LMFDB - Embedded newform 312.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,4,Mod(1,312)] 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("312.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,12,0,44] 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:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.55744$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} +26.0000 q^{11} -13.0000 q^{13} -21.3446 q^{15} +16.2298 q^{17} +35.5744 q^{19} +26.6554 q^{21} +153.379 q^{23} -74.3785 q^{25} +27.0000 q^{27} +223.379 q^{29} +126.723 q^{31} +78.0000 q^{33} -63.2166 q^{35} +217.608 q^{37} -39.0000 q^{39} +105.804 q^{41} -183.608 q^{43} -64.0339 q^{45} +96.6893 q^{47} -264.055 q^{49} +48.6893 q^{51} +386.460 q^{53} -184.987 q^{55} +106.723 q^{57} +34.5273 q^{59} +274.919 q^{61} +79.9661 q^{63} +92.4934 q^{65} +93.9397 q^{67} +460.136 q^{69} -741.812 q^{71} +640.689 q^{73} -223.136 q^{75} +231.013 q^{77} -182.945 q^{79} +81.0000 q^{81} -288.825 q^{83} -115.473 q^{85} +670.136 q^{87} +963.115 q^{89} -115.507 q^{91} +380.169 q^{93} -253.107 q^{95} -481.932 q^{97} +234.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35}+ \cdots + 468 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 + 52 * q^11 - 26 * q^13 + 36 * q^15 - 20 * q^17 - 60 * q^19 + 132 * q^21 - 8 * q^23 + 166 * q^25 + 54 * q^27 + 132 * q^29 - 140 * q^31 + 156 * q^33 + 608 * q^35 + 68 * q^37 - 78 * q^39 + 28 * q^41 + 108 * q^45 + 36 * q^47 + 626 * q^49 - 60 * q^51 + 668 * q^53 + 312 * q^55 - 180 * q^57 - 508 * q^59 + 340 * q^61 + 396 * q^63 - 156 * q^65 - 940 * q^67 - 24 * q^69 + 300 * q^71 + 1124 * q^73 + 498 * q^75 + 1144 * q^77 - 1520 * q^79 + 162 * q^81 + 524 * q^83 - 808 * q^85 + 396 * q^87 + 1900 * q^89 - 572 * q^91 - 420 * q^93 - 2080 * q^95 - 1436 * q^97 + 468 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−7.11488	−0.636374	−0.318187	−	0.948028i	$-0.603074\pi$
$5$	−7.11488	−0.636374	−0.318187	+	0.948028i	$0.603074\pi$
$6$	0	0
$7$	8.88512	0.479752	0.239876	−	0.970804i	$-0.422893\pi$
$7$	8.88512	0.479752	0.239876	+	0.970804i	$0.422893\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	26.0000	0.712663	0.356332	−	0.934360i	$-0.384027\pi$
$11$	26.0000	0.712663	0.356332	+	0.934360i	$0.384027\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−21.3446	−0.367411
$16$	0	0
$17$	16.2298	0.231547	0.115773	−	0.993276i	$-0.463065\pi$
$17$	16.2298	0.231547	0.115773	+	0.993276i	$0.463065\pi$
$18$	0	0
$19$	35.5744	0.429543	0.214772	−	0.976664i	$-0.431099\pi$
$19$	35.5744	0.429543	0.214772	+	0.976664i	$0.431099\pi$
$20$	0	0
$21$	26.6554	0.276985
$22$	0	0
$23$	153.379	1.39051	0.695253	−	0.718765i	$-0.255292\pi$
$23$	153.379	1.39051	0.695253	+	0.718765i	$0.255292\pi$
$24$	0	0
$25$	−74.3785	−0.595028
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	223.379	1.43036	0.715178	−	0.698942i	$-0.246345\pi$
$29$	223.379	1.43036	0.715178	+	0.698942i	$0.246345\pi$
$30$	0	0
$31$	126.723	0.734198	0.367099	−	0.930182i	$-0.380351\pi$
$31$	126.723	0.734198	0.367099	+	0.930182i	$0.380351\pi$
$32$	0	0
$33$	78.0000	0.411456
$34$	0	0
$35$	−63.2166	−0.305301
$36$	0	0
$37$	217.608	0.966881	0.483440	−	0.875377i	$-0.339387\pi$
$37$	217.608	0.966881	0.483440	+	0.875377i	$0.339387\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	105.804	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$41$	105.804	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$42$	0	0
$43$	−183.608	−0.651163	−0.325581	−	0.945514i	$-0.605560\pi$
$43$	−183.608	−0.651163	−0.325581	+	0.945514i	$0.605560\pi$
$44$	0	0
$45$	−64.0339	−0.212125
$46$	0	0
$47$	96.6893	0.300076	0.150038	−	0.988680i	$-0.452060\pi$
$47$	96.6893	0.300076	0.150038	+	0.988680i	$0.452060\pi$
$48$	0	0
$49$	−264.055	−0.769838
$50$	0	0
$51$	48.6893	0.133684
$52$	0	0
$53$	386.460	1.00159	0.500795	−	0.865566i	$-0.333041\pi$
$53$	386.460	1.00159	0.500795	+	0.865566i	$0.333041\pi$
$54$	0	0
$55$	−184.987	−0.453520
$56$	0	0
$57$	106.723	0.247997
$58$	0	0
$59$	34.5273	0.0761876	0.0380938	−	0.999274i	$-0.487871\pi$
$59$	34.5273	0.0761876	0.0380938	+	0.999274i	$0.487871\pi$
$60$	0	0
$61$	274.919	0.577045	0.288523	−	0.957473i	$-0.406836\pi$
$61$	274.919	0.577045	0.288523	+	0.957473i	$0.406836\pi$
$62$	0	0
$63$	79.9661	0.159917
$64$	0	0
$65$	92.4934	0.176498
$66$	0	0
$67$	93.9397	0.171292	0.0856460	−	0.996326i	$-0.472705\pi$
$67$	93.9397	0.171292	0.0856460	+	0.996326i	$0.472705\pi$
$68$	0	0
$69$	460.136	0.802809
$70$	0	0
$71$	−741.812	−1.23996	−0.619978	−	0.784619i	$-0.712858\pi$
$71$	−741.812	−1.23996	−0.619978	+	0.784619i	$0.712858\pi$
$72$	0	0
$73$	640.689	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$73$	640.689	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$74$	0	0
$75$	−223.136	−0.343540
$76$	0	0
$77$	231.013	0.341901
$78$	0	0
$79$	−182.945	−0.260544	−0.130272	−	0.991478i	$-0.541585\pi$
$79$	−182.945	−0.260544	−0.130272	+	0.991478i	$0.541585\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−288.825	−0.381960	−0.190980	−	0.981594i	$-0.561166\pi$
$83$	−288.825	−0.381960	−0.190980	+	0.981594i	$0.561166\pi$
$84$	0	0
$85$	−115.473	−0.147350
$86$	0	0
$87$	670.136	0.825817
$88$	0	0
$89$	963.115	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$89$	963.115	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$90$	0	0
$91$	−115.507	−0.133059
$92$	0	0
$93$	380.169	0.423890
$94$	0	0
$95$	−253.107	−0.273350
$96$	0	0
$97$	−481.932	−0.504462	−0.252231	−	0.967667i	$-0.581164\pi$
$97$	−481.932	−0.504462	−0.252231	+	0.967667i	$0.581164\pi$
$98$	0	0
$99$	234.000	0.237554
$100$	0	0
$101$	1138.85	1.12198	0.560990	−	0.827823i	$-0.310421\pi$
$101$	1138.85	1.12198	0.560990	+	0.827823i	$0.310421\pi$
$102$	0	0
$103$	−1090.96	−1.04365	−0.521823	−	0.853054i	$-0.674748\pi$
$103$	−1090.96	−1.04365	−0.521823	+	0.853054i	$0.674748\pi$
$104$	0	0
$105$	−189.650	−0.176266
$106$	0	0
$107$	124.595	0.112571	0.0562853	−	0.998415i	$-0.482074\pi$
$107$	124.595	0.112571	0.0562853	+	0.998415i	$0.482074\pi$
$108$	0	0
$109$	−1400.89	−1.23102	−0.615510	−	0.788129i	$-0.711050\pi$
$109$	−1400.89	−1.23102	−0.615510	+	0.788129i	$0.711050\pi$
$110$	0	0
$111$	652.825	0.558229
$112$	0	0
$113$	−541.812	−0.451056	−0.225528	−	0.974237i	$-0.572411\pi$
$113$	−541.812	−0.451056	−0.225528	+	0.974237i	$0.572411\pi$
$114$	0	0
$115$	−1091.27	−0.884882
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	144.203	0.111085
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	317.412	0.232684
$124$	0	0
$125$	1418.55	1.01503
$126$	0	0
$127$	−1552.07	−1.08444	−0.542219	−	0.840237i	$-0.682416\pi$
$127$	−1552.07	−1.08444	−0.542219	+	0.840237i	$0.682416\pi$
$128$	0	0
$129$	−550.825	−0.375949
$130$	0	0
$131$	−2281.97	−1.52196	−0.760981	−	0.648774i	$-0.775282\pi$
$131$	−2281.97	−1.52196	−0.760981	+	0.648774i	$0.775282\pi$
$132$	0	0
$133$	316.083	0.206074
$134$	0	0
$135$	−192.102	−0.122470
$136$	0	0
$137$	−1443.06	−0.899921	−0.449960	−	0.893049i	$-0.648562\pi$
$137$	−1443.06	−0.899921	−0.449960	+	0.893049i	$0.648562\pi$
$138$	0	0
$139$	−3060.41	−1.86748	−0.933742	−	0.357946i	$-0.883477\pi$
$139$	−3060.41	−1.86748	−0.933742	+	0.357946i	$0.883477\pi$
$140$	0	0
$141$	290.068	0.173249
$142$	0	0
$143$	−338.000	−0.197657
$144$	0	0
$145$	−1589.31	−0.910242
$146$	0	0
$147$	−792.164	−0.444466
$148$	0	0
$149$	302.520	0.166331	0.0831657	−	0.996536i	$-0.473497\pi$
$149$	302.520	0.166331	0.0831657	+	0.996536i	$0.473497\pi$
$150$	0	0
$151$	383.574	0.206721	0.103360	−	0.994644i	$-0.467040\pi$
$151$	383.574	0.206721	0.103360	+	0.994644i	$0.467040\pi$
$152$	0	0
$153$	146.068	0.0771822
$154$	0	0
$155$	−901.620	−0.467225
$156$	0	0
$157$	−2583.38	−1.31322	−0.656612	−	0.754229i	$-0.728011\pi$
$157$	−2583.38	−1.31322	−0.656612	+	0.754229i	$0.728011\pi$
$158$	0	0
$159$	1159.38	0.578269
$160$	0	0
$161$	1362.79	0.667097
$162$	0	0
$163$	1303.45	0.626346	0.313173	−	0.949696i	$-0.398608\pi$
$163$	1303.45	0.626346	0.313173	+	0.949696i	$0.398608\pi$
$164$	0	0
$165$	−554.960	−0.261840
$166$	0	0
$167$	1040.04	0.481921	0.240961	−	0.970535i	$-0.422538\pi$
$167$	1040.04	0.481921	0.240961	+	0.970535i	$0.422538\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	320.169	0.143181
$172$	0	0
$173$	2989.10	1.31362	0.656811	−	0.754055i	$-0.271905\pi$
$173$	2989.10	1.31362	0.656811	+	0.754055i	$0.271905\pi$
$174$	0	0
$175$	−660.862	−0.285466
$176$	0	0
$177$	103.582	0.0439870
$178$	0	0
$179$	−1786.06	−0.745788	−0.372894	−	0.927874i	$-0.621635\pi$
$179$	−1786.06	−0.745788	−0.372894	+	0.927874i	$0.621635\pi$
$180$	0	0
$181$	−1320.84	−0.542416	−0.271208	−	0.962521i	$-0.587423\pi$
$181$	−1320.84	−0.542416	−0.271208	+	0.962521i	$0.587423\pi$
$182$	0	0
$183$	824.757	0.333157
$184$	0	0
$185$	−1548.26	−0.615298
$186$	0	0
$187$	421.974	0.165015
$188$	0	0
$189$	239.898	0.0923282
$190$	0	0
$191$	−436.595	−0.165397	−0.0826987	−	0.996575i	$-0.526354\pi$
$191$	−436.595	−0.165397	−0.0826987	+	0.996575i	$0.526354\pi$
$192$	0	0
$193$	−2402.19	−0.895924	−0.447962	−	0.894053i	$-0.647850\pi$
$193$	−2402.19	−0.895924	−0.447962	+	0.894053i	$0.647850\pi$
$194$	0	0
$195$	277.480	0.101901
$196$	0	0
$197$	1129.94	0.408654	0.204327	−	0.978903i	$-0.434499\pi$
$197$	1129.94	0.408654	0.204327	+	0.978903i	$0.434499\pi$
$198$	0	0
$199$	1859.61	0.662433	0.331216	−	0.943555i	$-0.392541\pi$
$199$	1859.61	0.662433	0.331216	+	0.943555i	$0.392541\pi$
$200$	0	0
$201$	281.819	0.0988955
$202$	0	0
$203$	1984.75	0.686216
$204$	0	0
$205$	−752.783	−0.256472
$206$	0	0
$207$	1380.41	0.463502
$208$	0	0
$209$	924.934	0.306120
$210$	0	0
$211$	859.729	0.280503	0.140252	−	0.990116i	$-0.455209\pi$
$211$	859.729	0.280503	0.140252	+	0.990116i	$0.455209\pi$
$212$	0	0
$213$	−2225.43	−0.715889
$214$	0	0
$215$	1306.35	0.414383
$216$	0	0
$217$	1125.95	0.352233
$218$	0	0
$219$	1922.07	0.593065
$220$	0	0
$221$	−210.987	−0.0642195
$222$	0	0
$223$	265.345	0.0796807	0.0398403	−	0.999206i	$-0.487315\pi$
$223$	265.345	0.0796807	0.0398403	+	0.999206i	$0.487315\pi$
$224$	0	0
$225$	−669.407	−0.198343
$226$	0	0
$227$	1519.43	0.444265	0.222133	−	0.975016i	$-0.428698\pi$
$227$	1519.43	0.444265	0.222133	+	0.975016i	$0.428698\pi$
$228$	0	0
$229$	2069.00	0.597045	0.298522	−	0.954403i	$-0.403506\pi$
$229$	2069.00	0.597045	0.298522	+	0.954403i	$0.403506\pi$
$230$	0	0
$231$	693.040	0.197397
$232$	0	0
$233$	−4624.94	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$233$	−4624.94	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$234$	0	0
$235$	−687.932	−0.190961
$236$	0	0
$237$	−548.836	−0.150425
$238$	0	0
$239$	2098.35	0.567913	0.283957	−	0.958837i	$-0.408353\pi$
$239$	2098.35	0.567913	0.283957	+	0.958837i	$0.408353\pi$
$240$	0	0
$241$	−1345.96	−0.359755	−0.179878	−	0.983689i	$-0.557570\pi$
$241$	−1345.96	−0.359755	−0.179878	+	0.983689i	$0.557570\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1878.72	0.489905
$246$	0	0
$247$	−462.467	−0.119134
$248$	0	0
$249$	−866.475	−0.220524
$250$	0	0
$251$	4117.24	1.03537	0.517685	−	0.855571i	$-0.326794\pi$
$251$	4117.24	1.03537	0.517685	+	0.855571i	$0.326794\pi$
$252$	0	0
$253$	3987.84	0.990962
$254$	0	0
$255$	−346.418	−0.0850727
$256$	0	0
$257$	3393.64	0.823694	0.411847	−	0.911253i	$-0.364884\pi$
$257$	3393.64	0.823694	0.411847	+	0.911253i	$0.364884\pi$
$258$	0	0
$259$	1933.48	0.463862
$260$	0	0
$261$	2010.41	0.476786
$262$	0	0
$263$	−6335.19	−1.48534	−0.742671	−	0.669657i	$-0.766441\pi$
$263$	−6335.19	−1.48534	−0.742671	+	0.669657i	$0.766441\pi$
$264$	0	0
$265$	−2749.61	−0.637386
$266$	0	0
$267$	2889.34	0.662266
$268$	0	0
$269$	619.702	0.140461	0.0702303	−	0.997531i	$-0.477627\pi$
$269$	619.702	0.140461	0.0702303	+	0.997531i	$0.477627\pi$
$270$	0	0
$271$	5165.75	1.15792	0.578961	−	0.815355i	$-0.303458\pi$
$271$	5165.75	1.15792	0.578961	+	0.815355i	$0.303458\pi$
$272$	0	0
$273$	−346.520	−0.0768217
$274$	0	0
$275$	−1933.84	−0.424055
$276$	0	0
$277$	3534.38	0.766643	0.383322	−	0.923615i	$-0.374780\pi$
$277$	3534.38	0.766643	0.383322	+	0.923615i	$0.374780\pi$
$278$	0	0
$279$	1140.51	0.244733
$280$	0	0
$281$	4115.81	0.873767	0.436883	−	0.899518i	$-0.356082\pi$
$281$	4115.81	0.873767	0.436883	+	0.899518i	$0.356082\pi$
$282$	0	0
$283$	3944.56	0.828550	0.414275	−	0.910152i	$-0.364035\pi$
$283$	3944.56	0.828550	0.414275	+	0.910152i	$0.364035\pi$
$284$	0	0
$285$	−759.322	−0.157819
$286$	0	0
$287$	940.083	0.193350
$288$	0	0
$289$	−4649.60	−0.946386
$290$	0	0
$291$	−1445.80	−0.291251
$292$	0	0
$293$	772.441	0.154015	0.0770076	−	0.997031i	$-0.475463\pi$
$293$	772.441	0.154015	0.0770076	+	0.997031i	$0.475463\pi$
$294$	0	0
$295$	−245.657	−0.0484838
$296$	0	0
$297$	702.000	0.137152
$298$	0	0
$299$	−1993.92	−0.385657
$300$	0	0
$301$	−1631.38	−0.312396
$302$	0	0
$303$	3416.55	0.647775
$304$	0	0
$305$	−1956.01	−0.367217
$306$	0	0
$307$	4885.74	0.908286	0.454143	−	0.890929i	$-0.349946\pi$
$307$	4885.74	0.908286	0.454143	+	0.890929i	$0.349946\pi$
$308$	0	0
$309$	−3272.88	−0.602549
$310$	0	0
$311$	177.544	0.0323717	0.0161859	−	0.999869i	$-0.494848\pi$
$311$	177.544	0.0323717	0.0161859	+	0.999869i	$0.494848\pi$
$312$	0	0
$313$	−6677.03	−1.20578	−0.602888	−	0.797826i	$-0.705983\pi$
$313$	−6677.03	−1.20578	−0.602888	+	0.797826i	$0.705983\pi$
$314$	0	0
$315$	−568.949	−0.101767
$316$	0	0
$317$	1806.53	0.320079	0.160040	−	0.987111i	$-0.448838\pi$
$317$	1806.53	0.320079	0.160040	+	0.987111i	$0.448838\pi$
$318$	0	0
$319$	5807.84	1.01936
$320$	0	0
$321$	373.785	0.0649927
$322$	0	0
$323$	577.364	0.0994593
$324$	0	0
$325$	966.921	0.165031
$326$	0	0
$327$	−4202.68	−0.710730
$328$	0	0
$329$	859.096	0.143962
$330$	0	0
$331$	11447.2	1.90088	0.950442	−	0.310901i	$-0.100631\pi$
$331$	11447.2	1.90088	0.950442	+	0.310901i	$0.100631\pi$
$332$	0	0
$333$	1958.47	0.322294
$334$	0	0
$335$	−668.370	−0.109006
$336$	0	0
$337$	−821.752	−0.132830	−0.0664149	−	0.997792i	$-0.521156\pi$
$337$	−821.752	−0.132830	−0.0664149	+	0.997792i	$0.521156\pi$
$338$	0	0
$339$	−1625.43	−0.260417
$340$	0	0
$341$	3294.80	0.523236
$342$	0	0
$343$	−5393.75	−0.849083
$344$	0	0
$345$	−3273.81	−0.510887
$346$	0	0
$347$	−4819.55	−0.745611	−0.372805	−	0.927910i	$-0.621604\pi$
$347$	−4819.55	−0.745611	−0.372805	+	0.927910i	$0.621604\pi$
$348$	0	0
$349$	7922.72	1.21517	0.607584	−	0.794255i	$-0.292139\pi$
$349$	7922.72	1.21517	0.607584	+	0.794255i	$0.292139\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−2860.73	−0.431336	−0.215668	−	0.976467i	$-0.569193\pi$
$353$	−2860.73	−0.431336	−0.215668	+	0.976467i	$0.569193\pi$
$354$	0	0
$355$	5277.90	0.789075
$356$	0	0
$357$	432.610	0.0641349
$358$	0	0
$359$	−11785.8	−1.73267	−0.866334	−	0.499464i	$-0.833530\pi$
$359$	−11785.8	−1.73267	−0.866334	+	0.499464i	$0.833530\pi$
$360$	0	0
$361$	−5593.46	−0.815493
$362$	0	0
$363$	−1965.00	−0.284121
$364$	0	0
$365$	−4558.43	−0.653696
$366$	0	0
$367$	2017.19	0.286911	0.143456	−	0.989657i	$-0.454179\pi$
$367$	2017.19	0.286911	0.143456	+	0.989657i	$0.454179\pi$
$368$	0	0
$369$	952.237	0.134340
$370$	0	0
$371$	3433.74	0.480515
$372$	0	0
$373$	5148.21	0.714650	0.357325	−	0.933980i	$-0.383689\pi$
$373$	5148.21	0.714650	0.357325	+	0.933980i	$0.383689\pi$
$374$	0	0
$375$	4255.66	0.586030
$376$	0	0
$377$	−2903.92	−0.396710
$378$	0	0
$379$	−3419.10	−0.463397	−0.231698	−	0.972788i	$-0.574428\pi$
$379$	−3419.10	−0.463397	−0.231698	+	0.972788i	$0.574428\pi$
$380$	0	0
$381$	−4656.20	−0.626101
$382$	0	0
$383$	−5427.04	−0.724044	−0.362022	−	0.932170i	$-0.617913\pi$
$383$	−5427.04	−0.724044	−0.362022	+	0.932170i	$0.617913\pi$
$384$	0	0
$385$	−1643.63	−0.217577
$386$	0	0
$387$	−1652.47	−0.217054
$388$	0	0
$389$	6843.88	0.892027	0.446014	−	0.895026i	$-0.352843\pi$
$389$	6843.88	0.892027	0.446014	+	0.895026i	$0.352843\pi$
$390$	0	0
$391$	2489.30	0.321967
$392$	0	0
$393$	−6845.92	−0.878705
$394$	0	0
$395$	1301.63	0.165803
$396$	0	0
$397$	−8382.62	−1.05973	−0.529863	−	0.848083i	$-0.677757\pi$
$397$	−8382.62	−1.05973	−0.529863	+	0.848083i	$0.677757\pi$
$398$	0	0
$399$	948.248	0.118977
$400$	0	0
$401$	3525.30	0.439015	0.219508	−	0.975611i	$-0.429555\pi$
$401$	3525.30	0.439015	0.219508	+	0.975611i	$0.429555\pi$
$402$	0	0
$403$	−1647.40	−0.203630
$404$	0	0
$405$	−576.305	−0.0707082
$406$	0	0
$407$	5657.82	0.689060
$408$	0	0
$409$	940.772	0.113736	0.0568682	−	0.998382i	$-0.481889\pi$
$409$	940.772	0.113736	0.0568682	+	0.998382i	$0.481889\pi$
$410$	0	0
$411$	−4329.19	−0.519569
$412$	0	0
$413$	306.779	0.0365511
$414$	0	0
$415$	2054.95	0.243069
$416$	0	0
$417$	−9181.22	−1.07819
$418$	0	0
$419$	532.791	0.0621206	0.0310603	−	0.999518i	$-0.490112\pi$
$419$	532.791	0.0621206	0.0310603	+	0.999518i	$0.490112\pi$
$420$	0	0
$421$	14425.3	1.66994	0.834970	−	0.550295i	$-0.185485\pi$
$421$	14425.3	1.66994	0.834970	+	0.550295i	$0.185485\pi$
$422$	0	0
$423$	870.203	0.100025
$424$	0	0
$425$	−1207.15	−0.137777
$426$	0	0
$427$	2442.69	0.276838
$428$	0	0
$429$	−1014.00	−0.114117
$430$	0	0
$431$	−15224.9	−1.70152	−0.850762	−	0.525551i	$-0.823859\pi$
$431$	−15224.9	−1.70152	−0.850762	+	0.525551i	$0.823859\pi$
$432$	0	0
$433$	10115.5	1.12268	0.561338	−	0.827587i	$-0.310287\pi$
$433$	10115.5	1.12268	0.561338	+	0.827587i	$0.310287\pi$
$434$	0	0
$435$	−4767.93	−0.525528
$436$	0	0
$437$	5456.35	0.597283
$438$	0	0
$439$	−15925.0	−1.73134	−0.865669	−	0.500617i	$-0.833106\pi$
$439$	−15925.0	−1.73134	−0.865669	+	0.500617i	$0.833106\pi$
$440$	0	0
$441$	−2376.49	−0.256613
$442$	0	0
$443$	2387.09	0.256014	0.128007	−	0.991773i	$-0.459142\pi$
$443$	2387.09	0.256014	0.128007	+	0.991773i	$0.459142\pi$
$444$	0	0
$445$	−6852.44	−0.729971
$446$	0	0
$447$	907.559	0.0960315
$448$	0	0
$449$	−13985.3	−1.46995	−0.734973	−	0.678096i	$-0.762805\pi$
$449$	−13985.3	−1.46995	−0.734973	+	0.678096i	$0.762805\pi$
$450$	0	0
$451$	2750.91	0.287218
$452$	0	0
$453$	1150.72	0.119350
$454$	0	0
$455$	821.815	0.0846754
$456$	0	0
$457$	−6520.34	−0.667414	−0.333707	−	0.942677i	$-0.608300\pi$
$457$	−6520.34	−0.667414	−0.333707	+	0.942677i	$0.608300\pi$
$458$	0	0
$459$	438.203	0.0445612
$460$	0	0
$461$	6543.86	0.661124	0.330562	−	0.943784i	$-0.392762\pi$
$461$	6543.86	0.661124	0.330562	+	0.943784i	$0.392762\pi$
$462$	0	0
$463$	13172.7	1.32221	0.661107	−	0.750292i	$-0.270087\pi$
$463$	13172.7	1.32221	0.661107	+	0.750292i	$0.270087\pi$
$464$	0	0
$465$	−2704.86	−0.269752
$466$	0	0
$467$	−3166.27	−0.313742	−0.156871	−	0.987619i	$-0.550141\pi$
$467$	−3166.27	−0.313742	−0.156871	+	0.987619i	$0.550141\pi$
$468$	0	0
$469$	834.666	0.0821776
$470$	0	0
$471$	−7750.14	−0.758190
$472$	0	0
$473$	−4773.82	−0.464060
$474$	0	0
$475$	−2645.97	−0.255590
$476$	0	0
$477$	3478.14	0.333863
$478$	0	0
$479$	−6470.71	−0.617232	−0.308616	−	0.951187i	$-0.599866\pi$
$479$	−6470.71	−0.617232	−0.308616	+	0.951187i	$0.599866\pi$
$480$	0	0
$481$	−2828.91	−0.268164
$482$	0	0
$483$	4088.36	0.385149
$484$	0	0
$485$	3428.89	0.321026
$486$	0	0
$487$	7620.36	0.709058	0.354529	−	0.935045i	$-0.384641\pi$
$487$	7620.36	0.709058	0.354529	+	0.935045i	$0.384641\pi$
$488$	0	0
$489$	3910.36	0.361621
$490$	0	0
$491$	−14472.0	−1.33017	−0.665084	−	0.746769i	$-0.731604\pi$
$491$	−14472.0	−1.33017	−0.665084	+	0.746769i	$0.731604\pi$
$492$	0	0
$493$	3625.38	0.331194
$494$	0	0
$495$	−1664.88	−0.151173
$496$	0	0
$497$	−6591.09	−0.594871
$498$	0	0
$499$	−12338.1	−1.10687	−0.553436	−	0.832892i	$-0.686684\pi$
$499$	−12338.1	−1.10687	−0.553436	+	0.832892i	$0.686684\pi$
$500$	0	0
$501$	3120.12	0.278237
$502$	0	0
$503$	−12491.1	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$503$	−12491.1	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$504$	0	0
$505$	−8102.79	−0.713999
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−21986.9	−1.91464	−0.957318	−	0.289036i	$-0.906665\pi$
$509$	−21986.9	−1.91464	−0.957318	+	0.289036i	$0.906665\pi$
$510$	0	0
$511$	5692.60	0.492810
$512$	0	0
$513$	960.508	0.0826657
$514$	0	0
$515$	7762.05	0.664149
$516$	0	0
$517$	2513.92	0.213853
$518$	0	0
$519$	8967.29	0.758421
$520$	0	0
$521$	−6385.80	−0.536981	−0.268491	−	0.963282i	$-0.586525\pi$
$521$	−6385.80	−0.536981	−0.268491	+	0.963282i	$0.586525\pi$
$522$	0	0
$523$	−11858.5	−0.991467	−0.495733	−	0.868475i	$-0.665101\pi$
$523$	−11858.5	−0.991467	−0.495733	+	0.868475i	$0.665101\pi$
$524$	0	0
$525$	−1982.59	−0.164814
$526$	0	0
$527$	2056.69	0.170001
$528$	0	0
$529$	11358.0	0.933506
$530$	0	0
$531$	310.746	0.0253959
$532$	0	0
$533$	−1375.45	−0.111778
$534$	0	0
$535$	−886.479	−0.0716370
$536$	0	0
$537$	−5358.17	−0.430581
$538$	0	0
$539$	−6865.42	−0.548636
$540$	0	0
$541$	19726.4	1.56766	0.783829	−	0.620976i	$-0.213264\pi$
$541$	19726.4	1.56766	0.783829	+	0.620976i	$0.213264\pi$
$542$	0	0
$543$	−3962.52	−0.313164
$544$	0	0
$545$	9967.18	0.783389
$546$	0	0
$547$	−9070.36	−0.708996	−0.354498	−	0.935057i	$-0.615348\pi$
$547$	−9070.36	−0.708996	−0.354498	+	0.935057i	$0.615348\pi$
$548$	0	0
$549$	2474.27	0.192348
$550$	0	0
$551$	7946.55	0.614400
$552$	0	0
$553$	−1625.49	−0.124996
$554$	0	0
$555$	−4644.77	−0.355242
$556$	0	0
$557$	21557.9	1.63992	0.819961	−	0.572419i	$-0.193995\pi$
$557$	21557.9	1.63992	0.819961	+	0.572419i	$0.193995\pi$
$558$	0	0
$559$	2386.91	0.180600
$560$	0	0
$561$	1265.92	0.0952713
$562$	0	0
$563$	22899.8	1.71423	0.857113	−	0.515128i	$-0.172255\pi$
$563$	22899.8	1.71423	0.857113	+	0.515128i	$0.172255\pi$
$564$	0	0
$565$	3854.92	0.287040
$566$	0	0
$567$	719.695	0.0533057
$568$	0	0
$569$	−14009.0	−1.03214	−0.516071	−	0.856546i	$-0.672606\pi$
$569$	−14009.0	−1.03214	−0.516071	+	0.856546i	$0.672606\pi$
$570$	0	0
$571$	−22800.2	−1.67103	−0.835517	−	0.549465i	$-0.814832\pi$
$571$	−22800.2	−1.67103	−0.835517	+	0.549465i	$0.814832\pi$
$572$	0	0
$573$	−1309.79	−0.0954923
$574$	0	0
$575$	−11408.1	−0.827390
$576$	0	0
$577$	16722.7	1.20655	0.603273	−	0.797535i	$-0.293863\pi$
$577$	16722.7	1.20655	0.603273	+	0.797535i	$0.293863\pi$
$578$	0	0
$579$	−7206.57	−0.517262
$580$	0	0
$581$	−2566.24	−0.183246
$582$	0	0
$583$	10047.9	0.713797
$584$	0	0
$585$	832.441	0.0588328
$586$	0	0
$587$	4728.47	0.332479	0.166239	−	0.986085i	$-0.446838\pi$
$587$	4728.47	0.332479	0.166239	+	0.986085i	$0.446838\pi$
$588$	0	0
$589$	4508.10	0.315370
$590$	0	0
$591$	3389.82	0.235937
$592$	0	0
$593$	15903.8	1.10133	0.550667	−	0.834725i	$-0.314374\pi$
$593$	15903.8	1.10133	0.550667	+	0.834725i	$0.314374\pi$
$594$	0	0
$595$	−1025.99	−0.0706915
$596$	0	0
$597$	5578.82	0.382456
$598$	0	0
$599$	−7515.22	−0.512627	−0.256313	−	0.966594i	$-0.582508\pi$
$599$	−7515.22	−0.512627	−0.256313	+	0.966594i	$0.582508\pi$
$600$	0	0
$601$	26704.5	1.81247	0.906237	−	0.422769i	$-0.138942\pi$
$601$	26704.5	1.81247	0.906237	+	0.422769i	$0.138942\pi$
$602$	0	0
$603$	845.457	0.0570973
$604$	0	0
$605$	4660.24	0.313167
$606$	0	0
$607$	−1113.76	−0.0744748	−0.0372374	−	0.999306i	$-0.511856\pi$
$607$	−1113.76	−0.0744748	−0.0372374	+	0.999306i	$0.511856\pi$
$608$	0	0
$609$	5954.24	0.396187
$610$	0	0
$611$	−1256.96	−0.0832261
$612$	0	0
$613$	−12195.8	−0.803559	−0.401780	−	0.915736i	$-0.631608\pi$
$613$	−12195.8	−0.803559	−0.401780	+	0.915736i	$0.631608\pi$
$614$	0	0
$615$	−2258.35	−0.148074
$616$	0	0
$617$	−447.009	−0.0291668	−0.0145834	−	0.999894i	$-0.504642\pi$
$617$	−447.009	−0.0291668	−0.0145834	+	0.999894i	$0.504642\pi$
$618$	0	0
$619$	−22476.3	−1.45945	−0.729725	−	0.683740i	$-0.760352\pi$
$619$	−22476.3	−1.45945	−0.729725	+	0.683740i	$0.760352\pi$
$620$	0	0
$621$	4141.22	0.267603
$622$	0	0
$623$	8557.39	0.550313
$624$	0	0
$625$	−795.520	−0.0509132
$626$	0	0
$627$	2774.80	0.176738
$628$	0	0
$629$	3531.73	0.223878
$630$	0	0
$631$	−7165.04	−0.452037	−0.226019	−	0.974123i	$-0.572571\pi$
$631$	−7165.04	−0.452037	−0.226019	+	0.974123i	$0.572571\pi$
$632$	0	0
$633$	2579.19	0.161949
$634$	0	0
$635$	11042.8	0.690109
$636$	0	0
$637$	3432.71	0.213515
$638$	0	0
$639$	−6676.30	−0.413319
$640$	0	0
$641$	12463.4	0.767980	0.383990	−	0.923337i	$-0.374550\pi$
$641$	12463.4	0.767980	0.383990	+	0.923337i	$0.374550\pi$
$642$	0	0
$643$	11009.8	0.675249	0.337625	−	0.941281i	$-0.390377\pi$
$643$	11009.8	0.675249	0.337625	+	0.941281i	$0.390377\pi$
$644$	0	0
$645$	3919.05	0.239244
$646$	0	0
$647$	−30825.9	−1.87309	−0.936547	−	0.350541i	$-0.885998\pi$
$647$	−30825.9	−1.87309	−0.936547	+	0.350541i	$0.885998\pi$
$648$	0	0
$649$	897.710	0.0542961
$650$	0	0
$651$	3377.85	0.203362
$652$	0	0
$653$	15675.9	0.939426	0.469713	−	0.882819i	$-0.344357\pi$
$653$	15675.9	0.939426	0.469713	+	0.882819i	$0.344357\pi$
$654$	0	0
$655$	16236.0	0.968537
$656$	0	0
$657$	5766.20	0.342406
$658$	0	0
$659$	22568.8	1.33408	0.667038	−	0.745024i	$-0.267562\pi$
$659$	22568.8	1.33408	0.667038	+	0.745024i	$0.267562\pi$
$660$	0	0
$661$	31243.5	1.83847	0.919237	−	0.393705i	$-0.128807\pi$
$661$	31243.5	1.83847	0.919237	+	0.393705i	$0.128807\pi$
$662$	0	0
$663$	−632.960	−0.0370771
$664$	0	0
$665$	−2248.89	−0.131140
$666$	0	0
$667$	34261.5	1.98892
$668$	0	0
$669$	796.034	0.0460037
$670$	0	0
$671$	7147.89	0.411239
$672$	0	0
$673$	−19818.1	−1.13511	−0.567556	−	0.823335i	$-0.692111\pi$
$673$	−19818.1	−1.13511	−0.567556	+	0.823335i	$0.692111\pi$
$674$	0	0
$675$	−2008.22	−0.114513
$676$	0	0
$677$	3421.40	0.194232	0.0971159	−	0.995273i	$-0.469038\pi$
$677$	3421.40	0.194232	0.0971159	+	0.995273i	$0.469038\pi$
$678$	0	0
$679$	−4282.03	−0.242016
$680$	0	0
$681$	4558.29	0.256497
$682$	0	0
$683$	11002.1	0.616373	0.308186	−	0.951326i	$-0.400278\pi$
$683$	11002.1	0.616373	0.308186	+	0.951326i	$0.400278\pi$
$684$	0	0
$685$	10267.2	0.572686
$686$	0	0
$687$	6206.99	0.344704
$688$	0	0
$689$	−5023.97	−0.277791
$690$	0	0
$691$	−9817.44	−0.540482	−0.270241	−	0.962793i	$-0.587103\pi$
$691$	−9817.44	−0.540482	−0.270241	+	0.962793i	$0.587103\pi$
$692$	0	0
$693$	2079.12	0.113967
$694$	0	0
$695$	21774.4	1.18842
$696$	0	0
$697$	1717.18	0.0933180
$698$	0	0
$699$	−13874.8	−0.750778
$700$	0	0
$701$	−5776.14	−0.311215	−0.155608	−	0.987819i	$-0.549734\pi$
$701$	−5776.14	−0.311215	−0.155608	+	0.987819i	$0.549734\pi$
$702$	0	0
$703$	7741.28	0.415317
$704$	0	0
$705$	−2063.80	−0.110251
$706$	0	0
$707$	10118.8	0.538271
$708$	0	0
$709$	−35654.9	−1.88864	−0.944322	−	0.329022i	$-0.893281\pi$
$709$	−35654.9	−1.88864	−0.944322	+	0.329022i	$0.893281\pi$
$710$	0	0
$711$	−1646.51	−0.0868480
$712$	0	0
$713$	19436.6	1.02091
$714$	0	0
$715$	2404.83	0.125784
$716$	0	0
$717$	6295.06	0.327885
$718$	0	0
$719$	14353.1	0.744478	0.372239	−	0.928137i	$-0.378590\pi$
$719$	14353.1	0.744478	0.372239	+	0.928137i	$0.378590\pi$
$720$	0	0
$721$	−9693.32	−0.500691
$722$	0	0
$723$	−4037.89	−0.207705
$724$	0	0
$725$	−16614.6	−0.851103
$726$	0	0
$727$	−28167.8	−1.43698	−0.718491	−	0.695536i	$-0.755167\pi$
$727$	−28167.8	−1.43698	−0.718491	+	0.695536i	$0.755167\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2979.92	−0.150775
$732$	0	0
$733$	−23945.2	−1.20660	−0.603299	−	0.797515i	$-0.706147\pi$
$733$	−23945.2	−1.20660	−0.603299	+	0.797515i	$0.706147\pi$
$734$	0	0
$735$	5636.15	0.282847
$736$	0	0
$737$	2442.43	0.122073
$738$	0	0
$739$	−4021.49	−0.200180	−0.100090	−	0.994978i	$-0.531913\pi$
$739$	−4021.49	−0.200180	−0.100090	+	0.994978i	$0.531913\pi$
$740$	0	0
$741$	−1387.40	−0.0687820
$742$	0	0
$743$	13584.9	0.670770	0.335385	−	0.942081i	$-0.391134\pi$
$743$	13584.9	0.670770	0.335385	+	0.942081i	$0.391134\pi$
$744$	0	0
$745$	−2152.39	−0.105849
$746$	0	0
$747$	−2599.42	−0.127320
$748$	0	0
$749$	1107.04	0.0540060
$750$	0	0
$751$	3537.89	0.171903	0.0859517	−	0.996299i	$-0.472607\pi$
$751$	3537.89	0.171903	0.0859517	+	0.996299i	$0.472607\pi$
$752$	0	0
$753$	12351.7	0.597772
$754$	0	0
$755$	−2729.08	−0.131552
$756$	0	0
$757$	−4421.46	−0.212286	−0.106143	−	0.994351i	$-0.533850\pi$
$757$	−4421.46	−0.212286	−0.106143	+	0.994351i	$0.533850\pi$
$758$	0	0
$759$	11963.5	0.572132
$760$	0	0
$761$	1093.63	0.0520947	0.0260473	−	0.999661i	$-0.491708\pi$
$761$	1093.63	0.0520947	0.0260473	+	0.999661i	$0.491708\pi$
$762$	0	0
$763$	−12447.1	−0.590584
$764$	0	0
$765$	−1039.25	−0.0491168
$766$	0	0
$767$	−448.855	−0.0211307
$768$	0	0
$769$	−15814.2	−0.741580	−0.370790	−	0.928717i	$-0.620913\pi$
$769$	−15814.2	−0.741580	−0.370790	+	0.928717i	$0.620913\pi$
$770$	0	0
$771$	10180.9	0.475560
$772$	0	0
$773$	−37380.8	−1.73932	−0.869659	−	0.493653i	$-0.835661\pi$
$773$	−37380.8	−1.73932	−0.869659	+	0.493653i	$0.835661\pi$
$774$	0	0
$775$	−9425.48	−0.436869
$776$	0	0
$777$	5800.43	0.267811
$778$	0	0
$779$	3763.92	0.173115
$780$	0	0
$781$	−19287.1	−0.883671
$782$	0	0
$783$	6031.22	0.275272
$784$	0	0
$785$	18380.4	0.835701
$786$	0	0
$787$	1979.23	0.0896469	0.0448234	−	0.998995i	$-0.485727\pi$
$787$	1979.23	0.0896469	0.0448234	+	0.998995i	$0.485727\pi$
$788$	0	0
$789$	−19005.6	−0.857562
$790$	0	0
$791$	−4814.06	−0.216395
$792$	0	0
$793$	−3573.95	−0.160044
$794$	0	0
$795$	−8248.84	−0.367995
$796$	0	0
$797$	−23575.5	−1.04779	−0.523894	−	0.851783i	$-0.675521\pi$
$797$	−23575.5	−1.04779	−0.523894	+	0.851783i	$0.675521\pi$
$798$	0	0
$799$	1569.24	0.0694816
$800$	0	0
$801$	8668.03	0.382359
$802$	0	0
$803$	16657.9	0.732061
$804$	0	0
$805$	−9696.06	−0.424523
$806$	0	0
$807$	1859.11	0.0810950
$808$	0	0
$809$	−39145.7	−1.70122	−0.850611	−	0.525796i	$-0.823768\pi$
$809$	−39145.7	−1.70122	−0.850611	+	0.525796i	$0.823768\pi$
$810$	0	0
$811$	−39342.9	−1.70347	−0.851735	−	0.523973i	$-0.824449\pi$
$811$	−39342.9	−1.70347	−0.851735	+	0.523973i	$0.824449\pi$
$812$	0	0
$813$	15497.3	0.668527
$814$	0	0
$815$	−9273.91	−0.398590
$816$	0	0
$817$	−6531.75	−0.279703
$818$	0	0
$819$	−1039.56	−0.0443530
$820$	0	0
$821$	216.569	0.00920622	0.00460311	−	0.999989i	$-0.498535\pi$
$821$	216.569	0.00920622	0.00460311	+	0.999989i	$0.498535\pi$
$822$	0	0
$823$	45061.7	1.90857	0.954284	−	0.298901i	$-0.0966201\pi$
$823$	45061.7	1.90857	0.954284	+	0.298901i	$0.0966201\pi$
$824$	0	0
$825$	−5801.52	−0.244828
$826$	0	0
$827$	1228.67	0.0516625	0.0258313	−	0.999666i	$-0.491777\pi$
$827$	1228.67	0.0516625	0.0258313	+	0.999666i	$0.491777\pi$
$828$	0	0
$829$	−29560.3	−1.23845	−0.619224	−	0.785215i	$-0.712553\pi$
$829$	−29560.3	−1.23845	−0.619224	+	0.785215i	$0.712553\pi$
$830$	0	0
$831$	10603.1	0.442622
$832$	0	0
$833$	−4285.54	−0.178254
$834$	0	0
$835$	−7399.77	−0.306682
$836$	0	0
$837$	3421.53	0.141297
$838$	0	0
$839$	23680.0	0.974402	0.487201	−	0.873290i	$-0.338018\pi$
$839$	23680.0	0.974402	0.487201	+	0.873290i	$0.338018\pi$
$840$	0	0
$841$	25509.0	1.04592
$842$	0	0
$843$	12347.4	0.504470
$844$	0	0
$845$	−1202.41	−0.0489518
$846$	0	0
$847$	−5819.76	−0.236091
$848$	0	0
$849$	11833.7	0.478363
$850$	0	0
$851$	33376.4	1.34445
$852$	0	0
$853$	15779.1	0.633374	0.316687	−	0.948530i	$-0.397430\pi$
$853$	15779.1	0.633374	0.316687	+	0.948530i	$0.397430\pi$
$854$	0	0
$855$	−2277.97	−0.0911167
$856$	0	0
$857$	−6759.13	−0.269414	−0.134707	−	0.990886i	$-0.543009\pi$
$857$	−6759.13	−0.269414	−0.134707	+	0.990886i	$0.543009\pi$
$858$	0	0
$859$	−20653.1	−0.820344	−0.410172	−	0.912008i	$-0.634531\pi$
$859$	−20653.1	−0.820344	−0.410172	+	0.912008i	$0.634531\pi$
$860$	0	0
$861$	2820.25	0.111630
$862$	0	0
$863$	6087.19	0.240105	0.120052	−	0.992768i	$-0.461694\pi$
$863$	6087.19	0.240105	0.120052	+	0.992768i	$0.461694\pi$
$864$	0	0
$865$	−21267.1	−0.835955
$866$	0	0
$867$	−13948.8	−0.546396
$868$	0	0
$869$	−4756.58	−0.185680
$870$	0	0
$871$	−1221.22	−0.0475078
$872$	0	0
$873$	−4337.39	−0.168154
$874$	0	0
$875$	12604.0	0.486964
$876$	0	0
$877$	−6841.87	−0.263436	−0.131718	−	0.991287i	$-0.542049\pi$
$877$	−6841.87	−0.263436	−0.131718	+	0.991287i	$0.542049\pi$
$878$	0	0
$879$	2317.32	0.0889207
$880$	0	0
$881$	9416.61	0.360106	0.180053	−	0.983657i	$-0.442373\pi$
$881$	9416.61	0.360106	0.180053	+	0.983657i	$0.442373\pi$
$882$	0	0
$883$	−21561.3	−0.821740	−0.410870	−	0.911694i	$-0.634775\pi$
$883$	−21561.3	−0.821740	−0.410870	+	0.911694i	$0.634775\pi$
$884$	0	0
$885$	−736.972	−0.0279922
$886$	0	0
$887$	−25672.9	−0.971828	−0.485914	−	0.874007i	$-0.661513\pi$
$887$	−25672.9	−0.971828	−0.485914	+	0.874007i	$0.661513\pi$
$888$	0	0
$889$	−13790.3	−0.520261
$890$	0	0
$891$	2106.00	0.0791848
$892$	0	0
$893$	3439.66	0.128896
$894$	0	0
$895$	12707.6	0.474600
$896$	0	0
$897$	−5981.76	−0.222659
$898$	0	0
$899$	28307.2	1.05017
$900$	0	0
$901$	6272.14	0.231915
$902$	0	0
$903$	−4894.15	−0.180362
$904$	0	0
$905$	9397.61	0.345179
$906$	0	0
$907$	−17612.3	−0.644770	−0.322385	−	0.946609i	$-0.604485\pi$
$907$	−17612.3	−0.644770	−0.322385	+	0.946609i	$0.604485\pi$
$908$	0	0
$909$	10249.7	0.373993
$910$	0	0
$911$	−8830.95	−0.321166	−0.160583	−	0.987022i	$-0.551337\pi$
$911$	−8830.95	−0.321166	−0.160583	+	0.987022i	$0.551337\pi$
$912$	0	0
$913$	−7509.45	−0.272209
$914$	0	0
$915$	−5868.04	−0.212013
$916$	0	0
$917$	−20275.6	−0.730163
$918$	0	0
$919$	−30983.7	−1.11214	−0.556071	−	0.831135i	$-0.687692\pi$
$919$	−30983.7	−1.11214	−0.556071	+	0.831135i	$0.687692\pi$
$920$	0	0
$921$	14657.2	0.524399
$922$	0	0
$923$	9643.55	0.343902
$924$	0	0
$925$	−16185.4	−0.575321
$926$	0	0
$927$	−9818.64	−0.347882
$928$	0	0
$929$	−15131.2	−0.534378	−0.267189	−	0.963644i	$-0.586095\pi$
$929$	−15131.2	−0.534378	−0.267189	+	0.963644i	$0.586095\pi$
$930$	0	0
$931$	−9393.58	−0.330679
$932$	0	0
$933$	532.632	0.0186898
$934$	0	0
$935$	−3002.29	−0.105011
$936$	0	0
$937$	40264.3	1.40382	0.701909	−	0.712267i	$-0.252331\pi$
$937$	40264.3	1.40382	0.701909	+	0.712267i	$0.252331\pi$
$938$	0	0
$939$	−20031.1	−0.696155
$940$	0	0
$941$	−36254.5	−1.25597	−0.627983	−	0.778227i	$-0.716119\pi$
$941$	−36254.5	−1.25597	−0.627983	+	0.778227i	$0.716119\pi$
$942$	0	0
$943$	16228.1	0.560402
$944$	0	0
$945$	−1706.85	−0.0587553
$946$	0	0
$947$	47707.4	1.63705	0.818524	−	0.574473i	$-0.194793\pi$
$947$	47707.4	1.63705	0.818524	+	0.574473i	$0.194793\pi$
$948$	0	0
$949$	−8328.96	−0.284899
$950$	0	0
$951$	5419.60	0.184798
$952$	0	0
$953$	−28934.4	−0.983502	−0.491751	−	0.870736i	$-0.663643\pi$
$953$	−28934.4	−0.983502	−0.491751	+	0.870736i	$0.663643\pi$
$954$	0	0
$955$	3106.32	0.105255
$956$	0	0
$957$	17423.5	0.588529
$958$	0	0
$959$	−12821.8	−0.431738
$960$	0	0
$961$	−13732.2	−0.460953
$962$	0	0
$963$	1121.36	0.0375236
$964$	0	0
$965$	17091.3	0.570143
$966$	0	0
$967$	9108.15	0.302894	0.151447	−	0.988465i	$-0.451607\pi$
$967$	9108.15	0.302894	0.151447	+	0.988465i	$0.451607\pi$
$968$	0	0
$969$	1732.09	0.0574229
$970$	0	0
$971$	−34912.3	−1.15385	−0.576925	−	0.816797i	$-0.695748\pi$
$971$	−34912.3	−1.15385	−0.576925	+	0.816797i	$0.695748\pi$
$972$	0	0
$973$	−27192.1	−0.895928
$974$	0	0
$975$	2900.76	0.0952808
$976$	0	0
$977$	−41510.7	−1.35931	−0.679655	−	0.733532i	$-0.737870\pi$
$977$	−41510.7	−1.35931	−0.679655	+	0.733532i	$0.737870\pi$
$978$	0	0
$979$	25041.0	0.817480
$980$	0	0
$981$	−12608.0	−0.410340
$982$	0	0
$983$	−18673.8	−0.605901	−0.302950	−	0.953006i	$-0.597972\pi$
$983$	−18673.8	−0.605901	−0.302950	+	0.953006i	$0.597972\pi$
$984$	0	0
$985$	−8039.38	−0.260057
$986$	0	0
$987$	2577.29	0.0831165
$988$	0	0
$989$	−28161.6	−0.905446
$990$	0	0
$991$	−5954.58	−0.190871	−0.0954356	−	0.995436i	$-0.530424\pi$
$991$	−5954.58	−0.190871	−0.0954356	+	0.995436i	$0.530424\pi$
$992$	0	0
$993$	34341.5	1.09748
$994$	0	0
$995$	−13230.9	−0.421555
$996$	0	0
$997$	−8918.13	−0.283290	−0.141645	−	0.989918i	$-0.545239\pi$
$997$	−8918.13	−0.283290	−0.141645	+	0.989918i	$0.545239\pi$
$998$	0	0
$999$	5875.42	0.186076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.f.1.1	✓	2
3.2	odd	2		936.4.a.c.1.2		2
4.3	odd	2		624.4.a.l.1.1		2
8.3	odd	2		2496.4.a.bd.1.2		2
8.5	even	2		2496.4.a.u.1.2		2
12.11	even	2		1872.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.f.1.1	✓	2	1.1	even	1	trivial
624.4.a.l.1.1		2	4.3	odd	2
936.4.a.c.1.2		2	3.2	odd	2
1872.4.a.v.1.2		2	12.11	even	2
2496.4.a.u.1.2		2	8.5	even	2
2496.4.a.bd.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} +26.0000 q^{11} -13.0000 q^{13} -21.3446 q^{15} +16.2298 q^{17} +35.5744 q^{19} +26.6554 q^{21} +153.379 q^{23} -74.3785 q^{25} +27.0000 q^{27} +223.379 q^{29} +126.723 q^{31} +78.0000 q^{33} -63.2166 q^{35} +217.608 q^{37} -39.0000 q^{39} +105.804 q^{41} -183.608 q^{43} -64.0339 q^{45} +96.6893 q^{47} -264.055 q^{49} +48.6893 q^{51} +386.460 q^{53} -184.987 q^{55} +106.723 q^{57} +34.5273 q^{59} +274.919 q^{61} +79.9661 q^{63} +92.4934 q^{65} +93.9397 q^{67} +460.136 q^{69} -741.812 q^{71} +640.689 q^{73} -223.136 q^{75} +231.013 q^{77} -182.945 q^{79} +81.0000 q^{81} -288.825 q^{83} -115.473 q^{85} +670.136 q^{87} +963.115 q^{89} -115.507 q^{91} +380.169 q^{93} -253.107 q^{95} -481.932 q^{97} +234.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35}+ \cdots + 468 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 + 52 * q^11 - 26 * q^13 + 36 * q^15 - 20 * q^17 - 60 * q^19 + 132 * q^21 - 8 * q^23 + 166 * q^25 + 54 * q^27 + 132 * q^29 - 140 * q^31 + 156 * q^33 + 608 * q^35 + 68 * q^37 - 78 * q^39 + 28 * q^41 + 108 * q^45 + 36 * q^47 + 626 * q^49 - 60 * q^51 + 668 * q^53 + 312 * q^55 - 180 * q^57 - 508 * q^59 + 340 * q^61 + 396 * q^63 - 156 * q^65 - 940 * q^67 - 24 * q^69 + 300 * q^71 + 1124 * q^73 + 498 * q^75 + 1144 * q^77 - 1520 * q^79 + 162 * q^81 + 524 * q^83 - 808 * q^85 + 396 * q^87 + 1900 * q^89 - 572 * q^91 - 420 * q^93 - 2080 * q^95 - 1436 * q^97 + 468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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