LMFDB - Embedded newform 32.18.b.a.17.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,18,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, 18, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 18 $
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:	$58.6310679503$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + \cdots + 21\!\cdots\!00 $ x^16 - 8x^15 + 83403052x^14 - 583821224x^13 + 2709232900692534x^12 - 16255389814479656x^11 + 44023334982114936772524x^10 - 220116525902848632203992x^9 + 375637035261256983364255923921x^8 - 1502546820346051325629336482512x^7 + 1580449203685347736142710540597573024x^6 - 4741342352143096486478474620456939520x^5 + 2671768262327077521053591460599154325518592x^4 - 5343528622418654441129876221219442847805440x^3 + 1662520959198696999402497079485716800591945908224x^2 - 1662518287435176013550209293272621607067374059520*x + 210991858777295491315579293406297919761107107446784000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{240}\cdot 3^{14}\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.9
Root			$0.500000 - 409.736i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.18.b.a.17.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1638.94i q^{3} -1.21254e6i q^{5} -1.76580e7 q^{7} +1.26454e8 q^{9} +O(q^{10})$	$q+1638.94i q^{3} -1.21254e6i q^{5} -1.76580e7 q^{7} +1.26454e8 q^{9} +7.26817e8i q^{11} +3.16645e9i q^{13} +1.98729e9 q^{15} +4.56030e10 q^{17} -1.55789e10i q^{19} -2.89404e10i q^{21} -4.09250e11 q^{23} -7.07318e11 q^{25} +4.18904e11i q^{27} -3.11338e12i q^{29} +6.57077e12 q^{31} -1.19121e12 q^{33} +2.14111e13i q^{35} -1.36898e13i q^{37} -5.18964e12 q^{39} -2.81830e12 q^{41} -1.36024e13i q^{43} -1.53331e14i q^{45} -6.84132e11 q^{47} +7.91743e13 q^{49} +7.47407e13i q^{51} +2.47869e14i q^{53} +8.81297e14 q^{55} +2.55329e13 q^{57} -2.03296e15i q^{59} -8.92495e14i q^{61} -2.23292e15 q^{63} +3.83946e15 q^{65} -5.06693e15i q^{67} -6.70737e14i q^{69} +6.85113e15 q^{71} +1.37250e15 q^{73} -1.15925e15i q^{75} -1.28341e16i q^{77} +1.78556e16 q^{79} +1.56437e16 q^{81} +1.48017e15i q^{83} -5.52955e16i q^{85} +5.10265e15 q^{87} -3.78700e16 q^{89} -5.59132e16i q^{91} +1.07691e16i q^{93} -1.88900e16 q^{95} +1.28098e17 q^{97} +9.19090e16i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) $ 16 * q - 11529600 * q^7 - 602654096 * q^9	$ 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) $ 16 * q - 11529600 * q^7 - 602654096 * q^9 + 9993282176 * q^15 - 7489125600 * q^17 - 746845345920 * q^23 - 1809682431664 * q^25 + 318979758592 * q^31 + 5633526177600 * q^33 + 18457706051456 * q^39 + 7482251536032 * q^41 + 376698804821760 * q^47 + 127691292101520 * q^49 - 2209036687713152 * q^55 - 190521298294720 * q^57 + 8131096607338880 * q^63 + 2385987975356160 * q^65 - 9025926285576576 * q^71 + 11332002046118560 * q^73 + 45299671392008448 * q^79 + 20101901999290832 * q^81 - 25965768920837760 * q^87 - 69879174608766048 * q^89 - 93790444358203776 * q^95 + 95593398602180640 * 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
$3$	1638.94i	0.144223i	0.997397	+	0.0721113i	$0.0229737\pi$
$3$	1638.94i	0.144223i	−0.997397	+	0.0721113i	$0.977026\pi$
$4$	0	0
$5$	− 1.21254e6i	− 1.38820i	−0.719879	−	0.694100i	$-0.755803\pi$
$5$	− 1.21254e6i	− 1.38820i	0.719879	−	0.694100i	$-0.244197\pi$
$6$	0	0
$7$	−1.76580e7	−1.15773	−0.578866	−	0.815423i	$-0.696505\pi$
$7$	−1.76580e7	−1.15773	−0.578866	+	0.815423i	$0.696505\pi$
$8$	0	0
$9$	1.26454e8	0.979200
$10$	0	0
$11$	7.26817e8i	1.02232i	0.859485	+	0.511161i	$0.170784\pi$
$11$	7.26817e8i	1.02232i	−0.859485	+	0.511161i	$0.829216\pi$
$12$	0	0
$13$	3.16645e9i	1.07660i	0.842753	+	0.538300i	$0.180933\pi$
$13$	3.16645e9i	1.07660i	−0.842753	+	0.538300i	$0.819067\pi$
$14$	0	0
$15$	1.98729e9	0.200210
$16$	0	0
$17$	4.56030e10	1.58554	0.792771	−	0.609520i	$-0.208638\pi$
$17$	4.56030e10	1.58554	0.792771	+	0.609520i	$0.208638\pi$
$18$	0	0
$19$	− 1.55789e10i	− 0.210441i	−0.994449	−	0.105220i	$-0.966445\pi$
$19$	− 1.55789e10i	− 0.210441i	0.994449	−	0.105220i	$-0.0335548\pi$
$20$	0	0
$21$	− 2.89404e10i	− 0.166971i
$22$	0	0
$23$	−4.09250e11	−1.08969	−0.544843	−	0.838538i	$-0.683411\pi$
$23$	−4.09250e11	−1.08969	−0.544843	+	0.838538i	$0.683411\pi$
$24$	0	0
$25$	−7.07318e11	−0.927096
$26$	0	0
$27$	4.18904e11i	0.285445i
$28$	0	0
$29$	− 3.11338e12i	− 1.15571i	−0.816139	−	0.577855i	$-0.803890\pi$
$29$	− 3.11338e12i	− 1.15571i	0.816139	−	0.577855i	$-0.196110\pi$
$30$	0	0
$31$	6.57077e12	1.38370	0.691850	−	0.722041i	$-0.256796\pi$
$31$	6.57077e12	1.38370	0.691850	+	0.722041i	$0.256796\pi$
$32$	0	0
$33$	−1.19121e12	−0.147442
$34$	0	0
$35$	2.14111e13i	1.60716i
$36$	0	0
$37$	− 1.36898e13i	− 0.640739i	−0.947293	−	0.320369i	$-0.896193\pi$
$37$	− 1.36898e13i	− 0.640739i	0.947293	−	0.320369i	$-0.103807\pi$
$38$	0	0
$39$	−5.18964e12	−0.155270
$40$	0	0
$41$	−2.81830e12	−0.0551220	−0.0275610	−	0.999620i	$-0.508774\pi$
$41$	−2.81830e12	−0.0551220	−0.0275610	+	0.999620i	$0.508774\pi$
$42$	0	0
$43$	− 1.36024e13i	− 0.177474i	−0.996055	−	0.0887369i	$-0.971717\pi$
$43$	− 1.36024e13i	− 0.177474i	0.996055	−	0.0887369i	$-0.0282830\pi$
$44$	0	0
$45$	− 1.53331e14i	− 1.35932i
$46$	0	0
$47$	−6.84132e11	−0.00419091	−0.00209545	−	0.999998i	$-0.500667\pi$
$47$	−6.84132e11	−0.00419091	−0.00209545	+	0.999998i	$0.500667\pi$
$48$	0	0
$49$	7.91743e13	0.340343
$50$	0	0
$51$	7.47407e13i	0.228671i
$52$	0	0
$53$	2.47869e14i	0.546862i	0.961892	+	0.273431i	$0.0881585\pi$
$53$	2.47869e14i	0.546862i	−0.961892	+	0.273431i	$0.911841\pi$
$54$	0	0
$55$	8.81297e14	1.41919
$56$	0	0
$57$	2.55329e13	0.0303503
$58$	0	0
$59$	− 2.03296e15i	− 1.80255i	−0.433245	−	0.901276i	$-0.642632\pi$
$59$	− 2.03296e15i	− 1.80255i	0.433245	−	0.901276i	$-0.357368\pi$
$60$	0	0
$61$	− 8.92495e14i	− 0.596077i	−0.954554	−	0.298038i	$-0.903668\pi$
$61$	− 8.92495e14i	− 0.596077i	0.954554	−	0.298038i	$-0.0963324\pi$
$62$	0	0
$63$	−2.23292e15	−1.13365
$64$	0	0
$65$	3.83946e15	1.49454
$66$	0	0
$67$	− 5.06693e15i	− 1.52444i	−0.647321	−	0.762218i	$-0.724111\pi$
$67$	− 5.06693e15i	− 1.52444i	0.647321	−	0.762218i	$-0.275889\pi$
$68$	0	0
$69$	− 6.70737e14i	− 0.157157i
$70$	0	0
$71$	6.85113e15	1.25912	0.629559	−	0.776953i	$-0.283236\pi$
$71$	6.85113e15	1.25912	0.629559	+	0.776953i	$0.283236\pi$
$72$	0	0
$73$	1.37250e15	0.199191	0.0995954	−	0.995028i	$-0.468245\pi$
$73$	1.37250e15	0.199191	0.0995954	+	0.995028i	$0.468245\pi$
$74$	0	0
$75$	− 1.15925e15i	− 0.133708i
$76$	0	0
$77$	− 1.28341e16i	− 1.18357i
$78$	0	0
$79$	1.78556e16	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$79$	1.78556e16	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$80$	0	0
$81$	1.56437e16	0.938032
$82$	0	0
$83$	1.48017e15i	0.0721354i	0.999349	+	0.0360677i	$0.0114832\pi$
$83$	1.48017e15i	0.0721354i	−0.999349	+	0.0360677i	$0.988517\pi$
$84$	0	0
$85$	− 5.52955e16i	− 2.20105i
$86$	0	0
$87$	5.10265e15	0.166680
$88$	0	0
$89$	−3.78700e16	−1.01972	−0.509859	−	0.860258i	$-0.670303\pi$
$89$	−3.78700e16	−1.01972	−0.509859	+	0.860258i	$0.670303\pi$
$90$	0	0
$91$	− 5.59132e16i	− 1.24642i
$92$	0	0
$93$	1.07691e16i	0.199561i
$94$	0	0
$95$	−1.88900e16	−0.292134
$96$	0	0
$97$	1.28098e17	1.65952	0.829760	−	0.558121i	$-0.188477\pi$
$97$	1.28098e17	1.65952	0.829760	+	0.558121i	$0.188477\pi$
$98$	0	0
$99$	9.19090e16i	1.00106i
$100$	0	0
$101$	6.82139e16i	0.626818i	0.949618	+	0.313409i	$0.101471\pi$
$101$	6.82139e16i	0.626818i	−0.949618	+	0.313409i	$0.898529\pi$
$102$	0	0
$103$	3.08715e16	0.240127	0.120063	−	0.992766i	$-0.461690\pi$
$103$	3.08715e16	0.240127	0.120063	+	0.992766i	$0.461690\pi$
$104$	0	0
$105$	−3.50915e16	−0.231789
$106$	0	0
$107$	− 2.71861e17i	− 1.52962i	−0.644253	−	0.764812i	$-0.722832\pi$
$107$	− 2.71861e17i	− 1.52962i	0.644253	−	0.764812i	$-0.277168\pi$
$108$	0	0
$109$	1.32808e17i	0.638407i	0.947686	+	0.319203i	$0.103415\pi$
$109$	1.32808e17i	0.638407i	−0.947686	+	0.319203i	$0.896585\pi$
$110$	0	0
$111$	2.24367e16	0.0924090
$112$	0	0
$113$	−3.78953e17	−1.34097	−0.670484	−	0.741924i	$-0.733913\pi$
$113$	−3.78953e17	−1.34097	−0.670484	+	0.741924i	$0.733913\pi$
$114$	0	0
$115$	4.96232e17i	1.51270i
$116$	0	0
$117$	4.00411e17i	1.05421i
$118$	0	0
$119$	−8.05257e17	−1.83563
$120$	0	0
$121$	−2.28165e16	−0.0451413
$122$	0	0
$123$	− 4.61904e15i	− 0.00794983i
$124$	0	0
$125$	− 6.74428e16i	− 0.101205i
$126$	0	0
$127$	2.56390e17	0.336179	0.168089	−	0.985772i	$-0.446240\pi$
$127$	2.56390e17	0.336179	0.168089	+	0.985772i	$0.446240\pi$
$128$	0	0
$129$	2.22936e16	0.0255957
$130$	0	0
$131$	1.61862e18i	1.63057i	0.579060	+	0.815285i	$0.303420\pi$
$131$	1.61862e18i	1.63057i	−0.579060	+	0.815285i	$0.696580\pi$
$132$	0	0
$133$	2.75092e17i	0.243634i
$134$	0	0
$135$	5.07939e17	0.396255
$136$	0	0
$137$	−3.32494e17	−0.228907	−0.114453	−	0.993429i	$-0.536512\pi$
$137$	−3.32494e17	−0.228907	−0.114453	+	0.993429i	$0.536512\pi$
$138$	0	0
$139$	6.37994e17i	0.388321i	0.980970	+	0.194161i	$0.0621983\pi$
$139$	6.37994e17i	0.388321i	−0.980970	+	0.194161i	$0.937802\pi$
$140$	0	0
$141$	− 1.12125e15i	0 0.000604423i
$142$	0	0
$143$	−2.30143e18	−1.10063
$144$	0	0
$145$	−3.77510e18	−1.60436
$146$	0	0
$147$	1.29762e17i	0.0490852i
$148$	0	0
$149$	− 2.77142e18i	− 0.934585i	−0.884103	−	0.467293i	$-0.845229\pi$
$149$	− 2.77142e18i	− 0.934585i	0.884103	−	0.467293i	$-0.154771\pi$
$150$	0	0
$151$	4.43299e18	1.33473	0.667364	−	0.744732i	$-0.267423\pi$
$151$	4.43299e18	1.33473	0.667364	+	0.744732i	$0.267423\pi$
$152$	0	0
$153$	5.76668e18	1.55256
$154$	0	0
$155$	− 7.96734e18i	− 1.92085i
$156$	0	0
$157$	3.50626e18i	0.758048i	0.925387	+	0.379024i	$0.123740\pi$
$157$	3.50626e18i	0.758048i	−0.925387	+	0.379024i	$0.876260\pi$
$158$	0	0
$159$	−4.06244e17	−0.0788698
$160$	0	0
$161$	7.22653e18	1.26157
$162$	0	0
$163$	− 2.12490e18i	− 0.333998i	−0.985957	−	0.166999i	$-0.946592\pi$
$163$	− 2.12490e18i	− 0.333998i	0.985957	−	0.166999i	$-0.0534076\pi$
$164$	0	0
$165$	1.44439e18i	0.204679i
$166$	0	0
$167$	−3.81083e18	−0.487450	−0.243725	−	0.969844i	$-0.578369\pi$
$167$	−3.81083e18	−0.487450	−0.243725	+	0.969844i	$0.578369\pi$
$168$	0	0
$169$	−1.37601e18	−0.159069
$170$	0	0
$171$	− 1.97001e18i	− 0.206064i
$172$	0	0
$173$	− 8.51874e18i	− 0.807204i	−0.914935	−	0.403602i	$-0.867758\pi$
$173$	− 8.51874e18i	− 0.807204i	0.914935	−	0.403602i	$-0.132242\pi$
$174$	0	0
$175$	1.24898e19	1.07333
$176$	0	0
$177$	3.33191e18	0.259969
$178$	0	0
$179$	− 1.98628e19i	− 1.40860i	−0.709900	−	0.704302i	$-0.751260\pi$
$179$	− 1.98628e19i	− 1.40860i	0.709900	−	0.704302i	$-0.248740\pi$
$180$	0	0
$181$	− 2.34430e19i	− 1.51267i	−0.654182	−	0.756337i	$-0.726987\pi$
$181$	− 2.34430e19i	− 1.51267i	0.654182	−	0.756337i	$-0.273013\pi$
$182$	0	0
$183$	1.46275e18	0.0859677
$184$	0	0
$185$	−1.65994e19	−0.889473
$186$	0	0
$187$	3.31451e19i	1.62093i
$188$	0	0
$189$	− 7.39701e18i	− 0.330469i
$190$	0	0
$191$	1.50710e19	0.615686	0.307843	−	0.951437i	$-0.400393\pi$
$191$	1.50710e19	0.615686	0.307843	+	0.951437i	$0.400393\pi$
$192$	0	0
$193$	9.30819e18	0.348039	0.174020	−	0.984742i	$-0.444324\pi$
$193$	9.30819e18	0.348039	0.174020	+	0.984742i	$0.444324\pi$
$194$	0	0
$195$	6.29265e18i	0.215546i
$196$	0	0
$197$	4.15016e19i	1.30347i	0.758445	+	0.651737i	$0.225959\pi$
$197$	4.15016e19i	1.30347i	−0.758445	+	0.651737i	$0.774041\pi$
$198$	0	0
$199$	−1.63945e19	−0.472550	−0.236275	−	0.971686i	$-0.575927\pi$
$199$	−1.63945e19	−0.472550	−0.236275	+	0.971686i	$0.575927\pi$
$200$	0	0
$201$	8.30440e18	0.219858
$202$	0	0
$203$	5.49760e19i	1.33800i
$204$	0	0
$205$	3.41731e18i	0.0765203i
$206$	0	0
$207$	−5.17513e19	−1.06702
$208$	0	0
$209$	1.13230e19	0.215138
$210$	0	0
$211$	− 7.37209e19i	− 1.29178i	−0.763429	−	0.645892i	$-0.776486\pi$
$211$	− 7.37209e19i	− 1.29178i	0.763429	−	0.645892i	$-0.223514\pi$
$212$	0	0
$213$	1.12286e19i	0.181593i
$214$	0	0
$215$	−1.64935e19	−0.246369
$216$	0	0
$217$	−1.16027e20	−1.60195
$218$	0	0
$219$	2.24946e18i	0.0287278i
$220$	0	0
$221$	1.44400e20i	1.70699i
$222$	0	0
$223$	−3.21618e19	−0.352167	−0.176083	−	0.984375i	$-0.556343\pi$
$223$	−3.21618e19	−0.352167	−0.176083	+	0.984375i	$0.556343\pi$
$224$	0	0
$225$	−8.94433e19	−0.907813
$226$	0	0
$227$	9.66411e19i	0.909792i	0.890545	+	0.454896i	$0.150323\pi$
$227$	9.66411e19i	0.909792i	−0.890545	+	0.454896i	$0.849677\pi$
$228$	0	0
$229$	− 3.07436e16i	0 0.000268629i	−1.00000		0.000134314i	$-0.999957\pi$
$229$	− 3.07436e16i	0 0.000268629i	1.00000		0.000134314i	$-4.27536e-5\pi$
$230$	0	0
$231$	2.10344e19	0.170698
$232$	0	0
$233$	2.21535e20	1.67077	0.835385	−	0.549666i	$-0.185245\pi$
$233$	2.21535e20	1.67077	0.835385	+	0.549666i	$0.185245\pi$
$234$	0	0
$235$	8.29539e17i	0.00581781i
$236$	0	0
$237$	2.92643e19i	0.190975i
$238$	0	0
$239$	4.51465e18	0.0274310	0.0137155	−	0.999906i	$-0.495634\pi$
$239$	4.51465e18	0.0274310	0.0137155	+	0.999906i	$0.495634\pi$
$240$	0	0
$241$	−1.81683e19	−0.102842	−0.0514209	−	0.998677i	$-0.516375\pi$
$241$	−1.81683e19	−0.102842	−0.0514209	+	0.998677i	$0.516375\pi$
$242$	0	0
$243$	7.97365e19i	0.420731i
$244$	0	0
$245$	− 9.60021e19i	− 0.472464i
$246$	0	0
$247$	4.93298e19	0.226561
$248$	0	0
$249$	−2.42592e18	−0.0104035
$250$	0	0
$251$	− 3.26192e20i	− 1.30691i	−0.756964	−	0.653456i	$-0.773318\pi$
$251$	− 3.26192e20i	− 1.30691i	0.756964	−	0.653456i	$-0.226682\pi$
$252$	0	0
$253$	− 2.97450e20i	− 1.11401i
$254$	0	0
$255$	9.06262e19	0.317441
$256$	0	0
$257$	−1.22717e20	−0.402228	−0.201114	−	0.979568i	$-0.564456\pi$
$257$	−1.22717e20	−0.402228	−0.201114	+	0.979568i	$0.564456\pi$
$258$	0	0
$259$	2.41734e20i	0.741804i
$260$	0	0
$261$	− 3.93699e20i	− 1.13167i
$262$	0	0
$263$	−3.35733e18	−0.00904420	−0.00452210	−	0.999990i	$-0.501439\pi$
$263$	−3.35733e18	−0.00904420	−0.00452210	+	0.999990i	$0.501439\pi$
$264$	0	0
$265$	3.00552e20	0.759153
$266$	0	0
$267$	− 6.20668e19i	− 0.147066i
$268$	0	0
$269$	− 2.42496e20i	− 0.539275i	−0.962962	−	0.269637i	$-0.913096\pi$
$269$	− 2.42496e20i	− 0.539275i	0.962962	−	0.269637i	$-0.0869038\pi$
$270$	0	0
$271$	2.38962e20	0.498988	0.249494	−	0.968376i	$-0.419736\pi$
$271$	2.38962e20	0.498988	0.249494	+	0.968376i	$0.419736\pi$
$272$	0	0
$273$	9.16386e19	0.179761
$274$	0	0
$275$	− 5.14091e20i	− 0.947791i
$276$	0	0
$277$	5.05700e20i	0.876628i	0.898822	+	0.438314i	$0.144424\pi$
$277$	5.05700e20i	0.876628i	−0.898822	+	0.438314i	$0.855576\pi$
$278$	0	0
$279$	8.30901e20	1.35492
$280$	0	0
$281$	−2.23986e20	−0.343730	−0.171865	−	0.985120i	$-0.554979\pi$
$281$	−2.23986e20	−0.343730	−0.171865	+	0.985120i	$0.554979\pi$
$282$	0	0
$283$	− 5.21843e20i	− 0.753971i	−0.926219	−	0.376986i	$-0.876961\pi$
$283$	− 5.21843e20i	− 0.753971i	0.926219	−	0.376986i	$-0.123039\pi$
$284$	0	0
$285$	− 3.09597e19i	− 0.0421323i
$286$	0	0
$287$	4.97656e19	0.0638165
$288$	0	0
$289$	1.25239e21	1.51394
$290$	0	0
$291$	2.09945e20i	0.239340i
$292$	0	0
$293$	4.94662e20i	0.532027i	0.963969	+	0.266014i	$0.0857066\pi$
$293$	4.94662e20i	0.532027i	−0.963969	+	0.266014i	$0.914293\pi$
$294$	0	0
$295$	−2.46506e21	−2.50230
$296$	0	0
$297$	−3.04467e20	−0.291817
$298$	0	0
$299$	− 1.29587e21i	− 1.17316i
$300$	0	0
$301$	2.40191e20i	0.205467i
$302$	0	0
$303$	−1.11799e20	−0.0904012
$304$	0	0
$305$	−1.08219e21	−0.827473
$306$	0	0
$307$	1.86234e21i	1.34705i	0.739166	+	0.673523i	$0.235220\pi$
$307$	1.86234e21i	1.34705i	−0.739166	+	0.673523i	$0.764780\pi$
$308$	0	0
$309$	5.05965e19i	0.0346317i
$310$	0	0
$311$	6.47522e20	0.419557	0.209779	−	0.977749i	$-0.432726\pi$
$311$	6.47522e20	0.419557	0.209779	+	0.977749i	$0.432726\pi$
$312$	0	0
$313$	−1.62295e21	−0.995817	−0.497909	−	0.867230i	$-0.665898\pi$
$313$	−1.62295e21	−0.995817	−0.497909	+	0.867230i	$0.665898\pi$
$314$	0	0
$315$	2.70751e21i	1.57373i
$316$	0	0
$317$	2.11436e21i	1.16460i	0.812976	+	0.582298i	$0.197846\pi$
$317$	2.11436e21i	1.16460i	−0.812976	+	0.582298i	$0.802154\pi$
$318$	0	0
$319$	2.26286e21	1.18151
$320$	0	0
$321$	4.45565e20	0.220606
$322$	0	0
$323$	− 7.10443e20i	− 0.333663i
$324$	0	0
$325$	− 2.23969e21i	− 0.998113i
$326$	0	0
$327$	−2.17664e20	−0.0920726
$328$	0	0
$329$	1.20804e19	0.00485195
$330$	0	0
$331$	4.36509e21i	1.66516i	0.553908	+	0.832578i	$0.313136\pi$
$331$	4.36509e21i	1.66516i	−0.553908	+	0.832578i	$0.686864\pi$
$332$	0	0
$333$	− 1.73112e21i	− 0.627411i
$334$	0	0
$335$	−6.14386e21	−2.11622
$336$	0	0
$337$	−2.59518e21	−0.849792	−0.424896	−	0.905242i	$-0.639689\pi$
$337$	−2.59518e21	−0.849792	−0.424896	+	0.905242i	$0.639689\pi$
$338$	0	0
$339$	− 6.21082e20i	− 0.193398i
$340$	0	0
$341$	4.77575e21i	1.41459i
$342$	0	0
$343$	2.70973e21	0.763706
$344$	0	0
$345$	−8.13296e20	−0.218166
$346$	0	0
$347$	− 7.40818e20i	− 0.189195i	−0.995516	−	0.0945977i	$-0.969844\pi$
$347$	− 7.40818e20i	− 0.189195i	0.995516	−	0.0945977i	$-0.0301565\pi$
$348$	0	0
$349$	− 1.81507e20i	− 0.0441446i	−0.999756	−	0.0220723i	$-0.992974\pi$
$349$	− 1.81507e20i	− 0.0441446i	0.999756	−	0.0220723i	$-0.00702640\pi$
$350$	0	0
$351$	−1.32644e21	−0.307310
$352$	0	0
$353$	3.26526e21	0.720830	0.360415	−	0.932792i	$-0.382635\pi$
$353$	3.26526e21	0.720830	0.360415	+	0.932792i	$0.382635\pi$
$354$	0	0
$355$	− 8.30728e21i	− 1.74791i
$356$	0	0
$357$	− 1.31977e21i	− 0.264739i
$358$	0	0
$359$	−2.19419e21	−0.419732	−0.209866	−	0.977730i	$-0.567303\pi$
$359$	−2.19419e21	−0.419732	−0.209866	+	0.977730i	$0.567303\pi$
$360$	0	0
$361$	5.23769e21	0.955715
$362$	0	0
$363$	− 3.73950e19i	− 0.00651040i
$364$	0	0
$365$	− 1.66422e21i	− 0.276516i
$366$	0	0
$367$	1.08265e22	1.71723	0.858615	−	0.512621i	$-0.171325\pi$
$367$	1.08265e22	1.71723	0.858615	+	0.512621i	$0.171325\pi$
$368$	0	0
$369$	−3.56386e20	−0.0539755
$370$	0	0
$371$	− 4.37687e21i	− 0.633120i
$372$	0	0
$373$	5.20618e20i	0.0719439i	0.999353	+	0.0359719i	$0.0114527\pi$
$373$	5.20618e20i	0.0719439i	−0.999353	+	0.0359719i	$0.988547\pi$
$374$	0	0
$375$	1.10535e20	0.0145960
$376$	0	0
$377$	9.85837e21	1.24424
$378$	0	0
$379$	2.99720e21i	0.361645i	0.983516	+	0.180823i	$0.0578760\pi$
$379$	2.99720e21i	0.361645i	−0.983516	+	0.180823i	$0.942124\pi$
$380$	0	0
$381$	4.20209e20i	0.0484845i
$382$	0	0
$383$	1.22679e22	1.35388	0.676939	−	0.736039i	$-0.263306\pi$
$383$	1.22679e22	1.35388	0.676939	+	0.736039i	$0.263306\pi$
$384$	0	0
$385$	−1.55619e22	−1.64304
$386$	0	0
$387$	− 1.72008e21i	− 0.173782i
$388$	0	0
$389$	− 1.60002e21i	− 0.154722i	−0.997003	−	0.0773612i	$-0.975351\pi$
$389$	− 1.60002e21i	− 0.154722i	0.997003	−	0.0773612i	$-0.0246495\pi$
$390$	0	0
$391$	−1.86630e22	−1.72774
$392$	0	0
$393$	−2.65283e21	−0.235165
$394$	0	0
$395$	− 2.16506e22i	− 1.83821i
$396$	0	0
$397$	− 2.44521e22i	− 1.98883i	−0.105546	−	0.994414i	$-0.533659\pi$
$397$	− 2.44521e22i	− 1.98883i	0.105546	−	0.994414i	$-0.466341\pi$
$398$	0	0
$399$	−4.50859e20	−0.0351375
$400$	0	0
$401$	2.46119e22	1.83830	0.919152	−	0.393903i	$-0.128875\pi$
$401$	2.46119e22	1.83830	0.919152	+	0.393903i	$0.128875\pi$
$402$	0	0
$403$	2.08061e22i	1.48969i
$404$	0	0
$405$	− 1.89687e22i	− 1.30218i
$406$	0	0
$407$	9.94995e21	0.655041
$408$	0	0
$409$	−5.67464e21	−0.358336	−0.179168	−	0.983819i	$-0.557341\pi$
$409$	−5.67464e21	−0.358336	−0.179168	+	0.983819i	$0.557341\pi$
$410$	0	0
$411$	− 5.44938e20i	− 0.0330135i
$412$	0	0
$413$	3.58981e22i	2.08687i
$414$	0	0
$415$	1.79477e21	0.100138
$416$	0	0
$417$	−1.04564e21	−0.0560047
$418$	0	0
$419$	2.70146e22i	1.38925i	0.719374	+	0.694623i	$0.244429\pi$
$419$	2.70146e22i	1.38925i	−0.719374	+	0.694623i	$0.755571\pi$
$420$	0	0
$421$	1.26169e22i	0.623094i	0.950231	+	0.311547i	$0.100847\pi$
$421$	1.26169e22i	0.623094i	−0.950231	+	0.311547i	$0.899153\pi$
$422$	0	0
$423$	−8.65113e19	−0.00410374
$424$	0	0
$425$	−3.22558e22	−1.46995
$426$	0	0
$427$	1.57597e22i	0.690097i
$428$	0	0
$429$	− 3.77192e21i	− 0.158736i
$430$	0	0
$431$	−4.26128e21	−0.172378	−0.0861892	−	0.996279i	$-0.527469\pi$
$431$	−4.26128e21	−0.172378	−0.0861892	+	0.996279i	$0.527469\pi$
$432$	0	0
$433$	−1.62172e21	−0.0630707	−0.0315353	−	0.999503i	$-0.510040\pi$
$433$	−1.62172e21	−0.0630707	−0.0315353	+	0.999503i	$0.510040\pi$
$434$	0	0
$435$	− 6.18718e21i	− 0.231384i
$436$	0	0
$437$	6.37565e21i	0.229315i
$438$	0	0
$439$	−1.56475e22	−0.541375	−0.270687	−	0.962667i	$-0.587251\pi$
$439$	−1.56475e22	−0.541375	−0.270687	+	0.962667i	$0.587251\pi$
$440$	0	0
$441$	1.00119e22	0.333264
$442$	0	0
$443$	− 3.08815e22i	− 0.989161i	−0.869132	−	0.494581i	$-0.835322\pi$
$443$	− 3.08815e22i	− 0.989161i	0.869132	−	0.494581i	$-0.164678\pi$
$444$	0	0
$445$	4.59190e22i	1.41557i
$446$	0	0
$447$	4.54220e21	0.134788
$448$	0	0
$449$	−1.44678e22	−0.413340	−0.206670	−	0.978411i	$-0.566263\pi$
$449$	−1.44678e22	−0.413340	−0.206670	+	0.978411i	$0.566263\pi$
$450$	0	0
$451$	− 2.04839e21i	− 0.0563524i
$452$	0	0
$453$	7.26542e21i	0.192498i
$454$	0	0
$455$	−6.77971e22	−1.73027
$456$	0	0
$457$	−1.09366e22	−0.268903	−0.134452	−	0.990920i	$-0.542927\pi$
$457$	−1.09366e22	−0.268903	−0.134452	+	0.990920i	$0.542927\pi$
$458$	0	0
$459$	1.91033e22i	0.452585i
$460$	0	0
$461$	− 3.78938e22i	− 0.865187i	−0.901589	−	0.432594i	$-0.857598\pi$
$461$	− 3.78938e22i	− 0.865187i	0.901589	−	0.432594i	$-0.142402\pi$
$462$	0	0
$463$	−5.34510e21	−0.117630	−0.0588149	−	0.998269i	$-0.518732\pi$
$463$	−5.34510e21	−0.117630	−0.0588149	+	0.998269i	$0.518732\pi$
$464$	0	0
$465$	1.30580e22	0.277030
$466$	0	0
$467$	2.81699e21i	0.0576224i	0.999585	+	0.0288112i	$0.00917216\pi$
$467$	2.81699e21i	0.0576224i	−0.999585	+	0.0288112i	$0.990828\pi$
$468$	0	0
$469$	8.94718e22i	1.76489i
$470$	0	0
$471$	−5.74655e21	−0.109328
$472$	0	0
$473$	9.88647e21	0.181435
$474$	0	0
$475$	1.10192e22i	0.195099i
$476$	0	0
$477$	3.13441e22i	0.535487i
$478$	0	0
$479$	9.05318e22	1.49262	0.746310	−	0.665598i	$-0.231823\pi$
$479$	9.05318e22	1.49262	0.746310	+	0.665598i	$0.231823\pi$
$480$	0	0
$481$	4.33480e22	0.689820
$482$	0	0
$483$	1.18439e22i	0.181946i
$484$	0	0
$485$	− 1.55324e23i	− 2.30374i
$486$	0	0
$487$	3.07462e22	0.440347	0.220173	−	0.975461i	$-0.429338\pi$
$487$	3.07462e22	0.440347	0.220173	+	0.975461i	$0.429338\pi$
$488$	0	0
$489$	3.48259e21	0.0481700
$490$	0	0
$491$	1.94168e22i	0.259409i	0.991553	+	0.129704i	$0.0414028\pi$
$491$	1.94168e22i	0.259409i	−0.991553	+	0.129704i	$0.958597\pi$
$492$	0	0
$493$	− 1.41979e23i	− 1.83243i
$494$	0	0
$495$	1.11444e23	1.38967
$496$	0	0
$497$	−1.20977e23	−1.45772
$498$	0	0
$499$	− 7.79434e22i	− 0.907664i	−0.891087	−	0.453832i	$-0.850057\pi$
$499$	− 7.79434e22i	− 0.907664i	0.891087	−	0.453832i	$-0.149943\pi$
$500$	0	0
$501$	− 6.24573e21i	− 0.0703012i
$502$	0	0
$503$	−1.18157e23	−1.28568	−0.642838	−	0.766002i	$-0.722243\pi$
$503$	−1.18157e23	−1.28568	−0.642838	+	0.766002i	$0.722243\pi$
$504$	0	0
$505$	8.27122e22	0.870148
$506$	0	0
$507$	− 2.25521e21i	− 0.0229413i
$508$	0	0
$509$	2.87932e21i	0.0283262i	0.999900	+	0.0141631i	$0.00450841\pi$
$509$	2.87932e21i	0.0283262i	−0.999900	+	0.0141631i	$0.995492\pi$
$510$	0	0
$511$	−2.42357e22	−0.230610
$512$	0	0
$513$	6.52605e21	0.0600694
$514$	0	0
$515$	− 3.74329e22i	− 0.333344i
$516$	0	0
$517$	− 4.97239e20i	− 0.00428446i
$518$	0	0
$519$	1.39617e22	0.116417
$520$	0	0
$521$	1.28391e23	1.03613	0.518066	−	0.855340i	$-0.326652\pi$
$521$	1.28391e23	1.03613	0.518066	+	0.855340i	$0.326652\pi$
$522$	0	0
$523$	1.89027e22i	0.147659i	0.997271	+	0.0738293i	$0.0235220\pi$
$523$	1.89027e22i	0.147659i	−0.997271	+	0.0738293i	$0.976478\pi$
$524$	0	0
$525$	2.04701e22i	0.154798i
$526$	0	0
$527$	2.99647e23	2.19391
$528$	0	0
$529$	2.64353e22	0.187418
$530$	0	0
$531$	− 2.57077e23i	− 1.76506i
$532$	0	0
$533$	− 8.92403e21i	− 0.0593444i
$534$	0	0
$535$	−3.29643e23	−2.12342
$536$	0	0
$537$	3.25539e22	0.203152
$538$	0	0
$539$	5.75452e22i	0.347940i
$540$	0	0
$541$	3.25660e23i	1.90804i	0.299744	+	0.954020i	$0.403099\pi$
$541$	3.25660e23i	1.90804i	−0.299744	+	0.954020i	$0.596901\pi$
$542$	0	0
$543$	3.84218e22	0.218162
$544$	0	0
$545$	1.61035e23	0.886236
$546$	0	0
$547$	4.90295e22i	0.261556i	0.991412	+	0.130778i	$0.0417476\pi$
$547$	4.90295e22i	0.261556i	−0.991412	+	0.130778i	$0.958252\pi$
$548$	0	0
$549$	− 1.12860e23i	− 0.583679i
$550$	0	0
$551$	−4.85029e22	−0.243209
$552$	0	0
$553$	−3.15294e23	−1.53303
$554$	0	0
$555$	− 2.72055e22i	− 0.128282i
$556$	0	0
$557$	− 2.60011e23i	− 1.18911i	−0.804054	−	0.594556i	$-0.797328\pi$
$557$	− 2.60011e23i	− 1.18911i	0.804054	−	0.594556i	$-0.202672\pi$
$558$	0	0
$559$	4.30714e22	0.191068
$560$	0	0
$561$	−5.43228e22	−0.233775
$562$	0	0
$563$	1.80701e23i	0.754467i	0.926118	+	0.377233i	$0.123125\pi$
$563$	1.80701e23i	0.754467i	−0.926118	+	0.377233i	$0.876875\pi$
$564$	0	0
$565$	4.59496e23i	1.86153i
$566$	0	0
$567$	−2.76237e23	−1.08599
$568$	0	0
$569$	−3.54026e23	−1.35077	−0.675384	−	0.737467i	$-0.736022\pi$
$569$	−3.54026e23	−1.35077	−0.675384	+	0.737467i	$0.736022\pi$
$570$	0	0
$571$	1.77850e23i	0.658637i	0.944219	+	0.329318i	$0.106819\pi$
$571$	1.77850e23i	0.658637i	−0.944219	+	0.329318i	$0.893181\pi$
$572$	0	0
$573$	2.47006e22i	0.0887958i
$574$	0	0
$575$	2.89470e23	1.01024
$576$	0	0
$577$	−3.55869e23	−1.20586	−0.602928	−	0.797796i	$-0.705999\pi$
$577$	−3.55869e23	−1.20586	−0.602928	+	0.797796i	$0.705999\pi$
$578$	0	0
$579$	1.52556e22i	0.0501951i
$580$	0	0
$581$	− 2.61369e22i	− 0.0835134i
$582$	0	0
$583$	−1.80156e23	−0.559069
$584$	0	0
$585$	4.85515e23	1.46345
$586$	0	0
$587$	2.22604e23i	0.651791i	0.945406	+	0.325896i	$0.105666\pi$
$587$	2.22604e23i	0.651791i	−0.945406	+	0.325896i	$0.894334\pi$
$588$	0	0
$589$	− 1.02365e23i	− 0.291187i
$590$	0	0
$591$	−6.80188e22	−0.187990
$592$	0	0
$593$	6.01922e23	1.61650	0.808250	−	0.588840i	$-0.200415\pi$
$593$	6.01922e23	1.61650	0.808250	+	0.588840i	$0.200415\pi$
$594$	0	0
$595$	9.76408e23i	2.54822i
$596$	0	0
$597$	− 2.68697e22i	− 0.0681523i
$598$	0	0
$599$	−2.83674e23	−0.699346	−0.349673	−	0.936872i	$-0.613707\pi$
$599$	−2.83674e23	−0.699346	−0.349673	+	0.936872i	$0.613707\pi$
$600$	0	0
$601$	−4.18303e23	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$601$	−4.18303e23	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$602$	0	0
$603$	− 6.40734e23i	− 1.49273i
$604$	0	0
$605$	2.76660e22i	0.0626651i
$606$	0	0
$607$	2.31638e23	0.510159	0.255080	−	0.966920i	$-0.417898\pi$
$607$	2.31638e23	0.510159	0.255080	+	0.966920i	$0.417898\pi$
$608$	0	0
$609$	−9.01026e22	−0.192970
$610$	0	0
$611$	− 2.16627e21i	− 0.00451193i
$612$	0	0
$613$	2.79613e23i	0.566425i	0.959057	+	0.283213i	$0.0914002\pi$
$613$	2.79613e23i	0.566425i	−0.959057	+	0.283213i	$0.908600\pi$
$614$	0	0
$615$	−5.60078e21	−0.0110360
$616$	0	0
$617$	−1.73444e23	−0.332457	−0.166229	−	0.986087i	$-0.553159\pi$
$617$	−1.73444e23	−0.332457	−0.166229	+	0.986087i	$0.553159\pi$
$618$	0	0
$619$	− 6.77105e23i	− 1.26266i	−0.775515	−	0.631329i	$-0.782510\pi$
$619$	− 6.77105e23i	− 1.26266i	0.775515	−	0.631329i	$-0.217490\pi$
$620$	0	0
$621$	− 1.71436e23i	− 0.311046i
$622$	0	0
$623$	6.68708e23	1.18056
$624$	0	0
$625$	−6.21418e23	−1.06759
$626$	0	0
$627$	1.85577e22i	0.0310278i
$628$	0	0
$629$	− 6.24294e23i	− 1.01592i
$630$	0	0
$631$	−1.02867e24	−1.62940	−0.814698	−	0.579885i	$-0.803097\pi$
$631$	−1.02867e24	−1.62940	−0.814698	+	0.579885i	$0.803097\pi$
$632$	0	0
$633$	1.20824e23	0.186304
$634$	0	0
$635$	− 3.10884e23i	− 0.466683i
$636$	0	0
$637$	2.50702e23i	0.366414i
$638$	0	0
$639$	8.66353e23	1.23293
$640$	0	0
$641$	7.31083e23	1.01315	0.506575	−	0.862196i	$-0.330911\pi$
$641$	7.31083e23	1.01315	0.506575	+	0.862196i	$0.330911\pi$
$642$	0	0
$643$	− 5.31860e23i	− 0.717801i	−0.933376	−	0.358901i	$-0.883152\pi$
$643$	− 5.31860e23i	− 0.717801i	0.933376	−	0.358901i	$-0.116848\pi$
$644$	0	0
$645$	− 2.70319e22i	− 0.0355319i
$646$	0	0
$647$	−7.87216e23	−1.00788	−0.503939	−	0.863740i	$-0.668116\pi$
$647$	−7.87216e23	−1.00788	−0.503939	+	0.863740i	$0.668116\pi$
$648$	0	0
$649$	1.47759e24	1.84279
$650$	0	0
$651$	− 1.90161e23i	− 0.231038i
$652$	0	0
$653$	2.77252e22i	0.0328180i	0.999865	+	0.0164090i	$0.00522338\pi$
$653$	2.77252e22i	0.0328180i	−0.999865	+	0.0164090i	$0.994777\pi$
$654$	0	0
$655$	1.96265e24	2.26356
$656$	0	0
$657$	1.73559e23	0.195048
$658$	0	0
$659$	− 1.48992e24i	− 1.63169i	−0.578273	−	0.815843i	$-0.696273\pi$
$659$	− 1.48992e24i	− 1.63169i	0.578273	−	0.815843i	$-0.303727\pi$
$660$	0	0
$661$	− 2.01553e22i	− 0.0215118i	−0.999942	−	0.0107559i	$-0.996576\pi$
$661$	− 2.01553e22i	− 0.0215118i	0.999942	−	0.0107559i	$-0.00342377\pi$
$662$	0	0
$663$	−2.36663e23	−0.246187
$664$	0	0
$665$	3.33560e23	0.338213
$666$	0	0
$667$	1.27415e24i	1.25936i
$668$	0	0
$669$	− 5.27113e22i	− 0.0507904i
$670$	0	0
$671$	6.48681e23	0.609382
$672$	0	0
$673$	−7.70622e23	−0.705851	−0.352926	−	0.935651i	$-0.614813\pi$
$673$	−7.70622e23	−0.705851	−0.352926	+	0.935651i	$0.614813\pi$
$674$	0	0
$675$	− 2.96299e23i	− 0.264635i
$676$	0	0
$677$	− 8.03721e23i	− 0.700006i	−0.936749	−	0.350003i	$-0.886181\pi$
$677$	− 8.03721e23i	− 0.700006i	0.936749	−	0.350003i	$-0.113819\pi$
$678$	0	0
$679$	−2.26195e24	−1.92128
$680$	0	0
$681$	−1.58389e23	−0.131212
$682$	0	0
$683$	3.12264e23i	0.252317i	0.992010	+	0.126158i	$0.0402647\pi$
$683$	3.12264e23i	0.252317i	−0.992010	+	0.126158i	$0.959735\pi$
$684$	0	0
$685$	4.03163e23i	0.317768i
$686$	0	0
$687$	5.03869e19	3.87423e−5 0
$688$	0	0
$689$	−7.84867e23	−0.588752
$690$	0	0
$691$	1.46577e24i	1.07276i	0.843977	+	0.536379i	$0.180208\pi$
$691$	1.46577e24i	1.07276i	−0.843977	+	0.536379i	$0.819792\pi$
$692$	0	0
$693$	− 1.62293e24i	− 1.15896i
$694$	0	0
$695$	7.73595e23	0.539067
$696$	0	0
$697$	−1.28523e23	−0.0873982
$698$	0	0
$699$	3.63083e23i	0.240963i
$700$	0	0
$701$	− 2.04249e24i	− 1.32299i	−0.749950	−	0.661495i	$-0.769922\pi$
$701$	− 2.04249e24i	− 1.32299i	0.749950	−	0.661495i	$-0.230078\pi$
$702$	0	0
$703$	−2.13271e23	−0.134838
$704$	0	0
$705$	−1.35957e21	−0.000839060 0
$706$	0	0
$707$	− 1.20452e24i	− 0.725687i
$708$	0	0
$709$	2.96920e24i	1.74641i	0.487352	+	0.873206i	$0.337963\pi$
$709$	2.96920e24i	1.74641i	−0.487352	+	0.873206i	$0.662037\pi$
$710$	0	0
$711$	2.25791e24	1.29663
$712$	0	0
$713$	−2.68909e24	−1.50780
$714$	0	0
$715$	2.79058e24i	1.52790i
$716$	0	0
$717$	7.39925e21i	0.00395617i
$718$	0	0
$719$	−4.13376e23	−0.215849	−0.107924	−	0.994159i	$-0.534420\pi$
$719$	−4.13376e23	−0.215849	−0.107924	+	0.994159i	$0.534420\pi$
$720$	0	0
$721$	−5.45128e23	−0.278003
$722$	0	0
$723$	− 2.97768e22i	− 0.0148321i
$724$	0	0
$725$	2.20215e24i	1.07146i
$726$	0	0
$727$	−2.92465e24	−1.39005	−0.695027	−	0.718983i	$-0.744608\pi$
$727$	−2.92465e24	−1.39005	−0.695027	+	0.718983i	$0.744608\pi$
$728$	0	0
$729$	1.88955e24	0.877353
$730$	0	0
$731$	− 6.20311e23i	− 0.281392i
$732$	0	0
$733$	3.06846e24i	1.35999i	0.733216	+	0.679996i	$0.238018\pi$
$733$	3.06846e24i	1.35999i	−0.733216	+	0.679996i	$0.761982\pi$
$734$	0	0
$735$	1.57342e23	0.0681400
$736$	0	0
$737$	3.68273e24	1.55846
$738$	0	0
$739$	3.06955e24i	1.26939i	0.772761	+	0.634697i	$0.218875\pi$
$739$	3.06955e24i	1.26939i	−0.772761	+	0.634697i	$0.781125\pi$
$740$	0	0
$741$	8.08486e22i	0.0326752i
$742$	0	0
$743$	−4.34334e24	−1.71561	−0.857805	−	0.513975i	$-0.828172\pi$
$743$	−4.34334e24	−1.71561	−0.857805	+	0.513975i	$0.828172\pi$
$744$	0	0
$745$	−3.36046e24	−1.29739
$746$	0	0
$747$	1.87174e23i	0.0706350i
$748$	0	0
$749$	4.80052e24i	1.77090i
$750$	0	0
$751$	−7.59956e23	−0.274062	−0.137031	−	0.990567i	$-0.543756\pi$
$751$	−7.59956e23	−0.274062	−0.137031	+	0.990567i	$0.543756\pi$
$752$	0	0
$753$	5.34610e23	0.188486
$754$	0	0
$755$	− 5.37519e24i	− 1.85287i
$756$	0	0
$757$	− 2.36037e24i	− 0.795544i	−0.917484	−	0.397772i	$-0.869783\pi$
$757$	− 2.36037e24i	− 0.795544i	0.917484	−	0.397772i	$-0.130217\pi$
$758$	0	0
$759$	4.87503e23	0.160665
$760$	0	0
$761$	−3.99857e23	−0.128865	−0.0644326	−	0.997922i	$-0.520524\pi$
$761$	−3.99857e23	−0.128865	−0.0644326	+	0.997922i	$0.520524\pi$
$762$	0	0
$763$	− 2.34511e24i	− 0.739104i
$764$	0	0
$765$	− 6.99234e24i	− 2.15526i
$766$	0	0
$767$	6.43729e24	1.94063
$768$	0	0
$769$	−3.40521e24	−1.00408	−0.502042	−	0.864843i	$-0.667418\pi$
$769$	−3.40521e24	−1.00408	−0.502042	+	0.864843i	$0.667418\pi$
$770$	0	0
$771$	− 2.01126e23i	− 0.0580103i
$772$	0	0
$773$	9.20037e23i	0.259585i	0.991541	+	0.129793i	$0.0414311\pi$
$773$	9.20037e23i	0.259585i	−0.991541	+	0.129793i	$0.958569\pi$
$774$	0	0
$775$	−4.64763e24	−1.28282
$776$	0	0
$777$	−3.96188e23	−0.106985
$778$	0	0
$779$	4.39060e22i	0.0115999i
$780$	0	0
$781$	4.97952e24i	1.28722i
$782$	0	0
$783$	1.30421e24	0.329892
$784$	0	0
$785$	4.25148e24	1.05232
$786$	0	0
$787$	7.43784e24i	1.80161i	0.434220	+	0.900807i	$0.357024\pi$
$787$	7.43784e24i	1.80161i	−0.434220	+	0.900807i	$0.642976\pi$
$788$	0	0
$789$	− 5.50247e21i	− 0.00130438i
$790$	0	0
$791$	6.69154e24	1.55248
$792$	0	0
$793$	2.82605e24	0.641737
$794$	0	0
$795$	4.92587e23i	0.109487i
$796$	0	0
$797$	− 6.58511e24i	− 1.43274i	−0.697721	−	0.716369i	$-0.745803\pi$
$797$	− 6.58511e24i	− 1.43274i	0.697721	−	0.716369i	$-0.254197\pi$
$798$	0	0
$799$	−3.11985e22	−0.00664486
$800$	0	0
$801$	−4.78881e24	−0.998508
$802$	0	0
$803$	9.97560e23i	0.203637i
$804$	0	0
$805$	− 8.76247e24i	− 1.75130i
$806$	0	0
$807$	3.97437e23	0.0777755
$808$	0	0
$809$	6.49816e24	1.24517	0.622584	−	0.782553i	$-0.286083\pi$
$809$	6.49816e24	1.24517	0.622584	+	0.782553i	$0.286083\pi$
$810$	0	0
$811$	− 5.42750e24i	− 1.01841i	−0.860645	−	0.509205i	$-0.829940\pi$
$811$	− 5.42750e24i	− 1.01841i	0.860645	−	0.509205i	$-0.170060\pi$
$812$	0	0
$813$	3.91645e23i	0.0719653i
$814$	0	0
$815$	−2.57653e24	−0.463655
$816$	0	0
$817$	−2.11910e23	−0.0373477
$818$	0	0
$819$	− 7.07045e24i	− 1.22049i
$820$	0	0
$821$	− 1.44666e24i	− 0.244595i	−0.992493	−	0.122298i	$-0.960974\pi$
$821$	− 1.44666e24i	− 0.244595i	0.992493	−	0.122298i	$-0.0390263\pi$
$822$	0	0
$823$	5.56998e24	0.922476	0.461238	−	0.887276i	$-0.347405\pi$
$823$	5.56998e24	0.922476	0.461238	+	0.887276i	$0.347405\pi$
$824$	0	0
$825$	8.42566e23	0.136693
$826$	0	0
$827$	− 5.21332e24i	− 0.828548i	−0.910152	−	0.414274i	$-0.864036\pi$
$827$	− 5.21332e24i	− 0.828548i	0.910152	−	0.414274i	$-0.135964\pi$
$828$	0	0
$829$	− 3.83347e24i	− 0.596869i	−0.954430	−	0.298434i	$-0.903536\pi$
$829$	− 3.83347e24i	− 0.596869i	0.954430	−	0.298434i	$-0.0964644\pi$
$830$	0	0
$831$	−8.28814e23	−0.126429
$832$	0	0
$833$	3.61058e24	0.539628
$834$	0	0
$835$	4.62079e24i	0.676677i
$836$	0	0
$837$	2.75252e24i	0.394971i
$838$	0	0
$839$	−4.72339e24	−0.664167	−0.332083	−	0.943250i	$-0.607752\pi$
$839$	−4.72339e24	−0.664167	−0.332083	+	0.943250i	$0.607752\pi$
$840$	0	0
$841$	−2.43599e24	−0.335667
$842$	0	0
$843$	− 3.67101e23i	− 0.0495737i
$844$	0	0
$845$	1.66847e24i	0.220819i
$846$	0	0
$847$	4.02894e23	0.0522616
$848$	0	0
$849$	8.55270e23	0.108740
$850$	0	0
$851$	5.60253e24i	0.698205i
$852$	0	0
$853$	7.83079e24i	0.956619i	0.878191	+	0.478309i	$0.158750\pi$
$853$	7.83079e24i	0.956619i	−0.878191	+	0.478309i	$0.841250\pi$
$854$	0	0
$855$	−2.38872e24	−0.286057
$856$	0	0
$857$	−4.64085e24	−0.544830	−0.272415	−	0.962180i	$-0.587822\pi$
$857$	−4.64085e24	−0.544830	−0.272415	+	0.962180i	$0.587822\pi$
$858$	0	0
$859$	− 6.90901e24i	− 0.795195i	−0.917560	−	0.397598i	$-0.869844\pi$
$859$	− 6.90901e24i	− 0.795195i	0.917560	−	0.397598i	$-0.130156\pi$
$860$	0	0
$861$	8.15629e22i	0.00920378i
$862$	0	0
$863$	−4.17194e24	−0.461579	−0.230790	−	0.973004i	$-0.574131\pi$
$863$	−4.17194e24	−0.461579	−0.230790	+	0.973004i	$0.574131\pi$
$864$	0	0
$865$	−1.03293e25	−1.12056
$866$	0	0
$867$	2.05260e24i	0.218344i
$868$	0	0
$869$	1.29778e25i	1.35373i
$870$	0	0
$871$	1.60442e25	1.64121
$872$	0	0
$873$	1.61985e25	1.62500
$874$	0	0
$875$	1.19090e24i	0.117168i
$876$	0	0
$877$	2.21294e24i	0.213537i	0.994284	+	0.106768i	$0.0340504\pi$
$877$	2.21294e24i	0.213537i	−0.994284	+	0.106768i	$0.965950\pi$
$878$	0	0
$879$	−8.10723e23	−0.0767303
$880$	0	0
$881$	−1.30646e24	−0.121283	−0.0606415	−	0.998160i	$-0.519315\pi$
$881$	−1.30646e24	−0.121283	−0.0606415	+	0.998160i	$0.519315\pi$
$882$	0	0
$883$	2.46757e24i	0.224700i	0.993669	+	0.112350i	$0.0358378\pi$
$883$	2.46757e24i	0.224700i	−0.993669	+	0.112350i	$0.964162\pi$
$884$	0	0
$885$	− 4.04008e24i	− 0.360888i
$886$	0	0
$887$	−2.19751e24	−0.192567	−0.0962833	−	0.995354i	$-0.530695\pi$
$887$	−2.19751e24	−0.192567	−0.0962833	+	0.995354i	$0.530695\pi$
$888$	0	0
$889$	−4.52734e24	−0.389205
$890$	0	0
$891$	1.13701e25i	0.958971i
$892$	0	0
$893$	1.06580e22i	0 0.000881939i
$894$	0	0
$895$	−2.40844e25	−1.95542
$896$	0	0
$897$	2.12386e24	0.169196
$898$	0	0
$899$	− 2.04573e25i	− 1.59916i
$900$	0	0
$901$	1.13036e25i	0.867073i
$902$	0	0
$903$	−3.93660e23	−0.0296330
$904$	0	0
$905$	−2.84256e25	−2.09989
$906$	0	0
$907$	2.67764e25i	1.94129i	0.240512	+	0.970646i	$0.422685\pi$
$907$	2.67764e25i	1.94129i	−0.240512	+	0.970646i	$0.577315\pi$
$908$	0	0
$909$	8.62592e24i	0.613780i
$910$	0	0
$911$	−2.35546e25	−1.64501	−0.822506	−	0.568757i	$-0.807424\pi$
$911$	−2.35546e25	−1.64501	−0.822506	+	0.568757i	$0.807424\pi$
$912$	0	0
$913$	−1.07581e24	−0.0737456
$914$	0	0
$915$	− 1.77364e24i	− 0.119340i
$916$	0	0
$917$	− 2.85816e25i	− 1.88776i
$918$	0	0
$919$	5.71735e24	0.370692	0.185346	−	0.982673i	$-0.440659\pi$
$919$	5.71735e24	0.370692	0.185346	+	0.982673i	$0.440659\pi$
$920$	0	0
$921$	−3.05227e24	−0.194274
$922$	0	0
$923$	2.16938e25i	1.35557i
$924$	0	0
$925$	9.68302e24i	0.594027i
$926$	0	0
$927$	3.90382e24	0.235132
$928$	0	0
$929$	2.14170e25	1.26656	0.633278	−	0.773925i	$-0.281709\pi$
$929$	2.14170e25	1.26656	0.633278	+	0.773925i	$0.281709\pi$
$930$	0	0
$931$	− 1.23345e24i	− 0.0716222i
$932$	0	0
$933$	1.06125e24i	0.0605096i
$934$	0	0
$935$	4.01898e25	2.25018
$936$	0	0
$937$	1.03504e25	0.569076	0.284538	−	0.958665i	$-0.408160\pi$
$937$	1.03504e25	0.569076	0.284538	+	0.958665i	$0.408160\pi$
$938$	0	0
$939$	− 2.65993e24i	− 0.143619i
$940$	0	0
$941$	− 1.49549e25i	− 0.792997i	−0.918035	−	0.396498i	$-0.870225\pi$
$941$	− 1.49549e25i	− 0.792997i	0.918035	−	0.396498i	$-0.129775\pi$
$942$	0	0
$943$	1.15339e24	0.0600657
$944$	0	0
$945$	−8.96918e24	−0.458757
$946$	0	0
$947$	2.05139e25i	1.03056i	0.857021	+	0.515281i	$0.172312\pi$
$947$	2.05139e25i	1.03056i	−0.857021	+	0.515281i	$0.827688\pi$
$948$	0	0
$949$	4.34597e24i	0.214449i
$950$	0	0
$951$	−3.46531e24	−0.167961
$952$	0	0
$953$	−1.91249e25	−0.910563	−0.455281	−	0.890348i	$-0.650461\pi$
$953$	−1.91249e25	−0.910563	−0.455281	+	0.890348i	$0.650461\pi$
$954$	0	0
$955$	− 1.82743e25i	− 0.854695i
$956$	0	0
$957$	3.70870e24i	0.170400i
$958$	0	0
$959$	5.87117e24	0.265013
$960$	0	0
$961$	2.06250e25	0.914627
$962$	0	0
$963$	− 3.43779e25i	− 1.49781i
$964$	0	0
$965$	− 1.12866e25i	− 0.483148i
$966$	0	0
$967$	8.63419e24	0.363159	0.181579	−	0.983376i	$-0.441879\pi$
$967$	8.63419e24	0.363159	0.181579	+	0.983376i	$0.441879\pi$
$968$	0	0
$969$	1.16438e24	0.0481217
$970$	0	0
$971$	− 7.31893e24i	− 0.297224i	−0.988896	−	0.148612i	$-0.952519\pi$
$971$	− 7.31893e24i	− 0.297224i	0.988896	−	0.148612i	$-0.0474806\pi$
$972$	0	0
$973$	− 1.12657e25i	− 0.449572i
$974$	0	0
$975$	3.67073e24	0.143950
$976$	0	0
$977$	−1.91796e25	−0.739154	−0.369577	−	0.929200i	$-0.620497\pi$
$977$	−1.91796e25	−0.739154	−0.369577	+	0.929200i	$0.620497\pi$
$978$	0	0
$979$	− 2.75246e25i	− 1.04248i
$980$	0	0
$981$	1.67940e25i	0.625128i
$982$	0	0
$983$	−9.21150e24	−0.336996	−0.168498	−	0.985702i	$-0.553892\pi$
$983$	−9.21150e24	−0.336996	−0.168498	+	0.985702i	$0.553892\pi$
$984$	0	0
$985$	5.03225e25	1.80948
$986$	0	0
$987$	1.97991e22i	0 0.000699760i
$988$	0	0
$989$	5.56678e24i	0.193391i
$990$	0	0
$991$	−5.52241e25	−1.88583	−0.942915	−	0.333034i	$-0.891927\pi$
$991$	−5.52241e25	−1.88583	−0.942915	+	0.333034i	$0.891927\pi$
$992$	0	0
$993$	−7.15413e24	−0.240153
$994$	0	0
$995$	1.98790e25i	0.655993i
$996$	0	0
$997$	− 3.29322e25i	− 1.06835i	−0.845375	−	0.534174i	$-0.820623\pi$
$997$	− 3.29322e25i	− 1.06835i	0.845375	−	0.534174i	$-0.179377\pi$
$998$	0	0
$999$	5.73470e24	0.182896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.18.b.a.17.9		16
4.3	odd	2		8.18.b.a.5.12	yes	16
8.3	odd	2		8.18.b.a.5.11	✓	16
8.5	even	2	inner	32.18.b.a.17.8		16
12.11	even	2		72.18.d.b.37.5		16
24.11	even	2		72.18.d.b.37.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.b.a.5.11	✓	16	8.3	odd	2
8.18.b.a.5.12	yes	16	4.3	odd	2
32.18.b.a.17.8		16	8.5	even	2	inner
32.18.b.a.17.9		16	1.1	even	1	trivial
72.18.d.b.37.5		16	12.11	even	2
72.18.d.b.37.6		16	24.11	even	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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1638.94i q^{3} -1.21254e6i q^{5} -1.76580e7 q^{7} +1.26454e8 q^{9} +O(q^{10})\)	\(q+1638.94i q^{3} -1.21254e6i q^{5} -1.76580e7 q^{7} +1.26454e8 q^{9} +7.26817e8i q^{11} +3.16645e9i q^{13} +1.98729e9 q^{15} +4.56030e10 q^{17} -1.55789e10i q^{19} -2.89404e10i q^{21} -4.09250e11 q^{23} -7.07318e11 q^{25} +4.18904e11i q^{27} -3.11338e12i q^{29} +6.57077e12 q^{31} -1.19121e12 q^{33} +2.14111e13i q^{35} -1.36898e13i q^{37} -5.18964e12 q^{39} -2.81830e12 q^{41} -1.36024e13i q^{43} -1.53331e14i q^{45} -6.84132e11 q^{47} +7.91743e13 q^{49} +7.47407e13i q^{51} +2.47869e14i q^{53} +8.81297e14 q^{55} +2.55329e13 q^{57} -2.03296e15i q^{59} -8.92495e14i q^{61} -2.23292e15 q^{63} +3.83946e15 q^{65} -5.06693e15i q^{67} -6.70737e14i q^{69} +6.85113e15 q^{71} +1.37250e15 q^{73} -1.15925e15i q^{75} -1.28341e16i q^{77} +1.78556e16 q^{79} +1.56437e16 q^{81} +1.48017e15i q^{83} -5.52955e16i q^{85} +5.10265e15 q^{87} -3.78700e16 q^{89} -5.59132e16i q^{91} +1.07691e16i q^{93} -1.88900e16 q^{95} +1.28098e17 q^{97} +9.19090e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) 16 * q - 11529600 * q^7 - 602654096 * q^9	\( 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) \) 16 * q - 11529600 * q^7 - 602654096 * q^9 + 9993282176 * q^15 - 7489125600 * q^17 - 746845345920 * q^23 - 1809682431664 * q^25 + 318979758592 * q^31 + 5633526177600 * q^33 + 18457706051456 * q^39 + 7482251536032 * q^41 + 376698804821760 * q^47 + 127691292101520 * q^49 - 2209036687713152 * q^55 - 190521298294720 * q^57 + 8131096607338880 * q^63 + 2385987975356160 * q^65 - 9025926285576576 * q^71 + 11332002046118560 * q^73 + 45299671392008448 * q^79 + 20101901999290832 * q^81 - 25965768920837760 * q^87 - 69879174608766048 * q^89 - 93790444358203776 * q^95 + 95593398602180640 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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