LMFDB - Embedded newform 105.8.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [105,8,Mod(1,105)] 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("105.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.8004276758$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+18.0000 q^{2} -27.0000 q^{3} +196.000 q^{4} -125.000 q^{5} -486.000 q^{6} +343.000 q^{7} +1224.00 q^{8} +729.000 q^{9} -2250.00 q^{10} -8016.00 q^{11} -5292.00 q^{12} -1786.00 q^{13} +6174.00 q^{14} +3375.00 q^{15} -3056.00 q^{16} +8358.00 q^{17} +13122.0 q^{18} -5884.00 q^{19} -24500.0 q^{20} -9261.00 q^{21} -144288. q^{22} -77700.0 q^{23} -33048.0 q^{24} +15625.0 q^{25} -32148.0 q^{26} -19683.0 q^{27} +67228.0 q^{28} +155742. q^{29} +60750.0 q^{30} -310000. q^{31} -211680. q^{32} +216432. q^{33} +150444. q^{34} -42875.0 q^{35} +142884. q^{36} -433618. q^{37} -105912. q^{38} +48222.0 q^{39} -153000. q^{40} +357942. q^{41} -166698. q^{42} -724492. q^{43} -1.57114e6 q^{44} -91125.0 q^{45} -1.39860e6 q^{46} +175320. q^{47} +82512.0 q^{48} +117649. q^{49} +281250. q^{50} -225666. q^{51} -350056. q^{52} +132198. q^{53} -354294. q^{54} +1.00200e6 q^{55} +419832. q^{56} +158868. q^{57} +2.80336e6 q^{58} +2.64863e6 q^{59} +661500. q^{60} +835478. q^{61} -5.58000e6 q^{62} +250047. q^{63} -3.41907e6 q^{64} +223250. q^{65} +3.89578e6 q^{66} +3.48631e6 q^{67} +1.63817e6 q^{68} +2.09790e6 q^{69} -771750. q^{70} -2.87226e6 q^{71} +892296. q^{72} +5.95188e6 q^{73} -7.80512e6 q^{74} -421875. q^{75} -1.15326e6 q^{76} -2.74949e6 q^{77} +867996. q^{78} -1.68090e6 q^{79} +382000. q^{80} +531441. q^{81} +6.44296e6 q^{82} +3.57752e6 q^{83} -1.81516e6 q^{84} -1.04475e6 q^{85} -1.30409e7 q^{86} -4.20503e6 q^{87} -9.81158e6 q^{88} -6.25483e6 q^{89} -1.64025e6 q^{90} -612598. q^{91} -1.52292e7 q^{92} +8.37000e6 q^{93} +3.15576e6 q^{94} +735500. q^{95} +5.71536e6 q^{96} -5.25705e6 q^{97} +2.11768e6 q^{98} -5.84366e6 q^{99} +O(q^{100})\)

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 $

$2$	18.0000	1.59099	0.795495	−	0.605960i	$-0.207211\pi$
$2$	18.0000	1.59099	0.795495	+	0.605960i	$0.207211\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	196.000	1.53125
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	−486.000	−0.918559
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	1224.00	0.845214
$9$	729.000	0.333333
$10$	−2250.00	−0.711512
$11$	−8016.00	−1.81586	−0.907932	−	0.419118i	$-0.862340\pi$
$11$	−8016.00	−1.81586	−0.907932	+	0.419118i	$0.862340\pi$
$12$	−5292.00	−0.884068
$13$	−1786.00	−0.225465	−0.112733	−	0.993625i	$-0.535960\pi$
$13$	−1786.00	−0.225465	−0.112733	+	0.993625i	$0.535960\pi$
$14$	6174.00	0.601338
$15$	3375.00	0.258199
$16$	−3056.00	−0.186523
$17$	8358.00	0.412602	0.206301	−	0.978489i	$-0.433857\pi$
$17$	8358.00	0.412602	0.206301	+	0.978489i	$0.433857\pi$
$18$	13122.0	0.530330
$19$	−5884.00	−0.196805	−0.0984023	−	0.995147i	$-0.531373\pi$
$19$	−5884.00	−0.196805	−0.0984023	+	0.995147i	$0.531373\pi$
$20$	−24500.0	−0.684796
$21$	−9261.00	−0.218218
$22$	−144288.	−2.88902
$23$	−77700.0	−1.33160	−0.665800	−	0.746131i	$-0.731909\pi$
$23$	−77700.0	−1.33160	−0.665800	+	0.746131i	$0.731909\pi$
$24$	−33048.0	−0.487984
$25$	15625.0	0.200000
$26$	−32148.0	−0.358713
$27$	−19683.0	−0.192450
$28$	67228.0	0.578758
$29$	155742.	1.18580	0.592902	−	0.805275i	$-0.297982\pi$
$29$	155742.	1.18580	0.592902	+	0.805275i	$0.297982\pi$
$30$	60750.0	0.410792
$31$	−310000.	−1.86894	−0.934471	−	0.356040i	$-0.884127\pi$
$31$	−310000.	−1.86894	−0.934471	+	0.356040i	$0.884127\pi$
$32$	−211680.	−1.14197
$33$	216432.	1.04839
$34$	150444.	0.656445
$35$	−42875.0	−0.169031
$36$	142884.	0.510417
$37$	−433618.	−1.40735	−0.703674	−	0.710523i	$-0.748458\pi$
$37$	−433618.	−1.40735	−0.703674	+	0.710523i	$0.748458\pi$
$38$	−105912.	−0.313114
$39$	48222.0	0.130172
$40$	−153000.	−0.377991
$41$	357942.	0.811090	0.405545	−	0.914075i	$-0.367082\pi$
$41$	357942.	0.811090	0.405545	+	0.914075i	$0.367082\pi$
$42$	−166698.	−0.347183
$43$	−724492.	−1.38961	−0.694807	−	0.719197i	$-0.744510\pi$
$43$	−724492.	−1.38961	−0.694807	+	0.719197i	$0.744510\pi$
$44$	−1.57114e6	−2.78054
$45$	−91125.0	−0.149071
$46$	−1.39860e6	−2.11856
$47$	175320.	0.246314	0.123157	−	0.992387i	$-0.460698\pi$
$47$	175320.	0.246314	0.123157	+	0.992387i	$0.460698\pi$
$48$	82512.0	0.107689
$49$	117649.	0.142857
$50$	281250.	0.318198
$51$	−225666.	−0.238216
$52$	−350056.	−0.345244
$53$	132198.	0.121972	0.0609859	−	0.998139i	$-0.480576\pi$
$53$	132198.	0.121972	0.0609859	+	0.998139i	$0.480576\pi$
$54$	−354294.	−0.306186
$55$	1.00200e6	0.812079
$56$	419832.	0.319461
$57$	158868.	0.113625
$58$	2.80336e6	1.88660
$59$	2.64863e6	1.67895	0.839477	−	0.543395i	$-0.182861\pi$
$59$	2.64863e6	1.67895	0.839477	+	0.543395i	$0.182861\pi$
$60$	661500.	0.395367
$61$	835478.	0.471282	0.235641	−	0.971840i	$-0.424281\pi$
$61$	835478.	0.471282	0.235641	+	0.971840i	$0.424281\pi$
$62$	−5.58000e6	−2.97347
$63$	250047.	0.125988
$64$	−3.41907e6	−1.63034
$65$	223250.	0.100831
$66$	3.89578e6	1.66798
$67$	3.48631e6	1.41613	0.708066	−	0.706146i	$-0.249568\pi$
$67$	3.48631e6	1.41613	0.708066	+	0.706146i	$0.249568\pi$
$68$	1.63817e6	0.631797
$69$	2.09790e6	0.768799
$70$	−771750.	−0.268926
$71$	−2.87226e6	−0.952400	−0.476200	−	0.879337i	$-0.657986\pi$
$71$	−2.87226e6	−0.952400	−0.476200	+	0.879337i	$0.657986\pi$
$72$	892296.	0.281738
$73$	5.95188e6	1.79071	0.895353	−	0.445357i	$-0.146923\pi$
$73$	5.95188e6	1.79071	0.895353	+	0.445357i	$0.146923\pi$
$74$	−7.80512e6	−2.23908
$75$	−421875.	−0.115470
$76$	−1.15326e6	−0.301357
$77$	−2.74949e6	−0.686332
$78$	867996.	0.207103
$79$	−1.68090e6	−0.383573	−0.191787	−	0.981437i	$-0.561428\pi$
$79$	−1.68090e6	−0.383573	−0.191787	+	0.981437i	$0.561428\pi$
$80$	382000.	0.0834158
$81$	531441.	0.111111
$82$	6.44296e6	1.29044
$83$	3.57752e6	0.686767	0.343383	−	0.939195i	$-0.388427\pi$
$83$	3.57752e6	0.686767	0.343383	+	0.939195i	$0.388427\pi$
$84$	−1.81516e6	−0.334146
$85$	−1.04475e6	−0.184521
$86$	−1.30409e7	−2.21086
$87$	−4.20503e6	−0.684624
$88$	−9.81158e6	−1.53479
$89$	−6.25483e6	−0.940481	−0.470241	−	0.882538i	$-0.655833\pi$
$89$	−6.25483e6	−0.940481	−0.470241	+	0.882538i	$0.655833\pi$
$90$	−1.64025e6	−0.237171
$91$	−612598.	−0.0852179
$92$	−1.52292e7	−2.03901
$93$	8.37000e6	1.07903
$94$	3.15576e6	0.391883
$95$	735500.	0.0880137
$96$	5.71536e6	0.659317
$97$	−5.25705e6	−0.584846	−0.292423	−	0.956289i	$-0.594461\pi$
$97$	−5.25705e6	−0.584846	−0.292423	+	0.956289i	$0.594461\pi$
$98$	2.11768e6	0.227284
$99$	−5.84366e6	−0.605288
$100$	3.06250e6	0.306250
$101$	1.50250e7	1.45107	0.725535	−	0.688185i	$-0.241592\pi$
$101$	1.50250e7	1.45107	0.725535	+	0.688185i	$0.241592\pi$
$102$	−4.06199e6	−0.378999
$103$	−1.60066e7	−1.44334	−0.721669	−	0.692238i	$-0.756625\pi$
$103$	−1.60066e7	−1.44334	−0.721669	+	0.692238i	$0.756625\pi$
$104$	−2.18606e6	−0.190566
$105$	1.15762e6	0.0975900
$106$	2.37956e6	0.194056
$107$	5.30105e6	0.418330	0.209165	−	0.977880i	$-0.432926\pi$
$107$	5.30105e6	0.418330	0.209165	+	0.977880i	$0.432926\pi$
$108$	−3.85787e6	−0.294689
$109$	−6.46340e6	−0.478045	−0.239022	−	0.971014i	$-0.576827\pi$
$109$	−6.46340e6	−0.478045	−0.239022	+	0.971014i	$0.576827\pi$
$110$	1.80360e7	1.29201
$111$	1.17077e7	0.812532
$112$	−1.04821e6	−0.0704992
$113$	−1.48333e7	−0.967082	−0.483541	−	0.875322i	$-0.660650\pi$
$113$	−1.48333e7	−0.967082	−0.483541	+	0.875322i	$0.660650\pi$
$114$	2.85962e6	0.180776
$115$	9.71250e6	0.595509
$116$	3.05254e7	1.81576
$117$	−1.30199e6	−0.0751551
$118$	4.76753e7	2.67120
$119$	2.86679e6	0.155949
$120$	4.13100e6	0.218233
$121$	4.47691e7	2.29736
$122$	1.50386e7	0.749805
$123$	−9.66443e6	−0.468283
$124$	−6.07600e7	−2.86182
$125$	−1.95312e6	−0.0894427
$126$	4.50085e6	0.200446
$127$	1.35024e7	0.584920	0.292460	−	0.956278i	$-0.405526\pi$
$127$	1.35024e7	0.584920	0.292460	+	0.956278i	$0.405526\pi$
$128$	−3.44483e7	−1.45189
$129$	1.95613e7	0.802294
$130$	4.01850e6	0.160421
$131$	1.22470e7	0.475972	0.237986	−	0.971269i	$-0.423513\pi$
$131$	1.22470e7	0.475972	0.237986	+	0.971269i	$0.423513\pi$
$132$	4.24207e7	1.60535
$133$	−2.01821e6	−0.0743851
$134$	6.27535e7	2.25305
$135$	2.46038e6	0.0860663
$136$	1.02302e7	0.348737
$137$	1.24000e7	0.412002	0.206001	−	0.978552i	$-0.433955\pi$
$137$	1.24000e7	0.412002	0.206001	+	0.978552i	$0.433955\pi$
$138$	3.77622e7	1.22315
$139$	−5.95433e6	−0.188053	−0.0940267	−	0.995570i	$-0.529974\pi$
$139$	−5.95433e6	−0.188053	−0.0940267	+	0.995570i	$0.529974\pi$
$140$	−8.40350e6	−0.258828
$141$	−4.73364e6	−0.142209
$142$	−5.17007e7	−1.51526
$143$	1.43166e7	0.409414
$144$	−2.22782e6	−0.0621745
$145$	−1.94678e7	−0.530307
$146$	1.07134e8	2.84900
$147$	−3.17652e6	−0.0824786
$148$	−8.49891e7	−2.15500
$149$	−3.10990e7	−0.770185	−0.385092	−	0.922878i	$-0.625830\pi$
$149$	−3.10990e7	−0.770185	−0.385092	+	0.922878i	$0.625830\pi$
$150$	−7.59375e6	−0.183712
$151$	−1.43273e7	−0.338645	−0.169322	−	0.985561i	$-0.554158\pi$
$151$	−1.43273e7	−0.338645	−0.169322	+	0.985561i	$0.554158\pi$
$152$	−7.20202e6	−0.166342
$153$	6.09298e6	0.137534
$154$	−4.94908e7	−1.09195
$155$	3.87500e7	0.835816
$156$	9.45151e6	0.199327
$157$	−5.81072e7	−1.19834	−0.599171	−	0.800621i	$-0.704503\pi$
$157$	−5.81072e7	−1.19834	−0.599171	+	0.800621i	$0.704503\pi$
$158$	−3.02563e7	−0.610261
$159$	−3.56935e6	−0.0704204
$160$	2.64600e7	0.510705
$161$	−2.66511e7	−0.503297
$162$	9.56594e6	0.176777
$163$	−2.36004e7	−0.426838	−0.213419	−	0.976961i	$-0.568460\pi$
$163$	−2.36004e7	−0.426838	−0.213419	+	0.976961i	$0.568460\pi$
$164$	7.01566e7	1.24198
$165$	−2.70540e7	−0.468854
$166$	6.43954e7	1.09264
$167$	2.65801e7	0.441621	0.220810	−	0.975317i	$-0.429130\pi$
$167$	2.65801e7	0.441621	0.220810	+	0.975317i	$0.429130\pi$
$168$	−1.13355e7	−0.184441
$169$	−5.95587e7	−0.949165
$170$	−1.88055e7	−0.293571
$171$	−4.28944e6	−0.0656015
$172$	−1.42000e8	−2.12785
$173$	−7.09671e7	−1.04207	−0.521033	−	0.853536i	$-0.674453\pi$
$173$	−7.09671e7	−1.04207	−0.521033	+	0.853536i	$0.674453\pi$
$174$	−7.56906e7	−1.08923
$175$	5.35938e6	0.0755929
$176$	2.44969e7	0.338701
$177$	−7.15130e7	−0.969345
$178$	−1.12587e8	−1.49630
$179$	1.15061e8	1.49949	0.749746	−	0.661726i	$-0.230176\pi$
$179$	1.15061e8	1.49949	0.749746	+	0.661726i	$0.230176\pi$
$180$	−1.78605e7	−0.228265
$181$	−2.26091e6	−0.0283405	−0.0141702	−	0.999900i	$-0.504511\pi$
$181$	−2.26091e6	−0.0283405	−0.0141702	+	0.999900i	$0.504511\pi$
$182$	−1.10268e7	−0.135581
$183$	−2.25579e7	−0.272095
$184$	−9.51048e7	−1.12549
$185$	5.42022e7	0.629385
$186$	1.50660e8	1.71673
$187$	−6.69977e7	−0.749229
$188$	3.43627e7	0.377168
$189$	−6.75127e6	−0.0727393
$190$	1.32390e7	0.140029
$191$	−8.07135e7	−0.838165	−0.419083	−	0.907948i	$-0.637648\pi$
$191$	−8.07135e7	−0.838165	−0.419083	+	0.907948i	$0.637648\pi$
$192$	9.23149e7	0.941278
$193$	7.07939e7	0.708835	0.354418	−	0.935087i	$-0.384679\pi$
$193$	7.07939e7	0.708835	0.354418	+	0.935087i	$0.384679\pi$
$194$	−9.46270e7	−0.930484
$195$	−6.02775e6	−0.0582149
$196$	2.30592e7	0.218750
$197$	7.53473e7	0.702160	0.351080	−	0.936345i	$-0.385815\pi$
$197$	7.53473e7	0.702160	0.351080	+	0.936345i	$0.385815\pi$
$198$	−1.05186e8	−0.963007
$199$	−1.57565e8	−1.41734	−0.708670	−	0.705540i	$-0.750705\pi$
$199$	−1.57565e8	−1.41734	−0.708670	+	0.705540i	$0.750705\pi$
$200$	1.91250e7	0.169043
$201$	−9.41303e7	−0.817605
$202$	2.70449e8	2.30864
$203$	5.34195e7	0.448192
$204$	−4.42305e7	−0.364768
$205$	−4.47428e7	−0.362731
$206$	−2.88118e8	−2.29634
$207$	−5.66433e7	−0.443866
$208$	5.45802e6	0.0420546
$209$	4.71661e7	0.357370
$210$	2.08372e7	0.155265
$211$	−2.22118e8	−1.62778	−0.813890	−	0.581019i	$-0.802654\pi$
$211$	−2.22118e8	−1.62778	−0.813890	+	0.581019i	$0.802654\pi$
$212$	2.59108e7	0.186769
$213$	7.75510e7	0.549869
$214$	9.54189e7	0.665558
$215$	9.05615e7	0.621454
$216$	−2.40920e7	−0.162661
$217$	−1.06330e8	−0.706394
$218$	−1.16341e8	−0.760564
$219$	−1.60701e8	−1.03386
$220$	1.96392e8	1.24350
$221$	−1.49274e7	−0.0930274
$222$	2.10738e8	1.29273
$223$	1.40215e8	0.846694	0.423347	−	0.905968i	$-0.360855\pi$
$223$	1.40215e8	0.846694	0.423347	+	0.905968i	$0.360855\pi$
$224$	−7.26062e7	−0.431624
$225$	1.13906e7	0.0666667
$226$	−2.66999e8	−1.53862
$227$	−1.13463e8	−0.643820	−0.321910	−	0.946770i	$-0.604325\pi$
$227$	−1.13463e8	−0.643820	−0.321910	+	0.946770i	$0.604325\pi$
$228$	3.11381e7	0.173988
$229$	−1.00120e8	−0.550931	−0.275465	−	0.961311i	$-0.588832\pi$
$229$	−1.00120e8	−0.550931	−0.275465	+	0.961311i	$0.588832\pi$
$230$	1.74825e8	0.947450
$231$	7.42362e7	0.396254
$232$	1.90628e8	1.00226
$233$	−3.72064e8	−1.92696	−0.963478	−	0.267786i	$-0.913708\pi$
$233$	−3.72064e8	−1.92696	−0.963478	+	0.267786i	$0.913708\pi$
$234$	−2.34359e7	−0.119571
$235$	−2.19150e7	−0.110155
$236$	5.19131e8	2.57090
$237$	4.53844e7	0.221456
$238$	5.16023e7	0.248113
$239$	−9.50953e7	−0.450574	−0.225287	−	0.974292i	$-0.572332\pi$
$239$	−9.50953e7	−0.450574	−0.225287	+	0.974292i	$0.572332\pi$
$240$	−1.03140e7	−0.0481601
$241$	−2.86367e8	−1.31784	−0.658921	−	0.752212i	$-0.728987\pi$
$241$	−2.86367e8	−1.31784	−0.658921	+	0.752212i	$0.728987\pi$
$242$	8.05844e8	3.65508
$243$	−1.43489e7	−0.0641500
$244$	1.63754e8	0.721650
$245$	−1.47061e7	−0.0638877
$246$	−1.73960e8	−0.745034
$247$	1.05088e7	0.0443726
$248$	−3.79440e8	−1.57965
$249$	−9.65931e7	−0.396505
$250$	−3.51562e7	−0.142302
$251$	5.53423e7	0.220902	0.110451	−	0.993882i	$-0.464771\pi$
$251$	5.53423e7	0.220902	0.110451	+	0.993882i	$0.464771\pi$
$252$	4.90092e7	0.192919
$253$	6.22843e8	2.41800
$254$	2.43042e8	0.930602
$255$	2.82082e7	0.106533
$256$	−1.82427e8	−0.679595
$257$	−5.97749e7	−0.219661	−0.109830	−	0.993950i	$-0.535031\pi$
$257$	−5.97749e7	−0.219661	−0.109830	+	0.993950i	$0.535031\pi$
$258$	3.52103e8	1.27644
$259$	−1.48731e8	−0.531927
$260$	4.37570e7	0.154398
$261$	1.13536e8	0.395268
$262$	2.20447e8	0.757267
$263$	−1.23562e8	−0.418833	−0.209416	−	0.977827i	$-0.567156\pi$
$263$	−1.23562e8	−0.418833	−0.209416	+	0.977827i	$0.567156\pi$
$264$	2.64913e8	0.886113
$265$	−1.65248e7	−0.0545474
$266$	−3.63278e7	−0.118346
$267$	1.68880e8	0.542987
$268$	6.83316e8	2.16845
$269$	4.29280e8	1.34465	0.672323	−	0.740258i	$-0.265297\pi$
$269$	4.29280e8	1.34465	0.672323	+	0.740258i	$0.265297\pi$
$270$	4.42868e7	0.136931
$271$	−5.48961e8	−1.67552	−0.837759	−	0.546040i	$-0.816135\pi$
$271$	−5.48961e8	−1.67552	−0.837759	+	0.546040i	$0.816135\pi$
$272$	−2.55420e7	−0.0769599
$273$	1.65401e7	0.0492006
$274$	2.23200e8	0.655491
$275$	−1.25250e8	−0.363173
$276$	4.11188e8	1.17722
$277$	3.32836e8	0.940915	0.470458	−	0.882423i	$-0.344089\pi$
$277$	3.32836e8	0.940915	0.470458	+	0.882423i	$0.344089\pi$
$278$	−1.07178e8	−0.299191
$279$	−2.25990e8	−0.622981
$280$	−5.24790e7	−0.142867
$281$	−4.54457e8	−1.22186	−0.610928	−	0.791686i	$-0.709204\pi$
$281$	−4.54457e8	−1.22186	−0.610928	+	0.791686i	$0.709204\pi$
$282$	−8.52055e7	−0.226254
$283$	8.21849e7	0.215546	0.107773	−	0.994176i	$-0.465628\pi$
$283$	8.21849e7	0.215546	0.107773	+	0.994176i	$0.465628\pi$
$284$	−5.62963e8	−1.45836
$285$	−1.98585e7	−0.0508147
$286$	2.57698e8	0.651374
$287$	1.22774e8	0.306563
$288$	−1.54315e8	−0.380657
$289$	−3.40483e8	−0.829760
$290$	−3.50420e8	−0.843714
$291$	1.41940e8	0.337661
$292$	1.16657e9	2.74202
$293$	3.71449e8	0.862705	0.431353	−	0.902183i	$-0.358036\pi$
$293$	3.71449e8	0.862705	0.431353	+	0.902183i	$0.358036\pi$
$294$	−5.71774e7	−0.131223
$295$	−3.31078e8	−0.750851
$296$	−5.30748e8	−1.18951
$297$	1.57779e8	0.349463
$298$	−5.59782e8	−1.22536
$299$	1.38772e8	0.300229
$300$	−8.26875e7	−0.176814
$301$	−2.48501e8	−0.525225
$302$	−2.57891e8	−0.538780
$303$	−4.05674e8	−0.837776
$304$	1.79815e7	0.0367087
$305$	−1.04435e8	−0.210764
$306$	1.09674e8	0.218815
$307$	−4.59953e8	−0.907255	−0.453628	−	0.891191i	$-0.649870\pi$
$307$	−4.59953e8	−0.907255	−0.453628	+	0.891191i	$0.649870\pi$
$308$	−5.38900e8	−1.05095
$309$	4.32178e8	0.833312
$310$	6.97500e8	1.32978
$311$	−4.94973e8	−0.933082	−0.466541	−	0.884499i	$-0.654500\pi$
$311$	−4.94973e8	−0.933082	−0.466541	+	0.884499i	$0.654500\pi$
$312$	5.90237e7	0.110024
$313$	3.59666e8	0.662971	0.331485	−	0.943460i	$-0.392450\pi$
$313$	3.59666e8	0.662971	0.331485	+	0.943460i	$0.392450\pi$
$314$	−1.04593e9	−1.90655
$315$	−3.12559e7	−0.0563436
$316$	−3.29457e8	−0.587346
$317$	1.84942e8	0.326083	0.163041	−	0.986619i	$-0.447870\pi$
$317$	1.84942e8	0.326083	0.163041	+	0.986619i	$0.447870\pi$
$318$	−6.42482e7	−0.112038
$319$	−1.24843e9	−2.15326
$320$	4.27384e8	0.729110
$321$	−1.43128e8	−0.241523
$322$	−4.79720e8	−0.800741
$323$	−4.91785e7	−0.0812019
$324$	1.04162e8	0.170139
$325$	−2.79063e7	−0.0450931
$326$	−4.24808e8	−0.679095
$327$	1.74512e8	0.275999
$328$	4.38121e8	0.685544
$329$	6.01348e7	0.0930979
$330$	−4.86972e8	−0.745942
$331$	1.02518e9	1.55383	0.776915	−	0.629605i	$-0.216783\pi$
$331$	1.02518e9	1.55383	0.776915	+	0.629605i	$0.216783\pi$
$332$	7.01195e8	1.05161
$333$	−3.16108e8	−0.469116
$334$	4.78443e8	0.702615
$335$	−4.35788e8	−0.633314
$336$	2.83016e7	0.0407028
$337$	1.13073e9	1.60936	0.804680	−	0.593709i	$-0.202337\pi$
$337$	1.13073e9	1.60936	0.804680	+	0.593709i	$0.202337\pi$
$338$	−1.07206e9	−1.51011
$339$	4.00499e8	0.558345
$340$	−2.04771e8	−0.282548
$341$	2.48496e9	3.39374
$342$	−7.72098e7	−0.104371
$343$	4.03536e7	0.0539949
$344$	−8.86778e8	−1.17452
$345$	−2.62238e8	−0.343817
$346$	−1.27741e9	−1.65792
$347$	6.28829e8	0.807940	0.403970	−	0.914772i	$-0.367630\pi$
$347$	6.28829e8	0.807940	0.403970	+	0.914772i	$0.367630\pi$
$348$	−8.24187e8	−1.04833
$349$	9.05467e8	1.14021	0.570103	−	0.821573i	$-0.306903\pi$
$349$	9.05467e8	1.14021	0.570103	+	0.821573i	$0.306903\pi$
$350$	9.64688e7	0.120268
$351$	3.51538e7	0.0433908
$352$	1.69683e9	2.07366
$353$	−1.13858e8	−0.137769	−0.0688843	−	0.997625i	$-0.521944\pi$
$353$	−1.13858e8	−0.137769	−0.0688843	+	0.997625i	$0.521944\pi$
$354$	−1.28723e9	−1.54222
$355$	3.59033e8	0.425926
$356$	−1.22595e9	−1.44011
$357$	−7.74034e7	−0.0900371
$358$	2.07111e9	2.38568
$359$	2.71654e8	0.309874	0.154937	−	0.987924i	$-0.450483\pi$
$359$	2.71654e8	0.309874	0.154937	+	0.987924i	$0.450483\pi$
$360$	−1.11537e8	−0.125997
$361$	−8.59250e8	−0.961268
$362$	−4.06963e7	−0.0450895
$363$	−1.20877e9	−1.32638
$364$	−1.20069e8	−0.130490
$365$	−7.43985e8	−0.800828
$366$	−4.06042e8	−0.432900
$367$	1.35093e9	1.42659	0.713297	−	0.700862i	$-0.247201\pi$
$367$	1.35093e9	1.42659	0.713297	+	0.700862i	$0.247201\pi$
$368$	2.37451e8	0.248374
$369$	2.60940e8	0.270363
$370$	9.75640e8	1.00135
$371$	4.53439e7	0.0461010
$372$	1.64052e9	1.65227
$373$	−1.09767e9	−1.09520	−0.547599	−	0.836741i	$-0.684458\pi$
$373$	−1.09767e9	−1.09520	−0.547599	+	0.836741i	$0.684458\pi$
$374$	−1.20596e9	−1.19202
$375$	5.27344e7	0.0516398
$376$	2.14592e8	0.208188
$377$	−2.78155e8	−0.267358
$378$	−1.21523e8	−0.115728
$379$	−4.31536e8	−0.407174	−0.203587	−	0.979057i	$-0.565260\pi$
$379$	−4.31536e8	−0.407174	−0.203587	+	0.979057i	$0.565260\pi$
$380$	1.44158e8	0.134771
$381$	−3.64564e8	−0.337704
$382$	−1.45284e9	−1.33351
$383$	5.29318e8	0.481416	0.240708	−	0.970598i	$-0.422620\pi$
$383$	5.29318e8	0.481416	0.240708	+	0.970598i	$0.422620\pi$
$384$	9.30103e8	0.838246
$385$	3.43686e8	0.306937
$386$	1.27429e9	1.12775
$387$	−5.28155e8	−0.463204
$388$	−1.03038e9	−0.895545
$389$	−1.27010e7	−0.0109399	−0.00546996	−	0.999985i	$-0.501741\pi$
$389$	−1.27010e7	−0.0109399	−0.00546996	+	0.999985i	$0.501741\pi$
$390$	−1.08500e8	−0.0926193
$391$	−6.49417e8	−0.549420
$392$	1.44002e8	0.120745
$393$	−3.30670e8	−0.274803
$394$	1.35625e9	1.11713
$395$	2.10113e8	0.171539
$396$	−1.14536e9	−0.926847
$397$	−1.94087e9	−1.55679	−0.778393	−	0.627777i	$-0.783965\pi$
$397$	−1.94087e9	−1.55679	−0.778393	+	0.627777i	$0.783965\pi$
$398$	−2.83617e9	−2.25498
$399$	5.44917e7	0.0429463
$400$	−4.77500e7	−0.0373047
$401$	1.29335e9	1.00164	0.500819	−	0.865552i	$-0.333032\pi$
$401$	1.29335e9	1.00164	0.500819	+	0.865552i	$0.333032\pi$
$402$	−1.69435e9	−1.30080
$403$	5.53660e8	0.421382
$404$	2.94489e9	2.22195
$405$	−6.64301e7	−0.0496904
$406$	9.61551e8	0.713069
$407$	3.47588e9	2.55555
$408$	−2.76215e8	−0.201343
$409$	−8.27345e8	−0.597937	−0.298968	−	0.954263i	$-0.596642\pi$
$409$	−8.27345e8	−0.597937	−0.298968	+	0.954263i	$0.596642\pi$
$410$	−8.05370e8	−0.577101
$411$	−3.34799e8	−0.237869
$412$	−3.13729e9	−2.21011
$413$	9.08479e8	0.634585
$414$	−1.01958e9	−0.706187
$415$	−4.47190e8	−0.307131
$416$	3.78060e8	0.257475
$417$	1.60767e8	0.108573
$418$	8.48991e8	0.568573
$419$	−1.68808e9	−1.12110	−0.560550	−	0.828120i	$-0.689410\pi$
$419$	−1.68808e9	−1.12110	−0.560550	+	0.828120i	$0.689410\pi$
$420$	2.26895e8	0.149435
$421$	1.24673e9	0.814301	0.407151	−	0.913361i	$-0.366522\pi$
$421$	1.24673e9	0.814301	0.407151	+	0.913361i	$0.366522\pi$
$422$	−3.99813e9	−2.58978
$423$	1.27808e8	0.0821046
$424$	1.61810e8	0.103092
$425$	1.30594e8	0.0825204
$426$	1.39592e9	0.874836
$427$	2.86569e8	0.178128
$428$	1.03901e9	0.640567
$429$	−3.86548e8	−0.236375
$430$	1.63011e9	0.988727
$431$	1.41807e9	0.853152	0.426576	−	0.904452i	$-0.359720\pi$
$431$	1.41807e9	0.853152	0.426576	+	0.904452i	$0.359720\pi$
$432$	6.01512e7	0.0358965
$433$	2.11064e9	1.24942	0.624708	−	0.780858i	$-0.285218\pi$
$433$	2.11064e9	1.24942	0.624708	+	0.780858i	$0.285218\pi$
$434$	−1.91394e9	−1.12387
$435$	5.25629e8	0.306173
$436$	−1.26683e9	−0.732006
$437$	4.57187e8	0.262065
$438$	−2.89261e9	−1.64487
$439$	−2.43638e9	−1.37442	−0.687209	−	0.726460i	$-0.741164\pi$
$439$	−2.43638e9	−1.37442	−0.687209	+	0.726460i	$0.741164\pi$
$440$	1.22645e9	0.686380
$441$	8.57661e7	0.0476190
$442$	−2.68693e8	−0.148006
$443$	2.17599e9	1.18917	0.594585	−	0.804033i	$-0.297316\pi$
$443$	2.17599e9	1.18917	0.594585	+	0.804033i	$0.297316\pi$
$444$	2.29471e9	1.24419
$445$	7.81853e8	0.420596
$446$	2.52386e9	1.34708
$447$	8.39674e8	0.444666
$448$	−1.17274e9	−0.616211
$449$	−3.88948e8	−0.202782	−0.101391	−	0.994847i	$-0.532329\pi$
$449$	−3.88948e8	−0.202782	−0.101391	+	0.994847i	$0.532329\pi$
$450$	2.05031e8	0.106066
$451$	−2.86926e9	−1.47283
$452$	−2.90733e9	−1.48084
$453$	3.86836e8	0.195517
$454$	−2.04234e9	−1.02431
$455$	7.65747e7	0.0381106
$456$	1.94454e8	0.0960375
$457$	−3.97621e9	−1.94878	−0.974390	−	0.224864i	$-0.927806\pi$
$457$	−3.97621e9	−1.94878	−0.974390	+	0.224864i	$0.927806\pi$
$458$	−1.80216e9	−0.876526
$459$	−1.64511e8	−0.0794053
$460$	1.90365e9	0.911874
$461$	3.61720e9	1.71957	0.859783	−	0.510659i	$-0.170599\pi$
$461$	3.61720e9	1.71957	0.859783	+	0.510659i	$0.170599\pi$
$462$	1.33625e9	0.630436
$463$	2.41022e9	1.12855	0.564277	−	0.825585i	$-0.309155\pi$
$463$	2.41022e9	1.12855	0.564277	+	0.825585i	$0.309155\pi$
$464$	−4.75948e8	−0.221180
$465$	−1.04625e9	−0.482559
$466$	−6.69715e9	−3.06577
$467$	1.75978e9	0.799555	0.399777	−	0.916612i	$-0.369087\pi$
$467$	1.75978e9	0.799555	0.399777	+	0.916612i	$0.369087\pi$
$468$	−2.55191e8	−0.115081
$469$	1.19580e9	0.535248
$470$	−3.94470e8	−0.175255
$471$	1.56889e9	0.691863
$472$	3.24192e9	1.41908
$473$	5.80753e9	2.52335
$474$	8.16919e8	0.352334
$475$	−9.19375e7	−0.0393609
$476$	5.61892e8	0.238797
$477$	9.63723e7	0.0406573
$478$	−1.71172e9	−0.716859
$479$	−4.18797e9	−1.74112	−0.870562	−	0.492059i	$-0.836245\pi$
$479$	−4.18797e9	−1.74112	−0.870562	+	0.492059i	$0.836245\pi$
$480$	−7.14420e8	−0.294856
$481$	7.74442e8	0.317308
$482$	−5.15461e9	−2.09667
$483$	7.19580e8	0.290579
$484$	8.77474e9	3.51784
$485$	6.57132e8	0.261551
$486$	−2.58280e8	−0.102062
$487$	−3.65982e9	−1.43585	−0.717924	−	0.696122i	$-0.754907\pi$
$487$	−3.65982e9	−1.43585	−0.717924	+	0.696122i	$0.754907\pi$
$488$	1.02263e9	0.398334
$489$	6.37212e8	0.246435
$490$	−2.64710e8	−0.101645
$491$	−3.63614e9	−1.38629	−0.693146	−	0.720797i	$-0.743776\pi$
$491$	−3.63614e9	−1.38629	−0.693146	+	0.720797i	$0.743776\pi$
$492$	−1.89423e9	−0.717059
$493$	1.30169e9	0.489265
$494$	1.89159e8	0.0705964
$495$	7.30458e8	0.270693
$496$	9.47360e8	0.348601
$497$	−9.85185e8	−0.359973
$498$	−1.73868e9	−0.630836
$499$	6.51843e8	0.234850	0.117425	−	0.993082i	$-0.462536\pi$
$499$	6.51843e8	0.234850	0.117425	+	0.993082i	$0.462536\pi$
$500$	−3.82812e8	−0.136959
$501$	−7.17664e8	−0.254970
$502$	9.96161e8	0.351452
$503$	−7.87014e8	−0.275737	−0.137868	−	0.990451i	$-0.544025\pi$
$503$	−7.87014e8	−0.275737	−0.137868	+	0.990451i	$0.544025\pi$
$504$	3.06058e8	0.106487
$505$	−1.87812e9	−0.648938
$506$	1.12112e10	3.84702
$507$	1.60809e9	0.548001
$508$	2.64646e9	0.895659
$509$	−6.08921e8	−0.204667	−0.102334	−	0.994750i	$-0.532631\pi$
$509$	−6.08921e8	−0.204667	−0.102334	+	0.994750i	$0.532631\pi$
$510$	5.07748e8	0.169493
$511$	2.04150e9	0.676823
$512$	1.12568e9	0.370656
$513$	1.15815e8	0.0378750
$514$	−1.07595e9	−0.349478
$515$	2.00082e9	0.645481
$516$	3.83401e9	1.22851
$517$	−1.40537e9	−0.447273
$518$	−2.67716e9	−0.846291
$519$	1.91611e9	0.601637
$520$	2.73258e8	0.0852239
$521$	−1.12396e9	−0.348193	−0.174096	−	0.984729i	$-0.555700\pi$
$521$	−1.12396e9	−0.348193	−0.174096	+	0.984729i	$0.555700\pi$
$522$	2.04365e9	0.628867
$523$	4.13165e9	1.26290	0.631448	−	0.775418i	$-0.282461\pi$
$523$	4.13165e9	1.26290	0.631448	+	0.775418i	$0.282461\pi$
$524$	2.40042e9	0.728833
$525$	−1.44703e8	−0.0436436
$526$	−2.22412e9	−0.666359
$527$	−2.59098e9	−0.771129
$528$	−6.61416e8	−0.195549
$529$	2.63246e9	0.773157
$530$	−2.97446e8	−0.0867844
$531$	1.93085e9	0.559651
$532$	−3.95570e8	−0.113902
$533$	−6.39284e8	−0.182873
$534$	3.03985e9	0.863887
$535$	−6.62631e8	−0.187083
$536$	4.26724e9	1.19693
$537$	−3.10666e9	−0.865732
$538$	7.72705e9	2.13932
$539$	−9.43074e8	−0.259409
$540$	4.82234e8	0.131789
$541$	−5.15257e9	−1.39905	−0.699525	−	0.714608i	$-0.746605\pi$
$541$	−5.15257e9	−1.39905	−0.699525	+	0.714608i	$0.746605\pi$
$542$	−9.88131e9	−2.66573
$543$	6.10445e7	0.0163624
$544$	−1.76922e9	−0.471179
$545$	8.07925e8	0.213788
$546$	2.97723e8	0.0782776
$547$	−6.96991e8	−0.182084	−0.0910420	−	0.995847i	$-0.529020\pi$
$547$	−6.96991e8	−0.182084	−0.0910420	+	0.995847i	$0.529020\pi$
$548$	2.43040e9	0.630877
$549$	6.09063e8	0.157094
$550$	−2.25450e9	−0.577804
$551$	−9.16386e8	−0.233371
$552$	2.56783e9	0.649800
$553$	−5.76550e8	−0.144977
$554$	5.99104e9	1.49699
$555$	−1.46346e9	−0.363375
$556$	−1.16705e9	−0.287957
$557$	2.88908e9	0.708381	0.354191	−	0.935173i	$-0.384756\pi$
$557$	2.88908e9	0.708381	0.354191	+	0.935173i	$0.384756\pi$
$558$	−4.06782e9	−0.991156
$559$	1.29394e9	0.313310
$560$	1.31026e8	0.0315282
$561$	1.80894e9	0.432567
$562$	−8.18022e9	−1.94396
$563$	−5.01488e9	−1.18435	−0.592176	−	0.805809i	$-0.701731\pi$
$563$	−5.01488e9	−1.18435	−0.592176	+	0.805809i	$0.701731\pi$
$564$	−9.27793e8	−0.217758
$565$	1.85416e9	0.432492
$566$	1.47933e9	0.342931
$567$	1.82284e8	0.0419961
$568$	−3.51565e9	−0.804982
$569$	1.91122e9	0.434929	0.217464	−	0.976068i	$-0.430221\pi$
$569$	1.91122e9	0.434929	0.217464	+	0.976068i	$0.430221\pi$
$570$	−3.57453e8	−0.0808457
$571$	3.73938e8	0.0840568	0.0420284	−	0.999116i	$-0.486618\pi$
$571$	3.73938e8	0.0840568	0.0420284	+	0.999116i	$0.486618\pi$
$572$	2.80605e9	0.626916
$573$	2.17927e9	0.483915
$574$	2.20993e9	0.487739
$575$	−1.21406e9	−0.266320
$576$	−2.49250e9	−0.543447
$577$	1.91539e9	0.415089	0.207545	−	0.978226i	$-0.433453\pi$
$577$	1.91539e9	0.415089	0.207545	+	0.978226i	$0.433453\pi$
$578$	−6.12869e9	−1.32014
$579$	−1.91144e9	−0.409246
$580$	−3.81568e9	−0.812033
$581$	1.22709e9	0.259573
$582$	2.55493e9	0.537215
$583$	−1.05970e9	−0.221484
$584$	7.28510e9	1.51353
$585$	1.62749e8	0.0336104
$586$	6.68608e9	1.37256
$587$	−4.10449e9	−0.837578	−0.418789	−	0.908084i	$-0.637545\pi$
$587$	−4.10449e9	−0.837578	−0.418789	+	0.908084i	$0.637545\pi$
$588$	−6.22599e8	−0.126295
$589$	1.82404e9	0.367816
$590$	−5.95941e9	−1.19460
$591$	−2.03438e9	−0.405392
$592$	1.32514e9	0.262503
$593$	−3.09117e9	−0.608740	−0.304370	−	0.952554i	$-0.598446\pi$
$593$	−3.09117e9	−0.608740	−0.304370	+	0.952554i	$0.598446\pi$
$594$	2.84002e9	0.555993
$595$	−3.58349e8	−0.0697424
$596$	−6.09541e9	−1.17935
$597$	4.25426e9	0.818302
$598$	2.49790e9	0.477662
$599$	−9.69612e9	−1.84333	−0.921667	−	0.387982i	$-0.873172\pi$
$599$	−9.69612e9	−1.84333	−0.921667	+	0.387982i	$0.873172\pi$
$600$	−5.16375e8	−0.0975969
$601$	2.12239e9	0.398808	0.199404	−	0.979917i	$-0.436099\pi$
$601$	2.12239e9	0.398808	0.199404	+	0.979917i	$0.436099\pi$
$602$	−4.47301e9	−0.835627
$603$	2.54152e9	0.472044
$604$	−2.80815e9	−0.518550
$605$	−5.59614e9	−1.02741
$606$	−7.30213e9	−1.33289
$607$	2.11212e9	0.383316	0.191658	−	0.981462i	$-0.438613\pi$
$607$	2.11212e9	0.383316	0.191658	+	0.981462i	$0.438613\pi$
$608$	1.24553e9	0.224745
$609$	−1.44233e9	−0.258764
$610$	−1.87983e9	−0.335323
$611$	−3.13122e8	−0.0555352
$612$	1.19422e9	0.210599
$613$	−2.41937e9	−0.424220	−0.212110	−	0.977246i	$-0.568034\pi$
$613$	−2.41937e9	−0.424220	−0.212110	+	0.977246i	$0.568034\pi$
$614$	−8.27916e9	−1.44343
$615$	1.20805e9	0.209423
$616$	−3.36537e9	−0.580097
$617$	−3.53370e9	−0.605664	−0.302832	−	0.953044i	$-0.597932\pi$
$617$	−3.53370e9	−0.605664	−0.302832	+	0.953044i	$0.597932\pi$
$618$	7.77920e9	1.32579
$619$	−1.71721e8	−0.0291009	−0.0145505	−	0.999894i	$-0.504632\pi$
$619$	−1.71721e8	−0.0291009	−0.0145505	+	0.999894i	$0.504632\pi$
$620$	7.59500e9	1.27984
$621$	1.52937e9	0.256266
$622$	−8.90951e9	−1.48452
$623$	−2.14541e9	−0.355469
$624$	−1.47366e8	−0.0242802
$625$	2.44141e8	0.0400000
$626$	6.47399e9	1.05478
$627$	−1.27349e9	−0.206328
$628$	−1.13890e10	−1.83496
$629$	−3.62418e9	−0.580674
$630$	−5.62606e8	−0.0896421
$631$	−1.02893e10	−1.63036	−0.815180	−	0.579208i	$-0.803362\pi$
$631$	−1.02893e10	−1.63036	−0.815180	+	0.579208i	$0.803362\pi$
$632$	−2.05743e9	−0.324201
$633$	5.99719e9	0.939799
$634$	3.32895e9	0.518794
$635$	−1.68780e9	−0.261584
$636$	−6.99592e8	−0.107831
$637$	−2.10121e8	−0.0322093
$638$	−2.24717e10	−3.42581
$639$	−2.09388e9	−0.317467
$640$	4.30603e9	0.649303
$641$	1.10606e10	1.65873	0.829363	−	0.558710i	$-0.188703\pi$
$641$	1.10606e10	1.65873	0.829363	+	0.558710i	$0.188703\pi$
$642$	−2.57631e9	−0.384260
$643$	−1.75155e9	−0.259827	−0.129913	−	0.991525i	$-0.541470\pi$
$643$	−1.75155e9	−0.259827	−0.129913	+	0.991525i	$0.541470\pi$
$644$	−5.22362e9	−0.770674
$645$	−2.44516e9	−0.358797
$646$	−8.85212e8	−0.129191
$647$	3.04738e8	0.0442346	0.0221173	−	0.999755i	$-0.492959\pi$
$647$	3.04738e8	0.0442346	0.0221173	+	0.999755i	$0.492959\pi$
$648$	6.50484e8	0.0939126
$649$	−2.12314e10	−3.04875
$650$	−5.02312e8	−0.0717426
$651$	2.87091e9	0.407837
$652$	−4.62568e9	−0.653596
$653$	1.91092e9	0.268562	0.134281	−	0.990943i	$-0.457127\pi$
$653$	1.91092e9	0.268562	0.134281	+	0.990943i	$0.457127\pi$
$654$	3.14121e9	0.439112
$655$	−1.53088e9	−0.212861
$656$	−1.09387e9	−0.151287
$657$	4.33892e9	0.596902
$658$	1.08243e9	0.148118
$659$	2.66917e9	0.363310	0.181655	−	0.983362i	$-0.441855\pi$
$659$	2.66917e9	0.363310	0.181655	+	0.983362i	$0.441855\pi$
$660$	−5.30258e9	−0.717933
$661$	−1.20889e9	−0.162810	−0.0814051	−	0.996681i	$-0.525941\pi$
$661$	−1.20889e9	−0.162810	−0.0814051	+	0.996681i	$0.525941\pi$
$662$	1.84533e10	2.47213
$663$	4.03039e8	0.0537094
$664$	4.37889e9	0.580465
$665$	2.52277e8	0.0332660
$666$	−5.68994e9	−0.746359
$667$	−1.21012e10	−1.57902
$668$	5.20971e9	0.676232
$669$	−3.78579e9	−0.488839
$670$	−7.84419e9	−1.00760
$671$	−6.69719e9	−0.855783
$672$	1.96037e9	0.249198
$673$	2.65330e9	0.335531	0.167766	−	0.985827i	$-0.446345\pi$
$673$	2.65330e9	0.335531	0.167766	+	0.985827i	$0.446345\pi$
$674$	2.03531e10	2.56048
$675$	−3.07547e8	−0.0384900
$676$	−1.16735e10	−1.45341
$677$	9.29262e9	1.15101	0.575504	−	0.817799i	$-0.304806\pi$
$677$	9.29262e9	1.15101	0.575504	+	0.817799i	$0.304806\pi$
$678$	7.20898e9	0.888322
$679$	−1.80317e9	−0.221051
$680$	−1.27877e9	−0.155960
$681$	3.06351e9	0.371710
$682$	4.47293e10	5.39941
$683$	1.52018e10	1.82567	0.912837	−	0.408325i	$-0.133887\pi$
$683$	1.52018e10	1.82567	0.912837	+	0.408325i	$0.133887\pi$
$684$	−8.40729e8	−0.100452
$685$	−1.55000e9	−0.184253
$686$	7.26365e8	0.0859054
$687$	2.70324e9	0.318080
$688$	2.21405e9	0.259195
$689$	−2.36106e8	−0.0275004
$690$	−4.72028e9	−0.547010
$691$	3.49520e9	0.402994	0.201497	−	0.979489i	$-0.435419\pi$
$691$	3.49520e9	0.402994	0.201497	+	0.979489i	$0.435419\pi$
$692$	−1.39095e10	−1.59566
$693$	−2.00438e9	−0.228777
$694$	1.13189e10	1.28542
$695$	7.44292e8	0.0841000
$696$	−5.14696e9	−0.578654
$697$	2.99168e9	0.334657
$698$	1.62984e10	1.81406
$699$	1.00457e10	1.11253
$700$	1.05044e9	0.115752
$701$	−1.34756e10	−1.47752	−0.738761	−	0.673968i	$-0.764589\pi$
$701$	−1.34756e10	−1.47752	−0.738761	+	0.673968i	$0.764589\pi$
$702$	6.32769e8	0.0690344
$703$	2.55141e9	0.276972
$704$	2.74073e10	2.96048
$705$	5.91705e8	0.0635980
$706$	−2.04944e9	−0.219189
$707$	5.15356e9	0.548453
$708$	−1.40165e10	−1.48431
$709$	−1.37164e10	−1.44537	−0.722683	−	0.691179i	$-0.757092\pi$
$709$	−1.37164e10	−1.44537	−0.722683	+	0.691179i	$0.757092\pi$
$710$	6.46258e9	0.677645
$711$	−1.22538e9	−0.127858
$712$	−7.65591e9	−0.794908
$713$	2.40870e10	2.48868
$714$	−1.39326e9	−0.143248
$715$	−1.78957e9	−0.183096
$716$	2.25520e10	2.29610
$717$	2.56757e9	0.260139
$718$	4.88977e9	0.493006
$719$	1.52367e10	1.52876	0.764378	−	0.644768i	$-0.223046\pi$
$719$	1.52367e10	1.52876	0.764378	+	0.644768i	$0.223046\pi$
$720$	2.78478e8	0.0278053
$721$	−5.49026e9	−0.545531
$722$	−1.54665e10	−1.52937
$723$	7.73191e9	0.760857
$724$	−4.43138e8	−0.0433964
$725$	2.43347e9	0.237161
$726$	−2.17578e10	−2.11026
$727$	8.49661e9	0.820116	0.410058	−	0.912059i	$-0.365508\pi$
$727$	8.49661e9	0.820116	0.410058	+	0.912059i	$0.365508\pi$
$728$	−7.49820e8	−0.0720273
$729$	3.87420e8	0.0370370
$730$	−1.33917e10	−1.27411
$731$	−6.05530e9	−0.573357
$732$	−4.42135e9	−0.416645
$733$	1.02983e10	0.965828	0.482914	−	0.875668i	$-0.339578\pi$
$733$	1.02983e10	0.965828	0.482914	+	0.875668i	$0.339578\pi$
$734$	2.43167e10	2.26970
$735$	3.97065e8	0.0368856
$736$	1.64475e10	1.52065
$737$	−2.79462e10	−2.57150
$738$	4.69691e9	0.430146
$739$	2.01922e10	1.84047	0.920235	−	0.391367i	$-0.127998\pi$
$739$	2.01922e10	1.84047	0.920235	+	0.391367i	$0.127998\pi$
$740$	1.06236e10	0.963746
$741$	−2.83738e8	−0.0256185
$742$	8.16190e8	0.0733462
$743$	1.20966e10	1.08194	0.540971	−	0.841042i	$-0.318057\pi$
$743$	1.20966e10	1.08194	0.540971	+	0.841042i	$0.318057\pi$
$744$	1.02449e10	0.912014
$745$	3.88738e9	0.344437
$746$	−1.97581e10	−1.74245
$747$	2.60801e9	0.228922
$748$	−1.31316e10	−1.14726
$749$	1.81826e9	0.158114
$750$	9.49219e8	0.0821584
$751$	−3.49566e9	−0.301154	−0.150577	−	0.988598i	$-0.548113\pi$
$751$	−3.49566e9	−0.301154	−0.150577	+	0.988598i	$0.548113\pi$
$752$	−5.35778e8	−0.0459433
$753$	−1.49424e9	−0.127538
$754$	−5.00679e9	−0.425363
$755$	1.79091e9	0.151446
$756$	−1.32325e9	−0.111382
$757$	1.27420e10	1.06758	0.533792	−	0.845616i	$-0.320767\pi$
$757$	1.27420e10	1.06758	0.533792	+	0.845616i	$0.320767\pi$
$758$	−7.76765e9	−0.647810
$759$	−1.68168e10	−1.39603
$760$	9.00252e8	0.0743903
$761$	5.66077e9	0.465617	0.232809	−	0.972523i	$-0.425208\pi$
$761$	5.66077e9	0.465617	0.232809	+	0.972523i	$0.425208\pi$
$762$	−6.56215e9	−0.537284
$763$	−2.21695e9	−0.180684
$764$	−1.58199e10	−1.28344
$765$	−7.61623e8	−0.0615070
$766$	9.52772e9	0.765928
$767$	−4.73045e9	−0.378546
$768$	4.92554e9	0.392364
$769$	1.39747e10	1.10816	0.554078	−	0.832465i	$-0.313071\pi$
$769$	1.39747e10	1.10816	0.554078	+	0.832465i	$0.313071\pi$
$770$	6.18635e9	0.488334
$771$	1.61392e9	0.126821
$772$	1.38756e10	1.08540
$773$	−1.26246e10	−0.983083	−0.491541	−	0.870854i	$-0.663566\pi$
$773$	−1.26246e10	−0.983083	−0.491541	+	0.870854i	$0.663566\pi$
$774$	−9.50678e9	−0.736954
$775$	−4.84375e9	−0.373788
$776$	−6.43463e9	−0.494320
$777$	4.01574e9	0.307108
$778$	−2.28618e8	−0.0174053
$779$	−2.10613e9	−0.159626
$780$	−1.18144e9	−0.0891416
$781$	2.30240e10	1.72943
$782$	−1.16895e10	−0.874122
$783$	−3.06547e9	−0.228208
$784$	−3.59535e8	−0.0266462
$785$	7.26339e9	0.535915
$786$	−5.95206e9	−0.437209
$787$	−2.29786e10	−1.68040	−0.840199	−	0.542278i	$-0.817562\pi$
$787$	−2.29786e10	−1.68040	−0.840199	+	0.542278i	$0.817562\pi$
$788$	1.47681e10	1.07518
$789$	3.33618e9	0.241813
$790$	3.78203e9	0.272917
$791$	−5.08782e9	−0.365523
$792$	−7.15264e9	−0.511598
$793$	−1.49216e9	−0.106258
$794$	−3.49356e10	−2.47683
$795$	4.46168e8	0.0314930
$796$	−3.08828e10	−2.17030
$797$	5.78646e8	0.0404864	0.0202432	−	0.999795i	$-0.493556\pi$
$797$	5.78646e8	0.0404864	0.0202432	+	0.999795i	$0.493556\pi$
$798$	9.80851e8	0.0683271
$799$	1.46532e9	0.101630
$800$	−3.30750e9	−0.228394
$801$	−4.55977e9	−0.313494
$802$	2.32803e10	1.59360
$803$	−4.77103e10	−3.25168
$804$	−1.84495e10	−1.25196
$805$	3.33139e9	0.225081
$806$	9.96588e9	0.670414
$807$	−1.15906e10	−0.776332
$808$	1.83905e10	1.22646
$809$	−9.57879e9	−0.636050	−0.318025	−	0.948082i	$-0.603020\pi$
$809$	−9.57879e9	−0.636050	−0.318025	+	0.948082i	$0.603020\pi$
$810$	−1.19574e9	−0.0790569
$811$	−2.25648e10	−1.48545	−0.742727	−	0.669595i	$-0.766468\pi$
$811$	−2.25648e10	−1.48545	−0.742727	+	0.669595i	$0.766468\pi$
$812$	1.04702e10	0.686293
$813$	1.48220e10	0.967361
$814$	6.25659e10	4.06586
$815$	2.95005e9	0.190888
$816$	6.89635e8	0.0444328
$817$	4.26291e9	0.273482
$818$	−1.48922e10	−0.951311
$819$	−4.46584e8	−0.0284060
$820$	−8.76958e9	−0.555431
$821$	−2.33524e10	−1.47275	−0.736377	−	0.676572i	$-0.763465\pi$
$821$	−2.33524e10	−1.47275	−0.736377	+	0.676572i	$0.763465\pi$
$822$	−6.02639e9	−0.378448
$823$	−1.48433e10	−0.928180	−0.464090	−	0.885788i	$-0.653619\pi$
$823$	−1.48433e10	−0.928180	−0.464090	+	0.885788i	$0.653619\pi$
$824$	−1.95920e10	−1.21993
$825$	3.38175e9	0.209678
$826$	1.63526e10	1.00962
$827$	−2.18669e10	−1.34436	−0.672182	−	0.740386i	$-0.734643\pi$
$827$	−2.18669e10	−1.34436	−0.672182	+	0.740386i	$0.734643\pi$
$828$	−1.11021e10	−0.679671
$829$	−1.60225e10	−0.976764	−0.488382	−	0.872630i	$-0.662413\pi$
$829$	−1.60225e10	−0.976764	−0.488382	+	0.872630i	$0.662413\pi$
$830$	−8.04943e9	−0.488643
$831$	−8.98656e9	−0.543238
$832$	6.10646e9	0.367585
$833$	9.83310e8	0.0589431
$834$	2.89381e9	0.172738
$835$	−3.32252e9	−0.197499
$836$	9.24456e9	0.547223
$837$	6.10173e9	0.359678
$838$	−3.03855e10	−1.78366
$839$	−7.37475e9	−0.431102	−0.215551	−	0.976493i	$-0.569155\pi$
$839$	−7.37475e9	−0.431102	−0.215551	+	0.976493i	$0.569155\pi$
$840$	1.41693e9	0.0824844
$841$	7.00569e9	0.406130
$842$	2.24411e10	1.29555
$843$	1.22703e10	0.705439
$844$	−4.35352e10	−2.49254
$845$	7.44484e9	0.424480
$846$	2.30055e9	0.130628
$847$	1.53558e10	0.868321
$848$	−4.03997e8	−0.0227506
$849$	−2.21899e9	−0.124445
$850$	2.35069e9	0.131289
$851$	3.36921e10	1.87402
$852$	1.52000e10	0.841986
$853$	2.99861e10	1.65424	0.827119	−	0.562027i	$-0.189978\pi$
$853$	2.99861e10	1.65424	0.827119	+	0.562027i	$0.189978\pi$
$854$	5.15824e9	0.283400
$855$	5.36179e8	0.0293379
$856$	6.48848e9	0.353578
$857$	3.07594e10	1.66934	0.834671	−	0.550749i	$-0.185658\pi$
$857$	3.07594e10	1.66934	0.834671	+	0.550749i	$0.185658\pi$
$858$	−6.95786e9	−0.376071
$859$	−1.66508e10	−0.896312	−0.448156	−	0.893955i	$-0.647919\pi$
$859$	−1.66508e10	−0.896312	−0.448156	+	0.893955i	$0.647919\pi$
$860$	1.77501e10	0.951601
$861$	−3.31490e9	−0.176994
$862$	2.55252e10	1.35736
$863$	1.98562e10	1.05162	0.525809	−	0.850603i	$-0.323763\pi$
$863$	1.98562e10	1.05162	0.525809	+	0.850603i	$0.323763\pi$
$864$	4.16650e9	0.219772
$865$	8.87088e9	0.466026
$866$	3.79916e10	1.98781
$867$	9.19303e9	0.479062
$868$	−2.08407e10	−1.08167
$869$	1.34741e10	0.696516
$870$	9.46133e9	0.487119
$871$	−6.22655e9	−0.319289
$872$	−7.91120e9	−0.404050
$873$	−3.83239e9	−0.194949
$874$	8.22936e9	0.416943
$875$	−6.69922e8	−0.0338062
$876$	−3.14974e10	−1.58311
$877$	−6.35922e9	−0.318350	−0.159175	−	0.987250i	$-0.550883\pi$
$877$	−6.35922e9	−0.318350	−0.159175	+	0.987250i	$0.550883\pi$
$878$	−4.38548e10	−2.18669
$879$	−1.00291e10	−0.498083
$880$	−3.06211e9	−0.151472
$881$	−1.37251e10	−0.676237	−0.338118	−	0.941104i	$-0.609790\pi$
$881$	−1.37251e10	−0.676237	−0.338118	+	0.941104i	$0.609790\pi$
$882$	1.54379e9	0.0757614
$883$	2.43465e9	0.119007	0.0595036	−	0.998228i	$-0.481048\pi$
$883$	2.43465e9	0.119007	0.0595036	+	0.998228i	$0.481048\pi$
$884$	−2.92577e9	−0.142448
$885$	8.93912e9	0.433504
$886$	3.91678e10	1.89196
$887$	−3.32171e10	−1.59819	−0.799097	−	0.601202i	$-0.794689\pi$
$887$	−3.32171e10	−1.59819	−0.799097	+	0.601202i	$0.794689\pi$
$888$	1.43302e10	0.686763
$889$	4.63131e9	0.221079
$890$	1.40734e10	0.669164
$891$	−4.26003e9	−0.201763
$892$	2.74821e10	1.29650
$893$	−1.03158e9	−0.0484757
$894$	1.51141e10	0.707460
$895$	−1.43827e10	−0.670593
$896$	−1.18158e10	−0.548761
$897$	−3.74685e9	−0.173338
$898$	−7.00106e9	−0.322624
$899$	−4.82800e10	−2.21620
$900$	2.23256e9	0.102083
$901$	1.10491e9	0.0503258
$902$	−5.16467e10	−2.34326
$903$	6.70952e9	0.303239
$904$	−1.81560e10	−0.817391
$905$	2.82613e8	0.0126743
$906$	6.96305e9	0.311065
$907$	−1.71277e10	−0.762210	−0.381105	−	0.924532i	$-0.624456\pi$
$907$	−1.71277e10	−0.762210	−0.381105	+	0.924532i	$0.624456\pi$
$908$	−2.22388e10	−0.985850
$909$	1.09532e10	0.483690
$910$	1.37835e9	0.0606336
$911$	3.08592e10	1.35229	0.676146	−	0.736768i	$-0.263649\pi$
$911$	3.08592e10	1.35229	0.676146	+	0.736768i	$0.263649\pi$
$912$	−4.85501e8	−0.0211938
$913$	−2.86774e10	−1.24707
$914$	−7.15718e10	−3.10049
$915$	2.81974e9	0.121684
$916$	−1.96235e10	−0.843613
$917$	4.20074e9	0.179901
$918$	−2.96119e9	−0.126333
$919$	1.23061e10	0.523018	0.261509	−	0.965201i	$-0.415780\pi$
$919$	1.23061e10	0.523018	0.261509	+	0.965201i	$0.415780\pi$
$920$	1.18881e10	0.503333
$921$	1.24187e10	0.523804
$922$	6.51096e10	2.73581
$923$	5.12986e9	0.214733
$924$	1.45503e10	0.606764
$925$	−6.77528e9	−0.281469
$926$	4.33839e10	1.79552
$927$	−1.16688e10	−0.481113
$928$	−3.29675e10	−1.35415
$929$	−2.84458e10	−1.16403	−0.582013	−	0.813179i	$-0.697735\pi$
$929$	−2.84458e10	−1.16403	−0.582013	+	0.813179i	$0.697735\pi$
$930$	−1.88325e10	−0.767746
$931$	−6.92247e8	−0.0281149
$932$	−7.29246e10	−2.95065
$933$	1.33643e10	0.538715
$934$	3.16760e10	1.27208
$935$	8.37472e9	0.335065
$936$	−1.59364e9	−0.0635221
$937$	−4.08626e10	−1.62270	−0.811349	−	0.584562i	$-0.801266\pi$
$937$	−4.08626e10	−1.62270	−0.811349	+	0.584562i	$0.801266\pi$
$938$	2.15245e10	0.851574
$939$	−9.71098e9	−0.382766
$940$	−4.29534e9	−0.168675
$941$	1.52622e10	0.597108	0.298554	−	0.954393i	$-0.403496\pi$
$941$	1.52622e10	0.597108	0.298554	+	0.954393i	$0.403496\pi$
$942$	2.82401e10	1.10075
$943$	−2.78121e10	−1.08005
$944$	−8.09421e9	−0.313164
$945$	8.43909e8	0.0325300
$946$	1.04536e11	4.01462
$947$	−2.32859e10	−0.890980	−0.445490	−	0.895287i	$-0.646970\pi$
$947$	−2.32859e10	−0.890980	−0.445490	+	0.895287i	$0.646970\pi$
$948$	8.89534e9	0.339105
$949$	−1.06301e10	−0.403742
$950$	−1.65487e9	−0.0626228
$951$	−4.99343e9	−0.188264
$952$	3.50896e9	0.131810
$953$	2.98085e10	1.11562	0.557809	−	0.829970i	$-0.311642\pi$
$953$	2.98085e10	1.11562	0.557809	+	0.829970i	$0.311642\pi$
$954$	1.73470e9	0.0646853
$955$	1.00892e10	0.374839
$956$	−1.86387e10	−0.689942
$957$	3.37076e10	1.24318
$958$	−7.53835e10	−2.77011
$959$	4.25319e9	0.155722
$960$	−1.15394e10	−0.420952
$961$	6.85874e10	2.49294
$962$	1.39400e10	0.504834
$963$	3.86446e9	0.139443
$964$	−5.61280e10	−2.01795
$965$	−8.84924e9	−0.317001
$966$	1.29524e10	0.462308
$967$	−7.13076e9	−0.253596	−0.126798	−	0.991929i	$-0.540470\pi$
$967$	−7.13076e9	−0.253596	−0.126798	+	0.991929i	$0.540470\pi$
$968$	5.47974e10	1.94176
$969$	1.32782e9	0.0468819
$970$	1.18284e10	0.416125
$971$	−4.32969e10	−1.51771	−0.758857	−	0.651258i	$-0.774242\pi$
$971$	−4.32969e10	−1.51771	−0.758857	+	0.651258i	$0.774242\pi$
$972$	−2.81239e9	−0.0982297
$973$	−2.04234e9	−0.0710775
$974$	−6.58767e10	−2.28442
$975$	7.53469e8	0.0260345
$976$	−2.55322e9	−0.0879051
$977$	2.77442e10	0.951789	0.475895	−	0.879502i	$-0.342124\pi$
$977$	2.77442e10	0.951789	0.475895	+	0.879502i	$0.342124\pi$
$978$	1.14698e10	0.392076
$979$	5.01387e10	1.70779
$980$	−2.88240e9	−0.0978280
$981$	−4.71182e9	−0.159348
$982$	−6.54505e10	−2.20558
$983$	−3.01277e9	−0.101164	−0.0505822	−	0.998720i	$-0.516108\pi$
$983$	−3.01277e9	−0.101164	−0.0505822	+	0.998720i	$0.516108\pi$
$984$	−1.18293e10	−0.395799
$985$	−9.41842e9	−0.314016
$986$	2.34304e10	0.778415
$987$	−1.62364e9	−0.0537501
$988$	2.05973e9	0.0679455
$989$	5.62930e10	1.85041
$990$	1.31482e10	0.430670
$991$	−2.67845e10	−0.874232	−0.437116	−	0.899405i	$-0.644000\pi$
$991$	−2.67845e10	−0.874232	−0.437116	+	0.899405i	$0.644000\pi$
$992$	6.56208e10	2.13428
$993$	−2.76800e10	−0.897105
$994$	−1.77333e10	−0.572714
$995$	1.96956e10	0.633854
$996$	−1.89323e10	−0.607148
$997$	−2.46784e10	−0.788649	−0.394325	−	0.918971i	$-0.629021\pi$
$997$	−2.46784e10	−0.788649	−0.394325	+	0.918971i	$0.629021\pi$
$998$	1.17332e10	0.373645
$999$	8.53490e9	0.270844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.8.a.b.1.1	✓	1
3.2	odd	2		315.8.a.a.1.1		1
5.4	even	2		525.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.b.1.1	✓	1	1.1	even	1	trivial
315.8.a.a.1.1		1	3.2	odd	2
525.8.a.a.1.1		1	5.4	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:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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