LMFDB - Embedded newform 1216.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,4,Mod(1,1216)] 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("1216.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.15207$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.15207 q^{3} +8.30413 q^{5} -35.1521 q^{7} +24.1521 q^{9} +18.3041 q^{11} +40.0645 q^{13} +59.3917 q^{15} -125.281 q^{17} -19.0000 q^{19} -251.410 q^{21} -8.97688 q^{23} -56.0413 q^{25} -20.3686 q^{27} -153.410 q^{29} +114.433 q^{31} +130.912 q^{33} -291.908 q^{35} -83.5669 q^{37} +286.544 q^{39} -355.088 q^{41} +467.299 q^{43} +200.562 q^{45} -166.083 q^{47} +892.668 q^{49} -896.018 q^{51} -258.369 q^{53} +152.000 q^{55} -135.889 q^{57} -371.797 q^{59} +47.3090 q^{61} -848.995 q^{63} +332.701 q^{65} -755.539 q^{67} -64.2032 q^{69} -349.345 q^{71} +54.8479 q^{73} -400.811 q^{75} -643.428 q^{77} -438.820 q^{79} -797.783 q^{81} +1073.09 q^{83} -1040.35 q^{85} -1097.20 q^{87} -501.521 q^{89} -1408.35 q^{91} +818.433 q^{93} -157.779 q^{95} -1437.56 q^{97} +442.083 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35}+ \cdots + 352 q^{99}+O(q^{100}) $ 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9 + 10 * q^11 - 13 * q^13 + 172 * q^15 - 51 * q^17 - 38 * q^19 - 117 * q^21 + 155 * q^23 + 154 * q^25 + 79 * q^27 + 79 * q^29 + 16 * q^31 + 182 * q^33 + 108 * q^35 - 380 * q^37 + 613 * q^39 - 790 * q^41 + 296 * q^43 + 2 * q^45 + 200 * q^47 + 1027 * q^49 - 1353 * q^51 - 397 * q^53 + 304 * q^55 - 19 * q^57 + 201 * q^59 + 680 * q^61 - 1086 * q^63 + 1304 * q^65 - 939 * q^67 - 1073 * q^69 - 406 * q^71 + 123 * q^73 - 1693 * q^75 - 462 * q^77 - 106 * q^79 - 1702 * q^81 + 2226 * q^83 - 2400 * q^85 - 2527 * q^87 - 870 * q^89 - 249 * q^91 + 1424 * q^93 + 190 * q^95 - 1864 * q^97 + 352 * 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
$3$	7.15207	1.37642	0.688208	−	0.725513i	$-0.258398\pi$
$3$	7.15207	1.37642	0.688208	+	0.725513i	$0.258398\pi$
$4$	0	0
$5$	8.30413	0.742744	0.371372	−	0.928484i	$-0.378887\pi$
$5$	8.30413	0.742744	0.371372	+	0.928484i	$0.378887\pi$
$6$	0	0
$7$	−35.1521	−1.89803	−0.949017	−	0.315226i	$-0.897920\pi$
$7$	−35.1521	−1.89803	−0.949017	+	0.315226i	$0.897920\pi$
$8$	0	0
$9$	24.1521	0.894521
$10$	0	0
$11$	18.3041	0.501719	0.250859	−	0.968024i	$-0.419287\pi$
$11$	18.3041	0.501719	0.250859	+	0.968024i	$0.419287\pi$
$12$	0	0
$13$	40.0645	0.854760	0.427380	−	0.904072i	$-0.359437\pi$
$13$	40.0645	0.854760	0.427380	+	0.904072i	$0.359437\pi$
$14$	0	0
$15$	59.3917	1.02233
$16$	0	0
$17$	−125.281	−1.78736	−0.893680	−	0.448706i	$-0.851885\pi$
$17$	−125.281	−1.78736	−0.893680	+	0.448706i	$0.851885\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−251.410	−2.61248
$22$	0	0
$23$	−8.97688	−0.0813830	−0.0406915	−	0.999172i	$-0.512956\pi$
$23$	−8.97688	−0.0813830	−0.0406915	+	0.999172i	$0.512956\pi$
$24$	0	0
$25$	−56.0413	−0.448331
$26$	0	0
$27$	−20.3686	−0.145183
$28$	0	0
$29$	−153.410	−0.982328	−0.491164	−	0.871067i	$-0.663428\pi$
$29$	−153.410	−0.982328	−0.491164	+	0.871067i	$0.663428\pi$
$30$	0	0
$31$	114.433	0.662993	0.331497	−	0.943456i	$-0.392446\pi$
$31$	114.433	0.662993	0.331497	+	0.943456i	$0.392446\pi$
$32$	0	0
$33$	130.912	0.690573
$34$	0	0
$35$	−291.908	−1.40975
$36$	0	0
$37$	−83.5669	−0.371306	−0.185653	−	0.982615i	$-0.559440\pi$
$37$	−83.5669	−0.371306	−0.185653	+	0.982615i	$0.559440\pi$
$38$	0	0
$39$	286.544	1.17651
$40$	0	0
$41$	−355.088	−1.35257	−0.676285	−	0.736640i	$-0.736411\pi$
$41$	−355.088	−1.35257	−0.676285	+	0.736640i	$0.736411\pi$
$42$	0	0
$43$	467.299	1.65727	0.828633	−	0.559792i	$-0.189119\pi$
$43$	467.299	1.65727	0.828633	+	0.559792i	$0.189119\pi$
$44$	0	0
$45$	200.562	0.664400
$46$	0	0
$47$	−166.083	−0.515439	−0.257720	−	0.966220i	$-0.582971\pi$
$47$	−166.083	−0.515439	−0.257720	+	0.966220i	$0.582971\pi$
$48$	0	0
$49$	892.668	2.60253
$50$	0	0
$51$	−896.018	−2.46015
$52$	0	0
$53$	−258.369	−0.669616	−0.334808	−	0.942286i	$-0.608671\pi$
$53$	−258.369	−0.669616	−0.334808	+	0.942286i	$0.608671\pi$
$54$	0	0
$55$	152.000	0.372649
$56$	0	0
$57$	−135.889	−0.315771
$58$	0	0
$59$	−371.797	−0.820404	−0.410202	−	0.911995i	$-0.634542\pi$
$59$	−371.797	−0.820404	−0.410202	+	0.911995i	$0.634542\pi$
$60$	0	0
$61$	47.3090	0.0993000	0.0496500	−	0.998767i	$-0.484189\pi$
$61$	47.3090	0.0993000	0.0496500	+	0.998767i	$0.484189\pi$
$62$	0	0
$63$	−848.995	−1.69783
$64$	0	0
$65$	332.701	0.634868
$66$	0	0
$67$	−755.539	−1.37767	−0.688834	−	0.724919i	$-0.741877\pi$
$67$	−755.539	−1.37767	−0.688834	+	0.724919i	$0.741877\pi$
$68$	0	0
$69$	−64.2032	−0.112017
$70$	0	0
$71$	−349.345	−0.583939	−0.291970	−	0.956428i	$-0.594311\pi$
$71$	−349.345	−0.583939	−0.291970	+	0.956428i	$0.594311\pi$
$72$	0	0
$73$	54.8479	0.0879379	0.0439689	−	0.999033i	$-0.486000\pi$
$73$	54.8479	0.0879379	0.0439689	+	0.999033i	$0.486000\pi$
$74$	0	0
$75$	−400.811	−0.617090
$76$	0	0
$77$	−643.428	−0.952279
$78$	0	0
$79$	−438.820	−0.624951	−0.312475	−	0.949926i	$-0.601158\pi$
$79$	−438.820	−0.624951	−0.312475	+	0.949926i	$0.601158\pi$
$80$	0	0
$81$	−797.783	−1.09435
$82$	0	0
$83$	1073.09	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$83$	1073.09	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$84$	0	0
$85$	−1040.35	−1.32755
$86$	0	0
$87$	−1097.20	−1.35209
$88$	0	0
$89$	−501.521	−0.597316	−0.298658	−	0.954360i	$-0.596539\pi$
$89$	−501.521	−0.597316	−0.298658	+	0.954360i	$0.596539\pi$
$90$	0	0
$91$	−1408.35	−1.62236
$92$	0	0
$93$	818.433	0.912554
$94$	0	0
$95$	−157.779	−0.170397
$96$	0	0
$97$	−1437.56	−1.50476	−0.752380	−	0.658729i	$-0.771094\pi$
$97$	−1437.56	−1.50476	−0.752380	+	0.658729i	$0.771094\pi$
$98$	0	0
$99$	442.083	0.448798
$100$	0	0
$101$	−395.124	−0.389270	−0.194635	−	0.980876i	$-0.562352\pi$
$101$	−395.124	−0.389270	−0.194635	+	0.980876i	$0.562352\pi$
$102$	0	0
$103$	−1285.68	−1.22992	−0.614958	−	0.788559i	$-0.710827\pi$
$103$	−1285.68	−1.22992	−0.614958	+	0.788559i	$0.710827\pi$
$104$	0	0
$105$	−2087.74	−1.94041
$106$	0	0
$107$	1203.10	1.08699	0.543494	−	0.839413i	$-0.317101\pi$
$107$	1203.10	1.08699	0.543494	+	0.839413i	$0.317101\pi$
$108$	0	0
$109$	1333.74	1.17201	0.586005	−	0.810307i	$-0.300700\pi$
$109$	1333.74	1.17201	0.586005	+	0.810307i	$0.300700\pi$
$110$	0	0
$111$	−597.676	−0.511071
$112$	0	0
$113$	836.506	0.696388	0.348194	−	0.937422i	$-0.386795\pi$
$113$	836.506	0.696388	0.348194	+	0.937422i	$0.386795\pi$
$114$	0	0
$115$	−74.5452	−0.0604467
$116$	0	0
$117$	967.640	0.764601
$118$	0	0
$119$	4403.89	3.39247
$120$	0	0
$121$	−995.959	−0.748278
$122$	0	0
$123$	−2539.61	−1.86170
$124$	0	0
$125$	−1503.39	−1.07574
$126$	0	0
$127$	385.954	0.269668	0.134834	−	0.990868i	$-0.456950\pi$
$127$	385.954	0.269668	0.134834	+	0.990868i	$0.456950\pi$
$128$	0	0
$129$	3342.16	2.28109
$130$	0	0
$131$	1737.90	1.15909	0.579545	−	0.814940i	$-0.303230\pi$
$131$	1737.90	1.15909	0.579545	+	0.814940i	$0.303230\pi$
$132$	0	0
$133$	667.889	0.435439
$134$	0	0
$135$	−169.144	−0.107834
$136$	0	0
$137$	41.7603	0.0260425	0.0130213	−	0.999915i	$-0.495855\pi$
$137$	41.7603	0.0260425	0.0130213	+	0.999915i	$0.495855\pi$
$138$	0	0
$139$	1536.79	0.937763	0.468882	−	0.883261i	$-0.344657\pi$
$139$	1536.79	0.937763	0.468882	+	0.883261i	$0.344657\pi$
$140$	0	0
$141$	−1187.83	−0.709459
$142$	0	0
$143$	733.345	0.428849
$144$	0	0
$145$	−1273.94	−0.729619
$146$	0	0
$147$	6384.42	3.58216
$148$	0	0
$149$	−1656.65	−0.910862	−0.455431	−	0.890271i	$-0.650515\pi$
$149$	−1656.65	−0.910862	−0.455431	+	0.890271i	$0.650515\pi$
$150$	0	0
$151$	714.985	0.385329	0.192664	−	0.981265i	$-0.438287\pi$
$151$	714.985	0.385329	0.192664	+	0.981265i	$0.438287\pi$
$152$	0	0
$153$	−3025.80	−1.59883
$154$	0	0
$155$	950.268	0.492434
$156$	0	0
$157$	−1684.23	−0.856153	−0.428077	−	0.903742i	$-0.640809\pi$
$157$	−1684.23	−0.856153	−0.428077	+	0.903742i	$0.640809\pi$
$158$	0	0
$159$	−1847.87	−0.921670
$160$	0	0
$161$	315.556	0.154468
$162$	0	0
$163$	702.175	0.337415	0.168707	−	0.985666i	$-0.446041\pi$
$163$	702.175	0.337415	0.168707	+	0.985666i	$0.446041\pi$
$164$	0	0
$165$	1087.11	0.512920
$166$	0	0
$167$	282.506	0.130904	0.0654520	−	0.997856i	$-0.479151\pi$
$167$	282.506	0.130904	0.0654520	+	0.997856i	$0.479151\pi$
$168$	0	0
$169$	−591.838	−0.269385
$170$	0	0
$171$	−458.889	−0.205217
$172$	0	0
$173$	−2183.44	−0.959558	−0.479779	−	0.877389i	$-0.659283\pi$
$173$	−2183.44	−0.959558	−0.479779	+	0.877389i	$0.659283\pi$
$174$	0	0
$175$	1969.97	0.850947
$176$	0	0
$177$	−2659.12	−1.12922
$178$	0	0
$179$	−3198.51	−1.33557	−0.667786	−	0.744353i	$-0.732758\pi$
$179$	−3198.51	−1.33557	−0.667786	+	0.744353i	$0.732758\pi$
$180$	0	0
$181$	2151.05	0.883350	0.441675	−	0.897175i	$-0.354384\pi$
$181$	2151.05	0.883350	0.441675	+	0.897175i	$0.354384\pi$
$182$	0	0
$183$	338.357	0.136678
$184$	0	0
$185$	−693.951	−0.275785
$186$	0	0
$187$	−2293.16	−0.896751
$188$	0	0
$189$	715.999	0.275562
$190$	0	0
$191$	4435.52	1.68033	0.840165	−	0.542331i	$-0.182458\pi$
$191$	4435.52	1.68033	0.840165	+	0.542331i	$0.182458\pi$
$192$	0	0
$193$	2720.60	1.01468	0.507339	−	0.861746i	$-0.330629\pi$
$193$	2720.60	1.01468	0.507339	+	0.861746i	$0.330629\pi$
$194$	0	0
$195$	2379.50	0.873843
$196$	0	0
$197$	−1254.08	−0.453549	−0.226775	−	0.973947i	$-0.572818\pi$
$197$	−1254.08	−0.453549	−0.226775	+	0.973947i	$0.572818\pi$
$198$	0	0
$199$	−4155.19	−1.48017	−0.740084	−	0.672515i	$-0.765214\pi$
$199$	−4155.19	−1.48017	−0.740084	+	0.672515i	$0.765214\pi$
$200$	0	0
$201$	−5403.67	−1.89624
$202$	0	0
$203$	5392.68	1.86449
$204$	0	0
$205$	−2948.70	−1.00461
$206$	0	0
$207$	−216.810	−0.0727988
$208$	0	0
$209$	−347.779	−0.115102
$210$	0	0
$211$	−633.437	−0.206671	−0.103335	−	0.994647i	$-0.532952\pi$
$211$	−633.437	−0.206671	−0.103335	+	0.994647i	$0.532952\pi$
$212$	0	0
$213$	−2498.54	−0.803743
$214$	0	0
$215$	3880.52	1.23093
$216$	0	0
$217$	−4022.56	−1.25838
$218$	0	0
$219$	392.276	0.121039
$220$	0	0
$221$	−5019.32	−1.52776
$222$	0	0
$223$	4798.34	1.44090	0.720449	−	0.693507i	$-0.243936\pi$
$223$	4798.34	1.44090	0.720449	+	0.693507i	$0.243936\pi$
$224$	0	0
$225$	−1353.51	−0.401041
$226$	0	0
$227$	641.770	0.187647	0.0938233	−	0.995589i	$-0.470091\pi$
$227$	641.770	0.187647	0.0938233	+	0.995589i	$0.470091\pi$
$228$	0	0
$229$	1231.34	0.355325	0.177662	−	0.984091i	$-0.443146\pi$
$229$	1231.34	0.355325	0.177662	+	0.984091i	$0.443146\pi$
$230$	0	0
$231$	−4601.84	−1.31073
$232$	0	0
$233$	226.701	0.0637410	0.0318705	−	0.999492i	$-0.489854\pi$
$233$	226.701	0.0637410	0.0318705	+	0.999492i	$0.489854\pi$
$234$	0	0
$235$	−1379.17	−0.382840
$236$	0	0
$237$	−3138.47	−0.860192
$238$	0	0
$239$	−2433.27	−0.658557	−0.329278	−	0.944233i	$-0.606805\pi$
$239$	−2433.27	−0.658557	−0.329278	+	0.944233i	$0.606805\pi$
$240$	0	0
$241$	−901.336	−0.240913	−0.120457	−	0.992719i	$-0.538436\pi$
$241$	−901.336	−0.240913	−0.120457	+	0.992719i	$0.538436\pi$
$242$	0	0
$243$	−5155.85	−1.36110
$244$	0	0
$245$	7412.83	1.93301
$246$	0	0
$247$	−761.225	−0.196095
$248$	0	0
$249$	7674.79	1.95329
$250$	0	0
$251$	−5675.92	−1.42734	−0.713668	−	0.700484i	$-0.752967\pi$
$251$	−5675.92	−1.42734	−0.713668	+	0.700484i	$0.752967\pi$
$252$	0	0
$253$	−164.314	−0.0408313
$254$	0	0
$255$	−7440.66	−1.82726
$256$	0	0
$257$	2035.88	0.494143	0.247072	−	0.968997i	$-0.420532\pi$
$257$	2035.88	0.494143	0.247072	+	0.968997i	$0.420532\pi$
$258$	0	0
$259$	2937.55	0.704751
$260$	0	0
$261$	−3705.17	−0.878713
$262$	0	0
$263$	−1924.20	−0.451147	−0.225573	−	0.974226i	$-0.572426\pi$
$263$	−1924.20	−0.451147	−0.225573	+	0.974226i	$0.572426\pi$
$264$	0	0
$265$	−2145.53	−0.497354
$266$	0	0
$267$	−3586.91	−0.822155
$268$	0	0
$269$	829.280	0.187963	0.0939815	−	0.995574i	$-0.470041\pi$
$269$	829.280	0.187963	0.0939815	+	0.995574i	$0.470041\pi$
$270$	0	0
$271$	1223.13	0.274169	0.137085	−	0.990559i	$-0.456227\pi$
$271$	1223.13	0.274169	0.137085	+	0.990559i	$0.456227\pi$
$272$	0	0
$273$	−10072.6	−2.23305
$274$	0	0
$275$	−1025.79	−0.224936
$276$	0	0
$277$	−1622.81	−0.352004	−0.176002	−	0.984390i	$-0.556317\pi$
$277$	−1622.81	−0.352004	−0.176002	+	0.984390i	$0.556317\pi$
$278$	0	0
$279$	2763.80	0.593061
$280$	0	0
$281$	3618.96	0.768288	0.384144	−	0.923273i	$-0.374497\pi$
$281$	3618.96	0.768288	0.384144	+	0.923273i	$0.374497\pi$
$282$	0	0
$283$	2102.43	0.441613	0.220807	−	0.975318i	$-0.429131\pi$
$283$	2102.43	0.441613	0.220807	+	0.975318i	$0.429131\pi$
$284$	0	0
$285$	−1128.44	−0.234537
$286$	0	0
$287$	12482.1	2.56722
$288$	0	0
$289$	10782.3	2.19465
$290$	0	0
$291$	−10281.5	−2.07118
$292$	0	0
$293$	5383.99	1.07350	0.536751	−	0.843741i	$-0.319652\pi$
$293$	5383.99	1.07350	0.536751	+	0.843741i	$0.319652\pi$
$294$	0	0
$295$	−3087.45	−0.609350
$296$	0	0
$297$	−372.830	−0.0728410
$298$	0	0
$299$	−359.654	−0.0695629
$300$	0	0
$301$	−16426.5	−3.14555
$302$	0	0
$303$	−2825.95	−0.535798
$304$	0	0
$305$	392.861	0.0737545
$306$	0	0
$307$	3692.85	0.686521	0.343260	−	0.939240i	$-0.388469\pi$
$307$	3692.85	0.686521	0.343260	+	0.939240i	$0.388469\pi$
$308$	0	0
$309$	−9195.24	−1.69288
$310$	0	0
$311$	4427.67	0.807299	0.403649	−	0.914914i	$-0.367742\pi$
$311$	4427.67	0.807299	0.403649	+	0.914914i	$0.367742\pi$
$312$	0	0
$313$	−7356.47	−1.32847	−0.664236	−	0.747523i	$-0.731243\pi$
$313$	−7356.47	−1.32847	−0.664236	+	0.747523i	$0.731243\pi$
$314$	0	0
$315$	−7050.17	−1.26105
$316$	0	0
$317$	−1612.77	−0.285747	−0.142874	−	0.989741i	$-0.545634\pi$
$317$	−1612.77	−0.285747	−0.142874	+	0.989741i	$0.545634\pi$
$318$	0	0
$319$	−2808.04	−0.492852
$320$	0	0
$321$	8604.62	1.49615
$322$	0	0
$323$	2380.34	0.410048
$324$	0	0
$325$	−2245.27	−0.383215
$326$	0	0
$327$	9539.00	1.61317
$328$	0	0
$329$	5838.15	0.978321
$330$	0	0
$331$	−2262.78	−0.375752	−0.187876	−	0.982193i	$-0.560160\pi$
$331$	−2262.78	−0.375752	−0.187876	+	0.982193i	$0.560160\pi$
$332$	0	0
$333$	−2018.31	−0.332141
$334$	0	0
$335$	−6274.10	−1.02326
$336$	0	0
$337$	−2037.63	−0.329367	−0.164683	−	0.986346i	$-0.552660\pi$
$337$	−2037.63	−0.329367	−0.164683	+	0.986346i	$0.552660\pi$
$338$	0	0
$339$	5982.75	0.958520
$340$	0	0
$341$	2094.60	0.332636
$342$	0	0
$343$	−19322.0	−3.04166
$344$	0	0
$345$	−533.152	−0.0831998
$346$	0	0
$347$	−1844.33	−0.285328	−0.142664	−	0.989771i	$-0.545567\pi$
$347$	−1844.33	−0.285328	−0.142664	+	0.989771i	$0.545567\pi$
$348$	0	0
$349$	10156.9	1.55784	0.778918	−	0.627125i	$-0.215769\pi$
$349$	10156.9	1.55784	0.778918	+	0.627125i	$0.215769\pi$
$350$	0	0
$351$	−816.057	−0.124097
$352$	0	0
$353$	1905.43	0.287296	0.143648	−	0.989629i	$-0.454117\pi$
$353$	1905.43	0.287296	0.143648	+	0.989629i	$0.454117\pi$
$354$	0	0
$355$	−2901.01	−0.433718
$356$	0	0
$357$	31496.9	4.66945
$358$	0	0
$359$	2496.50	0.367020	0.183510	−	0.983018i	$-0.441254\pi$
$359$	2496.50	0.367020	0.183510	+	0.983018i	$0.441254\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−7123.16	−1.02994
$364$	0	0
$365$	455.465	0.0653154
$366$	0	0
$367$	8748.42	1.24432	0.622158	−	0.782892i	$-0.286256\pi$
$367$	8748.42	1.24432	0.622158	+	0.782892i	$0.286256\pi$
$368$	0	0
$369$	−8576.10	−1.20990
$370$	0	0
$371$	9082.19	1.27095
$372$	0	0
$373$	−1460.62	−0.202756	−0.101378	−	0.994848i	$-0.532325\pi$
$373$	−1460.62	−0.202756	−0.101378	+	0.994848i	$0.532325\pi$
$374$	0	0
$375$	−10752.4	−1.48067
$376$	0	0
$377$	−6146.29	−0.839655
$378$	0	0
$379$	10581.0	1.43406	0.717029	−	0.697043i	$-0.245501\pi$
$379$	10581.0	1.43406	0.717029	+	0.697043i	$0.245501\pi$
$380$	0	0
$381$	2760.37	0.371176
$382$	0	0
$383$	2932.54	0.391242	0.195621	−	0.980680i	$-0.437328\pi$
$383$	2932.54	0.391242	0.195621	+	0.980680i	$0.437328\pi$
$384$	0	0
$385$	−5343.11	−0.707300
$386$	0	0
$387$	11286.2	1.48246
$388$	0	0
$389$	3631.19	0.473287	0.236644	−	0.971597i	$-0.423953\pi$
$389$	3631.19	0.473287	0.236644	+	0.971597i	$0.423953\pi$
$390$	0	0
$391$	1124.63	0.145461
$392$	0	0
$393$	12429.6	1.59539
$394$	0	0
$395$	−3644.02	−0.464179
$396$	0	0
$397$	2005.32	0.253512	0.126756	−	0.991934i	$-0.459544\pi$
$397$	2005.32	0.253512	0.126756	+	0.991934i	$0.459544\pi$
$398$	0	0
$399$	4776.79	0.599345
$400$	0	0
$401$	−8187.30	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$401$	−8187.30	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$402$	0	0
$403$	4584.70	0.566700
$404$	0	0
$405$	−6624.90	−0.812825
$406$	0	0
$407$	−1529.62	−0.186291
$408$	0	0
$409$	−15565.0	−1.88175	−0.940877	−	0.338747i	$-0.889997\pi$
$409$	−15565.0	−1.88175	−0.940877	+	0.338747i	$0.889997\pi$
$410$	0	0
$411$	298.673	0.0358454
$412$	0	0
$413$	13069.4	1.55715
$414$	0	0
$415$	8911.06	1.05404
$416$	0	0
$417$	10991.2	1.29075
$418$	0	0
$419$	−1839.83	−0.214514	−0.107257	−	0.994231i	$-0.534207\pi$
$419$	−1839.83	−0.214514	−0.107257	+	0.994231i	$0.534207\pi$
$420$	0	0
$421$	−4658.28	−0.539266	−0.269633	−	0.962963i	$-0.586902\pi$
$421$	−4658.28	−0.539266	−0.269633	+	0.962963i	$0.586902\pi$
$422$	0	0
$423$	−4011.24	−0.461071
$424$	0	0
$425$	7020.92	0.801328
$426$	0	0
$427$	−1663.01	−0.188475
$428$	0	0
$429$	5244.94	0.590275
$430$	0	0
$431$	−2820.47	−0.315214	−0.157607	−	0.987502i	$-0.550378\pi$
$431$	−2820.47	−0.315214	−0.157607	+	0.987502i	$0.550378\pi$
$432$	0	0
$433$	8981.39	0.996809	0.498404	−	0.866945i	$-0.333920\pi$
$433$	8981.39	0.996809	0.498404	+	0.866945i	$0.333920\pi$
$434$	0	0
$435$	−9111.28	−1.00426
$436$	0	0
$437$	170.561	0.0186705
$438$	0	0
$439$	−14131.1	−1.53631	−0.768157	−	0.640261i	$-0.778826\pi$
$439$	−14131.1	−1.53631	−0.768157	+	0.640261i	$0.778826\pi$
$440$	0	0
$441$	21559.8	2.32802
$442$	0	0
$443$	1659.59	0.177990	0.0889951	−	0.996032i	$-0.471634\pi$
$443$	1659.59	0.177990	0.0889951	+	0.996032i	$0.471634\pi$
$444$	0	0
$445$	−4164.70	−0.443653
$446$	0	0
$447$	−11848.5	−1.25372
$448$	0	0
$449$	−13732.6	−1.44338	−0.721692	−	0.692214i	$-0.756635\pi$
$449$	−13732.6	−1.44338	−0.721692	+	0.692214i	$0.756635\pi$
$450$	0	0
$451$	−6499.57	−0.678609
$452$	0	0
$453$	5113.62	0.530373
$454$	0	0
$455$	−11695.1	−1.20500
$456$	0	0
$457$	−5273.37	−0.539776	−0.269888	−	0.962892i	$-0.586987\pi$
$457$	−5273.37	−0.539776	−0.269888	+	0.962892i	$0.586987\pi$
$458$	0	0
$459$	2551.80	0.259494
$460$	0	0
$461$	−16402.4	−1.65713	−0.828565	−	0.559893i	$-0.810842\pi$
$461$	−16402.4	−1.65713	−0.828565	+	0.559893i	$0.810842\pi$
$462$	0	0
$463$	5296.03	0.531593	0.265796	−	0.964029i	$-0.414365\pi$
$463$	5296.03	0.531593	0.265796	+	0.964029i	$0.414365\pi$
$464$	0	0
$465$	6796.38	0.677795
$466$	0	0
$467$	3470.94	0.343931	0.171966	−	0.985103i	$-0.444988\pi$
$467$	3470.94	0.343931	0.171966	+	0.985103i	$0.444988\pi$
$468$	0	0
$469$	26558.8	2.61486
$470$	0	0
$471$	−12045.7	−1.17842
$472$	0	0
$473$	8553.51	0.831481
$474$	0	0
$475$	1064.79	0.102854
$476$	0	0
$477$	−6240.14	−0.598986
$478$	0	0
$479$	16294.7	1.55433	0.777163	−	0.629300i	$-0.216658\pi$
$479$	16294.7	1.55433	0.777163	+	0.629300i	$0.216658\pi$
$480$	0	0
$481$	−3348.06	−0.317378
$482$	0	0
$483$	2256.88	0.212612
$484$	0	0
$485$	−11937.7	−1.11765
$486$	0	0
$487$	2245.82	0.208969	0.104485	−	0.994527i	$-0.466681\pi$
$487$	2245.82	0.208969	0.104485	+	0.994527i	$0.466681\pi$
$488$	0	0
$489$	5022.00	0.464423
$490$	0	0
$491$	−7578.64	−0.696577	−0.348288	−	0.937387i	$-0.613237\pi$
$491$	−7578.64	−0.696577	−0.348288	+	0.937387i	$0.613237\pi$
$492$	0	0
$493$	19219.4	1.75577
$494$	0	0
$495$	3671.11	0.333342
$496$	0	0
$497$	12280.2	1.10834
$498$	0	0
$499$	−10558.4	−0.947209	−0.473604	−	0.880738i	$-0.657047\pi$
$499$	−10558.4	−0.947209	−0.473604	+	0.880738i	$0.657047\pi$
$500$	0	0
$501$	2020.50	0.180178
$502$	0	0
$503$	5816.42	0.515590	0.257795	−	0.966200i	$-0.417004\pi$
$503$	5816.42	0.515590	0.257795	+	0.966200i	$0.417004\pi$
$504$	0	0
$505$	−3281.16	−0.289128
$506$	0	0
$507$	−4232.87	−0.370785
$508$	0	0
$509$	−18051.0	−1.57190	−0.785950	−	0.618290i	$-0.787826\pi$
$509$	−18051.0	−1.57190	−0.785950	+	0.618290i	$0.787826\pi$
$510$	0	0
$511$	−1928.02	−0.166909
$512$	0	0
$513$	387.004	0.0333073
$514$	0	0
$515$	−10676.4	−0.913514
$516$	0	0
$517$	−3040.00	−0.258606
$518$	0	0
$519$	−15616.1	−1.32075
$520$	0	0
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5	−	1.00000i	$-0.500009\pi$
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5		1.00000i	$0.500009\pi$
$522$	0	0
$523$	16912.1	1.41399	0.706994	−	0.707219i	$-0.250051\pi$
$523$	16912.1	1.41399	0.706994	+	0.707219i	$0.250051\pi$
$524$	0	0
$525$	14089.4	1.17126
$526$	0	0
$527$	−14336.3	−1.18501
$528$	0	0
$529$	−12086.4	−0.993377
$530$	0	0
$531$	−8979.66	−0.733868
$532$	0	0
$533$	−14226.4	−1.15612
$534$	0	0
$535$	9990.67	0.807354
$536$	0	0
$537$	−22875.9	−1.83830
$538$	0	0
$539$	16339.5	1.30574
$540$	0	0
$541$	8893.75	0.706788	0.353394	−	0.935475i	$-0.385028\pi$
$541$	8893.75	0.706788	0.353394	+	0.935475i	$0.385028\pi$
$542$	0	0
$543$	15384.5	1.21586
$544$	0	0
$545$	11075.6	0.870505
$546$	0	0
$547$	16972.8	1.32670	0.663351	−	0.748308i	$-0.269134\pi$
$547$	16972.8	1.32670	0.663351	+	0.748308i	$0.269134\pi$
$548$	0	0
$549$	1142.61	0.0888260
$550$	0	0
$551$	2914.79	0.225362
$552$	0	0
$553$	15425.4	1.18618
$554$	0	0
$555$	−4963.18	−0.379595
$556$	0	0
$557$	17045.8	1.29669	0.648343	−	0.761348i	$-0.275462\pi$
$557$	17045.8	1.29669	0.648343	+	0.761348i	$0.275462\pi$
$558$	0	0
$559$	18722.1	1.41657
$560$	0	0
$561$	−16400.8	−1.23430
$562$	0	0
$563$	−6973.83	−0.522046	−0.261023	−	0.965333i	$-0.584060\pi$
$563$	−6973.83	−0.522046	−0.261023	+	0.965333i	$0.584060\pi$
$564$	0	0
$565$	6946.46	0.517238
$566$	0	0
$567$	28043.7	2.07712
$568$	0	0
$569$	−21644.0	−1.59466	−0.797332	−	0.603541i	$-0.793756\pi$
$569$	−21644.0	−1.59466	−0.797332	+	0.603541i	$0.793756\pi$
$570$	0	0
$571$	−13891.0	−1.01807	−0.509035	−	0.860746i	$-0.669998\pi$
$571$	−13891.0	−1.01807	−0.509035	+	0.860746i	$0.669998\pi$
$572$	0	0
$573$	31723.1	2.31283
$574$	0	0
$575$	503.076	0.0364865
$576$	0	0
$577$	−2386.14	−0.172160	−0.0860798	−	0.996288i	$-0.527434\pi$
$577$	−2386.14	−0.172160	−0.0860798	+	0.996288i	$0.527434\pi$
$578$	0	0
$579$	19457.9	1.39662
$580$	0	0
$581$	−37721.2	−2.69353
$582$	0	0
$583$	−4729.21	−0.335959
$584$	0	0
$585$	8035.41	0.567903
$586$	0	0
$587$	−826.851	−0.0581393	−0.0290697	−	0.999577i	$-0.509254\pi$
$587$	−826.851	−0.0581393	−0.0290697	+	0.999577i	$0.509254\pi$
$588$	0	0
$589$	−2174.23	−0.152101
$590$	0	0
$591$	−8969.23	−0.624272
$592$	0	0
$593$	9789.50	0.677920	0.338960	−	0.940801i	$-0.389925\pi$
$593$	9789.50	0.677920	0.338960	+	0.940801i	$0.389925\pi$
$594$	0	0
$595$	36570.5	2.51974
$596$	0	0
$597$	−29718.2	−2.03733
$598$	0	0
$599$	23555.7	1.60678	0.803389	−	0.595455i	$-0.203028\pi$
$599$	23555.7	1.60678	0.803389	+	0.595455i	$0.203028\pi$
$600$	0	0
$601$	12263.4	0.832334	0.416167	−	0.909288i	$-0.363373\pi$
$601$	12263.4	0.832334	0.416167	+	0.909288i	$0.363373\pi$
$602$	0	0
$603$	−18247.8	−1.23235
$604$	0	0
$605$	−8270.57	−0.555780
$606$	0	0
$607$	−464.109	−0.0310340	−0.0155170	−	0.999880i	$-0.504939\pi$
$607$	−464.109	−0.0310340	−0.0155170	+	0.999880i	$0.504939\pi$
$608$	0	0
$609$	38568.8	2.56632
$610$	0	0
$611$	−6654.02	−0.440577
$612$	0	0
$613$	−11526.0	−0.759427	−0.379714	−	0.925104i	$-0.623977\pi$
$613$	−11526.0	−0.759427	−0.379714	+	0.925104i	$0.623977\pi$
$614$	0	0
$615$	−21089.3	−1.38277
$616$	0	0
$617$	17632.5	1.15050	0.575249	−	0.817978i	$-0.304905\pi$
$617$	17632.5	1.15050	0.575249	+	0.817978i	$0.304905\pi$
$618$	0	0
$619$	16538.0	1.07386	0.536928	−	0.843628i	$-0.319585\pi$
$619$	16538.0	1.07386	0.536928	+	0.843628i	$0.319585\pi$
$620$	0	0
$621$	182.846	0.0118154
$622$	0	0
$623$	17629.5	1.13372
$624$	0	0
$625$	−5479.20	−0.350669
$626$	0	0
$627$	−2487.34	−0.158428
$628$	0	0
$629$	10469.3	0.663657
$630$	0	0
$631$	8881.84	0.560349	0.280175	−	0.959949i	$-0.409608\pi$
$631$	8881.84	0.560349	0.280175	+	0.959949i	$0.409608\pi$
$632$	0	0
$633$	−4530.38	−0.284465
$634$	0	0
$635$	3205.01	0.200295
$636$	0	0
$637$	35764.3	2.22454
$638$	0	0
$639$	−8437.42	−0.522346
$640$	0	0
$641$	22459.3	1.38391	0.691956	−	0.721939i	$-0.256749\pi$
$641$	22459.3	1.38391	0.691956	+	0.721939i	$0.256749\pi$
$642$	0	0
$643$	19685.1	1.20731	0.603657	−	0.797244i	$-0.293710\pi$
$643$	19685.1	1.20731	0.603657	+	0.797244i	$0.293710\pi$
$644$	0	0
$645$	27753.7	1.69427
$646$	0	0
$647$	11755.6	0.714312	0.357156	−	0.934045i	$-0.383746\pi$
$647$	11755.6	0.714312	0.357156	+	0.934045i	$0.383746\pi$
$648$	0	0
$649$	−6805.42	−0.411612
$650$	0	0
$651$	−28769.6	−1.73206
$652$	0	0
$653$	−18350.7	−1.09972	−0.549862	−	0.835256i	$-0.685320\pi$
$653$	−18350.7	−1.09972	−0.549862	+	0.835256i	$0.685320\pi$
$654$	0	0
$655$	14431.7	0.860908
$656$	0	0
$657$	1324.69	0.0786623
$658$	0	0
$659$	14617.0	0.864034	0.432017	−	0.901866i	$-0.357802\pi$
$659$	14617.0	0.864034	0.432017	+	0.901866i	$0.357802\pi$
$660$	0	0
$661$	2932.29	0.172546	0.0862729	−	0.996272i	$-0.472504\pi$
$661$	2932.29	0.172546	0.0862729	+	0.996272i	$0.472504\pi$
$662$	0	0
$663$	−35898.5	−2.10284
$664$	0	0
$665$	5546.24	0.323420
$666$	0	0
$667$	1377.14	0.0799448
$668$	0	0
$669$	34318.0	1.98328
$670$	0	0
$671$	865.951	0.0498207
$672$	0	0
$673$	21271.4	1.21835	0.609177	−	0.793034i	$-0.291500\pi$
$673$	21271.4	1.21835	0.609177	+	0.793034i	$0.291500\pi$
$674$	0	0
$675$	1141.48	0.0650900
$676$	0	0
$677$	32878.2	1.86649	0.933245	−	0.359241i	$-0.116964\pi$
$677$	32878.2	1.86649	0.933245	+	0.359241i	$0.116964\pi$
$678$	0	0
$679$	50533.1	2.85609
$680$	0	0
$681$	4589.98	0.258280
$682$	0	0
$683$	−7771.01	−0.435358	−0.217679	−	0.976020i	$-0.569849\pi$
$683$	−7771.01	−0.435358	−0.217679	+	0.976020i	$0.569849\pi$
$684$	0	0
$685$	346.783	0.0193429
$686$	0	0
$687$	8806.65	0.489075
$688$	0	0
$689$	−10351.4	−0.572361
$690$	0	0
$691$	3655.65	0.201255	0.100628	−	0.994924i	$-0.467915\pi$
$691$	3655.65	0.201255	0.100628	+	0.994924i	$0.467915\pi$
$692$	0	0
$693$	−15540.1	−0.851833
$694$	0	0
$695$	12761.7	0.696518
$696$	0	0
$697$	44485.7	2.41753
$698$	0	0
$699$	1621.38	0.0877342
$700$	0	0
$701$	28207.0	1.51977	0.759887	−	0.650055i	$-0.225254\pi$
$701$	28207.0	1.51977	0.759887	+	0.650055i	$0.225254\pi$
$702$	0	0
$703$	1587.77	0.0851834
$704$	0	0
$705$	−9863.94	−0.526947
$706$	0	0
$707$	13889.4	0.738848
$708$	0	0
$709$	−36417.7	−1.92905	−0.964524	−	0.263997i	$-0.914959\pi$
$709$	−36417.7	−1.92905	−0.964524	+	0.263997i	$0.914959\pi$
$710$	0	0
$711$	−10598.4	−0.559031
$712$	0	0
$713$	−1027.25	−0.0539563
$714$	0	0
$715$	6089.80	0.318525
$716$	0	0
$717$	−17402.9	−0.906448
$718$	0	0
$719$	−27227.6	−1.41227	−0.706133	−	0.708079i	$-0.749562\pi$
$719$	−27227.6	−1.41227	−0.706133	+	0.708079i	$0.749562\pi$
$720$	0	0
$721$	45194.2	2.33442
$722$	0	0
$723$	−6446.41	−0.331597
$724$	0	0
$725$	8597.30	0.440408
$726$	0	0
$727$	−27647.2	−1.41043	−0.705213	−	0.708996i	$-0.749149\pi$
$727$	−27647.2	−1.41043	−0.705213	+	0.708996i	$0.749149\pi$
$728$	0	0
$729$	−15334.8	−0.779090
$730$	0	0
$731$	−58543.7	−2.96213
$732$	0	0
$733$	32139.8	1.61952	0.809761	−	0.586760i	$-0.199597\pi$
$733$	32139.8	1.61952	0.809761	+	0.586760i	$0.199597\pi$
$734$	0	0
$735$	53017.1	2.66063
$736$	0	0
$737$	−13829.5	−0.691202
$738$	0	0
$739$	16224.9	0.807635	0.403818	−	0.914839i	$-0.367683\pi$
$739$	16224.9	0.807635	0.403818	+	0.914839i	$0.367683\pi$
$740$	0	0
$741$	−5444.33	−0.269909
$742$	0	0
$743$	−22463.2	−1.10915	−0.554573	−	0.832135i	$-0.687118\pi$
$743$	−22463.2	−1.10915	−0.554573	+	0.832135i	$0.687118\pi$
$744$	0	0
$745$	−13757.1	−0.676538
$746$	0	0
$747$	25917.3	1.26943
$748$	0	0
$749$	−42291.3	−2.06314
$750$	0	0
$751$	−9310.24	−0.452377	−0.226189	−	0.974083i	$-0.572627\pi$
$751$	−9310.24	−0.452377	−0.226189	+	0.974083i	$0.572627\pi$
$752$	0	0
$753$	−40594.6	−1.96461
$754$	0	0
$755$	5937.33	0.286201
$756$	0	0
$757$	−28707.1	−1.37830	−0.689152	−	0.724617i	$-0.742017\pi$
$757$	−28707.1	−1.37830	−0.689152	+	0.724617i	$0.742017\pi$
$758$	0	0
$759$	−1175.18	−0.0562009
$760$	0	0
$761$	−20820.3	−0.991768	−0.495884	−	0.868389i	$-0.665156\pi$
$761$	−20820.3	−0.991768	−0.495884	+	0.868389i	$0.665156\pi$
$762$	0	0
$763$	−46883.7	−2.22452
$764$	0	0
$765$	−25126.6	−1.18752
$766$	0	0
$767$	−14895.8	−0.701249
$768$	0	0
$769$	28026.2	1.31424	0.657120	−	0.753786i	$-0.271775\pi$
$769$	28026.2	1.31424	0.657120	+	0.753786i	$0.271775\pi$
$770$	0	0
$771$	14560.8	0.680146
$772$	0	0
$773$	16405.1	0.763326	0.381663	−	0.924302i	$-0.375352\pi$
$773$	16405.1	0.763326	0.381663	+	0.924302i	$0.375352\pi$
$774$	0	0
$775$	−6412.98	−0.297240
$776$	0	0
$777$	21009.6	0.970030
$778$	0	0
$779$	6746.66	0.310301
$780$	0	0
$781$	−6394.47	−0.292973
$782$	0	0
$783$	3124.75	0.142617
$784$	0	0
$785$	−13986.1	−0.635903
$786$	0	0
$787$	−28468.8	−1.28946	−0.644730	−	0.764411i	$-0.723030\pi$
$787$	−28468.8	−1.28946	−0.644730	+	0.764411i	$0.723030\pi$
$788$	0	0
$789$	−13762.0	−0.620965
$790$	0	0
$791$	−29404.9	−1.32177
$792$	0	0
$793$	1895.41	0.0848777
$794$	0	0
$795$	−15345.0	−0.684566
$796$	0	0
$797$	−7256.68	−0.322515	−0.161258	−	0.986912i	$-0.551555\pi$
$797$	−7256.68	−0.322515	−0.161258	+	0.986912i	$0.551555\pi$
$798$	0	0
$799$	20807.0	0.921275
$800$	0	0
$801$	−12112.8	−0.534311
$802$	0	0
$803$	1003.94	0.0441201
$804$	0	0
$805$	2620.42	0.114730
$806$	0	0
$807$	5931.06	0.258715
$808$	0	0
$809$	10647.1	0.462711	0.231356	−	0.972869i	$-0.425684\pi$
$809$	10647.1	0.462711	0.231356	+	0.972869i	$0.425684\pi$
$810$	0	0
$811$	−27076.9	−1.17238	−0.586190	−	0.810174i	$-0.699373\pi$
$811$	−27076.9	−1.17238	−0.586190	+	0.810174i	$0.699373\pi$
$812$	0	0
$813$	8747.91	0.377371
$814$	0	0
$815$	5830.96	0.250613
$816$	0	0
$817$	−8878.69	−0.380203
$818$	0	0
$819$	−34014.5	−1.45124
$820$	0	0
$821$	−1880.29	−0.0799301	−0.0399650	−	0.999201i	$-0.512725\pi$
$821$	−1880.29	−0.0799301	−0.0399650	+	0.999201i	$0.512725\pi$
$822$	0	0
$823$	−36904.5	−1.56307	−0.781537	−	0.623859i	$-0.785564\pi$
$823$	−36904.5	−1.56307	−0.781537	+	0.623859i	$0.785564\pi$
$824$	0	0
$825$	−7336.51	−0.309605
$826$	0	0
$827$	−23091.4	−0.970939	−0.485470	−	0.874254i	$-0.661351\pi$
$827$	−23091.4	−0.970939	−0.485470	+	0.874254i	$0.661351\pi$
$828$	0	0
$829$	15527.1	0.650515	0.325258	−	0.945625i	$-0.394549\pi$
$829$	15527.1	0.650515	0.325258	+	0.945625i	$0.394549\pi$
$830$	0	0
$831$	−11606.4	−0.484504
$832$	0	0
$833$	−111834.	−4.65166
$834$	0	0
$835$	2345.97	0.0972282
$836$	0	0
$837$	−2330.84	−0.0962553
$838$	0	0
$839$	−25695.1	−1.05732	−0.528661	−	0.848833i	$-0.677306\pi$
$839$	−25695.1	−1.05732	−0.528661	+	0.848833i	$0.677306\pi$
$840$	0	0
$841$	−854.386	−0.0350316
$842$	0	0
$843$	25883.0	1.05748
$844$	0	0
$845$	−4914.70	−0.200084
$846$	0	0
$847$	35010.0	1.42026
$848$	0	0
$849$	15036.7	0.607843
$850$	0	0
$851$	750.170	0.0302180
$852$	0	0
$853$	4534.78	0.182026	0.0910129	−	0.995850i	$-0.470990\pi$
$853$	4534.78	0.182026	0.0910129	+	0.995850i	$0.470990\pi$
$854$	0	0
$855$	−3810.68	−0.152424
$856$	0	0
$857$	48754.1	1.94330	0.971651	−	0.236422i	$-0.0759747\pi$
$857$	48754.1	1.94330	0.971651	+	0.236422i	$0.0759747\pi$
$858$	0	0
$859$	−11730.5	−0.465937	−0.232968	−	0.972484i	$-0.574844\pi$
$859$	−11730.5	−0.465937	−0.232968	+	0.972484i	$0.574844\pi$
$860$	0	0
$861$	89272.6	3.53357
$862$	0	0
$863$	−9907.21	−0.390783	−0.195391	−	0.980725i	$-0.562598\pi$
$863$	−9907.21	−0.390783	−0.195391	+	0.980725i	$0.562598\pi$
$864$	0	0
$865$	−18131.5	−0.712706
$866$	0	0
$867$	77116.0	3.02076
$868$	0	0
$869$	−8032.22	−0.313549
$870$	0	0
$871$	−30270.3	−1.17758
$872$	0	0
$873$	−34720.0	−1.34604
$874$	0	0
$875$	52847.3	2.04179
$876$	0	0
$877$	−23741.6	−0.914136	−0.457068	−	0.889432i	$-0.651100\pi$
$877$	−23741.6	−0.914136	−0.457068	+	0.889432i	$0.651100\pi$
$878$	0	0
$879$	38506.7	1.47758
$880$	0	0
$881$	−24381.3	−0.932378	−0.466189	−	0.884685i	$-0.654373\pi$
$881$	−24381.3	−0.932378	−0.466189	+	0.884685i	$0.654373\pi$
$882$	0	0
$883$	343.893	0.0131064	0.00655319	−	0.999979i	$-0.497914\pi$
$883$	343.893	0.0131064	0.00655319	+	0.999979i	$0.497914\pi$
$884$	0	0
$885$	−22081.7	−0.838719
$886$	0	0
$887$	−33805.3	−1.27967	−0.639837	−	0.768511i	$-0.720998\pi$
$887$	−33805.3	−1.27967	−0.639837	+	0.768511i	$0.720998\pi$
$888$	0	0
$889$	−13567.1	−0.511839
$890$	0	0
$891$	−14602.7	−0.549057
$892$	0	0
$893$	3155.57	0.118250
$894$	0	0
$895$	−26560.8	−0.991989
$896$	0	0
$897$	−2572.27	−0.0957475
$898$	0	0
$899$	−17555.2	−0.651277
$900$	0	0
$901$	32368.7	1.19684
$902$	0	0
$903$	−117484.	−4.32958
$904$	0	0
$905$	17862.6	0.656103
$906$	0	0
$907$	−26456.3	−0.968540	−0.484270	−	0.874919i	$-0.660915\pi$
$907$	−26456.3	−0.968540	−0.484270	+	0.874919i	$0.660915\pi$
$908$	0	0
$909$	−9543.06	−0.348211
$910$	0	0
$911$	310.509	0.0112927	0.00564633	−	0.999984i	$-0.498203\pi$
$911$	310.509	0.0112927	0.00564633	+	0.999984i	$0.498203\pi$
$912$	0	0
$913$	19641.9	0.711997
$914$	0	0
$915$	2809.77	0.101517
$916$	0	0
$917$	−61090.7	−2.19999
$918$	0	0
$919$	17316.8	0.621575	0.310787	−	0.950479i	$-0.399407\pi$
$919$	17316.8	0.621575	0.310787	+	0.950479i	$0.399407\pi$
$920$	0	0
$921$	26411.5	0.944938
$922$	0	0
$923$	−13996.3	−0.499128
$924$	0	0
$925$	4683.20	0.166468
$926$	0	0
$927$	−31051.7	−1.10019
$928$	0	0
$929$	−36473.5	−1.28811	−0.644056	−	0.764978i	$-0.722750\pi$
$929$	−36473.5	−1.28811	−0.644056	+	0.764978i	$0.722750\pi$
$930$	0	0
$931$	−16960.7	−0.597061
$932$	0	0
$933$	31667.0	1.11118
$934$	0	0
$935$	−19042.7	−0.666057
$936$	0	0
$937$	4777.88	0.166581	0.0832905	−	0.996525i	$-0.473457\pi$
$937$	4777.88	0.166581	0.0832905	+	0.996525i	$0.473457\pi$
$938$	0	0
$939$	−52613.9	−1.82853
$940$	0	0
$941$	−31366.5	−1.08663	−0.543315	−	0.839529i	$-0.682831\pi$
$941$	−31366.5	−1.08663	−0.543315	+	0.839529i	$0.682831\pi$
$942$	0	0
$943$	3187.58	0.110076
$944$	0	0
$945$	5945.75	0.204672
$946$	0	0
$947$	26496.1	0.909195	0.454597	−	0.890697i	$-0.349783\pi$
$947$	26496.1	0.909195	0.454597	+	0.890697i	$0.349783\pi$
$948$	0	0
$949$	2197.45	0.0751658
$950$	0	0
$951$	−11534.6	−0.393307
$952$	0	0
$953$	−11928.3	−0.405451	−0.202725	−	0.979236i	$-0.564980\pi$
$953$	−11928.3	−0.405451	−0.202725	+	0.979236i	$0.564980\pi$
$954$	0	0
$955$	36833.1	1.24806
$956$	0	0
$957$	−20083.3	−0.678370
$958$	0	0
$959$	−1467.96	−0.0494296
$960$	0	0
$961$	−16696.1	−0.560440
$962$	0	0
$963$	29057.3	0.972333
$964$	0	0
$965$	22592.2	0.753647
$966$	0	0
$967$	58320.0	1.93945	0.969723	−	0.244209i	$-0.0785282\pi$
$967$	58320.0	1.93945	0.969723	+	0.244209i	$0.0785282\pi$
$968$	0	0
$969$	17024.3	0.564397
$970$	0	0
$971$	−27526.8	−0.909760	−0.454880	−	0.890553i	$-0.650318\pi$
$971$	−27526.8	−0.909760	−0.454880	+	0.890553i	$0.650318\pi$
$972$	0	0
$973$	−54021.5	−1.77991
$974$	0	0
$975$	−16058.3	−0.527464
$976$	0	0
$977$	−36194.1	−1.18521	−0.592605	−	0.805493i	$-0.701901\pi$
$977$	−36194.1	−1.18521	−0.592605	+	0.805493i	$0.701901\pi$
$978$	0	0
$979$	−9179.90	−0.299684
$980$	0	0
$981$	32212.6	1.04839
$982$	0	0
$983$	58081.3	1.88454	0.942271	−	0.334850i	$-0.108686\pi$
$983$	58081.3	1.88454	0.942271	+	0.334850i	$0.108686\pi$
$984$	0	0
$985$	−10414.0	−0.336871
$986$	0	0
$987$	41754.8	1.34658
$988$	0	0
$989$	−4194.89	−0.134873
$990$	0	0
$991$	−54921.4	−1.76048	−0.880240	−	0.474529i	$-0.842618\pi$
$991$	−54921.4	−1.76048	−0.880240	+	0.474529i	$0.842618\pi$
$992$	0	0
$993$	−16183.6	−0.517190
$994$	0	0
$995$	−34505.2	−1.09939
$996$	0	0
$997$	33999.5	1.08001	0.540007	−	0.841661i	$-0.318422\pi$
$997$	33999.5	1.08001	0.540007	+	0.841661i	$0.318422\pi$
$998$	0	0
$999$	1702.14	0.0539073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.l.1.2		2
4.3	odd	2		1216.4.a.j.1.1		2
8.3	odd	2		38.4.a.b.1.2	✓	2
8.5	even	2		304.4.a.d.1.1		2
24.11	even	2		342.4.a.k.1.2		2
40.3	even	4		950.4.b.g.799.4		4
40.19	odd	2		950.4.a.h.1.1		2
40.27	even	4		950.4.b.g.799.1		4
56.27	even	2		1862.4.a.b.1.1		2
152.75	even	2		722.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.2	✓	2	8.3	odd	2
304.4.a.d.1.1		2	8.5	even	2
342.4.a.k.1.2		2	24.11	even	2
722.4.a.i.1.1		2	152.75	even	2
950.4.a.h.1.1		2	40.19	odd	2
950.4.b.g.799.1		4	40.27	even	4
950.4.b.g.799.4		4	40.3	even	4
1216.4.a.j.1.1		2	4.3	odd	2
1216.4.a.l.1.2		2	1.1	even	1	trivial
1862.4.a.b.1.1		2	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+7.15207 q^{3} +8.30413 q^{5} -35.1521 q^{7} +24.1521 q^{9} +18.3041 q^{11} +40.0645 q^{13} +59.3917 q^{15} -125.281 q^{17} -19.0000 q^{19} -251.410 q^{21} -8.97688 q^{23} -56.0413 q^{25} -20.3686 q^{27} -153.410 q^{29} +114.433 q^{31} +130.912 q^{33} -291.908 q^{35} -83.5669 q^{37} +286.544 q^{39} -355.088 q^{41} +467.299 q^{43} +200.562 q^{45} -166.083 q^{47} +892.668 q^{49} -896.018 q^{51} -258.369 q^{53} +152.000 q^{55} -135.889 q^{57} -371.797 q^{59} +47.3090 q^{61} -848.995 q^{63} +332.701 q^{65} -755.539 q^{67} -64.2032 q^{69} -349.345 q^{71} +54.8479 q^{73} -400.811 q^{75} -643.428 q^{77} -438.820 q^{79} -797.783 q^{81} +1073.09 q^{83} -1040.35 q^{85} -1097.20 q^{87} -501.521 q^{89} -1408.35 q^{91} +818.433 q^{93} -157.779 q^{95} -1437.56 q^{97} +442.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35}+ \cdots + 352 q^{99}+O(q^{100}) \) 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9 + 10 * q^11 - 13 * q^13 + 172 * q^15 - 51 * q^17 - 38 * q^19 - 117 * q^21 + 155 * q^23 + 154 * q^25 + 79 * q^27 + 79 * q^29 + 16 * q^31 + 182 * q^33 + 108 * q^35 - 380 * q^37 + 613 * q^39 - 790 * q^41 + 296 * q^43 + 2 * q^45 + 200 * q^47 + 1027 * q^49 - 1353 * q^51 - 397 * q^53 + 304 * q^55 - 19 * q^57 + 201 * q^59 + 680 * q^61 - 1086 * q^63 + 1304 * q^65 - 939 * q^67 - 1073 * q^69 - 406 * q^71 + 123 * q^73 - 1693 * q^75 - 462 * q^77 - 106 * q^79 - 1702 * q^81 + 2226 * q^83 - 2400 * q^85 - 2527 * q^87 - 870 * q^89 - 249 * q^91 + 1424 * q^93 + 190 * q^95 - 1864 * q^97 + 352 * q^99

Introduction

L-functions

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Database

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