LMFDB - Embedded newform 1274.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1274,2,Mod(1,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1274 = 2 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1274.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,-2,2,-4,-2,0,2,0,-4,2,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.1729412175$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1274.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} -0.585786 q^{10} -0.414214 q^{11} -2.41421 q^{12} +1.00000 q^{13} +1.41421 q^{15} +1.00000 q^{16} -4.00000 q^{17} +2.82843 q^{18} -0.585786 q^{19} -0.585786 q^{20} -0.414214 q^{22} +5.24264 q^{23} -2.41421 q^{24} -4.65685 q^{25} +1.00000 q^{26} +0.414214 q^{27} -3.41421 q^{29} +1.41421 q^{30} -5.58579 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} +2.82843 q^{36} -0.171573 q^{37} -0.585786 q^{38} -2.41421 q^{39} -0.585786 q^{40} -7.00000 q^{41} -6.58579 q^{43} -0.414214 q^{44} -1.65685 q^{45} +5.24264 q^{46} -1.24264 q^{47} -2.41421 q^{48} -4.65685 q^{50} +9.65685 q^{51} +1.00000 q^{52} -11.8995 q^{53} +0.414214 q^{54} +0.242641 q^{55} +1.41421 q^{57} -3.41421 q^{58} +4.58579 q^{59} +1.41421 q^{60} -2.17157 q^{61} -5.58579 q^{62} +1.00000 q^{64} -0.585786 q^{65} +1.00000 q^{66} -14.4142 q^{67} -4.00000 q^{68} -12.6569 q^{69} -0.242641 q^{71} +2.82843 q^{72} +3.00000 q^{73} -0.171573 q^{74} +11.2426 q^{75} -0.585786 q^{76} -2.41421 q^{78} -5.58579 q^{79} -0.585786 q^{80} -9.48528 q^{81} -7.00000 q^{82} +9.89949 q^{83} +2.34315 q^{85} -6.58579 q^{86} +8.24264 q^{87} -0.414214 q^{88} -18.1421 q^{89} -1.65685 q^{90} +5.24264 q^{92} +13.4853 q^{93} -1.24264 q^{94} +0.343146 q^{95} -2.41421 q^{96} -8.31371 q^{97} -1.17157 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} - 4 q^{10} + 2 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 - 4 * q^10 + 2 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^26 - 2 * q^27 - 4 * q^29 - 14 * q^31 + 2 * q^32 + 2 * q^33 - 8 * q^34 - 6 * q^37 - 4 * q^38 - 2 * q^39 - 4 * q^40 - 14 * q^41 - 16 * q^43 + 2 * q^44 + 8 * q^45 + 2 * q^46 + 6 * q^47 - 2 * q^48 + 2 * q^50 + 8 * q^51 + 2 * q^52 - 4 * q^53 - 2 * q^54 - 8 * q^55 - 4 * q^58 + 12 * q^59 - 10 * q^61 - 14 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 26 * q^67 - 8 * q^68 - 14 * q^69 + 8 * q^71 + 6 * q^73 - 6 * q^74 + 14 * q^75 - 4 * q^76 - 2 * q^78 - 14 * q^79 - 4 * q^80 - 2 * q^81 - 14 * q^82 + 16 * q^85 - 16 * q^86 + 8 * q^87 + 2 * q^88 - 8 * q^89 + 8 * q^90 + 2 * q^92 + 10 * q^93 + 6 * q^94 + 12 * q^95 - 2 * q^96 + 6 * q^97 - 8 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	1.00000	0.500000
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	−2.41421	−0.985599
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	−0.585786	−0.185242
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	−2.41421	−0.696923
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.41421	0.365148
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	2.82843	0.666667
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	−0.585786	−0.130986
$21$	0	0
$22$	−0.414214	−0.0883106
$23$	5.24264	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$23$	5.24264	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$24$	−2.41421	−0.492799
$25$	−4.65685	−0.931371
$26$	1.00000	0.196116
$27$	0.414214	0.0797154
$28$	0	0
$29$	−3.41421	−0.634004	−0.317002	−	0.948425i	$-0.602676\pi$
$29$	−3.41421	−0.634004	−0.317002	+	0.948425i	$0.602676\pi$
$30$	1.41421	0.258199
$31$	−5.58579	−1.00324	−0.501618	−	0.865089i	$-0.667262\pi$
$31$	−5.58579	−1.00324	−0.501618	+	0.865089i	$0.667262\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	2.82843	0.471405
$37$	−0.171573	−0.0282064	−0.0141032	−	0.999901i	$-0.504489\pi$
$37$	−0.171573	−0.0282064	−0.0141032	+	0.999901i	$0.504489\pi$
$38$	−0.585786	−0.0950271
$39$	−2.41421	−0.386584
$40$	−0.585786	−0.0926210
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	−6.58579	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$43$	−6.58579	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$44$	−0.414214	−0.0624450
$45$	−1.65685	−0.246989
$46$	5.24264	0.772985
$47$	−1.24264	−0.181258	−0.0906289	−	0.995885i	$-0.528888\pi$
$47$	−1.24264	−0.181258	−0.0906289	+	0.995885i	$0.528888\pi$
$48$	−2.41421	−0.348462
$49$	0	0
$50$	−4.65685	−0.658579
$51$	9.65685	1.35223
$52$	1.00000	0.138675
$53$	−11.8995	−1.63452	−0.817261	−	0.576268i	$-0.804508\pi$
$53$	−11.8995	−1.63452	−0.817261	+	0.576268i	$0.804508\pi$
$54$	0.414214	0.0563673
$55$	0.242641	0.0327177
$56$	0	0
$57$	1.41421	0.187317
$58$	−3.41421	−0.448308
$59$	4.58579	0.597019	0.298509	−	0.954407i	$-0.403511\pi$
$59$	4.58579	0.597019	0.298509	+	0.954407i	$0.403511\pi$
$60$	1.41421	0.182574
$61$	−2.17157	−0.278041	−0.139021	−	0.990289i	$-0.544395\pi$
$61$	−2.17157	−0.278041	−0.139021	+	0.990289i	$0.544395\pi$
$62$	−5.58579	−0.709396
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.585786	−0.0726579
$66$	1.00000	0.123091
$67$	−14.4142	−1.76098	−0.880488	−	0.474068i	$-0.842785\pi$
$67$	−14.4142	−1.76098	−0.880488	+	0.474068i	$0.842785\pi$
$68$	−4.00000	−0.485071
$69$	−12.6569	−1.52371
$70$	0	0
$71$	−0.242641	−0.0287962	−0.0143981	−	0.999896i	$-0.504583\pi$
$71$	−0.242641	−0.0287962	−0.0143981	+	0.999896i	$0.504583\pi$
$72$	2.82843	0.333333
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−0.171573	−0.0199449
$75$	11.2426	1.29819
$76$	−0.585786	−0.0671943
$77$	0	0
$78$	−2.41421	−0.273356
$79$	−5.58579	−0.628450	−0.314225	−	0.949349i	$-0.601745\pi$
$79$	−5.58579	−0.628450	−0.314225	+	0.949349i	$0.601745\pi$
$80$	−0.585786	−0.0654929
$81$	−9.48528	−1.05392
$82$	−7.00000	−0.773021
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	2.34315	0.254150
$86$	−6.58579	−0.710164
$87$	8.24264	0.883704
$88$	−0.414214	−0.0441553
$89$	−18.1421	−1.92306	−0.961531	−	0.274696i	$-0.911423\pi$
$89$	−18.1421	−1.92306	−0.961531	+	0.274696i	$0.911423\pi$
$90$	−1.65685	−0.174648
$91$	0	0
$92$	5.24264	0.546583
$93$	13.4853	1.39836
$94$	−1.24264	−0.128169
$95$	0.343146	0.0352060
$96$	−2.41421	−0.246400
$97$	−8.31371	−0.844129	−0.422065	−	0.906566i	$-0.638694\pi$
$97$	−8.31371	−0.844129	−0.422065	+	0.906566i	$0.638694\pi$
$98$	0	0
$99$	−1.17157	−0.117748
$100$	−4.65685	−0.465685
$101$	18.3137	1.82228	0.911141	−	0.412095i	$-0.135203\pi$
$101$	18.3137	1.82228	0.911141	+	0.412095i	$0.135203\pi$
$102$	9.65685	0.956171
$103$	−10.4853	−1.03315	−0.516573	−	0.856243i	$-0.672792\pi$
$103$	−10.4853	−1.03315	−0.516573	+	0.856243i	$0.672792\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−11.8995	−1.15578
$107$	12.2426	1.18354	0.591770	−	0.806107i	$-0.298429\pi$
$107$	12.2426	1.18354	0.591770	+	0.806107i	$0.298429\pi$
$108$	0.414214	0.0398577
$109$	−3.31371	−0.317396	−0.158698	−	0.987327i	$-0.550730\pi$
$109$	−3.31371	−0.317396	−0.158698	+	0.987327i	$0.550730\pi$
$110$	0.242641	0.0231349
$111$	0.414214	0.0393154
$112$	0	0
$113$	−13.9706	−1.31424	−0.657120	−	0.753786i	$-0.728226\pi$
$113$	−13.9706	−1.31424	−0.657120	+	0.753786i	$0.728226\pi$
$114$	1.41421	0.132453
$115$	−3.07107	−0.286379
$116$	−3.41421	−0.317002
$117$	2.82843	0.261488
$118$	4.58579	0.422156
$119$	0	0
$120$	1.41421	0.129099
$121$	−10.8284	−0.984402
$122$	−2.17157	−0.196605
$123$	16.8995	1.52378
$124$	−5.58579	−0.501618
$125$	5.65685	0.505964
$126$	0	0
$127$	12.5563	1.11420	0.557098	−	0.830447i	$-0.311915\pi$
$127$	12.5563	1.11420	0.557098	+	0.830447i	$0.311915\pi$
$128$	1.00000	0.0883883
$129$	15.8995	1.39987
$130$	−0.585786	−0.0513769
$131$	15.6569	1.36795	0.683973	−	0.729507i	$-0.260251\pi$
$131$	15.6569	1.36795	0.683973	+	0.729507i	$0.260251\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−14.4142	−1.24520
$135$	−0.242641	−0.0208832
$136$	−4.00000	−0.342997
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−12.6569	−1.07742
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−0.242641	−0.0203620
$143$	−0.414214	−0.0346383
$144$	2.82843	0.235702
$145$	2.00000	0.166091
$146$	3.00000	0.248282
$147$	0	0
$148$	−0.171573	−0.0141032
$149$	4.31371	0.353393	0.176696	−	0.984265i	$-0.443459\pi$
$149$	4.31371	0.353393	0.176696	+	0.984265i	$0.443459\pi$
$150$	11.2426	0.917958
$151$	20.9706	1.70656	0.853280	−	0.521453i	$-0.174610\pi$
$151$	20.9706	1.70656	0.853280	+	0.521453i	$0.174610\pi$
$152$	−0.585786	−0.0475136
$153$	−11.3137	−0.914659
$154$	0	0
$155$	3.27208	0.262820
$156$	−2.41421	−0.193292
$157$	11.9706	0.955355	0.477677	−	0.878535i	$-0.341479\pi$
$157$	11.9706	0.955355	0.477677	+	0.878535i	$0.341479\pi$
$158$	−5.58579	−0.444381
$159$	28.7279	2.27827
$160$	−0.585786	−0.0463105
$161$	0	0
$162$	−9.48528	−0.745234
$163$	−5.31371	−0.416202	−0.208101	−	0.978107i	$-0.566728\pi$
$163$	−5.31371	−0.416202	−0.208101	+	0.978107i	$0.566728\pi$
$164$	−7.00000	−0.546608
$165$	−0.585786	−0.0456034
$166$	9.89949	0.768350
$167$	10.8284	0.837929	0.418964	−	0.908003i	$-0.362393\pi$
$167$	10.8284	0.837929	0.418964	+	0.908003i	$0.362393\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.34315	0.179711
$171$	−1.65685	−0.126703
$172$	−6.58579	−0.502162
$173$	−13.6569	−1.03831	−0.519156	−	0.854680i	$-0.673754\pi$
$173$	−13.6569	−1.03831	−0.519156	+	0.854680i	$0.673754\pi$
$174$	8.24264	0.624873
$175$	0	0
$176$	−0.414214	−0.0312225
$177$	−11.0711	−0.832152
$178$	−18.1421	−1.35981
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	−1.65685	−0.123495
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	5.24264	0.387547
$184$	5.24264	0.386493
$185$	0.100505	0.00738928
$186$	13.4853	0.988789
$187$	1.65685	0.121161
$188$	−1.24264	−0.0906289
$189$	0	0
$190$	0.343146	0.0248944
$191$	0.485281	0.0351137	0.0175569	−	0.999846i	$-0.494411\pi$
$191$	0.485281	0.0351137	0.0175569	+	0.999846i	$0.494411\pi$
$192$	−2.41421	−0.174231
$193$	20.9706	1.50949	0.754747	−	0.656016i	$-0.227760\pi$
$193$	20.9706	1.50949	0.754747	+	0.656016i	$0.227760\pi$
$194$	−8.31371	−0.596889
$195$	1.41421	0.101274
$196$	0	0
$197$	10.3137	0.734821	0.367411	−	0.930059i	$-0.380244\pi$
$197$	10.3137	0.734821	0.367411	+	0.930059i	$0.380244\pi$
$198$	−1.17157	−0.0832601
$199$	−24.7279	−1.75292	−0.876458	−	0.481478i	$-0.840100\pi$
$199$	−24.7279	−1.75292	−0.876458	+	0.481478i	$0.840100\pi$
$200$	−4.65685	−0.329289
$201$	34.7990	2.45453
$202$	18.3137	1.28855
$203$	0	0
$204$	9.65685	0.676115
$205$	4.10051	0.286392
$206$	−10.4853	−0.730544
$207$	14.8284	1.03065
$208$	1.00000	0.0693375
$209$	0.242641	0.0167838
$210$	0	0
$211$	−9.75736	−0.671724	−0.335862	−	0.941911i	$-0.609028\pi$
$211$	−9.75736	−0.671724	−0.335862	+	0.941911i	$0.609028\pi$
$212$	−11.8995	−0.817261
$213$	0.585786	0.0401374
$214$	12.2426	0.836890
$215$	3.85786	0.263104
$216$	0.414214	0.0281837
$217$	0	0
$218$	−3.31371	−0.224433
$219$	−7.24264	−0.489412
$220$	0.242641	0.0163588
$221$	−4.00000	−0.269069
$222$	0.414214	0.0278002
$223$	23.2426	1.55644	0.778221	−	0.627990i	$-0.216122\pi$
$223$	23.2426	1.55644	0.778221	+	0.627990i	$0.216122\pi$
$224$	0	0
$225$	−13.1716	−0.878105
$226$	−13.9706	−0.929308
$227$	−12.9706	−0.860886	−0.430443	−	0.902618i	$-0.641643\pi$
$227$	−12.9706	−0.860886	−0.430443	+	0.902618i	$0.641643\pi$
$228$	1.41421	0.0936586
$229$	−15.8995	−1.05067	−0.525334	−	0.850896i	$-0.676060\pi$
$229$	−15.8995	−1.05067	−0.525334	+	0.850896i	$0.676060\pi$
$230$	−3.07107	−0.202500
$231$	0	0
$232$	−3.41421	−0.224154
$233$	21.4853	1.40755	0.703774	−	0.710424i	$-0.251497\pi$
$233$	21.4853	1.40755	0.703774	+	0.710424i	$0.251497\pi$
$234$	2.82843	0.184900
$235$	0.727922	0.0474844
$236$	4.58579	0.298509
$237$	13.4853	0.875963
$238$	0	0
$239$	21.7990	1.41006	0.705030	−	0.709178i	$-0.250934\pi$
$239$	21.7990	1.41006	0.705030	+	0.709178i	$0.250934\pi$
$240$	1.41421	0.0912871
$241$	−14.8284	−0.955183	−0.477591	−	0.878582i	$-0.658490\pi$
$241$	−14.8284	−0.955183	−0.477591	+	0.878582i	$0.658490\pi$
$242$	−10.8284	−0.696078
$243$	21.6569	1.38929
$244$	−2.17157	−0.139021
$245$	0	0
$246$	16.8995	1.07747
$247$	−0.585786	−0.0372727
$248$	−5.58579	−0.354698
$249$	−23.8995	−1.51457
$250$	5.65685	0.357771
$251$	−13.7279	−0.866499	−0.433249	−	0.901274i	$-0.642633\pi$
$251$	−13.7279	−0.866499	−0.433249	+	0.901274i	$0.642633\pi$
$252$	0	0
$253$	−2.17157	−0.136526
$254$	12.5563	0.787855
$255$	−5.65685	−0.354246
$256$	1.00000	0.0625000
$257$	16.9706	1.05859	0.529297	−	0.848436i	$-0.322456\pi$
$257$	16.9706	1.05859	0.529297	+	0.848436i	$0.322456\pi$
$258$	15.8995	0.989859
$259$	0	0
$260$	−0.585786	−0.0363289
$261$	−9.65685	−0.597744
$262$	15.6569	0.967284
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	1.00000	0.0615457
$265$	6.97056	0.428198
$266$	0	0
$267$	43.7990	2.68045
$268$	−14.4142	−0.880488
$269$	13.8284	0.843134	0.421567	−	0.906797i	$-0.361480\pi$
$269$	13.8284	0.843134	0.421567	+	0.906797i	$0.361480\pi$
$270$	−0.242641	−0.0147666
$271$	−24.2132	−1.47085	−0.735424	−	0.677608i	$-0.763017\pi$
$271$	−24.2132	−1.47085	−0.735424	+	0.677608i	$0.763017\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−2.00000	−0.120824
$275$	1.92893	0.116319
$276$	−12.6569	−0.761853
$277$	15.5563	0.934690	0.467345	−	0.884075i	$-0.345211\pi$
$277$	15.5563	0.934690	0.467345	+	0.884075i	$0.345211\pi$
$278$	6.34315	0.380437
$279$	−15.7990	−0.945861
$280$	0	0
$281$	−31.4142	−1.87401	−0.937007	−	0.349309i	$-0.886416\pi$
$281$	−31.4142	−1.87401	−0.937007	+	0.349309i	$0.886416\pi$
$282$	3.00000	0.178647
$283$	6.55635	0.389735	0.194867	−	0.980830i	$-0.437572\pi$
$283$	6.55635	0.389735	0.194867	+	0.980830i	$0.437572\pi$
$284$	−0.242641	−0.0143981
$285$	−0.828427	−0.0490718
$286$	−0.414214	−0.0244930
$287$	0	0
$288$	2.82843	0.166667
$289$	−1.00000	−0.0588235
$290$	2.00000	0.117444
$291$	20.0711	1.17659
$292$	3.00000	0.175562
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−2.68629	−0.156402
$296$	−0.171573	−0.00997247
$297$	−0.171573	−0.00995567
$298$	4.31371	0.249886
$299$	5.24264	0.303190
$300$	11.2426	0.649094
$301$	0	0
$302$	20.9706	1.20672
$303$	−44.2132	−2.53998
$304$	−0.585786	−0.0335972
$305$	1.27208	0.0728390
$306$	−11.3137	−0.646762
$307$	27.0711	1.54503	0.772514	−	0.634998i	$-0.218999\pi$
$307$	27.0711	1.54503	0.772514	+	0.634998i	$0.218999\pi$
$308$	0	0
$309$	25.3137	1.44005
$310$	3.27208	0.185842
$311$	2.72792	0.154686	0.0773431	−	0.997005i	$-0.475356\pi$
$311$	2.72792	0.154686	0.0773431	+	0.997005i	$0.475356\pi$
$312$	−2.41421	−0.136678
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	11.9706	0.675538
$315$	0	0
$316$	−5.58579	−0.314225
$317$	18.3137	1.02860	0.514300	−	0.857610i	$-0.328052\pi$
$317$	18.3137	1.02860	0.514300	+	0.857610i	$0.328052\pi$
$318$	28.7279	1.61098
$319$	1.41421	0.0791808
$320$	−0.585786	−0.0327465
$321$	−29.5563	−1.64967
$322$	0	0
$323$	2.34315	0.130376
$324$	−9.48528	−0.526960
$325$	−4.65685	−0.258316
$326$	−5.31371	−0.294299
$327$	8.00000	0.442401
$328$	−7.00000	−0.386510
$329$	0	0
$330$	−0.585786	−0.0322465
$331$	−26.6985	−1.46748	−0.733741	−	0.679430i	$-0.762227\pi$
$331$	−26.6985	−1.46748	−0.733741	+	0.679430i	$0.762227\pi$
$332$	9.89949	0.543305
$333$	−0.485281	−0.0265933
$334$	10.8284	0.592505
$335$	8.44365	0.461326
$336$	0	0
$337$	30.6569	1.66999	0.834993	−	0.550261i	$-0.185472\pi$
$337$	30.6569	1.66999	0.834993	+	0.550261i	$0.185472\pi$
$338$	1.00000	0.0543928
$339$	33.7279	1.83185
$340$	2.34315	0.127075
$341$	2.31371	0.125294
$342$	−1.65685	−0.0895924
$343$	0	0
$344$	−6.58579	−0.355082
$345$	7.41421	0.399168
$346$	−13.6569	−0.734197
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	8.24264	0.441852
$349$	3.55635	0.190367	0.0951835	−	0.995460i	$-0.469656\pi$
$349$	3.55635	0.190367	0.0951835	+	0.995460i	$0.469656\pi$
$350$	0	0
$351$	0.414214	0.0221091
$352$	−0.414214	−0.0220777
$353$	6.17157	0.328480	0.164240	−	0.986420i	$-0.447483\pi$
$353$	6.17157	0.328480	0.164240	+	0.986420i	$0.447483\pi$
$354$	−11.0711	−0.588421
$355$	0.142136	0.00754378
$356$	−18.1421	−0.961531
$357$	0	0
$358$	−6.48528	−0.342758
$359$	−7.55635	−0.398809	−0.199404	−	0.979917i	$-0.563901\pi$
$359$	−7.55635	−0.398809	−0.199404	+	0.979917i	$0.563901\pi$
$360$	−1.65685	−0.0873239
$361$	−18.6569	−0.981940
$362$	−7.00000	−0.367912
$363$	26.1421	1.37211
$364$	0	0
$365$	−1.75736	−0.0919844
$366$	5.24264	0.274037
$367$	−4.92893	−0.257288	−0.128644	−	0.991691i	$-0.541062\pi$
$367$	−4.92893	−0.257288	−0.128644	+	0.991691i	$0.541062\pi$
$368$	5.24264	0.273292
$369$	−19.7990	−1.03069
$370$	0.100505	0.00522501
$371$	0	0
$372$	13.4853	0.699179
$373$	−23.4142	−1.21234	−0.606171	−	0.795334i	$-0.707295\pi$
$373$	−23.4142	−1.21234	−0.606171	+	0.795334i	$0.707295\pi$
$374$	1.65685	0.0856739
$375$	−13.6569	−0.705237
$376$	−1.24264	−0.0640843
$377$	−3.41421	−0.175841
$378$	0	0
$379$	5.17157	0.265646	0.132823	−	0.991140i	$-0.457596\pi$
$379$	5.17157	0.265646	0.132823	+	0.991140i	$0.457596\pi$
$380$	0.343146	0.0176030
$381$	−30.3137	−1.55302
$382$	0.485281	0.0248292
$383$	−6.27208	−0.320488	−0.160244	−	0.987077i	$-0.551228\pi$
$383$	−6.27208	−0.320488	−0.160244	+	0.987077i	$0.551228\pi$
$384$	−2.41421	−0.123200
$385$	0	0
$386$	20.9706	1.06737
$387$	−18.6274	−0.946885
$388$	−8.31371	−0.422065
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	1.41421	0.0716115
$391$	−20.9706	−1.06053
$392$	0	0
$393$	−37.7990	−1.90671
$394$	10.3137	0.519597
$395$	3.27208	0.164636
$396$	−1.17157	−0.0588738
$397$	−37.9411	−1.90421	−0.952105	−	0.305770i	$-0.901086\pi$
$397$	−37.9411	−1.90421	−0.952105	+	0.305770i	$0.901086\pi$
$398$	−24.7279	−1.23950
$399$	0	0
$400$	−4.65685	−0.232843
$401$	−17.0711	−0.852488	−0.426244	−	0.904608i	$-0.640164\pi$
$401$	−17.0711	−0.852488	−0.426244	+	0.904608i	$0.640164\pi$
$402$	34.7990	1.73562
$403$	−5.58579	−0.278248
$404$	18.3137	0.911141
$405$	5.55635	0.276097
$406$	0	0
$407$	0.0710678	0.00352270
$408$	9.65685	0.478086
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	4.10051	0.202510
$411$	4.82843	0.238169
$412$	−10.4853	−0.516573
$413$	0	0
$414$	14.8284	0.728777
$415$	−5.79899	−0.284661
$416$	1.00000	0.0490290
$417$	−15.3137	−0.749916
$418$	0.242641	0.0118679
$419$	23.2426	1.13548	0.567739	−	0.823209i	$-0.307818\pi$
$419$	23.2426	1.13548	0.567739	+	0.823209i	$0.307818\pi$
$420$	0	0
$421$	7.68629	0.374607	0.187303	−	0.982302i	$-0.440025\pi$
$421$	7.68629	0.374607	0.187303	+	0.982302i	$0.440025\pi$
$422$	−9.75736	−0.474981
$423$	−3.51472	−0.170891
$424$	−11.8995	−0.577891
$425$	18.6274	0.903562
$426$	0.585786	0.0283814
$427$	0	0
$428$	12.2426	0.591770
$429$	1.00000	0.0482805
$430$	3.85786	0.186043
$431$	−28.1421	−1.35556	−0.677779	−	0.735265i	$-0.737058\pi$
$431$	−28.1421	−1.35556	−0.677779	+	0.735265i	$0.737058\pi$
$432$	0.414214	0.0199289
$433$	−7.65685	−0.367965	−0.183982	−	0.982930i	$-0.558899\pi$
$433$	−7.65685	−0.367965	−0.183982	+	0.982930i	$0.558899\pi$
$434$	0	0
$435$	−4.82843	−0.231505
$436$	−3.31371	−0.158698
$437$	−3.07107	−0.146909
$438$	−7.24264	−0.346067
$439$	−0.585786	−0.0279581	−0.0139790	−	0.999902i	$-0.504450\pi$
$439$	−0.585786	−0.0279581	−0.0139790	+	0.999902i	$0.504450\pi$
$440$	0.242641	0.0115674
$441$	0	0
$442$	−4.00000	−0.190261
$443$	10.3848	0.493396	0.246698	−	0.969092i	$-0.420654\pi$
$443$	10.3848	0.493396	0.246698	+	0.969092i	$0.420654\pi$
$444$	0.414214	0.0196577
$445$	10.6274	0.503788
$446$	23.2426	1.10057
$447$	−10.4142	−0.492575
$448$	0	0
$449$	1.61522	0.0762271	0.0381136	−	0.999273i	$-0.487865\pi$
$449$	1.61522	0.0762271	0.0381136	+	0.999273i	$0.487865\pi$
$450$	−13.1716	−0.620914
$451$	2.89949	0.136532
$452$	−13.9706	−0.657120
$453$	−50.6274	−2.37868
$454$	−12.9706	−0.608739
$455$	0	0
$456$	1.41421	0.0662266
$457$	−24.7279	−1.15672	−0.578362	−	0.815780i	$-0.696308\pi$
$457$	−24.7279	−1.15672	−0.578362	+	0.815780i	$0.696308\pi$
$458$	−15.8995	−0.742935
$459$	−1.65685	−0.0773353
$460$	−3.07107	−0.143189
$461$	−34.7279	−1.61744	−0.808720	−	0.588193i	$-0.799839\pi$
$461$	−34.7279	−1.61744	−0.808720	+	0.588193i	$0.799839\pi$
$462$	0	0
$463$	−36.2843	−1.68627	−0.843137	−	0.537700i	$-0.819293\pi$
$463$	−36.2843	−1.68627	−0.843137	+	0.537700i	$0.819293\pi$
$464$	−3.41421	−0.158501
$465$	−7.89949	−0.366330
$466$	21.4853	0.995286
$467$	−18.1421	−0.839518	−0.419759	−	0.907636i	$-0.637885\pi$
$467$	−18.1421	−0.839518	−0.419759	+	0.907636i	$0.637885\pi$
$468$	2.82843	0.130744
$469$	0	0
$470$	0.727922	0.0335765
$471$	−28.8995	−1.33162
$472$	4.58579	0.211078
$473$	2.72792	0.125430
$474$	13.4853	0.619399
$475$	2.72792	0.125166
$476$	0	0
$477$	−33.6569	−1.54104
$478$	21.7990	0.997063
$479$	20.2843	0.926812	0.463406	−	0.886146i	$-0.346627\pi$
$479$	20.2843	0.926812	0.463406	+	0.886146i	$0.346627\pi$
$480$	1.41421	0.0645497
$481$	−0.171573	−0.00782305
$482$	−14.8284	−0.675416
$483$	0	0
$484$	−10.8284	−0.492201
$485$	4.87006	0.221138
$486$	21.6569	0.982375
$487$	40.3848	1.83001	0.915004	−	0.403444i	$-0.132187\pi$
$487$	40.3848	1.83001	0.915004	+	0.403444i	$0.132187\pi$
$488$	−2.17157	−0.0983025
$489$	12.8284	0.580122
$490$	0	0
$491$	−26.3848	−1.19073	−0.595364	−	0.803456i	$-0.702992\pi$
$491$	−26.3848	−1.19073	−0.595364	+	0.803456i	$0.702992\pi$
$492$	16.8995	0.761888
$493$	13.6569	0.615074
$494$	−0.585786	−0.0263558
$495$	0.686292	0.0308465
$496$	−5.58579	−0.250809
$497$	0	0
$498$	−23.8995	−1.07096
$499$	12.8995	0.577461	0.288730	−	0.957410i	$-0.406767\pi$
$499$	12.8995	0.577461	0.288730	+	0.957410i	$0.406767\pi$
$500$	5.65685	0.252982
$501$	−26.1421	−1.16794
$502$	−13.7279	−0.612707
$503$	−6.34315	−0.282827	−0.141413	−	0.989951i	$-0.545165\pi$
$503$	−6.34315	−0.282827	−0.141413	+	0.989951i	$0.545165\pi$
$504$	0	0
$505$	−10.7279	−0.477386
$506$	−2.17157	−0.0965382
$507$	−2.41421	−0.107219
$508$	12.5563	0.557098
$509$	−6.38478	−0.283000	−0.141500	−	0.989938i	$-0.545193\pi$
$509$	−6.38478	−0.283000	−0.141500	+	0.989938i	$0.545193\pi$
$510$	−5.65685	−0.250490
$511$	0	0
$512$	1.00000	0.0441942
$513$	−0.242641	−0.0107128
$514$	16.9706	0.748539
$515$	6.14214	0.270655
$516$	15.8995	0.699936
$517$	0.514719	0.0226373
$518$	0	0
$519$	32.9706	1.44725
$520$	−0.585786	−0.0256884
$521$	−26.5858	−1.16474	−0.582372	−	0.812922i	$-0.697875\pi$
$521$	−26.5858	−1.16474	−0.582372	+	0.812922i	$0.697875\pi$
$522$	−9.65685	−0.422669
$523$	12.2132	0.534046	0.267023	−	0.963690i	$-0.413960\pi$
$523$	12.2132	0.534046	0.267023	+	0.963690i	$0.413960\pi$
$524$	15.6569	0.683973
$525$	0	0
$526$	−16.0000	−0.697633
$527$	22.3431	0.973283
$528$	1.00000	0.0435194
$529$	4.48528	0.195012
$530$	6.97056	0.302782
$531$	12.9706	0.562874
$532$	0	0
$533$	−7.00000	−0.303204
$534$	43.7990	1.89537
$535$	−7.17157	−0.310054
$536$	−14.4142	−0.622599
$537$	15.6569	0.675643
$538$	13.8284	0.596186
$539$	0	0
$540$	−0.242641	−0.0104416
$541$	43.7990	1.88307	0.941533	−	0.336921i	$-0.109386\pi$
$541$	43.7990	1.88307	0.941533	+	0.336921i	$0.109386\pi$
$542$	−24.2132	−1.04005
$543$	16.8995	0.725227
$544$	−4.00000	−0.171499
$545$	1.94113	0.0831487
$546$	0	0
$547$	1.61522	0.0690620	0.0345310	−	0.999404i	$-0.489006\pi$
$547$	1.61522	0.0690620	0.0345310	+	0.999404i	$0.489006\pi$
$548$	−2.00000	−0.0854358
$549$	−6.14214	−0.262140
$550$	1.92893	0.0822499
$551$	2.00000	0.0852029
$552$	−12.6569	−0.538711
$553$	0	0
$554$	15.5563	0.660926
$555$	−0.242641	−0.0102995
$556$	6.34315	0.269009
$557$	6.31371	0.267520	0.133760	−	0.991014i	$-0.457295\pi$
$557$	6.31371	0.267520	0.133760	+	0.991014i	$0.457295\pi$
$558$	−15.7990	−0.668825
$559$	−6.58579	−0.278549
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−31.4142	−1.32513
$563$	13.4437	0.566582	0.283291	−	0.959034i	$-0.408574\pi$
$563$	13.4437	0.566582	0.283291	+	0.959034i	$0.408574\pi$
$564$	3.00000	0.126323
$565$	8.18377	0.344294
$566$	6.55635	0.275584
$567$	0	0
$568$	−0.242641	−0.0101810
$569$	31.7696	1.33185	0.665924	−	0.746019i	$-0.268037\pi$
$569$	31.7696	1.33185	0.665924	+	0.746019i	$0.268037\pi$
$570$	−0.828427	−0.0346990
$571$	35.8995	1.50235	0.751174	−	0.660105i	$-0.229488\pi$
$571$	35.8995	1.50235	0.751174	+	0.660105i	$0.229488\pi$
$572$	−0.414214	−0.0173191
$573$	−1.17157	−0.0489432
$574$	0	0
$575$	−24.4142	−1.01814
$576$	2.82843	0.117851
$577$	−25.6569	−1.06811	−0.534054	−	0.845450i	$-0.679332\pi$
$577$	−25.6569	−1.06811	−0.534054	+	0.845450i	$0.679332\pi$
$578$	−1.00000	−0.0415945
$579$	−50.6274	−2.10400
$580$	2.00000	0.0830455
$581$	0	0
$582$	20.0711	0.831973
$583$	4.92893	0.204136
$584$	3.00000	0.124141
$585$	−1.65685	−0.0685025
$586$	14.0000	0.578335
$587$	40.1421	1.65684	0.828422	−	0.560105i	$-0.189239\pi$
$587$	40.1421	1.65684	0.828422	+	0.560105i	$0.189239\pi$
$588$	0	0
$589$	3.27208	0.134824
$590$	−2.68629	−0.110593
$591$	−24.8995	−1.02423
$592$	−0.171573	−0.00705160
$593$	12.4853	0.512709	0.256355	−	0.966583i	$-0.417479\pi$
$593$	12.4853	0.512709	0.256355	+	0.966583i	$0.417479\pi$
$594$	−0.171573	−0.00703972
$595$	0	0
$596$	4.31371	0.176696
$597$	59.6985	2.44330
$598$	5.24264	0.214388
$599$	−14.4142	−0.588949	−0.294474	−	0.955659i	$-0.595145\pi$
$599$	−14.4142	−0.588949	−0.294474	+	0.955659i	$0.595145\pi$
$600$	11.2426	0.458979
$601$	−22.5858	−0.921293	−0.460647	−	0.887584i	$-0.652382\pi$
$601$	−22.5858	−0.921293	−0.460647	+	0.887584i	$0.652382\pi$
$602$	0	0
$603$	−40.7696	−1.66026
$604$	20.9706	0.853280
$605$	6.34315	0.257886
$606$	−44.2132	−1.79604
$607$	−28.5858	−1.16026	−0.580131	−	0.814523i	$-0.696999\pi$
$607$	−28.5858	−1.16026	−0.580131	+	0.814523i	$0.696999\pi$
$608$	−0.585786	−0.0237568
$609$	0	0
$610$	1.27208	0.0515049
$611$	−1.24264	−0.0502719
$612$	−11.3137	−0.457330
$613$	43.2843	1.74824	0.874118	−	0.485714i	$-0.161440\pi$
$613$	43.2843	1.74824	0.874118	+	0.485714i	$0.161440\pi$
$614$	27.0711	1.09250
$615$	−9.89949	−0.399186
$616$	0	0
$617$	2.38478	0.0960075	0.0480037	−	0.998847i	$-0.484714\pi$
$617$	2.38478	0.0960075	0.0480037	+	0.998847i	$0.484714\pi$
$618$	25.3137	1.01827
$619$	−4.38478	−0.176239	−0.0881195	−	0.996110i	$-0.528086\pi$
$619$	−4.38478	−0.176239	−0.0881195	+	0.996110i	$0.528086\pi$
$620$	3.27208	0.131410
$621$	2.17157	0.0871422
$622$	2.72792	0.109380
$623$	0	0
$624$	−2.41421	−0.0966459
$625$	19.9706	0.798823
$626$	−6.00000	−0.239808
$627$	−0.585786	−0.0233941
$628$	11.9706	0.477677
$629$	0.686292	0.0273642
$630$	0	0
$631$	21.7990	0.867804	0.433902	−	0.900960i	$-0.357136\pi$
$631$	21.7990	0.867804	0.433902	+	0.900960i	$0.357136\pi$
$632$	−5.58579	−0.222191
$633$	23.5563	0.936281
$634$	18.3137	0.727330
$635$	−7.35534	−0.291888
$636$	28.7279	1.13914
$637$	0	0
$638$	1.41421	0.0559893
$639$	−0.686292	−0.0271493
$640$	−0.585786	−0.0231552
$641$	−4.51472	−0.178321	−0.0891603	−	0.996017i	$-0.528418\pi$
$641$	−4.51472	−0.178321	−0.0891603	+	0.996017i	$0.528418\pi$
$642$	−29.5563	−1.16650
$643$	−24.2843	−0.957678	−0.478839	−	0.877903i	$-0.658942\pi$
$643$	−24.2843	−0.957678	−0.478839	+	0.877903i	$0.658942\pi$
$644$	0	0
$645$	−9.31371	−0.366727
$646$	2.34315	0.0921898
$647$	12.2426	0.481308	0.240654	−	0.970611i	$-0.422638\pi$
$647$	12.2426	0.481308	0.240654	+	0.970611i	$0.422638\pi$
$648$	−9.48528	−0.372617
$649$	−1.89949	−0.0745617
$650$	−4.65685	−0.182657
$651$	0	0
$652$	−5.31371	−0.208101
$653$	13.5563	0.530501	0.265250	−	0.964180i	$-0.414545\pi$
$653$	13.5563	0.530501	0.265250	+	0.964180i	$0.414545\pi$
$654$	8.00000	0.312825
$655$	−9.17157	−0.358363
$656$	−7.00000	−0.273304
$657$	8.48528	0.331042
$658$	0	0
$659$	−20.5858	−0.801908	−0.400954	−	0.916098i	$-0.631321\pi$
$659$	−20.5858	−0.801908	−0.400954	+	0.916098i	$0.631321\pi$
$660$	−0.585786	−0.0228017
$661$	−14.4437	−0.561793	−0.280896	−	0.959738i	$-0.590632\pi$
$661$	−14.4437	−0.561793	−0.280896	+	0.959738i	$0.590632\pi$
$662$	−26.6985	−1.03767
$663$	9.65685	0.375041
$664$	9.89949	0.384175
$665$	0	0
$666$	−0.485281	−0.0188043
$667$	−17.8995	−0.693071
$668$	10.8284	0.418964
$669$	−56.1127	−2.16944
$670$	8.44365	0.326207
$671$	0.899495	0.0347246
$672$	0	0
$673$	−33.7696	−1.30172	−0.650860	−	0.759198i	$-0.725592\pi$
$673$	−33.7696	−1.30172	−0.650860	+	0.759198i	$0.725592\pi$
$674$	30.6569	1.18086
$675$	−1.92893	−0.0742446
$676$	1.00000	0.0384615
$677$	−30.1716	−1.15959	−0.579794	−	0.814763i	$-0.696867\pi$
$677$	−30.1716	−1.15959	−0.579794	+	0.814763i	$0.696867\pi$
$678$	33.7279	1.29531
$679$	0	0
$680$	2.34315	0.0898555
$681$	31.3137	1.19994
$682$	2.31371	0.0885965
$683$	28.8995	1.10581	0.552904	−	0.833245i	$-0.313520\pi$
$683$	28.8995	1.10581	0.552904	+	0.833245i	$0.313520\pi$
$684$	−1.65685	−0.0633514
$685$	1.17157	0.0447635
$686$	0	0
$687$	38.3848	1.46447
$688$	−6.58579	−0.251081
$689$	−11.8995	−0.453335
$690$	7.41421	0.282254
$691$	−4.68629	−0.178275	−0.0891375	−	0.996019i	$-0.528411\pi$
$691$	−4.68629	−0.178275	−0.0891375	+	0.996019i	$0.528411\pi$
$692$	−13.6569	−0.519156
$693$	0	0
$694$	12.0000	0.455514
$695$	−3.71573	−0.140946
$696$	8.24264	0.312436
$697$	28.0000	1.06058
$698$	3.55635	0.134610
$699$	−51.8701	−1.96191
$700$	0	0
$701$	41.7574	1.57715	0.788577	−	0.614936i	$-0.210818\pi$
$701$	41.7574	1.57715	0.788577	+	0.614936i	$0.210818\pi$
$702$	0.414214	0.0156335
$703$	0.100505	0.00379062
$704$	−0.414214	−0.0156113
$705$	−1.75736	−0.0661860
$706$	6.17157	0.232270
$707$	0	0
$708$	−11.0711	−0.416076
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	0.142136	0.00533425
$711$	−15.7990	−0.592508
$712$	−18.1421	−0.679905
$713$	−29.2843	−1.09670
$714$	0	0
$715$	0.242641	0.00907425
$716$	−6.48528	−0.242366
$717$	−52.6274	−1.96541
$718$	−7.55635	−0.282000
$719$	44.4264	1.65683	0.828413	−	0.560118i	$-0.189244\pi$
$719$	44.4264	1.65683	0.828413	+	0.560118i	$0.189244\pi$
$720$	−1.65685	−0.0617473
$721$	0	0
$722$	−18.6569	−0.694336
$723$	35.7990	1.33138
$724$	−7.00000	−0.260153
$725$	15.8995	0.590492
$726$	26.1421	0.970226
$727$	18.4853	0.685581	0.342791	−	0.939412i	$-0.388628\pi$
$727$	18.4853	0.685581	0.342791	+	0.939412i	$0.388628\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	−1.75736	−0.0650428
$731$	26.3431	0.974336
$732$	5.24264	0.193774
$733$	−28.9706	−1.07005	−0.535026	−	0.844836i	$-0.679698\pi$
$733$	−28.9706	−1.07005	−0.535026	+	0.844836i	$0.679698\pi$
$734$	−4.92893	−0.181930
$735$	0	0
$736$	5.24264	0.193246
$737$	5.97056	0.219929
$738$	−19.7990	−0.728811
$739$	−14.1421	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$739$	−14.1421	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$740$	0.100505	0.00369464
$741$	1.41421	0.0519524
$742$	0	0
$743$	22.3431	0.819691	0.409845	−	0.912155i	$-0.365583\pi$
$743$	22.3431	0.819691	0.409845	+	0.912155i	$0.365583\pi$
$744$	13.4853	0.494394
$745$	−2.52691	−0.0925789
$746$	−23.4142	−0.857255
$747$	28.0000	1.02447
$748$	1.65685	0.0605806
$749$	0	0
$750$	−13.6569	−0.498678
$751$	−45.7279	−1.66864	−0.834318	−	0.551284i	$-0.814138\pi$
$751$	−45.7279	−1.66864	−0.834318	+	0.551284i	$0.814138\pi$
$752$	−1.24264	−0.0453144
$753$	33.1421	1.20777
$754$	−3.41421	−0.124338
$755$	−12.2843	−0.447070
$756$	0	0
$757$	−10.6863	−0.388400	−0.194200	−	0.980962i	$-0.562211\pi$
$757$	−10.6863	−0.388400	−0.194200	+	0.980962i	$0.562211\pi$
$758$	5.17157	0.187840
$759$	5.24264	0.190296
$760$	0.343146	0.0124472
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	−30.3137	−1.09815
$763$	0	0
$764$	0.485281	0.0175569
$765$	6.62742	0.239615
$766$	−6.27208	−0.226619
$767$	4.58579	0.165583
$768$	−2.41421	−0.0871154
$769$	−38.1716	−1.37650	−0.688251	−	0.725473i	$-0.741621\pi$
$769$	−38.1716	−1.37650	−0.688251	+	0.725473i	$0.741621\pi$
$770$	0	0
$771$	−40.9706	−1.47552
$772$	20.9706	0.754747
$773$	−2.68629	−0.0966192	−0.0483096	−	0.998832i	$-0.515383\pi$
$773$	−2.68629	−0.0966192	−0.0483096	+	0.998832i	$0.515383\pi$
$774$	−18.6274	−0.669549
$775$	26.0122	0.934386
$776$	−8.31371	−0.298445
$777$	0	0
$778$	12.0000	0.430221
$779$	4.10051	0.146916
$780$	1.41421	0.0506370
$781$	0.100505	0.00359635
$782$	−20.9706	−0.749906
$783$	−1.41421	−0.0505399
$784$	0	0
$785$	−7.01219	−0.250276
$786$	−37.7990	−1.34825
$787$	20.2843	0.723056	0.361528	−	0.932361i	$-0.382255\pi$
$787$	20.2843	0.723056	0.361528	+	0.932361i	$0.382255\pi$
$788$	10.3137	0.367411
$789$	38.6274	1.37517
$790$	3.27208	0.116415
$791$	0	0
$792$	−1.17157	−0.0416300
$793$	−2.17157	−0.0771148
$794$	−37.9411	−1.34648
$795$	−16.8284	−0.596843
$796$	−24.7279	−0.876458
$797$	−15.9706	−0.565706	−0.282853	−	0.959163i	$-0.591281\pi$
$797$	−15.9706	−0.565706	−0.282853	+	0.959163i	$0.591281\pi$
$798$	0	0
$799$	4.97056	0.175846
$800$	−4.65685	−0.164645
$801$	−51.3137	−1.81308
$802$	−17.0711	−0.602800
$803$	−1.24264	−0.0438518
$804$	34.7990	1.22727
$805$	0	0
$806$	−5.58579	−0.196751
$807$	−33.3848	−1.17520
$808$	18.3137	0.644274
$809$	22.8284	0.802605	0.401302	−	0.915946i	$-0.368558\pi$
$809$	22.8284	0.802605	0.401302	+	0.915946i	$0.368558\pi$
$810$	5.55635	0.195230
$811$	−21.6569	−0.760475	−0.380238	−	0.924889i	$-0.624158\pi$
$811$	−21.6569	−0.760475	−0.380238	+	0.924889i	$0.624158\pi$
$812$	0	0
$813$	58.4558	2.05014
$814$	0.0710678	0.00249093
$815$	3.11270	0.109033
$816$	9.65685	0.338058
$817$	3.85786	0.134970
$818$	−18.0000	−0.629355
$819$	0	0
$820$	4.10051	0.143196
$821$	−17.4558	−0.609213	−0.304607	−	0.952478i	$-0.598525\pi$
$821$	−17.4558	−0.609213	−0.304607	+	0.952478i	$0.598525\pi$
$822$	4.82843	0.168411
$823$	−30.2721	−1.05522	−0.527609	−	0.849487i	$-0.676911\pi$
$823$	−30.2721	−1.05522	−0.527609	+	0.849487i	$0.676911\pi$
$824$	−10.4853	−0.365272
$825$	−4.65685	−0.162131
$826$	0	0
$827$	7.79899	0.271197	0.135599	−	0.990764i	$-0.456704\pi$
$827$	7.79899	0.271197	0.135599	+	0.990764i	$0.456704\pi$
$828$	14.8284	0.515323
$829$	43.7990	1.52120	0.760601	−	0.649220i	$-0.224904\pi$
$829$	43.7990	1.52120	0.760601	+	0.649220i	$0.224904\pi$
$830$	−5.79899	−0.201286
$831$	−37.5563	−1.30282
$832$	1.00000	0.0346688
$833$	0	0
$834$	−15.3137	−0.530270
$835$	−6.34315	−0.219514
$836$	0.242641	0.00839190
$837$	−2.31371	−0.0799735
$838$	23.2426	0.802904
$839$	−6.61522	−0.228383	−0.114191	−	0.993459i	$-0.536428\pi$
$839$	−6.61522	−0.228383	−0.114191	+	0.993459i	$0.536428\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	7.68629	0.264887
$843$	75.8406	2.61209
$844$	−9.75736	−0.335862
$845$	−0.585786	−0.0201517
$846$	−3.51472	−0.120839
$847$	0	0
$848$	−11.8995	−0.408630
$849$	−15.8284	−0.543230
$850$	18.6274	0.638915
$851$	−0.899495	−0.0308343
$852$	0.585786	0.0200687
$853$	−10.4437	−0.357584	−0.178792	−	0.983887i	$-0.557219\pi$
$853$	−10.4437	−0.357584	−0.178792	+	0.983887i	$0.557219\pi$
$854$	0	0
$855$	0.970563	0.0331925
$856$	12.2426	0.418445
$857$	7.75736	0.264986	0.132493	−	0.991184i	$-0.457702\pi$
$857$	7.75736	0.264986	0.132493	+	0.991184i	$0.457702\pi$
$858$	1.00000	0.0341394
$859$	−30.5563	−1.04257	−0.521285	−	0.853383i	$-0.674547\pi$
$859$	−30.5563	−1.04257	−0.521285	+	0.853383i	$0.674547\pi$
$860$	3.85786	0.131552
$861$	0	0
$862$	−28.1421	−0.958525
$863$	−34.6274	−1.17873	−0.589365	−	0.807867i	$-0.700622\pi$
$863$	−34.6274	−1.17873	−0.589365	+	0.807867i	$0.700622\pi$
$864$	0.414214	0.0140918
$865$	8.00000	0.272008
$866$	−7.65685	−0.260190
$867$	2.41421	0.0819910
$868$	0	0
$869$	2.31371	0.0784872
$870$	−4.82843	−0.163699
$871$	−14.4142	−0.488407
$872$	−3.31371	−0.112216
$873$	−23.5147	−0.795853
$874$	−3.07107	−0.103880
$875$	0	0
$876$	−7.24264	−0.244706
$877$	12.5147	0.422592	0.211296	−	0.977422i	$-0.432232\pi$
$877$	12.5147	0.422592	0.211296	+	0.977422i	$0.432232\pi$
$878$	−0.585786	−0.0197693
$879$	−33.7990	−1.14001
$880$	0.242641	0.00817942
$881$	−5.02944	−0.169446	−0.0847230	−	0.996405i	$-0.527001\pi$
$881$	−5.02944	−0.169446	−0.0847230	+	0.996405i	$0.527001\pi$
$882$	0	0
$883$	−26.7696	−0.900867	−0.450434	−	0.892810i	$-0.648731\pi$
$883$	−26.7696	−0.900867	−0.450434	+	0.892810i	$0.648731\pi$
$884$	−4.00000	−0.134535
$885$	6.48528	0.218000
$886$	10.3848	0.348883
$887$	1.27208	0.0427122	0.0213561	−	0.999772i	$-0.493202\pi$
$887$	1.27208	0.0427122	0.0213561	+	0.999772i	$0.493202\pi$
$888$	0.414214	0.0139001
$889$	0	0
$890$	10.6274	0.356232
$891$	3.92893	0.131624
$892$	23.2426	0.778221
$893$	0.727922	0.0243590
$894$	−10.4142	−0.348303
$895$	3.79899	0.126986
$896$	0	0
$897$	−12.6569	−0.422600
$898$	1.61522	0.0539007
$899$	19.0711	0.636056
$900$	−13.1716	−0.439052
$901$	47.5980	1.58572
$902$	2.89949	0.0965426
$903$	0	0
$904$	−13.9706	−0.464654
$905$	4.10051	0.136305
$906$	−50.6274	−1.68198
$907$	48.4264	1.60797	0.803986	−	0.594648i	$-0.202709\pi$
$907$	48.4264	1.60797	0.803986	+	0.594648i	$0.202709\pi$
$908$	−12.9706	−0.430443
$909$	51.7990	1.71806
$910$	0	0
$911$	−1.17157	−0.0388159	−0.0194080	−	0.999812i	$-0.506178\pi$
$911$	−1.17157	−0.0388159	−0.0194080	+	0.999812i	$0.506178\pi$
$912$	1.41421	0.0468293
$913$	−4.10051	−0.135707
$914$	−24.7279	−0.817927
$915$	−3.07107	−0.101526
$916$	−15.8995	−0.525334
$917$	0	0
$918$	−1.65685	−0.0546843
$919$	−37.5269	−1.23790	−0.618949	−	0.785431i	$-0.712441\pi$
$919$	−37.5269	−1.23790	−0.618949	+	0.785431i	$0.712441\pi$
$920$	−3.07107	−0.101250
$921$	−65.3553	−2.15353
$922$	−34.7279	−1.14370
$923$	−0.242641	−0.00798662
$924$	0	0
$925$	0.798990	0.0262706
$926$	−36.2843	−1.19238
$927$	−29.6569	−0.974059
$928$	−3.41421	−0.112077
$929$	59.0833	1.93846	0.969229	−	0.246159i	$-0.0791685\pi$
$929$	59.0833	1.93846	0.969229	+	0.246159i	$0.0791685\pi$
$930$	−7.89949	−0.259035
$931$	0	0
$932$	21.4853	0.703774
$933$	−6.58579	−0.215609
$934$	−18.1421	−0.593629
$935$	−0.970563	−0.0317408
$936$	2.82843	0.0924500
$937$	−30.6274	−1.00055	−0.500277	−	0.865865i	$-0.666769\pi$
$937$	−30.6274	−1.00055	−0.500277	+	0.865865i	$0.666769\pi$
$938$	0	0
$939$	14.4853	0.472709
$940$	0.727922	0.0237422
$941$	23.0711	0.752095	0.376048	−	0.926600i	$-0.377283\pi$
$941$	23.0711	0.752095	0.376048	+	0.926600i	$0.377283\pi$
$942$	−28.8995	−0.941596
$943$	−36.6985	−1.19507
$944$	4.58579	0.149255
$945$	0	0
$946$	2.72792	0.0886924
$947$	−8.48528	−0.275735	−0.137867	−	0.990451i	$-0.544025\pi$
$947$	−8.48528	−0.275735	−0.137867	+	0.990451i	$0.544025\pi$
$948$	13.4853	0.437982
$949$	3.00000	0.0973841
$950$	2.72792	0.0885055
$951$	−44.2132	−1.43371
$952$	0	0
$953$	20.4853	0.663583	0.331792	−	0.943353i	$-0.392347\pi$
$953$	20.4853	0.663583	0.331792	+	0.943353i	$0.392347\pi$
$954$	−33.6569	−1.08968
$955$	−0.284271	−0.00919880
$956$	21.7990	0.705030
$957$	−3.41421	−0.110366
$958$	20.2843	0.655355
$959$	0	0
$960$	1.41421	0.0456435
$961$	0.201010	0.00648420
$962$	−0.171573	−0.00553173
$963$	34.6274	1.11585
$964$	−14.8284	−0.477591
$965$	−12.2843	−0.395445
$966$	0	0
$967$	37.1127	1.19346	0.596732	−	0.802441i	$-0.296466\pi$
$967$	37.1127	1.19346	0.596732	+	0.802441i	$0.296466\pi$
$968$	−10.8284	−0.348039
$969$	−5.65685	−0.181724
$970$	4.87006	0.156368
$971$	−24.2132	−0.777039	−0.388519	−	0.921441i	$-0.627013\pi$
$971$	−24.2132	−0.777039	−0.388519	+	0.921441i	$0.627013\pi$
$972$	21.6569	0.694644
$973$	0	0
$974$	40.3848	1.29401
$975$	11.2426	0.360053
$976$	−2.17157	−0.0695104
$977$	56.6274	1.81167	0.905836	−	0.423629i	$-0.139244\pi$
$977$	56.6274	1.81167	0.905836	+	0.423629i	$0.139244\pi$
$978$	12.8284	0.410208
$979$	7.51472	0.240171
$980$	0	0
$981$	−9.37258	−0.299244
$982$	−26.3848	−0.841972
$983$	59.6569	1.90276	0.951379	−	0.308022i	$-0.0996672\pi$
$983$	59.6569	1.90276	0.951379	+	0.308022i	$0.0996672\pi$
$984$	16.8995	0.538736
$985$	−6.04163	−0.192502
$986$	13.6569	0.434923
$987$	0	0
$988$	−0.585786	−0.0186363
$989$	−34.5269	−1.09789
$990$	0.686292	0.0218118
$991$	−47.6690	−1.51426	−0.757129	−	0.653266i	$-0.773398\pi$
$991$	−47.6690	−1.51426	−0.757129	+	0.653266i	$0.773398\pi$
$992$	−5.58579	−0.177349
$993$	64.4558	2.04544
$994$	0	0
$995$	14.4853	0.459214
$996$	−23.8995	−0.757284
$997$	−23.1421	−0.732919	−0.366459	−	0.930434i	$-0.619430\pi$
$997$	−23.1421	−0.732919	−0.366459	+	0.930434i	$0.619430\pi$
$998$	12.8995	0.408326
$999$	−0.0710678	−0.00224849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1274.2.a.p.1.1	✓	2
7.2	even	3		1274.2.f.w.1145.2		4
7.3	odd	6		1274.2.f.v.79.1		4
7.4	even	3		1274.2.f.w.79.2		4
7.5	odd	6		1274.2.f.v.1145.1		4
7.6	odd	2		1274.2.a.q.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1274.2.a.p.1.1	✓	2	1.1	even	1	trivial
1274.2.a.q.1.2	yes	2	7.6	odd	2
1274.2.f.v.79.1		4	7.3	odd	6
1274.2.f.v.1145.1		4	7.5	odd	6
1274.2.f.w.79.2		4	7.4	even	3
1274.2.f.w.1145.2		4	7.2	even	3

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:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} -0.585786 q^{10} -0.414214 q^{11} -2.41421 q^{12} +1.00000 q^{13} +1.41421 q^{15} +1.00000 q^{16} -4.00000 q^{17} +2.82843 q^{18} -0.585786 q^{19} -0.585786 q^{20} -0.414214 q^{22} +5.24264 q^{23} -2.41421 q^{24} -4.65685 q^{25} +1.00000 q^{26} +0.414214 q^{27} -3.41421 q^{29} +1.41421 q^{30} -5.58579 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} +2.82843 q^{36} -0.171573 q^{37} -0.585786 q^{38} -2.41421 q^{39} -0.585786 q^{40} -7.00000 q^{41} -6.58579 q^{43} -0.414214 q^{44} -1.65685 q^{45} +5.24264 q^{46} -1.24264 q^{47} -2.41421 q^{48} -4.65685 q^{50} +9.65685 q^{51} +1.00000 q^{52} -11.8995 q^{53} +0.414214 q^{54} +0.242641 q^{55} +1.41421 q^{57} -3.41421 q^{58} +4.58579 q^{59} +1.41421 q^{60} -2.17157 q^{61} -5.58579 q^{62} +1.00000 q^{64} -0.585786 q^{65} +1.00000 q^{66} -14.4142 q^{67} -4.00000 q^{68} -12.6569 q^{69} -0.242641 q^{71} +2.82843 q^{72} +3.00000 q^{73} -0.171573 q^{74} +11.2426 q^{75} -0.585786 q^{76} -2.41421 q^{78} -5.58579 q^{79} -0.585786 q^{80} -9.48528 q^{81} -7.00000 q^{82} +9.89949 q^{83} +2.34315 q^{85} -6.58579 q^{86} +8.24264 q^{87} -0.414214 q^{88} -18.1421 q^{89} -1.65685 q^{90} +5.24264 q^{92} +13.4853 q^{93} -1.24264 q^{94} +0.343146 q^{95} -2.41421 q^{96} -8.31371 q^{97} -1.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} - 4 q^{10} + 2 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 - 4 * q^10 + 2 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 + 2 * q^26 - 2 * q^27 - 4 * q^29 - 14 * q^31 + 2 * q^32 + 2 * q^33 - 8 * q^34 - 6 * q^37 - 4 * q^38 - 2 * q^39 - 4 * q^40 - 14 * q^41 - 16 * q^43 + 2 * q^44 + 8 * q^45 + 2 * q^46 + 6 * q^47 - 2 * q^48 + 2 * q^50 + 8 * q^51 + 2 * q^52 - 4 * q^53 - 2 * q^54 - 8 * q^55 - 4 * q^58 + 12 * q^59 - 10 * q^61 - 14 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 26 * q^67 - 8 * q^68 - 14 * q^69 + 8 * q^71 + 6 * q^73 - 6 * q^74 + 14 * q^75 - 4 * q^76 - 2 * q^78 - 14 * q^79 - 4 * q^80 - 2 * q^81 - 14 * q^82 + 16 * q^85 - 16 * q^86 + 8 * q^87 + 2 * q^88 - 8 * q^89 + 8 * q^90 + 2 * q^92 + 10 * q^93 + 6 * q^94 + 12 * q^95 - 2 * q^96 + 6 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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