LMFDB - Embedded newform 4842.2.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 33x^{6} + 352x^{4} - 18x^{3} - 1229x^{2} + 178x + 108 $ x^8 - 33x^6 + 352x^4 - 18x^3 - 1229x^2 + 178*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1614)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.386375$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.386375 q^{5} +2.17487 q^{7} +1.00000 q^{8} -0.386375 q^{10} +2.08072 q^{11} +4.23070 q^{13} +2.17487 q^{14} +1.00000 q^{16} +1.79530 q^{17} +7.85768 q^{19} -0.386375 q^{20} +2.08072 q^{22} +4.06238 q^{23} -4.85071 q^{25} +4.23070 q^{26} +2.17487 q^{28} -6.13152 q^{29} +1.40766 q^{31} +1.00000 q^{32} +1.79530 q^{34} -0.840314 q^{35} -5.65299 q^{37} +7.85768 q^{38} -0.386375 q^{40} -8.47974 q^{41} +8.11821 q^{43} +2.08072 q^{44} +4.06238 q^{46} +2.74687 q^{47} -2.26996 q^{49} -4.85071 q^{50} +4.23070 q^{52} -9.83467 q^{53} -0.803938 q^{55} +2.17487 q^{56} -6.13152 q^{58} +5.01958 q^{59} -0.526805 q^{61} +1.40766 q^{62} +1.00000 q^{64} -1.63463 q^{65} -2.58263 q^{67} +1.79530 q^{68} -0.840314 q^{70} +2.23823 q^{71} -3.05440 q^{73} -5.65299 q^{74} +7.85768 q^{76} +4.52529 q^{77} +2.53118 q^{79} -0.386375 q^{80} -8.47974 q^{82} -5.95111 q^{83} -0.693660 q^{85} +8.11821 q^{86} +2.08072 q^{88} +5.61534 q^{89} +9.20120 q^{91} +4.06238 q^{92} +2.74687 q^{94} -3.03601 q^{95} +4.99827 q^{97} -2.26996 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.386375	−0.172792	−0.0863960	−	0.996261i	$-0.527535\pi$
$5$	−0.386375	−0.172792	−0.0863960	+	0.996261i	$0.527535\pi$
$6$	0	0
$7$	2.17487	0.822022	0.411011	−	0.911630i	$-0.365176\pi$
$7$	2.17487	0.822022	0.411011	+	0.911630i	$0.365176\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.386375	−0.122182
$11$	2.08072	0.627361	0.313680	−	0.949529i	$-0.398438\pi$
$11$	2.08072	0.627361	0.313680	+	0.949529i	$0.398438\pi$
$12$	0	0
$13$	4.23070	1.17338	0.586692	−	0.809810i	$-0.300430\pi$
$13$	4.23070	1.17338	0.586692	+	0.809810i	$0.300430\pi$
$14$	2.17487	0.581258
$15$	0	0
$16$	1.00000	0.250000
$17$	1.79530	0.435425	0.217713	−	0.976013i	$-0.430140\pi$
$17$	1.79530	0.435425	0.217713	+	0.976013i	$0.430140\pi$
$18$	0	0
$19$	7.85768	1.80268	0.901338	−	0.433116i	$-0.142586\pi$
$19$	7.85768	1.80268	0.901338	+	0.433116i	$0.142586\pi$
$20$	−0.386375	−0.0863960
$21$	0	0
$22$	2.08072	0.443611
$23$	4.06238	0.847065	0.423532	−	0.905881i	$-0.360790\pi$
$23$	4.06238	0.847065	0.423532	+	0.905881i	$0.360790\pi$
$24$	0	0
$25$	−4.85071	−0.970143
$26$	4.23070	0.829708
$27$	0	0
$28$	2.17487	0.411011
$29$	−6.13152	−1.13859	−0.569297	−	0.822132i	$-0.692785\pi$
$29$	−6.13152	−1.13859	−0.569297	+	0.822132i	$0.692785\pi$
$30$	0	0
$31$	1.40766	0.252823	0.126412	−	0.991978i	$-0.459654\pi$
$31$	1.40766	0.252823	0.126412	+	0.991978i	$0.459654\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.79530	0.307892
$35$	−0.840314	−0.142039
$36$	0	0
$37$	−5.65299	−0.929345	−0.464673	−	0.885483i	$-0.653828\pi$
$37$	−5.65299	−0.929345	−0.464673	+	0.885483i	$0.653828\pi$
$38$	7.85768	1.27468
$39$	0	0
$40$	−0.386375	−0.0610912
$41$	−8.47974	−1.32431	−0.662156	−	0.749366i	$-0.730358\pi$
$41$	−8.47974	−1.32431	−0.662156	+	0.749366i	$0.730358\pi$
$42$	0	0
$43$	8.11821	1.23802	0.619008	−	0.785385i	$-0.287535\pi$
$43$	8.11821	1.23802	0.619008	+	0.785385i	$0.287535\pi$
$44$	2.08072	0.313680
$45$	0	0
$46$	4.06238	0.598965
$47$	2.74687	0.400673	0.200336	−	0.979727i	$-0.435797\pi$
$47$	2.74687	0.400673	0.200336	+	0.979727i	$0.435797\pi$
$48$	0	0
$49$	−2.26996	−0.324279
$50$	−4.85071	−0.685995
$51$	0	0
$52$	4.23070	0.586692
$53$	−9.83467	−1.35090	−0.675448	−	0.737408i	$-0.736050\pi$
$53$	−9.83467	−1.35090	−0.675448	+	0.737408i	$0.736050\pi$
$54$	0	0
$55$	−0.803938	−0.108403
$56$	2.17487	0.290629
$57$	0	0
$58$	−6.13152	−0.805108
$59$	5.01958	0.653494	0.326747	−	0.945112i	$-0.394047\pi$
$59$	5.01958	0.653494	0.326747	+	0.945112i	$0.394047\pi$
$60$	0	0
$61$	−0.526805	−0.0674504	−0.0337252	−	0.999431i	$-0.510737\pi$
$61$	−0.526805	−0.0674504	−0.0337252	+	0.999431i	$0.510737\pi$
$62$	1.40766	0.178773
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.63463	−0.202751
$66$	0	0
$67$	−2.58263	−0.315519	−0.157760	−	0.987478i	$-0.550427\pi$
$67$	−2.58263	−0.315519	−0.157760	+	0.987478i	$0.550427\pi$
$68$	1.79530	0.217713
$69$	0	0
$70$	−0.840314	−0.100437
$71$	2.23823	0.265630	0.132815	−	0.991141i	$-0.457598\pi$
$71$	2.23823	0.265630	0.132815	+	0.991141i	$0.457598\pi$
$72$	0	0
$73$	−3.05440	−0.357491	−0.178745	−	0.983895i	$-0.557204\pi$
$73$	−3.05440	−0.357491	−0.178745	+	0.983895i	$0.557204\pi$
$74$	−5.65299	−0.657146
$75$	0	0
$76$	7.85768	0.901338
$77$	4.52529	0.515704
$78$	0	0
$79$	2.53118	0.284780	0.142390	−	0.989811i	$-0.454521\pi$
$79$	2.53118	0.284780	0.142390	+	0.989811i	$0.454521\pi$
$80$	−0.386375	−0.0431980
$81$	0	0
$82$	−8.47974	−0.936430
$83$	−5.95111	−0.653219	−0.326610	−	0.945159i	$-0.605906\pi$
$83$	−5.95111	−0.653219	−0.326610	+	0.945159i	$0.605906\pi$
$84$	0	0
$85$	−0.693660	−0.0752380
$86$	8.11821	0.875409
$87$	0	0
$88$	2.08072	0.221805
$89$	5.61534	0.595225	0.297613	−	0.954687i	$-0.403810\pi$
$89$	5.61534	0.595225	0.297613	+	0.954687i	$0.403810\pi$
$90$	0	0
$91$	9.20120	0.964548
$92$	4.06238	0.423532
$93$	0	0
$94$	2.74687	0.283318
$95$	−3.03601	−0.311488
$96$	0	0
$97$	4.99827	0.507498	0.253749	−	0.967270i	$-0.418336\pi$
$97$	4.99827	0.507498	0.253749	+	0.967270i	$0.418336\pi$
$98$	−2.26996	−0.229300
$99$	0	0
$100$	−4.85071	−0.485071
$101$	−4.15185	−0.413125	−0.206562	−	0.978433i	$-0.566228\pi$
$101$	−4.15185	−0.413125	−0.206562	+	0.978433i	$0.566228\pi$
$102$	0	0
$103$	0.906787	0.0893484	0.0446742	−	0.999002i	$-0.485775\pi$
$103$	0.906787	0.0893484	0.0446742	+	0.999002i	$0.485775\pi$
$104$	4.23070	0.414854
$105$	0	0
$106$	−9.83467	−0.955228
$107$	8.41797	0.813796	0.406898	−	0.913474i	$-0.366610\pi$
$107$	8.41797	0.813796	0.406898	+	0.913474i	$0.366610\pi$
$108$	0	0
$109$	15.5055	1.48516	0.742580	−	0.669758i	$-0.233602\pi$
$109$	15.5055	1.48516	0.742580	+	0.669758i	$0.233602\pi$
$110$	−0.803938	−0.0766525
$111$	0	0
$112$	2.17487	0.205506
$113$	−6.25209	−0.588147	−0.294074	−	0.955783i	$-0.595011\pi$
$113$	−6.25209	−0.588147	−0.294074	+	0.955783i	$0.595011\pi$
$114$	0	0
$115$	−1.56960	−0.146366
$116$	−6.13152	−0.569297
$117$	0	0
$118$	5.01958	0.462090
$119$	3.90455	0.357929
$120$	0	0
$121$	−6.67061	−0.606419
$122$	−0.526805	−0.0476947
$123$	0	0
$124$	1.40766	0.126412
$125$	3.80607	0.340425
$126$	0	0
$127$	−3.62484	−0.321653	−0.160826	−	0.986983i	$-0.551416\pi$
$127$	−3.62484	−0.321653	−0.160826	+	0.986983i	$0.551416\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.63463	−0.143367
$131$	0.359586	0.0314171	0.0157086	−	0.999877i	$-0.495000\pi$
$131$	0.359586	0.0314171	0.0157086	+	0.999877i	$0.495000\pi$
$132$	0	0
$133$	17.0894	1.48184
$134$	−2.58263	−0.223106
$135$	0	0
$136$	1.79530	0.153946
$137$	8.02051	0.685238	0.342619	−	0.939474i	$-0.388686\pi$
$137$	8.02051	0.685238	0.342619	+	0.939474i	$0.388686\pi$
$138$	0	0
$139$	8.37714	0.710539	0.355270	−	0.934764i	$-0.384389\pi$
$139$	8.37714	0.710539	0.355270	+	0.934764i	$0.384389\pi$
$140$	−0.840314	−0.0710195
$141$	0	0
$142$	2.23823	0.187828
$143$	8.80289	0.736135
$144$	0	0
$145$	2.36907	0.196740
$146$	−3.05440	−0.252784
$147$	0	0
$148$	−5.65299	−0.464673
$149$	−1.95478	−0.160142	−0.0800708	−	0.996789i	$-0.525515\pi$
$149$	−1.95478	−0.160142	−0.0800708	+	0.996789i	$0.525515\pi$
$150$	0	0
$151$	9.43437	0.767758	0.383879	−	0.923383i	$-0.374588\pi$
$151$	9.43437	0.767758	0.383879	+	0.923383i	$0.374588\pi$
$152$	7.85768	0.637342
$153$	0	0
$154$	4.52529	0.364658
$155$	−0.543885	−0.0436859
$156$	0	0
$157$	14.2014	1.13339	0.566696	−	0.823927i	$-0.308221\pi$
$157$	14.2014	1.13339	0.566696	+	0.823927i	$0.308221\pi$
$158$	2.53118	0.201370
$159$	0	0
$160$	−0.386375	−0.0305456
$161$	8.83514	0.696306
$162$	0	0
$163$	−0.0974752	−0.00763485	−0.00381742	−	0.999993i	$-0.501215\pi$
$163$	−0.0974752	−0.00763485	−0.00381742	+	0.999993i	$0.501215\pi$
$164$	−8.47974	−0.662156
$165$	0	0
$166$	−5.95111	−0.461896
$167$	10.3290	0.799280	0.399640	−	0.916672i	$-0.369135\pi$
$167$	10.3290	0.799280	0.399640	+	0.916672i	$0.369135\pi$
$168$	0	0
$169$	4.89879	0.376830
$170$	−0.693660	−0.0532013
$171$	0	0
$172$	8.11821	0.619008
$173$	2.63105	0.200035	0.100017	−	0.994986i	$-0.468110\pi$
$173$	2.63105	0.200035	0.100017	+	0.994986i	$0.468110\pi$
$174$	0	0
$175$	−10.5497	−0.797479
$176$	2.08072	0.156840
$177$	0	0
$178$	5.61534	0.420888
$179$	−12.4561	−0.931010	−0.465505	−	0.885045i	$-0.654127\pi$
$179$	−12.4561	−0.931010	−0.465505	+	0.885045i	$0.654127\pi$
$180$	0	0
$181$	15.3173	1.13852	0.569261	−	0.822157i	$-0.307229\pi$
$181$	15.3173	1.13852	0.569261	+	0.822157i	$0.307229\pi$
$182$	9.20120	0.682038
$183$	0	0
$184$	4.06238	0.299483
$185$	2.18417	0.160584
$186$	0	0
$187$	3.73552	0.273169
$188$	2.74687	0.200336
$189$	0	0
$190$	−3.03601	−0.220255
$191$	4.43481	0.320892	0.160446	−	0.987045i	$-0.448707\pi$
$191$	4.43481	0.320892	0.160446	+	0.987045i	$0.448707\pi$
$192$	0	0
$193$	−4.69578	−0.338010	−0.169005	−	0.985615i	$-0.554055\pi$
$193$	−4.69578	−0.338010	−0.169005	+	0.985615i	$0.554055\pi$
$194$	4.99827	0.358855
$195$	0	0
$196$	−2.26996	−0.162140
$197$	4.93353	0.351500	0.175750	−	0.984435i	$-0.443765\pi$
$197$	4.93353	0.351500	0.175750	+	0.984435i	$0.443765\pi$
$198$	0	0
$199$	−19.3169	−1.36934	−0.684671	−	0.728853i	$-0.740054\pi$
$199$	−19.3169	−1.36934	−0.684671	+	0.728853i	$0.740054\pi$
$200$	−4.85071	−0.342997
$201$	0	0
$202$	−4.15185	−0.292123
$203$	−13.3352	−0.935950
$204$	0	0
$205$	3.27636	0.228831
$206$	0.906787	0.0631788
$207$	0	0
$208$	4.23070	0.293346
$209$	16.3496	1.13093
$210$	0	0
$211$	1.23621	0.0851040	0.0425520	−	0.999094i	$-0.486451\pi$
$211$	1.23621	0.0851040	0.0425520	+	0.999094i	$0.486451\pi$
$212$	−9.83467	−0.675448
$213$	0	0
$214$	8.41797	0.575441
$215$	−3.13667	−0.213919
$216$	0	0
$217$	3.06147	0.207826
$218$	15.5055	1.05017
$219$	0	0
$220$	−0.803938	−0.0542015
$221$	7.59538	0.510921
$222$	0	0
$223$	13.4472	0.900489	0.450244	−	0.892905i	$-0.351337\pi$
$223$	13.4472	0.900489	0.450244	+	0.892905i	$0.351337\pi$
$224$	2.17487	0.145314
$225$	0	0
$226$	−6.25209	−0.415883
$227$	−17.3344	−1.15053	−0.575263	−	0.817968i	$-0.695100\pi$
$227$	−17.3344	−1.15053	−0.575263	+	0.817968i	$0.695100\pi$
$228$	0	0
$229$	−24.9415	−1.64818	−0.824090	−	0.566459i	$-0.808313\pi$
$229$	−24.9415	−1.64818	−0.824090	+	0.566459i	$0.808313\pi$
$230$	−1.56960	−0.103496
$231$	0	0
$232$	−6.13152	−0.402554
$233$	−18.0979	−1.18563	−0.592816	−	0.805338i	$-0.701984\pi$
$233$	−18.0979	−1.18563	−0.592816	+	0.805338i	$0.701984\pi$
$234$	0	0
$235$	−1.06132	−0.0692331
$236$	5.01958	0.326747
$237$	0	0
$238$	3.90455	0.253094
$239$	−14.6655	−0.948633	−0.474317	−	0.880354i	$-0.657305\pi$
$239$	−14.6655	−0.948633	−0.474317	+	0.880354i	$0.657305\pi$
$240$	0	0
$241$	−0.0577845	−0.00372223	−0.00186111	−	0.999998i	$-0.500592\pi$
$241$	−0.0577845	−0.00372223	−0.00186111	+	0.999998i	$0.500592\pi$
$242$	−6.67061	−0.428803
$243$	0	0
$244$	−0.526805	−0.0337252
$245$	0.877054	0.0560329
$246$	0	0
$247$	33.2435	2.11523
$248$	1.40766	0.0893865
$249$	0	0
$250$	3.80607	0.240717
$251$	24.5122	1.54720	0.773599	−	0.633675i	$-0.218454\pi$
$251$	24.5122	1.54720	0.773599	+	0.633675i	$0.218454\pi$
$252$	0	0
$253$	8.45268	0.531415
$254$	−3.62484	−0.227443
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.99285	0.186689	0.0933445	−	0.995634i	$-0.470244\pi$
$257$	2.99285	0.186689	0.0933445	+	0.995634i	$0.470244\pi$
$258$	0	0
$259$	−12.2945	−0.763943
$260$	−1.63463	−0.101376
$261$	0	0
$262$	0.359586	0.0222153
$263$	−22.6403	−1.39606	−0.698031	−	0.716068i	$-0.745940\pi$
$263$	−22.6403	−1.39606	−0.698031	+	0.716068i	$0.745940\pi$
$264$	0	0
$265$	3.79987	0.233424
$266$	17.0894	1.04782
$267$	0	0
$268$	−2.58263	−0.157760
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−0.615051	−0.0373617	−0.0186808	−	0.999825i	$-0.505947\pi$
$271$	−0.615051	−0.0373617	−0.0186808	+	0.999825i	$0.505947\pi$
$272$	1.79530	0.108856
$273$	0	0
$274$	8.02051	0.484537
$275$	−10.0930	−0.608629
$276$	0	0
$277$	−4.21492	−0.253250	−0.126625	−	0.991951i	$-0.540414\pi$
$277$	−4.21492	−0.253250	−0.126625	+	0.991951i	$0.540414\pi$
$278$	8.37714	0.502427
$279$	0	0
$280$	−0.840314	−0.0502184
$281$	2.85454	0.170288	0.0851439	−	0.996369i	$-0.472865\pi$
$281$	2.85454	0.170288	0.0851439	+	0.996369i	$0.472865\pi$
$282$	0	0
$283$	−28.2027	−1.67647	−0.838237	−	0.545306i	$-0.816413\pi$
$283$	−28.2027	−1.67647	−0.838237	+	0.545306i	$0.816413\pi$
$284$	2.23823	0.132815
$285$	0	0
$286$	8.80289	0.520526
$287$	−18.4423	−1.08861
$288$	0	0
$289$	−13.7769	−0.810405
$290$	2.36907	0.139116
$291$	0	0
$292$	−3.05440	−0.178745
$293$	6.52394	0.381133	0.190566	−	0.981674i	$-0.438968\pi$
$293$	6.52394	0.381133	0.190566	+	0.981674i	$0.438968\pi$
$294$	0	0
$295$	−1.93944	−0.112919
$296$	−5.65299	−0.328573
$297$	0	0
$298$	−1.95478	−0.113237
$299$	17.1867	0.993933
$300$	0	0
$301$	17.6560	1.01768
$302$	9.43437	0.542887
$303$	0	0
$304$	7.85768	0.450669
$305$	0.203544	0.0116549
$306$	0	0
$307$	−22.8518	−1.30422	−0.652111	−	0.758123i	$-0.726116\pi$
$307$	−22.8518	−1.30422	−0.652111	+	0.758123i	$0.726116\pi$
$308$	4.52529	0.257852
$309$	0	0
$310$	−0.543885	−0.0308906
$311$	16.4962	0.935412	0.467706	−	0.883884i	$-0.345081\pi$
$311$	16.4962	0.935412	0.467706	+	0.883884i	$0.345081\pi$
$312$	0	0
$313$	−1.78952	−0.101149	−0.0505747	−	0.998720i	$-0.516105\pi$
$313$	−1.78952	−0.101149	−0.0505747	+	0.998720i	$0.516105\pi$
$314$	14.2014	0.801429
$315$	0	0
$316$	2.53118	0.142390
$317$	−5.57591	−0.313174	−0.156587	−	0.987664i	$-0.550049\pi$
$317$	−5.57591	−0.313174	−0.156587	+	0.987664i	$0.550049\pi$
$318$	0	0
$319$	−12.7580	−0.714309
$320$	−0.386375	−0.0215990
$321$	0	0
$322$	8.83514	0.492363
$323$	14.1069	0.784930
$324$	0	0
$325$	−20.5219	−1.13835
$326$	−0.0974752	−0.00539865
$327$	0	0
$328$	−8.47974	−0.468215
$329$	5.97408	0.329362
$330$	0	0
$331$	−27.6569	−1.52016	−0.760080	−	0.649829i	$-0.774840\pi$
$331$	−27.6569	−1.52016	−0.760080	+	0.649829i	$0.774840\pi$
$332$	−5.95111	−0.326610
$333$	0	0
$334$	10.3290	0.565176
$335$	0.997865	0.0545192
$336$	0	0
$337$	17.2086	0.937412	0.468706	−	0.883354i	$-0.344720\pi$
$337$	17.2086	0.937412	0.468706	+	0.883354i	$0.344720\pi$
$338$	4.89879	0.266459
$339$	0	0
$340$	−0.693660	−0.0376190
$341$	2.92895	0.158611
$342$	0	0
$343$	−20.1609	−1.08859
$344$	8.11821	0.437705
$345$	0	0
$346$	2.63105	0.141446
$347$	26.4564	1.42026	0.710128	−	0.704073i	$-0.248637\pi$
$347$	26.4564	1.42026	0.710128	+	0.704073i	$0.248637\pi$
$348$	0	0
$349$	−2.31155	−0.123734	−0.0618671	−	0.998084i	$-0.519705\pi$
$349$	−2.31155	−0.123734	−0.0618671	+	0.998084i	$0.519705\pi$
$350$	−10.5497	−0.563903
$351$	0	0
$352$	2.08072	0.110903
$353$	6.16551	0.328157	0.164079	−	0.986447i	$-0.447535\pi$
$353$	6.16551	0.328157	0.164079	+	0.986447i	$0.447535\pi$
$354$	0	0
$355$	−0.864797	−0.0458987
$356$	5.61534	0.297613
$357$	0	0
$358$	−12.4561	−0.658323
$359$	23.8004	1.25614	0.628068	−	0.778158i	$-0.283846\pi$
$359$	23.8004	1.25614	0.628068	+	0.778158i	$0.283846\pi$
$360$	0	0
$361$	42.7432	2.24964
$362$	15.3173	0.805057
$363$	0	0
$364$	9.20120	0.482274
$365$	1.18014	0.0617716
$366$	0	0
$367$	−37.6968	−1.96776	−0.983879	−	0.178836i	$-0.942767\pi$
$367$	−37.6968	−1.96776	−0.983879	+	0.178836i	$0.942767\pi$
$368$	4.06238	0.211766
$369$	0	0
$370$	2.18417	0.113550
$371$	−21.3891	−1.11047
$372$	0	0
$373$	−9.78397	−0.506595	−0.253297	−	0.967388i	$-0.581515\pi$
$373$	−9.78397	−0.506595	−0.253297	+	0.967388i	$0.581515\pi$
$374$	3.73552	0.193159
$375$	0	0
$376$	2.74687	0.141659
$377$	−25.9406	−1.33601
$378$	0	0
$379$	14.3303	0.736099	0.368050	−	0.929806i	$-0.380026\pi$
$379$	14.3303	0.736099	0.368050	+	0.929806i	$0.380026\pi$
$380$	−3.03601	−0.155744
$381$	0	0
$382$	4.43481	0.226905
$383$	23.7847	1.21534	0.607671	−	0.794189i	$-0.292104\pi$
$383$	23.7847	1.21534	0.607671	+	0.794189i	$0.292104\pi$
$384$	0	0
$385$	−1.74846	−0.0891096
$386$	−4.69578	−0.239009
$387$	0	0
$388$	4.99827	0.253749
$389$	8.41143	0.426476	0.213238	−	0.977000i	$-0.431599\pi$
$389$	8.41143	0.426476	0.213238	+	0.977000i	$0.431599\pi$
$390$	0	0
$391$	7.29321	0.368833
$392$	−2.26996	−0.114650
$393$	0	0
$394$	4.93353	0.248548
$395$	−0.977983	−0.0492077
$396$	0	0
$397$	−4.79537	−0.240673	−0.120336	−	0.992733i	$-0.538397\pi$
$397$	−4.79537	−0.240673	−0.120336	+	0.992733i	$0.538397\pi$
$398$	−19.3169	−0.968270
$399$	0	0
$400$	−4.85071	−0.242536
$401$	−16.2879	−0.813380	−0.406690	−	0.913566i	$-0.633317\pi$
$401$	−16.2879	−0.813380	−0.406690	+	0.913566i	$0.633317\pi$
$402$	0	0
$403$	5.95538	0.296659
$404$	−4.15185	−0.206562
$405$	0	0
$406$	−13.3352	−0.661817
$407$	−11.7623	−0.583035
$408$	0	0
$409$	5.13956	0.254135	0.127067	−	0.991894i	$-0.459444\pi$
$409$	5.13956	0.254135	0.127067	+	0.991894i	$0.459444\pi$
$410$	3.27636	0.161808
$411$	0	0
$412$	0.906787	0.0446742
$413$	10.9169	0.537187
$414$	0	0
$415$	2.29936	0.112871
$416$	4.23070	0.207427
$417$	0	0
$418$	16.3496	0.799687
$419$	6.07355	0.296713	0.148356	−	0.988934i	$-0.452602\pi$
$419$	6.07355	0.296713	0.148356	+	0.988934i	$0.452602\pi$
$420$	0	0
$421$	−35.2344	−1.71722	−0.858610	−	0.512630i	$-0.828671\pi$
$421$	−35.2344	−1.71722	−0.858610	+	0.512630i	$0.828671\pi$
$422$	1.23621	0.0601776
$423$	0	0
$424$	−9.83467	−0.477614
$425$	−8.70850	−0.422424
$426$	0	0
$427$	−1.14573	−0.0554458
$428$	8.41797	0.406898
$429$	0	0
$430$	−3.13667	−0.151264
$431$	25.3935	1.22316	0.611580	−	0.791183i	$-0.290534\pi$
$431$	25.3935	1.22316	0.611580	+	0.791183i	$0.290534\pi$
$432$	0	0
$433$	−30.3331	−1.45772	−0.728858	−	0.684664i	$-0.759949\pi$
$433$	−30.3331	−1.45772	−0.728858	+	0.684664i	$0.759949\pi$
$434$	3.06147	0.146955
$435$	0	0
$436$	15.5055	0.742580
$437$	31.9209	1.52698
$438$	0	0
$439$	10.6888	0.510148	0.255074	−	0.966922i	$-0.417900\pi$
$439$	10.6888	0.510148	0.255074	+	0.966922i	$0.417900\pi$
$440$	−0.803938	−0.0383262
$441$	0	0
$442$	7.59538	0.361276
$443$	−3.86159	−0.183470	−0.0917348	−	0.995783i	$-0.529241\pi$
$443$	−3.86159	−0.183470	−0.0917348	+	0.995783i	$0.529241\pi$
$444$	0	0
$445$	−2.16963	−0.102850
$446$	13.4472	0.636742
$447$	0	0
$448$	2.17487	0.102753
$449$	3.93355	0.185636	0.0928179	−	0.995683i	$-0.470413\pi$
$449$	3.93355	0.185636	0.0928179	+	0.995683i	$0.470413\pi$
$450$	0	0
$451$	−17.6440	−0.830821
$452$	−6.25209	−0.294074
$453$	0	0
$454$	−17.3344	−0.813545
$455$	−3.55511	−0.166666
$456$	0	0
$457$	−18.4341	−0.862311	−0.431156	−	0.902278i	$-0.641894\pi$
$457$	−18.4341	−0.862311	−0.431156	+	0.902278i	$0.641894\pi$
$458$	−24.9415	−1.16544
$459$	0	0
$460$	−1.56960	−0.0731831
$461$	4.61243	0.214822	0.107411	−	0.994215i	$-0.465744\pi$
$461$	4.61243	0.214822	0.107411	+	0.994215i	$0.465744\pi$
$462$	0	0
$463$	−11.2551	−0.523067	−0.261533	−	0.965194i	$-0.584228\pi$
$463$	−11.2551	−0.523067	−0.261533	+	0.965194i	$0.584228\pi$
$464$	−6.13152	−0.284649
$465$	0	0
$466$	−18.0979	−0.838368
$467$	20.1395	0.931946	0.465973	−	0.884799i	$-0.345704\pi$
$467$	20.1395	0.931946	0.465973	+	0.884799i	$0.345704\pi$
$468$	0	0
$469$	−5.61689	−0.259364
$470$	−1.06132	−0.0489552
$471$	0	0
$472$	5.01958	0.231045
$473$	16.8917	0.776682
$474$	0	0
$475$	−38.1154	−1.74885
$476$	3.90455	0.178965
$477$	0	0
$478$	−14.6655	−0.670785
$479$	−16.0705	−0.734278	−0.367139	−	0.930166i	$-0.619663\pi$
$479$	−16.0705	−0.734278	−0.367139	+	0.930166i	$0.619663\pi$
$480$	0	0
$481$	−23.9161	−1.09048
$482$	−0.0577845	−0.00263201
$483$	0	0
$484$	−6.67061	−0.303209
$485$	−1.93121	−0.0876916
$486$	0	0
$487$	−36.7241	−1.66413	−0.832064	−	0.554679i	$-0.812841\pi$
$487$	−36.7241	−1.66413	−0.832064	+	0.554679i	$0.812841\pi$
$488$	−0.526805	−0.0238473
$489$	0	0
$490$	0.877054	0.0396213
$491$	29.1923	1.31743	0.658716	−	0.752392i	$-0.271100\pi$
$491$	29.1923	1.31743	0.658716	+	0.752392i	$0.271100\pi$
$492$	0	0
$493$	−11.0079	−0.495773
$494$	33.2435	1.49569
$495$	0	0
$496$	1.40766	0.0632058
$497$	4.86786	0.218353
$498$	0	0
$499$	28.9874	1.29766	0.648828	−	0.760935i	$-0.275259\pi$
$499$	28.9874	1.29766	0.648828	+	0.760935i	$0.275259\pi$
$500$	3.80607	0.170213
$501$	0	0
$502$	24.5122	1.09403
$503$	42.8530	1.91072	0.955361	−	0.295440i	$-0.0954663\pi$
$503$	42.8530	1.91072	0.955361	+	0.295440i	$0.0954663\pi$
$504$	0	0
$505$	1.60417	0.0713847
$506$	8.45268	0.375767
$507$	0	0
$508$	−3.62484	−0.160826
$509$	6.93676	0.307466	0.153733	−	0.988112i	$-0.450870\pi$
$509$	6.93676	0.307466	0.153733	+	0.988112i	$0.450870\pi$
$510$	0	0
$511$	−6.64292	−0.293866
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.99285	0.132009
$515$	−0.350360	−0.0154387
$516$	0	0
$517$	5.71547	0.251366
$518$	−12.2945	−0.540189
$519$	0	0
$520$	−1.63463	−0.0716835
$521$	−22.8021	−0.998977	−0.499489	−	0.866320i	$-0.666479\pi$
$521$	−22.8021	−0.998977	−0.499489	+	0.866320i	$0.666479\pi$
$522$	0	0
$523$	−5.50512	−0.240722	−0.120361	−	0.992730i	$-0.538405\pi$
$523$	−5.50512	−0.240722	−0.120361	+	0.992730i	$0.538405\pi$
$524$	0.359586	0.0157086
$525$	0	0
$526$	−22.6403	−0.987164
$527$	2.52718	0.110086
$528$	0	0
$529$	−6.49706	−0.282481
$530$	3.79987	0.165056
$531$	0	0
$532$	17.0894	0.740920
$533$	−35.8752	−1.55393
$534$	0	0
$535$	−3.25249	−0.140618
$536$	−2.58263	−0.111553
$537$	0	0
$538$	1.00000	0.0431131
$539$	−4.72314	−0.203440
$540$	0	0
$541$	21.7806	0.936423	0.468212	−	0.883616i	$-0.344899\pi$
$541$	21.7806	0.936423	0.468212	+	0.883616i	$0.344899\pi$
$542$	−0.615051	−0.0264187
$543$	0	0
$544$	1.79530	0.0769730
$545$	−5.99094	−0.256624
$546$	0	0
$547$	12.0368	0.514655	0.257327	−	0.966324i	$-0.417158\pi$
$547$	12.0368	0.514655	0.257327	+	0.966324i	$0.417158\pi$
$548$	8.02051	0.342619
$549$	0	0
$550$	−10.0930	−0.430366
$551$	−48.1795	−2.05252
$552$	0	0
$553$	5.50497	0.234095
$554$	−4.21492	−0.179075
$555$	0	0
$556$	8.37714	0.355270
$557$	−24.2067	−1.02567	−0.512835	−	0.858487i	$-0.671405\pi$
$557$	−24.2067	−1.02567	−0.512835	+	0.858487i	$0.671405\pi$
$558$	0	0
$559$	34.3457	1.45267
$560$	−0.840314	−0.0355097
$561$	0	0
$562$	2.85454	0.120412
$563$	39.3050	1.65651	0.828253	−	0.560354i	$-0.189335\pi$
$563$	39.3050	1.65651	0.828253	+	0.560354i	$0.189335\pi$
$564$	0	0
$565$	2.41565	0.101627
$566$	−28.2027	−1.18545
$567$	0	0
$568$	2.23823	0.0939142
$569$	−17.2522	−0.723249	−0.361624	−	0.932324i	$-0.617778\pi$
$569$	−17.2522	−0.723249	−0.361624	+	0.932324i	$0.617778\pi$
$570$	0	0
$571$	−25.0795	−1.04954	−0.524772	−	0.851243i	$-0.675850\pi$
$571$	−25.0795	−1.04954	−0.524772	+	0.851243i	$0.675850\pi$
$572$	8.80289	0.368067
$573$	0	0
$574$	−18.4423	−0.769766
$575$	−19.7054	−0.821774
$576$	0	0
$577$	40.9505	1.70479	0.852397	−	0.522895i	$-0.175148\pi$
$577$	40.9505	1.70479	0.852397	+	0.522895i	$0.175148\pi$
$578$	−13.7769	−0.573043
$579$	0	0
$580$	2.36907	0.0983701
$581$	−12.9429	−0.536961
$582$	0	0
$583$	−20.4632	−0.847499
$584$	−3.05440	−0.126392
$585$	0	0
$586$	6.52394	0.269502
$587$	−9.21317	−0.380268	−0.190134	−	0.981758i	$-0.560892\pi$
$587$	−9.21317	−0.380268	−0.190134	+	0.981758i	$0.560892\pi$
$588$	0	0
$589$	11.0609	0.455758
$590$	−1.93944	−0.0798455
$591$	0	0
$592$	−5.65299	−0.232336
$593$	9.46289	0.388594	0.194297	−	0.980943i	$-0.437757\pi$
$593$	9.46289	0.388594	0.194297	+	0.980943i	$0.437757\pi$
$594$	0	0
$595$	−1.50862	−0.0618473
$596$	−1.95478	−0.0800708
$597$	0	0
$598$	17.1867	0.702816
$599$	−16.6993	−0.682315	−0.341157	−	0.940006i	$-0.610819\pi$
$599$	−16.6993	−0.682315	−0.341157	+	0.940006i	$0.610819\pi$
$600$	0	0
$601$	44.4943	1.81496	0.907479	−	0.420097i	$-0.138004\pi$
$601$	44.4943	1.81496	0.907479	+	0.420097i	$0.138004\pi$
$602$	17.6560	0.719606
$603$	0	0
$604$	9.43437	0.383879
$605$	2.57735	0.104784
$606$	0	0
$607$	−27.0420	−1.09760	−0.548801	−	0.835953i	$-0.684915\pi$
$607$	−27.0420	−1.09760	−0.548801	+	0.835953i	$0.684915\pi$
$608$	7.85768	0.318671
$609$	0	0
$610$	0.203544	0.00824126
$611$	11.6212	0.470143
$612$	0	0
$613$	−7.52962	−0.304118	−0.152059	−	0.988371i	$-0.548590\pi$
$613$	−7.52962	−0.304118	−0.152059	+	0.988371i	$0.548590\pi$
$614$	−22.8518	−0.922224
$615$	0	0
$616$	4.52529	0.182329
$617$	−25.0161	−1.00711	−0.503555	−	0.863963i	$-0.667975\pi$
$617$	−25.0161	−1.00711	−0.503555	+	0.863963i	$0.667975\pi$
$618$	0	0
$619$	0.146043	0.00586997	0.00293499	−	0.999996i	$-0.499066\pi$
$619$	0.146043	0.00586997	0.00293499	+	0.999996i	$0.499066\pi$
$620$	−0.543885	−0.0218429
$621$	0	0
$622$	16.4962	0.661437
$623$	12.2126	0.489288
$624$	0	0
$625$	22.7830	0.911320
$626$	−1.78952	−0.0715235
$627$	0	0
$628$	14.2014	0.566696
$629$	−10.1488	−0.404660
$630$	0	0
$631$	−17.2298	−0.685908	−0.342954	−	0.939352i	$-0.611428\pi$
$631$	−17.2298	−0.685908	−0.342954	+	0.939352i	$0.611428\pi$
$632$	2.53118	0.100685
$633$	0	0
$634$	−5.57591	−0.221448
$635$	1.40055	0.0555790
$636$	0	0
$637$	−9.60349	−0.380504
$638$	−12.7580	−0.505093
$639$	0	0
$640$	−0.386375	−0.0152728
$641$	−27.4878	−1.08570	−0.542852	−	0.839828i	$-0.682656\pi$
$641$	−27.4878	−1.08570	−0.542852	+	0.839828i	$0.682656\pi$
$642$	0	0
$643$	−9.36764	−0.369423	−0.184712	−	0.982793i	$-0.559135\pi$
$643$	−9.36764	−0.369423	−0.184712	+	0.982793i	$0.559135\pi$
$644$	8.83514	0.348153
$645$	0	0
$646$	14.1069	0.555030
$647$	−10.0619	−0.395574	−0.197787	−	0.980245i	$-0.563375\pi$
$647$	−10.0619	−0.395574	−0.197787	+	0.980245i	$0.563375\pi$
$648$	0	0
$649$	10.4443	0.409976
$650$	−20.5219	−0.804935
$651$	0	0
$652$	−0.0974752	−0.00381742
$653$	29.0482	1.13675	0.568373	−	0.822771i	$-0.307573\pi$
$653$	29.0482	1.13675	0.568373	+	0.822771i	$0.307573\pi$
$654$	0	0
$655$	−0.138935	−0.00542863
$656$	−8.47974	−0.331078
$657$	0	0
$658$	5.97408	0.232894
$659$	−34.5012	−1.34398	−0.671989	−	0.740561i	$-0.734560\pi$
$659$	−34.5012	−1.34398	−0.671989	+	0.740561i	$0.734560\pi$
$660$	0	0
$661$	−19.5744	−0.761358	−0.380679	−	0.924707i	$-0.624310\pi$
$661$	−19.5744	−0.761358	−0.380679	+	0.924707i	$0.624310\pi$
$662$	−27.6569	−1.07492
$663$	0	0
$664$	−5.95111	−0.230948
$665$	−6.60292	−0.256050
$666$	0	0
$667$	−24.9086	−0.964464
$668$	10.3290	0.399640
$669$	0	0
$670$	0.997865	0.0385509
$671$	−1.09613	−0.0423157
$672$	0	0
$673$	18.4387	0.710758	0.355379	−	0.934722i	$-0.384352\pi$
$673$	18.4387	0.710758	0.355379	+	0.934722i	$0.384352\pi$
$674$	17.2086	0.662850
$675$	0	0
$676$	4.89879	0.188415
$677$	−6.59421	−0.253436	−0.126718	−	0.991939i	$-0.540444\pi$
$677$	−6.59421	−0.253436	−0.126718	+	0.991939i	$0.540444\pi$
$678$	0	0
$679$	10.8706	0.417174
$680$	−0.693660	−0.0266006
$681$	0	0
$682$	2.92895	0.112155
$683$	−21.4509	−0.820795	−0.410397	−	0.911907i	$-0.634610\pi$
$683$	−21.4509	−0.820795	−0.410397	+	0.911907i	$0.634610\pi$
$684$	0	0
$685$	−3.09892	−0.118404
$686$	−20.1609	−0.769747
$687$	0	0
$688$	8.11821	0.309504
$689$	−41.6075	−1.58512
$690$	0	0
$691$	−17.6165	−0.670164	−0.335082	−	0.942189i	$-0.608764\pi$
$691$	−17.6165	−0.670164	−0.335082	+	0.942189i	$0.608764\pi$
$692$	2.63105	0.100017
$693$	0	0
$694$	26.4564	1.00427
$695$	−3.23671	−0.122776
$696$	0	0
$697$	−15.2237	−0.576639
$698$	−2.31155	−0.0874933
$699$	0	0
$700$	−10.5497	−0.398740
$701$	−24.4735	−0.924353	−0.462176	−	0.886788i	$-0.652931\pi$
$701$	−24.4735	−0.924353	−0.462176	+	0.886788i	$0.652931\pi$
$702$	0	0
$703$	−44.4194	−1.67531
$704$	2.08072	0.0784201
$705$	0	0
$706$	6.16551	0.232042
$707$	−9.02973	−0.339598
$708$	0	0
$709$	0.934758	0.0351056	0.0175528	−	0.999846i	$-0.494412\pi$
$709$	0.934758	0.0351056	0.0175528	+	0.999846i	$0.494412\pi$
$710$	−0.864797	−0.0324553
$711$	0	0
$712$	5.61534	0.210444
$713$	5.71845	0.214158
$714$	0	0
$715$	−3.40122	−0.127198
$716$	−12.4561	−0.465505
$717$	0	0
$718$	23.8004	0.888223
$719$	21.0007	0.783195	0.391597	−	0.920137i	$-0.371923\pi$
$719$	21.0007	0.783195	0.391597	+	0.920137i	$0.371923\pi$
$720$	0	0
$721$	1.97214	0.0734463
$722$	42.7432	1.59074
$723$	0	0
$724$	15.3173	0.569261
$725$	29.7423	1.10460
$726$	0	0
$727$	−7.99152	−0.296389	−0.148195	−	0.988958i	$-0.547346\pi$
$727$	−7.99152	−0.296389	−0.148195	+	0.988958i	$0.547346\pi$
$728$	9.20120	0.341019
$729$	0	0
$730$	1.18014	0.0436791
$731$	14.5747	0.539063
$732$	0	0
$733$	15.5065	0.572745	0.286372	−	0.958118i	$-0.407551\pi$
$733$	15.5065	0.572745	0.286372	+	0.958118i	$0.407551\pi$
$734$	−37.6968	−1.39141
$735$	0	0
$736$	4.06238	0.149741
$737$	−5.37374	−0.197944
$738$	0	0
$739$	22.7579	0.837162	0.418581	−	0.908179i	$-0.362528\pi$
$739$	22.7579	0.837162	0.418581	+	0.908179i	$0.362528\pi$
$740$	2.18417	0.0802918
$741$	0	0
$742$	−21.3891	−0.785218
$743$	42.6471	1.56457	0.782285	−	0.622921i	$-0.214054\pi$
$743$	42.6471	1.56457	0.782285	+	0.622921i	$0.214054\pi$
$744$	0	0
$745$	0.755277	0.0276712
$746$	−9.78397	−0.358217
$747$	0	0
$748$	3.73552	0.136584
$749$	18.3080	0.668958
$750$	0	0
$751$	18.1698	0.663025	0.331512	−	0.943451i	$-0.392441\pi$
$751$	18.1698	0.663025	0.331512	+	0.943451i	$0.392441\pi$
$752$	2.74687	0.100168
$753$	0	0
$754$	−25.9406	−0.944701
$755$	−3.64520	−0.132663
$756$	0	0
$757$	−17.0878	−0.621065	−0.310532	−	0.950563i	$-0.600507\pi$
$757$	−17.0878	−0.621065	−0.310532	+	0.950563i	$0.600507\pi$
$758$	14.3303	0.520501
$759$	0	0
$760$	−3.03601	−0.110128
$761$	−28.3484	−1.02763	−0.513815	−	0.857901i	$-0.671768\pi$
$761$	−28.3484	−1.02763	−0.513815	+	0.857901i	$0.671768\pi$
$762$	0	0
$763$	33.7224	1.22083
$764$	4.43481	0.160446
$765$	0	0
$766$	23.7847	0.859376
$767$	21.2363	0.766800
$768$	0	0
$769$	−30.5696	−1.10237	−0.551184	−	0.834384i	$-0.685824\pi$
$769$	−30.5696	−1.10237	−0.551184	+	0.834384i	$0.685824\pi$
$770$	−1.74846	−0.0630100
$771$	0	0
$772$	−4.69578	−0.169005
$773$	−9.69200	−0.348597	−0.174299	−	0.984693i	$-0.555766\pi$
$773$	−9.69200	−0.348597	−0.174299	+	0.984693i	$0.555766\pi$
$774$	0	0
$775$	−6.82816	−0.245275
$776$	4.99827	0.179427
$777$	0	0
$778$	8.41143	0.301564
$779$	−66.6311	−2.38731
$780$	0	0
$781$	4.65714	0.166646
$782$	7.29321	0.260804
$783$	0	0
$784$	−2.26996	−0.0810698
$785$	−5.48705	−0.195841
$786$	0	0
$787$	−37.7666	−1.34623	−0.673117	−	0.739536i	$-0.735045\pi$
$787$	−37.7666	−1.34623	−0.673117	+	0.739536i	$0.735045\pi$
$788$	4.93353	0.175750
$789$	0	0
$790$	−0.977983	−0.0347951
$791$	−13.5975	−0.483470
$792$	0	0
$793$	−2.22875	−0.0791453
$794$	−4.79537	−0.170181
$795$	0	0
$796$	−19.3169	−0.684671
$797$	−41.6753	−1.47621	−0.738107	−	0.674684i	$-0.764280\pi$
$797$	−41.6753	−1.47621	−0.738107	+	0.674684i	$0.764280\pi$
$798$	0	0
$799$	4.93147	0.174463
$800$	−4.85071	−0.171499
$801$	0	0
$802$	−16.2879	−0.575146
$803$	−6.35536	−0.224276
$804$	0	0
$805$	−3.41367	−0.120316
$806$	5.95538	0.209769
$807$	0	0
$808$	−4.15185	−0.146062
$809$	−28.2635	−0.993691	−0.496845	−	0.867839i	$-0.665508\pi$
$809$	−28.2635	−0.993691	−0.496845	+	0.867839i	$0.665508\pi$
$810$	0	0
$811$	42.7084	1.49969	0.749847	−	0.661611i	$-0.230127\pi$
$811$	42.7084	1.49969	0.749847	+	0.661611i	$0.230127\pi$
$812$	−13.3352	−0.467975
$813$	0	0
$814$	−11.7623	−0.412268
$815$	0.0376620	0.00131924
$816$	0	0
$817$	63.7903	2.23174
$818$	5.13956	0.179701
$819$	0	0
$820$	3.27636	0.114415
$821$	26.1422	0.912370	0.456185	−	0.889885i	$-0.349216\pi$
$821$	26.1422	0.912370	0.456185	+	0.889885i	$0.349216\pi$
$822$	0	0
$823$	14.9839	0.522307	0.261153	−	0.965297i	$-0.415897\pi$
$823$	14.9839	0.522307	0.261153	+	0.965297i	$0.415897\pi$
$824$	0.906787	0.0315894
$825$	0	0
$826$	10.9169	0.379848
$827$	10.9582	0.381053	0.190527	−	0.981682i	$-0.438980\pi$
$827$	10.9582	0.381053	0.190527	+	0.981682i	$0.438980\pi$
$828$	0	0
$829$	42.1685	1.46457	0.732287	−	0.680996i	$-0.238453\pi$
$829$	42.1685	1.46457	0.732287	+	0.680996i	$0.238453\pi$
$830$	2.29936	0.0798119
$831$	0	0
$832$	4.23070	0.146673
$833$	−4.07526	−0.141199
$834$	0	0
$835$	−3.99086	−0.138109
$836$	16.3496	0.565464
$837$	0	0
$838$	6.07355	0.209807
$839$	−43.2210	−1.49216	−0.746078	−	0.665859i	$-0.768065\pi$
$839$	−43.2210	−1.49216	−0.746078	+	0.665859i	$0.768065\pi$
$840$	0	0
$841$	8.59553	0.296398
$842$	−35.2344	−1.21426
$843$	0	0
$844$	1.23621	0.0425520
$845$	−1.89277	−0.0651133
$846$	0	0
$847$	−14.5077	−0.498490
$848$	−9.83467	−0.337724
$849$	0	0
$850$	−8.70850	−0.298699
$851$	−22.9646	−0.787216
$852$	0	0
$853$	9.53071	0.326325	0.163163	−	0.986599i	$-0.447830\pi$
$853$	9.53071	0.326325	0.163163	+	0.986599i	$0.447830\pi$
$854$	−1.14573	−0.0392061
$855$	0	0
$856$	8.41797	0.287720
$857$	30.8695	1.05448	0.527240	−	0.849716i	$-0.323227\pi$
$857$	30.8695	1.05448	0.527240	+	0.849716i	$0.323227\pi$
$858$	0	0
$859$	−25.0554	−0.854879	−0.427440	−	0.904044i	$-0.640584\pi$
$859$	−25.0554	−0.854879	−0.427440	+	0.904044i	$0.640584\pi$
$860$	−3.13667	−0.106960
$861$	0	0
$862$	25.3935	0.864905
$863$	−29.0959	−0.990435	−0.495218	−	0.868769i	$-0.664912\pi$
$863$	−29.0959	−0.990435	−0.495218	+	0.868769i	$0.664912\pi$
$864$	0	0
$865$	−1.01657	−0.0345644
$866$	−30.3331	−1.03076
$867$	0	0
$868$	3.06147	0.103913
$869$	5.26667	0.178660
$870$	0	0
$871$	−10.9263	−0.370225
$872$	15.5055	0.525083
$873$	0	0
$874$	31.9209	1.07974
$875$	8.27769	0.279837
$876$	0	0
$877$	−56.6609	−1.91330	−0.956651	−	0.291238i	$-0.905933\pi$
$877$	−56.6609	−1.91330	−0.956651	+	0.291238i	$0.905933\pi$
$878$	10.6888	0.360729
$879$	0	0
$880$	−0.803938	−0.0271007
$881$	−37.8256	−1.27438	−0.637188	−	0.770708i	$-0.719903\pi$
$881$	−37.8256	−1.27438	−0.637188	+	0.770708i	$0.719903\pi$
$882$	0	0
$883$	−2.51756	−0.0847225	−0.0423613	−	0.999102i	$-0.513488\pi$
$883$	−2.51756	−0.0847225	−0.0423613	+	0.999102i	$0.513488\pi$
$884$	7.59538	0.255460
$885$	0	0
$886$	−3.86159	−0.129733
$887$	−22.2716	−0.747806	−0.373903	−	0.927468i	$-0.621981\pi$
$887$	−22.2716	−0.747806	−0.373903	+	0.927468i	$0.621981\pi$
$888$	0	0
$889$	−7.88354	−0.264406
$890$	−2.16963	−0.0727261
$891$	0	0
$892$	13.4472	0.450244
$893$	21.5841	0.722283
$894$	0	0
$895$	4.81271	0.160871
$896$	2.17487	0.0726572
$897$	0	0
$898$	3.93355	0.131264
$899$	−8.63110	−0.287863
$900$	0	0
$901$	−17.6562	−0.588214
$902$	−17.6440	−0.587479
$903$	0	0
$904$	−6.25209	−0.207941
$905$	−5.91820	−0.196728
$906$	0	0
$907$	12.7169	0.422256	0.211128	−	0.977458i	$-0.432286\pi$
$907$	12.7169	0.422256	0.211128	+	0.977458i	$0.432286\pi$
$908$	−17.3344	−0.575263
$909$	0	0
$910$	−3.55511	−0.117851
$911$	54.6544	1.81078	0.905391	−	0.424579i	$-0.139578\pi$
$911$	54.6544	1.81078	0.905391	+	0.424579i	$0.139578\pi$
$912$	0	0
$913$	−12.3826	−0.409804
$914$	−18.4341	−0.609746
$915$	0	0
$916$	−24.9415	−0.824090
$917$	0.782051	0.0258256
$918$	0	0
$919$	−21.0761	−0.695237	−0.347618	−	0.937636i	$-0.613010\pi$
$919$	−21.0761	−0.695237	−0.347618	+	0.937636i	$0.613010\pi$
$920$	−1.56960	−0.0517482
$921$	0	0
$922$	4.61243	0.151902
$923$	9.46929	0.311685
$924$	0	0
$925$	27.4210	0.901598
$926$	−11.2551	−0.369864
$927$	0	0
$928$	−6.13152	−0.201277
$929$	0.720989	0.0236549	0.0118274	−	0.999930i	$-0.496235\pi$
$929$	0.720989	0.0236549	0.0118274	+	0.999930i	$0.496235\pi$
$930$	0	0
$931$	−17.8366	−0.584571
$932$	−18.0979	−0.592816
$933$	0	0
$934$	20.1395	0.658985
$935$	−1.44331	−0.0472014
$936$	0	0
$937$	17.9047	0.584920	0.292460	−	0.956278i	$-0.405526\pi$
$937$	17.9047	0.584920	0.292460	+	0.956278i	$0.405526\pi$
$938$	−5.61689	−0.183398
$939$	0	0
$940$	−1.06132	−0.0346165
$941$	−14.2034	−0.463019	−0.231509	−	0.972833i	$-0.574366\pi$
$941$	−14.2034	−0.463019	−0.231509	+	0.972833i	$0.574366\pi$
$942$	0	0
$943$	−34.4479	−1.12178
$944$	5.01958	0.163374
$945$	0	0
$946$	16.8917	0.549197
$947$	25.5050	0.828800	0.414400	−	0.910095i	$-0.363991\pi$
$947$	25.5050	0.828800	0.414400	+	0.910095i	$0.363991\pi$
$948$	0	0
$949$	−12.9223	−0.419474
$950$	−38.1154	−1.23663
$951$	0	0
$952$	3.90455	0.126547
$953$	31.9636	1.03540	0.517702	−	0.855561i	$-0.326788\pi$
$953$	31.9636	1.03540	0.517702	+	0.855561i	$0.326788\pi$
$954$	0	0
$955$	−1.71350	−0.0554475
$956$	−14.6655	−0.474317
$957$	0	0
$958$	−16.0705	−0.519213
$959$	17.4435	0.563281
$960$	0	0
$961$	−29.0185	−0.936080
$962$	−23.9161	−0.771085
$963$	0	0
$964$	−0.0577845	−0.00186111
$965$	1.81433	0.0584054
$966$	0	0
$967$	18.3313	0.589495	0.294748	−	0.955575i	$-0.404764\pi$
$967$	18.3313	0.589495	0.294748	+	0.955575i	$0.404764\pi$
$968$	−6.67061	−0.214401
$969$	0	0
$970$	−1.93121	−0.0620073
$971$	−10.0164	−0.321441	−0.160720	−	0.987000i	$-0.551382\pi$
$971$	−10.0164	−0.321441	−0.160720	+	0.987000i	$0.551382\pi$
$972$	0	0
$973$	18.2192	0.584079
$974$	−36.7241	−1.17672
$975$	0	0
$976$	−0.526805	−0.0168626
$977$	−30.4337	−0.973662	−0.486831	−	0.873496i	$-0.661847\pi$
$977$	−30.4337	−0.973662	−0.486831	+	0.873496i	$0.661847\pi$
$978$	0	0
$979$	11.6840	0.373421
$980$	0.877054	0.0280165
$981$	0	0
$982$	29.1923	0.931565
$983$	−29.9963	−0.956735	−0.478367	−	0.878160i	$-0.658771\pi$
$983$	−29.9963	−0.956735	−0.478367	+	0.878160i	$0.658771\pi$
$984$	0	0
$985$	−1.90619	−0.0607363
$986$	−11.0079	−0.350564
$987$	0	0
$988$	33.2435	1.05762
$989$	32.9793	1.04868
$990$	0	0
$991$	−11.8038	−0.374960	−0.187480	−	0.982268i	$-0.560032\pi$
$991$	−11.8038	−0.374960	−0.187480	+	0.982268i	$0.560032\pi$
$992$	1.40766	0.0446933
$993$	0	0
$994$	4.86786	0.154399
$995$	7.46358	0.236611
$996$	0	0
$997$	−47.4659	−1.50326	−0.751631	−	0.659584i	$-0.770732\pi$
$997$	−47.4659	−1.50326	−0.751631	+	0.659584i	$0.770732\pi$
$998$	28.9874	0.917581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.q.1.4		8
3.2	odd	2		1614.2.a.i.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1614.2.a.i.1.5	✓	8	3.2	odd	2
4842.2.a.q.1.4		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.386375 q^{5} +2.17487 q^{7} +1.00000 q^{8} -0.386375 q^{10} +2.08072 q^{11} +4.23070 q^{13} +2.17487 q^{14} +1.00000 q^{16} +1.79530 q^{17} +7.85768 q^{19} -0.386375 q^{20} +2.08072 q^{22} +4.06238 q^{23} -4.85071 q^{25} +4.23070 q^{26} +2.17487 q^{28} -6.13152 q^{29} +1.40766 q^{31} +1.00000 q^{32} +1.79530 q^{34} -0.840314 q^{35} -5.65299 q^{37} +7.85768 q^{38} -0.386375 q^{40} -8.47974 q^{41} +8.11821 q^{43} +2.08072 q^{44} +4.06238 q^{46} +2.74687 q^{47} -2.26996 q^{49} -4.85071 q^{50} +4.23070 q^{52} -9.83467 q^{53} -0.803938 q^{55} +2.17487 q^{56} -6.13152 q^{58} +5.01958 q^{59} -0.526805 q^{61} +1.40766 q^{62} +1.00000 q^{64} -1.63463 q^{65} -2.58263 q^{67} +1.79530 q^{68} -0.840314 q^{70} +2.23823 q^{71} -3.05440 q^{73} -5.65299 q^{74} +7.85768 q^{76} +4.52529 q^{77} +2.53118 q^{79} -0.386375 q^{80} -8.47974 q^{82} -5.95111 q^{83} -0.693660 q^{85} +8.11821 q^{86} +2.08072 q^{88} +5.61534 q^{89} +9.20120 q^{91} +4.06238 q^{92} +2.74687 q^{94} -3.03601 q^{95} +4.99827 q^{97} -2.26996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more