LMFDB - Embedded newform 315.10.a.j.1.4

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			1.4
Root			$-21.5262$ 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)

$f(q)$	$=$	\(q+21.5262 q^{2} -48.6242 q^{4} +625.000 q^{5} +2401.00 q^{7} -12068.1 q^{8} +13453.9 q^{10} -7264.45 q^{11} -104986. q^{13} +51684.3 q^{14} -234884. q^{16} -377735. q^{17} +699532. q^{19} -30390.1 q^{20} -156376. q^{22} +141129. q^{23} +390625. q^{25} -2.25995e6 q^{26} -116747. q^{28} -4.83936e6 q^{29} +9.21059e6 q^{31} +1.12271e6 q^{32} -8.13118e6 q^{34} +1.50062e6 q^{35} +1.62065e7 q^{37} +1.50582e7 q^{38} -7.54256e6 q^{40} +2.26946e6 q^{41} -5.43389e6 q^{43} +353228. q^{44} +3.03796e6 q^{46} -5.34021e7 q^{47} +5.76480e6 q^{49} +8.40866e6 q^{50} +5.10488e6 q^{52} +3.47332e7 q^{53} -4.54028e6 q^{55} -2.89755e7 q^{56} -1.04173e8 q^{58} +1.69116e8 q^{59} -4.86165e7 q^{61} +1.98269e8 q^{62} +1.44428e8 q^{64} -6.56165e7 q^{65} -3.24123e7 q^{67} +1.83671e7 q^{68} +3.23027e7 q^{70} +1.15162e8 q^{71} -2.66399e8 q^{73} +3.48865e8 q^{74} -3.40142e7 q^{76} -1.74419e7 q^{77} +2.75318e8 q^{79} -1.46803e8 q^{80} +4.88527e7 q^{82} -4.29752e8 q^{83} -2.36084e8 q^{85} -1.16971e8 q^{86} +8.76680e7 q^{88} +8.82031e8 q^{89} -2.52072e8 q^{91} -6.86227e6 q^{92} -1.14954e9 q^{94} +4.37207e8 q^{95} -3.64161e8 q^{97} +1.24094e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	21.5262	0.951331	0.475666	−	0.879626i	$-0.342207\pi$
$2$	21.5262	0.951331	0.475666	+	0.879626i	$0.342207\pi$
$3$	0	0
$3$	0	0
$4$	−48.6242	−0.0949692
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−12068.1	−1.04168
$9$	0	0
$10$	13453.9	0.425448
$11$	−7264.45	−0.149601	−0.0748007	−	0.997199i	$-0.523832\pi$
$11$	−7264.45	−0.149601	−0.0748007	+	0.997199i	$0.523832\pi$
$12$	0	0
$13$	−104986.	−1.01950	−0.509751	−	0.860322i	$-0.670262\pi$
$13$	−104986.	−1.01950	−0.509751	+	0.860322i	$0.670262\pi$
$14$	51684.3	0.359569
$15$	0	0
$16$	−234884.	−0.896012
$17$	−377735.	−1.09690	−0.548450	−	0.836184i	$-0.684782\pi$
$17$	−377735.	−1.09690	−0.548450	+	0.836184i	$0.684782\pi$
$18$	0	0
$19$	699532.	1.23145	0.615724	−	0.787962i	$-0.288864\pi$
$19$	699532.	1.23145	0.615724	+	0.787962i	$0.288864\pi$
$20$	−30390.1	−0.0424715
$21$	0	0
$22$	−156376.	−0.142320
$23$	141129.	0.105157	0.0525787	−	0.998617i	$-0.483256\pi$
$23$	141129.	0.105157	0.0525787	+	0.998617i	$0.483256\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−2.25995e6	−0.969883
$27$	0	0
$28$	−116747.	−0.0358950
$29$	−4.83936e6	−1.27056	−0.635282	−	0.772280i	$-0.719116\pi$
$29$	−4.83936e6	−1.27056	−0.635282	+	0.772280i	$0.719116\pi$
$30$	0	0
$31$	9.21059e6	1.79126	0.895632	−	0.444795i	$-0.146723\pi$
$31$	9.21059e6	1.79126	0.895632	+	0.444795i	$0.146723\pi$
$32$	1.12271e6	0.189275
$33$	0	0
$34$	−8.13118e6	−1.04351
$35$	1.50062e6	0.169031
$36$	0	0
$37$	1.62065e7	1.42162	0.710809	−	0.703385i	$-0.248329\pi$
$37$	1.62065e7	1.42162	0.710809	+	0.703385i	$0.248329\pi$
$38$	1.50582e7	1.17151
$39$	0	0
$40$	−7.54256e6	−0.465853
$41$	2.26946e6	0.125428	0.0627140	−	0.998032i	$-0.480024\pi$
$41$	2.26946e6	0.125428	0.0627140	+	0.998032i	$0.480024\pi$
$42$	0	0
$43$	−5.43389e6	−0.242383	−0.121192	−	0.992629i	$-0.538672\pi$
$43$	−5.43389e6	−0.242383	−0.121192	+	0.992629i	$0.538672\pi$
$44$	353228.	0.0142075
$45$	0	0
$46$	3.03796e6	0.100040
$47$	−5.34021e7	−1.59631	−0.798156	−	0.602450i	$-0.794191\pi$
$47$	−5.34021e7	−1.59631	−0.798156	+	0.602450i	$0.794191\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	8.40866e6	0.190266
$51$	0	0
$52$	5.10488e6	0.0968212
$53$	3.47332e7	0.604649	0.302324	−	0.953205i	$-0.402237\pi$
$53$	3.47332e7	0.604649	0.302324	+	0.953205i	$0.402237\pi$
$54$	0	0
$55$	−4.54028e6	−0.0669037
$56$	−2.89755e7	−0.393717
$57$	0	0
$58$	−1.04173e8	−1.20873
$59$	1.69116e8	1.81698	0.908489	−	0.417909i	$-0.137237\pi$
$59$	1.69116e8	1.81698	0.908489	+	0.417909i	$0.137237\pi$
$60$	0	0
$61$	−4.86165e7	−0.449572	−0.224786	−	0.974408i	$-0.572168\pi$
$61$	−4.86165e7	−0.449572	−0.224786	+	0.974408i	$0.572168\pi$
$62$	1.98269e8	1.70409
$63$	0	0
$64$	1.44428e8	1.07607
$65$	−6.56165e7	−0.455935
$66$	0	0
$67$	−3.24123e7	−0.196505	−0.0982524	−	0.995162i	$-0.531325\pi$
$67$	−3.24123e7	−0.196505	−0.0982524	+	0.995162i	$0.531325\pi$
$68$	1.83671e7	0.104172
$69$	0	0
$70$	3.23027e7	0.160804
$71$	1.15162e8	0.537832	0.268916	−	0.963164i	$-0.413335\pi$
$71$	1.15162e8	0.537832	0.268916	+	0.963164i	$0.413335\pi$
$72$	0	0
$73$	−2.66399e8	−1.09794	−0.548971	−	0.835841i	$-0.684980\pi$
$73$	−2.66399e8	−1.09794	−0.548971	+	0.835841i	$0.684980\pi$
$74$	3.48865e8	1.35243
$75$	0	0
$76$	−3.40142e7	−0.116950
$77$	−1.74419e7	−0.0565440
$78$	0	0
$79$	2.75318e8	0.795267	0.397633	−	0.917544i	$-0.369832\pi$
$79$	2.75318e8	0.795267	0.397633	+	0.917544i	$0.369832\pi$
$80$	−1.46803e8	−0.400709
$81$	0	0
$82$	4.88527e7	0.119324
$83$	−4.29752e8	−0.993953	−0.496977	−	0.867764i	$-0.665557\pi$
$83$	−4.29752e8	−0.993953	−0.496977	+	0.867764i	$0.665557\pi$
$84$	0	0
$85$	−2.36084e8	−0.490548
$86$	−1.16971e8	−0.230587
$87$	0	0
$88$	8.76680e7	0.155836
$89$	8.82031e8	1.49015	0.745073	−	0.666983i	$-0.232415\pi$
$89$	8.82031e8	1.49015	0.745073	+	0.666983i	$0.232415\pi$
$90$	0	0
$91$	−2.52072e8	−0.385335
$92$	−6.86227e6	−0.00998672
$93$	0	0
$94$	−1.14954e9	−1.51862
$95$	4.37207e8	0.550720
$96$	0	0
$97$	−3.64161e8	−0.417658	−0.208829	−	0.977952i	$-0.566965\pi$
$97$	−3.64161e8	−0.417658	−0.208829	+	0.977952i	$0.566965\pi$
$98$	1.24094e8	0.135904
$99$	0	0
$100$	−1.89938e7	−0.0189938
$101$	3.97117e8	0.379728	0.189864	−	0.981810i	$-0.439195\pi$
$101$	3.97117e8	0.379728	0.189864	+	0.981810i	$0.439195\pi$
$102$	0	0
$103$	1.05690e9	0.925269	0.462635	−	0.886549i	$-0.346904\pi$
$103$	1.05690e9	0.925269	0.462635	+	0.886549i	$0.346904\pi$
$104$	1.26698e9	1.06199
$105$	0	0
$106$	7.47672e8	0.575221
$107$	−8.84940e8	−0.652660	−0.326330	−	0.945256i	$-0.605812\pi$
$107$	−8.84940e8	−0.652660	−0.326330	+	0.945256i	$0.605812\pi$
$108$	0	0
$109$	1.11883e9	0.759181	0.379591	−	0.925155i	$-0.376065\pi$
$109$	1.11883e9	0.759181	0.379591	+	0.925155i	$0.376065\pi$
$110$	−9.77348e7	−0.0636476
$111$	0	0
$112$	−5.63957e8	−0.338661
$113$	2.24657e9	1.29619	0.648093	−	0.761561i	$-0.275567\pi$
$113$	2.24657e9	1.29619	0.648093	+	0.761561i	$0.275567\pi$
$114$	0	0
$115$	8.82054e7	0.0470278
$116$	2.35310e8	0.120664
$117$	0	0
$118$	3.64041e9	1.72855
$119$	−9.06941e8	−0.414589
$120$	0	0
$121$	−2.30518e9	−0.977619
$122$	−1.04653e9	−0.427692
$123$	0	0
$124$	−4.47858e8	−0.170115
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	7.06347e7	0.0240936	0.0120468	−	0.999927i	$-0.496165\pi$
$127$	7.06347e7	0.0240936	0.0120468	+	0.999927i	$0.496165\pi$
$128$	2.53416e9	0.834428
$129$	0	0
$130$	−1.41247e9	−0.433745
$131$	6.11002e9	1.81268	0.906341	−	0.422547i	$-0.138864\pi$
$131$	6.11002e9	1.81268	0.906341	+	0.422547i	$0.138864\pi$
$132$	0	0
$133$	1.67958e9	0.465444
$134$	−6.97712e8	−0.186941
$135$	0	0
$136$	4.55854e9	1.14262
$137$	2.08461e9	0.505572	0.252786	−	0.967522i	$-0.418653\pi$
$137$	2.08461e9	0.505572	0.252786	+	0.967522i	$0.418653\pi$
$138$	0	0
$139$	8.29470e9	1.88467	0.942333	−	0.334678i	$-0.108628\pi$
$139$	8.29470e9	1.88467	0.942333	+	0.334678i	$0.108628\pi$
$140$	−7.29667e7	−0.0160527
$141$	0	0
$142$	2.47900e9	0.511656
$143$	7.62668e8	0.152519
$144$	0	0
$145$	−3.02460e9	−0.568214
$146$	−5.73454e9	−1.04451
$147$	0	0
$148$	−7.88031e8	−0.135010
$149$	1.74044e9	0.289281	0.144640	−	0.989484i	$-0.453797\pi$
$149$	1.74044e9	0.289281	0.144640	+	0.989484i	$0.453797\pi$
$150$	0	0
$151$	1.10925e10	1.73634	0.868169	−	0.496269i	$-0.165297\pi$
$151$	1.10925e10	1.73634	0.868169	+	0.496269i	$0.165297\pi$
$152$	−8.44201e9	−1.28277
$153$	0	0
$154$	−3.75458e8	−0.0537920
$155$	5.75662e9	0.801078
$156$	0	0
$157$	2.35680e9	0.309581	0.154791	−	0.987947i	$-0.450530\pi$
$157$	2.35680e9	0.309581	0.154791	+	0.987947i	$0.450530\pi$
$158$	5.92654e9	0.756562
$159$	0	0
$160$	7.01693e8	0.0846461
$161$	3.38850e8	0.0397458
$162$	0	0
$163$	9.52524e9	1.05689	0.528447	−	0.848966i	$-0.322774\pi$
$163$	9.52524e9	1.05689	0.528447	+	0.848966i	$0.322774\pi$
$164$	−1.10351e8	−0.0119118
$165$	0	0
$166$	−9.25090e9	−0.945579
$167$	9.96994e9	0.991902	0.495951	−	0.868351i	$-0.334820\pi$
$167$	9.96994e9	0.991902	0.495951	+	0.868351i	$0.334820\pi$
$168$	0	0
$169$	4.17631e8	0.0393824
$170$	−5.08199e9	−0.466674
$171$	0	0
$172$	2.64219e8	0.0230189
$173$	−1.39303e10	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$173$	−1.39303e10	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	1.70630e9	0.134045
$177$	0	0
$178$	1.89867e10	1.41762
$179$	1.63551e10	1.19073	0.595365	−	0.803455i	$-0.297007\pi$
$179$	1.63551e10	1.19073	0.595365	+	0.803455i	$0.297007\pi$
$180$	0	0
$181$	−2.74412e10	−1.90042	−0.950210	−	0.311610i	$-0.899132\pi$
$181$	−2.74412e10	−1.90042	−0.950210	+	0.311610i	$0.899132\pi$
$182$	−5.42615e9	−0.366581
$183$	0	0
$184$	−1.70315e9	−0.109540
$185$	1.01291e10	0.635767
$186$	0	0
$187$	2.74403e9	0.164098
$188$	2.59664e9	0.151601
$189$	0	0
$190$	9.41140e9	0.523917
$191$	−1.36618e9	−0.0742777	−0.0371389	−	0.999310i	$-0.511824\pi$
$191$	−1.36618e9	−0.0742777	−0.0371389	+	0.999310i	$0.511824\pi$
$192$	0	0
$193$	−1.59049e10	−0.825131	−0.412565	−	0.910928i	$-0.635367\pi$
$193$	−1.59049e10	−0.825131	−0.412565	+	0.910928i	$0.635367\pi$
$194$	−7.83900e9	−0.397331
$195$	0	0
$196$	−2.80309e8	−0.0135670
$197$	−2.27159e10	−1.07457	−0.537283	−	0.843402i	$-0.680549\pi$
$197$	−2.27159e10	−1.07457	−0.537283	+	0.843402i	$0.680549\pi$
$198$	0	0
$199$	2.06061e10	0.931443	0.465721	−	0.884931i	$-0.345795\pi$
$199$	2.06061e10	0.931443	0.465721	+	0.884931i	$0.345795\pi$
$200$	−4.71410e9	−0.208336
$201$	0	0
$202$	8.54842e9	0.361247
$203$	−1.16193e10	−0.480228
$204$	0	0
$205$	1.41841e9	0.0560931
$206$	2.27511e10	0.880237
$207$	0	0
$208$	2.46596e10	0.913485
$209$	−5.08171e9	−0.184226
$210$	0	0
$211$	4.28352e10	1.48775	0.743874	−	0.668320i	$-0.232986\pi$
$211$	4.28352e10	1.48775	0.743874	+	0.668320i	$0.232986\pi$
$212$	−1.68887e9	−0.0574230
$213$	0	0
$214$	−1.90494e10	−0.620896
$215$	−3.39618e9	−0.108397
$216$	0	0
$217$	2.21146e10	0.677035
$218$	2.40842e10	0.722233
$219$	0	0
$220$	2.20768e8	0.00635380
$221$	3.96570e10	1.11829
$222$	0	0
$223$	−8.41935e9	−0.227985	−0.113993	−	0.993482i	$-0.536364\pi$
$223$	−8.41935e9	−0.227985	−0.113993	+	0.993482i	$0.536364\pi$
$224$	2.69562e9	0.0715390
$225$	0	0
$226$	4.83601e10	1.23310
$227$	3.35674e10	0.839077	0.419538	−	0.907738i	$-0.362192\pi$
$227$	3.35674e10	0.839077	0.419538	+	0.907738i	$0.362192\pi$
$228$	0	0
$229$	8.22103e9	0.197545	0.0987726	−	0.995110i	$-0.468508\pi$
$229$	8.22103e9	0.197545	0.0987726	+	0.995110i	$0.468508\pi$
$230$	1.89872e9	0.0447390
$231$	0	0
$232$	5.84018e10	1.32352
$233$	3.09446e10	0.687832	0.343916	−	0.939000i	$-0.388246\pi$
$233$	3.09446e10	0.687832	0.343916	+	0.939000i	$0.388246\pi$
$234$	0	0
$235$	−3.33763e10	−0.713893
$236$	−8.22312e9	−0.172557
$237$	0	0
$238$	−1.95230e10	−0.394411
$239$	−2.32714e9	−0.0461351	−0.0230676	−	0.999734i	$-0.507343\pi$
$239$	−2.32714e9	−0.0461351	−0.0230676	+	0.999734i	$0.507343\pi$
$240$	0	0
$241$	−7.26614e10	−1.38748	−0.693740	−	0.720225i	$-0.744038\pi$
$241$	−7.26614e10	−1.38748	−0.693740	+	0.720225i	$0.744038\pi$
$242$	−4.96216e10	−0.930040
$243$	0	0
$244$	2.36394e9	0.0426955
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−7.34413e10	−1.25546
$248$	−1.11154e11	−1.86592
$249$	0	0
$250$	5.25541e9	0.0850896
$251$	−6.59838e10	−1.04931	−0.524657	−	0.851314i	$-0.675806\pi$
$251$	−6.59838e10	−1.04931	−0.524657	+	0.851314i	$0.675806\pi$
$252$	0	0
$253$	−1.02522e9	−0.0157317
$254$	1.52049e9	0.0229210
$255$	0	0
$256$	−1.93965e10	−0.282257
$257$	−1.16248e11	−1.66221	−0.831106	−	0.556114i	$-0.812292\pi$
$257$	−1.16248e11	−1.66221	−0.831106	+	0.556114i	$0.812292\pi$
$258$	0	0
$259$	3.89119e10	0.537321
$260$	3.19055e9	0.0432998
$261$	0	0
$262$	1.31525e11	1.72446
$263$	1.31792e11	1.69858	0.849292	−	0.527924i	$-0.177029\pi$
$263$	1.31792e11	1.69858	0.849292	+	0.527924i	$0.177029\pi$
$264$	0	0
$265$	2.17082e10	0.270407
$266$	3.61548e10	0.442791
$267$	0	0
$268$	1.57602e9	0.0186619
$269$	−1.32552e10	−0.154348	−0.0771740	−	0.997018i	$-0.524590\pi$
$269$	−1.32552e10	−0.154348	−0.0771740	+	0.997018i	$0.524590\pi$
$270$	0	0
$271$	−9.35557e10	−1.05368	−0.526840	−	0.849965i	$-0.676623\pi$
$271$	−9.35557e10	−1.05368	−0.526840	+	0.849965i	$0.676623\pi$
$272$	8.87238e10	0.982834
$273$	0	0
$274$	4.48737e10	0.480966
$275$	−2.83767e9	−0.0299203
$276$	0	0
$277$	8.01061e10	0.817535	0.408768	−	0.912639i	$-0.365959\pi$
$277$	8.01061e10	0.817535	0.408768	+	0.912639i	$0.365959\pi$
$278$	1.78553e11	1.79294
$279$	0	0
$280$	−1.81097e10	−0.176076
$281$	9.66301e10	0.924557	0.462279	−	0.886735i	$-0.347032\pi$
$281$	9.66301e10	0.924557	0.462279	+	0.886735i	$0.347032\pi$
$282$	0	0
$283$	−8.32923e10	−0.771909	−0.385954	−	0.922518i	$-0.626128\pi$
$283$	−8.32923e10	−0.771909	−0.385954	+	0.922518i	$0.626128\pi$
$284$	−5.59966e9	−0.0510775
$285$	0	0
$286$	1.64173e10	0.145096
$287$	5.44897e9	0.0474073
$288$	0	0
$289$	2.40956e10	0.203187
$290$	−6.51080e10	−0.540559
$291$	0	0
$292$	1.29534e10	0.104271
$293$	1.06756e11	0.846226	0.423113	−	0.906077i	$-0.360937\pi$
$293$	1.06756e11	0.846226	0.423113	+	0.906077i	$0.360937\pi$
$294$	0	0
$295$	1.05697e11	0.812577
$296$	−1.95582e11	−1.48087
$297$	0	0
$298$	3.74649e10	0.275202
$299$	−1.48166e10	−0.107208
$300$	0	0
$301$	−1.30468e10	−0.0916122
$302$	2.38780e11	1.65183
$303$	0	0
$304$	−1.64309e11	−1.10339
$305$	−3.03853e10	−0.201055
$306$	0	0
$307$	−1.85260e10	−0.119031	−0.0595153	−	0.998227i	$-0.518956\pi$
$307$	−1.85260e10	−0.119031	−0.0595153	+	0.998227i	$0.518956\pi$
$308$	8.48101e8	0.00536994
$309$	0	0
$310$	1.23918e11	0.762090
$311$	2.47643e11	1.50108	0.750541	−	0.660824i	$-0.229793\pi$
$311$	2.47643e11	1.50108	0.750541	+	0.660824i	$0.229793\pi$
$312$	0	0
$313$	2.50393e11	1.47459	0.737296	−	0.675569i	$-0.236102\pi$
$313$	2.50393e11	1.47459	0.737296	+	0.675569i	$0.236102\pi$
$314$	5.07329e10	0.294514
$315$	0	0
$316$	−1.33871e10	−0.0755259
$317$	−2.40990e11	−1.34040	−0.670198	−	0.742182i	$-0.733791\pi$
$317$	−2.40990e11	−1.34040	−0.670198	+	0.742182i	$0.733791\pi$
$318$	0	0
$319$	3.51552e10	0.190078
$320$	9.02677e10	0.481235
$321$	0	0
$322$	7.29414e9	0.0378114
$323$	−2.64237e11	−1.35077
$324$	0	0
$325$	−4.10103e10	−0.203900
$326$	2.05042e11	1.00546
$327$	0	0
$328$	−2.73880e10	−0.130656
$329$	−1.28218e11	−0.603350
$330$	0	0
$331$	3.58008e11	1.63933	0.819665	−	0.572844i	$-0.194160\pi$
$331$	3.58008e11	1.63933	0.819665	+	0.572844i	$0.194160\pi$
$332$	2.08963e10	0.0943950
$333$	0	0
$334$	2.14615e11	0.943627
$335$	−2.02577e10	−0.0878796
$336$	0	0
$337$	6.23655e10	0.263397	0.131698	−	0.991290i	$-0.457957\pi$
$337$	6.23655e10	0.263397	0.131698	+	0.991290i	$0.457957\pi$
$338$	8.98999e9	0.0374657
$339$	0	0
$340$	1.14794e10	0.0465870
$341$	−6.69098e10	−0.267976
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	6.55766e10	0.252485
$345$	0	0
$346$	−2.99865e11	−1.12482
$347$	−2.06428e11	−0.764341	−0.382170	−	0.924092i	$-0.624823\pi$
$347$	−2.06428e11	−0.764341	−0.382170	+	0.924092i	$0.624823\pi$
$348$	0	0
$349$	−3.94645e11	−1.42394	−0.711971	−	0.702208i	$-0.752197\pi$
$349$	−3.94645e11	−1.42394	−0.711971	+	0.702208i	$0.752197\pi$
$350$	2.01892e10	0.0719139
$351$	0	0
$352$	−8.15586e9	−0.0283157
$353$	−1.04492e11	−0.358175	−0.179087	−	0.983833i	$-0.557314\pi$
$353$	−1.04492e11	−0.358175	−0.179087	+	0.983833i	$0.557314\pi$
$354$	0	0
$355$	7.19762e10	0.240526
$356$	−4.28881e10	−0.141518
$357$	0	0
$358$	3.52062e11	1.13278
$359$	5.35779e11	1.70240	0.851198	−	0.524845i	$-0.175877\pi$
$359$	5.35779e11	1.70240	0.851198	+	0.524845i	$0.175877\pi$
$360$	0	0
$361$	1.66657e11	0.516465
$362$	−5.90704e11	−1.80793
$363$	0	0
$364$	1.22568e10	0.0365950
$365$	−1.66499e11	−0.491015
$366$	0	0
$367$	5.94309e11	1.71008	0.855038	−	0.518566i	$-0.173534\pi$
$367$	5.94309e11	1.71008	0.855038	+	0.518566i	$0.173534\pi$
$368$	−3.31489e10	−0.0942223
$369$	0	0
$370$	2.18041e11	0.604825
$371$	8.33944e10	0.228536
$372$	0	0
$373$	−1.14988e11	−0.307582	−0.153791	−	0.988103i	$-0.549148\pi$
$373$	−1.14988e11	−0.307582	−0.153791	+	0.988103i	$0.549148\pi$
$374$	5.90685e10	0.156111
$375$	0	0
$376$	6.44461e11	1.66284
$377$	5.08066e11	1.29534
$378$	0	0
$379$	−2.64315e11	−0.658030	−0.329015	−	0.944325i	$-0.606717\pi$
$379$	−2.64315e11	−0.658030	−0.329015	+	0.944325i	$0.606717\pi$
$380$	−2.12589e10	−0.0523015
$381$	0	0
$382$	−2.94087e10	−0.0706627
$383$	−3.02019e10	−0.0717199	−0.0358599	−	0.999357i	$-0.511417\pi$
$383$	−3.02019e10	−0.0717199	−0.0358599	+	0.999357i	$0.511417\pi$
$384$	0	0
$385$	−1.09012e10	−0.0252872
$386$	−3.42371e11	−0.784972
$387$	0	0
$388$	1.77071e10	0.0396647
$389$	1.40060e11	0.310129	0.155064	−	0.987904i	$-0.450442\pi$
$389$	1.40060e11	0.310129	0.155064	+	0.987904i	$0.450442\pi$
$390$	0	0
$391$	−5.33092e10	−0.115347
$392$	−6.95701e10	−0.148811
$393$	0	0
$394$	−4.88987e11	−1.02227
$395$	1.72074e11	0.355654
$396$	0	0
$397$	1.00518e11	0.203090	0.101545	−	0.994831i	$-0.467621\pi$
$397$	1.00518e11	0.203090	0.101545	+	0.994831i	$0.467621\pi$
$398$	4.43570e11	0.886111
$399$	0	0
$400$	−9.17516e10	−0.179202
$401$	−6.92168e11	−1.33679	−0.668393	−	0.743808i	$-0.733018\pi$
$401$	−6.92168e11	−1.33679	−0.668393	+	0.743808i	$0.733018\pi$
$402$	0	0
$403$	−9.66986e11	−1.82620
$404$	−1.93095e10	−0.0360625
$405$	0	0
$406$	−2.50119e11	−0.456856
$407$	−1.17732e11	−0.212676
$408$	0	0
$409$	−4.19973e11	−0.742108	−0.371054	−	0.928611i	$-0.621003\pi$
$409$	−4.19973e11	−0.742108	−0.371054	+	0.928611i	$0.621003\pi$
$410$	3.05329e10	0.0533631
$411$	0	0
$412$	−5.13912e10	−0.0878721
$413$	4.06047e11	0.686753
$414$	0	0
$415$	−2.68595e11	−0.444509
$416$	−1.17869e11	−0.192966
$417$	0	0
$418$	−1.09390e11	−0.175260
$419$	3.54108e11	0.561271	0.280636	−	0.959814i	$-0.409455\pi$
$419$	3.54108e11	0.561271	0.280636	+	0.959814i	$0.409455\pi$
$420$	0	0
$421$	−9.73431e11	−1.51020	−0.755102	−	0.655607i	$-0.772413\pi$
$421$	−9.73431e11	−1.51020	−0.755102	+	0.655607i	$0.772413\pi$
$422$	9.22077e11	1.41534
$423$	0	0
$424$	−4.19163e11	−0.629850
$425$	−1.47553e11	−0.219380
$426$	0	0
$427$	−1.16728e11	−0.169922
$428$	4.30295e10	0.0619826
$429$	0	0
$430$	−7.31067e10	−0.103122
$431$	−1.86273e11	−0.260017	−0.130008	−	0.991513i	$-0.541500\pi$
$431$	−1.86273e11	−0.260017	−0.130008	+	0.991513i	$0.541500\pi$
$432$	0	0
$433$	1.32179e12	1.80703	0.903517	−	0.428552i	$-0.140976\pi$
$433$	1.32179e12	1.80703	0.903517	+	0.428552i	$0.140976\pi$
$434$	4.76043e11	0.644084
$435$	0	0
$436$	−5.44023e10	−0.0720988
$437$	9.87240e10	0.129496
$438$	0	0
$439$	−2.89150e11	−0.371563	−0.185782	−	0.982591i	$-0.559482\pi$
$439$	−2.89150e11	−0.371563	−0.185782	+	0.982591i	$0.559482\pi$
$440$	5.47925e10	0.0696922
$441$	0	0
$442$	8.53663e11	1.06386
$443$	3.94183e11	0.486275	0.243137	−	0.969992i	$-0.421823\pi$
$443$	3.94183e11	0.486275	0.243137	+	0.969992i	$0.421823\pi$
$444$	0	0
$445$	5.51269e11	0.666414
$446$	−1.81236e11	−0.216890
$447$	0	0
$448$	3.46772e11	0.406718
$449$	−1.11883e11	−0.129914	−0.0649571	−	0.997888i	$-0.520691\pi$
$449$	−1.11883e11	−0.129914	−0.0649571	+	0.997888i	$0.520691\pi$
$450$	0	0
$451$	−1.64863e10	−0.0187642
$452$	−1.09238e11	−0.123098
$453$	0	0
$454$	7.22578e11	0.798240
$455$	−1.57545e11	−0.172327
$456$	0	0
$457$	−1.20520e12	−1.29251	−0.646256	−	0.763120i	$-0.723666\pi$
$457$	−1.20520e12	−1.29251	−0.646256	+	0.763120i	$0.723666\pi$
$458$	1.76967e11	0.187931
$459$	0	0
$460$	−4.28892e9	−0.00446620
$461$	7.10906e11	0.733091	0.366545	−	0.930400i	$-0.380540\pi$
$461$	7.10906e11	0.733091	0.366545	+	0.930400i	$0.380540\pi$
$462$	0	0
$463$	−1.42691e12	−1.44305	−0.721527	−	0.692387i	$-0.756559\pi$
$463$	−1.42691e12	−1.44305	−0.721527	+	0.692387i	$0.756559\pi$
$464$	1.13669e12	1.13844
$465$	0	0
$466$	6.66118e11	0.654356
$467$	−1.17577e12	−1.14392	−0.571961	−	0.820281i	$-0.693817\pi$
$467$	−1.17577e12	−1.14392	−0.571961	+	0.820281i	$0.693817\pi$
$468$	0	0
$469$	−7.78219e10	−0.0742718
$470$	−7.18464e11	−0.679148
$471$	0	0
$472$	−2.04090e12	−1.89271
$473$	3.94742e10	0.0362609
$474$	0	0
$475$	2.73255e11	0.246290
$476$	4.40993e10	0.0393732
$477$	0	0
$478$	−5.00944e10	−0.0438898
$479$	−2.60353e11	−0.225971	−0.112985	−	0.993597i	$-0.536041\pi$
$479$	−2.60353e11	−0.225971	−0.112985	+	0.993597i	$0.536041\pi$
$480$	0	0
$481$	−1.70147e12	−1.44934
$482$	−1.56412e12	−1.31995
$483$	0	0
$484$	1.12087e11	0.0928437
$485$	−2.27601e11	−0.186782
$486$	0	0
$487$	−1.30615e12	−1.05223	−0.526117	−	0.850412i	$-0.676352\pi$
$487$	−1.30615e12	−1.05223	−0.526117	+	0.850412i	$0.676352\pi$
$488$	5.86708e11	0.468309
$489$	0	0
$490$	7.75588e10	0.0607783
$491$	7.95997e11	0.618080	0.309040	−	0.951049i	$-0.399992\pi$
$491$	7.95997e11	0.618080	0.309040	+	0.951049i	$0.399992\pi$
$492$	0	0
$493$	1.82799e12	1.39368
$494$	−1.58091e12	−1.19436
$495$	0	0
$496$	−2.16342e12	−1.60499
$497$	2.76504e11	0.203281
$498$	0	0
$499$	−1.93563e11	−0.139756	−0.0698779	−	0.997556i	$-0.522261\pi$
$499$	−1.93563e11	−0.139756	−0.0698779	+	0.997556i	$0.522261\pi$
$500$	−1.18712e10	−0.00849430
$501$	0	0
$502$	−1.42038e12	−0.998245
$503$	−1.96053e12	−1.36558	−0.682789	−	0.730616i	$-0.739233\pi$
$503$	−1.96053e12	−1.36558	−0.682789	+	0.730616i	$0.739233\pi$
$504$	0	0
$505$	2.48198e11	0.169820
$506$	−2.20691e10	−0.0149660
$507$	0	0
$508$	−3.43456e9	−0.00228815
$509$	−1.24135e12	−0.819720	−0.409860	−	0.912149i	$-0.634422\pi$
$509$	−1.24135e12	−0.819720	−0.409860	+	0.912149i	$0.634422\pi$
$510$	0	0
$511$	−6.39624e11	−0.414983
$512$	−1.71502e12	−1.10295
$513$	0	0
$514$	−2.50237e12	−1.58131
$515$	6.60565e11	0.413793
$516$	0	0
$517$	3.87937e11	0.238811
$518$	8.37625e11	0.511170
$519$	0	0
$520$	7.91865e11	0.474937
$521$	−1.79601e12	−1.06792	−0.533961	−	0.845509i	$-0.679297\pi$
$521$	−1.79601e12	−1.06792	−0.533961	+	0.845509i	$0.679297\pi$
$522$	0	0
$523$	1.13682e12	0.664407	0.332203	−	0.943208i	$-0.392208\pi$
$523$	1.13682e12	0.664407	0.332203	+	0.943208i	$0.392208\pi$
$524$	−2.97095e11	−0.172149
$525$	0	0
$526$	2.83697e12	1.61592
$527$	−3.47916e12	−1.96484
$528$	0	0
$529$	−1.78124e12	−0.988942
$530$	4.67295e11	0.257247
$531$	0	0
$532$	−8.16681e10	−0.0442028
$533$	−2.38262e11	−0.127874
$534$	0	0
$535$	−5.53088e11	−0.291878
$536$	3.91154e11	0.204695
$537$	0	0
$538$	−2.85334e11	−0.146836
$539$	−4.18781e10	−0.0213716
$540$	0	0
$541$	−2.94272e12	−1.47693	−0.738467	−	0.674290i	$-0.764450\pi$
$541$	−2.94272e12	−1.47693	−0.738467	+	0.674290i	$0.764450\pi$
$542$	−2.01390e12	−1.00240
$543$	0	0
$544$	−4.24086e11	−0.207615
$545$	6.99270e11	0.339516
$546$	0	0
$547$	−2.05780e12	−0.982788	−0.491394	−	0.870937i	$-0.663512\pi$
$547$	−2.05780e12	−0.982788	−0.491394	+	0.870937i	$0.663512\pi$
$548$	−1.01363e11	−0.0480138
$549$	0	0
$550$	−6.10842e10	−0.0284641
$551$	−3.38528e12	−1.56463
$552$	0	0
$553$	6.61039e11	0.300583
$554$	1.72438e12	0.777747
$555$	0	0
$556$	−4.03324e11	−0.178985
$557$	−2.98049e11	−0.131202	−0.0656008	−	0.997846i	$-0.520896\pi$
$557$	−2.98049e11	−0.131202	−0.0656008	+	0.997846i	$0.520896\pi$
$558$	0	0
$559$	5.70484e11	0.247110
$560$	−3.52473e11	−0.151454
$561$	0	0
$562$	2.08007e12	0.879560
$563$	2.06345e12	0.865577	0.432788	−	0.901495i	$-0.357530\pi$
$563$	2.06345e12	0.865577	0.432788	+	0.901495i	$0.357530\pi$
$564$	0	0
$565$	1.40411e12	0.579672
$566$	−1.79296e12	−0.734341
$567$	0	0
$568$	−1.38979e12	−0.560248
$569$	8.41905e11	0.336711	0.168356	−	0.985726i	$-0.446154\pi$
$569$	8.41905e11	0.336711	0.168356	+	0.985726i	$0.446154\pi$
$570$	0	0
$571$	−3.64457e12	−1.43478	−0.717388	−	0.696674i	$-0.754663\pi$
$571$	−3.64457e12	−1.43478	−0.717388	+	0.696674i	$0.754663\pi$
$572$	−3.70841e10	−0.0144846
$573$	0	0
$574$	1.17295e11	0.0451001
$575$	5.51284e10	0.0210315
$576$	0	0
$577$	2.62373e12	0.985434	0.492717	−	0.870190i	$-0.336004\pi$
$577$	2.62373e12	0.985434	0.492717	+	0.870190i	$0.336004\pi$
$578$	5.18685e11	0.193299
$579$	0	0
$580$	1.47069e11	0.0539628
$581$	−1.03183e12	−0.375679
$582$	0	0
$583$	−2.52317e11	−0.0904563
$584$	3.21492e12	1.14370
$585$	0	0
$586$	2.29804e12	0.805041
$587$	7.13908e11	0.248182	0.124091	−	0.992271i	$-0.460399\pi$
$587$	7.13908e11	0.248182	0.124091	+	0.992271i	$0.460399\pi$
$588$	0	0
$589$	6.44310e12	2.20585
$590$	2.27526e12	0.773030
$591$	0	0
$592$	−3.80666e12	−1.27379
$593$	5.80259e11	0.192697	0.0963487	−	0.995348i	$-0.469284\pi$
$593$	5.80259e11	0.192697	0.0963487	+	0.995348i	$0.469284\pi$
$594$	0	0
$595$	−5.66838e11	−0.185410
$596$	−8.46273e10	−0.0274728
$597$	0	0
$598$	−3.18944e11	−0.101990
$599$	−5.55108e12	−1.76180	−0.880901	−	0.473300i	$-0.843063\pi$
$599$	−5.55108e12	−1.76180	−0.880901	+	0.473300i	$0.843063\pi$
$600$	0	0
$601$	−8.68746e11	−0.271618	−0.135809	−	0.990735i	$-0.543363\pi$
$601$	−8.68746e11	−0.271618	−0.135809	+	0.990735i	$0.543363\pi$
$602$	−2.80847e11	−0.0871536
$603$	0	0
$604$	−5.39366e11	−0.164899
$605$	−1.44073e12	−0.437205
$606$	0	0
$607$	5.48983e12	1.64138	0.820691	−	0.571372i	$-0.193589\pi$
$607$	5.48983e12	1.64138	0.820691	+	0.571372i	$0.193589\pi$
$608$	7.85370e11	0.233082
$609$	0	0
$610$	−6.54079e11	−0.191270
$611$	5.60649e12	1.62744
$612$	0	0
$613$	2.53621e12	0.725460	0.362730	−	0.931894i	$-0.381845\pi$
$613$	2.53621e12	0.725460	0.362730	+	0.931894i	$0.381845\pi$
$614$	−3.98794e11	−0.113238
$615$	0	0
$616$	2.10491e11	0.0589006
$617$	−3.02344e12	−0.839881	−0.419941	−	0.907552i	$-0.637949\pi$
$617$	−3.02344e12	−0.839881	−0.419941	+	0.907552i	$0.637949\pi$
$618$	0	0
$619$	1.35752e12	0.371653	0.185826	−	0.982583i	$-0.440504\pi$
$619$	1.35752e12	0.371653	0.185826	+	0.982583i	$0.440504\pi$
$620$	−2.79911e11	−0.0760777
$621$	0	0
$622$	5.33080e12	1.42803
$623$	2.11776e12	0.563222
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	5.38999e12	1.40283
$627$	0	0
$628$	−1.14598e11	−0.0294007
$629$	−6.12177e12	−1.55937
$630$	0	0
$631$	−4.10270e12	−1.03024	−0.515120	−	0.857118i	$-0.672253\pi$
$631$	−4.10270e12	−1.03024	−0.515120	+	0.857118i	$0.672253\pi$
$632$	−3.32256e12	−0.828412
$633$	0	0
$634$	−5.18760e12	−1.27516
$635$	4.41467e10	0.0107750
$636$	0	0
$637$	−6.05225e11	−0.145643
$638$	7.56758e11	0.180827
$639$	0	0
$640$	1.58385e12	0.373168
$641$	1.81016e12	0.423503	0.211751	−	0.977324i	$-0.432083\pi$
$641$	1.81016e12	0.423503	0.211751	+	0.977324i	$0.432083\pi$
$642$	0	0
$643$	1.56540e12	0.361141	0.180570	−	0.983562i	$-0.442206\pi$
$643$	1.56540e12	0.361141	0.180570	+	0.983562i	$0.442206\pi$
$644$	−1.64763e10	−0.00377463
$645$	0	0
$646$	−5.68802e12	−1.28503
$647$	7.12668e12	1.59889	0.799444	−	0.600740i	$-0.205127\pi$
$647$	7.12668e12	1.59889	0.799444	+	0.600740i	$0.205127\pi$
$648$	0	0
$649$	−1.22853e12	−0.271822
$650$	−8.82794e11	−0.193977
$651$	0	0
$652$	−4.63158e11	−0.100372
$653$	2.05343e12	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$653$	2.05343e12	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$654$	0	0
$655$	3.81876e12	0.810656
$656$	−5.33059e11	−0.112385
$657$	0	0
$658$	−2.76005e12	−0.573985
$659$	1.63048e12	0.336768	0.168384	−	0.985721i	$-0.446145\pi$
$659$	1.63048e12	0.336768	0.168384	+	0.985721i	$0.446145\pi$
$660$	0	0
$661$	1.17276e12	0.238948	0.119474	−	0.992837i	$-0.461879\pi$
$661$	1.17276e12	0.238948	0.119474	+	0.992837i	$0.461879\pi$
$662$	7.70653e12	1.55954
$663$	0	0
$664$	5.18628e12	1.03538
$665$	1.04973e12	0.208153
$666$	0	0
$667$	−6.82972e11	−0.133609
$668$	−4.84781e11	−0.0942001
$669$	0	0
$670$	−4.36070e11	−0.0836026
$671$	3.53172e11	0.0672566
$672$	0	0
$673$	5.67765e12	1.06684	0.533422	−	0.845849i	$-0.320906\pi$
$673$	5.67765e12	1.06684	0.533422	+	0.845849i	$0.320906\pi$
$674$	1.34249e12	0.250577
$675$	0	0
$676$	−2.03070e10	−0.00374012
$677$	−1.03850e12	−0.190002	−0.0950012	−	0.995477i	$-0.530285\pi$
$677$	−1.03850e12	−0.190002	−0.0950012	+	0.995477i	$0.530285\pi$
$678$	0	0
$679$	−8.74351e11	−0.157860
$680$	2.84908e12	0.510993
$681$	0	0
$682$	−1.44031e12	−0.254934
$683$	−2.03661e12	−0.358108	−0.179054	−	0.983839i	$-0.557304\pi$
$683$	−2.03661e12	−0.358108	−0.179054	+	0.983839i	$0.557304\pi$
$684$	0	0
$685$	1.30288e12	0.226099
$686$	2.97950e11	0.0513670
$687$	0	0
$688$	1.27633e12	0.217178
$689$	−3.64651e12	−0.616440
$690$	0	0
$691$	8.06979e12	1.34652	0.673258	−	0.739408i	$-0.264895\pi$
$691$	8.06979e12	1.34652	0.673258	+	0.739408i	$0.264895\pi$
$692$	6.77349e11	0.112288
$693$	0	0
$694$	−4.44361e12	−0.727141
$695$	5.18419e12	0.842848
$696$	0	0
$697$	−8.57253e11	−0.137582
$698$	−8.49520e12	−1.35464
$699$	0	0
$700$	−4.56042e10	−0.00717900
$701$	6.63031e12	1.03706	0.518528	−	0.855060i	$-0.326480\pi$
$701$	6.63031e12	1.03706	0.518528	+	0.855060i	$0.326480\pi$
$702$	0	0
$703$	1.13370e13	1.75065
$704$	−1.04919e12	−0.160982
$705$	0	0
$706$	−2.24930e12	−0.340743
$707$	9.53479e11	0.143524
$708$	0	0
$709$	−1.05445e12	−0.156718	−0.0783591	−	0.996925i	$-0.524968\pi$
$709$	−1.05445e12	−0.156718	−0.0783591	+	0.996925i	$0.524968\pi$
$710$	1.54937e12	0.228820
$711$	0	0
$712$	−1.06444e13	−1.55225
$713$	1.29988e12	0.188365
$714$	0	0
$715$	4.76667e11	0.0682084
$716$	−7.95252e11	−0.113083
$717$	0	0
$718$	1.15333e13	1.61954
$719$	−6.47058e12	−0.902949	−0.451474	−	0.892284i	$-0.649102\pi$
$719$	−6.47058e12	−0.902949	−0.451474	+	0.892284i	$0.649102\pi$
$720$	0	0
$721$	2.53763e12	0.349719
$722$	3.58748e12	0.491329
$723$	0	0
$724$	1.33431e12	0.180481
$725$	−1.89037e12	−0.254113
$726$	0	0
$727$	6.57148e12	0.872485	0.436243	−	0.899829i	$-0.356309\pi$
$727$	6.57148e12	0.872485	0.436243	+	0.899829i	$0.356309\pi$
$728$	3.04203e12	0.401395
$729$	0	0
$730$	−3.58409e12	−0.467117
$731$	2.05257e12	0.265870
$732$	0	0
$733$	−1.49411e12	−0.191168	−0.0955839	−	0.995421i	$-0.530472\pi$
$733$	−1.49411e12	−0.191168	−0.0955839	+	0.995421i	$0.530472\pi$
$734$	1.27932e13	1.62685
$735$	0	0
$736$	1.58446e11	0.0199036
$737$	2.35457e11	0.0293974
$738$	0	0
$739$	−5.22401e12	−0.644324	−0.322162	−	0.946685i	$-0.604410\pi$
$739$	−5.22401e12	−0.644324	−0.322162	+	0.946685i	$0.604410\pi$
$740$	−4.92519e11	−0.0603783
$741$	0	0
$742$	1.79516e12	0.217413
$743$	6.71821e12	0.808731	0.404365	−	0.914597i	$-0.367492\pi$
$743$	6.71821e12	0.808731	0.404365	+	0.914597i	$0.367492\pi$
$744$	0	0
$745$	1.08777e12	0.129370
$746$	−2.47524e12	−0.292613
$747$	0	0
$748$	−1.33426e11	−0.0155842
$749$	−2.12474e12	−0.246682
$750$	0	0
$751$	4.56125e12	0.523244	0.261622	−	0.965170i	$-0.415743\pi$
$751$	4.56125e12	0.523244	0.261622	+	0.965170i	$0.415743\pi$
$752$	1.25433e13	1.43031
$753$	0	0
$754$	1.09367e13	1.23230
$755$	6.93283e12	0.776514
$756$	0	0
$757$	8.41083e11	0.0930910	0.0465455	−	0.998916i	$-0.485179\pi$
$757$	8.41083e11	0.0930910	0.0465455	+	0.998916i	$0.485179\pi$
$758$	−5.68969e12	−0.626004
$759$	0	0
$760$	−5.27626e12	−0.573673
$761$	−4.81444e11	−0.0520373	−0.0260186	−	0.999661i	$-0.508283\pi$
$761$	−4.81444e11	−0.0520373	−0.0260186	+	0.999661i	$0.508283\pi$
$762$	0	0
$763$	2.68631e12	0.286944
$764$	6.64296e10	0.00705410
$765$	0	0
$766$	−6.50130e11	−0.0682293
$767$	−1.77548e13	−1.85241
$768$	0	0
$769$	−1.26367e12	−0.130306	−0.0651531	−	0.997875i	$-0.520754\pi$
$769$	−1.26367e12	−0.130306	−0.0651531	+	0.997875i	$0.520754\pi$
$770$	−2.34661e11	−0.0240565
$771$	0	0
$772$	7.73363e11	0.0783620
$773$	1.45857e13	1.46933	0.734667	−	0.678428i	$-0.237338\pi$
$773$	1.45857e13	1.46933	0.734667	+	0.678428i	$0.237338\pi$
$774$	0	0
$775$	3.59789e12	0.358253
$776$	4.39473e12	0.435065
$777$	0	0
$778$	3.01496e12	0.295035
$779$	1.58756e12	0.154458
$780$	0	0
$781$	−8.36588e11	−0.0804604
$782$	−1.14754e12	−0.109733
$783$	0	0
$784$	−1.35406e12	−0.128002
$785$	1.47300e12	0.138449
$786$	0	0
$787$	−1.20629e13	−1.12089	−0.560446	−	0.828191i	$-0.689370\pi$
$787$	−1.20629e13	−1.12089	−0.560446	+	0.828191i	$0.689370\pi$
$788$	1.10455e12	0.102051
$789$	0	0
$790$	3.70409e12	0.338345
$791$	5.39402e12	0.489912
$792$	0	0
$793$	5.10407e12	0.458339
$794$	2.16377e12	0.193205
$795$	0	0
$796$	−1.00195e12	−0.0884584
$797$	1.28438e13	1.12754	0.563769	−	0.825933i	$-0.309351\pi$
$797$	1.28438e13	1.12754	0.563769	+	0.825933i	$0.309351\pi$
$798$	0	0
$799$	2.01718e13	1.75099
$800$	4.38558e11	0.0378549
$801$	0	0
$802$	−1.48997e13	−1.27173
$803$	1.93524e12	0.164254
$804$	0	0
$805$	2.11781e11	0.0177749
$806$	−2.08155e13	−1.73732
$807$	0	0
$808$	−4.79245e12	−0.395555
$809$	2.39566e12	0.196633	0.0983166	−	0.995155i	$-0.468654\pi$
$809$	2.39566e12	0.196633	0.0983166	+	0.995155i	$0.468654\pi$
$810$	0	0
$811$	−4.17696e11	−0.0339052	−0.0169526	−	0.999856i	$-0.505396\pi$
$811$	−4.17696e11	−0.0339052	−0.0169526	+	0.999856i	$0.505396\pi$
$812$	5.64979e11	0.0456069
$813$	0	0
$814$	−2.53431e12	−0.202325
$815$	5.95328e12	0.472658
$816$	0	0
$817$	−3.80118e12	−0.298482
$818$	−9.04042e12	−0.705990
$819$	0	0
$820$	−6.89691e10	−0.00532712
$821$	−4.80827e12	−0.369355	−0.184678	−	0.982799i	$-0.559124\pi$
$821$	−4.80827e12	−0.369355	−0.184678	+	0.982799i	$0.559124\pi$
$822$	0	0
$823$	−8.70346e11	−0.0661291	−0.0330645	−	0.999453i	$-0.510527\pi$
$823$	−8.70346e11	−0.0661291	−0.0330645	+	0.999453i	$0.510527\pi$
$824$	−1.27548e13	−0.963833
$825$	0	0
$826$	8.74063e12	0.653330
$827$	−1.29605e13	−0.963492	−0.481746	−	0.876311i	$-0.659997\pi$
$827$	−1.29605e13	−0.963492	−0.481746	+	0.876311i	$0.659997\pi$
$828$	0	0
$829$	2.82712e12	0.207897	0.103949	−	0.994583i	$-0.466852\pi$
$829$	2.82712e12	0.207897	0.103949	+	0.994583i	$0.466852\pi$
$830$	−5.78181e12	−0.422876
$831$	0	0
$832$	−1.51630e13	−1.09706
$833$	−2.17756e12	−0.156700
$834$	0	0
$835$	6.23121e12	0.443592
$836$	2.47094e11	0.0174958
$837$	0	0
$838$	7.62259e12	0.533955
$839$	2.42321e13	1.68835	0.844175	−	0.536068i	$-0.180091\pi$
$839$	2.42321e13	1.68835	0.844175	+	0.536068i	$0.180091\pi$
$840$	0	0
$841$	8.91222e12	0.614333
$842$	−2.09542e13	−1.43670
$843$	0	0
$844$	−2.08283e12	−0.141290
$845$	2.61019e11	0.0176124
$846$	0	0
$847$	−5.53473e12	−0.369505
$848$	−8.15827e12	−0.541772
$849$	0	0
$850$	−3.17624e12	−0.208703
$851$	2.28721e12	0.149494
$852$	0	0
$853$	2.32764e13	1.50538	0.752689	−	0.658376i	$-0.228756\pi$
$853$	2.32764e13	1.50538	0.752689	+	0.658376i	$0.228756\pi$
$854$	−2.51271e12	−0.161652
$855$	0	0
$856$	1.06795e13	0.679862
$857$	−5.15214e11	−0.0326268	−0.0163134	−	0.999867i	$-0.505193\pi$
$857$	−5.15214e11	−0.0326268	−0.0163134	+	0.999867i	$0.505193\pi$
$858$	0	0
$859$	2.02157e13	1.26683	0.633415	−	0.773812i	$-0.281653\pi$
$859$	2.02157e13	1.26683	0.633415	+	0.773812i	$0.281653\pi$
$860$	1.65137e11	0.0102944
$861$	0	0
$862$	−4.00974e12	−0.247362
$863$	5.68124e12	0.348654	0.174327	−	0.984688i	$-0.444225\pi$
$863$	5.68124e12	0.348654	0.174327	+	0.984688i	$0.444225\pi$
$864$	0	0
$865$	−8.70642e12	−0.528770
$866$	2.84530e13	1.71909
$867$	0	0
$868$	−1.07531e12	−0.0642974
$869$	−2.00003e12	−0.118973
$870$	0	0
$871$	3.40285e12	0.200337
$872$	−1.35022e13	−0.790823
$873$	0	0
$874$	2.12515e12	0.123194
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−1.01078e13	−0.576977	−0.288488	−	0.957483i	$-0.593153\pi$
$877$	−1.01078e13	−0.576977	−0.288488	+	0.957483i	$0.593153\pi$
$878$	−6.22429e12	−0.353480
$879$	0	0
$880$	1.06644e12	0.0599465
$881$	2.95519e13	1.65270	0.826349	−	0.563159i	$-0.190414\pi$
$881$	2.95519e13	1.65270	0.826349	+	0.563159i	$0.190414\pi$
$882$	0	0
$883$	−2.03644e13	−1.12733	−0.563663	−	0.826005i	$-0.690608\pi$
$883$	−2.03644e13	−1.12733	−0.563663	+	0.826005i	$0.690608\pi$
$884$	−1.92829e12	−0.106203
$885$	0	0
$886$	8.48526e12	0.462608
$887$	−1.62796e13	−0.883054	−0.441527	−	0.897248i	$-0.645563\pi$
$887$	−1.62796e13	−0.883054	−0.441527	+	0.897248i	$0.645563\pi$
$888$	0	0
$889$	1.69594e11	0.00910651
$890$	1.18667e13	0.633980
$891$	0	0
$892$	4.09385e11	0.0216516
$893$	−3.73565e13	−1.96578
$894$	0	0
$895$	1.02219e13	0.532511
$896$	6.08452e12	0.315384
$897$	0	0
$898$	−2.40842e12	−0.123591
$899$	−4.45733e13	−2.27592
$900$	0	0
$901$	−1.31199e13	−0.663239
$902$	−3.54888e11	−0.0178510
$903$	0	0
$904$	−2.71118e13	−1.35021
$905$	−1.71508e13	−0.849894
$906$	0	0
$907$	−9.56727e12	−0.469413	−0.234706	−	0.972066i	$-0.575413\pi$
$907$	−9.56727e12	−0.469413	−0.234706	+	0.972066i	$0.575413\pi$
$908$	−1.63219e12	−0.0796864
$909$	0	0
$910$	−3.39134e12	−0.163940
$911$	−2.09743e13	−1.00892	−0.504458	−	0.863437i	$-0.668307\pi$
$911$	−2.09743e13	−1.00892	−0.504458	+	0.863437i	$0.668307\pi$
$912$	0	0
$913$	3.12191e12	0.148697
$914$	−2.59432e13	−1.22961
$915$	0	0
$916$	−3.99741e11	−0.0187607
$917$	1.46702e13	0.685129
$918$	0	0
$919$	4.57719e12	0.211680	0.105840	−	0.994383i	$-0.466247\pi$
$919$	4.57719e12	0.211680	0.105840	+	0.994383i	$0.466247\pi$
$920$	−1.06447e12	−0.0489879
$921$	0	0
$922$	1.53031e13	0.697412
$923$	−1.20904e13	−0.548320
$924$	0	0
$925$	6.33068e12	0.284324
$926$	−3.07159e13	−1.37282
$927$	0	0
$928$	−5.43319e12	−0.240485
$929$	1.61221e13	0.710151	0.355075	−	0.934838i	$-0.384455\pi$
$929$	1.61221e13	0.710151	0.355075	+	0.934838i	$0.384455\pi$
$930$	0	0
$931$	4.03266e12	0.175921
$932$	−1.50466e12	−0.0653229
$933$	0	0
$934$	−2.53098e13	−1.08825
$935$	1.71502e12	0.0733866
$936$	0	0
$937$	−4.01643e13	−1.70220	−0.851102	−	0.525000i	$-0.824065\pi$
$937$	−4.01643e13	−1.70220	−0.851102	+	0.525000i	$0.824065\pi$
$938$	−1.67521e12	−0.0706571
$939$	0	0
$940$	1.62290e12	0.0677978
$941$	2.70938e13	1.12646	0.563231	−	0.826300i	$-0.309558\pi$
$941$	2.70938e13	1.12646	0.563231	+	0.826300i	$0.309558\pi$
$942$	0	0
$943$	3.20286e11	0.0131897
$944$	−3.97226e13	−1.62803
$945$	0	0
$946$	8.49728e11	0.0344961
$947$	2.97937e13	1.20379	0.601894	−	0.798576i	$-0.294413\pi$
$947$	2.97937e13	1.20379	0.601894	+	0.798576i	$0.294413\pi$
$948$	0	0
$949$	2.79682e13	1.11935
$950$	5.88212e12	0.234303
$951$	0	0
$952$	1.09450e13	0.431868
$953$	4.34409e13	1.70601	0.853003	−	0.521906i	$-0.174779\pi$
$953$	4.34409e13	1.70601	0.853003	+	0.521906i	$0.174779\pi$
$954$	0	0
$955$	−8.53864e11	−0.0332180
$956$	1.13155e11	0.00438142
$957$	0	0
$958$	−5.60440e12	−0.214973
$959$	5.00516e12	0.191088
$960$	0	0
$961$	5.83953e13	2.20863
$962$	−3.66260e13	−1.37880
$963$	0	0
$964$	3.53310e12	0.131768
$965$	−9.94055e12	−0.369010
$966$	0	0
$967$	−3.35056e13	−1.23225	−0.616123	−	0.787650i	$-0.711298\pi$
$967$	−3.35056e13	−1.23225	−0.616123	+	0.787650i	$0.711298\pi$
$968$	2.78191e13	1.01836
$969$	0	0
$970$	−4.89937e12	−0.177692
$971$	−4.36552e13	−1.57598	−0.787988	−	0.615690i	$-0.788877\pi$
$971$	−4.36552e13	−1.57598	−0.787988	+	0.615690i	$0.788877\pi$
$972$	0	0
$973$	1.99156e13	0.712336
$974$	−2.81163e13	−1.00102
$975$	0	0
$976$	1.14192e13	0.402822
$977$	6.90081e12	0.242312	0.121156	−	0.992633i	$-0.461340\pi$
$977$	6.90081e12	0.242312	0.121156	+	0.992633i	$0.461340\pi$
$978$	0	0
$979$	−6.40747e12	−0.222928
$980$	−1.75193e11	−0.00606736
$981$	0	0
$982$	1.71348e13	0.587999
$983$	3.55554e13	1.21455	0.607274	−	0.794492i	$-0.292263\pi$
$983$	3.55554e13	1.21455	0.607274	+	0.794492i	$0.292263\pi$
$984$	0	0
$985$	−1.41975e13	−0.480560
$986$	3.93497e13	1.32585
$987$	0	0
$988$	3.57103e12	0.119230
$989$	−7.66877e11	−0.0254884
$990$	0	0
$991$	−4.59127e13	−1.51217	−0.756086	−	0.654472i	$-0.772891\pi$
$991$	−4.59127e13	−1.51217	−0.756086	+	0.654472i	$0.772891\pi$
$992$	1.03408e13	0.339041
$993$	0	0
$994$	5.95207e12	0.193388
$995$	1.28788e13	0.416554
$996$	0	0
$997$	2.33439e13	0.748249	0.374124	−	0.927378i	$-0.377943\pi$
$997$	2.33439e13	0.748249	0.374124	+	0.927378i	$0.377943\pi$
$998$	−4.16667e12	−0.132954
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.j.1.4		5
3.2	odd	2		35.10.a.d.1.2	✓	5
15.2	even	4		175.10.b.f.99.4		10
15.8	even	4		175.10.b.f.99.7		10
15.14	odd	2		175.10.a.f.1.4		5
21.20	even	2		245.10.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.d.1.2	✓	5	3.2	odd	2
175.10.a.f.1.4		5	15.14	odd	2
175.10.b.f.99.4		10	15.2	even	4
175.10.b.f.99.7		10	15.8	even	4
245.10.a.f.1.2		5	21.20	even	2
315.10.a.j.1.4		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)

\(f(q)\)	\(=\)	\(q+21.5262 q^{2} -48.6242 q^{4} +625.000 q^{5} +2401.00 q^{7} -12068.1 q^{8} +13453.9 q^{10} -7264.45 q^{11} -104986. q^{13} +51684.3 q^{14} -234884. q^{16} -377735. q^{17} +699532. q^{19} -30390.1 q^{20} -156376. q^{22} +141129. q^{23} +390625. q^{25} -2.25995e6 q^{26} -116747. q^{28} -4.83936e6 q^{29} +9.21059e6 q^{31} +1.12271e6 q^{32} -8.13118e6 q^{34} +1.50062e6 q^{35} +1.62065e7 q^{37} +1.50582e7 q^{38} -7.54256e6 q^{40} +2.26946e6 q^{41} -5.43389e6 q^{43} +353228. q^{44} +3.03796e6 q^{46} -5.34021e7 q^{47} +5.76480e6 q^{49} +8.40866e6 q^{50} +5.10488e6 q^{52} +3.47332e7 q^{53} -4.54028e6 q^{55} -2.89755e7 q^{56} -1.04173e8 q^{58} +1.69116e8 q^{59} -4.86165e7 q^{61} +1.98269e8 q^{62} +1.44428e8 q^{64} -6.56165e7 q^{65} -3.24123e7 q^{67} +1.83671e7 q^{68} +3.23027e7 q^{70} +1.15162e8 q^{71} -2.66399e8 q^{73} +3.48865e8 q^{74} -3.40142e7 q^{76} -1.74419e7 q^{77} +2.75318e8 q^{79} -1.46803e8 q^{80} +4.88527e7 q^{82} -4.29752e8 q^{83} -2.36084e8 q^{85} -1.16971e8 q^{86} +8.76680e7 q^{88} +8.82031e8 q^{89} -2.52072e8 q^{91} -6.86227e6 q^{92} -1.14954e9 q^{94} +4.37207e8 q^{95} -3.64161e8 q^{97} +1.24094e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) \) 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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