LMFDB - Embedded newform 1216.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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			1.1
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)

$f(q)$	$=$	$q-7.15207 q^{3} +8.30413 q^{5} +35.1521 q^{7} +24.1521 q^{9} +O(q^{10})$	$q-7.15207 q^{3} +8.30413 q^{5} +35.1521 q^{7} +24.1521 q^{9} -18.3041 q^{11} +40.0645 q^{13} -59.3917 q^{15} -125.281 q^{17} +19.0000 q^{19} -251.410 q^{21} +8.97688 q^{23} -56.0413 q^{25} +20.3686 q^{27} -153.410 q^{29} -114.433 q^{31} +130.912 q^{33} +291.908 q^{35} -83.5669 q^{37} -286.544 q^{39} -355.088 q^{41} -467.299 q^{43} +200.562 q^{45} +166.083 q^{47} +892.668 q^{49} +896.018 q^{51} -258.369 q^{53} -152.000 q^{55} -135.889 q^{57} +371.797 q^{59} +47.3090 q^{61} +848.995 q^{63} +332.701 q^{65} +755.539 q^{67} -64.2032 q^{69} +349.345 q^{71} +54.8479 q^{73} +400.811 q^{75} -643.428 q^{77} +438.820 q^{79} -797.783 q^{81} -1073.09 q^{83} -1040.35 q^{85} +1097.20 q^{87} -501.521 q^{89} +1408.35 q^{91} +818.433 q^{93} +157.779 q^{95} -1437.56 q^{97} -442.083 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 10 q^{5} + 57 q^{7} + 35 q^{9}+O(q^{10}) $ 2 * q - q^3 - 10 * q^5 + 57 * q^7 + 35 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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	$-0.741602\pi$
$3$	−7.15207	−1.37642	−0.688208	+	0.725513i	$0.741602\pi$
$4$	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	$-0.102080\pi$
$7$	35.1521	1.89803	0.949017	+	0.315226i	$0.102080\pi$
$8$	0	0
$9$	24.1521	0.894521
$10$	0	0
$11$	−18.3041	−0.501719	−0.250859	−	0.968024i	$-0.580713\pi$
$11$	−18.3041	−0.501719	−0.250859	+	0.968024i	$0.580713\pi$
$12$	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	$-0.487044\pi$
$23$	8.97688	0.0813830	0.0406915	+	0.999172i	$0.487044\pi$
$24$	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	$-0.607554\pi$
$31$	−114.433	−0.662993	−0.331497	+	0.943456i	$0.607554\pi$
$32$	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	$-0.810881\pi$
$43$	−467.299	−1.65727	−0.828633	+	0.559792i	$0.810881\pi$
$44$	0	0
$45$	200.562	0.664400
$46$	0	0
$47$	166.083	0.515439	0.257720	−	0.966220i	$-0.417029\pi$
$47$	166.083	0.515439	0.257720	+	0.966220i	$0.417029\pi$
$48$	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	$-0.365458\pi$
$59$	371.797	0.820404	0.410202	+	0.911995i	$0.365458\pi$
$60$	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	$-0.258123\pi$
$67$	755.539	1.37767	0.688834	+	0.724919i	$0.258123\pi$
$68$	0	0
$69$	−64.2032	−0.112017
$70$	0	0
$71$	349.345	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$71$	349.345	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$72$	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	$-0.398842\pi$
$79$	438.820	0.624951	0.312475	+	0.949926i	$0.398842\pi$
$80$	0	0
$81$	−797.783	−1.09435
$82$	0	0
$83$	−1073.09	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$83$	−1073.09	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$84$	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	$-0.289173\pi$
$103$	1285.68	1.22992	0.614958	+	0.788559i	$0.289173\pi$
$104$	0	0
$105$	−2087.74	−1.94041
$106$	0	0
$107$	−1203.10	−1.08699	−0.543494	−	0.839413i	$-0.682899\pi$
$107$	−1203.10	−1.08699	−0.543494	+	0.839413i	$0.682899\pi$
$108$	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	$-0.543050\pi$
$127$	−385.954	−0.269668	−0.134834	+	0.990868i	$0.543050\pi$
$128$	0	0
$129$	3342.16	2.28109
$130$	0	0
$131$	−1737.90	−1.15909	−0.579545	−	0.814940i	$-0.696770\pi$
$131$	−1737.90	−1.15909	−0.579545	+	0.814940i	$0.696770\pi$
$132$	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	$-0.655343\pi$
$139$	−1536.79	−0.937763	−0.468882	+	0.883261i	$0.655343\pi$
$140$	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	$-0.561713\pi$
$151$	−714.985	−0.385329	−0.192664	+	0.981265i	$0.561713\pi$
$152$	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	$-0.553959\pi$
$163$	−702.175	−0.337415	−0.168707	+	0.985666i	$0.553959\pi$
$164$	0	0
$165$	1087.11	0.512920
$166$	0	0
$167$	−282.506	−0.130904	−0.0654520	−	0.997856i	$-0.520849\pi$
$167$	−282.506	−0.130904	−0.0654520	+	0.997856i	$0.520849\pi$
$168$	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	$-0.267242\pi$
$179$	3198.51	1.33557	0.667786	+	0.744353i	$0.267242\pi$
$180$	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	$-0.817542\pi$
$191$	−4435.52	−1.68033	−0.840165	+	0.542331i	$0.817542\pi$
$192$	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	$-0.234786\pi$
$199$	4155.19	1.48017	0.740084	+	0.672515i	$0.234786\pi$
$200$	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	$-0.467048\pi$
$211$	633.437	0.206671	0.103335	+	0.994647i	$0.467048\pi$
$212$	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	$-0.756064\pi$
$223$	−4798.34	−1.44090	−0.720449	+	0.693507i	$0.756064\pi$
$224$	0	0
$225$	−1353.51	−0.401041
$226$	0	0
$227$	−641.770	−0.187647	−0.0938233	−	0.995589i	$-0.529909\pi$
$227$	−641.770	−0.187647	−0.0938233	+	0.995589i	$0.529909\pi$
$228$	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	$-0.393195\pi$
$239$	2433.27	0.658557	0.329278	+	0.944233i	$0.393195\pi$
$240$	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	$-0.247033\pi$
$251$	5675.92	1.42734	0.713668	+	0.700484i	$0.247033\pi$
$252$	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	$-0.427574\pi$
$263$	1924.20	0.451147	0.225573	+	0.974226i	$0.427574\pi$
$264$	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	$-0.543773\pi$
$271$	−1223.13	−0.274169	−0.137085	+	0.990559i	$0.543773\pi$
$272$	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	$-0.570869\pi$
$283$	−2102.43	−0.441613	−0.220807	+	0.975318i	$0.570869\pi$
$284$	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	$-0.611531\pi$
$307$	−3692.85	−0.686521	−0.343260	+	0.939240i	$0.611531\pi$
$308$	0	0
$309$	−9195.24	−1.69288
$310$	0	0
$311$	−4427.67	−0.807299	−0.403649	−	0.914914i	$-0.632258\pi$
$311$	−4427.67	−0.807299	−0.403649	+	0.914914i	$0.632258\pi$
$312$	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	$-0.439840\pi$
$331$	2262.78	0.375752	0.187876	+	0.982193i	$0.439840\pi$
$332$	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	$-0.454433\pi$
$347$	1844.33	0.285328	0.142664	+	0.989771i	$0.454433\pi$
$348$	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	$-0.558746\pi$
$359$	−2496.50	−0.367020	−0.183510	+	0.983018i	$0.558746\pi$
$360$	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	$-0.713744\pi$
$367$	−8748.42	−1.24432	−0.622158	+	0.782892i	$0.713744\pi$
$368$	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	$-0.754499\pi$
$379$	−10581.0	−1.43406	−0.717029	+	0.697043i	$0.754499\pi$
$380$	0	0
$381$	2760.37	0.371176
$382$	0	0
$383$	−2932.54	−0.391242	−0.195621	−	0.980680i	$-0.562672\pi$
$383$	−2932.54	−0.391242	−0.195621	+	0.980680i	$0.562672\pi$
$384$	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	$-0.465793\pi$
$419$	1839.83	0.214514	0.107257	+	0.994231i	$0.465793\pi$
$420$	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	$-0.449622\pi$
$431$	2820.47	0.315214	0.157607	+	0.987502i	$0.449622\pi$
$432$	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	$-0.221174\pi$
$439$	14131.1	1.53631	0.768157	+	0.640261i	$0.221174\pi$
$440$	0	0
$441$	21559.8	2.32802
$442$	0	0
$443$	−1659.59	−0.177990	−0.0889951	−	0.996032i	$-0.528366\pi$
$443$	−1659.59	−0.177990	−0.0889951	+	0.996032i	$0.528366\pi$
$444$	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	$-0.585635\pi$
$463$	−5296.03	−0.531593	−0.265796	+	0.964029i	$0.585635\pi$
$464$	0	0
$465$	6796.38	0.677795
$466$	0	0
$467$	−3470.94	−0.343931	−0.171966	−	0.985103i	$-0.555012\pi$
$467$	−3470.94	−0.343931	−0.171966	+	0.985103i	$0.555012\pi$
$468$	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	$-0.783342\pi$
$479$	−16294.7	−1.55433	−0.777163	+	0.629300i	$0.783342\pi$
$480$	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	$-0.533319\pi$
$487$	−2245.82	−0.208969	−0.104485	+	0.994527i	$0.533319\pi$
$488$	0	0
$489$	5022.00	0.464423
$490$	0	0
$491$	7578.64	0.696577	0.348288	−	0.937387i	$-0.386763\pi$
$491$	7578.64	0.696577	0.348288	+	0.937387i	$0.386763\pi$
$492$	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	$-0.342953\pi$
$499$	10558.4	0.947209	0.473604	+	0.880738i	$0.342953\pi$
$500$	0	0
$501$	2020.50	0.180178
$502$	0	0
$503$	−5816.42	−0.515590	−0.257795	−	0.966200i	$-0.582996\pi$
$503$	−5816.42	−0.515590	−0.257795	+	0.966200i	$0.582996\pi$
$504$	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	$-0.749949\pi$
$523$	−16912.1	−1.41399	−0.706994	+	0.707219i	$0.749949\pi$
$524$	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	$-0.730866\pi$
$547$	−16972.8	−1.32670	−0.663351	+	0.748308i	$0.730866\pi$
$548$	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	$-0.415940\pi$
$563$	6973.83	0.522046	0.261023	+	0.965333i	$0.415940\pi$
$564$	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	$-0.330002\pi$
$571$	13891.0	1.01807	0.509035	+	0.860746i	$0.330002\pi$
$572$	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	$-0.490746\pi$
$587$	826.851	0.0581393	0.0290697	+	0.999577i	$0.490746\pi$
$588$	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	$-0.796972\pi$
$599$	−23555.7	−1.60678	−0.803389	+	0.595455i	$0.796972\pi$
$600$	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	$-0.495061\pi$
$607$	464.109	0.0310340	0.0155170	+	0.999880i	$0.495061\pi$
$608$	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	$-0.680415\pi$
$619$	−16538.0	−1.07386	−0.536928	+	0.843628i	$0.680415\pi$
$620$	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	$-0.590392\pi$
$631$	−8881.84	−0.560349	−0.280175	+	0.959949i	$0.590392\pi$
$632$	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	$-0.706290\pi$
$643$	−19685.1	−1.20731	−0.603657	+	0.797244i	$0.706290\pi$
$644$	0	0
$645$	27753.7	1.69427
$646$	0	0
$647$	−11755.6	−0.714312	−0.357156	−	0.934045i	$-0.616254\pi$
$647$	−11755.6	−0.714312	−0.357156	+	0.934045i	$0.616254\pi$
$648$	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	$-0.642198\pi$
$659$	−14617.0	−0.864034	−0.432017	+	0.901866i	$0.642198\pi$
$660$	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	$-0.430151\pi$
$683$	7771.01	0.435358	0.217679	+	0.976020i	$0.430151\pi$
$684$	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	$-0.532085\pi$
$691$	−3655.65	−0.201255	−0.100628	+	0.994924i	$0.532085\pi$
$692$	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	$-0.250438\pi$
$719$	27227.6	1.41227	0.706133	+	0.708079i	$0.250438\pi$
$720$	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	$-0.250851\pi$
$727$	27647.2	1.41043	0.705213	+	0.708996i	$0.250851\pi$
$728$	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	$-0.632317\pi$
$739$	−16224.9	−0.807635	−0.403818	+	0.914839i	$0.632317\pi$
$740$	0	0
$741$	−5444.33	−0.269909
$742$	0	0
$743$	22463.2	1.10915	0.554573	−	0.832135i	$-0.312882\pi$
$743$	22463.2	1.10915	0.554573	+	0.832135i	$0.312882\pi$
$744$	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	$-0.427373\pi$
$751$	9310.24	0.452377	0.226189	+	0.974083i	$0.427373\pi$
$752$	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	$-0.276970\pi$
$787$	28468.8	1.28946	0.644730	+	0.764411i	$0.276970\pi$
$788$	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	$-0.300627\pi$
$811$	27076.9	1.17238	0.586190	+	0.810174i	$0.300627\pi$
$812$	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	$-0.214436\pi$
$823$	36904.5	1.56307	0.781537	+	0.623859i	$0.214436\pi$
$824$	0	0
$825$	−7336.51	−0.309605
$826$	0	0
$827$	23091.4	0.970939	0.485470	−	0.874254i	$-0.338649\pi$
$827$	23091.4	0.970939	0.485470	+	0.874254i	$0.338649\pi$
$828$	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	$-0.322694\pi$
$839$	25695.1	1.05732	0.528661	+	0.848833i	$0.322694\pi$
$840$	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	$-0.425156\pi$
$859$	11730.5	0.465937	0.232968	+	0.972484i	$0.425156\pi$
$860$	0	0
$861$	89272.6	3.53357
$862$	0	0
$863$	9907.21	0.390783	0.195391	−	0.980725i	$-0.437402\pi$
$863$	9907.21	0.390783	0.195391	+	0.980725i	$0.437402\pi$
$864$	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	$-0.502086\pi$
$883$	−343.893	−0.0131064	−0.00655319	+	0.999979i	$0.502086\pi$
$884$	0	0
$885$	−22081.7	−0.838719
$886$	0	0
$887$	33805.3	1.27967	0.639837	−	0.768511i	$-0.279002\pi$
$887$	33805.3	1.27967	0.639837	+	0.768511i	$0.279002\pi$
$888$	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	$-0.339085\pi$
$907$	26456.3	0.968540	0.484270	+	0.874919i	$0.339085\pi$
$908$	0	0
$909$	−9543.06	−0.348211
$910$	0	0
$911$	−310.509	−0.0112927	−0.00564633	−	0.999984i	$-0.501797\pi$
$911$	−310.509	−0.0112927	−0.00564633	+	0.999984i	$0.501797\pi$
$912$	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	$-0.600593\pi$
$919$	−17316.8	−0.621575	−0.310787	+	0.950479i	$0.600593\pi$
$920$	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	$-0.650217\pi$
$947$	−26496.1	−0.909195	−0.454597	+	0.890697i	$0.650217\pi$
$948$	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	$-0.921472\pi$
$967$	−58320.0	−1.93945	−0.969723	+	0.244209i	$0.921472\pi$
$968$	0	0
$969$	17024.3	0.564397
$970$	0	0
$971$	27526.8	0.909760	0.454880	−	0.890553i	$-0.349682\pi$
$971$	27526.8	0.909760	0.454880	+	0.890553i	$0.349682\pi$
$972$	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	$-0.891314\pi$
$983$	−58081.3	−1.88454	−0.942271	+	0.334850i	$0.891314\pi$
$984$	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	$-0.157382\pi$
$991$	54921.4	1.76048	0.880240	+	0.474529i	$0.157382\pi$
$992$	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.j.1.1		2
4.3	odd	2		1216.4.a.l.1.2		2
8.3	odd	2		304.4.a.d.1.1		2
8.5	even	2		38.4.a.b.1.2	✓	2
24.5	odd	2		342.4.a.k.1.2		2
40.13	odd	4		950.4.b.g.799.4		4
40.29	even	2		950.4.a.h.1.1		2
40.37	odd	4		950.4.b.g.799.1		4
56.13	odd	2		1862.4.a.b.1.1		2
152.37	odd	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	8.5	even	2
304.4.a.d.1.1		2	8.3	odd	2
342.4.a.k.1.2		2	24.5	odd	2
722.4.a.i.1.1		2	152.37	odd	2
950.4.a.h.1.1		2	40.29	even	2
950.4.b.g.799.1		4	40.37	odd	4
950.4.b.g.799.4		4	40.13	odd	4
1216.4.a.j.1.1		2	1.1	even	1	trivial
1216.4.a.l.1.2		2	4.3	odd	2
1862.4.a.b.1.1		2	56.13	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 $\F_{q}$

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

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

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