LMFDB - Embedded newform 312.4.a.e.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,-4,0,-20] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -7.29150 q^{5} +5.87451 q^{7} +9.00000 q^{9} -51.1660 q^{11} -13.0000 q^{13} -21.8745 q^{15} -73.7490 q^{17} +59.9555 q^{19} +17.6235 q^{21} +69.8301 q^{23} -71.8340 q^{25} +27.0000 q^{27} -294.826 q^{29} -334.450 q^{31} -153.498 q^{33} -42.8340 q^{35} +261.409 q^{37} -39.0000 q^{39} +222.701 q^{41} +79.2470 q^{43} -65.6235 q^{45} -584.405 q^{47} -308.490 q^{49} -221.247 q^{51} +465.158 q^{53} +373.077 q^{55} +179.867 q^{57} -530.089 q^{59} +548.332 q^{61} +52.8706 q^{63} +94.7895 q^{65} -384.959 q^{67} +209.490 q^{69} -307.004 q^{71} -844.891 q^{73} -215.502 q^{75} -300.575 q^{77} +30.1699 q^{79} +81.0000 q^{81} +19.5633 q^{83} +537.741 q^{85} -884.478 q^{87} -513.017 q^{89} -76.3686 q^{91} -1003.35 q^{93} -437.166 q^{95} +787.903 q^{97} -460.494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9} - 60 q^{11} - 26 q^{13} - 12 q^{15} - 84 q^{17} - 60 q^{19} - 60 q^{21} - 72 q^{23} - 186 q^{25} + 54 q^{27} - 124 q^{29} - 108 q^{31} - 180 q^{33} - 128 q^{35}+ \cdots - 540 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 4 * q^5 - 20 * q^7 + 18 * q^9 - 60 * q^11 - 26 * q^13 - 12 * q^15 - 84 * q^17 - 60 * q^19 - 60 * q^21 - 72 * q^23 - 186 * q^25 + 54 * q^27 - 124 * q^29 - 108 * q^31 - 180 * q^33 - 128 * q^35 + 36 * q^37 - 78 * q^39 - 52 * q^41 - 32 * q^43 - 36 * q^45 - 428 * q^47 + 18 * q^49 - 252 * q^51 + 380 * q^53 + 344 * q^55 - 180 * q^57 - 1420 * q^59 + 1012 * q^61 - 180 * q^63 + 52 * q^65 - 844 * q^67 - 216 * q^69 - 868 * q^71 - 60 * q^73 - 558 * q^75 - 72 * q^77 + 272 * q^79 + 162 * q^81 - 1252 * q^83 + 504 * q^85 - 372 * q^87 + 572 * q^89 + 260 * q^91 - 324 * q^93 - 832 * q^95 + 708 * q^97 - 540 * 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
$5$	−7.29150	−0.652172	−0.326086	−	0.945340i	$-0.605730\pi$
$5$	−7.29150	−0.652172	−0.326086	+	0.945340i	$0.605730\pi$
$6$	0	0
$7$	5.87451	0.317194	0.158597	−	0.987343i	$-0.449303\pi$
$7$	5.87451	0.317194	0.158597	+	0.987343i	$0.449303\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−51.1660	−1.40247	−0.701233	−	0.712932i	$-0.747367\pi$
$11$	−51.1660	−1.40247	−0.701233	+	0.712932i	$0.747367\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−21.8745	−0.376532
$16$	0	0
$17$	−73.7490	−1.05216	−0.526081	−	0.850434i	$-0.676339\pi$
$17$	−73.7490	−1.05216	−0.526081	+	0.850434i	$0.676339\pi$
$18$	0	0
$19$	59.9555	0.723934	0.361967	−	0.932191i	$-0.382105\pi$
$19$	59.9555	0.723934	0.361967	+	0.932191i	$0.382105\pi$
$20$	0	0
$21$	17.6235	0.183132
$22$	0	0
$23$	69.8301	0.633068	0.316534	−	0.948581i	$-0.397481\pi$
$23$	69.8301	0.633068	0.316534	+	0.948581i	$0.397481\pi$
$24$	0	0
$25$	−71.8340	−0.574672
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−294.826	−1.88786	−0.943928	−	0.330151i	$-0.892900\pi$
$29$	−294.826	−1.88786	−0.943928	+	0.330151i	$0.892900\pi$
$30$	0	0
$31$	−334.450	−1.93771	−0.968854	−	0.247634i	$-0.920347\pi$
$31$	−334.450	−1.93771	−0.968854	+	0.247634i	$0.920347\pi$
$32$	0	0
$33$	−153.498	−0.809714
$34$	0	0
$35$	−42.8340	−0.206865
$36$	0	0
$37$	261.409	1.16150	0.580749	−	0.814083i	$-0.302760\pi$
$37$	261.409	1.16150	0.580749	+	0.814083i	$0.302760\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	222.701	0.848293	0.424146	−	0.905594i	$-0.360574\pi$
$41$	222.701	0.848293	0.424146	+	0.905594i	$0.360574\pi$
$42$	0	0
$43$	79.2470	0.281048	0.140524	−	0.990077i	$-0.455121\pi$
$43$	79.2470	0.281048	0.140524	+	0.990077i	$0.455121\pi$
$44$	0	0
$45$	−65.6235	−0.217391
$46$	0	0
$47$	−584.405	−1.81371	−0.906854	−	0.421445i	$-0.861523\pi$
$47$	−584.405	−1.81371	−0.906854	+	0.421445i	$0.861523\pi$
$48$	0	0
$49$	−308.490	−0.899388
$50$	0	0
$51$	−221.247	−0.607466
$52$	0	0
$53$	465.158	1.20555	0.602777	−	0.797910i	$-0.294061\pi$
$53$	465.158	1.20555	0.602777	+	0.797910i	$0.294061\pi$
$54$	0	0
$55$	373.077	0.914649
$56$	0	0
$57$	179.867	0.417963
$58$	0	0
$59$	−530.089	−1.16969	−0.584845	−	0.811145i	$-0.698845\pi$
$59$	−530.089	−1.16969	−0.584845	+	0.811145i	$0.698845\pi$
$60$	0	0
$61$	548.332	1.15093	0.575465	−	0.817826i	$-0.304821\pi$
$61$	548.332	1.15093	0.575465	+	0.817826i	$0.304821\pi$
$62$	0	0
$63$	52.8706	0.105731
$64$	0	0
$65$	94.7895	0.180880
$66$	0	0
$67$	−384.959	−0.701945	−0.350972	−	0.936386i	$-0.614149\pi$
$67$	−384.959	−0.701945	−0.350972	+	0.936386i	$0.614149\pi$
$68$	0	0
$69$	209.490	0.365502
$70$	0	0
$71$	−307.004	−0.513164	−0.256582	−	0.966522i	$-0.582596\pi$
$71$	−307.004	−0.513164	−0.256582	+	0.966522i	$0.582596\pi$
$72$	0	0
$73$	−844.891	−1.35462	−0.677309	−	0.735699i	$-0.736854\pi$
$73$	−844.891	−1.35462	−0.677309	+	0.735699i	$0.736854\pi$
$74$	0	0
$75$	−215.502	−0.331787
$76$	0	0
$77$	−300.575	−0.444853
$78$	0	0
$79$	30.1699	0.0429669	0.0214834	−	0.999769i	$-0.493161\pi$
$79$	30.1699	0.0429669	0.0214834	+	0.999769i	$0.493161\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	19.5633	0.0258717	0.0129359	−	0.999916i	$-0.495882\pi$
$83$	19.5633	0.0258717	0.0129359	+	0.999916i	$0.495882\pi$
$84$	0	0
$85$	537.741	0.686191
$86$	0	0
$87$	−884.478	−1.08995
$88$	0	0
$89$	−513.017	−0.611008	−0.305504	−	0.952191i	$-0.598825\pi$
$89$	−513.017	−0.611008	−0.305504	+	0.952191i	$0.598825\pi$
$90$	0	0
$91$	−76.3686	−0.0879737
$92$	0	0
$93$	−1003.35	−1.11874
$94$	0	0
$95$	−437.166	−0.472129
$96$	0	0
$97$	787.903	0.824737	0.412368	−	0.911017i	$-0.364702\pi$
$97$	787.903	0.824737	0.412368	+	0.911017i	$0.364702\pi$
$98$	0	0
$99$	−460.494	−0.467489
$100$	0	0
$101$	1391.73	1.37111	0.685554	−	0.728022i	$-0.259560\pi$
$101$	1391.73	1.37111	0.685554	+	0.728022i	$0.259560\pi$
$102$	0	0
$103$	1428.58	1.36662	0.683309	−	0.730129i	$-0.260540\pi$
$103$	1428.58	1.36662	0.683309	+	0.730129i	$0.260540\pi$
$104$	0	0
$105$	−128.502	−0.119433
$106$	0	0
$107$	−1363.30	−1.23173	−0.615867	−	0.787850i	$-0.711194\pi$
$107$	−1363.30	−1.23173	−0.615867	+	0.787850i	$0.711194\pi$
$108$	0	0
$109$	979.182	0.860446	0.430223	−	0.902723i	$-0.358435\pi$
$109$	979.182	0.860446	0.430223	+	0.902723i	$0.358435\pi$
$110$	0	0
$111$	784.227	0.670591
$112$	0	0
$113$	1495.83	1.24527	0.622637	−	0.782511i	$-0.286062\pi$
$113$	1495.83	1.24527	0.622637	+	0.782511i	$0.286062\pi$
$114$	0	0
$115$	−509.166	−0.412869
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−433.239	−0.333739
$120$	0	0
$121$	1286.96	0.966913
$122$	0	0
$123$	668.102	0.489762
$124$	0	0
$125$	1435.22	1.02696
$126$	0	0
$127$	2221.69	1.55231	0.776155	−	0.630542i	$-0.217167\pi$
$127$	2221.69	1.55231	0.776155	+	0.630542i	$0.217167\pi$
$128$	0	0
$129$	237.741	0.162263
$130$	0	0
$131$	−878.138	−0.585674	−0.292837	−	0.956162i	$-0.594599\pi$
$131$	−878.138	−0.585674	−0.292837	+	0.956162i	$0.594599\pi$
$132$	0	0
$133$	352.209	0.229627
$134$	0	0
$135$	−196.871	−0.125511
$136$	0	0
$137$	−1176.62	−0.733762	−0.366881	−	0.930268i	$-0.619574\pi$
$137$	−1176.62	−0.733762	−0.366881	+	0.930268i	$0.619574\pi$
$138$	0	0
$139$	359.514	0.219378	0.109689	−	0.993966i	$-0.465014\pi$
$139$	359.514	0.219378	0.109689	+	0.993966i	$0.465014\pi$
$140$	0	0
$141$	−1753.22	−1.04714
$142$	0	0
$143$	665.158	0.388974
$144$	0	0
$145$	2149.73	1.23121
$146$	0	0
$147$	−925.470	−0.519262
$148$	0	0
$149$	1226.94	0.674594	0.337297	−	0.941398i	$-0.390487\pi$
$149$	1226.94	0.674594	0.337297	+	0.941398i	$0.390487\pi$
$150$	0	0
$151$	−705.665	−0.380306	−0.190153	−	0.981754i	$-0.560898\pi$
$151$	−705.665	−0.380306	−0.190153	+	0.981754i	$0.560898\pi$
$152$	0	0
$153$	−663.741	−0.350721
$154$	0	0
$155$	2438.64	1.26372
$156$	0	0
$157$	2519.80	1.28090	0.640452	−	0.767998i	$-0.278747\pi$
$157$	2519.80	1.28090	0.640452	+	0.767998i	$0.278747\pi$
$158$	0	0
$159$	1395.47	0.696027
$160$	0	0
$161$	410.217	0.200805
$162$	0	0
$163$	1221.40	0.586919	0.293459	−	0.955972i	$-0.405193\pi$
$163$	1221.40	0.586919	0.293459	+	0.955972i	$0.405193\pi$
$164$	0	0
$165$	1119.23	0.528073
$166$	0	0
$167$	1099.52	0.509480	0.254740	−	0.967010i	$-0.418010\pi$
$167$	1099.52	0.509480	0.254740	+	0.967010i	$0.418010\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	539.600	0.241311
$172$	0	0
$173$	−1817.39	−0.798693	−0.399346	−	0.916800i	$-0.630763\pi$
$173$	−1817.39	−0.798693	−0.399346	+	0.916800i	$0.630763\pi$
$174$	0	0
$175$	−421.989	−0.182282
$176$	0	0
$177$	−1590.27	−0.675321
$178$	0	0
$179$	−4705.22	−1.96472	−0.982359	−	0.187005i	$-0.940122\pi$
$179$	−4705.22	−1.96472	−0.982359	+	0.187005i	$0.940122\pi$
$180$	0	0
$181$	1694.86	0.696010	0.348005	−	0.937493i	$-0.386859\pi$
$181$	1694.86	0.696010	0.348005	+	0.937493i	$0.386859\pi$
$182$	0	0
$183$	1645.00	0.664490
$184$	0	0
$185$	−1906.07	−0.757496
$186$	0	0
$187$	3773.44	1.47562
$188$	0	0
$189$	158.612	0.0610439
$190$	0	0
$191$	−1839.68	−0.696933	−0.348467	−	0.937321i	$-0.613298\pi$
$191$	−1839.68	−0.696933	−0.348467	+	0.937321i	$0.613298\pi$
$192$	0	0
$193$	−1776.15	−0.662437	−0.331219	−	0.943554i	$-0.607460\pi$
$193$	−1776.15	−0.662437	−0.331219	+	0.943554i	$0.607460\pi$
$194$	0	0
$195$	284.369	0.104431
$196$	0	0
$197$	642.894	0.232509	0.116255	−	0.993219i	$-0.462911\pi$
$197$	642.894	0.232509	0.116255	+	0.993219i	$0.462911\pi$
$198$	0	0
$199$	−1086.88	−0.387169	−0.193584	−	0.981084i	$-0.562011\pi$
$199$	−1086.88	−0.387169	−0.193584	+	0.981084i	$0.562011\pi$
$200$	0	0
$201$	−1154.88	−0.405268
$202$	0	0
$203$	−1731.96	−0.598816
$204$	0	0
$205$	−1623.82	−0.553232
$206$	0	0
$207$	628.470	0.211023
$208$	0	0
$209$	−3067.69	−1.01529
$210$	0	0
$211$	3275.58	1.06872	0.534361	−	0.845256i	$-0.320552\pi$
$211$	3275.58	1.06872	0.534361	+	0.845256i	$0.320552\pi$
$212$	0	0
$213$	−921.012	−0.296276
$214$	0	0
$215$	−577.830	−0.183292
$216$	0	0
$217$	−1964.73	−0.614628
$218$	0	0
$219$	−2534.67	−0.782089
$220$	0	0
$221$	958.737	0.291817
$222$	0	0
$223$	−2075.20	−0.623164	−0.311582	−	0.950219i	$-0.600859\pi$
$223$	−2075.20	−0.623164	−0.311582	+	0.950219i	$0.600859\pi$
$224$	0	0
$225$	−646.506	−0.191557
$226$	0	0
$227$	2673.53	0.781712	0.390856	−	0.920452i	$-0.372179\pi$
$227$	2673.53	0.781712	0.390856	+	0.920452i	$0.372179\pi$
$228$	0	0
$229$	558.099	0.161049	0.0805245	−	0.996753i	$-0.474340\pi$
$229$	558.099	0.161049	0.0805245	+	0.996753i	$0.474340\pi$
$230$	0	0
$231$	−901.725	−0.256836
$232$	0	0
$233$	2436.99	0.685204	0.342602	−	0.939481i	$-0.388692\pi$
$233$	2436.99	0.685204	0.342602	+	0.939481i	$0.388692\pi$
$234$	0	0
$235$	4261.19	1.18285
$236$	0	0
$237$	90.5098	0.0248069
$238$	0	0
$239$	−968.672	−0.262168	−0.131084	−	0.991371i	$-0.541846\pi$
$239$	−968.672	−0.262168	−0.131084	+	0.991371i	$0.541846\pi$
$240$	0	0
$241$	−6872.52	−1.83692	−0.918460	−	0.395513i	$-0.870567\pi$
$241$	−6872.52	−1.83692	−0.918460	+	0.395513i	$0.870567\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	2249.36	0.586556
$246$	0	0
$247$	−779.422	−0.200783
$248$	0	0
$249$	58.6900	0.0149370
$250$	0	0
$251$	−3944.62	−0.991960	−0.495980	−	0.868334i	$-0.665191\pi$
$251$	−3944.62	−0.991960	−0.495980	+	0.868334i	$0.665191\pi$
$252$	0	0
$253$	−3572.93	−0.887857
$254$	0	0
$255$	1613.22	0.396172
$256$	0	0
$257$	5755.13	1.39687	0.698434	−	0.715674i	$-0.253881\pi$
$257$	5755.13	1.39687	0.698434	+	0.715674i	$0.253881\pi$
$258$	0	0
$259$	1535.65	0.368419
$260$	0	0
$261$	−2653.44	−0.629285
$262$	0	0
$263$	−4.67979	−0.00109722	−0.000548609	−	1.00000i	$-0.500175\pi$
$263$	−4.67979	−0.00109722	−0.000548609		1.00000i	$0.500175\pi$
$264$	0	0
$265$	−3391.70	−0.786229
$266$	0	0
$267$	−1539.05	−0.352765
$268$	0	0
$269$	3269.87	0.741143	0.370571	−	0.928804i	$-0.379162\pi$
$269$	3269.87	0.741143	0.370571	+	0.928804i	$0.379162\pi$
$270$	0	0
$271$	−2656.18	−0.595393	−0.297696	−	0.954661i	$-0.596218\pi$
$271$	−2656.18	−0.595393	−0.297696	+	0.954661i	$0.596218\pi$
$272$	0	0
$273$	−229.106	−0.0507916
$274$	0	0
$275$	3675.46	0.805958
$276$	0	0
$277$	−7909.63	−1.71568	−0.857840	−	0.513916i	$-0.828194\pi$
$277$	−7909.63	−1.71568	−0.857840	+	0.513916i	$0.828194\pi$
$278$	0	0
$279$	−3010.05	−0.645902
$280$	0	0
$281$	−7720.91	−1.63911	−0.819557	−	0.572998i	$-0.805780\pi$
$281$	−7720.91	−1.63911	−0.819557	+	0.572998i	$0.805780\pi$
$282$	0	0
$283$	1318.54	0.276957	0.138478	−	0.990365i	$-0.455779\pi$
$283$	1318.54	0.276957	0.138478	+	0.990365i	$0.455779\pi$
$284$	0	0
$285$	−1311.50	−0.272584
$286$	0	0
$287$	1308.26	0.269073
$288$	0	0
$289$	525.917	0.107046
$290$	0	0
$291$	2363.71	0.476162
$292$	0	0
$293$	−4038.77	−0.805281	−0.402641	−	0.915358i	$-0.631908\pi$
$293$	−4038.77	−0.805281	−0.402641	+	0.915358i	$0.631908\pi$
$294$	0	0
$295$	3865.14	0.762839
$296$	0	0
$297$	−1381.48	−0.269905
$298$	0	0
$299$	−907.791	−0.175582
$300$	0	0
$301$	465.537	0.0891466
$302$	0	0
$303$	4175.18	0.791609
$304$	0	0
$305$	−3998.16	−0.750604
$306$	0	0
$307$	−9694.98	−1.80235	−0.901175	−	0.433455i	$-0.857294\pi$
$307$	−9694.98	−1.80235	−0.901175	+	0.433455i	$0.857294\pi$
$308$	0	0
$309$	4285.73	0.789017
$310$	0	0
$311$	−7949.12	−1.44937	−0.724684	−	0.689082i	$-0.758014\pi$
$311$	−7949.12	−1.44937	−0.724684	+	0.689082i	$0.758014\pi$
$312$	0	0
$313$	5115.77	0.923835	0.461918	−	0.886923i	$-0.347162\pi$
$313$	5115.77	0.923835	0.461918	+	0.886923i	$0.347162\pi$
$314$	0	0
$315$	−385.506	−0.0689549
$316$	0	0
$317$	−1218.09	−0.215820	−0.107910	−	0.994161i	$-0.534416\pi$
$317$	−1218.09	−0.215820	−0.107910	+	0.994161i	$0.534416\pi$
$318$	0	0
$319$	15085.1	2.64766
$320$	0	0
$321$	−4089.91	−0.711142
$322$	0	0
$323$	−4421.66	−0.761696
$324$	0	0
$325$	933.842	0.159385
$326$	0	0
$327$	2937.55	0.496779
$328$	0	0
$329$	−3433.09	−0.575296
$330$	0	0
$331$	−9888.33	−1.64203	−0.821015	−	0.570907i	$-0.806592\pi$
$331$	−9888.33	−1.64203	−0.821015	+	0.570907i	$0.806592\pi$
$332$	0	0
$333$	2352.68	0.387166
$334$	0	0
$335$	2806.93	0.457788
$336$	0	0
$337$	−9544.60	−1.54281	−0.771406	−	0.636344i	$-0.780446\pi$
$337$	−9544.60	−1.54281	−0.771406	+	0.636344i	$0.780446\pi$
$338$	0	0
$339$	4487.49	0.718959
$340$	0	0
$341$	17112.5	2.71757
$342$	0	0
$343$	−3827.18	−0.602474
$344$	0	0
$345$	−1527.50	−0.238370
$346$	0	0
$347$	4547.76	0.703563	0.351782	−	0.936082i	$-0.385576\pi$
$347$	4547.76	0.703563	0.351782	+	0.936082i	$0.385576\pi$
$348$	0	0
$349$	1513.12	0.232079	0.116039	−	0.993245i	$-0.462980\pi$
$349$	1513.12	0.232079	0.116039	+	0.993245i	$0.462980\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	61.1114	0.00921425	0.00460713	−	0.999989i	$-0.498534\pi$
$353$	61.1114	0.00921425	0.00460713	+	0.999989i	$0.498534\pi$
$354$	0	0
$355$	2238.52	0.334671
$356$	0	0
$357$	−1299.72	−0.192684
$358$	0	0
$359$	−1065.57	−0.156653	−0.0783265	−	0.996928i	$-0.524958\pi$
$359$	−1065.57	−0.156653	−0.0783265	+	0.996928i	$0.524958\pi$
$360$	0	0
$361$	−3264.33	−0.475920
$362$	0	0
$363$	3860.88	0.558247
$364$	0	0
$365$	6160.53	0.883443
$366$	0	0
$367$	−8962.02	−1.27470	−0.637348	−	0.770576i	$-0.719969\pi$
$367$	−8962.02	−1.27470	−0.637348	+	0.770576i	$0.719969\pi$
$368$	0	0
$369$	2004.31	0.282764
$370$	0	0
$371$	2732.58	0.382394
$372$	0	0
$373$	1250.29	0.173560	0.0867798	−	0.996228i	$-0.472342\pi$
$373$	1250.29	0.173560	0.0867798	+	0.996228i	$0.472342\pi$
$374$	0	0
$375$	4305.65	0.592914
$376$	0	0
$377$	3832.74	0.523597
$378$	0	0
$379$	8367.05	1.13400	0.567001	−	0.823717i	$-0.308104\pi$
$379$	8367.05	1.13400	0.567001	+	0.823717i	$0.308104\pi$
$380$	0	0
$381$	6665.08	0.896227
$382$	0	0
$383$	−6241.47	−0.832700	−0.416350	−	0.909204i	$-0.636691\pi$
$383$	−6241.47	−0.832700	−0.416350	+	0.909204i	$0.636691\pi$
$384$	0	0
$385$	2191.64	0.290121
$386$	0	0
$387$	713.223	0.0936826
$388$	0	0
$389$	2651.76	0.345628	0.172814	−	0.984954i	$-0.444714\pi$
$389$	2651.76	0.345628	0.172814	+	0.984954i	$0.444714\pi$
$390$	0	0
$391$	−5149.90	−0.666091
$392$	0	0
$393$	−2634.42	−0.338139
$394$	0	0
$395$	−219.984	−0.0280218
$396$	0	0
$397$	−11460.4	−1.44882	−0.724408	−	0.689372i	$-0.757887\pi$
$397$	−11460.4	−1.44882	−0.724408	+	0.689372i	$0.757887\pi$
$398$	0	0
$399$	1056.63	0.132575
$400$	0	0
$401$	−6271.25	−0.780976	−0.390488	−	0.920608i	$-0.627694\pi$
$401$	−6271.25	−0.780976	−0.390488	+	0.920608i	$0.627694\pi$
$402$	0	0
$403$	4347.85	0.537423
$404$	0	0
$405$	−590.612	−0.0724635
$406$	0	0
$407$	−13375.3	−1.62896
$408$	0	0
$409$	−4432.82	−0.535915	−0.267957	−	0.963431i	$-0.586349\pi$
$409$	−4432.82	−0.535915	−0.267957	+	0.963431i	$0.586349\pi$
$410$	0	0
$411$	−3529.86	−0.423638
$412$	0	0
$413$	−3114.01	−0.371018
$414$	0	0
$415$	−142.646	−0.0168728
$416$	0	0
$417$	1078.54	0.126658
$418$	0	0
$419$	11383.5	1.32726	0.663630	−	0.748061i	$-0.269015\pi$
$419$	11383.5	1.32726	0.663630	+	0.748061i	$0.269015\pi$
$420$	0	0
$421$	7422.33	0.859245	0.429623	−	0.903009i	$-0.358647\pi$
$421$	7422.33	0.859245	0.429623	+	0.903009i	$0.358647\pi$
$422$	0	0
$423$	−5259.65	−0.604569
$424$	0	0
$425$	5297.69	0.604648
$426$	0	0
$427$	3221.18	0.365068
$428$	0	0
$429$	1995.47	0.224574
$430$	0	0
$431$	17065.8	1.90727	0.953633	−	0.300971i	$-0.0973107\pi$
$431$	17065.8	1.90727	0.953633	+	0.300971i	$0.0973107\pi$
$432$	0	0
$433$	−13703.0	−1.52084	−0.760420	−	0.649432i	$-0.775007\pi$
$433$	−13703.0	−1.52084	−0.760420	+	0.649432i	$0.775007\pi$
$434$	0	0
$435$	6449.18	0.710838
$436$	0	0
$437$	4186.70	0.458300
$438$	0	0
$439$	8990.23	0.977404	0.488702	−	0.872451i	$-0.337471\pi$
$439$	8990.23	0.977404	0.488702	+	0.872451i	$0.337471\pi$
$440$	0	0
$441$	−2776.41	−0.299796
$442$	0	0
$443$	11047.9	1.18488	0.592438	−	0.805616i	$-0.298165\pi$
$443$	11047.9	1.18488	0.592438	+	0.805616i	$0.298165\pi$
$444$	0	0
$445$	3740.66	0.398482
$446$	0	0
$447$	3680.81	0.389477
$448$	0	0
$449$	14780.2	1.55350	0.776750	−	0.629809i	$-0.216867\pi$
$449$	14780.2	1.55350	0.776750	+	0.629809i	$0.216867\pi$
$450$	0	0
$451$	−11394.7	−1.18970
$452$	0	0
$453$	−2117.00	−0.219570
$454$	0	0
$455$	556.842	0.0573740
$456$	0	0
$457$	407.385	0.0416995	0.0208498	−	0.999783i	$-0.493363\pi$
$457$	407.385	0.0416995	0.0208498	+	0.999783i	$0.493363\pi$
$458$	0	0
$459$	−1991.22	−0.202489
$460$	0	0
$461$	13198.6	1.33345	0.666725	−	0.745303i	$-0.267695\pi$
$461$	13198.6	1.33345	0.666725	+	0.745303i	$0.267695\pi$
$462$	0	0
$463$	−534.058	−0.0536064	−0.0268032	−	0.999641i	$-0.508533\pi$
$463$	−534.058	−0.0536064	−0.0268032	+	0.999641i	$0.508533\pi$
$464$	0	0
$465$	7315.92	0.729608
$466$	0	0
$467$	5150.44	0.510351	0.255176	−	0.966895i	$-0.417867\pi$
$467$	5150.44	0.510351	0.255176	+	0.966895i	$0.417867\pi$
$468$	0	0
$469$	−2261.45	−0.222652
$470$	0	0
$471$	7559.40	0.739530
$472$	0	0
$473$	−4054.76	−0.394160
$474$	0	0
$475$	−4306.85	−0.416025
$476$	0	0
$477$	4186.42	0.401851
$478$	0	0
$479$	−20668.2	−1.97151	−0.985756	−	0.168179i	$-0.946211\pi$
$479$	−20668.2	−1.97151	−0.985756	+	0.168179i	$0.946211\pi$
$480$	0	0
$481$	−3398.32	−0.322141
$482$	0	0
$483$	1230.65	0.115935
$484$	0	0
$485$	−5745.00	−0.537870
$486$	0	0
$487$	3239.32	0.301412	0.150706	−	0.988579i	$-0.451845\pi$
$487$	3239.32	0.301412	0.150706	+	0.988579i	$0.451845\pi$
$488$	0	0
$489$	3664.21	0.338858
$490$	0	0
$491$	−4704.78	−0.432432	−0.216216	−	0.976346i	$-0.569371\pi$
$491$	−4704.78	−0.432432	−0.216216	+	0.976346i	$0.569371\pi$
$492$	0	0
$493$	21743.1	1.98633
$494$	0	0
$495$	3357.69	0.304883
$496$	0	0
$497$	−1803.50	−0.162772
$498$	0	0
$499$	−12173.8	−1.09213	−0.546065	−	0.837743i	$-0.683875\pi$
$499$	−12173.8	−1.09213	−0.546065	+	0.837743i	$0.683875\pi$
$500$	0	0
$501$	3298.55	0.294148
$502$	0	0
$503$	−10090.3	−0.894439	−0.447219	−	0.894424i	$-0.647586\pi$
$503$	−10090.3	−0.894439	−0.447219	+	0.894424i	$0.647586\pi$
$504$	0	0
$505$	−10147.8	−0.894198
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	12946.5	1.12739	0.563696	−	0.825982i	$-0.309379\pi$
$509$	12946.5	1.12739	0.563696	+	0.825982i	$0.309379\pi$
$510$	0	0
$511$	−4963.32	−0.429676
$512$	0	0
$513$	1618.80	0.139321
$514$	0	0
$515$	−10416.5	−0.891270
$516$	0	0
$517$	29901.7	2.54366
$518$	0	0
$519$	−5452.18	−0.461126
$520$	0	0
$521$	−2267.25	−0.190653	−0.0953265	−	0.995446i	$-0.530390\pi$
$521$	−2267.25	−0.190653	−0.0953265	+	0.995446i	$0.530390\pi$
$522$	0	0
$523$	8850.80	0.739997	0.369998	−	0.929032i	$-0.379358\pi$
$523$	8850.80	0.739997	0.369998	+	0.929032i	$0.379358\pi$
$524$	0	0
$525$	−1265.97	−0.105241
$526$	0	0
$527$	24665.3	2.03878
$528$	0	0
$529$	−7290.76	−0.599224
$530$	0	0
$531$	−4770.80	−0.389897
$532$	0	0
$533$	−2895.11	−0.235274
$534$	0	0
$535$	9940.54	0.803303
$536$	0	0
$537$	−14115.7	−1.13433
$538$	0	0
$539$	15784.2	1.26136
$540$	0	0
$541$	−6720.57	−0.534085	−0.267042	−	0.963685i	$-0.586046\pi$
$541$	−6720.57	−0.534085	−0.267042	+	0.963685i	$0.586046\pi$
$542$	0	0
$543$	5084.57	0.401841
$544$	0	0
$545$	−7139.71	−0.561159
$546$	0	0
$547$	−3251.85	−0.254185	−0.127093	−	0.991891i	$-0.540565\pi$
$547$	−3251.85	−0.254185	−0.127093	+	0.991891i	$0.540565\pi$
$548$	0	0
$549$	4934.99	0.383643
$550$	0	0
$551$	−17676.5	−1.36668
$552$	0	0
$553$	177.234	0.0136288
$554$	0	0
$555$	−5718.20	−0.437340
$556$	0	0
$557$	4215.22	0.320655	0.160327	−	0.987064i	$-0.448745\pi$
$557$	4215.22	0.320655	0.160327	+	0.987064i	$0.448745\pi$
$558$	0	0
$559$	−1030.21	−0.0779487
$560$	0	0
$561$	11320.3	0.851951
$562$	0	0
$563$	15802.9	1.18297	0.591486	−	0.806316i	$-0.298542\pi$
$563$	15802.9	1.18297	0.591486	+	0.806316i	$0.298542\pi$
$564$	0	0
$565$	−10906.8	−0.812132
$566$	0	0
$567$	475.835	0.0352437
$568$	0	0
$569$	12964.1	0.955155	0.477578	−	0.878590i	$-0.341515\pi$
$569$	12964.1	0.955155	0.477578	+	0.878590i	$0.341515\pi$
$570$	0	0
$571$	−13555.7	−0.993499	−0.496750	−	0.867894i	$-0.665473\pi$
$571$	−13555.7	−0.993499	−0.496750	+	0.867894i	$0.665473\pi$
$572$	0	0
$573$	−5519.03	−0.402375
$574$	0	0
$575$	−5016.17	−0.363807
$576$	0	0
$577$	25349.5	1.82897	0.914483	−	0.404624i	$-0.132598\pi$
$577$	25349.5	1.82897	0.914483	+	0.404624i	$0.132598\pi$
$578$	0	0
$579$	−5328.46	−0.382458
$580$	0	0
$581$	114.925	0.00820635
$582$	0	0
$583$	−23800.3	−1.69075
$584$	0	0
$585$	853.106	0.0602933
$586$	0	0
$587$	−18013.3	−1.26659	−0.633296	−	0.773910i	$-0.718298\pi$
$587$	−18013.3	−1.26659	−0.633296	+	0.773910i	$0.718298\pi$
$588$	0	0
$589$	−20052.1	−1.40277
$590$	0	0
$591$	1928.68	0.134239
$592$	0	0
$593$	21521.6	1.49036	0.745181	−	0.666862i	$-0.232363\pi$
$593$	21521.6	1.49036	0.745181	+	0.666862i	$0.232363\pi$
$594$	0	0
$595$	3158.96	0.217655
$596$	0	0
$597$	−3260.63	−0.223532
$598$	0	0
$599$	10081.8	0.687700	0.343850	−	0.939025i	$-0.388269\pi$
$599$	10081.8	0.687700	0.343850	+	0.939025i	$0.388269\pi$
$600$	0	0
$601$	−5180.24	−0.351591	−0.175796	−	0.984427i	$-0.556250\pi$
$601$	−5180.24	−0.351591	−0.175796	+	0.984427i	$0.556250\pi$
$602$	0	0
$603$	−3464.64	−0.233982
$604$	0	0
$605$	−9383.88	−0.630593
$606$	0	0
$607$	−24166.4	−1.61595	−0.807976	−	0.589215i	$-0.799437\pi$
$607$	−24166.4	−1.61595	−0.807976	+	0.589215i	$0.799437\pi$
$608$	0	0
$609$	−5195.88	−0.345727
$610$	0	0
$611$	7597.27	0.503032
$612$	0	0
$613$	23768.7	1.56608	0.783040	−	0.621971i	$-0.213668\pi$
$613$	23768.7	1.56608	0.783040	+	0.621971i	$0.213668\pi$
$614$	0	0
$615$	−4871.47	−0.319409
$616$	0	0
$617$	−20306.1	−1.32495	−0.662475	−	0.749084i	$-0.730494\pi$
$617$	−20306.1	−1.32495	−0.662475	+	0.749084i	$0.730494\pi$
$618$	0	0
$619$	−16062.5	−1.04299	−0.521493	−	0.853256i	$-0.674625\pi$
$619$	−16062.5	−1.04299	−0.521493	+	0.853256i	$0.674625\pi$
$620$	0	0
$621$	1885.41	0.121834
$622$	0	0
$623$	−3013.72	−0.193808
$624$	0	0
$625$	−1485.63	−0.0950803
$626$	0	0
$627$	−9203.06	−0.586180
$628$	0	0
$629$	−19278.7	−1.22208
$630$	0	0
$631$	6958.72	0.439021	0.219511	−	0.975610i	$-0.429554\pi$
$631$	6958.72	0.439021	0.219511	+	0.975610i	$0.429554\pi$
$632$	0	0
$633$	9826.74	0.617027
$634$	0	0
$635$	−16199.5	−1.01237
$636$	0	0
$637$	4010.37	0.249445
$638$	0	0
$639$	−2763.04	−0.171055
$640$	0	0
$641$	−15200.3	−0.936625	−0.468312	−	0.883563i	$-0.655138\pi$
$641$	−15200.3	−0.936625	−0.468312	+	0.883563i	$0.655138\pi$
$642$	0	0
$643$	−27963.3	−1.71503	−0.857516	−	0.514457i	$-0.827994\pi$
$643$	−27963.3	−1.71503	−0.857516	+	0.514457i	$0.827994\pi$
$644$	0	0
$645$	−1733.49	−0.105823
$646$	0	0
$647$	3952.07	0.240142	0.120071	−	0.992765i	$-0.461688\pi$
$647$	3952.07	0.240142	0.120071	+	0.992765i	$0.461688\pi$
$648$	0	0
$649$	27122.5	1.64045
$650$	0	0
$651$	−5894.18	−0.354856
$652$	0	0
$653$	−3772.15	−0.226058	−0.113029	−	0.993592i	$-0.536055\pi$
$653$	−3772.15	−0.226058	−0.113029	+	0.993592i	$0.536055\pi$
$654$	0	0
$655$	6402.95	0.381960
$656$	0	0
$657$	−7604.02	−0.451539
$658$	0	0
$659$	7305.22	0.431822	0.215911	−	0.976413i	$-0.430728\pi$
$659$	7305.22	0.431822	0.215911	+	0.976413i	$0.430728\pi$
$660$	0	0
$661$	−10985.6	−0.646431	−0.323216	−	0.946325i	$-0.604764\pi$
$661$	−10985.6	−0.646431	−0.323216	+	0.946325i	$0.604764\pi$
$662$	0	0
$663$	2876.21	0.168481
$664$	0	0
$665$	−2568.14	−0.149756
$666$	0	0
$667$	−20587.7	−1.19514
$668$	0	0
$669$	−6225.59	−0.359784
$670$	0	0
$671$	−28056.0	−1.61414
$672$	0	0
$673$	−9236.54	−0.529038	−0.264519	−	0.964380i	$-0.585213\pi$
$673$	−9236.54	−0.529038	−0.264519	+	0.964380i	$0.585213\pi$
$674$	0	0
$675$	−1939.52	−0.110596
$676$	0	0
$677$	−1763.78	−0.100129	−0.0500646	−	0.998746i	$-0.515943\pi$
$677$	−1763.78	−0.100129	−0.0500646	+	0.998746i	$0.515943\pi$
$678$	0	0
$679$	4628.54	0.261601
$680$	0	0
$681$	8020.60	0.451322
$682$	0	0
$683$	−9580.46	−0.536729	−0.268365	−	0.963317i	$-0.586483\pi$
$683$	−9580.46	−0.536729	−0.268365	+	0.963317i	$0.586483\pi$
$684$	0	0
$685$	8579.32	0.478539
$686$	0	0
$687$	1674.30	0.0929817
$688$	0	0
$689$	−6047.06	−0.334361
$690$	0	0
$691$	10720.2	0.590179	0.295090	−	0.955470i	$-0.404651\pi$
$691$	10720.2	0.590179	0.295090	+	0.955470i	$0.404651\pi$
$692$	0	0
$693$	−2705.18	−0.148284
$694$	0	0
$695$	−2621.40	−0.143072
$696$	0	0
$697$	−16424.0	−0.892542
$698$	0	0
$699$	7310.97	0.395603
$700$	0	0
$701$	−9060.31	−0.488164	−0.244082	−	0.969755i	$-0.578487\pi$
$701$	−9060.31	−0.488164	−0.244082	+	0.969755i	$0.578487\pi$
$702$	0	0
$703$	15672.9	0.840847
$704$	0	0
$705$	12783.6	0.682918
$706$	0	0
$707$	8175.70	0.434907
$708$	0	0
$709$	26681.4	1.41332	0.706658	−	0.707556i	$-0.250202\pi$
$709$	26681.4	1.41332	0.706658	+	0.707556i	$0.250202\pi$
$710$	0	0
$711$	271.530	0.0143223
$712$	0	0
$713$	−23354.6	−1.22670
$714$	0	0
$715$	−4850.00	−0.253678
$716$	0	0
$717$	−2906.02	−0.151363
$718$	0	0
$719$	−491.820	−0.0255102	−0.0127551	−	0.999919i	$-0.504060\pi$
$719$	−491.820	−0.0255102	−0.0127551	+	0.999919i	$0.504060\pi$
$720$	0	0
$721$	8392.18	0.433483
$722$	0	0
$723$	−20617.6	−1.06055
$724$	0	0
$725$	21178.5	1.08490
$726$	0	0
$727$	12358.2	0.630452	0.315226	−	0.949017i	$-0.397920\pi$
$727$	12358.2	0.630452	0.315226	+	0.949017i	$0.397920\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−5844.39	−0.295708
$732$	0	0
$733$	−24037.3	−1.21124	−0.605620	−	0.795754i	$-0.707075\pi$
$733$	−24037.3	−1.21124	−0.605620	+	0.795754i	$0.707075\pi$
$734$	0	0
$735$	6748.07	0.338648
$736$	0	0
$737$	19696.8	0.984454
$738$	0	0
$739$	21651.5	1.07776	0.538879	−	0.842383i	$-0.318848\pi$
$739$	21651.5	1.07776	0.538879	+	0.842383i	$0.318848\pi$
$740$	0	0
$741$	−2338.27	−0.115922
$742$	0	0
$743$	−2580.05	−0.127393	−0.0636964	−	0.997969i	$-0.520289\pi$
$743$	−2580.05	−0.127393	−0.0636964	+	0.997969i	$0.520289\pi$
$744$	0	0
$745$	−8946.21	−0.439951
$746$	0	0
$747$	176.070	0.00862391
$748$	0	0
$749$	−8008.74	−0.390698
$750$	0	0
$751$	38542.3	1.87274	0.936371	−	0.351012i	$-0.114162\pi$
$751$	38542.3	1.87274	0.936371	+	0.351012i	$0.114162\pi$
$752$	0	0
$753$	−11833.9	−0.572708
$754$	0	0
$755$	5145.36	0.248025
$756$	0	0
$757$	997.651	0.0478999	0.0239500	−	0.999713i	$-0.492376\pi$
$757$	997.651	0.0478999	0.0239500	+	0.999713i	$0.492376\pi$
$758$	0	0
$759$	−10718.8	−0.512605
$760$	0	0
$761$	−29051.7	−1.38387	−0.691933	−	0.721962i	$-0.743241\pi$
$761$	−29051.7	−1.38387	−0.691933	+	0.721962i	$0.743241\pi$
$762$	0	0
$763$	5752.21	0.272928
$764$	0	0
$765$	4839.67	0.228730
$766$	0	0
$767$	6891.16	0.324414
$768$	0	0
$769$	−8935.69	−0.419024	−0.209512	−	0.977806i	$-0.567188\pi$
$769$	−8935.69	−0.419024	−0.209512	+	0.977806i	$0.567188\pi$
$770$	0	0
$771$	17265.4	0.806482
$772$	0	0
$773$	−22239.3	−1.03479	−0.517395	−	0.855747i	$-0.673098\pi$
$773$	−22239.3	−1.03479	−0.517395	+	0.855747i	$0.673098\pi$
$774$	0	0
$775$	24024.9	1.11355
$776$	0	0
$777$	4606.95	0.212707
$778$	0	0
$779$	13352.1	0.614108
$780$	0	0
$781$	15708.2	0.719696
$782$	0	0
$783$	−7960.31	−0.363318
$784$	0	0
$785$	−18373.1	−0.835369
$786$	0	0
$787$	36698.1	1.66219	0.831096	−	0.556129i	$-0.187714\pi$
$787$	36698.1	1.66219	0.831096	+	0.556129i	$0.187714\pi$
$788$	0	0
$789$	−14.0394	−0.000633479 0
$790$	0	0
$791$	8787.27	0.394993
$792$	0	0
$793$	−7128.32	−0.319211
$794$	0	0
$795$	−10175.1	−0.453929
$796$	0	0
$797$	−35811.1	−1.59159	−0.795794	−	0.605568i	$-0.792946\pi$
$797$	−35811.1	−1.59159	−0.795794	+	0.605568i	$0.792946\pi$
$798$	0	0
$799$	43099.3	1.90832
$800$	0	0
$801$	−4617.15	−0.203669
$802$	0	0
$803$	43229.7	1.89981
$804$	0	0
$805$	−2991.10	−0.130960
$806$	0	0
$807$	9809.61	0.427899
$808$	0	0
$809$	28481.2	1.23776	0.618879	−	0.785486i	$-0.287587\pi$
$809$	28481.2	1.23776	0.618879	+	0.785486i	$0.287587\pi$
$810$	0	0
$811$	−18290.5	−0.791944	−0.395972	−	0.918263i	$-0.629592\pi$
$811$	−18290.5	−0.791944	−0.395972	+	0.918263i	$0.629592\pi$
$812$	0	0
$813$	−7968.54	−0.343750
$814$	0	0
$815$	−8905.87	−0.382772
$816$	0	0
$817$	4751.30	0.203460
$818$	0	0
$819$	−687.317	−0.0293246
$820$	0	0
$821$	−21577.5	−0.917248	−0.458624	−	0.888631i	$-0.651657\pi$
$821$	−21577.5	−0.917248	−0.458624	+	0.888631i	$0.651657\pi$
$822$	0	0
$823$	−25940.2	−1.09869	−0.549344	−	0.835596i	$-0.685122\pi$
$823$	−25940.2	−1.09869	−0.549344	+	0.835596i	$0.685122\pi$
$824$	0	0
$825$	11026.4	0.465320
$826$	0	0
$827$	−3112.03	−0.130853	−0.0654267	−	0.997857i	$-0.520841\pi$
$827$	−3112.03	−0.130853	−0.0654267	+	0.997857i	$0.520841\pi$
$828$	0	0
$829$	−3729.80	−0.156262	−0.0781312	−	0.996943i	$-0.524895\pi$
$829$	−3729.80	−0.156262	−0.0781312	+	0.996943i	$0.524895\pi$
$830$	0	0
$831$	−23728.9	−0.990549
$832$	0	0
$833$	22750.8	0.946303
$834$	0	0
$835$	−8017.12	−0.332268
$836$	0	0
$837$	−9030.14	−0.372912
$838$	0	0
$839$	−22919.3	−0.943102	−0.471551	−	0.881839i	$-0.656306\pi$
$839$	−22919.3	−0.943102	−0.471551	+	0.881839i	$0.656306\pi$
$840$	0	0
$841$	62533.4	2.56400
$842$	0	0
$843$	−23162.7	−0.946343
$844$	0	0
$845$	−1232.26	−0.0501671
$846$	0	0
$847$	7560.26	0.306698
$848$	0	0
$849$	3955.61	0.159901
$850$	0	0
$851$	18254.2	0.735307
$852$	0	0
$853$	10092.6	0.405115	0.202557	−	0.979270i	$-0.435075\pi$
$853$	10092.6	0.405115	0.202557	+	0.979270i	$0.435075\pi$
$854$	0	0
$855$	−3934.49	−0.157376
$856$	0	0
$857$	−14803.0	−0.590036	−0.295018	−	0.955492i	$-0.595326\pi$
$857$	−14803.0	−0.590036	−0.295018	+	0.955492i	$0.595326\pi$
$858$	0	0
$859$	7164.74	0.284584	0.142292	−	0.989825i	$-0.454553\pi$
$859$	7164.74	0.284584	0.142292	+	0.989825i	$0.454553\pi$
$860$	0	0
$861$	3924.77	0.155349
$862$	0	0
$863$	20504.4	0.808782	0.404391	−	0.914586i	$-0.367483\pi$
$863$	20504.4	0.808782	0.404391	+	0.914586i	$0.367483\pi$
$864$	0	0
$865$	13251.5	0.520885
$866$	0	0
$867$	1577.75	0.0618031
$868$	0	0
$869$	−1543.68	−0.0602596
$870$	0	0
$871$	5004.47	0.194684
$872$	0	0
$873$	7091.13	0.274912
$874$	0	0
$875$	8431.19	0.325744
$876$	0	0
$877$	−23980.1	−0.923318	−0.461659	−	0.887057i	$-0.652746\pi$
$877$	−23980.1	−0.923318	−0.461659	+	0.887057i	$0.652746\pi$
$878$	0	0
$879$	−12116.3	−0.464929
$880$	0	0
$881$	32577.9	1.24583	0.622915	−	0.782289i	$-0.285948\pi$
$881$	32577.9	1.24583	0.622915	+	0.782289i	$0.285948\pi$
$882$	0	0
$883$	−5201.59	−0.198242	−0.0991208	−	0.995075i	$-0.531603\pi$
$883$	−5201.59	−0.198242	−0.0991208	+	0.995075i	$0.531603\pi$
$884$	0	0
$885$	11595.4	0.440425
$886$	0	0
$887$	10719.5	0.405779	0.202890	−	0.979202i	$-0.434967\pi$
$887$	10719.5	0.405779	0.202890	+	0.979202i	$0.434967\pi$
$888$	0	0
$889$	13051.4	0.492383
$890$	0	0
$891$	−4144.45	−0.155830
$892$	0	0
$893$	−35038.3	−1.31300
$894$	0	0
$895$	34308.1	1.28133
$896$	0	0
$897$	−2723.37	−0.101372
$898$	0	0
$899$	98604.5	3.65811
$900$	0	0
$901$	−34305.0	−1.26844
$902$	0	0
$903$	1396.61	0.0514688
$904$	0	0
$905$	−12358.1	−0.453918
$906$	0	0
$907$	−18995.9	−0.695424	−0.347712	−	0.937601i	$-0.613041\pi$
$907$	−18995.9	−0.695424	−0.347712	+	0.937601i	$0.613041\pi$
$908$	0	0
$909$	12525.5	0.457036
$910$	0	0
$911$	−8475.13	−0.308226	−0.154113	−	0.988053i	$-0.549252\pi$
$911$	−8475.13	−0.308226	−0.154113	+	0.988053i	$0.549252\pi$
$912$	0	0
$913$	−1000.98	−0.0362842
$914$	0	0
$915$	−11994.5	−0.433361
$916$	0	0
$917$	−5158.63	−0.185772
$918$	0	0
$919$	−41161.3	−1.47746	−0.738731	−	0.674000i	$-0.764575\pi$
$919$	−41161.3	−1.47746	−0.738731	+	0.674000i	$0.764575\pi$
$920$	0	0
$921$	−29084.9	−1.04059
$922$	0	0
$923$	3991.05	0.142326
$924$	0	0
$925$	−18778.1	−0.667480
$926$	0	0
$927$	12857.2	0.455539
$928$	0	0
$929$	−24181.6	−0.854007	−0.427004	−	0.904250i	$-0.640431\pi$
$929$	−24181.6	−0.854007	−0.427004	+	0.904250i	$0.640431\pi$
$930$	0	0
$931$	−18495.7	−0.651098
$932$	0	0
$933$	−23847.4	−0.836792
$934$	0	0
$935$	−27514.1	−0.962360
$936$	0	0
$937$	3815.36	0.133023	0.0665114	−	0.997786i	$-0.478813\pi$
$937$	3815.36	0.133023	0.0665114	+	0.997786i	$0.478813\pi$
$938$	0	0
$939$	15347.3	0.533377
$940$	0	0
$941$	34381.2	1.19107	0.595534	−	0.803330i	$-0.296940\pi$
$941$	34381.2	1.19107	0.595534	+	0.803330i	$0.296940\pi$
$942$	0	0
$943$	15551.2	0.537027
$944$	0	0
$945$	−1156.52	−0.0398111
$946$	0	0
$947$	−20897.6	−0.717087	−0.358543	−	0.933513i	$-0.616726\pi$
$947$	−20897.6	−0.717087	−0.358543	+	0.933513i	$0.616726\pi$
$948$	0	0
$949$	10983.6	0.375703
$950$	0	0
$951$	−3654.28	−0.124604
$952$	0	0
$953$	−21656.9	−0.736136	−0.368068	−	0.929799i	$-0.619981\pi$
$953$	−21656.9	−0.736136	−0.368068	+	0.929799i	$0.619981\pi$
$954$	0	0
$955$	13414.0	0.454520
$956$	0	0
$957$	45255.2	1.52862
$958$	0	0
$959$	−6912.06	−0.232745
$960$	0	0
$961$	82065.6	2.75471
$962$	0	0
$963$	−12269.7	−0.410578
$964$	0	0
$965$	12950.8	0.432023
$966$	0	0
$967$	38712.2	1.28738	0.643691	−	0.765285i	$-0.277402\pi$
$967$	38712.2	1.28738	0.643691	+	0.765285i	$0.277402\pi$
$968$	0	0
$969$	−13265.0	−0.439766
$970$	0	0
$971$	11532.4	0.381147	0.190573	−	0.981673i	$-0.438965\pi$
$971$	11532.4	0.381147	0.190573	+	0.981673i	$0.438965\pi$
$972$	0	0
$973$	2111.97	0.0695853
$974$	0	0
$975$	2801.53	0.0920212
$976$	0	0
$977$	26477.6	0.867036	0.433518	−	0.901145i	$-0.357272\pi$
$977$	26477.6	0.867036	0.433518	+	0.901145i	$0.357272\pi$
$978$	0	0
$979$	26249.0	0.856918
$980$	0	0
$981$	8812.64	0.286815
$982$	0	0
$983$	−23993.5	−0.778508	−0.389254	−	0.921131i	$-0.627267\pi$
$983$	−23993.5	−0.778508	−0.389254	+	0.921131i	$0.627267\pi$
$984$	0	0
$985$	−4687.66	−0.151636
$986$	0	0
$987$	−10299.3	−0.332148
$988$	0	0
$989$	5533.83	0.177923
$990$	0	0
$991$	−9436.25	−0.302475	−0.151237	−	0.988497i	$-0.548326\pi$
$991$	−9436.25	−0.302475	−0.151237	+	0.988497i	$0.548326\pi$
$992$	0	0
$993$	−29665.0	−0.948026
$994$	0	0
$995$	7924.96	0.252501
$996$	0	0
$997$	−31197.1	−0.990995	−0.495498	−	0.868609i	$-0.665014\pi$
$997$	−31197.1	−0.990995	−0.495498	+	0.868609i	$0.665014\pi$
$998$	0	0
$999$	7058.05	0.223530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.e.1.1	✓	2
3.2	odd	2		936.4.a.f.1.2		2
4.3	odd	2		624.4.a.k.1.1		2
8.3	odd	2		2496.4.a.bh.1.2		2
8.5	even	2		2496.4.a.y.1.2		2
12.11	even	2		1872.4.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.e.1.1	✓	2	1.1	even	1	trivial
624.4.a.k.1.1		2	4.3	odd	2
936.4.a.f.1.2		2	3.2	odd	2
1872.4.a.bf.1.2		2	12.11	even	2
2496.4.a.y.1.2		2	8.5	even	2
2496.4.a.bh.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -7.29150 q^{5} +5.87451 q^{7} +9.00000 q^{9} -51.1660 q^{11} -13.0000 q^{13} -21.8745 q^{15} -73.7490 q^{17} +59.9555 q^{19} +17.6235 q^{21} +69.8301 q^{23} -71.8340 q^{25} +27.0000 q^{27} -294.826 q^{29} -334.450 q^{31} -153.498 q^{33} -42.8340 q^{35} +261.409 q^{37} -39.0000 q^{39} +222.701 q^{41} +79.2470 q^{43} -65.6235 q^{45} -584.405 q^{47} -308.490 q^{49} -221.247 q^{51} +465.158 q^{53} +373.077 q^{55} +179.867 q^{57} -530.089 q^{59} +548.332 q^{61} +52.8706 q^{63} +94.7895 q^{65} -384.959 q^{67} +209.490 q^{69} -307.004 q^{71} -844.891 q^{73} -215.502 q^{75} -300.575 q^{77} +30.1699 q^{79} +81.0000 q^{81} +19.5633 q^{83} +537.741 q^{85} -884.478 q^{87} -513.017 q^{89} -76.3686 q^{91} -1003.35 q^{93} -437.166 q^{95} +787.903 q^{97} -460.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 4 q^{5} - 20 q^{7} + 18 q^{9} - 60 q^{11} - 26 q^{13} - 12 q^{15} - 84 q^{17} - 60 q^{19} - 60 q^{21} - 72 q^{23} - 186 q^{25} + 54 q^{27} - 124 q^{29} - 108 q^{31} - 180 q^{33} - 128 q^{35}+ \cdots - 540 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 4 * q^5 - 20 * q^7 + 18 * q^9 - 60 * q^11 - 26 * q^13 - 12 * q^15 - 84 * q^17 - 60 * q^19 - 60 * q^21 - 72 * q^23 - 186 * q^25 + 54 * q^27 - 124 * q^29 - 108 * q^31 - 180 * q^33 - 128 * q^35 + 36 * q^37 - 78 * q^39 - 52 * q^41 - 32 * q^43 - 36 * q^45 - 428 * q^47 + 18 * q^49 - 252 * q^51 + 380 * q^53 + 344 * q^55 - 180 * q^57 - 1420 * q^59 + 1012 * q^61 - 180 * q^63 + 52 * q^65 - 844 * q^67 - 216 * q^69 - 868 * q^71 - 60 * q^73 - 558 * q^75 - 72 * q^77 + 272 * q^79 + 162 * q^81 - 1252 * q^83 + 504 * q^85 - 372 * q^87 + 572 * q^89 + 260 * q^91 - 324 * q^93 - 832 * q^95 + 708 * q^97 - 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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