LMFDB - Embedded newform 315.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,Mod(1,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$98.4012830275$
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	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-18.0000 q^{2} +196.000 q^{4} +125.000 q^{5} +343.000 q^{7} -1224.00 q^{8} +O(q^{10})$

\(q-18.0000 q^{2} +196.000 q^{4} +125.000 q^{5} +343.000 q^{7} -1224.00 q^{8} -2250.00 q^{10} +8016.00 q^{11} -1786.00 q^{13} -6174.00 q^{14} -3056.00 q^{16} -8358.00 q^{17} -5884.00 q^{19} +24500.0 q^{20} -144288. q^{22} +77700.0 q^{23} +15625.0 q^{25} +32148.0 q^{26} +67228.0 q^{28} -155742. q^{29} -310000. q^{31} +211680. q^{32} +150444. q^{34} +42875.0 q^{35} -433618. q^{37} +105912. q^{38} -153000. q^{40} -357942. q^{41} -724492. q^{43} +1.57114e6 q^{44} -1.39860e6 q^{46} -175320. q^{47} +117649. q^{49} -281250. q^{50} -350056. q^{52} -132198. q^{53} +1.00200e6 q^{55} -419832. q^{56} +2.80336e6 q^{58} -2.64863e6 q^{59} +835478. q^{61} +5.58000e6 q^{62} -3.41907e6 q^{64} -223250. q^{65} +3.48631e6 q^{67} -1.63817e6 q^{68} -771750. q^{70} +2.87226e6 q^{71} +5.95188e6 q^{73} +7.80512e6 q^{74} -1.15326e6 q^{76} +2.74949e6 q^{77} -1.68090e6 q^{79} -382000. q^{80} +6.44296e6 q^{82} -3.57752e6 q^{83} -1.04475e6 q^{85} +1.30409e7 q^{86} -9.81158e6 q^{88} +6.25483e6 q^{89} -612598. q^{91} +1.52292e7 q^{92} +3.15576e6 q^{94} -735500. q^{95} -5.25705e6 q^{97} -2.11768e6 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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	$0.792789\pi$
$3$	0	0
$3$	0	0
$4$	196.000	1.53125
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−1224.00	−0.845214
$9$	0	0
$10$	−2250.00	−0.711512
$11$	8016.00	1.81586	0.907932	−	0.419118i	$-0.137660\pi$
$11$	8016.00	1.81586	0.907932	+	0.419118i	$0.137660\pi$
$12$	0	0
$13$	−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
$15$	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	$0.566143\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−144288.	−2.88902
$23$	77700.0	1.33160	0.665800	−	0.746131i	$-0.268091\pi$
$23$	77700.0	1.33160	0.665800	+	0.746131i	$0.268091\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	32148.0	0.358713
$27$	0	0
$28$	67228.0	0.578758
$29$	−155742.	−1.18580	−0.592902	−	0.805275i	$-0.702018\pi$
$29$	−155742.	−1.18580	−0.592902	+	0.805275i	$0.702018\pi$
$30$	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
$33$	0	0
$34$	150444.	0.656445
$35$	42875.0	0.169031
$36$	0	0
$37$	−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
$39$	0	0
$40$	−153000.	−0.377991
$41$	−357942.	−0.811090	−0.405545	−	0.914075i	$-0.632918\pi$
$41$	−357942.	−0.811090	−0.405545	+	0.914075i	$0.632918\pi$
$42$	0	0
$43$	−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
$45$	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	$0.539302\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−281250.	−0.318198
$51$	0	0
$52$	−350056.	−0.345244
$53$	−132198.	−0.121972	−0.0609859	−	0.998139i	$-0.519424\pi$
$53$	−132198.	−0.121972	−0.0609859	+	0.998139i	$0.519424\pi$
$54$	0	0
$55$	1.00200e6	0.812079
$56$	−419832.	−0.319461
$57$	0	0
$58$	2.80336e6	1.88660
$59$	−2.64863e6	−1.67895	−0.839477	−	0.543395i	$-0.817139\pi$
$59$	−2.64863e6	−1.67895	−0.839477	+	0.543395i	$0.817139\pi$
$60$	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
$63$	0	0
$64$	−3.41907e6	−1.63034
$65$	−223250.	−0.100831
$66$	0	0
$67$	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
$69$	0	0
$70$	−771750.	−0.268926
$71$	2.87226e6	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$71$	2.87226e6	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$72$	0	0
$73$	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
$75$	0	0
$76$	−1.15326e6	−0.301357
$77$	2.74949e6	0.686332
$78$	0	0
$79$	−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
$81$	0	0
$82$	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	$0.611573\pi$
$84$	0	0
$85$	−1.04475e6	−0.184521
$86$	1.30409e7	2.21086
$87$	0	0
$88$	−9.81158e6	−1.53479
$89$	6.25483e6	0.940481	0.470241	−	0.882538i	$-0.344167\pi$
$89$	6.25483e6	0.940481	0.470241	+	0.882538i	$0.344167\pi$
$90$	0	0
$91$	−612598.	−0.0852179
$92$	1.52292e7	2.03901
$93$	0	0
$94$	3.15576e6	0.391883
$95$	−735500.	−0.0880137
$96$	0	0
$97$	−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
$99$	0	0
$100$	3.06250e6	0.306250
$101$	−1.50250e7	−1.45107	−0.725535	−	0.688185i	$-0.758408\pi$
$101$	−1.50250e7	−1.45107	−0.725535	+	0.688185i	$0.758408\pi$
$102$	0	0
$103$	−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
$105$	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	$0.567074\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	−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	$0.339350\pi$
$114$	0	0
$115$	9.71250e6	0.595509
$116$	−3.05254e7	−1.81576
$117$	0	0
$118$	4.76753e7	2.67120
$119$	−2.86679e6	−0.155949
$120$	0	0
$121$	4.47691e7	2.29736
$122$	−1.50386e7	−0.749805
$123$	0	0
$124$	−6.07600e7	−2.86182
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	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
$129$	0	0
$130$	4.01850e6	0.160421
$131$	−1.22470e7	−0.475972	−0.237986	−	0.971269i	$-0.576487\pi$
$131$	−1.22470e7	−0.475972	−0.237986	+	0.971269i	$0.576487\pi$
$132$	0	0
$133$	−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	$0.566045\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−5.17007e7	−1.51526
$143$	−1.43166e7	−0.409414
$144$	0	0
$145$	−1.94678e7	−0.530307
$146$	−1.07134e8	−2.84900
$147$	0	0
$148$	−8.49891e7	−2.15500
$149$	3.10990e7	0.770185	0.385092	−	0.922878i	$-0.374170\pi$
$149$	3.10990e7	0.770185	0.385092	+	0.922878i	$0.374170\pi$
$150$	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
$153$	0	0
$154$	−4.94908e7	−1.09195
$155$	−3.87500e7	−0.835816
$156$	0	0
$157$	−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
$159$	0	0
$160$	2.64600e7	0.510705
$161$	2.66511e7	0.503297
$162$	0	0
$163$	−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
$165$	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	$0.570870\pi$
$168$	0	0
$169$	−5.95587e7	−0.949165
$170$	1.88055e7	0.293571
$171$	0	0
$172$	−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	$0.325547\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	−2.44969e7	−0.338701
$177$	0	0
$178$	−1.12587e8	−1.49630
$179$	−1.15061e8	−1.49949	−0.749746	−	0.661726i	$-0.769824\pi$
$179$	−1.15061e8	−1.49949	−0.749746	+	0.661726i	$0.769824\pi$
$180$	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
$183$	0	0
$184$	−9.51048e7	−1.12549
$185$	−5.42022e7	−0.629385
$186$	0	0
$187$	−6.69977e7	−0.749229
$188$	−3.43627e7	−0.377168
$189$	0	0
$190$	1.32390e7	0.140029
$191$	8.07135e7	0.838165	0.419083	−	0.907948i	$-0.362352\pi$
$191$	8.07135e7	0.838165	0.419083	+	0.907948i	$0.362352\pi$
$192$	0	0
$193$	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
$195$	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	$0.614185\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	2.70449e8	2.30864
$203$	−5.34195e7	−0.448192
$204$	0	0
$205$	−4.47428e7	−0.362731
$206$	2.88118e8	2.29634
$207$	0	0
$208$	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
$213$	0	0
$214$	9.54189e7	0.665558
$215$	−9.05615e7	−0.621454
$216$	0	0
$217$	−1.06330e8	−0.706394
$218$	1.16341e8	0.760564
$219$	0	0
$220$	1.96392e8	1.24350
$221$	1.49274e7	0.0930274
$222$	0	0
$223$	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
$225$	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	$0.395675\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	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	$0.0862921\pi$
$234$	0	0
$235$	−2.19150e7	−0.110155
$236$	−5.19131e8	−2.57090
$237$	0	0
$238$	5.16023e7	0.248113
$239$	9.50953e7	0.450574	0.225287	−	0.974292i	$-0.427668\pi$
$239$	9.50953e7	0.450574	0.225287	+	0.974292i	$0.427668\pi$
$240$	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
$243$	0	0
$244$	1.63754e8	0.721650
$245$	1.47061e7	0.0638877
$246$	0	0
$247$	1.05088e7	0.0443726
$248$	3.79440e8	1.57965
$249$	0	0
$250$	−3.51562e7	−0.142302
$251$	−5.53423e7	−0.220902	−0.110451	−	0.993882i	$-0.535229\pi$
$251$	−5.53423e7	−0.220902	−0.110451	+	0.993882i	$0.535229\pi$
$252$	0	0
$253$	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	$0.464969\pi$
$258$	0	0
$259$	−1.48731e8	−0.531927
$260$	−4.37570e7	−0.154398
$261$	0	0
$262$	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	$0.432844\pi$
$264$	0	0
$265$	−1.65248e7	−0.0545474
$266$	3.63278e7	0.118346
$267$	0	0
$268$	6.83316e8	2.16845
$269$	−4.29280e8	−1.34465	−0.672323	−	0.740258i	$-0.734703\pi$
$269$	−4.29280e8	−1.34465	−0.672323	+	0.740258i	$0.734703\pi$
$270$	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
$273$	0	0
$274$	2.23200e8	0.655491
$275$	1.25250e8	0.363173
$276$	0	0
$277$	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
$279$	0	0
$280$	−5.24790e7	−0.142867
$281$	4.54457e8	1.22186	0.610928	−	0.791686i	$-0.290796\pi$
$281$	4.54457e8	1.22186	0.610928	+	0.791686i	$0.290796\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	2.57698e8	0.651374
$287$	−1.22774e8	−0.306563
$288$	0	0
$289$	−3.40483e8	−0.829760
$290$	3.50420e8	0.843714
$291$	0	0
$292$	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	$0.641964\pi$
$294$	0	0
$295$	−3.31078e8	−0.750851
$296$	5.30748e8	1.18951
$297$	0	0
$298$	−5.59782e8	−1.22536
$299$	−1.38772e8	−0.300229
$300$	0	0
$301$	−2.48501e8	−0.525225
$302$	2.57891e8	0.538780
$303$	0	0
$304$	1.79815e7	0.0367087
$305$	1.04435e8	0.210764
$306$	0	0
$307$	−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
$309$	0	0
$310$	6.97500e8	1.32978
$311$	4.94973e8	0.933082	0.466541	−	0.884499i	$-0.345500\pi$
$311$	4.94973e8	0.933082	0.466541	+	0.884499i	$0.345500\pi$
$312$	0	0
$313$	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
$315$	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	$0.552130\pi$
$318$	0	0
$319$	−1.24843e9	−2.15326
$320$	−4.27384e8	−0.729110
$321$	0	0
$322$	−4.79720e8	−0.800741
$323$	4.91785e7	0.0812019
$324$	0	0
$325$	−2.79063e7	−0.0450931
$326$	4.24808e8	0.679095
$327$	0	0
$328$	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
$333$	0	0
$334$	4.78443e8	0.702615
$335$	4.35788e8	0.633314
$336$	0	0
$337$	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
$339$	0	0
$340$	−2.04771e8	−0.282548
$341$	−2.48496e9	−3.39374
$342$	0	0
$343$	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	$0.632370\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	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	$0.478056\pi$
$354$	0	0
$355$	3.59033e8	0.425926
$356$	1.22595e9	1.44011
$357$	0	0
$358$	2.07111e9	2.38568
$359$	−2.71654e8	−0.309874	−0.154937	−	0.987924i	$-0.549517\pi$
$359$	−2.71654e8	−0.309874	−0.154937	+	0.987924i	$0.549517\pi$
$360$	0	0
$361$	−8.59250e8	−0.961268
$362$	4.06963e7	0.0450895
$363$	0	0
$364$	−1.20069e8	−0.130490
$365$	7.43985e8	0.800828
$366$	0	0
$367$	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
$369$	0	0
$370$	9.75640e8	1.00135
$371$	−4.53439e7	−0.0461010
$372$	0	0
$373$	−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
$375$	0	0
$376$	2.14592e8	0.208188
$377$	2.78155e8	0.267358
$378$	0	0
$379$	−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
$381$	0	0
$382$	−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	$0.577380\pi$
$384$	0	0
$385$	3.43686e8	0.306937
$386$	−1.27429e9	−1.12775
$387$	0	0
$388$	−1.03038e9	−0.895545
$389$	1.27010e7	0.0109399	0.00546996	−	0.999985i	$-0.498259\pi$
$389$	1.27010e7	0.0109399	0.00546996	+	0.999985i	$0.498259\pi$
$390$	0	0
$391$	−6.49417e8	−0.549420
$392$	−1.44002e8	−0.120745
$393$	0	0
$394$	1.35625e9	1.11713
$395$	−2.10113e8	−0.171539
$396$	0	0
$397$	−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
$399$	0	0
$400$	−4.77500e7	−0.0373047
$401$	−1.29335e9	−1.00164	−0.500819	−	0.865552i	$-0.666968\pi$
$401$	−1.29335e9	−1.00164	−0.500819	+	0.865552i	$0.666968\pi$
$402$	0	0
$403$	5.53660e8	0.421382
$404$	−2.94489e9	−2.22195
$405$	0	0
$406$	9.61551e8	0.713069
$407$	−3.47588e9	−2.55555
$408$	0	0
$409$	−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
$411$	0	0
$412$	−3.13729e9	−2.21011
$413$	−9.08479e8	−0.634585
$414$	0	0
$415$	−4.47190e8	−0.307131
$416$	−3.78060e8	−0.257475
$417$	0	0
$418$	8.48991e8	0.568573
$419$	1.68808e9	1.12110	0.560550	−	0.828120i	$-0.310590\pi$
$419$	1.68808e9	1.12110	0.560550	+	0.828120i	$0.310590\pi$
$420$	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
$423$	0	0
$424$	1.61810e8	0.103092
$425$	−1.30594e8	−0.0825204
$426$	0	0
$427$	2.86569e8	0.178128
$428$	−1.03901e9	−0.640567
$429$	0	0
$430$	1.63011e9	0.988727
$431$	−1.41807e9	−0.853152	−0.426576	−	0.904452i	$-0.640280\pi$
$431$	−1.41807e9	−0.853152	−0.426576	+	0.904452i	$0.640280\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	−1.26683e9	−0.732006
$437$	−4.57187e8	−0.262065
$438$	0	0
$439$	−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
$441$	0	0
$442$	−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	$0.702684\pi$
$444$	0	0
$445$	7.81853e8	0.420596
$446$	−2.52386e9	−1.34708
$447$	0	0
$448$	−1.17274e9	−0.616211
$449$	3.88948e8	0.202782	0.101391	−	0.994847i	$-0.467671\pi$
$449$	3.88948e8	0.202782	0.101391	+	0.994847i	$0.467671\pi$
$450$	0	0
$451$	−2.86926e9	−1.47283
$452$	2.90733e9	1.48084
$453$	0	0
$454$	−2.04234e9	−1.02431
$455$	−7.65747e7	−0.0381106
$456$	0	0
$457$	−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
$459$	0	0
$460$	1.90365e9	0.911874
$461$	−3.61720e9	−1.71957	−0.859783	−	0.510659i	$-0.829401\pi$
$461$	−3.61720e9	−1.71957	−0.859783	+	0.510659i	$0.829401\pi$
$462$	0	0
$463$	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
$465$	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	$0.630913\pi$
$468$	0	0
$469$	1.19580e9	0.535248
$470$	3.94470e8	0.175255
$471$	0	0
$472$	3.24192e9	1.41908
$473$	−5.80753e9	−2.52335
$474$	0	0
$475$	−9.19375e7	−0.0393609
$476$	−5.61892e8	−0.238797
$477$	0	0
$478$	−1.71172e9	−0.716859
$479$	4.18797e9	1.74112	0.870562	−	0.492059i	$-0.163755\pi$
$479$	4.18797e9	1.74112	0.870562	+	0.492059i	$0.163755\pi$
$480$	0	0
$481$	7.74442e8	0.317308
$482$	5.15461e9	2.09667
$483$	0	0
$484$	8.77474e9	3.51784
$485$	−6.57132e8	−0.261551
$486$	0	0
$487$	−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
$489$	0	0
$490$	−2.64710e8	−0.101645
$491$	3.63614e9	1.38629	0.693146	−	0.720797i	$-0.256224\pi$
$491$	3.63614e9	1.38629	0.693146	+	0.720797i	$0.256224\pi$
$492$	0	0
$493$	1.30169e9	0.489265
$494$	−1.89159e8	−0.0705964
$495$	0	0
$496$	9.47360e8	0.348601
$497$	9.85185e8	0.359973
$498$	0	0
$499$	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
$501$	0	0
$502$	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	$0.455975\pi$
$504$	0	0
$505$	−1.87812e9	−0.648938
$506$	−1.12112e10	−3.84702
$507$	0	0
$508$	2.64646e9	0.895659
$509$	6.08921e8	0.204667	0.102334	−	0.994750i	$-0.467369\pi$
$509$	6.08921e8	0.204667	0.102334	+	0.994750i	$0.467369\pi$
$510$	0	0
$511$	2.04150e9	0.676823
$512$	−1.12568e9	−0.370656
$513$	0	0
$514$	−1.07595e9	−0.349478
$515$	−2.00082e9	−0.645481
$516$	0	0
$517$	−1.40537e9	−0.447273
$518$	2.67716e9	0.846291
$519$	0	0
$520$	2.73258e8	0.0852239
$521$	1.12396e9	0.348193	0.174096	−	0.984729i	$-0.444300\pi$
$521$	1.12396e9	0.348193	0.174096	+	0.984729i	$0.444300\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	−2.22412e9	−0.666359
$527$	2.59098e9	0.771129
$528$	0	0
$529$	2.63246e9	0.773157
$530$	2.97446e8	0.0867844
$531$	0	0
$532$	−3.95570e8	−0.113902
$533$	6.39284e8	0.182873
$534$	0	0
$535$	−6.62631e8	−0.187083
$536$	−4.26724e9	−1.19693
$537$	0	0
$538$	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
$543$	0	0
$544$	−1.76922e9	−0.471179
$545$	−8.07925e8	−0.213788
$546$	0	0
$547$	−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
$549$	0	0
$550$	−2.25450e9	−0.577804
$551$	9.16386e8	0.233371
$552$	0	0
$553$	−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	$0.615244\pi$
$558$	0	0
$559$	1.29394e9	0.313310
$560$	−1.31026e8	−0.0315282
$561$	0	0
$562$	−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	$0.298269\pi$
$564$	0	0
$565$	1.85416e9	0.432492
$566$	−1.47933e9	−0.342931
$567$	0	0
$568$	−3.51565e9	−0.804982
$569$	−1.91122e9	−0.434929	−0.217464	−	0.976068i	$-0.569779\pi$
$569$	−1.91122e9	−0.434929	−0.217464	+	0.976068i	$0.569779\pi$
$570$	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
$573$	0	0
$574$	2.20993e9	0.487739
$575$	1.21406e9	0.266320
$576$	0	0
$577$	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
$579$	0	0
$580$	−3.81568e9	−0.812033
$581$	−1.22709e9	−0.259573
$582$	0	0
$583$	−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	$0.362455\pi$
$588$	0	0
$589$	1.82404e9	0.367816
$590$	5.95941e9	1.19460
$591$	0	0
$592$	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	$0.401554\pi$
$594$	0	0
$595$	−3.58349e8	−0.0697424
$596$	6.09541e9	1.17935
$597$	0	0
$598$	2.49790e9	0.477662
$599$	9.69612e9	1.84333	0.921667	−	0.387982i	$-0.126828\pi$
$599$	9.69612e9	1.84333	0.921667	+	0.387982i	$0.126828\pi$
$600$	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
$603$	0	0
$604$	−2.80815e9	−0.518550
$605$	5.59614e9	1.02741
$606$	0	0
$607$	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
$609$	0	0
$610$	−1.87983e9	−0.335323
$611$	3.13122e8	0.0555352
$612$	0	0
$613$	−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
$615$	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	$0.402068\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	−8.90951e9	−1.48452
$623$	2.14541e9	0.355469
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−6.47399e9	−1.05478
$627$	0	0
$628$	−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
$633$	0	0
$634$	3.32895e9	0.518794
$635$	1.68780e9	0.261584
$636$	0	0
$637$	−2.10121e8	−0.0322093
$638$	2.24717e10	3.42581
$639$	0	0
$640$	4.30603e9	0.649303
$641$	−1.10606e10	−1.65873	−0.829363	−	0.558710i	$-0.811297\pi$
$641$	−1.10606e10	−1.65873	−0.829363	+	0.558710i	$0.811297\pi$
$642$	0	0
$643$	−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
$645$	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	$0.507041\pi$
$648$	0	0
$649$	−2.12314e10	−3.04875
$650$	5.02312e8	0.0717426
$651$	0	0
$652$	−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	$0.542873\pi$
$654$	0	0
$655$	−1.53088e9	−0.212861
$656$	1.09387e9	0.151287
$657$	0	0
$658$	1.08243e9	0.148118
$659$	−2.66917e9	−0.363310	−0.181655	−	0.983362i	$-0.558145\pi$
$659$	−2.66917e9	−0.363310	−0.181655	+	0.983362i	$0.558145\pi$
$660$	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
$663$	0	0
$664$	4.37889e9	0.580465
$665$	−2.52277e8	−0.0332660
$666$	0	0
$667$	−1.21012e10	−1.57902
$668$	−5.20971e9	−0.676232
$669$	0	0
$670$	−7.84419e9	−1.00760
$671$	6.69719e9	0.855783
$672$	0	0
$673$	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
$675$	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	$0.695194\pi$
$678$	0	0
$679$	−1.80317e9	−0.221051
$680$	1.27877e9	0.155960
$681$	0	0
$682$	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	$0.866113\pi$
$684$	0	0
$685$	−1.55000e9	−0.184253
$686$	−7.26365e8	−0.0859054
$687$	0	0
$688$	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
$693$	0	0
$694$	1.13189e10	1.28542
$695$	−7.44292e8	−0.0841000
$696$	0	0
$697$	2.99168e9	0.334657
$698$	−1.62984e10	−1.81406
$699$	0	0
$700$	1.05044e9	0.115752
$701$	1.34756e10	1.47752	0.738761	−	0.673968i	$-0.235411\pi$
$701$	1.34756e10	1.47752	0.738761	+	0.673968i	$0.235411\pi$
$702$	0	0
$703$	2.55141e9	0.276972
$704$	−2.74073e10	−2.96048
$705$	0	0
$706$	−2.04944e9	−0.219189
$707$	−5.15356e9	−0.548453
$708$	0	0
$709$	−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
$711$	0	0
$712$	−7.65591e9	−0.794908
$713$	−2.40870e10	−2.48868
$714$	0	0
$715$	−1.78957e9	−0.183096
$716$	−2.25520e10	−2.29610
$717$	0	0
$718$	4.88977e9	0.493006
$719$	−1.52367e10	−1.52876	−0.764378	−	0.644768i	$-0.776954\pi$
$719$	−1.52367e10	−1.52876	−0.764378	+	0.644768i	$0.776954\pi$
$720$	0	0
$721$	−5.49026e9	−0.545531
$722$	1.54665e10	1.52937
$723$	0	0
$724$	−4.43138e8	−0.0433964
$725$	−2.43347e9	−0.237161
$726$	0	0
$727$	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
$729$	0	0
$730$	−1.33917e10	−1.27411
$731$	6.05530e9	0.573357
$732$	0	0
$733$	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
$735$	0	0
$736$	1.64475e10	1.52065
$737$	2.79462e10	2.57150
$738$	0	0
$739$	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
$741$	0	0
$742$	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	$0.681943\pi$
$744$	0	0
$745$	3.88738e9	0.344437
$746$	1.97581e10	1.74245
$747$	0	0
$748$	−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
$753$	0	0
$754$	−5.00679e9	−0.425363
$755$	−1.79091e9	−0.151446
$756$	0	0
$757$	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
$759$	0	0
$760$	9.00252e8	0.0743903
$761$	−5.66077e9	−0.465617	−0.232809	−	0.972523i	$-0.574792\pi$
$761$	−5.66077e9	−0.465617	−0.232809	+	0.972523i	$0.574792\pi$
$762$	0	0
$763$	−2.21695e9	−0.180684
$764$	1.58199e10	1.28344
$765$	0	0
$766$	9.52772e9	0.765928
$767$	4.73045e9	0.378546
$768$	0	0
$769$	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
$771$	0	0
$772$	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	$0.336434\pi$
$774$	0	0
$775$	−4.84375e9	−0.373788
$776$	6.43463e9	0.494320
$777$	0	0
$778$	−2.28618e8	−0.0174053
$779$	2.10613e9	0.159626
$780$	0	0
$781$	2.30240e10	1.72943
$782$	1.16895e10	0.874122
$783$	0	0
$784$	−3.59535e8	−0.0266462
$785$	−7.26339e9	−0.535915
$786$	0	0
$787$	−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
$789$	0	0
$790$	3.78203e9	0.272917
$791$	5.08782e9	0.365523
$792$	0	0
$793$	−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	$0.506444\pi$
$798$	0	0
$799$	1.46532e9	0.101630
$800$	3.30750e9	0.228394
$801$	0	0
$802$	2.32803e10	1.59360
$803$	4.77103e10	3.25168
$804$	0	0
$805$	3.33139e9	0.225081
$806$	−9.96588e9	−0.670414
$807$	0	0
$808$	1.83905e10	1.22646
$809$	9.57879e9	0.636050	0.318025	−	0.948082i	$-0.396980\pi$
$809$	9.57879e9	0.636050	0.318025	+	0.948082i	$0.396980\pi$
$810$	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
$813$	0	0
$814$	6.25659e10	4.06586
$815$	−2.95005e9	−0.190888
$816$	0	0
$817$	4.26291e9	0.273482
$818$	1.48922e10	0.951311
$819$	0	0
$820$	−8.76958e9	−0.555431
$821$	2.33524e10	1.47275	0.736377	−	0.676572i	$-0.236535\pi$
$821$	2.33524e10	1.47275	0.736377	+	0.676572i	$0.236535\pi$
$822$	0	0
$823$	−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
$825$	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	$0.265357\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	6.10646e9	0.367585
$833$	−9.83310e8	−0.0589431
$834$	0	0
$835$	−3.32252e9	−0.197499
$836$	−9.24456e9	−0.547223
$837$	0	0
$838$	−3.03855e10	−1.78366
$839$	7.37475e9	0.431102	0.215551	−	0.976493i	$-0.430845\pi$
$839$	7.37475e9	0.431102	0.215551	+	0.976493i	$0.430845\pi$
$840$	0	0
$841$	7.00569e9	0.406130
$842$	−2.24411e10	−1.29555
$843$	0	0
$844$	−4.35352e10	−2.49254
$845$	−7.44484e9	−0.424480
$846$	0	0
$847$	1.53558e10	0.868321
$848$	4.03997e8	0.0227506
$849$	0	0
$850$	2.35069e9	0.131289
$851$	−3.36921e10	−1.87402
$852$	0	0
$853$	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
$855$	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	$0.814342\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	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	$0.676237\pi$
$864$	0	0
$865$	8.87088e9	0.466026
$866$	−3.79916e10	−1.98781
$867$	0	0
$868$	−2.08407e10	−1.08167
$869$	−1.34741e10	−0.696516
$870$	0	0
$871$	−6.22655e9	−0.319289
$872$	7.91120e9	0.404050
$873$	0	0
$874$	8.22936e9	0.416943
$875$	6.69922e8	0.0338062
$876$	0	0
$877$	−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
$879$	0	0
$880$	−3.06211e9	−0.151472
$881$	1.37251e10	0.676237	0.338118	−	0.941104i	$-0.390210\pi$
$881$	1.37251e10	0.676237	0.338118	+	0.941104i	$0.390210\pi$
$882$	0	0
$883$	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
$885$	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	$0.205311\pi$
$888$	0	0
$889$	4.63131e9	0.221079
$890$	−1.40734e10	−0.669164
$891$	0	0
$892$	2.74821e10	1.29650
$893$	1.03158e9	0.0484757
$894$	0	0
$895$	−1.43827e10	−0.670593
$896$	1.18158e10	0.548761
$897$	0	0
$898$	−7.00106e9	−0.322624
$899$	4.82800e10	2.21620
$900$	0	0
$901$	1.10491e9	0.0503258
$902$	5.16467e10	2.34326
$903$	0	0
$904$	−1.81560e10	−0.817391
$905$	−2.82613e8	−0.0126743
$906$	0	0
$907$	−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
$909$	0	0
$910$	1.37835e9	0.0606336
$911$	−3.08592e10	−1.35229	−0.676146	−	0.736768i	$-0.736351\pi$
$911$	−3.08592e10	−1.35229	−0.676146	+	0.736768i	$0.736351\pi$
$912$	0	0
$913$	−2.86774e10	−1.24707
$914$	7.15718e10	3.10049
$915$	0	0
$916$	−1.96235e10	−0.843613
$917$	−4.20074e9	−0.179901
$918$	0	0
$919$	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
$921$	0	0
$922$	6.51096e10	2.73581
$923$	−5.12986e9	−0.214733
$924$	0	0
$925$	−6.77528e9	−0.281469
$926$	−4.33839e10	−1.79552
$927$	0	0
$928$	−3.29675e10	−1.35415
$929$	2.84458e10	1.16403	0.582013	−	0.813179i	$-0.302265\pi$
$929$	2.84458e10	1.16403	0.582013	+	0.813179i	$0.302265\pi$
$930$	0	0
$931$	−6.92247e8	−0.0281149
$932$	7.29246e10	2.95065
$933$	0	0
$934$	3.16760e10	1.27208
$935$	−8.37472e9	−0.335065
$936$	0	0
$937$	−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
$939$	0	0
$940$	−4.29534e9	−0.168675
$941$	−1.52622e10	−0.597108	−0.298554	−	0.954393i	$-0.596504\pi$
$941$	−1.52622e10	−0.597108	−0.298554	+	0.954393i	$0.596504\pi$
$942$	0	0
$943$	−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	$0.353030\pi$
$948$	0	0
$949$	−1.06301e10	−0.403742
$950$	1.65487e9	0.0626228
$951$	0	0
$952$	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	$0.688358\pi$
$954$	0	0
$955$	1.00892e10	0.374839
$956$	1.86387e10	0.689942
$957$	0	0
$958$	−7.53835e10	−2.77011
$959$	−4.25319e9	−0.155722
$960$	0	0
$961$	6.85874e10	2.49294
$962$	−1.39400e10	−0.504834
$963$	0	0
$964$	−5.61280e10	−2.01795
$965$	8.84924e9	0.317001
$966$	0	0
$967$	−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
$969$	0	0
$970$	1.18284e10	0.416125
$971$	4.32969e10	1.51771	0.758857	−	0.651258i	$-0.225758\pi$
$971$	4.32969e10	1.51771	0.758857	+	0.651258i	$0.225758\pi$
$972$	0	0
$973$	−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	$0.657876\pi$
$978$	0	0
$979$	5.01387e10	1.70779
$980$	2.88240e9	0.0978280
$981$	0	0
$982$	−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	$0.483892\pi$
$984$	0	0
$985$	−9.41842e9	−0.314016
$986$	−2.34304e10	−0.778415
$987$	0	0
$988$	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
$993$	0	0
$994$	−1.77333e10	−0.572714
$995$	−1.96956e10	−0.633854
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.a.1.1		1
3.2	odd	2		105.8.a.b.1.1	✓	1
15.14	odd	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	3.2	odd	2
315.8.a.a.1.1		1	1.1	even	1	trivial
525.8.a.a.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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