LMFDB - Embedded newform 525.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,8,Mod(1,525)] 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("525.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	525.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:	$164.002138379$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

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

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	$-0.792789\pi$
$2$	−18.0000	−1.59099	−0.795495	+	0.605960i	$0.792789\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	196.000	1.53125
$5$	0	0
$5$	0	0
$6$	−486.000	−0.918559
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−1224.00	−0.845214
$9$	729.000	0.333333
$10$	0	0
$11$	−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	$-0.464040\pi$
$13$	1786.00	0.225465	0.112733	+	0.993625i	$0.464040\pi$
$14$	6174.00	0.601338
$15$	0	0
$16$	−3056.00	−0.186523
$17$	−8358.00	−0.412602	−0.206301	−	0.978489i	$-0.566143\pi$
$17$	−8358.00	−0.412602	−0.206301	+	0.978489i	$0.566143\pi$
$18$	−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	$0.531373\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	144288.	2.88902
$23$	77700.0	1.33160	0.665800	−	0.746131i	$-0.268091\pi$
$23$	77700.0	1.33160	0.665800	+	0.746131i	$0.268091\pi$
$24$	−33048.0	−0.487984
$25$	0	0
$26$	−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	$0.297982\pi$
$30$	0	0
$31$	−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
$35$	0	0
$36$	142884.	0.510417
$37$	433618.	1.40735	0.703674	−	0.710523i	$-0.251542\pi$
$37$	433618.	1.40735	0.703674	+	0.710523i	$0.251542\pi$
$38$	105912.	0.313114
$39$	48222.0	0.130172
$40$	0	0
$41$	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	$-0.255490\pi$
$43$	724492.	1.38961	0.694807	+	0.719197i	$0.255490\pi$
$44$	−1.57114e6	−2.78054
$45$	0	0
$46$	−1.39860e6	−2.11856
$47$	−175320.	−0.246314	−0.123157	−	0.992387i	$-0.539302\pi$
$47$	−175320.	−0.246314	−0.123157	+	0.992387i	$0.539302\pi$
$48$	−82512.0	−0.107689
$49$	117649.	0.142857
$50$	0	0
$51$	−225666.	−0.238216
$52$	350056.	0.345244
$53$	−132198.	−0.121972	−0.0609859	−	0.998139i	$-0.519424\pi$
$53$	−132198.	−0.121972	−0.0609859	+	0.998139i	$0.519424\pi$
$54$	−354294.	−0.306186
$55$	0	0
$56$	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	$0.182861\pi$
$60$	0	0
$61$	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
$65$	0	0
$66$	3.89578e6	1.66798
$67$	−3.48631e6	−1.41613	−0.708066	−	0.706146i	$-0.750432\pi$
$67$	−3.48631e6	−1.41613	−0.708066	+	0.706146i	$0.750432\pi$
$68$	−1.63817e6	−0.631797
$69$	2.09790e6	0.768799
$70$	0	0
$71$	−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	$-0.853077\pi$
$73$	−5.95188e6	−1.79071	−0.895353	+	0.445357i	$0.853077\pi$
$74$	−7.80512e6	−2.23908
$75$	0	0
$76$	−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	$0.561428\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−6.44296e6	−1.29044
$83$	−3.57752e6	−0.686767	−0.343383	−	0.939195i	$-0.611573\pi$
$83$	−3.57752e6	−0.686767	−0.343383	+	0.939195i	$0.611573\pi$
$84$	−1.81516e6	−0.334146
$85$	0	0
$86$	−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	$0.655833\pi$
$90$	0	0
$91$	−612598.	−0.0852179
$92$	1.52292e7	2.03901
$93$	−8.37000e6	−1.07903
$94$	3.15576e6	0.391883
$95$	0	0
$96$	5.71536e6	0.659317
$97$	5.25705e6	0.584846	0.292423	−	0.956289i	$-0.405539\pi$
$97$	5.25705e6	0.584846	0.292423	+	0.956289i	$0.405539\pi$
$98$	−2.11768e6	−0.227284
$99$	−5.84366e6	−0.605288
$100$	0	0
$101$	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	$-0.243375\pi$
$103$	1.60066e7	1.44334	0.721669	+	0.692238i	$0.243375\pi$
$104$	−2.18606e6	−0.190566
$105$	0	0
$106$	2.37956e6	0.194056
$107$	−5.30105e6	−0.418330	−0.209165	−	0.977880i	$-0.567074\pi$
$107$	−5.30105e6	−0.418330	−0.209165	+	0.977880i	$0.567074\pi$
$108$	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	$0.576827\pi$
$110$	0	0
$111$	1.17077e7	0.812532
$112$	1.04821e6	0.0704992
$113$	1.48333e7	0.967082	0.483541	−	0.875322i	$-0.339350\pi$
$113$	1.48333e7	0.967082	0.483541	+	0.875322i	$0.339350\pi$
$114$	2.85962e6	0.180776
$115$	0	0
$116$	3.05254e7	1.81576
$117$	1.30199e6	0.0751551
$118$	−4.76753e7	−2.67120
$119$	2.86679e6	0.155949
$120$	0	0
$121$	4.47691e7	2.29736
$122$	−1.50386e7	−0.749805
$123$	9.66443e6	0.468283
$124$	−6.07600e7	−2.86182
$125$	0	0
$126$	4.50085e6	0.200446
$127$	−1.35024e7	−0.584920	−0.292460	−	0.956278i	$-0.594474\pi$
$127$	−1.35024e7	−0.584920	−0.292460	+	0.956278i	$0.594474\pi$
$128$	3.44483e7	1.45189
$129$	1.95613e7	0.802294
$130$	0	0
$131$	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
$135$	0	0
$136$	1.02302e7	0.348737
$137$	−1.24000e7	−0.412002	−0.206001	−	0.978552i	$-0.566045\pi$
$137$	−1.24000e7	−0.412002	−0.206001	+	0.978552i	$0.566045\pi$
$138$	−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	$0.529974\pi$
$140$	0	0
$141$	−4.73364e6	−0.142209
$142$	5.17007e7	1.51526
$143$	−1.43166e7	−0.409414
$144$	−2.22782e6	−0.0621745
$145$	0	0
$146$	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	$0.625830\pi$
$150$	0	0
$151$	−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
$155$	0	0
$156$	9.45151e6	0.199327
$157$	5.81072e7	1.19834	0.599171	−	0.800621i	$-0.295497\pi$
$157$	5.81072e7	1.19834	0.599171	+	0.800621i	$0.295497\pi$
$158$	3.02563e7	0.610261
$159$	−3.56935e6	−0.0704204
$160$	0	0
$161$	−2.66511e7	−0.503297
$162$	−9.56594e6	−0.176777
$163$	2.36004e7	0.426838	0.213419	−	0.976961i	$-0.431540\pi$
$163$	2.36004e7	0.426838	0.213419	+	0.976961i	$0.431540\pi$
$164$	7.01566e7	1.24198
$165$	0	0
$166$	6.43954e7	1.09264
$167$	−2.65801e7	−0.441621	−0.220810	−	0.975317i	$-0.570870\pi$
$167$	−2.65801e7	−0.441621	−0.220810	+	0.975317i	$0.570870\pi$
$168$	1.13355e7	0.184441
$169$	−5.95587e7	−0.949165
$170$	0	0
$171$	−4.28944e6	−0.0656015
$172$	1.42000e8	2.12785
$173$	7.09671e7	1.04207	0.521033	−	0.853536i	$-0.325547\pi$
$173$	7.09671e7	1.04207	0.521033	+	0.853536i	$0.325547\pi$
$174$	−7.56906e7	−1.08923
$175$	0	0
$176$	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	$0.230176\pi$
$180$	0	0
$181$	−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
$185$	0	0
$186$	1.50660e8	1.71673
$187$	6.69977e7	0.749229
$188$	−3.43627e7	−0.377168
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	−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	$-0.615321\pi$
$193$	−7.07939e7	−0.708835	−0.354418	+	0.935087i	$0.615321\pi$
$194$	−9.46270e7	−0.930484
$195$	0	0
$196$	2.30592e7	0.218750
$197$	−7.53473e7	−0.702160	−0.351080	−	0.936345i	$-0.614185\pi$
$197$	−7.53473e7	−0.702160	−0.351080	+	0.936345i	$0.614185\pi$
$198$	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	$0.750705\pi$
$200$	0	0
$201$	−9.41303e7	−0.817605
$202$	−2.70449e8	−2.30864
$203$	−5.34195e7	−0.448192
$204$	−4.42305e7	−0.364768
$205$	0	0
$206$	−2.88118e8	−2.29634
$207$	5.66433e7	0.443866
$208$	−5.45802e6	−0.0420546
$209$	4.71661e7	0.357370
$210$	0	0
$211$	−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
$215$	0	0
$216$	−2.40920e7	−0.162661
$217$	1.06330e8	0.706394
$218$	1.16341e8	0.760564
$219$	−1.60701e8	−1.03386
$220$	0	0
$221$	−1.49274e7	−0.0930274
$222$	−2.10738e8	−1.29273
$223$	−1.40215e8	−0.846694	−0.423347	−	0.905968i	$-0.639145\pi$
$223$	−1.40215e8	−0.846694	−0.423347	+	0.905968i	$0.639145\pi$
$224$	−7.26062e7	−0.431624
$225$	0	0
$226$	−2.66999e8	−1.53862
$227$	1.13463e8	0.643820	0.321910	−	0.946770i	$-0.395675\pi$
$227$	1.13463e8	0.643820	0.321910	+	0.946770i	$0.395675\pi$
$228$	−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	$0.588832\pi$
$230$	0	0
$231$	7.42362e7	0.396254
$232$	−1.90628e8	−1.00226
$233$	3.72064e8	1.92696	0.963478	−	0.267786i	$-0.0862921\pi$
$233$	3.72064e8	1.92696	0.963478	+	0.267786i	$0.0862921\pi$
$234$	−2.34359e7	−0.119571
$235$	0	0
$236$	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	$0.572332\pi$
$240$	0	0
$241$	−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
$245$	0	0
$246$	−1.73960e8	−0.745034
$247$	−1.05088e7	−0.0443726
$248$	3.79440e8	1.57965
$249$	−9.65931e7	−0.396505
$250$	0	0
$251$	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
$255$	0	0
$256$	−1.82427e8	−0.679595
$257$	5.97749e7	0.219661	0.109830	−	0.993950i	$-0.464969\pi$
$257$	5.97749e7	0.219661	0.109830	+	0.993950i	$0.464969\pi$
$258$	−3.52103e8	−1.27644
$259$	−1.48731e8	−0.531927
$260$	0	0
$261$	1.13536e8	0.395268
$262$	−2.20447e8	−0.757267
$263$	1.23562e8	0.418833	0.209416	−	0.977827i	$-0.432844\pi$
$263$	1.23562e8	0.418833	0.209416	+	0.977827i	$0.432844\pi$
$264$	2.64913e8	0.886113
$265$	0	0
$266$	−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	$0.265297\pi$
$270$	0	0
$271$	−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
$275$	0	0
$276$	4.11188e8	1.17722
$277$	−3.32836e8	−0.940915	−0.470458	−	0.882423i	$-0.655911\pi$
$277$	−3.32836e8	−0.940915	−0.470458	+	0.882423i	$0.655911\pi$
$278$	1.07178e8	0.299191
$279$	−2.25990e8	−0.622981
$280$	0	0
$281$	−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	$-0.534372\pi$
$283$	−8.21849e7	−0.215546	−0.107773	+	0.994176i	$0.534372\pi$
$284$	−5.62963e8	−1.45836
$285$	0	0
$286$	2.57698e8	0.651374
$287$	−1.22774e8	−0.306563
$288$	1.54315e8	0.380657
$289$	−3.40483e8	−0.829760
$290$	0	0
$291$	1.41940e8	0.337661
$292$	−1.16657e9	−2.74202
$293$	−3.71449e8	−0.862705	−0.431353	−	0.902183i	$-0.641964\pi$
$293$	−3.71449e8	−0.862705	−0.431353	+	0.902183i	$0.641964\pi$
$294$	−5.71774e7	−0.131223
$295$	0	0
$296$	−5.30748e8	−1.18951
$297$	−1.57779e8	−0.349463
$298$	5.59782e8	1.22536
$299$	1.38772e8	0.300229
$300$	0	0
$301$	−2.48501e8	−0.525225
$302$	2.57891e8	0.538780
$303$	4.05674e8	0.837776
$304$	1.79815e7	0.0367087
$305$	0	0
$306$	1.09674e8	0.218815
$307$	4.59953e8	0.907255	0.453628	−	0.891191i	$-0.350130\pi$
$307$	4.59953e8	0.907255	0.453628	+	0.891191i	$0.350130\pi$
$308$	5.38900e8	1.05095
$309$	4.32178e8	0.833312
$310$	0	0
$311$	−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	$-0.607550\pi$
$313$	−3.59666e8	−0.662971	−0.331485	+	0.943460i	$0.607550\pi$
$314$	−1.04593e9	−1.90655
$315$	0	0
$316$	−3.29457e8	−0.587346
$317$	−1.84942e8	−0.326083	−0.163041	−	0.986619i	$-0.552130\pi$
$317$	−1.84942e8	−0.326083	−0.163041	+	0.986619i	$0.552130\pi$
$318$	6.42482e7	0.112038
$319$	−1.24843e9	−2.15326
$320$	0	0
$321$	−1.43128e8	−0.241523
$322$	4.79720e8	0.800741
$323$	4.91785e7	0.0812019
$324$	1.04162e8	0.170139
$325$	0	0
$326$	−4.24808e8	−0.679095
$327$	−1.74512e8	−0.275999
$328$	−4.38121e8	−0.685544
$329$	6.01348e7	0.0930979
$330$	0	0
$331$	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
$335$	0	0
$336$	2.83016e7	0.0407028
$337$	−1.13073e9	−1.60936	−0.804680	−	0.593709i	$-0.797663\pi$
$337$	−1.13073e9	−1.60936	−0.804680	+	0.593709i	$0.797663\pi$
$338$	1.07206e9	1.51011
$339$	4.00499e8	0.558345
$340$	0	0
$341$	2.48496e9	3.39374
$342$	7.72098e7	0.104371
$343$	−4.03536e7	−0.0539949
$344$	−8.86778e8	−1.17452
$345$	0	0
$346$	−1.27741e9	−1.65792
$347$	−6.28829e8	−0.807940	−0.403970	−	0.914772i	$-0.632370\pi$
$347$	−6.28829e8	−0.807940	−0.403970	+	0.914772i	$0.632370\pi$
$348$	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	$0.306903\pi$
$350$	0	0
$351$	3.51538e7	0.0433908
$352$	−1.69683e9	−2.07366
$353$	1.13858e8	0.137769	0.0688843	−	0.997625i	$-0.478056\pi$
$353$	1.13858e8	0.137769	0.0688843	+	0.997625i	$0.478056\pi$
$354$	−1.28723e9	−1.54222
$355$	0	0
$356$	−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	$0.450483\pi$
$360$	0	0
$361$	−8.59250e8	−0.961268
$362$	4.06963e7	0.0450895
$363$	1.20877e9	1.32638
$364$	−1.20069e8	−0.130490
$365$	0	0
$366$	−4.06042e8	−0.432900
$367$	−1.35093e9	−1.42659	−0.713297	−	0.700862i	$-0.752799\pi$
$367$	−1.35093e9	−1.42659	−0.713297	+	0.700862i	$0.752799\pi$
$368$	−2.37451e8	−0.248374
$369$	2.60940e8	0.270363
$370$	0	0
$371$	4.53439e7	0.0461010
$372$	−1.64052e9	−1.65227
$373$	1.09767e9	1.09520	0.547599	−	0.836741i	$-0.315542\pi$
$373$	1.09767e9	1.09520	0.547599	+	0.836741i	$0.315542\pi$
$374$	−1.20596e9	−1.19202
$375$	0	0
$376$	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	$0.565260\pi$
$380$	0	0
$381$	−3.64564e8	−0.337704
$382$	1.45284e9	1.33351
$383$	−5.29318e8	−0.481416	−0.240708	−	0.970598i	$-0.577380\pi$
$383$	−5.29318e8	−0.481416	−0.240708	+	0.970598i	$0.577380\pi$
$384$	9.30103e8	0.838246
$385$	0	0
$386$	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	$0.501741\pi$
$390$	0	0
$391$	−6.49417e8	−0.549420
$392$	−1.44002e8	−0.120745
$393$	3.30670e8	0.274803
$394$	1.35625e9	1.11713
$395$	0	0
$396$	−1.14536e9	−0.926847
$397$	1.94087e9	1.55679	0.778393	−	0.627777i	$-0.216035\pi$
$397$	1.94087e9	1.55679	0.778393	+	0.627777i	$0.216035\pi$
$398$	2.83617e9	2.25498
$399$	5.44917e7	0.0429463
$400$	0	0
$401$	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
$405$	0	0
$406$	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	$0.596642\pi$
$410$	0	0
$411$	−3.34799e8	−0.237869
$412$	3.13729e9	2.21011
$413$	−9.08479e8	−0.634585
$414$	−1.01958e9	−0.706187
$415$	0	0
$416$	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	$0.689410\pi$
$420$	0	0
$421$	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
$425$	0	0
$426$	1.39592e9	0.874836
$427$	−2.86569e8	−0.178128
$428$	−1.03901e9	−0.640567
$429$	−3.86548e8	−0.236375
$430$	0	0
$431$	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	$-0.714782\pi$
$433$	−2.11064e9	−1.24942	−0.624708	+	0.780858i	$0.714782\pi$
$434$	−1.91394e9	−1.12387
$435$	0	0
$436$	−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	$0.741164\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	2.68693e8	0.148006
$443$	−2.17599e9	−1.18917	−0.594585	−	0.804033i	$-0.702684\pi$
$443$	−2.17599e9	−1.18917	−0.594585	+	0.804033i	$0.702684\pi$
$444$	2.29471e9	1.24419
$445$	0	0
$446$	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	$0.532329\pi$
$450$	0	0
$451$	−2.86926e9	−1.47283
$452$	2.90733e9	1.48084
$453$	−3.86836e8	−0.195517
$454$	−2.04234e9	−1.02431
$455$	0	0
$456$	1.94454e8	0.0960375
$457$	3.97621e9	1.94878	0.974390	−	0.224864i	$-0.0721937\pi$
$457$	3.97621e9	1.94878	0.974390	+	0.224864i	$0.0721937\pi$
$458$	1.80216e9	0.876526
$459$	−1.64511e8	−0.0794053
$460$	0	0
$461$	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	$-0.690845\pi$
$463$	−2.41022e9	−1.12855	−0.564277	+	0.825585i	$0.690845\pi$
$464$	−4.75948e8	−0.221180
$465$	0	0
$466$	−6.69715e9	−3.06577
$467$	−1.75978e9	−0.799555	−0.399777	−	0.916612i	$-0.630913\pi$
$467$	−1.75978e9	−0.799555	−0.399777	+	0.916612i	$0.630913\pi$
$468$	2.55191e8	0.115081
$469$	1.19580e9	0.535248
$470$	0	0
$471$	1.56889e9	0.691863
$472$	−3.24192e9	−1.41908
$473$	−5.80753e9	−2.52335
$474$	8.16919e8	0.352334
$475$	0	0
$476$	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	$0.836245\pi$
$480$	0	0
$481$	7.74442e8	0.317308
$482$	5.15461e9	2.09667
$483$	−7.19580e8	−0.290579
$484$	8.77474e9	3.51784
$485$	0	0
$486$	−2.58280e8	−0.102062
$487$	3.65982e9	1.43585	0.717924	−	0.696122i	$-0.245093\pi$
$487$	3.65982e9	1.43585	0.717924	+	0.696122i	$0.245093\pi$
$488$	−1.02263e9	−0.398334
$489$	6.37212e8	0.246435
$490$	0	0
$491$	−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
$495$	0	0
$496$	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	$0.462536\pi$
$500$	0	0
$501$	−7.17664e8	−0.254970
$502$	−9.96161e8	−0.351452
$503$	7.87014e8	0.275737	0.137868	−	0.990451i	$-0.455975\pi$
$503$	7.87014e8	0.275737	0.137868	+	0.990451i	$0.455975\pi$
$504$	3.06058e8	0.106487
$505$	0	0
$506$	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	$0.532631\pi$
$510$	0	0
$511$	2.04150e9	0.676823
$512$	−1.12568e9	−0.370656
$513$	−1.15815e8	−0.0378750
$514$	−1.07595e9	−0.349478
$515$	0	0
$516$	3.83401e9	1.22851
$517$	1.40537e9	0.447273
$518$	2.67716e9	0.846291
$519$	1.91611e9	0.601637
$520$	0	0
$521$	−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	$-0.717539\pi$
$523$	−4.13165e9	−1.26290	−0.631448	+	0.775418i	$0.717539\pi$
$524$	2.40042e9	0.728833
$525$	0	0
$526$	−2.22412e9	−0.666359
$527$	2.59098e9	0.771129
$528$	6.61416e8	0.195549
$529$	2.63246e9	0.773157
$530$	0	0
$531$	1.93085e9	0.559651
$532$	3.95570e8	0.113902
$533$	6.39284e8	0.182873
$534$	3.03985e9	0.863887
$535$	0	0
$536$	4.26724e9	1.19693
$537$	3.10666e9	0.865732
$538$	−7.72705e9	−2.13932
$539$	−9.43074e8	−0.259409
$540$	0	0
$541$	−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
$545$	0	0
$546$	2.97723e8	0.0782776
$547$	6.96991e8	0.182084	0.0910420	−	0.995847i	$-0.470980\pi$
$547$	6.96991e8	0.182084	0.0910420	+	0.995847i	$0.470980\pi$
$548$	−2.43040e9	−0.630877
$549$	6.09063e8	0.157094
$550$	0	0
$551$	−9.16386e8	−0.233371
$552$	−2.56783e9	−0.649800
$553$	5.76550e8	0.144977
$554$	5.99104e9	1.49699
$555$	0	0
$556$	−1.16705e9	−0.287957
$557$	−2.88908e9	−0.708381	−0.354191	−	0.935173i	$-0.615244\pi$
$557$	−2.88908e9	−0.708381	−0.354191	+	0.935173i	$0.615244\pi$
$558$	4.06782e9	0.991156
$559$	1.29394e9	0.313310
$560$	0	0
$561$	1.80894e9	0.432567
$562$	8.18022e9	1.94396
$563$	5.01488e9	1.18435	0.592176	−	0.805809i	$-0.298269\pi$
$563$	5.01488e9	1.18435	0.592176	+	0.805809i	$0.298269\pi$
$564$	−9.27793e8	−0.217758
$565$	0	0
$566$	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	$0.430221\pi$
$570$	0	0
$571$	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
$575$	0	0
$576$	−2.49250e9	−0.543447
$577$	−1.91539e9	−0.415089	−0.207545	−	0.978226i	$-0.566547\pi$
$577$	−1.91539e9	−0.415089	−0.207545	+	0.978226i	$0.566547\pi$
$578$	6.12869e9	1.32014
$579$	−1.91144e9	−0.409246
$580$	0	0
$581$	1.22709e9	0.259573
$582$	−2.55493e9	−0.537215
$583$	1.05970e9	0.221484
$584$	7.28510e9	1.51353
$585$	0	0
$586$	6.68608e9	1.37256
$587$	4.10449e9	0.837578	0.418789	−	0.908084i	$-0.362455\pi$
$587$	4.10449e9	0.837578	0.418789	+	0.908084i	$0.362455\pi$
$588$	6.22599e8	0.126295
$589$	1.82404e9	0.367816
$590$	0	0
$591$	−2.03438e9	−0.405392
$592$	−1.32514e9	−0.262503
$593$	3.09117e9	0.608740	0.304370	−	0.952554i	$-0.401554\pi$
$593$	3.09117e9	0.608740	0.304370	+	0.952554i	$0.401554\pi$
$594$	2.84002e9	0.555993
$595$	0	0
$596$	−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	$0.873172\pi$
$600$	0	0
$601$	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
$605$	0	0
$606$	−7.30213e9	−1.33289
$607$	−2.11212e9	−0.383316	−0.191658	−	0.981462i	$-0.561387\pi$
$607$	−2.11212e9	−0.383316	−0.191658	+	0.981462i	$0.561387\pi$
$608$	−1.24553e9	−0.224745
$609$	−1.44233e9	−0.258764
$610$	0	0
$611$	−3.13122e8	−0.0555352
$612$	−1.19422e9	−0.210599
$613$	2.41937e9	0.424220	0.212110	−	0.977246i	$-0.431966\pi$
$613$	2.41937e9	0.424220	0.212110	+	0.977246i	$0.431966\pi$
$614$	−8.27916e9	−1.44343
$615$	0	0
$616$	−3.36537e9	−0.580097
$617$	3.53370e9	0.605664	0.302832	−	0.953044i	$-0.402068\pi$
$617$	3.53370e9	0.605664	0.302832	+	0.953044i	$0.402068\pi$
$618$	−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	$0.504632\pi$
$620$	0	0
$621$	1.52937e9	0.256266
$622$	8.90951e9	1.48452
$623$	2.14541e9	0.355469
$624$	−1.47366e8	−0.0242802
$625$	0	0
$626$	6.47399e9	1.05478
$627$	1.27349e9	0.206328
$628$	1.13890e10	1.83496
$629$	−3.62418e9	−0.580674
$630$	0	0
$631$	−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
$635$	0	0
$636$	−6.99592e8	−0.107831
$637$	2.10121e8	0.0322093
$638$	2.24717e10	3.42581
$639$	−2.09388e9	−0.317467
$640$	0	0
$641$	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	$-0.458530\pi$
$643$	1.75155e9	0.259827	0.129913	+	0.991525i	$0.458530\pi$
$644$	−5.22362e9	−0.770674
$645$	0	0
$646$	−8.85212e8	−0.129191
$647$	−3.04738e8	−0.0442346	−0.0221173	−	0.999755i	$-0.507041\pi$
$647$	−3.04738e8	−0.0442346	−0.0221173	+	0.999755i	$0.507041\pi$
$648$	−6.50484e8	−0.0939126
$649$	−2.12314e10	−3.04875
$650$	0	0
$651$	2.87091e9	0.407837
$652$	4.62568e9	0.653596
$653$	−1.91092e9	−0.268562	−0.134281	−	0.990943i	$-0.542873\pi$
$653$	−1.91092e9	−0.268562	−0.134281	+	0.990943i	$0.542873\pi$
$654$	3.14121e9	0.439112
$655$	0	0
$656$	−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	$0.441855\pi$
$660$	0	0
$661$	−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
$665$	0	0
$666$	−5.68994e9	−0.746359
$667$	1.21012e10	1.57902
$668$	−5.20971e9	−0.676232
$669$	−3.78579e9	−0.488839
$670$	0	0
$671$	−6.69719e9	−0.855783
$672$	−1.96037e9	−0.249198
$673$	−2.65330e9	−0.335531	−0.167766	−	0.985827i	$-0.553655\pi$
$673$	−2.65330e9	−0.335531	−0.167766	+	0.985827i	$0.553655\pi$
$674$	2.03531e10	2.56048
$675$	0	0
$676$	−1.16735e10	−1.45341
$677$	−9.29262e9	−1.15101	−0.575504	−	0.817799i	$-0.695194\pi$
$677$	−9.29262e9	−1.15101	−0.575504	+	0.817799i	$0.695194\pi$
$678$	−7.20898e9	−0.888322
$679$	−1.80317e9	−0.221051
$680$	0	0
$681$	3.06351e9	0.371710
$682$	−4.47293e10	−5.39941
$683$	−1.52018e10	−1.82567	−0.912837	−	0.408325i	$-0.866113\pi$
$683$	−1.52018e10	−1.82567	−0.912837	+	0.408325i	$0.866113\pi$
$684$	−8.40729e8	−0.100452
$685$	0	0
$686$	7.26365e8	0.0859054
$687$	−2.70324e9	−0.318080
$688$	−2.21405e9	−0.259195
$689$	−2.36106e8	−0.0275004
$690$	0	0
$691$	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
$695$	0	0
$696$	−5.14696e9	−0.578654
$697$	−2.99168e9	−0.334657
$698$	−1.62984e10	−1.81406
$699$	1.00457e10	1.11253
$700$	0	0
$701$	−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
$705$	0	0
$706$	−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	$0.757092\pi$
$710$	0	0
$711$	−1.22538e9	−0.127858
$712$	7.65591e9	0.794908
$713$	−2.40870e10	−2.48868
$714$	−1.39326e9	−0.143248
$715$	0	0
$716$	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	$0.223046\pi$
$720$	0	0
$721$	−5.49026e9	−0.545531
$722$	1.54665e10	1.52937
$723$	−7.73191e9	−0.760857
$724$	−4.43138e8	−0.0433964
$725$	0	0
$726$	−2.17578e10	−2.11026
$727$	−8.49661e9	−0.820116	−0.410058	−	0.912059i	$-0.634492\pi$
$727$	−8.49661e9	−0.820116	−0.410058	+	0.912059i	$0.634492\pi$
$728$	7.49820e8	0.0720273
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−6.05530e9	−0.573357
$732$	4.42135e9	0.416645
$733$	−1.02983e10	−0.965828	−0.482914	−	0.875668i	$-0.660422\pi$
$733$	−1.02983e10	−0.965828	−0.482914	+	0.875668i	$0.660422\pi$
$734$	2.43167e10	2.26970
$735$	0	0
$736$	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	$0.127998\pi$
$740$	0	0
$741$	−2.83738e8	−0.0256185
$742$	−8.16190e8	−0.0733462
$743$	−1.20966e10	−1.08194	−0.540971	−	0.841042i	$-0.681943\pi$
$743$	−1.20966e10	−1.08194	−0.540971	+	0.841042i	$0.681943\pi$
$744$	1.02449e10	0.912014
$745$	0	0
$746$	−1.97581e10	−1.74245
$747$	−2.60801e9	−0.228922
$748$	1.31316e10	1.14726
$749$	1.81826e9	0.158114
$750$	0	0
$751$	−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
$755$	0	0
$756$	−1.32325e9	−0.111382
$757$	−1.27420e10	−1.06758	−0.533792	−	0.845616i	$-0.679233\pi$
$757$	−1.27420e10	−1.06758	−0.533792	+	0.845616i	$0.679233\pi$
$758$	7.76765e9	0.647810
$759$	−1.68168e10	−1.39603
$760$	0	0
$761$	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
$765$	0	0
$766$	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	$0.313071\pi$
$770$	0	0
$771$	1.61392e9	0.126821
$772$	−1.38756e10	−1.08540
$773$	1.26246e10	0.983083	0.491541	−	0.870854i	$-0.336434\pi$
$773$	1.26246e10	0.983083	0.491541	+	0.870854i	$0.336434\pi$
$774$	−9.50678e9	−0.736954
$775$	0	0
$776$	−6.43463e9	−0.494320
$777$	−4.01574e9	−0.307108
$778$	2.28618e8	0.0174053
$779$	−2.10613e9	−0.159626
$780$	0	0
$781$	2.30240e10	1.72943
$782$	1.16895e10	0.874122
$783$	3.06547e9	0.228208
$784$	−3.59535e8	−0.0266462
$785$	0	0
$786$	−5.95206e9	−0.437209
$787$	2.29786e10	1.68040	0.840199	−	0.542278i	$-0.182438\pi$
$787$	2.29786e10	1.68040	0.840199	+	0.542278i	$0.182438\pi$
$788$	−1.47681e10	−1.07518
$789$	3.33618e9	0.241813
$790$	0	0
$791$	−5.08782e9	−0.365523
$792$	7.15264e9	0.511598
$793$	1.49216e9	0.106258
$794$	−3.49356e10	−2.47683
$795$	0	0
$796$	−3.08828e10	−2.17030
$797$	−5.78646e8	−0.0404864	−0.0202432	−	0.999795i	$-0.506444\pi$
$797$	−5.78646e8	−0.0404864	−0.0202432	+	0.999795i	$0.506444\pi$
$798$	−9.80851e8	−0.0683271
$799$	1.46532e9	0.101630
$800$	0	0
$801$	−4.55977e9	−0.313494
$802$	−2.32803e10	−1.59360
$803$	4.77103e10	3.25168
$804$	−1.84495e10	−1.25196
$805$	0	0
$806$	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	$0.603020\pi$
$810$	0	0
$811$	−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
$815$	0	0
$816$	6.89635e8	0.0444328
$817$	−4.26291e9	−0.273482
$818$	1.48922e10	0.951311
$819$	−4.46584e8	−0.0284060
$820$	0	0
$821$	−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	$-0.346381\pi$
$823$	1.48433e10	0.928180	0.464090	+	0.885788i	$0.346381\pi$
$824$	−1.95920e10	−1.21993
$825$	0	0
$826$	1.63526e10	1.00962
$827$	2.18669e10	1.34436	0.672182	−	0.740386i	$-0.265357\pi$
$827$	2.18669e10	1.34436	0.672182	+	0.740386i	$0.265357\pi$
$828$	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	$0.662413\pi$
$830$	0	0
$831$	−8.98656e9	−0.543238
$832$	−6.10646e9	−0.367585
$833$	−9.83310e8	−0.0589431
$834$	2.89381e9	0.172738
$835$	0	0
$836$	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	$0.569155\pi$
$840$	0	0
$841$	7.00569e9	0.406130
$842$	−2.24411e10	−1.29555
$843$	−1.22703e10	−0.705439
$844$	−4.35352e10	−2.49254
$845$	0	0
$846$	2.30055e9	0.130628
$847$	−1.53558e10	−0.868321
$848$	4.03997e8	0.0227506
$849$	−2.21899e9	−0.124445
$850$	0	0
$851$	3.36921e10	1.87402
$852$	−1.52000e10	−0.841986
$853$	−2.99861e10	−1.65424	−0.827119	−	0.562027i	$-0.810022\pi$
$853$	−2.99861e10	−1.65424	−0.827119	+	0.562027i	$0.810022\pi$
$854$	5.15824e9	0.283400
$855$	0	0
$856$	6.48848e9	0.353578
$857$	−3.07594e10	−1.66934	−0.834671	−	0.550749i	$-0.814342\pi$
$857$	−3.07594e10	−1.66934	−0.834671	+	0.550749i	$0.814342\pi$
$858$	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	$0.647919\pi$
$860$	0	0
$861$	−3.31490e9	−0.176994
$862$	−2.55252e10	−1.35736
$863$	−1.98562e10	−1.05162	−0.525809	−	0.850603i	$-0.676237\pi$
$863$	−1.98562e10	−1.05162	−0.525809	+	0.850603i	$0.676237\pi$
$864$	4.16650e9	0.219772
$865$	0	0
$866$	3.79916e10	1.98781
$867$	−9.19303e9	−0.479062
$868$	2.08407e10	1.08167
$869$	1.34741e10	0.696516
$870$	0	0
$871$	−6.22655e9	−0.319289
$872$	7.91120e9	0.404050
$873$	3.83239e9	0.194949
$874$	8.22936e9	0.416943
$875$	0	0
$876$	−3.14974e10	−1.58311
$877$	6.35922e9	0.318350	0.159175	−	0.987250i	$-0.449117\pi$
$877$	6.35922e9	0.318350	0.159175	+	0.987250i	$0.449117\pi$
$878$	4.38548e10	2.18669
$879$	−1.00291e10	−0.498083
$880$	0	0
$881$	−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	$-0.518952\pi$
$883$	−2.43465e9	−0.119007	−0.0595036	+	0.998228i	$0.518952\pi$
$884$	−2.92577e9	−0.142448
$885$	0	0
$886$	3.91678e10	1.89196
$887$	3.32171e10	1.59819	0.799097	−	0.601202i	$-0.205311\pi$
$887$	3.32171e10	1.59819	0.799097	+	0.601202i	$0.205311\pi$
$888$	−1.43302e10	−0.686763
$889$	4.63131e9	0.221079
$890$	0	0
$891$	−4.26003e9	−0.201763
$892$	−2.74821e10	−1.29650
$893$	1.03158e9	0.0484757
$894$	1.51141e10	0.707460
$895$	0	0
$896$	−1.18158e10	−0.548761
$897$	3.74685e9	0.173338
$898$	7.00106e9	0.322624
$899$	−4.82800e10	−2.21620
$900$	0	0
$901$	1.10491e9	0.0503258
$902$	5.16467e10	2.34326
$903$	−6.70952e9	−0.303239
$904$	−1.81560e10	−0.817391
$905$	0	0
$906$	6.96305e9	0.311065
$907$	1.71277e10	0.762210	0.381105	−	0.924532i	$-0.375544\pi$
$907$	1.71277e10	0.762210	0.381105	+	0.924532i	$0.375544\pi$
$908$	2.22388e10	0.985850
$909$	1.09532e10	0.483690
$910$	0	0
$911$	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
$915$	0	0
$916$	−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	$0.415780\pi$
$920$	0	0
$921$	1.24187e10	0.523804
$922$	−6.51096e10	−2.73581
$923$	−5.12986e9	−0.214733
$924$	1.45503e10	0.606764
$925$	0	0
$926$	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	$0.697735\pi$
$930$	0	0
$931$	−6.92247e8	−0.0281149
$932$	7.29246e10	2.95065
$933$	−1.33643e10	−0.538715
$934$	3.16760e10	1.27208
$935$	0	0
$936$	−1.59364e9	−0.0635221
$937$	4.08626e10	1.62270	0.811349	−	0.584562i	$-0.198734\pi$
$937$	4.08626e10	1.62270	0.811349	+	0.584562i	$0.198734\pi$
$938$	−2.15245e10	−0.851574
$939$	−9.71098e9	−0.382766
$940$	0	0
$941$	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
$945$	0	0
$946$	1.04536e11	4.01462
$947$	2.32859e10	0.890980	0.445490	−	0.895287i	$-0.353030\pi$
$947$	2.32859e10	0.890980	0.445490	+	0.895287i	$0.353030\pi$
$948$	−8.89534e9	−0.339105
$949$	−1.06301e10	−0.403742
$950$	0	0
$951$	−4.99343e9	−0.188264
$952$	−3.50896e9	−0.131810
$953$	−2.98085e10	−1.11562	−0.557809	−	0.829970i	$-0.688358\pi$
$953$	−2.98085e10	−1.11562	−0.557809	+	0.829970i	$0.688358\pi$
$954$	1.73470e9	0.0646853
$955$	0	0
$956$	−1.86387e10	−0.689942
$957$	−3.37076e10	−1.24318
$958$	7.53835e10	2.77011
$959$	4.25319e9	0.155722
$960$	0	0
$961$	6.85874e10	2.49294
$962$	−1.39400e10	−0.504834
$963$	−3.86446e9	−0.139443
$964$	−5.61280e10	−2.01795
$965$	0	0
$966$	1.29524e10	0.462308
$967$	7.13076e9	0.253596	0.126798	−	0.991929i	$-0.459530\pi$
$967$	7.13076e9	0.253596	0.126798	+	0.991929i	$0.459530\pi$
$968$	−5.47974e10	−1.94176
$969$	1.32782e9	0.0468819
$970$	0	0
$971$	−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
$975$	0	0
$976$	−2.55322e9	−0.0879051
$977$	−2.77442e10	−0.951789	−0.475895	−	0.879502i	$-0.657876\pi$
$977$	−2.77442e10	−0.951789	−0.475895	+	0.879502i	$0.657876\pi$
$978$	−1.14698e10	−0.392076
$979$	5.01387e10	1.70779
$980$	0	0
$981$	−4.71182e9	−0.159348
$982$	6.54505e10	2.20558
$983$	3.01277e9	0.101164	0.0505822	−	0.998720i	$-0.483892\pi$
$983$	3.01277e9	0.101164	0.0505822	+	0.998720i	$0.483892\pi$
$984$	−1.18293e10	−0.395799
$985$	0	0
$986$	2.34304e10	0.778415
$987$	1.62364e9	0.0537501
$988$	−2.05973e9	−0.0679455
$989$	5.62930e10	1.85041
$990$	0	0
$991$	−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
$995$	0	0
$996$	−1.89323e10	−0.607148
$997$	2.46784e10	0.788649	0.394325	−	0.918971i	$-0.370979\pi$
$997$	2.46784e10	0.788649	0.394325	+	0.918971i	$0.370979\pi$
$998$	−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	525.8.a.a.1.1		1
5.4	even	2		105.8.a.b.1.1	✓	1
15.14	odd	2		315.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.b.1.1	✓	1	5.4	even	2
315.8.a.a.1.1		1	15.14	odd	2
525.8.a.a.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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