LMFDB - Embedded newform 32.18.b.a.17.13

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			17.13
Root			$0.500000 - 3446.66i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.18.b.a.17.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13786.7i q^{3} -96356.3i q^{5} +1.47728e7 q^{7} -6.09318e7 q^{9} +O(q^{10})$	$q+13786.7i q^{3} -96356.3i q^{5} +1.47728e7 q^{7} -6.09318e7 q^{9} +6.68721e8i q^{11} +1.72473e8i q^{13} +1.32843e9 q^{15} +1.92991e10 q^{17} -1.11328e11i q^{19} +2.03668e11i q^{21} +5.92040e11 q^{23} +7.53655e11 q^{25} +9.40365e11i q^{27} -2.95085e12i q^{29} +7.53978e12 q^{31} -9.21942e12 q^{33} -1.42346e12i q^{35} +2.20235e13i q^{37} -2.37783e12 q^{39} -8.98214e13 q^{41} -1.45802e13i q^{43} +5.87116e12i q^{45} +7.43032e13 q^{47} -1.43935e13 q^{49} +2.66070e14i q^{51} +2.84157e14i q^{53} +6.44354e13 q^{55} +1.53484e15 q^{57} +1.09328e15i q^{59} +4.14860e14i q^{61} -9.00136e14 q^{63} +1.66189e13 q^{65} +2.20200e15i q^{67} +8.16225e15i q^{69} -6.67796e15 q^{71} -1.64289e15 q^{73} +1.03904e16i q^{75} +9.87891e15i q^{77} +1.46317e16 q^{79} -2.08332e16 q^{81} +3.05535e16i q^{83} -1.85959e15i q^{85} +4.06824e16 q^{87} +4.48498e16 q^{89} +2.54792e15i q^{91} +1.03948e17i q^{93} -1.07272e16 q^{95} -3.26514e16 q^{97} -4.07464e16i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	13786.7i	1.21319i	0.795011	+	0.606594i	$0.207465\pi$
$3$	13786.7i	1.21319i	−0.795011	+	0.606594i	$0.792535\pi$
$4$	0	0
$5$	− 96356.3i	− 0.110315i	−0.998478	−	0.0551575i	$-0.982434\pi$
$5$	− 96356.3i	− 0.110315i	0.998478	−	0.0551575i	$-0.0175661\pi$
$6$	0	0
$7$	1.47728e7	0.968570	0.484285	−	0.874910i	$-0.339080\pi$
$7$	1.47728e7	0.968570	0.484285	+	0.874910i	$0.339080\pi$
$8$	0	0
$9$	−6.09318e7	−0.471827
$10$	0	0
$11$	6.68721e8i	0.940604i	0.882505	+	0.470302i	$0.155855\pi$
$11$	6.68721e8i	0.940604i	−0.882505	+	0.470302i	$0.844145\pi$
$12$	0	0
$13$	1.72473e8i	0.0586413i	0.999570	+	0.0293206i	$0.00933439\pi$
$13$	1.72473e8i	0.0586413i	−0.999570	+	0.0293206i	$0.990666\pi$
$14$	0	0
$15$	1.32843e9	0.133833
$16$	0	0
$17$	1.92991e10	0.670998	0.335499	−	0.942041i	$-0.391095\pi$
$17$	1.92991e10	0.670998	0.335499	+	0.942041i	$0.391095\pi$
$18$	0	0
$19$	− 1.11328e11i	− 1.50383i	−0.659259	−	0.751916i	$-0.729130\pi$
$19$	− 1.11328e11i	− 1.50383i	0.659259	−	0.751916i	$-0.270870\pi$
$20$	0	0
$21$	2.03668e11i	1.17506i
$22$	0	0
$23$	5.92040e11	1.57639	0.788196	−	0.615424i	$-0.211015\pi$
$23$	5.92040e11	1.57639	0.788196	+	0.615424i	$0.211015\pi$
$24$	0	0
$25$	7.53655e11	0.987831
$26$	0	0
$27$	9.40365e11i	0.640774i
$28$	0	0
$29$	− 2.95085e12i	− 1.09538i	−0.836682	−	0.547689i	$-0.815508\pi$
$29$	− 2.95085e12i	− 1.09538i	0.836682	−	0.547689i	$-0.184492\pi$
$30$	0	0
$31$	7.53978e12	1.58776	0.793879	−	0.608075i	$-0.208058\pi$
$31$	7.53978e12	1.58776	0.793879	+	0.608075i	$0.208058\pi$
$32$	0	0
$33$	−9.21942e12	−1.14113
$34$	0	0
$35$	− 1.42346e12i	− 0.106848i
$36$	0	0
$37$	2.20235e13i	1.03079i	0.856952	+	0.515396i	$0.172355\pi$
$37$	2.20235e13i	1.03079i	−0.856952	+	0.515396i	$0.827645\pi$
$38$	0	0
$39$	−2.37783e12	−0.0711430
$40$	0	0
$41$	−8.98214e13	−1.75678	−0.878389	−	0.477946i	$-0.841382\pi$
$41$	−8.98214e13	−1.75678	−0.878389	+	0.477946i	$0.841382\pi$
$42$	0	0
$43$	− 1.45802e13i	− 0.190231i	−0.995466	−	0.0951157i	$-0.969678\pi$
$43$	− 1.45802e13i	− 0.190231i	0.995466	−	0.0951157i	$-0.0303221\pi$
$44$	0	0
$45$	5.87116e12i	0.0520496i
$46$	0	0
$47$	7.43032e13	0.455172	0.227586	−	0.973758i	$-0.426917\pi$
$47$	7.43032e13	0.455172	0.227586	+	0.973758i	$0.426917\pi$
$48$	0	0
$49$	−1.43935e13	−0.0618727
$50$	0	0
$51$	2.66070e14i	0.814047i
$52$	0	0
$53$	2.84157e14i	0.626922i	0.949601	+	0.313461i	$0.101488\pi$
$53$	2.84157e14i	0.626922i	−0.949601	+	0.313461i	$0.898512\pi$
$54$	0	0
$55$	6.44354e13	0.103763
$56$	0	0
$57$	1.53484e15	1.82443
$58$	0	0
$59$	1.09328e15i	0.969373i	0.874688	+	0.484687i	$0.161066\pi$
$59$	1.09328e15i	0.969373i	−0.874688	+	0.484687i	$0.838934\pi$
$60$	0	0
$61$	4.14860e14i	0.277075i	0.990357	+	0.138538i	$0.0442402\pi$
$61$	4.14860e14i	0.277075i	−0.990357	+	0.138538i	$0.955760\pi$
$62$	0	0
$63$	−9.00136e14	−0.456997
$64$	0	0
$65$	1.66189e13	0.00646902
$66$	0	0
$67$	2.20200e15i	0.662492i	0.943544	+	0.331246i	$0.107469\pi$
$67$	2.20200e15i	0.662492i	−0.943544	+	0.331246i	$0.892531\pi$
$68$	0	0
$69$	8.16225e15i	1.91246i
$70$	0	0
$71$	−6.67796e15	−1.22729	−0.613646	−	0.789581i	$-0.710298\pi$
$71$	−6.67796e15	−1.22729	−0.613646	+	0.789581i	$0.710298\pi$
$72$	0	0
$73$	−1.64289e15	−0.238432	−0.119216	−	0.992868i	$-0.538038\pi$
$73$	−1.64289e15	−0.238432	−0.119216	+	0.992868i	$0.538038\pi$
$74$	0	0
$75$	1.03904e16i	1.19842i
$76$	0	0
$77$	9.87891e15i	0.911041i
$78$	0	0
$79$	1.46317e16	1.08509	0.542543	−	0.840028i	$-0.317461\pi$
$79$	1.46317e16	1.08509	0.542543	+	0.840028i	$0.317461\pi$
$80$	0	0
$81$	−2.08332e16	−1.24921
$82$	0	0
$83$	3.05535e16i	1.48901i	0.667618	+	0.744504i	$0.267314\pi$
$83$	3.05535e16i	1.48901i	−0.667618	+	0.744504i	$0.732686\pi$
$84$	0	0
$85$	− 1.85959e15i	− 0.0740211i
$86$	0	0
$87$	4.06824e16	1.32890
$88$	0	0
$89$	4.48498e16	1.20766	0.603831	−	0.797112i	$-0.293640\pi$
$89$	4.48498e16	1.20766	0.603831	+	0.797112i	$0.293640\pi$
$90$	0	0
$91$	2.54792e15i	0.0567982i
$92$	0	0
$93$	1.03948e17i	1.92625i
$94$	0	0
$95$	−1.07272e16	−0.165895
$96$	0	0
$97$	−3.26514e16	−0.423002	−0.211501	−	0.977378i	$-0.567835\pi$
$97$	−3.26514e16	−0.423002	−0.211501	+	0.977378i	$0.567835\pi$
$98$	0	0
$99$	− 4.07464e16i	− 0.443802i
$100$	0	0
$101$	− 4.25294e16i	− 0.390803i	−0.980723	−	0.195402i	$-0.937399\pi$
$101$	− 4.25294e16i	− 0.390803i	0.980723	−	0.195402i	$-0.0626010\pi$
$102$	0	0
$103$	−1.23310e17	−0.959140	−0.479570	−	0.877504i	$-0.659207\pi$
$103$	−1.23310e17	−0.959140	−0.479570	+	0.877504i	$0.659207\pi$
$104$	0	0
$105$	1.96247e16	0.129627
$106$	0	0
$107$	6.12850e16i	0.344820i	0.985025	+	0.172410i	$0.0551554\pi$
$107$	6.12850e16i	0.344820i	−0.985025	+	0.172410i	$0.944845\pi$
$108$	0	0
$109$	− 3.63163e17i	− 1.74573i	−0.487962	−	0.872865i	$-0.662260\pi$
$109$	− 3.63163e17i	− 1.74573i	0.487962	−	0.872865i	$-0.337740\pi$
$110$	0	0
$111$	−3.03630e17	−1.25054
$112$	0	0
$113$	6.33950e16	0.224330	0.112165	−	0.993690i	$-0.464221\pi$
$113$	6.33950e16	0.224330	0.112165	+	0.993690i	$0.464221\pi$
$114$	0	0
$115$	− 5.70467e16i	− 0.173900i
$116$	0	0
$117$	− 1.05091e16i	− 0.0276685i
$118$	0	0
$119$	2.85102e17	0.649908
$120$	0	0
$121$	5.82597e16	0.115264
$122$	0	0
$123$	− 1.23834e18i	− 2.13130i
$124$	0	0
$125$	− 1.46133e17i	− 0.219288i
$126$	0	0
$127$	8.96529e17	1.17553	0.587764	−	0.809033i	$-0.300009\pi$
$127$	8.96529e17	1.17553	0.587764	+	0.809033i	$0.300009\pi$
$128$	0	0
$129$	2.01013e17	0.230787
$130$	0	0
$131$	− 7.31204e17i	− 0.736601i	−0.929707	−	0.368301i	$-0.879940\pi$
$131$	− 7.31204e17i	− 0.736601i	0.929707	−	0.368301i	$-0.120060\pi$
$132$	0	0
$133$	− 1.64463e18i	− 1.45657i
$134$	0	0
$135$	9.06101e16	0.0706870
$136$	0	0
$137$	−1.67484e18	−1.15305	−0.576524	−	0.817080i	$-0.695591\pi$
$137$	−1.67484e18	−1.15305	−0.576524	+	0.817080i	$0.695591\pi$
$138$	0	0
$139$	1.51826e18i	0.924105i	0.886853	+	0.462053i	$0.152887\pi$
$139$	1.51826e18i	0.924105i	−0.886853	+	0.462053i	$0.847113\pi$
$140$	0	0
$141$	1.02439e18i	0.552209i
$142$	0	0
$143$	−1.15336e17	−0.0551582
$144$	0	0
$145$	−2.84333e17	−0.120837
$146$	0	0
$147$	− 1.98438e17i	− 0.0750632i
$148$	0	0
$149$	− 9.25567e17i	− 0.312122i	−0.987747	−	0.156061i	$-0.950120\pi$
$149$	− 9.25567e17i	− 0.312122i	0.987747	−	0.156061i	$-0.0498796\pi$
$150$	0	0
$151$	−2.49950e18	−0.752574	−0.376287	−	0.926503i	$-0.622799\pi$
$151$	−2.49950e18	−0.752574	−0.376287	+	0.926503i	$0.622799\pi$
$152$	0	0
$153$	−1.17593e18	−0.316595
$154$	0	0
$155$	− 7.26505e17i	− 0.175154i
$156$	0	0
$157$	5.60339e18i	1.21145i	0.795676	+	0.605723i	$0.207116\pi$
$157$	5.60339e18i	1.21145i	−0.795676	+	0.605723i	$0.792884\pi$
$158$	0	0
$159$	−3.91757e18	−0.760574
$160$	0	0
$161$	8.74611e18	1.52685
$162$	0	0
$163$	9.07691e18i	1.42673i	0.700790	+	0.713367i	$0.252831\pi$
$163$	9.07691e18i	1.42673i	−0.700790	+	0.713367i	$0.747169\pi$
$164$	0	0
$165$	8.88349e17i	0.125884i
$166$	0	0
$167$	3.89443e18	0.498143	0.249071	−	0.968485i	$-0.419875\pi$
$167$	3.89443e18	0.498143	0.249071	+	0.968485i	$0.419875\pi$
$168$	0	0
$169$	8.62067e18	0.996561
$170$	0	0
$171$	6.78342e18i	0.709548i
$172$	0	0
$173$	1.88076e19i	1.78214i	0.453868	+	0.891069i	$0.350044\pi$
$173$	1.88076e19i	1.78214i	−0.453868	+	0.891069i	$0.649956\pi$
$174$	0	0
$175$	1.11336e19	0.956783
$176$	0	0
$177$	−1.50727e19	−1.17603
$178$	0	0
$179$	− 3.45942e18i	− 0.245331i	−0.992448	−	0.122665i	$-0.960856\pi$
$179$	− 3.45942e18i	− 0.245331i	0.992448	−	0.122665i	$-0.0391442\pi$
$180$	0	0
$181$	− 1.78211e19i	− 1.14992i	−0.818182	−	0.574959i	$-0.805018\pi$
$181$	− 1.78211e19i	− 1.14992i	0.818182	−	0.574959i	$-0.194982\pi$
$182$	0	0
$183$	−5.71953e18	−0.336144
$184$	0	0
$185$	2.12210e18	0.113712
$186$	0	0
$187$	1.29057e19i	0.631143i
$188$	0	0
$189$	1.38919e19i	0.620634i
$190$	0	0
$191$	−6.12664e18	−0.250287	−0.125144	−	0.992139i	$-0.539939\pi$
$191$	−6.12664e18	−0.250287	−0.125144	+	0.992139i	$0.539939\pi$
$192$	0	0
$193$	−3.57527e19	−1.33682	−0.668408	−	0.743795i	$-0.733024\pi$
$193$	−3.57527e19	−1.33682	−0.668408	+	0.743795i	$0.733024\pi$
$194$	0	0
$195$	2.29119e17i	0.00784814i
$196$	0	0
$197$	− 1.53763e19i	− 0.482936i	−0.970409	−	0.241468i	$-0.922371\pi$
$197$	− 1.53763e19i	− 0.482936i	0.970409	−	0.241468i	$-0.0776290\pi$
$198$	0	0
$199$	2.52712e19	0.728407	0.364204	−	0.931319i	$-0.381341\pi$
$199$	2.52712e19	0.728407	0.364204	+	0.931319i	$0.381341\pi$
$200$	0	0
$201$	−3.03582e19	−0.803728
$202$	0	0
$203$	− 4.35925e19i	− 1.06095i
$204$	0	0
$205$	8.65485e18i	0.193799i
$206$	0	0
$207$	−3.60741e19	−0.743784
$208$	0	0
$209$	7.44474e19	1.41451
$210$	0	0
$211$	− 7.63891e19i	− 1.33854i	−0.743020	−	0.669269i	$-0.766607\pi$
$211$	− 7.63891e19i	− 1.33854i	0.743020	−	0.669269i	$-0.233393\pi$
$212$	0	0
$213$	− 9.20667e19i	− 1.48894i
$214$	0	0
$215$	−1.40490e18	−0.0209854
$216$	0	0
$217$	1.11384e20	1.53785
$218$	0	0
$219$	− 2.26500e19i	− 0.289263i
$220$	0	0
$221$	3.32858e18i	0.0393482i
$222$	0	0
$223$	5.35584e19	0.586457	0.293229	−	0.956042i	$-0.405270\pi$
$223$	5.35584e19	0.586457	0.293229	+	0.956042i	$0.405270\pi$
$224$	0	0
$225$	−4.59216e19	−0.466085
$226$	0	0
$227$	8.47585e19i	0.797928i	0.916967	+	0.398964i	$0.130630\pi$
$227$	8.47585e19i	0.797928i	−0.916967	+	0.398964i	$0.869370\pi$
$228$	0	0
$229$	− 2.17239e20i	− 1.89817i	−0.315016	−	0.949086i	$-0.602010\pi$
$229$	− 2.17239e20i	− 1.89817i	0.315016	−	0.949086i	$-0.397990\pi$
$230$	0	0
$231$	−1.36197e20	−1.10526
$232$	0	0
$233$	−4.79641e19	−0.361736	−0.180868	−	0.983507i	$-0.557891\pi$
$233$	−4.79641e19	−0.361736	−0.180868	+	0.983507i	$0.557891\pi$
$234$	0	0
$235$	− 7.15957e18i	− 0.0502123i
$236$	0	0
$237$	2.01722e20i	1.31641i
$238$	0	0
$239$	6.27917e18	0.0381523	0.0190761	−	0.999818i	$-0.493928\pi$
$239$	6.27917e18	0.0381523	0.0190761	+	0.999818i	$0.493928\pi$
$240$	0	0
$241$	1.50770e19	0.0853436	0.0426718	−	0.999089i	$-0.486413\pi$
$241$	1.50770e19	0.0853436	0.0426718	+	0.999089i	$0.486413\pi$
$242$	0	0
$243$	− 1.65782e20i	− 0.874749i
$244$	0	0
$245$	1.38690e18i	0.00682549i
$246$	0	0
$247$	1.92011e19	0.0881866
$248$	0	0
$249$	−4.21231e20	−1.80645
$250$	0	0
$251$	6.51149e18i	0.0260888i	0.999915	+	0.0130444i	$0.00415227\pi$
$251$	6.51149e18i	0.0260888i	−0.999915	+	0.0130444i	$0.995848\pi$
$252$	0	0
$253$	3.95909e20i	1.48276i
$254$	0	0
$255$	2.56375e19	0.0898016
$256$	0	0
$257$	2.25404e20	0.738805	0.369402	−	0.929269i	$-0.379562\pi$
$257$	2.25404e20	0.738805	0.369402	+	0.929269i	$0.379562\pi$
$258$	0	0
$259$	3.25349e20i	0.998394i
$260$	0	0
$261$	1.79801e20i	0.516829i
$262$	0	0
$263$	4.46257e20	1.20216	0.601079	−	0.799190i	$-0.294738\pi$
$263$	4.46257e20	1.20216	0.601079	+	0.799190i	$0.294738\pi$
$264$	0	0
$265$	2.73803e19	0.0691589
$266$	0	0
$267$	6.18329e20i	1.46512i
$268$	0	0
$269$	− 4.92825e20i	− 1.09597i	−0.836489	−	0.547984i	$-0.815395\pi$
$269$	− 4.92825e20i	− 1.09597i	0.836489	−	0.547984i	$-0.184605\pi$
$270$	0	0
$271$	3.92364e20	0.819313	0.409657	−	0.912240i	$-0.365649\pi$
$271$	3.92364e20	0.819313	0.409657	+	0.912240i	$0.365649\pi$
$272$	0	0
$273$	−3.51273e19	−0.0689069
$274$	0	0
$275$	5.03985e20i	0.929158i
$276$	0	0
$277$	7.33443e20i	1.27142i	0.771929	+	0.635709i	$0.219292\pi$
$277$	7.33443e20i	1.27142i	−0.771929	+	0.635709i	$0.780708\pi$
$278$	0	0
$279$	−4.59413e20	−0.749147
$280$	0	0
$281$	1.18919e21	1.82493	0.912467	−	0.409149i	$-0.134174\pi$
$281$	1.18919e21	1.82493	0.912467	+	0.409149i	$0.134174\pi$
$282$	0	0
$283$	− 6.56438e20i	− 0.948438i	−0.880407	−	0.474219i	$-0.842731\pi$
$283$	− 6.56438e20i	− 0.948438i	0.880407	−	0.474219i	$-0.157269\pi$
$284$	0	0
$285$	− 1.47892e20i	− 0.201262i
$286$	0	0
$287$	−1.32692e21	−1.70156
$288$	0	0
$289$	−4.54785e20	−0.549762
$290$	0	0
$291$	− 4.50154e20i	− 0.513181i
$292$	0	0
$293$	6.98093e20i	0.750825i	0.926858	+	0.375412i	$0.122499\pi$
$293$	6.98093e20i	0.750825i	−0.926858	+	0.375412i	$0.877501\pi$
$294$	0	0
$295$	1.05345e20	0.106936
$296$	0	0
$297$	−6.28842e20	−0.602714
$298$	0	0
$299$	1.02111e20i	0.0924417i
$300$	0	0
$301$	− 2.15391e20i	− 0.184252i
$302$	0	0
$303$	5.86339e20	0.474118
$304$	0	0
$305$	3.99743e19	0.0305656
$306$	0	0
$307$	− 3.85295e20i	− 0.278687i	−0.990244	−	0.139344i	$-0.955501\pi$
$307$	− 3.85295e20i	− 0.278687i	0.990244	−	0.139344i	$-0.0444992\pi$
$308$	0	0
$309$	− 1.70003e21i	− 1.16362i
$310$	0	0
$311$	−2.10525e21	−1.36408	−0.682041	−	0.731313i	$-0.738908\pi$
$311$	−2.10525e21	−1.36408	−0.682041	+	0.731313i	$0.738908\pi$
$312$	0	0
$313$	−2.88772e20	−0.177186	−0.0885928	−	0.996068i	$-0.528237\pi$
$313$	−2.88772e20	−0.177186	−0.0885928	+	0.996068i	$0.528237\pi$
$314$	0	0
$315$	8.67338e19i	0.0504137i
$316$	0	0
$317$	− 2.62929e21i	− 1.44822i	−0.689683	−	0.724111i	$-0.742250\pi$
$317$	− 2.62929e21i	− 1.44822i	0.689683	−	0.724111i	$-0.257750\pi$
$318$	0	0
$319$	1.97329e21	1.03032
$320$	0	0
$321$	−8.44915e20	−0.418331
$322$	0	0
$323$	− 2.14853e21i	− 1.00907i
$324$	0	0
$325$	1.29985e20i	0.0579277i
$326$	0	0
$327$	5.00681e21	2.11790
$328$	0	0
$329$	1.09767e21	0.440866
$330$	0	0
$331$	− 3.08298e21i	− 1.17607i	−0.808836	−	0.588035i	$-0.799902\pi$
$331$	− 3.08298e21i	− 1.17607i	0.808836	−	0.588035i	$-0.200098\pi$
$332$	0	0
$333$	− 1.34193e21i	− 0.486355i
$334$	0	0
$335$	2.12176e20	0.0730829
$336$	0	0
$337$	−1.59524e21	−0.522363	−0.261181	−	0.965290i	$-0.584112\pi$
$337$	−1.59524e21	−0.522363	−0.261181	+	0.965290i	$0.584112\pi$
$338$	0	0
$339$	8.74005e20i	0.272155i
$340$	0	0
$341$	5.04201e21i	1.49345i
$342$	0	0
$343$	−3.64925e21	−1.02850
$344$	0	0
$345$	7.86484e20	0.210973
$346$	0	0
$347$	1.54612e21i	0.394859i	0.980317	+	0.197430i	$0.0632594\pi$
$347$	1.54612e21i	0.394859i	−0.980317	+	0.197430i	$0.936741\pi$
$348$	0	0
$349$	− 5.41385e21i	− 1.31671i	−0.752708	−	0.658355i	$-0.771253\pi$
$349$	− 5.41385e21i	− 1.31671i	0.752708	−	0.658355i	$-0.228747\pi$
$350$	0	0
$351$	−1.62188e20	−0.0375758
$352$	0	0
$353$	3.93297e21	0.868232	0.434116	−	0.900857i	$-0.357061\pi$
$353$	3.93297e21	0.868232	0.434116	+	0.900857i	$0.357061\pi$
$354$	0	0
$355$	6.43463e20i	0.135389i
$356$	0	0
$357$	3.93061e21i	0.788461i
$358$	0	0
$359$	−7.90394e21	−1.51196	−0.755981	−	0.654593i	$-0.772840\pi$
$359$	−7.90394e21	−1.51196	−0.755981	+	0.654593i	$0.772840\pi$
$360$	0	0
$361$	−6.91355e21	−1.26151
$362$	0	0
$363$	8.03207e20i	0.139837i
$364$	0	0
$365$	1.58303e20i	0.0263026i
$366$	0	0
$367$	−5.10700e20	−0.0810036	−0.0405018	−	0.999179i	$-0.512896\pi$
$367$	−5.10700e20	−0.0810036	−0.0405018	+	0.999179i	$0.512896\pi$
$368$	0	0
$369$	5.47298e21	0.828896
$370$	0	0
$371$	4.19781e21i	0.607217i
$372$	0	0
$373$	3.13000e21i	0.432533i	0.976334	+	0.216266i	$0.0693880\pi$
$373$	3.13000e21i	0.432533i	−0.976334	+	0.216266i	$0.930612\pi$
$374$	0	0
$375$	2.01469e21	0.266037
$376$	0	0
$377$	5.08943e20	0.0642344
$378$	0	0
$379$	1.31559e22i	1.58741i	0.608305	+	0.793704i	$0.291850\pi$
$379$	1.31559e22i	1.58741i	−0.608305	+	0.793704i	$0.708150\pi$
$380$	0	0
$381$	1.23601e22i	1.42614i
$382$	0	0
$383$	−3.30408e21	−0.364637	−0.182319	−	0.983239i	$-0.558360\pi$
$383$	−3.30408e21	−0.364637	−0.182319	+	0.983239i	$0.558360\pi$
$384$	0	0
$385$	9.51895e20	0.100502
$386$	0	0
$387$	8.88400e20i	0.0897564i
$388$	0	0
$389$	− 7.70023e20i	− 0.0744615i	−0.999307	−	0.0372308i	$-0.988146\pi$
$389$	− 7.70023e20i	− 0.0744615i	0.999307	−	0.0372308i	$-0.0118537\pi$
$390$	0	0
$391$	1.14258e22	1.05776
$392$	0	0
$393$	1.00809e22	0.893637
$394$	0	0
$395$	− 1.40986e21i	− 0.119701i
$396$	0	0
$397$	− 5.29693e21i	− 0.430829i	−0.976523	−	0.215415i	$-0.930890\pi$
$397$	− 5.29693e21i	− 0.430829i	0.976523	−	0.215415i	$-0.0691103\pi$
$398$	0	0
$399$	2.26740e22	1.76709
$400$	0	0
$401$	−4.56356e21	−0.340861	−0.170430	−	0.985370i	$-0.554516\pi$
$401$	−4.56356e21	−0.340861	−0.170430	+	0.985370i	$0.554516\pi$
$402$	0	0
$403$	1.30041e21i	0.0931082i
$404$	0	0
$405$	2.00741e21i	0.137806i
$406$	0	0
$407$	−1.47275e22	−0.969567
$408$	0	0
$409$	−1.58699e22	−1.00214	−0.501069	−	0.865407i	$-0.667060\pi$
$409$	−1.58699e22	−1.00214	−0.501069	+	0.865407i	$0.667060\pi$
$410$	0	0
$411$	− 2.30904e22i	− 1.39887i
$412$	0	0
$413$	1.61509e22i	0.938906i
$414$	0	0
$415$	2.94402e21	0.164260
$416$	0	0
$417$	−2.09318e22	−1.12111
$418$	0	0
$419$	2.63907e21i	0.135716i	0.997695	+	0.0678581i	$0.0216165\pi$
$419$	2.63907e21i	0.135716i	−0.997695	+	0.0678581i	$0.978383\pi$
$420$	0	0
$421$	− 1.44643e22i	− 0.714329i	−0.934041	−	0.357165i	$-0.883743\pi$
$421$	− 1.44643e22i	− 0.714329i	0.934041	−	0.357165i	$-0.116257\pi$
$422$	0	0
$423$	−4.52743e21	−0.214762
$424$	0	0
$425$	1.45449e22	0.662832
$426$	0	0
$427$	6.12866e21i	0.268367i
$428$	0	0
$429$	− 1.59010e21i	− 0.0669174i
$430$	0	0
$431$	−1.52835e22	−0.618253	−0.309127	−	0.951021i	$-0.600037\pi$
$431$	−1.52835e22	−0.618253	−0.309127	+	0.951021i	$0.600037\pi$
$432$	0	0
$433$	−2.10323e21	−0.0817975	−0.0408988	−	0.999163i	$-0.513022\pi$
$433$	−2.10323e21	−0.0817975	−0.0408988	+	0.999163i	$0.513022\pi$
$434$	0	0
$435$	− 3.92000e21i	− 0.146598i
$436$	0	0
$437$	− 6.59107e22i	− 2.37063i
$438$	0	0
$439$	−3.28337e22	−1.13598	−0.567991	−	0.823035i	$-0.692279\pi$
$439$	−3.28337e22	−1.13598	−0.567991	+	0.823035i	$0.692279\pi$
$440$	0	0
$441$	8.77020e20	0.0291932
$442$	0	0
$443$	− 9.12162e21i	− 0.292173i	−0.989272	−	0.146087i	$-0.953332\pi$
$443$	− 9.12162e21i	− 0.292173i	0.989272	−	0.146087i	$-0.0466678\pi$
$444$	0	0
$445$	− 4.32156e21i	− 0.133223i
$446$	0	0
$447$	1.27605e22	0.378663
$448$	0	0
$449$	2.31577e22	0.661610	0.330805	−	0.943699i	$-0.392680\pi$
$449$	2.31577e22	0.661610	0.330805	+	0.943699i	$0.392680\pi$
$450$	0	0
$451$	− 6.00654e22i	− 1.65243i
$452$	0	0
$453$	− 3.44598e22i	− 0.913014i
$454$	0	0
$455$	2.45508e20	0.00626570
$456$	0	0
$457$	3.81496e22	0.937999	0.468999	−	0.883198i	$-0.344615\pi$
$457$	3.81496e22	0.937999	0.468999	+	0.883198i	$0.344615\pi$
$458$	0	0
$459$	1.81482e22i	0.429958i
$460$	0	0
$461$	6.60861e22i	1.50887i	0.656373	+	0.754436i	$0.272090\pi$
$461$	6.60861e22i	1.50887i	−0.656373	+	0.754436i	$0.727910\pi$
$462$	0	0
$463$	6.02518e22	1.32596	0.662981	−	0.748636i	$-0.269291\pi$
$463$	6.02518e22	1.32596	0.662981	+	0.748636i	$0.269291\pi$
$464$	0	0
$465$	1.00161e22	0.212494
$466$	0	0
$467$	3.05437e21i	0.0624782i	0.999512	+	0.0312391i	$0.00994534\pi$
$467$	3.05437e21i	0.0624782i	−0.999512	+	0.0312391i	$0.990055\pi$
$468$	0	0
$469$	3.25298e22i	0.641670i
$470$	0	0
$471$	−7.72520e22	−1.46971
$472$	0	0
$473$	9.75010e21	0.178933
$474$	0	0
$475$	− 8.39030e22i	− 1.48553i
$476$	0	0
$477$	− 1.73142e22i	− 0.295799i
$478$	0	0
$479$	8.21420e22	1.35430	0.677148	−	0.735847i	$-0.263216\pi$
$479$	8.21420e22	1.35430	0.677148	+	0.735847i	$0.263216\pi$
$480$	0	0
$481$	−3.79846e21	−0.0604469
$482$	0	0
$483$	1.20580e23i	1.85235i
$484$	0	0
$485$	3.14617e21i	0.0466635i
$486$	0	0
$487$	−4.20663e22	−0.602473	−0.301237	−	0.953549i	$-0.597399\pi$
$487$	−4.20663e22	−0.602473	−0.301237	+	0.953549i	$0.597399\pi$
$488$	0	0
$489$	−1.25140e23	−1.73090
$490$	0	0
$491$	8.80209e22i	1.17596i	0.808875	+	0.587980i	$0.200077\pi$
$491$	8.80209e22i	1.17596i	−0.808875	+	0.587980i	$0.799923\pi$
$492$	0	0
$493$	− 5.69487e22i	− 0.734996i
$494$	0	0
$495$	−3.92617e21	−0.0489581
$496$	0	0
$497$	−9.86525e22	−1.18872
$498$	0	0
$499$	− 2.77007e22i	− 0.322579i	−0.986907	−	0.161290i	$-0.948435\pi$
$499$	− 2.77007e22i	− 0.322579i	0.986907	−	0.161290i	$-0.0515654\pi$
$500$	0	0
$501$	5.36912e22i	0.604341i
$502$	0	0
$503$	1.28825e23	1.40176	0.700878	−	0.713281i	$-0.252792\pi$
$503$	1.28825e23	1.40176	0.700878	+	0.713281i	$0.252792\pi$
$504$	0	0
$505$	−4.09798e21	−0.0431115
$506$	0	0
$507$	1.18850e23i	1.20902i
$508$	0	0
$509$	1.41955e23i	1.39653i	0.715839	+	0.698265i	$0.246044\pi$
$509$	1.41955e23i	1.39653i	−0.715839	+	0.698265i	$0.753956\pi$
$510$	0	0
$511$	−2.42702e22	−0.230938
$512$	0	0
$513$	1.04689e23	0.963615
$514$	0	0
$515$	1.18817e22i	0.105808i
$516$	0	0
$517$	4.96881e22i	0.428137i
$518$	0	0
$519$	−2.59294e23	−2.16207
$520$	0	0
$521$	−1.60040e22	−0.129154	−0.0645772	−	0.997913i	$-0.520570\pi$
$521$	−1.60040e22	−0.129154	−0.0645772	+	0.997913i	$0.520570\pi$
$522$	0	0
$523$	− 4.32799e22i	− 0.338082i	−0.985609	−	0.169041i	$-0.945933\pi$
$523$	− 4.32799e22i	− 0.338082i	0.985609	−	0.169041i	$-0.0540670\pi$
$524$	0	0
$525$	1.53496e23i	1.16076i
$526$	0	0
$527$	1.45511e23	1.06538
$528$	0	0
$529$	2.09461e23	1.48501
$530$	0	0
$531$	− 6.66158e22i	− 0.457376i
$532$	0	0
$533$	− 1.54918e22i	− 0.103020i
$534$	0	0
$535$	5.90519e21	0.0380388
$536$	0	0
$537$	4.76938e22	0.297633
$538$	0	0
$539$	− 9.62521e21i	− 0.0581977i
$540$	0	0
$541$	− 2.24293e23i	− 1.31413i	−0.753833	−	0.657066i	$-0.771797\pi$
$541$	− 2.24293e23i	− 1.31413i	0.753833	−	0.657066i	$-0.228203\pi$
$542$	0	0
$543$	2.45693e23	1.39507
$544$	0	0
$545$	−3.49931e22	−0.192580
$546$	0	0
$547$	7.07789e22i	0.377582i	0.982017	+	0.188791i	$0.0604569\pi$
$547$	7.07789e22i	0.377582i	−0.982017	+	0.188791i	$0.939543\pi$
$548$	0	0
$549$	− 2.52781e22i	− 0.130732i
$550$	0	0
$551$	−3.28512e23	−1.64726
$552$	0	0
$553$	2.16152e23	1.05098
$554$	0	0
$555$	2.92566e22i	0.137954i
$556$	0	0
$557$	− 2.94239e23i	− 1.34565i	−0.739803	−	0.672823i	$-0.765081\pi$
$557$	− 2.94239e23i	− 1.34565i	0.739803	−	0.672823i	$-0.234919\pi$
$558$	0	0
$559$	2.51470e21	0.0111554
$560$	0	0
$561$	−1.77926e23	−0.765696
$562$	0	0
$563$	9.39183e22i	0.392129i	0.980591	+	0.196064i	$0.0628162\pi$
$563$	9.39183e22i	0.392129i	−0.980591	+	0.196064i	$0.937184\pi$
$564$	0	0
$565$	− 6.10850e21i	− 0.0247470i
$566$	0	0
$567$	−3.07766e23	−1.20994
$568$	0	0
$569$	−7.54187e22	−0.287756	−0.143878	−	0.989595i	$-0.545957\pi$
$569$	−7.54187e22	−0.287756	−0.143878	+	0.989595i	$0.545957\pi$
$570$	0	0
$571$	− 3.40566e23i	− 1.26123i	−0.776095	−	0.630616i	$-0.782802\pi$
$571$	− 3.40566e23i	− 1.26123i	0.776095	−	0.630616i	$-0.217198\pi$
$572$	0	0
$573$	− 8.44659e22i	− 0.303646i
$574$	0	0
$575$	4.46194e23	1.55721
$576$	0	0
$577$	−4.78683e23	−1.62201	−0.811006	−	0.585038i	$-0.801079\pi$
$577$	−4.78683e23	−1.62201	−0.811006	+	0.585038i	$0.801079\pi$
$578$	0	0
$579$	− 4.92910e23i	− 1.62181i
$580$	0	0
$581$	4.51362e23i	1.44221i
$582$	0	0
$583$	−1.90022e23	−0.589685
$584$	0	0
$585$	−1.01262e21	−0.00305226
$586$	0	0
$587$	− 1.81919e22i	− 0.0532666i	−0.999645	−	0.0266333i	$-0.991521\pi$
$587$	− 1.81919e22i	− 0.0532666i	0.999645	−	0.0266333i	$-0.00847865\pi$
$588$	0	0
$589$	− 8.39390e23i	− 2.38772i
$590$	0	0
$591$	2.11988e23	0.585893
$592$	0	0
$593$	−9.82089e22	−0.263746	−0.131873	−	0.991267i	$-0.542099\pi$
$593$	−9.82089e22	−0.263746	−0.131873	+	0.991267i	$0.542099\pi$
$594$	0	0
$595$	− 2.74714e22i	− 0.0716946i
$596$	0	0
$597$	3.48405e23i	0.883695i
$598$	0	0
$599$	−2.49917e22	−0.0616124	−0.0308062	−	0.999525i	$-0.509807\pi$
$599$	−2.49917e22	−0.0616124	−0.0308062	+	0.999525i	$0.509807\pi$
$600$	0	0
$601$	−1.67913e23	−0.402394	−0.201197	−	0.979551i	$-0.564483\pi$
$601$	−1.67913e23	−0.402394	−0.201197	+	0.979551i	$0.564483\pi$
$602$	0	0
$603$	− 1.34172e23i	− 0.312582i
$604$	0	0
$605$	− 5.61369e21i	− 0.0127153i
$606$	0	0
$607$	3.57610e23	0.787600	0.393800	−	0.919196i	$-0.371160\pi$
$607$	3.57610e23	0.787600	0.393800	+	0.919196i	$0.371160\pi$
$608$	0	0
$609$	6.00994e23	1.28713
$610$	0	0
$611$	1.28153e22i	0.0266919i
$612$	0	0
$613$	− 5.01122e23i	− 1.01515i	−0.861608	−	0.507575i	$-0.830542\pi$
$613$	− 5.01122e23i	− 1.01515i	0.861608	−	0.507575i	$-0.169458\pi$
$614$	0	0
$615$	−1.19322e23	−0.235115
$616$	0	0
$617$	−1.62324e23	−0.311141	−0.155571	−	0.987825i	$-0.549722\pi$
$617$	−1.62324e23	−0.311141	−0.155571	+	0.987825i	$0.549722\pi$
$618$	0	0
$619$	3.38341e23i	0.630934i	0.948937	+	0.315467i	$0.102161\pi$
$619$	3.38341e23i	0.630934i	−0.948937	+	0.315467i	$0.897839\pi$
$620$	0	0
$621$	5.56734e23i	1.01011i
$622$	0	0
$623$	6.62559e23	1.16970
$624$	0	0
$625$	5.60912e23	0.963640
$626$	0	0
$627$	1.02638e24i	1.71607i
$628$	0	0
$629$	4.25033e23i	0.691659i
$630$	0	0
$631$	−1.07823e24	−1.70789	−0.853947	−	0.520359i	$-0.825798\pi$
$631$	−1.07823e24	−1.70789	−0.853947	+	0.520359i	$0.825798\pi$
$632$	0	0
$633$	1.05315e24	1.62390
$634$	0	0
$635$	− 8.63862e22i	− 0.129678i
$636$	0	0
$637$	− 2.48249e21i	− 0.00362829i
$638$	0	0
$639$	4.06900e23	0.579069
$640$	0	0
$641$	−8.18063e23	−1.13369	−0.566844	−	0.823825i	$-0.691836\pi$
$641$	−8.18063e23	−1.13369	−0.566844	+	0.823825i	$0.691836\pi$
$642$	0	0
$643$	− 1.03008e24i	− 1.39020i	−0.718913	−	0.695100i	$-0.755360\pi$
$643$	− 1.03008e24i	− 1.39020i	0.718913	−	0.695100i	$-0.244640\pi$
$644$	0	0
$645$	− 1.93688e22i	− 0.0254593i
$646$	0	0
$647$	−1.16066e24	−1.48600	−0.742998	−	0.669293i	$-0.766597\pi$
$647$	−1.16066e24	−1.48600	−0.742998	+	0.669293i	$0.766597\pi$
$648$	0	0
$649$	−7.31102e23	−0.911797
$650$	0	0
$651$	1.53561e24i	1.86571i
$652$	0	0
$653$	1.25728e24i	1.48823i	0.668054	+	0.744113i	$0.267128\pi$
$653$	1.25728e24i	1.48823i	−0.668054	+	0.744113i	$0.732872\pi$
$654$	0	0
$655$	−7.04561e22	−0.0812582
$656$	0	0
$657$	1.00104e23	0.112499
$658$	0	0
$659$	1.29456e24i	1.41774i	0.705338	+	0.708871i	$0.250795\pi$
$659$	1.29456e24i	1.41774i	−0.705338	+	0.708871i	$0.749205\pi$
$660$	0	0
$661$	− 8.22393e23i	− 0.877742i	−0.898550	−	0.438871i	$-0.855378\pi$
$661$	− 8.22393e23i	− 0.877742i	0.898550	−	0.438871i	$-0.144622\pi$
$662$	0	0
$663$	−4.58900e22	−0.0477367
$664$	0	0
$665$	−1.58471e23	−0.160681
$666$	0	0
$667$	− 1.74702e24i	− 1.72675i
$668$	0	0
$669$	7.38391e23i	0.711483i
$670$	0	0
$671$	−2.77425e23	−0.260618
$672$	0	0
$673$	1.06449e24	0.975016	0.487508	−	0.873118i	$-0.337906\pi$
$673$	1.06449e24	0.975016	0.487508	+	0.873118i	$0.337906\pi$
$674$	0	0
$675$	7.08711e23i	0.632976i
$676$	0	0
$677$	1.14609e24i	0.998198i	0.866545	+	0.499099i	$0.166336\pi$
$677$	1.14609e24i	0.998198i	−0.866545	+	0.499099i	$0.833664\pi$
$678$	0	0
$679$	−4.82355e23	−0.409707
$680$	0	0
$681$	−1.16854e24	−0.968037
$682$	0	0
$683$	1.21554e24i	0.982186i	0.871107	+	0.491093i	$0.163402\pi$
$683$	1.21554e24i	0.982186i	−0.871107	+	0.491093i	$0.836598\pi$
$684$	0	0
$685$	1.61381e23i	0.127199i
$686$	0	0
$687$	2.99500e24	2.30284
$688$	0	0
$689$	−4.90095e22	−0.0367635
$690$	0	0
$691$	− 1.76793e24i	− 1.29390i	−0.762531	−	0.646952i	$-0.776043\pi$
$691$	− 1.76793e24i	− 1.29390i	0.762531	−	0.646952i	$-0.223957\pi$
$692$	0	0
$693$	− 6.01940e23i	− 0.429854i
$694$	0	0
$695$	1.46294e23	0.101943
$696$	0	0
$697$	−1.73347e24	−1.17879
$698$	0	0
$699$	− 6.61265e23i	− 0.438854i
$700$	0	0
$701$	− 1.00329e24i	− 0.649868i	−0.945737	−	0.324934i	$-0.894658\pi$
$701$	− 1.00329e24i	− 0.649868i	0.945737	−	0.324934i	$-0.105342\pi$
$702$	0	0
$703$	2.45183e24	1.55014
$704$	0	0
$705$	9.87066e22	0.0609170
$706$	0	0
$707$	− 6.28281e23i	− 0.378520i
$708$	0	0
$709$	− 2.79760e24i	− 1.64548i	−0.568418	−	0.822740i	$-0.692444\pi$
$709$	− 2.79760e24i	− 1.64548i	0.568418	−	0.822740i	$-0.307556\pi$
$710$	0	0
$711$	−8.91536e23	−0.511973
$712$	0	0
$713$	4.46385e24	2.50293
$714$	0	0
$715$	1.11134e22i	0.00608479i
$716$	0	0
$717$	8.65688e22i	0.0462859i
$718$	0	0
$719$	−1.18105e24	−0.616699	−0.308349	−	0.951273i	$-0.599777\pi$
$719$	−1.18105e24	−0.616699	−0.308349	+	0.951273i	$0.599777\pi$
$720$	0	0
$721$	−1.82164e24	−0.928994
$722$	0	0
$723$	2.07862e23i	0.103538i
$724$	0	0
$725$	− 2.22392e24i	− 1.08205i
$726$	0	0
$727$	−2.11470e24	−1.00509	−0.502547	−	0.864550i	$-0.667604\pi$
$727$	−2.11470e24	−1.00509	−0.502547	+	0.864550i	$0.667604\pi$
$728$	0	0
$729$	−4.04830e23	−0.187970
$730$	0	0
$731$	− 2.81385e23i	− 0.127645i
$732$	0	0
$733$	1.52276e24i	0.674912i	0.941341	+	0.337456i	$0.109566\pi$
$733$	1.52276e24i	0.674912i	−0.941341	+	0.337456i	$0.890434\pi$
$734$	0	0
$735$	−1.91207e22	−0.00828061
$736$	0	0
$737$	−1.47252e24	−0.623143
$738$	0	0
$739$	− 2.35464e24i	− 0.973750i	−0.873472	−	0.486875i	$-0.838137\pi$
$739$	− 2.35464e24i	− 0.973750i	0.873472	−	0.486875i	$-0.161863\pi$
$740$	0	0
$741$	2.64719e23i	0.106987i
$742$	0	0
$743$	−1.71431e24	−0.677150	−0.338575	−	0.940939i	$-0.609945\pi$
$743$	−1.71431e24	−0.677150	−0.338575	+	0.940939i	$0.609945\pi$
$744$	0	0
$745$	−8.91841e22	−0.0344318
$746$	0	0
$747$	− 1.86168e24i	− 0.702554i
$748$	0	0
$749$	9.05354e23i	0.333982i
$750$	0	0
$751$	3.84431e23	0.138637	0.0693184	−	0.997595i	$-0.477918\pi$
$751$	3.84431e23	0.138637	0.0693184	+	0.997595i	$0.477918\pi$
$752$	0	0
$753$	−8.97716e22	−0.0316506
$754$	0	0
$755$	2.40843e23i	0.0830203i
$756$	0	0
$757$	− 1.62822e24i	− 0.548781i	−0.961618	−	0.274390i	$-0.911524\pi$
$757$	− 1.62822e24i	− 0.548781i	0.961618	−	0.274390i	$-0.0884760\pi$
$758$	0	0
$759$	−5.45827e24	−1.79887
$760$	0	0
$761$	2.88866e24	0.930950	0.465475	−	0.885061i	$-0.345884\pi$
$761$	2.88866e24	0.930950	0.465475	+	0.885061i	$0.345884\pi$
$762$	0	0
$763$	− 5.36496e24i	− 1.69086i
$764$	0	0
$765$	1.13308e23i	0.0349252i
$766$	0	0
$767$	−1.88562e23	−0.0568453
$768$	0	0
$769$	3.36233e24	0.991440	0.495720	−	0.868482i	$-0.334904\pi$
$769$	3.36233e24	0.991440	0.495720	+	0.868482i	$0.334904\pi$
$770$	0	0
$771$	3.10757e24i	0.896310i
$772$	0	0
$773$	− 7.95637e23i	− 0.224486i	−0.993681	−	0.112243i	$-0.964197\pi$
$773$	− 7.95637e23i	− 0.224486i	0.993681	−	0.112243i	$-0.0358035\pi$
$774$	0	0
$775$	5.68239e24	1.56844
$776$	0	0
$777$	−4.48548e24	−1.21124
$778$	0	0
$779$	9.99964e24i	2.64190i
$780$	0	0
$781$	− 4.46569e24i	− 1.15440i
$782$	0	0
$783$	2.77488e24	0.701890
$784$	0	0
$785$	5.39922e23	0.133641
$786$	0	0
$787$	1.65941e24i	0.401947i	0.979597	+	0.200973i	$0.0644104\pi$
$787$	1.65941e24i	0.401947i	−0.979597	+	0.200973i	$0.935590\pi$
$788$	0	0
$789$	6.15240e24i	1.45844i
$790$	0	0
$791$	9.36524e23	0.217280
$792$	0	0
$793$	−7.15522e22	−0.0162480
$794$	0	0
$795$	3.77483e23i	0.0839028i
$796$	0	0
$797$	4.56354e24i	0.992902i	0.868065	+	0.496451i	$0.165364\pi$
$797$	4.56354e24i	0.992902i	−0.868065	+	0.496451i	$0.834636\pi$
$798$	0	0
$799$	1.43398e24	0.305419
$800$	0	0
$801$	−2.73278e24	−0.569807
$802$	0	0
$803$	− 1.09863e24i	− 0.224270i
$804$	0	0
$805$	− 8.42743e23i	− 0.168434i
$806$	0	0
$807$	6.79440e24	1.32962
$808$	0	0
$809$	4.31106e24	0.826080	0.413040	−	0.910713i	$-0.364467\pi$
$809$	4.31106e24	0.826080	0.413040	+	0.910713i	$0.364467\pi$
$810$	0	0
$811$	3.06428e24i	0.574978i	0.957784	+	0.287489i	$0.0928204\pi$
$811$	3.06428e24i	0.574978i	−0.957784	+	0.287489i	$0.907180\pi$
$812$	0	0
$813$	5.40939e24i	0.993982i
$814$	0	0
$815$	8.74617e23	0.157390
$816$	0	0
$817$	−1.62319e24	−0.286076
$818$	0	0
$819$	− 1.55250e23i	− 0.0267989i
$820$	0	0
$821$	6.52742e24i	1.10363i	0.833965	+	0.551817i	$0.186065\pi$
$821$	6.52742e24i	1.10363i	−0.833965	+	0.551817i	$0.813935\pi$
$822$	0	0
$823$	−2.74865e24	−0.455219	−0.227610	−	0.973752i	$-0.573091\pi$
$823$	−2.74865e24	−0.455219	−0.227610	+	0.973752i	$0.573091\pi$
$824$	0	0
$825$	−6.94826e24	−1.12724
$826$	0	0
$827$	− 1.09366e25i	− 1.73814i	−0.494690	−	0.869069i	$-0.664718\pi$
$827$	− 1.09366e25i	− 1.73814i	0.494690	−	0.869069i	$-0.335282\pi$
$828$	0	0
$829$	− 4.30001e24i	− 0.669509i	−0.942305	−	0.334754i	$-0.891347\pi$
$829$	− 4.30001e24i	− 0.669509i	0.942305	−	0.334754i	$-0.108653\pi$
$830$	0	0
$831$	−1.01117e25	−1.54247
$832$	0	0
$833$	−2.77781e23	−0.0415164
$834$	0	0
$835$	− 3.75253e23i	− 0.0549526i
$836$	0	0
$837$	7.09015e24i	1.01739i
$838$	0	0
$839$	−4.56684e24	−0.642154	−0.321077	−	0.947053i	$-0.604045\pi$
$839$	−4.56684e24	−0.642154	−0.321077	+	0.947053i	$0.604045\pi$
$840$	0	0
$841$	−1.45037e24	−0.199854
$842$	0	0
$843$	1.63950e25i	2.21399i
$844$	0	0
$845$	− 8.30655e23i	− 0.109936i
$846$	0	0
$847$	8.60662e23	0.111641
$848$	0	0
$849$	9.05009e24	1.15063
$850$	0	0
$851$	1.30388e25i	1.62493i
$852$	0	0
$853$	− 5.04765e24i	− 0.616627i	−0.951285	−	0.308314i	$-0.900235\pi$
$853$	− 5.04765e24i	− 0.616627i	0.951285	−	0.308314i	$-0.0997647\pi$
$854$	0	0
$855$	6.53625e23	0.0782739
$856$	0	0
$857$	−3.32276e24	−0.390088	−0.195044	−	0.980795i	$-0.562485\pi$
$857$	−3.32276e24	−0.390088	−0.195044	+	0.980795i	$0.562485\pi$
$858$	0	0
$859$	− 1.10498e25i	− 1.27178i	−0.771780	−	0.635889i	$-0.780633\pi$
$859$	− 1.10498e25i	− 1.27178i	0.771780	−	0.635889i	$-0.219367\pi$
$860$	0	0
$861$	− 1.82938e25i	− 2.06432i
$862$	0	0
$863$	−2.29368e23	−0.0253770	−0.0126885	−	0.999919i	$-0.504039\pi$
$863$	−2.29368e23	−0.0253770	−0.0126885	+	0.999919i	$0.504039\pi$
$864$	0	0
$865$	1.81223e24	0.196597
$866$	0	0
$867$	− 6.26997e24i	− 0.666965i
$868$	0	0
$869$	9.78452e24i	1.02064i
$870$	0	0
$871$	−3.79786e23	−0.0388494
$872$	0	0
$873$	1.98951e24	0.199584
$874$	0	0
$875$	− 2.15881e24i	− 0.212395i
$876$	0	0
$877$	− 1.16909e25i	− 1.12811i	−0.825738	−	0.564054i	$-0.809241\pi$
$877$	− 1.16909e25i	− 1.12811i	0.825738	−	0.564054i	$-0.190759\pi$
$878$	0	0
$879$	−9.62437e24	−0.910892
$880$	0	0
$881$	6.35170e24	0.589651	0.294825	−	0.955551i	$-0.404738\pi$
$881$	6.35170e24	0.589651	0.294825	+	0.955551i	$0.404738\pi$
$882$	0	0
$883$	− 5.61888e24i	− 0.511663i	−0.966721	−	0.255831i	$-0.917651\pi$
$883$	− 5.61888e24i	− 0.511663i	0.966721	−	0.255831i	$-0.0823492\pi$
$884$	0	0
$885$	1.45235e24i	0.129734i
$886$	0	0
$887$	1.44844e25	1.26926	0.634630	−	0.772816i	$-0.281153\pi$
$887$	1.44844e25	1.26926	0.634630	+	0.772816i	$0.281153\pi$
$888$	0	0
$889$	1.32443e25	1.13858
$890$	0	0
$891$	− 1.39316e25i	− 1.17501i
$892$	0	0
$893$	− 8.27203e24i	− 0.684502i
$894$	0	0
$895$	−3.33336e23	−0.0270637
$896$	0	0
$897$	−1.40777e24	−0.112149
$898$	0	0
$899$	− 2.22488e25i	− 1.73920i
$900$	0	0
$901$	5.48397e24i	0.420663i
$902$	0	0
$903$	2.96953e24	0.223533
$904$	0	0
$905$	−1.71717e24	−0.126853
$906$	0	0
$907$	1.33655e25i	0.969000i	0.874791	+	0.484500i	$0.160999\pi$
$907$	1.33655e25i	0.969000i	−0.874791	+	0.484500i	$0.839001\pi$
$908$	0	0
$909$	2.59140e24i	0.184391i
$910$	0	0
$911$	6.62275e23	0.0462522	0.0231261	−	0.999733i	$-0.492638\pi$
$911$	6.62275e23	0.0462522	0.0231261	+	0.999733i	$0.492638\pi$
$912$	0	0
$913$	−2.04317e25	−1.40057
$914$	0	0
$915$	5.51112e23i	0.0370818i
$916$	0	0
$917$	− 1.08020e25i	− 0.713450i
$918$	0	0
$919$	1.15477e25	0.748708	0.374354	−	0.927286i	$-0.377864\pi$
$919$	1.15477e25	0.748708	0.374354	+	0.927286i	$0.377864\pi$
$920$	0	0
$921$	5.31193e24	0.338100
$922$	0	0
$923$	− 1.15177e24i	− 0.0719700i
$924$	0	0
$925$	1.65981e25i	1.01825i
$926$	0	0
$927$	7.51350e24	0.452548
$928$	0	0
$929$	−2.55502e25	−1.51099	−0.755494	−	0.655155i	$-0.772603\pi$
$929$	−2.55502e25	−1.51099	−0.755494	+	0.655155i	$0.772603\pi$
$930$	0	0
$931$	1.60240e24i	0.0930460i
$932$	0	0
$933$	− 2.90244e25i	− 1.65489i
$934$	0	0
$935$	1.24354e24	0.0696246
$936$	0	0
$937$	−1.85797e25	−1.02153	−0.510765	−	0.859720i	$-0.670638\pi$
$937$	−1.85797e25	−1.02153	−0.510765	+	0.859720i	$0.670638\pi$
$938$	0	0
$939$	− 3.98120e24i	− 0.214960i
$940$	0	0
$941$	− 2.94823e24i	− 0.156333i	−0.996940	−	0.0781664i	$-0.975093\pi$
$941$	− 2.94823e24i	− 0.156333i	0.996940	−	0.0781664i	$-0.0249066\pi$
$942$	0	0
$943$	−5.31778e25	−2.76937
$944$	0	0
$945$	1.33857e24	0.0684653
$946$	0	0
$947$	2.14283e25i	1.07650i	0.842786	+	0.538249i	$0.180914\pi$
$947$	2.14283e25i	1.07650i	−0.842786	+	0.538249i	$0.819086\pi$
$948$	0	0
$949$	− 2.83355e23i	− 0.0139819i
$950$	0	0
$951$	3.62492e25	1.75697
$952$	0	0
$953$	−1.03026e24	−0.0490520	−0.0245260	−	0.999699i	$-0.507808\pi$
$953$	−1.03026e24	−0.0490520	−0.0245260	+	0.999699i	$0.507808\pi$
$954$	0	0
$955$	5.90340e23i	0.0276105i
$956$	0	0
$957$	2.72051e25i	1.24997i
$958$	0	0
$959$	−2.47421e25	−1.11681
$960$	0	0
$961$	3.42982e25	1.52098
$962$	0	0
$963$	− 3.73421e24i	− 0.162695i
$964$	0	0
$965$	3.44500e24i	0.147471i
$966$	0	0
$967$	2.08697e25	0.877790	0.438895	−	0.898538i	$-0.355370\pi$
$967$	2.08697e25	0.877790	0.438895	+	0.898538i	$0.355370\pi$
$968$	0	0
$969$	2.96211e25	1.22419
$970$	0	0
$971$	4.19124e24i	0.170208i	0.996372	+	0.0851038i	$0.0271222\pi$
$971$	4.19124e24i	0.170208i	−0.996372	+	0.0851038i	$0.972878\pi$
$972$	0	0
$973$	2.24291e25i	0.895060i
$974$	0	0
$975$	−1.79206e24	−0.0702772
$976$	0	0
$977$	−2.32588e25	−0.896362	−0.448181	−	0.893943i	$-0.647928\pi$
$977$	−2.32588e25	−0.896362	−0.448181	+	0.893943i	$0.647928\pi$
$978$	0	0
$979$	2.99920e25i	1.13593i
$980$	0	0
$981$	2.21282e25i	0.823682i
$982$	0	0
$983$	8.66584e24	0.317034	0.158517	−	0.987356i	$-0.449329\pi$
$983$	8.66584e24	0.317034	0.158517	+	0.987356i	$0.449329\pi$
$984$	0	0
$985$	−1.48161e24	−0.0532752
$986$	0	0
$987$	1.51332e25i	0.534853i
$988$	0	0
$989$	− 8.63208e24i	− 0.299879i
$990$	0	0
$991$	−6.64102e24	−0.226782	−0.113391	−	0.993550i	$-0.536171\pi$
$991$	−6.64102e24	−0.226782	−0.113391	+	0.993550i	$0.536171\pi$
$992$	0	0
$993$	4.25040e25	1.42679
$994$	0	0
$995$	− 2.43503e24i	− 0.0803543i
$996$	0	0
$997$	1.58692e25i	0.514808i	0.966304	+	0.257404i	$0.0828671\pi$
$997$	1.58692e25i	0.514808i	−0.966304	+	0.257404i	$0.917133\pi$
$998$	0	0
$999$	−2.07101e25	−0.660504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.18.b.a.17.13		16
4.3	odd	2		8.18.b.a.5.7	✓	16
8.3	odd	2		8.18.b.a.5.8	yes	16
8.5	even	2	inner	32.18.b.a.17.4		16
12.11	even	2		72.18.d.b.37.10		16
24.11	even	2		72.18.d.b.37.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.18.b.a.5.7	✓	16	4.3	odd	2
8.18.b.a.5.8	yes	16	8.3	odd	2
32.18.b.a.17.4		16	8.5	even	2	inner
32.18.b.a.17.13		16	1.1	even	1	trivial
72.18.d.b.37.9		16	24.11	even	2
72.18.d.b.37.10		16	12.11	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)

\(f(q)\)	\(=\)	\(q+13786.7i q^{3} -96356.3i q^{5} +1.47728e7 q^{7} -6.09318e7 q^{9} +O(q^{10})\)	\(q+13786.7i q^{3} -96356.3i q^{5} +1.47728e7 q^{7} -6.09318e7 q^{9} +6.68721e8i q^{11} +1.72473e8i q^{13} +1.32843e9 q^{15} +1.92991e10 q^{17} -1.11328e11i q^{19} +2.03668e11i q^{21} +5.92040e11 q^{23} +7.53655e11 q^{25} +9.40365e11i q^{27} -2.95085e12i q^{29} +7.53978e12 q^{31} -9.21942e12 q^{33} -1.42346e12i q^{35} +2.20235e13i q^{37} -2.37783e12 q^{39} -8.98214e13 q^{41} -1.45802e13i q^{43} +5.87116e12i q^{45} +7.43032e13 q^{47} -1.43935e13 q^{49} +2.66070e14i q^{51} +2.84157e14i q^{53} +6.44354e13 q^{55} +1.53484e15 q^{57} +1.09328e15i q^{59} +4.14860e14i q^{61} -9.00136e14 q^{63} +1.66189e13 q^{65} +2.20200e15i q^{67} +8.16225e15i q^{69} -6.67796e15 q^{71} -1.64289e15 q^{73} +1.03904e16i q^{75} +9.87891e15i q^{77} +1.46317e16 q^{79} -2.08332e16 q^{81} +3.05535e16i q^{83} -1.85959e15i q^{85} +4.06824e16 q^{87} +4.48498e16 q^{89} +2.54792e15i q^{91} +1.03948e17i q^{93} -1.07272e16 q^{95} -3.26514e16 q^{97} -4.07464e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) 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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