LMFDB - Embedded newform 32.8.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,8,Mod(1,32)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("32.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	32.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:	$9.99632081549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+42.5964 q^{3} +314.772 q^{5} -84.3502 q^{7} -372.543 q^{9} +O(q^{10})$	$q+42.5964 q^{3} +314.772 q^{5} -84.3502 q^{7} -372.543 q^{9} +8314.73 q^{11} -7321.57 q^{13} +13408.1 q^{15} +20753.1 q^{17} +29237.6 q^{19} -3593.02 q^{21} +75428.6 q^{23} +20956.1 q^{25} -109027. q^{27} -233826. q^{29} -237072. q^{31} +354178. q^{33} -26551.0 q^{35} -150010. q^{37} -311873. q^{39} +277638. q^{41} -286846. q^{43} -117266. q^{45} -67658.6 q^{47} -816428. q^{49} +884010. q^{51} -261185. q^{53} +2.61724e6 q^{55} +1.24542e6 q^{57} -104918. q^{59} +1.39344e6 q^{61} +31424.1 q^{63} -2.30462e6 q^{65} +1.26321e6 q^{67} +3.21299e6 q^{69} -2.53669e6 q^{71} -4.10497e6 q^{73} +892656. q^{75} -701349. q^{77} -1.34557e6 q^{79} -3.82943e6 q^{81} -8.19658e6 q^{83} +6.53250e6 q^{85} -9.96016e6 q^{87} +8.90083e6 q^{89} +617576. q^{91} -1.00984e7 q^{93} +9.20315e6 q^{95} +1.81427e6 q^{97} -3.09760e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9}+O(q^{10}) $ 2 * q - 16 * q^3 - 180 * q^5 + 1248 * q^7 + 874 * q^9	$ 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9} + 9040 q^{11} - 2500 q^{13} + 42400 q^{15} + 17220 q^{17} + 74160 q^{19} - 81664 q^{21} + 19104 q^{23} + 187630 q^{25} - 53920 q^{27} - 245028 q^{29} - 251520 q^{31} + 311680 q^{33} - 685760 q^{35} - 530740 q^{37} - 594400 q^{39} + 726900 q^{41} + 44496 q^{43} - 734020 q^{45} + 494912 q^{47} + 135186 q^{49} + 1091040 q^{51} + 1319340 q^{53} + 2258400 q^{55} - 1386880 q^{57} + 2618000 q^{59} + 836700 q^{61} + 1692256 q^{63} - 4690200 q^{65} + 2374128 q^{67} + 6513408 q^{69} - 2836000 q^{71} - 171180 q^{73} - 8873840 q^{75} + 264960 q^{77} - 2498880 q^{79} - 9784718 q^{81} - 9531984 q^{83} + 8280600 q^{85} - 9303776 q^{87} + 7318068 q^{89} + 7041600 q^{91} - 9251840 q^{93} - 13023200 q^{95} - 9316060 q^{97} - 2193520 q^{99}+O(q^{100}) $ 2 * q - 16 * q^3 - 180 * q^5 + 1248 * q^7 + 874 * q^9 + 9040 * q^11 - 2500 * q^13 + 42400 * q^15 + 17220 * q^17 + 74160 * q^19 - 81664 * q^21 + 19104 * q^23 + 187630 * q^25 - 53920 * q^27 - 245028 * q^29 - 251520 * q^31 + 311680 * q^33 - 685760 * q^35 - 530740 * q^37 - 594400 * q^39 + 726900 * q^41 + 44496 * q^43 - 734020 * q^45 + 494912 * q^47 + 135186 * q^49 + 1091040 * q^51 + 1319340 * q^53 + 2258400 * q^55 - 1386880 * q^57 + 2618000 * q^59 + 836700 * q^61 + 1692256 * q^63 - 4690200 * q^65 + 2374128 * q^67 + 6513408 * q^69 - 2836000 * q^71 - 171180 * q^73 - 8873840 * q^75 + 264960 * q^77 - 2498880 * q^79 - 9784718 * q^81 - 9531984 * q^83 + 8280600 * q^85 - 9303776 * q^87 + 7318068 * q^89 + 7041600 * q^91 - 9251840 * q^93 - 13023200 * q^95 - 9316060 * q^97 - 2193520 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	42.5964	0.910854	0.455427	−	0.890273i	$-0.349487\pi$
$3$	42.5964	0.910854	0.455427	+	0.890273i	$0.349487\pi$
$4$	0	0
$5$	314.772	1.12616	0.563080	−	0.826402i	$-0.309616\pi$
$5$	314.772	1.12616	0.563080	+	0.826402i	$0.309616\pi$
$6$	0	0
$7$	−84.3502	−0.0929486	−0.0464743	−	0.998919i	$-0.514799\pi$
$7$	−84.3502	−0.0929486	−0.0464743	+	0.998919i	$0.514799\pi$
$8$	0	0
$9$	−372.543	−0.170344
$10$	0	0
$11$	8314.73	1.88354	0.941768	−	0.336263i	$-0.109163\pi$
$11$	8314.73	1.88354	0.941768	+	0.336263i	$0.109163\pi$
$12$	0	0
$13$	−7321.57	−0.924278	−0.462139	−	0.886807i	$-0.652918\pi$
$13$	−7321.57	−0.924278	−0.462139	+	0.886807i	$0.652918\pi$
$14$	0	0
$15$	13408.1	1.02577
$16$	0	0
$17$	20753.1	1.02450	0.512251	−	0.858836i	$-0.328812\pi$
$17$	20753.1	1.02450	0.512251	+	0.858836i	$0.328812\pi$
$18$	0	0
$19$	29237.6	0.977920	0.488960	−	0.872306i	$-0.337376\pi$
$19$	29237.6	0.977920	0.488960	+	0.872306i	$0.337376\pi$
$20$	0	0
$21$	−3593.02	−0.0846627
$22$	0	0
$23$	75428.6	1.29267	0.646336	−	0.763053i	$-0.276300\pi$
$23$	75428.6	1.29267	0.646336	+	0.763053i	$0.276300\pi$
$24$	0	0
$25$	20956.1	0.268238
$26$	0	0
$27$	−109027.	−1.06601
$28$	0	0
$29$	−233826.	−1.78033	−0.890164	−	0.455640i	$-0.849411\pi$
$29$	−233826.	−1.78033	−0.890164	+	0.455640i	$0.849411\pi$
$30$	0	0
$31$	−237072.	−1.42927	−0.714636	−	0.699497i	$-0.753407\pi$
$31$	−237072.	−1.42927	−0.714636	+	0.699497i	$0.753407\pi$
$32$	0	0
$33$	354178.	1.71563
$34$	0	0
$35$	−26551.0	−0.104675
$36$	0	0
$37$	−150010.	−0.486872	−0.243436	−	0.969917i	$-0.578275\pi$
$37$	−150010.	−0.486872	−0.243436	+	0.969917i	$0.578275\pi$
$38$	0	0
$39$	−311873.	−0.841883
$40$	0	0
$41$	277638.	0.629124	0.314562	−	0.949237i	$-0.398142\pi$
$41$	277638.	0.629124	0.314562	+	0.949237i	$0.398142\pi$
$42$	0	0
$43$	−286846.	−0.550185	−0.275092	−	0.961418i	$-0.588708\pi$
$43$	−286846.	−0.550185	−0.275092	+	0.961418i	$0.588708\pi$
$44$	0	0
$45$	−117266.	−0.191835
$46$	0	0
$47$	−67658.6	−0.0950563	−0.0475281	−	0.998870i	$-0.515134\pi$
$47$	−67658.6	−0.0950563	−0.0475281	+	0.998870i	$0.515134\pi$
$48$	0	0
$49$	−816428.	−0.991361
$50$	0	0
$51$	884010.	0.933172
$52$	0	0
$53$	−261185.	−0.240981	−0.120491	−	0.992714i	$-0.538447\pi$
$53$	−261185.	−0.240981	−0.120491	+	0.992714i	$0.538447\pi$
$54$	0	0
$55$	2.61724e6	2.12116
$56$	0	0
$57$	1.24542e6	0.890743
$58$	0	0
$59$	−104918.	−0.0665068	−0.0332534	−	0.999447i	$-0.510587\pi$
$59$	−104918.	−0.0665068	−0.0332534	+	0.999447i	$0.510587\pi$
$60$	0	0
$61$	1.39344e6	0.786023	0.393012	−	0.919534i	$-0.371433\pi$
$61$	1.39344e6	0.786023	0.393012	+	0.919534i	$0.371433\pi$
$62$	0	0
$63$	31424.1	0.0158333
$64$	0	0
$65$	−2.30462e6	−1.04089
$66$	0	0
$67$	1.26321e6	0.513115	0.256557	−	0.966529i	$-0.417412\pi$
$67$	1.26321e6	0.513115	0.256557	+	0.966529i	$0.417412\pi$
$68$	0	0
$69$	3.21299e6	1.17744
$70$	0	0
$71$	−2.53669e6	−0.841129	−0.420565	−	0.907263i	$-0.638168\pi$
$71$	−2.53669e6	−0.841129	−0.420565	+	0.907263i	$0.638168\pi$
$72$	0	0
$73$	−4.10497e6	−1.23504	−0.617519	−	0.786556i	$-0.711862\pi$
$73$	−4.10497e6	−1.23504	−0.617519	+	0.786556i	$0.711862\pi$
$74$	0	0
$75$	892656.	0.244326
$76$	0	0
$77$	−701349.	−0.175072
$78$	0	0
$79$	−1.34557e6	−0.307052	−0.153526	−	0.988145i	$-0.549063\pi$
$79$	−1.34557e6	−0.307052	−0.153526	+	0.988145i	$0.549063\pi$
$80$	0	0
$81$	−3.82943e6	−0.800638
$82$	0	0
$83$	−8.19658e6	−1.57347	−0.786737	−	0.617289i	$-0.788231\pi$
$83$	−8.19658e6	−1.57347	−0.786737	+	0.617289i	$0.788231\pi$
$84$	0	0
$85$	6.53250e6	1.15375
$86$	0	0
$87$	−9.96016e6	−1.62162
$88$	0	0
$89$	8.90083e6	1.33834	0.669168	−	0.743111i	$-0.266651\pi$
$89$	8.90083e6	1.33834	0.669168	+	0.743111i	$0.266651\pi$
$90$	0	0
$91$	617576.	0.0859104
$92$	0	0
$93$	−1.00984e7	−1.30186
$94$	0	0
$95$	9.20315e6	1.10130
$96$	0	0
$97$	1.81427e6	0.201837	0.100918	−	0.994895i	$-0.467822\pi$
$97$	1.81427e6	0.201837	0.100918	+	0.994895i	$0.467822\pi$
$98$	0	0
$99$	−3.09760e6	−0.320850
$100$	0	0
$101$	4.41804e6	0.426683	0.213341	−	0.976978i	$-0.431565\pi$
$101$	4.41804e6	0.426683	0.213341	+	0.976978i	$0.431565\pi$
$102$	0	0
$103$	1.50740e7	1.35925	0.679623	−	0.733562i	$-0.262143\pi$
$103$	1.50740e7	1.35925	0.679623	+	0.733562i	$0.262143\pi$
$104$	0	0
$105$	−1.13098e6	−0.0953438
$106$	0	0
$107$	4.70769e6	0.371505	0.185753	−	0.982597i	$-0.440528\pi$
$107$	4.70769e6	0.371505	0.185753	+	0.982597i	$0.440528\pi$
$108$	0	0
$109$	−1.82524e7	−1.34998	−0.674990	−	0.737827i	$-0.735852\pi$
$109$	−1.82524e7	−1.34998	−0.674990	+	0.737827i	$0.735852\pi$
$110$	0	0
$111$	−6.38990e6	−0.443469
$112$	0	0
$113$	−6.03326e6	−0.393348	−0.196674	−	0.980469i	$-0.563014\pi$
$113$	−6.03326e6	−0.393348	−0.196674	+	0.980469i	$0.563014\pi$
$114$	0	0
$115$	2.37428e7	1.45576
$116$	0	0
$117$	2.72760e6	0.157446
$118$	0	0
$119$	−1.75053e6	−0.0952260
$120$	0	0
$121$	4.96476e7	2.54771
$122$	0	0
$123$	1.18264e7	0.573040
$124$	0	0
$125$	−1.79951e7	−0.824081
$126$	0	0
$127$	2.53140e7	1.09660	0.548300	−	0.836282i	$-0.315275\pi$
$127$	2.53140e7	1.09660	0.548300	+	0.836282i	$0.315275\pi$
$128$	0	0
$129$	−1.22186e7	−0.501138
$130$	0	0
$131$	−1.78779e7	−0.694811	−0.347406	−	0.937715i	$-0.612937\pi$
$131$	−1.78779e7	−0.694811	−0.347406	+	0.937715i	$0.612937\pi$
$132$	0	0
$133$	−2.46619e6	−0.0908963
$134$	0	0
$135$	−3.43187e7	−1.20050
$136$	0	0
$137$	2.20101e7	0.731308	0.365654	−	0.930751i	$-0.380845\pi$
$137$	2.20101e7	0.731308	0.365654	+	0.930751i	$0.380845\pi$
$138$	0	0
$139$	−1.32216e7	−0.417574	−0.208787	−	0.977961i	$-0.566952\pi$
$139$	−1.32216e7	−0.417574	−0.208787	+	0.977961i	$0.566952\pi$
$140$	0	0
$141$	−2.88202e6	−0.0865824
$142$	0	0
$143$	−6.08769e7	−1.74091
$144$	0	0
$145$	−7.36018e7	−2.00494
$146$	0	0
$147$	−3.47769e7	−0.902985
$148$	0	0
$149$	3.07211e7	0.760825	0.380412	−	0.924817i	$-0.375782\pi$
$149$	3.07211e7	0.760825	0.380412	+	0.924817i	$0.375782\pi$
$150$	0	0
$151$	7.45933e7	1.76311	0.881557	−	0.472077i	$-0.156496\pi$
$151$	7.45933e7	1.76311	0.881557	+	0.472077i	$0.156496\pi$
$152$	0	0
$153$	−7.73144e6	−0.174518
$154$	0	0
$155$	−7.46236e7	−1.60959
$156$	0	0
$157$	1.86070e7	0.383733	0.191866	−	0.981421i	$-0.438546\pi$
$157$	1.86070e7	0.383733	0.191866	+	0.981421i	$0.438546\pi$
$158$	0	0
$159$	−1.11256e7	−0.219499
$160$	0	0
$161$	−6.36241e6	−0.120152
$162$	0	0
$163$	8.72485e7	1.57798	0.788990	−	0.614406i	$-0.210604\pi$
$163$	8.72485e7	1.57798	0.788990	+	0.614406i	$0.210604\pi$
$164$	0	0
$165$	1.11485e8	1.93207
$166$	0	0
$167$	−3.27780e7	−0.544597	−0.272299	−	0.962213i	$-0.587784\pi$
$167$	−3.27780e7	−0.544597	−0.272299	+	0.962213i	$0.587784\pi$
$168$	0	0
$169$	−9.14308e6	−0.145710
$170$	0	0
$171$	−1.08922e7	−0.166583
$172$	0	0
$173$	−9.61813e7	−1.41231	−0.706154	−	0.708058i	$-0.749571\pi$
$173$	−9.61813e7	−1.41231	−0.706154	+	0.708058i	$0.749571\pi$
$174$	0	0
$175$	−1.76765e6	−0.0249324
$176$	0	0
$177$	−4.46912e6	−0.0605780
$178$	0	0
$179$	6.29897e7	0.820888	0.410444	−	0.911886i	$-0.365374\pi$
$179$	6.29897e7	0.820888	0.410444	+	0.911886i	$0.365374\pi$
$180$	0	0
$181$	5.14863e7	0.645382	0.322691	−	0.946504i	$-0.395412\pi$
$181$	5.14863e7	0.645382	0.322691	+	0.946504i	$0.395412\pi$
$182$	0	0
$183$	5.93558e7	0.715953
$184$	0	0
$185$	−4.72189e7	−0.548296
$186$	0	0
$187$	1.72557e8	1.92969
$188$	0	0
$189$	9.19649e6	0.0990845
$190$	0	0
$191$	1.09113e8	1.13308	0.566540	−	0.824034i	$-0.308282\pi$
$191$	1.09113e8	1.13308	0.566540	+	0.824034i	$0.308282\pi$
$192$	0	0
$193$	−1.03233e8	−1.03364	−0.516820	−	0.856094i	$-0.672884\pi$
$193$	−1.03233e8	−1.03364	−0.516820	+	0.856094i	$0.672884\pi$
$194$	0	0
$195$	−9.81687e7	−0.948095
$196$	0	0
$197$	−3.69454e7	−0.344293	−0.172147	−	0.985071i	$-0.555070\pi$
$197$	−3.69454e7	−0.344293	−0.172147	+	0.985071i	$0.555070\pi$
$198$	0	0
$199$	−5.46990e7	−0.492032	−0.246016	−	0.969266i	$-0.579122\pi$
$199$	−5.46990e7	−0.492032	−0.246016	+	0.969266i	$0.579122\pi$
$200$	0	0
$201$	5.38083e7	0.467373
$202$	0	0
$203$	1.97233e7	0.165479
$204$	0	0
$205$	8.73927e7	0.708495
$206$	0	0
$207$	−2.81004e7	−0.220199
$208$	0	0
$209$	2.43102e8	1.84195
$210$	0	0
$211$	−1.30886e8	−0.959193	−0.479597	−	0.877489i	$-0.659217\pi$
$211$	−1.30886e8	−0.959193	−0.479597	+	0.877489i	$0.659217\pi$
$212$	0	0
$213$	−1.08054e8	−0.766146
$214$	0	0
$215$	−9.02909e7	−0.619597
$216$	0	0
$217$	1.99971e7	0.132849
$218$	0	0
$219$	−1.74857e8	−1.12494
$220$	0	0
$221$	−1.51946e8	−0.946925
$222$	0	0
$223$	9.63402e7	0.581755	0.290878	−	0.956760i	$-0.406053\pi$
$223$	9.63402e7	0.581755	0.290878	+	0.956760i	$0.406053\pi$
$224$	0	0
$225$	−7.80706e6	−0.0456929
$226$	0	0
$227$	−4.54282e7	−0.257771	−0.128886	−	0.991659i	$-0.541140\pi$
$227$	−4.54282e7	−0.257771	−0.128886	+	0.991659i	$0.541140\pi$
$228$	0	0
$229$	−1.81318e8	−0.997739	−0.498869	−	0.866677i	$-0.666251\pi$
$229$	−1.81318e8	−0.997739	−0.498869	+	0.866677i	$0.666251\pi$
$230$	0	0
$231$	−2.98750e7	−0.159465
$232$	0	0
$233$	2.68643e8	1.39133	0.695665	−	0.718366i	$-0.255110\pi$
$233$	2.68643e8	1.39133	0.695665	+	0.718366i	$0.255110\pi$
$234$	0	0
$235$	−2.12970e7	−0.107049
$236$	0	0
$237$	−5.73166e7	−0.279680
$238$	0	0
$239$	−3.16821e8	−1.50114	−0.750570	−	0.660791i	$-0.770221\pi$
$239$	−3.16821e8	−1.50114	−0.750570	+	0.660791i	$0.770221\pi$
$240$	0	0
$241$	−4.18186e7	−0.192446	−0.0962232	−	0.995360i	$-0.530676\pi$
$241$	−4.18186e7	−0.192446	−0.0962232	+	0.995360i	$0.530676\pi$
$242$	0	0
$243$	7.53229e7	0.336748
$244$	0	0
$245$	−2.56988e8	−1.11643
$246$	0	0
$247$	−2.14065e8	−0.903870
$248$	0	0
$249$	−3.49145e8	−1.43321
$250$	0	0
$251$	1.49974e8	0.598629	0.299315	−	0.954154i	$-0.403242\pi$
$251$	1.49974e8	0.598629	0.299315	+	0.954154i	$0.403242\pi$
$252$	0	0
$253$	6.27168e8	2.43479
$254$	0	0
$255$	2.78261e8	1.05090
$256$	0	0
$257$	3.18732e8	1.17128	0.585638	−	0.810573i	$-0.300844\pi$
$257$	3.18732e8	1.17128	0.585638	+	0.810573i	$0.300844\pi$
$258$	0	0
$259$	1.26534e7	0.0452541
$260$	0	0
$261$	8.71103e7	0.303269
$262$	0	0
$263$	3.77283e8	1.27886	0.639429	−	0.768850i	$-0.279171\pi$
$263$	3.77283e8	1.27886	0.639429	+	0.768850i	$0.279171\pi$
$264$	0	0
$265$	−8.22137e7	−0.271384
$266$	0	0
$267$	3.79144e8	1.21903
$268$	0	0
$269$	−1.41747e8	−0.443997	−0.221998	−	0.975047i	$-0.571258\pi$
$269$	−1.41747e8	−0.443997	−0.221998	+	0.975047i	$0.571258\pi$
$270$	0	0
$271$	1.58260e8	0.483035	0.241517	−	0.970396i	$-0.422355\pi$
$271$	1.58260e8	0.483035	0.241517	+	0.970396i	$0.422355\pi$
$272$	0	0
$273$	2.63065e7	0.0782518
$274$	0	0
$275$	1.74245e8	0.505237
$276$	0	0
$277$	−2.20643e8	−0.623750	−0.311875	−	0.950123i	$-0.600957\pi$
$277$	−2.20643e8	−0.623750	−0.311875	+	0.950123i	$0.600957\pi$
$278$	0	0
$279$	8.83196e7	0.243468
$280$	0	0
$281$	−3.64491e8	−0.979974	−0.489987	−	0.871730i	$-0.662998\pi$
$281$	−3.64491e8	−0.979974	−0.489987	+	0.871730i	$0.662998\pi$
$282$	0	0
$283$	−2.59408e8	−0.680348	−0.340174	−	0.940362i	$-0.610486\pi$
$283$	−2.59408e8	−0.680348	−0.340174	+	0.940362i	$0.610486\pi$
$284$	0	0
$285$	3.92021e8	1.00312
$286$	0	0
$287$	−2.34189e7	−0.0584762
$288$	0	0
$289$	2.03544e7	0.0496039
$290$	0	0
$291$	7.72813e7	0.183844
$292$	0	0
$293$	8.41077e7	0.195344	0.0976718	−	0.995219i	$-0.468860\pi$
$293$	8.41077e7	0.195344	0.0976718	+	0.995219i	$0.468860\pi$
$294$	0	0
$295$	−3.30251e7	−0.0748974
$296$	0	0
$297$	−9.06534e8	−2.00787
$298$	0	0
$299$	−5.52256e8	−1.19479
$300$	0	0
$301$	2.41955e7	0.0511389
$302$	0	0
$303$	1.88193e8	0.388646
$304$	0	0
$305$	4.38617e8	0.885188
$306$	0	0
$307$	−1.64273e8	−0.324027	−0.162014	−	0.986789i	$-0.551799\pi$
$307$	−1.64273e8	−0.324027	−0.162014	+	0.986789i	$0.551799\pi$
$308$	0	0
$309$	6.42098e8	1.23808
$310$	0	0
$311$	5.51963e8	1.04052	0.520258	−	0.854009i	$-0.325836\pi$
$311$	5.51963e8	1.04052	0.520258	+	0.854009i	$0.325836\pi$
$312$	0	0
$313$	−2.89945e8	−0.534454	−0.267227	−	0.963634i	$-0.586107\pi$
$313$	−2.89945e8	−0.534454	−0.267227	+	0.963634i	$0.586107\pi$
$314$	0	0
$315$	9.89141e6	0.0178308
$316$	0	0
$317$	4.89405e8	0.862900	0.431450	−	0.902137i	$-0.358002\pi$
$317$	4.89405e8	0.862900	0.431450	+	0.902137i	$0.358002\pi$
$318$	0	0
$319$	−1.94420e9	−3.35331
$320$	0	0
$321$	2.00531e8	0.338387
$322$	0	0
$323$	6.06771e8	1.00188
$324$	0	0
$325$	−1.53432e8	−0.247927
$326$	0	0
$327$	−7.77488e8	−1.22964
$328$	0	0
$329$	5.70702e6	0.00883535
$330$	0	0
$331$	−8.95022e8	−1.35655	−0.678275	−	0.734808i	$-0.737272\pi$
$331$	−8.95022e8	−1.35655	−0.678275	+	0.734808i	$0.737272\pi$
$332$	0	0
$333$	5.58852e7	0.0829358
$334$	0	0
$335$	3.97623e8	0.577850
$336$	0	0
$337$	−3.02469e8	−0.430503	−0.215251	−	0.976559i	$-0.569057\pi$
$337$	−3.02469e8	−0.430503	−0.215251	+	0.976559i	$0.569057\pi$
$338$	0	0
$339$	−2.56995e8	−0.358283
$340$	0	0
$341$	−1.97119e9	−2.69208
$342$	0	0
$343$	1.38332e8	0.185094
$344$	0	0
$345$	1.01136e9	1.32598
$346$	0	0
$347$	6.84204e8	0.879088	0.439544	−	0.898221i	$-0.355140\pi$
$347$	6.84204e8	0.879088	0.439544	+	0.898221i	$0.355140\pi$
$348$	0	0
$349$	2.89192e8	0.364164	0.182082	−	0.983283i	$-0.441716\pi$
$349$	2.89192e8	0.364164	0.182082	+	0.983283i	$0.441716\pi$
$350$	0	0
$351$	7.98252e8	0.985293
$352$	0	0
$353$	4.51179e8	0.545930	0.272965	−	0.962024i	$-0.411996\pi$
$353$	4.51179e8	0.545930	0.272965	+	0.962024i	$0.411996\pi$
$354$	0	0
$355$	−7.98477e8	−0.947247
$356$	0	0
$357$	−7.45664e7	−0.0867370
$358$	0	0
$359$	7.90263e7	0.0901449	0.0450724	−	0.998984i	$-0.485648\pi$
$359$	7.90263e7	0.0901449	0.0450724	+	0.998984i	$0.485648\pi$
$360$	0	0
$361$	−3.90373e7	−0.0436722
$362$	0	0
$363$	2.11481e9	2.32059
$364$	0	0
$365$	−1.29213e9	−1.39085
$366$	0	0
$367$	1.35691e9	1.43291	0.716454	−	0.697634i	$-0.245764\pi$
$367$	1.35691e9	1.43291	0.716454	+	0.697634i	$0.245764\pi$
$368$	0	0
$369$	−1.03432e8	−0.107168
$370$	0	0
$371$	2.20310e7	0.0223989
$372$	0	0
$373$	6.43282e8	0.641831	0.320916	−	0.947108i	$-0.396009\pi$
$373$	6.43282e8	0.641831	0.320916	+	0.947108i	$0.396009\pi$
$374$	0	0
$375$	−7.66529e8	−0.750618
$376$	0	0
$377$	1.71198e9	1.64552
$378$	0	0
$379$	−1.07948e9	−1.01854	−0.509268	−	0.860608i	$-0.670084\pi$
$379$	−1.07948e9	−1.01854	−0.509268	+	0.860608i	$0.670084\pi$
$380$	0	0
$381$	1.07829e9	0.998843
$382$	0	0
$383$	8.12870e8	0.739308	0.369654	−	0.929170i	$-0.379476\pi$
$383$	8.12870e8	0.739308	0.369654	+	0.929170i	$0.379476\pi$
$384$	0	0
$385$	−2.20765e8	−0.197159
$386$	0	0
$387$	1.06862e8	0.0937209
$388$	0	0
$389$	6.42047e7	0.0553023	0.0276511	−	0.999618i	$-0.491197\pi$
$389$	6.42047e7	0.0553023	0.0276511	+	0.999618i	$0.491197\pi$
$390$	0	0
$391$	1.56538e9	1.32435
$392$	0	0
$393$	−7.61535e8	−0.632872
$394$	0	0
$395$	−4.23548e8	−0.345790
$396$	0	0
$397$	−1.63802e9	−1.31387	−0.656936	−	0.753947i	$-0.728148\pi$
$397$	−1.63802e9	−1.31387	−0.656936	+	0.753947i	$0.728148\pi$
$398$	0	0
$399$	−1.05051e8	−0.0827933
$400$	0	0
$401$	4.95766e8	0.383947	0.191974	−	0.981400i	$-0.438511\pi$
$401$	4.95766e8	0.383947	0.191974	+	0.981400i	$0.438511\pi$
$402$	0	0
$403$	1.73574e9	1.32104
$404$	0	0
$405$	−1.20540e9	−0.901648
$406$	0	0
$407$	−1.24729e9	−0.917040
$408$	0	0
$409$	−1.19355e8	−0.0862596	−0.0431298	−	0.999069i	$-0.513733\pi$
$409$	−1.19355e8	−0.0862596	−0.0431298	+	0.999069i	$0.513733\pi$
$410$	0	0
$411$	9.37552e8	0.666115
$412$	0	0
$413$	8.84982e6	0.00618172
$414$	0	0
$415$	−2.58005e9	−1.77198
$416$	0	0
$417$	−5.63195e8	−0.380349
$418$	0	0
$419$	−1.78395e9	−1.18477	−0.592383	−	0.805656i	$-0.701813\pi$
$419$	−1.78395e9	−1.18477	−0.592383	+	0.805656i	$0.701813\pi$
$420$	0	0
$421$	2.75686e9	1.80064	0.900322	−	0.435224i	$-0.143331\pi$
$421$	2.75686e9	1.80064	0.900322	+	0.435224i	$0.143331\pi$
$422$	0	0
$423$	2.52058e7	0.0161923
$424$	0	0
$425$	4.34905e8	0.274811
$426$	0	0
$427$	−1.17537e8	−0.0730598
$428$	0	0
$429$	−2.59314e9	−1.58572
$430$	0	0
$431$	−7.91965e8	−0.476470	−0.238235	−	0.971208i	$-0.576569\pi$
$431$	−7.91965e8	−0.476470	−0.238235	+	0.971208i	$0.576569\pi$
$432$	0	0
$433$	−1.77410e9	−1.05019	−0.525097	−	0.851042i	$-0.675971\pi$
$433$	−1.77410e9	−1.05019	−0.525097	+	0.851042i	$0.675971\pi$
$434$	0	0
$435$	−3.13518e9	−1.82620
$436$	0	0
$437$	2.20535e9	1.26413
$438$	0	0
$439$	−2.13058e9	−1.20191	−0.600955	−	0.799283i	$-0.705213\pi$
$439$	−2.13058e9	−1.20191	−0.600955	+	0.799283i	$0.705213\pi$
$440$	0	0
$441$	3.04155e8	0.168873
$442$	0	0
$443$	−9.71953e7	−0.0531168	−0.0265584	−	0.999647i	$-0.508455\pi$
$443$	−9.71953e7	−0.0531168	−0.0265584	+	0.999647i	$0.508455\pi$
$444$	0	0
$445$	2.80173e9	1.50718
$446$	0	0
$447$	1.30861e9	0.693001
$448$	0	0
$449$	−1.12639e9	−0.587257	−0.293628	−	0.955920i	$-0.594863\pi$
$449$	−1.12639e9	−0.587257	−0.293628	+	0.955920i	$0.594863\pi$
$450$	0	0
$451$	2.30849e9	1.18498
$452$	0	0
$453$	3.17741e9	1.60594
$454$	0	0
$455$	1.94395e8	0.0967489
$456$	0	0
$457$	1.48680e8	0.0728695	0.0364348	−	0.999336i	$-0.488400\pi$
$457$	1.48680e8	0.0728695	0.0364348	+	0.999336i	$0.488400\pi$
$458$	0	0
$459$	−2.26266e9	−1.09213
$460$	0	0
$461$	−1.82070e9	−0.865538	−0.432769	−	0.901505i	$-0.642463\pi$
$461$	−1.82070e9	−0.865538	−0.432769	+	0.901505i	$0.642463\pi$
$462$	0	0
$463$	2.68931e9	1.25924	0.629619	−	0.776904i	$-0.283211\pi$
$463$	2.68931e9	1.25924	0.629619	+	0.776904i	$0.283211\pi$
$464$	0	0
$465$	−3.17870e9	−1.46610
$466$	0	0
$467$	1.06001e9	0.481615	0.240807	−	0.970573i	$-0.422588\pi$
$467$	1.06001e9	0.481615	0.240807	+	0.970573i	$0.422588\pi$
$468$	0	0
$469$	−1.06552e8	−0.0476933
$470$	0	0
$471$	7.92594e8	0.349524
$472$	0	0
$473$	−2.38505e9	−1.03629
$474$	0	0
$475$	6.12706e8	0.262316
$476$	0	0
$477$	9.73028e7	0.0410498
$478$	0	0
$479$	−1.92366e9	−0.799751	−0.399875	−	0.916569i	$-0.630947\pi$
$479$	−1.92366e9	−0.799751	−0.399875	+	0.916569i	$0.630947\pi$
$480$	0	0
$481$	1.09831e9	0.450005
$482$	0	0
$483$	−2.71016e8	−0.109441
$484$	0	0
$485$	5.71080e8	0.227301
$486$	0	0
$487$	3.66385e9	1.43743	0.718714	−	0.695305i	$-0.244731\pi$
$487$	3.66385e9	1.43743	0.718714	+	0.695305i	$0.244731\pi$
$488$	0	0
$489$	3.71648e9	1.43731
$490$	0	0
$491$	6.96407e8	0.265508	0.132754	−	0.991149i	$-0.457618\pi$
$491$	6.96407e8	0.265508	0.132754	+	0.991149i	$0.457618\pi$
$492$	0	0
$493$	−4.85263e9	−1.82395
$494$	0	0
$495$	−9.75035e8	−0.361328
$496$	0	0
$497$	2.13970e8	0.0781818
$498$	0	0
$499$	2.06766e9	0.744951	0.372476	−	0.928042i	$-0.378509\pi$
$499$	2.06766e9	0.744951	0.372476	+	0.928042i	$0.378509\pi$
$500$	0	0
$501$	−1.39623e9	−0.496049
$502$	0	0
$503$	4.58378e9	1.60596	0.802982	−	0.596004i	$-0.203246\pi$
$503$	4.58378e9	1.60596	0.802982	+	0.596004i	$0.203246\pi$
$504$	0	0
$505$	1.39067e9	0.480513
$506$	0	0
$507$	−3.89463e8	−0.132721
$508$	0	0
$509$	5.29729e9	1.78050	0.890250	−	0.455473i	$-0.150530\pi$
$509$	5.29729e9	1.78050	0.890250	+	0.455473i	$0.150530\pi$
$510$	0	0
$511$	3.46255e8	0.114795
$512$	0	0
$513$	−3.18770e9	−1.04248
$514$	0	0
$515$	4.74486e9	1.53073
$516$	0	0
$517$	−5.62564e8	−0.179042
$518$	0	0
$519$	−4.09698e9	−1.28641
$520$	0	0
$521$	4.18704e9	1.29710	0.648552	−	0.761170i	$-0.275375\pi$
$521$	4.18704e9	1.29710	0.648552	+	0.761170i	$0.275375\pi$
$522$	0	0
$523$	3.00098e9	0.917291	0.458645	−	0.888619i	$-0.348335\pi$
$523$	3.00098e9	0.917291	0.458645	+	0.888619i	$0.348335\pi$
$524$	0	0
$525$	−7.52957e7	−0.0227098
$526$	0	0
$527$	−4.91999e9	−1.46429
$528$	0	0
$529$	2.28464e9	0.671002
$530$	0	0
$531$	3.90863e7	0.0113291
$532$	0	0
$533$	−2.03275e9	−0.581485
$534$	0	0
$535$	1.48185e9	0.418374
$536$	0	0
$537$	2.68314e9	0.747710
$538$	0	0
$539$	−6.78838e9	−1.86726
$540$	0	0
$541$	−3.79319e9	−1.02995	−0.514973	−	0.857206i	$-0.672198\pi$
$541$	−3.79319e9	−1.02995	−0.514973	+	0.857206i	$0.672198\pi$
$542$	0	0
$543$	2.19313e9	0.587849
$544$	0	0
$545$	−5.74534e9	−1.52030
$546$	0	0
$547$	−5.01604e9	−1.31040	−0.655202	−	0.755454i	$-0.727417\pi$
$547$	−5.01604e9	−1.31040	−0.655202	+	0.755454i	$0.727417\pi$
$548$	0	0
$549$	−5.19118e8	−0.133895
$550$	0	0
$551$	−6.83650e9	−1.74102
$552$	0	0
$553$	1.13499e8	0.0285401
$554$	0	0
$555$	−2.01136e9	−0.499418
$556$	0	0
$557$	−4.91682e9	−1.20557	−0.602783	−	0.797905i	$-0.705941\pi$
$557$	−4.91682e9	−1.20557	−0.602783	+	0.797905i	$0.705941\pi$
$558$	0	0
$559$	2.10016e9	0.508524
$560$	0	0
$561$	7.35031e9	1.75766
$562$	0	0
$563$	−3.99617e9	−0.943767	−0.471883	−	0.881661i	$-0.656426\pi$
$563$	−3.99617e9	−0.943767	−0.471883	+	0.881661i	$0.656426\pi$
$564$	0	0
$565$	−1.89910e9	−0.442974
$566$	0	0
$567$	3.23013e8	0.0744182
$568$	0	0
$569$	−5.16407e9	−1.17517	−0.587583	−	0.809164i	$-0.699920\pi$
$569$	−5.16407e9	−1.17517	−0.587583	+	0.809164i	$0.699920\pi$
$570$	0	0
$571$	8.05729e9	1.81118	0.905592	−	0.424149i	$-0.139427\pi$
$571$	8.05729e9	1.81118	0.905592	+	0.424149i	$0.139427\pi$
$572$	0	0
$573$	4.64783e9	1.03207
$574$	0	0
$575$	1.58069e9	0.346744
$576$	0	0
$577$	7.25217e9	1.57164	0.785819	−	0.618457i	$-0.212242\pi$
$577$	7.25217e9	1.57164	0.785819	+	0.618457i	$0.212242\pi$
$578$	0	0
$579$	−4.39737e9	−0.941495
$580$	0	0
$581$	6.91383e8	0.146252
$582$	0	0
$583$	−2.17169e9	−0.453897
$584$	0	0
$585$	8.58571e8	0.177309
$586$	0	0
$587$	−1.61963e9	−0.330508	−0.165254	−	0.986251i	$-0.552844\pi$
$587$	−1.61963e9	−0.330508	−0.165254	+	0.986251i	$0.552844\pi$
$588$	0	0
$589$	−6.93141e9	−1.39771
$590$	0	0
$591$	−1.57374e9	−0.313601
$592$	0	0
$593$	6.39908e8	0.126016	0.0630081	−	0.998013i	$-0.479931\pi$
$593$	6.39908e8	0.126016	0.0630081	+	0.998013i	$0.479931\pi$
$594$	0	0
$595$	−5.51018e8	−0.107240
$596$	0	0
$597$	−2.32998e9	−0.448170
$598$	0	0
$599$	8.84431e8	0.168140	0.0840698	−	0.996460i	$-0.473208\pi$
$599$	8.84431e8	0.168140	0.0840698	+	0.996460i	$0.473208\pi$
$600$	0	0
$601$	−2.91639e9	−0.548005	−0.274003	−	0.961729i	$-0.588348\pi$
$601$	−2.91639e9	−0.548005	−0.274003	+	0.961729i	$0.588348\pi$
$602$	0	0
$603$	−4.70601e8	−0.0874062
$604$	0	0
$605$	1.56277e10	2.86913
$606$	0	0
$607$	5.62992e9	1.02174	0.510872	−	0.859657i	$-0.329323\pi$
$607$	5.62992e9	1.02174	0.510872	+	0.859657i	$0.329323\pi$
$608$	0	0
$609$	8.40142e8	0.150727
$610$	0	0
$611$	4.95368e8	0.0878584
$612$	0	0
$613$	9.65351e9	1.69267	0.846337	−	0.532647i	$-0.178803\pi$
$613$	9.65351e9	1.69267	0.846337	+	0.532647i	$0.178803\pi$
$614$	0	0
$615$	3.72262e9	0.645335
$616$	0	0
$617$	−9.65184e9	−1.65429	−0.827146	−	0.561987i	$-0.810037\pi$
$617$	−9.65184e9	−1.65429	−0.827146	+	0.561987i	$0.810037\pi$
$618$	0	0
$619$	3.81375e9	0.646301	0.323150	−	0.946348i	$-0.395258\pi$
$619$	3.81375e9	0.646301	0.323150	+	0.946348i	$0.395258\pi$
$620$	0	0
$621$	−8.22378e9	−1.37801
$622$	0	0
$623$	−7.50786e8	−0.124397
$624$	0	0
$625$	−7.30155e9	−1.19629
$626$	0	0
$627$	1.03553e10	1.67775
$628$	0	0
$629$	−3.11318e9	−0.498801
$630$	0	0
$631$	−3.12153e9	−0.494612	−0.247306	−	0.968937i	$-0.579545\pi$
$631$	−3.12153e9	−0.494612	−0.247306	+	0.968937i	$0.579545\pi$
$632$	0	0
$633$	−5.57530e9	−0.873685
$634$	0	0
$635$	7.96814e9	1.23495
$636$	0	0
$637$	5.97754e9	0.916293
$638$	0	0
$639$	9.45025e8	0.143282
$640$	0	0
$641$	3.39915e9	0.509761	0.254881	−	0.966973i	$-0.417964\pi$
$641$	3.39915e9	0.509761	0.254881	+	0.966973i	$0.417964\pi$
$642$	0	0
$643$	5.74851e9	0.852740	0.426370	−	0.904549i	$-0.359792\pi$
$643$	5.74851e9	0.852740	0.426370	+	0.904549i	$0.359792\pi$
$644$	0	0
$645$	−3.84607e9	−0.564362
$646$	0	0
$647$	7.87801e9	1.14354	0.571770	−	0.820414i	$-0.306257\pi$
$647$	7.87801e9	1.14354	0.571770	+	0.820414i	$0.306257\pi$
$648$	0	0
$649$	−8.72362e8	−0.125268
$650$	0	0
$651$	8.51805e8	0.121006
$652$	0	0
$653$	1.27731e10	1.79514	0.897571	−	0.440869i	$-0.145330\pi$
$653$	1.27731e10	1.79514	0.897571	+	0.440869i	$0.145330\pi$
$654$	0	0
$655$	−5.62745e9	−0.782469
$656$	0	0
$657$	1.52928e9	0.210382
$658$	0	0
$659$	−5.30294e9	−0.721802	−0.360901	−	0.932604i	$-0.617531\pi$
$659$	−5.30294e9	−0.721802	−0.360901	+	0.932604i	$0.617531\pi$
$660$	0	0
$661$	−7.37711e9	−0.993530	−0.496765	−	0.867885i	$-0.665479\pi$
$661$	−7.37711e9	−0.993530	−0.496765	+	0.867885i	$0.665479\pi$
$662$	0	0
$663$	−6.47235e9	−0.862510
$664$	0	0
$665$	−7.76287e8	−0.102364
$666$	0	0
$667$	−1.76372e10	−2.30138
$668$	0	0
$669$	4.10375e9	0.529894
$670$	0	0
$671$	1.15861e10	1.48050
$672$	0	0
$673$	3.46946e9	0.438742	0.219371	−	0.975641i	$-0.429599\pi$
$673$	3.46946e9	0.438742	0.219371	+	0.975641i	$0.429599\pi$
$674$	0	0
$675$	−2.28479e9	−0.285946
$676$	0	0
$677$	−3.20369e9	−0.396817	−0.198409	−	0.980119i	$-0.563577\pi$
$677$	−3.20369e9	−0.396817	−0.198409	+	0.980119i	$0.563577\pi$
$678$	0	0
$679$	−1.53034e8	−0.0187604
$680$	0	0
$681$	−1.93508e9	−0.234792
$682$	0	0
$683$	3.92573e9	0.471463	0.235732	−	0.971818i	$-0.424251\pi$
$683$	3.92573e9	0.471463	0.235732	+	0.971818i	$0.424251\pi$
$684$	0	0
$685$	6.92815e9	0.823570
$686$	0	0
$687$	−7.72350e9	−0.908795
$688$	0	0
$689$	1.91229e9	0.222734
$690$	0	0
$691$	−1.18034e10	−1.36092	−0.680460	−	0.732785i	$-0.738220\pi$
$691$	−1.18034e10	−1.36092	−0.680460	+	0.732785i	$0.738220\pi$
$692$	0	0
$693$	2.61283e8	0.0298225
$694$	0	0
$695$	−4.16179e9	−0.470255
$696$	0	0
$697$	5.76187e9	0.644538
$698$	0	0
$699$	1.14432e10	1.26730
$700$	0	0
$701$	1.70281e9	0.186704	0.0933520	−	0.995633i	$-0.470242\pi$
$701$	1.70281e9	0.186704	0.0933520	+	0.995633i	$0.470242\pi$
$702$	0	0
$703$	−4.38593e9	−0.476122
$704$	0	0
$705$	−9.07177e8	−0.0975057
$706$	0	0
$707$	−3.72663e8	−0.0396596
$708$	0	0
$709$	−4.08680e9	−0.430647	−0.215324	−	0.976543i	$-0.569081\pi$
$709$	−4.08680e9	−0.430647	−0.215324	+	0.976543i	$0.569081\pi$
$710$	0	0
$711$	5.01284e8	0.0523046
$712$	0	0
$713$	−1.78820e10	−1.84758
$714$	0	0
$715$	−1.91623e10	−1.96055
$716$	0	0
$717$	−1.34954e10	−1.36732
$718$	0	0
$719$	3.06705e8	0.0307730	0.0153865	−	0.999882i	$-0.495102\pi$
$719$	3.06705e8	0.0307730	0.0153865	+	0.999882i	$0.495102\pi$
$720$	0	0
$721$	−1.27149e9	−0.126340
$722$	0	0
$723$	−1.78132e9	−0.175291
$724$	0	0
$725$	−4.90009e9	−0.477552
$726$	0	0
$727$	7.24420e9	0.699230	0.349615	−	0.936893i	$-0.386312\pi$
$727$	7.24420e9	0.699230	0.349615	+	0.936893i	$0.386312\pi$
$728$	0	0
$729$	1.15835e10	1.10737
$730$	0	0
$731$	−5.95295e9	−0.563665
$732$	0	0
$733$	7.67542e9	0.719844	0.359922	−	0.932982i	$-0.382803\pi$
$733$	7.67542e9	0.719844	0.359922	+	0.932982i	$0.382803\pi$
$734$	0	0
$735$	−1.09468e10	−1.01691
$736$	0	0
$737$	1.05033e10	0.966470
$738$	0	0
$739$	8.23548e9	0.750643	0.375322	−	0.926895i	$-0.377532\pi$
$739$	8.23548e9	0.750643	0.375322	+	0.926895i	$0.377532\pi$
$740$	0	0
$741$	−9.11840e9	−0.823294
$742$	0	0
$743$	−7.14950e9	−0.639462	−0.319731	−	0.947508i	$-0.603593\pi$
$743$	−7.14950e9	−0.639462	−0.319731	+	0.947508i	$0.603593\pi$
$744$	0	0
$745$	9.67012e9	0.856811
$746$	0	0
$747$	3.05358e9	0.268032
$748$	0	0
$749$	−3.97095e8	−0.0345309
$750$	0	0
$751$	−1.27608e10	−1.09935	−0.549677	−	0.835377i	$-0.685249\pi$
$751$	−1.27608e10	−1.09935	−0.549677	+	0.835377i	$0.685249\pi$
$752$	0	0
$753$	6.38836e9	0.545264
$754$	0	0
$755$	2.34799e10	1.98555
$756$	0	0
$757$	−1.56902e10	−1.31460	−0.657298	−	0.753631i	$-0.728301\pi$
$757$	−1.56902e10	−1.31460	−0.657298	+	0.753631i	$0.728301\pi$
$758$	0	0
$759$	2.67151e10	2.21774
$760$	0	0
$761$	6.91116e9	0.568467	0.284233	−	0.958755i	$-0.408261\pi$
$761$	6.91116e9	0.568467	0.284233	+	0.958755i	$0.408261\pi$
$762$	0	0
$763$	1.53959e9	0.125479
$764$	0	0
$765$	−2.43364e9	−0.196535
$766$	0	0
$767$	7.68162e8	0.0614708
$768$	0	0
$769$	9.20668e9	0.730064	0.365032	−	0.930995i	$-0.381058\pi$
$769$	9.20668e9	0.730064	0.365032	+	0.930995i	$0.381058\pi$
$770$	0	0
$771$	1.35768e10	1.06686
$772$	0	0
$773$	1.23288e10	0.960044	0.480022	−	0.877256i	$-0.340629\pi$
$773$	1.23288e10	0.960044	0.480022	+	0.877256i	$0.340629\pi$
$774$	0	0
$775$	−4.96811e9	−0.383385
$776$	0	0
$777$	5.38989e8	0.0412198
$778$	0	0
$779$	8.11747e9	0.615233
$780$	0	0
$781$	−2.10919e10	−1.58430
$782$	0	0
$783$	2.54935e10	1.89785
$784$	0	0
$785$	5.85697e9	0.432145
$786$	0	0
$787$	1.81216e8	0.0132521	0.00662606	−	0.999978i	$-0.497891\pi$
$787$	1.81216e8	0.0132521	0.00662606	+	0.999978i	$0.497891\pi$
$788$	0	0
$789$	1.60709e10	1.16485
$790$	0	0
$791$	5.08906e8	0.0365612
$792$	0	0
$793$	−1.02022e10	−0.726504
$794$	0	0
$795$	−3.50201e9	−0.247191
$796$	0	0
$797$	−5.31512e8	−0.0371885	−0.0185943	−	0.999827i	$-0.505919\pi$
$797$	−5.31512e8	−0.0371885	−0.0185943	+	0.999827i	$0.505919\pi$
$798$	0	0
$799$	−1.40413e9	−0.0973853
$800$	0	0
$801$	−3.31594e9	−0.227978
$802$	0	0
$803$	−3.41317e10	−2.32624
$804$	0	0
$805$	−2.00271e9	−0.135311
$806$	0	0
$807$	−6.03790e9	−0.404416
$808$	0	0
$809$	−2.84667e10	−1.89024	−0.945120	−	0.326724i	$-0.894055\pi$
$809$	−2.84667e10	−1.89024	−0.945120	+	0.326724i	$0.894055\pi$
$810$	0	0
$811$	1.36237e10	0.896853	0.448426	−	0.893820i	$-0.351985\pi$
$811$	1.36237e10	0.896853	0.448426	+	0.893820i	$0.351985\pi$
$812$	0	0
$813$	6.74131e9	0.439974
$814$	0	0
$815$	2.74633e10	1.77706
$816$	0	0
$817$	−8.38666e9	−0.538037
$818$	0	0
$819$	−2.30074e8	−0.0146343
$820$	0	0
$821$	2.16780e9	0.136716	0.0683578	−	0.997661i	$-0.478224\pi$
$821$	2.16780e9	0.136716	0.0683578	+	0.997661i	$0.478224\pi$
$822$	0	0
$823$	−2.18959e10	−1.36919	−0.684593	−	0.728925i	$-0.740020\pi$
$823$	−2.18959e10	−1.36919	−0.684593	+	0.728925i	$0.740020\pi$
$824$	0	0
$825$	7.42220e9	0.460197
$826$	0	0
$827$	1.26499e10	0.777710	0.388855	−	0.921299i	$-0.372871\pi$
$827$	1.26499e10	0.777710	0.388855	+	0.921299i	$0.372871\pi$
$828$	0	0
$829$	−1.06942e10	−0.651937	−0.325969	−	0.945381i	$-0.605690\pi$
$829$	−1.06942e10	−0.651937	−0.325969	+	0.945381i	$0.605690\pi$
$830$	0	0
$831$	−9.39860e9	−0.568145
$832$	0	0
$833$	−1.69435e10	−1.01565
$834$	0	0
$835$	−1.03176e10	−0.613304
$836$	0	0
$837$	2.58474e10	1.52362
$838$	0	0
$839$	1.24683e10	0.728855	0.364428	−	0.931232i	$-0.381265\pi$
$839$	1.24683e10	0.728855	0.364428	+	0.931232i	$0.381265\pi$
$840$	0	0
$841$	3.74248e10	2.16957
$842$	0	0
$843$	−1.55260e10	−0.892613
$844$	0	0
$845$	−2.87798e9	−0.164093
$846$	0	0
$847$	−4.18779e9	−0.236806
$848$	0	0
$849$	−1.10499e10	−0.619698
$850$	0	0
$851$	−1.13150e10	−0.629366
$852$	0	0
$853$	2.50444e10	1.38162	0.690810	−	0.723037i	$-0.257254\pi$
$853$	2.50444e10	1.38162	0.690810	+	0.723037i	$0.257254\pi$
$854$	0	0
$855$	−3.42857e9	−0.187599
$856$	0	0
$857$	−1.45853e9	−0.0791559	−0.0395779	−	0.999216i	$-0.512601\pi$
$857$	−1.45853e9	−0.0791559	−0.0395779	+	0.999216i	$0.512601\pi$
$858$	0	0
$859$	9.13497e8	0.0491735	0.0245868	−	0.999698i	$-0.492173\pi$
$859$	9.13497e8	0.0491735	0.0245868	+	0.999698i	$0.492173\pi$
$860$	0	0
$861$	−9.97560e8	−0.0532633
$862$	0	0
$863$	−2.97408e10	−1.57513	−0.787563	−	0.616235i	$-0.788657\pi$
$863$	−2.97408e10	−1.57513	−0.787563	+	0.616235i	$0.788657\pi$
$864$	0	0
$865$	−3.02751e10	−1.59049
$866$	0	0
$867$	8.67025e8	0.0451819
$868$	0	0
$869$	−1.11881e10	−0.578344
$870$	0	0
$871$	−9.24870e9	−0.474261
$872$	0	0
$873$	−6.75893e8	−0.0343817
$874$	0	0
$875$	1.51789e9	0.0765972
$876$	0	0
$877$	2.22211e10	1.11241	0.556207	−	0.831044i	$-0.312256\pi$
$877$	2.22211e10	1.11241	0.556207	+	0.831044i	$0.312256\pi$
$878$	0	0
$879$	3.58269e9	0.177930
$880$	0	0
$881$	1.95120e10	0.961358	0.480679	−	0.876897i	$-0.340390\pi$
$881$	1.95120e10	0.961358	0.480679	+	0.876897i	$0.340390\pi$
$882$	0	0
$883$	−2.63500e10	−1.28801	−0.644004	−	0.765022i	$-0.722728\pi$
$883$	−2.63500e10	−1.28801	−0.644004	+	0.765022i	$0.722728\pi$
$884$	0	0
$885$	−1.40675e9	−0.0682206
$886$	0	0
$887$	−3.55439e10	−1.71014	−0.855072	−	0.518510i	$-0.826487\pi$
$887$	−3.55439e10	−1.71014	−0.855072	+	0.518510i	$0.826487\pi$
$888$	0	0
$889$	−2.13524e9	−0.101928
$890$	0	0
$891$	−3.18407e10	−1.50803
$892$	0	0
$893$	−1.97817e9	−0.0929574
$894$	0	0
$895$	1.98274e10	0.924452
$896$	0	0
$897$	−2.35241e10	−1.08828
$898$	0	0
$899$	5.54337e10	2.54457
$900$	0	0
$901$	−5.42042e9	−0.246886
$902$	0	0
$903$	1.03064e9	0.0465801
$904$	0	0
$905$	1.62064e10	0.726804
$906$	0	0
$907$	2.57011e10	1.14374	0.571868	−	0.820345i	$-0.306219\pi$
$907$	2.57011e10	1.14374	0.571868	+	0.820345i	$0.306219\pi$
$908$	0	0
$909$	−1.64591e9	−0.0726830
$910$	0	0
$911$	−5.61618e9	−0.246109	−0.123054	−	0.992400i	$-0.539269\pi$
$911$	−5.61618e9	−0.246109	−0.123054	+	0.992400i	$0.539269\pi$
$912$	0	0
$913$	−6.81524e10	−2.96369
$914$	0	0
$915$	1.86835e10	0.806278
$916$	0	0
$917$	1.50800e9	0.0645818
$918$	0	0
$919$	1.22621e10	0.521147	0.260574	−	0.965454i	$-0.416088\pi$
$919$	1.22621e10	0.521147	0.260574	+	0.965454i	$0.416088\pi$
$920$	0	0
$921$	−6.99745e9	−0.295142
$922$	0	0
$923$	1.85725e10	0.777437
$924$	0	0
$925$	−3.14363e9	−0.130598
$926$	0	0
$927$	−5.61571e9	−0.231540
$928$	0	0
$929$	2.81141e10	1.15046	0.575228	−	0.817993i	$-0.304913\pi$
$929$	2.81141e10	1.15046	0.575228	+	0.817993i	$0.304913\pi$
$930$	0	0
$931$	−2.38704e10	−0.969471
$932$	0	0
$933$	2.35117e10	0.947758
$934$	0	0
$935$	5.43160e10	2.17314
$936$	0	0
$937$	4.24338e10	1.68509	0.842546	−	0.538625i	$-0.181056\pi$
$937$	4.24338e10	1.68509	0.842546	+	0.538625i	$0.181056\pi$
$938$	0	0
$939$	−1.23506e10	−0.486810
$940$	0	0
$941$	3.89941e9	0.152558	0.0762791	−	0.997087i	$-0.475696\pi$
$941$	3.89941e9	0.152558	0.0762791	+	0.997087i	$0.475696\pi$
$942$	0	0
$943$	2.09419e10	0.813251
$944$	0	0
$945$	2.89479e9	0.111585
$946$	0	0
$947$	−3.27153e10	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$947$	−3.27153e10	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$948$	0	0
$949$	3.00548e10	1.14152
$950$	0	0
$951$	2.08469e10	0.785976
$952$	0	0
$953$	−2.89349e9	−0.108292	−0.0541461	−	0.998533i	$-0.517244\pi$
$953$	−2.89349e9	−0.108292	−0.0541461	+	0.998533i	$0.517244\pi$
$954$	0	0
$955$	3.43457e10	1.27603
$956$	0	0
$957$	−8.28161e10	−3.05438
$958$	0	0
$959$	−1.85656e9	−0.0679740
$960$	0	0
$961$	2.86906e10	1.04282
$962$	0	0
$963$	−1.75382e9	−0.0632838
$964$	0	0
$965$	−3.24949e10	−1.16404
$966$	0	0
$967$	−8.74961e8	−0.0311169	−0.0155584	−	0.999879i	$-0.504953\pi$
$967$	−8.74961e8	−0.0311169	−0.0155584	+	0.999879i	$0.504953\pi$
$968$	0	0
$969$	2.58463e10	0.912568
$970$	0	0
$971$	−2.78873e10	−0.977549	−0.488775	−	0.872410i	$-0.662556\pi$
$971$	−2.78873e10	−0.977549	−0.488775	+	0.872410i	$0.662556\pi$
$972$	0	0
$973$	1.11525e9	0.0388129
$974$	0	0
$975$	−6.53565e9	−0.225825
$976$	0	0
$977$	4.90191e10	1.68165	0.840823	−	0.541310i	$-0.182071\pi$
$977$	4.90191e10	1.68165	0.840823	+	0.541310i	$0.182071\pi$
$978$	0	0
$979$	7.40080e10	2.52080
$980$	0	0
$981$	6.79981e9	0.229962
$982$	0	0
$983$	−4.72491e10	−1.58656	−0.793280	−	0.608857i	$-0.791628\pi$
$983$	−4.72491e10	−1.58656	−0.793280	+	0.608857i	$0.791628\pi$
$984$	0	0
$985$	−1.16294e10	−0.387730
$986$	0	0
$987$	2.43099e8	0.00804772
$988$	0	0
$989$	−2.16364e10	−0.711209
$990$	0	0
$991$	4.34968e10	1.41971	0.709855	−	0.704348i	$-0.248761\pi$
$991$	4.34968e10	1.41971	0.709855	+	0.704348i	$0.248761\pi$
$992$	0	0
$993$	−3.81248e10	−1.23562
$994$	0	0
$995$	−1.72177e10	−0.554108
$996$	0	0
$997$	8.30555e9	0.265421	0.132710	−	0.991155i	$-0.457632\pi$
$997$	8.30555e9	0.265421	0.132710	+	0.991155i	$0.457632\pi$
$998$	0	0
$999$	1.63552e10	0.519012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.a.b.1.2	✓	2
3.2	odd	2		288.8.a.o.1.1		2
4.3	odd	2		32.8.a.d.1.1	yes	2
8.3	odd	2		64.8.a.h.1.2		2
8.5	even	2		64.8.a.j.1.1		2
12.11	even	2		288.8.a.n.1.1		2
16.3	odd	4		256.8.b.j.129.2		4
16.5	even	4		256.8.b.h.129.2		4
16.11	odd	4		256.8.b.j.129.3		4
16.13	even	4		256.8.b.h.129.3		4
24.5	odd	2		576.8.a.bf.1.2		2
24.11	even	2		576.8.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.a.b.1.2	✓	2	1.1	even	1	trivial
32.8.a.d.1.1	yes	2	4.3	odd	2
64.8.a.h.1.2		2	8.3	odd	2
64.8.a.j.1.1		2	8.5	even	2
256.8.b.h.129.2		4	16.5	even	4
256.8.b.h.129.3		4	16.13	even	4
256.8.b.j.129.2		4	16.3	odd	4
256.8.b.j.129.3		4	16.11	odd	4
288.8.a.n.1.1		2	12.11	even	2
288.8.a.o.1.1		2	3.2	odd	2
576.8.a.be.1.2		2	24.11	even	2
576.8.a.bf.1.2		2	24.5	odd	2

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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+42.5964 q^{3} +314.772 q^{5} -84.3502 q^{7} -372.543 q^{9} +O(q^{10})\)	\(q+42.5964 q^{3} +314.772 q^{5} -84.3502 q^{7} -372.543 q^{9} +8314.73 q^{11} -7321.57 q^{13} +13408.1 q^{15} +20753.1 q^{17} +29237.6 q^{19} -3593.02 q^{21} +75428.6 q^{23} +20956.1 q^{25} -109027. q^{27} -233826. q^{29} -237072. q^{31} +354178. q^{33} -26551.0 q^{35} -150010. q^{37} -311873. q^{39} +277638. q^{41} -286846. q^{43} -117266. q^{45} -67658.6 q^{47} -816428. q^{49} +884010. q^{51} -261185. q^{53} +2.61724e6 q^{55} +1.24542e6 q^{57} -104918. q^{59} +1.39344e6 q^{61} +31424.1 q^{63} -2.30462e6 q^{65} +1.26321e6 q^{67} +3.21299e6 q^{69} -2.53669e6 q^{71} -4.10497e6 q^{73} +892656. q^{75} -701349. q^{77} -1.34557e6 q^{79} -3.82943e6 q^{81} -8.19658e6 q^{83} +6.53250e6 q^{85} -9.96016e6 q^{87} +8.90083e6 q^{89} +617576. q^{91} -1.00984e7 q^{93} +9.20315e6 q^{95} +1.81427e6 q^{97} -3.09760e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9}+O(q^{10}) \) 2 * q - 16 * q^3 - 180 * q^5 + 1248 * q^7 + 874 * q^9	\( 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9} + 9040 q^{11} - 2500 q^{13} + 42400 q^{15} + 17220 q^{17} + 74160 q^{19} - 81664 q^{21} + 19104 q^{23} + 187630 q^{25} - 53920 q^{27} - 245028 q^{29} - 251520 q^{31} + 311680 q^{33} - 685760 q^{35} - 530740 q^{37} - 594400 q^{39} + 726900 q^{41} + 44496 q^{43} - 734020 q^{45} + 494912 q^{47} + 135186 q^{49} + 1091040 q^{51} + 1319340 q^{53} + 2258400 q^{55} - 1386880 q^{57} + 2618000 q^{59} + 836700 q^{61} + 1692256 q^{63} - 4690200 q^{65} + 2374128 q^{67} + 6513408 q^{69} - 2836000 q^{71} - 171180 q^{73} - 8873840 q^{75} + 264960 q^{77} - 2498880 q^{79} - 9784718 q^{81} - 9531984 q^{83} + 8280600 q^{85} - 9303776 q^{87} + 7318068 q^{89} + 7041600 q^{91} - 9251840 q^{93} - 13023200 q^{95} - 9316060 q^{97} - 2193520 q^{99}+O(q^{100}) \) 2 * q - 16 * q^3 - 180 * q^5 + 1248 * q^7 + 874 * q^9 + 9040 * q^11 - 2500 * q^13 + 42400 * q^15 + 17220 * q^17 + 74160 * q^19 - 81664 * q^21 + 19104 * q^23 + 187630 * q^25 - 53920 * q^27 - 245028 * q^29 - 251520 * q^31 + 311680 * q^33 - 685760 * q^35 - 530740 * q^37 - 594400 * q^39 + 726900 * q^41 + 44496 * q^43 - 734020 * q^45 + 494912 * q^47 + 135186 * q^49 + 1091040 * q^51 + 1319340 * q^53 + 2258400 * q^55 - 1386880 * q^57 + 2618000 * q^59 + 836700 * q^61 + 1692256 * q^63 - 4690200 * q^65 + 2374128 * q^67 + 6513408 * q^69 - 2836000 * q^71 - 171180 * q^73 - 8873840 * q^75 + 264960 * q^77 - 2498880 * q^79 - 9784718 * q^81 - 9531984 * q^83 + 8280600 * q^85 - 9303776 * q^87 + 7318068 * q^89 + 7041600 * q^91 - 9251840 * q^93 - 13023200 * q^95 - 9316060 * q^97 - 2193520 * q^99

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