LMFDB - Embedded newform 392.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,4,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	392.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:	$23.1287487223$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{113})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 59x^{2} + 60x + 674 $ x^4 - 2x^3 - 59x^2 + 60*x + 674
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			$-3.40086$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.63797 q^{3} +5.91838 q^{5} +65.8904 q^{9} +O(q^{10})$	$q-9.63797 q^{3} +5.91838 q^{5} +65.8904 q^{9} -18.3699 q^{11} -90.9844 q^{13} -57.0412 q^{15} +91.9239 q^{17} -124.247 q^{19} +65.0412 q^{23} -89.9728 q^{25} -374.825 q^{27} +83.4110 q^{29} -165.308 q^{31} +177.048 q^{33} +310.973 q^{37} +876.904 q^{39} +181.504 q^{41} +82.3699 q^{43} +389.965 q^{45} +367.909 q^{47} -885.959 q^{51} +52.6574 q^{53} -108.720 q^{55} +1197.49 q^{57} +561.278 q^{59} +276.305 q^{61} -538.480 q^{65} -60.1368 q^{67} -626.865 q^{69} -155.534 q^{71} +731.478 q^{73} +867.155 q^{75} +290.384 q^{79} +1833.51 q^{81} -43.5791 q^{83} +544.041 q^{85} -803.913 q^{87} -393.093 q^{89} +1593.23 q^{93} -735.344 q^{95} +521.748 q^{97} -1210.40 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 136 q^{9}+O(q^{10}) $ 4 * q + 136 * q^9	$ 4 q + 136 q^{9} - 116 q^{11} + 112 q^{15} - 80 q^{23} + 448 q^{25} + 36 q^{29} + 436 q^{37} + 2232 q^{39} + 372 q^{43} - 780 q^{51} + 976 q^{53} + 3812 q^{57} + 780 q^{65} - 1176 q^{67} - 2408 q^{71} + 56 q^{79} + 2104 q^{81} + 4940 q^{85} + 2376 q^{93} + 4032 q^{95} - 2588 q^{99}+O(q^{100}) $ 4 * q + 136 * q^9 - 116 * q^11 + 112 * q^15 - 80 * q^23 + 448 * q^25 + 36 * q^29 + 436 * q^37 + 2232 * q^39 + 372 * q^43 - 780 * q^51 + 976 * q^53 + 3812 * q^57 + 780 * q^65 - 1176 * q^67 - 2408 * q^71 + 56 * q^79 + 2104 * q^81 + 4940 * q^85 + 2376 * q^93 + 4032 * q^95 - 2588 * 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
$3$	−9.63797	−1.85483	−0.927414	−	0.374037i	$-0.877973\pi$
$3$	−9.63797	−1.85483	−0.927414	+	0.374037i	$0.877973\pi$
$4$	0	0
$5$	5.91838	0.529356	0.264678	−	0.964337i	$-0.414734\pi$
$5$	5.91838	0.529356	0.264678	+	0.964337i	$0.414734\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	65.8904	2.44039
$10$	0	0
$11$	−18.3699	−0.503520	−0.251760	−	0.967790i	$-0.581009\pi$
$11$	−18.3699	−0.503520	−0.251760	+	0.967790i	$0.581009\pi$
$12$	0	0
$13$	−90.9844	−1.94112	−0.970558	−	0.240866i	$-0.922569\pi$
$13$	−90.9844	−1.94112	−0.970558	+	0.240866i	$0.922569\pi$
$14$	0	0
$15$	−57.0412	−0.981864
$16$	0	0
$17$	91.9239	1.31146	0.655730	−	0.754996i	$-0.272361\pi$
$17$	91.9239	1.31146	0.655730	+	0.754996i	$0.272361\pi$
$18$	0	0
$19$	−124.247	−1.50023	−0.750114	−	0.661309i	$-0.770001\pi$
$19$	−124.247	−1.50023	−0.750114	+	0.661309i	$0.770001\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	65.0412	0.589653	0.294827	−	0.955551i	$-0.404738\pi$
$23$	65.0412	0.589653	0.294827	+	0.955551i	$0.404738\pi$
$24$	0	0
$25$	−89.9728	−0.719782
$26$	0	0
$27$	−374.825	−2.67167
$28$	0	0
$29$	83.4110	0.534105	0.267052	−	0.963682i	$-0.413950\pi$
$29$	83.4110	0.534105	0.267052	+	0.963682i	$0.413950\pi$
$30$	0	0
$31$	−165.308	−0.957748	−0.478874	−	0.877884i	$-0.658955\pi$
$31$	−165.308	−0.957748	−0.478874	+	0.877884i	$0.658955\pi$
$32$	0	0
$33$	177.048	0.933943
$34$	0	0
$35$	0	0
$36$	0	0
$37$	310.973	1.38172	0.690860	−	0.722989i	$-0.257232\pi$
$37$	310.973	1.38172	0.690860	+	0.722989i	$0.257232\pi$
$38$	0	0
$39$	876.904	3.60044
$40$	0	0
$41$	181.504	0.691369	0.345685	−	0.938351i	$-0.387647\pi$
$41$	181.504	0.691369	0.345685	+	0.938351i	$0.387647\pi$
$42$	0	0
$43$	82.3699	0.292123	0.146061	−	0.989276i	$-0.453340\pi$
$43$	82.3699	0.292123	0.146061	+	0.989276i	$0.453340\pi$
$44$	0	0
$45$	389.965	1.29183
$46$	0	0
$47$	367.909	1.14181	0.570904	−	0.821017i	$-0.306593\pi$
$47$	367.909	1.14181	0.570904	+	0.821017i	$0.306593\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−885.959	−2.43253
$52$	0	0
$53$	52.6574	0.136473	0.0682363	−	0.997669i	$-0.478263\pi$
$53$	52.6574	0.136473	0.0682363	+	0.997669i	$0.478263\pi$
$54$	0	0
$55$	−108.720	−0.266541
$56$	0	0
$57$	1197.49	2.78266
$58$	0	0
$59$	561.278	1.23851	0.619256	−	0.785189i	$-0.287434\pi$
$59$	561.278	1.23851	0.619256	+	0.785189i	$0.287434\pi$
$60$	0	0
$61$	276.305	0.579954	0.289977	−	0.957034i	$-0.406352\pi$
$61$	276.305	0.579954	0.289977	+	0.957034i	$0.406352\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−538.480	−1.02754
$66$	0	0
$67$	−60.1368	−0.109655	−0.0548275	−	0.998496i	$-0.517461\pi$
$67$	−60.1368	−0.109655	−0.0548275	+	0.998496i	$0.517461\pi$
$68$	0	0
$69$	−626.865	−1.09370
$70$	0	0
$71$	−155.534	−0.259979	−0.129989	−	0.991515i	$-0.541494\pi$
$71$	−155.534	−0.259979	−0.129989	+	0.991515i	$0.541494\pi$
$72$	0	0
$73$	731.478	1.17278	0.586391	−	0.810028i	$-0.300548\pi$
$73$	731.478	1.17278	0.586391	+	0.810028i	$0.300548\pi$
$74$	0	0
$75$	867.155	1.33507
$76$	0	0
$77$	0	0
$78$	0	0
$79$	290.384	0.413554	0.206777	−	0.978388i	$-0.433703\pi$
$79$	290.384	0.413554	0.206777	+	0.978388i	$0.433703\pi$
$80$	0	0
$81$	1833.51	2.51510
$82$	0	0
$83$	−43.5791	−0.0576317	−0.0288158	−	0.999585i	$-0.509174\pi$
$83$	−43.5791	−0.0576317	−0.0288158	+	0.999585i	$0.509174\pi$
$84$	0	0
$85$	544.041	0.694229
$86$	0	0
$87$	−803.913	−0.990672
$88$	0	0
$89$	−393.093	−0.468177	−0.234089	−	0.972215i	$-0.575211\pi$
$89$	−393.093	−0.468177	−0.234089	+	0.972215i	$0.575211\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1593.23	1.77646
$94$	0	0
$95$	−735.344	−0.794155
$96$	0	0
$97$	521.748	0.546139	0.273070	−	0.961994i	$-0.411961\pi$
$97$	521.748	0.546139	0.273070	+	0.961994i	$0.411961\pi$
$98$	0	0
$99$	−1210.40	−1.22878
$100$	0	0
$101$	−753.089	−0.741932	−0.370966	−	0.928646i	$-0.620973\pi$
$101$	−753.089	−0.741932	−0.370966	+	0.928646i	$0.620973\pi$
$102$	0	0
$103$	930.922	0.890548	0.445274	−	0.895394i	$-0.353106\pi$
$103$	930.922	0.890548	0.445274	+	0.895394i	$0.353106\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1683.04	1.52062	0.760308	−	0.649563i	$-0.225048\pi$
$107$	1683.04	1.52062	0.760308	+	0.649563i	$0.225048\pi$
$108$	0	0
$109$	−783.905	−0.688848	−0.344424	−	0.938814i	$-0.611926\pi$
$109$	−783.905	−0.688848	−0.344424	+	0.938814i	$0.611926\pi$
$110$	0	0
$111$	−2997.15	−2.56285
$112$	0	0
$113$	−628.205	−0.522979	−0.261489	−	0.965206i	$-0.584214\pi$
$113$	−628.205	−0.522979	−0.261489	+	0.965206i	$0.584214\pi$
$114$	0	0
$115$	384.938	0.312136
$116$	0	0
$117$	−5995.00	−4.73708
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−993.548	−0.746468
$122$	0	0
$123$	−1749.33	−1.28237
$124$	0	0
$125$	−1272.29	−0.910377
$126$	0	0
$127$	328.466	0.229501	0.114751	−	0.993394i	$-0.463393\pi$
$127$	328.466	0.229501	0.114751	+	0.993394i	$0.463393\pi$
$128$	0	0
$129$	−793.878	−0.541838
$130$	0	0
$131$	2412.79	1.60921	0.804606	−	0.593809i	$-0.202376\pi$
$131$	2412.79	1.60921	0.804606	+	0.593809i	$0.202376\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2218.36	−1.41426
$136$	0	0
$137$	−175.699	−0.109569	−0.0547845	−	0.998498i	$-0.517447\pi$
$137$	−175.699	−0.109569	−0.0547845	+	0.998498i	$0.517447\pi$
$138$	0	0
$139$	1489.47	0.908885	0.454442	−	0.890776i	$-0.349839\pi$
$139$	1489.47	0.908885	0.454442	+	0.890776i	$0.349839\pi$
$140$	0	0
$141$	−3545.89	−2.11786
$142$	0	0
$143$	1671.37	0.977391
$144$	0	0
$145$	493.658	0.282732
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3034.33	1.66834	0.834168	−	0.551511i	$-0.185949\pi$
$149$	3034.33	1.66834	0.834168	+	0.551511i	$0.185949\pi$
$150$	0	0
$151$	−2114.82	−1.13975	−0.569873	−	0.821732i	$-0.693008\pi$
$151$	−2114.82	−1.13975	−0.569873	+	0.821732i	$0.693008\pi$
$152$	0	0
$153$	6056.90	3.20047
$154$	0	0
$155$	−978.356	−0.506990
$156$	0	0
$157$	−275.704	−0.140150	−0.0700752	−	0.997542i	$-0.522324\pi$
$157$	−275.704	−0.140150	−0.0700752	+	0.997542i	$0.522324\pi$
$158$	0	0
$159$	−507.510	−0.253133
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1219.25	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$163$	1219.25	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$164$	0	0
$165$	1047.84	0.494388
$166$	0	0
$167$	2781.99	1.28908	0.644542	−	0.764569i	$-0.277048\pi$
$167$	2781.99	1.28908	0.644542	+	0.764569i	$0.277048\pi$
$168$	0	0
$169$	6081.15	2.76794
$170$	0	0
$171$	−8186.72	−3.66114
$172$	0	0
$173$	2707.05	1.18967	0.594835	−	0.803847i	$-0.297217\pi$
$173$	2707.05	1.18967	0.594835	+	0.803847i	$0.297217\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5409.58	−2.29723
$178$	0	0
$179$	−1721.73	−0.718926	−0.359463	−	0.933159i	$-0.617040\pi$
$179$	−1721.73	−0.718926	−0.359463	+	0.933159i	$0.617040\pi$
$180$	0	0
$181$	624.549	0.256477	0.128238	−	0.991743i	$-0.459068\pi$
$181$	624.549	0.256477	0.128238	+	0.991743i	$0.459068\pi$
$182$	0	0
$183$	−2663.01	−1.07571
$184$	0	0
$185$	1840.46	0.731421
$186$	0	0
$187$	−1688.63	−0.660346
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4789.78	−1.81454	−0.907269	−	0.420552i	$-0.861836\pi$
$191$	−4789.78	−1.81454	−0.907269	+	0.420552i	$0.861836\pi$
$192$	0	0
$193$	−1436.23	−0.535660	−0.267830	−	0.963466i	$-0.586307\pi$
$193$	−1436.23	−0.535660	−0.267830	+	0.963466i	$0.586307\pi$
$194$	0	0
$195$	5189.85	1.90591
$196$	0	0
$197$	2508.66	0.907282	0.453641	−	0.891184i	$-0.350125\pi$
$197$	2508.66	0.907282	0.453641	+	0.891184i	$0.350125\pi$
$198$	0	0
$199$	4093.76	1.45829	0.729143	−	0.684361i	$-0.239919\pi$
$199$	4093.76	1.45829	0.729143	+	0.684361i	$0.239919\pi$
$200$	0	0
$201$	579.597	0.203391
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1074.21	0.365981
$206$	0	0
$207$	4285.59	1.43898
$208$	0	0
$209$	2282.41	0.755395
$210$	0	0
$211$	−2308.68	−0.753253	−0.376626	−	0.926365i	$-0.622916\pi$
$211$	−2308.68	−0.753253	−0.376626	+	0.926365i	$0.622916\pi$
$212$	0	0
$213$	1499.03	0.482215
$214$	0	0
$215$	487.496	0.154637
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7049.96	−2.17531
$220$	0	0
$221$	−8363.64	−2.54570
$222$	0	0
$223$	−775.802	−0.232967	−0.116483	−	0.993193i	$-0.537162\pi$
$223$	−775.802	−0.232967	−0.116483	+	0.993193i	$0.537162\pi$
$224$	0	0
$225$	−5928.35	−1.75655
$226$	0	0
$227$	−178.238	−0.0521150	−0.0260575	−	0.999660i	$-0.508295\pi$
$227$	−178.238	−0.0521150	−0.0260575	+	0.999660i	$0.508295\pi$
$228$	0	0
$229$	−2609.81	−0.753106	−0.376553	−	0.926395i	$-0.622891\pi$
$229$	−2609.81	−0.753106	−0.376553	+	0.926395i	$0.622891\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1575.84	0.443075	0.221538	−	0.975152i	$-0.428892\pi$
$233$	1575.84	0.443075	0.221538	+	0.975152i	$0.428892\pi$
$234$	0	0
$235$	2177.42	0.604423
$236$	0	0
$237$	−2798.71	−0.767071
$238$	0	0
$239$	620.547	0.167949	0.0839746	−	0.996468i	$-0.473239\pi$
$239$	620.547	0.167949	0.0839746	+	0.996468i	$0.473239\pi$
$240$	0	0
$241$	−2552.79	−0.682323	−0.341161	−	0.940005i	$-0.610820\pi$
$241$	−2552.79	−0.682323	−0.341161	+	0.940005i	$0.610820\pi$
$242$	0	0
$243$	−7551.02	−1.99341
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11304.6	2.91212
$248$	0	0
$249$	420.014	0.106897
$250$	0	0
$251$	−4057.62	−1.02038	−0.510189	−	0.860062i	$-0.670425\pi$
$251$	−4057.62	−1.02038	−0.510189	+	0.860062i	$0.670425\pi$
$252$	0	0
$253$	−1194.80	−0.296902
$254$	0	0
$255$	−5243.45	−1.28768
$256$	0	0
$257$	5161.13	1.25269	0.626347	−	0.779545i	$-0.284549\pi$
$257$	5161.13	1.25269	0.626347	+	0.779545i	$0.284549\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5495.99	1.30342
$262$	0	0
$263$	7119.65	1.66926	0.834632	−	0.550808i	$-0.185680\pi$
$263$	7119.65	1.66926	0.834632	+	0.550808i	$0.185680\pi$
$264$	0	0
$265$	311.646	0.0722426
$266$	0	0
$267$	3788.62	0.868389
$268$	0	0
$269$	−6002.71	−1.36056	−0.680282	−	0.732951i	$-0.738143\pi$
$269$	−6002.71	−1.36056	−0.680282	+	0.732951i	$0.738143\pi$
$270$	0	0
$271$	−793.085	−0.177773	−0.0888864	−	0.996042i	$-0.528331\pi$
$271$	−793.085	−0.177773	−0.0888864	+	0.996042i	$0.528331\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1652.79	0.362425
$276$	0	0
$277$	−4054.33	−0.879427	−0.439713	−	0.898138i	$-0.644920\pi$
$277$	−4054.33	−0.879427	−0.439713	+	0.898138i	$0.644920\pi$
$278$	0	0
$279$	−10892.2	−2.33728
$280$	0	0
$281$	2990.91	0.634955	0.317478	−	0.948266i	$-0.397164\pi$
$281$	2990.91	0.634955	0.317478	+	0.948266i	$0.397164\pi$
$282$	0	0
$283$	5716.11	1.20066	0.600331	−	0.799752i	$-0.295035\pi$
$283$	5716.11	1.20066	0.600331	+	0.799752i	$0.295035\pi$
$284$	0	0
$285$	7087.22	1.47302
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3537.00	0.719927
$290$	0	0
$291$	−5028.59	−1.01299
$292$	0	0
$293$	−5301.47	−1.05705	−0.528524	−	0.848918i	$-0.677254\pi$
$293$	−5301.47	−1.05705	−0.528524	+	0.848918i	$0.677254\pi$
$294$	0	0
$295$	3321.86	0.655613
$296$	0	0
$297$	6885.48	1.34524
$298$	0	0
$299$	−5917.73	−1.14459
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7258.25	1.37616
$304$	0	0
$305$	1635.28	0.307002
$306$	0	0
$307$	5978.40	1.11142	0.555709	−	0.831377i	$-0.312447\pi$
$307$	5978.40	1.11142	0.555709	+	0.831377i	$0.312447\pi$
$308$	0	0
$309$	−8972.19	−1.65181
$310$	0	0
$311$	4850.71	0.884433	0.442217	−	0.896908i	$-0.354192\pi$
$311$	4850.71	0.884433	0.442217	+	0.896908i	$0.354192\pi$
$312$	0	0
$313$	−4455.34	−0.804571	−0.402285	−	0.915514i	$-0.631784\pi$
$313$	−4455.34	−0.804571	−0.402285	+	0.915514i	$0.631784\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10261.9	−1.81818	−0.909091	−	0.416597i	$-0.863223\pi$
$317$	−10261.9	−1.81818	−0.909091	+	0.416597i	$0.863223\pi$
$318$	0	0
$319$	−1532.25	−0.268932
$320$	0	0
$321$	−16221.1	−2.82048
$322$	0	0
$323$	−11421.3	−1.96749
$324$	0	0
$325$	8186.11	1.39718
$326$	0	0
$327$	7555.25	1.27770
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3087.11	−0.512638	−0.256319	−	0.966592i	$-0.582510\pi$
$331$	−3087.11	−0.512638	−0.256319	+	0.966592i	$0.582510\pi$
$332$	0	0
$333$	20490.1	3.37193
$334$	0	0
$335$	−355.912	−0.0580465
$336$	0	0
$337$	4588.57	0.741707	0.370853	−	0.928691i	$-0.379065\pi$
$337$	4588.57	0.741707	0.370853	+	0.928691i	$0.379065\pi$
$338$	0	0
$339$	6054.62	0.970035
$340$	0	0
$341$	3036.68	0.482245
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3710.02	−0.578959
$346$	0	0
$347$	−9645.58	−1.49222	−0.746112	−	0.665820i	$-0.768082\pi$
$347$	−9645.58	−1.49222	−0.746112	+	0.665820i	$0.768082\pi$
$348$	0	0
$349$	2612.70	0.400730	0.200365	−	0.979721i	$-0.435787\pi$
$349$	2612.70	0.400730	0.200365	+	0.979721i	$0.435787\pi$
$350$	0	0
$351$	34103.2	5.18602
$352$	0	0
$353$	−8688.08	−1.30997	−0.654985	−	0.755642i	$-0.727325\pi$
$353$	−8688.08	−1.30997	−0.654985	+	0.755642i	$0.727325\pi$
$354$	0	0
$355$	−920.509	−0.137621
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2738.82	−0.402645	−0.201322	−	0.979525i	$-0.564524\pi$
$359$	−2738.82	−0.402645	−0.201322	+	0.979525i	$0.564524\pi$
$360$	0	0
$361$	8578.44	1.25068
$362$	0	0
$363$	9575.79	1.38457
$364$	0	0
$365$	4329.17	0.620819
$366$	0	0
$367$	−1968.59	−0.279999	−0.139999	−	0.990152i	$-0.544710\pi$
$367$	−1968.59	−0.279999	−0.139999	+	0.990152i	$0.544710\pi$
$368$	0	0
$369$	11959.4	1.68721
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9915.60	−1.37643	−0.688217	−	0.725504i	$-0.741607\pi$
$373$	−9915.60	−1.37643	−0.688217	+	0.725504i	$0.741607\pi$
$374$	0	0
$375$	12262.3	1.68859
$376$	0	0
$377$	−7589.10	−1.03676
$378$	0	0
$379$	10933.4	1.48183	0.740913	−	0.671601i	$-0.234393\pi$
$379$	10933.4	1.48183	0.740913	+	0.671601i	$0.234393\pi$
$380$	0	0
$381$	−3165.75	−0.425685
$382$	0	0
$383$	12009.7	1.60226	0.801131	−	0.598489i	$-0.204232\pi$
$383$	12009.7	1.60226	0.801131	+	0.598489i	$0.204232\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5427.39	0.712893
$388$	0	0
$389$	9242.27	1.20463	0.602316	−	0.798258i	$-0.294245\pi$
$389$	9242.27	1.20463	0.602316	+	0.798258i	$0.294245\pi$
$390$	0	0
$391$	5978.84	0.773306
$392$	0	0
$393$	−23254.4	−2.98481
$394$	0	0
$395$	1718.60	0.218917
$396$	0	0
$397$	−3702.46	−0.468063	−0.234032	−	0.972229i	$-0.575192\pi$
$397$	−3702.46	−0.468063	−0.234032	+	0.972229i	$0.575192\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3194.78	0.397855	0.198927	−	0.980014i	$-0.436254\pi$
$401$	3194.78	0.397855	0.198927	+	0.980014i	$0.436254\pi$
$402$	0	0
$403$	15040.4	1.85910
$404$	0	0
$405$	10851.4	1.33138
$406$	0	0
$407$	−5712.52	−0.695723
$408$	0	0
$409$	−6217.59	−0.751687	−0.375843	−	0.926683i	$-0.622647\pi$
$409$	−6217.59	−0.751687	−0.375843	+	0.926683i	$0.622647\pi$
$410$	0	0
$411$	1693.38	0.203231
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−257.918	−0.0305077
$416$	0	0
$417$	−14355.4	−1.68583
$418$	0	0
$419$	290.002	0.0338127	0.0169063	−	0.999857i	$-0.494618\pi$
$419$	290.002	0.0338127	0.0169063	+	0.999857i	$0.494618\pi$
$420$	0	0
$421$	16109.3	1.86489	0.932445	−	0.361311i	$-0.117671\pi$
$421$	16109.3	1.86489	0.932445	+	0.361311i	$0.117671\pi$
$422$	0	0
$423$	24241.7	2.78645
$424$	0	0
$425$	−8270.65	−0.943965
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−16108.6	−1.81289
$430$	0	0
$431$	−71.3997	−0.00797958	−0.00398979	−	0.999992i	$-0.501270\pi$
$431$	−71.3997	−0.00797958	−0.00398979	+	0.999992i	$0.501270\pi$
$432$	0	0
$433$	2473.42	0.274515	0.137258	−	0.990535i	$-0.456171\pi$
$433$	2473.42	0.274515	0.137258	+	0.990535i	$0.456171\pi$
$434$	0	0
$435$	−4757.86	−0.524418
$436$	0	0
$437$	−8081.20	−0.884614
$438$	0	0
$439$	−13051.8	−1.41898	−0.709488	−	0.704718i	$-0.751074\pi$
$439$	−13051.8	−1.41898	−0.709488	+	0.704718i	$0.751074\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5104.36	0.547439	0.273719	−	0.961810i	$-0.411746\pi$
$443$	5104.36	0.547439	0.273719	+	0.961810i	$0.411746\pi$
$444$	0	0
$445$	−2326.47	−0.247833
$446$	0	0
$447$	−29244.8	−3.09447
$448$	0	0
$449$	−7106.77	−0.746969	−0.373485	−	0.927636i	$-0.621837\pi$
$449$	−7106.77	−0.746969	−0.373485	+	0.927636i	$0.621837\pi$
$450$	0	0
$451$	−3334.20	−0.348118
$452$	0	0
$453$	20382.6	2.11403
$454$	0	0
$455$	0	0
$456$	0	0
$457$	942.949	0.0965192	0.0482596	−	0.998835i	$-0.484633\pi$
$457$	942.949	0.0965192	0.0482596	+	0.998835i	$0.484633\pi$
$458$	0	0
$459$	−34455.4	−3.50379
$460$	0	0
$461$	1906.71	0.192634	0.0963172	−	0.995351i	$-0.469294\pi$
$461$	1906.71	0.192634	0.0963172	+	0.995351i	$0.469294\pi$
$462$	0	0
$463$	−4896.83	−0.491523	−0.245761	−	0.969330i	$-0.579038\pi$
$463$	−4896.83	−0.491523	−0.245761	+	0.969330i	$0.579038\pi$
$464$	0	0
$465$	9429.36	0.940379
$466$	0	0
$467$	−363.895	−0.0360580	−0.0180290	−	0.999837i	$-0.505739\pi$
$467$	−363.895	−0.0360580	−0.0180290	+	0.999837i	$0.505739\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2657.23	0.259955
$472$	0	0
$473$	−1513.12	−0.147090
$474$	0	0
$475$	11178.9	1.07984
$476$	0	0
$477$	3469.62	0.333046
$478$	0	0
$479$	5223.85	0.498296	0.249148	−	0.968465i	$-0.419849\pi$
$479$	5223.85	0.498296	0.249148	+	0.968465i	$0.419849\pi$
$480$	0	0
$481$	−28293.7	−2.68208
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3087.90	0.289102
$486$	0	0
$487$	12133.2	1.12896	0.564482	−	0.825445i	$-0.309076\pi$
$487$	12133.2	1.12896	0.564482	+	0.825445i	$0.309076\pi$
$488$	0	0
$489$	−11751.1	−1.08671
$490$	0	0
$491$	2875.76	0.264320	0.132160	−	0.991228i	$-0.457809\pi$
$491$	2875.76	0.264320	0.132160	+	0.991228i	$0.457809\pi$
$492$	0	0
$493$	7667.46	0.700457
$494$	0	0
$495$	−7163.59	−0.650464
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15643.9	1.40344	0.701719	−	0.712454i	$-0.252416\pi$
$499$	15643.9	1.40344	0.701719	+	0.712454i	$0.252416\pi$
$500$	0	0
$501$	−26812.8	−2.39103
$502$	0	0
$503$	3530.58	0.312964	0.156482	−	0.987681i	$-0.449985\pi$
$503$	3530.58	0.312964	0.156482	+	0.987681i	$0.449985\pi$
$504$	0	0
$505$	−4457.07	−0.392746
$506$	0	0
$507$	−58610.0	−5.13404
$508$	0	0
$509$	6053.60	0.527153	0.263577	−	0.964638i	$-0.415098\pi$
$509$	6053.60	0.527153	0.263577	+	0.964638i	$0.415098\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	46571.0	4.00811
$514$	0	0
$515$	5509.55	0.471417
$516$	0	0
$517$	−6758.43	−0.574923
$518$	0	0
$519$	−26090.4	−2.20663
$520$	0	0
$521$	−20876.4	−1.75549	−0.877746	−	0.479126i	$-0.840954\pi$
$521$	−20876.4	−1.75549	−0.877746	+	0.479126i	$0.840954\pi$
$522$	0	0
$523$	−19017.4	−1.59000	−0.795001	−	0.606608i	$-0.792530\pi$
$523$	−19017.4	−1.59000	−0.795001	+	0.606608i	$0.792530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15195.8	−1.25605
$528$	0	0
$529$	−7936.65	−0.652309
$530$	0	0
$531$	36982.9	3.02245
$532$	0	0
$533$	−16514.0	−1.34203
$534$	0	0
$535$	9960.89	0.804947
$536$	0	0
$537$	16593.9	1.33348
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14070.7	1.11820	0.559099	−	0.829101i	$-0.311147\pi$
$541$	14070.7	1.11820	0.559099	+	0.829101i	$0.311147\pi$
$542$	0	0
$543$	−6019.38	−0.475721
$544$	0	0
$545$	−4639.45	−0.364646
$546$	0	0
$547$	−16099.5	−1.25844	−0.629218	−	0.777229i	$-0.716625\pi$
$547$	−16099.5	−1.25844	−0.629218	+	0.777229i	$0.716625\pi$
$548$	0	0
$549$	18205.8	1.41531
$550$	0	0
$551$	−10363.6	−0.801279
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−17738.2	−1.35666
$556$	0	0
$557$	12327.2	0.937740	0.468870	−	0.883267i	$-0.344661\pi$
$557$	12327.2	0.937740	0.468870	+	0.883267i	$0.344661\pi$
$558$	0	0
$559$	−7494.37	−0.567045
$560$	0	0
$561$	16274.9	1.22483
$562$	0	0
$563$	4492.34	0.336287	0.168143	−	0.985763i	$-0.446223\pi$
$563$	4492.34	0.336287	0.168143	+	0.985763i	$0.446223\pi$
$564$	0	0
$565$	−3717.96	−0.276842
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16895.1	−1.24478	−0.622390	−	0.782707i	$-0.713838\pi$
$569$	−16895.1	−1.24478	−0.622390	+	0.782707i	$0.713838\pi$
$570$	0	0
$571$	−88.1151	−0.00645797	−0.00322899	−	0.999995i	$-0.501028\pi$
$571$	−88.1151	−0.00645797	−0.00322899	+	0.999995i	$0.501028\pi$
$572$	0	0
$573$	46163.8	3.36565
$574$	0	0
$575$	−5851.93	−0.424422
$576$	0	0
$577$	8172.92	0.589676	0.294838	−	0.955547i	$-0.404734\pi$
$577$	8172.92	0.589676	0.294838	+	0.955547i	$0.404734\pi$
$578$	0	0
$579$	13842.4	0.993557
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−967.308	−0.0687167
$584$	0	0
$585$	−35480.7	−2.50760
$586$	0	0
$587$	−18301.4	−1.28685	−0.643425	−	0.765509i	$-0.722487\pi$
$587$	−18301.4	−1.28685	−0.643425	+	0.765509i	$0.722487\pi$
$588$	0	0
$589$	20539.1	1.43684
$590$	0	0
$591$	−24178.4	−1.68285
$592$	0	0
$593$	−2069.18	−0.143290	−0.0716452	−	0.997430i	$-0.522825\pi$
$593$	−2069.18	−0.143290	−0.0716452	+	0.997430i	$0.522825\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39455.6	−2.70487
$598$	0	0
$599$	13615.6	0.928747	0.464373	−	0.885639i	$-0.346280\pi$
$599$	13615.6	0.928747	0.464373	+	0.885639i	$0.346280\pi$
$600$	0	0
$601$	−1041.12	−0.0706624	−0.0353312	−	0.999376i	$-0.511249\pi$
$601$	−1041.12	−0.0706624	−0.0353312	+	0.999376i	$0.511249\pi$
$602$	0	0
$603$	−3962.44	−0.267600
$604$	0	0
$605$	−5880.20	−0.395147
$606$	0	0
$607$	−15496.0	−1.03618	−0.518090	−	0.855326i	$-0.673357\pi$
$607$	−15496.0	−1.03618	−0.518090	+	0.855326i	$0.673357\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−33473.9	−2.21638
$612$	0	0
$613$	936.457	0.0617017	0.0308509	−	0.999524i	$-0.490178\pi$
$613$	936.457	0.0617017	0.0308509	+	0.999524i	$0.490178\pi$
$614$	0	0
$615$	−10353.2	−0.678831
$616$	0	0
$617$	−798.142	−0.0520778	−0.0260389	−	0.999661i	$-0.508289\pi$
$617$	−798.142	−0.0520778	−0.0260389	+	0.999661i	$0.508289\pi$
$618$	0	0
$619$	−999.034	−0.0648701	−0.0324350	−	0.999474i	$-0.510326\pi$
$619$	−999.034	−0.0648701	−0.0324350	+	0.999474i	$0.510326\pi$
$620$	0	0
$621$	−24379.0	−1.57536
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3716.70	0.237869
$626$	0	0
$627$	−21997.8	−1.40113
$628$	0	0
$629$	28585.8	1.81207
$630$	0	0
$631$	25618.7	1.61626	0.808132	−	0.589001i	$-0.200479\pi$
$631$	25618.7	1.61626	0.808132	+	0.589001i	$0.200479\pi$
$632$	0	0
$633$	22251.0	1.39715
$634$	0	0
$635$	1943.99	0.121488
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10248.2	−0.634448
$640$	0	0
$641$	18351.4	1.13079	0.565394	−	0.824821i	$-0.308724\pi$
$641$	18351.4	1.13079	0.565394	+	0.824821i	$0.308724\pi$
$642$	0	0
$643$	22068.9	1.35352	0.676759	−	0.736205i	$-0.263384\pi$
$643$	22068.9	1.35352	0.676759	+	0.736205i	$0.263384\pi$
$644$	0	0
$645$	−4698.47	−0.286825
$646$	0	0
$647$	6409.50	0.389464	0.194732	−	0.980856i	$-0.437616\pi$
$647$	6409.50	0.389464	0.194732	+	0.980856i	$0.437616\pi$
$648$	0	0
$649$	−10310.6	−0.623615
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21098.8	1.26441	0.632206	−	0.774800i	$-0.282150\pi$
$653$	21098.8	1.26441	0.632206	+	0.774800i	$0.282150\pi$
$654$	0	0
$655$	14279.8	0.851846
$656$	0	0
$657$	48197.4	2.86204
$658$	0	0
$659$	−29509.0	−1.74432	−0.872159	−	0.489222i	$-0.837281\pi$
$659$	−29509.0	−1.74432	−0.872159	+	0.489222i	$0.837281\pi$
$660$	0	0
$661$	−21717.9	−1.27795	−0.638976	−	0.769226i	$-0.720642\pi$
$661$	−21717.9	−1.27795	−0.638976	+	0.769226i	$0.720642\pi$
$662$	0	0
$663$	80608.5	4.72183
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5425.15	0.314937
$668$	0	0
$669$	7477.16	0.432113
$670$	0	0
$671$	−5075.67	−0.292018
$672$	0	0
$673$	3168.78	0.181497	0.0907486	−	0.995874i	$-0.471074\pi$
$673$	3168.78	0.181497	0.0907486	+	0.995874i	$0.471074\pi$
$674$	0	0
$675$	33724.0	1.92302
$676$	0	0
$677$	22586.8	1.28225	0.641124	−	0.767437i	$-0.278468\pi$
$677$	22586.8	1.28225	0.641124	+	0.767437i	$0.278468\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1717.86	0.0966643
$682$	0	0
$683$	−7130.82	−0.399492	−0.199746	−	0.979848i	$-0.564012\pi$
$683$	−7130.82	−0.399492	−0.199746	+	0.979848i	$0.564012\pi$
$684$	0	0
$685$	−1039.85	−0.0580010
$686$	0	0
$687$	25153.3	1.39688
$688$	0	0
$689$	−4791.00	−0.264909
$690$	0	0
$691$	11414.6	0.628408	0.314204	−	0.949355i	$-0.398262\pi$
$691$	11414.6	0.628408	0.314204	+	0.949355i	$0.398262\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8815.24	0.481124
$696$	0	0
$697$	16684.5	0.906703
$698$	0	0
$699$	−15187.9	−0.821829
$700$	0	0
$701$	10005.8	0.539106	0.269553	−	0.962986i	$-0.413124\pi$
$701$	10005.8	0.539106	0.269553	+	0.962986i	$0.413124\pi$
$702$	0	0
$703$	−38637.6	−2.07289
$704$	0	0
$705$	−20985.9	−1.12110
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9540.20	−0.505345	−0.252673	−	0.967552i	$-0.581310\pi$
$709$	−9540.20	−0.505345	−0.252673	+	0.967552i	$0.581310\pi$
$710$	0	0
$711$	19133.5	1.00923
$712$	0	0
$713$	−10751.8	−0.564739
$714$	0	0
$715$	9891.80	0.517388
$716$	0	0
$717$	−5980.81	−0.311517
$718$	0	0
$719$	−5713.54	−0.296355	−0.148177	−	0.988961i	$-0.547341\pi$
$719$	−5713.54	−0.296355	−0.148177	+	0.988961i	$0.547341\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24603.7	1.26559
$724$	0	0
$725$	−7504.72	−0.384439
$726$	0	0
$727$	18615.1	0.949649	0.474824	−	0.880081i	$-0.342512\pi$
$727$	18615.1	0.949649	0.474824	+	0.880081i	$0.342512\pi$
$728$	0	0
$729$	23271.8	1.18233
$730$	0	0
$731$	7571.76	0.383107
$732$	0	0
$733$	−26452.2	−1.33293	−0.666463	−	0.745538i	$-0.732192\pi$
$733$	−26452.2	−1.33293	−0.666463	+	0.745538i	$0.732192\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1104.70	0.0552134
$738$	0	0
$739$	−22916.1	−1.14071	−0.570353	−	0.821400i	$-0.693193\pi$
$739$	−22916.1	−1.14071	−0.570353	+	0.821400i	$0.693193\pi$
$740$	0	0
$741$	−108953.	−5.40148
$742$	0	0
$743$	−24313.3	−1.20050	−0.600248	−	0.799814i	$-0.704931\pi$
$743$	−24313.3	−1.20050	−0.600248	+	0.799814i	$0.704931\pi$
$744$	0	0
$745$	17958.3	0.883143
$746$	0	0
$747$	−2871.45	−0.140644
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5618.48	−0.272997	−0.136499	−	0.990640i	$-0.543585\pi$
$751$	−5618.48	−0.272997	−0.136499	+	0.990640i	$0.543585\pi$
$752$	0	0
$753$	39107.3	1.89263
$754$	0	0
$755$	−12516.3	−0.603332
$756$	0	0
$757$	−2786.08	−0.133767	−0.0668837	−	0.997761i	$-0.521306\pi$
$757$	−2786.08	−0.133767	−0.0668837	+	0.997761i	$0.521306\pi$
$758$	0	0
$759$	11515.4	0.550702
$760$	0	0
$761$	6814.43	0.324603	0.162302	−	0.986741i	$-0.448108\pi$
$761$	6814.43	0.324603	0.162302	+	0.986741i	$0.448108\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	35847.1	1.69419
$766$	0	0
$767$	−51067.5	−2.40410
$768$	0	0
$769$	−13274.3	−0.622474	−0.311237	−	0.950332i	$-0.600743\pi$
$769$	−13274.3	−0.622474	−0.311237	+	0.950332i	$0.600743\pi$
$770$	0	0
$771$	−49742.8	−2.32353
$772$	0	0
$773$	18651.4	0.867845	0.433922	−	0.900950i	$-0.357129\pi$
$773$	18651.4	0.867845	0.433922	+	0.900950i	$0.357129\pi$
$774$	0	0
$775$	14873.2	0.689370
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−22551.4	−1.03721
$780$	0	0
$781$	2857.13	0.130904
$782$	0	0
$783$	−31264.5	−1.42695
$784$	0	0
$785$	−1631.72	−0.0741895
$786$	0	0
$787$	35028.4	1.58657	0.793284	−	0.608852i	$-0.208370\pi$
$787$	35028.4	1.58657	0.793284	+	0.608852i	$0.208370\pi$
$788$	0	0
$789$	−68618.9	−3.09620
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25139.4	−1.12576
$794$	0	0
$795$	−3003.64	−0.133998
$796$	0	0
$797$	1170.81	0.0520352	0.0260176	−	0.999661i	$-0.491717\pi$
$797$	1170.81	0.0520352	0.0260176	+	0.999661i	$0.491717\pi$
$798$	0	0
$799$	33819.6	1.49744
$800$	0	0
$801$	−25901.1	−1.14253
$802$	0	0
$803$	−13437.1	−0.590519
$804$	0	0
$805$	0	0
$806$	0	0
$807$	57853.9	2.52361
$808$	0	0
$809$	13354.8	0.580381	0.290190	−	0.956969i	$-0.406281\pi$
$809$	13354.8	0.580381	0.290190	+	0.956969i	$0.406281\pi$
$810$	0	0
$811$	−12327.2	−0.533745	−0.266872	−	0.963732i	$-0.585990\pi$
$811$	−12327.2	−0.533745	−0.266872	+	0.963732i	$0.585990\pi$
$812$	0	0
$813$	7643.72	0.329738
$814$	0	0
$815$	7215.96	0.310140
$816$	0	0
$817$	−10234.2	−0.438251
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45453.9	1.93222	0.966109	−	0.258134i	$-0.0831076\pi$
$821$	45453.9	1.93222	0.966109	+	0.258134i	$0.0831076\pi$
$822$	0	0
$823$	9286.25	0.393315	0.196657	−	0.980472i	$-0.436991\pi$
$823$	9286.25	0.393315	0.196657	+	0.980472i	$0.436991\pi$
$824$	0	0
$825$	−15929.5	−0.672235
$826$	0	0
$827$	22224.4	0.934485	0.467242	−	0.884129i	$-0.345248\pi$
$827$	22224.4	0.934485	0.467242	+	0.884129i	$0.345248\pi$
$828$	0	0
$829$	44036.7	1.84495	0.922473	−	0.386063i	$-0.126165\pi$
$829$	44036.7	1.84495	0.922473	+	0.386063i	$0.126165\pi$
$830$	0	0
$831$	39075.5	1.63119
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16464.9	0.682384
$836$	0	0
$837$	61961.6	2.55879
$838$	0	0
$839$	−37521.9	−1.54398	−0.771991	−	0.635633i	$-0.780739\pi$
$839$	−37521.9	−1.54398	−0.771991	+	0.635633i	$0.780739\pi$
$840$	0	0
$841$	−17431.6	−0.714732
$842$	0	0
$843$	−28826.3	−1.17773
$844$	0	0
$845$	35990.6	1.46522
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−55091.7	−2.22702
$850$	0	0
$851$	20226.0	0.814735
$852$	0	0
$853$	34441.3	1.38247	0.691235	−	0.722630i	$-0.257067\pi$
$853$	34441.3	1.38247	0.691235	+	0.722630i	$0.257067\pi$
$854$	0	0
$855$	−48452.1	−1.93804
$856$	0	0
$857$	−24970.8	−0.995318	−0.497659	−	0.867373i	$-0.665807\pi$
$857$	−24970.8	−0.995318	−0.497659	+	0.867373i	$0.665807\pi$
$858$	0	0
$859$	17355.5	0.689364	0.344682	−	0.938720i	$-0.387987\pi$
$859$	17355.5	0.689364	0.344682	+	0.938720i	$0.387987\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26568.0	1.04795	0.523977	−	0.851732i	$-0.324448\pi$
$863$	26568.0	1.04795	0.523977	+	0.851732i	$0.324448\pi$
$864$	0	0
$865$	16021.3	0.629760
$866$	0	0
$867$	−34089.5	−1.33534
$868$	0	0
$869$	−5334.31	−0.208232
$870$	0	0
$871$	5471.51	0.212853
$872$	0	0
$873$	34378.2	1.33279
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47235.3	1.81873	0.909363	−	0.416003i	$-0.136569\pi$
$877$	47235.3	1.81873	0.909363	+	0.416003i	$0.136569\pi$
$878$	0	0
$879$	51095.4	1.96064
$880$	0	0
$881$	29598.0	1.13188	0.565938	−	0.824448i	$-0.308514\pi$
$881$	29598.0	1.13188	0.565938	+	0.824448i	$0.308514\pi$
$882$	0	0
$883$	42952.5	1.63700	0.818498	−	0.574509i	$-0.194807\pi$
$883$	42952.5	1.63700	0.818498	+	0.574509i	$0.194807\pi$
$884$	0	0
$885$	−32016.0	−1.21605
$886$	0	0
$887$	−33497.9	−1.26804	−0.634018	−	0.773318i	$-0.718596\pi$
$887$	−33497.9	−1.26804	−0.634018	+	0.773318i	$0.718596\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−33681.3	−1.26640
$892$	0	0
$893$	−45711.7	−1.71297
$894$	0	0
$895$	−10189.8	−0.380568
$896$	0	0
$897$	57034.9	2.12301
$898$	0	0
$899$	−13788.5	−0.511538
$900$	0	0
$901$	4840.47	0.178978
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3696.32	0.135768
$906$	0	0
$907$	13751.1	0.503417	0.251708	−	0.967803i	$-0.419008\pi$
$907$	13751.1	0.503417	0.251708	+	0.967803i	$0.419008\pi$
$908$	0	0
$909$	−49621.4	−1.81060
$910$	0	0
$911$	38706.9	1.40770	0.703851	−	0.710348i	$-0.251462\pi$
$911$	38706.9	1.40770	0.703851	+	0.710348i	$0.251462\pi$
$912$	0	0
$913$	800.542	0.0290187
$914$	0	0
$915$	−15760.7	−0.569436
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−23292.4	−0.836068	−0.418034	−	0.908431i	$-0.637281\pi$
$919$	−23292.4	−0.836068	−0.418034	+	0.908431i	$0.637281\pi$
$920$	0	0
$921$	−57619.6	−2.06149
$922$	0	0
$923$	14151.1	0.504649
$924$	0	0
$925$	−27979.1	−0.994537
$926$	0	0
$927$	61338.8	2.17328
$928$	0	0
$929$	42925.3	1.51597	0.757984	−	0.652273i	$-0.226184\pi$
$929$	42925.3	1.51597	0.757984	+	0.652273i	$0.226184\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−46751.0	−1.64047
$934$	0	0
$935$	−9993.95	−0.349558
$936$	0	0
$937$	44869.6	1.56438	0.782191	−	0.623039i	$-0.214102\pi$
$937$	44869.6	1.56438	0.782191	+	0.623039i	$0.214102\pi$
$938$	0	0
$939$	42940.4	1.49234
$940$	0	0
$941$	−12243.4	−0.424148	−0.212074	−	0.977254i	$-0.568022\pi$
$941$	−12243.4	−0.424148	−0.212074	+	0.977254i	$0.568022\pi$
$942$	0	0
$943$	11805.2	0.407668
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42027.1	−1.44213	−0.721065	−	0.692868i	$-0.756347\pi$
$947$	−42027.1	−1.44213	−0.721065	+	0.692868i	$0.756347\pi$
$948$	0	0
$949$	−66553.1	−2.27651
$950$	0	0
$951$	98903.6	3.37242
$952$	0	0
$953$	15757.3	0.535601	0.267801	−	0.963474i	$-0.413703\pi$
$953$	15757.3	0.535601	0.267801	+	0.963474i	$0.413703\pi$
$954$	0	0
$955$	−28347.8	−0.960536
$956$	0	0
$957$	14767.8	0.498823
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2464.25	−0.0827180
$962$	0	0
$963$	110896.	3.71089
$964$	0	0
$965$	−8500.17	−0.283555
$966$	0	0
$967$	−25863.4	−0.860093	−0.430046	−	0.902807i	$-0.641503\pi$
$967$	−25863.4	−0.860093	−0.430046	+	0.902807i	$0.641503\pi$
$968$	0	0
$969$	110078.	3.64935
$970$	0	0
$971$	47845.6	1.58129	0.790647	−	0.612272i	$-0.209744\pi$
$971$	47845.6	1.58129	0.790647	+	0.612272i	$0.209744\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−78897.5	−2.59153
$976$	0	0
$977$	54722.6	1.79195	0.895973	−	0.444108i	$-0.146479\pi$
$977$	54722.6	1.79195	0.895973	+	0.444108i	$0.146479\pi$
$978$	0	0
$979$	7221.06	0.235737
$980$	0	0
$981$	−51651.8	−1.68106
$982$	0	0
$983$	30062.6	0.975430	0.487715	−	0.873003i	$-0.337831\pi$
$983$	30062.6	0.975430	0.487715	+	0.873003i	$0.337831\pi$
$984$	0	0
$985$	14847.2	0.480275
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5357.43	0.172251
$990$	0	0
$991$	−37286.6	−1.19521	−0.597603	−	0.801792i	$-0.703880\pi$
$991$	−37286.6	−1.19521	−0.597603	+	0.801792i	$0.703880\pi$
$992$	0	0
$993$	29753.5	0.950855
$994$	0	0
$995$	24228.4	0.771953
$996$	0	0
$997$	58934.5	1.87209	0.936045	−	0.351881i	$-0.114458\pi$
$997$	58934.5	1.87209	0.936045	+	0.351881i	$0.114458\pi$
$998$	0	0
$999$	−116560.	−3.69150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.4.a.n.1.1	✓	4
4.3	odd	2		784.4.a.bh.1.4		4
7.2	even	3		392.4.i.o.361.4		8
7.3	odd	6		392.4.i.o.177.1		8
7.4	even	3		392.4.i.o.177.4		8
7.5	odd	6		392.4.i.o.361.1		8
7.6	odd	2	inner	392.4.a.n.1.4	yes	4
28.27	even	2		784.4.a.bh.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.n.1.1	✓	4	1.1	even	1	trivial
392.4.a.n.1.4	yes	4	7.6	odd	2	inner
392.4.i.o.177.1		8	7.3	odd	6
392.4.i.o.177.4		8	7.4	even	3
392.4.i.o.361.1		8	7.5	odd	6
392.4.i.o.361.4		8	7.2	even	3
784.4.a.bh.1.1		4	28.27	even	2
784.4.a.bh.1.4		4	4.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q-9.63797 q^{3} +5.91838 q^{5} +65.8904 q^{9} +O(q^{10})\)	\(q-9.63797 q^{3} +5.91838 q^{5} +65.8904 q^{9} -18.3699 q^{11} -90.9844 q^{13} -57.0412 q^{15} +91.9239 q^{17} -124.247 q^{19} +65.0412 q^{23} -89.9728 q^{25} -374.825 q^{27} +83.4110 q^{29} -165.308 q^{31} +177.048 q^{33} +310.973 q^{37} +876.904 q^{39} +181.504 q^{41} +82.3699 q^{43} +389.965 q^{45} +367.909 q^{47} -885.959 q^{51} +52.6574 q^{53} -108.720 q^{55} +1197.49 q^{57} +561.278 q^{59} +276.305 q^{61} -538.480 q^{65} -60.1368 q^{67} -626.865 q^{69} -155.534 q^{71} +731.478 q^{73} +867.155 q^{75} +290.384 q^{79} +1833.51 q^{81} -43.5791 q^{83} +544.041 q^{85} -803.913 q^{87} -393.093 q^{89} +1593.23 q^{93} -735.344 q^{95} +521.748 q^{97} -1210.40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 136 q^{9}+O(q^{10}) \) 4 * q + 136 * q^9	\( 4 q + 136 q^{9} - 116 q^{11} + 112 q^{15} - 80 q^{23} + 448 q^{25} + 36 q^{29} + 436 q^{37} + 2232 q^{39} + 372 q^{43} - 780 q^{51} + 976 q^{53} + 3812 q^{57} + 780 q^{65} - 1176 q^{67} - 2408 q^{71} + 56 q^{79} + 2104 q^{81} + 4940 q^{85} + 2376 q^{93} + 4032 q^{95} - 2588 q^{99}+O(q^{100}) \) 4 * q + 136 * q^9 - 116 * q^11 + 112 * q^15 - 80 * q^23 + 448 * q^25 + 36 * q^29 + 436 * q^37 + 2232 * q^39 + 372 * q^43 - 780 * q^51 + 976 * q^53 + 3812 * q^57 + 780 * q^65 - 1176 * q^67 - 2408 * q^71 + 56 * q^79 + 2104 * q^81 + 4940 * q^85 + 2376 * q^93 + 4032 * q^95 - 2588 * q^99

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