LMFDB - Embedded newform 2793.2.a.bc.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2793,2,Mod(1,2793)] 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("2793.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2793.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.3022172845$
Analytic rank:	$0$
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			1.2
Root			$0.396339$ of defining polynomial
Character	$\chi$	$=$	2793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.396339 q^{2} -1.00000 q^{3} -1.84292 q^{4} -0.842916 q^{5} +0.396339 q^{6} +1.52310 q^{8} +1.00000 q^{9} +0.334080 q^{10} -0.0359789 q^{11} +1.84292 q^{12} +1.96402 q^{13} +0.842916 q^{15} +3.08217 q^{16} +3.25351 q^{17} -0.396339 q^{18} +1.00000 q^{19} +1.55342 q^{20} +0.0142598 q^{22} +1.28384 q^{23} -1.52310 q^{24} -4.28949 q^{25} -0.778417 q^{26} -1.00000 q^{27} -7.06045 q^{29} -0.334080 q^{30} -3.61953 q^{31} -4.26777 q^{32} +0.0359789 q^{33} -1.28949 q^{34} -1.84292 q^{36} -2.16274 q^{37} -0.396339 q^{38} -1.96402 q^{39} -1.28384 q^{40} +0.130805 q^{41} -4.50884 q^{43} +0.0663061 q^{44} -0.842916 q^{45} -0.508836 q^{46} -7.98554 q^{47} -3.08217 q^{48} +1.70009 q^{50} -3.25351 q^{51} -3.61953 q^{52} +2.98169 q^{53} +0.396339 q^{54} +0.0303272 q^{55} -1.00000 q^{57} +2.79833 q^{58} +12.9799 q^{59} -1.55342 q^{60} +2.58759 q^{61} +1.43456 q^{62} -4.47286 q^{64} -1.65550 q^{65} -0.0142598 q^{66} +4.51905 q^{67} -5.99595 q^{68} -1.28384 q^{69} -2.64985 q^{71} +1.52310 q^{72} +6.21297 q^{73} +0.857176 q^{74} +4.28949 q^{75} -1.84292 q^{76} +0.778417 q^{78} +10.3728 q^{79} -2.59801 q^{80} +1.00000 q^{81} -0.0518429 q^{82} +10.2752 q^{83} -2.74244 q^{85} +1.78703 q^{86} +7.06045 q^{87} -0.0547993 q^{88} +8.37281 q^{89} +0.334080 q^{90} -2.36601 q^{92} +3.61953 q^{93} +3.16498 q^{94} -0.842916 q^{95} +4.26777 q^{96} -2.55162 q^{97} -0.0359789 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{8} + 4 q^{9} - 3 q^{10} - 2 q^{11} + 6 q^{13} - 4 q^{15} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 12 q^{20} - 6 q^{22} + 5 q^{23} + 3 q^{24} - 4 q^{25} - 6 q^{26} - 4 q^{27} - 4 q^{29}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^8 + 4 * q^9 - 3 * q^10 - 2 * q^11 + 6 * q^13 - 4 * q^15 - 4 * q^16 - 2 * q^17 + 4 * q^19 + 12 * q^20 - 6 * q^22 + 5 * q^23 + 3 * q^24 - 4 * q^25 - 6 * q^26 - 4 * q^27 - 4 * q^29 + 3 * q^30 + 17 * q^31 + 4 * q^32 + 2 * q^33 + 8 * q^34 - 3 * q^37 - 6 * q^39 - 5 * q^40 + 7 * q^41 - 15 * q^43 + 17 * q^44 + 4 * q^45 + q^46 + 16 * q^47 + 4 * q^48 - 14 * q^50 + 2 * q^51 + 17 * q^52 + 4 * q^53 + 15 * q^55 - 4 * q^57 - 5 * q^58 + 17 * q^59 - 12 * q^60 - 9 * q^61 - 7 * q^62 - 13 * q^64 + 23 * q^65 + 6 * q^66 - 5 * q^67 - 10 * q^68 - 5 * q^69 + 6 * q^71 - 3 * q^72 + 15 * q^73 - 10 * q^74 + 4 * q^75 + 6 * q^78 + 5 * q^79 - 25 * q^80 + 4 * q^81 + 31 * q^82 + 34 * q^83 - 12 * q^85 + 17 * q^86 + 4 * q^87 + 11 * q^88 - 3 * q^89 - 3 * q^90 + 7 * q^92 - 17 * q^93 - 6 * q^94 + 4 * q^95 - 4 * q^96 + 11 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.396339	−0.280254	−0.140127	−	0.990134i	$-0.544751\pi$
$2$	−0.396339	−0.280254	−0.140127	+	0.990134i	$0.544751\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.84292	−0.921458
$5$	−0.842916	−0.376963	−0.188482	−	0.982077i	$-0.560357\pi$
$5$	−0.842916	−0.376963	−0.188482	+	0.982077i	$0.560357\pi$
$6$	0.396339	0.161805
$7$	0	0
$7$	0	0
$8$	1.52310	0.538496
$9$	1.00000	0.333333
$10$	0.334080	0.105645
$11$	−0.0359789	−0.0108480	−0.00542402	−	0.999985i	$-0.501727\pi$
$11$	−0.0359789	−0.0108480	−0.00542402	+	0.999985i	$0.501727\pi$
$12$	1.84292	0.532004
$13$	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
$15$	0.842916	0.217640
$16$	3.08217	0.770543
$17$	3.25351	0.789093	0.394547	−	0.918876i	$-0.370902\pi$
$17$	3.25351	0.789093	0.394547	+	0.918876i	$0.370902\pi$
$18$	−0.396339	−0.0934179
$19$	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	$-0.457266\pi$
$23$	1.28384	0.267699	0.133850	+	0.991002i	$0.457266\pi$
$24$	−1.52310	−0.310901
$25$	−4.28949	−0.857899
$26$	−0.778417	−0.152660
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.06045	−1.31109	−0.655546	−	0.755155i	$-0.727562\pi$
$29$	−7.06045	−1.31109	−0.655546	+	0.755155i	$0.727562\pi$
$30$	−0.334080	−0.0609944
$31$	−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
$33$	0.0359789	0.00626312
$34$	−1.28949	−0.221146
$35$	0	0
$36$	−1.84292	−0.307153
$37$	−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
$39$	−1.96402	−0.314495
$40$	−1.28384	−0.202993
$41$	0.130805	0.0204282	0.0102141	−	0.999948i	$-0.496749\pi$
$41$	0.130805	0.0204282	0.0102141	+	0.999948i	$0.496749\pi$
$42$	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
$45$	−0.842916	−0.125654
$46$	−0.508836	−0.0750237
$47$	−7.98554	−1.16481	−0.582405	−	0.812899i	$-0.697888\pi$
$47$	−7.98554	−1.16481	−0.582405	+	0.812899i	$0.697888\pi$
$48$	−3.08217	−0.444873
$49$	0	0
$50$	1.70009	0.240429
$51$	−3.25351	−0.455583
$52$	−3.61953	−0.501938
$53$	2.98169	0.409567	0.204783	−	0.978807i	$-0.434351\pi$
$53$	2.98169	0.409567	0.204783	+	0.978807i	$0.434351\pi$
$54$	0.396339	0.0539348
$55$	0.0303272	0.00408931
$56$	0	0
$57$	−1.00000	−0.132453
$58$	2.79833	0.367439
$59$	12.9799	1.68984	0.844919	−	0.534895i	$-0.179649\pi$
$59$	12.9799	1.68984	0.844919	+	0.534895i	$0.179649\pi$
$60$	−1.55342	−0.200546
$61$	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
$66$	−0.0142598	−0.00175526
$67$	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
$69$	−1.28384	−0.154556
$70$	0	0
$71$	−2.64985	−0.314480	−0.157240	−	0.987560i	$-0.550260\pi$
$71$	−2.64985	−0.314480	−0.157240	+	0.987560i	$0.550260\pi$
$72$	1.52310	0.179499
$73$	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
$75$	4.28949	0.495308
$76$	−1.84292	−0.211397
$77$	0	0
$78$	0.778417	0.0881384
$79$	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
$81$	1.00000	0.111111
$82$	−0.0518429	−0.00572509
$83$	10.2752	1.12785	0.563927	−	0.825825i	$-0.309290\pi$
$83$	10.2752	1.12785	0.563927	+	0.825825i	$0.309290\pi$
$84$	0	0
$85$	−2.74244	−0.297459
$86$	1.78703	0.192700
$87$	7.06045	0.756960
$88$	−0.0547993	−0.00584162
$89$	8.37281	0.887516	0.443758	−	0.896147i	$-0.353645\pi$
$89$	8.37281	0.887516	0.443758	+	0.896147i	$0.353645\pi$
$90$	0.334080	0.0352151
$91$	0	0
$92$	−2.36601	−0.246674
$93$	3.61953	0.375327
$94$	3.16498	0.326442
$95$	−0.842916	−0.0864813
$96$	4.26777	0.435578
$97$	−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
$99$	−0.0359789	−0.00361601
$100$	7.90517	0.790517
$101$	7.13017	0.709478	0.354739	−	0.934965i	$-0.384570\pi$
$101$	7.13017	0.709478	0.354739	+	0.934965i	$0.384570\pi$
$102$	1.28949	0.127679
$103$	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	$-0.363380\pi$
$107$	8.60931	0.832294	0.416147	+	0.909297i	$0.363380\pi$
$108$	1.84292	0.177335
$109$	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
$111$	2.16274	0.205278
$112$	0	0
$113$	−15.2784	−1.43727	−0.718637	−	0.695385i	$-0.755234\pi$
$113$	−15.2784	−1.43727	−0.718637	+	0.695385i	$0.755234\pi$
$114$	0.396339	0.0371205
$115$	−1.08217	−0.100913
$116$	13.0118	1.20812
$117$	1.96402	0.181574
$118$	−5.14443	−0.473583
$119$	0	0
$120$	1.28384	0.117198
$121$	−10.9987	−0.999882
$122$	−1.02556	−0.0928501
$123$	−0.130805	−0.0117943
$124$	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
$129$	4.50884	0.396981
$130$	0.656140	0.0575473
$131$	−16.9821	−1.48374	−0.741868	−	0.670546i	$-0.766060\pi$
$131$	−16.9821	−1.48374	−0.741868	+	0.670546i	$0.766060\pi$
$132$	−0.0663061	−0.00577120
$133$	0	0
$134$	−1.79107	−0.154725
$135$	0.842916	0.0725466
$136$	4.95541	0.424923
$137$	−2.24221	−0.191565	−0.0957825	−	0.995402i	$-0.530535\pi$
$137$	−2.24221	−0.191565	−0.0957825	+	0.995402i	$0.530535\pi$
$138$	0.508836	0.0433150
$139$	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
$141$	7.98554	0.672504
$142$	1.05024	0.0881341
$143$	−0.0706633	−0.00590916
$144$	3.08217	0.256848
$145$	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	$-0.325364\pi$
$149$	12.7320	1.04305	0.521524	+	0.853237i	$0.325364\pi$
$150$	−1.70009	−0.138812
$151$	−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
$153$	3.25351	0.263031
$154$	0	0
$155$	3.05096	0.245059
$156$	3.61953	0.289794
$157$	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
$159$	−2.98169	−0.236464
$160$	3.59737	0.284397
$161$	0	0
$162$	−0.396339	−0.0311393
$163$	−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
$165$	−0.0303272	−0.00236097
$166$	−4.07247	−0.316085
$167$	22.6354	1.75158	0.875790	−	0.482693i	$-0.160341\pi$
$167$	22.6354	1.75158	0.875790	+	0.482693i	$0.160341\pi$
$168$	0	0
$169$	−9.14262	−0.703279
$170$	1.08693	0.0833640
$171$	1.00000	0.0764719
$172$	8.30940	0.633586
$173$	6.88282	0.523291	0.261646	−	0.965164i	$-0.415735\pi$
$173$	6.88282	0.523291	0.261646	+	0.965164i	$0.415735\pi$
$174$	−2.79833	−0.212141
$175$	0	0
$176$	−0.110893	−0.00835888
$177$	−12.9799	−0.975628
$178$	−3.31847	−0.248730
$179$	0.620878	0.0464066	0.0232033	−	0.999731i	$-0.492614\pi$
$179$	0.620878	0.0464066	0.0232033	+	0.999731i	$0.492614\pi$
$180$	1.55342	0.115785
$181$	−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
$183$	−2.58759	−0.191280
$184$	1.95541	0.144155
$185$	1.82300	0.134030
$186$	−1.43456	−0.105187
$187$	−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	$-0.269088\pi$
$191$	18.3384	1.32692	0.663459	+	0.748212i	$0.269088\pi$
$192$	4.47286	0.322801
$193$	−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
$195$	1.65550	0.118553
$196$	0	0
$197$	7.77205	0.553736	0.276868	−	0.960908i	$-0.410704\pi$
$197$	7.77205	0.553736	0.276868	+	0.960908i	$0.410704\pi$
$198$	0.0142598	0.00101340
$199$	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
$201$	−4.51905	−0.318749
$202$	−2.82596	−0.198834
$203$	0	0
$204$	5.99595	0.419801
$205$	−0.110257	−0.00770070
$206$	−1.15304	−0.0803359
$207$	1.28384	0.0892331
$208$	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
$213$	2.64985	0.181565
$214$	−3.41220	−0.233253
$215$	3.80057	0.259197
$216$	−1.52310	−0.103634
$217$	0	0
$218$	−0.227870	−0.0154333
$219$	−6.21297	−0.419834
$220$	−0.0558904	−0.00376813
$221$	6.38997	0.429836
$222$	−0.857176	−0.0575299
$223$	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
$225$	−4.28949	−0.285966
$226$	6.05544	0.402802
$227$	−6.50362	−0.431660	−0.215830	−	0.976431i	$-0.569246\pi$
$227$	−6.50362	−0.431660	−0.215830	+	0.976431i	$0.569246\pi$
$228$	1.84292	0.122050
$229$	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	$-0.239626\pi$
$233$	22.2790	1.45954	0.729771	+	0.683691i	$0.239626\pi$
$234$	−0.778417	−0.0508867
$235$	6.73113	0.439091
$236$	−23.9208	−1.55711
$237$	−10.3728	−0.673787
$238$	0	0
$239$	26.5221	1.71557	0.857784	−	0.514009i	$-0.171840\pi$
$239$	26.5221	1.71557	0.857784	+	0.514009i	$0.171840\pi$
$240$	2.59801	0.167701
$241$	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
$243$	−1.00000	−0.0641500
$244$	−4.76872	−0.305286
$245$	0	0
$246$	0.0518429	0.00330538
$247$	1.96402	0.124968
$248$	−5.51288	−0.350068
$249$	−10.2752	−0.651166
$250$	−3.10343	−0.196278
$251$	3.32163	0.209659	0.104830	−	0.994490i	$-0.466570\pi$
$251$	3.32163	0.209659	0.104830	+	0.994490i	$0.466570\pi$
$252$	0	0
$253$	−0.0461912	−0.00290401
$254$	−0.405199	−0.0254244
$255$	2.74244	0.171738
$256$	4.86013	0.303758
$257$	30.1189	1.87877	0.939383	−	0.342869i	$-0.111399\pi$
$257$	30.1189	1.87877	0.939383	+	0.342869i	$0.111399\pi$
$258$	−1.78703	−0.111255
$259$	0	0
$260$	3.05096	0.189212
$261$	−7.06045	−0.437031
$262$	6.73067	0.415822
$263$	−25.5666	−1.57651	−0.788253	−	0.615351i	$-0.789014\pi$
$263$	−25.5666	−1.57651	−0.788253	+	0.615351i	$0.789014\pi$
$264$	0.0547993	0.00337266
$265$	−2.51332	−0.154392
$266$	0	0
$267$	−8.37281	−0.512408
$268$	−8.32822	−0.508727
$269$	−6.35130	−0.387245	−0.193623	−	0.981076i	$-0.562024\pi$
$269$	−6.35130	−0.387245	−0.193623	+	0.981076i	$0.562024\pi$
$270$	−0.334080	−0.0203315
$271$	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
$276$	2.36601	0.142417
$277$	−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
$279$	−3.61953	−0.216695
$280$	0	0
$281$	16.7309	0.998084	0.499042	−	0.866578i	$-0.333685\pi$
$281$	16.7309	0.998084	0.499042	+	0.866578i	$0.333685\pi$
$282$	−3.16498	−0.188472
$283$	−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
$285$	0.842916	0.0499300
$286$	0.0280066	0.00165606
$287$	0	0
$288$	−4.26777	−0.251481
$289$	−6.41465	−0.377332
$290$	−2.35876	−0.138511
$291$	2.55162	0.149578
$292$	−11.4500	−0.670060
$293$	−12.1831	−0.711744	−0.355872	−	0.934535i	$-0.615816\pi$
$293$	−12.1831	−0.711744	−0.355872	+	0.934535i	$0.615816\pi$
$294$	0	0
$295$	−10.9409	−0.637007
$296$	−3.29405	−0.191463
$297$	0.0359789	0.00208771
$298$	−5.04619	−0.292318
$299$	2.52149	0.145822
$300$	−7.90517	−0.456405
$301$	0	0
$302$	4.68744	0.269732
$303$	−7.13017	−0.409617
$304$	3.08217	0.176775
$305$	−2.18112	−0.124891
$306$	−1.28949	−0.0737154
$307$	−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
$309$	−2.90922	−0.165500
$310$	−1.20921	−0.0686786
$311$	9.88730	0.560657	0.280329	−	0.959904i	$-0.409557\pi$
$311$	9.88730	0.560657	0.280329	+	0.959904i	$0.409557\pi$
$312$	−2.99139	−0.169354
$313$	−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	$-0.230277\pi$
$317$	26.6902	1.49907	0.749535	+	0.661965i	$0.230277\pi$
$318$	1.18176	0.0662698
$319$	0.254027	0.0142228
$320$	3.77024	0.210763
$321$	−8.60931	−0.480525
$322$	0	0
$323$	3.25351	0.181030
$324$	−1.84292	−0.102384
$325$	−8.42465	−0.467316
$326$	7.15367	0.396205
$327$	−0.574938	−0.0317942
$328$	0.199228	0.0110005
$329$	0	0
$330$	0.0120198	0.000661670 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
$333$	−2.16274	−0.118517
$334$	−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
$339$	15.2784	0.829811
$340$	5.05408	0.274096
$341$	0.130226	0.00705216
$342$	−0.396339	−0.0214315
$343$	0	0
$344$	−6.86739	−0.370265
$345$	1.08217	0.0582621
$346$	−2.72793	−0.146654
$347$	28.2087	1.51432	0.757162	−	0.653227i	$-0.226585\pi$
$347$	28.2087	1.51432	0.757162	+	0.653227i	$0.226585\pi$
$348$	−13.0118	−0.697507
$349$	−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
$351$	−1.96402	−0.104832
$352$	0.153550	0.00818423
$353$	−20.2454	−1.07755	−0.538776	−	0.842449i	$-0.681113\pi$
$353$	−20.2454	−1.07755	−0.538776	+	0.842449i	$0.681113\pi$
$354$	5.14443	0.273423
$355$	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	$-0.387108\pi$
$359$	13.1598	0.694547	0.347273	+	0.937764i	$0.387108\pi$
$360$	−1.28384	−0.0676644
$361$	1.00000	0.0526316
$362$	3.69014	0.193949
$363$	10.9987	0.577282
$364$	0	0
$365$	−5.23701	−0.274118
$366$	1.02556	0.0536070
$367$	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
$369$	0.130805	0.00680941
$370$	−0.722527	−0.0375624
$371$	0	0
$372$	−6.67048	−0.345848
$373$	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
$375$	−7.83026	−0.404353
$376$	−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
$381$	−1.02235	−0.0523768
$382$	−7.26821	−0.371874
$383$	31.6289	1.61616	0.808081	−	0.589072i	$-0.200507\pi$
$383$	31.6289	1.61616	0.808081	+	0.589072i	$0.200507\pi$
$384$	−10.3083	−0.526044
$385$	0	0
$386$	6.02449	0.306639
$387$	−4.50884	−0.229197
$388$	4.70241	0.238729
$389$	−4.33747	−0.219918	−0.109959	−	0.993936i	$-0.535072\pi$
$389$	−4.33747	−0.219918	−0.109959	+	0.993936i	$0.535072\pi$
$390$	−0.656140	−0.0332249
$391$	4.17700	0.211240
$392$	0	0
$393$	16.9821	0.856635
$394$	−3.08036	−0.155186
$395$	−8.74341	−0.439929
$396$	0.0663061	0.00333200
$397$	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	$-0.308565\pi$
$401$	22.6605	1.13161	0.565806	+	0.824539i	$0.308565\pi$
$402$	1.79107	0.0893306
$403$	−7.10882	−0.354116
$404$	−13.1403	−0.653754
$405$	−0.842916	−0.0418848
$406$	0	0
$407$	0.0778128	0.00385704
$408$	−4.95541	−0.245330
$409$	−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
$411$	2.24221	0.110600
$412$	−5.36145	−0.264140
$413$	0	0
$414$	−0.508836	−0.0250079
$415$	−8.66116	−0.425159
$416$	−8.38200	−0.410961
$417$	−16.8301	−0.824171
$418$	0.0142598	0.000697471 0
$419$	29.9706	1.46416	0.732081	−	0.681218i	$-0.238549\pi$
$419$	29.9706	1.46416	0.732081	+	0.681218i	$0.238549\pi$
$420$	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
$423$	−7.98554	−0.388270
$424$	4.54140	0.220550
$425$	−13.9559	−0.676962
$426$	−1.05024	−0.0508842
$427$	0	0
$428$	−15.8662	−0.766924
$429$	0.0706633	0.00341166
$430$	−1.50631	−0.0726408
$431$	20.8757	1.00555	0.502774	−	0.864418i	$-0.332313\pi$
$431$	20.8757	1.00555	0.502774	+	0.864418i	$0.332313\pi$
$432$	−3.08217	−0.148291
$433$	−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
$435$	−5.95137	−0.285346
$436$	−1.05956	−0.0507439
$437$	1.28384	0.0614145
$438$	2.46244	0.117660
$439$	−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	$-0.648053\pi$
$443$	−18.8810	−0.897064	−0.448532	+	0.893767i	$0.648053\pi$
$444$	−3.98574	−0.189155
$445$	−7.05758	−0.334561
$446$	−6.23271	−0.295128
$447$	−12.7320	−0.602204
$448$	0	0
$449$	19.9572	0.941839	0.470920	−	0.882176i	$-0.343922\pi$
$449$	19.9572	0.941839	0.470920	+	0.882176i	$0.343922\pi$
$450$	1.70009	0.0801431
$451$	−0.00470620	−0.000221606 0
$452$	28.1569	1.32439
$453$	11.8268	0.555674
$454$	2.57763	0.120974
$455$	0	0
$456$	−1.52310	−0.0713255
$457$	−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
$459$	−3.25351	−0.151861
$460$	1.99435	0.0929870
$461$	−27.9766	−1.30300	−0.651500	−	0.758649i	$-0.725860\pi$
$461$	−27.9766	−1.30300	−0.651500	+	0.758649i	$0.725860\pi$
$462$	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
$465$	−3.05096	−0.141485
$466$	−8.83001	−0.409042
$467$	−8.63931	−0.399780	−0.199890	−	0.979818i	$-0.564058\pi$
$467$	−8.63931	−0.399780	−0.199890	+	0.979818i	$0.564058\pi$
$468$	−3.61953	−0.167313
$469$	0	0
$470$	−2.66781	−0.123057
$471$	−7.37686	−0.339908
$472$	19.7696	0.909970
$473$	0.162223	0.00745902
$474$	4.11115	0.188831
$475$	−4.28949	−0.196815
$476$	0	0
$477$	2.98169	0.136522
$478$	−10.5117	−0.480794
$479$	14.8403	0.678069	0.339035	−	0.940774i	$-0.389900\pi$
$479$	14.8403	0.678069	0.339035	+	0.940774i	$0.389900\pi$
$480$	−3.59737	−0.164197
$481$	−4.24766	−0.193677
$482$	−4.39166	−0.200034
$483$	0	0
$484$	20.2697	0.921349
$485$	2.15080	0.0976626
$486$	0.396339	0.0179783
$487$	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
$489$	18.0494	0.816222
$490$	0	0
$491$	19.1867	0.865883	0.432942	−	0.901422i	$-0.357476\pi$
$491$	19.1867	0.865883	0.432942	+	0.901422i	$0.357476\pi$
$492$	0.241062	0.0108679
$493$	−22.9713	−1.03457
$494$	−0.778417	−0.0350226
$495$	0.0303272	0.00136310
$496$	−11.1560	−0.500919
$497$	0	0
$498$	4.07247	0.182492
$499$	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
$501$	−22.6354	−1.01127
$502$	−1.31649	−0.0587578
$503$	16.7355	0.746199	0.373099	−	0.927791i	$-0.378295\pi$
$503$	16.7355	0.746199	0.373099	+	0.927791i	$0.378295\pi$
$504$	0	0
$505$	−6.01013	−0.267447
$506$	0.0183073	0.000813861 0
$507$	9.14262	0.406038
$508$	−1.88411	−0.0835940
$509$	−23.5184	−1.04243	−0.521217	−	0.853424i	$-0.674522\pi$
$509$	−23.5184	−1.04243	−0.521217	+	0.853424i	$0.674522\pi$
$510$	−1.08693	−0.0481302
$511$	0	0
$512$	−22.5429	−0.996264
$513$	−1.00000	−0.0441511
$514$	−11.9373	−0.526531
$515$	−2.45223	−0.108058
$516$	−8.30940	−0.365801
$517$	0.287311	0.0126359
$518$	0	0
$519$	−6.88282	−0.302122
$520$	−2.52149	−0.110575
$521$	6.05480	0.265266	0.132633	−	0.991165i	$-0.457657\pi$
$521$	6.05480	0.265266	0.132633	+	0.991165i	$0.457657\pi$
$522$	2.79833	0.122480
$523$	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
$528$	0.110893	0.00482600
$529$	−21.3518	−0.928337
$530$	0.996124	0.0432689
$531$	12.9799	0.563279
$532$	0	0
$533$	0.256903	0.0111277
$534$	3.31847	0.143604
$535$	−7.25693	−0.313744
$536$	6.88294	0.297298
$537$	−0.620878	−0.0267928
$538$	2.51726	0.108527
$539$	0	0
$540$	−1.55342	−0.0668487
$541$	−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
$543$	9.31058	0.399555
$544$	−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
$549$	2.58759	0.110436
$550$	−0.0611674	−0.00260819
$551$	−7.06045	−0.300785
$552$	−1.95541	−0.0832279
$553$	0	0
$554$	2.32876	0.0989396
$555$	−1.82300	−0.0773822
$556$	−31.0164	−1.31539
$557$	15.2693	0.646979	0.323490	−	0.946232i	$-0.395144\pi$
$557$	15.2693	0.646979	0.323490	+	0.946232i	$0.395144\pi$
$558$	1.43456	0.0607297
$559$	−8.85545	−0.374546
$560$	0	0
$561$	0.117058	0.00494218
$562$	−6.63111	−0.279717
$563$	−30.5738	−1.28853	−0.644267	−	0.764801i	$-0.722837\pi$
$563$	−30.5738	−1.28853	−0.644267	+	0.764801i	$0.722837\pi$
$564$	−14.7167	−0.619684
$565$	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	$-0.558003\pi$
$569$	−8.64529	−0.362429	−0.181215	+	0.983444i	$0.558003\pi$
$570$	−0.334080	−0.0139931
$571$	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
$573$	−18.3384	−0.766097
$574$	0	0
$575$	−5.50703	−0.229659
$576$	−4.47286	−0.186369
$577$	−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
$579$	15.2004	0.631706
$580$	−10.9679	−0.455416
$581$	0	0
$582$	−1.01130	−0.0419199
$583$	−0.107278	−0.00444300
$584$	9.46295	0.391580
$585$	−1.65550	−0.0684467
$586$	4.82863	0.199469
$587$	19.9966	0.825350	0.412675	−	0.910878i	$-0.364595\pi$
$587$	19.9966	0.825350	0.412675	+	0.910878i	$0.364595\pi$
$588$	0	0
$589$	−3.61953	−0.149140
$590$	4.33632	0.178523
$591$	−7.77205	−0.319699
$592$	−6.66592	−0.273968
$593$	11.1740	0.458863	0.229431	−	0.973325i	$-0.426313\pi$
$593$	11.1740	0.458863	0.229431	+	0.973325i	$0.426313\pi$
$594$	−0.0142598	−0.000585087 0
$595$	0	0
$596$	−23.4640	−0.961125
$597$	−25.7262	−1.05290
$598$	−0.999364	−0.0408670
$599$	−24.0539	−0.982815	−0.491407	−	0.870930i	$-0.663517\pi$
$599$	−24.0539	−0.982815	−0.491407	+	0.870930i	$0.663517\pi$
$600$	6.53331	0.266721
$601$	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
$603$	4.51905	0.184030
$604$	21.7959	0.886862
$605$	9.27098	0.376919
$606$	2.82596	0.114797
$607$	−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
$612$	−5.99595	−0.242372
$613$	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
$615$	0.110257	0.00444600
$616$	0	0
$617$	25.0627	1.00899	0.504493	−	0.863416i	$-0.331679\pi$
$617$	25.0627	1.00899	0.504493	+	0.863416i	$0.331679\pi$
$618$	1.15304	0.0463819
$619$	−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
$621$	−1.28384	−0.0515188
$622$	−3.91872	−0.157126
$623$	0	0
$624$	−6.05345	−0.242332
$625$	14.8472	0.593889
$626$	0.544863	0.0217771
$627$	0.0359789	0.00143686
$628$	−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
$633$	−17.2470	−0.685508
$634$	−10.5784	−0.420120
$635$	−0.861759	−0.0341979
$636$	5.49501	0.217891
$637$	0	0
$638$	−0.100681	−0.00398599
$639$	−2.64985	−0.104827
$640$	−8.68904	−0.343464
$641$	49.3339	1.94857	0.974286	−	0.225313i	$-0.0723404\pi$
$641$	49.3339	1.94857	0.974286	+	0.225313i	$0.0723404\pi$
$642$	3.41220	0.134669
$643$	−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
$645$	−3.80057	−0.149647
$646$	−1.28949	−0.0507344
$647$	−35.1832	−1.38319	−0.691597	−	0.722283i	$-0.743093\pi$
$647$	−35.1832	−1.38319	−0.691597	+	0.722283i	$0.743093\pi$
$648$	1.52310	0.0598328
$649$	−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	$-0.684737\pi$
$653$	−28.0240	−1.09666	−0.548332	+	0.836261i	$0.684737\pi$
$654$	0.227870	0.00891043
$655$	14.3145	0.559314
$656$	0.403162	0.0157408
$657$	6.21297	0.242391
$658$	0	0
$659$	30.0543	1.17075	0.585374	−	0.810763i	$-0.300948\pi$
$659$	30.0543	1.17075	0.585374	+	0.810763i	$0.300948\pi$
$660$	0.0558904	0.00217553
$661$	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
$663$	−6.38997	−0.248166
$664$	15.6502	0.607344
$665$	0	0
$666$	0.857176	0.0332149
$667$	−9.06450	−0.350979
$668$	−41.7151	−1.61401
$669$	−15.7257	−0.607992
$670$	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
$675$	4.28949	0.165103
$676$	16.8491	0.648042
$677$	−15.1655	−0.582858	−0.291429	−	0.956592i	$-0.594131\pi$
$677$	−15.1655	−0.582858	−0.291429	+	0.956592i	$0.594131\pi$
$678$	−6.05544	−0.232558
$679$	0	0
$680$	−4.17700	−0.160180
$681$	6.50362	0.249219
$682$	−0.0516138	−0.00197639
$683$	29.7816	1.13956	0.569781	−	0.821796i	$-0.307028\pi$
$683$	29.7816	1.13956	0.569781	+	0.821796i	$0.307028\pi$
$684$	−1.84292	−0.0704656
$685$	1.88999	0.0722130
$686$	0	0
$687$	−7.87261	−0.300359
$688$	−13.8970	−0.529818
$689$	5.85611	0.223100
$690$	−0.428906	−0.0163282
$691$	−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
$696$	10.7537	0.407620
$697$	0.425575	0.0161198
$698$	1.71443	0.0648922
$699$	−22.2790	−0.842667
$700$	0	0
$701$	−3.66534	−0.138438	−0.0692190	−	0.997601i	$-0.522051\pi$
$701$	−3.66534	−0.138438	−0.0692190	+	0.997601i	$0.522051\pi$
$702$	0.778417	0.0293795
$703$	−2.16274	−0.0815691
$704$	0.160928	0.00606522
$705$	−6.73113	−0.253509
$706$	8.02402	0.301988
$707$	0	0
$708$	23.9208	0.899000
$709$	−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
$711$	10.3728	0.389011
$712$	12.7526	0.477924
$713$	−4.64690	−0.174028
$714$	0	0
$715$	0.0595632	0.00222754
$716$	−1.14423	−0.0427617
$717$	−26.5221	−0.990484
$718$	−5.21573	−0.194649
$719$	−37.9567	−1.41555	−0.707773	−	0.706440i	$-0.750300\pi$
$719$	−37.9567	−1.41555	−0.707773	+	0.706440i	$0.750300\pi$
$720$	−2.59801	−0.0968221
$721$	0	0
$722$	−0.396339	−0.0147502
$723$	−11.0806	−0.412091
$724$	17.1586	0.637695
$725$	30.2858	1.12478
$726$	−4.35921	−0.161785
$727$	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
$729$	1.00000	0.0370370
$730$	2.07563	0.0768225
$731$	−14.6696	−0.542573
$732$	4.76872	0.176257
$733$	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
$738$	−0.0518429	−0.00190836
$739$	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
$741$	−1.96402	−0.0721501
$742$	0	0
$743$	20.8522	0.764992	0.382496	−	0.923957i	$-0.375065\pi$
$743$	20.8522	0.764992	0.382496	+	0.923957i	$0.375065\pi$
$744$	5.51288	0.202112
$745$	−10.7320	−0.393191
$746$	−0.978461	−0.0358240
$747$	10.2752	0.375951
$748$	0.215728	0.00788779
$749$	0	0
$750$	3.10343	0.113321
$751$	−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
$753$	−3.32163	−0.121047
$754$	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
$759$	0.0461912	0.00167663
$760$	−1.28384	−0.0465698
$761$	−11.5744	−0.419570	−0.209785	−	0.977748i	$-0.567276\pi$
$761$	−11.5744	−0.419570	−0.209785	+	0.977748i	$0.567276\pi$
$762$	0.405199	0.0146788
$763$	0	0
$764$	−33.7961	−1.22270
$765$	−2.74244	−0.0991531
$766$	−12.5358	−0.452935
$767$	25.4928	0.920491
$768$	−4.86013	−0.175375
$769$	−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
$771$	−30.1189	−1.08471
$772$	28.0130	1.00821
$773$	43.3738	1.56005	0.780023	−	0.625751i	$-0.215207\pi$
$773$	43.3738	1.56005	0.780023	+	0.625751i	$0.215207\pi$
$774$	1.78703	0.0642333
$775$	15.5259	0.557708
$776$	−3.88635	−0.139512
$777$	0	0
$778$	1.71911	0.0616329
$779$	0.130805	0.00468656
$780$	−3.05096	−0.109242
$781$	0.0953388	0.00341149
$782$	−1.65550	−0.0592007
$783$	7.06045	0.252320
$784$	0	0
$785$	−6.21807	−0.221932
$786$	−6.73067	−0.240075
$787$	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
$789$	25.5666	0.910196
$790$	3.46535	0.123292
$791$	0	0
$792$	−0.0547993	−0.00194721
$793$	5.08209	0.180470
$794$	−0.541222	−0.0192073
$795$	2.51332	0.0891381
$796$	−47.4112	−1.68044
$797$	−44.1477	−1.56379	−0.781897	−	0.623408i	$-0.785748\pi$
$797$	−44.1477	−1.56379	−0.781897	+	0.623408i	$0.785748\pi$
$798$	0	0
$799$	−25.9811	−0.919144
$800$	18.3066	0.647236
$801$	8.37281	0.295839
$802$	−8.98123	−0.317138
$803$	−0.223536	−0.00788841
$804$	8.32822	0.293714
$805$	0	0
$806$	2.81750	0.0992422
$807$	6.35130	0.223576
$808$	10.8599	0.382051
$809$	−23.8993	−0.840253	−0.420127	−	0.907465i	$-0.638014\pi$
$809$	−23.8993	−0.840253	−0.420127	+	0.907465i	$0.638014\pi$
$810$	0.334080	0.0117384
$811$	−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
$813$	−7.86174	−0.275723
$814$	−0.0308402	−0.00108095
$815$	15.2141	0.532928
$816$	−10.0279	−0.351046
$817$	−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	$-0.623055\pi$
$821$	−21.6062	−0.754060	−0.377030	+	0.926201i	$0.623055\pi$
$822$	−0.888674	−0.0309961
$823$	−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
$825$	−0.154331	−0.00537312
$826$	0	0
$827$	46.0507	1.60134	0.800669	−	0.599106i	$-0.204477\pi$
$827$	46.0507	1.60134	0.800669	+	0.599106i	$0.204477\pi$
$828$	−2.36601	−0.0822246
$829$	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
$831$	5.87569	0.203825
$832$	−8.78479	−0.304558
$833$	0	0
$834$	6.67040	0.230977
$835$	−19.0797	−0.660281
$836$	0.0663061	0.00229324
$837$	3.61953	0.125109
$838$	−11.8785	−0.410337
$839$	12.5994	0.434980	0.217490	−	0.976063i	$-0.430213\pi$
$839$	12.5994	0.434980	0.217490	+	0.976063i	$0.430213\pi$
$840$	0	0
$841$	20.8500	0.718964
$842$	11.4786	0.395578
$843$	−16.7309	−0.576244
$844$	−31.7848	−1.09408
$845$	7.70646	0.265110
$846$	3.16498	0.108814
$847$	0	0
$848$	9.19008	0.315589
$849$	20.8957	0.717139
$850$	5.53127	0.189721
$851$	−2.77661	−0.0951810
$852$	−4.88346	−0.167304
$853$	−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
$855$	−0.842916	−0.0288271
$856$	13.1128	0.448187
$857$	50.4223	1.72239	0.861197	−	0.508271i	$-0.169715\pi$
$857$	50.4223	1.72239	0.861197	+	0.508271i	$0.169715\pi$
$858$	−0.0280066	−0.000956129 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	$-0.311976\pi$
$863$	32.7221	1.11387	0.556937	+	0.830555i	$0.311976\pi$
$864$	4.26777	0.145193
$865$	−5.80164	−0.197262
$866$	2.77939	0.0944474
$867$	6.41465	0.217853
$868$	0	0
$869$	−0.373202	−0.0126600
$870$	2.35876	0.0799693
$871$	8.87551	0.300735
$872$	0.875686	0.0296545
$873$	−2.55162	−0.0863591
$874$	−0.508836	−0.0172116
$875$	0	0
$876$	11.4500	0.386859
$877$	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
$879$	12.1831	0.410926
$880$	0.0934735	0.00315099
$881$	−40.9688	−1.38027	−0.690137	−	0.723679i	$-0.742450\pi$
$881$	−40.9688	−1.38027	−0.690137	+	0.723679i	$0.742450\pi$
$882$	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
$885$	10.9409	0.367776
$886$	7.48327	0.251405
$887$	6.28855	0.211149	0.105574	−	0.994411i	$-0.466332\pi$
$887$	6.28855	0.211149	0.105574	+	0.994411i	$0.466332\pi$
$888$	3.29405	0.110541
$889$	0	0
$890$	2.79719	0.0937620
$891$	−0.0359789	−0.00120534
$892$	−28.9812	−0.970363
$893$	−7.98554	−0.267226
$894$	5.04619	0.168770
$895$	−0.523348	−0.0174936
$896$	0	0
$897$	−2.52149	−0.0841901
$898$	−7.90982	−0.263954
$899$	25.5555	0.852323
$900$	7.90517	0.263506
$901$	9.70098	0.323186
$902$	0.00186525	6.21060e−5 0
$903$	0	0
$904$	−23.2705	−0.773966
$905$	7.84803	0.260877
$906$	−4.68744	−0.155730
$907$	−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
$909$	7.13017	0.236493
$910$	0	0
$911$	−48.0342	−1.59144	−0.795722	−	0.605662i	$-0.792909\pi$
$911$	−48.0342	−1.59144	−0.795722	+	0.605662i	$0.792909\pi$
$912$	−3.08217	−0.102061
$913$	−0.369691	−0.0122350
$914$	8.54406	0.282612
$915$	2.18112	0.0721057
$916$	−14.5086	−0.479376
$917$	0	0
$918$	1.28949	0.0425596
$919$	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
$921$	25.2971	0.833566
$922$	11.0882	0.365171
$923$	−5.20437	−0.171304
$924$	0	0
$925$	9.27704	0.305027
$926$	−15.3974	−0.505990
$927$	2.90922	0.0955514
$928$	30.1324	0.989145
$929$	−5.72639	−0.187877	−0.0939384	−	0.995578i	$-0.529946\pi$
$929$	−5.72639	−0.187877	−0.0939384	+	0.995578i	$0.529946\pi$
$930$	1.20921	0.0396516
$931$	0	0
$932$	−41.0582	−1.34491
$933$	−9.88730	−0.323696
$934$	3.42409	0.112040
$935$	0.0986699	0.00322685
$936$	2.99139	0.0977767
$937$	−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
$939$	1.37474	0.0448630
$940$	−12.4049	−0.404604
$941$	6.15731	0.200723	0.100361	−	0.994951i	$-0.468000\pi$
$941$	6.15731	0.200723	0.100361	+	0.994951i	$0.468000\pi$
$942$	2.92373	0.0952604
$943$	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	$-0.635714\pi$
$947$	−25.4531	−0.827113	−0.413557	+	0.910478i	$0.635714\pi$
$948$	19.1162	0.620866
$949$	12.2024	0.396107
$950$	1.70009	0.0551582
$951$	−26.6902	−0.865489
$952$	0	0
$953$	11.3961	0.369157	0.184579	−	0.982818i	$-0.440908\pi$
$953$	11.3961	0.369157	0.184579	+	0.982818i	$0.440908\pi$
$954$	−1.18176	−0.0382609
$955$	−15.4577	−0.500200
$956$	−48.8779	−1.58082
$957$	−0.254027	−0.00821153
$958$	−5.88177	−0.190031
$959$	0	0
$960$	−3.77024	−0.121684
$961$	−17.8990	−0.577388
$962$	1.68351	0.0542786
$963$	8.60931	0.277431
$964$	−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
$969$	−3.25351	−0.104518
$970$	−0.852444	−0.0273703
$971$	−0.694382	−0.0222838	−0.0111419	−	0.999938i	$-0.503547\pi$
$971$	−0.694382	−0.0222838	−0.0111419	+	0.999938i	$0.503547\pi$
$972$	1.84292	0.0591116
$973$	0	0
$974$	−7.09615	−0.227375
$975$	8.42465	0.269805
$976$	7.97541	0.255286
$977$	−36.7435	−1.17553	−0.587765	−	0.809032i	$-0.699992\pi$
$977$	−36.7435	−1.17553	−0.587765	+	0.809032i	$0.699992\pi$
$978$	−7.15367	−0.228749
$979$	−0.301244	−0.00962781
$980$	0	0
$981$	0.574938	0.0183564
$982$	−7.60443	−0.242667
$983$	−13.3577	−0.426045	−0.213022	−	0.977047i	$-0.568331\pi$
$983$	−13.3577	−0.426045	−0.213022	+	0.977047i	$0.568331\pi$
$984$	−0.199228	−0.00635115
$985$	−6.55118	−0.208738
$986$	9.10440	0.289943
$987$	0	0
$988$	−3.61953	−0.115152
$989$	−5.78863	−0.184068
$990$	−0.0120198	−0.000382015 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
$993$	36.3735	1.15428
$994$	0	0
$995$	−21.6850	−0.687460
$996$	18.9364	0.600022
$997$	−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
$999$	2.16274	0.0684259

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2793.a (trivial)

\(f(q)\)	\(=\)	\(q-0.396339 q^{2} -1.00000 q^{3} -1.84292 q^{4} -0.842916 q^{5} +0.396339 q^{6} +1.52310 q^{8} +1.00000 q^{9} +0.334080 q^{10} -0.0359789 q^{11} +1.84292 q^{12} +1.96402 q^{13} +0.842916 q^{15} +3.08217 q^{16} +3.25351 q^{17} -0.396339 q^{18} +1.00000 q^{19} +1.55342 q^{20} +0.0142598 q^{22} +1.28384 q^{23} -1.52310 q^{24} -4.28949 q^{25} -0.778417 q^{26} -1.00000 q^{27} -7.06045 q^{29} -0.334080 q^{30} -3.61953 q^{31} -4.26777 q^{32} +0.0359789 q^{33} -1.28949 q^{34} -1.84292 q^{36} -2.16274 q^{37} -0.396339 q^{38} -1.96402 q^{39} -1.28384 q^{40} +0.130805 q^{41} -4.50884 q^{43} +0.0663061 q^{44} -0.842916 q^{45} -0.508836 q^{46} -7.98554 q^{47} -3.08217 q^{48} +1.70009 q^{50} -3.25351 q^{51} -3.61953 q^{52} +2.98169 q^{53} +0.396339 q^{54} +0.0303272 q^{55} -1.00000 q^{57} +2.79833 q^{58} +12.9799 q^{59} -1.55342 q^{60} +2.58759 q^{61} +1.43456 q^{62} -4.47286 q^{64} -1.65550 q^{65} -0.0142598 q^{66} +4.51905 q^{67} -5.99595 q^{68} -1.28384 q^{69} -2.64985 q^{71} +1.52310 q^{72} +6.21297 q^{73} +0.857176 q^{74} +4.28949 q^{75} -1.84292 q^{76} +0.778417 q^{78} +10.3728 q^{79} -2.59801 q^{80} +1.00000 q^{81} -0.0518429 q^{82} +10.2752 q^{83} -2.74244 q^{85} +1.78703 q^{86} +7.06045 q^{87} -0.0547993 q^{88} +8.37281 q^{89} +0.334080 q^{90} -2.36601 q^{92} +3.61953 q^{93} +3.16498 q^{94} -0.842916 q^{95} +4.26777 q^{96} -2.55162 q^{97} -0.0359789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{8} + 4 q^{9} - 3 q^{10} - 2 q^{11} + 6 q^{13} - 4 q^{15} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 12 q^{20} - 6 q^{22} + 5 q^{23} + 3 q^{24} - 4 q^{25} - 6 q^{26} - 4 q^{27} - 4 q^{29}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^8 + 4 * q^9 - 3 * q^10 - 2 * q^11 + 6 * q^13 - 4 * q^15 - 4 * q^16 - 2 * q^17 + 4 * q^19 + 12 * q^20 - 6 * q^22 + 5 * q^23 + 3 * q^24 - 4 * q^25 - 6 * q^26 - 4 * q^27 - 4 * q^29 + 3 * q^30 + 17 * q^31 + 4 * q^32 + 2 * q^33 + 8 * q^34 - 3 * q^37 - 6 * q^39 - 5 * q^40 + 7 * q^41 - 15 * q^43 + 17 * q^44 + 4 * q^45 + q^46 + 16 * q^47 + 4 * q^48 - 14 * q^50 + 2 * q^51 + 17 * q^52 + 4 * q^53 + 15 * q^55 - 4 * q^57 - 5 * q^58 + 17 * q^59 - 12 * q^60 - 9 * q^61 - 7 * q^62 - 13 * q^64 + 23 * q^65 + 6 * q^66 - 5 * q^67 - 10 * q^68 - 5 * q^69 + 6 * q^71 - 3 * q^72 + 15 * q^73 - 10 * q^74 + 4 * q^75 + 6 * q^78 + 5 * q^79 - 25 * q^80 + 4 * q^81 + 31 * q^82 + 34 * q^83 - 12 * q^85 + 17 * q^86 + 4 * q^87 + 11 * q^88 - 3 * q^89 - 3 * q^90 + 7 * q^92 - 17 * q^93 - 6 * q^94 + 4 * q^95 - 4 * q^96 + 11 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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