LMFDB - Embedded newform 18.26.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,26,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 26, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	18.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.2794203914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{106705}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26676 $ x^2 - x - 26676
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{5}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-162.829$ of defining polynomial
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4096.00 q^{2} +1.67772e7 q^{4} +1.37041e8 q^{5} -3.04153e10 q^{7} -6.87195e10 q^{8} -5.61320e11 q^{10} -2.58704e12 q^{11} -9.57327e13 q^{13} +1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15 q^{17} +4.95030e15 q^{19} +2.29916e15 q^{20} +1.05965e16 q^{22} +1.07650e16 q^{23} -2.79243e17 q^{25} +3.92121e17 q^{26} -5.10284e17 q^{28} +1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18 q^{32} -6.74549e18 q^{34} -4.16814e18 q^{35} +1.01944e19 q^{37} -2.02764e19 q^{38} -9.41738e18 q^{40} -1.58687e20 q^{41} +1.83575e20 q^{43} -4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21 q^{47} -4.15978e20 q^{49} +1.14378e21 q^{50} -1.60613e21 q^{52} +1.99903e21 q^{53} -3.54531e20 q^{55} +2.09012e21 q^{56} -5.60091e21 q^{58} +4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22 q^{62} +4.72237e21 q^{64} -1.31193e22 q^{65} +8.67051e22 q^{67} +2.76295e22 q^{68} +1.70727e22 q^{70} +5.13159e22 q^{71} +3.49147e22 q^{73} -4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22 q^{77} +2.91588e23 q^{79} +3.85736e22 q^{80} +6.49981e23 q^{82} +1.64916e24 q^{83} +2.25686e23 q^{85} -7.51925e23 q^{86} +1.77780e23 q^{88} -8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23 q^{92} +5.74273e24 q^{94} +6.78393e23 q^{95} +1.00608e25 q^{97} +1.70384e24 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) $ 2 * q - 8192 * q^2 + 33554432 * q^4 - 741953100 * q^5 - 376536944 * q^7 - 137438953472 * q^8 + 3039039897600 * q^10 - 8323034610264 * q^11 - 106467053152292 * q^13 + 1542295322624 * q^14 + 562949953421312 * q^16 - 1327878920113956 * q^17 - 477079242949400 * q^19 - 12447907420569600 * q^20 + 34091149763641344 * q^22 + 115305025781473872 * q^23 + 195364218352298750 * q^25 + 436089049711788032 * q^26 - 6317241641467904 * q^28 - 1724412645206435580 * q^29 - 8688082288351126976 * q^31 - 2305843009213693952 * q^32 + 5438992056786763776 * q^34 - 30572039779548136800 * q^35 - 34908364049750170484 * q^37 + 1954116579120742400 * q^38 + 50986628794653081600 * q^40 - 83014324355953468884 * q^41 + 44539608583471901848 * q^43 - 139637349431874945024 * q^44 - 472289385600916979712 * q^46 - 1869360732054674599776 * q^47 - 854718643011973772046 * q^49 - 800211838371015680000 * q^50 - 1786220747619483779072 * q^52 + 3242170458210097794132 * q^53 + 4687370283547568509200 * q^55 + 25875421763452534784 * q^56 + 7063194194765560135680 * q^58 + 17413675430632049453640 * q^59 + 33975859953090212810044 * q^61 + 35586385053086216093696 * q^62 + 9444732965739290427392 * q^64 - 3683906908205094047400 * q^65 + 33012908196969622380616 * q^67 - 22278111464598584426496 * q^68 + 125223074937029168332800 * q^70 + 278348008145600813615856 * q^71 + 313442631796571689392148 * q^73 + 142984659147776698302464 * q^74 - 8004061508078560870400 * q^76 - 93616352280445975603392 * q^77 + 928546182288781647817120 * q^79 - 208841231542899022233600 * q^80 + 340026672561985408548864 * q^82 + 452542629950725008985752 * q^83 + 2840452690831654686931800 * q^85 - 182434236757900909969408 * q^86 + 571954583272959774818304 * q^88 + 2346904179674200701244620 * q^89 + 2589295264446210583737824 * q^91 + 1934497323421355948900352 * q^92 + 7656901558495947160682496 * q^94 + 5449023740708681753370000 * q^95 + 13645482361777595497674436 * q^97 + 3500927561777044570300416 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4096.00	−0.707107
$2$	−4096.00	−0.707107
$3$	0	0
$3$	0	0
$4$	1.67772e7	0.500000
$5$	1.37041e8	0.251030	0.125515	−	0.992092i	$-0.459942\pi$
$5$	1.37041e8	0.251030	0.125515	+	0.992092i	$0.459942\pi$
$6$	0	0
$7$	−3.04153e10	−0.830552	−0.415276	−	0.909696i	$-0.636315\pi$
$7$	−3.04153e10	−0.830552	−0.415276	+	0.909696i	$0.636315\pi$
$8$	−6.87195e10	−0.353553
$9$	0	0
$10$	−5.61320e11	−0.177505
$11$	−2.58704e12	−0.248539	−0.124270	−	0.992248i	$-0.539659\pi$
$11$	−2.58704e12	−0.248539	−0.124270	+	0.992248i	$0.539659\pi$
$12$	0	0
$13$	−9.57327e13	−1.13964	−0.569821	−	0.821769i	$-0.692988\pi$
$13$	−9.57327e13	−1.13964	−0.569821	+	0.821769i	$0.692988\pi$
$14$	1.24581e14	0.587289
$15$	0	0
$16$	2.81475e14	0.250000
$17$	1.64685e15	0.685555	0.342777	−	0.939417i	$-0.388632\pi$
$17$	1.64685e15	0.685555	0.342777	+	0.939417i	$0.388632\pi$
$18$	0	0
$19$	4.95030e15	0.513111	0.256555	−	0.966530i	$-0.417412\pi$
$19$	4.95030e15	0.513111	0.256555	+	0.966530i	$0.417412\pi$
$20$	2.29916e15	0.125515
$21$	0	0
$22$	1.05965e16	0.175744
$23$	1.07650e16	0.102427	0.0512136	−	0.998688i	$-0.483691\pi$
$23$	1.07650e16	0.102427	0.0512136	+	0.998688i	$0.483691\pi$
$24$	0	0
$25$	−2.79243e17	−0.936984
$26$	3.92121e17	0.805849
$27$	0	0
$28$	−5.10284e17	−0.415276
$29$	1.36741e18	0.717668	0.358834	−	0.933401i	$-0.383174\pi$
$29$	1.36741e18	0.717668	0.358834	+	0.933401i	$0.383174\pi$
$30$	0	0
$31$	−4.42000e18	−1.00786	−0.503931	−	0.863744i	$-0.668113\pi$
$31$	−4.42000e18	−1.00786	−0.503931	+	0.863744i	$0.668113\pi$
$32$	−1.15292e18	−0.176777
$33$	0	0
$34$	−6.74549e18	−0.484760
$35$	−4.16814e18	−0.208493
$36$	0	0
$37$	1.01944e19	0.254590	0.127295	−	0.991865i	$-0.459371\pi$
$37$	1.01944e19	0.254590	0.127295	+	0.991865i	$0.459371\pi$
$38$	−2.02764e19	−0.362824
$39$	0	0
$40$	−9.41738e18	−0.0887524
$41$	−1.58687e20	−1.09835	−0.549177	−	0.835706i	$-0.685059\pi$
$41$	−1.58687e20	−1.09835	−0.549177	+	0.835706i	$0.685059\pi$
$42$	0	0
$43$	1.83575e20	0.700582	0.350291	−	0.936641i	$-0.386083\pi$
$43$	1.83575e20	0.700582	0.350291	+	0.936641i	$0.386083\pi$
$44$	−4.34034e19	−0.124270
$45$	0	0
$46$	−4.40934e19	−0.0724270
$47$	−1.40203e21	−1.76009	−0.880045	−	0.474890i	$-0.842488\pi$
$47$	−1.40203e21	−1.76009	−0.880045	+	0.474890i	$0.842488\pi$
$48$	0	0
$49$	−4.15978e20	−0.310184
$50$	1.14378e21	0.662548
$51$	0	0
$52$	−1.60613e21	−0.569821
$53$	1.99903e21	0.558946	0.279473	−	0.960154i	$-0.409840\pi$
$53$	1.99903e21	0.558946	0.279473	+	0.960154i	$0.409840\pi$
$54$	0	0
$55$	−3.54531e20	−0.0623908
$56$	2.09012e21	0.293644
$57$	0	0
$58$	−5.60091e21	−0.507468
$59$	4.16691e21	0.304905	0.152452	−	0.988311i	$-0.451283\pi$
$59$	4.16691e21	0.304905	0.152452	+	0.988311i	$0.451283\pi$
$60$	0	0
$61$	3.42128e22	1.65031	0.825156	−	0.564904i	$-0.191087\pi$
$61$	3.42128e22	1.65031	0.825156	+	0.564904i	$0.191087\pi$
$62$	1.81043e22	0.712666
$63$	0	0
$64$	4.72237e21	0.125000
$65$	−1.31193e22	−0.286084
$66$	0	0
$67$	8.67051e22	1.29452	0.647261	−	0.762268i	$-0.275914\pi$
$67$	8.67051e22	1.29452	0.647261	+	0.762268i	$0.275914\pi$
$68$	2.76295e22	0.342777
$69$	0	0
$70$	1.70727e22	0.147427
$71$	5.13159e22	0.371127	0.185564	−	0.982632i	$-0.440589\pi$
$71$	5.13159e22	0.371127	0.185564	+	0.982632i	$0.440589\pi$
$72$	0	0
$73$	3.49147e22	0.178432	0.0892159	−	0.996012i	$-0.471564\pi$
$73$	3.49147e22	0.178432	0.0892159	+	0.996012i	$0.471564\pi$
$74$	−4.17563e22	−0.180022
$75$	0	0
$76$	8.30522e22	0.256555
$77$	7.86857e22	0.206425
$78$	0	0
$79$	2.91588e23	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$79$	2.91588e23	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$80$	3.85736e22	0.0627574
$81$	0	0
$82$	6.49981e23	0.776654
$83$	1.64916e24	1.69351	0.846753	−	0.531986i	$-0.178554\pi$
$83$	1.64916e24	1.69351	0.846753	+	0.531986i	$0.178554\pi$
$84$	0	0
$85$	2.25686e23	0.172095
$86$	−7.51925e23	−0.495386
$87$	0	0
$88$	1.77780e23	0.0878720
$89$	−8.74435e23	−0.375277	−0.187639	−	0.982238i	$-0.560083\pi$
$89$	−8.74435e23	−0.375277	−0.187639	+	0.982238i	$0.560083\pi$
$90$	0	0
$91$	2.91174e24	0.946532
$92$	1.80607e23	0.0512136
$93$	0	0
$94$	5.74273e24	1.24457
$95$	6.78393e23	0.128806
$96$	0	0
$97$	1.00608e25	1.47227	0.736134	−	0.676835i	$-0.236649\pi$
$97$	1.00608e25	1.47227	0.736134	+	0.676835i	$0.236649\pi$
$98$	1.70384e24	0.219333
$99$	0	0
$100$	−4.68492e24	−0.468492
$101$	1.42151e25	1.25526	0.627628	−	0.778513i	$-0.284026\pi$
$101$	1.42151e25	1.25526	0.627628	+	0.778513i	$0.284026\pi$
$102$	0	0
$103$	1.03211e25	0.713283	0.356642	−	0.934241i	$-0.383922\pi$
$103$	1.03211e25	0.713283	0.356642	+	0.934241i	$0.383922\pi$
$104$	6.57870e24	0.402924
$105$	0	0
$106$	−8.18801e24	−0.395235
$107$	1.74396e25	0.748582	0.374291	−	0.927311i	$-0.377886\pi$
$107$	1.74396e25	0.748582	0.374291	+	0.927311i	$0.377886\pi$
$108$	0	0
$109$	−1.65117e25	−0.562292	−0.281146	−	0.959665i	$-0.590714\pi$
$109$	−1.65117e25	−0.562292	−0.281146	+	0.959665i	$0.590714\pi$
$110$	1.45216e24	0.0441169
$111$	0	0
$112$	−8.56115e24	−0.207638
$113$	−6.96039e25	−1.51061	−0.755306	−	0.655372i	$-0.772512\pi$
$113$	−6.96039e25	−1.51061	−0.755306	+	0.655372i	$0.772512\pi$
$114$	0	0
$115$	1.47524e24	0.0257123
$116$	2.29413e25	0.358834
$117$	0	0
$118$	−1.70677e25	−0.215600
$119$	−5.00894e25	−0.569389
$120$	0	0
$121$	−1.01654e26	−0.938228
$122$	−1.40136e26	−1.16695
$123$	0	0
$124$	−7.41552e25	−0.503931
$125$	−7.91091e25	−0.486241
$126$	0	0
$127$	−6.18922e25	−0.311953	−0.155977	−	0.987761i	$-0.549852\pi$
$127$	−6.18922e25	−0.311953	−0.155977	+	0.987761i	$0.549852\pi$
$128$	−1.93428e25	−0.0883883
$129$	0	0
$130$	5.37367e25	0.202292
$131$	1.39682e26	0.477805	0.238902	−	0.971044i	$-0.423212\pi$
$131$	1.39682e26	0.477805	0.238902	+	0.971044i	$0.423212\pi$
$132$	0	0
$133$	−1.50565e26	−0.426165
$134$	−3.55144e26	−0.915366
$135$	0	0
$136$	−1.13171e26	−0.242380
$137$	6.92418e26	1.35320	0.676599	−	0.736352i	$-0.263453\pi$
$137$	6.92418e26	1.35320	0.676599	+	0.736352i	$0.263453\pi$
$138$	0	0
$139$	−8.06682e26	−1.31528	−0.657639	−	0.753333i	$-0.728445\pi$
$139$	−8.06682e26	−1.31528	−0.657639	+	0.753333i	$0.728445\pi$
$140$	−6.99298e25	−0.104247
$141$	0	0
$142$	−2.10190e26	−0.262427
$143$	2.47665e26	0.283246
$144$	0	0
$145$	1.87391e26	0.180156
$146$	−1.43010e26	−0.126170
$147$	0	0
$148$	1.71034e26	0.127295
$149$	4.24039e26	0.290120	0.145060	−	0.989423i	$-0.453663\pi$
$149$	4.24039e26	0.290120	0.145060	+	0.989423i	$0.453663\pi$
$150$	0	0
$151$	2.51382e27	1.45587	0.727934	−	0.685647i	$-0.240481\pi$
$151$	2.51382e27	1.45587	0.727934	+	0.685647i	$0.240481\pi$
$152$	−3.40182e26	−0.181412
$153$	0	0
$154$	−3.22297e26	−0.145964
$155$	−6.05720e26	−0.253003
$156$	0	0
$157$	4.75882e27	1.69338	0.846689	−	0.532088i	$-0.178593\pi$
$157$	4.75882e27	1.69338	0.846689	+	0.532088i	$0.178593\pi$
$158$	−1.19434e27	−0.392569
$159$	0	0
$160$	−1.57997e26	−0.0443762
$161$	−3.27420e26	−0.0850712
$162$	0	0
$163$	7.59899e26	0.169204	0.0846020	−	0.996415i	$-0.473038\pi$
$163$	7.59899e26	0.169204	0.0846020	+	0.996415i	$0.473038\pi$
$164$	−2.66232e27	−0.549177
$165$	0	0
$166$	−6.75496e27	−1.19749
$167$	1.00207e28	1.64794	0.823970	−	0.566634i	$-0.191755\pi$
$167$	1.00207e28	1.64794	0.823970	+	0.566634i	$0.191755\pi$
$168$	0	0
$169$	2.10835e27	0.298785
$170$	−9.24408e26	−0.121689
$171$	0	0
$172$	3.07989e27	0.350291
$173$	7.89989e27	0.835689	0.417845	−	0.908518i	$-0.362786\pi$
$173$	7.89989e27	0.835689	0.417845	+	0.908518i	$0.362786\pi$
$174$	0	0
$175$	8.49326e27	0.778214
$176$	−7.28188e26	−0.0621349
$177$	0	0
$178$	3.58168e27	0.265361
$179$	8.08087e27	0.558207	0.279103	−	0.960261i	$-0.409963\pi$
$179$	8.08087e27	0.558207	0.279103	+	0.960261i	$0.409963\pi$
$180$	0	0
$181$	1.80674e27	0.108621	0.0543104	−	0.998524i	$-0.482704\pi$
$181$	1.80674e27	0.108621	0.0543104	+	0.998524i	$0.482704\pi$
$182$	−1.19265e28	−0.669299
$183$	0	0
$184$	−7.39765e26	−0.0362135
$185$	1.39705e27	0.0639096
$186$	0	0
$187$	−4.26047e27	−0.170387
$188$	−2.35222e28	−0.880045
$189$	0	0
$190$	−2.77870e27	−0.0910796
$191$	4.93091e28	1.51360	0.756798	−	0.653649i	$-0.226763\pi$
$191$	4.93091e28	1.51360	0.756798	+	0.653649i	$0.226763\pi$
$192$	0	0
$193$	1.42443e28	0.383861	0.191931	−	0.981408i	$-0.438525\pi$
$193$	1.42443e28	0.383861	0.191931	+	0.981408i	$0.438525\pi$
$194$	−4.12092e28	−1.04105
$195$	0	0
$196$	−6.97895e27	−0.155092
$197$	−3.59073e28	−0.748781	−0.374390	−	0.927271i	$-0.622148\pi$
$197$	−3.59073e28	−0.748781	−0.374390	+	0.927271i	$0.622148\pi$
$198$	0	0
$199$	−8.11996e28	−1.49242	−0.746209	−	0.665712i	$-0.768128\pi$
$199$	−8.11996e28	−1.49242	−0.746209	+	0.665712i	$0.768128\pi$
$200$	1.91894e28	0.331274
$201$	0	0
$202$	−5.82250e28	−0.887601
$203$	−4.15902e28	−0.596060
$204$	0	0
$205$	−2.17466e28	−0.275720
$206$	−4.22753e28	−0.504367
$207$	0	0
$208$	−2.69464e28	−0.284911
$209$	−1.28066e28	−0.127528
$210$	0	0
$211$	−1.03285e29	−0.913075	−0.456537	−	0.889704i	$-0.650911\pi$
$211$	−1.03285e29	−0.913075	−0.456537	+	0.889704i	$0.650911\pi$
$212$	3.35381e28	0.279473
$213$	0	0
$214$	−7.14326e28	−0.529327
$215$	2.51573e28	0.175867
$216$	0	0
$217$	1.34436e29	0.837081
$218$	6.76321e28	0.397600
$219$	0	0
$220$	−5.94804e27	−0.0311954
$221$	−1.57657e29	−0.781287
$222$	0	0
$223$	1.15542e29	0.511597	0.255799	−	0.966730i	$-0.417662\pi$
$223$	1.15542e29	0.511597	0.255799	+	0.966730i	$0.417662\pi$
$224$	3.50665e28	0.146822
$225$	0	0
$226$	2.85098e29	1.06816
$227$	9.01151e28	0.319502	0.159751	−	0.987157i	$-0.448931\pi$
$227$	9.01151e28	0.319502	0.159751	+	0.987157i	$0.448931\pi$
$228$	0	0
$229$	3.69415e29	1.17374	0.586869	−	0.809682i	$-0.300360\pi$
$229$	3.69415e29	1.17374	0.586869	+	0.809682i	$0.300360\pi$
$230$	−6.04260e27	−0.0181813
$231$	0	0
$232$	−9.39676e28	−0.253734
$233$	−4.10753e28	−0.105107	−0.0525536	−	0.998618i	$-0.516736\pi$
$233$	−4.10753e28	−0.105107	−0.0525536	+	0.998618i	$0.516736\pi$
$234$	0	0
$235$	−1.92136e29	−0.441835
$236$	6.99092e28	0.152452
$237$	0	0
$238$	2.05166e29	0.402618
$239$	6.49097e29	1.20875	0.604374	−	0.796701i	$-0.293423\pi$
$239$	6.49097e29	1.20875	0.604374	+	0.796701i	$0.293423\pi$
$240$	0	0
$241$	1.68781e29	0.283212	0.141606	−	0.989923i	$-0.454773\pi$
$241$	1.68781e29	0.283212	0.141606	+	0.989923i	$0.454773\pi$
$242$	4.16376e29	0.663427
$243$	0	0
$244$	5.73996e29	0.825156
$245$	−5.70060e28	−0.0778653
$246$	0	0
$247$	−4.73905e29	−0.584763
$248$	3.03740e29	0.356333
$249$	0	0
$250$	3.24031e29	0.343824
$251$	−2.65029e29	−0.267530	−0.133765	−	0.991013i	$-0.542707\pi$
$251$	−2.65029e29	−0.267530	−0.133765	+	0.991013i	$0.542707\pi$
$252$	0	0
$253$	−2.78495e28	−0.0254572
$254$	2.53510e29	0.220584
$255$	0	0
$256$	7.92282e28	0.0625000
$257$	−1.11592e30	−0.838433	−0.419217	−	0.907886i	$-0.637695\pi$
$257$	−1.11592e30	−0.838433	−0.419217	+	0.907886i	$0.637695\pi$
$258$	0	0
$259$	−3.10066e29	−0.211450
$260$	−2.20105e29	−0.143042
$261$	0	0
$262$	−5.72139e29	−0.337859
$263$	−1.53721e30	−0.865541	−0.432770	−	0.901504i	$-0.642464\pi$
$263$	−1.53721e30	−0.865541	−0.432770	+	0.901504i	$0.642464\pi$
$264$	0	0
$265$	2.73948e29	0.140312
$266$	6.16713e29	0.301344
$267$	0	0
$268$	1.45467e30	0.647261
$269$	−2.34612e30	−0.996431	−0.498216	−	0.867053i	$-0.666011\pi$
$269$	−2.34612e30	−0.996431	−0.498216	+	0.867053i	$0.666011\pi$
$270$	0	0
$271$	−1.54578e30	−0.598454	−0.299227	−	0.954182i	$-0.596729\pi$
$271$	−1.54578e30	−0.598454	−0.299227	+	0.954182i	$0.596729\pi$
$272$	4.63547e29	0.171389
$273$	0	0
$274$	−2.83614e30	−0.956855
$275$	7.22414e29	0.232877
$276$	0	0
$277$	−1.21614e30	−0.358085	−0.179042	−	0.983841i	$-0.557300\pi$
$277$	−1.21614e30	−0.358085	−0.179042	+	0.983841i	$0.557300\pi$
$278$	3.30417e30	0.930042
$279$	0	0
$280$	2.86432e29	0.0737135
$281$	−5.09885e30	−1.25500	−0.627499	−	0.778617i	$-0.715921\pi$
$281$	−5.09885e30	−1.25500	−0.627499	+	0.778617i	$0.715921\pi$
$282$	0	0
$283$	−1.56288e30	−0.352042	−0.176021	−	0.984386i	$-0.556323\pi$
$283$	−1.56288e30	−0.352042	−0.176021	+	0.984386i	$0.556323\pi$
$284$	8.60939e29	0.185564
$285$	0	0
$286$	−1.01444e30	−0.200285
$287$	4.82651e30	0.912240
$288$	0	0
$289$	−3.05852e30	−0.530015
$290$	−7.67553e29	−0.127389
$291$	0	0
$292$	5.85771e29	0.0892159
$293$	4.17112e30	0.608706	0.304353	−	0.952559i	$-0.401560\pi$
$293$	4.17112e30	0.608706	0.304353	+	0.952559i	$0.401560\pi$
$294$	0	0
$295$	5.71038e29	0.0765402
$296$	−7.00555e29	−0.0900110
$297$	0	0
$298$	−1.73686e30	−0.205146
$299$	−1.03056e30	−0.116730
$300$	0	0
$301$	−5.58350e30	−0.581870
$302$	−1.02966e31	−1.02945
$303$	0	0
$304$	1.39338e30	0.128278
$305$	4.68856e30	0.414278
$306$	0	0
$307$	7.69994e30	0.626986	0.313493	−	0.949591i	$-0.398501\pi$
$307$	7.69994e30	0.626986	0.313493	+	0.949591i	$0.398501\pi$
$308$	1.32013e30	0.103212
$309$	0	0
$310$	2.48103e30	0.178900
$311$	−1.54260e31	−1.06843	−0.534217	−	0.845347i	$-0.679394\pi$
$311$	−1.54260e31	−1.06843	−0.534217	+	0.845347i	$0.679394\pi$
$312$	0	0
$313$	1.62517e31	1.03895	0.519476	−	0.854485i	$-0.326127\pi$
$313$	1.62517e31	1.03895	0.519476	+	0.854485i	$0.326127\pi$
$314$	−1.94921e31	−1.19740
$315$	0	0
$316$	4.89203e30	0.277588
$317$	2.29098e31	1.24963	0.624814	−	0.780773i	$-0.285175\pi$
$317$	2.29098e31	1.24963	0.624814	+	0.780773i	$0.285175\pi$
$318$	0	0
$319$	−3.53755e30	−0.178369
$320$	6.47157e29	0.0313787
$321$	0	0
$322$	1.34111e30	0.0601544
$323$	8.15239e30	0.351765
$324$	0	0
$325$	2.67327e31	1.06783
$326$	−3.11255e30	−0.119645
$327$	0	0
$328$	1.09049e31	0.388327
$329$	4.26433e31	1.46185
$330$	0	0
$331$	3.45199e31	1.09703	0.548517	−	0.836140i	$-0.315193\pi$
$331$	3.45199e31	1.09703	0.548517	+	0.836140i	$0.315193\pi$
$332$	2.76683e31	0.846753
$333$	0	0
$334$	−4.10447e31	−1.16527
$335$	1.18821e31	0.324964
$336$	0	0
$337$	−5.27735e31	−1.33981	−0.669904	−	0.742448i	$-0.733665\pi$
$337$	−5.27735e31	−1.33981	−0.669904	+	0.742448i	$0.733665\pi$
$338$	−8.63579e30	−0.211273
$339$	0	0
$340$	3.78638e30	0.0860473
$341$	1.14347e31	0.250493
$342$	0	0
$343$	5.34411e31	1.08818
$344$	−1.26152e31	−0.247693
$345$	0	0
$346$	−3.23580e31	−0.590922
$347$	−4.40093e30	−0.0775222	−0.0387611	−	0.999249i	$-0.512341\pi$
$347$	−4.40093e30	−0.0775222	−0.0387611	+	0.999249i	$0.512341\pi$
$348$	0	0
$349$	2.98069e31	0.488651	0.244326	−	0.969693i	$-0.421433\pi$
$349$	2.98069e31	0.488651	0.244326	+	0.969693i	$0.421433\pi$
$350$	−3.47884e31	−0.550280
$351$	0	0
$352$	2.98266e30	0.0439360
$353$	−4.46084e31	−0.634209	−0.317105	−	0.948391i	$-0.602711\pi$
$353$	−4.46084e31	−0.634209	−0.317105	+	0.948391i	$0.602711\pi$
$354$	0	0
$355$	7.03238e30	0.0931640
$356$	−1.46706e31	−0.187639
$357$	0	0
$358$	−3.30992e31	−0.394712
$359$	−4.38054e31	−0.504483	−0.252242	−	0.967664i	$-0.581168\pi$
$359$	−4.38054e31	−0.504483	−0.252242	+	0.967664i	$0.581168\pi$
$360$	0	0
$361$	−6.85711e31	−0.736717
$362$	−7.40039e30	−0.0768065
$363$	0	0
$364$	4.88509e31	0.473266
$365$	4.78474e30	0.0447917
$366$	0	0
$367$	−7.76735e31	−0.679121	−0.339560	−	0.940584i	$-0.610278\pi$
$367$	−7.76735e31	−0.679121	−0.339560	+	0.940584i	$0.610278\pi$
$368$	3.03008e30	0.0256068
$369$	0	0
$370$	−5.72232e30	−0.0451909
$371$	−6.08010e31	−0.464234
$372$	0	0
$373$	−2.37205e32	−1.69342	−0.846708	−	0.532058i	$-0.821419\pi$
$373$	−2.37205e32	−1.69342	−0.846708	+	0.532058i	$0.821419\pi$
$374$	1.74509e31	0.120482
$375$	0	0
$376$	9.63470e31	0.622286
$377$	−1.30906e32	−0.817884
$378$	0	0
$379$	−1.38944e32	−0.812551	−0.406276	−	0.913751i	$-0.633173\pi$
$379$	−1.38944e32	−0.812551	−0.406276	+	0.913751i	$0.633173\pi$
$380$	1.13815e31	0.0644030
$381$	0	0
$382$	−2.01970e32	−1.07027
$383$	3.79085e31	0.194425	0.0972125	−	0.995264i	$-0.469007\pi$
$383$	3.79085e31	0.194425	0.0972125	+	0.995264i	$0.469007\pi$
$384$	0	0
$385$	1.07832e31	0.0518188
$386$	−5.83446e31	−0.271431
$387$	0	0
$388$	1.68793e32	0.736134
$389$	2.20209e32	0.929963	0.464982	−	0.885320i	$-0.346061\pi$
$389$	2.20209e32	0.929963	0.464982	+	0.885320i	$0.346061\pi$
$390$	0	0
$391$	1.77283e31	0.0702195
$392$	2.85858e31	0.109667
$393$	0	0
$394$	1.47076e32	0.529468
$395$	3.99595e31	0.139366
$396$	0	0
$397$	2.18941e32	0.716877	0.358439	−	0.933553i	$-0.383309\pi$
$397$	2.18941e32	0.716877	0.358439	+	0.933553i	$0.383309\pi$
$398$	3.32594e32	1.05530
$399$	0	0
$400$	−7.85999e31	−0.234246
$401$	−2.99036e32	−0.863810	−0.431905	−	0.901919i	$-0.642158\pi$
$401$	−2.99036e32	−0.863810	−0.431905	+	0.901919i	$0.642158\pi$
$402$	0	0
$403$	4.23138e32	1.14860
$404$	2.38490e32	0.627628
$405$	0	0
$406$	1.70353e32	0.421478
$407$	−2.63734e31	−0.0632755
$408$	0	0
$409$	7.12330e31	0.160746	0.0803730	−	0.996765i	$-0.474389\pi$
$409$	7.12330e31	0.160746	0.0803730	+	0.996765i	$0.474389\pi$
$410$	8.90740e31	0.194963
$411$	0	0
$412$	1.73160e32	0.356642
$413$	−1.26738e32	−0.253239
$414$	0	0
$415$	2.26003e32	0.425121
$416$	1.10372e32	0.201462
$417$	0	0
$418$	5.24560e31	0.0901761
$419$	−2.94461e32	−0.491307	−0.245654	−	0.969358i	$-0.579003\pi$
$419$	−2.94461e32	−0.491307	−0.245654	+	0.969358i	$0.579003\pi$
$420$	0	0
$421$	1.01892e33	1.60182	0.800908	−	0.598787i	$-0.204350\pi$
$421$	1.01892e33	1.60182	0.800908	+	0.598787i	$0.204350\pi$
$422$	4.23055e32	0.645641
$423$	0	0
$424$	−1.37372e32	−0.197617
$425$	−4.59871e32	−0.642354
$426$	0	0
$427$	−1.04059e33	−1.37067
$428$	2.92588e32	0.374291
$429$	0	0
$430$	−1.03044e32	−0.124357
$431$	6.79497e32	0.796565	0.398283	−	0.917263i	$-0.369606\pi$
$431$	6.79497e32	0.796565	0.398283	+	0.917263i	$0.369606\pi$
$432$	0	0
$433$	−7.24164e32	−0.801195	−0.400597	−	0.916254i	$-0.631197\pi$
$433$	−7.24164e32	−0.801195	−0.400597	+	0.916254i	$0.631197\pi$
$434$	−5.50648e32	−0.591906
$435$	0	0
$436$	−2.77021e32	−0.281146
$437$	5.32899e31	0.0525566
$438$	0	0
$439$	−9.13459e32	−0.850908	−0.425454	−	0.904980i	$-0.639886\pi$
$439$	−9.13459e32	−0.850908	−0.425454	+	0.904980i	$0.639886\pi$
$440$	2.43632e31	0.0220585
$441$	0	0
$442$	6.45764e32	0.552453
$443$	3.52555e32	0.293211	0.146605	−	0.989195i	$-0.453165\pi$
$443$	3.52555e32	0.293211	0.146605	+	0.989195i	$0.453165\pi$
$444$	0	0
$445$	−1.19833e32	−0.0942058
$446$	−4.73259e32	−0.361754
$447$	0	0
$448$	−1.43632e32	−0.103819
$449$	−3.70918e32	−0.260735	−0.130367	−	0.991466i	$-0.541616\pi$
$449$	−3.70918e32	−0.260735	−0.130367	+	0.991466i	$0.541616\pi$
$450$	0	0
$451$	4.10530e32	0.272984
$452$	−1.16776e33	−0.755306
$453$	0	0
$454$	−3.69111e32	−0.225922
$455$	3.99028e32	0.237608
$456$	0	0
$457$	1.76958e33	0.997512	0.498756	−	0.866742i	$-0.333790\pi$
$457$	1.76958e33	0.997512	0.498756	+	0.866742i	$0.333790\pi$
$458$	−1.51313e33	−0.829959
$459$	0	0
$460$	2.47505e31	0.0128561
$461$	−7.01018e32	−0.354379	−0.177189	−	0.984177i	$-0.556701\pi$
$461$	−7.01018e32	−0.354379	−0.177189	+	0.984177i	$0.556701\pi$
$462$	0	0
$463$	−9.20521e32	−0.440830	−0.220415	−	0.975406i	$-0.570741\pi$
$463$	−9.20521e32	−0.440830	−0.220415	+	0.975406i	$0.570741\pi$
$464$	3.84891e32	0.179417
$465$	0	0
$466$	1.68245e32	0.0743219
$467$	1.84600e33	0.793908	0.396954	−	0.917838i	$-0.370067\pi$
$467$	1.84600e33	0.793908	0.396954	+	0.917838i	$0.370067\pi$
$468$	0	0
$469$	−2.63716e33	−1.07517
$470$	7.86988e32	0.312425
$471$	0	0
$472$	−2.86348e32	−0.107800
$473$	−4.74918e32	−0.174122
$474$	0	0
$475$	−1.38234e33	−0.480777
$476$	−8.40360e32	−0.284694
$477$	0	0
$478$	−2.65870e33	−0.854714
$479$	3.88481e32	0.121668	0.0608339	−	0.998148i	$-0.480624\pi$
$479$	3.88481e32	0.121668	0.0608339	+	0.998148i	$0.480624\pi$
$480$	0	0
$481$	−9.75939e32	−0.290141
$482$	−6.91328e32	−0.200261
$483$	0	0
$484$	−1.70548e33	−0.469114
$485$	1.37875e33	0.369583
$486$	0	0
$487$	4.26102e33	1.08493	0.542465	−	0.840078i	$-0.317491\pi$
$487$	4.26102e33	1.08493	0.542465	+	0.840078i	$0.317491\pi$
$488$	−2.35109e33	−0.583474
$489$	0	0
$490$	2.33496e32	0.0550591
$491$	4.52548e33	1.04027	0.520134	−	0.854084i	$-0.325882\pi$
$491$	4.52548e33	1.04027	0.520134	+	0.854084i	$0.325882\pi$
$492$	0	0
$493$	2.25191e33	0.492000
$494$	1.94112e33	0.413490
$495$	0	0
$496$	−1.24412e33	−0.251965
$497$	−1.56079e33	−0.308241
$498$	0	0
$499$	4.45020e33	0.835840	0.417920	−	0.908484i	$-0.362759\pi$
$499$	4.45020e33	0.835840	0.417920	+	0.908484i	$0.362759\pi$
$500$	−1.32723e33	−0.243120
$501$	0	0
$502$	1.08556e33	0.189172
$503$	−2.29032e33	−0.389311	−0.194656	−	0.980872i	$-0.562359\pi$
$503$	−2.29032e33	−0.389311	−0.194656	+	0.980872i	$0.562359\pi$
$504$	0	0
$505$	1.94805e33	0.315107
$506$	1.14072e32	0.0180010
$507$	0	0
$508$	−1.03838e33	−0.155977
$509$	−5.35889e33	−0.785421	−0.392711	−	0.919662i	$-0.628463\pi$
$509$	−5.35889e33	−0.785421	−0.392711	+	0.919662i	$0.628463\pi$
$510$	0	0
$511$	−1.06194e33	−0.148197
$512$	−3.24519e32	−0.0441942
$513$	0	0
$514$	4.57080e33	0.592862
$515$	1.41442e33	0.179055
$516$	0	0
$517$	3.62712e33	0.437452
$518$	1.27003e33	0.149518
$519$	0	0
$520$	9.01552e32	0.101146
$521$	1.14090e34	1.24961	0.624807	−	0.780779i	$-0.285178\pi$
$521$	1.14090e34	1.24961	0.624807	+	0.780779i	$0.285178\pi$
$522$	0	0
$523$	1.37717e34	1.43786	0.718928	−	0.695085i	$-0.244633\pi$
$523$	1.37717e34	1.43786	0.718928	+	0.695085i	$0.244633\pi$
$524$	2.34348e33	0.238902
$525$	0	0
$526$	6.29643e33	0.612030
$527$	−7.27906e33	−0.690944
$528$	0	0
$529$	−1.09299e34	−0.989509
$530$	−1.12209e33	−0.0992156
$531$	0	0
$532$	−2.52606e33	−0.213083
$533$	1.51915e34	1.25173
$534$	0	0
$535$	2.38994e33	0.187916
$536$	−5.95833e33	−0.457683
$537$	0	0
$538$	9.60972e33	0.704583
$539$	1.07615e33	0.0770929
$540$	0	0
$541$	−1.58682e34	−1.08533	−0.542667	−	0.839948i	$-0.682585\pi$
$541$	−1.58682e34	−1.08533	−0.542667	+	0.839948i	$0.682585\pi$
$542$	6.33151e33	0.423171
$543$	0	0
$544$	−1.89869e33	−0.121190
$545$	−2.26278e33	−0.141152
$546$	0	0
$547$	−8.29322e33	−0.494176	−0.247088	−	0.968993i	$-0.579474\pi$
$547$	−8.29322e33	−0.494176	−0.247088	+	0.968993i	$0.579474\pi$
$548$	1.16168e34	0.676599
$549$	0	0
$550$	−2.95901e33	−0.164669
$551$	6.76908e33	0.368243
$552$	0	0
$553$	−8.86874e33	−0.461103
$554$	4.98130e33	0.253204
$555$	0	0
$556$	−1.35339e34	−0.657639
$557$	−3.56496e34	−1.69381	−0.846905	−	0.531744i	$-0.821537\pi$
$557$	−3.56496e34	−1.69381	−0.846905	+	0.531744i	$0.821537\pi$
$558$	0	0
$559$	−1.75742e34	−0.798413
$560$	−1.17323e33	−0.0521233
$561$	0	0
$562$	2.08849e34	0.887417
$563$	−1.62059e34	−0.673471	−0.336735	−	0.941599i	$-0.609323\pi$
$563$	−1.62059e34	−0.673471	−0.336735	+	0.941599i	$0.609323\pi$
$564$	0	0
$565$	−9.53858e33	−0.379209
$566$	6.40155e33	0.248931
$567$	0	0
$568$	−3.52641e33	−0.131213
$569$	−4.19926e34	−1.52851	−0.764257	−	0.644912i	$-0.776894\pi$
$569$	−4.19926e34	−1.52851	−0.764257	+	0.644912i	$0.776894\pi$
$570$	0	0
$571$	2.27922e34	0.794026	0.397013	−	0.917813i	$-0.370047\pi$
$571$	2.27922e34	0.794026	0.397013	+	0.917813i	$0.370047\pi$
$572$	4.15513e33	0.141623
$573$	0	0
$574$	−1.97694e34	−0.645051
$575$	−3.00605e33	−0.0959727
$576$	0	0
$577$	−1.30064e34	−0.397613	−0.198807	−	0.980039i	$-0.563707\pi$
$577$	−1.30064e34	−0.397613	−0.198807	+	0.980039i	$0.563707\pi$
$578$	1.25277e34	0.374777
$579$	0	0
$580$	3.14390e33	0.0900780
$581$	−5.01597e34	−1.40655
$582$	0	0
$583$	−5.17157e33	−0.138920
$584$	−2.39932e33	−0.0630851
$585$	0	0
$586$	−1.70849e34	−0.430420
$587$	−3.13619e34	−0.773440	−0.386720	−	0.922197i	$-0.626392\pi$
$587$	−3.13619e34	−0.773440	−0.386720	+	0.922197i	$0.626392\pi$
$588$	0	0
$589$	−2.18803e34	−0.517145
$590$	−2.33897e33	−0.0541221
$591$	0	0
$592$	2.86947e33	0.0636474
$593$	6.43753e34	1.39809	0.699046	−	0.715077i	$-0.253608\pi$
$593$	6.43753e34	1.39809	0.699046	+	0.715077i	$0.253608\pi$
$594$	0	0
$595$	−6.86429e33	−0.142933
$596$	7.11419e33	0.145060
$597$	0	0
$598$	4.22118e33	0.0825409
$599$	9.82124e34	1.88075	0.940375	−	0.340140i	$-0.110474\pi$
$599$	9.82124e34	1.88075	0.940375	+	0.340140i	$0.110474\pi$
$600$	0	0
$601$	1.10144e34	0.202316	0.101158	−	0.994870i	$-0.467745\pi$
$601$	1.10144e34	0.202316	0.101158	+	0.994870i	$0.467745\pi$
$602$	2.28700e34	0.411444
$603$	0	0
$604$	4.21749e34	0.727934
$605$	−1.39308e34	−0.235523
$606$	0	0
$607$	9.06735e34	1.47103	0.735516	−	0.677508i	$-0.236940\pi$
$607$	9.06735e34	1.47103	0.735516	+	0.677508i	$0.236940\pi$
$608$	−5.70730e33	−0.0907060
$609$	0	0
$610$	−1.92043e34	−0.292938
$611$	1.34220e35	2.00587
$612$	0	0
$613$	2.62922e34	0.377199	0.188599	−	0.982054i	$-0.439605\pi$
$613$	2.62922e34	0.377199	0.188599	+	0.982054i	$0.439605\pi$
$614$	−3.15390e34	−0.443346
$615$	0	0
$616$	−5.40724e33	−0.0729822
$617$	−6.11422e34	−0.808679	−0.404340	−	0.914609i	$-0.632499\pi$
$617$	−6.11422e34	−0.808679	−0.404340	+	0.914609i	$0.632499\pi$
$618$	0	0
$619$	−4.66687e34	−0.592779	−0.296389	−	0.955067i	$-0.595783\pi$
$619$	−4.66687e34	−0.592779	−0.296389	+	0.955067i	$0.595783\pi$
$620$	−1.01623e34	−0.126502
$621$	0	0
$622$	6.31848e34	0.755497
$623$	2.65962e34	0.311687
$624$	0	0
$625$	7.23797e34	0.814923
$626$	−6.65672e34	−0.734651
$627$	0	0
$628$	7.98398e34	0.846689
$629$	1.67886e34	0.174535
$630$	0	0
$631$	−7.88083e33	−0.0787417	−0.0393709	−	0.999225i	$-0.512535\pi$
$631$	−7.88083e33	−0.0787417	−0.0393709	+	0.999225i	$0.512535\pi$
$632$	−2.00378e34	−0.196284
$633$	0	0
$634$	−9.38386e34	−0.883621
$635$	−8.48176e33	−0.0783095
$636$	0	0
$637$	3.98227e34	0.353499
$638$	1.44898e34	0.126126
$639$	0	0
$640$	−2.65076e33	−0.0221881
$641$	−3.14908e34	−0.258499	−0.129250	−	0.991612i	$-0.541257\pi$
$641$	−3.14908e34	−0.258499	−0.129250	+	0.991612i	$0.541257\pi$
$642$	0	0
$643$	8.32480e34	0.657259	0.328630	−	0.944459i	$-0.393413\pi$
$643$	8.32480e34	0.657259	0.328630	+	0.944459i	$0.393413\pi$
$644$	−5.49320e33	−0.0425356
$645$	0	0
$646$	−3.33922e34	−0.248736
$647$	1.33124e35	0.972641	0.486321	−	0.873780i	$-0.338339\pi$
$647$	1.33124e35	0.972641	0.486321	+	0.873780i	$0.338339\pi$
$648$	0	0
$649$	−1.07800e34	−0.0757809
$650$	−1.09497e35	−0.755068
$651$	0	0
$652$	1.27490e34	0.0846020
$653$	−2.06756e35	−1.34599	−0.672997	−	0.739645i	$-0.734993\pi$
$653$	−2.06756e35	−1.34599	−0.672997	+	0.739645i	$0.734993\pi$
$654$	0	0
$655$	1.91422e34	0.119943
$656$	−4.46664e34	−0.274589
$657$	0	0
$658$	−1.74667e35	−1.03368
$659$	2.57849e35	1.49726	0.748632	−	0.662986i	$-0.230711\pi$
$659$	2.57849e35	1.49726	0.748632	+	0.662986i	$0.230711\pi$
$660$	0	0
$661$	−3.31671e34	−0.185434	−0.0927170	−	0.995692i	$-0.529555\pi$
$661$	−3.31671e34	−0.185434	−0.0927170	+	0.995692i	$0.529555\pi$
$662$	−1.41394e35	−0.775720
$663$	0	0
$664$	−1.13329e35	−0.598745
$665$	−2.06335e34	−0.106980
$666$	0	0
$667$	1.47201e34	0.0735087
$668$	1.68119e35	0.823970
$669$	0	0
$670$	−4.86693e34	−0.229784
$671$	−8.85101e34	−0.410168
$672$	0	0
$673$	−2.87165e35	−1.28216	−0.641082	−	0.767472i	$-0.721514\pi$
$673$	−2.87165e35	−1.28216	−0.641082	+	0.767472i	$0.721514\pi$
$674$	2.16160e35	0.947387
$675$	0	0
$676$	3.53722e34	0.149392
$677$	−3.20244e35	−1.32777	−0.663885	−	0.747835i	$-0.731093\pi$
$677$	−3.20244e35	−1.32777	−0.663885	+	0.747835i	$0.731093\pi$
$678$	0	0
$679$	−3.06003e35	−1.22280
$680$	−1.55090e34	−0.0608446
$681$	0	0
$682$	−4.68366e34	−0.177125
$683$	4.34050e35	1.61169	0.805843	−	0.592129i	$-0.201712\pi$
$683$	4.34050e35	1.61169	0.805843	+	0.592129i	$0.201712\pi$
$684$	0	0
$685$	9.48896e34	0.339693
$686$	−2.18895e35	−0.769456
$687$	0	0
$688$	5.16719e34	0.175145
$689$	−1.91372e35	−0.636999
$690$	0	0
$691$	3.59659e35	1.15456	0.577279	−	0.816547i	$-0.304114\pi$
$691$	3.59659e35	1.15456	0.577279	+	0.816547i	$0.304114\pi$
$692$	1.32538e35	0.417845
$693$	0	0
$694$	1.80262e34	0.0548165
$695$	−1.10548e35	−0.330174
$696$	0	0
$697$	−2.61333e35	−0.752982
$698$	−1.22089e35	−0.345529
$699$	0	0
$700$	1.42493e35	0.389107
$701$	4.01671e34	0.107745	0.0538723	−	0.998548i	$-0.482844\pi$
$701$	4.01671e34	0.107745	0.0538723	+	0.998548i	$0.482844\pi$
$702$	0	0
$703$	5.04654e34	0.130633
$704$	−1.22170e34	−0.0310674
$705$	0	0
$706$	1.82716e35	0.448454
$707$	−4.32357e35	−1.04256
$708$	0	0
$709$	−2.37128e34	−0.0551957	−0.0275979	−	0.999619i	$-0.508786\pi$
$709$	−2.37128e34	−0.0551957	−0.0275979	+	0.999619i	$0.508786\pi$
$710$	−2.88046e34	−0.0658769
$711$	0	0
$712$	6.00907e34	0.132681
$713$	−4.75812e34	−0.103232
$714$	0	0
$715$	3.39402e34	0.0711032
$716$	1.35574e35	0.279103
$717$	0	0
$718$	1.79427e35	0.356724
$719$	8.41947e35	1.64503	0.822514	−	0.568744i	$-0.192571\pi$
$719$	8.41947e35	1.64503	0.822514	+	0.568744i	$0.192571\pi$
$720$	0	0
$721$	−3.13920e35	−0.592419
$722$	2.80867e35	0.520938
$723$	0	0
$724$	3.03120e34	0.0543104
$725$	−3.81839e35	−0.672443
$726$	0	0
$727$	8.07486e35	1.37390	0.686950	−	0.726705i	$-0.258949\pi$
$727$	8.07486e35	1.37390	0.686950	+	0.726705i	$0.258949\pi$
$728$	−2.00093e35	−0.334650
$729$	0	0
$730$	−1.95983e34	−0.0316725
$731$	3.02321e35	0.480287
$732$	0	0
$733$	−9.32041e35	−1.43099	−0.715493	−	0.698620i	$-0.753798\pi$
$733$	−9.32041e35	−1.43099	−0.715493	+	0.698620i	$0.753798\pi$
$734$	3.18151e35	0.480211
$735$	0	0
$736$	−1.24112e34	−0.0181068
$737$	−2.24310e35	−0.321740
$738$	0	0
$739$	−9.24490e34	−0.128188	−0.0640939	−	0.997944i	$-0.520416\pi$
$739$	−9.24490e34	−0.128188	−0.0640939	+	0.997944i	$0.520416\pi$
$740$	2.34386e34	0.0319548
$741$	0	0
$742$	2.49041e35	0.328263
$743$	−3.86129e35	−0.500463	−0.250232	−	0.968186i	$-0.580507\pi$
$743$	−3.86129e35	−0.500463	−0.250232	+	0.968186i	$0.580507\pi$
$744$	0	0
$745$	5.81106e34	0.0728287
$746$	9.71592e35	1.19743
$747$	0	0
$748$	−7.14788e34	−0.0851937
$749$	−5.30431e35	−0.621736
$750$	0	0
$751$	5.31819e35	0.602927	0.301463	−	0.953478i	$-0.402525\pi$
$751$	5.31819e35	0.602927	0.301463	+	0.953478i	$0.402525\pi$
$752$	−3.94637e35	−0.440023
$753$	0	0
$754$	5.36190e35	0.578332
$755$	3.44496e35	0.365466
$756$	0	0
$757$	1.46803e36	1.50674	0.753368	−	0.657599i	$-0.228428\pi$
$757$	1.46803e36	1.50674	0.753368	+	0.657599i	$0.228428\pi$
$758$	5.69116e35	0.574560
$759$	0	0
$760$	−4.66188e34	−0.0455398
$761$	−1.26153e36	−1.21225	−0.606123	−	0.795371i	$-0.707276\pi$
$761$	−1.26153e36	−1.21225	−0.606123	+	0.795371i	$0.707276\pi$
$762$	0	0
$763$	5.02210e35	0.467012
$764$	8.27269e35	0.756798
$765$	0	0
$766$	−1.55273e35	−0.137479
$767$	−3.98910e35	−0.347482
$768$	0	0
$769$	−9.78483e35	−0.825039	−0.412519	−	0.910949i	$-0.635351\pi$
$769$	−9.78483e35	−0.825039	−0.412519	+	0.910949i	$0.635351\pi$
$770$	−4.41678e34	−0.0366414
$771$	0	0
$772$	2.38980e35	0.191931
$773$	1.97217e36	1.55848	0.779238	−	0.626728i	$-0.215606\pi$
$773$	1.97217e36	1.55848	0.779238	+	0.626728i	$0.215606\pi$
$774$	0	0
$775$	1.23425e36	0.944350
$776$	−6.91375e35	−0.520526
$777$	0	0
$778$	−9.01978e35	−0.657583
$779$	−7.85547e35	−0.563578
$780$	0	0
$781$	−1.32757e35	−0.0922398
$782$	−7.26151e34	−0.0496527
$783$	0	0
$784$	−1.17087e35	−0.0775459
$785$	6.52153e35	0.425088
$786$	0	0
$787$	−2.25189e35	−0.142188	−0.0710940	−	0.997470i	$-0.522649\pi$
$787$	−2.25189e35	−0.142188	−0.0710940	+	0.997470i	$0.522649\pi$
$788$	−6.02424e35	−0.374390
$789$	0	0
$790$	−1.63674e35	−0.0985465
$791$	2.11702e36	1.25464
$792$	0	0
$793$	−3.27529e36	−1.88077
$794$	−8.96781e35	−0.506909
$795$	0	0
$796$	−1.36230e36	−0.746209
$797$	−3.69113e36	−1.99035	−0.995177	−	0.0980978i	$-0.968724\pi$
$797$	−3.69113e36	−1.99035	−0.995177	+	0.0980978i	$0.968724\pi$
$798$	0	0
$799$	−2.30893e36	−1.20664
$800$	3.21945e35	0.165637
$801$	0	0
$802$	1.22485e36	0.610806
$803$	−9.03258e34	−0.0443473
$804$	0	0
$805$	−4.48700e34	−0.0213554
$806$	−1.73317e36	−0.812184
$807$	0	0
$808$	−9.76854e35	−0.443800
$809$	2.93815e36	1.31437	0.657185	−	0.753730i	$-0.271747\pi$
$809$	2.93815e36	1.31437	0.657185	+	0.753730i	$0.271747\pi$
$810$	0	0
$811$	−2.44583e36	−1.06088	−0.530439	−	0.847723i	$-0.677973\pi$
$811$	−2.44583e36	−1.06088	−0.530439	+	0.847723i	$0.677973\pi$
$812$	−6.97767e35	−0.298030
$813$	0	0
$814$	1.08025e35	0.0447426
$815$	1.04137e35	0.0424752
$816$	0	0
$817$	9.08753e35	0.359476
$818$	−2.91771e35	−0.113665
$819$	0	0
$820$	−3.64847e35	−0.137860
$821$	−3.33815e36	−1.24227	−0.621136	−	0.783703i	$-0.713328\pi$
$821$	−3.33815e36	−1.24227	−0.621136	+	0.783703i	$0.713328\pi$
$822$	0	0
$823$	2.54036e36	0.917058	0.458529	−	0.888679i	$-0.348377\pi$
$823$	2.54036e36	0.917058	0.458529	+	0.888679i	$0.348377\pi$
$824$	−7.09263e35	−0.252184
$825$	0	0
$826$	5.19119e35	0.179067
$827$	3.63276e36	1.23429	0.617146	−	0.786849i	$-0.288289\pi$
$827$	3.63276e36	1.23429	0.617146	+	0.786849i	$0.288289\pi$
$828$	0	0
$829$	1.07727e36	0.355135	0.177567	−	0.984109i	$-0.443177\pi$
$829$	1.07727e36	0.355135	0.177567	+	0.984109i	$0.443177\pi$
$830$	−9.25706e35	−0.300606
$831$	0	0
$832$	−4.52085e35	−0.142455
$833$	−6.85052e35	−0.212648
$834$	0	0
$835$	1.37324e36	0.413682
$836$	−2.14860e35	−0.0637641
$837$	0	0
$838$	1.20611e36	0.347407
$839$	2.96808e36	0.842269	0.421134	−	0.906998i	$-0.361632\pi$
$839$	2.96808e36	0.842269	0.421134	+	0.906998i	$0.361632\pi$
$840$	0	0
$841$	−1.76056e36	−0.484953
$842$	−4.17349e36	−1.13266
$843$	0	0
$844$	−1.73283e36	−0.456537
$845$	2.88930e35	0.0750039
$846$	0	0
$847$	3.09185e36	0.779247
$848$	5.62676e35	0.139737
$849$	0	0
$850$	1.88363e36	0.454213
$851$	1.09743e35	0.0260769
$852$	0	0
$853$	6.13786e36	1.41630	0.708148	−	0.706064i	$-0.249531\pi$
$853$	6.13786e36	1.41630	0.708148	+	0.706064i	$0.249531\pi$
$854$	4.26227e36	0.969210
$855$	0	0
$856$	−1.19844e36	−0.264664
$857$	−3.16481e36	−0.688791	−0.344395	−	0.938825i	$-0.611916\pi$
$857$	−3.16481e36	−0.688791	−0.344395	+	0.938825i	$0.611916\pi$
$858$	0	0
$859$	5.65370e36	1.19514	0.597569	−	0.801818i	$-0.296134\pi$
$859$	5.65370e36	1.19514	0.597569	+	0.801818i	$0.296134\pi$
$860$	4.22070e35	0.0879334
$861$	0	0
$862$	−2.78322e36	−0.563257
$863$	8.00828e36	1.59736	0.798682	−	0.601753i	$-0.205531\pi$
$863$	8.00828e36	1.59736	0.798682	+	0.601753i	$0.205531\pi$
$864$	0	0
$865$	1.08261e36	0.209783
$866$	2.96617e36	0.566530
$867$	0	0
$868$	2.25545e36	0.418540
$869$	−7.54351e35	−0.137983
$870$	0	0
$871$	−8.30052e36	−1.47529
$872$	1.13468e36	0.198800
$873$	0	0
$874$	−2.18275e35	−0.0371631
$875$	2.40613e36	0.403848
$876$	0	0
$877$	3.29560e36	0.537575	0.268788	−	0.963199i	$-0.413377\pi$
$877$	3.29560e36	0.537575	0.268788	+	0.963199i	$0.413377\pi$
$878$	3.74153e36	0.601683
$879$	0	0
$880$	−9.97916e34	−0.0155977
$881$	8.05386e36	1.24109	0.620547	−	0.784169i	$-0.286910\pi$
$881$	8.05386e36	1.24109	0.620547	+	0.784169i	$0.286910\pi$
$882$	0	0
$883$	−4.91849e35	−0.0736754	−0.0368377	−	0.999321i	$-0.511728\pi$
$883$	−4.91849e35	−0.0736754	−0.0368377	+	0.999321i	$0.511728\pi$
$884$	−2.64505e36	−0.390644
$885$	0	0
$886$	−1.44406e36	−0.207331
$887$	−7.79484e36	−1.10347	−0.551737	−	0.834018i	$-0.686035\pi$
$887$	−7.79484e36	−1.10347	−0.551737	+	0.834018i	$0.686035\pi$
$888$	0	0
$889$	1.88247e36	0.259093
$890$	4.90837e35	0.0666135
$891$	0	0
$892$	1.93847e36	0.255799
$893$	−6.94048e36	−0.903121
$894$	0	0
$895$	1.10741e36	0.140127
$896$	5.88318e35	0.0734111
$897$	0	0
$898$	1.51928e36	0.184367
$899$	−6.04394e36	−0.723309
$900$	0	0
$901$	3.29209e36	0.383188
$902$	−1.68153e36	−0.193029
$903$	0	0
$904$	4.78314e36	0.534082
$905$	2.47597e35	0.0272670
$906$	0	0
$907$	−1.31417e37	−1.40786	−0.703932	−	0.710268i	$-0.748574\pi$
$907$	−1.31417e37	−1.40786	−0.703932	+	0.710268i	$0.748574\pi$
$908$	1.51188e36	0.159751
$909$	0	0
$910$	−1.63442e36	−0.168014
$911$	−1.44767e37	−1.46788	−0.733940	−	0.679214i	$-0.762321\pi$
$911$	−1.44767e37	−1.46788	−0.733940	+	0.679214i	$0.762321\pi$
$912$	0	0
$913$	−4.26645e36	−0.420903
$914$	−7.24821e36	−0.705347
$915$	0	0
$916$	6.19776e36	0.586869
$917$	−4.24848e36	−0.396842
$918$	0	0
$919$	−1.03191e37	−0.937990	−0.468995	−	0.883201i	$-0.655384\pi$
$919$	−1.03191e37	−0.937990	−0.468995	+	0.883201i	$0.655384\pi$
$920$	−1.01378e35	−0.00909067
$921$	0	0
$922$	2.87137e36	0.250584
$923$	−4.91262e36	−0.422953
$924$	0	0
$925$	−2.84672e36	−0.238546
$926$	3.77045e36	0.311714
$927$	0	0
$928$	−1.57651e36	−0.126867
$929$	2.17912e37	1.73015	0.865075	−	0.501642i	$-0.167271\pi$
$929$	2.17912e37	1.73015	0.865075	+	0.501642i	$0.167271\pi$
$930$	0	0
$931$	−2.05921e36	−0.159159
$932$	−6.89130e35	−0.0525536
$933$	0	0
$934$	−7.56121e36	−0.561378
$935$	−5.83858e35	−0.0427723
$936$	0	0
$937$	−2.08020e37	−1.48375	−0.741873	−	0.670540i	$-0.766062\pi$
$937$	−2.08020e37	−1.48375	−0.741873	+	0.670540i	$0.766062\pi$
$938$	1.08018e37	0.760258
$939$	0	0
$940$	−3.22350e36	−0.220917
$941$	−2.39454e37	−1.61939	−0.809695	−	0.586851i	$-0.800368\pi$
$941$	−2.39454e37	−1.61939	−0.809695	+	0.586851i	$0.800368\pi$
$942$	0	0
$943$	−1.70826e36	−0.112501
$944$	1.17288e36	0.0762262
$945$	0	0
$946$	1.94526e36	0.123123
$947$	8.06134e36	0.503539	0.251769	−	0.967787i	$-0.418988\pi$
$947$	8.06134e36	0.503539	0.251769	+	0.967787i	$0.418988\pi$
$948$	0	0
$949$	−3.34248e36	−0.203348
$950$	5.66205e36	0.339960
$951$	0	0
$952$	3.44212e36	0.201309
$953$	1.92589e37	1.11166	0.555829	−	0.831297i	$-0.312401\pi$
$953$	1.92589e37	1.11166	0.555829	+	0.831297i	$0.312401\pi$
$954$	0	0
$955$	6.75736e36	0.379957
$956$	1.08900e37	0.604374
$957$	0	0
$958$	−1.59122e36	−0.0860321
$959$	−2.10601e37	−1.12390
$960$	0	0
$961$	3.03575e35	0.0157842
$962$	3.99745e36	0.205161
$963$	0	0
$964$	2.83168e36	0.141606
$965$	1.95205e36	0.0963606
$966$	0	0
$967$	2.29324e37	1.10311	0.551554	−	0.834139i	$-0.314035\pi$
$967$	2.29324e37	1.10311	0.551554	+	0.834139i	$0.314035\pi$
$968$	6.98563e36	0.331714
$969$	0	0
$970$	−5.64734e36	−0.261335
$971$	−2.10199e37	−0.960265	−0.480132	−	0.877196i	$-0.659411\pi$
$971$	−2.10199e37	−0.960265	−0.480132	+	0.877196i	$0.659411\pi$
$972$	0	0
$973$	2.45355e37	1.09241
$974$	−1.74531e37	−0.767161
$975$	0	0
$976$	9.63006e36	0.412578
$977$	3.52592e37	1.49139	0.745694	−	0.666289i	$-0.232118\pi$
$977$	3.52592e37	1.49139	0.745694	+	0.666289i	$0.232118\pi$
$978$	0	0
$979$	2.26220e36	0.0932712
$980$	−9.56401e35	−0.0389327
$981$	0	0
$982$	−1.85363e37	−0.735581
$983$	−7.31962e36	−0.286794	−0.143397	−	0.989665i	$-0.545803\pi$
$983$	−7.31962e36	−0.286794	−0.143397	+	0.989665i	$0.545803\pi$
$984$	0	0
$985$	−4.92076e36	−0.187966
$986$	−9.22384e36	−0.347897
$987$	0	0
$988$	−7.95081e36	−0.292381
$989$	1.97619e36	0.0717587
$990$	0	0
$991$	1.16985e37	0.414200	0.207100	−	0.978320i	$-0.433597\pi$
$991$	1.16985e37	0.414200	0.207100	+	0.978320i	$0.433597\pi$
$992$	5.09591e36	0.178166
$993$	0	0
$994$	6.39300e36	0.217959
$995$	−1.11277e37	−0.374641
$996$	0	0
$997$	2.33411e37	0.766358	0.383179	−	0.923674i	$-0.374829\pi$
$997$	2.33411e37	0.766358	0.383179	+	0.923674i	$0.374829\pi$
$998$	−1.82280e37	−0.591028
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.26.a.e.1.2		2
3.2	odd	2		2.26.a.b.1.2	✓	2
12.11	even	2		16.26.a.c.1.1		2
15.2	even	4		50.26.b.e.49.3		4
15.8	even	4		50.26.b.e.49.2		4
15.14	odd	2		50.26.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.26.a.b.1.2	✓	2	3.2	odd	2
16.26.a.c.1.1		2	12.11	even	2
18.26.a.e.1.2		2	1.1	even	1	trivial
50.26.a.c.1.1		2	15.14	odd	2
50.26.b.e.49.2		4	15.8	even	4
50.26.b.e.49.3		4	15.2	even	4

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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

\(f(q)\)	\(=\)	\(q-4096.00 q^{2} +1.67772e7 q^{4} +1.37041e8 q^{5} -3.04153e10 q^{7} -6.87195e10 q^{8} -5.61320e11 q^{10} -2.58704e12 q^{11} -9.57327e13 q^{13} +1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15 q^{17} +4.95030e15 q^{19} +2.29916e15 q^{20} +1.05965e16 q^{22} +1.07650e16 q^{23} -2.79243e17 q^{25} +3.92121e17 q^{26} -5.10284e17 q^{28} +1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18 q^{32} -6.74549e18 q^{34} -4.16814e18 q^{35} +1.01944e19 q^{37} -2.02764e19 q^{38} -9.41738e18 q^{40} -1.58687e20 q^{41} +1.83575e20 q^{43} -4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21 q^{47} -4.15978e20 q^{49} +1.14378e21 q^{50} -1.60613e21 q^{52} +1.99903e21 q^{53} -3.54531e20 q^{55} +2.09012e21 q^{56} -5.60091e21 q^{58} +4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22 q^{62} +4.72237e21 q^{64} -1.31193e22 q^{65} +8.67051e22 q^{67} +2.76295e22 q^{68} +1.70727e22 q^{70} +5.13159e22 q^{71} +3.49147e22 q^{73} -4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22 q^{77} +2.91588e23 q^{79} +3.85736e22 q^{80} +6.49981e23 q^{82} +1.64916e24 q^{83} +2.25686e23 q^{85} -7.51925e23 q^{86} +1.77780e23 q^{88} -8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23 q^{92} +5.74273e24 q^{94} +6.78393e23 q^{95} +1.00608e25 q^{97} +1.70384e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 2 * q - 8192 * q^2 + 33554432 * q^4 - 741953100 * q^5 - 376536944 * q^7 - 137438953472 * q^8 + 3039039897600 * q^10 - 8323034610264 * q^11 - 106467053152292 * q^13 + 1542295322624 * q^14 + 562949953421312 * q^16 - 1327878920113956 * q^17 - 477079242949400 * q^19 - 12447907420569600 * q^20 + 34091149763641344 * q^22 + 115305025781473872 * q^23 + 195364218352298750 * q^25 + 436089049711788032 * q^26 - 6317241641467904 * q^28 - 1724412645206435580 * q^29 - 8688082288351126976 * q^31 - 2305843009213693952 * q^32 + 5438992056786763776 * q^34 - 30572039779548136800 * q^35 - 34908364049750170484 * q^37 + 1954116579120742400 * q^38 + 50986628794653081600 * q^40 - 83014324355953468884 * q^41 + 44539608583471901848 * q^43 - 139637349431874945024 * q^44 - 472289385600916979712 * q^46 - 1869360732054674599776 * q^47 - 854718643011973772046 * q^49 - 800211838371015680000 * q^50 - 1786220747619483779072 * q^52 + 3242170458210097794132 * q^53 + 4687370283547568509200 * q^55 + 25875421763452534784 * q^56 + 7063194194765560135680 * q^58 + 17413675430632049453640 * q^59 + 33975859953090212810044 * q^61 + 35586385053086216093696 * q^62 + 9444732965739290427392 * q^64 - 3683906908205094047400 * q^65 + 33012908196969622380616 * q^67 - 22278111464598584426496 * q^68 + 125223074937029168332800 * q^70 + 278348008145600813615856 * q^71 + 313442631796571689392148 * q^73 + 142984659147776698302464 * q^74 - 8004061508078560870400 * q^76 - 93616352280445975603392 * q^77 + 928546182288781647817120 * q^79 - 208841231542899022233600 * q^80 + 340026672561985408548864 * q^82 + 452542629950725008985752 * q^83 + 2840452690831654686931800 * q^85 - 182434236757900909969408 * q^86 + 571954583272959774818304 * q^88 + 2346904179674200701244620 * q^89 + 2589295264446210583737824 * q^91 + 1934497323421355948900352 * q^92 + 7656901558495947160682496 * q^94 + 5449023740708681753370000 * q^95 + 13645482361777595497674436 * q^97 + 3500927561777044570300416 * q^98

Introduction

L-functions

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Database

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