	−1.09608	−0.548038	−	0.836453i	$-0.684625\pi$
$131$	−12.5452	−1.09608	−0.548038	+	0.836453i	$0.684625\pi$
$132$	0	0
$133$	17.3296	1.50267
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.87026	−0.586966	−0.293483	−	0.955964i	$-0.594814\pi$
$137$	−6.87026	−0.586966	−0.293483	+	0.955964i	$0.594814\pi$
$138$	0	0
$139$	−19.6022	−1.66264	−0.831319	−	0.555796i	$-0.812414\pi$
$139$	−19.6022	−1.66264	−0.831319	+	0.555796i	$0.812414\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.24835	0.522513
$144$	0	0
$145$	2.45936	0.204239
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.8263	−1.78808	−0.894041	−	0.447985i	$-0.852142\pi$
$149$	−21.8263	−1.78808	−0.894041	+	0.447985i	$0.852142\pi$
$150$	0	0
$151$	−1.37809	−0.112147	−0.0560736	−	0.998427i	$-0.517858\pi$
$151$	−1.37809	−0.112147	−0.0560736	+	0.998427i	$0.517858\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.707712	0.0568448
$156$	0	0
$157$	−14.6264	−1.16732	−0.583658	−	0.811999i	$-0.698379\pi$
$157$	−14.6264	−1.16732	−0.583658	+	0.811999i	$0.698379\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.52093	−0.671543
$162$	0	0
$163$	−13.5780	−1.06351	−0.531754	−	0.846899i	$-0.678467\pi$
$163$	−13.5780	−1.06351	−0.531754	+	0.846899i	$0.678467\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.16707	−0.554605	−0.277302	−	0.960783i	$-0.589440\pi$
$167$	−7.16707	−0.554605	−0.277302	+	0.960783i	$0.589440\pi$
$168$	0	0
$169$	26.0419	2.00322
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.0373	−1.37135	−0.685677	−	0.727906i	$-0.740494\pi$
$173$	−18.0373	−1.37135	−0.685677	+	0.727906i	$0.740494\pi$
$174$	0	0
$175$	−16.0616	−1.21414
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.2044	−1.51015	−0.755074	−	0.655639i	$-0.772399\pi$
$179$	−20.2044	−1.51015	−0.755074	+	0.655639i	$0.772399\pi$
$180$	0	0
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.32962	−0.244799
$186$	0	0
$187$	4.89449	0.357921
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.62644	−0.624187	−0.312094	−	0.950051i	$-0.601030\pi$
$191$	−8.62644	−0.624187	−0.312094	+	0.950051i	$0.601030\pi$
$192$	0	0
$193$	15.8748	1.14269	0.571346	−	0.820709i	$-0.306421\pi$
$193$	15.8748	1.14269	0.571346	+	0.820709i	$0.306421\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.8990	−1.13276	−0.566379	−	0.824145i	$-0.691656\pi$
$197$	−15.8990	−1.13276	−0.566379	+	0.824145i	$0.691656\pi$
$198$	0	0
$199$	0.296815	0.0210406	0.0105203	−	0.999945i	$-0.496651\pi$
$199$	0.296815	0.0210406	0.0105203	+	0.999945i	$0.496651\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−17.9561	−1.26027
$204$	0	0
$205$	−0.199883	−0.0139604
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.16707	−0.357414
$210$	0	0
$211$	−4.85716	−0.334381	−0.167190	−	0.985925i	$-0.553469\pi$
$211$	−4.85716	−0.334381	−0.167190	+	0.985925i	$0.553469\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.83293	0.329603
$216$	0	0
$217$	−5.16707	−0.350764
$218$	0	0
$219$	0	0
$220$	0	0
$221$	30.5825	2.05720
$222$	0	0
$223$	11.8748	0.795194	0.397597	−	0.917560i	$-0.369844\pi$
$223$	11.8748	0.795194	0.397597	+	0.917560i	$0.369844\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.0858	−0.802163	−0.401081	−	0.916042i	$-0.631366\pi$
$227$	−12.0858	−0.802163	−0.401081	+	0.916042i	$0.631366\pi$
$228$	0	0
$229$	−6.71224	−0.443557	−0.221779	−	0.975097i	$-0.571186\pi$
$229$	−6.71224	−0.443557	−0.221779	+	0.975097i	$0.571186\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.62191	−0.564840	−0.282420	−	0.959291i	$-0.591137\pi$
$233$	−8.62191	−0.564840	−0.282420	+	0.959291i	$0.591137\pi$
$234$	0	0
$235$	4.03733	0.263367
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.79351	0.439436	0.219718	−	0.975563i	$-0.429486\pi$
$239$	6.79351	0.439436	0.219718	+	0.975563i	$0.429486\pi$
$240$	0	0
$241$	18.1716	1.17054	0.585268	−	0.810840i	$-0.300989\pi$
$241$	18.1716	1.17054	0.585268	+	0.810840i	$0.300989\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.95153	−0.124679
$246$	0	0
$247$	−32.2857	−2.05429
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.0747	1.70894	0.854469	−	0.519502i	$-0.173883\pi$
$251$	27.0747	1.70894	0.854469	+	0.519502i	$0.173883\pi$
$252$	0	0
$253$	2.54064	0.159728
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.0373	1.49941	0.749704	−	0.661773i	$-0.230196\pi$
$257$	24.0373	1.49941	0.749704	+	0.661773i	$0.230196\pi$
$258$	0	0
$259$	24.3099	1.51054
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.9934	1.66448	0.832242	−	0.554413i	$-0.187057\pi$
$263$	26.9934	1.66448	0.832242	+	0.554413i	$0.187057\pi$
$264$	0	0
$265$	−0.325096	−0.0199705
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.2902	1.29809	0.649043	−	0.760751i	$-0.275169\pi$
$269$	21.2902	1.29809	0.649043	+	0.760751i	$0.275169\pi$
$270$	0	0
$271$	9.26806	0.562994	0.281497	−	0.959562i	$-0.409169\pi$
$271$	9.26806	0.562994	0.281497	+	0.959562i	$0.409169\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.78899	0.288787
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.9364	1.42792	0.713962	−	0.700185i	$-0.246899\pi$
$281$	23.9364	1.42792	0.713962	+	0.700185i	$0.246899\pi$
$282$	0	0
$283$	−30.1999	−1.79520	−0.897598	−	0.440814i	$-0.854690\pi$
$283$	−30.1999	−1.79520	−0.897598	+	0.440814i	$0.854690\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.45936	0.0861435
$288$	0	0
$289$	6.95606	0.409180
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.7450	1.79614	0.898072	−	0.439848i	$-0.144968\pi$
$293$	30.7450	1.79614	0.898072	+	0.439848i	$0.144968\pi$
$294$	0	0
$295$	5.09241	0.296491
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.8748	0.918063
$300$	0	0
$301$	−35.2857	−2.03383
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.96719	−0.169901
$306$	0	0
$307$	4.55826	0.260154	0.130077	−	0.991504i	$-0.458478\pi$
$307$	4.55826	0.260154	0.130077	+	0.991504i	$0.458478\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.00453	0.113666	0.0568331	−	0.998384i	$-0.481900\pi$
$311$	2.00453	0.113666	0.0568331	+	0.998384i	$0.481900\pi$
$312$	0	0
$313$	−21.7496	−1.22936	−0.614679	−	0.788777i	$-0.710715\pi$
$313$	−21.7496	−1.22936	−0.614679	+	0.788777i	$0.710715\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.29229	0.0725821	0.0362911	−	0.999341i	$-0.488446\pi$
$317$	1.29229	0.0725821	0.0362911	+	0.999341i	$0.488446\pi$
$318$	0	0
$319$	5.35386	0.299758
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.2902	−1.40718
$324$	0	0
$325$	29.9233	1.65984
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−29.4770	−1.62512
$330$	0	0
$331$	−5.21554	−0.286672	−0.143336	−	0.989674i	$-0.545783\pi$
$331$	−5.21554	−0.286672	−0.143336	+	0.989674i	$0.545783\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.96719	0.162115
$336$	0	0
$337$	0.621911	0.0338777	0.0169388	−	0.999857i	$-0.494608\pi$
$337$	0.621911	0.0338777	0.0169388	+	0.999857i	$0.494608\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.54064	0.0834302
$342$	0	0
$343$	−9.22864	−0.498300
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.03281	−0.0554440	−0.0277220	−	0.999616i	$-0.508825\pi$
$347$	−1.03281	−0.0554440	−0.0277220	+	0.999616i	$0.508825\pi$
$348$	0	0
$349$	0.947008	0.0506922	0.0253461	−	0.999679i	$-0.491931\pi$
$349$	0.947008	0.0506922	0.0253461	+	0.999679i	$0.491931\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.123135	−0.00655380	−0.00327690	−	0.999995i	$-0.501043\pi$
$353$	−0.123135	−0.00655380	−0.00327690	+	0.999995i	$0.501043\pi$
$354$	0	0
$355$	−3.00661	−0.159574
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.7032	0.828782	0.414391	−	0.910099i	$-0.363995\pi$
$359$	15.7032	0.828782	0.414391	+	0.910099i	$0.363995\pi$
$360$	0	0
$361$	7.69866	0.405193
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.99339	−0.261366
$366$	0	0
$367$	−18.4482	−0.962990	−0.481495	−	0.876449i	$-0.659906\pi$
$367$	−18.4482	−0.962990	−0.481495	+	0.876449i	$0.659906\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.37356	0.123229
$372$	0	0
$373$	30.1605	1.56165	0.780825	−	0.624750i	$-0.214799\pi$
$373$	30.1605	1.56165	0.780825	+	0.624750i	$0.214799\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.4528	1.72290
$378$	0	0
$379$	−4.67038	−0.239901	−0.119951	−	0.992780i	$-0.538274\pi$
$379$	−4.67038	−0.239901	−0.119951	+	0.992780i	$0.538274\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.45484	0.0743387	0.0371693	−	0.999309i	$-0.488166\pi$
$383$	1.45484	0.0743387	0.0371693	+	0.999309i	$0.488166\pi$
$384$	0	0
$385$	1.54064	0.0785181
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.74052	0.392460	0.196230	−	0.980558i	$-0.437130\pi$
$389$	7.74052	0.392460	0.196230	+	0.980558i	$0.437130\pi$
$390$	0	0
$391$	12.4351	0.628872
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.336228	0.0169175
$396$	0	0
$397$	14.1716	0.711252	0.355626	−	0.934628i	$-0.384268\pi$
$397$	14.1716	0.711252	0.355626	+	0.934628i	$0.384268\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.5295	−0.975257	−0.487628	−	0.873051i	$-0.662138\pi$
$401$	−19.5295	−0.975257	−0.487628	+	0.873051i	$0.662138\pi$
$402$	0	0
$403$	9.62644	0.479527
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.24835	−0.359287
$408$	0	0
$409$	−32.0747	−1.58599	−0.792995	−	0.609228i	$-0.791479\pi$
$409$	−32.0747	−1.58599	−0.792995	+	0.609228i	$0.791479\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−37.1802	−1.82952
$414$	0	0
$415$	−7.19536	−0.353206
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.3760	1.43511	0.717556	−	0.696501i	$-0.245261\pi$
$419$	29.3760	1.43511	0.717556	+	0.696501i	$0.245261\pi$
$420$	0	0
$421$	−4.12974	−0.201271	−0.100636	−	0.994923i	$-0.532088\pi$
$421$	−4.12974	−0.201271	−0.100636	+	0.994923i	$0.532088\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.4397	1.13699
$426$	0	0
$427$	21.6638	1.04838
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.2483	0.493646	0.246823	−	0.969061i	$-0.420613\pi$
$431$	10.2483	0.493646	0.246823	+	0.969061i	$0.420613\pi$
$432$	0	0
$433$	−19.4684	−0.935592	−0.467796	−	0.883836i	$-0.654952\pi$
$433$	−19.4684	−0.935592	−0.467796	+	0.883836i	$0.654952\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.1277	−0.627981
$438$	0	0
$439$	23.0086	1.09814	0.549070	−	0.835777i	$-0.314982\pi$
$439$	23.0086	1.09814	0.549070	+	0.835777i	$0.314982\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.24382	0.0590958	0.0295479	−	0.999563i	$-0.490593\pi$
$443$	1.24382	0.0590958	0.0295479	+	0.999563i	$0.490593\pi$
$444$	0	0
$445$	8.07467	0.382776
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.2483	−1.23874	−0.619368	−	0.785101i	$-0.712611\pi$
$449$	−26.2483	−1.23874	−0.619368	+	0.785101i	$0.712611\pi$
$450$	0	0
$451$	−0.435130	−0.0204895
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.62644	0.451294
$456$	0	0
$457$	16.1252	0.754306	0.377153	−	0.926151i	$-0.376903\pi$
$457$	16.1252	0.754306	0.377153	+	0.926151i	$0.376903\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.3715	1.32139	0.660696	−	0.750654i	$-0.270261\pi$
$461$	28.3715	1.32139	0.660696	+	0.750654i	$0.270261\pi$
$462$	0	0
$463$	8.82179	0.409984	0.204992	−	0.978764i	$-0.434283\pi$
$463$	8.82179	0.409984	0.204992	+	0.978764i	$0.434283\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.5406	1.78345	0.891724	−	0.452580i	$-0.149496\pi$
$467$	38.5406	1.78345	0.891724	+	0.452580i	$0.149496\pi$
$468$	0	0
$469$	−21.6638	−1.00034
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.5209	0.483753
$474$	0	0
$475$	−24.7450	−1.13538
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.9121	−0.635661	−0.317831	−	0.948148i	$-0.602954\pi$
$479$	−13.9121	−0.635661	−0.317831	+	0.948148i	$0.602954\pi$
$480$	0	0
$481$	−45.2902	−2.06506
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.92325	−0.178146
$486$	0	0
$487$	−24.1716	−1.09532	−0.547660	−	0.836701i	$-0.684481\pi$
$487$	−24.1716	−1.09532	−0.547660	+	0.836701i	$0.684481\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.3736	−0.738929	−0.369464	−	0.929245i	$-0.620459\pi$
$491$	−16.3736	−0.738929	−0.369464	+	0.929245i	$0.620459\pi$
$492$	0	0
$493$	26.2044	1.18019
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.9515	0.984661
$498$	0	0
$499$	−18.0858	−0.809632	−0.404816	−	0.914398i	$-0.632664\pi$
$499$	−18.0858	−0.809632	−0.404816	+	0.914398i	$0.632664\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	37.9606	1.69258	0.846289	−	0.532723i	$-0.178831\pi$
$503$	37.9606	1.69258	0.846289	+	0.532723i	$0.178831\pi$
$504$	0	0
$505$	0.507829	0.0225981
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.33623	0.103551	0.0517757	−	0.998659i	$-0.483512\pi$
$509$	2.33623	0.103551	0.0517757	+	0.998659i	$0.483512\pi$
$510$	0	0
$511$	36.4573	1.61278
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.27663	−0.100320
$516$	0	0
$517$	8.78899	0.386539
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.16707	−0.401617	−0.200808	−	0.979631i	$-0.564357\pi$
$521$	−9.16707	−0.401617	−0.200808	+	0.979631i	$0.564357\pi$
$522$	0	0
$523$	−17.7738	−0.777194	−0.388597	−	0.921408i	$-0.627040\pi$
$523$	−17.7738	−0.777194	−0.388597	+	0.921408i	$0.627040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.54064	0.328475
$528$	0	0
$529$	−16.5452	−0.719355
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.71884	−0.117766
$534$	0	0
$535$	−0.285683	−0.0123512
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.24835	−0.182989
$540$	0	0
$541$	38.1322	1.63943	0.819715	−	0.572771i	$-0.194132\pi$
$541$	38.1322	1.63943	0.819715	+	0.572771i	$0.194132\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.98434	0.299176
$546$	0	0
$547$	−36.1474	−1.54555	−0.772775	−	0.634680i	$-0.781132\pi$
$547$	−36.1474	−1.54555	−0.772775	+	0.634680i	$0.781132\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.6638	−1.17852
$552$	0	0
$553$	−2.45484	−0.104390
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.3584	1.54055	0.770277	−	0.637709i	$-0.220118\pi$
$557$	36.3584	1.54055	0.770277	+	0.637709i	$0.220118\pi$
$558$	0	0
$559$	65.7384	2.78044
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.7824	1.29732	0.648661	−	0.761077i	$-0.275329\pi$
$563$	30.7824	1.29732	0.648661	+	0.761077i	$0.275329\pi$
$564$	0	0
$565$	−7.19536	−0.302711
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.21554	0.218647	0.109323	−	0.994006i	$-0.465132\pi$
$569$	5.21554	0.218647	0.109323	+	0.994006i	$0.465132\pi$
$570$	0	0
$571$	−21.1913	−0.886829	−0.443414	−	0.896317i	$-0.646233\pi$
$571$	−21.1913	−0.886829	−0.443414	+	0.896317i	$0.646233\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.1671	0.507402
$576$	0	0
$577$	−26.0813	−1.08578	−0.542889	−	0.839805i	$-0.682670\pi$
$577$	−26.0813	−1.08578	−0.542889	+	0.839805i	$0.682670\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	52.5340	2.17948
$582$	0	0
$583$	−0.707712	−0.0293104
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.96267	0.122282	0.0611411	−	0.998129i	$-0.480526\pi$
$587$	2.96267	0.122282	0.0611411	+	0.998129i	$0.480526\pi$
$588$	0	0
$589$	−7.96059	−0.328010
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.13832	−0.169940	−0.0849701	−	0.996383i	$-0.527079\pi$
$593$	−4.13832	−0.169940	−0.0849701	+	0.996383i	$0.527079\pi$
$594$	0	0
$595$	7.54064	0.309136
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.4684	−0.958893	−0.479447	−	0.877571i	$-0.659163\pi$
$599$	−23.4684	−0.958893	−0.479447	+	0.877571i	$0.659163\pi$
$600$	0	0
$601$	−1.66377	−0.0678667	−0.0339333	−	0.999424i	$-0.510803\pi$
$601$	−1.66377	−0.0678667	−0.0339333	+	0.999424i	$0.510803\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.459363	−0.0186758
$606$	0	0
$607$	41.6638	1.69108	0.845540	−	0.533912i	$-0.179279\pi$
$607$	41.6638	1.69108	0.845540	+	0.533912i	$0.179279\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	54.9166	2.22169
$612$	0	0
$613$	−0.239296	−0.00966508	−0.00483254	−	0.999988i	$-0.501538\pi$
$613$	−0.239296	−0.00966508	−0.00483254	+	0.999988i	$0.501538\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.62852	0.146079	0.0730393	−	0.997329i	$-0.476730\pi$
$617$	3.62852	0.146079	0.0730393	+	0.997329i	$0.476730\pi$
$618$	0	0
$619$	−8.73599	−0.351129	−0.175565	−	0.984468i	$-0.556175\pi$
$619$	−8.73599	−0.351129	−0.175565	+	0.984468i	$0.556175\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−58.9540	−2.36194
$624$	0	0
$625$	21.8793	0.875172
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−35.4770	−1.41456
$630$	0	0
$631$	−19.5012	−0.776331	−0.388166	−	0.921590i	$-0.626891\pi$
$631$	−19.5012	−0.776331	−0.388166	+	0.921590i	$0.626891\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.20441	−0.206531
$636$	0	0
$637$	−26.5452	−1.05176
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.70319	0.383253	0.191626	−	0.981468i	$-0.438624\pi$
$641$	9.70319	0.383253	0.191626	+	0.981468i	$0.438624\pi$
$642$	0	0
$643$	−46.7824	−1.84492	−0.922458	−	0.386096i	$-0.873823\pi$
$643$	−46.7824	−1.84492	−0.922458	+	0.386096i	$0.873823\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.6592	−0.419058	−0.209529	−	0.977802i	$-0.567193\pi$
$647$	−10.6592	−0.419058	−0.209529	+	0.977802i	$0.567193\pi$
$648$	0	0
$649$	11.0858	0.435156
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.8375	0.854566	0.427283	−	0.904118i	$-0.359471\pi$
$653$	21.8375	0.854566	0.427283	+	0.904118i	$0.359471\pi$
$654$	0	0
$655$	5.76278	0.225171
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.0858	−0.392887	−0.196444	−	0.980515i	$-0.562939\pi$
$659$	−10.0858	−0.392887	−0.196444	+	0.980515i	$0.562939\pi$
$660$	0	0
$661$	−5.70771	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$661$	−5.70771	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.96059	−0.308698
$666$	0	0
$667$	13.6022	0.526679
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.45936	−0.249361
$672$	0	0
$673$	23.2812	0.897423	0.448711	−	0.893677i	$-0.351883\pi$
$673$	23.2812	0.897423	0.448711	+	0.893677i	$0.351883\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.1847	0.852627	0.426314	−	0.904575i	$-0.359812\pi$
$677$	22.1847	0.852627	0.426314	+	0.904575i	$0.359812\pi$
$678$	0	0
$679$	28.6441	1.09926
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.4548	−0.400043	−0.200022	−	0.979791i	$-0.564101\pi$
$683$	−10.4548	−0.400043	−0.200022	+	0.979791i	$0.564101\pi$
$684$	0	0
$685$	3.15594	0.120582
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.42203	−0.168466
$690$	0	0
$691$	31.6355	1.20347	0.601735	−	0.798696i	$-0.294476\pi$
$691$	31.6355	1.20347	0.601735	+	0.798696i	$0.294476\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.00453	0.341561
$696$	0	0
$697$	−2.12974	−0.0806697
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.3009	−0.842292	−0.421146	−	0.906993i	$-0.638372\pi$
$701$	−22.3009	−0.842292	−0.421146	+	0.906993i	$0.638372\pi$
$702$	0	0
$703$	37.4528	1.41256
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.70771	−0.139443
$708$	0	0
$709$	−15.2483	−0.572664	−0.286332	−	0.958131i	$-0.592436\pi$
$709$	−15.2483	−0.572664	−0.286332	+	0.958131i	$0.592436\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.91420	0.146588
$714$	0	0
$715$	−2.87026	−0.107342
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.4573	0.762928	0.381464	−	0.924384i	$-0.375420\pi$
$719$	20.4573	0.762928	0.381464	+	0.924384i	$0.375420\pi$
$720$	0	0
$721$	16.6219	0.619032
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25.6395	0.952229
$726$	0	0
$727$	6.28776	0.233200	0.116600	−	0.993179i	$-0.462800\pi$
$727$	6.28776	0.233200	0.116600	+	0.993179i	$0.462800\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	51.4946	1.90460
$732$	0	0
$733$	47.0398	1.73745	0.868727	−	0.495291i	$-0.164939\pi$
$733$	47.0398	1.73745	0.868727	+	0.495291i	$0.164939\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.45936	0.237934
$738$	0	0
$739$	−27.0176	−0.993859	−0.496930	−	0.867791i	$-0.665539\pi$
$739$	−27.0176	−0.993859	−0.496930	+	0.867791i	$0.665539\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.0024	−1.43086	−0.715431	−	0.698684i	$-0.753769\pi$
$743$	−39.0024	−1.43086	−0.715431	+	0.698684i	$0.753769\pi$
$744$	0	0
$745$	10.0262	0.367332
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.08580	0.0762135
$750$	0	0
$751$	−27.8263	−1.01540	−0.507698	−	0.861535i	$-0.669504\pi$
$751$	−27.8263	−1.01540	−0.507698	+	0.861535i	$0.669504\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.633043	0.0230388
$756$	0	0
$757$	20.0813	0.729866	0.364933	−	0.931034i	$-0.381092\pi$
$757$	20.0813	0.729866	0.364933	+	0.931034i	$0.381092\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.1277	0.838377	0.419189	−	0.907899i	$-0.362315\pi$
$761$	23.1277	0.838377	0.419189	+	0.907899i	$0.362315\pi$
$762$	0	0
$763$	−50.9934	−1.84608
$764$	0	0
$765$	0	0
$766$	0	0
$767$	69.2679	2.50112
$768$	0	0
$769$	−28.2372	−1.01826	−0.509130	−	0.860690i	$-0.670033\pi$
$769$	−28.2372	−1.01826	−0.509130	+	0.860690i	$0.670033\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.5340	−1.24210	−0.621051	−	0.783770i	$-0.713294\pi$
$773$	−34.5340	−1.24210	−0.621051	+	0.783770i	$0.713294\pi$
$774$	0	0
$775$	7.37809	0.265029
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.24835	0.0805554
$780$	0	0
$781$	−6.54516	−0.234204
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.71884	0.239806
$786$	0	0
$787$	−14.1605	−0.504766	−0.252383	−	0.967627i	$-0.581214\pi$
$787$	−14.1605	−0.504766	−0.252383	+	0.967627i	$0.581214\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	52.5340	1.86790
$792$	0	0
$793$	−40.3604	−1.43324
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.3387	0.543324	0.271662	−	0.962393i	$-0.412427\pi$
$797$	15.3387	0.543324	0.271662	+	0.962393i	$0.412427\pi$
$798$	0	0
$799$	43.0176	1.52185
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.8703	−0.383603
$804$	0	0
$805$	3.91420	0.137957
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.6001	1.11100	0.555501	−	0.831516i	$-0.312527\pi$
$809$	31.6001	1.11100	0.555501	+	0.831516i	$0.312527\pi$
$810$	0	0
$811$	4.98029	0.174882	0.0874409	−	0.996170i	$-0.472131\pi$
$811$	4.98029	0.174882	0.0874409	+	0.996170i	$0.472131\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.23722	0.218480
$816$	0	0
$817$	−54.3624	−1.90190
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.9692	−0.871430	−0.435715	−	0.900085i	$-0.643504\pi$
$821$	−24.9692	−0.871430	−0.435715	+	0.900085i	$0.643504\pi$
$822$	0	0
$823$	29.1297	1.01540	0.507699	−	0.861534i	$-0.330496\pi$
$823$	29.1297	1.01540	0.507699	+	0.861534i	$0.330496\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.3230	−1.12398	−0.561991	−	0.827144i	$-0.689964\pi$
$827$	−32.3230	−1.12398	−0.561991	+	0.827144i	$0.689964\pi$
$828$	0	0
$829$	−51.9076	−1.80283	−0.901413	−	0.432961i	$-0.857469\pi$
$829$	−51.9076	−1.80283	−0.901413	+	0.432961i	$0.857469\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.7935	−0.720452
$834$	0	0
$835$	3.29229	0.113934
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.4266	−0.429012	−0.214506	−	0.976723i	$-0.568814\pi$
$839$	−12.4266	−0.429012	−0.214506	+	0.976723i	$0.568814\pi$
$840$	0	0
$841$	−0.336228	−0.0115941
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.9627	−0.411528
$846$	0	0
$847$	3.35386	0.115240
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.4154	−0.631273
$852$	0	0
$853$	20.7077	0.709019	0.354509	−	0.935052i	$-0.384648\pi$
$853$	20.7077	0.709019	0.354509	+	0.935052i	$0.384648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.8834	−0.918318	−0.459159	−	0.888354i	$-0.651849\pi$
$857$	−26.8834	−0.918318	−0.459159	+	0.888354i	$0.651849\pi$
$858$	0	0
$859$	−4.19988	−0.143298	−0.0716491	−	0.997430i	$-0.522826\pi$
$859$	−4.19988	−0.143298	−0.0716491	+	0.997430i	$0.522826\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.66132	0.0565522	0.0282761	−	0.999600i	$-0.490998\pi$
$863$	1.66132	0.0565522	0.0282761	+	0.999600i	$0.490998\pi$
$864$	0	0
$865$	8.28568	0.281722
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.731945	0.0248295
$870$	0	0
$871$	40.3604	1.36756
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.0813	0.509840
$876$	0	0
$877$	12.1231	0.409369	0.204685	−	0.978828i	$-0.434383\pi$
$877$	12.1231	0.409369	0.204685	+	0.978828i	$0.434383\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.0858	0.339799	0.169900	−	0.985461i	$-0.445656\pi$
$881$	10.0858	0.339799	0.169900	+	0.985461i	$0.445656\pi$
$882$	0	0
$883$	−31.1297	−1.04760	−0.523800	−	0.851842i	$-0.675486\pi$
$883$	−31.1297	−1.04760	−0.523800	+	0.851842i	$0.675486\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.6704	−0.694043	−0.347022	−	0.937857i	$-0.612807\pi$
$887$	−20.6704	−0.694043	−0.347022	+	0.937857i	$0.612807\pi$
$888$	0	0
$889$	37.9979	1.27441
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−45.4133	−1.51970
$894$	0	0
$895$	9.28116	0.310235
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.24835	0.275098
$900$	0	0
$901$	−3.46389	−0.115399
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.37809	−0.0458092
$906$	0	0
$907$	−34.0858	−1.13180	−0.565900	−	0.824474i	$-0.691471\pi$
$907$	−34.0858	−1.13180	−0.565900	+	0.824474i	$0.691471\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.48764	0.0492879	0.0246439	−	0.999696i	$-0.492155\pi$
$911$	1.48764	0.0492879	0.0246439	+	0.999696i	$0.492155\pi$
$912$	0	0
$913$	−15.6638	−0.518395
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−42.0747	−1.38943
$918$	0	0
$919$	35.9384	1.18550	0.592750	−	0.805387i	$-0.298042\pi$
$919$	35.9384	1.18550	0.592750	+	0.805387i	$0.298042\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−40.8965	−1.34612
$924$	0	0
$925$	−34.7122	−1.14133
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.88592	−0.258729	−0.129364	−	0.991597i	$-0.541294\pi$
$929$	−7.88592	−0.258729	−0.129364	+	0.991597i	$0.541294\pi$
$930$	0	0
$931$	21.9515	0.719432
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.24835	−0.0735289
$936$	0	0
$937$	26.3230	0.859935	0.429968	−	0.902844i	$-0.358525\pi$
$937$	26.3230	0.859935	0.429968	+	0.902844i	$0.358525\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.7365	−0.838985	−0.419492	−	0.907759i	$-0.637792\pi$
$941$	−25.7365	−0.838985	−0.419492	+	0.907759i	$0.637792\pi$
$942$	0	0
$943$	−1.10551	−0.0360003
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.8793	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$947$	−36.8793	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$948$	0	0
$949$	−67.9212	−2.20481
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.2705	1.43406	0.717031	−	0.697041i	$-0.245501\pi$
$953$	44.2705	1.43406	0.717031	+	0.697041i	$0.245501\pi$
$954$	0	0
$955$	3.96267	0.128229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23.0419	−0.744060
$960$	0	0
$961$	−28.6264	−0.923433
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.29229	−0.234747
$966$	0	0
$967$	−16.5956	−0.533678	−0.266839	−	0.963741i	$-0.585979\pi$
$967$	−16.5956	−0.533678	−0.266839	+	0.963741i	$0.585979\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.4088	0.719133	0.359567	−	0.933119i	$-0.382925\pi$
$971$	22.4088	0.719133	0.359567	+	0.933119i	$0.382925\pi$
$972$	0	0
$973$	−65.7430	−2.10762
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.4967	−0.335819	−0.167910	−	0.985802i	$-0.553702\pi$
$977$	−10.4967	−0.335819	−0.167910	+	0.985802i	$0.553702\pi$
$978$	0	0
$979$	17.5780	0.561794
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.0771	1.62911	0.814553	−	0.580088i	$-0.196982\pi$
$983$	51.0771	1.62911	0.814553	+	0.580088i	$0.196982\pi$
$984$	0	0
$985$	7.30342	0.232706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.7299	0.849960
$990$	0	0
$991$	10.1625	0.322824	0.161412	−	0.986887i	$-0.448395\pi$
$991$	10.1625	0.322824	0.161412	+	0.986887i	$0.448395\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.136346	−0.00432245
$996$	0	0
$997$	53.3760	1.69044	0.845218	−	0.534422i	$-0.179471\pi$
$997$	53.3760	1.69044	0.845218	+	0.534422i	$0.179471\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.l.1.2	✓	3
3.2	odd	2		2376.2.a.q.1.2	yes	3
4.3	odd	2		4752.2.a.bh.1.2		3
12.11	even	2		4752.2.a.bp.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.l.1.2	✓	3	1.1	even	1	trivial
2376.2.a.q.1.2	yes	3	3.2	odd	2
4752.2.a.bh.1.2		3	4.3	odd	2
4752.2.a.bp.1.2		3	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)

\(f(q)\)	\(=\)	\(q-0.459363 q^{5} +3.35386 q^{7} -1.00000 q^{11} -6.24835 q^{13} -4.89449 q^{17} +5.16707 q^{19} -2.54064 q^{23} -4.78899 q^{25} -5.35386 q^{29} -1.54064 q^{31} -1.54064 q^{35} +7.24835 q^{37} +0.435130 q^{41} -10.5209 q^{43} -8.78899 q^{47} +4.24835 q^{49} +0.707712 q^{53} +0.459363 q^{55} -11.0858 q^{59} +6.45936 q^{61} +2.87026 q^{65} -6.45936 q^{67} +6.54516 q^{71} +10.8703 q^{73} -3.35386 q^{77} -0.731945 q^{79} +15.6638 q^{83} +2.24835 q^{85} -17.5780 q^{89} -20.9561 q^{91} -2.37356 q^{95} +8.54064 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{17} - 2 q^{19} - 7 q^{23} + 5 q^{25} - 7 q^{29} - 4 q^{31} - 4 q^{35} + 3 q^{37} - 9 q^{41} - 5 q^{43} - 7 q^{47} - 6 q^{49} - 16 q^{53} + 2 q^{55} - 17 q^{59} + 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 + q^7 - 3 * q^11 - 5 * q^17 - 2 * q^19 - 7 * q^23 + 5 * q^25 - 7 * q^29 - 4 * q^31 - 4 * q^35 + 3 * q^37 - 9 * q^41 - 5 * q^43 - 7 * q^47 - 6 * q^49 - 16 * q^53 + 2 * q^55 - 17 * q^59 + 20 * q^61 - 12 * q^65 - 20 * q^67 + 4 * q^71 + 12 * q^73 - q^77 + 5 * q^79 - 8 * q^83 - 12 * q^85 - 14 * q^89 - 26 * q^91 - 24 * q^95 + 25 * q^97

Introduction

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