LMFDB - Embedded newform 32.10.b.a.17.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,10,Mod(17,32)]

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, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$16.4811467572$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 $ x^8 - x^7 + 59x^6 - 313x^5 - 315x^4 - 92091x^3 + 1261649x^2 - 16074123x + 251007534
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{56}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.7
Root			$3.68032 - 10.3002i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.10.b.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+150.106i q^{3} +292.339i q^{5} +9955.46 q^{7} -2848.66 q^{9} +O(q^{10})$	$q+150.106i q^{3} +292.339i q^{5} +9955.46 q^{7} -2848.66 q^{9} +65874.1i q^{11} -44990.6i q^{13} -43881.7 q^{15} -469098. q^{17} +438793. i q^{19} +1.49437e6i q^{21} -1.14082e6 q^{23} +1.86766e6 q^{25} +2.52693e6i q^{27} +5.39024e6i q^{29} -1.85181e6 q^{31} -9.88806e6 q^{33} +2.91037e6i q^{35} -1.45757e7i q^{37} +6.75334e6 q^{39} +5.45239e6 q^{41} -5.79053e6i q^{43} -832774. i q^{45} +1.69791e7 q^{47} +5.87576e7 q^{49} -7.04142e7i q^{51} -4.94827e7i q^{53} -1.92575e7 q^{55} -6.58652e7 q^{57} -4.70612e7i q^{59} +7.39468e7i q^{61} -2.83597e7 q^{63} +1.31525e7 q^{65} +2.37554e8i q^{67} -1.71243e8i q^{69} +6.33394e7 q^{71} -2.73992e7 q^{73} +2.80346e8i q^{75} +6.55807e8i q^{77} +1.20924e8 q^{79} -4.35376e8 q^{81} -1.31132e8i q^{83} -1.37136e8i q^{85} -8.09104e8 q^{87} +6.90156e8 q^{89} -4.47903e8i q^{91} -2.77967e8i q^{93} -1.28276e8 q^{95} +1.17879e8 q^{97} -1.87653e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4800 q^{7} - 39368 q^{9}+O(q^{10}) $ 8 * q - 4800 * q^7 - 39368 * q^9	$ 8 q - 4800 q^{7} - 39368 q^{9} + 163136 q^{15} - 102000 q^{17} - 3412032 q^{23} - 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} + 17590208 q^{39} - 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} - 7056832 q^{55} + 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} - 560234688 q^{71} - 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} - 540527424 q^{87} + 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) $ 8 * q - 4800 * q^7 - 39368 * q^9 + 163136 * q^15 - 102000 * q^17 - 3412032 * q^23 - 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 + 17590208 * q^39 - 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 - 7056832 * q^55 + 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 - 560234688 * q^71 - 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 - 540527424 * q^87 + 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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
$3$	150.106i	1.06992i	0.844878	+	0.534960i	$0.179673\pi$
$3$	150.106i	1.06992i	−0.844878	+	0.534960i	$0.820327\pi$
$4$	0	0
$5$	292.339i	0.209181i	0.994515	+	0.104590i	$0.0333531\pi$
$5$	292.339i	0.209181i	−0.994515	+	0.104590i	$0.966647\pi$
$6$	0	0
$7$	9955.46	1.56718	0.783592	−	0.621275i	$-0.213385\pi$
$7$	9955.46	1.56718	0.783592	+	0.621275i	$0.213385\pi$
$8$	0	0
$9$	−2848.66	−0.144727
$10$	0	0
$11$	65874.1i	1.35659i	0.734791	+	0.678293i	$0.237280\pi$
$11$	65874.1i	1.35659i	−0.734791	+	0.678293i	$0.762720\pi$
$12$	0	0
$13$	− 44990.6i	− 0.436895i	−0.975849	−	0.218447i	$-0.929901\pi$
$13$	− 44990.6i	− 0.436895i	0.975849	−	0.218447i	$-0.0700992\pi$
$14$	0	0
$15$	−43881.7	−0.223806
$16$	0	0
$17$	−469098.	−1.36221	−0.681104	−	0.732187i	$-0.738500\pi$
$17$	−469098.	−1.36221	−0.681104	+	0.732187i	$0.738500\pi$
$18$	0	0
$19$	438793.i	0.772447i	0.922405	+	0.386223i	$0.126221\pi$
$19$	438793.i	0.772447i	−0.922405	+	0.386223i	$0.873779\pi$
$20$	0	0
$21$	1.49437e6i	1.67676i
$22$	0	0
$23$	−1.14082e6	−0.850041	−0.425021	−	0.905184i	$-0.639733\pi$
$23$	−1.14082e6	−0.850041	−0.425021	+	0.905184i	$0.639733\pi$
$24$	0	0
$25$	1.86766e6	0.956243
$26$	0	0
$27$	2.52693e6i	0.915073i
$28$	0	0
$29$	5.39024e6i	1.41520i	0.706615	+	0.707598i	$0.250221\pi$
$29$	5.39024e6i	1.41520i	−0.706615	+	0.707598i	$0.749779\pi$
$30$	0	0
$31$	−1.85181e6	−0.360138	−0.180069	−	0.983654i	$-0.557632\pi$
$31$	−1.85181e6	−0.360138	−0.180069	+	0.983654i	$0.557632\pi$
$32$	0	0
$33$	−9.88806e6	−1.45144
$34$	0	0
$35$	2.91037e6i	0.327825i
$36$	0	0
$37$	− 1.45757e7i	− 1.27856i	−0.768972	−	0.639282i	$-0.779232\pi$
$37$	− 1.45757e7i	− 1.27856i	0.768972	−	0.639282i	$-0.220768\pi$
$38$	0	0
$39$	6.75334e6	0.467442
$40$	0	0
$41$	5.45239e6	0.301342	0.150671	−	0.988584i	$-0.451857\pi$
$41$	5.45239e6	0.301342	0.150671	+	0.988584i	$0.451857\pi$
$42$	0	0
$43$	− 5.79053e6i	− 0.258292i	−0.991626	−	0.129146i	$-0.958776\pi$
$43$	− 5.79053e6i	− 0.258292i	0.991626	−	0.129146i	$-0.0412235\pi$
$44$	0	0
$45$	− 832774.i	− 0.0302741i
$46$	0	0
$47$	1.69791e7	0.507544	0.253772	−	0.967264i	$-0.418329\pi$
$47$	1.69791e7	0.507544	0.253772	+	0.967264i	$0.418329\pi$
$48$	0	0
$49$	5.87576e7	1.45607
$50$	0	0
$51$	− 7.04142e7i	− 1.45745i
$52$	0	0
$53$	− 4.94827e7i	− 0.861415i	−0.902492	−	0.430707i	$-0.858264\pi$
$53$	− 4.94827e7i	− 0.861415i	0.902492	−	0.430707i	$-0.141736\pi$
$54$	0	0
$55$	−1.92575e7	−0.283772
$56$	0	0
$57$	−6.58652e7	−0.826455
$58$	0	0
$59$	− 4.70612e7i	− 0.505625i	−0.967515	−	0.252813i	$-0.918644\pi$
$59$	− 4.70612e7i	− 0.505625i	0.967515	−	0.252813i	$-0.0813556\pi$
$60$	0	0
$61$	7.39468e7i	0.683810i	0.939735	+	0.341905i	$0.111072\pi$
$61$	7.39468e7i	0.683810i	−0.939735	+	0.341905i	$0.888928\pi$
$62$	0	0
$63$	−2.83597e7	−0.226814
$64$	0	0
$65$	1.31525e7	0.0913900
$66$	0	0
$67$	2.37554e8i	1.44021i	0.693866	+	0.720104i	$0.255906\pi$
$67$	2.37554e8i	1.44021i	−0.693866	+	0.720104i	$0.744094\pi$
$68$	0	0
$69$	− 1.71243e8i	− 0.909476i
$70$	0	0
$71$	6.33394e7	0.295809	0.147904	−	0.989002i	$-0.452747\pi$
$71$	6.33394e7	0.295809	0.147904	+	0.989002i	$0.452747\pi$
$72$	0	0
$73$	−2.73992e7	−0.112924	−0.0564618	−	0.998405i	$-0.517982\pi$
$73$	−2.73992e7	−0.112924	−0.0564618	+	0.998405i	$0.517982\pi$
$74$	0	0
$75$	2.80346e8i	1.02310i
$76$	0	0
$77$	6.55807e8i	2.12602i
$78$	0	0
$79$	1.20924e8	0.349294	0.174647	−	0.984631i	$-0.444122\pi$
$79$	1.20924e8	0.349294	0.174647	+	0.984631i	$0.444122\pi$
$80$	0	0
$81$	−4.35376e8	−1.12378
$82$	0	0
$83$	− 1.31132e8i	− 0.303290i	−0.988435	−	0.151645i	$-0.951543\pi$
$83$	− 1.31132e8i	− 0.303290i	0.988435	−	0.151645i	$-0.0484570\pi$
$84$	0	0
$85$	− 1.37136e8i	− 0.284947i
$86$	0	0
$87$	−8.09104e8	−1.51415
$88$	0	0
$89$	6.90156e8	1.16598	0.582991	−	0.812478i	$-0.301882\pi$
$89$	6.90156e8	1.16598	0.582991	+	0.812478i	$0.301882\pi$
$90$	0	0
$91$	− 4.47903e8i	− 0.684695i
$92$	0	0
$93$	− 2.77967e8i	− 0.385319i
$94$	0	0
$95$	−1.28276e8	−0.161581
$96$	0	0
$97$	1.17879e8	0.135196	0.0675982	−	0.997713i	$-0.478466\pi$
$97$	1.17879e8	0.135196	0.0675982	+	0.997713i	$0.478466\pi$
$98$	0	0
$99$	− 1.87653e8i	− 0.196335i
$100$	0	0
$101$	9.55963e7i	0.0914103i	0.998955	+	0.0457051i	$0.0145535\pi$
$101$	9.55963e7i	0.0914103i	−0.998955	+	0.0457051i	$0.985447\pi$
$102$	0	0
$103$	1.83082e9	1.60280	0.801399	−	0.598130i	$-0.204090\pi$
$103$	1.83082e9	1.60280	0.801399	+	0.598130i	$0.204090\pi$
$104$	0	0
$105$	−4.36862e8	−0.350746
$106$	0	0
$107$	− 3.98905e8i	− 0.294200i	−0.989122	−	0.147100i	$-0.953006\pi$
$107$	− 3.98905e8i	− 0.294200i	0.989122	−	0.147100i	$-0.0469939\pi$
$108$	0	0
$109$	− 2.62653e9i	− 1.78223i	−0.453782	−	0.891113i	$-0.649926\pi$
$109$	− 2.62653e9i	− 1.78223i	0.453782	−	0.891113i	$-0.350074\pi$
$110$	0	0
$111$	2.18790e9	1.36796
$112$	0	0
$113$	1.25718e9	0.725343	0.362672	−	0.931917i	$-0.381865\pi$
$113$	1.25718e9	0.725343	0.362672	+	0.931917i	$0.381865\pi$
$114$	0	0
$115$	− 3.33505e8i	− 0.177812i
$116$	0	0
$117$	1.28163e8i	0.0632305i
$118$	0	0
$119$	−4.67009e9	−2.13483
$120$	0	0
$121$	−1.98144e9	−0.840326
$122$	0	0
$123$	8.18434e8i	0.322412i
$124$	0	0
$125$	1.11696e9i	0.409208i
$126$	0	0
$127$	2.41608e8	0.0824129	0.0412065	−	0.999151i	$-0.486880\pi$
$127$	2.41608e8	0.0824129	0.0412065	+	0.999151i	$0.486880\pi$
$128$	0	0
$129$	8.69191e8	0.276351
$130$	0	0
$131$	− 1.35260e9i	− 0.401281i	−0.979665	−	0.200641i	$-0.935698\pi$
$131$	− 1.35260e9i	− 0.401281i	0.979665	−	0.200641i	$-0.0643023\pi$
$132$	0	0
$133$	4.36839e9i	1.21057i
$134$	0	0
$135$	−7.38719e8	−0.191416
$136$	0	0
$137$	6.09544e8	0.147830	0.0739150	−	0.997265i	$-0.476451\pi$
$137$	6.09544e8	0.147830	0.0739150	+	0.997265i	$0.476451\pi$
$138$	0	0
$139$	2.13256e9i	0.484546i	0.970208	+	0.242273i	$0.0778929\pi$
$139$	2.13256e9i	0.484546i	−0.970208	+	0.242273i	$0.922107\pi$
$140$	0	0
$141$	2.54865e9i	0.543031i
$142$	0	0
$143$	2.96372e9	0.592686
$144$	0	0
$145$	−1.57578e9	−0.296032
$146$	0	0
$147$	8.81984e9i	1.55787i
$148$	0	0
$149$	5.25087e9i	0.872757i	0.899763	+	0.436378i	$0.143739\pi$
$149$	5.25087e9i	0.872757i	−0.899763	+	0.436378i	$0.856261\pi$
$150$	0	0
$151$	−5.75614e9	−0.901022	−0.450511	−	0.892771i	$-0.648758\pi$
$151$	−5.75614e9	−0.901022	−0.450511	+	0.892771i	$0.648758\pi$
$152$	0	0
$153$	1.33630e9	0.197148
$154$	0	0
$155$	− 5.41356e8i	− 0.0753339i
$156$	0	0
$157$	− 1.51315e10i	− 1.98763i	−0.111070	−	0.993813i	$-0.535428\pi$
$157$	− 1.51315e10i	− 1.98763i	0.111070	−	0.993813i	$-0.464572\pi$
$158$	0	0
$159$	7.42763e9	0.921644
$160$	0	0
$161$	−1.13573e10	−1.33217
$162$	0	0
$163$	− 1.00347e10i	− 1.11343i	−0.830705	−	0.556713i	$-0.812062\pi$
$163$	− 1.00347e10i	− 1.11343i	0.830705	−	0.556713i	$-0.187938\pi$
$164$	0	0
$165$	− 2.89066e9i	− 0.303613i
$166$	0	0
$167$	−9.12880e9	−0.908217	−0.454108	−	0.890946i	$-0.650042\pi$
$167$	−9.12880e9	−0.908217	−0.454108	+	0.890946i	$0.650042\pi$
$168$	0	0
$169$	8.58034e9	0.809123
$170$	0	0
$171$	− 1.24997e9i	− 0.111794i
$172$	0	0
$173$	6.47969e9i	0.549980i	0.961447	+	0.274990i	$0.0886746\pi$
$173$	6.47969e9i	0.549980i	−0.961447	+	0.274990i	$0.911325\pi$
$174$	0	0
$175$	1.85934e10	1.49861
$176$	0	0
$177$	7.06414e9	0.540978
$178$	0	0
$179$	− 1.37068e9i	− 0.0997923i	−0.998754	−	0.0498962i	$-0.984111\pi$
$179$	− 1.37068e9i	− 0.0997923i	0.998754	−	0.0498962i	$-0.0158890\pi$
$180$	0	0
$181$	− 1.85728e10i	− 1.28625i	−0.765763	−	0.643123i	$-0.777638\pi$
$181$	− 1.85728e10i	− 1.28625i	0.765763	−	0.643123i	$-0.222362\pi$
$182$	0	0
$183$	−1.10998e10	−0.731621
$184$	0	0
$185$	4.26105e9	0.267451
$186$	0	0
$187$	− 3.09014e10i	− 1.84795i
$188$	0	0
$189$	2.51567e10i	1.43409i
$190$	0	0
$191$	1.73970e10	0.945852	0.472926	−	0.881102i	$-0.343198\pi$
$191$	1.73970e10	0.945852	0.472926	+	0.881102i	$0.343198\pi$
$192$	0	0
$193$	3.11955e10	1.61839	0.809197	−	0.587538i	$-0.199903\pi$
$193$	3.11955e10	1.61839	0.809197	+	0.587538i	$0.199903\pi$
$194$	0	0
$195$	1.97426e9i	0.0977799i
$196$	0	0
$197$	− 1.86636e10i	− 0.882872i	−0.897293	−	0.441436i	$-0.854469\pi$
$197$	− 1.86636e10i	− 0.882872i	0.897293	−	0.441436i	$-0.145531\pi$
$198$	0	0
$199$	−3.27467e10	−1.48023	−0.740115	−	0.672481i	$-0.765229\pi$
$199$	−3.27467e10	−1.48023	−0.740115	+	0.672481i	$0.765229\pi$
$200$	0	0
$201$	−3.56581e10	−1.54091
$202$	0	0
$203$	5.36623e10i	2.21788i
$204$	0	0
$205$	1.59395e9i	0.0630349i
$206$	0	0
$207$	3.24980e9	0.123024
$208$	0	0
$209$	−2.89051e10	−1.04789
$210$	0	0
$211$	− 6.63427e9i	− 0.230421i	−0.993341	−	0.115210i	$-0.963246\pi$
$211$	− 6.63427e9i	− 0.230421i	0.993341	−	0.115210i	$-0.0367543\pi$
$212$	0	0
$213$	9.50759e9i	0.316491i
$214$	0	0
$215$	1.69280e9	0.0540296
$216$	0	0
$217$	−1.84356e10	−0.564403
$218$	0	0
$219$	− 4.11277e9i	− 0.120819i
$220$	0	0
$221$	2.11050e10i	0.595142i
$222$	0	0
$223$	6.34852e10	1.71910	0.859549	−	0.511053i	$-0.170744\pi$
$223$	6.34852e10	1.71910	0.859549	+	0.511053i	$0.170744\pi$
$224$	0	0
$225$	−5.32034e9	−0.138394
$226$	0	0
$227$	5.09567e10i	1.27375i	0.770966	+	0.636877i	$0.219774\pi$
$227$	5.09567e10i	1.27375i	−0.770966	+	0.636877i	$0.780226\pi$
$228$	0	0
$229$	1.53720e10i	0.369377i	0.982797	+	0.184689i	$0.0591277\pi$
$229$	1.53720e10i	0.369377i	−0.982797	+	0.184689i	$0.940872\pi$
$230$	0	0
$231$	−9.84402e10	−2.27467
$232$	0	0
$233$	5.78808e10	1.28657	0.643284	−	0.765628i	$-0.277571\pi$
$233$	5.78808e10	1.28657	0.643284	+	0.765628i	$0.277571\pi$
$234$	0	0
$235$	4.96365e9i	0.106168i
$236$	0	0
$237$	1.81514e10i	0.373716i
$238$	0	0
$239$	5.44879e9	0.108021	0.0540106	−	0.998540i	$-0.482800\pi$
$239$	5.44879e9	0.108021	0.0540106	+	0.998540i	$0.482800\pi$
$240$	0	0
$241$	3.35748e10	0.641116	0.320558	−	0.947229i	$-0.396130\pi$
$241$	3.35748e10	0.641116	0.320558	+	0.947229i	$0.396130\pi$
$242$	0	0
$243$	− 1.56148e10i	− 0.287282i
$244$	0	0
$245$	1.71771e10i	0.304581i
$246$	0	0
$247$	1.97416e10	0.337478
$248$	0	0
$249$	1.96836e10	0.324495
$250$	0	0
$251$	− 8.11855e10i	− 1.29106i	−0.763734	−	0.645531i	$-0.776636\pi$
$251$	− 8.11855e10i	− 1.29106i	0.763734	−	0.645531i	$-0.223364\pi$
$252$	0	0
$253$	− 7.51501e10i	− 1.15315i
$254$	0	0
$255$	2.05848e10	0.304871
$256$	0	0
$257$	−1.18704e11	−1.69733	−0.848666	−	0.528930i	$-0.822594\pi$
$257$	−1.18704e11	−1.69733	−0.848666	+	0.528930i	$0.822594\pi$
$258$	0	0
$259$	− 1.45108e11i	− 2.00375i
$260$	0	0
$261$	− 1.53550e10i	− 0.204817i
$262$	0	0
$263$	−1.49616e10	−0.192832	−0.0964158	−	0.995341i	$-0.530738\pi$
$263$	−1.49616e10	−0.192832	−0.0964158	+	0.995341i	$0.530738\pi$
$264$	0	0
$265$	1.44657e10	0.180191
$266$	0	0
$267$	1.03596e11i	1.24751i
$268$	0	0
$269$	− 6.60058e10i	− 0.768594i	−0.923210	−	0.384297i	$-0.874444\pi$
$269$	− 6.60058e10i	− 0.768594i	0.923210	−	0.384297i	$-0.125556\pi$
$270$	0	0
$271$	5.96531e10	0.671849	0.335924	−	0.941889i	$-0.390951\pi$
$271$	5.96531e10	0.671849	0.335924	+	0.941889i	$0.390951\pi$
$272$	0	0
$273$	6.72326e10	0.732568
$274$	0	0
$275$	1.23031e11i	1.29723i
$276$	0	0
$277$	− 8.76886e10i	− 0.894920i	−0.894304	−	0.447460i	$-0.852329\pi$
$277$	− 8.76886e10i	− 0.894920i	0.894304	−	0.447460i	$-0.147671\pi$
$278$	0	0
$279$	5.27519e9	0.0521217
$280$	0	0
$281$	2.36402e10	0.226189	0.113095	−	0.993584i	$-0.463924\pi$
$281$	2.36402e10	0.226189	0.113095	+	0.993584i	$0.463924\pi$
$282$	0	0
$283$	− 1.27110e11i	− 1.17799i	−0.808138	−	0.588993i	$-0.799525\pi$
$283$	− 1.27110e11i	− 1.17799i	0.808138	−	0.588993i	$-0.200475\pi$
$284$	0	0
$285$	− 1.92550e10i	− 0.172878i
$286$	0	0
$287$	5.42811e10	0.472259
$288$	0	0
$289$	1.01465e11	0.855610
$290$	0	0
$291$	1.76943e10i	0.144649i
$292$	0	0
$293$	1.89715e11i	1.50383i	0.659261	+	0.751914i	$0.270869\pi$
$293$	1.89715e11i	1.50383i	−0.659261	+	0.751914i	$0.729131\pi$
$294$	0	0
$295$	1.37578e10	0.105767
$296$	0	0
$297$	−1.66459e11	−1.24138
$298$	0	0
$299$	5.13260e10i	0.371379i
$300$	0	0
$301$	− 5.76474e10i	− 0.404791i
$302$	0	0
$303$	−1.43495e10	−0.0978016
$304$	0	0
$305$	−2.16175e10	−0.143040
$306$	0	0
$307$	1.07930e11i	0.693457i	0.937966	+	0.346729i	$0.112707\pi$
$307$	1.07930e11i	0.693457i	−0.937966	+	0.346729i	$0.887293\pi$
$308$	0	0
$309$	2.74816e11i	1.71486i
$310$	0	0
$311$	−2.31164e11	−1.40120	−0.700598	−	0.713556i	$-0.747083\pi$
$311$	−2.31164e11	−1.40120	−0.700598	+	0.713556i	$0.747083\pi$
$312$	0	0
$313$	−1.20963e11	−0.712367	−0.356183	−	0.934416i	$-0.615922\pi$
$313$	−1.20963e11	−0.712367	−0.356183	+	0.934416i	$0.615922\pi$
$314$	0	0
$315$	− 8.29065e9i	− 0.0474451i
$316$	0	0
$317$	5.78132e10i	0.321559i	0.986990	+	0.160779i	$0.0514008\pi$
$317$	5.78132e10i	0.321559i	−0.986990	+	0.160779i	$0.948599\pi$
$318$	0	0
$319$	−3.55077e11	−1.91984
$320$	0	0
$321$	5.98779e10	0.314770
$322$	0	0
$323$	− 2.05837e11i	− 1.05223i
$324$	0	0
$325$	− 8.40273e10i	− 0.417778i
$326$	0	0
$327$	3.94256e11	1.90684
$328$	0	0
$329$	1.69035e11	0.795416
$330$	0	0
$331$	2.91884e11i	1.33655i	0.743915	+	0.668274i	$0.232967\pi$
$331$	2.91884e11i	1.33655i	−0.743915	+	0.668274i	$0.767033\pi$
$332$	0	0
$333$	4.15213e10i	0.185043i
$334$	0	0
$335$	−6.94462e10	−0.301264
$336$	0	0
$337$	−1.08867e11	−0.459791	−0.229895	−	0.973215i	$-0.573838\pi$
$337$	−1.08867e11	−0.459791	−0.229895	+	0.973215i	$0.573838\pi$
$338$	0	0
$339$	1.88709e11i	0.776059i
$340$	0	0
$341$	− 1.21986e11i	− 0.488558i
$342$	0	0
$343$	1.83220e11	0.714743
$344$	0	0
$345$	5.00609e10	0.190245
$346$	0	0
$347$	− 1.21916e11i	− 0.451416i	−0.974195	−	0.225708i	$-0.927530\pi$
$347$	− 1.21916e11i	− 0.451416i	0.974195	−	0.225708i	$-0.0724696\pi$
$348$	0	0
$349$	2.36404e11i	0.852983i	0.904492	+	0.426491i	$0.140251\pi$
$349$	2.36404e11i	0.852983i	−0.904492	+	0.426491i	$0.859749\pi$
$350$	0	0
$351$	1.13688e11	0.399791
$352$	0	0
$353$	−1.76095e11	−0.603616	−0.301808	−	0.953369i	$-0.597590\pi$
$353$	−1.76095e11	−0.603616	−0.301808	+	0.953369i	$0.597590\pi$
$354$	0	0
$355$	1.85166e10i	0.0618775i
$356$	0	0
$357$	− 7.01006e11i	− 2.28410i
$358$	0	0
$359$	2.58619e11	0.821743	0.410872	−	0.911693i	$-0.365224\pi$
$359$	2.58619e11	0.821743	0.410872	+	0.911693i	$0.365224\pi$
$360$	0	0
$361$	1.30148e11	0.403326
$362$	0	0
$363$	− 2.97426e11i	− 0.899081i
$364$	0	0
$365$	− 8.00985e9i	− 0.0236214i
$366$	0	0
$367$	−2.80806e10	−0.0807997	−0.0403998	−	0.999184i	$-0.512863\pi$
$367$	−2.80806e10	−0.0807997	−0.0403998	+	0.999184i	$0.512863\pi$
$368$	0	0
$369$	−1.55320e10	−0.0436123
$370$	0	0
$371$	− 4.92624e11i	− 1.35000i
$372$	0	0
$373$	− 2.27140e11i	− 0.607582i	−0.952739	−	0.303791i	$-0.901748\pi$
$373$	− 2.27140e11i	− 0.607582i	0.952739	−	0.303791i	$-0.0982524\pi$
$374$	0	0
$375$	−1.67663e11	−0.437820
$376$	0	0
$377$	2.42510e11	0.618292
$378$	0	0
$379$	− 7.70802e10i	− 0.191896i	−0.995386	−	0.0959481i	$-0.969412\pi$
$379$	− 7.70802e10i	− 0.191896i	0.995386	−	0.0959481i	$-0.0305883\pi$
$380$	0	0
$381$	3.62667e10i	0.0881752i
$382$	0	0
$383$	−6.86276e11	−1.62969	−0.814844	−	0.579680i	$-0.803178\pi$
$383$	−6.86276e11	−1.62969	−0.814844	+	0.579680i	$0.803178\pi$
$384$	0	0
$385$	−1.91718e11	−0.444722
$386$	0	0
$387$	1.64953e10i	0.0373818i
$388$	0	0
$389$	3.92836e11i	0.869837i	0.900470	+	0.434919i	$0.143223\pi$
$389$	3.92836e11i	0.869837i	−0.900470	+	0.434919i	$0.856777\pi$
$390$	0	0
$391$	5.35154e11	1.15793
$392$	0	0
$393$	2.03033e11	0.429338
$394$	0	0
$395$	3.53508e10i	0.0730655i
$396$	0	0
$397$	3.27648e11i	0.661988i	0.943633	+	0.330994i	$0.107384\pi$
$397$	3.27648e11i	0.661988i	−0.943633	+	0.330994i	$0.892616\pi$
$398$	0	0
$399$	−6.55719e11	−1.29521
$400$	0	0
$401$	8.62754e11	1.66624	0.833120	−	0.553093i	$-0.186553\pi$
$401$	8.62754e11	1.66624	0.833120	+	0.553093i	$0.186553\pi$
$402$	0	0
$403$	8.33142e10i	0.157343i
$404$	0	0
$405$	− 1.27277e11i	− 0.235073i
$406$	0	0
$407$	9.60162e11	1.73448
$408$	0	0
$409$	−7.29199e11	−1.28852	−0.644260	−	0.764807i	$-0.722834\pi$
$409$	−7.29199e11	−1.28852	−0.644260	+	0.764807i	$0.722834\pi$
$410$	0	0
$411$	9.14959e10i	0.158166i
$412$	0	0
$413$	− 4.68516e11i	− 0.792408i
$414$	0	0
$415$	3.83350e10	0.0634423
$416$	0	0
$417$	−3.20109e11	−0.518425
$418$	0	0
$419$	9.24005e11i	1.46457i	0.680997	+	0.732286i	$0.261547\pi$
$419$	9.24005e11i	1.46457i	−0.680997	+	0.732286i	$0.738453\pi$
$420$	0	0
$421$	9.91623e11i	1.53843i	0.638991	+	0.769214i	$0.279352\pi$
$421$	9.91623e11i	1.53843i	−0.638991	+	0.769214i	$0.720648\pi$
$422$	0	0
$423$	−4.83677e10	−0.0734554
$424$	0	0
$425$	−8.76117e11	−1.30260
$426$	0	0
$427$	7.36175e11i	1.07166i
$428$	0	0
$429$	4.44870e11i	0.634126i
$430$	0	0
$431$	−9.73542e10	−0.135896	−0.0679481	−	0.997689i	$-0.521645\pi$
$431$	−9.73542e10	−0.135896	−0.0679481	+	0.997689i	$0.521645\pi$
$432$	0	0
$433$	1.47830e11	0.202100	0.101050	−	0.994881i	$-0.467780\pi$
$433$	1.47830e11	0.202100	0.101050	+	0.994881i	$0.467780\pi$
$434$	0	0
$435$	− 2.36533e11i	− 0.316730i
$436$	0	0
$437$	− 5.00582e11i	− 0.656611i
$438$	0	0
$439$	−4.93675e11	−0.634381	−0.317191	−	0.948362i	$-0.602740\pi$
$439$	−4.93675e11	−0.634381	−0.317191	+	0.948362i	$0.602740\pi$
$440$	0	0
$441$	−1.67381e11	−0.210732
$442$	0	0
$443$	3.53853e10i	0.0436522i	0.999762	+	0.0218261i	$0.00694801\pi$
$443$	3.53853e10i	0.0436522i	−0.999762	+	0.0218261i	$0.993052\pi$
$444$	0	0
$445$	2.01759e11i	0.243901i
$446$	0	0
$447$	−7.88185e11	−0.933779
$448$	0	0
$449$	−2.44259e11	−0.283623	−0.141812	−	0.989894i	$-0.545293\pi$
$449$	−2.44259e11	−0.283623	−0.141812	+	0.989894i	$0.545293\pi$
$450$	0	0
$451$	3.59171e11i	0.408796i
$452$	0	0
$453$	− 8.64029e11i	− 0.964021i
$454$	0	0
$455$	1.30939e11	0.143225
$456$	0	0
$457$	1.16957e12	1.25431	0.627154	−	0.778895i	$-0.284220\pi$
$457$	1.16957e12	1.25431	0.627154	+	0.778895i	$0.284220\pi$
$458$	0	0
$459$	− 1.18538e12i	− 1.24652i
$460$	0	0
$461$	− 8.92772e11i	− 0.920633i	−0.887755	−	0.460317i	$-0.847736\pi$
$461$	− 8.92772e11i	− 0.920633i	0.887755	−	0.460317i	$-0.152264\pi$
$462$	0	0
$463$	−7.55549e11	−0.764097	−0.382048	−	0.924142i	$-0.624781\pi$
$463$	−7.55549e11	−0.764097	−0.382048	+	0.924142i	$0.624781\pi$
$464$	0	0
$465$	8.12606e10	0.0806012
$466$	0	0
$467$	− 7.06860e11i	− 0.687713i	−0.939022	−	0.343857i	$-0.888267\pi$
$467$	− 7.06860e11i	− 0.687713i	0.939022	−	0.343857i	$-0.111733\pi$
$468$	0	0
$469$	2.36496e12i	2.25707i
$470$	0	0
$471$	2.27133e12	2.12660
$472$	0	0
$473$	3.81446e11	0.350395
$474$	0	0
$475$	8.19517e11i	0.738647i
$476$	0	0
$477$	1.40960e11i	0.124670i
$478$	0	0
$479$	1.32511e12	1.15012	0.575061	−	0.818111i	$-0.304978\pi$
$479$	1.32511e12	1.15012	0.575061	+	0.818111i	$0.304978\pi$
$480$	0	0
$481$	−6.55771e11	−0.558598
$482$	0	0
$483$	− 1.70480e12i	− 1.42532i
$484$	0	0
$485$	3.44607e10i	0.0282805i
$486$	0	0
$487$	−1.69922e12	−1.36889	−0.684446	−	0.729064i	$-0.739956\pi$
$487$	−1.69922e12	−1.36889	−0.684446	+	0.729064i	$0.739956\pi$
$488$	0	0
$489$	1.50627e12	1.19128
$490$	0	0
$491$	− 1.61014e12i	− 1.25025i	−0.780525	−	0.625125i	$-0.785048\pi$
$491$	− 1.61014e12i	− 1.25025i	0.780525	−	0.625125i	$-0.214952\pi$
$492$	0	0
$493$	− 2.52855e12i	− 1.92779i
$494$	0	0
$495$	5.48582e10	0.0410694
$496$	0	0
$497$	6.30572e11	0.463587
$498$	0	0
$499$	1.12530e12i	0.812489i	0.913764	+	0.406244i	$0.133162\pi$
$499$	1.12530e12i	0.812489i	−0.913764	+	0.406244i	$0.866838\pi$
$500$	0	0
$501$	− 1.37028e12i	− 0.971718i
$502$	0	0
$503$	3.27201e11	0.227908	0.113954	−	0.993486i	$-0.463648\pi$
$503$	3.27201e11	0.227908	0.113954	+	0.993486i	$0.463648\pi$
$504$	0	0
$505$	−2.79465e10	−0.0191213
$506$	0	0
$507$	1.28796e12i	0.865696i
$508$	0	0
$509$	− 1.78571e12i	− 1.17918i	−0.807701	−	0.589592i	$-0.799289\pi$
$509$	− 1.78571e12i	− 1.17918i	0.807701	−	0.589592i	$-0.200711\pi$
$510$	0	0
$511$	−2.72772e11	−0.176972
$512$	0	0
$513$	−1.10880e12	−0.706845
$514$	0	0
$515$	5.35220e11i	0.335274i
$516$	0	0
$517$	1.11848e12i	0.688528i
$518$	0	0
$519$	−9.72637e11	−0.588434
$520$	0	0
$521$	−2.62743e12	−1.56229	−0.781145	−	0.624350i	$-0.785364\pi$
$521$	−2.62743e12	−1.56229	−0.781145	+	0.624350i	$0.785364\pi$
$522$	0	0
$523$	− 1.74460e12i	− 1.01962i	−0.860288	−	0.509808i	$-0.829716\pi$
$523$	− 1.74460e12i	− 1.01962i	0.860288	−	0.509808i	$-0.170284\pi$
$524$	0	0
$525$	2.79098e12i	1.60339i
$526$	0	0
$527$	8.68681e11	0.490583
$528$	0	0
$529$	−4.99693e11	−0.277430
$530$	0	0
$531$	1.34061e11i	0.0731777i
$532$	0	0
$533$	− 2.45307e11i	− 0.131655i
$534$	0	0
$535$	1.16616e11	0.0615410
$536$	0	0
$537$	2.05746e11	0.106770
$538$	0	0
$539$	3.87060e12i	1.97528i
$540$	0	0
$541$	4.60286e11i	0.231015i	0.993307	+	0.115508i	$0.0368495\pi$
$541$	4.60286e11i	0.231015i	−0.993307	+	0.115508i	$0.963151\pi$
$542$	0	0
$543$	2.78788e12	1.37618
$544$	0	0
$545$	7.67836e11	0.372807
$546$	0	0
$547$	− 5.60408e11i	− 0.267646i	−0.991005	−	0.133823i	$-0.957275\pi$
$547$	− 5.60408e11i	− 0.267646i	0.991005	−	0.133823i	$-0.0427254\pi$
$548$	0	0
$549$	− 2.10650e11i	− 0.0989658i
$550$	0	0
$551$	−2.36520e12	−1.09316
$552$	0	0
$553$	1.20386e12	0.547408
$554$	0	0
$555$	6.39607e11i	0.286151i
$556$	0	0
$557$	3.67343e12i	1.61705i	0.588462	+	0.808525i	$0.299734\pi$
$557$	3.67343e12i	1.61705i	−0.588462	+	0.808525i	$0.700266\pi$
$558$	0	0
$559$	−2.60520e11	−0.112846
$560$	0	0
$561$	4.63847e12	1.97716
$562$	0	0
$563$	− 7.28222e11i	− 0.305475i	−0.988267	−	0.152738i	$-0.951191\pi$
$563$	− 7.28222e11i	− 0.305475i	0.988267	−	0.152738i	$-0.0488089\pi$
$564$	0	0
$565$	3.67522e11i	0.151728i
$566$	0	0
$567$	−4.33437e12	−1.76117
$568$	0	0
$569$	−3.21622e12	−1.28629	−0.643147	−	0.765742i	$-0.722372\pi$
$569$	−3.21622e12	−1.28629	−0.643147	+	0.765742i	$0.722372\pi$
$570$	0	0
$571$	− 9.17863e11i	− 0.361339i	−0.983544	−	0.180670i	$-0.942174\pi$
$571$	− 9.17863e11i	− 0.361339i	0.983544	−	0.180670i	$-0.0578265\pi$
$572$	0	0
$573$	2.61138e12i	1.01199i
$574$	0	0
$575$	−2.13066e12	−0.812846
$576$	0	0
$577$	−3.29707e12	−1.23833	−0.619167	−	0.785260i	$-0.712529\pi$
$577$	−3.29707e12	−1.23833	−0.619167	+	0.785260i	$0.712529\pi$
$578$	0	0
$579$	4.68262e12i	1.73155i
$580$	0	0
$581$	− 1.30548e12i	− 0.475311i
$582$	0	0
$583$	3.25963e12	1.16858
$584$	0	0
$585$	−3.74671e10	−0.0132266
$586$	0	0
$587$	− 4.56218e12i	− 1.58599i	−0.609227	−	0.792996i	$-0.708520\pi$
$587$	− 4.56218e12i	− 1.58599i	0.609227	−	0.792996i	$-0.291480\pi$
$588$	0	0
$589$	− 8.12562e11i	− 0.278187i
$590$	0	0
$591$	2.80151e12	0.944602
$592$	0	0
$593$	3.34603e12	1.11118	0.555590	−	0.831457i	$-0.312493\pi$
$593$	3.34603e12	1.11118	0.555590	+	0.831457i	$0.312493\pi$
$594$	0	0
$595$	− 1.36525e12i	− 0.446565i
$596$	0	0
$597$	− 4.91546e12i	− 1.58373i
$598$	0	0
$599$	−3.92344e12	−1.24522	−0.622610	−	0.782532i	$-0.713928\pi$
$599$	−3.92344e12	−1.24522	−0.622610	+	0.782532i	$0.713928\pi$
$600$	0	0
$601$	−1.68334e12	−0.526305	−0.263153	−	0.964754i	$-0.584762\pi$
$601$	−1.68334e12	−0.526305	−0.263153	+	0.964754i	$0.584762\pi$
$602$	0	0
$603$	− 6.76710e11i	− 0.208437i
$604$	0	0
$605$	− 5.79253e11i	− 0.175780i
$606$	0	0
$607$	4.88789e12	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$607$	4.88789e12	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$608$	0	0
$609$	−8.05501e12	−2.37295
$610$	0	0
$611$	− 7.63900e11i	− 0.221744i
$612$	0	0
$613$	− 3.80258e12i	− 1.08769i	−0.839185	−	0.543846i	$-0.816967\pi$
$613$	− 3.80258e12i	− 1.08769i	0.839185	−	0.543846i	$-0.183033\pi$
$614$	0	0
$615$	−2.39260e11	−0.0674423
$616$	0	0
$617$	7.15842e12	1.98854	0.994270	−	0.106902i	$-0.0340932\pi$
$617$	7.15842e12	1.98854	0.994270	+	0.106902i	$0.0340932\pi$
$618$	0	0
$619$	4.99966e12i	1.36878i	0.729118	+	0.684388i	$0.239930\pi$
$619$	4.99966e12i	1.36878i	−0.729118	+	0.684388i	$0.760070\pi$
$620$	0	0
$621$	− 2.88276e12i	− 0.777850i
$622$	0	0
$623$	6.87082e12	1.82731
$624$	0	0
$625$	3.32125e12	0.870645
$626$	0	0
$627$	− 4.33881e12i	− 1.12116i
$628$	0	0
$629$	6.83744e12i	1.74167i
$630$	0	0
$631$	−3.31885e12	−0.833403	−0.416702	−	0.909043i	$-0.636814\pi$
$631$	−3.31885e12	−0.833403	−0.416702	+	0.909043i	$0.636814\pi$
$632$	0	0
$633$	9.95840e11	0.246532
$634$	0	0
$635$	7.06315e10i	0.0172392i
$636$	0	0
$637$	− 2.64354e12i	− 0.636149i
$638$	0	0
$639$	−1.80432e11	−0.0428115
$640$	0	0
$641$	−6.76534e11	−0.158281	−0.0791404	−	0.996863i	$-0.525218\pi$
$641$	−6.76534e11	−0.158281	−0.0791404	+	0.996863i	$0.525218\pi$
$642$	0	0
$643$	− 2.51359e12i	− 0.579889i	−0.957044	−	0.289944i	$-0.906363\pi$
$643$	− 2.51359e12i	− 0.579889i	0.957044	−	0.289944i	$-0.0936368\pi$
$644$	0	0
$645$	2.54098e11i	0.0578073i
$646$	0	0
$647$	−1.44698e12	−0.324634	−0.162317	−	0.986739i	$-0.551897\pi$
$647$	−1.44698e12	−0.324634	−0.162317	+	0.986739i	$0.551897\pi$
$648$	0	0
$649$	3.10011e12	0.685924
$650$	0	0
$651$	− 2.76729e12i	− 0.603866i
$652$	0	0
$653$	− 3.39613e12i	− 0.730928i	−0.930825	−	0.365464i	$-0.880910\pi$
$653$	− 3.39613e12i	− 0.730928i	0.930825	−	0.365464i	$-0.119090\pi$
$654$	0	0
$655$	3.95418e11	0.0839402
$656$	0	0
$657$	7.80510e10	0.0163431
$658$	0	0
$659$	2.73212e12i	0.564308i	0.959369	+	0.282154i	$0.0910489\pi$
$659$	2.73212e12i	0.564308i	−0.959369	+	0.282154i	$0.908951\pi$
$660$	0	0
$661$	− 3.29057e12i	− 0.670448i	−0.942138	−	0.335224i	$-0.891188\pi$
$661$	− 3.29057e12i	− 0.670448i	0.942138	−	0.335224i	$-0.108812\pi$
$662$	0	0
$663$	−3.16798e12	−0.636754
$664$	0	0
$665$	−1.27705e12	−0.253227
$666$	0	0
$667$	− 6.14926e12i	− 1.20298i
$668$	0	0
$669$	9.52948e12i	1.83930i
$670$	0	0
$671$	−4.87118e12	−0.927647
$672$	0	0
$673$	3.00500e11	0.0564646	0.0282323	−	0.999601i	$-0.491012\pi$
$673$	3.00500e11	0.0564646	0.0282323	+	0.999601i	$0.491012\pi$
$674$	0	0
$675$	4.71945e12i	0.875033i
$676$	0	0
$677$	8.63566e12i	1.57996i	0.613132	+	0.789981i	$0.289909\pi$
$677$	8.63566e12i	1.57996i	−0.613132	+	0.789981i	$0.710091\pi$
$678$	0	0
$679$	1.17354e12	0.211878
$680$	0	0
$681$	−7.64889e12	−1.36281
$682$	0	0
$683$	4.86440e12i	0.855335i	0.903936	+	0.427668i	$0.140665\pi$
$683$	4.86440e12i	0.855335i	−0.903936	+	0.427668i	$0.859335\pi$
$684$	0	0
$685$	1.78193e11i	0.0309232i
$686$	0	0
$687$	−2.30742e12	−0.395204
$688$	0	0
$689$	−2.22626e12	−0.376348
$690$	0	0
$691$	− 8.78244e11i	− 0.146543i	−0.997312	−	0.0732713i	$-0.976656\pi$
$691$	− 8.78244e11i	− 0.146543i	0.997312	−	0.0732713i	$-0.0233439\pi$
$692$	0	0
$693$	− 1.86817e12i	− 0.307693i
$694$	0	0
$695$	−6.23430e11	−0.101358
$696$	0	0
$697$	−2.55771e12	−0.410490
$698$	0	0
$699$	8.68822e12i	1.37652i
$700$	0	0
$701$	− 3.76755e12i	− 0.589288i	−0.955607	−	0.294644i	$-0.904799\pi$
$701$	− 3.76755e12i	− 0.589288i	0.955607	−	0.294644i	$-0.0952011\pi$
$702$	0	0
$703$	6.39573e12	0.987622
$704$	0	0
$705$	−7.45071e11	−0.113592
$706$	0	0
$707$	9.51706e11i	0.143257i
$708$	0	0
$709$	6.49276e11i	0.0964986i	0.998835	+	0.0482493i	$0.0153642\pi$
$709$	6.49276e11i	0.0964986i	−0.998835	+	0.0482493i	$0.984636\pi$
$710$	0	0
$711$	−3.44472e11	−0.0505523
$712$	0	0
$713$	2.11257e12	0.306132
$714$	0	0
$715$	8.66409e11i	0.123978i
$716$	0	0
$717$	8.17893e11i	0.115574i
$718$	0	0
$719$	3.35865e12	0.468689	0.234344	−	0.972154i	$-0.424706\pi$
$719$	3.35865e12	0.468689	0.234344	+	0.972154i	$0.424706\pi$
$720$	0	0
$721$	1.82267e13	2.51188
$722$	0	0
$723$	5.03976e12i	0.685942i
$724$	0	0
$725$	1.00671e13i	1.35327i
$726$	0	0
$727$	−7.42904e11	−0.0986342	−0.0493171	−	0.998783i	$-0.515704\pi$
$727$	−7.42904e11	−0.0986342	−0.0493171	+	0.998783i	$0.515704\pi$
$728$	0	0
$729$	−6.22563e12	−0.816413
$730$	0	0
$731$	2.71633e12i	0.351847i
$732$	0	0
$733$	− 1.31480e13i	− 1.68226i	−0.540832	−	0.841130i	$-0.681891\pi$
$733$	− 1.31480e13i	− 1.68226i	0.540832	−	0.841130i	$-0.318109\pi$
$734$	0	0
$735$	−2.57838e12	−0.325877
$736$	0	0
$737$	−1.56486e13	−1.95377
$738$	0	0
$739$	− 4.12361e12i	− 0.508602i	−0.967125	−	0.254301i	$-0.918155\pi$
$739$	− 4.12361e12i	− 0.508602i	0.967125	−	0.254301i	$-0.0818454\pi$
$740$	0	0
$741$	2.96332e12i	0.361074i
$742$	0	0
$743$	8.36687e11	0.100719	0.0503597	−	0.998731i	$-0.483963\pi$
$743$	8.36687e11	0.100719	0.0503597	+	0.998731i	$0.483963\pi$
$744$	0	0
$745$	−1.53503e12	−0.182564
$746$	0	0
$747$	3.73551e11i	0.0438942i
$748$	0	0
$749$	− 3.97129e12i	− 0.461066i
$750$	0	0
$751$	−5.28606e12	−0.606390	−0.303195	−	0.952928i	$-0.598053\pi$
$751$	−5.28606e12	−0.606390	−0.303195	+	0.952928i	$0.598053\pi$
$752$	0	0
$753$	1.21864e13	1.38133
$754$	0	0
$755$	− 1.68274e12i	− 0.188476i
$756$	0	0
$757$	1.04605e12i	0.115777i	0.998323	+	0.0578884i	$0.0184368\pi$
$757$	1.04605e12i	0.115777i	−0.998323	+	0.0578884i	$0.981563\pi$
$758$	0	0
$759$	1.12804e13	1.23378
$760$	0	0
$761$	−6.96339e11	−0.0752644	−0.0376322	−	0.999292i	$-0.511982\pi$
$761$	−6.96339e11	−0.0752644	−0.0376322	+	0.999292i	$0.511982\pi$
$762$	0	0
$763$	− 2.61483e13i	− 2.79308i
$764$	0	0
$765$	3.90653e11i	0.0412396i
$766$	0	0
$767$	−2.11731e12	−0.220905
$768$	0	0
$769$	−1.48499e13	−1.53128	−0.765642	−	0.643267i	$-0.777579\pi$
$769$	−1.48499e13	−1.53128	−0.765642	+	0.643267i	$0.777579\pi$
$770$	0	0
$771$	− 1.78181e13i	− 1.81601i
$772$	0	0
$773$	− 1.26481e13i	− 1.27414i	−0.770804	−	0.637072i	$-0.780145\pi$
$773$	− 1.26481e13i	− 1.27414i	0.770804	−	0.637072i	$-0.219855\pi$
$774$	0	0
$775$	−3.45856e12	−0.344380
$776$	0	0
$777$	2.17815e13	2.14385
$778$	0	0
$779$	2.39247e12i	0.232771i
$780$	0	0
$781$	4.17242e12i	0.401290i
$782$	0	0
$783$	−1.36207e13	−1.29501
$784$	0	0
$785$	4.42353e12	0.415773
$786$	0	0
$787$	1.55699e13i	1.44677i	0.690446	+	0.723383i	$0.257414\pi$
$787$	1.55699e13i	1.44677i	−0.690446	+	0.723383i	$0.742586\pi$
$788$	0	0
$789$	− 2.24582e12i	− 0.206314i
$790$	0	0
$791$	1.25158e13	1.13675
$792$	0	0
$793$	3.32691e12	0.298753
$794$	0	0
$795$	2.17139e12i	0.192790i
$796$	0	0
$797$	1.00331e12i	0.0880792i	0.999030	+	0.0440396i	$0.0140228\pi$
$797$	1.00331e12i	0.0880792i	−0.999030	+	0.0440396i	$0.985977\pi$
$798$	0	0
$799$	−7.96486e12	−0.691381
$800$	0	0
$801$	−1.96602e12	−0.168749
$802$	0	0
$803$	− 1.80490e12i	− 0.153191i
$804$	0	0
$805$	− 3.32019e12i	− 0.278665i
$806$	0	0
$807$	9.90784e12	0.822333
$808$	0	0
$809$	1.16269e13	0.954327	0.477164	−	0.878815i	$-0.341665\pi$
$809$	1.16269e13	0.954327	0.477164	+	0.878815i	$0.341665\pi$
$810$	0	0
$811$	1.45881e13i	1.18415i	0.805884	+	0.592074i	$0.201691\pi$
$811$	1.45881e13i	1.18415i	−0.805884	+	0.592074i	$0.798309\pi$
$812$	0	0
$813$	8.95427e12i	0.718824i
$814$	0	0
$815$	2.93354e12	0.232907
$816$	0	0
$817$	2.54085e12	0.199517
$818$	0	0
$819$	1.27592e12i	0.0990939i
$820$	0	0
$821$	− 2.01173e13i	− 1.54534i	−0.634805	−	0.772672i	$-0.718920\pi$
$821$	− 2.01173e13i	− 1.54534i	0.634805	−	0.772672i	$-0.281080\pi$
$822$	0	0
$823$	1.80557e13	1.37188	0.685939	−	0.727659i	$-0.259392\pi$
$823$	1.80557e13	1.37188	0.685939	+	0.727659i	$0.259392\pi$
$824$	0	0
$825$	−1.84676e13	−1.38793
$826$	0	0
$827$	− 5.41935e12i	− 0.402877i	−0.979501	−	0.201438i	$-0.935438\pi$
$827$	− 5.41935e12i	− 0.402877i	0.979501	−	0.201438i	$-0.0645616\pi$
$828$	0	0
$829$	1.93367e13i	1.42196i	0.703213	+	0.710980i	$0.251748\pi$
$829$	1.93367e13i	1.42196i	−0.703213	+	0.710980i	$0.748252\pi$
$830$	0	0
$831$	1.31625e13	0.957492
$832$	0	0
$833$	−2.75631e13	−1.98347
$834$	0	0
$835$	− 2.66870e12i	− 0.189981i
$836$	0	0
$837$	− 4.67939e12i	− 0.329553i
$838$	0	0
$839$	−1.70460e13	−1.18766	−0.593831	−	0.804590i	$-0.702385\pi$
$839$	−1.70460e13	−1.18766	−0.593831	+	0.804590i	$0.702385\pi$
$840$	0	0
$841$	−1.45475e13	−1.00278
$842$	0	0
$843$	3.54852e12i	0.242004i
$844$	0	0
$845$	2.50837e12i	0.169253i
$846$	0	0
$847$	−1.97262e13	−1.31695
$848$	0	0
$849$	1.90799e13	1.26035
$850$	0	0
$851$	1.66282e13i	1.08683i
$852$	0	0
$853$	1.12979e13i	0.730677i	0.930875	+	0.365339i	$0.119047\pi$
$853$	1.12979e13i	0.730677i	−0.930875	+	0.365339i	$0.880953\pi$
$854$	0	0
$855$	3.65416e11	0.0233851
$856$	0	0
$857$	−2.94210e12	−0.186313	−0.0931567	−	0.995651i	$-0.529696\pi$
$857$	−2.94210e12	−0.186313	−0.0931567	+	0.995651i	$0.529696\pi$
$858$	0	0
$859$	7.27946e12i	0.456173i	0.973641	+	0.228087i	$0.0732470\pi$
$859$	7.27946e12i	0.456173i	−0.973641	+	0.228087i	$0.926753\pi$
$860$	0	0
$861$	8.14789e12i	0.505278i
$862$	0	0
$863$	−2.79331e13	−1.71424	−0.857118	−	0.515120i	$-0.827747\pi$
$863$	−2.79331e13	−1.71424	−0.857118	+	0.515120i	$0.827747\pi$
$864$	0	0
$865$	−1.89427e12	−0.115045
$866$	0	0
$867$	1.52305e13i	0.915434i
$868$	0	0
$869$	7.96576e12i	0.473847i
$870$	0	0
$871$	1.06877e13	0.629220
$872$	0	0
$873$	−3.35798e11	−0.0195666
$874$	0	0
$875$	1.11199e13i	0.641305i
$876$	0	0
$877$	− 7.87352e12i	− 0.449439i	−0.974424	−	0.224719i	$-0.927853\pi$
$877$	− 7.87352e12i	− 0.449439i	0.974424	−	0.224719i	$-0.0721465\pi$
$878$	0	0
$879$	−2.84773e13	−1.60897
$880$	0	0
$881$	2.87110e13	1.60567	0.802837	−	0.596198i	$-0.203323\pi$
$881$	2.87110e13	1.60567	0.802837	+	0.596198i	$0.203323\pi$
$882$	0	0
$883$	− 2.78143e13i	− 1.53973i	−0.638207	−	0.769865i	$-0.720324\pi$
$883$	− 2.78143e13i	− 1.53973i	0.638207	−	0.769865i	$-0.279676\pi$
$884$	0	0
$885$	2.06512e12i	0.113162i
$886$	0	0
$887$	2.72103e13	1.47597	0.737985	−	0.674817i	$-0.235777\pi$
$887$	2.72103e13	1.47597	0.737985	+	0.674817i	$0.235777\pi$
$888$	0	0
$889$	2.40532e12	0.129156
$890$	0	0
$891$	− 2.86800e13i	− 1.52451i
$892$	0	0
$893$	7.45030e12i	0.392051i
$894$	0	0
$895$	4.00703e11	0.0208746
$896$	0	0
$897$	−7.70432e12	−0.397345
$898$	0	0
$899$	− 9.98170e12i	− 0.509666i
$900$	0	0
$901$	2.32123e13i	1.17343i
$902$	0	0
$903$	8.65320e12	0.433094
$904$	0	0
$905$	5.42955e12	0.269058
$906$	0	0
$907$	2.32280e13i	1.13967i	0.821760	+	0.569834i	$0.192993\pi$
$907$	2.32280e13i	1.13967i	−0.821760	+	0.569834i	$0.807007\pi$
$908$	0	0
$909$	− 2.72322e11i	− 0.0132295i
$910$	0	0
$911$	3.33123e13	1.60240	0.801201	−	0.598396i	$-0.204195\pi$
$911$	3.33123e13	1.60240	0.801201	+	0.598396i	$0.204195\pi$
$912$	0	0
$913$	8.63820e12	0.411438
$914$	0	0
$915$	− 3.24491e12i	− 0.153041i
$916$	0	0
$917$	− 1.34658e13i	− 0.628882i
$918$	0	0
$919$	−1.19970e13	−0.554822	−0.277411	−	0.960751i	$-0.589476\pi$
$919$	−1.19970e13	−0.554822	−0.277411	+	0.960751i	$0.589476\pi$
$920$	0	0
$921$	−1.62009e13	−0.741943
$922$	0	0
$923$	− 2.84968e12i	− 0.129237i
$924$	0	0
$925$	− 2.72225e13i	− 1.22262i
$926$	0	0
$927$	−5.21539e12	−0.231968
$928$	0	0
$929$	2.87066e11	0.0126448	0.00632238	−	0.999980i	$-0.497988\pi$
$929$	2.87066e11	0.0126448	0.00632238	+	0.999980i	$0.497988\pi$
$930$	0	0
$931$	2.57824e13i	1.12473i
$932$	0	0
$933$	− 3.46990e13i	− 1.49917i
$934$	0	0
$935$	9.03367e12	0.386556
$936$	0	0
$937$	4.23141e12	0.179332	0.0896659	−	0.995972i	$-0.471420\pi$
$937$	4.23141e12	0.179332	0.0896659	+	0.995972i	$0.471420\pi$
$938$	0	0
$939$	− 1.81572e13i	− 0.762175i
$940$	0	0
$941$	− 6.44282e10i	− 0.00267869i	−0.999999	−	0.00133935i	$-0.999574\pi$
$941$	− 6.44282e10i	− 0.00267869i	0.999999	−	0.00133935i	$-0.000426327\pi$
$942$	0	0
$943$	−6.22017e12	−0.256153
$944$	0	0
$945$	−7.35429e12	−0.299983
$946$	0	0
$947$	− 4.05102e13i	− 1.63678i	−0.574667	−	0.818388i	$-0.694868\pi$
$947$	− 4.05102e13i	− 1.63678i	0.574667	−	0.818388i	$-0.305132\pi$
$948$	0	0
$949$	1.23271e12i	0.0493358i
$950$	0	0
$951$	−8.67808e12	−0.344042
$952$	0	0
$953$	8.30603e11	0.0326194	0.0163097	−	0.999867i	$-0.494808\pi$
$953$	8.30603e11	0.0326194	0.0163097	+	0.999867i	$0.494808\pi$
$954$	0	0
$955$	5.08581e12i	0.197854i
$956$	0	0
$957$	− 5.32990e13i	− 2.05407i
$958$	0	0
$959$	6.06829e12	0.231677
$960$	0	0
$961$	−2.30104e13	−0.870300
$962$	0	0
$963$	1.13635e12i	0.0425787i
$964$	0	0
$965$	9.11966e12i	0.338537i
$966$	0	0
$967$	2.92619e13	1.07618	0.538088	−	0.842889i	$-0.319147\pi$
$967$	2.92619e13	1.07618	0.538088	+	0.842889i	$0.319147\pi$
$968$	0	0
$969$	3.08972e13	1.12580
$970$	0	0
$971$	− 4.12204e13i	− 1.48808i	−0.668136	−	0.744039i	$-0.732907\pi$
$971$	− 4.12204e13i	− 1.48808i	0.668136	−	0.744039i	$-0.267093\pi$
$972$	0	0
$973$	2.12306e13i	0.759373i
$974$	0	0
$975$	1.26130e13	0.446989
$976$	0	0
$977$	6.53429e12	0.229442	0.114721	−	0.993398i	$-0.463403\pi$
$977$	6.53429e12	0.229442	0.114721	+	0.993398i	$0.463403\pi$
$978$	0	0
$979$	4.54634e13i	1.58176i
$980$	0	0
$981$	7.48209e12i	0.257936i
$982$	0	0
$983$	−4.98970e13	−1.70445	−0.852224	−	0.523178i	$-0.824746\pi$
$983$	−4.98970e13	−1.70445	−0.852224	+	0.523178i	$0.824746\pi$
$984$	0	0
$985$	5.45610e12	0.184680
$986$	0	0
$987$	2.53730e13i	0.851031i
$988$	0	0
$989$	6.60593e12i	0.219559i
$990$	0	0
$991$	−3.03676e13	−1.00018	−0.500090	−	0.865973i	$-0.666700\pi$
$991$	−3.03676e13	−1.00018	−0.500090	+	0.865973i	$0.666700\pi$
$992$	0	0
$993$	−4.38134e13	−1.43000
$994$	0	0
$995$	− 9.57314e12i	− 0.309635i
$996$	0	0
$997$	3.60157e13i	1.15442i	0.816596	+	0.577209i	$0.195858\pi$
$997$	3.60157e13i	1.15442i	−0.816596	+	0.577209i	$0.804142\pi$
$998$	0	0
$999$	3.68318e13	1.16998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.10.b.a.17.7		8
3.2	odd	2		288.10.d.b.145.4		8
4.3	odd	2		8.10.b.a.5.3	✓	8
8.3	odd	2		8.10.b.a.5.4	yes	8
8.5	even	2	inner	32.10.b.a.17.2		8
12.11	even	2		72.10.d.b.37.6		8
16.3	odd	4		256.10.a.p.1.7		8
16.5	even	4		256.10.a.s.1.7		8
16.11	odd	4		256.10.a.p.1.2		8
16.13	even	4		256.10.a.s.1.2		8
24.5	odd	2		288.10.d.b.145.5		8
24.11	even	2		72.10.d.b.37.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.b.a.5.3	✓	8	4.3	odd	2
8.10.b.a.5.4	yes	8	8.3	odd	2
32.10.b.a.17.2		8	8.5	even	2	inner
32.10.b.a.17.7		8	1.1	even	1	trivial
72.10.d.b.37.5		8	24.11	even	2
72.10.d.b.37.6		8	12.11	even	2
256.10.a.p.1.2		8	16.11	odd	4
256.10.a.p.1.7		8	16.3	odd	4
256.10.a.s.1.2		8	16.13	even	4
256.10.a.s.1.7		8	16.5	even	4
288.10.d.b.145.4		8	3.2	odd	2
288.10.d.b.145.5		8	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+150.106i q^{3} +292.339i q^{5} +9955.46 q^{7} -2848.66 q^{9} +O(q^{10})\)	\(q+150.106i q^{3} +292.339i q^{5} +9955.46 q^{7} -2848.66 q^{9} +65874.1i q^{11} -44990.6i q^{13} -43881.7 q^{15} -469098. q^{17} +438793. i q^{19} +1.49437e6i q^{21} -1.14082e6 q^{23} +1.86766e6 q^{25} +2.52693e6i q^{27} +5.39024e6i q^{29} -1.85181e6 q^{31} -9.88806e6 q^{33} +2.91037e6i q^{35} -1.45757e7i q^{37} +6.75334e6 q^{39} +5.45239e6 q^{41} -5.79053e6i q^{43} -832774. i q^{45} +1.69791e7 q^{47} +5.87576e7 q^{49} -7.04142e7i q^{51} -4.94827e7i q^{53} -1.92575e7 q^{55} -6.58652e7 q^{57} -4.70612e7i q^{59} +7.39468e7i q^{61} -2.83597e7 q^{63} +1.31525e7 q^{65} +2.37554e8i q^{67} -1.71243e8i q^{69} +6.33394e7 q^{71} -2.73992e7 q^{73} +2.80346e8i q^{75} +6.55807e8i q^{77} +1.20924e8 q^{79} -4.35376e8 q^{81} -1.31132e8i q^{83} -1.37136e8i q^{85} -8.09104e8 q^{87} +6.90156e8 q^{89} -4.47903e8i q^{91} -2.77967e8i q^{93} -1.28276e8 q^{95} +1.17879e8 q^{97} -1.87653e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4800 q^{7} - 39368 q^{9}+O(q^{10}) \) 8 * q - 4800 * q^7 - 39368 * q^9	\( 8 q - 4800 q^{7} - 39368 q^{9} + 163136 q^{15} - 102000 q^{17} - 3412032 q^{23} - 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} + 17590208 q^{39} - 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} - 7056832 q^{55} + 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} - 560234688 q^{71} - 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} - 540527424 q^{87} + 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) \) 8 * q - 4800 * q^7 - 39368 * q^9 + 163136 * q^15 - 102000 * q^17 - 3412032 * q^23 - 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 + 17590208 * q^39 - 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 - 7056832 * q^55 + 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 - 560234688 * q^71 - 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 - 540527424 * q^87 + 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

Introduction

L-functions

Modular forms

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Database

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