LMFDB - Embedded newform 2376.2.a.l.1.3

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			1.3
Root			$-1.87740$ 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)

$f(q)$	$=$	$q+2.27945 q^{5} -1.87740 q^{7} -1.00000 q^{11} +1.47536 q^{13} -2.40205 q^{17} -8.03426 q^{19} -5.27945 q^{23} +0.195903 q^{25} -0.122596 q^{29} -4.27945 q^{31} -4.27945 q^{35} -0.475355 q^{37} +0.681501 q^{41} +7.91166 q^{43} -3.80410 q^{47} -3.47536 q^{49} -9.75481 q^{53} -2.27945 q^{55} +7.59316 q^{59} +3.72055 q^{61} +3.36300 q^{65} -3.72055 q^{67} -14.8726 q^{71} +11.3630 q^{73} +1.87740 q^{77} +12.7158 q^{79} -12.9850 q^{83} -5.47536 q^{85} -7.60819 q^{89} -2.76984 q^{91} -18.3137 q^{95} +11.2795 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.27945	1.01940	0.509701	−	0.860352i	$-0.329756\pi$
$5$	2.27945	1.01940	0.509701	+	0.860352i	$0.329756\pi$
$6$	0	0
$7$	−1.87740	−0.709592	−0.354796	−	0.934944i	$-0.615450\pi$
$7$	−1.87740	−0.709592	−0.354796	+	0.934944i	$0.615450\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.47536	0.409190	0.204595	−	0.978847i	$-0.434412\pi$
$13$	1.47536	0.409190	0.204595	+	0.978847i	$0.434412\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.40205	−0.582582	−0.291291	−	0.956634i	$-0.594085\pi$
$17$	−2.40205	−0.582582	−0.291291	+	0.956634i	$0.594085\pi$
$18$	0	0
$19$	−8.03426	−1.84319	−0.921593	−	0.388158i	$-0.873111\pi$
$19$	−8.03426	−1.84319	−0.921593	+	0.388158i	$0.873111\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.27945	−1.10084	−0.550421	−	0.834887i	$-0.685533\pi$
$23$	−5.27945	−1.10084	−0.550421	+	0.834887i	$0.685533\pi$
$24$	0	0
$25$	0.195903	0.0391806
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.122596	−0.0227656	−0.0113828	−	0.999935i	$-0.503623\pi$
$29$	−0.122596	−0.0227656	−0.0113828	+	0.999935i	$0.503623\pi$
$30$	0	0
$31$	−4.27945	−0.768612	−0.384306	−	0.923206i	$-0.625559\pi$
$31$	−4.27945	−0.768612	−0.384306	+	0.923206i	$0.625559\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.27945	−0.723359
$36$	0	0
$37$	−0.475355	−0.0781479	−0.0390740	−	0.999236i	$-0.512441\pi$
$37$	−0.475355	−0.0781479	−0.0390740	+	0.999236i	$0.512441\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.681501	0.106433	0.0532163	−	0.998583i	$-0.483053\pi$
$41$	0.681501	0.106433	0.0532163	+	0.998583i	$0.483053\pi$
$42$	0	0
$43$	7.91166	1.20652	0.603259	−	0.797546i	$-0.293869\pi$
$43$	7.91166	1.20652	0.603259	+	0.797546i	$0.293869\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.80410	−0.554885	−0.277442	−	0.960742i	$-0.589487\pi$
$47$	−3.80410	−0.554885	−0.277442	+	0.960742i	$0.589487\pi$
$48$	0	0
$49$	−3.47536	−0.496479
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.75481	−1.33993	−0.669963	−	0.742395i	$-0.733690\pi$
$53$	−9.75481	−1.33993	−0.669963	+	0.742395i	$0.733690\pi$
$54$	0	0
$55$	−2.27945	−0.307361
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.59316	0.988546	0.494273	−	0.869307i	$-0.335434\pi$
$59$	7.59316	0.988546	0.494273	+	0.869307i	$0.335434\pi$
$60$	0	0
$61$	3.72055	0.476367	0.238184	−	0.971220i	$-0.423448\pi$
$61$	3.72055	0.476367	0.238184	+	0.971220i	$0.423448\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.36300	0.417129
$66$	0	0
$67$	−3.72055	−0.454537	−0.227269	−	0.973832i	$-0.572980\pi$
$67$	−3.72055	−0.454537	−0.227269	+	0.973832i	$0.572980\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.8726	−1.76505	−0.882527	−	0.470261i	$-0.844160\pi$
$71$	−14.8726	−1.76505	−0.882527	+	0.470261i	$0.844160\pi$
$72$	0	0
$73$	11.3630	1.32994	0.664969	−	0.746871i	$-0.268445\pi$
$73$	11.3630	1.32994	0.664969	+	0.746871i	$0.268445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.87740	0.213950
$78$	0	0
$79$	12.7158	1.43063	0.715317	−	0.698800i	$-0.246282\pi$
$79$	12.7158	1.43063	0.715317	+	0.698800i	$0.246282\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.9850	−1.42529	−0.712643	−	0.701527i	$-0.752502\pi$
$83$	−12.9850	−1.42529	−0.712643	+	0.701527i	$0.752502\pi$
$84$	0	0
$85$	−5.47536	−0.593886
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.60819	−0.806467	−0.403233	−	0.915097i	$-0.632114\pi$
$89$	−7.60819	−0.806467	−0.403233	+	0.915097i	$0.632114\pi$
$90$	0	0
$91$	−2.76984	−0.290358
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−18.3137	−1.87895
$96$	0	0
$97$	11.2795	1.14525	0.572627	−	0.819816i	$-0.305924\pi$
$97$	11.2795	1.14525	0.572627	+	0.819816i	$0.305924\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.59795	−0.358010	−0.179005	−	0.983848i	$-0.557288\pi$
$101$	−3.59795	−0.358010	−0.179005	+	0.983848i	$0.557288\pi$
$102$	0	0
$103$	−13.2302	−1.30361	−0.651803	−	0.758388i	$-0.725987\pi$
$103$	−13.2302	−1.30361	−0.651803	+	0.758388i	$0.725987\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.83836	0.854436	0.427218	−	0.904149i	$-0.359494\pi$
$107$	8.83836	0.854436	0.427218	+	0.904149i	$0.359494\pi$
$108$	0	0
$109$	10.7055	1.02540	0.512701	−	0.858567i	$-0.328645\pi$
$109$	10.7055	1.02540	0.512701	+	0.858567i	$0.328645\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.9850	−1.22152	−0.610762	−	0.791815i	$-0.709137\pi$
$113$	−12.9850	−1.22152	−0.610762	+	0.791815i	$0.709137\pi$
$114$	0	0
$115$	−12.0343	−1.12220
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.50961	0.413396
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.9507	−0.979461
$126$	0	0
$127$	9.08355	0.806035	0.403017	−	0.915192i	$-0.367961\pi$
$127$	9.08355	0.806035	0.403017	+	0.915192i	$0.367961\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.87262	0.775204	0.387602	−	0.921827i	$-0.373303\pi$
$131$	8.87262	0.775204	0.387602	+	0.921827i	$0.373303\pi$
$132$	0	0
$133$	15.0835	1.30791
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.36300	−0.629064	−0.314532	−	0.949247i	$-0.601848\pi$
$137$	−7.36300	−0.629064	−0.314532	+	0.949247i	$0.601848\pi$
$138$	0	0
$139$	−6.64724	−0.563812	−0.281906	−	0.959442i	$-0.590967\pi$
$139$	−6.64724	−0.563812	−0.281906	+	0.959442i	$0.590967\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.47536	−0.123375
$144$	0	0
$145$	−0.279452	−0.0232073
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.13284	−0.338575	−0.169288	−	0.985567i	$-0.554147\pi$
$149$	−4.13284	−0.338575	−0.169288	+	0.985567i	$0.554147\pi$
$150$	0	0
$151$	6.83836	0.556498	0.278249	−	0.960509i	$-0.410246\pi$
$151$	6.83836	0.556498	0.278249	+	0.960509i	$0.410246\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.75481	−0.783525
$156$	0	0
$157$	1.31371	0.104846	0.0524228	−	0.998625i	$-0.483306\pi$
$157$	1.31371	0.104846	0.0524228	+	0.998625i	$0.483306\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.91166	0.781149
$162$	0	0
$163$	−3.60819	−0.282616	−0.141308	−	0.989966i	$-0.545131\pi$
$163$	−3.60819	−0.282616	−0.141308	+	0.989966i	$0.545131\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.03426	0.466945	0.233473	−	0.972363i	$-0.424991\pi$
$167$	6.03426	0.466945	0.233473	+	0.972363i	$0.424991\pi$
$168$	0	0
$169$	−10.8233	−0.832564
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.32874	−0.405137	−0.202568	−	0.979268i	$-0.564929\pi$
$173$	−5.32874	−0.405137	−0.202568	+	0.979268i	$0.564929\pi$
$174$	0	0
$175$	−0.367789	−0.0278022
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.70552	0.426450	0.213225	−	0.977003i	$-0.431603\pi$
$179$	5.70552	0.426450	0.213225	+	0.977003i	$0.431603\pi$
$180$	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
$185$	−1.08355	−0.0796641
$186$	0	0
$187$	2.40205	0.175655
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.31371	0.529202	0.264601	−	0.964358i	$-0.414760\pi$
$191$	7.31371	0.529202	0.264601	+	0.964358i	$0.414760\pi$
$192$	0	0
$193$	−7.78907	−0.560669	−0.280335	−	0.959902i	$-0.590445\pi$
$193$	−7.78907	−0.560669	−0.280335	+	0.959902i	$0.590445\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.7500	0.765907	0.382954	−	0.923768i	$-0.374907\pi$
$197$	10.7500	0.765907	0.382954	+	0.923768i	$0.374907\pi$
$198$	0	0
$199$	−13.3973	−0.949707	−0.474853	−	0.880065i	$-0.657499\pi$
$199$	−13.3973	−0.949707	−0.474853	+	0.880065i	$0.657499\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.230163	0.0161543
$204$	0	0
$205$	1.55345	0.108498
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.03426	0.555741
$210$	0	0
$211$	−15.0733	−1.03769	−0.518845	−	0.854869i	$-0.673638\pi$
$211$	−15.0733	−1.03769	−0.518845	+	0.854869i	$0.673638\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.0343	1.22993
$216$	0	0
$217$	8.03426	0.545401
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.54387	−0.238387
$222$	0	0
$223$	−11.7891	−0.789454	−0.394727	−	0.918798i	$-0.629161\pi$
$223$	−11.7891	−0.789454	−0.394727	+	0.918798i	$0.629161\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.59316	0.437604	0.218802	−	0.975769i	$-0.429785\pi$
$227$	6.59316	0.437604	0.218802	+	0.975769i	$0.429785\pi$
$228$	0	0
$229$	27.9069	1.84414	0.922069	−	0.387025i	$-0.126497\pi$
$229$	27.9069	1.84414	0.922069	+	0.387025i	$0.126497\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.8384	−1.10312	−0.551559	−	0.834136i	$-0.685967\pi$
$233$	−16.8384	−1.10312	−0.551559	+	0.834136i	$0.685967\pi$
$234$	0	0
$235$	−8.67126	−0.565651
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.3480	−1.44557	−0.722785	−	0.691073i	$-0.757138\pi$
$239$	−22.3480	−1.44557	−0.722785	+	0.691073i	$0.757138\pi$
$240$	0	0
$241$	−19.1863	−1.23590	−0.617950	−	0.786217i	$-0.712037\pi$
$241$	−19.1863	−1.23590	−0.617950	+	0.786217i	$0.712037\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.92191	−0.506112
$246$	0	0
$247$	−11.8534	−0.754213
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.65748	0.104619	0.0523097	−	0.998631i	$-0.483342\pi$
$251$	1.65748	0.104619	0.0523097	+	0.998631i	$0.483342\pi$
$252$	0	0
$253$	5.27945	0.331916
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.3287	0.706668	0.353334	−	0.935497i	$-0.385048\pi$
$257$	11.3287	0.706668	0.353334	+	0.935497i	$0.385048\pi$
$258$	0	0
$259$	0.892434	0.0554531
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.90142	−0.240572	−0.120286	−	0.992739i	$-0.538381\pi$
$263$	−3.90142	−0.240572	−0.120286	+	0.992739i	$0.538381\pi$
$264$	0	0
$265$	−22.2356	−1.36592
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.2987	−1.42055	−0.710273	−	0.703926i	$-0.751428\pi$
$269$	−23.2987	−1.42055	−0.710273	+	0.703926i	$0.751428\pi$
$270$	0	0
$271$	22.7158	1.37988	0.689942	−	0.723865i	$-0.257636\pi$
$271$	22.7158	1.37988	0.689942	+	0.723865i	$0.257636\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.195903	−0.0118134
$276$	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
$281$	−15.4213	−0.919956	−0.459978	−	0.887930i	$-0.652143\pi$
$281$	−15.4213	−0.919956	−0.459978	+	0.887930i	$0.652143\pi$
$282$	0	0
$283$	−28.4466	−1.69097	−0.845486	−	0.533998i	$-0.820689\pi$
$283$	−28.4466	−1.69097	−0.845486	+	0.533998i	$0.820689\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.27945	−0.0755237
$288$	0	0
$289$	−11.2302	−0.660598
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.57393	0.442474	0.221237	−	0.975220i	$-0.428991\pi$
$293$	7.57393	0.442474	0.221237	+	0.975220i	$0.428991\pi$
$294$	0	0
$295$	17.3083	1.00773
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.78907	−0.450453
$300$	0	0
$301$	−14.8534	−0.856135
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.48081	0.485610
$306$	0	0
$307$	−26.5829	−1.51717	−0.758584	−	0.651576i	$-0.774108\pi$
$307$	−26.5829	−1.51717	−0.758584	+	0.651576i	$0.774108\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.1521	−1.25613	−0.628064	−	0.778161i	$-0.716153\pi$
$311$	−22.1521	−1.25613	−0.628064	+	0.778161i	$0.716153\pi$
$312$	0	0
$313$	25.5781	1.44576	0.722881	−	0.690973i	$-0.242818\pi$
$313$	25.5781	1.44576	0.722881	+	0.690973i	$0.242818\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.7548	0.660216	0.330108	−	0.943943i	$-0.392915\pi$
$317$	11.7548	0.660216	0.330108	+	0.943943i	$0.392915\pi$
$318$	0	0
$319$	0.122596	0.00686407
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.2987	1.07381
$324$	0	0
$325$	0.289026	0.0160323
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.14183	0.393742
$330$	0	0
$331$	13.9562	0.767100	0.383550	−	0.923520i	$-0.374701\pi$
$331$	13.9562	0.767100	0.383550	+	0.923520i	$0.374701\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.48081	−0.463356
$336$	0	0
$337$	8.83836	0.481456	0.240728	−	0.970593i	$-0.422614\pi$
$337$	8.83836	0.481456	0.240728	+	0.970593i	$0.422614\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.27945	0.231745
$342$	0	0
$343$	19.6665	1.06189
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.4808	−0.670005	−0.335003	−	0.942217i	$-0.608737\pi$
$347$	−12.4808	−0.670005	−0.335003	+	0.942217i	$0.608737\pi$
$348$	0	0
$349$	31.0740	1.66335	0.831676	−	0.555261i	$-0.187382\pi$
$349$	31.0740	1.66335	0.831676	+	0.555261i	$0.187382\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.2644	1.66404	0.832019	−	0.554748i	$-0.187185\pi$
$353$	31.2644	1.66404	0.832019	+	0.554748i	$0.187185\pi$
$354$	0	0
$355$	−33.9014	−1.79930
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.3973	1.55153	0.775764	−	0.631023i	$-0.217365\pi$
$359$	29.3973	1.55153	0.775764	+	0.631023i	$0.217365\pi$
$360$	0	0
$361$	45.5493	2.39733
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.9014	1.35574
$366$	0	0
$367$	−8.97120	−0.468293	−0.234146	−	0.972201i	$-0.575229\pi$
$367$	−8.97120	−0.468293	−0.234146	+	0.972201i	$0.575229\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.3137	0.950801
$372$	0	0
$373$	−13.9357	−0.721562	−0.360781	−	0.932651i	$-0.617490\pi$
$373$	−13.9357	−0.721562	−0.360781	+	0.932651i	$0.617490\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.180873	−0.00931543
$378$	0	0
$379$	−6.91645	−0.355274	−0.177637	−	0.984096i	$-0.556845\pi$
$379$	−6.91645	−0.355274	−0.177637	+	0.984096i	$0.556845\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.8726	1.16874	0.584368	−	0.811489i	$-0.301342\pi$
$383$	22.8726	1.16874	0.584368	+	0.811489i	$0.301342\pi$
$384$	0	0
$385$	4.27945	0.218101
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.72600	0.442426	0.221213	−	0.975226i	$-0.428998\pi$
$389$	8.72600	0.442426	0.221213	+	0.975226i	$0.428998\pi$
$390$	0	0
$391$	12.6815	0.641331
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.9850	1.45839
$396$	0	0
$397$	−23.1863	−1.16369	−0.581844	−	0.813300i	$-0.697669\pi$
$397$	−23.1863	−1.16369	−0.581844	+	0.813300i	$0.697669\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.5301	−0.775536	−0.387768	−	0.921757i	$-0.626754\pi$
$401$	−15.5301	−0.775536	−0.387768	+	0.921757i	$0.626754\pi$
$402$	0	0
$403$	−6.31371	−0.314508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.475355	0.0235625
$408$	0	0
$409$	−6.65748	−0.329191	−0.164596	−	0.986361i	$-0.552632\pi$
$409$	−6.65748	−0.329191	−0.164596	+	0.986361i	$0.552632\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.2554	−0.701464
$414$	0	0
$415$	−29.5986	−1.45294
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.8918	−1.65573	−0.827863	−	0.560931i	$-0.810443\pi$
$419$	−33.8918	−1.65573	−0.827863	+	0.560931i	$0.810443\pi$
$420$	0	0
$421$	−3.63700	−0.177256	−0.0886282	−	0.996065i	$-0.528248\pi$
$421$	−3.63700	−0.177256	−0.0886282	+	0.996065i	$0.528248\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.470568	−0.0228259
$426$	0	0
$427$	−6.98497	−0.338026
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.52464	0.121608	0.0608039	−	0.998150i	$-0.480634\pi$
$431$	2.52464	0.121608	0.0608039	+	0.998150i	$0.480634\pi$
$432$	0	0
$433$	31.5836	1.51781	0.758905	−	0.651201i	$-0.225735\pi$
$433$	31.5836	1.51781	0.758905	+	0.651201i	$0.225735\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	42.4165	2.02906
$438$	0	0
$439$	37.4418	1.78700	0.893499	−	0.449065i	$-0.148243\pi$
$439$	37.4418	1.78700	0.893499	+	0.449065i	$0.148243\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.6767	0.839846	0.419923	−	0.907560i	$-0.362057\pi$
$443$	17.6767	0.839846	0.419923	+	0.907560i	$0.362057\pi$
$444$	0	0
$445$	−17.3425	−0.822114
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.5246	−0.874232	−0.437116	−	0.899405i	$-0.644000\pi$
$449$	−18.5246	−0.874232	−0.437116	+	0.899405i	$0.644000\pi$
$450$	0	0
$451$	−0.681501	−0.0320906
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.31371	−0.295991
$456$	0	0
$457$	39.7891	1.86125	0.930627	−	0.365969i	$-0.119262\pi$
$457$	39.7891	1.86125	0.930627	+	0.365969i	$0.119262\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.7398	−0.500201	−0.250101	−	0.968220i	$-0.580464\pi$
$461$	−10.7398	−0.500201	−0.250101	+	0.968220i	$0.580464\pi$
$462$	0	0
$463$	15.2849	0.710350	0.355175	−	0.934800i	$-0.384421\pi$
$463$	15.2849	0.710350	0.355175	+	0.934800i	$0.384421\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.2795	1.91019	0.955093	−	0.296307i	$-0.0957553\pi$
$467$	41.2795	1.91019	0.955093	+	0.296307i	$0.0957553\pi$
$468$	0	0
$469$	6.98497	0.322536
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.91166	−0.363779
$474$	0	0
$475$	−1.57393	−0.0722171
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.4603	1.02624	0.513119	−	0.858317i	$-0.328490\pi$
$479$	22.4603	1.02624	0.513119	+	0.858317i	$0.328490\pi$
$480$	0	0
$481$	−0.701318	−0.0319773
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.7110	1.16748
$486$	0	0
$487$	13.1863	0.597530	0.298765	−	0.954327i	$-0.403425\pi$
$487$	13.1863	0.597530	0.298765	+	0.954327i	$0.403425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.3137	−1.45830	−0.729149	−	0.684355i	$-0.760084\pi$
$491$	−32.3137	−1.45830	−0.729149	+	0.684355i	$0.760084\pi$
$492$	0	0
$493$	0.294482	0.0132628
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.9219	1.25247
$498$	0	0
$499$	0.593164	0.0265537	0.0132768	−	0.999912i	$-0.495774\pi$
$499$	0.593164	0.0265537	0.0132768	+	0.999912i	$0.495774\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.38223	−0.195394	−0.0976970	−	0.995216i	$-0.531148\pi$
$503$	−4.38223	−0.195394	−0.0976970	+	0.995216i	$0.531148\pi$
$504$	0	0
$505$	−8.20136	−0.364956
$506$	0	0
$507$	0	0
$508$	0	0
$509$	30.9850	1.37338	0.686692	−	0.726948i	$-0.259062\pi$
$509$	30.9850	1.37338	0.686692	+	0.726948i	$0.259062\pi$
$510$	0	0
$511$	−21.3329	−0.943714
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−30.1575	−1.32890
$516$	0	0
$517$	3.80410	0.167304
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.03426	0.176744	0.0883721	−	0.996088i	$-0.471834\pi$
$521$	4.03426	0.176744	0.0883721	+	0.996088i	$0.471834\pi$
$522$	0	0
$523$	32.5391	1.42283	0.711417	−	0.702770i	$-0.248054\pi$
$523$	32.5391	1.42283	0.711417	+	0.702770i	$0.248054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.2795	0.447780
$528$	0	0
$529$	4.87262	0.211853
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.00546	0.0435511
$534$	0	0
$535$	20.1466	0.871014
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.47536	0.149694
$540$	0	0
$541$	−41.5686	−1.78717	−0.893586	−	0.448892i	$-0.851819\pi$
$541$	−41.5686	−1.78717	−0.893586	+	0.448892i	$0.851819\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	24.4027	1.04530
$546$	0	0
$547$	−1.77462	−0.0758775	−0.0379387	−	0.999280i	$-0.512079\pi$
$547$	−1.77462	−0.0758775	−0.0379387	+	0.999280i	$0.512079\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.984970	0.0419611
$552$	0	0
$553$	−23.8726	−1.01517
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.97053	0.295351	0.147675	−	0.989036i	$-0.452821\pi$
$557$	6.97053	0.295351	0.147675	+	0.989036i	$0.452821\pi$
$558$	0	0
$559$	11.6725	0.493695
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.09732	−0.214827	−0.107413	−	0.994214i	$-0.534257\pi$
$563$	−5.09732	−0.214827	−0.107413	+	0.994214i	$0.534257\pi$
$564$	0	0
$565$	−29.5986	−1.24522
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.9562	−0.585073	−0.292536	−	0.956254i	$-0.594499\pi$
$569$	−13.9562	−0.585073	−0.292536	+	0.956254i	$0.594499\pi$
$570$	0	0
$571$	−5.00479	−0.209444	−0.104722	−	0.994502i	$-0.533395\pi$
$571$	−5.00479	−0.209444	−0.104722	+	0.994502i	$0.533395\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.03426	−0.0431316
$576$	0	0
$577$	−31.5589	−1.31381	−0.656907	−	0.753971i	$-0.728136\pi$
$577$	−31.5589	−1.31381	−0.656907	+	0.753971i	$0.728136\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.3780	1.01137
$582$	0	0
$583$	9.75481	0.404003
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.6713	0.646822	0.323411	−	0.946259i	$-0.395170\pi$
$587$	15.6713	0.646822	0.323411	+	0.946259i	$0.395170\pi$
$588$	0	0
$589$	34.3822	1.41670
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18.0788	−0.742406	−0.371203	−	0.928552i	$-0.621055\pi$
$593$	−18.0788	−0.742406	−0.371203	+	0.928552i	$0.621055\pi$
$594$	0	0
$595$	10.2795	0.421416
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.5836	1.12703	0.563517	−	0.826104i	$-0.309448\pi$
$599$	27.5836	1.12703	0.563517	+	0.826104i	$0.309448\pi$
$600$	0	0
$601$	26.9850	1.10074	0.550370	−	0.834921i	$-0.314487\pi$
$601$	26.9850	1.10074	0.550370	+	0.834921i	$0.314487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.27945	0.0926729
$606$	0	0
$607$	13.0150	0.528264	0.264132	−	0.964487i	$-0.414915\pi$
$607$	13.0150	0.528264	0.264132	+	0.964487i	$0.414915\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.61239	−0.227053
$612$	0	0
$613$	−40.8288	−1.64906	−0.824530	−	0.565819i	$-0.808560\pi$
$613$	−40.8288	−1.64906	−0.824530	+	0.565819i	$0.808560\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.7398	1.72064	0.860319	−	0.509755i	$-0.170264\pi$
$617$	42.7398	1.72064	0.860319	+	0.509755i	$0.170264\pi$
$618$	0	0
$619$	−33.8781	−1.36167	−0.680837	−	0.732435i	$-0.738384\pi$
$619$	−33.8781	−1.36167	−0.680837	+	0.732435i	$0.738384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.2837	0.572262
$624$	0	0
$625$	−25.9411	−1.03765
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.14183	0.0455276
$630$	0	0
$631$	20.1028	0.800279	0.400140	−	0.916454i	$-0.368962\pi$
$631$	20.1028	0.800279	0.400140	+	0.916454i	$0.368962\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20.7055	0.821673
$636$	0	0
$637$	−5.12738	−0.203154
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.3973	0.924136	0.462068	−	0.886844i	$-0.347108\pi$
$641$	23.3973	0.924136	0.462068	+	0.886844i	$0.347108\pi$
$642$	0	0
$643$	−10.9027	−0.429960	−0.214980	−	0.976618i	$-0.568969\pi$
$643$	−10.9027	−0.429960	−0.214980	+	0.976618i	$0.568969\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.16710	−0.242454	−0.121227	−	0.992625i	$-0.538683\pi$
$647$	−6.16710	−0.242454	−0.121227	+	0.992625i	$0.538683\pi$
$648$	0	0
$649$	−7.59316	−0.298058
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.8822	0.425853	0.212927	−	0.977068i	$-0.431701\pi$
$653$	10.8822	0.425853	0.212927	+	0.977068i	$0.431701\pi$
$654$	0	0
$655$	20.2247	0.790245
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.59316	0.334742	0.167371	−	0.985894i	$-0.446472\pi$
$659$	8.59316	0.334742	0.167371	+	0.985894i	$0.446472\pi$
$660$	0	0
$661$	4.75481	0.184941	0.0924703	−	0.995715i	$-0.470524\pi$
$661$	4.75481	0.184941	0.0924703	+	0.995715i	$0.470524\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34.3822	1.33329
$666$	0	0
$667$	0.647241	0.0250613
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.72055	−0.143630
$672$	0	0
$673$	27.0055	1.04098	0.520492	−	0.853867i	$-0.325749\pi$
$673$	27.0055	1.04098	0.520492	+	0.853867i	$0.325749\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.8966	−0.956855	−0.478428	−	0.878127i	$-0.658793\pi$
$677$	−24.8966	−0.956855	−0.478428	+	0.878127i	$0.658793\pi$
$678$	0	0
$679$	−21.1761	−0.812664
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.8726	−1.21957	−0.609786	−	0.792566i	$-0.708745\pi$
$683$	−31.8726	−1.21957	−0.609786	+	0.792566i	$0.708745\pi$
$684$	0	0
$685$	−16.7836	−0.641269
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.3918	−0.548284
$690$	0	0
$691$	−32.6179	−1.24084	−0.620421	−	0.784269i	$-0.713038\pi$
$691$	−32.6179	−1.24084	−0.620421	+	0.784269i	$0.713038\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.1521	−0.574751
$696$	0	0
$697$	−1.63700	−0.0620057
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−47.1966	−1.78259	−0.891295	−	0.453424i	$-0.850202\pi$
$701$	−47.1966	−1.78259	−0.891295	+	0.453424i	$0.850202\pi$
$702$	0	0
$703$	3.81913	0.144041
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.75481	0.254041
$708$	0	0
$709$	−7.52464	−0.282594	−0.141297	−	0.989967i	$-0.545127\pi$
$709$	−7.52464	−0.282594	−0.141297	+	0.989967i	$0.545127\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.5932	0.846121
$714$	0	0
$715$	−3.36300	−0.125769
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−37.3329	−1.39228	−0.696142	−	0.717904i	$-0.745102\pi$
$719$	−37.3329	−1.39228	−0.696142	+	0.717904i	$0.745102\pi$
$720$	0	0
$721$	24.8384	0.925029
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.0240170	−0.000891967 0
$726$	0	0
$727$	40.9069	1.51715	0.758576	−	0.651585i	$-0.225895\pi$
$727$	40.9069	1.51715	0.758576	+	0.651585i	$0.225895\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.0042	−0.702896
$732$	0	0
$733$	−44.8768	−1.65756	−0.828782	−	0.559572i	$-0.810965\pi$
$733$	−44.8768	−1.65756	−0.828782	+	0.559572i	$0.810965\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.72055	0.137048
$738$	0	0
$739$	6.86237	0.252437	0.126218	−	0.992002i	$-0.459716\pi$
$739$	6.86237	0.252437	0.126218	+	0.992002i	$0.459716\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.2056	1.47500	0.737499	−	0.675348i	$-0.236006\pi$
$743$	40.2056	1.47500	0.737499	+	0.675348i	$0.236006\pi$
$744$	0	0
$745$	−9.42061	−0.345144
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−16.5932	−0.606301
$750$	0	0
$751$	−10.1328	−0.369753	−0.184876	−	0.982762i	$-0.559188\pi$
$751$	−10.1328	−0.369753	−0.184876	+	0.982762i	$0.559188\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.5877	0.567295
$756$	0	0
$757$	25.5589	0.928954	0.464477	−	0.885585i	$-0.346242\pi$
$757$	25.5589	0.928954	0.464477	+	0.885585i	$0.346242\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.4165	−1.17510	−0.587548	−	0.809189i	$-0.699907\pi$
$761$	−32.4165	−1.17510	−0.587548	+	0.809189i	$0.699907\pi$
$762$	0	0
$763$	−20.0986	−0.727617
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.2026	0.404503
$768$	0	0
$769$	−13.7753	−0.496750	−0.248375	−	0.968664i	$-0.579896\pi$
$769$	−13.7753	−0.496750	−0.248375	+	0.968664i	$0.579896\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.37803	−0.229402	−0.114701	−	0.993400i	$-0.536591\pi$
$773$	−6.37803	−0.229402	−0.114701	+	0.993400i	$0.536591\pi$
$774$	0	0
$775$	−0.838357	−0.0301147
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.47536	−0.196175
$780$	0	0
$781$	14.8726	0.532184
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.99454	0.106880
$786$	0	0
$787$	29.9357	1.06709	0.533546	−	0.845771i	$-0.320859\pi$
$787$	29.9357	1.06709	0.533546	+	0.845771i	$0.320859\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.3780	0.866783
$792$	0	0
$793$	5.48913	0.194925
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.2206	−1.24758	−0.623789	−	0.781593i	$-0.714407\pi$
$797$	−35.2206	−1.24758	−0.623789	+	0.781593i	$0.714407\pi$
$798$	0	0
$799$	9.13763	0.323266
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.3630	−0.400992
$804$	0	0
$805$	22.5932	0.796304
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.4062	−1.27998	−0.639988	−	0.768385i	$-0.721061\pi$
$809$	−36.4062	−1.27998	−0.639988	+	0.768385i	$0.721061\pi$
$810$	0	0
$811$	−16.1911	−0.568547	−0.284273	−	0.958743i	$-0.591752\pi$
$811$	−16.1911	−0.568547	−0.284273	+	0.958743i	$0.591752\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.22471	−0.288099
$816$	0	0
$817$	−63.5644	−2.22384
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.94047	0.102623	0.0513115	−	0.998683i	$-0.483660\pi$
$821$	2.94047	0.102623	0.0513115	+	0.998683i	$0.483660\pi$
$822$	0	0
$823$	28.6370	0.998223	0.499112	−	0.866538i	$-0.333660\pi$
$823$	28.6370	0.998223	0.499112	+	0.866538i	$0.333660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.817871	0.0284402	0.0142201	−	0.999899i	$-0.495473\pi$
$827$	0.817871	0.0284402	0.0142201	+	0.999899i	$0.495473\pi$
$828$	0	0
$829$	−39.6917	−1.37855	−0.689276	−	0.724499i	$-0.742071\pi$
$829$	−39.6917	−1.37855	−0.689276	+	0.724499i	$0.742071\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.34797	0.289240
$834$	0	0
$835$	13.7548	0.476005
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.76026	0.0607710	0.0303855	−	0.999538i	$-0.490326\pi$
$839$	1.76026	0.0607710	0.0303855	+	0.999538i	$0.490326\pi$
$840$	0	0
$841$	−28.9850	−0.999482
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.6713	−0.848717
$846$	0	0
$847$	−1.87740	−0.0645084
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.50961	0.0860285
$852$	0	0
$853$	10.2452	0.350789	0.175394	−	0.984498i	$-0.443880\pi$
$853$	10.2452	0.350789	0.175394	+	0.984498i	$0.443880\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.6527	−0.603005	−0.301502	−	0.953465i	$-0.597488\pi$
$857$	−17.6527	−0.603005	−0.301502	+	0.953465i	$0.597488\pi$
$858$	0	0
$859$	−2.44655	−0.0834753	−0.0417376	−	0.999129i	$-0.513289\pi$
$859$	−2.44655	−0.0834753	−0.0417376	+	0.999129i	$0.513289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.2206	1.77761	0.888805	−	0.458286i	$-0.151536\pi$
$863$	52.2206	1.77761	0.888805	+	0.458286i	$0.151536\pi$
$864$	0	0
$865$	−12.1466	−0.412997
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.7158	−0.431353
$870$	0	0
$871$	−5.48913	−0.185992
$872$	0	0
$873$	0	0
$874$	0	0
$875$	20.5589	0.695018
$876$	0	0
$877$	−19.2644	−0.650513	−0.325257	−	0.945626i	$-0.605451\pi$
$877$	−19.2644	−0.650513	−0.325257	+	0.945626i	$0.605451\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.59316	−0.289511	−0.144756	−	0.989467i	$-0.546240\pi$
$881$	−8.59316	−0.289511	−0.144756	+	0.989467i	$0.546240\pi$
$882$	0	0
$883$	−30.6370	−1.03102	−0.515509	−	0.856884i	$-0.672397\pi$
$883$	−30.6370	−1.03102	−0.515509	+	0.856884i	$0.672397\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.9165	−0.769459	−0.384730	−	0.923029i	$-0.625705\pi$
$887$	−22.9165	−0.769459	−0.384730	+	0.923029i	$0.625705\pi$
$888$	0	0
$889$	−17.0535	−0.571956
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.5631	1.02276
$894$	0	0
$895$	13.0055	0.434724
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.524645	0.0174979
$900$	0	0
$901$	23.4315	0.780617
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.83836	0.227315
$906$	0	0
$907$	−15.4068	−0.511576	−0.255788	−	0.966733i	$-0.582335\pi$
$907$	−15.4068	−0.511576	−0.255788	+	0.966733i	$0.582335\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.3534	1.13818	0.569090	−	0.822275i	$-0.307296\pi$
$911$	34.3534	1.13818	0.569090	+	0.822275i	$0.307296\pi$
$912$	0	0
$913$	12.9850	0.429740
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.6575	−0.550079
$918$	0	0
$919$	51.6322	1.70319	0.851595	−	0.524201i	$-0.175636\pi$
$919$	51.6322	1.70319	0.851595	+	0.524201i	$0.175636\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.9424	−0.722243
$924$	0	0
$925$	−0.0931234	−0.00306188
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.03972	0.296583	0.148292	−	0.988944i	$-0.452623\pi$
$929$	9.03972	0.296583	0.148292	+	0.988944i	$0.452623\pi$
$930$	0	0
$931$	27.9219	0.915104
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.47536	0.179063
$936$	0	0
$937$	−6.81787	−0.222730	−0.111365	−	0.993780i	$-0.535522\pi$
$937$	−6.81787	−0.222730	−0.111365	+	0.993780i	$0.535522\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.8678	0.386880	0.193440	−	0.981112i	$-0.438035\pi$
$941$	11.8678	0.386880	0.193440	+	0.981112i	$0.438035\pi$
$942$	0	0
$943$	−3.59795	−0.117165
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.9411	0.355539	0.177770	−	0.984072i	$-0.443112\pi$
$947$	10.9411	0.355539	0.177770	+	0.984072i	$0.443112\pi$
$948$	0	0
$949$	16.7645	0.544198
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.4898	−0.696123	−0.348061	−	0.937472i	$-0.613160\pi$
$953$	−21.4898	−0.696123	−0.348061	+	0.937472i	$0.613160\pi$
$954$	0	0
$955$	16.6713	0.539469
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.8233	0.446379
$960$	0	0
$961$	−12.6863	−0.409235
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.7548	−0.571547
$966$	0	0
$967$	27.2542	0.876435	0.438218	−	0.898869i	$-0.355610\pi$
$967$	27.2542	0.876435	0.438218	+	0.898869i	$0.355610\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.4110	−0.943845	−0.471923	−	0.881640i	$-0.656440\pi$
$971$	−29.4110	−0.943845	−0.471923	+	0.881640i	$0.656440\pi$
$972$	0	0
$973$	12.4796	0.400076
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.95071	0.158387	0.0791936	−	0.996859i	$-0.474765\pi$
$977$	4.95071	0.158387	0.0791936	+	0.996859i	$0.474765\pi$
$978$	0	0
$979$	7.60819	0.243159
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.5481	−1.70792	−0.853959	−	0.520340i	$-0.825805\pi$
$983$	−53.5481	−1.70792	−0.853959	+	0.520340i	$0.825805\pi$
$984$	0	0
$985$	24.5042	0.780767
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.7693	−1.32818
$990$	0	0
$991$	21.1178	0.670829	0.335415	−	0.942071i	$-0.391124\pi$
$991$	21.1178	0.670829	0.335415	+	0.942071i	$0.391124\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−30.5384	−0.968133
$996$	0	0
$997$	−9.89185	−0.313278	−0.156639	−	0.987656i	$-0.550066\pi$
$997$	−9.89185	−0.313278	−0.156639	+	0.987656i	$0.550066\pi$
$998$	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.1.3	✓	3
3.2	odd	2		2376.2.a.q.1.1	yes	3
4.3	odd	2		4752.2.a.bh.1.3		3
12.11	even	2		4752.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.l.1.3	✓	3	1.1	even	1	trivial
2376.2.a.q.1.1	yes	3	3.2	odd	2
4752.2.a.bh.1.3		3	4.3	odd	2
4752.2.a.bp.1.1		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)

\(f(q)\)	\(=\)	\(q+2.27945 q^{5} -1.87740 q^{7} -1.00000 q^{11} +1.47536 q^{13} -2.40205 q^{17} -8.03426 q^{19} -5.27945 q^{23} +0.195903 q^{25} -0.122596 q^{29} -4.27945 q^{31} -4.27945 q^{35} -0.475355 q^{37} +0.681501 q^{41} +7.91166 q^{43} -3.80410 q^{47} -3.47536 q^{49} -9.75481 q^{53} -2.27945 q^{55} +7.59316 q^{59} +3.72055 q^{61} +3.36300 q^{65} -3.72055 q^{67} -14.8726 q^{71} +11.3630 q^{73} +1.87740 q^{77} +12.7158 q^{79} -12.9850 q^{83} -5.47536 q^{85} -7.60819 q^{89} -2.76984 q^{91} -18.3137 q^{95} +11.2795 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{17} - 2 q^{19} - 7 q^{23} + 5 q^{25} - 7 q^{29} - 4 q^{31} - 4 q^{35} + 3 q^{37} - 9 q^{41} - 5 q^{43} - 7 q^{47} - 6 q^{49} - 16 q^{53} + 2 q^{55} - 17 q^{59} + 20 q^{61}+ \cdots + 25 q^{97}+O(q^{100}) \) 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

L-functions

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