LMFDB - Embedded newform 315.10.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,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, 10, names="a")

//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);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
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:	$162.236288392$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 1694x^{3} + 7420x^{2} + 583584x - 2721600 $ x^5 - 2x^4 - 1694x^3 + 7420x^2 + 583584x - 2721600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.68049$ 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)

$f(q)$	$=$	\(q-4.68049 q^{2} -490.093 q^{4} +625.000 q^{5} +2401.00 q^{7} +4690.29 q^{8} -2925.31 q^{10} -61293.3 q^{11} +127788. q^{13} -11237.9 q^{14} +228975. q^{16} +374440. q^{17} +351367. q^{19} -306308. q^{20} +286883. q^{22} -1.55822e6 q^{23} +390625. q^{25} -598109. q^{26} -1.17671e6 q^{28} -3.12932e6 q^{29} +6.54089e6 q^{31} -3.47314e6 q^{32} -1.75256e6 q^{34} +1.50062e6 q^{35} -9.25714e6 q^{37} -1.64457e6 q^{38} +2.93143e6 q^{40} +8.05013e6 q^{41} +1.58627e7 q^{43} +3.00394e7 q^{44} +7.29324e6 q^{46} +8.88373e6 q^{47} +5.76480e6 q^{49} -1.82832e6 q^{50} -6.26278e7 q^{52} +5.68387e7 q^{53} -3.83083e7 q^{55} +1.12614e7 q^{56} +1.46467e7 q^{58} -8.14197e7 q^{59} -2.04282e8 q^{61} -3.06146e7 q^{62} -1.00979e8 q^{64} +7.98673e7 q^{65} +8.88281e7 q^{67} -1.83510e8 q^{68} -7.02366e6 q^{70} -2.24233e8 q^{71} +1.91441e8 q^{73} +4.33280e7 q^{74} -1.72203e8 q^{76} -1.47165e8 q^{77} +2.39411e7 q^{79} +1.43109e8 q^{80} -3.76785e7 q^{82} -4.05879e8 q^{83} +2.34025e8 q^{85} -7.42452e7 q^{86} -2.87483e8 q^{88} -2.03470e8 q^{89} +3.06818e8 q^{91} +7.63674e8 q^{92} -4.15802e7 q^{94} +2.19604e8 q^{95} +9.77776e8 q^{97} -2.69821e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 832 * q^4 + 3125 * q^5 + 12005 * q^7 + 14136 * q^8 - 1250 * q^10 - 10312 * q^11 + 158638 * q^13 - 4802 * q^14 - 526696 * q^16 - 31614 * q^17 + 1655376 * q^19 + 520000 * q^20 + 3659464 * q^22 - 796104 * q^23 + 1953125 * q^25 - 10219060 * q^26 + 1997632 * q^28 + 3353726 * q^29 + 2678120 * q^31 + 130400 * q^32 + 21548612 * q^34 + 7503125 * q^35 + 50994846 * q^37 + 34907600 * q^38 + 8835000 * q^40 - 6330194 * q^41 + 6149468 * q^43 + 71225892 * q^44 - 142845472 * q^46 + 6897780 * q^47 + 28824005 * q^49 - 781250 * q^50 - 22199468 * q^52 + 70886738 * q^53 - 6445000 * q^55 + 33940536 * q^56 - 31955924 * q^58 + 152335168 * q^59 + 257015698 * q^61 + 215582768 * q^62 - 193988432 * q^64 + 99148750 * q^65 + 133467828 * q^67 + 268709380 * q^68 - 3001250 * q^70 - 522788960 * q^71 + 159370858 * q^73 + 244178868 * q^74 - 345056616 * q^76 - 24759112 * q^77 + 464174900 * q^79 - 329185000 * q^80 + 308947620 * q^82 - 2207636832 * q^83 - 19758750 * q^85 - 694560616 * q^86 + 930528736 * q^88 + 1106708326 * q^89 + 380889838 * q^91 - 205160216 * q^92 - 3602501744 * q^94 + 1034610000 * q^95 + 1956142254 * q^97 - 11529602 * 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 $

$2$	−4.68049	−0.206850	−0.103425	−	0.994637i	$-0.532980\pi$
$2$	−4.68049	−0.206850	−0.103425	+	0.994637i	$0.532980\pi$
$3$	0	0
$3$	0	0
$4$	−490.093	−0.957213
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	4690.29	0.404850
$9$	0	0
$10$	−2925.31	−0.0925063
$11$	−61293.3	−1.26225	−0.631126	−	0.775680i	$-0.717407\pi$
$11$	−61293.3	−1.26225	−0.631126	+	0.775680i	$0.717407\pi$
$12$	0	0
$13$	127788.	1.24092	0.620460	−	0.784238i	$-0.286946\pi$
$13$	127788.	1.24092	0.620460	+	0.784238i	$0.286946\pi$
$14$	−11237.9	−0.0781821
$15$	0	0
$16$	228975.	0.873470
$17$	374440.	1.08733	0.543666	−	0.839302i	$-0.317036\pi$
$17$	374440.	1.08733	0.543666	+	0.839302i	$0.317036\pi$
$18$	0	0
$19$	351367.	0.618543	0.309272	−	0.950974i	$-0.399915\pi$
$19$	351367.	0.618543	0.309272	+	0.950974i	$0.399915\pi$
$20$	−306308.	−0.428079
$21$	0	0
$22$	286883.	0.261097
$23$	−1.55822e6	−1.16106	−0.580529	−	0.814239i	$-0.697154\pi$
$23$	−1.55822e6	−1.16106	−0.580529	+	0.814239i	$0.697154\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−598109.	−0.256685
$27$	0	0
$28$	−1.17671e6	−0.361792
$29$	−3.12932e6	−0.821597	−0.410798	−	0.911726i	$-0.634750\pi$
$29$	−3.12932e6	−0.821597	−0.410798	+	0.911726i	$0.634750\pi$
$30$	0	0
$31$	6.54089e6	1.27206	0.636032	−	0.771662i	$-0.280574\pi$
$31$	6.54089e6	1.27206	0.636032	+	0.771662i	$0.280574\pi$
$32$	−3.47314e6	−0.585528
$33$	0	0
$34$	−1.75256e6	−0.224915
$35$	1.50062e6	0.169031
$36$	0	0
$37$	−9.25714e6	−0.812025	−0.406012	−	0.913868i	$-0.633081\pi$
$37$	−9.25714e6	−0.812025	−0.406012	+	0.913868i	$0.633081\pi$
$38$	−1.64457e6	−0.127946
$39$	0	0
$40$	2.93143e6	0.181055
$41$	8.05013e6	0.444913	0.222457	−	0.974943i	$-0.428592\pi$
$41$	8.05013e6	0.444913	0.222457	+	0.974943i	$0.428592\pi$
$42$	0	0
$43$	1.58627e7	0.707570	0.353785	−	0.935327i	$-0.384895\pi$
$43$	1.58627e7	0.707570	0.353785	+	0.935327i	$0.384895\pi$
$44$	3.00394e7	1.20824
$45$	0	0
$46$	7.29324e6	0.240165
$47$	8.88373e6	0.265555	0.132778	−	0.991146i	$-0.457610\pi$
$47$	8.88373e6	0.265555	0.132778	+	0.991146i	$0.457610\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.82832e6	−0.0413701
$51$	0	0
$52$	−6.26278e7	−1.18782
$53$	5.68387e7	0.989471	0.494735	−	0.869044i	$-0.335265\pi$
$53$	5.68387e7	0.989471	0.494735	+	0.869044i	$0.335265\pi$
$54$	0	0
$55$	−3.83083e7	−0.564496
$56$	1.12614e7	0.153019
$57$	0	0
$58$	1.46467e7	0.169948
$59$	−8.14197e7	−0.874773	−0.437387	−	0.899274i	$-0.644096\pi$
$59$	−8.14197e7	−0.874773	−0.437387	+	0.899274i	$0.644096\pi$
$60$	0	0
$61$	−2.04282e8	−1.88906	−0.944531	−	0.328421i	$-0.893483\pi$
$61$	−2.04282e8	−1.88906	−0.944531	+	0.328421i	$0.893483\pi$
$62$	−3.06146e7	−0.263127
$63$	0	0
$64$	−1.00979e8	−0.752353
$65$	7.98673e7	0.554956
$66$	0	0
$67$	8.88281e7	0.538535	0.269267	−	0.963065i	$-0.413218\pi$
$67$	8.88281e7	0.538535	0.269267	+	0.963065i	$0.413218\pi$
$68$	−1.83510e8	−1.04081
$69$	0	0
$70$	−7.02366e6	−0.0349641
$71$	−2.24233e8	−1.04722	−0.523609	−	0.851959i	$-0.675415\pi$
$71$	−2.24233e8	−1.04722	−0.523609	+	0.851959i	$0.675415\pi$
$72$	0	0
$73$	1.91441e8	0.789010	0.394505	−	0.918894i	$-0.370916\pi$
$73$	1.91441e8	0.789010	0.394505	+	0.918894i	$0.370916\pi$
$74$	4.33280e7	0.167968
$75$	0	0
$76$	−1.72203e8	−0.592077
$77$	−1.47165e8	−0.477086
$78$	0	0
$79$	2.39411e7	0.0691549	0.0345774	−	0.999402i	$-0.488991\pi$
$79$	2.39411e7	0.0691549	0.0345774	+	0.999402i	$0.488991\pi$
$80$	1.43109e8	0.390627
$81$	0	0
$82$	−3.76785e7	−0.0920305
$83$	−4.05879e8	−0.938741	−0.469370	−	0.883001i	$-0.655519\pi$
$83$	−4.05879e8	−0.938741	−0.469370	+	0.883001i	$0.655519\pi$
$84$	0	0
$85$	2.34025e8	0.486270
$86$	−7.42452e7	−0.146361
$87$	0	0
$88$	−2.87483e8	−0.511023
$89$	−2.03470e8	−0.343752	−0.171876	−	0.985119i	$-0.554983\pi$
$89$	−2.03470e8	−0.343752	−0.171876	+	0.985119i	$0.554983\pi$
$90$	0	0
$91$	3.06818e8	0.469024
$92$	7.63674e8	1.11138
$93$	0	0
$94$	−4.15802e7	−0.0549302
$95$	2.19604e8	0.276621
$96$	0	0
$97$	9.77776e8	1.12142	0.560708	−	0.828014i	$-0.310529\pi$
$97$	9.77776e8	1.12142	0.560708	+	0.828014i	$0.310529\pi$
$98$	−2.69821e7	−0.0295501
$99$	0	0
$100$	−1.91443e8	−0.191443
$101$	−1.21059e9	−1.15758	−0.578792	−	0.815475i	$-0.696476\pi$
$101$	−1.21059e9	−1.15758	−0.578792	+	0.815475i	$0.696476\pi$
$102$	0	0
$103$	1.67608e9	1.46733	0.733666	−	0.679511i	$-0.237808\pi$
$103$	1.67608e9	1.46733	0.733666	+	0.679511i	$0.237808\pi$
$104$	5.99361e8	0.502387
$105$	0	0
$106$	−2.66033e8	−0.204672
$107$	1.18347e9	0.872831	0.436416	−	0.899745i	$-0.356248\pi$
$107$	1.18347e9	0.872831	0.436416	+	0.899745i	$0.356248\pi$
$108$	0	0
$109$	−2.42544e9	−1.64578	−0.822888	−	0.568203i	$-0.807639\pi$
$109$	−2.42544e9	−1.64578	−0.822888	+	0.568203i	$0.807639\pi$
$110$	1.79302e8	0.116766
$111$	0	0
$112$	5.49768e8	0.330140
$113$	−1.29095e9	−0.744828	−0.372414	−	0.928067i	$-0.621470\pi$
$113$	−1.29095e9	−0.744828	−0.372414	+	0.928067i	$0.621470\pi$
$114$	0	0
$115$	−9.73889e8	−0.519241
$116$	1.53366e9	0.786443
$117$	0	0
$118$	3.81084e8	0.180947
$119$	8.99030e8	0.410973
$120$	0	0
$121$	1.39892e9	0.593280
$122$	9.56141e8	0.390753
$123$	0	0
$124$	−3.20564e9	−1.21764
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	4.57285e9	1.55980	0.779902	−	0.625902i	$-0.215269\pi$
$127$	4.57285e9	1.55980	0.779902	+	0.625902i	$0.215269\pi$
$128$	2.25088e9	0.741152
$129$	0	0
$130$	−3.73818e8	−0.114793
$131$	−1.97356e9	−0.585504	−0.292752	−	0.956188i	$-0.594571\pi$
$131$	−1.97356e9	−0.585504	−0.292752	+	0.956188i	$0.594571\pi$
$132$	0	0
$133$	8.43632e8	0.233787
$134$	−4.15759e8	−0.111396
$135$	0	0
$136$	1.75623e9	0.440207
$137$	−2.91623e9	−0.707260	−0.353630	−	0.935385i	$-0.615053\pi$
$137$	−2.91623e9	−0.707260	−0.353630	+	0.935385i	$0.615053\pi$
$138$	0	0
$139$	1.68574e8	0.0383023	0.0191512	−	0.999817i	$-0.493904\pi$
$139$	1.68574e8	0.0383023	0.0191512	+	0.999817i	$0.493904\pi$
$140$	−7.35446e8	−0.161799
$141$	0	0
$142$	1.04952e9	0.216617
$143$	−7.83253e9	−1.56635
$144$	0	0
$145$	−1.95582e9	−0.367429
$146$	−8.96039e8	−0.163207
$147$	0	0
$148$	4.53686e9	0.777281
$149$	1.03146e10	1.71440	0.857202	−	0.514980i	$-0.172201\pi$
$149$	1.03146e10	1.71440	0.857202	+	0.514980i	$0.172201\pi$
$150$	0	0
$151$	−6.51350e9	−1.01957	−0.509786	−	0.860301i	$-0.670276\pi$
$151$	−6.51350e9	−1.01957	−0.509786	+	0.860301i	$0.670276\pi$
$152$	1.64801e9	0.250417
$153$	0	0
$154$	6.88805e8	0.0986855
$155$	4.08806e9	0.568885
$156$	0	0
$157$	1.06861e10	1.40369	0.701843	−	0.712332i	$-0.252361\pi$
$157$	1.06861e10	1.40369	0.701843	+	0.712332i	$0.252361\pi$
$158$	−1.12056e8	−0.0143047
$159$	0	0
$160$	−2.17071e9	−0.261856
$161$	−3.74129e9	−0.438839
$162$	0	0
$163$	1.42299e10	1.57891	0.789457	−	0.613806i	$-0.210362\pi$
$163$	1.42299e10	1.57891	0.789457	+	0.613806i	$0.210362\pi$
$164$	−3.94531e9	−0.425877
$165$	0	0
$166$	1.89971e9	0.194179
$167$	−2.78646e9	−0.277223	−0.138611	−	0.990347i	$-0.544264\pi$
$167$	−2.78646e9	−0.277223	−0.138611	+	0.990347i	$0.544264\pi$
$168$	0	0
$169$	5.72518e9	0.539882
$170$	−1.09535e9	−0.100585
$171$	0	0
$172$	−7.77420e9	−0.677295
$173$	1.71904e10	1.45908	0.729540	−	0.683938i	$-0.239734\pi$
$173$	1.71904e10	1.45908	0.729540	+	0.683938i	$0.239734\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−1.40346e10	−1.10254
$177$	0	0
$178$	9.52340e8	0.0711053
$179$	2.31226e10	1.68344	0.841719	−	0.539915i	$-0.181544\pi$
$179$	2.31226e10	1.68344	0.841719	+	0.539915i	$0.181544\pi$
$180$	0	0
$181$	2.28943e10	1.58553	0.792763	−	0.609531i	$-0.208642\pi$
$181$	2.28943e10	1.58553	0.792763	+	0.609531i	$0.208642\pi$
$182$	−1.43606e9	−0.0970177
$183$	0	0
$184$	−7.30851e9	−0.470055
$185$	−5.78572e9	−0.363149
$186$	0	0
$187$	−2.29507e10	−1.37249
$188$	−4.35386e9	−0.254193
$189$	0	0
$190$	−1.02786e9	−0.0572191
$191$	1.66664e10	0.906131	0.453065	−	0.891477i	$-0.350330\pi$
$191$	1.66664e10	0.906131	0.453065	+	0.891477i	$0.350330\pi$
$192$	0	0
$193$	−2.30952e10	−1.19816	−0.599079	−	0.800690i	$-0.704466\pi$
$193$	−2.30952e10	−1.19816	−0.599079	+	0.800690i	$0.704466\pi$
$194$	−4.57647e9	−0.231965
$195$	0	0
$196$	−2.82529e9	−0.136745
$197$	2.42166e10	1.14555	0.572776	−	0.819712i	$-0.305866\pi$
$197$	2.42166e10	1.14555	0.572776	+	0.819712i	$0.305866\pi$
$198$	0	0
$199$	−2.18408e10	−0.987257	−0.493628	−	0.869673i	$-0.664330\pi$
$199$	−2.18408e10	−0.987257	−0.493628	+	0.869673i	$0.664330\pi$
$200$	1.83214e9	0.0809700
$201$	0	0
$202$	5.66617e9	0.239447
$203$	−7.51349e9	−0.310534
$204$	0	0
$205$	5.03133e9	0.198971
$206$	−7.84489e9	−0.303518
$207$	0	0
$208$	2.92601e10	1.08391
$209$	−2.15365e10	−0.780757
$210$	0	0
$211$	7.72507e9	0.268307	0.134153	−	0.990961i	$-0.457169\pi$
$211$	7.72507e9	0.268307	0.134153	+	0.990961i	$0.457169\pi$
$212$	−2.78563e10	−0.947134
$213$	0	0
$214$	−5.53922e9	−0.180545
$215$	9.91419e9	0.316435
$216$	0	0
$217$	1.57047e10	0.480795
$218$	1.13522e10	0.340429
$219$	0	0
$220$	1.87746e10	0.540343
$221$	4.78488e10	1.34929
$222$	0	0
$223$	2.17022e10	0.587668	0.293834	−	0.955856i	$-0.405069\pi$
$223$	2.17022e10	0.587668	0.293834	+	0.955856i	$0.405069\pi$
$224$	−8.33901e9	−0.221309
$225$	0	0
$226$	6.04227e9	0.154068
$227$	−4.79669e10	−1.19902	−0.599508	−	0.800369i	$-0.704637\pi$
$227$	−4.79669e10	−1.19902	−0.599508	+	0.800369i	$0.704637\pi$
$228$	0	0
$229$	4.12350e10	0.990846	0.495423	−	0.868652i	$-0.335013\pi$
$229$	4.12350e10	0.990846	0.495423	+	0.868652i	$0.335013\pi$
$230$	4.55828e9	0.107405
$231$	0	0
$232$	−1.46774e10	−0.332624
$233$	7.25284e10	1.61215	0.806077	−	0.591811i	$-0.201587\pi$
$233$	7.25284e10	1.61215	0.806077	+	0.591811i	$0.201587\pi$
$234$	0	0
$235$	5.55233e9	0.118760
$236$	3.99032e10	0.837344
$237$	0	0
$238$	−4.20790e9	−0.0850099
$239$	−8.27096e10	−1.63970	−0.819852	−	0.572576i	$-0.805944\pi$
$239$	−8.27096e10	−1.63970	−0.819852	+	0.572576i	$0.805944\pi$
$240$	0	0
$241$	5.66304e10	1.08137	0.540683	−	0.841226i	$-0.318166\pi$
$241$	5.66304e10	1.08137	0.540683	+	0.841226i	$0.318166\pi$
$242$	−6.54765e9	−0.122720
$243$	0	0
$244$	1.00117e11	1.80824
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	4.49004e10	0.767562
$248$	3.06786e10	0.514996
$249$	0	0
$250$	−1.14270e9	−0.0185013
$251$	4.37000e10	0.694944	0.347472	−	0.937690i	$-0.387040\pi$
$251$	4.37000e10	0.694944	0.347472	+	0.937690i	$0.387040\pi$
$252$	0	0
$253$	9.55086e10	1.46555
$254$	−2.14032e10	−0.322646
$255$	0	0
$256$	4.11661e10	0.599045
$257$	1.04178e11	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$257$	1.04178e11	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$258$	0	0
$259$	−2.22264e10	−0.306917
$260$	−3.91424e10	−0.531211
$261$	0	0
$262$	9.23723e9	0.121112
$263$	1.04346e11	1.34485	0.672424	−	0.740166i	$-0.265253\pi$
$263$	1.04346e11	1.34485	0.672424	+	0.740166i	$0.265253\pi$
$264$	0	0
$265$	3.55242e10	0.442505
$266$	−3.94861e9	−0.0483590
$267$	0	0
$268$	−4.35340e10	−0.515493
$269$	−9.25092e10	−1.07721	−0.538604	−	0.842559i	$-0.681048\pi$
$269$	−9.25092e10	−1.07721	−0.538604	+	0.842559i	$0.681048\pi$
$270$	0	0
$271$	−5.41823e10	−0.610233	−0.305117	−	0.952315i	$-0.598695\pi$
$271$	−5.41823e10	−0.610233	−0.305117	+	0.952315i	$0.598695\pi$
$272$	8.57373e10	0.949751
$273$	0	0
$274$	1.36494e10	0.146297
$275$	−2.39427e10	−0.252450
$276$	0	0
$277$	1.18973e11	1.21420	0.607099	−	0.794627i	$-0.292333\pi$
$277$	1.18973e11	1.21420	0.607099	+	0.794627i	$0.292333\pi$
$278$	−7.89011e8	−0.00792285
$279$	0	0
$280$	7.03836e9	0.0684322
$281$	7.86550e10	0.752572	0.376286	−	0.926504i	$-0.377201\pi$
$281$	7.86550e10	0.752572	0.376286	+	0.926504i	$0.377201\pi$
$282$	0	0
$283$	−8.35155e10	−0.773977	−0.386989	−	0.922085i	$-0.626485\pi$
$283$	−8.35155e10	−0.773977	−0.386989	+	0.922085i	$0.626485\pi$
$284$	1.09895e11	1.00241
$285$	0	0
$286$	3.66601e10	0.324001
$287$	1.93284e10	0.168161
$288$	0	0
$289$	2.16174e10	0.182290
$290$	9.15421e9	0.0760029
$291$	0	0
$292$	−9.38240e10	−0.755251
$293$	6.91745e10	0.548330	0.274165	−	0.961683i	$-0.411599\pi$
$293$	6.91745e10	0.548330	0.274165	+	0.961683i	$0.411599\pi$
$294$	0	0
$295$	−5.08873e10	−0.391210
$296$	−4.34187e10	−0.328748
$297$	0	0
$298$	−4.82773e10	−0.354625
$299$	−1.99122e11	−1.44078
$300$	0	0
$301$	3.80864e10	0.267436
$302$	3.04864e10	0.210899
$303$	0	0
$304$	8.04542e10	0.540278
$305$	−1.27676e11	−0.844814
$306$	0	0
$307$	1.72821e10	0.111039	0.0555194	−	0.998458i	$-0.482319\pi$
$307$	1.72821e10	0.111039	0.0555194	+	0.998458i	$0.482319\pi$
$308$	7.21247e10	0.456673
$309$	0	0
$310$	−1.91341e10	−0.117674
$311$	2.24570e11	1.36123	0.680614	−	0.732642i	$-0.261713\pi$
$311$	2.24570e11	1.36123	0.680614	+	0.732642i	$0.261713\pi$
$312$	0	0
$313$	6.15640e10	0.362558	0.181279	−	0.983432i	$-0.441976\pi$
$313$	6.15640e10	0.362558	0.181279	+	0.983432i	$0.441976\pi$
$314$	−5.00161e10	−0.290353
$315$	0	0
$316$	−1.17334e10	−0.0661959
$317$	1.03866e11	0.577705	0.288853	−	0.957374i	$-0.406726\pi$
$317$	1.03866e11	0.577705	0.288853	+	0.957374i	$0.406726\pi$
$318$	0	0
$319$	1.91806e11	1.03706
$320$	−6.31119e10	−0.336462
$321$	0	0
$322$	1.75111e10	0.0907740
$323$	1.31566e11	0.672561
$324$	0	0
$325$	4.99170e10	0.248184
$326$	−6.66030e10	−0.326599
$327$	0	0
$328$	3.77574e10	0.180123
$329$	2.13298e10	0.100371
$330$	0	0
$331$	1.19600e11	0.547651	0.273826	−	0.961779i	$-0.411711\pi$
$331$	1.19600e11	0.547651	0.273826	+	0.961779i	$0.411711\pi$
$332$	1.98919e11	0.898575
$333$	0	0
$334$	1.30420e10	0.0573436
$335$	5.55176e10	0.240840
$336$	0	0
$337$	1.78854e11	0.755379	0.377690	−	0.925932i	$-0.376719\pi$
$337$	1.78854e11	0.755379	0.377690	+	0.925932i	$0.376719\pi$
$338$	−2.67967e10	−0.111675
$339$	0	0
$340$	−1.14694e11	−0.465464
$341$	−4.00913e11	−1.60567
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	7.44006e10	0.286460
$345$	0	0
$346$	−8.04596e10	−0.301811
$347$	−2.90904e11	−1.07713	−0.538563	−	0.842585i	$-0.681033\pi$
$347$	−2.90904e11	−1.07713	−0.538563	+	0.842585i	$0.681033\pi$
$348$	0	0
$349$	−1.20156e11	−0.433543	−0.216772	−	0.976222i	$-0.569553\pi$
$349$	−1.20156e11	−0.433543	−0.216772	+	0.976222i	$0.569553\pi$
$350$	−4.38979e9	−0.0156364
$351$	0	0
$352$	2.12880e11	0.739084
$353$	−2.68637e11	−0.920831	−0.460415	−	0.887704i	$-0.652300\pi$
$353$	−2.68637e11	−0.920831	−0.460415	+	0.887704i	$0.652300\pi$
$354$	0	0
$355$	−1.40146e11	−0.468330
$356$	9.97193e10	0.329044
$357$	0	0
$358$	−1.08225e11	−0.348220
$359$	−4.23529e11	−1.34573	−0.672865	−	0.739765i	$-0.734937\pi$
$359$	−4.23529e11	−1.34573	−0.672865	+	0.739765i	$0.734937\pi$
$360$	0	0
$361$	−1.99229e11	−0.617405
$362$	−1.07156e11	−0.327966
$363$	0	0
$364$	−1.50369e11	−0.448956
$365$	1.19651e11	0.352856
$366$	0	0
$367$	−3.14338e11	−0.904480	−0.452240	−	0.891896i	$-0.649375\pi$
$367$	−3.14338e11	−0.904480	−0.452240	+	0.891896i	$0.649375\pi$
$368$	−3.56794e11	−1.01415
$369$	0	0
$370$	2.70800e10	0.0751174
$371$	1.36470e11	0.373985
$372$	0	0
$373$	4.22978e11	1.13143	0.565715	−	0.824601i	$-0.308600\pi$
$373$	4.22978e11	1.13143	0.565715	+	0.824601i	$0.308600\pi$
$374$	1.07420e11	0.283899
$375$	0	0
$376$	4.16673e10	0.107510
$377$	−3.99888e11	−1.01954
$378$	0	0
$379$	−4.54590e11	−1.13173	−0.565866	−	0.824497i	$-0.691458\pi$
$379$	−4.54590e11	−1.13173	−0.565866	+	0.824497i	$0.691458\pi$
$380$	−1.07627e11	−0.264785
$381$	0	0
$382$	−7.80068e10	−0.187433
$383$	2.56165e11	0.608311	0.304155	−	0.952622i	$-0.401626\pi$
$383$	2.56165e11	0.608311	0.304155	+	0.952622i	$0.401626\pi$
$384$	0	0
$385$	−9.19783e10	−0.213360
$386$	1.08097e11	0.247839
$387$	0	0
$388$	−4.79201e11	−1.07343
$389$	−4.07620e9	−0.00902572	−0.00451286	−	0.999990i	$-0.501436\pi$
$389$	−4.07620e9	−0.00902572	−0.00451286	+	0.999990i	$0.501436\pi$
$390$	0	0
$391$	−5.83461e11	−1.26246
$392$	2.70386e10	0.0578357
$393$	0	0
$394$	−1.13346e11	−0.236958
$395$	1.49632e10	0.0309270
$396$	0	0
$397$	5.25740e11	1.06222	0.531109	−	0.847303i	$-0.321775\pi$
$397$	5.25740e11	1.06222	0.531109	+	0.847303i	$0.321775\pi$
$398$	1.02226e11	0.204214
$399$	0	0
$400$	8.94433e10	0.174694
$401$	4.96583e11	0.959051	0.479526	−	0.877528i	$-0.340809\pi$
$401$	4.96583e11	0.959051	0.479526	+	0.877528i	$0.340809\pi$
$402$	0	0
$403$	8.35845e11	1.57853
$404$	5.93304e11	1.10805
$405$	0	0
$406$	3.51668e10	0.0642341
$407$	5.67401e11	1.02498
$408$	0	0
$409$	−6.22845e11	−1.10059	−0.550294	−	0.834971i	$-0.685484\pi$
$409$	−6.22845e11	−1.10059	−0.550294	+	0.834971i	$0.685484\pi$
$410$	−2.35491e10	−0.0411573
$411$	0	0
$412$	−8.21437e11	−1.40455
$413$	−1.95489e11	−0.330633
$414$	0	0
$415$	−2.53675e11	−0.419818
$416$	−4.43824e11	−0.726593
$417$	0	0
$418$	1.00801e11	0.161500
$419$	9.02432e11	1.43038	0.715190	−	0.698930i	$-0.246340\pi$
$419$	9.02432e11	1.43038	0.715190	+	0.698930i	$0.246340\pi$
$420$	0	0
$421$	1.13768e12	1.76502	0.882509	−	0.470296i	$-0.155853\pi$
$421$	1.13768e12	1.76502	0.882509	+	0.470296i	$0.155853\pi$
$422$	−3.61571e10	−0.0554993
$423$	0	0
$424$	2.66590e11	0.400587
$425$	1.46266e11	0.217466
$426$	0	0
$427$	−4.90482e11	−0.713999
$428$	−5.80010e11	−0.835485
$429$	0	0
$430$	−4.64033e10	−0.0654547
$431$	−4.68899e11	−0.654533	−0.327266	−	0.944932i	$-0.606127\pi$
$431$	−4.68899e11	−0.654533	−0.327266	+	0.944932i	$0.606127\pi$
$432$	0	0
$433$	3.39136e11	0.463638	0.231819	−	0.972759i	$-0.425532\pi$
$433$	3.39136e11	0.463638	0.231819	+	0.972759i	$0.425532\pi$
$434$	−7.35056e10	−0.0994527
$435$	0	0
$436$	1.18869e12	1.57536
$437$	−5.47508e11	−0.718165
$438$	0	0
$439$	−8.34074e11	−1.07180	−0.535901	−	0.844281i	$-0.680028\pi$
$439$	−8.34074e11	−1.07180	−0.535901	+	0.844281i	$0.680028\pi$
$440$	−1.79677e11	−0.228536
$441$	0	0
$442$	−2.23956e11	−0.279101
$443$	8.87244e11	1.09453	0.547263	−	0.836961i	$-0.315670\pi$
$443$	8.87244e11	1.09453	0.547263	+	0.836961i	$0.315670\pi$
$444$	0	0
$445$	−1.27169e11	−0.153731
$446$	−1.01577e11	−0.121559
$447$	0	0
$448$	−2.42451e11	−0.284363
$449$	2.68409e9	0.00311665	0.00155832	−	0.999999i	$-0.499504\pi$
$449$	2.68409e9	0.00311665	0.00155832	+	0.999999i	$0.499504\pi$
$450$	0	0
$451$	−4.93419e11	−0.561593
$452$	6.32685e11	0.712959
$453$	0	0
$454$	2.24509e11	0.248017
$455$	1.91761e11	0.209754
$456$	0	0
$457$	9.37032e11	1.00492	0.502460	−	0.864600i	$-0.332428\pi$
$457$	9.37032e11	1.00492	0.502460	+	0.864600i	$0.332428\pi$
$458$	−1.93000e11	−0.204957
$459$	0	0
$460$	4.77296e11	0.497024
$461$	2.76095e11	0.284711	0.142356	−	0.989816i	$-0.454532\pi$
$461$	2.76095e11	0.284711	0.142356	+	0.989816i	$0.454532\pi$
$462$	0	0
$463$	−1.47542e11	−0.149211	−0.0746055	−	0.997213i	$-0.523770\pi$
$463$	−1.47542e11	−0.149211	−0.0746055	+	0.997213i	$0.523770\pi$
$464$	−7.16535e11	−0.717640
$465$	0	0
$466$	−3.39468e11	−0.333475
$467$	−1.62968e12	−1.58554	−0.792770	−	0.609521i	$-0.791362\pi$
$467$	−1.62968e12	−1.58554	−0.792770	+	0.609521i	$0.791362\pi$
$468$	0	0
$469$	2.13276e11	0.203547
$470$	−2.59876e10	−0.0245656
$471$	0	0
$472$	−3.81882e11	−0.354152
$473$	−9.72278e11	−0.893131
$474$	0	0
$475$	1.37253e11	0.123709
$476$	−4.40609e11	−0.393388
$477$	0	0
$478$	3.87121e11	0.339173
$479$	−1.04475e12	−0.906779	−0.453389	−	0.891313i	$-0.649785\pi$
$479$	−1.04475e12	−0.906779	−0.453389	+	0.891313i	$0.649785\pi$
$480$	0	0
$481$	−1.18295e12	−1.00766
$482$	−2.65058e11	−0.223681
$483$	0	0
$484$	−6.85603e11	−0.567895
$485$	6.11110e11	0.501512
$486$	0	0
$487$	−6.55197e11	−0.527827	−0.263913	−	0.964546i	$-0.585013\pi$
$487$	−6.55197e11	−0.527827	−0.263913	+	0.964546i	$0.585013\pi$
$488$	−9.58142e11	−0.764787
$489$	0	0
$490$	−1.68638e10	−0.0132152
$491$	1.69481e12	1.31600	0.657998	−	0.753019i	$-0.271403\pi$
$491$	1.69481e12	1.31600	0.657998	+	0.753019i	$0.271403\pi$
$492$	0	0
$493$	−1.17174e12	−0.893348
$494$	−2.10156e11	−0.158771
$495$	0	0
$496$	1.49770e12	1.11111
$497$	−5.38383e11	−0.395811
$498$	0	0
$499$	−9.93558e11	−0.717366	−0.358683	−	0.933459i	$-0.616774\pi$
$499$	−9.93558e11	−0.717366	−0.358683	+	0.933459i	$0.616774\pi$
$500$	−1.19652e11	−0.0856157
$501$	0	0
$502$	−2.04537e11	−0.143749
$503$	−2.36079e11	−0.164438	−0.0822188	−	0.996614i	$-0.526201\pi$
$503$	−2.36079e11	−0.164438	−0.0822188	+	0.996614i	$0.526201\pi$
$504$	0	0
$505$	−7.56621e11	−0.517687
$506$	−4.47027e11	−0.303149
$507$	0	0
$508$	−2.24112e12	−1.49306
$509$	−6.12223e11	−0.404277	−0.202139	−	0.979357i	$-0.564789\pi$
$509$	−6.12223e11	−0.404277	−0.202139	+	0.979357i	$0.564789\pi$
$510$	0	0
$511$	4.59651e11	0.298218
$512$	−1.34513e12	−0.865065
$513$	0	0
$514$	−4.87605e11	−0.308130
$515$	1.04755e12	0.656211
$516$	0	0
$517$	−5.44514e11	−0.335198
$518$	1.04030e11	0.0634858
$519$	0	0
$520$	3.74600e11	0.224674
$521$	3.50205e11	0.208235	0.104117	−	0.994565i	$-0.466798\pi$
$521$	3.50205e11	0.208235	0.104117	+	0.994565i	$0.466798\pi$
$522$	0	0
$523$	−2.61442e12	−1.52798	−0.763989	−	0.645229i	$-0.776762\pi$
$523$	−2.61442e12	−1.52798	−0.763989	+	0.645229i	$0.776762\pi$
$524$	9.67229e11	0.560452
$525$	0	0
$526$	−4.88388e11	−0.278182
$527$	2.44917e12	1.38316
$528$	0	0
$529$	6.26903e11	0.348057
$530$	−1.66271e11	−0.0915323
$531$	0	0
$532$	−4.13458e11	−0.223784
$533$	1.02871e12	0.552102
$534$	0	0
$535$	7.39669e11	0.390342
$536$	4.16629e11	0.218026
$537$	0	0
$538$	4.32988e11	0.222821
$539$	−3.53344e11	−0.180322
$540$	0	0
$541$	−1.47804e12	−0.741821	−0.370911	−	0.928669i	$-0.620954\pi$
$541$	−1.47804e12	−0.741821	−0.370911	+	0.928669i	$0.620954\pi$
$542$	2.53600e11	0.126227
$543$	0	0
$544$	−1.30048e12	−0.636663
$545$	−1.51590e12	−0.736014
$546$	0	0
$547$	7.35485e11	0.351262	0.175631	−	0.984456i	$-0.443804\pi$
$547$	7.35485e11	0.351262	0.175631	+	0.984456i	$0.443804\pi$
$548$	1.42922e12	0.676999
$549$	0	0
$550$	1.12064e11	0.0522195
$551$	−1.09954e12	−0.508193
$552$	0	0
$553$	5.74827e10	0.0261381
$554$	−5.56852e11	−0.251157
$555$	0	0
$556$	−8.26172e10	−0.0366635
$557$	9.41535e11	0.414465	0.207233	−	0.978292i	$-0.433554\pi$
$557$	9.41535e11	0.414465	0.207233	+	0.978292i	$0.433554\pi$
$558$	0	0
$559$	2.02706e12	0.878037
$560$	3.43605e11	0.147643
$561$	0	0
$562$	−3.68144e11	−0.155670
$563$	−3.00632e12	−1.26110	−0.630548	−	0.776151i	$-0.717170\pi$
$563$	−3.00632e12	−1.26110	−0.630548	+	0.776151i	$0.717170\pi$
$564$	0	0
$565$	−8.06843e11	−0.333097
$566$	3.90893e11	0.160097
$567$	0	0
$568$	−1.05172e12	−0.423966
$569$	−3.42996e12	−1.37178	−0.685889	−	0.727706i	$-0.740587\pi$
$569$	−3.42996e12	−1.37178	−0.685889	+	0.727706i	$0.740587\pi$
$570$	0	0
$571$	−3.90075e12	−1.53563	−0.767814	−	0.640673i	$-0.778656\pi$
$571$	−3.90075e12	−1.53563	−0.767814	+	0.640673i	$0.778656\pi$
$572$	3.83867e12	1.49933
$573$	0	0
$574$	−9.04662e10	−0.0347843
$575$	−6.08680e11	−0.232212
$576$	0	0
$577$	6.26050e11	0.235135	0.117568	−	0.993065i	$-0.462490\pi$
$577$	6.26050e11	0.235135	0.117568	+	0.993065i	$0.462490\pi$
$578$	−1.01180e11	−0.0377069
$579$	0	0
$580$	9.58535e11	0.351708
$581$	−9.74517e11	−0.354811
$582$	0	0
$583$	−3.48383e12	−1.24896
$584$	8.97914e11	0.319431
$585$	0	0
$586$	−3.23771e11	−0.113422
$587$	−1.68145e12	−0.584538	−0.292269	−	0.956336i	$-0.594410\pi$
$587$	−1.68145e12	−0.584538	−0.292269	+	0.956336i	$0.594410\pi$
$588$	0	0
$589$	2.29825e12	0.786827
$590$	2.38177e11	0.0809220
$591$	0	0
$592$	−2.11965e12	−0.709279
$593$	1.58522e12	0.526434	0.263217	−	0.964737i	$-0.415217\pi$
$593$	1.58522e12	0.526434	0.263217	+	0.964737i	$0.415217\pi$
$594$	0	0
$595$	5.61894e11	0.183793
$596$	−5.05510e12	−1.64105
$597$	0	0
$598$	9.31986e11	0.298026
$599$	4.78087e12	1.51735	0.758675	−	0.651469i	$-0.225847\pi$
$599$	4.78087e12	1.51735	0.758675	+	0.651469i	$0.225847\pi$
$600$	0	0
$601$	2.13180e10	0.00666516	0.00333258	−	0.999994i	$-0.498939\pi$
$601$	2.13180e10	0.00666516	0.00333258	+	0.999994i	$0.498939\pi$
$602$	−1.78263e11	−0.0553193
$603$	0	0
$604$	3.19222e12	0.975948
$605$	8.74327e11	0.265323
$606$	0	0
$607$	3.71643e12	1.11116	0.555580	−	0.831463i	$-0.312496\pi$
$607$	3.71643e12	1.11116	0.555580	+	0.831463i	$0.312496\pi$
$608$	−1.22035e12	−0.362174
$609$	0	0
$610$	5.97588e11	0.174750
$611$	1.13523e12	0.329533
$612$	0	0
$613$	2.28399e12	0.653315	0.326657	−	0.945143i	$-0.394078\pi$
$613$	2.28399e12	0.653315	0.326657	+	0.945143i	$0.394078\pi$
$614$	−8.08889e10	−0.0229684
$615$	0	0
$616$	−6.90247e11	−0.193149
$617$	−4.11730e12	−1.14374	−0.571872	−	0.820343i	$-0.693783\pi$
$617$	−4.11730e12	−1.14374	−0.571872	+	0.820343i	$0.693783\pi$
$618$	0	0
$619$	5.99191e12	1.64043	0.820214	−	0.572057i	$-0.193854\pi$
$619$	5.99191e12	1.64043	0.820214	+	0.572057i	$0.193854\pi$
$620$	−2.00353e12	−0.544544
$621$	0	0
$622$	−1.05110e12	−0.281571
$623$	−4.88532e11	−0.129926
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−2.88150e11	−0.0749952
$627$	0	0
$628$	−5.23717e12	−1.34363
$629$	−3.46625e12	−0.882940
$630$	0	0
$631$	4.14396e12	1.04060	0.520300	−	0.853984i	$-0.325820\pi$
$631$	4.14396e12	1.04060	0.520300	+	0.853984i	$0.325820\pi$
$632$	1.12291e11	0.0279974
$633$	0	0
$634$	−4.86143e11	−0.119498
$635$	2.85803e12	0.697565
$636$	0	0
$637$	7.36670e11	0.177274
$638$	−8.97747e11	−0.214517
$639$	0	0
$640$	1.40680e12	0.331453
$641$	3.87177e12	0.905833	0.452917	−	0.891553i	$-0.350383\pi$
$641$	3.87177e12	0.905833	0.452917	+	0.891553i	$0.350383\pi$
$642$	0	0
$643$	6.81081e12	1.57127	0.785633	−	0.618693i	$-0.212338\pi$
$643$	6.81081e12	1.57127	0.785633	+	0.618693i	$0.212338\pi$
$644$	1.83358e12	0.420062
$645$	0	0
$646$	−6.15793e11	−0.139120
$647$	−2.40797e12	−0.540233	−0.270117	−	0.962828i	$-0.587062\pi$
$647$	−2.40797e12	−0.540233	−0.270117	+	0.962828i	$0.587062\pi$
$648$	0	0
$649$	4.99048e12	1.10418
$650$	−2.33636e11	−0.0513370
$651$	0	0
$652$	−6.97399e12	−1.51136
$653$	4.90892e12	1.05652	0.528259	−	0.849083i	$-0.322845\pi$
$653$	4.90892e12	1.05652	0.528259	+	0.849083i	$0.322845\pi$
$654$	0	0
$655$	−1.23348e12	−0.261845
$656$	1.84328e12	0.388618
$657$	0	0
$658$	−9.98341e10	−0.0207617
$659$	3.36518e11	0.0695062	0.0347531	−	0.999396i	$-0.488936\pi$
$659$	3.36518e11	0.0695062	0.0347531	+	0.999396i	$0.488936\pi$
$660$	0	0
$661$	−5.58058e11	−0.113703	−0.0568516	−	0.998383i	$-0.518106\pi$
$661$	−5.58058e11	−0.113703	−0.0568516	+	0.998383i	$0.518106\pi$
$662$	−5.59785e11	−0.113282
$663$	0	0
$664$	−1.90369e12	−0.380049
$665$	5.27270e11	0.104553
$666$	0	0
$667$	4.87617e12	0.953922
$668$	1.36563e12	0.265361
$669$	0	0
$670$	−2.59849e11	−0.0498179
$671$	1.25211e13	2.38447
$672$	0	0
$673$	−3.26341e12	−0.613202	−0.306601	−	0.951838i	$-0.599192\pi$
$673$	−3.26341e12	−0.613202	−0.306601	+	0.951838i	$0.599192\pi$
$674$	−8.37126e11	−0.156250
$675$	0	0
$676$	−2.80587e12	−0.516782
$677$	2.71325e12	0.496410	0.248205	−	0.968708i	$-0.420159\pi$
$677$	2.71325e12	0.496410	0.248205	+	0.968708i	$0.420159\pi$
$678$	0	0
$679$	2.34764e12	0.423855
$680$	1.09764e12	0.196866
$681$	0	0
$682$	1.87647e12	0.332133
$683$	−2.42887e12	−0.427082	−0.213541	−	0.976934i	$-0.568500\pi$
$683$	−2.42887e12	−0.427082	−0.213541	+	0.976934i	$0.568500\pi$
$684$	0	0
$685$	−1.82264e12	−0.316296
$686$	−6.47840e10	−0.0111689
$687$	0	0
$688$	3.63216e12	0.618041
$689$	7.26329e12	1.22785
$690$	0	0
$691$	5.97125e12	0.996355	0.498178	−	0.867075i	$-0.334003\pi$
$691$	5.97125e12	0.996355	0.498178	+	0.867075i	$0.334003\pi$
$692$	−8.42491e12	−1.39665
$693$	0	0
$694$	1.36157e12	0.222804
$695$	1.05359e11	0.0171293
$696$	0	0
$697$	3.01429e12	0.483768
$698$	5.62391e11	0.0896786
$699$	0	0
$700$	−4.59654e11	−0.0723585
$701$	4.18398e12	0.654423	0.327212	−	0.944951i	$-0.393891\pi$
$701$	4.18398e12	0.654423	0.327212	+	0.944951i	$0.393891\pi$
$702$	0	0
$703$	−3.25266e12	−0.502272
$704$	6.18934e12	0.949659
$705$	0	0
$706$	1.25735e12	0.190474
$707$	−2.90664e12	−0.437525
$708$	0	0
$709$	−8.83893e12	−1.31369	−0.656843	−	0.754027i	$-0.728109\pi$
$709$	−8.83893e12	−1.31369	−0.656843	+	0.754027i	$0.728109\pi$
$710$	6.55950e11	0.0968742
$711$	0	0
$712$	−9.54333e11	−0.139168
$713$	−1.01922e13	−1.47694
$714$	0	0
$715$	−4.89533e12	−0.700495
$716$	−1.13322e13	−1.61141
$717$	0	0
$718$	1.98232e12	0.278365
$719$	2.92264e12	0.407845	0.203922	−	0.978987i	$-0.434631\pi$
$719$	2.92264e12	0.407845	0.203922	+	0.978987i	$0.434631\pi$
$720$	0	0
$721$	4.02428e12	0.554599
$722$	9.32489e11	0.127710
$723$	0	0
$724$	−1.12203e13	−1.51769
$725$	−1.22239e12	−0.164319
$726$	0	0
$727$	−1.21723e13	−1.61610	−0.808048	−	0.589117i	$-0.799476\pi$
$727$	−1.21723e13	−1.61610	−0.808048	+	0.589117i	$0.799476\pi$
$728$	1.43906e12	0.189884
$729$	0	0
$730$	−5.60024e11	−0.0729884
$731$	5.93963e12	0.769363
$732$	0	0
$733$	1.08103e13	1.38315	0.691574	−	0.722305i	$-0.256918\pi$
$733$	1.08103e13	1.38315	0.691574	+	0.722305i	$0.256918\pi$
$734$	1.47125e12	0.187092
$735$	0	0
$736$	5.41192e12	0.679832
$737$	−5.44457e12	−0.679767
$738$	0	0
$739$	9.83935e12	1.21357	0.606787	−	0.794864i	$-0.292458\pi$
$739$	9.83935e12	1.21357	0.606787	+	0.794864i	$0.292458\pi$
$740$	2.83554e12	0.347610
$741$	0	0
$742$	−6.38745e11	−0.0773589
$743$	1.84462e12	0.222053	0.111027	−	0.993817i	$-0.464586\pi$
$743$	1.84462e12	0.222053	0.111027	+	0.993817i	$0.464586\pi$
$744$	0	0
$745$	6.44661e12	0.766705
$746$	−1.97974e12	−0.234037
$747$	0	0
$748$	1.12480e13	1.31376
$749$	2.84151e12	0.329899
$750$	0	0
$751$	1.55156e13	1.77987	0.889936	−	0.456085i	$-0.150749\pi$
$751$	1.55156e13	1.77987	0.889936	+	0.456085i	$0.150749\pi$
$752$	2.03415e12	0.231955
$753$	0	0
$754$	1.87167e12	0.210891
$755$	−4.07094e12	−0.455967
$756$	0	0
$757$	−1.10594e12	−0.122406	−0.0612029	−	0.998125i	$-0.519494\pi$
$757$	−1.10594e12	−0.122406	−0.0612029	+	0.998125i	$0.519494\pi$
$758$	2.12770e12	0.234099
$759$	0	0
$760$	1.03001e12	0.111990
$761$	−1.94850e12	−0.210606	−0.105303	−	0.994440i	$-0.533581\pi$
$761$	−1.94850e12	−0.210606	−0.105303	+	0.994440i	$0.533581\pi$
$762$	0	0
$763$	−5.82347e12	−0.622045
$764$	−8.16807e12	−0.867360
$765$	0	0
$766$	−1.19898e12	−0.125829
$767$	−1.04044e13	−1.08552
$768$	0	0
$769$	−1.56599e13	−1.61481	−0.807405	−	0.589998i	$-0.799129\pi$
$769$	−1.56599e13	−1.61481	−0.807405	+	0.589998i	$0.799129\pi$
$770$	4.30503e11	0.0441335
$771$	0	0
$772$	1.13188e13	1.14689
$773$	−1.42340e13	−1.43390	−0.716949	−	0.697126i	$-0.754462\pi$
$773$	−1.42340e13	−1.43390	−0.716949	+	0.697126i	$0.754462\pi$
$774$	0	0
$775$	2.55504e12	0.254413
$776$	4.58605e12	0.454005
$777$	0	0
$778$	1.90786e10	0.00186697
$779$	2.82855e12	0.275198
$780$	0	0
$781$	1.37440e13	1.32185
$782$	2.73088e12	0.261139
$783$	0	0
$784$	1.31999e12	0.124781
$785$	6.67880e12	0.627747
$786$	0	0
$787$	1.81218e13	1.68390	0.841949	−	0.539556i	$-0.181408\pi$
$787$	1.81218e13	1.68390	0.841949	+	0.539556i	$0.181408\pi$
$788$	−1.18684e13	−1.09654
$789$	0	0
$790$	−7.00351e10	−0.00639726
$791$	−3.09957e12	−0.281518
$792$	0	0
$793$	−2.61047e13	−2.34418
$794$	−2.46072e12	−0.219720
$795$	0	0
$796$	1.07040e13	0.945015
$797$	2.21087e13	1.94089	0.970446	−	0.241319i	$-0.0775799\pi$
$797$	2.21087e13	1.94089	0.970446	+	0.241319i	$0.0775799\pi$
$798$	0	0
$799$	3.32643e12	0.288747
$800$	−1.35670e12	−0.117106
$801$	0	0
$802$	−2.32425e12	−0.198380
$803$	−1.17341e13	−0.995930
$804$	0	0
$805$	−2.33831e12	−0.196255
$806$	−3.91216e12	−0.326520
$807$	0	0
$808$	−5.67803e12	−0.468648
$809$	−1.80822e12	−0.148417	−0.0742084	−	0.997243i	$-0.523643\pi$
$809$	−1.80822e12	−0.148417	−0.0742084	+	0.997243i	$0.523643\pi$
$810$	0	0
$811$	−1.59062e13	−1.29113	−0.645567	−	0.763703i	$-0.723379\pi$
$811$	−1.59062e13	−1.29113	−0.645567	+	0.763703i	$0.723379\pi$
$812$	3.68231e12	0.297247
$813$	0	0
$814$	−2.65572e12	−0.212017
$815$	8.89370e12	0.706112
$816$	0	0
$817$	5.57363e12	0.437662
$818$	2.91522e12	0.227657
$819$	0	0
$820$	−2.46582e12	−0.190458
$821$	1.49953e13	1.15189	0.575945	−	0.817489i	$-0.304634\pi$
$821$	1.49953e13	1.15189	0.575945	+	0.817489i	$0.304634\pi$
$822$	0	0
$823$	2.41121e13	1.83204	0.916020	−	0.401132i	$-0.131383\pi$
$823$	2.41121e13	1.83204	0.916020	+	0.401132i	$0.131383\pi$
$824$	7.86131e12	0.594049
$825$	0	0
$826$	9.14983e11	0.0683916
$827$	3.51697e12	0.261453	0.130727	−	0.991418i	$-0.458269\pi$
$827$	3.51697e12	0.261453	0.130727	+	0.991418i	$0.458269\pi$
$828$	0	0
$829$	−2.21258e13	−1.62706	−0.813531	−	0.581522i	$-0.802457\pi$
$829$	−2.21258e13	−1.62706	−0.813531	+	0.581522i	$0.802457\pi$
$830$	1.18732e12	0.0868394
$831$	0	0
$832$	−1.29039e13	−0.933610
$833$	2.15857e12	0.155333
$834$	0	0
$835$	−1.74154e12	−0.123978
$836$	1.05549e13	0.747351
$837$	0	0
$838$	−4.22382e12	−0.295875
$839$	−4.84005e12	−0.337226	−0.168613	−	0.985682i	$-0.553929\pi$
$839$	−4.84005e12	−0.337226	−0.168613	+	0.985682i	$0.553929\pi$
$840$	0	0
$841$	−4.71452e12	−0.324979
$842$	−5.32488e12	−0.365095
$843$	0	0
$844$	−3.78600e12	−0.256827
$845$	3.57824e12	0.241443
$846$	0	0
$847$	3.35882e12	0.224239
$848$	1.30146e13	0.864272
$849$	0	0
$850$	−6.84595e11	−0.0449830
$851$	1.44247e13	0.942808
$852$	0	0
$853$	2.65572e13	1.71756	0.858779	−	0.512345i	$-0.171223\pi$
$853$	2.65572e13	1.71756	0.858779	+	0.512345i	$0.171223\pi$
$854$	2.29569e12	0.147691
$855$	0	0
$856$	5.55081e12	0.353366
$857$	1.86512e13	1.18112	0.590560	−	0.806994i	$-0.298907\pi$
$857$	1.86512e13	1.18112	0.590560	+	0.806994i	$0.298907\pi$
$858$	0	0
$859$	1.19252e13	0.747305	0.373652	−	0.927569i	$-0.378105\pi$
$859$	1.19252e13	0.747305	0.373652	+	0.927569i	$0.378105\pi$
$860$	−4.85888e12	−0.302895
$861$	0	0
$862$	2.19468e12	0.135390
$863$	1.97885e13	1.21441	0.607205	−	0.794545i	$-0.292291\pi$
$863$	1.97885e13	1.21441	0.607205	+	0.794545i	$0.292291\pi$
$864$	0	0
$865$	1.07440e13	0.652521
$866$	−1.58732e12	−0.0959037
$867$	0	0
$868$	−7.69675e12	−0.460223
$869$	−1.46743e12	−0.0872909
$870$	0	0
$871$	1.13511e13	0.668279
$872$	−1.13760e13	−0.666293
$873$	0	0
$874$	2.56261e12	0.148553
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−3.21002e12	−0.183235	−0.0916177	−	0.995794i	$-0.529204\pi$
$877$	−3.21002e12	−0.183235	−0.0916177	+	0.995794i	$0.529204\pi$
$878$	3.90387e12	0.221702
$879$	0	0
$880$	−8.77164e12	−0.493070
$881$	3.43067e13	1.91862	0.959308	−	0.282363i	$-0.0911182\pi$
$881$	3.43067e13	1.91862	0.959308	+	0.282363i	$0.0911182\pi$
$882$	0	0
$883$	−2.27401e13	−1.25884	−0.629418	−	0.777067i	$-0.716707\pi$
$883$	−2.27401e13	−1.25884	−0.629418	+	0.777067i	$0.716707\pi$
$884$	−2.34504e13	−1.29156
$885$	0	0
$886$	−4.15274e12	−0.226403
$887$	−3.40833e13	−1.84878	−0.924390	−	0.381448i	$-0.875426\pi$
$887$	−3.40833e13	−1.84878	−0.924390	+	0.381448i	$0.875426\pi$
$888$	0	0
$889$	1.09794e13	0.589550
$890$	5.95212e11	0.0317993
$891$	0	0
$892$	−1.06361e13	−0.562524
$893$	3.12145e12	0.164257
$894$	0	0
$895$	1.44516e13	0.752857
$896$	5.40436e12	0.280129
$897$	0	0
$898$	−1.25628e10	−0.000644680 0
$899$	−2.04685e13	−1.04512
$900$	0	0
$901$	2.12827e13	1.07588
$902$	2.30944e12	0.116166
$903$	0	0
$904$	−6.05492e12	−0.301544
$905$	1.43089e13	0.709068
$906$	0	0
$907$	−2.39990e13	−1.17750	−0.588750	−	0.808315i	$-0.700380\pi$
$907$	−2.39990e13	−1.17750	−0.588750	+	0.808315i	$0.700380\pi$
$908$	2.35082e13	1.14771
$909$	0	0
$910$	−8.97537e11	−0.0433876
$911$	−1.81078e13	−0.871029	−0.435515	−	0.900182i	$-0.643434\pi$
$911$	−1.81078e13	−0.871029	−0.435515	+	0.900182i	$0.643434\pi$
$912$	0	0
$913$	2.48777e13	1.18493
$914$	−4.38577e12	−0.207868
$915$	0	0
$916$	−2.02090e13	−0.948451
$917$	−4.73852e12	−0.221300
$918$	0	0
$919$	−1.93468e12	−0.0894724	−0.0447362	−	0.998999i	$-0.514245\pi$
$919$	−1.93468e12	−0.0894724	−0.0447362	+	0.998999i	$0.514245\pi$
$920$	−4.56782e12	−0.210215
$921$	0	0
$922$	−1.29226e12	−0.0588926
$923$	−2.86542e13	−1.29951
$924$	0	0
$925$	−3.61607e12	−0.162405
$926$	6.90568e11	0.0308644
$927$	0	0
$928$	1.08686e13	0.481068
$929$	−2.56977e13	−1.13194	−0.565971	−	0.824425i	$-0.691499\pi$
$929$	−2.56977e13	−1.13194	−0.565971	+	0.824425i	$0.691499\pi$
$930$	0	0
$931$	2.02556e12	0.0883633
$932$	−3.55457e13	−1.54317
$933$	0	0
$934$	7.62771e12	0.327969
$935$	−1.43442e13	−0.613795
$936$	0	0
$937$	2.26453e13	0.959731	0.479865	−	0.877342i	$-0.340686\pi$
$937$	2.26453e13	0.959731	0.479865	+	0.877342i	$0.340686\pi$
$938$	−9.98237e11	−0.0421038
$939$	0	0
$940$	−2.72116e12	−0.113679
$941$	−2.28097e13	−0.948345	−0.474173	−	0.880432i	$-0.657253\pi$
$941$	−2.28097e13	−0.948345	−0.474173	+	0.880432i	$0.657253\pi$
$942$	0	0
$943$	−1.25439e13	−0.516570
$944$	−1.86431e13	−0.764088
$945$	0	0
$946$	4.55074e12	0.184745
$947$	−3.24218e12	−0.130997	−0.0654987	−	0.997853i	$-0.520864\pi$
$947$	−3.24218e12	−0.130997	−0.0654987	+	0.997853i	$0.520864\pi$
$948$	0	0
$949$	2.44638e13	0.979099
$950$	−6.42410e11	−0.0255892
$951$	0	0
$952$	4.21671e12	0.166382
$953$	4.96092e13	1.94825	0.974124	−	0.226014i	$-0.0725697\pi$
$953$	4.96092e13	1.94825	0.974124	+	0.226014i	$0.0725697\pi$
$954$	0	0
$955$	1.04165e13	0.405234
$956$	4.05354e13	1.56955
$957$	0	0
$958$	4.88993e12	0.187567
$959$	−7.00187e12	−0.267319
$960$	0	0
$961$	1.63436e13	0.618149
$962$	5.53678e12	0.208434
$963$	0	0
$964$	−2.77542e13	−1.03510
$965$	−1.44345e13	−0.535832
$966$	0	0
$967$	−2.45015e13	−0.901100	−0.450550	−	0.892751i	$-0.648772\pi$
$967$	−2.45015e13	−0.901100	−0.450550	+	0.892751i	$0.648772\pi$
$968$	6.56135e12	0.240190
$969$	0	0
$970$	−2.86029e12	−0.103738
$971$	1.85016e13	0.667917	0.333958	−	0.942588i	$-0.391616\pi$
$971$	1.85016e13	0.667917	0.333958	+	0.942588i	$0.391616\pi$
$972$	0	0
$973$	4.04747e11	0.0144769
$974$	3.06664e12	0.109181
$975$	0	0
$976$	−4.67755e13	−1.65004
$977$	2.19913e13	0.772193	0.386096	−	0.922458i	$-0.373823\pi$
$977$	2.19913e13	0.772193	0.386096	+	0.922458i	$0.373823\pi$
$978$	0	0
$979$	1.24714e13	0.433902
$980$	−1.76581e12	−0.0611541
$981$	0	0
$982$	−7.93255e12	−0.272214
$983$	−1.25865e12	−0.0429946	−0.0214973	−	0.999769i	$-0.506843\pi$
$983$	−1.25865e12	−0.0429946	−0.0214973	+	0.999769i	$0.506843\pi$
$984$	0	0
$985$	1.51354e13	0.512307
$986$	5.48432e12	0.184789
$987$	0	0
$988$	−2.20054e13	−0.734721
$989$	−2.47176e13	−0.821530
$990$	0	0
$991$	−5.19617e12	−0.171140	−0.0855701	−	0.996332i	$-0.527271\pi$
$991$	−5.19617e12	−0.171140	−0.0855701	+	0.996332i	$0.527271\pi$
$992$	−2.27174e13	−0.744829
$993$	0	0
$994$	2.51990e12	0.0818736
$995$	−1.36505e13	−0.441515
$996$	0	0
$997$	5.63563e12	0.180640	0.0903200	−	0.995913i	$-0.471211\pi$
$997$	5.63563e12	0.180640	0.0903200	+	0.995913i	$0.471211\pi$
$998$	4.65034e12	0.148387
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.j.1.3		5
3.2	odd	2		35.10.a.d.1.3	✓	5
15.2	even	4		175.10.b.f.99.6		10
15.8	even	4		175.10.b.f.99.5		10
15.14	odd	2		175.10.a.f.1.3		5
21.20	even	2		245.10.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.d.1.3	✓	5	3.2	odd	2
175.10.a.f.1.3		5	15.14	odd	2
175.10.b.f.99.5		10	15.8	even	4
175.10.b.f.99.6		10	15.2	even	4
245.10.a.f.1.3		5	21.20	even	2
315.10.a.j.1.3		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.68049 q^{2} -490.093 q^{4} +625.000 q^{5} +2401.00 q^{7} +4690.29 q^{8} -2925.31 q^{10} -61293.3 q^{11} +127788. q^{13} -11237.9 q^{14} +228975. q^{16} +374440. q^{17} +351367. q^{19} -306308. q^{20} +286883. q^{22} -1.55822e6 q^{23} +390625. q^{25} -598109. q^{26} -1.17671e6 q^{28} -3.12932e6 q^{29} +6.54089e6 q^{31} -3.47314e6 q^{32} -1.75256e6 q^{34} +1.50062e6 q^{35} -9.25714e6 q^{37} -1.64457e6 q^{38} +2.93143e6 q^{40} +8.05013e6 q^{41} +1.58627e7 q^{43} +3.00394e7 q^{44} +7.29324e6 q^{46} +8.88373e6 q^{47} +5.76480e6 q^{49} -1.82832e6 q^{50} -6.26278e7 q^{52} +5.68387e7 q^{53} -3.83083e7 q^{55} +1.12614e7 q^{56} +1.46467e7 q^{58} -8.14197e7 q^{59} -2.04282e8 q^{61} -3.06146e7 q^{62} -1.00979e8 q^{64} +7.98673e7 q^{65} +8.88281e7 q^{67} -1.83510e8 q^{68} -7.02366e6 q^{70} -2.24233e8 q^{71} +1.91441e8 q^{73} +4.33280e7 q^{74} -1.72203e8 q^{76} -1.47165e8 q^{77} +2.39411e7 q^{79} +1.43109e8 q^{80} -3.76785e7 q^{82} -4.05879e8 q^{83} +2.34025e8 q^{85} -7.42452e7 q^{86} -2.87483e8 q^{88} -2.03470e8 q^{89} +3.06818e8 q^{91} +7.63674e8 q^{92} -4.15802e7 q^{94} +2.19604e8 q^{95} +9.77776e8 q^{97} -2.69821e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 832 * q^4 + 3125 * q^5 + 12005 * q^7 + 14136 * q^8 - 1250 * q^10 - 10312 * q^11 + 158638 * q^13 - 4802 * q^14 - 526696 * q^16 - 31614 * q^17 + 1655376 * q^19 + 520000 * q^20 + 3659464 * q^22 - 796104 * q^23 + 1953125 * q^25 - 10219060 * q^26 + 1997632 * q^28 + 3353726 * q^29 + 2678120 * q^31 + 130400 * q^32 + 21548612 * q^34 + 7503125 * q^35 + 50994846 * q^37 + 34907600 * q^38 + 8835000 * q^40 - 6330194 * q^41 + 6149468 * q^43 + 71225892 * q^44 - 142845472 * q^46 + 6897780 * q^47 + 28824005 * q^49 - 781250 * q^50 - 22199468 * q^52 + 70886738 * q^53 - 6445000 * q^55 + 33940536 * q^56 - 31955924 * q^58 + 152335168 * q^59 + 257015698 * q^61 + 215582768 * q^62 - 193988432 * q^64 + 99148750 * q^65 + 133467828 * q^67 + 268709380 * q^68 - 3001250 * q^70 - 522788960 * q^71 + 159370858 * q^73 + 244178868 * q^74 - 345056616 * q^76 - 24759112 * q^77 + 464174900 * q^79 - 329185000 * q^80 + 308947620 * q^82 - 2207636832 * q^83 - 19758750 * q^85 - 694560616 * q^86 + 930528736 * q^88 + 1106708326 * q^89 + 380889838 * q^91 - 205160216 * q^92 - 3602501744 * q^94 + 1034610000 * q^95 + 1956142254 * q^97 - 11529602 * q^98

Introduction

L-functions

Modular forms

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Database

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