LMFDB - Embedded newform 315.10.a.j.1.2

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			1.2
Root			$22.7661$ 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)

$f(q)$	$=$	\(q-22.7661 q^{2} +6.29742 q^{4} +625.000 q^{5} +2401.00 q^{7} +11512.9 q^{8} -14228.8 q^{10} +21227.5 q^{11} +47399.7 q^{13} -54661.5 q^{14} -265329. q^{16} -569500. q^{17} +1.08959e6 q^{19} +3935.89 q^{20} -483268. q^{22} +1.34776e6 q^{23} +390625. q^{25} -1.07911e6 q^{26} +15120.1 q^{28} +6.23203e6 q^{29} -413669. q^{31} +145906. q^{32} +1.29653e7 q^{34} +1.50062e6 q^{35} +1.28411e7 q^{37} -2.48057e7 q^{38} +7.19556e6 q^{40} -3.90722e6 q^{41} +1.27760e7 q^{43} +133678. q^{44} -3.06834e7 q^{46} +3.61213e7 q^{47} +5.76480e6 q^{49} -8.89303e6 q^{50} +298496. q^{52} -1.10032e8 q^{53} +1.32672e7 q^{55} +2.76425e7 q^{56} -1.41879e8 q^{58} +8.68650e6 q^{59} +1.06957e8 q^{61} +9.41766e6 q^{62} +1.32527e8 q^{64} +2.96248e7 q^{65} +1.89423e8 q^{67} -3.58638e6 q^{68} -3.41634e7 q^{70} -9.10989e7 q^{71} -9.84572e7 q^{73} -2.92341e8 q^{74} +6.86157e6 q^{76} +5.09672e7 q^{77} -2.41368e8 q^{79} -1.65830e8 q^{80} +8.89525e7 q^{82} -4.28456e8 q^{83} -3.55937e8 q^{85} -2.90861e8 q^{86} +2.44390e8 q^{88} -5.29736e7 q^{89} +1.13807e8 q^{91} +8.48743e6 q^{92} -8.22342e8 q^{94} +6.80991e8 q^{95} +3.01374e8 q^{97} -1.31242e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−22.7661	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$2$	−22.7661	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$3$	0	0
$3$	0	0
$4$	6.29742	0.0122996
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	11512.9	0.993756
$9$	0	0
$10$	−14228.8	−0.449955
$11$	21227.5	0.437151	0.218576	−	0.975820i	$-0.429859\pi$
$11$	21227.5	0.437151	0.218576	+	0.975820i	$0.429859\pi$
$12$	0	0
$13$	47399.7	0.460289	0.230144	−	0.973156i	$-0.426080\pi$
$13$	47399.7	0.460289	0.230144	+	0.973156i	$0.426080\pi$
$14$	−54661.5	−0.380282
$15$	0	0
$16$	−265329.	−1.01215
$17$	−569500.	−1.65376	−0.826882	−	0.562376i	$-0.809887\pi$
$17$	−569500.	−1.65376	−0.826882	+	0.562376i	$0.809887\pi$
$18$	0	0
$19$	1.08959e6	1.91810	0.959048	−	0.283245i	$-0.0914109\pi$
$19$	1.08959e6	1.91810	0.959048	+	0.283245i	$0.0914109\pi$
$20$	3935.89	0.00550057
$21$	0	0
$22$	−483268.	−0.439831
$23$	1.34776e6	1.00424	0.502121	−	0.864797i	$-0.332553\pi$
$23$	1.34776e6	1.00424	0.502121	+	0.864797i	$0.332553\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−1.07911e6	−0.463111
$27$	0	0
$28$	15120.1	0.00464883
$29$	6.23203e6	1.63621	0.818104	−	0.575071i	$-0.195026\pi$
$29$	6.23203e6	1.63621	0.818104	+	0.575071i	$0.195026\pi$
$30$	0	0
$31$	−413669.	−0.0804499	−0.0402250	−	0.999191i	$-0.512807\pi$
$31$	−413669.	−0.0804499	−0.0402250	+	0.999191i	$0.512807\pi$
$32$	145906.	0.0245979
$33$	0	0
$34$	1.29653e7	1.66390
$35$	1.50062e6	0.169031
$36$	0	0
$37$	1.28411e7	1.12640	0.563200	−	0.826320i	$-0.309570\pi$
$37$	1.28411e7	1.12640	0.563200	+	0.826320i	$0.309570\pi$
$38$	−2.48057e7	−1.92986
$39$	0	0
$40$	7.19556e6	0.444421
$41$	−3.90722e6	−0.215944	−0.107972	−	0.994154i	$-0.534436\pi$
$41$	−3.90722e6	−0.215944	−0.107972	+	0.994154i	$0.534436\pi$
$42$	0	0
$43$	1.27760e7	0.569886	0.284943	−	0.958544i	$-0.408025\pi$
$43$	1.27760e7	0.569886	0.284943	+	0.958544i	$0.408025\pi$
$44$	133678.	0.00537680
$45$	0	0
$46$	−3.06834e7	−1.01040
$47$	3.61213e7	1.07975	0.539874	−	0.841746i	$-0.318472\pi$
$47$	3.61213e7	1.07975	0.539874	+	0.841746i	$0.318472\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−8.89303e6	−0.201226
$51$	0	0
$52$	298496.	0.00566139
$53$	−1.10032e8	−1.91548	−0.957740	−	0.287635i	$-0.907131\pi$
$53$	−1.10032e8	−1.91548	−0.957740	+	0.287635i	$0.907131\pi$
$54$	0	0
$55$	1.32672e7	0.195500
$56$	2.76425e7	0.375604
$57$	0	0
$58$	−1.41879e8	−1.64624
$59$	8.68650e6	0.0933277	0.0466639	−	0.998911i	$-0.485141\pi$
$59$	8.68650e6	0.0933277	0.0466639	+	0.998911i	$0.485141\pi$
$60$	0	0
$61$	1.06957e8	0.989066	0.494533	−	0.869159i	$-0.335339\pi$
$61$	1.06957e8	0.989066	0.494533	+	0.869159i	$0.335339\pi$
$62$	9.41766e6	0.0809432
$63$	0	0
$64$	1.32527e8	0.987400
$65$	2.96248e7	0.205847
$66$	0	0
$67$	1.89423e8	1.14841	0.574204	−	0.818712i	$-0.305312\pi$
$67$	1.89423e8	1.14841	0.574204	+	0.818712i	$0.305312\pi$
$68$	−3.58638e6	−0.0203407
$69$	0	0
$70$	−3.41634e7	−0.170067
$71$	−9.10989e7	−0.425452	−0.212726	−	0.977112i	$-0.568234\pi$
$71$	−9.10989e7	−0.425452	−0.212726	+	0.977112i	$0.568234\pi$
$72$	0	0
$73$	−9.84572e7	−0.405784	−0.202892	−	0.979201i	$-0.565034\pi$
$73$	−9.84572e7	−0.405784	−0.202892	+	0.979201i	$0.565034\pi$
$74$	−2.92341e8	−1.13331
$75$	0	0
$76$	6.86157e6	0.0235919
$77$	5.09672e7	0.165228
$78$	0	0
$79$	−2.41368e8	−0.697200	−0.348600	−	0.937272i	$-0.613343\pi$
$79$	−2.41368e8	−0.697200	−0.348600	+	0.937272i	$0.613343\pi$
$80$	−1.65830e8	−0.452647
$81$	0	0
$82$	8.89525e7	0.217268
$83$	−4.28456e8	−0.990956	−0.495478	−	0.868620i	$-0.665007\pi$
$83$	−4.28456e8	−0.990956	−0.495478	+	0.868620i	$0.665007\pi$
$84$	0	0
$85$	−3.55937e8	−0.739586
$86$	−2.90861e8	−0.573380
$87$	0	0
$88$	2.44390e8	0.434422
$89$	−5.29736e7	−0.0894962	−0.0447481	−	0.998998i	$-0.514249\pi$
$89$	−5.29736e7	−0.0894962	−0.0447481	+	0.998998i	$0.514249\pi$
$90$	0	0
$91$	1.13807e8	0.173973
$92$	8.48743e6	0.0123518
$93$	0	0
$94$	−8.22342e8	−1.08637
$95$	6.80991e8	0.857798
$96$	0	0
$97$	3.01374e8	0.345647	0.172824	−	0.984953i	$-0.444711\pi$
$97$	3.01374e8	0.345647	0.172824	+	0.984953i	$0.444711\pi$
$98$	−1.31242e8	−0.143733
$99$	0	0
$100$	2.45993e6	0.00245993
$101$	−1.68070e9	−1.60710	−0.803552	−	0.595234i	$-0.797059\pi$
$101$	−1.68070e9	−1.60710	−0.803552	+	0.595234i	$0.797059\pi$
$102$	0	0
$103$	4.68604e8	0.410241	0.205120	−	0.978737i	$-0.434241\pi$
$103$	4.68604e8	0.410241	0.205120	+	0.978737i	$0.434241\pi$
$104$	5.45708e8	0.457415
$105$	0	0
$106$	2.50500e9	1.92722
$107$	−5.15481e8	−0.380177	−0.190089	−	0.981767i	$-0.560877\pi$
$107$	−5.15481e8	−0.380177	−0.190089	+	0.981767i	$0.560877\pi$
$108$	0	0
$109$	5.36760e8	0.364218	0.182109	−	0.983278i	$-0.441708\pi$
$109$	5.36760e8	0.364218	0.182109	+	0.983278i	$0.441708\pi$
$110$	−3.02043e8	−0.196699
$111$	0	0
$112$	−6.37054e8	−0.382556
$113$	5.40168e8	0.311656	0.155828	−	0.987784i	$-0.450195\pi$
$113$	5.40168e8	0.311656	0.155828	+	0.987784i	$0.450195\pi$
$114$	0	0
$115$	8.42353e8	0.449111
$116$	3.92457e7	0.0201248
$117$	0	0
$118$	−1.97758e8	−0.0938999
$119$	−1.36737e9	−0.625064
$120$	0	0
$121$	−1.90734e9	−0.808899
$122$	−2.43500e9	−0.995130
$123$	0	0
$124$	−2.60505e6	−0.000989505 0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	8.96674e8	0.305857	0.152928	−	0.988237i	$-0.451130\pi$
$127$	8.96674e8	0.305857	0.152928	+	0.988237i	$0.451130\pi$
$128$	−3.09182e9	−1.01805
$129$	0	0
$130$	−6.74443e8	−0.207109
$131$	2.92350e9	0.867325	0.433663	−	0.901075i	$-0.357221\pi$
$131$	2.92350e9	0.867325	0.433663	+	0.901075i	$0.357221\pi$
$132$	0	0
$133$	2.61610e9	0.724972
$134$	−4.31243e9	−1.15545
$135$	0	0
$136$	−6.55659e9	−1.64344
$137$	−4.52069e9	−1.09638	−0.548191	−	0.836353i	$-0.684683\pi$
$137$	−4.52069e9	−1.09638	−0.548191	+	0.836353i	$0.684683\pi$
$138$	0	0
$139$	−7.23687e9	−1.64431	−0.822156	−	0.569262i	$-0.807229\pi$
$139$	−7.23687e9	−1.64431	−0.822156	+	0.569262i	$0.807229\pi$
$140$	9.45006e6	0.00207902
$141$	0	0
$142$	2.07397e9	0.428061
$143$	1.00618e9	0.201216
$144$	0	0
$145$	3.89502e9	0.731734
$146$	2.24149e9	0.408272
$147$	0	0
$148$	8.08655e7	0.0138543
$149$	−3.83905e9	−0.638096	−0.319048	−	0.947739i	$-0.603363\pi$
$149$	−3.83905e9	−0.638096	−0.319048	+	0.947739i	$0.603363\pi$
$150$	0	0
$151$	−6.14269e9	−0.961528	−0.480764	−	0.876850i	$-0.659641\pi$
$151$	−6.14269e9	−0.961528	−0.480764	+	0.876850i	$0.659641\pi$
$152$	1.25443e10	1.90612
$153$	0	0
$154$	−1.16033e9	−0.166241
$155$	−2.58543e8	−0.0359783
$156$	0	0
$157$	9.31680e9	1.22382	0.611911	−	0.790926i	$-0.290401\pi$
$157$	9.31680e9	1.22382	0.611911	+	0.790926i	$0.290401\pi$
$158$	5.49501e9	0.701474
$159$	0	0
$160$	9.11912e7	0.0110005
$161$	3.23598e9	0.379568
$162$	0	0
$163$	6.53147e9	0.724714	0.362357	−	0.932039i	$-0.381972\pi$
$163$	6.53147e9	0.724714	0.362357	+	0.932039i	$0.381972\pi$
$164$	−2.46054e7	−0.00265603
$165$	0	0
$166$	9.75429e9	0.997032
$167$	−2.73053e9	−0.271659	−0.135829	−	0.990732i	$-0.543370\pi$
$167$	−2.73053e9	−0.271659	−0.135829	+	0.990732i	$0.543370\pi$
$168$	0	0
$169$	−8.35777e9	−0.788134
$170$	8.10332e9	0.744120
$171$	0	0
$172$	8.04560e7	0.00700939
$173$	−3.68616e8	−0.0312872	−0.0156436	−	0.999878i	$-0.504980\pi$
$173$	−3.68616e8	−0.0312872	−0.0156436	+	0.999878i	$0.504980\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−5.63226e9	−0.442462
$177$	0	0
$178$	1.20600e9	0.0900449
$179$	1.97250e10	1.43608	0.718040	−	0.696001i	$-0.245039\pi$
$179$	1.97250e10	1.43608	0.718040	+	0.696001i	$0.245039\pi$
$180$	0	0
$181$	−3.11426e9	−0.215676	−0.107838	−	0.994168i	$-0.534393\pi$
$181$	−3.11426e9	−0.215676	−0.107838	+	0.994168i	$0.534393\pi$
$182$	−2.59094e9	−0.175039
$183$	0	0
$184$	1.55167e10	0.997972
$185$	8.02566e9	0.503742
$186$	0	0
$187$	−1.20891e10	−0.722945
$188$	2.27471e8	0.0132805
$189$	0	0
$190$	−1.55035e10	−0.863057
$191$	3.17752e10	1.72758	0.863790	−	0.503852i	$-0.168084\pi$
$191$	3.17752e10	1.72758	0.863790	+	0.503852i	$0.168084\pi$
$192$	0	0
$193$	−2.21167e10	−1.14739	−0.573696	−	0.819068i	$-0.694491\pi$
$193$	−2.21167e10	−1.14739	−0.573696	+	0.819068i	$0.694491\pi$
$194$	−6.86112e9	−0.347766
$195$	0	0
$196$	3.63034e7	0.00175709
$197$	−2.15871e10	−1.02116	−0.510582	−	0.859829i	$-0.670570\pi$
$197$	−2.15871e10	−1.02116	−0.510582	+	0.859829i	$0.670570\pi$
$198$	0	0
$199$	2.78732e10	1.25993	0.629966	−	0.776622i	$-0.283069\pi$
$199$	2.78732e10	1.25993	0.629966	+	0.776622i	$0.283069\pi$
$200$	4.49723e9	0.198751
$201$	0	0
$202$	3.82631e10	1.61696
$203$	1.49631e10	0.618428
$204$	0	0
$205$	−2.44202e9	−0.0965731
$206$	−1.06683e10	−0.412756
$207$	0	0
$208$	−1.25765e10	−0.465881
$209$	2.31292e10	0.838498
$210$	0	0
$211$	1.31458e10	0.456579	0.228290	−	0.973593i	$-0.426687\pi$
$211$	1.31458e10	0.456579	0.228290	+	0.973593i	$0.426687\pi$
$212$	−6.92917e8	−0.0235597
$213$	0	0
$214$	1.17355e10	0.382508
$215$	7.98502e9	0.254861
$216$	0	0
$217$	−9.93220e8	−0.0304072
$218$	−1.22200e10	−0.366451
$219$	0	0
$220$	8.35490e7	0.00240458
$221$	−2.69941e10	−0.761209
$222$	0	0
$223$	5.58951e10	1.51357	0.756784	−	0.653665i	$-0.226769\pi$
$223$	5.58951e10	1.51357	0.756784	+	0.653665i	$0.226769\pi$
$224$	3.50320e8	0.00929713
$225$	0	0
$226$	−1.22975e10	−0.313567
$227$	5.98856e10	1.49695	0.748473	−	0.663165i	$-0.230787\pi$
$227$	5.98856e10	1.49695	0.748473	+	0.663165i	$0.230787\pi$
$228$	0	0
$229$	6.62019e10	1.59078	0.795391	−	0.606097i	$-0.207266\pi$
$229$	6.62019e10	1.59078	0.795391	+	0.606097i	$0.207266\pi$
$230$	−1.91771e10	−0.451864
$231$	0	0
$232$	7.17487e10	1.62599
$233$	−6.50041e10	−1.44490	−0.722452	−	0.691421i	$-0.756985\pi$
$233$	−6.50041e10	−1.44490	−0.722452	+	0.691421i	$0.756985\pi$
$234$	0	0
$235$	2.25758e10	0.482878
$236$	5.47025e7	0.00114790
$237$	0	0
$238$	3.11297e10	0.628896
$239$	−9.18633e9	−0.182117	−0.0910587	−	0.995846i	$-0.529025\pi$
$239$	−9.18633e9	−0.182117	−0.0910587	+	0.995846i	$0.529025\pi$
$240$	0	0
$241$	1.49904e10	0.286244	0.143122	−	0.989705i	$-0.454286\pi$
$241$	1.49904e10	0.286244	0.143122	+	0.989705i	$0.454286\pi$
$242$	4.34228e10	0.813858
$243$	0	0
$244$	6.73553e8	0.0121652
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	5.16460e10	0.882878
$248$	−4.76253e9	−0.0799476
$249$	0	0
$250$	−5.55814e9	−0.0899911
$251$	1.03095e11	1.63947	0.819737	−	0.572741i	$-0.194120\pi$
$251$	1.03095e11	1.63947	0.819737	+	0.572741i	$0.194120\pi$
$252$	0	0
$253$	2.86097e10	0.439006
$254$	−2.04138e10	−0.307732
$255$	0	0
$256$	2.53529e9	0.0368934
$257$	−7.34759e10	−1.05062	−0.525310	−	0.850911i	$-0.676051\pi$
$257$	−7.34759e10	−1.05062	−0.525310	+	0.850911i	$0.676051\pi$
$258$	0	0
$259$	3.08314e10	0.425739
$260$	1.86560e8	0.00253185
$261$	0	0
$262$	−6.65568e10	−0.872643
$263$	8.95222e10	1.15380	0.576899	−	0.816815i	$-0.304263\pi$
$263$	8.95222e10	1.15380	0.576899	+	0.816815i	$0.304263\pi$
$264$	0	0
$265$	−6.87700e10	−0.856629
$266$	−5.95584e10	−0.729417
$267$	0	0
$268$	1.19288e9	0.0141250
$269$	−1.00270e11	−1.16758	−0.583791	−	0.811904i	$-0.698431\pi$
$269$	−1.00270e11	−1.16758	−0.583791	+	0.811904i	$0.698431\pi$
$270$	0	0
$271$	1.05822e11	1.19183	0.595915	−	0.803048i	$-0.296790\pi$
$271$	1.05822e11	1.19183	0.595915	+	0.803048i	$0.296790\pi$
$272$	1.51105e11	1.67385
$273$	0	0
$274$	1.02919e11	1.10310
$275$	8.29199e9	0.0874303
$276$	0	0
$277$	−1.20160e10	−0.122631	−0.0613156	−	0.998118i	$-0.519530\pi$
$277$	−1.20160e10	−0.122631	−0.0613156	+	0.998118i	$0.519530\pi$
$278$	1.64756e11	1.65439
$279$	0	0
$280$	1.72765e10	0.167975
$281$	2.03641e11	1.94844	0.974218	−	0.225610i	$-0.0724375\pi$
$281$	2.03641e11	1.94844	0.974218	+	0.225610i	$0.0724375\pi$
$282$	0	0
$283$	1.52674e11	1.41490	0.707449	−	0.706765i	$-0.249846\pi$
$283$	1.52674e11	1.41490	0.707449	+	0.706765i	$0.249846\pi$
$284$	−5.73688e8	−0.00523291
$285$	0	0
$286$	−2.29068e10	−0.202449
$287$	−9.38125e9	−0.0816191
$288$	0	0
$289$	2.05742e11	1.73493
$290$	−8.86745e10	−0.736220
$291$	0	0
$292$	−6.20026e8	−0.00499099
$293$	1.91313e11	1.51649	0.758244	−	0.651970i	$-0.226057\pi$
$293$	1.91313e11	1.51649	0.758244	+	0.651970i	$0.226057\pi$
$294$	0	0
$295$	5.42906e9	0.0417374
$296$	1.47838e11	1.11937
$297$	0	0
$298$	8.74004e10	0.642008
$299$	6.38836e10	0.462242
$300$	0	0
$301$	3.06753e10	0.215397
$302$	1.39845e11	0.967423
$303$	0	0
$304$	−2.89098e11	−1.94140
$305$	6.68482e10	0.442324
$306$	0	0
$307$	−7.89027e10	−0.506955	−0.253477	−	0.967341i	$-0.581574\pi$
$307$	−7.89027e10	−0.506955	−0.253477	+	0.967341i	$0.581574\pi$
$308$	3.20962e8	0.00203224
$309$	0	0
$310$	5.88603e9	0.0361989
$311$	−1.17336e11	−0.711230	−0.355615	−	0.934633i	$-0.615729\pi$
$311$	−1.17336e11	−0.711230	−0.355615	+	0.934633i	$0.615729\pi$
$312$	0	0
$313$	−1.77136e11	−1.04318	−0.521588	−	0.853198i	$-0.674660\pi$
$313$	−1.77136e11	−1.04318	−0.521588	+	0.853198i	$0.674660\pi$
$314$	−2.12108e11	−1.23133
$315$	0	0
$316$	−1.51999e9	−0.00857531
$317$	−3.17183e11	−1.76418	−0.882091	−	0.471080i	$-0.843864\pi$
$317$	−3.17183e11	−1.76418	−0.882091	+	0.471080i	$0.843864\pi$
$318$	0	0
$319$	1.32290e11	0.715270
$320$	8.28291e10	0.441579
$321$	0	0
$322$	−7.36708e10	−0.381895
$323$	−6.20519e11	−3.17208
$324$	0	0
$325$	1.85155e10	0.0920578
$326$	−1.48696e11	−0.729157
$327$	0	0
$328$	−4.49835e10	−0.214596
$329$	8.67272e10	0.408107
$330$	0	0
$331$	−7.29998e10	−0.334269	−0.167134	−	0.985934i	$-0.553451\pi$
$331$	−7.29998e10	−0.334269	−0.167134	+	0.985934i	$0.553451\pi$
$332$	−2.69816e9	−0.0121884
$333$	0	0
$334$	6.21637e10	0.273324
$335$	1.18389e11	0.513584
$336$	0	0
$337$	−1.07231e11	−0.452882	−0.226441	−	0.974025i	$-0.572709\pi$
$337$	−1.07231e11	−0.452882	−0.226441	+	0.974025i	$0.572709\pi$
$338$	1.90274e11	0.792966
$339$	0	0
$340$	−2.24149e9	−0.00909664
$341$	−8.78116e9	−0.0351688
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	1.47089e11	0.566328
$345$	0	0
$346$	8.39196e9	0.0314790
$347$	2.66192e11	0.985627	0.492813	−	0.870135i	$-0.335969\pi$
$347$	2.66192e11	0.985627	0.492813	+	0.870135i	$0.335969\pi$
$348$	0	0
$349$	7.30078e10	0.263424	0.131712	−	0.991288i	$-0.457953\pi$
$349$	7.30078e10	0.263424	0.131712	+	0.991288i	$0.457953\pi$
$350$	−2.13522e10	−0.0760564
$351$	0	0
$352$	3.09722e9	0.0107530
$353$	−1.03915e11	−0.356198	−0.178099	−	0.984013i	$-0.556995\pi$
$353$	−1.03915e11	−0.356198	−0.178099	+	0.984013i	$0.556995\pi$
$354$	0	0
$355$	−5.69368e10	−0.190268
$356$	−3.33597e8	−0.00110077
$357$	0	0
$358$	−4.49063e11	−1.44489
$359$	−8.28712e10	−0.263317	−0.131658	−	0.991295i	$-0.542030\pi$
$359$	−8.28712e10	−0.263317	−0.131658	+	0.991295i	$0.542030\pi$
$360$	0	0
$361$	8.64509e11	2.67909
$362$	7.08997e10	0.216998
$363$	0	0
$364$	7.16688e8	0.00213980
$365$	−6.15357e10	−0.181472
$366$	0	0
$367$	−1.99606e11	−0.574348	−0.287174	−	0.957878i	$-0.592716\pi$
$367$	−1.99606e11	−0.574348	−0.287174	+	0.957878i	$0.592716\pi$
$368$	−3.57600e11	−1.01644
$369$	0	0
$370$	−1.82713e11	−0.506830
$371$	−2.64187e11	−0.723983
$372$	0	0
$373$	4.49796e10	0.120317	0.0601583	−	0.998189i	$-0.480839\pi$
$373$	4.49796e10	0.120317	0.0601583	+	0.998189i	$0.480839\pi$
$374$	2.75221e11	0.727377
$375$	0	0
$376$	4.15861e11	1.07301
$377$	2.95396e11	0.753128
$378$	0	0
$379$	−1.89077e9	−0.00470718	−0.00235359	−	0.999997i	$-0.500749\pi$
$379$	−1.89077e9	−0.00470718	−0.00235359	+	0.999997i	$0.500749\pi$
$380$	4.28848e9	0.0105506
$381$	0	0
$382$	−7.23399e11	−1.73817
$383$	−3.12549e11	−0.742205	−0.371103	−	0.928592i	$-0.621020\pi$
$383$	−3.12549e11	−0.742205	−0.371103	+	0.928592i	$0.621020\pi$
$384$	0	0
$385$	3.18545e10	0.0738920
$386$	5.03511e11	1.15443
$387$	0	0
$388$	1.89788e9	0.00425134
$389$	−2.84599e11	−0.630173	−0.315086	−	0.949063i	$-0.602033\pi$
$389$	−2.84599e11	−0.630173	−0.315086	+	0.949063i	$0.602033\pi$
$390$	0	0
$391$	−7.67551e11	−1.66078
$392$	6.63696e10	0.141965
$393$	0	0
$394$	4.91454e11	1.02742
$395$	−1.50855e11	−0.311797
$396$	0	0
$397$	−7.92352e11	−1.60089	−0.800444	−	0.599408i	$-0.795403\pi$
$397$	−7.92352e11	−1.60089	−0.800444	+	0.599408i	$0.795403\pi$
$398$	−6.34564e11	−1.26766
$399$	0	0
$400$	−1.03644e11	−0.202430
$401$	−6.62180e11	−1.27887	−0.639435	−	0.768845i	$-0.720832\pi$
$401$	−6.62180e11	−1.27887	−0.639435	+	0.768845i	$0.720832\pi$
$402$	0	0
$403$	−1.96078e10	−0.0370302
$404$	−1.05841e10	−0.0197668
$405$	0	0
$406$	−3.40652e11	−0.622220
$407$	2.72583e11	0.492407
$408$	0	0
$409$	1.09286e12	1.93112	0.965560	−	0.260182i	$-0.0837825\pi$
$409$	1.09286e12	1.93112	0.965560	+	0.260182i	$0.0837825\pi$
$410$	5.55953e10	0.0971652
$411$	0	0
$412$	2.95100e9	0.00504581
$413$	2.08563e10	0.0352746
$414$	0	0
$415$	−2.67785e11	−0.443169
$416$	6.91589e9	0.0113221
$417$	0	0
$418$	−5.26562e11	−0.843639
$419$	−3.74698e11	−0.593907	−0.296953	−	0.954892i	$-0.595971\pi$
$419$	−3.74698e11	−0.593907	−0.296953	+	0.954892i	$0.595971\pi$
$420$	0	0
$421$	7.35051e11	1.14038	0.570188	−	0.821514i	$-0.306870\pi$
$421$	7.35051e11	1.14038	0.570188	+	0.821514i	$0.306870\pi$
$422$	−2.99279e11	−0.459379
$423$	0	0
$424$	−1.26679e12	−1.90352
$425$	−2.22461e11	−0.330753
$426$	0	0
$427$	2.56804e11	0.373832
$428$	−3.24620e9	−0.00467604
$429$	0	0
$430$	−1.81788e11	−0.256423
$431$	5.29788e11	0.739528	0.369764	−	0.929126i	$-0.379439\pi$
$431$	5.29788e11	0.739528	0.369764	+	0.929126i	$0.379439\pi$
$432$	0	0
$433$	1.82162e11	0.249036	0.124518	−	0.992217i	$-0.460262\pi$
$433$	1.82162e11	0.249036	0.124518	+	0.992217i	$0.460262\pi$
$434$	2.26118e10	0.0305936
$435$	0	0
$436$	3.38020e9	0.00447975
$437$	1.46850e12	1.92623
$438$	0	0
$439$	−8.13895e11	−1.04587	−0.522936	−	0.852372i	$-0.675163\pi$
$439$	−8.13895e11	−1.04587	−0.522936	+	0.852372i	$0.675163\pi$
$440$	1.52744e11	0.194279
$441$	0	0
$442$	6.14552e11	0.765876
$443$	5.65176e11	0.697215	0.348607	−	0.937269i	$-0.386655\pi$
$443$	5.65176e11	0.697215	0.348607	+	0.937269i	$0.386655\pi$
$444$	0	0
$445$	−3.31085e10	−0.0400239
$446$	−1.27252e12	−1.52285
$447$	0	0
$448$	3.18196e11	0.373202
$449$	5.86919e11	0.681506	0.340753	−	0.940153i	$-0.389318\pi$
$449$	5.86919e11	0.681506	0.340753	+	0.940153i	$0.389318\pi$
$450$	0	0
$451$	−8.29406e10	−0.0944002
$452$	3.40166e9	0.00383326
$453$	0	0
$454$	−1.36337e12	−1.50612
$455$	7.11292e10	0.0778030
$456$	0	0
$457$	−2.06593e10	−0.0221561	−0.0110781	−	0.999939i	$-0.503526\pi$
$457$	−2.06593e10	−0.0221561	−0.0110781	+	0.999939i	$0.503526\pi$
$458$	−1.50716e12	−1.60054
$459$	0	0
$460$	5.30464e9	0.00552390
$461$	−2.59983e11	−0.268096	−0.134048	−	0.990975i	$-0.542798\pi$
$461$	−2.59983e11	−0.268096	−0.134048	+	0.990975i	$0.542798\pi$
$462$	0	0
$463$	−6.62434e11	−0.669928	−0.334964	−	0.942231i	$-0.608724\pi$
$463$	−6.62434e11	−0.669928	−0.334964	+	0.942231i	$0.608724\pi$
$464$	−1.65353e12	−1.65608
$465$	0	0
$466$	1.47989e12	1.45376
$467$	5.78561e11	0.562890	0.281445	−	0.959577i	$-0.409186\pi$
$467$	5.78561e11	0.562890	0.281445	+	0.959577i	$0.409186\pi$
$468$	0	0
$469$	4.54805e11	0.434058
$470$	−5.13964e11	−0.485839
$471$	0	0
$472$	1.00007e11	0.0927450
$473$	2.71203e11	0.249126
$474$	0	0
$475$	4.25619e11	0.383619
$476$	−8.61089e9	−0.00768806
$477$	0	0
$478$	2.09137e11	0.183234
$479$	−1.30509e12	−1.13274	−0.566370	−	0.824151i	$-0.691653\pi$
$479$	−1.30509e12	−1.13274	−0.566370	+	0.824151i	$0.691653\pi$
$480$	0	0
$481$	6.08662e11	0.518470
$482$	−3.41274e11	−0.287999
$483$	0	0
$484$	−1.20113e10	−0.00994916
$485$	1.88359e11	0.154578
$486$	0	0
$487$	2.26035e12	1.82094	0.910468	−	0.413579i	$-0.135722\pi$
$487$	2.26035e12	1.82094	0.910468	+	0.413579i	$0.135722\pi$
$488$	1.23139e12	0.982890
$489$	0	0
$490$	−8.20264e10	−0.0642794
$491$	2.05833e11	0.159827	0.0799133	−	0.996802i	$-0.474536\pi$
$491$	2.05833e11	0.159827	0.0799133	+	0.996802i	$0.474536\pi$
$492$	0	0
$493$	−3.54914e12	−2.70590
$494$	−1.17578e12	−0.888291
$495$	0	0
$496$	1.09758e11	0.0814273
$497$	−2.18729e11	−0.160806
$498$	0	0
$499$	−2.10713e11	−0.152139	−0.0760693	−	0.997103i	$-0.524237\pi$
$499$	−2.10713e11	−0.152139	−0.0760693	+	0.997103i	$0.524237\pi$
$500$	1.53746e9	0.00110011
$501$	0	0
$502$	−2.34707e12	−1.64953
$503$	−7.30487e10	−0.0508811	−0.0254406	−	0.999676i	$-0.508099\pi$
$503$	−7.30487e10	−0.0508811	−0.0254406	+	0.999676i	$0.508099\pi$
$504$	0	0
$505$	−1.05044e12	−0.718719
$506$	−6.51332e11	−0.441697
$507$	0	0
$508$	5.64673e9	0.00376193
$509$	1.52888e11	0.100958	0.0504792	−	0.998725i	$-0.483925\pi$
$509$	1.52888e11	0.100958	0.0504792	+	0.998725i	$0.483925\pi$
$510$	0	0
$511$	−2.36396e11	−0.153372
$512$	1.52529e12	0.980932
$513$	0	0
$514$	1.67276e12	1.05706
$515$	2.92878e11	0.183465
$516$	0	0
$517$	7.66764e11	0.472014
$518$	−7.01911e11	−0.428350
$519$	0	0
$520$	3.41067e11	0.204562
$521$	3.12005e11	0.185520	0.0927601	−	0.995688i	$-0.470431\pi$
$521$	3.12005e11	0.185520	0.0927601	+	0.995688i	$0.470431\pi$
$522$	0	0
$523$	7.13578e11	0.417046	0.208523	−	0.978017i	$-0.433134\pi$
$523$	7.13578e11	0.417046	0.208523	+	0.978017i	$0.433134\pi$
$524$	1.84105e10	0.0106678
$525$	0	0
$526$	−2.03808e12	−1.16087
$527$	2.35585e11	0.133045
$528$	0	0
$529$	1.53155e10	0.00850315
$530$	1.56563e12	0.861881
$531$	0	0
$532$	1.64746e10	0.00891689
$533$	−1.85201e11	−0.0993966
$534$	0	0
$535$	−3.22176e11	−0.170020
$536$	2.18081e12	1.14124
$537$	0	0
$538$	2.28277e12	1.17474
$539$	1.22372e11	0.0624502
$540$	0	0
$541$	−7.80203e9	−0.00391580	−0.00195790	−	0.999998i	$-0.500623\pi$
$541$	−7.80203e9	−0.00391580	−0.00195790	+	0.999998i	$0.500623\pi$
$542$	−2.40916e12	−1.19914
$543$	0	0
$544$	−8.30934e10	−0.0406791
$545$	3.35475e11	0.162883
$546$	0	0
$547$	−2.00379e12	−0.956994	−0.478497	−	0.878089i	$-0.658818\pi$
$547$	−2.00379e12	−0.956994	−0.478497	+	0.878089i	$0.658818\pi$
$548$	−2.84687e10	−0.0134851
$549$	0	0
$550$	−1.88777e11	−0.0879663
$551$	6.79033e12	3.13840
$552$	0	0
$553$	−5.79524e11	−0.263517
$554$	2.73558e11	0.123383
$555$	0	0
$556$	−4.55736e10	−0.0202245
$557$	−6.30817e11	−0.277687	−0.138843	−	0.990314i	$-0.544338\pi$
$557$	−6.30817e11	−0.277687	−0.138843	+	0.990314i	$0.544338\pi$
$558$	0	0
$559$	6.05580e11	0.262312
$560$	−3.98159e11	−0.171084
$561$	0	0
$562$	−4.63611e12	−1.96038
$563$	−2.72592e12	−1.14347	−0.571736	−	0.820438i	$-0.693730\pi$
$563$	−2.72592e12	−1.14347	−0.571736	+	0.820438i	$0.693730\pi$
$564$	0	0
$565$	3.37605e11	0.139377
$566$	−3.47579e12	−1.42357
$567$	0	0
$568$	−1.04881e12	−0.422796
$569$	3.54264e12	1.41684	0.708422	−	0.705789i	$-0.249408\pi$
$569$	3.54264e12	1.41684	0.708422	+	0.705789i	$0.249408\pi$
$570$	0	0
$571$	3.74623e12	1.47480	0.737398	−	0.675458i	$-0.236054\pi$
$571$	3.74623e12	1.47480	0.737398	+	0.675458i	$0.236054\pi$
$572$	6.33631e9	0.00247488
$573$	0	0
$574$	2.13575e11	0.0821196
$575$	5.26470e11	0.200849
$576$	0	0
$577$	3.74243e12	1.40560	0.702802	−	0.711386i	$-0.251932\pi$
$577$	3.74243e12	1.40560	0.702802	+	0.711386i	$0.251932\pi$
$578$	−4.68396e12	−1.74557
$579$	0	0
$580$	2.45285e10	0.00900007
$581$	−1.02872e12	−0.374546
$582$	0	0
$583$	−2.33570e12	−0.837354
$584$	−1.13353e12	−0.403250
$585$	0	0
$586$	−4.35545e12	−1.52579
$587$	2.19633e12	0.763532	0.381766	−	0.924259i	$-0.375316\pi$
$587$	2.19633e12	0.763532	0.381766	+	0.924259i	$0.375316\pi$
$588$	0	0
$589$	−4.50728e11	−0.154311
$590$	−1.23599e11	−0.0419933
$591$	0	0
$592$	−3.40710e12	−1.14008
$593$	5.88312e12	1.95372	0.976859	−	0.213884i	$-0.0686114\pi$
$593$	5.88312e12	1.95372	0.976859	+	0.213884i	$0.0686114\pi$
$594$	0	0
$595$	−8.54606e11	−0.279537
$596$	−2.41761e10	−0.00784835
$597$	0	0
$598$	−1.45438e12	−0.465076
$599$	1.32770e12	0.421385	0.210692	−	0.977552i	$-0.432428\pi$
$599$	1.32770e12	0.421385	0.210692	+	0.977552i	$0.432428\pi$
$600$	0	0
$601$	−5.90463e11	−0.184611	−0.0923055	−	0.995731i	$-0.529424\pi$
$601$	−5.90463e11	−0.184611	−0.0923055	+	0.995731i	$0.529424\pi$
$602$	−6.98357e11	−0.216717
$603$	0	0
$604$	−3.86830e10	−0.0118265
$605$	−1.19209e12	−0.361751
$606$	0	0
$607$	1.15860e12	0.346404	0.173202	−	0.984886i	$-0.444589\pi$
$607$	1.15860e12	0.346404	0.173202	+	0.984886i	$0.444589\pi$
$608$	1.58977e11	0.0471811
$609$	0	0
$610$	−1.52187e12	−0.445036
$611$	1.71214e12	0.496996
$612$	0	0
$613$	−4.56114e12	−1.30467	−0.652337	−	0.757929i	$-0.726211\pi$
$613$	−4.56114e12	−1.30467	−0.652337	+	0.757929i	$0.726211\pi$
$614$	1.79631e12	0.510063
$615$	0	0
$616$	5.86781e11	0.164196
$617$	5.85962e10	0.0162774	0.00813872	−	0.999967i	$-0.497409\pi$
$617$	5.85962e10	0.0162774	0.00813872	+	0.999967i	$0.497409\pi$
$618$	0	0
$619$	2.31615e12	0.634101	0.317050	−	0.948409i	$-0.397308\pi$
$619$	2.31615e12	0.634101	0.317050	+	0.948409i	$0.397308\pi$
$620$	−1.62815e9	−0.000442520 0
$621$	0	0
$622$	2.67129e12	0.715590
$623$	−1.27190e11	−0.0338264
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	4.03271e12	1.04957
$627$	0	0
$628$	5.86718e10	0.0150526
$629$	−7.31298e12	−1.86280
$630$	0	0
$631$	1.36072e12	0.341693	0.170847	−	0.985298i	$-0.445350\pi$
$631$	1.36072e12	0.341693	0.170847	+	0.985298i	$0.445350\pi$
$632$	−2.77884e12	−0.692847
$633$	0	0
$634$	7.22103e12	1.77500
$635$	5.60422e11	0.136783
$636$	0	0
$637$	2.73250e11	0.0657555
$638$	−3.01174e12	−0.719655
$639$	0	0
$640$	−1.93239e12	−0.455286
$641$	5.19542e12	1.21551	0.607756	−	0.794124i	$-0.292070\pi$
$641$	5.19542e12	1.21551	0.607756	+	0.794124i	$0.292070\pi$
$642$	0	0
$643$	−5.15183e12	−1.18854	−0.594268	−	0.804267i	$-0.702558\pi$
$643$	−5.15183e12	−1.18854	−0.594268	+	0.804267i	$0.702558\pi$
$644$	2.03783e10	0.00466855
$645$	0	0
$646$	1.41268e13	3.19152
$647$	9.10873e11	0.204357	0.102178	−	0.994766i	$-0.467419\pi$
$647$	9.10873e11	0.204357	0.102178	+	0.994766i	$0.467419\pi$
$648$	0	0
$649$	1.84393e11	0.0407983
$650$	−4.21527e11	−0.0926222
$651$	0	0
$652$	4.11314e10	0.00891372
$653$	3.27767e12	0.705433	0.352717	−	0.935730i	$-0.385258\pi$
$653$	3.27767e12	0.705433	0.352717	+	0.935730i	$0.385258\pi$
$654$	0	0
$655$	1.82719e12	0.387880
$656$	1.03670e12	0.218567
$657$	0	0
$658$	−1.97444e12	−0.410609
$659$	−2.27024e12	−0.468907	−0.234453	−	0.972127i	$-0.575330\pi$
$659$	−2.27024e12	−0.468907	−0.234453	+	0.972127i	$0.575330\pi$
$660$	0	0
$661$	−3.17394e12	−0.646684	−0.323342	−	0.946282i	$-0.604806\pi$
$661$	−3.17394e12	−0.646684	−0.323342	+	0.946282i	$0.604806\pi$
$662$	1.66192e12	0.336318
$663$	0	0
$664$	−4.93277e12	−0.984769
$665$	1.63506e12	0.324217
$666$	0	0
$667$	8.39930e12	1.64315
$668$	−1.71953e10	−0.00334130
$669$	0	0
$670$	−2.69527e12	−0.516733
$671$	2.27043e12	0.432371
$672$	0	0
$673$	−4.96227e12	−0.932423	−0.466212	−	0.884673i	$-0.654381\pi$
$673$	−4.96227e12	−0.932423	−0.466212	+	0.884673i	$0.654381\pi$
$674$	2.44124e12	0.455659
$675$	0	0
$676$	−5.26324e10	−0.00969377
$677$	−6.67904e12	−1.22198	−0.610991	−	0.791638i	$-0.709229\pi$
$677$	−6.67904e12	−1.22198	−0.610991	+	0.791638i	$0.709229\pi$
$678$	0	0
$679$	7.23599e11	0.130642
$680$	−4.09787e12	−0.734968
$681$	0	0
$682$	1.99913e11	0.0353844
$683$	−5.75109e12	−1.01125	−0.505623	−	0.862755i	$-0.668737\pi$
$683$	−5.75109e12	−1.01125	−0.505623	+	0.862755i	$0.668737\pi$
$684$	0	0
$685$	−2.82543e12	−0.490317
$686$	−3.15113e11	−0.0543260
$687$	0	0
$688$	−3.38985e12	−0.576809
$689$	−5.21548e12	−0.881674
$690$	0	0
$691$	4.84476e12	0.808391	0.404196	−	0.914673i	$-0.367552\pi$
$691$	4.84476e12	0.808391	0.404196	+	0.914673i	$0.367552\pi$
$692$	−2.32133e9	−0.000384821 0
$693$	0	0
$694$	−6.06017e12	−0.991669
$695$	−4.52305e12	−0.735359
$696$	0	0
$697$	2.22516e12	0.357120
$698$	−1.66211e12	−0.265039
$699$	0	0
$700$	5.90629e9	0.000929765 0
$701$	2.61772e12	0.409441	0.204721	−	0.978820i	$-0.434371\pi$
$701$	2.61772e12	0.409441	0.204721	+	0.978820i	$0.434371\pi$
$702$	0	0
$703$	1.39914e13	2.16054
$704$	2.81321e12	0.431643
$705$	0	0
$706$	2.36574e12	0.358382
$707$	−4.03536e12	−0.607429
$708$	0	0
$709$	2.83866e12	0.421897	0.210948	−	0.977497i	$-0.432345\pi$
$709$	2.83866e12	0.421897	0.210948	+	0.977497i	$0.432345\pi$
$710$	1.29623e12	0.191435
$711$	0	0
$712$	−6.09880e11	−0.0889373
$713$	−5.57529e11	−0.0807912
$714$	0	0
$715$	6.28861e11	0.0899865
$716$	1.24217e11	0.0176633
$717$	0	0
$718$	1.88666e12	0.264931
$719$	3.57178e12	0.498430	0.249215	−	0.968448i	$-0.419827\pi$
$719$	3.57178e12	0.498430	0.249215	+	0.968448i	$0.419827\pi$
$720$	0	0
$721$	1.12512e12	0.155056
$722$	−1.96815e13	−2.69552
$723$	0	0
$724$	−1.96118e10	−0.00265273
$725$	2.43439e12	0.327241
$726$	0	0
$727$	2.44104e12	0.324094	0.162047	−	0.986783i	$-0.448190\pi$
$727$	2.44104e12	0.324094	0.162047	+	0.986783i	$0.448190\pi$
$728$	1.31024e12	0.172887
$729$	0	0
$730$	1.40093e12	0.182585
$731$	−7.27595e12	−0.942457
$732$	0	0
$733$	−2.05203e12	−0.262553	−0.131276	−	0.991346i	$-0.541908\pi$
$733$	−2.05203e12	−0.262553	−0.131276	+	0.991346i	$0.541908\pi$
$734$	4.54425e12	0.577870
$735$	0	0
$736$	1.96647e11	0.0247023
$737$	4.02098e12	0.502028
$738$	0	0
$739$	5.67988e12	0.700550	0.350275	−	0.936647i	$-0.386088\pi$
$739$	5.67988e12	0.700550	0.350275	+	0.936647i	$0.386088\pi$
$740$	5.05409e10	0.00619584
$741$	0	0
$742$	6.01452e12	0.728422
$743$	1.78018e12	0.214296	0.107148	−	0.994243i	$-0.465828\pi$
$743$	1.78018e12	0.214296	0.107148	+	0.994243i	$0.465828\pi$
$744$	0	0
$745$	−2.39941e12	−0.285365
$746$	−1.02401e12	−0.121054
$747$	0	0
$748$	−7.61298e10	−0.00889196
$749$	−1.23767e12	−0.143693
$750$	0	0
$751$	4.66195e12	0.534796	0.267398	−	0.963586i	$-0.413836\pi$
$751$	4.66195e12	0.534796	0.267398	+	0.963586i	$0.413836\pi$
$752$	−9.58401e12	−1.09287
$753$	0	0
$754$	−6.72503e12	−0.757745
$755$	−3.83918e12	−0.430008
$756$	0	0
$757$	6.36622e12	0.704612	0.352306	−	0.935885i	$-0.385398\pi$
$757$	6.36622e12	0.704612	0.352306	+	0.935885i	$0.385398\pi$
$758$	4.30454e10	0.00473604
$759$	0	0
$760$	7.84018e12	0.852442
$761$	9.79260e12	1.05844	0.529221	−	0.848484i	$-0.322484\pi$
$761$	9.79260e12	1.05844	0.529221	+	0.848484i	$0.322484\pi$
$762$	0	0
$763$	1.28876e12	0.137661
$764$	2.00102e11	0.0212486
$765$	0	0
$766$	7.11554e12	0.746756
$767$	4.11737e11	0.0429577
$768$	0	0
$769$	1.05064e13	1.08339	0.541696	−	0.840575i	$-0.317783\pi$
$769$	1.05064e13	1.08339	0.541696	+	0.840575i	$0.317783\pi$
$770$	−7.25205e11	−0.0743451
$771$	0	0
$772$	−1.39278e11	−0.0141125
$773$	−6.50971e12	−0.655774	−0.327887	−	0.944717i	$-0.606337\pi$
$773$	−6.50971e12	−0.655774	−0.327887	+	0.944717i	$0.606337\pi$
$774$	0	0
$775$	−1.61590e11	−0.0160900
$776$	3.46969e12	0.343489
$777$	0	0
$778$	6.47921e12	0.634036
$779$	−4.25726e12	−0.414201
$780$	0	0
$781$	−1.93380e12	−0.185987
$782$	1.74742e13	1.67096
$783$	0	0
$784$	−1.52957e12	−0.144593
$785$	5.82300e12	0.547310
$786$	0	0
$787$	−1.85396e13	−1.72272	−0.861359	−	0.507996i	$-0.830386\pi$
$787$	−1.85396e13	−1.72272	−0.861359	+	0.507996i	$0.830386\pi$
$788$	−1.35943e11	−0.0125599
$789$	0	0
$790$	3.43438e12	0.313709
$791$	1.29694e12	0.117795
$792$	0	0
$793$	5.06973e12	0.455256
$794$	1.80388e13	1.61070
$795$	0	0
$796$	1.75529e11	0.0154967
$797$	4.13989e12	0.363435	0.181718	−	0.983351i	$-0.441834\pi$
$797$	4.13989e12	0.363435	0.181718	+	0.983351i	$0.441834\pi$
$798$	0	0
$799$	−2.05711e13	−1.78565
$800$	5.69945e10	0.00491958
$801$	0	0
$802$	1.50753e13	1.28671
$803$	−2.09000e12	−0.177389
$804$	0	0
$805$	2.02249e12	0.169748
$806$	4.46394e11	0.0372572
$807$	0	0
$808$	−1.93497e13	−1.59707
$809$	1.06411e13	0.873408	0.436704	−	0.899605i	$-0.356146\pi$
$809$	1.06411e13	0.873408	0.436704	+	0.899605i	$0.356146\pi$
$810$	0	0
$811$	1.16080e11	0.00942244	0.00471122	−	0.999989i	$-0.498500\pi$
$811$	1.16080e11	0.00942244	0.00471122	+	0.999989i	$0.498500\pi$
$812$	9.42288e10	0.00760644
$813$	0	0
$814$	−6.20567e12	−0.495426
$815$	4.08217e12	0.324102
$816$	0	0
$817$	1.39206e13	1.09310
$818$	−2.48802e13	−1.94296
$819$	0	0
$820$	−1.53784e10	−0.00118781
$821$	2.47649e13	1.90236	0.951179	−	0.308641i	$-0.0998741\pi$
$821$	2.47649e13	1.90236	0.951179	+	0.308641i	$0.0998741\pi$
$822$	0	0
$823$	7.59942e12	0.577406	0.288703	−	0.957419i	$-0.406776\pi$
$823$	7.59942e12	0.577406	0.288703	+	0.957419i	$0.406776\pi$
$824$	5.39499e12	0.407679
$825$	0	0
$826$	−4.74817e11	−0.0354908
$827$	9.92558e12	0.737872	0.368936	−	0.929455i	$-0.379722\pi$
$827$	9.92558e12	0.737872	0.368936	+	0.929455i	$0.379722\pi$
$828$	0	0
$829$	2.17549e12	0.159978	0.0799891	−	0.996796i	$-0.474511\pi$
$829$	2.17549e12	0.159978	0.0799891	+	0.996796i	$0.474511\pi$
$830$	6.09643e12	0.445886
$831$	0	0
$832$	6.28172e12	0.454489
$833$	−3.28305e12	−0.236252
$834$	0	0
$835$	−1.70658e12	−0.121489
$836$	1.45654e11	0.0103132
$837$	0	0
$838$	8.53043e12	0.597548
$839$	8.79323e12	0.612660	0.306330	−	0.951925i	$-0.400899\pi$
$839$	8.79323e12	0.612660	0.306330	+	0.951925i	$0.400899\pi$
$840$	0	0
$841$	2.43310e13	1.67717
$842$	−1.67343e13	−1.14737
$843$	0	0
$844$	8.27846e10	0.00561576
$845$	−5.22361e12	−0.352464
$846$	0	0
$847$	−4.57953e12	−0.305735
$848$	2.91946e13	1.93875
$849$	0	0
$850$	5.06458e12	0.332781
$851$	1.73067e13	1.13118
$852$	0	0
$853$	−2.98980e13	−1.93362	−0.966811	−	0.255494i	$-0.917762\pi$
$853$	−2.98980e13	−1.93362	−0.966811	+	0.255494i	$0.917762\pi$
$854$	−5.84643e12	−0.376124
$855$	0	0
$856$	−5.93468e12	−0.377803
$857$	−1.46014e13	−0.924654	−0.462327	−	0.886709i	$-0.652985\pi$
$857$	−1.46014e13	−0.924654	−0.462327	+	0.886709i	$0.652985\pi$
$858$	0	0
$859$	−6.23914e11	−0.0390981	−0.0195490	−	0.999809i	$-0.506223\pi$
$859$	−6.23914e11	−0.0390981	−0.0195490	+	0.999809i	$0.506223\pi$
$860$	5.02850e10	0.00313470
$861$	0	0
$862$	−1.20612e13	−0.744062
$863$	−2.27436e12	−0.139576	−0.0697879	−	0.997562i	$-0.522232\pi$
$863$	−2.27436e12	−0.139576	−0.0697879	+	0.997562i	$0.522232\pi$
$864$	0	0
$865$	−2.30385e11	−0.0139920
$866$	−4.14712e12	−0.250563
$867$	0	0
$868$	−6.25472e9	−0.000373998 0
$869$	−5.12363e12	−0.304782
$870$	0	0
$871$	8.97860e12	0.528600
$872$	6.17966e12	0.361943
$873$	0	0
$874$	−3.34322e13	−1.93804
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	2.75045e13	1.57002	0.785011	−	0.619482i	$-0.212657\pi$
$877$	2.75045e13	1.57002	0.785011	+	0.619482i	$0.212657\pi$
$878$	1.85293e13	1.05228
$879$	0	0
$880$	−3.52016e12	−0.197875
$881$	1.54100e13	0.861809	0.430904	−	0.902398i	$-0.358195\pi$
$881$	1.54100e13	0.861809	0.430904	+	0.902398i	$0.358195\pi$
$882$	0	0
$883$	−1.77905e13	−0.984838	−0.492419	−	0.870358i	$-0.663887\pi$
$883$	−1.77905e13	−0.984838	−0.492419	+	0.870358i	$0.663887\pi$
$884$	−1.69993e11	−0.00936260
$885$	0	0
$886$	−1.28669e13	−0.701489
$887$	1.07061e13	0.580734	0.290367	−	0.956915i	$-0.406223\pi$
$887$	1.07061e13	0.580734	0.290367	+	0.956915i	$0.406223\pi$
$888$	0	0
$889$	2.15292e12	0.115603
$890$	7.53753e11	0.0402693
$891$	0	0
$892$	3.51995e11	0.0186163
$893$	3.93572e13	2.07106
$894$	0	0
$895$	1.23281e13	0.642235
$896$	−7.42347e12	−0.384787
$897$	0	0
$898$	−1.33619e13	−0.685684
$899$	−2.57800e12	−0.131633
$900$	0	0
$901$	6.26632e13	3.16775
$902$	1.88824e12	0.0949789
$903$	0	0
$904$	6.21890e12	0.309710
$905$	−1.94641e12	−0.0964531
$906$	0	0
$907$	−9.46109e12	−0.464203	−0.232102	−	0.972692i	$-0.574560\pi$
$907$	−9.46109e12	−0.464203	−0.232102	+	0.972692i	$0.574560\pi$
$908$	3.77125e11	0.0184119
$909$	0	0
$910$	−1.61934e12	−0.0782800
$911$	−3.69156e13	−1.77573	−0.887865	−	0.460104i	$-0.847812\pi$
$911$	−3.69156e13	−1.77573	−0.887865	+	0.460104i	$0.847812\pi$
$912$	0	0
$913$	−9.09504e12	−0.433198
$914$	4.70334e11	0.0222920
$915$	0	0
$916$	4.16901e11	0.0195660
$917$	7.01932e12	0.327818
$918$	0	0
$919$	2.21297e13	1.02343	0.511713	−	0.859156i	$-0.329011\pi$
$919$	2.21297e13	1.02343	0.511713	+	0.859156i	$0.329011\pi$
$920$	9.69792e12	0.446307
$921$	0	0
$922$	5.91882e12	0.269740
$923$	−4.31806e12	−0.195831
$924$	0	0
$925$	5.01604e12	0.225280
$926$	1.50811e13	0.674035
$927$	0	0
$928$	9.09289e11	0.0402472
$929$	−1.24192e13	−0.547043	−0.273521	−	0.961866i	$-0.588188\pi$
$929$	−1.24192e13	−0.547043	−0.273521	+	0.961866i	$0.588188\pi$
$930$	0	0
$931$	6.28124e12	0.274014
$932$	−4.09358e11	−0.0177718
$933$	0	0
$934$	−1.31716e13	−0.566341
$935$	−7.55566e12	−0.323311
$936$	0	0
$937$	−3.54355e13	−1.50179	−0.750896	−	0.660420i	$-0.770378\pi$
$937$	−3.54355e13	−1.50179	−0.750896	+	0.660420i	$0.770378\pi$
$938$	−1.03542e13	−0.436719
$939$	0	0
$940$	1.42169e11	0.00593923
$941$	−1.16327e13	−0.483646	−0.241823	−	0.970320i	$-0.577745\pi$
$941$	−1.16327e13	−0.483646	−0.241823	+	0.970320i	$0.577745\pi$
$942$	0	0
$943$	−5.26602e12	−0.216860
$944$	−2.30478e12	−0.0944615
$945$	0	0
$946$	−6.17425e12	−0.250654
$947$	−3.20915e13	−1.29663	−0.648314	−	0.761373i	$-0.724526\pi$
$947$	−3.20915e13	−1.29663	−0.648314	+	0.761373i	$0.724526\pi$
$948$	0	0
$949$	−4.66684e12	−0.186778
$950$	−9.68971e12	−0.385971
$951$	0	0
$952$	−1.57424e13	−0.621161
$953$	−3.35135e12	−0.131614	−0.0658069	−	0.997832i	$-0.520962\pi$
$953$	−3.35135e12	−0.131614	−0.0658069	+	0.997832i	$0.520962\pi$
$954$	0	0
$955$	1.98595e13	0.772597
$956$	−5.78502e10	−0.00223998
$957$	0	0
$958$	2.97118e13	1.13968
$959$	−1.08542e13	−0.414394
$960$	0	0
$961$	−2.62685e13	−0.993528
$962$	−1.38569e13	−0.521648
$963$	0	0
$964$	9.44009e10	0.00352070
$965$	−1.38229e13	−0.513129
$966$	0	0
$967$	3.10299e13	1.14120	0.570599	−	0.821229i	$-0.306711\pi$
$967$	3.10299e13	1.14120	0.570599	+	0.821229i	$0.306711\pi$
$968$	−2.19590e13	−0.803848
$969$	0	0
$970$	−4.28820e12	−0.155526
$971$	1.95452e13	0.705591	0.352796	−	0.935700i	$-0.385231\pi$
$971$	1.95452e13	0.705591	0.352796	+	0.935700i	$0.385231\pi$
$972$	0	0
$973$	−1.73757e13	−0.621492
$974$	−5.14594e13	−1.83210
$975$	0	0
$976$	−2.83788e13	−1.00108
$977$	2.92875e13	1.02839	0.514194	−	0.857674i	$-0.328091\pi$
$977$	2.92875e13	1.02839	0.514194	+	0.857674i	$0.328091\pi$
$978$	0	0
$979$	−1.12450e12	−0.0391234
$980$	2.26896e10	0.000785795 0
$981$	0	0
$982$	−4.68603e12	−0.160806
$983$	−2.28350e13	−0.780029	−0.390014	−	0.920809i	$-0.627530\pi$
$983$	−2.28350e13	−0.780029	−0.390014	+	0.920809i	$0.627530\pi$
$984$	0	0
$985$	−1.34919e13	−0.456678
$986$	8.08002e13	2.72249
$987$	0	0
$988$	3.25236e11	0.0108591
$989$	1.72191e13	0.572304
$990$	0	0
$991$	−9.28345e11	−0.0305758	−0.0152879	−	0.999883i	$-0.504866\pi$
$991$	−9.28345e11	−0.0305758	−0.0152879	+	0.999883i	$0.504866\pi$
$992$	−6.03568e10	−0.00197890
$993$	0	0
$994$	4.97961e12	0.161792
$995$	1.74207e13	0.563459
$996$	0	0
$997$	−3.95911e13	−1.26902	−0.634511	−	0.772914i	$-0.718798\pi$
$997$	−3.95911e13	−1.26902	−0.634511	+	0.772914i	$0.718798\pi$
$998$	4.79713e12	0.153071
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.j.1.2		5
3.2	odd	2		35.10.a.d.1.4	✓	5
15.2	even	4		175.10.b.f.99.8		10
15.8	even	4		175.10.b.f.99.3		10
15.14	odd	2		175.10.a.f.1.2		5
21.20	even	2		245.10.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.d.1.4	✓	5	3.2	odd	2
175.10.a.f.1.2		5	15.14	odd	2
175.10.b.f.99.3		10	15.8	even	4
175.10.b.f.99.8		10	15.2	even	4
245.10.a.f.1.4		5	21.20	even	2
315.10.a.j.1.2		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)

\(f(q)\)	\(=\)	\(q-22.7661 q^{2} +6.29742 q^{4} +625.000 q^{5} +2401.00 q^{7} +11512.9 q^{8} -14228.8 q^{10} +21227.5 q^{11} +47399.7 q^{13} -54661.5 q^{14} -265329. q^{16} -569500. q^{17} +1.08959e6 q^{19} +3935.89 q^{20} -483268. q^{22} +1.34776e6 q^{23} +390625. q^{25} -1.07911e6 q^{26} +15120.1 q^{28} +6.23203e6 q^{29} -413669. q^{31} +145906. q^{32} +1.29653e7 q^{34} +1.50062e6 q^{35} +1.28411e7 q^{37} -2.48057e7 q^{38} +7.19556e6 q^{40} -3.90722e6 q^{41} +1.27760e7 q^{43} +133678. q^{44} -3.06834e7 q^{46} +3.61213e7 q^{47} +5.76480e6 q^{49} -8.89303e6 q^{50} +298496. q^{52} -1.10032e8 q^{53} +1.32672e7 q^{55} +2.76425e7 q^{56} -1.41879e8 q^{58} +8.68650e6 q^{59} +1.06957e8 q^{61} +9.41766e6 q^{62} +1.32527e8 q^{64} +2.96248e7 q^{65} +1.89423e8 q^{67} -3.58638e6 q^{68} -3.41634e7 q^{70} -9.10989e7 q^{71} -9.84572e7 q^{73} -2.92341e8 q^{74} +6.86157e6 q^{76} +5.09672e7 q^{77} -2.41368e8 q^{79} -1.65830e8 q^{80} +8.89525e7 q^{82} -4.28456e8 q^{83} -3.55937e8 q^{85} -2.90861e8 q^{86} +2.44390e8 q^{88} -5.29736e7 q^{89} +1.13807e8 q^{91} +8.48743e6 q^{92} -8.22342e8 q^{94} +6.80991e8 q^{95} +3.01374e8 q^{97} -1.31242e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) \) 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

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Representations

Groups

Database

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