LMFDB - Embedded newform 315.10.a.j.1.1

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			1.1
Root			$32.5415$ 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)

$f(q)$	$=$	\(q-32.5415 q^{2} +546.947 q^{4} +625.000 q^{5} +2401.00 q^{7} -1137.23 q^{8} -20338.4 q^{10} -38584.6 q^{11} +137768. q^{13} -78132.1 q^{14} -243030. q^{16} +18339.0 q^{17} -927311. q^{19} +341842. q^{20} +1.25560e6 q^{22} +1.39114e6 q^{23} +390625. q^{25} -4.48318e6 q^{26} +1.31322e6 q^{28} -200634. q^{29} -7.24758e6 q^{31} +8.49081e6 q^{32} -596776. q^{34} +1.50062e6 q^{35} +1.43970e7 q^{37} +3.01761e7 q^{38} -710767. q^{40} -9.75951e6 q^{41} -5.93288e6 q^{43} -2.11037e7 q^{44} -4.52699e7 q^{46} +3.11103e7 q^{47} +5.76480e6 q^{49} -1.27115e7 q^{50} +7.53519e7 q^{52} +6.15094e7 q^{53} -2.41154e7 q^{55} -2.73048e6 q^{56} +6.52893e6 q^{58} +7.30772e7 q^{59} +2.09425e8 q^{61} +2.35847e8 q^{62} -1.51872e8 q^{64} +8.61051e7 q^{65} -3.06068e8 q^{67} +1.00304e7 q^{68} -4.88325e7 q^{70} -1.80955e8 q^{71} +3.56483e8 q^{73} -4.68501e8 q^{74} -5.07190e8 q^{76} -9.26416e7 q^{77} +1.54673e8 q^{79} -1.51894e8 q^{80} +3.17589e8 q^{82} -7.36731e8 q^{83} +1.14618e7 q^{85} +1.93065e8 q^{86} +4.38794e7 q^{88} -2.13463e8 q^{89} +3.30781e8 q^{91} +7.60882e8 q^{92} -1.01238e9 q^{94} -5.79569e8 q^{95} -3.57772e8 q^{97} -1.87595e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−32.5415	−1.43814	−0.719072	−	0.694936i	$-0.755433\pi$
$2$	−32.5415	−1.43814	−0.719072	+	0.694936i	$0.755433\pi$
$3$	0	0
$3$	0	0
$4$	546.947	1.06826
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−1137.23	−0.0981617
$9$	0	0
$10$	−20338.4	−0.643157
$11$	−38584.6	−0.794597	−0.397298	−	0.917690i	$-0.630052\pi$
$11$	−38584.6	−0.794597	−0.397298	+	0.917690i	$0.630052\pi$
$12$	0	0
$13$	137768.	1.33784	0.668920	−	0.743335i	$-0.266757\pi$
$13$	137768.	1.33784	0.668920	+	0.743335i	$0.266757\pi$
$14$	−78132.1	−0.543567
$15$	0	0
$16$	−243030.	−0.927085
$17$	18339.0	0.0532543	0.0266271	−	0.999645i	$-0.491523\pi$
$17$	18339.0	0.0532543	0.0266271	+	0.999645i	$0.491523\pi$
$18$	0	0
$19$	−927311.	−1.63243	−0.816214	−	0.577750i	$-0.803931\pi$
$19$	−927311.	−1.63243	−0.816214	+	0.577750i	$0.803931\pi$
$20$	341842.	0.477739
$21$	0	0
$22$	1.25560e6	1.14274
$23$	1.39114e6	1.03657	0.518283	−	0.855209i	$-0.326571\pi$
$23$	1.39114e6	1.03657	0.518283	+	0.855209i	$0.326571\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−4.48318e6	−1.92400
$27$	0	0
$28$	1.31322e6	0.403763
$29$	−200634.	−0.0526761	−0.0263381	−	0.999653i	$-0.508385\pi$
$29$	−200634.	−0.0526761	−0.0263381	+	0.999653i	$0.508385\pi$
$30$	0	0
$31$	−7.24758e6	−1.40950	−0.704751	−	0.709455i	$-0.748941\pi$
$31$	−7.24758e6	−1.40950	−0.704751	+	0.709455i	$0.748941\pi$
$32$	8.49081e6	1.43144
$33$	0	0
$34$	−596776.	−0.0765873
$35$	1.50062e6	0.169031
$36$	0	0
$37$	1.43970e7	1.26289	0.631445	−	0.775421i	$-0.282462\pi$
$37$	1.43970e7	1.26289	0.631445	+	0.775421i	$0.282462\pi$
$38$	3.01761e7	2.34767
$39$	0	0
$40$	−710767.	−0.0438992
$41$	−9.75951e6	−0.539387	−0.269694	−	0.962946i	$-0.586922\pi$
$41$	−9.75951e6	−0.539387	−0.269694	+	0.962946i	$0.586922\pi$
$42$	0	0
$43$	−5.93288e6	−0.264641	−0.132321	−	0.991207i	$-0.542243\pi$
$43$	−5.93288e6	−0.264641	−0.132321	+	0.991207i	$0.542243\pi$
$44$	−2.11037e7	−0.848832
$45$	0	0
$46$	−4.52699e7	−1.49073
$47$	3.11103e7	0.929960	0.464980	−	0.885321i	$-0.346062\pi$
$47$	3.11103e7	0.929960	0.464980	+	0.885321i	$0.346062\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.27115e7	−0.287629
$51$	0	0
$52$	7.53519e7	1.42915
$53$	6.15094e7	1.07078	0.535390	−	0.844605i	$-0.320165\pi$
$53$	6.15094e7	1.07078	0.535390	+	0.844605i	$0.320165\pi$
$54$	0	0
$55$	−2.41154e7	−0.355354
$56$	−2.73048e6	−0.0371016
$57$	0	0
$58$	6.52893e6	0.0757558
$59$	7.30772e7	0.785141	0.392571	−	0.919722i	$-0.371586\pi$
$59$	7.30772e7	0.785141	0.392571	+	0.919722i	$0.371586\pi$
$60$	0	0
$61$	2.09425e8	1.93662	0.968312	−	0.249745i	$-0.0803466\pi$
$61$	2.09425e8	1.93662	0.968312	+	0.249745i	$0.0803466\pi$
$62$	2.35847e8	2.02706
$63$	0	0
$64$	−1.51872e8	−1.13153
$65$	8.61051e7	0.598300
$66$	0	0
$67$	−3.06068e8	−1.85559	−0.927794	−	0.373092i	$-0.878298\pi$
$67$	−3.06068e8	−1.85559	−0.927794	+	0.373092i	$0.878298\pi$
$68$	1.00304e7	0.0568892
$69$	0	0
$70$	−4.88325e7	−0.243091
$71$	−1.80955e8	−0.845098	−0.422549	−	0.906340i	$-0.638864\pi$
$71$	−1.80955e8	−0.845098	−0.422549	+	0.906340i	$0.638864\pi$
$72$	0	0
$73$	3.56483e8	1.46922	0.734609	−	0.678491i	$-0.237366\pi$
$73$	3.56483e8	1.46922	0.734609	+	0.678491i	$0.237366\pi$
$74$	−4.68501e8	−1.81622
$75$	0	0
$76$	−5.07190e8	−1.74385
$77$	−9.26416e7	−0.300329
$78$	0	0
$79$	1.54673e8	0.446777	0.223389	−	0.974729i	$-0.428288\pi$
$79$	1.54673e8	0.446777	0.223389	+	0.974729i	$0.428288\pi$
$80$	−1.51894e8	−0.414605
$81$	0	0
$82$	3.17589e8	0.775716
$83$	−7.36731e8	−1.70395	−0.851976	−	0.523580i	$-0.824596\pi$
$83$	−7.36731e8	−1.70395	−0.851976	+	0.523580i	$0.824596\pi$
$84$	0	0
$85$	1.14618e7	0.0238160
$86$	1.93065e8	0.380592
$87$	0	0
$88$	4.38794e7	0.0779990
$89$	−2.13463e8	−0.360634	−0.180317	−	0.983609i	$-0.557712\pi$
$89$	−2.13463e8	−0.360634	−0.180317	+	0.983609i	$0.557712\pi$
$90$	0	0
$91$	3.30781e8	0.505656
$92$	7.60882e8	1.10732
$93$	0	0
$94$	−1.01238e9	−1.33741
$95$	−5.79569e8	−0.730044
$96$	0	0
$97$	−3.57772e8	−0.410331	−0.205165	−	0.978727i	$-0.565773\pi$
$97$	−3.57772e8	−0.410331	−0.205165	+	0.978727i	$0.565773\pi$
$98$	−1.87595e8	−0.205449
$99$	0	0
$100$	2.13651e8	0.213651
$101$	9.31890e7	0.0891084	0.0445542	−	0.999007i	$-0.485813\pi$
$101$	9.31890e7	0.0891084	0.0445542	+	0.999007i	$0.485813\pi$
$102$	0	0
$103$	9.37886e8	0.821075	0.410537	−	0.911844i	$-0.365341\pi$
$103$	9.37886e8	0.821075	0.410537	+	0.911844i	$0.365341\pi$
$104$	−1.56674e8	−0.131325
$105$	0	0
$106$	−2.00161e9	−1.53993
$107$	1.76389e9	1.30090	0.650450	−	0.759549i	$-0.274580\pi$
$107$	1.76389e9	1.30090	0.650450	+	0.759549i	$0.274580\pi$
$108$	0	0
$109$	2.30117e7	0.0156145	0.00780727	−	0.999970i	$-0.497515\pi$
$109$	2.30117e7	0.0156145	0.00780727	+	0.999970i	$0.497515\pi$
$110$	7.84749e8	0.511051
$111$	0	0
$112$	−5.83515e8	−0.350405
$113$	−2.46240e9	−1.42071	−0.710356	−	0.703842i	$-0.751466\pi$
$113$	−2.46240e9	−1.42071	−0.710356	+	0.703842i	$0.751466\pi$
$114$	0	0
$115$	8.69465e8	0.463566
$116$	−1.09736e8	−0.0562716
$117$	0	0
$118$	−2.37804e9	−1.12915
$119$	4.40318e7	0.0201282
$120$	0	0
$121$	−8.69178e8	−0.368616
$122$	−6.81501e9	−2.78514
$123$	0	0
$124$	−3.96404e9	−1.50571
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−2.01587e9	−0.687615	−0.343807	−	0.939040i	$-0.611717\pi$
$127$	−2.01587e9	−0.687615	−0.343807	+	0.939040i	$0.611717\pi$
$128$	5.94845e8	0.195866
$129$	0	0
$130$	−2.80199e9	−0.860441
$131$	−6.57902e9	−1.95182	−0.975911	−	0.218167i	$-0.929992\pi$
$131$	−6.57902e9	−1.95182	−0.975911	+	0.218167i	$0.929992\pi$
$132$	0	0
$133$	−2.22647e9	−0.617000
$134$	9.95991e9	2.66860
$135$	0	0
$136$	−2.08555e7	−0.00522753
$137$	4.74874e9	1.15169	0.575845	−	0.817559i	$-0.304673\pi$
$137$	4.74874e9	1.15169	0.575845	+	0.817559i	$0.304673\pi$
$138$	0	0
$139$	1.25718e9	0.285648	0.142824	−	0.989748i	$-0.454382\pi$
$139$	1.25718e9	0.285648	0.142824	+	0.989748i	$0.454382\pi$
$140$	8.20762e8	0.180568
$141$	0	0
$142$	5.88853e9	1.21537
$143$	−5.31573e9	−1.06304
$144$	0	0
$145$	−1.25396e8	−0.0235575
$146$	−1.16005e10	−2.11295
$147$	0	0
$148$	7.87442e9	1.34909
$149$	−5.98036e9	−0.994006	−0.497003	−	0.867749i	$-0.665566\pi$
$149$	−5.98036e9	−0.994006	−0.497003	+	0.867749i	$0.665566\pi$
$150$	0	0
$151$	−1.53651e9	−0.240513	−0.120256	−	0.992743i	$-0.538372\pi$
$151$	−1.53651e9	−0.240513	−0.120256	+	0.992743i	$0.538372\pi$
$152$	1.05456e9	0.160242
$153$	0	0
$154$	3.01469e9	0.431917
$155$	−4.52974e9	−0.630348
$156$	0	0
$157$	−5.59824e7	−0.00735366	−0.00367683	−	0.999993i	$-0.501170\pi$
$157$	−5.59824e7	−0.00735366	−0.00367683	+	0.999993i	$0.501170\pi$
$158$	−5.03327e9	−0.642530
$159$	0	0
$160$	5.30675e9	0.640161
$161$	3.34014e9	0.391785
$162$	0	0
$163$	−6.71462e9	−0.745035	−0.372518	−	0.928025i	$-0.621505\pi$
$163$	−6.71462e9	−0.745035	−0.372518	+	0.928025i	$0.621505\pi$
$164$	−5.33793e9	−0.576203
$165$	0	0
$166$	2.39743e10	2.45053
$167$	−7.60507e9	−0.756622	−0.378311	−	0.925679i	$-0.623495\pi$
$167$	−7.60507e9	−0.756622	−0.378311	+	0.925679i	$0.623495\pi$
$168$	0	0
$169$	8.37558e9	0.789813
$170$	−3.72985e8	−0.0342509
$171$	0	0
$172$	−3.24497e9	−0.282704
$173$	1.27998e10	1.08642	0.543209	−	0.839598i	$-0.317209\pi$
$173$	1.27998e10	1.08642	0.543209	+	0.839598i	$0.317209\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	9.37720e9	0.736659
$177$	0	0
$178$	6.94639e9	0.518643
$179$	1.01190e10	0.736714	0.368357	−	0.929684i	$-0.379920\pi$
$179$	1.01190e10	0.736714	0.368357	+	0.929684i	$0.379920\pi$
$180$	0	0
$181$	1.65948e10	1.14926	0.574631	−	0.818413i	$-0.305146\pi$
$181$	1.65948e10	1.14926	0.574631	+	0.818413i	$0.305146\pi$
$182$	−1.07641e10	−0.727205
$183$	0	0
$184$	−1.58205e9	−0.101751
$185$	8.99815e9	0.564782
$186$	0	0
$187$	−7.07601e8	−0.0423157
$188$	1.70157e10	0.993435
$189$	0	0
$190$	1.88600e10	1.04991
$191$	−1.14204e10	−0.620912	−0.310456	−	0.950588i	$-0.600482\pi$
$191$	−1.14204e10	−0.620912	−0.310456	+	0.950588i	$0.600482\pi$
$192$	0	0
$193$	2.76955e10	1.43682	0.718408	−	0.695622i	$-0.244871\pi$
$193$	2.76955e10	1.43682	0.718408	+	0.695622i	$0.244871\pi$
$194$	1.16424e10	0.590114
$195$	0	0
$196$	3.15304e9	0.152608
$197$	−9.30317e9	−0.440081	−0.220041	−	0.975491i	$-0.570619\pi$
$197$	−9.30317e9	−0.440081	−0.220041	+	0.975491i	$0.570619\pi$
$198$	0	0
$199$	2.15043e10	0.972047	0.486023	−	0.873946i	$-0.338447\pi$
$199$	2.15043e10	0.972047	0.486023	+	0.873946i	$0.338447\pi$
$200$	−4.44229e8	−0.0196323
$201$	0	0
$202$	−3.03251e9	−0.128151
$203$	−4.81722e8	−0.0199097
$204$	0	0
$205$	−6.09969e9	−0.241221
$206$	−3.05202e10	−1.18082
$207$	0	0
$208$	−3.34818e10	−1.24029
$209$	3.57799e10	1.29712
$210$	0	0
$211$	1.60265e10	0.556633	0.278316	−	0.960489i	$-0.410224\pi$
$211$	1.60265e10	0.556633	0.278316	+	0.960489i	$0.410224\pi$
$212$	3.36424e10	1.14387
$213$	0	0
$214$	−5.73995e10	−1.87088
$215$	−3.70805e9	−0.118351
$216$	0	0
$217$	−1.74014e10	−0.532741
$218$	−7.48834e8	−0.0224559
$219$	0	0
$220$	−1.31898e10	−0.379609
$221$	2.52652e9	0.0712456
$222$	0	0
$223$	4.34766e10	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$223$	4.34766e10	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$224$	2.03864e10	0.541035
$225$	0	0
$226$	8.01302e10	2.04319
$227$	6.61741e10	1.65414	0.827068	−	0.562101i	$-0.190007\pi$
$227$	6.61741e10	1.65414	0.827068	+	0.562101i	$0.190007\pi$
$228$	0	0
$229$	4.51697e10	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$229$	4.51697e10	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$230$	−2.82937e10	−0.666675
$231$	0	0
$232$	2.28166e8	0.00517078
$233$	5.88888e10	1.30897	0.654487	−	0.756073i	$-0.272884\pi$
$233$	5.88888e10	1.30897	0.654487	+	0.756073i	$0.272884\pi$
$234$	0	0
$235$	1.94439e10	0.415891
$236$	3.99694e10	0.838732
$237$	0	0
$238$	−1.43286e9	−0.0289473
$239$	6.91640e10	1.37116	0.685582	−	0.727995i	$-0.259548\pi$
$239$	6.91640e10	1.37116	0.685582	+	0.727995i	$0.259548\pi$
$240$	0	0
$241$	−1.58739e10	−0.303115	−0.151557	−	0.988448i	$-0.548429\pi$
$241$	−1.58739e10	−0.303115	−0.151557	+	0.988448i	$0.548429\pi$
$242$	2.82843e10	0.530123
$243$	0	0
$244$	1.14545e11	2.06881
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−1.27754e11	−2.18393
$248$	8.24214e9	0.138359
$249$	0	0
$250$	−7.94469e9	−0.128631
$251$	−3.64132e10	−0.579064	−0.289532	−	0.957168i	$-0.593500\pi$
$251$	−3.64132e10	−0.579064	−0.289532	+	0.957168i	$0.593500\pi$
$252$	0	0
$253$	−5.36767e10	−0.823651
$254$	6.55993e10	0.988889
$255$	0	0
$256$	5.84013e10	0.849851
$257$	3.53518e10	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$257$	3.53518e10	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$258$	0	0
$259$	3.45673e10	0.477328
$260$	4.70949e10	0.639137
$261$	0	0
$262$	2.14091e11	2.80700
$263$	3.92117e10	0.505376	0.252688	−	0.967548i	$-0.418685\pi$
$263$	3.92117e10	0.505376	0.252688	+	0.967548i	$0.418685\pi$
$264$	0	0
$265$	3.84434e10	0.478867
$266$	7.24527e10	0.887334
$267$	0	0
$268$	−1.67403e11	−1.98224
$269$	7.55728e10	0.879995	0.439998	−	0.897999i	$-0.354979\pi$
$269$	7.55728e10	0.879995	0.439998	+	0.897999i	$0.354979\pi$
$270$	0	0
$271$	−1.40812e11	−1.58591	−0.792954	−	0.609282i	$-0.791458\pi$
$271$	−1.40812e11	−1.58591	−0.792954	+	0.609282i	$0.791458\pi$
$272$	−4.45691e9	−0.0493712
$273$	0	0
$274$	−1.54531e11	−1.65630
$275$	−1.50721e10	−0.158919
$276$	0	0
$277$	1.66935e11	1.70368	0.851839	−	0.523803i	$-0.175487\pi$
$277$	1.66935e11	1.70368	0.851839	+	0.523803i	$0.175487\pi$
$278$	−4.09105e10	−0.410802
$279$	0	0
$280$	−1.70655e9	−0.0165924
$281$	−3.51000e10	−0.335837	−0.167919	−	0.985801i	$-0.553705\pi$
$281$	−3.51000e10	−0.335837	−0.167919	+	0.985801i	$0.553705\pi$
$282$	0	0
$283$	−1.28495e11	−1.19082	−0.595410	−	0.803422i	$-0.703010\pi$
$283$	−1.28495e11	−1.19082	−0.595410	+	0.803422i	$0.703010\pi$
$284$	−9.89725e10	−0.902780
$285$	0	0
$286$	1.72982e11	1.52881
$287$	−2.34326e10	−0.203869
$288$	0	0
$289$	−1.18252e11	−0.997164
$290$	4.08058e9	0.0338790
$291$	0	0
$292$	1.94977e11	1.56950
$293$	−1.95418e11	−1.54903	−0.774514	−	0.632556i	$-0.782006\pi$
$293$	−1.95418e11	−1.54903	−0.774514	+	0.632556i	$0.782006\pi$
$294$	0	0
$295$	4.56732e10	0.351126
$296$	−1.63727e10	−0.123967
$297$	0	0
$298$	1.94610e11	1.42952
$299$	1.91655e11	1.38676
$300$	0	0
$301$	−1.42448e10	−0.100025
$302$	5.00002e10	0.345892
$303$	0	0
$304$	2.25364e11	1.51340
$305$	1.30891e11	0.866084
$306$	0	0
$307$	−7.89206e10	−0.507069	−0.253535	−	0.967326i	$-0.581593\pi$
$307$	−7.89206e10	−0.507069	−0.253535	+	0.967326i	$0.581593\pi$
$308$	−5.06700e10	−0.320829
$309$	0	0
$310$	1.47404e11	0.906531
$311$	−3.56793e10	−0.216269	−0.108135	−	0.994136i	$-0.534488\pi$
$311$	−3.56793e10	−0.216269	−0.108135	+	0.994136i	$0.534488\pi$
$312$	0	0
$313$	1.86669e11	1.09932	0.549659	−	0.835389i	$-0.314758\pi$
$313$	1.86669e11	1.09932	0.549659	+	0.835389i	$0.314758\pi$
$314$	1.82175e9	0.0105756
$315$	0	0
$316$	8.45977e10	0.477273
$317$	3.39631e10	0.188904	0.0944519	−	0.995529i	$-0.469890\pi$
$317$	3.39631e10	0.188904	0.0944519	+	0.995529i	$0.469890\pi$
$318$	0	0
$319$	7.74138e9	0.0418563
$320$	−9.49200e10	−0.506038
$321$	0	0
$322$	−1.08693e11	−0.563443
$323$	−1.70059e10	−0.0869338
$324$	0	0
$325$	5.38157e10	0.267568
$326$	2.18503e11	1.07147
$327$	0	0
$328$	1.10988e10	0.0529472
$329$	7.46959e10	0.351492
$330$	0	0
$331$	−1.44127e11	−0.659963	−0.329981	−	0.943987i	$-0.607043\pi$
$331$	−1.44127e11	−0.659963	−0.329981	+	0.943987i	$0.607043\pi$
$332$	−4.02953e11	−1.82026
$333$	0	0
$334$	2.47480e11	1.08813
$335$	−1.91293e11	−0.829845
$336$	0	0
$337$	3.62496e11	1.53098	0.765489	−	0.643449i	$-0.222497\pi$
$337$	3.62496e11	1.53098	0.765489	+	0.643449i	$0.222497\pi$
$338$	−2.72554e11	−1.13586
$339$	0	0
$340$	6.26902e9	0.0254416
$341$	2.79645e11	1.11999
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	6.74703e9	0.0259776
$345$	0	0
$346$	−4.16525e11	−1.56242
$347$	6.79594e10	0.251633	0.125816	−	0.992054i	$-0.459845\pi$
$347$	6.79594e10	0.251633	0.125816	+	0.992054i	$0.459845\pi$
$348$	0	0
$349$	−2.69779e11	−0.973407	−0.486704	−	0.873567i	$-0.661801\pi$
$349$	−2.69779e11	−0.973407	−0.486704	+	0.873567i	$0.661801\pi$
$350$	−3.05203e10	−0.108713
$351$	0	0
$352$	−3.27614e11	−1.13742
$353$	2.54313e11	0.871731	0.435865	−	0.900012i	$-0.356442\pi$
$353$	2.54313e11	0.871731	0.435865	+	0.900012i	$0.356442\pi$
$354$	0	0
$355$	−1.13097e11	−0.377939
$356$	−1.16753e11	−0.385249
$357$	0	0
$358$	−3.29287e11	−1.05950
$359$	2.09684e11	0.666254	0.333127	−	0.942882i	$-0.391896\pi$
$359$	2.09684e11	0.666254	0.333127	+	0.942882i	$0.391896\pi$
$360$	0	0
$361$	5.37217e11	1.66482
$362$	−5.40020e11	−1.65280
$363$	0	0
$364$	1.80920e11	0.540170
$365$	2.22802e11	0.657054
$366$	0	0
$367$	2.33213e11	0.671051	0.335525	−	0.942031i	$-0.391086\pi$
$367$	2.33213e11	0.671051	0.335525	+	0.942031i	$0.391086\pi$
$368$	−3.38089e11	−0.960985
$369$	0	0
$370$	−2.92813e11	−0.812237
$371$	1.47684e11	0.404716
$372$	0	0
$373$	−1.47353e11	−0.394157	−0.197078	−	0.980388i	$-0.563145\pi$
$373$	−1.47353e11	−0.394157	−0.197078	+	0.980388i	$0.563145\pi$
$374$	2.30264e10	0.0608560
$375$	0	0
$376$	−3.53795e10	−0.0912864
$377$	−2.76410e10	−0.0704722
$378$	0	0
$379$	−2.22389e11	−0.553651	−0.276825	−	0.960920i	$-0.589282\pi$
$379$	−2.22389e11	−0.553651	−0.276825	+	0.960920i	$0.589282\pi$
$380$	−3.16994e11	−0.779874
$381$	0	0
$382$	3.71635e11	0.892960
$383$	−5.03966e11	−1.19676	−0.598379	−	0.801213i	$-0.704188\pi$
$383$	−5.03966e11	−1.19676	−0.598379	+	0.801213i	$0.704188\pi$
$384$	0	0
$385$	−5.79010e10	−0.134311
$386$	−9.01252e11	−2.06635
$387$	0	0
$388$	−1.95683e11	−0.438338
$389$	−3.87228e10	−0.0857420	−0.0428710	−	0.999081i	$-0.513650\pi$
$389$	−3.87228e10	−0.0857420	−0.0428710	+	0.999081i	$0.513650\pi$
$390$	0	0
$391$	2.55121e10	0.0552015
$392$	−6.55588e9	−0.0140231
$393$	0	0
$394$	3.02739e11	0.632900
$395$	9.66703e10	0.199805
$396$	0	0
$397$	2.12003e11	0.428337	0.214168	−	0.976797i	$-0.431296\pi$
$397$	2.12003e11	0.428337	0.214168	+	0.976797i	$0.431296\pi$
$398$	−6.99783e11	−1.39794
$399$	0	0
$400$	−9.49335e10	−0.185417
$401$	2.81856e10	0.0544350	0.0272175	−	0.999630i	$-0.491335\pi$
$401$	2.81856e10	0.0544350	0.0272175	+	0.999630i	$0.491335\pi$
$402$	0	0
$403$	−9.98486e11	−1.88569
$404$	5.09694e10	0.0951905
$405$	0	0
$406$	1.56760e10	0.0286330
$407$	−5.55504e11	−1.00349
$408$	0	0
$409$	−5.47583e11	−0.967599	−0.483799	−	0.875179i	$-0.660744\pi$
$409$	−5.47583e11	−0.967599	−0.483799	+	0.875179i	$0.660744\pi$
$410$	1.98493e11	0.346911
$411$	0	0
$412$	5.12974e11	0.877118
$413$	1.75458e11	0.296756
$414$	0	0
$415$	−4.60457e11	−0.762031
$416$	1.16976e12	1.91504
$417$	0	0
$418$	−1.16433e12	−1.86545
$419$	−3.60982e10	−0.0572167	−0.0286083	−	0.999591i	$-0.509108\pi$
$419$	−3.60982e10	−0.0572167	−0.0286083	+	0.999591i	$0.509108\pi$
$420$	0	0
$421$	−7.12011e11	−1.10463	−0.552315	−	0.833635i	$-0.686256\pi$
$421$	−7.12011e11	−1.10463	−0.552315	+	0.833635i	$0.686256\pi$
$422$	−5.21527e11	−0.800518
$423$	0	0
$424$	−6.99501e10	−0.105109
$425$	7.16365e9	0.0106509
$426$	0	0
$427$	5.02831e11	0.731975
$428$	9.64754e11	1.38969
$429$	0	0
$430$	1.20665e11	0.170206
$431$	7.80249e11	1.08914	0.544572	−	0.838714i	$-0.316692\pi$
$431$	7.80249e11	1.08914	0.544572	+	0.838714i	$0.316692\pi$
$432$	0	0
$433$	−1.24062e11	−0.169607	−0.0848036	−	0.996398i	$-0.527026\pi$
$433$	−1.24062e11	−0.169607	−0.0848036	+	0.996398i	$0.527026\pi$
$434$	5.66269e11	0.766159
$435$	0	0
$436$	1.25862e10	0.0166803
$437$	−1.29002e12	−1.69212
$438$	0	0
$439$	1.53033e12	1.96650	0.983252	−	0.182252i	$-0.0583386\pi$
$439$	1.53033e12	1.96650	0.983252	+	0.182252i	$0.0583386\pi$
$440$	2.74246e10	0.0348822
$441$	0	0
$442$	−8.22168e10	−0.102461
$443$	1.37078e11	0.169103	0.0845515	−	0.996419i	$-0.473054\pi$
$443$	1.37078e11	0.169103	0.0845515	+	0.996419i	$0.473054\pi$
$444$	0	0
$445$	−1.33414e11	−0.161280
$446$	−1.41479e12	−1.69311
$447$	0	0
$448$	−3.64645e11	−0.427680
$449$	1.01260e12	1.17579	0.587894	−	0.808938i	$-0.299957\pi$
$449$	1.01260e12	1.17579	0.587894	+	0.808938i	$0.299957\pi$
$450$	0	0
$451$	3.76567e11	0.428595
$452$	−1.34680e12	−1.51768
$453$	0	0
$454$	−2.15340e12	−2.37889
$455$	2.06738e11	0.226136
$456$	0	0
$457$	−3.44289e11	−0.369233	−0.184616	−	0.982811i	$-0.559104\pi$
$457$	−3.44289e11	−0.369233	−0.184616	+	0.982811i	$0.559104\pi$
$458$	−1.46989e12	−1.56095
$459$	0	0
$460$	4.75551e11	0.495207
$461$	2.93939e11	0.303112	0.151556	−	0.988449i	$-0.451572\pi$
$461$	2.93939e11	0.303112	0.151556	+	0.988449i	$0.451572\pi$
$462$	0	0
$463$	1.08058e12	1.09280	0.546401	−	0.837524i	$-0.315997\pi$
$463$	1.08058e12	1.09280	0.546401	+	0.837524i	$0.315997\pi$
$464$	4.87601e10	0.0488352
$465$	0	0
$466$	−1.91633e12	−1.88249
$467$	1.94592e12	1.89321	0.946605	−	0.322395i	$-0.104488\pi$
$467$	1.94592e12	1.89321	0.946605	+	0.322395i	$0.104488\pi$
$468$	0	0
$469$	−7.34870e11	−0.701347
$470$	−6.32735e11	−0.598110
$471$	0	0
$472$	−8.31053e10	−0.0770708
$473$	2.28918e11	0.210283
$474$	0	0
$475$	−3.62231e11	−0.326486
$476$	2.40831e10	0.0215021
$477$	0	0
$478$	−2.25070e12	−1.97193
$479$	−2.08918e12	−1.81329	−0.906644	−	0.421897i	$-0.861365\pi$
$479$	−2.08918e12	−1.81329	−0.906644	+	0.421897i	$0.861365\pi$
$480$	0	0
$481$	1.98345e12	1.68954
$482$	5.16560e11	0.435922
$483$	0	0
$484$	−4.75394e11	−0.393776
$485$	−2.23608e11	−0.183505
$486$	0	0
$487$	6.80509e11	0.548218	0.274109	−	0.961699i	$-0.411617\pi$
$487$	6.80509e11	0.548218	0.274109	+	0.961699i	$0.411617\pi$
$488$	−2.38164e11	−0.190102
$489$	0	0
$490$	−1.17247e11	−0.0918796
$491$	1.00628e12	0.781360	0.390680	−	0.920527i	$-0.372240\pi$
$491$	1.00628e12	0.781360	0.390680	+	0.920527i	$0.372240\pi$
$492$	0	0
$493$	−3.67942e9	−0.00280523
$494$	4.15730e12	3.14080
$495$	0	0
$496$	1.76138e12	1.30673
$497$	−4.34472e11	−0.319417
$498$	0	0
$499$	−1.83579e12	−1.32547	−0.662735	−	0.748854i	$-0.730604\pi$
$499$	−1.83579e12	−1.32547	−0.662735	+	0.748854i	$0.730604\pi$
$500$	1.33532e11	0.0955477
$501$	0	0
$502$	1.18494e12	0.832777
$503$	1.62569e12	1.13236	0.566178	−	0.824283i	$-0.308422\pi$
$503$	1.62569e12	1.13236	0.566178	+	0.824283i	$0.308422\pi$
$504$	0	0
$505$	5.82431e10	0.0398505
$506$	1.74672e12	1.18453
$507$	0	0
$508$	−1.10257e12	−0.734549
$509$	1.68020e12	1.10951	0.554753	−	0.832015i	$-0.312813\pi$
$509$	1.68020e12	1.10951	0.554753	+	0.832015i	$0.312813\pi$
$510$	0	0
$511$	8.55916e11	0.555312
$512$	−2.20503e12	−1.41807
$513$	0	0
$514$	−1.15040e12	−0.726967
$515$	5.86179e11	0.367196
$516$	0	0
$517$	−1.20038e12	−0.738943
$518$	−1.12487e12	−0.686465
$519$	0	0
$520$	−9.79210e10	−0.0587301
$521$	1.35429e12	0.805268	0.402634	−	0.915361i	$-0.368095\pi$
$521$	1.35429e12	0.805268	0.402634	+	0.915361i	$0.368095\pi$
$522$	0	0
$523$	2.50673e11	0.146504	0.0732522	−	0.997313i	$-0.476662\pi$
$523$	2.50673e11	0.146504	0.0732522	+	0.997313i	$0.476662\pi$
$524$	−3.59838e12	−2.08505
$525$	0	0
$526$	−1.27600e12	−0.726803
$527$	−1.32913e11	−0.0750620
$528$	0	0
$529$	1.34129e11	0.0744681
$530$	−1.25100e12	−0.688679
$531$	0	0
$532$	−1.21776e12	−0.659114
$533$	−1.34455e12	−0.721613
$534$	0	0
$535$	1.10243e12	0.581780
$536$	3.48069e11	0.182148
$537$	0	0
$538$	−2.45925e12	−1.26556
$539$	−2.22432e11	−0.113514
$540$	0	0
$541$	1.76350e12	0.885091	0.442545	−	0.896746i	$-0.354076\pi$
$541$	1.76350e12	0.885091	0.442545	+	0.896746i	$0.354076\pi$
$542$	4.58223e12	2.28076
$543$	0	0
$544$	1.55713e11	0.0762305
$545$	1.43823e10	0.00698303
$546$	0	0
$547$	1.51530e12	0.723695	0.361847	−	0.932237i	$-0.382146\pi$
$547$	1.51530e12	0.723695	0.361847	+	0.932237i	$0.382146\pi$
$548$	2.59731e12	1.23030
$549$	0	0
$550$	4.90468e11	0.228549
$551$	1.86050e11	0.0859900
$552$	0	0
$553$	3.71369e11	0.168866
$554$	−5.43230e12	−2.45013
$555$	0	0
$556$	6.87611e11	0.305145
$557$	2.99849e12	1.31994	0.659970	−	0.751292i	$-0.270569\pi$
$557$	2.99849e12	1.31994	0.659970	+	0.751292i	$0.270569\pi$
$558$	0	0
$559$	−8.17362e11	−0.354047
$560$	−3.64697e11	−0.156706
$561$	0	0
$562$	1.14221e12	0.482982
$563$	3.32713e12	1.39567	0.697835	−	0.716259i	$-0.254147\pi$
$563$	3.32713e12	1.39567	0.697835	+	0.716259i	$0.254147\pi$
$564$	0	0
$565$	−1.53900e12	−0.635362
$566$	4.18140e12	1.71257
$567$	0	0
$568$	2.05786e11	0.0829562
$569$	−3.34495e12	−1.33778	−0.668890	−	0.743361i	$-0.733230\pi$
$569$	−3.34495e12	−1.33778	−0.668890	+	0.743361i	$0.733230\pi$
$570$	0	0
$571$	2.21613e12	0.872433	0.436217	−	0.899842i	$-0.356318\pi$
$571$	2.21613e12	0.872433	0.436217	+	0.899842i	$0.356318\pi$
$572$	−2.90742e12	−1.13560
$573$	0	0
$574$	7.62530e11	0.293193
$575$	5.43416e11	0.207313
$576$	0	0
$577$	2.78139e12	1.04465	0.522325	−	0.852746i	$-0.325065\pi$
$577$	2.78139e12	1.04465	0.522325	+	0.852746i	$0.325065\pi$
$578$	3.84808e12	1.43406
$579$	0	0
$580$	−6.85851e10	−0.0251654
$581$	−1.76889e12	−0.644034
$582$	0	0
$583$	−2.37331e12	−0.850838
$584$	−4.05402e11	−0.144221
$585$	0	0
$586$	6.35918e12	2.22773
$587$	1.65679e12	0.575966	0.287983	−	0.957636i	$-0.407015\pi$
$587$	1.65679e12	0.575966	0.287983	+	0.957636i	$0.407015\pi$
$588$	0	0
$589$	6.72076e12	2.30091
$590$	−1.48627e12	−0.504969
$591$	0	0
$592$	−3.49891e12	−1.17081
$593$	−5.08969e12	−1.69023	−0.845114	−	0.534586i	$-0.820467\pi$
$593$	−5.08969e12	−1.69023	−0.845114	+	0.534586i	$0.820467\pi$
$594$	0	0
$595$	2.75199e10	0.00900161
$596$	−3.27094e12	−1.06185
$597$	0	0
$598$	−6.23675e12	−1.99436
$599$	7.76622e11	0.246484	0.123242	−	0.992377i	$-0.460671\pi$
$599$	7.76622e11	0.246484	0.123242	+	0.992377i	$0.460671\pi$
$600$	0	0
$601$	−1.12953e11	−0.0353154	−0.0176577	−	0.999844i	$-0.505621\pi$
$601$	−1.12953e11	−0.0353154	−0.0176577	+	0.999844i	$0.505621\pi$
$602$	4.63548e11	0.143850
$603$	0	0
$604$	−8.40388e11	−0.256929
$605$	−5.43236e11	−0.164850
$606$	0	0
$607$	2.46011e12	0.735539	0.367770	−	0.929917i	$-0.380121\pi$
$607$	2.46011e12	0.735539	0.367770	+	0.929917i	$0.380121\pi$
$608$	−7.87362e12	−2.33673
$609$	0	0
$610$	−4.25938e12	−1.24555
$611$	4.28601e12	1.24414
$612$	0	0
$613$	5.49948e12	1.57307	0.786537	−	0.617543i	$-0.211872\pi$
$613$	5.49948e12	1.57307	0.786537	+	0.617543i	$0.211872\pi$
$614$	2.56819e12	0.729238
$615$	0	0
$616$	1.05354e11	0.0294808
$617$	−3.30419e12	−0.917872	−0.458936	−	0.888469i	$-0.651769\pi$
$617$	−3.30419e12	−0.917872	−0.458936	+	0.888469i	$0.651769\pi$
$618$	0	0
$619$	−6.99545e11	−0.191517	−0.0957586	−	0.995405i	$-0.530528\pi$
$619$	−6.99545e11	−0.191517	−0.0957586	+	0.995405i	$0.530528\pi$
$620$	−2.47753e12	−0.673373
$621$	0	0
$622$	1.16106e12	0.311026
$623$	−5.12524e11	−0.136307
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−6.07450e12	−1.58098
$627$	0	0
$628$	−3.06194e10	−0.00785559
$629$	2.64027e11	0.0672543
$630$	0	0
$631$	2.84950e12	0.715543	0.357772	−	0.933809i	$-0.383537\pi$
$631$	2.84950e12	0.715543	0.357772	+	0.933809i	$0.383537\pi$
$632$	−1.75898e11	−0.0438564
$633$	0	0
$634$	−1.10521e12	−0.271671
$635$	−1.25992e12	−0.307511
$636$	0	0
$637$	7.94206e11	0.191120
$638$	−2.51916e11	−0.0601953
$639$	0	0
$640$	3.71778e11	0.0875940
$641$	5.81049e12	1.35941	0.679706	−	0.733484i	$-0.262107\pi$
$641$	5.81049e12	1.35941	0.679706	+	0.733484i	$0.262107\pi$
$642$	0	0
$643$	3.50664e11	0.0808988	0.0404494	−	0.999182i	$-0.487121\pi$
$643$	3.50664e11	0.0808988	0.0404494	+	0.999182i	$0.487121\pi$
$644$	1.82688e12	0.418527
$645$	0	0
$646$	5.53397e11	0.125023
$647$	−1.57151e12	−0.352571	−0.176286	−	0.984339i	$-0.556408\pi$
$647$	−1.57151e12	−0.352571	−0.176286	+	0.984339i	$0.556408\pi$
$648$	0	0
$649$	−2.81965e12	−0.623871
$650$	−1.75124e12	−0.384801
$651$	0	0
$652$	−3.67254e12	−0.795889
$653$	1.58869e12	0.341925	0.170963	−	0.985278i	$-0.445312\pi$
$653$	1.58869e12	0.341925	0.170963	+	0.985278i	$0.445312\pi$
$654$	0	0
$655$	−4.11189e12	−0.872882
$656$	2.37185e12	0.500058
$657$	0	0
$658$	−2.43071e12	−0.505495
$659$	1.18042e12	0.243810	0.121905	−	0.992542i	$-0.461100\pi$
$659$	1.18042e12	0.243810	0.121905	+	0.992542i	$0.461100\pi$
$660$	0	0
$661$	−2.84810e12	−0.580294	−0.290147	−	0.956982i	$-0.593704\pi$
$661$	−2.84810e12	−0.580294	−0.290147	+	0.956982i	$0.593704\pi$
$662$	4.69011e12	0.949121
$663$	0	0
$664$	8.37830e11	0.167263
$665$	−1.39155e12	−0.275931
$666$	0	0
$667$	−2.79111e11	−0.0546022
$668$	−4.15957e12	−0.808266
$669$	0	0
$670$	6.22494e12	1.19344
$671$	−8.08059e12	−1.53883
$672$	0	0
$673$	−3.70205e12	−0.695624	−0.347812	−	0.937564i	$-0.613075\pi$
$673$	−3.70205e12	−0.695624	−0.347812	+	0.937564i	$0.613075\pi$
$674$	−1.17962e13	−2.20177
$675$	0	0
$676$	4.58100e12	0.843723
$677$	2.27798e12	0.416775	0.208388	−	0.978046i	$-0.433178\pi$
$677$	2.27798e12	0.416775	0.208388	+	0.978046i	$0.433178\pi$
$678$	0	0
$679$	−8.59012e11	−0.155090
$680$	−1.30347e10	−0.00233782
$681$	0	0
$682$	−9.10006e12	−1.61070
$683$	1.46467e12	0.257542	0.128771	−	0.991674i	$-0.458897\pi$
$683$	1.46467e12	0.257542	0.128771	+	0.991674i	$0.458897\pi$
$684$	0	0
$685$	2.96796e12	0.515052
$686$	−4.50416e11	−0.0776524
$687$	0	0
$688$	1.44187e12	0.245345
$689$	8.47404e12	1.43253
$690$	0	0
$691$	6.20353e12	1.03511	0.517557	−	0.855649i	$-0.326842\pi$
$691$	6.20353e12	1.03511	0.517557	+	0.855649i	$0.326842\pi$
$692$	7.00083e12	1.16057
$693$	0	0
$694$	−2.21150e12	−0.361884
$695$	7.85737e11	0.127746
$696$	0	0
$697$	−1.78979e11	−0.0287247
$698$	8.77902e12	1.39990
$699$	0	0
$700$	5.12976e11	0.0807526
$701$	4.91335e12	0.768505	0.384252	−	0.923228i	$-0.374459\pi$
$701$	4.91335e12	0.768505	0.384252	+	0.923228i	$0.374459\pi$
$702$	0	0
$703$	−1.33505e13	−2.06158
$704$	5.85992e12	0.899114
$705$	0	0
$706$	−8.27572e12	−1.25367
$707$	2.23747e11	0.0336798
$708$	0	0
$709$	−5.12256e12	−0.761340	−0.380670	−	0.924711i	$-0.624307\pi$
$709$	−5.12256e12	−0.761340	−0.380670	+	0.924711i	$0.624307\pi$
$710$	3.68033e12	0.543531
$711$	0	0
$712$	2.42755e11	0.0354005
$713$	−1.00824e13	−1.46104
$714$	0	0
$715$	−3.32233e12	−0.475407
$716$	5.53456e12	0.787000
$717$	0	0
$718$	−6.82342e12	−0.958168
$719$	−5.84180e12	−0.815205	−0.407602	−	0.913160i	$-0.633635\pi$
$719$	−5.84180e12	−0.815205	−0.407602	+	0.913160i	$0.633635\pi$
$720$	0	0
$721$	2.25187e12	0.310337
$722$	−1.74818e13	−2.39425
$723$	0	0
$724$	9.07649e12	1.22771
$725$	−7.83727e10	−0.0105352
$726$	0	0
$727$	3.28001e12	0.435482	0.217741	−	0.976007i	$-0.430131\pi$
$727$	3.28001e12	0.435482	0.217741	+	0.976007i	$0.430131\pi$
$728$	−3.76173e11	−0.0496360
$729$	0	0
$730$	−7.25031e12	−0.944938
$731$	−1.08803e11	−0.0140933
$732$	0	0
$733$	−5.27370e12	−0.674757	−0.337378	−	0.941369i	$-0.609540\pi$
$733$	−5.27370e12	−0.674757	−0.337378	+	0.941369i	$0.609540\pi$
$734$	−7.58909e12	−0.965067
$735$	0	0
$736$	1.18119e13	1.48378
$737$	1.18095e13	1.47444
$738$	0	0
$739$	2.97794e12	0.367295	0.183648	−	0.982992i	$-0.441209\pi$
$739$	2.97794e12	0.367295	0.183648	+	0.982992i	$0.441209\pi$
$740$	4.92151e12	0.603331
$741$	0	0
$742$	−4.80585e12	−0.582040
$743$	4.16813e11	0.0501754	0.0250877	−	0.999685i	$-0.492013\pi$
$743$	4.16813e11	0.0501754	0.0250877	+	0.999685i	$0.492013\pi$
$744$	0	0
$745$	−3.73772e12	−0.444533
$746$	4.79508e12	0.566854
$747$	0	0
$748$	−3.87020e11	−0.0452040
$749$	4.23510e12	0.491694
$750$	0	0
$751$	1.70758e13	1.95885	0.979423	−	0.201817i	$-0.0646847\pi$
$751$	1.70758e13	1.95885	0.979423	+	0.201817i	$0.0646847\pi$
$752$	−7.56074e12	−0.862152
$753$	0	0
$754$	8.99478e11	0.101349
$755$	−9.60317e11	−0.107561
$756$	0	0
$757$	1.86789e12	0.206738	0.103369	−	0.994643i	$-0.467038\pi$
$757$	1.86789e12	0.206738	0.103369	+	0.994643i	$0.467038\pi$
$758$	7.23685e12	0.796229
$759$	0	0
$760$	6.59101e11	0.0716624
$761$	1.21579e12	0.131409	0.0657047	−	0.997839i	$-0.479070\pi$
$761$	1.21579e12	0.131409	0.0657047	+	0.997839i	$0.479070\pi$
$762$	0	0
$763$	5.52511e10	0.00590174
$764$	−6.24634e12	−0.663293
$765$	0	0
$766$	1.63998e13	1.72111
$767$	1.00677e13	1.05039
$768$	0	0
$769$	−1.95773e12	−0.201876	−0.100938	−	0.994893i	$-0.532184\pi$
$769$	−1.95773e12	−0.201876	−0.100938	+	0.994893i	$0.532184\pi$
$770$	1.88418e12	0.193159
$771$	0	0
$772$	1.51480e13	1.53489
$773$	−1.13109e13	−1.13943	−0.569716	−	0.821842i	$-0.692947\pi$
$773$	−1.13109e13	−1.13943	−0.569716	+	0.821842i	$0.692947\pi$
$774$	0	0
$775$	−2.83109e12	−0.281900
$776$	4.06868e11	0.0402788
$777$	0	0
$778$	1.26010e12	0.123309
$779$	9.05010e12	0.880511
$780$	0	0
$781$	6.98205e12	0.671512
$782$	−8.30202e11	−0.0793877
$783$	0	0
$784$	−1.40102e12	−0.132441
$785$	−3.49890e10	−0.00328866
$786$	0	0
$787$	1.40494e13	1.30549	0.652744	−	0.757579i	$-0.273618\pi$
$787$	1.40494e13	1.30549	0.652744	+	0.757579i	$0.273618\pi$
$788$	−5.08834e12	−0.470119
$789$	0	0
$790$	−3.14579e12	−0.287348
$791$	−5.91223e12	−0.536979
$792$	0	0
$793$	2.88522e13	2.59089
$794$	−6.89890e12	−0.616010
$795$	0	0
$796$	1.17617e13	1.03839
$797$	−9.87211e12	−0.866657	−0.433329	−	0.901236i	$-0.642661\pi$
$797$	−9.87211e12	−0.866657	−0.433329	+	0.901236i	$0.642661\pi$
$798$	0	0
$799$	5.70531e11	0.0495243
$800$	3.31672e12	0.286289
$801$	0	0
$802$	−9.17201e11	−0.0782853
$803$	−1.37548e13	−1.16744
$804$	0	0
$805$	2.08759e12	0.175212
$806$	3.24922e13	2.71189
$807$	0	0
$808$	−1.05977e11	−0.00874703
$809$	−9.74950e11	−0.0800229	−0.0400114	−	0.999199i	$-0.512739\pi$
$809$	−9.74950e11	−0.0800229	−0.0400114	+	0.999199i	$0.512739\pi$
$810$	0	0
$811$	1.52765e13	1.24002	0.620011	−	0.784593i	$-0.287128\pi$
$811$	1.52765e13	1.24002	0.620011	+	0.784593i	$0.287128\pi$
$812$	−2.63477e11	−0.0212687
$813$	0	0
$814$	1.80769e13	1.44316
$815$	−4.19663e12	−0.333190
$816$	0	0
$817$	5.50162e12	0.432008
$818$	1.78192e13	1.39155
$819$	0	0
$820$	−3.33621e12	−0.257686
$821$	1.05334e13	0.809142	0.404571	−	0.914507i	$-0.367421\pi$
$821$	1.05334e13	0.809142	0.404571	+	0.914507i	$0.367421\pi$
$822$	0	0
$823$	7.09376e12	0.538986	0.269493	−	0.963002i	$-0.413144\pi$
$823$	7.09376e12	0.538986	0.269493	+	0.963002i	$0.413144\pi$
$824$	−1.06659e12	−0.0805981
$825$	0	0
$826$	−5.70967e12	−0.426777
$827$	−7.40831e12	−0.550737	−0.275368	−	0.961339i	$-0.588800\pi$
$827$	−7.40831e12	−0.550737	−0.275368	+	0.961339i	$0.588800\pi$
$828$	0	0
$829$	−2.61672e12	−0.192425	−0.0962126	−	0.995361i	$-0.530673\pi$
$829$	−2.61672e12	−0.192425	−0.0962126	+	0.995361i	$0.530673\pi$
$830$	1.49839e13	1.09591
$831$	0	0
$832$	−2.09231e13	−1.51381
$833$	1.05720e11	0.00760775
$834$	0	0
$835$	−4.75317e12	−0.338372
$836$	1.95697e13	1.38566
$837$	0	0
$838$	1.17469e12	0.0822858
$839$	−1.06924e13	−0.744985	−0.372492	−	0.928035i	$-0.621497\pi$
$839$	−1.06924e13	−0.744985	−0.372492	+	0.928035i	$0.621497\pi$
$840$	0	0
$841$	−1.44669e13	−0.997225
$842$	2.31699e13	1.58862
$843$	0	0
$844$	8.76567e12	0.594626
$845$	5.23474e12	0.353215
$846$	0	0
$847$	−2.08690e12	−0.139324
$848$	−1.49486e13	−0.992704
$849$	0	0
$850$	−2.33116e11	−0.0153175
$851$	2.00284e13	1.30907
$852$	0	0
$853$	−8.94214e12	−0.578324	−0.289162	−	0.957280i	$-0.593377\pi$
$853$	−8.94214e12	−0.578324	−0.289162	+	0.957280i	$0.593377\pi$
$854$	−1.63628e13	−1.05268
$855$	0	0
$856$	−2.00594e12	−0.127699
$857$	1.05093e12	0.0665520	0.0332760	−	0.999446i	$-0.489406\pi$
$857$	1.05093e12	0.0665520	0.0332760	+	0.999446i	$0.489406\pi$
$858$	0	0
$859$	5.88230e12	0.368619	0.184310	−	0.982868i	$-0.440995\pi$
$859$	5.88230e12	0.368619	0.184310	+	0.982868i	$0.440995\pi$
$860$	−2.02811e12	−0.126429
$861$	0	0
$862$	−2.53904e13	−1.56635
$863$	1.70026e13	1.04344	0.521718	−	0.853118i	$-0.325291\pi$
$863$	1.70026e13	1.04344	0.521718	+	0.853118i	$0.325291\pi$
$864$	0	0
$865$	7.99989e12	0.485861
$866$	4.03717e12	0.243920
$867$	0	0
$868$	−9.51767e12	−0.569104
$869$	−5.96798e12	−0.355008
$870$	0	0
$871$	−4.21665e13	−2.48248
$872$	−2.61695e10	−0.00153275
$873$	0	0
$874$	4.19792e13	2.43351
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−1.00896e13	−0.575940	−0.287970	−	0.957639i	$-0.592980\pi$
$877$	−1.00896e13	−0.575940	−0.287970	+	0.957639i	$0.592980\pi$
$878$	−4.97992e13	−2.82811
$879$	0	0
$880$	5.86075e12	0.329444
$881$	9.85298e12	0.551031	0.275515	−	0.961297i	$-0.411151\pi$
$881$	9.85298e12	0.551031	0.275515	+	0.961297i	$0.411151\pi$
$882$	0	0
$883$	2.89958e12	0.160514	0.0802568	−	0.996774i	$-0.474426\pi$
$883$	2.89958e12	0.160514	0.0802568	+	0.996774i	$0.474426\pi$
$884$	1.38188e12	0.0761086
$885$	0	0
$886$	−4.46072e12	−0.243194
$887$	1.46987e13	0.797302	0.398651	−	0.917103i	$-0.369478\pi$
$887$	1.46987e13	0.797302	0.398651	+	0.917103i	$0.369478\pi$
$888$	0	0
$889$	−4.84010e12	−0.259894
$890$	4.34149e12	0.231944
$891$	0	0
$892$	2.37794e13	1.25765
$893$	−2.88489e13	−1.51809
$894$	0	0
$895$	6.32438e12	0.329469
$896$	1.42822e12	0.0740304
$897$	0	0
$898$	−3.29515e13	−1.69095
$899$	1.45411e12	0.0742470
$900$	0	0
$901$	1.12802e12	0.0570236
$902$	−1.22540e13	−0.616381
$903$	0	0
$904$	2.80031e12	0.139460
$905$	1.03718e13	0.513966
$906$	0	0
$907$	3.52924e13	1.73160	0.865801	−	0.500388i	$-0.166809\pi$
$907$	3.52924e13	1.73160	0.865801	+	0.500388i	$0.166809\pi$
$908$	3.61937e13	1.76704
$909$	0	0
$910$	−6.72757e12	−0.325216
$911$	−5.03252e12	−0.242077	−0.121038	−	0.992648i	$-0.538622\pi$
$911$	−5.03252e12	−0.242077	−0.121038	+	0.992648i	$0.538622\pi$
$912$	0	0
$913$	2.84265e13	1.35396
$914$	1.12037e13	0.531009
$915$	0	0
$916$	2.47055e13	1.15948
$917$	−1.57962e13	−0.737720
$918$	0	0
$919$	−9.71752e12	−0.449403	−0.224701	−	0.974428i	$-0.572141\pi$
$919$	−9.71752e12	−0.449403	−0.224701	+	0.974428i	$0.572141\pi$
$920$	−9.88779e11	−0.0455044
$921$	0	0
$922$	−9.56522e12	−0.435919
$923$	−2.49298e13	−1.13060
$924$	0	0
$925$	5.62385e12	0.252578
$926$	−3.51636e13	−1.57161
$927$	0	0
$928$	−1.70355e12	−0.0754028
$929$	3.66898e13	1.61613	0.808063	−	0.589096i	$-0.200516\pi$
$929$	3.66898e13	1.61613	0.808063	+	0.589096i	$0.200516\pi$
$930$	0	0
$931$	−5.34576e12	−0.233204
$932$	3.22091e13	1.39832
$933$	0	0
$934$	−6.33231e13	−2.72271
$935$	−4.42251e11	−0.0189241
$936$	0	0
$937$	−3.33628e13	−1.41395	−0.706975	−	0.707239i	$-0.749941\pi$
$937$	−3.33628e13	−1.41395	−0.706975	+	0.707239i	$0.749941\pi$
$938$	2.39137e13	1.00864
$939$	0	0
$940$	1.06348e13	0.444278
$941$	1.83859e12	0.0764419	0.0382210	−	0.999269i	$-0.487831\pi$
$941$	1.83859e12	0.0764419	0.0382210	+	0.999269i	$0.487831\pi$
$942$	0	0
$943$	−1.35769e13	−0.559110
$944$	−1.77599e13	−0.727893
$945$	0	0
$946$	−7.44931e12	−0.302417
$947$	1.28031e13	0.517296	0.258648	−	0.965972i	$-0.416723\pi$
$947$	1.28031e13	0.517296	0.258648	+	0.965972i	$0.416723\pi$
$948$	0	0
$949$	4.91121e13	1.96558
$950$	1.17875e13	0.469533
$951$	0	0
$952$	−5.00742e10	−0.00197582
$953$	−2.18141e13	−0.856682	−0.428341	−	0.903617i	$-0.640902\pi$
$953$	−2.18141e13	−0.856682	−0.428341	+	0.903617i	$0.640902\pi$
$954$	0	0
$955$	−7.13773e12	−0.277680
$956$	3.78290e13	1.46475
$957$	0	0
$958$	6.79851e13	2.60777
$959$	1.14017e13	0.435298
$960$	0	0
$961$	2.60878e13	0.986694
$962$	−6.45445e13	−2.42981
$963$	0	0
$964$	−8.68218e12	−0.323804
$965$	1.73097e13	0.642564
$966$	0	0
$967$	−1.37098e13	−0.504212	−0.252106	−	0.967700i	$-0.581123\pi$
$967$	−1.37098e13	−0.504212	−0.252106	+	0.967700i	$0.581123\pi$
$968$	9.88452e11	0.0361840
$969$	0	0
$970$	7.27652e12	0.263907
$971$	2.48005e12	0.0895312	0.0447656	−	0.998998i	$-0.485746\pi$
$971$	2.48005e12	0.0895312	0.0447656	+	0.998998i	$0.485746\pi$
$972$	0	0
$973$	3.01849e12	0.107965
$974$	−2.21448e13	−0.788417
$975$	0	0
$976$	−5.08966e13	−1.79542
$977$	−1.24356e13	−0.436659	−0.218330	−	0.975875i	$-0.570061\pi$
$977$	−1.24356e13	−0.436659	−0.218330	+	0.975875i	$0.570061\pi$
$978$	0	0
$979$	8.23636e12	0.286559
$980$	1.97065e12	0.0682484
$981$	0	0
$982$	−3.27457e13	−1.12371
$983$	−2.26526e13	−0.773797	−0.386899	−	0.922122i	$-0.626454\pi$
$983$	−2.26526e13	−0.773797	−0.386899	+	0.922122i	$0.626454\pi$
$984$	0	0
$985$	−5.81448e12	−0.196810
$986$	1.19734e11	0.00403432
$987$	0	0
$988$	−6.98746e13	−2.33299
$989$	−8.25349e12	−0.274318
$990$	0	0
$991$	−2.61705e13	−0.861948	−0.430974	−	0.902364i	$-0.641830\pi$
$991$	−2.61705e13	−0.861948	−0.430974	+	0.902364i	$0.641830\pi$
$992$	−6.15378e13	−2.01762
$993$	0	0
$994$	1.41383e13	0.459367
$995$	1.34402e13	0.434713
$996$	0	0
$997$	−7.08868e12	−0.227215	−0.113608	−	0.993526i	$-0.536241\pi$
$997$	−7.08868e12	−0.227215	−0.113608	+	0.993526i	$0.536241\pi$
$998$	5.97392e13	1.90621
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.j.1.1		5
3.2	odd	2		35.10.a.d.1.5	✓	5
15.2	even	4		175.10.b.f.99.9		10
15.8	even	4		175.10.b.f.99.2		10
15.14	odd	2		175.10.a.f.1.1		5
21.20	even	2		245.10.a.f.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.d.1.5	✓	5	3.2	odd	2
175.10.a.f.1.1		5	15.14	odd	2
175.10.b.f.99.2		10	15.8	even	4
175.10.b.f.99.9		10	15.2	even	4
245.10.a.f.1.5		5	21.20	even	2
315.10.a.j.1.1		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)

\(f(q)\)	\(=\)	\(q-32.5415 q^{2} +546.947 q^{4} +625.000 q^{5} +2401.00 q^{7} -1137.23 q^{8} -20338.4 q^{10} -38584.6 q^{11} +137768. q^{13} -78132.1 q^{14} -243030. q^{16} +18339.0 q^{17} -927311. q^{19} +341842. q^{20} +1.25560e6 q^{22} +1.39114e6 q^{23} +390625. q^{25} -4.48318e6 q^{26} +1.31322e6 q^{28} -200634. q^{29} -7.24758e6 q^{31} +8.49081e6 q^{32} -596776. q^{34} +1.50062e6 q^{35} +1.43970e7 q^{37} +3.01761e7 q^{38} -710767. q^{40} -9.75951e6 q^{41} -5.93288e6 q^{43} -2.11037e7 q^{44} -4.52699e7 q^{46} +3.11103e7 q^{47} +5.76480e6 q^{49} -1.27115e7 q^{50} +7.53519e7 q^{52} +6.15094e7 q^{53} -2.41154e7 q^{55} -2.73048e6 q^{56} +6.52893e6 q^{58} +7.30772e7 q^{59} +2.09425e8 q^{61} +2.35847e8 q^{62} -1.51872e8 q^{64} +8.61051e7 q^{65} -3.06068e8 q^{67} +1.00304e7 q^{68} -4.88325e7 q^{70} -1.80955e8 q^{71} +3.56483e8 q^{73} -4.68501e8 q^{74} -5.07190e8 q^{76} -9.26416e7 q^{77} +1.54673e8 q^{79} -1.51894e8 q^{80} +3.17589e8 q^{82} -7.36731e8 q^{83} +1.14618e7 q^{85} +1.93065e8 q^{86} +4.38794e7 q^{88} -2.13463e8 q^{89} +3.30781e8 q^{91} +7.60882e8 q^{92} -1.01238e9 q^{94} -5.79569e8 q^{95} -3.57772e8 q^{97} -1.87595e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more