LMFDB - Embedded newform 16.26.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,26,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	16.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:	$63.3594847924$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$163.829$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75788e6 q^{3} -1.37041e8 q^{5} +3.04153e10 q^{7} +2.24285e12 q^{9} -2.58704e12 q^{11} -9.57327e13 q^{13} +2.40901e14 q^{15} -1.64685e15 q^{17} -4.95030e15 q^{19} -5.34664e16 q^{21} +1.07650e16 q^{23} -2.79243e17 q^{25} -2.45323e18 q^{27} -1.36741e18 q^{29} +4.42000e18 q^{31} +4.54771e18 q^{33} -4.16814e18 q^{35} +1.01944e19 q^{37} +1.68287e20 q^{39} +1.58687e20 q^{41} -1.83575e20 q^{43} -3.07362e20 q^{45} -1.40203e21 q^{47} -4.15978e20 q^{49} +2.89496e21 q^{51} -1.99903e21 q^{53} +3.54531e20 q^{55} +8.70202e21 q^{57} +4.16691e21 q^{59} +3.42128e22 q^{61} +6.82170e22 q^{63} +1.31193e22 q^{65} -8.67051e22 q^{67} -1.89236e22 q^{69} +5.13159e22 q^{71} +3.49147e22 q^{73} +4.90875e23 q^{75} -7.86857e22 q^{77} -2.91588e23 q^{79} +2.41214e24 q^{81} +1.64916e24 q^{83} +2.25686e23 q^{85} +2.40374e24 q^{87} +8.74435e23 q^{89} -2.91174e24 q^{91} -7.76982e24 q^{93} +6.78393e23 q^{95} +1.00608e25 q^{97} -5.80235e24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q - 379848 * q^3 + 741953100 * q^5 + 376536944 * q^7 + 3294531432666 * q^9 - 8323034610264 * q^11 - 106467053152292 * q^13 + 1452182413035600 * q^15 + 1327878920113956 * q^17 + 477079242949400 * q^19 - 94860791661752256 * q^21 + 115305025781473872 * q^23 + 195364218352298750 * q^25 - 2171569557205152720 * q^27 + 1724412645206435580 * q^29 + 8688082288351126976 * q^31 - 3356662204005020064 * q^33 - 30572039779548136800 * q^35 - 34908364049750170484 * q^37 + 153494373835331105808 * q^39 + 83014324355953468884 * q^41 - 44539608583471901848 * q^43 + 617059152350257182300 * q^45 - 1869360732054674599776 * q^47 - 854718643011973772046 * q^49 + 6994225992587553820656 * q^51 - 3242170458210097794132 * q^53 - 4687370283547568509200 * q^55 + 16181115057055601354400 * q^57 + 17413675430632049453640 * q^59 + 33975859953090212810044 * q^61 + 36625774166984600306352 * q^63 + 3683906908205094047400 * q^65 - 33012908196969622380616 * q^67 + 125135868082548386734272 * q^69 + 278348008145600813615856 * q^71 + 313442631796571689392148 * q^73 + 1144898988230230172205000 * q^75 + 93616352280445975603392 * q^77 - 928546182288781647817120 * q^79 + 1909195802481672047728242 * q^81 + 452542629950725008985752 * q^83 + 2840452690831654686931800 * q^85 + 6664364855011895770504080 * q^87 - 2346904179674200701244620 * q^89 - 2589295264446210583737824 * q^91 - 1888263951389289833389824 * q^93 + 5449023740708681753370000 * q^95 + 13645482361777595497674436 * q^97 - 11834785687944312803841912 * q^99

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
$3$	−1.75788e6	−1.90974	−0.954868	−	0.297031i	$-0.904004\pi$
$3$	−1.75788e6	−1.90974	−0.954868	+	0.297031i	$0.904004\pi$
$4$	0	0
$5$	−1.37041e8	−0.251030	−0.125515	−	0.992092i	$-0.540058\pi$
$5$	−1.37041e8	−0.251030	−0.125515	+	0.992092i	$0.540058\pi$
$6$	0	0
$7$	3.04153e10	0.830552	0.415276	−	0.909696i	$-0.363685\pi$
$7$	3.04153e10	0.830552	0.415276	+	0.909696i	$0.363685\pi$
$8$	0	0
$9$	2.24285e12	2.64709
$10$	0	0
$11$	−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	$0.692988\pi$
$14$	0	0
$15$	2.40901e14	0.479400
$16$	0	0
$17$	−1.64685e15	−0.685555	−0.342777	−	0.939417i	$-0.611368\pi$
$17$	−1.64685e15	−0.685555	−0.342777	+	0.939417i	$0.611368\pi$
$18$	0	0
$19$	−4.95030e15	−0.513111	−0.256555	−	0.966530i	$-0.582588\pi$
$19$	−4.95030e15	−0.513111	−0.256555	+	0.966530i	$0.582588\pi$
$20$	0	0
$21$	−5.34664e16	−1.58613
$22$	0	0
$23$	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
$26$	0	0
$27$	−2.45323e18	−3.14551
$28$	0	0
$29$	−1.36741e18	−0.717668	−0.358834	−	0.933401i	$-0.616826\pi$
$29$	−1.36741e18	−0.717668	−0.358834	+	0.933401i	$0.616826\pi$
$30$	0	0
$31$	4.42000e18	1.00786	0.503931	−	0.863744i	$-0.331887\pi$
$31$	4.42000e18	1.00786	0.503931	+	0.863744i	$0.331887\pi$
$32$	0	0
$33$	4.54771e18	0.474645
$34$	0	0
$35$	−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	$0.459371\pi$
$38$	0	0
$39$	1.68287e20	2.17642
$40$	0	0
$41$	1.58687e20	1.09835	0.549177	−	0.835706i	$-0.314941\pi$
$41$	1.58687e20	1.09835	0.549177	+	0.835706i	$0.314941\pi$
$42$	0	0
$43$	−1.83575e20	−0.700582	−0.350291	−	0.936641i	$-0.613917\pi$
$43$	−1.83575e20	−0.700582	−0.350291	+	0.936641i	$0.613917\pi$
$44$	0	0
$45$	−3.07362e20	−0.664499
$46$	0	0
$47$	−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
$50$	0	0
$51$	2.89496e21	1.30923
$52$	0	0
$53$	−1.99903e21	−0.558946	−0.279473	−	0.960154i	$-0.590160\pi$
$53$	−1.99903e21	−0.558946	−0.279473	+	0.960154i	$0.590160\pi$
$54$	0	0
$55$	3.54531e20	0.0623908
$56$	0	0
$57$	8.70202e21	0.979906
$58$	0	0
$59$	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	$0.191087\pi$
$62$	0	0
$63$	6.82170e22	2.19855
$64$	0	0
$65$	1.31193e22	0.286084
$66$	0	0
$67$	−8.67051e22	−1.29452	−0.647261	−	0.762268i	$-0.724086\pi$
$67$	−8.67051e22	−1.29452	−0.647261	+	0.762268i	$0.724086\pi$
$68$	0	0
$69$	−1.89236e22	−0.195609
$70$	0	0
$71$	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	$0.471564\pi$
$74$	0	0
$75$	4.90875e23	1.78939
$76$	0	0
$77$	−7.86857e22	−0.206425
$78$	0	0
$79$	−2.91588e23	−0.555176	−0.277588	−	0.960700i	$-0.589535\pi$
$79$	−2.91588e23	−0.555176	−0.277588	+	0.960700i	$0.589535\pi$
$80$	0	0
$81$	2.41214e24	3.36000
$82$	0	0
$83$	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
$86$	0	0
$87$	2.40374e24	1.37056
$88$	0	0
$89$	8.74435e23	0.375277	0.187639	−	0.982238i	$-0.439917\pi$
$89$	8.74435e23	0.375277	0.187639	+	0.982238i	$0.439917\pi$
$90$	0	0
$91$	−2.91174e24	−0.946532
$92$	0	0
$93$	−7.76982e24	−1.92475
$94$	0	0
$95$	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	$0.236649\pi$
$98$	0	0
$99$	−5.80235e24	−0.657907
$100$	0	0
$101$	−1.42151e25	−1.25526	−0.627628	−	0.778513i	$-0.715974\pi$
$101$	−1.42151e25	−1.25526	−0.627628	+	0.778513i	$0.715974\pi$
$102$	0	0
$103$	−1.03211e25	−0.713283	−0.356642	−	0.934241i	$-0.616078\pi$
$103$	−1.03211e25	−0.713283	−0.356642	+	0.934241i	$0.616078\pi$
$104$	0	0
$105$	7.32709e24	0.398167
$106$	0	0
$107$	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	$0.590714\pi$
$110$	0	0
$111$	−1.79205e25	−0.486199
$112$	0	0
$113$	6.96039e25	1.51061	0.755306	−	0.655372i	$-0.227488\pi$
$113$	6.96039e25	1.51061	0.755306	+	0.655372i	$0.227488\pi$
$114$	0	0
$115$	−1.47524e24	−0.0257123
$116$	0	0
$117$	−2.14714e26	−3.01674
$118$	0	0
$119$	−5.00894e25	−0.569389
$120$	0	0
$121$	−1.01654e26	−0.938228
$122$	0	0
$123$	−2.78952e26	−2.09757
$124$	0	0
$125$	7.91091e25	0.486241
$126$	0	0
$127$	6.18922e25	0.311953	0.155977	−	0.987761i	$-0.450148\pi$
$127$	6.18922e25	0.311953	0.155977	+	0.987761i	$0.450148\pi$
$128$	0	0
$129$	3.22703e26	1.33793
$130$	0	0
$131$	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
$134$	0	0
$135$	3.36193e26	0.789616
$136$	0	0
$137$	−6.92418e26	−1.35320	−0.676599	−	0.736352i	$-0.736547\pi$
$137$	−6.92418e26	−1.35320	−0.676599	+	0.736352i	$0.736547\pi$
$138$	0	0
$139$	8.06682e26	1.31528	0.657639	−	0.753333i	$-0.271555\pi$
$139$	8.06682e26	1.31528	0.657639	+	0.753333i	$0.271555\pi$
$140$	0	0
$141$	2.46460e27	3.36131
$142$	0	0
$143$	2.47665e26	0.283246
$144$	0	0
$145$	1.87391e26	0.180156
$146$	0	0
$147$	7.31239e26	0.592369
$148$	0	0
$149$	−4.24039e26	−0.290120	−0.145060	−	0.989423i	$-0.546337\pi$
$149$	−4.24039e26	−0.290120	−0.145060	+	0.989423i	$0.546337\pi$
$150$	0	0
$151$	−2.51382e27	−1.45587	−0.727934	−	0.685647i	$-0.759519\pi$
$151$	−2.51382e27	−1.45587	−0.727934	+	0.685647i	$0.759519\pi$
$152$	0	0
$153$	−3.69363e27	−1.81473
$154$	0	0
$155$	−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	$0.178593\pi$
$158$	0	0
$159$	3.51404e27	1.06744
$160$	0	0
$161$	3.27420e26	0.0850712
$162$	0	0
$163$	−7.59899e26	−0.169204	−0.0846020	−	0.996415i	$-0.526962\pi$
$163$	−7.59899e26	−0.169204	−0.0846020	+	0.996415i	$0.526962\pi$
$164$	0	0
$165$	−6.23222e26	−0.119150
$166$	0	0
$167$	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
$170$	0	0
$171$	−1.11028e28	−1.35825
$172$	0	0
$173$	−7.89989e27	−0.835689	−0.417845	−	0.908518i	$-0.637214\pi$
$173$	−7.89989e27	−0.835689	−0.417845	+	0.908518i	$0.637214\pi$
$174$	0	0
$175$	−8.49326e27	−0.778214
$176$	0	0
$177$	−7.32493e27	−0.582288
$178$	0	0
$179$	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	$0.482704\pi$
$182$	0	0
$183$	−6.01420e28	−3.15166
$184$	0	0
$185$	−1.39705e27	−0.0639096
$186$	0	0
$187$	4.26047e27	0.170387
$188$	0	0
$189$	−7.46157e28	−2.61251
$190$	0	0
$191$	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	$0.438525\pi$
$194$	0	0
$195$	−2.30621e28	−0.546345
$196$	0	0
$197$	3.59073e28	0.748781	0.374390	−	0.927271i	$-0.377852\pi$
$197$	3.59073e28	0.748781	0.374390	+	0.927271i	$0.377852\pi$
$198$	0	0
$199$	8.11996e28	1.49242	0.746209	−	0.665712i	$-0.231872\pi$
$199$	8.11996e28	1.49242	0.746209	+	0.665712i	$0.231872\pi$
$200$	0	0
$201$	1.52417e29	2.47220
$202$	0	0
$203$	−4.15902e28	−0.596060
$204$	0	0
$205$	−2.17466e28	−0.275720
$206$	0	0
$207$	2.41443e28	0.271134
$208$	0	0
$209$	1.28066e28	0.127528
$210$	0	0
$211$	1.03285e29	0.913075	0.456537	−	0.889704i	$-0.349089\pi$
$211$	1.03285e29	0.913075	0.456537	+	0.889704i	$0.349089\pi$
$212$	0	0
$213$	−9.02072e28	−0.708755
$214$	0	0
$215$	2.51573e28	0.175867
$216$	0	0
$217$	1.34436e29	0.837081
$218$	0	0
$219$	−6.13758e28	−0.340757
$220$	0	0
$221$	1.57657e29	0.781287
$222$	0	0
$223$	−1.15542e29	−0.511597	−0.255799	−	0.966730i	$-0.582338\pi$
$223$	−1.15542e29	−0.511597	−0.255799	+	0.966730i	$0.582338\pi$
$224$	0	0
$225$	−6.26300e29	−2.48028
$226$	0	0
$227$	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	$0.300360\pi$
$230$	0	0
$231$	1.38320e29	0.394217
$232$	0	0
$233$	4.10753e28	0.105107	0.0525536	−	0.998618i	$-0.483264\pi$
$233$	4.10753e28	0.105107	0.0525536	+	0.998618i	$0.483264\pi$
$234$	0	0
$235$	1.92136e29	0.441835
$236$	0	0
$237$	5.12576e29	1.06024
$238$	0	0
$239$	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	$0.454773\pi$
$242$	0	0
$243$	−2.16165e30	−3.27121
$244$	0	0
$245$	5.70060e28	0.0778653
$246$	0	0
$247$	4.73905e29	0.584763
$248$	0	0
$249$	−2.89903e30	−3.23415
$250$	0	0
$251$	−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
$254$	0	0
$255$	−3.96728e29	−0.328655
$256$	0	0
$257$	1.11592e30	0.838433	0.419217	−	0.907886i	$-0.362305\pi$
$257$	1.11592e30	0.838433	0.419217	+	0.907886i	$0.362305\pi$
$258$	0	0
$259$	3.10066e29	0.211450
$260$	0	0
$261$	−3.06689e30	−1.89973
$262$	0	0
$263$	−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
$266$	0	0
$267$	−1.53715e30	−0.716681
$268$	0	0
$269$	2.34612e30	0.996431	0.498216	−	0.867053i	$-0.333989\pi$
$269$	2.34612e30	0.996431	0.498216	+	0.867053i	$0.333989\pi$
$270$	0	0
$271$	1.54578e30	0.598454	0.299227	−	0.954182i	$-0.403271\pi$
$271$	1.54578e30	0.598454	0.299227	+	0.954182i	$0.403271\pi$
$272$	0	0
$273$	5.11849e30	1.80763
$274$	0	0
$275$	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	$0.557300\pi$
$278$	0	0
$279$	9.91339e30	2.66790
$280$	0	0
$281$	5.09885e30	1.25500	0.627499	−	0.778617i	$-0.284079\pi$
$281$	5.09885e30	1.25500	0.627499	+	0.778617i	$0.284079\pi$
$282$	0	0
$283$	1.56288e30	0.352042	0.176021	−	0.984386i	$-0.443677\pi$
$283$	1.56288e30	0.352042	0.176021	+	0.984386i	$0.443677\pi$
$284$	0	0
$285$	−1.19253e30	−0.245986
$286$	0	0
$287$	4.82651e30	0.912240
$288$	0	0
$289$	−3.05852e30	−0.530015
$290$	0	0
$291$	−1.76857e31	−2.81164
$292$	0	0
$293$	−4.17112e30	−0.608706	−0.304353	−	0.952559i	$-0.598440\pi$
$293$	−4.17112e30	−0.608706	−0.304353	+	0.952559i	$0.598440\pi$
$294$	0	0
$295$	−5.71038e29	−0.0765402
$296$	0	0
$297$	6.34661e30	0.781783
$298$	0	0
$299$	−1.03056e30	−0.116730
$300$	0	0
$301$	−5.58350e30	−0.581870
$302$	0	0
$303$	2.49884e31	2.39721
$304$	0	0
$305$	−4.68856e30	−0.414278
$306$	0	0
$307$	−7.69994e30	−0.626986	−0.313493	−	0.949591i	$-0.601499\pi$
$307$	−7.69994e30	−0.626986	−0.313493	+	0.949591i	$0.601499\pi$
$308$	0	0
$309$	1.81433e31	1.36218
$310$	0	0
$311$	−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	$0.326127\pi$
$314$	0	0
$315$	−9.34852e30	−0.551900
$316$	0	0
$317$	−2.29098e31	−1.24963	−0.624814	−	0.780773i	$-0.714825\pi$
$317$	−2.29098e31	−1.24963	−0.624814	+	0.780773i	$0.714825\pi$
$318$	0	0
$319$	3.53755e30	0.178369
$320$	0	0
$321$	−3.06567e31	−1.42959
$322$	0	0
$323$	8.15239e30	0.351765
$324$	0	0
$325$	2.67327e31	1.06783
$326$	0	0
$327$	2.90256e31	1.07383
$328$	0	0
$329$	−4.26433e31	−1.46185
$330$	0	0
$331$	−3.45199e31	−1.09703	−0.548517	−	0.836140i	$-0.684807\pi$
$331$	−3.45199e31	−1.09703	−0.548517	+	0.836140i	$0.684807\pi$
$332$	0	0
$333$	2.28645e31	0.673922
$334$	0	0
$335$	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	$0.733665\pi$
$338$	0	0
$339$	−1.22355e32	−2.88487
$340$	0	0
$341$	−1.14347e31	−0.250493
$342$	0	0
$343$	−5.34411e31	−1.08818
$344$	0	0
$345$	2.59330e30	0.0491037
$346$	0	0
$347$	−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	$0.421433\pi$
$350$	0	0
$351$	2.34854e32	3.58476
$352$	0	0
$353$	4.46084e31	0.634209	0.317105	−	0.948391i	$-0.397289\pi$
$353$	4.46084e31	0.634209	0.317105	+	0.948391i	$0.397289\pi$
$354$	0	0
$355$	−7.03238e30	−0.0931640
$356$	0	0
$357$	8.80511e31	1.08738
$358$	0	0
$359$	−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
$362$	0	0
$363$	1.78696e32	1.79177
$364$	0	0
$365$	−4.78474e30	−0.0447917
$366$	0	0
$367$	7.76735e31	0.679121	0.339560	−	0.940584i	$-0.389722\pi$
$367$	7.76735e31	0.679121	0.339560	+	0.940584i	$0.389722\pi$
$368$	0	0
$369$	3.55911e32	2.90744
$370$	0	0
$371$	−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	$0.821419\pi$
$374$	0	0
$375$	−1.39064e32	−0.928591
$376$	0	0
$377$	1.30906e32	0.817884
$378$	0	0
$379$	1.38944e32	0.812551	0.406276	−	0.913751i	$-0.366827\pi$
$379$	1.38944e32	0.812551	0.406276	+	0.913751i	$0.366827\pi$
$380$	0	0
$381$	−1.08799e32	−0.595748
$382$	0	0
$383$	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
$386$	0	0
$387$	−4.11732e32	−1.85450
$388$	0	0
$389$	−2.20209e32	−0.929963	−0.464982	−	0.885320i	$-0.653939\pi$
$389$	−2.20209e32	−0.929963	−0.464982	+	0.885320i	$0.653939\pi$
$390$	0	0
$391$	−1.77283e31	−0.0702195
$392$	0	0
$393$	−2.45545e32	−0.912481
$394$	0	0
$395$	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	$0.383309\pi$
$398$	0	0
$399$	2.64675e32	0.813863
$400$	0	0
$401$	2.99036e32	0.863810	0.431905	−	0.901919i	$-0.357842\pi$
$401$	2.99036e32	0.863810	0.431905	+	0.901919i	$0.357842\pi$
$402$	0	0
$403$	−4.23138e32	−1.14860
$404$	0	0
$405$	−3.30562e32	−0.843460
$406$	0	0
$407$	−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	$0.474389\pi$
$410$	0	0
$411$	1.21719e33	2.58425
$412$	0	0
$413$	1.26738e32	0.253239
$414$	0	0
$415$	−2.26003e32	−0.425121
$416$	0	0
$417$	−1.41805e33	−2.51183
$418$	0	0
$419$	−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	$0.204350\pi$
$422$	0	0
$423$	−3.14455e33	−4.65912
$424$	0	0
$425$	4.59871e32	0.642354
$426$	0	0
$427$	1.04059e33	1.37067
$428$	0	0
$429$	−4.35365e32	−0.540925
$430$	0	0
$431$	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	$0.631197\pi$
$434$	0	0
$435$	−3.29411e32	−0.344050
$436$	0	0
$437$	−5.32899e31	−0.0525566
$438$	0	0
$439$	9.13459e32	0.850908	0.425454	−	0.904980i	$-0.360114\pi$
$439$	9.13459e32	0.850908	0.425454	+	0.904980i	$0.360114\pi$
$440$	0	0
$441$	−9.32976e32	−0.821085
$442$	0	0
$443$	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
$446$	0	0
$447$	7.45409e32	0.554052
$448$	0	0
$449$	3.70918e32	0.260735	0.130367	−	0.991466i	$-0.458384\pi$
$449$	3.70918e32	0.260735	0.130367	+	0.991466i	$0.458384\pi$
$450$	0	0
$451$	−4.10530e32	−0.272984
$452$	0	0
$453$	4.41899e33	2.78032
$454$	0	0
$455$	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	$0.333790\pi$
$458$	0	0
$459$	4.04009e33	2.15642
$460$	0	0
$461$	7.01018e32	0.354379	0.177189	−	0.984177i	$-0.443299\pi$
$461$	7.01018e32	0.354379	0.177189	+	0.984177i	$0.443299\pi$
$462$	0	0
$463$	9.20521e32	0.440830	0.220415	−	0.975406i	$-0.429259\pi$
$463$	9.20521e32	0.440830	0.220415	+	0.975406i	$0.429259\pi$
$464$	0	0
$465$	1.06478e33	0.483169
$466$	0	0
$467$	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
$470$	0	0
$471$	−8.36543e33	−3.23390
$472$	0	0
$473$	4.74918e32	0.174122
$474$	0	0
$475$	1.38234e33	0.480777
$476$	0	0
$477$	−4.48351e33	−1.47958
$478$	0	0
$479$	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
$482$	0	0
$483$	−5.75566e32	−0.162463
$484$	0	0
$485$	−1.37875e33	−0.369583
$486$	0	0
$487$	−4.26102e33	−1.08493	−0.542465	−	0.840078i	$-0.682509\pi$
$487$	−4.26102e33	−1.08493	−0.542465	+	0.840078i	$0.682509\pi$
$488$	0	0
$489$	1.33581e33	0.323135
$490$	0	0
$491$	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
$494$	0	0
$495$	7.95160e32	0.165154
$496$	0	0
$497$	1.56079e33	0.308241
$498$	0	0
$499$	−4.45020e33	−0.835840	−0.417920	−	0.908484i	$-0.637241\pi$
$499$	−4.45020e33	−0.835840	−0.417920	+	0.908484i	$0.637241\pi$
$500$	0	0
$501$	−1.76151e34	−3.14713
$502$	0	0
$503$	−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
$506$	0	0
$507$	−3.70622e33	−0.570600
$508$	0	0
$509$	5.35889e33	0.785421	0.392711	−	0.919662i	$-0.371537\pi$
$509$	5.35889e33	0.785421	0.392711	+	0.919662i	$0.371537\pi$
$510$	0	0
$511$	1.06194e33	0.148197
$512$	0	0
$513$	1.21442e34	1.61400
$514$	0	0
$515$	1.41442e33	0.179055
$516$	0	0
$517$	3.62712e33	0.437452
$518$	0	0
$519$	1.38871e34	1.59595
$520$	0	0
$521$	−1.14090e34	−1.24961	−0.624807	−	0.780779i	$-0.714822\pi$
$521$	−1.14090e34	−1.24961	−0.624807	+	0.780779i	$0.714822\pi$
$522$	0	0
$523$	−1.37717e34	−1.43786	−0.718928	−	0.695085i	$-0.755367\pi$
$523$	−1.37717e34	−1.43786	−0.718928	+	0.695085i	$0.755367\pi$
$524$	0	0
$525$	1.49301e34	1.48618
$526$	0	0
$527$	−7.27906e33	−0.690944
$528$	0	0
$529$	−1.09299e34	−0.989509
$530$	0	0
$531$	9.34576e33	0.807111
$532$	0	0
$533$	−1.51915e34	−1.25173
$534$	0	0
$535$	−2.38994e33	−0.187916
$536$	0	0
$537$	−1.42052e34	−1.06603
$538$	0	0
$539$	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	$0.682585\pi$
$542$	0	0
$543$	−3.17603e33	−0.207437
$544$	0	0
$545$	2.26278e33	0.141152
$546$	0	0
$547$	8.29322e33	0.494176	0.247088	−	0.968993i	$-0.420526\pi$
$547$	8.29322e33	0.494176	0.247088	+	0.968993i	$0.420526\pi$
$548$	0	0
$549$	7.67343e34	4.36853
$550$	0	0
$551$	6.76908e33	0.368243
$552$	0	0
$553$	−8.86874e33	−0.461103
$554$	0	0
$555$	2.45585e33	0.122050
$556$	0	0
$557$	3.56496e34	1.69381	0.846905	−	0.531744i	$-0.178463\pi$
$557$	3.56496e34	1.69381	0.846905	+	0.531744i	$0.178463\pi$
$558$	0	0
$559$	1.75742e34	0.798413
$560$	0	0
$561$	−7.48939e33	−0.325395
$562$	0	0
$563$	−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
$566$	0	0
$567$	7.33659e34	2.79065
$568$	0	0
$569$	4.19926e34	1.52851	0.764257	−	0.644912i	$-0.223106\pi$
$569$	4.19926e34	1.52851	0.764257	+	0.644912i	$0.223106\pi$
$570$	0	0
$571$	−2.27922e34	−0.794026	−0.397013	−	0.917813i	$-0.629953\pi$
$571$	−2.27922e34	−0.794026	−0.397013	+	0.917813i	$0.629953\pi$
$572$	0	0
$573$	−8.66794e34	−2.89057
$574$	0	0
$575$	−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	$0.563707\pi$
$578$	0	0
$579$	−2.50397e34	−0.733074
$580$	0	0
$581$	5.01597e34	1.40655
$582$	0	0
$583$	5.17157e33	0.138920
$584$	0	0
$585$	2.94246e34	0.757291
$586$	0	0
$587$	−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
$590$	0	0
$591$	−6.31206e34	−1.42997
$592$	0	0
$593$	−6.43753e34	−1.39809	−0.699046	−	0.715077i	$-0.746392\pi$
$593$	−6.43753e34	−1.39809	−0.699046	+	0.715077i	$0.746392\pi$
$594$	0	0
$595$	6.86429e33	0.142933
$596$	0	0
$597$	−1.42739e35	−2.85012
$598$	0	0
$599$	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	$0.467745\pi$
$602$	0	0
$603$	−1.94467e35	−3.42672
$604$	0	0
$605$	1.39308e34	0.235523
$606$	0	0
$607$	−9.06735e34	−1.47103	−0.735516	−	0.677508i	$-0.763060\pi$
$607$	−9.06735e34	−1.47103	−0.735516	+	0.677508i	$0.763060\pi$
$608$	0	0
$609$	7.31105e34	1.13832
$610$	0	0
$611$	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	$0.439605\pi$
$614$	0	0
$615$	3.82279e34	0.526552
$616$	0	0
$617$	6.11422e34	0.808679	0.404340	−	0.914609i	$-0.367501\pi$
$617$	6.11422e34	0.808679	0.404340	+	0.914609i	$0.367501\pi$
$618$	0	0
$619$	4.66687e34	0.592779	0.296389	−	0.955067i	$-0.404217\pi$
$619$	4.66687e34	0.592779	0.296389	+	0.955067i	$0.404217\pi$
$620$	0	0
$621$	−2.64090e34	−0.322186
$622$	0	0
$623$	2.65962e34	0.311687
$624$	0	0
$625$	7.23797e34	0.814923
$626$	0	0
$627$	−2.25125e34	−0.243545
$628$	0	0
$629$	−1.67886e34	−0.174535
$630$	0	0
$631$	7.88083e33	0.0787417	0.0393709	−	0.999225i	$-0.487465\pi$
$631$	7.88083e33	0.0787417	0.0393709	+	0.999225i	$0.487465\pi$
$632$	0	0
$633$	−1.81562e35	−1.74373
$634$	0	0
$635$	−8.48176e33	−0.0783095
$636$	0	0
$637$	3.98227e34	0.353499
$638$	0	0
$639$	1.15094e35	0.982408
$640$	0	0
$641$	3.14908e34	0.258499	0.129250	−	0.991612i	$-0.458743\pi$
$641$	3.14908e34	0.258499	0.129250	+	0.991612i	$0.458743\pi$
$642$	0	0
$643$	−8.32480e34	−0.657259	−0.328630	−	0.944459i	$-0.606587\pi$
$643$	−8.32480e34	−0.657259	−0.328630	+	0.944459i	$0.606587\pi$
$644$	0	0
$645$	−4.42236e34	−0.335859
$646$	0	0
$647$	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
$650$	0	0
$651$	−2.36321e35	−1.59860
$652$	0	0
$653$	2.06756e35	1.34599	0.672997	−	0.739645i	$-0.265007\pi$
$653$	2.06756e35	1.34599	0.672997	+	0.739645i	$0.265007\pi$
$654$	0	0
$655$	−1.91422e34	−0.119943
$656$	0	0
$657$	7.83084e34	0.472325
$658$	0	0
$659$	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	$0.529555\pi$
$662$	0	0
$663$	−2.77142e35	−1.49205
$664$	0	0
$665$	2.06335e34	0.106980
$666$	0	0
$667$	−1.47201e34	−0.0735087
$668$	0	0
$669$	2.03109e35	0.977016
$670$	0	0
$671$	−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	$0.721514\pi$
$674$	0	0
$675$	6.85047e35	2.94729
$676$	0	0
$677$	3.20244e35	1.32777	0.663885	−	0.747835i	$-0.268907\pi$
$677$	3.20244e35	1.32777	0.663885	+	0.747835i	$0.268907\pi$
$678$	0	0
$679$	3.06003e35	1.22280
$680$	0	0
$681$	−1.58411e35	−0.610165
$682$	0	0
$683$	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
$686$	0	0
$687$	−6.49388e35	−2.24153
$688$	0	0
$689$	1.91372e35	0.636999
$690$	0	0
$691$	−3.59659e35	−1.15456	−0.577279	−	0.816547i	$-0.695886\pi$
$691$	−3.59659e35	−1.15456	−0.577279	+	0.816547i	$0.695886\pi$
$692$	0	0
$693$	−1.76480e35	−0.546425
$694$	0	0
$695$	−1.10548e35	−0.330174
$696$	0	0
$697$	−2.61333e35	−0.752982
$698$	0	0
$699$	−7.22055e34	−0.200727
$700$	0	0
$701$	−4.01671e34	−0.107745	−0.0538723	−	0.998548i	$-0.517156\pi$
$701$	−4.01671e34	−0.107745	−0.0538723	+	0.998548i	$0.517156\pi$
$702$	0	0
$703$	−5.04654e34	−0.130633
$704$	0	0
$705$	−3.37752e35	−0.843788
$706$	0	0
$707$	−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	$0.508786\pi$
$710$	0	0
$711$	−6.53988e35	−1.46960
$712$	0	0
$713$	4.75812e34	0.103232
$714$	0	0
$715$	−3.39402e34	−0.0711032
$716$	0	0
$717$	−1.14103e36	−2.30839
$718$	0	0
$719$	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
$722$	0	0
$723$	−2.96697e35	−0.540859
$724$	0	0
$725$	3.81839e35	0.672443
$726$	0	0
$727$	−8.07486e35	−1.37390	−0.686950	−	0.726705i	$-0.741051\pi$
$727$	−8.07486e35	−1.37390	−0.686950	+	0.726705i	$0.741051\pi$
$728$	0	0
$729$	1.75615e36	2.88714
$730$	0	0
$731$	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	$0.753798\pi$
$734$	0	0
$735$	−1.00210e35	−0.148702
$736$	0	0
$737$	2.24310e35	0.321740
$738$	0	0
$739$	9.24490e34	0.128188	0.0640939	−	0.997944i	$-0.479584\pi$
$739$	9.24490e34	0.128188	0.0640939	+	0.997944i	$0.479584\pi$
$740$	0	0
$741$	−8.33068e35	−1.11674
$742$	0	0
$743$	−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
$746$	0	0
$747$	3.69882e36	4.48287
$748$	0	0
$749$	5.30431e35	0.621736
$750$	0	0
$751$	−5.31819e35	−0.602927	−0.301463	−	0.953478i	$-0.597475\pi$
$751$	−5.31819e35	−0.602927	−0.301463	+	0.953478i	$0.597475\pi$
$752$	0	0
$753$	4.65889e35	0.510911
$754$	0	0
$755$	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	$0.228428\pi$
$758$	0	0
$759$	4.89561e34	0.0486166
$760$	0	0
$761$	1.26153e36	1.21225	0.606123	−	0.795371i	$-0.292724\pi$
$761$	1.26153e36	1.21225	0.606123	+	0.795371i	$0.292724\pi$
$762$	0	0
$763$	−5.02210e35	−0.467012
$764$	0	0
$765$	5.06179e35	0.455550
$766$	0	0
$767$	−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	$0.635351\pi$
$770$	0	0
$771$	−1.96165e36	−1.60119
$772$	0	0
$773$	−1.97217e36	−1.55848	−0.779238	−	0.626728i	$-0.784394\pi$
$773$	−1.97217e36	−1.55848	−0.779238	+	0.626728i	$0.784394\pi$
$774$	0	0
$775$	−1.23425e36	−0.944350
$776$	0	0
$777$	−5.45059e35	−0.403813
$778$	0	0
$779$	−7.85547e35	−0.563578
$780$	0	0
$781$	−1.32757e35	−0.0922398
$782$	0	0
$783$	3.35457e36	2.25743
$784$	0	0
$785$	−6.52153e35	−0.425088
$786$	0	0
$787$	2.25189e35	0.142188	0.0710940	−	0.997470i	$-0.477351\pi$
$787$	2.25189e35	0.142188	0.0710940	+	0.997470i	$0.477351\pi$
$788$	0	0
$789$	2.70224e36	1.65295
$790$	0	0
$791$	2.11702e36	1.25464
$792$	0	0
$793$	−3.27529e36	−1.88077
$794$	0	0
$795$	−4.81568e35	−0.267959
$796$	0	0
$797$	3.69113e36	1.99035	0.995177	−	0.0980978i	$-0.0312758\pi$
$797$	3.69113e36	1.99035	0.995177	+	0.0980978i	$0.0312758\pi$
$798$	0	0
$799$	2.30893e36	1.20664
$800$	0	0
$801$	1.96123e36	0.993393
$802$	0	0
$803$	−9.03258e34	−0.0443473
$804$	0	0
$805$	−4.48700e34	−0.0213554
$806$	0	0
$807$	−4.12420e36	−1.90292
$808$	0	0
$809$	−2.93815e36	−1.31437	−0.657185	−	0.753730i	$-0.728253\pi$
$809$	−2.93815e36	−1.31437	−0.657185	+	0.753730i	$0.728253\pi$
$810$	0	0
$811$	2.44583e36	1.06088	0.530439	−	0.847723i	$-0.322027\pi$
$811$	2.44583e36	1.06088	0.530439	+	0.847723i	$0.322027\pi$
$812$	0	0
$813$	−2.71729e36	−1.14289
$814$	0	0
$815$	1.04137e35	0.0424752
$816$	0	0
$817$	9.08753e35	0.359476
$818$	0	0
$819$	−6.53060e36	−2.50556
$820$	0	0
$821$	3.33815e36	1.24227	0.621136	−	0.783703i	$-0.286672\pi$
$821$	3.33815e36	1.24227	0.621136	+	0.783703i	$0.286672\pi$
$822$	0	0
$823$	−2.54036e36	−0.917058	−0.458529	−	0.888679i	$-0.651623\pi$
$823$	−2.54036e36	−0.917058	−0.458529	+	0.888679i	$0.651623\pi$
$824$	0	0
$825$	−1.26992e36	−0.444734
$826$	0	0
$827$	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	$0.443177\pi$
$830$	0	0
$831$	2.13782e36	0.683847
$832$	0	0
$833$	6.85052e35	0.212648
$834$	0	0
$835$	−1.37324e36	−0.413682
$836$	0	0
$837$	−1.08433e37	−3.17024
$838$	0	0
$839$	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
$842$	0	0
$843$	−8.96316e36	−2.39671
$844$	0	0
$845$	−2.88930e35	−0.0750039
$846$	0	0
$847$	−3.09185e36	−0.779247
$848$	0	0
$849$	−2.74735e36	−0.672308
$850$	0	0
$851$	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	$0.249531\pi$
$854$	0	0
$855$	1.52153e36	0.340961
$856$	0	0
$857$	3.16481e36	0.688791	0.344395	−	0.938825i	$-0.388084\pi$
$857$	3.16481e36	0.688791	0.344395	+	0.938825i	$0.388084\pi$
$858$	0	0
$859$	−5.65370e36	−1.19514	−0.597569	−	0.801818i	$-0.703866\pi$
$859$	−5.65370e36	−1.19514	−0.597569	+	0.801818i	$0.703866\pi$
$860$	0	0
$861$	−8.48442e36	−1.74214
$862$	0	0
$863$	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
$866$	0	0
$867$	5.37651e36	1.01219
$868$	0	0
$869$	7.54351e35	0.137983
$870$	0	0
$871$	8.30052e36	1.47529
$872$	0	0
$873$	2.25649e37	3.89723
$874$	0	0
$875$	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	$0.413377\pi$
$878$	0	0
$879$	7.33233e36	1.16247
$880$	0	0
$881$	−8.05386e36	−1.24109	−0.620547	−	0.784169i	$-0.713090\pi$
$881$	−8.05386e36	−1.24109	−0.620547	+	0.784169i	$0.713090\pi$
$882$	0	0
$883$	4.91849e35	0.0736754	0.0368377	−	0.999321i	$-0.488272\pi$
$883$	4.91849e35	0.0736754	0.0368377	+	0.999321i	$0.488272\pi$
$884$	0	0
$885$	1.00381e36	0.146172
$886$	0	0
$887$	−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
$890$	0	0
$891$	−6.24031e36	−0.835093
$892$	0	0
$893$	6.94048e36	0.903121
$894$	0	0
$895$	−1.10741e36	−0.140127
$896$	0	0
$897$	1.81160e36	0.222924
$898$	0	0
$899$	−6.04394e36	−0.723309
$900$	0	0
$901$	3.29209e36	0.383188
$902$	0	0
$903$	9.81513e36	1.11122
$904$	0	0
$905$	−2.47597e35	−0.0272670
$906$	0	0
$907$	1.31417e37	1.40786	0.703932	−	0.710268i	$-0.251426\pi$
$907$	1.31417e37	1.40786	0.703932	+	0.710268i	$0.251426\pi$
$908$	0	0
$909$	−3.18823e37	−3.32278
$910$	0	0
$911$	−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
$914$	0	0
$915$	8.24192e36	0.791161
$916$	0	0
$917$	4.24848e36	0.396842
$918$	0	0
$919$	1.03191e37	0.937990	0.468995	−	0.883201i	$-0.344616\pi$
$919$	1.03191e37	0.937990	0.468995	+	0.883201i	$0.344616\pi$
$920$	0	0
$921$	1.35356e37	1.19738
$922$	0	0
$923$	−4.91262e36	−0.422953
$924$	0	0
$925$	−2.84672e36	−0.238546
$926$	0	0
$927$	−2.31487e37	−1.88813
$928$	0	0
$929$	−2.17912e37	−1.73015	−0.865075	−	0.501642i	$-0.832729\pi$
$929$	−2.17912e37	−1.73015	−0.865075	+	0.501642i	$0.832729\pi$
$930$	0	0
$931$	2.05921e36	0.159159
$932$	0	0
$933$	2.71170e37	2.04043
$934$	0	0
$935$	−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	$0.766062\pi$
$938$	0	0
$939$	−2.85686e37	−1.98413
$940$	0	0
$941$	2.39454e37	1.61939	0.809695	−	0.586851i	$-0.199632\pi$
$941$	2.39454e37	1.61939	0.809695	+	0.586851i	$0.199632\pi$
$942$	0	0
$943$	1.70826e36	0.112501
$944$	0	0
$945$	1.02254e37	0.655817
$946$	0	0
$947$	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
$950$	0	0
$951$	4.02727e37	2.38646
$952$	0	0
$953$	−1.92589e37	−1.11166	−0.555829	−	0.831297i	$-0.687599\pi$
$953$	−1.92589e37	−1.11166	−0.555829	+	0.831297i	$0.687599\pi$
$954$	0	0
$955$	−6.75736e36	−0.379957
$956$	0	0
$957$	−6.21858e36	−0.340637
$958$	0	0
$959$	−2.10601e37	−1.12390
$960$	0	0
$961$	3.03575e35	0.0157842
$962$	0	0
$963$	3.91144e37	1.98157
$964$	0	0
$965$	−1.95205e36	−0.0963606
$966$	0	0
$967$	−2.29324e37	−1.10311	−0.551554	−	0.834139i	$-0.685965\pi$
$967$	−2.29324e37	−1.10311	−0.551554	+	0.834139i	$0.685965\pi$
$968$	0	0
$969$	−1.43309e37	−0.671779
$970$	0	0
$971$	−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
$974$	0	0
$975$	−4.69929e37	−2.03927
$976$	0	0
$977$	−3.52592e37	−1.49139	−0.745694	−	0.666289i	$-0.767882\pi$
$977$	−3.52592e37	−1.49139	−0.745694	+	0.666289i	$0.767882\pi$
$978$	0	0
$979$	−2.26220e36	−0.0932712
$980$	0	0
$981$	−3.70334e37	−1.48844
$982$	0	0
$983$	−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
$986$	0	0
$987$	7.49617e37	2.79174
$988$	0	0
$989$	−1.97619e36	−0.0717587
$990$	0	0
$991$	−1.16985e37	−0.414200	−0.207100	−	0.978320i	$-0.566403\pi$
$991$	−1.16985e37	−0.414200	−0.207100	+	0.978320i	$0.566403\pi$
$992$	0	0
$993$	6.06819e37	2.09504
$994$	0	0
$995$	−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	$0.374829\pi$
$998$	0	0
$999$	−2.50092e37	−0.800814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.26.a.c.1.1		2
4.3	odd	2		2.26.a.b.1.2	✓	2
12.11	even	2		18.26.a.e.1.2		2
20.3	even	4		50.26.b.e.49.2		4
20.7	even	4		50.26.b.e.49.3		4
20.19	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	4.3	odd	2
16.26.a.c.1.1		2	1.1	even	1	trivial
18.26.a.e.1.2		2	12.11	even	2
50.26.a.c.1.1		2	20.19	odd	2
50.26.b.e.49.2		4	20.3	even	4
50.26.b.e.49.3		4	20.7	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75788e6 q^{3} -1.37041e8 q^{5} +3.04153e10 q^{7} +2.24285e12 q^{9} -2.58704e12 q^{11} -9.57327e13 q^{13} +2.40901e14 q^{15} -1.64685e15 q^{17} -4.95030e15 q^{19} -5.34664e16 q^{21} +1.07650e16 q^{23} -2.79243e17 q^{25} -2.45323e18 q^{27} -1.36741e18 q^{29} +4.42000e18 q^{31} +4.54771e18 q^{33} -4.16814e18 q^{35} +1.01944e19 q^{37} +1.68287e20 q^{39} +1.58687e20 q^{41} -1.83575e20 q^{43} -3.07362e20 q^{45} -1.40203e21 q^{47} -4.15978e20 q^{49} +2.89496e21 q^{51} -1.99903e21 q^{53} +3.54531e20 q^{55} +8.70202e21 q^{57} +4.16691e21 q^{59} +3.42128e22 q^{61} +6.82170e22 q^{63} +1.31193e22 q^{65} -8.67051e22 q^{67} -1.89236e22 q^{69} +5.13159e22 q^{71} +3.49147e22 q^{73} +4.90875e23 q^{75} -7.86857e22 q^{77} -2.91588e23 q^{79} +2.41214e24 q^{81} +1.64916e24 q^{83} +2.25686e23 q^{85} +2.40374e24 q^{87} +8.74435e23 q^{89} -2.91174e24 q^{91} -7.76982e24 q^{93} +6.78393e23 q^{95} +1.00608e25 q^{97} -5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q - 379848 * q^3 + 741953100 * q^5 + 376536944 * q^7 + 3294531432666 * q^9 - 8323034610264 * q^11 - 106467053152292 * q^13 + 1452182413035600 * q^15 + 1327878920113956 * q^17 + 477079242949400 * q^19 - 94860791661752256 * q^21 + 115305025781473872 * q^23 + 195364218352298750 * q^25 - 2171569557205152720 * q^27 + 1724412645206435580 * q^29 + 8688082288351126976 * q^31 - 3356662204005020064 * q^33 - 30572039779548136800 * q^35 - 34908364049750170484 * q^37 + 153494373835331105808 * q^39 + 83014324355953468884 * q^41 - 44539608583471901848 * q^43 + 617059152350257182300 * q^45 - 1869360732054674599776 * q^47 - 854718643011973772046 * q^49 + 6994225992587553820656 * q^51 - 3242170458210097794132 * q^53 - 4687370283547568509200 * q^55 + 16181115057055601354400 * q^57 + 17413675430632049453640 * q^59 + 33975859953090212810044 * q^61 + 36625774166984600306352 * q^63 + 3683906908205094047400 * q^65 - 33012908196969622380616 * q^67 + 125135868082548386734272 * q^69 + 278348008145600813615856 * q^71 + 313442631796571689392148 * q^73 + 1144898988230230172205000 * q^75 + 93616352280445975603392 * q^77 - 928546182288781647817120 * q^79 + 1909195802481672047728242 * q^81 + 452542629950725008985752 * q^83 + 2840452690831654686931800 * q^85 + 6664364855011895770504080 * q^87 - 2346904179674200701244620 * q^89 - 2589295264446210583737824 * q^91 - 1888263951389289833389824 * q^93 + 5449023740708681753370000 * q^95 + 13645482361777595497674436 * q^97 - 11834785687944312803841912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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