LMFDB - Embedded newform 1872.4.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1872,4,Mod(1,1872)] 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("1872.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,8,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$110.451575531$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.69042$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+13.3808 q^{5} -15.3808 q^{7} +24.9042 q^{11} +13.0000 q^{13} +65.5233 q^{17} +73.1891 q^{19} +28.5700 q^{23} +54.0467 q^{25} +220.570 q^{29} -138.997 q^{31} -205.808 q^{35} -354.187 q^{37} +297.666 q^{41} +9.62171 q^{43} -219.666 q^{47} -106.430 q^{49} +189.238 q^{53} +333.238 q^{55} +329.376 q^{59} -838.757 q^{61} +173.951 q^{65} +386.997 q^{67} +664.521 q^{71} +248.280 q^{73} -383.047 q^{77} +1264.56 q^{79} -157.562 q^{83} +876.757 q^{85} -774.049 q^{89} -199.951 q^{91} +979.332 q^{95} +1054.94 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{5} - 12 q^{7} - 44 q^{11} + 26 q^{13} + 56 q^{17} - 60 q^{19} - 168 q^{23} - 42 q^{25} + 216 q^{29} + 116 q^{31} - 224 q^{35} - 108 q^{37} + 464 q^{41} + 432 q^{43} - 308 q^{47} - 438 q^{49}+ \cdots + 684 q^{97}+O(q^{100}) $ 2 * q + 8 * q^5 - 12 * q^7 - 44 * q^11 + 26 * q^13 + 56 * q^17 - 60 * q^19 - 168 * q^23 - 42 * q^25 + 216 * q^29 + 116 * q^31 - 224 * q^35 - 108 * q^37 + 464 * q^41 + 432 * q^43 - 308 * q^47 - 438 * q^49 + 416 * q^53 + 704 * q^55 - 148 * q^59 - 852 * q^61 + 104 * q^65 + 380 * q^67 + 860 * q^71 - 404 * q^73 - 616 * q^77 + 728 * q^79 + 1092 * q^83 + 928 * q^85 - 1792 * q^89 - 156 * q^91 + 1696 * q^95 + 684 * 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
$5$	13.3808	1.19682	0.598409	−	0.801191i	$-0.295800\pi$
$5$	13.3808	1.19682	0.598409	+	0.801191i	$0.295800\pi$
$6$	0	0
$7$	−15.3808	−0.830487	−0.415243	−	0.909710i	$-0.636304\pi$
$7$	−15.3808	−0.830487	−0.415243	+	0.909710i	$0.636304\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24.9042	0.682626	0.341313	−	0.939950i	$-0.389128\pi$
$11$	24.9042	0.682626	0.341313	+	0.939950i	$0.389128\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	65.5233	0.934808	0.467404	−	0.884044i	$-0.345189\pi$
$17$	65.5233	0.934808	0.467404	+	0.884044i	$0.345189\pi$
$18$	0	0
$19$	73.1891	0.883723	0.441862	−	0.897083i	$-0.354318\pi$
$19$	73.1891	0.883723	0.441862	+	0.897083i	$0.354318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	28.5700	0.259011	0.129505	−	0.991579i	$-0.458661\pi$
$23$	28.5700	0.259011	0.129505	+	0.991579i	$0.458661\pi$
$24$	0	0
$25$	54.0467	0.432373
$26$	0	0
$27$	0	0
$28$	0	0
$29$	220.570	1.41237	0.706186	−	0.708026i	$-0.250414\pi$
$29$	220.570	1.41237	0.706186	+	0.708026i	$0.250414\pi$
$30$	0	0
$31$	−138.997	−0.805312	−0.402656	−	0.915351i	$-0.631913\pi$
$31$	−138.997	−0.805312	−0.402656	+	0.915351i	$0.631913\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−205.808	−0.993942
$36$	0	0
$37$	−354.187	−1.57373	−0.786864	−	0.617127i	$-0.788297\pi$
$37$	−354.187	−1.57373	−0.786864	+	0.617127i	$0.788297\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	297.666	1.13384	0.566922	−	0.823772i	$-0.308134\pi$
$41$	297.666	1.13384	0.566922	+	0.823772i	$0.308134\pi$
$42$	0	0
$43$	9.62171	0.0341232	0.0170616	−	0.999854i	$-0.494569\pi$
$43$	9.62171	0.0341232	0.0170616	+	0.999854i	$0.494569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−219.666	−0.681735	−0.340868	−	0.940111i	$-0.610721\pi$
$47$	−219.666	−0.681735	−0.340868	+	0.940111i	$0.610721\pi$
$48$	0	0
$49$	−106.430	−0.310292
$50$	0	0
$51$	0	0
$52$	0	0
$53$	189.238	0.490451	0.245225	−	0.969466i	$-0.421138\pi$
$53$	189.238	0.490451	0.245225	+	0.969466i	$0.421138\pi$
$54$	0	0
$55$	333.238	0.816979
$56$	0	0
$57$	0	0
$58$	0	0
$59$	329.376	0.726798	0.363399	−	0.931634i	$-0.381616\pi$
$59$	329.376	0.726798	0.363399	+	0.931634i	$0.381616\pi$
$60$	0	0
$61$	−838.757	−1.76052	−0.880260	−	0.474491i	$-0.842632\pi$
$61$	−838.757	−1.76052	−0.880260	+	0.474491i	$0.842632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	173.951	0.331938
$66$	0	0
$67$	386.997	0.705661	0.352830	−	0.935687i	$-0.385219\pi$
$67$	386.997	0.705661	0.352830	+	0.935687i	$0.385219\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	664.521	1.11076	0.555381	−	0.831596i	$-0.312572\pi$
$71$	664.521	1.11076	0.555381	+	0.831596i	$0.312572\pi$
$72$	0	0
$73$	248.280	0.398068	0.199034	−	0.979993i	$-0.436220\pi$
$73$	248.280	0.398068	0.199034	+	0.979993i	$0.436220\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−383.047	−0.566912
$78$	0	0
$79$	1264.56	1.80094	0.900469	−	0.434920i	$-0.143223\pi$
$79$	1264.56	1.80094	0.900469	+	0.434920i	$0.143223\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−157.562	−0.208370	−0.104185	−	0.994558i	$-0.533223\pi$
$83$	−157.562	−0.208370	−0.104185	+	0.994558i	$0.533223\pi$
$84$	0	0
$85$	876.757	1.11880
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−774.049	−0.921899	−0.460950	−	0.887426i	$-0.652491\pi$
$89$	−774.049	−0.921899	−0.460950	+	0.887426i	$0.652491\pi$
$90$	0	0
$91$	−199.951	−0.230336
$92$	0	0
$93$	0	0
$94$	0	0
$95$	979.332	1.05766
$96$	0	0
$97$	1054.94	1.10426	0.552130	−	0.833758i	$-0.313815\pi$
$97$	1054.94	1.10426	0.552130	+	0.833758i	$0.313815\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−987.223	−0.972598	−0.486299	−	0.873793i	$-0.661653\pi$
$101$	−987.223	−0.972598	−0.486299	+	0.873793i	$0.661653\pi$
$102$	0	0
$103$	−1111.51	−1.06330	−0.531651	−	0.846963i	$-0.678428\pi$
$103$	−1111.51	−1.06330	−0.531651	+	0.846963i	$0.678428\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1390.85	1.25662	0.628311	−	0.777962i	$-0.283747\pi$
$107$	1390.85	1.25662	0.628311	+	0.777962i	$0.283747\pi$
$108$	0	0
$109$	739.523	0.649849	0.324924	−	0.945740i	$-0.394661\pi$
$109$	739.523	0.649849	0.324924	+	0.945740i	$0.394661\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	545.985	0.454530	0.227265	−	0.973833i	$-0.427022\pi$
$113$	545.985	0.454530	0.227265	+	0.973833i	$0.427022\pi$
$114$	0	0
$115$	382.290	0.309989
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1007.80	−0.776346
$120$	0	0
$121$	−710.783	−0.534022
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−949.415	−0.679346
$126$	0	0
$127$	1461.79	1.02136	0.510680	−	0.859771i	$-0.329394\pi$
$127$	1461.79	1.02136	0.510680	+	0.859771i	$0.329394\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1283.61	−0.856101	−0.428050	−	0.903755i	$-0.640799\pi$
$131$	−1283.61	−0.856101	−0.428050	+	0.903755i	$0.640799\pi$
$132$	0	0
$133$	−1125.71	−0.733921
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1705.65	1.06367	0.531836	−	0.846847i	$-0.321502\pi$
$137$	1705.65	1.06367	0.531836	+	0.846847i	$0.321502\pi$
$138$	0	0
$139$	2210.09	1.34862	0.674308	−	0.738450i	$-0.264442\pi$
$139$	2210.09	1.34862	0.674308	+	0.738450i	$0.264442\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	323.754	0.189326
$144$	0	0
$145$	2951.41	1.69035
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1794.71	−0.986766	−0.493383	−	0.869812i	$-0.664240\pi$
$149$	−1794.71	−0.986766	−0.493383	+	0.869812i	$0.664240\pi$
$150$	0	0
$151$	2141.10	1.15391	0.576955	−	0.816776i	$-0.304241\pi$
$151$	2141.10	1.15391	0.576955	+	0.816776i	$0.304241\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1859.90	−0.963812
$156$	0	0
$157$	1303.13	0.662427	0.331214	−	0.943556i	$-0.392542\pi$
$157$	1303.13	0.662427	0.331214	+	0.943556i	$0.392542\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−439.430	−0.215105
$162$	0	0
$163$	−1341.00	−0.644389	−0.322195	−	0.946673i	$-0.604421\pi$
$163$	−1341.00	−0.644389	−0.322195	+	0.946673i	$0.604421\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2816.70	1.30517	0.652583	−	0.757718i	$-0.273686\pi$
$167$	2816.70	1.30517	0.652583	+	0.757718i	$0.273686\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2006.66	0.881872	0.440936	−	0.897539i	$-0.354647\pi$
$173$	2006.66	0.881872	0.440936	+	0.897539i	$0.354647\pi$
$174$	0	0
$175$	−831.282	−0.359080
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3160.83	−1.31984	−0.659921	−	0.751335i	$-0.729410\pi$
$179$	−3160.83	−1.31984	−0.659921	+	0.751335i	$0.729410\pi$
$180$	0	0
$181$	930.010	0.381918	0.190959	−	0.981598i	$-0.438840\pi$
$181$	930.010	0.381918	0.190959	+	0.981598i	$0.438840\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4739.31	−1.88347
$186$	0	0
$187$	1631.80	0.638124
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2196.49	−0.832108	−0.416054	−	0.909340i	$-0.636587\pi$
$191$	−2196.49	−0.832108	−0.416054	+	0.909340i	$0.636587\pi$
$192$	0	0
$193$	1097.25	0.409233	0.204617	−	0.978842i	$-0.434405\pi$
$193$	1097.25	0.409233	0.204617	+	0.978842i	$0.434405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2531.28	0.915462	0.457731	−	0.889091i	$-0.348662\pi$
$197$	2531.28	0.915462	0.457731	+	0.889091i	$0.348662\pi$
$198$	0	0
$199$	−4170.92	−1.48577	−0.742886	−	0.669418i	$-0.766543\pi$
$199$	−4170.92	−1.48577	−0.742886	+	0.669418i	$0.766543\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3392.55	−1.17296
$204$	0	0
$205$	3983.02	1.35700
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1822.71	0.603252
$210$	0	0
$211$	3104.37	1.01286	0.506431	−	0.862281i	$-0.330965\pi$
$211$	3104.37	1.01286	0.506431	+	0.862281i	$0.330965\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	128.746	0.0408392
$216$	0	0
$217$	2137.90	0.668801
$218$	0	0
$219$	0	0
$220$	0	0
$221$	851.803	0.259269
$222$	0	0
$223$	1410.68	0.423615	0.211808	−	0.977311i	$-0.432065\pi$
$223$	1410.68	0.423615	0.211808	+	0.977311i	$0.432065\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3946.18	−1.15382	−0.576910	−	0.816808i	$-0.695742\pi$
$227$	−3946.18	−1.15382	−0.576910	+	0.816808i	$0.695742\pi$
$228$	0	0
$229$	−4161.02	−1.20073	−0.600366	−	0.799725i	$-0.704978\pi$
$229$	−4161.02	−1.20073	−0.600366	+	0.799725i	$0.704978\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1592.88	0.447866	0.223933	−	0.974605i	$-0.428110\pi$
$233$	1592.88	0.447866	0.223933	+	0.974605i	$0.428110\pi$
$234$	0	0
$235$	−2939.31	−0.815913
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4989.85	1.35049	0.675244	−	0.737595i	$-0.264039\pi$
$239$	4989.85	1.35049	0.675244	+	0.737595i	$0.264039\pi$
$240$	0	0
$241$	5053.80	1.35081	0.675403	−	0.737449i	$-0.263970\pi$
$241$	5053.80	1.35081	0.675403	+	0.737449i	$0.263970\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1424.12	−0.371363
$246$	0	0
$247$	951.459	0.245101
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4154.20	−1.04466	−0.522332	−	0.852742i	$-0.674938\pi$
$251$	−4154.20	−1.04466	−0.522332	+	0.852742i	$0.674938\pi$
$252$	0	0
$253$	711.511	0.176808
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1640.88	−0.398268	−0.199134	−	0.979972i	$-0.563813\pi$
$257$	−1640.88	−0.398268	−0.199134	+	0.979972i	$0.563813\pi$
$258$	0	0
$259$	5447.68	1.30696
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7542.74	1.76846	0.884231	−	0.467050i	$-0.154683\pi$
$263$	7542.74	1.76846	0.884231	+	0.467050i	$0.154683\pi$
$264$	0	0
$265$	2532.17	0.586980
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5816.13	1.31827	0.659137	−	0.752023i	$-0.270922\pi$
$269$	5816.13	1.31827	0.659137	+	0.752023i	$0.270922\pi$
$270$	0	0
$271$	−6943.50	−1.55641	−0.778205	−	0.628010i	$-0.783870\pi$
$271$	−6943.50	−1.55641	−0.778205	+	0.628010i	$0.783870\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1345.99	0.295149
$276$	0	0
$277$	6379.49	1.38378	0.691889	−	0.722004i	$-0.256779\pi$
$277$	6379.49	1.38378	0.691889	+	0.722004i	$0.256779\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8135.74	1.72718	0.863590	−	0.504195i	$-0.168211\pi$
$281$	8135.74	1.72718	0.863590	+	0.504195i	$0.168211\pi$
$282$	0	0
$283$	−2845.95	−0.597789	−0.298895	−	0.954286i	$-0.596618\pi$
$283$	−2845.95	−0.597789	−0.298895	+	0.954286i	$0.596618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4578.35	−0.941642
$288$	0	0
$289$	−619.694	−0.126133
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3824.64	−0.762587	−0.381294	−	0.924454i	$-0.624521\pi$
$293$	−3824.64	−0.762587	−0.381294	+	0.924454i	$0.624521\pi$
$294$	0	0
$295$	4407.32	0.869845
$296$	0	0
$297$	0	0
$298$	0	0
$299$	371.410	0.0718367
$300$	0	0
$301$	−147.990	−0.0283388
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11223.3	−2.10702
$306$	0	0
$307$	8984.03	1.67018	0.835091	−	0.550112i	$-0.185415\pi$
$307$	8984.03	1.67018	0.835091	+	0.550112i	$0.185415\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6923.35	1.26234	0.631169	−	0.775645i	$-0.282575\pi$
$311$	6923.35	1.26234	0.631169	+	0.775645i	$0.282575\pi$
$312$	0	0
$313$	3080.48	0.556290	0.278145	−	0.960539i	$-0.410280\pi$
$313$	3080.48	0.556290	0.278145	+	0.960539i	$0.410280\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3568.73	−0.632303	−0.316151	−	0.948709i	$-0.602391\pi$
$317$	−3568.73	−0.632303	−0.316151	+	0.948709i	$0.602391\pi$
$318$	0	0
$319$	5493.11	0.964123
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4795.60	0.826112
$324$	0	0
$325$	702.606	0.119919
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3378.64	0.566172
$330$	0	0
$331$	−2275.82	−0.377917	−0.188958	−	0.981985i	$-0.560511\pi$
$331$	−2275.82	−0.377917	−0.188958	+	0.981985i	$0.560511\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5178.35	0.844547
$336$	0	0
$337$	−4060.64	−0.656372	−0.328186	−	0.944613i	$-0.606437\pi$
$337$	−4060.64	−0.656372	−0.328186	+	0.944613i	$0.606437\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3461.61	−0.549727
$342$	0	0
$343$	6912.61	1.08818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8960.49	1.38624	0.693119	−	0.720824i	$-0.256236\pi$
$347$	8960.49	1.38624	0.693119	+	0.720824i	$0.256236\pi$
$348$	0	0
$349$	−4116.66	−0.631404	−0.315702	−	0.948858i	$-0.602240\pi$
$349$	−4116.66	−0.631404	−0.315702	+	0.948858i	$0.602240\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6883.40	−1.03787	−0.518933	−	0.854815i	$-0.673671\pi$
$353$	−6883.40	−1.03787	−0.518933	+	0.854815i	$0.673671\pi$
$354$	0	0
$355$	8891.84	1.32938
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10499.8	−1.54362	−0.771810	−	0.635854i	$-0.780648\pi$
$359$	−10499.8	−1.54362	−0.771810	+	0.635854i	$0.780648\pi$
$360$	0	0
$361$	−1502.35	−0.219033
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3322.19	0.476415
$366$	0	0
$367$	3612.78	0.513857	0.256928	−	0.966430i	$-0.417290\pi$
$367$	3612.78	0.513857	0.256928	+	0.966430i	$0.417290\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2910.64	−0.407313
$372$	0	0
$373$	12332.9	1.71199	0.855995	−	0.516984i	$-0.172945\pi$
$373$	12332.9	1.71199	0.855995	+	0.516984i	$0.172945\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2867.41	0.391722
$378$	0	0
$379$	6916.28	0.937375	0.468688	−	0.883364i	$-0.344727\pi$
$379$	6916.28	0.937375	0.468688	+	0.883364i	$0.344727\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11616.4	1.54980	0.774899	−	0.632085i	$-0.217801\pi$
$383$	11616.4	1.54980	0.774899	+	0.632085i	$0.217801\pi$
$384$	0	0
$385$	−5125.48	−0.678490
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14598.1	1.90271	0.951353	−	0.308103i	$-0.0996943\pi$
$389$	14598.1	1.90271	0.951353	+	0.308103i	$0.0996943\pi$
$390$	0	0
$391$	1872.00	0.242126
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16920.9	2.15539
$396$	0	0
$397$	2163.91	0.273560	0.136780	−	0.990601i	$-0.456325\pi$
$397$	2163.91	0.273560	0.136780	+	0.990601i	$0.456325\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5859.76	0.729731	0.364866	−	0.931060i	$-0.381115\pi$
$401$	5859.76	0.729731	0.364866	+	0.931060i	$0.381115\pi$
$402$	0	0
$403$	−1806.97	−0.223353
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8820.72	−1.07427
$408$	0	0
$409$	10750.4	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$409$	10750.4	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5066.07	−0.603596
$414$	0	0
$415$	−2108.32	−0.249381
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13174.7	1.53610	0.768048	−	0.640393i	$-0.221228\pi$
$419$	13174.7	1.53610	0.768048	+	0.640393i	$0.221228\pi$
$420$	0	0
$421$	879.574	0.101824	0.0509119	−	0.998703i	$-0.483787\pi$
$421$	879.574	0.101824	0.0509119	+	0.998703i	$0.483787\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3541.32	0.404186
$426$	0	0
$427$	12900.8	1.46209
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13181.9	−1.47321	−0.736603	−	0.676325i	$-0.763571\pi$
$431$	−13181.9	−1.47321	−0.736603	+	0.676325i	$0.763571\pi$
$432$	0	0
$433$	1731.91	0.192217	0.0961087	−	0.995371i	$-0.469360\pi$
$433$	1731.91	0.192217	0.0961087	+	0.995371i	$0.469360\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2091.01	0.228894
$438$	0	0
$439$	−6766.37	−0.735629	−0.367815	−	0.929899i	$-0.619894\pi$
$439$	−6766.37	−0.735629	−0.367815	+	0.929899i	$0.619894\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11292.9	−1.21116	−0.605579	−	0.795785i	$-0.707058\pi$
$443$	−11292.9	−1.21116	−0.605579	+	0.795785i	$0.707058\pi$
$444$	0	0
$445$	−10357.4	−1.10335
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7882.86	−0.828542	−0.414271	−	0.910154i	$-0.635963\pi$
$449$	−7882.86	−0.828542	−0.414271	+	0.910154i	$0.635963\pi$
$450$	0	0
$451$	7413.12	0.773991
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2675.51	−0.275670
$456$	0	0
$457$	3515.84	0.359877	0.179939	−	0.983678i	$-0.442410\pi$
$457$	3515.84	0.359877	0.179939	+	0.983678i	$0.442410\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8362.30	−0.844840	−0.422420	−	0.906400i	$-0.638819\pi$
$461$	−8362.30	−0.844840	−0.422420	+	0.906400i	$0.638819\pi$
$462$	0	0
$463$	−13064.9	−1.31140	−0.655702	−	0.755020i	$-0.727627\pi$
$463$	−13064.9	−1.31140	−0.655702	+	0.755020i	$0.727627\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16677.7	−1.65258	−0.826289	−	0.563247i	$-0.809552\pi$
$467$	−16677.7	−1.65258	−0.826289	+	0.563247i	$0.809552\pi$
$468$	0	0
$469$	−5952.34	−0.586042
$470$	0	0
$471$	0	0
$472$	0	0
$473$	239.620	0.0232934
$474$	0	0
$475$	3955.63	0.382098
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14040.4	−1.33929	−0.669645	−	0.742681i	$-0.733554\pi$
$479$	−14040.4	−1.33929	−0.669645	+	0.742681i	$0.733554\pi$
$480$	0	0
$481$	−4604.43	−0.436473
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14116.0	1.32160
$486$	0	0
$487$	−10312.0	−0.959507	−0.479754	−	0.877403i	$-0.659274\pi$
$487$	−10312.0	−0.959507	−0.479754	+	0.877403i	$0.659274\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14474.0	−1.33035	−0.665176	−	0.746687i	$-0.731643\pi$
$491$	−14474.0	−1.33035	−0.665176	+	0.746687i	$0.731643\pi$
$492$	0	0
$493$	14452.5	1.32030
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10220.9	−0.922473
$498$	0	0
$499$	5941.71	0.533041	0.266521	−	0.963829i	$-0.414126\pi$
$499$	5941.71	0.533041	0.266521	+	0.963829i	$0.414126\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5381.76	0.477059	0.238530	−	0.971135i	$-0.423335\pi$
$503$	5381.76	0.477059	0.238530	+	0.971135i	$0.423335\pi$
$504$	0	0
$505$	−13209.9	−1.16402
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9526.86	−0.829608	−0.414804	−	0.909911i	$-0.636150\pi$
$509$	−9526.86	−0.829608	−0.414804	+	0.909911i	$0.636150\pi$
$510$	0	0
$511$	−3818.75	−0.330590
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14872.9	−1.27258
$516$	0	0
$517$	−5470.59	−0.465370
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17924.5	1.50727	0.753633	−	0.657295i	$-0.228300\pi$
$521$	17924.5	1.50727	0.753633	+	0.657295i	$0.228300\pi$
$522$	0	0
$523$	−10371.1	−0.867107	−0.433554	−	0.901128i	$-0.642741\pi$
$523$	−10371.1	−0.867107	−0.433554	+	0.901128i	$0.642741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9107.58	−0.752813
$528$	0	0
$529$	−11350.8	−0.932913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3869.66	0.314472
$534$	0	0
$535$	18610.7	1.50395
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2650.55	−0.211813
$540$	0	0
$541$	20580.1	1.63550	0.817752	−	0.575570i	$-0.195220\pi$
$541$	20580.1	1.63550	0.817752	+	0.575570i	$0.195220\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9895.44	0.777750
$546$	0	0
$547$	4710.30	0.368186	0.184093	−	0.982909i	$-0.441065\pi$
$547$	4710.30	0.368186	0.184093	+	0.982909i	$0.441065\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16143.3	1.24815
$552$	0	0
$553$	−19450.0	−1.49566
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21727.0	−1.65279	−0.826395	−	0.563092i	$-0.809612\pi$
$557$	−21727.0	−1.65279	−0.826395	+	0.563092i	$0.809612\pi$
$558$	0	0
$559$	125.082	0.00946407
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8662.30	0.648441	0.324221	−	0.945982i	$-0.394898\pi$
$563$	8662.30	0.648441	0.324221	+	0.945982i	$0.394898\pi$
$564$	0	0
$565$	7305.73	0.543990
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6007.63	−0.442624	−0.221312	−	0.975203i	$-0.571034\pi$
$569$	−6007.63	−0.442624	−0.221312	+	0.975203i	$0.571034\pi$
$570$	0	0
$571$	−23283.4	−1.70644	−0.853222	−	0.521549i	$-0.825354\pi$
$571$	−23283.4	−1.70644	−0.853222	+	0.521549i	$0.825354\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1544.11	0.111989
$576$	0	0
$577$	12763.1	0.920859	0.460429	−	0.887696i	$-0.347695\pi$
$577$	12763.1	0.920859	0.460429	+	0.887696i	$0.347695\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2423.44	0.173049
$582$	0	0
$583$	4712.82	0.334794
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7683.28	0.540243	0.270122	−	0.962826i	$-0.412936\pi$
$587$	7683.28	0.540243	0.270122	+	0.962826i	$0.412936\pi$
$588$	0	0
$589$	−10173.1	−0.711673
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16500.6	−1.14266	−0.571332	−	0.820719i	$-0.693573\pi$
$593$	−16500.6	−1.14266	−0.571332	+	0.820719i	$0.693573\pi$
$594$	0	0
$595$	−13485.2	−0.929145
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3695.77	−0.252095	−0.126047	−	0.992024i	$-0.540229\pi$
$599$	−3695.77	−0.252095	−0.126047	+	0.992024i	$0.540229\pi$
$600$	0	0
$601$	−3407.15	−0.231249	−0.115624	−	0.993293i	$-0.536887\pi$
$601$	−3407.15	−0.231249	−0.115624	+	0.993293i	$0.536887\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9510.87	−0.639127
$606$	0	0
$607$	−23171.4	−1.54942	−0.774709	−	0.632318i	$-0.782104\pi$
$607$	−23171.4	−1.54942	−0.774709	+	0.632318i	$0.782104\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2855.66	−0.189079
$612$	0	0
$613$	−8605.59	−0.567009	−0.283504	−	0.958971i	$-0.591497\pi$
$613$	−8605.59	−0.567009	−0.283504	+	0.958971i	$0.591497\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13752.5	0.897332	0.448666	−	0.893699i	$-0.351899\pi$
$617$	13752.5	0.897332	0.448666	+	0.893699i	$0.351899\pi$
$618$	0	0
$619$	23090.4	1.49932	0.749662	−	0.661821i	$-0.230216\pi$
$619$	23090.4	1.49932	0.749662	+	0.661821i	$0.230216\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11905.5	0.765625
$624$	0	0
$625$	−19459.8	−1.24543
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−23207.5	−1.47113
$630$	0	0
$631$	−13189.8	−0.832134	−0.416067	−	0.909334i	$-0.636592\pi$
$631$	−13189.8	−0.832134	−0.416067	+	0.909334i	$0.636592\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19559.9	1.22238
$636$	0	0
$637$	−1383.59	−0.0860594
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6885.41	0.424271	0.212135	−	0.977240i	$-0.431958\pi$
$641$	6885.41	0.424271	0.212135	+	0.977240i	$0.431958\pi$
$642$	0	0
$643$	9846.82	0.603920	0.301960	−	0.953321i	$-0.402359\pi$
$643$	9846.82	0.603920	0.301960	+	0.953321i	$0.402359\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25559.8	−1.55310	−0.776552	−	0.630054i	$-0.783033\pi$
$647$	−25559.8	−1.55310	−0.776552	+	0.630054i	$0.783033\pi$
$648$	0	0
$649$	8202.83	0.496131
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9671.17	−0.579574	−0.289787	−	0.957091i	$-0.593585\pi$
$653$	−9671.17	−0.579574	−0.289787	+	0.957091i	$0.593585\pi$
$654$	0	0
$655$	−17175.7	−1.02460
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13462.2	0.795769	0.397884	−	0.917436i	$-0.369745\pi$
$659$	13462.2	0.795769	0.397884	+	0.917436i	$0.369745\pi$
$660$	0	0
$661$	−14414.7	−0.848210	−0.424105	−	0.905613i	$-0.639411\pi$
$661$	−14414.7	−0.848210	−0.424105	+	0.905613i	$0.639411\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15062.9	−0.878369
$666$	0	0
$667$	6301.68	0.365820
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20888.5	−1.20178
$672$	0	0
$673$	16234.3	0.929847	0.464924	−	0.885351i	$-0.346082\pi$
$673$	16234.3	0.929847	0.464924	+	0.885351i	$0.346082\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29789.5	1.69115	0.845573	−	0.533861i	$-0.179259\pi$
$677$	29789.5	1.69115	0.845573	+	0.533861i	$0.179259\pi$
$678$	0	0
$679$	−16225.9	−0.917074
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17283.7	0.968289	0.484144	−	0.874988i	$-0.339131\pi$
$683$	17283.7	0.968289	0.484144	+	0.874988i	$0.339131\pi$
$684$	0	0
$685$	22823.0	1.27302
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2460.10	0.136027
$690$	0	0
$691$	1931.87	0.106355	0.0531777	−	0.998585i	$-0.483065\pi$
$691$	1931.87	0.106355	0.0531777	+	0.998585i	$0.483065\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29572.9	1.61405
$696$	0	0
$697$	19504.1	1.05993
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23864.4	1.28580	0.642900	−	0.765950i	$-0.277731\pi$
$701$	23864.4	1.28580	0.642900	+	0.765950i	$0.277731\pi$
$702$	0	0
$703$	−25922.6	−1.39074
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15184.3	0.807730
$708$	0	0
$709$	5925.38	0.313868	0.156934	−	0.987609i	$-0.449839\pi$
$709$	5925.38	0.313868	0.156934	+	0.987609i	$0.449839\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3971.15	−0.208585
$714$	0	0
$715$	4332.10	0.226589
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2805.02	−0.145493	−0.0727465	−	0.997350i	$-0.523176\pi$
$719$	−2805.02	−0.145493	−0.0727465	+	0.997350i	$0.523176\pi$
$720$	0	0
$721$	17095.9	0.883059
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11921.1	0.610672
$726$	0	0
$727$	−23073.3	−1.17709	−0.588543	−	0.808466i	$-0.700298\pi$
$727$	−23073.3	−1.17709	−0.588543	+	0.808466i	$0.700298\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	630.446	0.0318986
$732$	0	0
$733$	−10658.0	−0.537056	−0.268528	−	0.963272i	$-0.586537\pi$
$733$	−10658.0	−0.537056	−0.268528	+	0.963272i	$0.586537\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9637.85	0.481702
$738$	0	0
$739$	−21525.4	−1.07148	−0.535742	−	0.844382i	$-0.679968\pi$
$739$	−21525.4	−1.07148	−0.535742	+	0.844382i	$0.679968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3723.86	−0.183870	−0.0919349	−	0.995765i	$-0.529305\pi$
$743$	−3723.86	−0.183870	−0.0919349	+	0.995765i	$0.529305\pi$
$744$	0	0
$745$	−24014.7	−1.18098
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−21392.4	−1.04361
$750$	0	0
$751$	−37456.8	−1.82000	−0.910000	−	0.414609i	$-0.863918\pi$
$751$	−37456.8	−1.82000	−0.910000	+	0.414609i	$0.863918\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28649.7	1.38102
$756$	0	0
$757$	−25551.8	−1.22681	−0.613405	−	0.789769i	$-0.710201\pi$
$757$	−25551.8	−1.22681	−0.613405	+	0.789769i	$0.710201\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1290.30	−0.0614629	−0.0307314	−	0.999528i	$-0.509784\pi$
$761$	−1290.30	−0.0614629	−0.0307314	+	0.999528i	$0.509784\pi$
$762$	0	0
$763$	−11374.5	−0.539691
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4281.88	0.201577
$768$	0	0
$769$	−31146.6	−1.46056	−0.730282	−	0.683145i	$-0.760612\pi$
$769$	−31146.6	−1.46056	−0.730282	+	0.683145i	$0.760612\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12497.6	−0.581513	−0.290756	−	0.956797i	$-0.593907\pi$
$773$	−12497.6	−0.581513	−0.290756	+	0.956797i	$0.593907\pi$
$774$	0	0
$775$	−7512.35	−0.348195
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21785.9	1.00200
$780$	0	0
$781$	16549.3	0.758235
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17437.0	0.792805
$786$	0	0
$787$	28306.9	1.28212	0.641061	−	0.767490i	$-0.278494\pi$
$787$	28306.9	1.28212	0.641061	+	0.767490i	$0.278494\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8397.70	−0.377481
$792$	0	0
$793$	−10903.8	−0.488281
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19084.9	−0.848206	−0.424103	−	0.905614i	$-0.639411\pi$
$797$	−19084.9	−0.848206	−0.424103	+	0.905614i	$0.639411\pi$
$798$	0	0
$799$	−14393.2	−0.637292
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6183.20	0.271732
$804$	0	0
$805$	−5879.94	−0.257442
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13587.0	−0.590475	−0.295237	−	0.955424i	$-0.595399\pi$
$809$	−13587.0	−0.590475	−0.295237	+	0.955424i	$0.595399\pi$
$810$	0	0
$811$	30426.8	1.31742	0.658711	−	0.752396i	$-0.271102\pi$
$811$	30426.8	1.31742	0.658711	+	0.752396i	$0.271102\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17943.7	−0.771216
$816$	0	0
$817$	704.204	0.0301554
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32312.9	1.37360	0.686801	−	0.726846i	$-0.259014\pi$
$821$	32312.9	1.37360	0.686801	+	0.726846i	$0.259014\pi$
$822$	0	0
$823$	19812.6	0.839153	0.419576	−	0.907720i	$-0.362179\pi$
$823$	19812.6	0.839153	0.419576	+	0.907720i	$0.362179\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36387.7	−1.53002	−0.765009	−	0.644019i	$-0.777266\pi$
$827$	−36387.7	−1.53002	−0.765009	+	0.644019i	$0.777266\pi$
$828$	0	0
$829$	−12723.8	−0.533071	−0.266535	−	0.963825i	$-0.585879\pi$
$829$	−12723.8	−0.533071	−0.266535	+	0.963825i	$0.585879\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6973.65	−0.290063
$834$	0	0
$835$	37689.8	1.56205
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19567.0	−0.805156	−0.402578	−	0.915386i	$-0.631886\pi$
$839$	−19567.0	−0.805156	−0.402578	+	0.915386i	$0.631886\pi$
$840$	0	0
$841$	24262.1	0.994797
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2261.36	0.0920629
$846$	0	0
$847$	10932.4	0.443498
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10119.1	−0.407613
$852$	0	0
$853$	−40260.3	−1.61605	−0.808023	−	0.589150i	$-0.799463\pi$
$853$	−40260.3	−1.61605	−0.808023	+	0.589150i	$0.799463\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20796.2	0.828919	0.414460	−	0.910068i	$-0.363971\pi$
$857$	20796.2	0.828919	0.414460	+	0.910068i	$0.363971\pi$
$858$	0	0
$859$	−8423.66	−0.334589	−0.167294	−	0.985907i	$-0.553503\pi$
$859$	−8423.66	−0.334589	−0.167294	+	0.985907i	$0.553503\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12566.4	0.495672	0.247836	−	0.968802i	$-0.420281\pi$
$863$	12566.4	0.495672	0.247836	+	0.968802i	$0.420281\pi$
$864$	0	0
$865$	26850.8	1.05544
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31492.8	1.22937
$870$	0	0
$871$	5030.97	0.195715
$872$	0	0
$873$	0	0
$874$	0	0
$875$	14602.8	0.564188
$876$	0	0
$877$	−34769.2	−1.33874	−0.669369	−	0.742930i	$-0.733436\pi$
$877$	−34769.2	−1.33874	−0.669369	+	0.742930i	$0.733436\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41031.4	−1.56911	−0.784553	−	0.620062i	$-0.787107\pi$
$881$	−41031.4	−1.56911	−0.784553	+	0.620062i	$0.787107\pi$
$882$	0	0
$883$	7638.17	0.291104	0.145552	−	0.989351i	$-0.453504\pi$
$883$	7638.17	0.291104	0.145552	+	0.989351i	$0.453504\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39173.3	−1.48287	−0.741437	−	0.671022i	$-0.765856\pi$
$887$	−39173.3	−1.48287	−0.741437	+	0.671022i	$0.765856\pi$
$888$	0	0
$889$	−22483.5	−0.848226
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16077.2	−0.602465
$894$	0	0
$895$	−42294.6	−1.57961
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30658.7	−1.13740
$900$	0	0
$901$	12399.5	0.458477
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12444.3	0.457086
$906$	0	0
$907$	6296.82	0.230521	0.115260	−	0.993335i	$-0.463230\pi$
$907$	6296.82	0.230521	0.115260	+	0.993335i	$0.463230\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1278.72	0.0465050	0.0232525	−	0.999730i	$-0.492598\pi$
$911$	1278.72	0.0465050	0.0232525	+	0.999730i	$0.492598\pi$
$912$	0	0
$913$	−3923.96	−0.142239
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19742.9	0.710981
$918$	0	0
$919$	−17427.0	−0.625533	−0.312766	−	0.949830i	$-0.601256\pi$
$919$	−17427.0	−0.625533	−0.312766	+	0.949830i	$0.601256\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8638.77	0.308070
$924$	0	0
$925$	−19142.6	−0.680438
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12008.2	0.424087	0.212044	−	0.977260i	$-0.431988\pi$
$929$	12008.2	0.424087	0.212044	+	0.977260i	$0.431988\pi$
$930$	0	0
$931$	−7789.52	−0.274212
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21834.9	0.763719
$936$	0	0
$937$	−5606.10	−0.195457	−0.0977286	−	0.995213i	$-0.531158\pi$
$937$	−5606.10	−0.195457	−0.0977286	+	0.995213i	$0.531158\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55487.3	1.92225	0.961124	−	0.276118i	$-0.0890481\pi$
$941$	55487.3	1.92225	0.961124	+	0.276118i	$0.0890481\pi$
$942$	0	0
$943$	8504.31	0.293678
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14663.7	0.503175	0.251587	−	0.967835i	$-0.419047\pi$
$947$	14663.7	0.503175	0.251587	+	0.967835i	$0.419047\pi$
$948$	0	0
$949$	3227.64	0.110404
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22363.5	0.760151	0.380076	−	0.924955i	$-0.375898\pi$
$953$	22363.5	0.760151	0.380076	+	0.924955i	$0.375898\pi$
$954$	0	0
$955$	−29390.9	−0.995881
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26234.2	−0.883366
$960$	0	0
$961$	−10470.7	−0.351472
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14682.2	0.489778
$966$	0	0
$967$	−10946.8	−0.364037	−0.182019	−	0.983295i	$-0.558263\pi$
$967$	−10946.8	−0.364037	−0.182019	+	0.983295i	$0.558263\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−166.215	−0.00549341	−0.00274670	−	0.999996i	$-0.500874\pi$
$971$	−166.215	−0.00549341	−0.00274670	+	0.999996i	$0.500874\pi$
$972$	0	0
$973$	−33993.1	−1.12001
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2570.82	0.0841840	0.0420920	−	0.999114i	$-0.486598\pi$
$977$	2570.82	0.0841840	0.0420920	+	0.999114i	$0.486598\pi$
$978$	0	0
$979$	−19277.0	−0.629313
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41497.3	1.34645	0.673224	−	0.739438i	$-0.264909\pi$
$983$	41497.3	1.34645	0.673224	+	0.739438i	$0.264909\pi$
$984$	0	0
$985$	33870.6	1.09564
$986$	0	0
$987$	0	0
$988$	0	0
$989$	274.892	0.00883828
$990$	0	0
$991$	−17238.5	−0.552572	−0.276286	−	0.961076i	$-0.589104\pi$
$991$	−17238.5	−0.552572	−0.276286	+	0.961076i	$0.589104\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−55810.3	−1.77820
$996$	0	0
$997$	−35192.2	−1.11790	−0.558952	−	0.829200i	$-0.688796\pi$
$997$	−35192.2	−1.11790	−0.558952	+	0.829200i	$0.688796\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bg.1.2		2
3.2	odd	2		1872.4.a.w.1.1		2
4.3	odd	2		234.4.a.m.1.2	yes	2
12.11	even	2		234.4.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.4.a.l.1.1	✓	2	12.11	even	2
234.4.a.m.1.2	yes	2	4.3	odd	2
1872.4.a.w.1.1		2	3.2	odd	2
1872.4.a.bg.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+13.3808 q^{5} -15.3808 q^{7} +24.9042 q^{11} +13.0000 q^{13} +65.5233 q^{17} +73.1891 q^{19} +28.5700 q^{23} +54.0467 q^{25} +220.570 q^{29} -138.997 q^{31} -205.808 q^{35} -354.187 q^{37} +297.666 q^{41} +9.62171 q^{43} -219.666 q^{47} -106.430 q^{49} +189.238 q^{53} +333.238 q^{55} +329.376 q^{59} -838.757 q^{61} +173.951 q^{65} +386.997 q^{67} +664.521 q^{71} +248.280 q^{73} -383.047 q^{77} +1264.56 q^{79} -157.562 q^{83} +876.757 q^{85} -774.049 q^{89} -199.951 q^{91} +979.332 q^{95} +1054.94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{5} - 12 q^{7} - 44 q^{11} + 26 q^{13} + 56 q^{17} - 60 q^{19} - 168 q^{23} - 42 q^{25} + 216 q^{29} + 116 q^{31} - 224 q^{35} - 108 q^{37} + 464 q^{41} + 432 q^{43} - 308 q^{47} - 438 q^{49}+ \cdots + 684 q^{97}+O(q^{100}) \) 2 * q + 8 * q^5 - 12 * q^7 - 44 * q^11 + 26 * q^13 + 56 * q^17 - 60 * q^19 - 168 * q^23 - 42 * q^25 + 216 * q^29 + 116 * q^31 - 224 * q^35 - 108 * q^37 + 464 * q^41 + 432 * q^43 - 308 * q^47 - 438 * q^49 + 416 * q^53 + 704 * q^55 - 148 * q^59 - 852 * q^61 + 104 * q^65 + 380 * q^67 + 860 * q^71 - 404 * q^73 - 616 * q^77 + 728 * q^79 + 1092 * q^83 + 928 * q^85 - 1792 * q^89 - 156 * q^91 + 1696 * q^95 + 684 * q^97

Introduction

L-functions

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