LMFDB - Embedded newform 315.10.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-5] 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:	$162.236288392$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1495x^{2} + 193x + 99862 $ x^4 - x^3 - 1495x^2 + 193x + 99862
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.32605$ of defining polynomial
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)

$f(q)$	$=$	\(q+7.32605 q^{2} -458.329 q^{4} +625.000 q^{5} -2401.00 q^{7} -7108.68 q^{8} +4578.78 q^{10} +25587.9 q^{11} -36704.8 q^{13} -17589.8 q^{14} +182586. q^{16} -191304. q^{17} +538840. q^{19} -286456. q^{20} +187458. q^{22} -1.45253e6 q^{23} +390625. q^{25} -268901. q^{26} +1.10045e6 q^{28} +995482. q^{29} +3.11544e6 q^{31} +4.97728e6 q^{32} -1.40150e6 q^{34} -1.50062e6 q^{35} +6.55984e6 q^{37} +3.94756e6 q^{38} -4.44292e6 q^{40} +2.64947e6 q^{41} -7.42236e6 q^{43} -1.17277e7 q^{44} -1.06413e7 q^{46} +1.64799e7 q^{47} +5.76480e6 q^{49} +2.86174e6 q^{50} +1.68229e7 q^{52} +4.36429e7 q^{53} +1.59924e7 q^{55} +1.70679e7 q^{56} +7.29294e6 q^{58} -5.20339e7 q^{59} -6.55307e7 q^{61} +2.28239e7 q^{62} -5.70203e7 q^{64} -2.29405e7 q^{65} +1.13208e8 q^{67} +8.76802e7 q^{68} -1.09936e7 q^{70} -3.26621e7 q^{71} -9.95220e7 q^{73} +4.80577e7 q^{74} -2.46966e8 q^{76} -6.14365e7 q^{77} -4.03718e8 q^{79} +1.14116e8 q^{80} +1.94101e7 q^{82} +5.47210e8 q^{83} -1.19565e8 q^{85} -5.43765e7 q^{86} -1.81896e8 q^{88} +5.33992e8 q^{89} +8.81282e7 q^{91} +6.65739e8 q^{92} +1.20732e8 q^{94} +3.36775e8 q^{95} +1.53704e8 q^{97} +4.22332e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * q^98

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 $

$2$	7.32605	0.323769	0.161884	−	0.986810i	$-0.448243\pi$
$2$	7.32605	0.323769	0.161884	+	0.986810i	$0.448243\pi$
$3$	0	0
$3$	0	0
$4$	−458.329	−0.895174
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−7108.68	−0.613598
$9$	0	0
$10$	4578.78	0.144794
$11$	25587.9	0.526947	0.263474	−	0.964667i	$-0.415132\pi$
$11$	25587.9	0.526947	0.263474	+	0.964667i	$0.415132\pi$
$12$	0	0
$13$	−36704.8	−0.356433	−0.178216	−	0.983991i	$-0.557033\pi$
$13$	−36704.8	−0.356433	−0.178216	+	0.983991i	$0.557033\pi$
$14$	−17589.8	−0.122373
$15$	0	0
$16$	182586.	0.696510
$17$	−191304.	−0.555526	−0.277763	−	0.960650i	$-0.589593\pi$
$17$	−191304.	−0.555526	−0.277763	+	0.960650i	$0.589593\pi$
$18$	0	0
$19$	538840.	0.948568	0.474284	−	0.880372i	$-0.342707\pi$
$19$	538840.	0.948568	0.474284	+	0.880372i	$0.342707\pi$
$20$	−286456.	−0.400334
$21$	0	0
$22$	187458.	0.170609
$23$	−1.45253e6	−1.08231	−0.541155	−	0.840923i	$-0.682013\pi$
$23$	−1.45253e6	−1.08231	−0.541155	+	0.840923i	$0.682013\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−268901.	−0.115402
$27$	0	0
$28$	1.10045e6	0.338344
$29$	995482.	0.261362	0.130681	−	0.991424i	$-0.458284\pi$
$29$	995482.	0.261362	0.130681	+	0.991424i	$0.458284\pi$
$30$	0	0
$31$	3.11544e6	0.605887	0.302944	−	0.953008i	$-0.402031\pi$
$31$	3.11544e6	0.605887	0.302944	+	0.953008i	$0.402031\pi$
$32$	4.97728e6	0.839106
$33$	0	0
$34$	−1.40150e6	−0.179862
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	6.55984e6	0.575421	0.287710	−	0.957717i	$-0.407106\pi$
$37$	6.55984e6	0.575421	0.287710	+	0.957717i	$0.407106\pi$
$38$	3.94756e6	0.307116
$39$	0	0
$40$	−4.44292e6	−0.274409
$41$	2.64947e6	0.146430	0.0732151	−	0.997316i	$-0.476674\pi$
$41$	2.64947e6	0.146430	0.0732151	+	0.997316i	$0.476674\pi$
$42$	0	0
$43$	−7.42236e6	−0.331081	−0.165540	−	0.986203i	$-0.552937\pi$
$43$	−7.42236e6	−0.331081	−0.165540	+	0.986203i	$0.552937\pi$
$44$	−1.17277e7	−0.471710
$45$	0	0
$46$	−1.06413e7	−0.350418
$47$	1.64799e7	0.492622	0.246311	−	0.969191i	$-0.420782\pi$
$47$	1.64799e7	0.492622	0.246311	+	0.969191i	$0.420782\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	2.86174e6	0.0647537
$51$	0	0
$52$	1.68229e7	0.319069
$53$	4.36429e7	0.759752	0.379876	−	0.925037i	$-0.375967\pi$
$53$	4.36429e7	0.759752	0.379876	+	0.925037i	$0.375967\pi$
$54$	0	0
$55$	1.59924e7	0.235658
$56$	1.70679e7	0.231918
$57$	0	0
$58$	7.29294e6	0.0846208
$59$	−5.20339e7	−0.559052	−0.279526	−	0.960138i	$-0.590177\pi$
$59$	−5.20339e7	−0.559052	−0.279526	+	0.960138i	$0.590177\pi$
$60$	0	0
$61$	−6.55307e7	−0.605983	−0.302992	−	0.952993i	$-0.597985\pi$
$61$	−6.55307e7	−0.605983	−0.302992	+	0.952993i	$0.597985\pi$
$62$	2.28239e7	0.196167
$63$	0	0
$64$	−5.70203e7	−0.424834
$65$	−2.29405e7	−0.159402
$66$	0	0
$67$	1.13208e8	0.686340	0.343170	−	0.939273i	$-0.388499\pi$
$67$	1.13208e8	0.686340	0.343170	+	0.939273i	$0.388499\pi$
$68$	8.76802e7	0.497292
$69$	0	0
$70$	−1.09936e7	−0.0547269
$71$	−3.26621e7	−0.152539	−0.0762696	−	0.997087i	$-0.524301\pi$
$71$	−3.26621e7	−0.152539	−0.0762696	+	0.997087i	$0.524301\pi$
$72$	0	0
$73$	−9.95220e7	−0.410172	−0.205086	−	0.978744i	$-0.565747\pi$
$73$	−9.95220e7	−0.410172	−0.205086	+	0.978744i	$0.565747\pi$
$74$	4.80577e7	0.186303
$75$	0	0
$76$	−2.46966e8	−0.849133
$77$	−6.14365e7	−0.199167
$78$	0	0
$79$	−4.03718e8	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$79$	−4.03718e8	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$80$	1.14116e8	0.311489
$81$	0	0
$82$	1.94101e7	0.0474095
$83$	5.47210e8	1.26562	0.632809	−	0.774308i	$-0.281902\pi$
$83$	5.47210e8	1.26562	0.632809	+	0.774308i	$0.281902\pi$
$84$	0	0
$85$	−1.19565e8	−0.248439
$86$	−5.43765e7	−0.107194
$87$	0	0
$88$	−1.81896e8	−0.323334
$89$	5.33992e8	0.902152	0.451076	−	0.892485i	$-0.351040\pi$
$89$	5.33992e8	0.902152	0.451076	+	0.892485i	$0.351040\pi$
$90$	0	0
$91$	8.81282e7	0.134719
$92$	6.65739e8	0.968855
$93$	0	0
$94$	1.20732e8	0.159495
$95$	3.36775e8	0.424212
$96$	0	0
$97$	1.53704e8	0.176283	0.0881417	−	0.996108i	$-0.471907\pi$
$97$	1.53704e8	0.176283	0.0881417	+	0.996108i	$0.471907\pi$
$98$	4.22332e7	0.0462526
$99$	0	0
$100$	−1.79035e8	−0.179035
$101$	−1.10800e9	−1.05948	−0.529742	−	0.848159i	$-0.677711\pi$
$101$	−1.10800e9	−1.05948	−0.529742	+	0.848159i	$0.677711\pi$
$102$	0	0
$103$	−6.16235e8	−0.539485	−0.269742	−	0.962933i	$-0.586939\pi$
$103$	−6.16235e8	−0.539485	−0.269742	+	0.962933i	$0.586939\pi$
$104$	2.60922e8	0.218706
$105$	0	0
$106$	3.19730e8	0.245984
$107$	1.14563e8	0.0844924	0.0422462	−	0.999107i	$-0.486549\pi$
$107$	1.14563e8	0.0844924	0.0422462	+	0.999107i	$0.486549\pi$
$108$	0	0
$109$	−3.96759e8	−0.269220	−0.134610	−	0.990899i	$-0.542978\pi$
$109$	−3.96759e8	−0.269220	−0.134610	+	0.990899i	$0.542978\pi$
$110$	1.17161e8	0.0762987
$111$	0	0
$112$	−4.38389e8	−0.263256
$113$	−5.19192e8	−0.299554	−0.149777	−	0.988720i	$-0.547856\pi$
$113$	−5.19192e8	−0.299554	−0.149777	+	0.988720i	$0.547856\pi$
$114$	0	0
$115$	−9.07834e8	−0.484023
$116$	−4.56258e8	−0.233964
$117$	0	0
$118$	−3.81203e8	−0.181004
$119$	4.59321e8	0.209969
$120$	0	0
$121$	−1.70321e9	−0.722326
$122$	−4.80081e8	−0.196198
$123$	0	0
$124$	−1.42790e9	−0.542375
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	3.12407e7	0.0106562	0.00532811	−	0.999986i	$-0.498304\pi$
$127$	3.12407e7	0.0106562	0.00532811	+	0.999986i	$0.498304\pi$
$128$	−2.96610e9	−0.976654
$129$	0	0
$130$	−1.68063e8	−0.0516092
$131$	−4.46739e9	−1.32536	−0.662678	−	0.748904i	$-0.730580\pi$
$131$	−4.46739e9	−1.32536	−0.662678	+	0.748904i	$0.730580\pi$
$132$	0	0
$133$	−1.29375e9	−0.358525
$134$	8.29365e8	0.222215
$135$	0	0
$136$	1.35992e9	0.340869
$137$	−4.16173e9	−1.00933	−0.504663	−	0.863317i	$-0.668383\pi$
$137$	−4.16173e9	−1.00933	−0.504663	+	0.863317i	$0.668383\pi$
$138$	0	0
$139$	−1.86095e9	−0.422833	−0.211417	−	0.977396i	$-0.567808\pi$
$139$	−1.86095e9	−0.422833	−0.211417	+	0.977396i	$0.567808\pi$
$140$	6.87780e8	0.151312
$141$	0	0
$142$	−2.39284e8	−0.0493874
$143$	−9.39198e8	−0.187821
$144$	0	0
$145$	6.22176e8	0.116885
$146$	−7.29103e8	−0.132801
$147$	0	0
$148$	−3.00657e9	−0.515102
$149$	−7.53898e9	−1.25307	−0.626534	−	0.779394i	$-0.715527\pi$
$149$	−7.53898e9	−1.25307	−0.626534	+	0.779394i	$0.715527\pi$
$150$	0	0
$151$	−4.79414e9	−0.750438	−0.375219	−	0.926936i	$-0.622433\pi$
$151$	−4.79414e9	−0.750438	−0.375219	+	0.926936i	$0.622433\pi$
$152$	−3.83044e9	−0.582039
$153$	0	0
$154$	−4.50087e8	−0.0644841
$155$	1.94715e9	0.270961
$156$	0	0
$157$	1.19187e10	1.56560	0.782799	−	0.622275i	$-0.213791\pi$
$157$	1.19187e10	1.56560	0.782799	+	0.622275i	$0.213791\pi$
$158$	−2.95765e9	−0.377564
$159$	0	0
$160$	3.11080e9	0.375260
$161$	3.48754e9	0.409074
$162$	0	0
$163$	9.79589e8	0.108693	0.0543463	−	0.998522i	$-0.482693\pi$
$163$	9.79589e8	0.108693	0.0543463	+	0.998522i	$0.482693\pi$
$164$	−1.21433e9	−0.131081
$165$	0	0
$166$	4.00888e9	0.409767
$167$	−6.22242e9	−0.619063	−0.309532	−	0.950889i	$-0.600172\pi$
$167$	−6.22242e9	−0.619063	−0.309532	+	0.950889i	$0.600172\pi$
$168$	0	0
$169$	−9.25726e9	−0.872956
$170$	−8.75939e8	−0.0804366
$171$	0	0
$172$	3.40188e9	0.296375
$173$	−5.09657e9	−0.432584	−0.216292	−	0.976329i	$-0.569396\pi$
$173$	−5.09657e9	−0.432584	−0.216292	+	0.976329i	$0.569396\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	4.67199e9	0.367024
$177$	0	0
$178$	3.91205e9	0.292089
$179$	6.15458e9	0.448084	0.224042	−	0.974579i	$-0.428075\pi$
$179$	6.15458e9	0.448084	0.224042	+	0.974579i	$0.428075\pi$
$180$	0	0
$181$	−1.10743e10	−0.766943	−0.383471	−	0.923553i	$-0.625271\pi$
$181$	−1.10743e10	−0.766943	−0.383471	+	0.923553i	$0.625271\pi$
$182$	6.45631e8	0.0436178
$183$	0	0
$184$	1.03256e10	0.664102
$185$	4.09990e9	0.257336
$186$	0	0
$187$	−4.89507e9	−0.292733
$188$	−7.55321e9	−0.440982
$189$	0	0
$190$	2.46723e9	0.137347
$191$	−3.22959e9	−0.175589	−0.0877945	−	0.996139i	$-0.527982\pi$
$191$	−3.22959e9	−0.175589	−0.0877945	+	0.996139i	$0.527982\pi$
$192$	0	0
$193$	2.06114e10	1.06930	0.534651	−	0.845073i	$-0.320443\pi$
$193$	2.06114e10	1.06930	0.534651	+	0.845073i	$0.320443\pi$
$194$	1.12604e9	0.0570750
$195$	0	0
$196$	−2.64218e9	−0.127882
$197$	6.31060e8	0.0298519	0.0149260	−	0.999889i	$-0.495249\pi$
$197$	6.31060e8	0.0298519	0.0149260	+	0.999889i	$0.495249\pi$
$198$	0	0
$199$	−2.03819e10	−0.921309	−0.460654	−	0.887580i	$-0.652385\pi$
$199$	−2.03819e10	−0.921309	−0.460654	+	0.887580i	$0.652385\pi$
$200$	−2.77683e9	−0.122720
$201$	0	0
$202$	−8.11728e9	−0.343028
$203$	−2.39015e9	−0.0987855
$204$	0	0
$205$	1.65592e9	0.0654856
$206$	−4.51457e9	−0.174668
$207$	0	0
$208$	−6.70178e9	−0.248259
$209$	1.37878e10	0.499845
$210$	0	0
$211$	−1.70385e10	−0.591780	−0.295890	−	0.955222i	$-0.595616\pi$
$211$	−1.70385e10	−0.591780	−0.295890	+	0.955222i	$0.595616\pi$
$212$	−2.00028e10	−0.680110
$213$	0	0
$214$	8.39294e8	0.0273560
$215$	−4.63897e9	−0.148064
$216$	0	0
$217$	−7.48017e9	−0.229004
$218$	−2.90667e9	−0.0871649
$219$	0	0
$220$	−7.32979e9	−0.210955
$221$	7.02178e9	0.198008
$222$	0	0
$223$	−5.09083e10	−1.37853	−0.689265	−	0.724509i	$-0.742067\pi$
$223$	−5.09083e10	−1.37853	−0.689265	+	0.724509i	$0.742067\pi$
$224$	−1.19504e10	−0.317152
$225$	0	0
$226$	−3.80362e9	−0.0969861
$227$	4.08124e9	0.102018	0.0510088	−	0.998698i	$-0.483756\pi$
$227$	4.08124e9	0.102018	0.0510088	+	0.998698i	$0.483756\pi$
$228$	0	0
$229$	−4.06683e10	−0.977230	−0.488615	−	0.872500i	$-0.662498\pi$
$229$	−4.06683e10	−0.977230	−0.488615	+	0.872500i	$0.662498\pi$
$230$	−6.65084e9	−0.156712
$231$	0	0
$232$	−7.07656e9	−0.160371
$233$	3.92653e10	0.872786	0.436393	−	0.899756i	$-0.356256\pi$
$233$	3.92653e10	0.872786	0.436393	+	0.899756i	$0.356256\pi$
$234$	0	0
$235$	1.02999e10	0.220307
$236$	2.38487e10	0.500449
$237$	0	0
$238$	3.36501e9	0.0679813
$239$	7.02454e10	1.39260	0.696302	−	0.717749i	$-0.254828\pi$
$239$	7.02454e10	1.39260	0.696302	+	0.717749i	$0.254828\pi$
$240$	0	0
$241$	−8.42447e10	−1.60867	−0.804333	−	0.594179i	$-0.797477\pi$
$241$	−8.42447e10	−1.60867	−0.804333	+	0.594179i	$0.797477\pi$
$242$	−1.24778e10	−0.233867
$243$	0	0
$244$	3.00346e10	0.542461
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−1.97780e10	−0.338101
$248$	−2.21467e10	−0.371771
$249$	0	0
$250$	1.78859e9	0.0289587
$251$	−6.01503e10	−0.956546	−0.478273	−	0.878211i	$-0.658737\pi$
$251$	−6.01503e10	−0.956546	−0.478273	+	0.878211i	$0.658737\pi$
$252$	0	0
$253$	−3.71673e10	−0.570320
$254$	2.28871e8	0.00345015
$255$	0	0
$256$	7.46462e9	0.108624
$257$	−5.99526e10	−0.857253	−0.428627	−	0.903482i	$-0.641002\pi$
$257$	−5.99526e10	−0.857253	−0.428627	+	0.903482i	$0.641002\pi$
$258$	0	0
$259$	−1.57502e10	−0.217489
$260$	1.05143e10	0.142692
$261$	0	0
$262$	−3.27283e10	−0.429109
$263$	−7.56878e10	−0.975495	−0.487747	−	0.872985i	$-0.662181\pi$
$263$	−7.56878e10	−0.975495	−0.487747	+	0.872985i	$0.662181\pi$
$264$	0	0
$265$	2.72768e10	0.339771
$266$	−9.47810e9	−0.116079
$267$	0	0
$268$	−5.18864e10	−0.614394
$269$	3.42802e10	0.399170	0.199585	−	0.979881i	$-0.436041\pi$
$269$	3.42802e10	0.399170	0.199585	+	0.979881i	$0.436041\pi$
$270$	0	0
$271$	−1.43425e11	−1.61534	−0.807669	−	0.589636i	$-0.799271\pi$
$271$	−1.43425e11	−1.61534	−0.807669	+	0.589636i	$0.799271\pi$
$272$	−3.49295e10	−0.386929
$273$	0	0
$274$	−3.04890e10	−0.326788
$275$	9.99527e9	0.105389
$276$	0	0
$277$	−6.59247e10	−0.672805	−0.336402	−	0.941718i	$-0.609210\pi$
$277$	−6.59247e10	−0.672805	−0.336402	+	0.941718i	$0.609210\pi$
$278$	−1.36334e10	−0.136900
$279$	0	0
$280$	1.06675e10	0.103717
$281$	8.77245e10	0.839348	0.419674	−	0.907675i	$-0.362144\pi$
$281$	8.77245e10	0.839348	0.419674	+	0.907675i	$0.362144\pi$
$282$	0	0
$283$	−1.95527e11	−1.81204	−0.906019	−	0.423238i	$-0.860894\pi$
$283$	−1.95527e11	−1.81204	−0.906019	+	0.423238i	$0.860894\pi$
$284$	1.49700e10	0.136549
$285$	0	0
$286$	−6.88061e9	−0.0608107
$287$	−6.36137e9	−0.0553454
$288$	0	0
$289$	−8.19906e10	−0.691391
$290$	4.55809e9	0.0378436
$291$	0	0
$292$	4.56138e10	0.367175
$293$	8.06431e10	0.639238	0.319619	−	0.947546i	$-0.396445\pi$
$293$	8.06431e10	0.639238	0.319619	+	0.947546i	$0.396445\pi$
$294$	0	0
$295$	−3.25212e10	−0.250016
$296$	−4.66318e10	−0.353077
$297$	0	0
$298$	−5.52309e10	−0.405704
$299$	5.33150e10	0.385771
$300$	0	0
$301$	1.78211e10	0.125137
$302$	−3.51221e10	−0.242968
$303$	0	0
$304$	9.83846e10	0.660687
$305$	−4.09567e10	−0.271004
$306$	0	0
$307$	9.63861e10	0.619287	0.309643	−	0.950853i	$-0.399790\pi$
$307$	9.63861e10	0.619287	0.309643	+	0.950853i	$0.399790\pi$
$308$	2.81581e10	0.178289
$309$	0	0
$310$	1.42649e10	0.0877287
$311$	−1.36856e11	−0.829552	−0.414776	−	0.909924i	$-0.636140\pi$
$311$	−1.36856e11	−0.829552	−0.414776	+	0.909924i	$0.636140\pi$
$312$	0	0
$313$	−2.53548e11	−1.49318	−0.746588	−	0.665287i	$-0.768309\pi$
$313$	−2.53548e11	−1.49318	−0.746588	+	0.665287i	$0.768309\pi$
$314$	8.73169e10	0.506891
$315$	0	0
$316$	1.85035e11	1.04391
$317$	4.49868e10	0.250218	0.125109	−	0.992143i	$-0.460072\pi$
$317$	4.49868e10	0.250218	0.125109	+	0.992143i	$0.460072\pi$
$318$	0	0
$319$	2.54723e10	0.137724
$320$	−3.56377e10	−0.189992
$321$	0	0
$322$	2.55499e10	0.132445
$323$	−1.03082e11	−0.526954
$324$	0	0
$325$	−1.43378e10	−0.0712866
$326$	7.17651e9	0.0351912
$327$	0	0
$328$	−1.88342e10	−0.0898493
$329$	−3.95682e10	−0.186194
$330$	0	0
$331$	1.30115e11	0.595799	0.297900	−	0.954597i	$-0.403714\pi$
$331$	1.30115e11	0.595799	0.297900	+	0.954597i	$0.403714\pi$
$332$	−2.50802e11	−1.13295
$333$	0	0
$334$	−4.55857e10	−0.200433
$335$	7.07548e10	0.306941
$336$	0	0
$337$	3.56450e11	1.50544	0.752720	−	0.658340i	$-0.228741\pi$
$337$	3.56450e11	1.50544	0.752720	+	0.658340i	$0.228741\pi$
$338$	−6.78191e10	−0.282636
$339$	0	0
$340$	5.48001e10	0.222396
$341$	7.97176e10	0.319271
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	5.27631e10	0.203150
$345$	0	0
$346$	−3.73377e10	−0.140057
$347$	−4.47907e11	−1.65846	−0.829230	−	0.558908i	$-0.811221\pi$
$347$	−4.47907e11	−1.65846	−0.829230	+	0.558908i	$0.811221\pi$
$348$	0	0
$349$	2.59553e11	0.936508	0.468254	−	0.883594i	$-0.344883\pi$
$349$	2.59553e11	0.936508	0.468254	+	0.883594i	$0.344883\pi$
$350$	−6.87103e9	−0.0244746
$351$	0	0
$352$	1.27358e11	0.442165
$353$	−1.78650e11	−0.612373	−0.306187	−	0.951971i	$-0.599053\pi$
$353$	−1.78650e11	−0.612373	−0.306187	+	0.951971i	$0.599053\pi$
$354$	0	0
$355$	−2.04138e10	−0.0682176
$356$	−2.44744e11	−0.807583
$357$	0	0
$358$	4.50887e10	0.145076
$359$	3.49176e11	1.10948	0.554739	−	0.832024i	$-0.312818\pi$
$359$	3.49176e11	1.10948	0.554739	+	0.832024i	$0.312818\pi$
$360$	0	0
$361$	−3.23396e10	−0.100220
$362$	−8.11309e10	−0.248312
$363$	0	0
$364$	−4.03917e10	−0.120597
$365$	−6.22013e10	−0.183435
$366$	0	0
$367$	3.79817e11	1.09289	0.546445	−	0.837495i	$-0.315981\pi$
$367$	3.79817e11	1.09289	0.546445	+	0.837495i	$0.315981\pi$
$368$	−2.65213e11	−0.753839
$369$	0	0
$370$	3.00361e10	0.0833173
$371$	−1.04787e11	−0.287159
$372$	0	0
$373$	5.48563e11	1.46736	0.733680	−	0.679495i	$-0.237801\pi$
$373$	5.48563e11	1.46736	0.733680	+	0.679495i	$0.237801\pi$
$374$	−3.58615e10	−0.0947777
$375$	0	0
$376$	−1.17150e11	−0.302272
$377$	−3.65389e10	−0.0931580
$378$	0	0
$379$	−6.68643e11	−1.66463	−0.832316	−	0.554302i	$-0.812985\pi$
$379$	−6.68643e11	−1.66463	−0.832316	+	0.554302i	$0.812985\pi$
$380$	−1.54354e11	−0.379744
$381$	0	0
$382$	−2.36601e10	−0.0568502
$383$	4.50094e11	1.06883	0.534415	−	0.845223i	$-0.320532\pi$
$383$	4.50094e11	1.06883	0.534415	+	0.845223i	$0.320532\pi$
$384$	0	0
$385$	−3.83978e10	−0.0890704
$386$	1.51000e11	0.346206
$387$	0	0
$388$	−7.04468e10	−0.157804
$389$	4.44303e11	0.983799	0.491900	−	0.870652i	$-0.336303\pi$
$389$	4.44303e11	0.983799	0.491900	+	0.870652i	$0.336303\pi$
$390$	0	0
$391$	2.77876e11	0.601251
$392$	−4.09801e10	−0.0876568
$393$	0	0
$394$	4.62317e9	0.00966512
$395$	−2.52323e11	−0.521520
$396$	0	0
$397$	−5.21351e11	−1.05335	−0.526675	−	0.850067i	$-0.676562\pi$
$397$	−5.21351e11	−1.05335	−0.526675	+	0.850067i	$0.676562\pi$
$398$	−1.49318e11	−0.298291
$399$	0	0
$400$	7.13227e10	0.139302
$401$	−1.28071e11	−0.247343	−0.123672	−	0.992323i	$-0.539467\pi$
$401$	−1.28071e11	−0.247343	−0.123672	+	0.992323i	$0.539467\pi$
$402$	0	0
$403$	−1.14352e11	−0.215958
$404$	5.07830e11	0.948423
$405$	0	0
$406$	−1.75104e10	−0.0319836
$407$	1.67852e11	0.303217
$408$	0	0
$409$	1.87684e10	0.0331645	0.0165822	−	0.999863i	$-0.494721\pi$
$409$	1.87684e10	0.0331645	0.0165822	+	0.999863i	$0.494721\pi$
$410$	1.21313e10	0.0212022
$411$	0	0
$412$	2.82438e11	0.482932
$413$	1.24933e11	0.211302
$414$	0	0
$415$	3.42006e11	0.566001
$416$	−1.82690e11	−0.299085
$417$	0	0
$418$	1.01010e11	0.161834
$419$	1.17317e12	1.85951	0.929755	−	0.368178i	$-0.120018\pi$
$419$	1.17317e12	1.85951	0.929755	+	0.368178i	$0.120018\pi$
$420$	0	0
$421$	−4.42477e11	−0.686470	−0.343235	−	0.939250i	$-0.611523\pi$
$421$	−4.42477e11	−0.686470	−0.343235	+	0.939250i	$0.611523\pi$
$422$	−1.24825e11	−0.191600
$423$	0	0
$424$	−3.10243e11	−0.466182
$425$	−7.47282e10	−0.111105
$426$	0	0
$427$	1.57339e11	0.229040
$428$	−5.25076e10	−0.0756354
$429$	0	0
$430$	−3.39853e10	−0.0479384
$431$	6.28009e11	0.876634	0.438317	−	0.898820i	$-0.355575\pi$
$431$	6.28009e11	0.876634	0.438317	+	0.898820i	$0.355575\pi$
$432$	0	0
$433$	−6.56192e11	−0.897089	−0.448545	−	0.893760i	$-0.648057\pi$
$433$	−6.56192e11	−0.897089	−0.448545	+	0.893760i	$0.648057\pi$
$434$	−5.48001e10	−0.0741443
$435$	0	0
$436$	1.81846e11	0.240999
$437$	−7.82683e11	−1.02664
$438$	0	0
$439$	−1.32064e12	−1.69705	−0.848527	−	0.529153i	$-0.822510\pi$
$439$	−1.32064e12	−1.69705	−0.848527	+	0.529153i	$0.822510\pi$
$440$	−1.13685e11	−0.144599
$441$	0	0
$442$	5.14419e10	0.0641086
$443$	−3.57477e11	−0.440992	−0.220496	−	0.975388i	$-0.570768\pi$
$443$	−3.57477e11	−0.440992	−0.220496	+	0.975388i	$0.570768\pi$
$444$	0	0
$445$	3.33745e11	0.403455
$446$	−3.72956e11	−0.446325
$447$	0	0
$448$	1.36906e11	0.160572
$449$	4.23987e11	0.492316	0.246158	−	0.969230i	$-0.420832\pi$
$449$	4.23987e11	0.492316	0.246158	+	0.969230i	$0.420832\pi$
$450$	0	0
$451$	6.77942e10	0.0771611
$452$	2.37961e11	0.268153
$453$	0	0
$454$	2.98993e10	0.0330301
$455$	5.50801e10	0.0602481
$456$	0	0
$457$	1.38696e12	1.48745	0.743724	−	0.668487i	$-0.233058\pi$
$457$	1.38696e12	1.48745	0.743724	+	0.668487i	$0.233058\pi$
$458$	−2.97938e11	−0.316396
$459$	0	0
$460$	4.16087e11	0.433285
$461$	2.51449e11	0.259296	0.129648	−	0.991560i	$-0.458615\pi$
$461$	2.51449e11	0.259296	0.129648	+	0.991560i	$0.458615\pi$
$462$	0	0
$463$	−1.11946e12	−1.13213	−0.566064	−	0.824362i	$-0.691534\pi$
$463$	−1.11946e12	−1.13213	−0.566064	+	0.824362i	$0.691534\pi$
$464$	1.81761e11	0.182041
$465$	0	0
$466$	2.87660e11	0.282581
$467$	1.54982e12	1.50784	0.753921	−	0.656965i	$-0.228160\pi$
$467$	1.54982e12	1.50784	0.753921	+	0.656965i	$0.228160\pi$
$468$	0	0
$469$	−2.71812e11	−0.259412
$470$	7.54577e10	0.0713285
$471$	0	0
$472$	3.69892e11	0.343033
$473$	−1.89922e11	−0.174462
$474$	0	0
$475$	2.10484e11	0.189714
$476$	−2.10520e11	−0.187959
$477$	0	0
$478$	5.14621e11	0.450881
$479$	−4.05300e11	−0.351777	−0.175888	−	0.984410i	$-0.556280\pi$
$479$	−4.05300e11	−0.351777	−0.175888	+	0.984410i	$0.556280\pi$
$480$	0	0
$481$	−2.40778e11	−0.205099
$482$	−6.17181e11	−0.520835
$483$	0	0
$484$	7.80630e11	0.646608
$485$	9.60648e10	0.0788363
$486$	0	0
$487$	1.75334e12	1.41249	0.706247	−	0.707965i	$-0.250387\pi$
$487$	1.75334e12	1.41249	0.706247	+	0.707965i	$0.250387\pi$
$488$	4.65837e11	0.371830
$489$	0	0
$490$	2.63957e10	0.0206848
$491$	1.09492e12	0.850190	0.425095	−	0.905149i	$-0.360241\pi$
$491$	1.09492e12	0.850190	0.425095	+	0.905149i	$0.360241\pi$
$492$	0	0
$493$	−1.90440e11	−0.145193
$494$	−1.44895e11	−0.109466
$495$	0	0
$496$	5.68836e11	0.422007
$497$	7.84217e10	0.0576544
$498$	0	0
$499$	3.79654e11	0.274117	0.137059	−	0.990563i	$-0.456235\pi$
$499$	3.79654e11	0.274117	0.137059	+	0.990563i	$0.456235\pi$
$500$	−1.11897e11	−0.0800668
$501$	0	0
$502$	−4.40664e11	−0.309699
$503$	2.58061e12	1.79749	0.898744	−	0.438474i	$-0.144481\pi$
$503$	2.58061e12	1.79749	0.898744	+	0.438474i	$0.144481\pi$
$504$	0	0
$505$	−6.92501e11	−0.473816
$506$	−2.72289e11	−0.184652
$507$	0	0
$508$	−1.43185e10	−0.00953918
$509$	2.91901e11	0.192755	0.0963776	−	0.995345i	$-0.469274\pi$
$509$	2.91901e11	0.192755	0.0963776	+	0.995345i	$0.469274\pi$
$510$	0	0
$511$	2.38952e11	0.155031
$512$	1.57333e12	1.01182
$513$	0	0
$514$	−4.39216e11	−0.277552
$515$	−3.85147e11	−0.241265
$516$	0	0
$517$	4.21685e11	0.259586
$518$	−1.15387e11	−0.0704160
$519$	0	0
$520$	1.63077e11	0.0978085
$521$	2.30490e12	1.37051	0.685256	−	0.728302i	$-0.259690\pi$
$521$	2.30490e12	1.37051	0.685256	+	0.728302i	$0.259690\pi$
$522$	0	0
$523$	1.90497e11	0.111335	0.0556674	−	0.998449i	$-0.482271\pi$
$523$	1.90497e11	0.111335	0.0556674	+	0.998449i	$0.482271\pi$
$524$	2.04753e12	1.18642
$525$	0	0
$526$	−5.54492e11	−0.315834
$527$	−5.95997e11	−0.336586
$528$	0	0
$529$	3.08705e11	0.171393
$530$	1.99831e11	0.110007
$531$	0	0
$532$	5.92965e11	0.320942
$533$	−9.72481e10	−0.0521926
$534$	0	0
$535$	7.16019e10	0.0377861
$536$	−8.04757e11	−0.421137
$537$	0	0
$538$	2.51138e11	0.129239
$539$	1.47509e11	0.0752782
$540$	0	0
$541$	3.60598e12	1.80982	0.904912	−	0.425600i	$-0.139937\pi$
$541$	3.60598e12	1.80982	0.904912	+	0.425600i	$0.139937\pi$
$542$	−1.05074e12	−0.522996
$543$	0	0
$544$	−9.52173e11	−0.466145
$545$	−2.47974e11	−0.120399
$546$	0	0
$547$	4.09380e12	1.95517	0.977583	−	0.210550i	$-0.0675256\pi$
$547$	4.09380e12	1.95517	0.977583	+	0.210550i	$0.0675256\pi$
$548$	1.90744e12	0.903521
$549$	0	0
$550$	7.32258e10	0.0341218
$551$	5.36405e11	0.247919
$552$	0	0
$553$	9.69326e11	0.440765
$554$	−4.82967e11	−0.217833
$555$	0	0
$556$	8.52930e11	0.378509
$557$	−3.73802e12	−1.64548	−0.822741	−	0.568416i	$-0.807556\pi$
$557$	−3.73802e12	−1.64548	−0.822741	+	0.568416i	$0.807556\pi$
$558$	0	0
$559$	2.72436e11	0.118008
$560$	−2.73993e11	−0.117732
$561$	0	0
$562$	6.42673e11	0.271755
$563$	1.94665e12	0.816583	0.408291	−	0.912852i	$-0.366125\pi$
$563$	1.94665e12	0.816583	0.408291	+	0.912852i	$0.366125\pi$
$564$	0	0
$565$	−3.24495e11	−0.133965
$566$	−1.43244e12	−0.586681
$567$	0	0
$568$	2.32184e11	0.0935977
$569$	−3.62490e12	−1.44974	−0.724870	−	0.688885i	$-0.758100\pi$
$569$	−3.62490e12	−1.44974	−0.724870	+	0.688885i	$0.758100\pi$
$570$	0	0
$571$	1.55260e12	0.611219	0.305610	−	0.952157i	$-0.401140\pi$
$571$	1.55260e12	0.611219	0.305610	+	0.952157i	$0.401140\pi$
$572$	4.30462e11	0.168133
$573$	0	0
$574$	−4.66037e10	−0.0179191
$575$	−5.67396e11	−0.216462
$576$	0	0
$577$	−3.77593e9	−0.00141818	−0.000709092	−	1.00000i	$-0.500226\pi$
$577$	−3.77593e9	−0.00141818	−0.000709092		1.00000i	$0.500226\pi$
$578$	−6.00667e11	−0.223851
$579$	0	0
$580$	−2.85161e11	−0.104632
$581$	−1.31385e12	−0.478358
$582$	0	0
$583$	1.11673e12	0.400349
$584$	7.07470e11	0.251681
$585$	0	0
$586$	5.90795e11	0.206965
$587$	7.75797e11	0.269697	0.134849	−	0.990866i	$-0.456945\pi$
$587$	7.75797e11	0.269697	0.134849	+	0.990866i	$0.456945\pi$
$588$	0	0
$589$	1.67872e12	0.574725
$590$	−2.38252e11	−0.0809473
$591$	0	0
$592$	1.19774e12	0.400787
$593$	1.81941e12	0.604204	0.302102	−	0.953276i	$-0.402312\pi$
$593$	1.81941e12	0.604204	0.302102	+	0.953276i	$0.402312\pi$
$594$	0	0
$595$	2.87076e11	0.0939010
$596$	3.45533e12	1.12171
$597$	0	0
$598$	3.90588e11	0.124900
$599$	−5.23034e12	−1.66001	−0.830003	−	0.557759i	$-0.811661\pi$
$599$	−5.23034e12	−1.66001	−0.830003	+	0.557759i	$0.811661\pi$
$600$	0	0
$601$	−1.40774e12	−0.440137	−0.220069	−	0.975484i	$-0.570628\pi$
$601$	−1.40774e12	−0.440137	−0.220069	+	0.975484i	$0.570628\pi$
$602$	1.30558e11	0.0405153
$603$	0	0
$604$	2.19730e12	0.671773
$605$	−1.06450e12	−0.323034
$606$	0	0
$607$	−2.91867e12	−0.872642	−0.436321	−	0.899791i	$-0.643719\pi$
$607$	−2.91867e12	−0.872642	−0.436321	+	0.899791i	$0.643719\pi$
$608$	2.68195e12	0.795949
$609$	0	0
$610$	−3.00051e11	−0.0877426
$611$	−6.04891e11	−0.175587
$612$	0	0
$613$	1.73168e11	0.0495331	0.0247665	−	0.999693i	$-0.492116\pi$
$613$	1.73168e11	0.0495331	0.0247665	+	0.999693i	$0.492116\pi$
$614$	7.06129e11	0.200506
$615$	0	0
$616$	4.36732e11	0.122209
$617$	−2.05458e12	−0.570741	−0.285371	−	0.958417i	$-0.592117\pi$
$617$	−2.05458e12	−0.570741	−0.285371	+	0.958417i	$0.592117\pi$
$618$	0	0
$619$	2.56114e12	0.701175	0.350587	−	0.936530i	$-0.385982\pi$
$619$	2.56114e12	0.701175	0.350587	+	0.936530i	$0.385982\pi$
$620$	−8.92436e11	−0.242557
$621$	0	0
$622$	−1.00262e12	−0.268583
$623$	−1.28212e12	−0.340981
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.85751e12	−0.483443
$627$	0	0
$628$	−5.46268e12	−1.40148
$629$	−1.25492e12	−0.319661
$630$	0	0
$631$	−6.93850e12	−1.74234	−0.871171	−	0.490979i	$-0.836639\pi$
$631$	−6.93850e12	−1.74234	−0.871171	+	0.490979i	$0.836639\pi$
$632$	2.86990e12	0.715549
$633$	0	0
$634$	3.29575e11	0.0810126
$635$	1.95254e10	0.00476561
$636$	0	0
$637$	−2.11596e11	−0.0509190
$638$	1.86611e11	0.0445907
$639$	0	0
$640$	−1.85381e12	−0.436773
$641$	−5.64214e12	−1.32003	−0.660013	−	0.751254i	$-0.729449\pi$
$641$	−5.64214e12	−1.32003	−0.660013	+	0.751254i	$0.729449\pi$
$642$	0	0
$643$	−4.29782e11	−0.0991513	−0.0495757	−	0.998770i	$-0.515787\pi$
$643$	−4.29782e11	−0.0991513	−0.0495757	+	0.998770i	$0.515787\pi$
$644$	−1.59844e12	−0.366193
$645$	0	0
$646$	−7.55185e11	−0.170611
$647$	9.46469e11	0.212342	0.106171	−	0.994348i	$-0.466141\pi$
$647$	9.46469e11	0.212342	0.106171	+	0.994348i	$0.466141\pi$
$648$	0	0
$649$	−1.33144e12	−0.294591
$650$	−1.05039e11	−0.0230803
$651$	0	0
$652$	−4.48974e11	−0.0972987
$653$	−6.81646e12	−1.46707	−0.733533	−	0.679654i	$-0.762130\pi$
$653$	−6.81646e12	−1.46707	−0.733533	+	0.679654i	$0.762130\pi$
$654$	0	0
$655$	−2.79212e12	−0.592717
$656$	4.83755e11	0.101990
$657$	0	0
$658$	−2.89878e11	−0.0602836
$659$	−8.07333e11	−0.166751	−0.0833754	−	0.996518i	$-0.526570\pi$
$659$	−8.07333e11	−0.166751	−0.0833754	+	0.996518i	$0.526570\pi$
$660$	0	0
$661$	−2.84512e12	−0.579687	−0.289844	−	0.957074i	$-0.593603\pi$
$661$	−2.84512e12	−0.579687	−0.289844	+	0.957074i	$0.593603\pi$
$662$	9.53226e11	0.192901
$663$	0	0
$664$	−3.88994e12	−0.776580
$665$	−8.08596e11	−0.160337
$666$	0	0
$667$	−1.44597e12	−0.282874
$668$	2.85191e12	0.554169
$669$	0	0
$670$	5.18353e11	0.0993778
$671$	−1.67679e12	−0.319321
$672$	0	0
$673$	−1.47475e12	−0.277110	−0.138555	−	0.990355i	$-0.544246\pi$
$673$	−1.47475e12	−0.277110	−0.138555	+	0.990355i	$0.544246\pi$
$674$	2.61137e12	0.487414
$675$	0	0
$676$	4.24287e12	0.781447
$677$	4.79081e12	0.876517	0.438258	−	0.898849i	$-0.355595\pi$
$677$	4.79081e12	0.876517	0.438258	+	0.898849i	$0.355595\pi$
$678$	0	0
$679$	−3.69042e11	−0.0666288
$680$	8.49949e11	0.152441
$681$	0	0
$682$	5.84014e11	0.103370
$683$	5.98600e11	0.105255	0.0526276	−	0.998614i	$-0.483240\pi$
$683$	5.98600e11	0.105255	0.0526276	+	0.998614i	$0.483240\pi$
$684$	0	0
$685$	−2.60108e12	−0.451384
$686$	−1.01402e11	−0.0174819
$687$	0	0
$688$	−1.35522e12	−0.230601
$689$	−1.60190e12	−0.270801
$690$	0	0
$691$	−4.79258e12	−0.799683	−0.399842	−	0.916584i	$-0.630935\pi$
$691$	−4.79258e12	−0.799683	−0.399842	+	0.916584i	$0.630935\pi$
$692$	2.33591e12	0.387238
$693$	0	0
$694$	−3.28139e12	−0.536957
$695$	−1.16310e12	−0.189097
$696$	0	0
$697$	−5.06854e11	−0.0813458
$698$	1.90150e12	0.303212
$699$	0	0
$700$	4.29863e11	0.0676688
$701$	5.88736e12	0.920851	0.460425	−	0.887698i	$-0.347697\pi$
$701$	5.88736e12	0.920851	0.460425	+	0.887698i	$0.347697\pi$
$702$	0	0
$703$	3.53470e12	0.545826
$704$	−1.45903e12	−0.223865
$705$	0	0
$706$	−1.30880e12	−0.198267
$707$	2.66031e12	0.400447
$708$	0	0
$709$	2.10448e12	0.312778	0.156389	−	0.987696i	$-0.450015\pi$
$709$	2.10448e12	0.312778	0.156389	+	0.987696i	$0.450015\pi$
$710$	−1.49553e11	−0.0220867
$711$	0	0
$712$	−3.79598e12	−0.553559
$713$	−4.52529e12	−0.655758
$714$	0	0
$715$	−5.86999e11	−0.0839963
$716$	−2.82082e12	−0.401113
$717$	0	0
$718$	2.55808e12	0.359214
$719$	3.03420e12	0.423414	0.211707	−	0.977333i	$-0.432098\pi$
$719$	3.03420e12	0.423414	0.211707	+	0.977333i	$0.432098\pi$
$720$	0	0
$721$	1.47958e12	0.203906
$722$	−2.36922e11	−0.0324480
$723$	0	0
$724$	5.07567e12	0.686547
$725$	3.88860e11	0.0522724
$726$	0	0
$727$	−5.06109e12	−0.671953	−0.335976	−	0.941870i	$-0.609066\pi$
$727$	−5.06109e12	−0.671953	−0.335976	+	0.941870i	$0.609066\pi$
$728$	−6.26475e11	−0.0826632
$729$	0	0
$730$	−4.55689e11	−0.0593904
$731$	1.41993e12	0.183924
$732$	0	0
$733$	−6.76795e12	−0.865943	−0.432971	−	0.901408i	$-0.642535\pi$
$733$	−6.76795e12	−0.865943	−0.432971	+	0.901408i	$0.642535\pi$
$734$	2.78256e12	0.353844
$735$	0	0
$736$	−7.22967e12	−0.908172
$737$	2.89675e12	0.361665
$738$	0	0
$739$	−7.58702e12	−0.935774	−0.467887	−	0.883788i	$-0.654985\pi$
$739$	−7.58702e12	−0.935774	−0.467887	+	0.883788i	$0.654985\pi$
$740$	−1.87910e12	−0.230361
$741$	0	0
$742$	−7.67671e11	−0.0929731
$743$	−1.43247e13	−1.72439	−0.862194	−	0.506578i	$-0.830910\pi$
$743$	−1.43247e13	−1.72439	−0.862194	+	0.506578i	$0.830910\pi$
$744$	0	0
$745$	−4.71186e12	−0.560389
$746$	4.01880e12	0.475085
$747$	0	0
$748$	2.24355e12	0.262047
$749$	−2.75066e11	−0.0319351
$750$	0	0
$751$	−6.36145e12	−0.729754	−0.364877	−	0.931056i	$-0.618889\pi$
$751$	−6.36145e12	−0.729754	−0.364877	+	0.931056i	$0.618889\pi$
$752$	3.00899e12	0.343116
$753$	0	0
$754$	−2.67686e11	−0.0301616
$755$	−2.99634e12	−0.335606
$756$	0	0
$757$	−6.68018e12	−0.739361	−0.369681	−	0.929159i	$-0.620533\pi$
$757$	−6.68018e12	−0.739361	−0.369681	+	0.929159i	$0.620533\pi$
$758$	−4.89851e12	−0.538955
$759$	0	0
$760$	−2.39402e12	−0.260296
$761$	−3.88867e12	−0.420310	−0.210155	−	0.977668i	$-0.567397\pi$
$761$	−3.88867e12	−0.420310	−0.210155	+	0.977668i	$0.567397\pi$
$762$	0	0
$763$	9.52617e11	0.101756
$764$	1.48022e12	0.157183
$765$	0	0
$766$	3.29741e12	0.346053
$767$	1.90989e12	0.199265
$768$	0	0
$769$	4.52279e12	0.466378	0.233189	−	0.972431i	$-0.425084\pi$
$769$	4.52279e12	0.466378	0.233189	+	0.972431i	$0.425084\pi$
$770$	−2.81304e11	−0.0288382
$771$	0	0
$772$	−9.44681e12	−0.957211
$773$	−3.10249e12	−0.312538	−0.156269	−	0.987715i	$-0.549947\pi$
$773$	−3.10249e12	−0.312538	−0.156269	+	0.987715i	$0.549947\pi$
$774$	0	0
$775$	1.21697e12	0.121177
$776$	−1.09263e12	−0.108167
$777$	0	0
$778$	3.25499e12	0.318523
$779$	1.42764e12	0.138899
$780$	0	0
$781$	−8.35754e11	−0.0803802
$782$	2.03573e12	0.194666
$783$	0	0
$784$	1.05257e12	0.0995015
$785$	7.44918e12	0.700157
$786$	0	0
$787$	−3.58114e12	−0.332763	−0.166381	−	0.986061i	$-0.553208\pi$
$787$	−3.58114e12	−0.332763	−0.166381	+	0.986061i	$0.553208\pi$
$788$	−2.89233e11	−0.0267227
$789$	0	0
$790$	−1.84853e12	−0.168852
$791$	1.24658e12	0.113221
$792$	0	0
$793$	2.40529e12	0.215992
$794$	−3.81944e12	−0.341042
$795$	0	0
$796$	9.34160e12	0.824731
$797$	1.80656e13	1.58595	0.792976	−	0.609253i	$-0.208531\pi$
$797$	1.80656e13	1.58595	0.792976	+	0.609253i	$0.208531\pi$
$798$	0	0
$799$	−3.15267e12	−0.273664
$800$	1.94425e12	0.167821
$801$	0	0
$802$	−9.38252e11	−0.0800820
$803$	−2.54656e12	−0.216139
$804$	0	0
$805$	2.17971e12	0.182944
$806$	−8.37745e11	−0.0699205
$807$	0	0
$808$	7.87643e12	0.650097
$809$	−1.39461e13	−1.14468	−0.572342	−	0.820015i	$-0.693965\pi$
$809$	−1.39461e13	−1.14468	−0.572342	+	0.820015i	$0.693965\pi$
$810$	0	0
$811$	1.97761e13	1.60526	0.802632	−	0.596475i	$-0.203433\pi$
$811$	1.97761e13	1.60526	0.802632	+	0.596475i	$0.203433\pi$
$812$	1.09548e12	0.0884302
$813$	0	0
$814$	1.22970e12	0.0981720
$815$	6.12243e11	0.0486088
$816$	0	0
$817$	−3.99946e12	−0.314052
$818$	1.37498e11	0.0107376
$819$	0	0
$820$	−7.58954e11	−0.0586210
$821$	−1.45623e13	−1.11863	−0.559315	−	0.828955i	$-0.688936\pi$
$821$	−1.45623e13	−1.11863	−0.559315	+	0.828955i	$0.688936\pi$
$822$	0	0
$823$	−1.18257e13	−0.898523	−0.449261	−	0.893400i	$-0.648313\pi$
$823$	−1.18257e13	−0.898523	−0.449261	+	0.893400i	$0.648313\pi$
$824$	4.38062e12	0.331026
$825$	0	0
$826$	9.15268e11	0.0684129
$827$	1.23993e13	0.921769	0.460885	−	0.887460i	$-0.347532\pi$
$827$	1.23993e13	0.921769	0.460885	+	0.887460i	$0.347532\pi$
$828$	0	0
$829$	1.37128e12	0.100840	0.0504198	−	0.998728i	$-0.483944\pi$
$829$	1.37128e12	0.100840	0.0504198	+	0.998728i	$0.483944\pi$
$830$	2.50555e12	0.183253
$831$	0	0
$832$	2.09292e12	0.151425
$833$	−1.10283e12	−0.0793608
$834$	0	0
$835$	−3.88901e12	−0.276853
$836$	−6.31933e12	−0.447448
$837$	0	0
$838$	8.59471e12	0.602051
$839$	−6.74670e12	−0.470070	−0.235035	−	0.971987i	$-0.575520\pi$
$839$	−6.74670e12	−0.470070	−0.235035	+	0.971987i	$0.575520\pi$
$840$	0	0
$841$	−1.35162e13	−0.931690
$842$	−3.24161e12	−0.222257
$843$	0	0
$844$	7.80923e12	0.529746
$845$	−5.78579e12	−0.390398
$846$	0	0
$847$	4.08940e12	0.273014
$848$	7.96858e12	0.529175
$849$	0	0
$850$	−5.47462e11	−0.0359723
$851$	−9.52840e12	−0.622783
$852$	0	0
$853$	−2.17444e13	−1.40630	−0.703150	−	0.711042i	$-0.748224\pi$
$853$	−2.17444e13	−1.40630	−0.703150	+	0.711042i	$0.748224\pi$
$854$	1.15267e12	0.0741560
$855$	0	0
$856$	−8.14391e11	−0.0518443
$857$	−1.39439e12	−0.0883020	−0.0441510	−	0.999025i	$-0.514058\pi$
$857$	−1.39439e12	−0.0883020	−0.0441510	+	0.999025i	$0.514058\pi$
$858$	0	0
$859$	1.86451e13	1.16841	0.584206	−	0.811606i	$-0.301406\pi$
$859$	1.86451e13	1.16841	0.584206	+	0.811606i	$0.301406\pi$
$860$	2.12618e12	0.132543
$861$	0	0
$862$	4.60082e12	0.283826
$863$	−9.55788e12	−0.586560	−0.293280	−	0.956027i	$-0.594747\pi$
$863$	−9.55788e12	−0.586560	−0.293280	+	0.956027i	$0.594747\pi$
$864$	0	0
$865$	−3.18536e12	−0.193458
$866$	−4.80729e12	−0.290449
$867$	0	0
$868$	3.42838e12	0.204998
$869$	−1.03303e13	−0.614502
$870$	0	0
$871$	−4.15527e12	−0.244634
$872$	2.82043e12	0.165193
$873$	0	0
$874$	−5.73397e12	−0.332395
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−1.79007e13	−1.02182	−0.510908	−	0.859636i	$-0.670691\pi$
$877$	−1.79007e13	−1.02182	−0.510908	+	0.859636i	$0.670691\pi$
$878$	−9.67510e12	−0.549453
$879$	0	0
$880$	2.91999e12	0.164138
$881$	−2.97665e13	−1.66470	−0.832350	−	0.554251i	$-0.813005\pi$
$881$	−2.97665e13	−1.66470	−0.832350	+	0.554251i	$0.813005\pi$
$882$	0	0
$883$	−1.88855e13	−1.04545	−0.522727	−	0.852500i	$-0.675085\pi$
$883$	−1.88855e13	−1.04545	−0.522727	+	0.852500i	$0.675085\pi$
$884$	−3.21828e12	−0.177251
$885$	0	0
$886$	−2.61889e12	−0.142779
$887$	2.66373e13	1.44489	0.722444	−	0.691429i	$-0.243019\pi$
$887$	2.66373e13	1.44489	0.722444	+	0.691429i	$0.243019\pi$
$888$	0	0
$889$	−7.50088e10	−0.00402768
$890$	2.44503e12	0.130626
$891$	0	0
$892$	2.33327e13	1.23402
$893$	8.88001e12	0.467285
$894$	0	0
$895$	3.84661e12	0.200389
$896$	7.12160e12	0.369140
$897$	0	0
$898$	3.10615e12	0.159396
$899$	3.10136e12	0.158356
$900$	0	0
$901$	−8.34906e12	−0.422062
$902$	4.96664e11	0.0249823
$903$	0	0
$904$	3.69077e12	0.183806
$905$	−6.92144e12	−0.342987
$906$	0	0
$907$	1.14021e13	0.559438	0.279719	−	0.960082i	$-0.409759\pi$
$907$	1.14021e13	0.559438	0.279719	+	0.960082i	$0.409759\pi$
$908$	−1.87055e12	−0.0913236
$909$	0	0
$910$	4.03520e11	0.0195065
$911$	−2.33859e13	−1.12492	−0.562459	−	0.826825i	$-0.690144\pi$
$911$	−2.33859e13	−1.12492	−0.562459	+	0.826825i	$0.690144\pi$
$912$	0	0
$913$	1.40019e13	0.666914
$914$	1.01609e13	0.481589
$915$	0	0
$916$	1.86395e13	0.874791
$917$	1.07262e13	0.500938
$918$	0	0
$919$	−1.49151e12	−0.0689775	−0.0344888	−	0.999405i	$-0.510980\pi$
$919$	−1.49151e12	−0.0689775	−0.0344888	+	0.999405i	$0.510980\pi$
$920$	6.45350e12	0.296996
$921$	0	0
$922$	1.84213e12	0.0839519
$923$	1.19886e12	0.0543700
$924$	0	0
$925$	2.56244e12	0.115084
$926$	−8.20124e12	−0.366547
$927$	0	0
$928$	4.95479e12	0.219310
$929$	1.92810e13	0.849296	0.424648	−	0.905359i	$-0.360398\pi$
$929$	1.92810e13	0.849296	0.424648	+	0.905359i	$0.360398\pi$
$930$	0	0
$931$	3.10630e12	0.135510
$932$	−1.79964e13	−0.781295
$933$	0	0
$934$	1.13541e13	0.488192
$935$	−3.05942e12	−0.130914
$936$	0	0
$937$	1.36005e13	0.576405	0.288202	−	0.957570i	$-0.406942\pi$
$937$	1.36005e13	0.576405	0.288202	+	0.957570i	$0.406942\pi$
$938$	−1.99131e12	−0.0839895
$939$	0	0
$940$	−4.72075e12	−0.197213
$941$	−1.29600e13	−0.538829	−0.269415	−	0.963024i	$-0.586830\pi$
$941$	−1.29600e13	−0.538829	−0.269415	+	0.963024i	$0.586830\pi$
$942$	0	0
$943$	−3.84844e12	−0.158483
$944$	−9.50067e12	−0.389386
$945$	0	0
$946$	−1.39138e12	−0.0564854
$947$	4.09790e13	1.65572	0.827859	−	0.560936i	$-0.189558\pi$
$947$	4.09790e13	1.65572	0.827859	+	0.560936i	$0.189558\pi$
$948$	0	0
$949$	3.65293e12	0.146199
$950$	1.54202e12	0.0614233
$951$	0	0
$952$	−3.26516e12	−0.128836
$953$	−1.38232e13	−0.542863	−0.271432	−	0.962458i	$-0.587497\pi$
$953$	−1.38232e13	−0.542863	−0.271432	+	0.962458i	$0.587497\pi$
$954$	0	0
$955$	−2.01850e12	−0.0785258
$956$	−3.21955e13	−1.24662
$957$	0	0
$958$	−2.96925e12	−0.113894
$959$	9.99231e12	0.381489
$960$	0	0
$961$	−1.67336e13	−0.632900
$962$	−1.76395e12	−0.0664046
$963$	0	0
$964$	3.86118e13	1.44004
$965$	1.28821e13	0.478206
$966$	0	0
$967$	−8.43960e12	−0.310387	−0.155193	−	0.987884i	$-0.549600\pi$
$967$	−8.43960e12	−0.310387	−0.155193	+	0.987884i	$0.549600\pi$
$968$	1.21076e13	0.443218
$969$	0	0
$970$	7.03775e11	0.0255247
$971$	−8.03600e12	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$971$	−8.03600e12	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$972$	0	0
$973$	4.46815e12	0.159816
$974$	1.28451e13	0.457321
$975$	0	0
$976$	−1.19650e13	−0.422074
$977$	2.02415e12	0.0710751	0.0355375	−	0.999368i	$-0.488686\pi$
$977$	2.02415e12	0.0710751	0.0355375	+	0.999368i	$0.488686\pi$
$978$	0	0
$979$	1.36637e13	0.475387
$980$	−1.65136e12	−0.0571906
$981$	0	0
$982$	8.02144e12	0.275265
$983$	5.22708e13	1.78554	0.892768	−	0.450518i	$-0.148761\pi$
$983$	5.22708e13	1.78554	0.892768	+	0.450518i	$0.148761\pi$
$984$	0	0
$985$	3.94412e11	0.0133502
$986$	−1.39517e12	−0.0470090
$987$	0	0
$988$	9.06483e12	0.302659
$989$	1.07812e13	0.358332
$990$	0	0
$991$	−4.94346e13	−1.62817	−0.814084	−	0.580747i	$-0.802760\pi$
$991$	−4.94346e13	−1.62817	−0.814084	+	0.580747i	$0.802760\pi$
$992$	1.55064e13	0.508404
$993$	0	0
$994$	5.74521e11	0.0186667
$995$	−1.27387e13	−0.412022
$996$	0	0
$997$	−3.50909e13	−1.12478	−0.562389	−	0.826873i	$-0.690118\pi$
$997$	−3.50909e13	−1.12478	−0.562389	+	0.826873i	$0.690118\pi$
$998$	2.78137e12	0.0887505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.d.1.3		4
3.2	odd	2		105.10.a.e.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.e.1.2	✓	4	3.2	odd	2
315.10.a.d.1.3		4	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+7.32605 q^{2} -458.329 q^{4} +625.000 q^{5} -2401.00 q^{7} -7108.68 q^{8} +4578.78 q^{10} +25587.9 q^{11} -36704.8 q^{13} -17589.8 q^{14} +182586. q^{16} -191304. q^{17} +538840. q^{19} -286456. q^{20} +187458. q^{22} -1.45253e6 q^{23} +390625. q^{25} -268901. q^{26} +1.10045e6 q^{28} +995482. q^{29} +3.11544e6 q^{31} +4.97728e6 q^{32} -1.40150e6 q^{34} -1.50062e6 q^{35} +6.55984e6 q^{37} +3.94756e6 q^{38} -4.44292e6 q^{40} +2.64947e6 q^{41} -7.42236e6 q^{43} -1.17277e7 q^{44} -1.06413e7 q^{46} +1.64799e7 q^{47} +5.76480e6 q^{49} +2.86174e6 q^{50} +1.68229e7 q^{52} +4.36429e7 q^{53} +1.59924e7 q^{55} +1.70679e7 q^{56} +7.29294e6 q^{58} -5.20339e7 q^{59} -6.55307e7 q^{61} +2.28239e7 q^{62} -5.70203e7 q^{64} -2.29405e7 q^{65} +1.13208e8 q^{67} +8.76802e7 q^{68} -1.09936e7 q^{70} -3.26621e7 q^{71} -9.95220e7 q^{73} +4.80577e7 q^{74} -2.46966e8 q^{76} -6.14365e7 q^{77} -4.03718e8 q^{79} +1.14116e8 q^{80} +1.94101e7 q^{82} +5.47210e8 q^{83} -1.19565e8 q^{85} -5.43765e7 q^{86} -1.81896e8 q^{88} +5.33992e8 q^{89} +8.81282e7 q^{91} +6.65739e8 q^{92} +1.20732e8 q^{94} +3.36775e8 q^{95} +1.53704e8 q^{97} +4.22332e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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