LMFDB - Embedded newform 8379.2.a.br.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,3,0,-3,2,0,6,0,0,-4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.396339$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.396339 q^{2} -1.84292 q^{4} +0.842916 q^{5} -1.52310 q^{8} +0.334080 q^{10} +0.0359789 q^{11} +1.96402 q^{13} +3.08217 q^{16} -3.25351 q^{17} +1.00000 q^{19} -1.55342 q^{20} +0.0142598 q^{22} -1.28384 q^{23} -4.28949 q^{25} +0.778417 q^{26} +7.06045 q^{29} -3.61953 q^{31} +4.26777 q^{32} -1.28949 q^{34} -2.16274 q^{37} +0.396339 q^{38} -1.28384 q^{40} -0.130805 q^{41} -4.50884 q^{43} -0.0663061 q^{44} -0.508836 q^{46} +7.98554 q^{47} -1.70009 q^{50} -3.61953 q^{52} -2.98169 q^{53} +0.0303272 q^{55} +2.79833 q^{58} -12.9799 q^{59} +2.58759 q^{61} -1.43456 q^{62} -4.47286 q^{64} +1.65550 q^{65} +4.51905 q^{67} +5.99595 q^{68} +2.64985 q^{71} +6.21297 q^{73} -0.857176 q^{74} -1.84292 q^{76} +10.3728 q^{79} +2.59801 q^{80} -0.0518429 q^{82} -10.2752 q^{83} -2.74244 q^{85} -1.78703 q^{86} -0.0547993 q^{88} -8.37281 q^{89} +2.36601 q^{92} +3.16498 q^{94} +0.842916 q^{95} -2.55162 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 3 q^{8} - 3 q^{10} + 2 q^{11} + 6 q^{13} - 4 q^{16} + 2 q^{17} + 4 q^{19} - 12 q^{20} - 6 q^{22} - 5 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{29} + 17 q^{31} - 4 q^{32} + 8 q^{34} - 3 q^{37}+ \cdots + 11 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 3 * q^8 - 3 * q^10 + 2 * q^11 + 6 * q^13 - 4 * q^16 + 2 * q^17 + 4 * q^19 - 12 * q^20 - 6 * q^22 - 5 * q^23 - 4 * q^25 + 6 * q^26 + 4 * q^29 + 17 * q^31 - 4 * q^32 + 8 * q^34 - 3 * q^37 - 5 * q^40 - 7 * q^41 - 15 * q^43 - 17 * q^44 + q^46 - 16 * q^47 + 14 * q^50 + 17 * q^52 - 4 * q^53 + 15 * q^55 - 5 * q^58 - 17 * q^59 - 9 * q^61 + 7 * q^62 - 13 * q^64 - 23 * q^65 - 5 * q^67 + 10 * q^68 - 6 * q^71 + 15 * q^73 + 10 * q^74 + 5 * q^79 + 25 * q^80 + 31 * q^82 - 34 * q^83 - 12 * q^85 - 17 * q^86 + 11 * q^88 + 3 * q^89 - 7 * q^92 - 6 * q^94 - 4 * q^95 + 11 * q^97

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 $

$2$	0.396339	0.280254	0.140127	−	0.990134i	$-0.455249\pi$
$2$	0.396339	0.280254	0.140127	+	0.990134i	$0.455249\pi$
$3$	0	0
$3$	0	0
$4$	−1.84292	−0.921458
$5$	0.842916	0.376963	0.188482	−	0.982077i	$-0.439643\pi$
$5$	0.842916	0.376963	0.188482	+	0.982077i	$0.439643\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.52310	−0.538496
$9$	0	0
$10$	0.334080	0.105645
$11$	0.0359789	0.0108480	0.00542402	−	0.999985i	$-0.498273\pi$
$11$	0.0359789	0.0108480	0.00542402	+	0.999985i	$0.498273\pi$
$12$	0	0
$13$	1.96402	0.544721	0.272361	−	0.962195i	$-0.412196\pi$
$13$	1.96402	0.544721	0.272361	+	0.962195i	$0.412196\pi$
$14$	0	0
$15$	0	0
$16$	3.08217	0.770543
$17$	−3.25351	−0.789093	−0.394547	−	0.918876i	$-0.629098\pi$
$17$	−3.25351	−0.789093	−0.394547	+	0.918876i	$0.629098\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.55342	−0.347356
$21$	0	0
$22$	0.0142598	0.00304020
$23$	−1.28384	−0.267699	−0.133850	−	0.991002i	$-0.542734\pi$
$23$	−1.28384	−0.267699	−0.133850	+	0.991002i	$0.542734\pi$
$24$	0	0
$25$	−4.28949	−0.857899
$26$	0.778417	0.152660
$27$	0	0
$28$	0	0
$29$	7.06045	1.31109	0.655546	−	0.755155i	$-0.272438\pi$
$29$	7.06045	1.31109	0.655546	+	0.755155i	$0.272438\pi$
$30$	0	0
$31$	−3.61953	−0.650086	−0.325043	−	0.945699i	$-0.605379\pi$
$31$	−3.61953	−0.650086	−0.325043	+	0.945699i	$0.605379\pi$
$32$	4.26777	0.754443
$33$	0	0
$34$	−1.28949	−0.221146
$35$	0	0
$36$	0	0
$37$	−2.16274	−0.355552	−0.177776	−	0.984071i	$-0.556890\pi$
$37$	−2.16274	−0.355552	−0.177776	+	0.984071i	$0.556890\pi$
$38$	0.396339	0.0642946
$39$	0	0
$40$	−1.28384	−0.202993
$41$	−0.130805	−0.0204282	−0.0102141	−	0.999948i	$-0.503251\pi$
$41$	−0.130805	−0.0204282	−0.0102141	+	0.999948i	$0.503251\pi$
$42$	0	0
$43$	−4.50884	−0.687591	−0.343796	−	0.939045i	$-0.611713\pi$
$43$	−4.50884	−0.687591	−0.343796	+	0.939045i	$0.611713\pi$
$44$	−0.0663061	−0.00999601
$45$	0	0
$46$	−0.508836	−0.0750237
$47$	7.98554	1.16481	0.582405	−	0.812899i	$-0.302112\pi$
$47$	7.98554	1.16481	0.582405	+	0.812899i	$0.302112\pi$
$48$	0	0
$49$	0	0
$50$	−1.70009	−0.240429
$51$	0	0
$52$	−3.61953	−0.501938
$53$	−2.98169	−0.409567	−0.204783	−	0.978807i	$-0.565649\pi$
$53$	−2.98169	−0.409567	−0.204783	+	0.978807i	$0.565649\pi$
$54$	0	0
$55$	0.0303272	0.00408931
$56$	0	0
$57$	0	0
$58$	2.79833	0.367439
$59$	−12.9799	−1.68984	−0.844919	−	0.534895i	$-0.820351\pi$
$59$	−12.9799	−1.68984	−0.844919	+	0.534895i	$0.820351\pi$
$60$	0	0
$61$	2.58759	0.331307	0.165654	−	0.986184i	$-0.447027\pi$
$61$	2.58759	0.331307	0.165654	+	0.986184i	$0.447027\pi$
$62$	−1.43456	−0.182189
$63$	0	0
$64$	−4.47286	−0.559107
$65$	1.65550	0.205340
$66$	0	0
$67$	4.51905	0.552090	0.276045	−	0.961145i	$-0.410976\pi$
$67$	4.51905	0.552090	0.276045	+	0.961145i	$0.410976\pi$
$68$	5.99595	0.727116
$69$	0	0
$70$	0	0
$71$	2.64985	0.314480	0.157240	−	0.987560i	$-0.449740\pi$
$71$	2.64985	0.314480	0.157240	+	0.987560i	$0.449740\pi$
$72$	0	0
$73$	6.21297	0.727174	0.363587	−	0.931560i	$-0.381552\pi$
$73$	6.21297	0.727174	0.363587	+	0.931560i	$0.381552\pi$
$74$	−0.857176	−0.0996446
$75$	0	0
$76$	−1.84292	−0.211397
$77$	0	0
$78$	0	0
$79$	10.3728	1.16703	0.583516	−	0.812101i	$-0.301676\pi$
$79$	10.3728	1.16703	0.583516	+	0.812101i	$0.301676\pi$
$80$	2.59801	0.290466
$81$	0	0
$82$	−0.0518429	−0.00572509
$83$	−10.2752	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$83$	−10.2752	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$84$	0	0
$85$	−2.74244	−0.297459
$86$	−1.78703	−0.192700
$87$	0	0
$88$	−0.0547993	−0.00584162
$89$	−8.37281	−0.887516	−0.443758	−	0.896147i	$-0.646355\pi$
$89$	−8.37281	−0.887516	−0.443758	+	0.896147i	$0.646355\pi$
$90$	0	0
$91$	0	0
$92$	2.36601	0.246674
$93$	0	0
$94$	3.16498	0.326442
$95$	0.842916	0.0864813
$96$	0	0
$97$	−2.55162	−0.259077	−0.129539	−	0.991574i	$-0.541350\pi$
$97$	−2.55162	−0.259077	−0.129539	+	0.991574i	$0.541350\pi$
$98$	0	0
$99$	0	0
$100$	7.90517	0.790517
$101$	−7.13017	−0.709478	−0.354739	−	0.934965i	$-0.615430\pi$
$101$	−7.13017	−0.709478	−0.354739	+	0.934965i	$0.615430\pi$
$102$	0	0
$103$	2.90922	0.286654	0.143327	−	0.989675i	$-0.454220\pi$
$103$	2.90922	0.286654	0.143327	+	0.989675i	$0.454220\pi$
$104$	−2.99139	−0.293330
$105$	0	0
$106$	−1.18176	−0.114783
$107$	−8.60931	−0.832294	−0.416147	−	0.909297i	$-0.636620\pi$
$107$	−8.60931	−0.832294	−0.416147	+	0.909297i	$0.636620\pi$
$108$	0	0
$109$	0.574938	0.0550691	0.0275346	−	0.999621i	$-0.491234\pi$
$109$	0.574938	0.0550691	0.0275346	+	0.999621i	$0.491234\pi$
$110$	0.0120198	0.00114605
$111$	0	0
$112$	0	0
$113$	15.2784	1.43727	0.718637	−	0.695385i	$-0.244766\pi$
$113$	15.2784	1.43727	0.718637	+	0.695385i	$0.244766\pi$
$114$	0	0
$115$	−1.08217	−0.100913
$116$	−13.0118	−1.20812
$117$	0	0
$118$	−5.14443	−0.473583
$119$	0	0
$120$	0	0
$121$	−10.9987	−0.999882
$122$	1.02556	0.0928501
$123$	0	0
$124$	6.67048	0.599027
$125$	−7.83026	−0.700360
$126$	0	0
$127$	1.02235	0.0907193	0.0453597	−	0.998971i	$-0.485557\pi$
$127$	1.02235	0.0907193	0.0453597	+	0.998971i	$0.485557\pi$
$128$	−10.3083	−0.911135
$129$	0	0
$130$	0.656140	0.0575473
$131$	16.9821	1.48374	0.741868	−	0.670546i	$-0.233940\pi$
$131$	16.9821	1.48374	0.741868	+	0.670546i	$0.233940\pi$
$132$	0	0
$133$	0	0
$134$	1.79107	0.154725
$135$	0	0
$136$	4.95541	0.424923
$137$	2.24221	0.191565	0.0957825	−	0.995402i	$-0.469465\pi$
$137$	2.24221	0.191565	0.0957825	+	0.995402i	$0.469465\pi$
$138$	0	0
$139$	16.8301	1.42751	0.713753	−	0.700397i	$-0.246994\pi$
$139$	16.8301	1.42751	0.713753	+	0.700397i	$0.246994\pi$
$140$	0	0
$141$	0	0
$142$	1.05024	0.0881341
$143$	0.0706633	0.00590916
$144$	0	0
$145$	5.95137	0.494234
$146$	2.46244	0.203793
$147$	0	0
$148$	3.98574	0.327626
$149$	−12.7320	−1.04305	−0.521524	−	0.853237i	$-0.674636\pi$
$149$	−12.7320	−1.04305	−0.521524	+	0.853237i	$0.674636\pi$
$150$	0	0
$151$	−11.8268	−0.962455	−0.481228	−	0.876596i	$-0.659809\pi$
$151$	−11.8268	−0.962455	−0.481228	+	0.876596i	$0.659809\pi$
$152$	−1.52310	−0.123539
$153$	0	0
$154$	0	0
$155$	−3.05096	−0.245059
$156$	0	0
$157$	7.37686	0.588737	0.294369	−	0.955692i	$-0.404891\pi$
$157$	7.37686	0.588737	0.294369	+	0.955692i	$0.404891\pi$
$158$	4.11115	0.327065
$159$	0	0
$160$	3.59737	0.284397
$161$	0	0
$162$	0	0
$163$	−18.0494	−1.41374	−0.706869	−	0.707344i	$-0.749893\pi$
$163$	−18.0494	−1.41374	−0.706869	+	0.707344i	$0.749893\pi$
$164$	0.241062	0.0188238
$165$	0	0
$166$	−4.07247	−0.316085
$167$	−22.6354	−1.75158	−0.875790	−	0.482693i	$-0.839659\pi$
$167$	−22.6354	−1.75158	−0.875790	+	0.482693i	$0.839659\pi$
$168$	0	0
$169$	−9.14262	−0.703279
$170$	−1.08693	−0.0833640
$171$	0	0
$172$	8.30940	0.633586
$173$	−6.88282	−0.523291	−0.261646	−	0.965164i	$-0.584265\pi$
$173$	−6.88282	−0.523291	−0.261646	+	0.965164i	$0.584265\pi$
$174$	0	0
$175$	0	0
$176$	0.110893	0.00835888
$177$	0	0
$178$	−3.31847	−0.248730
$179$	−0.620878	−0.0464066	−0.0232033	−	0.999731i	$-0.507386\pi$
$179$	−0.620878	−0.0464066	−0.0232033	+	0.999731i	$0.507386\pi$
$180$	0	0
$181$	−9.31058	−0.692050	−0.346025	−	0.938225i	$-0.612469\pi$
$181$	−9.31058	−0.692050	−0.346025	+	0.938225i	$0.612469\pi$
$182$	0	0
$183$	0	0
$184$	1.95541	0.144155
$185$	−1.82300	−0.134030
$186$	0	0
$187$	−0.117058	−0.00856012
$188$	−14.7167	−1.07332
$189$	0	0
$190$	0.334080	0.0242367
$191$	−18.3384	−1.32692	−0.663459	−	0.748212i	$-0.730912\pi$
$191$	−18.3384	−1.32692	−0.663459	+	0.748212i	$0.730912\pi$
$192$	0	0
$193$	−15.2004	−1.09415	−0.547074	−	0.837085i	$-0.684258\pi$
$193$	−15.2004	−1.09415	−0.547074	+	0.837085i	$0.684258\pi$
$194$	−1.01130	−0.0726074
$195$	0	0
$196$	0	0
$197$	−7.77205	−0.553736	−0.276868	−	0.960908i	$-0.589296\pi$
$197$	−7.77205	−0.553736	−0.276868	+	0.960908i	$0.589296\pi$
$198$	0	0
$199$	25.7262	1.82368	0.911840	−	0.410546i	$-0.134662\pi$
$199$	25.7262	1.82368	0.911840	+	0.410546i	$0.134662\pi$
$200$	6.53331	0.461975
$201$	0	0
$202$	−2.82596	−0.198834
$203$	0	0
$204$	0	0
$205$	−0.110257	−0.00770070
$206$	1.15304	0.0803359
$207$	0	0
$208$	6.05345	0.419731
$209$	0.0359789	0.00248871
$210$	0	0
$211$	17.2470	1.18733	0.593667	−	0.804711i	$-0.297680\pi$
$211$	17.2470	1.18733	0.593667	+	0.804711i	$0.297680\pi$
$212$	5.49501	0.377399
$213$	0	0
$214$	−3.41220	−0.233253
$215$	−3.80057	−0.259197
$216$	0	0
$217$	0	0
$218$	0.227870	0.0154333
$219$	0	0
$220$	−0.0558904	−0.00376813
$221$	−6.38997	−0.429836
$222$	0	0
$223$	15.7257	1.05307	0.526537	−	0.850152i	$-0.323490\pi$
$223$	15.7257	1.05307	0.526537	+	0.850152i	$0.323490\pi$
$224$	0	0
$225$	0	0
$226$	6.05544	0.402802
$227$	6.50362	0.431660	0.215830	−	0.976431i	$-0.430754\pi$
$227$	6.50362	0.431660	0.215830	+	0.976431i	$0.430754\pi$
$228$	0	0
$229$	7.87261	0.520237	0.260118	−	0.965577i	$-0.416238\pi$
$229$	7.87261	0.520237	0.260118	+	0.965577i	$0.416238\pi$
$230$	−0.428906	−0.0282812
$231$	0	0
$232$	−10.7537	−0.706018
$233$	−22.2790	−1.45954	−0.729771	−	0.683691i	$-0.760374\pi$
$233$	−22.2790	−1.45954	−0.729771	+	0.683691i	$0.760374\pi$
$234$	0	0
$235$	6.73113	0.439091
$236$	23.9208	1.55711
$237$	0	0
$238$	0	0
$239$	−26.5221	−1.71557	−0.857784	−	0.514009i	$-0.828160\pi$
$239$	−26.5221	−1.71557	−0.857784	+	0.514009i	$0.828160\pi$
$240$	0	0
$241$	11.0806	0.713762	0.356881	−	0.934150i	$-0.383840\pi$
$241$	11.0806	0.713762	0.356881	+	0.934150i	$0.383840\pi$
$242$	−4.35921	−0.280221
$243$	0	0
$244$	−4.76872	−0.305286
$245$	0	0
$246$	0	0
$247$	1.96402	0.124968
$248$	5.51288	0.350068
$249$	0	0
$250$	−3.10343	−0.196278
$251$	−3.32163	−0.209659	−0.104830	−	0.994490i	$-0.533430\pi$
$251$	−3.32163	−0.209659	−0.104830	+	0.994490i	$0.533430\pi$
$252$	0	0
$253$	−0.0461912	−0.00290401
$254$	0.405199	0.0254244
$255$	0	0
$256$	4.86013	0.303758
$257$	−30.1189	−1.87877	−0.939383	−	0.342869i	$-0.888601\pi$
$257$	−30.1189	−1.87877	−0.939383	+	0.342869i	$0.888601\pi$
$258$	0	0
$259$	0	0
$260$	−3.05096	−0.189212
$261$	0	0
$262$	6.73067	0.415822
$263$	25.5666	1.57651	0.788253	−	0.615351i	$-0.210986\pi$
$263$	25.5666	1.57651	0.788253	+	0.615351i	$0.210986\pi$
$264$	0	0
$265$	−2.51332	−0.154392
$266$	0	0
$267$	0	0
$268$	−8.32822	−0.508727
$269$	6.35130	0.387245	0.193623	−	0.981076i	$-0.437976\pi$
$269$	6.35130	0.387245	0.193623	+	0.981076i	$0.437976\pi$
$270$	0	0
$271$	7.86174	0.477566	0.238783	−	0.971073i	$-0.423251\pi$
$271$	7.86174	0.477566	0.238783	+	0.971073i	$0.423251\pi$
$272$	−10.0279	−0.608030
$273$	0	0
$274$	0.888674	0.0536868
$275$	−0.154331	−0.00930652
$276$	0	0
$277$	−5.87569	−0.353036	−0.176518	−	0.984297i	$-0.556483\pi$
$277$	−5.87569	−0.353036	−0.176518	+	0.984297i	$0.556483\pi$
$278$	6.67040	0.400064
$279$	0	0
$280$	0	0
$281$	−16.7309	−0.998084	−0.499042	−	0.866578i	$-0.666315\pi$
$281$	−16.7309	−0.998084	−0.499042	+	0.866578i	$0.666315\pi$
$282$	0	0
$283$	−20.8957	−1.24212	−0.621060	−	0.783763i	$-0.713298\pi$
$283$	−20.8957	−1.24212	−0.621060	+	0.783763i	$0.713298\pi$
$284$	−4.88346	−0.289780
$285$	0	0
$286$	0.0280066	0.00165606
$287$	0	0
$288$	0	0
$289$	−6.41465	−0.377332
$290$	2.35876	0.138511
$291$	0	0
$292$	−11.4500	−0.670060
$293$	12.1831	0.711744	0.355872	−	0.934535i	$-0.384184\pi$
$293$	12.1831	0.711744	0.355872	+	0.934535i	$0.384184\pi$
$294$	0	0
$295$	−10.9409	−0.637007
$296$	3.29405	0.191463
$297$	0	0
$298$	−5.04619	−0.292318
$299$	−2.52149	−0.145822
$300$	0	0
$301$	0	0
$302$	−4.68744	−0.269732
$303$	0	0
$304$	3.08217	0.176775
$305$	2.18112	0.124891
$306$	0	0
$307$	−25.2971	−1.44378	−0.721890	−	0.692008i	$-0.756726\pi$
$307$	−25.2971	−1.44378	−0.721890	+	0.692008i	$0.756726\pi$
$308$	0	0
$309$	0	0
$310$	−1.20921	−0.0686786
$311$	−9.88730	−0.560657	−0.280329	−	0.959904i	$-0.590443\pi$
$311$	−9.88730	−0.560657	−0.280329	+	0.959904i	$0.590443\pi$
$312$	0	0
$313$	−1.37474	−0.0777050	−0.0388525	−	0.999245i	$-0.512370\pi$
$313$	−1.37474	−0.0777050	−0.0388525	+	0.999245i	$0.512370\pi$
$314$	2.92373	0.164996
$315$	0	0
$316$	−19.1162	−1.07537
$317$	−26.6902	−1.49907	−0.749535	−	0.661965i	$-0.769723\pi$
$317$	−26.6902	−1.49907	−0.749535	+	0.661965i	$0.769723\pi$
$318$	0	0
$319$	0.254027	0.0142228
$320$	−3.77024	−0.210763
$321$	0	0
$322$	0	0
$323$	−3.25351	−0.181030
$324$	0	0
$325$	−8.42465	−0.467316
$326$	−7.15367	−0.396205
$327$	0	0
$328$	0.199228	0.0110005
$329$	0	0
$330$	0	0
$331$	−36.3735	−1.99927	−0.999635	−	0.0270266i	$-0.991396\pi$
$331$	−36.3735	−1.99927	−0.999635	+	0.0270266i	$0.991396\pi$
$332$	18.9364	1.03927
$333$	0	0
$334$	−8.97128	−0.490886
$335$	3.80918	0.208118
$336$	0	0
$337$	30.0191	1.63524	0.817622	−	0.575756i	$-0.195292\pi$
$337$	30.0191	1.63524	0.817622	+	0.575756i	$0.195292\pi$
$338$	−3.62357	−0.197096
$339$	0	0
$340$	5.05408	0.274096
$341$	−0.130226	−0.00705216
$342$	0	0
$343$	0	0
$344$	6.86739	0.370265
$345$	0	0
$346$	−2.72793	−0.146654
$347$	−28.2087	−1.51432	−0.757162	−	0.653227i	$-0.773415\pi$
$347$	−28.2087	−1.51432	−0.757162	+	0.653227i	$0.773415\pi$
$348$	0	0
$349$	−4.32568	−0.231548	−0.115774	−	0.993276i	$-0.536935\pi$
$349$	−4.32568	−0.231548	−0.115774	+	0.993276i	$0.536935\pi$
$350$	0	0
$351$	0	0
$352$	0.153550	0.00818423
$353$	20.2454	1.07755	0.538776	−	0.842449i	$-0.318887\pi$
$353$	20.2454	1.07755	0.538776	+	0.842449i	$0.318887\pi$
$354$	0	0
$355$	2.23360	0.118547
$356$	15.4304	0.817809
$357$	0	0
$358$	−0.246078	−0.0130056
$359$	−13.1598	−0.694547	−0.347273	−	0.937764i	$-0.612892\pi$
$359$	−13.1598	−0.694547	−0.347273	+	0.937764i	$0.612892\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.69014	−0.193949
$363$	0	0
$364$	0	0
$365$	5.23701	0.274118
$366$	0	0
$367$	24.4005	1.27370	0.636848	−	0.770990i	$-0.280238\pi$
$367$	24.4005	1.27370	0.636848	+	0.770990i	$0.280238\pi$
$368$	−3.95702	−0.206274
$369$	0	0
$370$	−0.722527	−0.0375624
$371$	0	0
$372$	0	0
$373$	2.46875	0.127827	0.0639136	−	0.997955i	$-0.479642\pi$
$373$	2.46875	0.127827	0.0639136	+	0.997955i	$0.479642\pi$
$374$	−0.0463945	−0.00239900
$375$	0	0
$376$	−12.1627	−0.627245
$377$	13.8669	0.714180
$378$	0	0
$379$	−18.7800	−0.964665	−0.482332	−	0.875988i	$-0.660210\pi$
$379$	−18.7800	−0.964665	−0.482332	+	0.875988i	$0.660210\pi$
$380$	−1.55342	−0.0796889
$381$	0	0
$382$	−7.26821	−0.371874
$383$	−31.6289	−1.61616	−0.808081	−	0.589072i	$-0.799493\pi$
$383$	−31.6289	−1.61616	−0.808081	+	0.589072i	$0.799493\pi$
$384$	0	0
$385$	0	0
$386$	−6.02449	−0.306639
$387$	0	0
$388$	4.70241	0.238729
$389$	4.33747	0.219918	0.109959	−	0.993936i	$-0.464928\pi$
$389$	4.33747	0.219918	0.109959	+	0.993936i	$0.464928\pi$
$390$	0	0
$391$	4.17700	0.211240
$392$	0	0
$393$	0	0
$394$	−3.08036	−0.155186
$395$	8.74341	0.439929
$396$	0	0
$397$	1.36556	0.0685353	0.0342676	−	0.999413i	$-0.489090\pi$
$397$	1.36556	0.0685353	0.0342676	+	0.999413i	$0.489090\pi$
$398$	10.1963	0.511093
$399$	0	0
$400$	−13.2209	−0.661047
$401$	−22.6605	−1.13161	−0.565806	−	0.824539i	$-0.691435\pi$
$401$	−22.6605	−1.13161	−0.565806	+	0.824539i	$0.691435\pi$
$402$	0	0
$403$	−7.10882	−0.354116
$404$	13.1403	0.653754
$405$	0	0
$406$	0	0
$407$	−0.0778128	−0.00385704
$408$	0	0
$409$	−26.7833	−1.32435	−0.662174	−	0.749350i	$-0.730366\pi$
$409$	−26.7833	−1.32435	−0.662174	+	0.749350i	$0.730366\pi$
$410$	−0.0436992	−0.00215815
$411$	0	0
$412$	−5.36145	−0.264140
$413$	0	0
$414$	0	0
$415$	−8.66116	−0.425159
$416$	8.38200	0.410961
$417$	0	0
$418$	0.0142598	0.000697471 0
$419$	−29.9706	−1.46416	−0.732081	−	0.681218i	$-0.761451\pi$
$419$	−29.9706	−1.46416	−0.732081	+	0.681218i	$0.761451\pi$
$420$	0	0
$421$	−28.9616	−1.41150	−0.705750	−	0.708460i	$-0.749390\pi$
$421$	−28.9616	−1.41150	−0.705750	+	0.708460i	$0.749390\pi$
$422$	6.83566	0.332755
$423$	0	0
$424$	4.54140	0.220550
$425$	13.9559	0.676962
$426$	0	0
$427$	0	0
$428$	15.8662	0.766924
$429$	0	0
$430$	−1.50631	−0.0726408
$431$	−20.8757	−1.00555	−0.502774	−	0.864418i	$-0.667687\pi$
$431$	−20.8757	−1.00555	−0.502774	+	0.864418i	$0.667687\pi$
$432$	0	0
$433$	−7.01266	−0.337007	−0.168503	−	0.985701i	$-0.553893\pi$
$433$	−7.01266	−0.337007	−0.168503	+	0.985701i	$0.553893\pi$
$434$	0	0
$435$	0	0
$436$	−1.05956	−0.0507439
$437$	−1.28384	−0.0614145
$438$	0	0
$439$	−1.74807	−0.0834307	−0.0417153	−	0.999130i	$-0.513282\pi$
$439$	−1.74807	−0.0834307	−0.0417153	+	0.999130i	$0.513282\pi$
$440$	−0.0461912	−0.00220208
$441$	0	0
$442$	−2.53259	−0.120463
$443$	18.8810	0.897064	0.448532	−	0.893767i	$-0.351947\pi$
$443$	18.8810	0.897064	0.448532	+	0.893767i	$0.351947\pi$
$444$	0	0
$445$	−7.05758	−0.334561
$446$	6.23271	0.295128
$447$	0	0
$448$	0	0
$449$	−19.9572	−0.941839	−0.470920	−	0.882176i	$-0.656078\pi$
$449$	−19.9572	−0.941839	−0.470920	+	0.882176i	$0.656078\pi$
$450$	0	0
$451$	−0.00470620	−0.000221606 0
$452$	−28.1569	−1.32439
$453$	0	0
$454$	2.57763	0.120974
$455$	0	0
$456$	0	0
$457$	−21.5575	−1.00842	−0.504208	−	0.863582i	$-0.668215\pi$
$457$	−21.5575	−1.00842	−0.504208	+	0.863582i	$0.668215\pi$
$458$	3.12022	0.145798
$459$	0	0
$460$	1.99435	0.0929870
$461$	27.9766	1.30300	0.651500	−	0.758649i	$-0.274140\pi$
$461$	27.9766	1.30300	0.651500	+	0.758649i	$0.274140\pi$
$462$	0	0
$463$	38.8491	1.80547	0.902736	−	0.430195i	$-0.141555\pi$
$463$	38.8491	1.80547	0.902736	+	0.430195i	$0.141555\pi$
$464$	21.7615	1.01025
$465$	0	0
$466$	−8.83001	−0.409042
$467$	8.63931	0.399780	0.199890	−	0.979818i	$-0.435942\pi$
$467$	8.63931	0.399780	0.199890	+	0.979818i	$0.435942\pi$
$468$	0	0
$469$	0	0
$470$	2.66781	0.123057
$471$	0	0
$472$	19.7696	0.909970
$473$	−0.162223	−0.00745902
$474$	0	0
$475$	−4.28949	−0.196815
$476$	0	0
$477$	0	0
$478$	−10.5117	−0.480794
$479$	−14.8403	−0.678069	−0.339035	−	0.940774i	$-0.610100\pi$
$479$	−14.8403	−0.678069	−0.339035	+	0.940774i	$0.610100\pi$
$480$	0	0
$481$	−4.24766	−0.193677
$482$	4.39166	0.200034
$483$	0	0
$484$	20.2697	0.921349
$485$	−2.15080	−0.0976626
$486$	0	0
$487$	17.9043	0.811319	0.405660	−	0.914024i	$-0.367042\pi$
$487$	17.9043	0.811319	0.405660	+	0.914024i	$0.367042\pi$
$488$	−3.94115	−0.178408
$489$	0	0
$490$	0	0
$491$	−19.1867	−0.865883	−0.432942	−	0.901422i	$-0.642524\pi$
$491$	−19.1867	−0.865883	−0.432942	+	0.901422i	$0.642524\pi$
$492$	0	0
$493$	−22.9713	−1.03457
$494$	0.778417	0.0350226
$495$	0	0
$496$	−11.1560	−0.500919
$497$	0	0
$498$	0	0
$499$	24.5843	1.10054	0.550272	−	0.834985i	$-0.314524\pi$
$499$	24.5843	1.10054	0.550272	+	0.834985i	$0.314524\pi$
$500$	14.4305	0.645352
$501$	0	0
$502$	−1.31649	−0.0587578
$503$	−16.7355	−0.746199	−0.373099	−	0.927791i	$-0.621705\pi$
$503$	−16.7355	−0.746199	−0.373099	+	0.927791i	$0.621705\pi$
$504$	0	0
$505$	−6.01013	−0.267447
$506$	−0.0183073	−0.000813861 0
$507$	0	0
$508$	−1.88411	−0.0835940
$509$	23.5184	1.04243	0.521217	−	0.853424i	$-0.325478\pi$
$509$	23.5184	1.04243	0.521217	+	0.853424i	$0.325478\pi$
$510$	0	0
$511$	0	0
$512$	22.5429	0.996264
$513$	0	0
$514$	−11.9373	−0.526531
$515$	2.45223	0.108058
$516$	0	0
$517$	0.287311	0.0126359
$518$	0	0
$519$	0	0
$520$	−2.52149	−0.110575
$521$	−6.05480	−0.265266	−0.132633	−	0.991165i	$-0.542343\pi$
$521$	−6.05480	−0.265266	−0.132633	+	0.991165i	$0.542343\pi$
$522$	0	0
$523$	37.2755	1.62995	0.814973	−	0.579499i	$-0.196752\pi$
$523$	37.2755	1.62995	0.814973	+	0.579499i	$0.196752\pi$
$524$	−31.2966	−1.36720
$525$	0	0
$526$	10.1330	0.441822
$527$	11.7762	0.512978
$528$	0	0
$529$	−21.3518	−0.928337
$530$	−0.996124	−0.0432689
$531$	0	0
$532$	0	0
$533$	−0.256903	−0.0111277
$534$	0	0
$535$	−7.25693	−0.313744
$536$	−6.88294	−0.297298
$537$	0	0
$538$	2.51726	0.108527
$539$	0	0
$540$	0	0
$541$	−4.52599	−0.194588	−0.0972938	−	0.995256i	$-0.531019\pi$
$541$	−4.52599	−0.194588	−0.0972938	+	0.995256i	$0.531019\pi$
$542$	3.11591	0.133840
$543$	0	0
$544$	−13.8853	−0.595326
$545$	0.484625	0.0207590
$546$	0	0
$547$	11.7047	0.500456	0.250228	−	0.968187i	$-0.419494\pi$
$547$	11.7047	0.500456	0.250228	+	0.968187i	$0.419494\pi$
$548$	−4.13221	−0.176519
$549$	0	0
$550$	−0.0611674	−0.00260819
$551$	7.06045	0.300785
$552$	0	0
$553$	0	0
$554$	−2.32876	−0.0989396
$555$	0	0
$556$	−31.0164	−1.31539
$557$	−15.2693	−0.646979	−0.323490	−	0.946232i	$-0.604856\pi$
$557$	−15.2693	−0.646979	−0.323490	+	0.946232i	$0.604856\pi$
$558$	0	0
$559$	−8.85545	−0.374546
$560$	0	0
$561$	0	0
$562$	−6.63111	−0.279717
$563$	30.5738	1.28853	0.644267	−	0.764801i	$-0.277163\pi$
$563$	30.5738	1.28853	0.644267	+	0.764801i	$0.277163\pi$
$564$	0	0
$565$	12.8784	0.541800
$566$	−8.28177	−0.348109
$567$	0	0
$568$	−4.03598	−0.169346
$569$	8.64529	0.362429	0.181215	−	0.983444i	$-0.441997\pi$
$569$	8.64529	0.362429	0.181215	+	0.983444i	$0.441997\pi$
$570$	0	0
$571$	1.17580	0.0492056	0.0246028	−	0.999697i	$-0.492168\pi$
$571$	1.17580	0.0492056	0.0246028	+	0.999697i	$0.492168\pi$
$572$	−0.130226	−0.00544504
$573$	0	0
$574$	0	0
$575$	5.50703	0.229659
$576$	0	0
$577$	−12.3411	−0.513768	−0.256884	−	0.966442i	$-0.582696\pi$
$577$	−12.3411	−0.513768	−0.256884	+	0.966442i	$0.582696\pi$
$578$	−2.54237	−0.105749
$579$	0	0
$580$	−10.9679	−0.455416
$581$	0	0
$582$	0	0
$583$	−0.107278	−0.00444300
$584$	−9.46295	−0.391580
$585$	0	0
$586$	4.82863	0.199469
$587$	−19.9966	−0.825350	−0.412675	−	0.910878i	$-0.635405\pi$
$587$	−19.9966	−0.825350	−0.412675	+	0.910878i	$0.635405\pi$
$588$	0	0
$589$	−3.61953	−0.149140
$590$	−4.33632	−0.178523
$591$	0	0
$592$	−6.66592	−0.273968
$593$	−11.1740	−0.458863	−0.229431	−	0.973325i	$-0.573687\pi$
$593$	−11.1740	−0.458863	−0.229431	+	0.973325i	$0.573687\pi$
$594$	0	0
$595$	0	0
$596$	23.4640	0.961125
$597$	0	0
$598$	−0.999364	−0.0408670
$599$	24.0539	0.982815	0.491407	−	0.870930i	$-0.336483\pi$
$599$	24.0539	0.982815	0.491407	+	0.870930i	$0.336483\pi$
$600$	0	0
$601$	32.2158	1.31411	0.657054	−	0.753843i	$-0.271802\pi$
$601$	32.2158	1.31411	0.657054	+	0.753843i	$0.271802\pi$
$602$	0	0
$603$	0	0
$604$	21.7959	0.886862
$605$	−9.27098	−0.376919
$606$	0	0
$607$	−2.25724	−0.0916184	−0.0458092	−	0.998950i	$-0.514587\pi$
$607$	−2.25724	−0.0916184	−0.0458092	+	0.998950i	$0.514587\pi$
$608$	4.26777	0.173081
$609$	0	0
$610$	0.864463	0.0350011
$611$	15.6838	0.634497
$612$	0	0
$613$	12.8871	0.520506	0.260253	−	0.965540i	$-0.416194\pi$
$613$	12.8871	0.520506	0.260253	+	0.965540i	$0.416194\pi$
$614$	−10.0262	−0.404624
$615$	0	0
$616$	0	0
$617$	−25.0627	−1.00899	−0.504493	−	0.863416i	$-0.668321\pi$
$617$	−25.0627	−1.00899	−0.504493	+	0.863416i	$0.668321\pi$
$618$	0	0
$619$	−16.8365	−0.676718	−0.338359	−	0.941017i	$-0.609872\pi$
$619$	−16.8365	−0.676718	−0.338359	+	0.941017i	$0.609872\pi$
$620$	5.62265	0.225811
$621$	0	0
$622$	−3.91872	−0.157126
$623$	0	0
$624$	0	0
$625$	14.8472	0.593889
$626$	−0.544863	−0.0217771
$627$	0	0
$628$	−13.5949	−0.542497
$629$	7.03649	0.280563
$630$	0	0
$631$	−25.8497	−1.02906	−0.514530	−	0.857472i	$-0.672034\pi$
$631$	−25.8497	−1.02906	−0.514530	+	0.857472i	$0.672034\pi$
$632$	−15.7988	−0.628442
$633$	0	0
$634$	−10.5784	−0.420120
$635$	0.861759	0.0341979
$636$	0	0
$637$	0	0
$638$	0.100681	0.00398599
$639$	0	0
$640$	−8.68904	−0.343464
$641$	−49.3339	−1.94857	−0.974286	−	0.225313i	$-0.927660\pi$
$641$	−49.3339	−1.94857	−0.974286	+	0.225313i	$0.927660\pi$
$642$	0	0
$643$	−2.55951	−0.100937	−0.0504686	−	0.998726i	$-0.516071\pi$
$643$	−2.55951	−0.100937	−0.0504686	+	0.998726i	$0.516071\pi$
$644$	0	0
$645$	0	0
$646$	−1.28949	−0.0507344
$647$	35.1832	1.38319	0.691597	−	0.722283i	$-0.256907\pi$
$647$	35.1832	1.38319	0.691597	+	0.722283i	$0.256907\pi$
$648$	0	0
$649$	−0.467002	−0.0183314
$650$	−3.33902	−0.130967
$651$	0	0
$652$	33.2635	1.30270
$653$	28.0240	1.09666	0.548332	−	0.836261i	$-0.315263\pi$
$653$	28.0240	1.09666	0.548332	+	0.836261i	$0.315263\pi$
$654$	0	0
$655$	14.3145	0.559314
$656$	−0.403162	−0.0157408
$657$	0	0
$658$	0	0
$659$	−30.0543	−1.17075	−0.585374	−	0.810763i	$-0.699052\pi$
$659$	−30.0543	−1.17075	−0.585374	+	0.810763i	$0.699052\pi$
$660$	0	0
$661$	8.39909	0.326687	0.163343	−	0.986569i	$-0.447772\pi$
$661$	8.39909	0.326687	0.163343	+	0.986569i	$0.447772\pi$
$662$	−14.4162	−0.560303
$663$	0	0
$664$	15.6502	0.607344
$665$	0	0
$666$	0	0
$667$	−9.06450	−0.350979
$668$	41.7151	1.61401
$669$	0	0
$670$	1.50972	0.0583257
$671$	0.0930988	0.00359404
$672$	0	0
$673$	12.5476	0.483673	0.241837	−	0.970317i	$-0.422250\pi$
$673$	12.5476	0.483673	0.241837	+	0.970317i	$0.422250\pi$
$674$	11.8977	0.458283
$675$	0	0
$676$	16.8491	0.648042
$677$	15.1655	0.582858	0.291429	−	0.956592i	$-0.405869\pi$
$677$	15.1655	0.582858	0.291429	+	0.956592i	$0.405869\pi$
$678$	0	0
$679$	0	0
$680$	4.17700	0.160180
$681$	0	0
$682$	−0.0516138	−0.00197639
$683$	−29.7816	−1.13956	−0.569781	−	0.821796i	$-0.692972\pi$
$683$	−29.7816	−1.13956	−0.569781	+	0.821796i	$0.692972\pi$
$684$	0	0
$685$	1.88999	0.0722130
$686$	0	0
$687$	0	0
$688$	−13.8970	−0.529818
$689$	−5.85611	−0.223100
$690$	0	0
$691$	−27.9518	−1.06334	−0.531668	−	0.846953i	$-0.678435\pi$
$691$	−27.9518	−1.06334	−0.531668	+	0.846953i	$0.678435\pi$
$692$	12.6845	0.482191
$693$	0	0
$694$	−11.1802	−0.424395
$695$	14.1863	0.538118
$696$	0	0
$697$	0.425575	0.0161198
$698$	−1.71443	−0.0648922
$699$	0	0
$700$	0	0
$701$	3.66534	0.138438	0.0692190	−	0.997601i	$-0.477949\pi$
$701$	3.66534	0.138438	0.0692190	+	0.997601i	$0.477949\pi$
$702$	0	0
$703$	−2.16274	−0.0815691
$704$	−0.160928	−0.00606522
$705$	0	0
$706$	8.02402	0.301988
$707$	0	0
$708$	0	0
$709$	−16.9617	−0.637010	−0.318505	−	0.947921i	$-0.603181\pi$
$709$	−16.9617	−0.637010	−0.318505	+	0.947921i	$0.603181\pi$
$710$	0.885263	0.0332233
$711$	0	0
$712$	12.7526	0.477924
$713$	4.64690	0.174028
$714$	0	0
$715$	0.0595632	0.00222754
$716$	1.14423	0.0427617
$717$	0	0
$718$	−5.21573	−0.194649
$719$	37.9567	1.41555	0.707773	−	0.706440i	$-0.249700\pi$
$719$	37.9567	1.41555	0.707773	+	0.706440i	$0.249700\pi$
$720$	0	0
$721$	0	0
$722$	0.396339	0.0147502
$723$	0	0
$724$	17.1586	0.637695
$725$	−30.2858	−1.12478
$726$	0	0
$727$	32.1265	1.19150	0.595752	−	0.803168i	$-0.296854\pi$
$727$	32.1265	1.19150	0.595752	+	0.803168i	$0.296854\pi$
$728$	0	0
$729$	0	0
$730$	2.07563	0.0768225
$731$	14.6696	0.542573
$732$	0	0
$733$	14.6218	0.540070	0.270035	−	0.962850i	$-0.412965\pi$
$733$	14.6218	0.540070	0.270035	+	0.962850i	$0.412965\pi$
$734$	9.67086	0.356958
$735$	0	0
$736$	−5.47914	−0.201964
$737$	0.162590	0.00598909
$738$	0	0
$739$	28.9192	1.06381	0.531904	−	0.846804i	$-0.321477\pi$
$739$	28.9192	1.06381	0.531904	+	0.846804i	$0.321477\pi$
$740$	3.35964	0.123503
$741$	0	0
$742$	0	0
$743$	−20.8522	−0.764992	−0.382496	−	0.923957i	$-0.624935\pi$
$743$	−20.8522	−0.764992	−0.382496	+	0.923957i	$0.624935\pi$
$744$	0	0
$745$	−10.7320	−0.393191
$746$	0.978461	0.0358240
$747$	0	0
$748$	0.215728	0.00788779
$749$	0	0
$750$	0	0
$751$	−32.9923	−1.20391	−0.601954	−	0.798531i	$-0.705611\pi$
$751$	−32.9923	−1.20391	−0.601954	+	0.798531i	$0.705611\pi$
$752$	24.6128	0.897536
$753$	0	0
$754$	5.49598	0.200152
$755$	−9.96904	−0.362810
$756$	0	0
$757$	14.8974	0.541455	0.270728	−	0.962656i	$-0.412736\pi$
$757$	14.8974	0.541455	0.270728	+	0.962656i	$0.412736\pi$
$758$	−7.44325	−0.270351
$759$	0	0
$760$	−1.28384	−0.0465698
$761$	11.5744	0.419570	0.209785	−	0.977748i	$-0.432724\pi$
$761$	11.5744	0.419570	0.209785	+	0.977748i	$0.432724\pi$
$762$	0	0
$763$	0	0
$764$	33.7961	1.22270
$765$	0	0
$766$	−12.5358	−0.452935
$767$	−25.4928	−0.920491
$768$	0	0
$769$	−52.4315	−1.89073	−0.945365	−	0.326015i	$-0.894294\pi$
$769$	−52.4315	−1.89073	−0.945365	+	0.326015i	$0.894294\pi$
$770$	0	0
$771$	0	0
$772$	28.0130	1.00821
$773$	−43.3738	−1.56005	−0.780023	−	0.625751i	$-0.784793\pi$
$773$	−43.3738	−1.56005	−0.780023	+	0.625751i	$0.784793\pi$
$774$	0	0
$775$	15.5259	0.557708
$776$	3.88635	0.139512
$777$	0	0
$778$	1.71911	0.0616329
$779$	−0.130805	−0.00468656
$780$	0	0
$781$	0.0953388	0.00341149
$782$	1.65550	0.0592007
$783$	0	0
$784$	0	0
$785$	6.21807	0.221932
$786$	0	0
$787$	21.3855	0.762312	0.381156	−	0.924511i	$-0.375526\pi$
$787$	21.3855	0.762312	0.381156	+	0.924511i	$0.375526\pi$
$788$	14.3232	0.510244
$789$	0	0
$790$	3.46535	0.123292
$791$	0	0
$792$	0	0
$793$	5.08209	0.180470
$794$	0.541222	0.0192073
$795$	0	0
$796$	−47.4112	−1.68044
$797$	44.1477	1.56379	0.781897	−	0.623408i	$-0.214252\pi$
$797$	44.1477	1.56379	0.781897	+	0.623408i	$0.214252\pi$
$798$	0	0
$799$	−25.9811	−0.919144
$800$	−18.3066	−0.647236
$801$	0	0
$802$	−8.98123	−0.317138
$803$	0.223536	0.00788841
$804$	0	0
$805$	0	0
$806$	−2.81750	−0.0992422
$807$	0	0
$808$	10.8599	0.382051
$809$	23.8993	0.840253	0.420127	−	0.907465i	$-0.361986\pi$
$809$	23.8993	0.840253	0.420127	+	0.907465i	$0.361986\pi$
$810$	0	0
$811$	−45.1710	−1.58617	−0.793083	−	0.609113i	$-0.791526\pi$
$811$	−45.1710	−1.58617	−0.793083	+	0.609113i	$0.791526\pi$
$812$	0	0
$813$	0	0
$814$	−0.0308402	−0.00108095
$815$	−15.2141	−0.532928
$816$	0	0
$817$	−4.50884	−0.157744
$818$	−10.6152	−0.371153
$819$	0	0
$820$	0.203195	0.00709587
$821$	21.6062	0.754060	0.377030	−	0.926201i	$-0.376945\pi$
$821$	21.6062	0.754060	0.377030	+	0.926201i	$0.376945\pi$
$822$	0	0
$823$	−7.31764	−0.255077	−0.127538	−	0.991834i	$-0.540708\pi$
$823$	−7.31764	−0.255077	−0.127538	+	0.991834i	$0.540708\pi$
$824$	−4.43102	−0.154362
$825$	0	0
$826$	0	0
$827$	−46.0507	−1.60134	−0.800669	−	0.599106i	$-0.795523\pi$
$827$	−46.0507	−1.60134	−0.800669	+	0.599106i	$0.795523\pi$
$828$	0	0
$829$	32.0446	1.11296	0.556478	−	0.830862i	$-0.312152\pi$
$829$	32.0446	1.11296	0.556478	+	0.830862i	$0.312152\pi$
$830$	−3.43275	−0.119152
$831$	0	0
$832$	−8.78479	−0.304558
$833$	0	0
$834$	0	0
$835$	−19.0797	−0.660281
$836$	−0.0663061	−0.00229324
$837$	0	0
$838$	−11.8785	−0.410337
$839$	−12.5994	−0.434980	−0.217490	−	0.976063i	$-0.569787\pi$
$839$	−12.5994	−0.434980	−0.217490	+	0.976063i	$0.569787\pi$
$840$	0	0
$841$	20.8500	0.718964
$842$	−11.4786	−0.395578
$843$	0	0
$844$	−31.7848	−1.09408
$845$	−7.70646	−0.265110
$846$	0	0
$847$	0	0
$848$	−9.19008	−0.315589
$849$	0	0
$850$	5.53127	0.189721
$851$	2.77661	0.0951810
$852$	0	0
$853$	−37.1154	−1.27081	−0.635403	−	0.772181i	$-0.719166\pi$
$853$	−37.1154	−1.27081	−0.635403	+	0.772181i	$0.719166\pi$
$854$	0	0
$855$	0	0
$856$	13.1128	0.448187
$857$	−50.4223	−1.72239	−0.861197	−	0.508271i	$-0.830285\pi$
$857$	−50.4223	−1.72239	−0.861197	+	0.508271i	$0.830285\pi$
$858$	0	0
$859$	9.84495	0.335905	0.167953	−	0.985795i	$-0.446284\pi$
$859$	9.84495	0.335905	0.167953	+	0.985795i	$0.446284\pi$
$860$	7.00413	0.238839
$861$	0	0
$862$	−8.27386	−0.281809
$863$	−32.7221	−1.11387	−0.556937	−	0.830555i	$-0.688024\pi$
$863$	−32.7221	−1.11387	−0.556937	+	0.830555i	$0.688024\pi$
$864$	0	0
$865$	−5.80164	−0.197262
$866$	−2.77939	−0.0944474
$867$	0	0
$868$	0	0
$869$	0.373202	0.0126600
$870$	0	0
$871$	8.87551	0.300735
$872$	−0.875686	−0.0296545
$873$	0	0
$874$	−0.508836	−0.0172116
$875$	0	0
$876$	0	0
$877$	6.87671	0.232210	0.116105	−	0.993237i	$-0.462959\pi$
$877$	6.87671	0.232210	0.116105	+	0.993237i	$0.462959\pi$
$878$	−0.692826	−0.0233818
$879$	0	0
$880$	0.0934735	0.00315099
$881$	40.9688	1.38027	0.690137	−	0.723679i	$-0.257550\pi$
$881$	40.9688	1.38027	0.690137	+	0.723679i	$0.257550\pi$
$882$	0	0
$883$	3.64432	0.122641	0.0613206	−	0.998118i	$-0.480469\pi$
$883$	3.64432	0.122641	0.0613206	+	0.998118i	$0.480469\pi$
$884$	11.7762	0.396076
$885$	0	0
$886$	7.48327	0.251405
$887$	−6.28855	−0.211149	−0.105574	−	0.994411i	$-0.533668\pi$
$887$	−6.28855	−0.211149	−0.105574	+	0.994411i	$0.533668\pi$
$888$	0	0
$889$	0	0
$890$	−2.79719	−0.0937620
$891$	0	0
$892$	−28.9812	−0.970363
$893$	7.98554	0.267226
$894$	0	0
$895$	−0.523348	−0.0174936
$896$	0	0
$897$	0	0
$898$	−7.90982	−0.263954
$899$	−25.5555	−0.852323
$900$	0	0
$901$	9.70098	0.323186
$902$	−0.00186525	−6.21060e−5 0
$903$	0	0
$904$	−23.2705	−0.773966
$905$	−7.84803	−0.260877
$906$	0	0
$907$	−51.6667	−1.71556	−0.857782	−	0.514013i	$-0.828158\pi$
$907$	−51.6667	−1.71556	−0.857782	+	0.514013i	$0.828158\pi$
$908$	−11.9856	−0.397757
$909$	0	0
$910$	0	0
$911$	48.0342	1.59144	0.795722	−	0.605662i	$-0.207091\pi$
$911$	48.0342	1.59144	0.795722	+	0.605662i	$0.207091\pi$
$912$	0	0
$913$	−0.369691	−0.0122350
$914$	−8.54406	−0.282612
$915$	0	0
$916$	−14.5086	−0.479376
$917$	0	0
$918$	0	0
$919$	23.7545	0.783589	0.391794	−	0.920053i	$-0.371854\pi$
$919$	23.7545	0.783589	0.391794	+	0.920053i	$0.371854\pi$
$920$	1.64825	0.0543411
$921$	0	0
$922$	11.0882	0.365171
$923$	5.20437	0.171304
$924$	0	0
$925$	9.27704	0.305027
$926$	15.3974	0.505990
$927$	0	0
$928$	30.1324	0.989145
$929$	5.72639	0.187877	0.0939384	−	0.995578i	$-0.470054\pi$
$929$	5.72639	0.187877	0.0939384	+	0.995578i	$0.470054\pi$
$930$	0	0
$931$	0	0
$932$	41.0582	1.34491
$933$	0	0
$934$	3.42409	0.112040
$935$	−0.0986699	−0.00322685
$936$	0	0
$937$	−24.1920	−0.790319	−0.395159	−	0.918613i	$-0.629311\pi$
$937$	−24.1920	−0.790319	−0.395159	+	0.918613i	$0.629311\pi$
$938$	0	0
$939$	0	0
$940$	−12.4049	−0.404604
$941$	−6.15731	−0.200723	−0.100361	−	0.994951i	$-0.532000\pi$
$941$	−6.15731	−0.200723	−0.100361	+	0.994951i	$0.532000\pi$
$942$	0	0
$943$	0.167932	0.00546863
$944$	−40.0062	−1.30209
$945$	0	0
$946$	−0.0642952	−0.00209042
$947$	25.4531	0.827113	0.413557	−	0.910478i	$-0.364286\pi$
$947$	25.4531	0.827113	0.413557	+	0.910478i	$0.364286\pi$
$948$	0	0
$949$	12.2024	0.396107
$950$	−1.70009	−0.0551582
$951$	0	0
$952$	0	0
$953$	−11.3961	−0.369157	−0.184579	−	0.982818i	$-0.559092\pi$
$953$	−11.3961	−0.369157	−0.184579	+	0.982818i	$0.559092\pi$
$954$	0	0
$955$	−15.4577	−0.500200
$956$	48.8779	1.58082
$957$	0	0
$958$	−5.88177	−0.190031
$959$	0	0
$960$	0	0
$961$	−17.8990	−0.577388
$962$	−1.68351	−0.0542786
$963$	0	0
$964$	−20.4205	−0.657702
$965$	−12.8126	−0.412453
$966$	0	0
$967$	45.9401	1.47733	0.738667	−	0.674071i	$-0.235456\pi$
$967$	45.9401	1.47733	0.738667	+	0.674071i	$0.235456\pi$
$968$	16.7521	0.538432
$969$	0	0
$970$	−0.852444	−0.0273703
$971$	0.694382	0.0222838	0.0111419	−	0.999938i	$-0.496453\pi$
$971$	0.694382	0.0222838	0.0111419	+	0.999938i	$0.496453\pi$
$972$	0	0
$973$	0	0
$974$	7.09615	0.227375
$975$	0	0
$976$	7.97541	0.255286
$977$	36.7435	1.17553	0.587765	−	0.809032i	$-0.300008\pi$
$977$	36.7435	1.17553	0.587765	+	0.809032i	$0.300008\pi$
$978$	0	0
$979$	−0.301244	−0.00962781
$980$	0	0
$981$	0	0
$982$	−7.60443	−0.242667
$983$	13.3577	0.426045	0.213022	−	0.977047i	$-0.431669\pi$
$983$	13.3577	0.426045	0.213022	+	0.977047i	$0.431669\pi$
$984$	0	0
$985$	−6.55118	−0.208738
$986$	−9.10440	−0.289943
$987$	0	0
$988$	−3.61953	−0.115152
$989$	5.78863	0.184068
$990$	0	0
$991$	40.3269	1.28103	0.640514	−	0.767947i	$-0.278722\pi$
$991$	40.3269	1.28103	0.640514	+	0.767947i	$0.278722\pi$
$992$	−15.4473	−0.490453
$993$	0	0
$994$	0	0
$995$	21.6850	0.687460
$996$	0	0
$997$	−7.89474	−0.250029	−0.125014	−	0.992155i	$-0.539898\pi$
$997$	−7.89474	−0.250029	−0.125014	+	0.992155i	$0.539898\pi$
$998$	9.74371	0.308432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.br.1.3		4
3.2	odd	2		2793.2.a.bc.1.2		4
7.2	even	3		1197.2.j.k.172.2		8
7.4	even	3		1197.2.j.k.856.2		8
7.6	odd	2		8379.2.a.bt.1.3		4
21.2	odd	6		399.2.j.d.172.3	yes	8
21.11	odd	6		399.2.j.d.58.3	✓	8
21.20	even	2		2793.2.a.bd.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.d.58.3	✓	8	21.11	odd	6
399.2.j.d.172.3	yes	8	21.2	odd	6
1197.2.j.k.172.2		8	7.2	even	3
1197.2.j.k.856.2		8	7.4	even	3
2793.2.a.bc.1.2		4	3.2	odd	2
2793.2.a.bd.1.2		4	21.20	even	2
8379.2.a.br.1.3		4	1.1	even	1	trivial
8379.2.a.bt.1.3		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+0.396339 q^{2} -1.84292 q^{4} +0.842916 q^{5} -1.52310 q^{8} +0.334080 q^{10} +0.0359789 q^{11} +1.96402 q^{13} +3.08217 q^{16} -3.25351 q^{17} +1.00000 q^{19} -1.55342 q^{20} +0.0142598 q^{22} -1.28384 q^{23} -4.28949 q^{25} +0.778417 q^{26} +7.06045 q^{29} -3.61953 q^{31} +4.26777 q^{32} -1.28949 q^{34} -2.16274 q^{37} +0.396339 q^{38} -1.28384 q^{40} -0.130805 q^{41} -4.50884 q^{43} -0.0663061 q^{44} -0.508836 q^{46} +7.98554 q^{47} -1.70009 q^{50} -3.61953 q^{52} -2.98169 q^{53} +0.0303272 q^{55} +2.79833 q^{58} -12.9799 q^{59} +2.58759 q^{61} -1.43456 q^{62} -4.47286 q^{64} +1.65550 q^{65} +4.51905 q^{67} +5.99595 q^{68} +2.64985 q^{71} +6.21297 q^{73} -0.857176 q^{74} -1.84292 q^{76} +10.3728 q^{79} +2.59801 q^{80} -0.0518429 q^{82} -10.2752 q^{83} -2.74244 q^{85} -1.78703 q^{86} -0.0547993 q^{88} -8.37281 q^{89} +2.36601 q^{92} +3.16498 q^{94} +0.842916 q^{95} -2.55162 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 3 q^{8} - 3 q^{10} + 2 q^{11} + 6 q^{13} - 4 q^{16} + 2 q^{17} + 4 q^{19} - 12 q^{20} - 6 q^{22} - 5 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{29} + 17 q^{31} - 4 q^{32} + 8 q^{34} - 3 q^{37}+ \cdots + 11 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 3 * q^8 - 3 * q^10 + 2 * q^11 + 6 * q^13 - 4 * q^16 + 2 * q^17 + 4 * q^19 - 12 * q^20 - 6 * q^22 - 5 * q^23 - 4 * q^25 + 6 * q^26 + 4 * q^29 + 17 * q^31 - 4 * q^32 + 8 * q^34 - 3 * q^37 - 5 * q^40 - 7 * q^41 - 15 * q^43 - 17 * q^44 + q^46 - 16 * q^47 + 14 * q^50 + 17 * q^52 - 4 * q^53 + 15 * q^55 - 5 * q^58 - 17 * q^59 - 9 * q^61 + 7 * q^62 - 13 * q^64 - 23 * q^65 - 5 * q^67 + 10 * q^68 - 6 * q^71 + 15 * q^73 + 10 * q^74 + 5 * q^79 + 25 * q^80 + 31 * q^82 - 34 * q^83 - 12 * q^85 - 17 * q^86 + 11 * q^88 + 3 * q^89 - 7 * q^92 - 6 * q^94 - 4 * q^95 + 11 * q^97

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