LMFDB - Embedded newform 16.26.a.c.1.2

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			1.2
Root			$-162.829$ 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)

$f(q)$	$=$	$q+1.37803e6 q^{3} +8.78994e8 q^{5} -3.00388e10 q^{7} +1.05168e12 q^{9} -5.73599e12 q^{11} -1.07343e13 q^{13} +1.21128e15 q^{15} +2.97473e15 q^{17} +5.42738e15 q^{19} -4.13944e16 q^{21} +1.04540e17 q^{23} +4.74607e17 q^{25} +2.81659e17 q^{27} +3.09182e18 q^{29} +4.26809e18 q^{31} -7.90437e18 q^{33} -2.64039e19 q^{35} -4.51028e19 q^{37} -1.47922e19 q^{39} -7.56724e19 q^{41} +1.39036e20 q^{43} +9.24421e20 q^{45} -4.67328e20 q^{47} -4.38741e20 q^{49} +4.09927e21 q^{51} -1.24315e21 q^{53} -5.04190e21 q^{55} +7.47909e21 q^{57} +1.32468e22 q^{59} -2.36984e20 q^{61} -3.15912e22 q^{63} -9.43539e21 q^{65} +5.36922e22 q^{67} +1.44059e23 q^{69} +2.27032e23 q^{71} +2.78528e23 q^{73} +6.54024e23 q^{75} +1.72302e23 q^{77} -6.36958e23 q^{79} -5.02942e23 q^{81} -1.19662e24 q^{83} +2.61477e24 q^{85} +4.26063e24 q^{87} -3.22134e24 q^{89} +3.22445e23 q^{91} +5.88156e24 q^{93} +4.77063e24 q^{95} +3.58465e24 q^{97} -6.03243e24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.37803e6	1.49707	0.748537	−	0.663093i	$-0.230757\pi$
$3$	1.37803e6	1.49707	0.748537	+	0.663093i	$0.230757\pi$
$4$	0	0
$5$	8.78994e8	1.61013	0.805065	−	0.593187i	$-0.202130\pi$
$5$	8.78994e8	1.61013	0.805065	+	0.593187i	$0.202130\pi$
$6$	0	0
$7$	−3.00388e10	−0.820270	−0.410135	−	0.912025i	$-0.634518\pi$
$7$	−3.00388e10	−0.820270	−0.410135	+	0.912025i	$0.634518\pi$
$8$	0	0
$9$	1.05168e12	1.24123
$10$	0	0
$11$	−5.73599e12	−0.551061	−0.275531	−	0.961292i	$-0.588854\pi$
$11$	−5.73599e12	−0.551061	−0.275531	+	0.961292i	$0.588854\pi$
$12$	0	0
$13$	−1.07343e13	−0.127786	−0.0638928	−	0.997957i	$-0.520352\pi$
$13$	−1.07343e13	−0.127786	−0.0638928	+	0.997957i	$0.520352\pi$
$14$	0	0
$15$	1.21128e15	2.41048
$16$	0	0
$17$	2.97473e15	1.23833	0.619164	−	0.785262i	$-0.287472\pi$
$17$	2.97473e15	1.23833	0.619164	+	0.785262i	$0.287472\pi$
$18$	0	0
$19$	5.42738e15	0.562561	0.281281	−	0.959626i	$-0.409241\pi$
$19$	5.42738e15	0.562561	0.281281	+	0.959626i	$0.409241\pi$
$20$	0	0
$21$	−4.13944e16	−1.22800
$22$	0	0
$23$	1.04540e17	0.994683	0.497341	−	0.867555i	$-0.334310\pi$
$23$	1.04540e17	0.994683	0.497341	+	0.867555i	$0.334310\pi$
$24$	0	0
$25$	4.74607e17	1.59252
$26$	0	0
$27$	2.81659e17	0.361141
$28$	0	0
$29$	3.09182e18	1.62270	0.811352	−	0.584558i	$-0.198732\pi$
$29$	3.09182e18	1.62270	0.811352	+	0.584558i	$0.198732\pi$
$30$	0	0
$31$	4.26809e18	0.973222	0.486611	−	0.873619i	$-0.338233\pi$
$31$	4.26809e18	0.973222	0.486611	+	0.873619i	$0.338233\pi$
$32$	0	0
$33$	−7.90437e18	−0.824980
$34$	0	0
$35$	−2.64039e19	−1.32074
$36$	0	0
$37$	−4.51028e19	−1.12637	−0.563186	−	0.826330i	$-0.690424\pi$
$37$	−4.51028e19	−1.12637	−0.563186	+	0.826330i	$0.690424\pi$
$38$	0	0
$39$	−1.47922e19	−0.191305
$40$	0	0
$41$	−7.56724e19	−0.523769	−0.261884	−	0.965099i	$-0.584344\pi$
$41$	−7.56724e19	−0.523769	−0.261884	+	0.965099i	$0.584344\pi$
$42$	0	0
$43$	1.39036e20	0.530605	0.265302	−	0.964165i	$-0.414528\pi$
$43$	1.39036e20	0.530605	0.265302	+	0.964165i	$0.414528\pi$
$44$	0	0
$45$	9.24421e20	1.99854
$46$	0	0
$47$	−4.67328e20	−0.586676	−0.293338	−	0.956009i	$-0.594766\pi$
$47$	−4.67328e20	−0.586676	−0.293338	+	0.956009i	$0.594766\pi$
$48$	0	0
$49$	−4.38741e20	−0.327158
$50$	0	0
$51$	4.09927e21	1.85387
$52$	0	0
$53$	−1.24315e21	−0.347595	−0.173797	−	0.984781i	$-0.555604\pi$
$53$	−1.24315e21	−0.347595	−0.173797	+	0.984781i	$0.555604\pi$
$54$	0	0
$55$	−5.04190e21	−0.887280
$56$	0	0
$57$	7.47909e21	0.842196
$58$	0	0
$59$	1.32468e22	0.969303	0.484652	−	0.874707i	$-0.338946\pi$
$59$	1.32468e22	0.969303	0.484652	+	0.874707i	$0.338946\pi$
$60$	0	0
$61$	−2.36984e20	−0.0114313	−0.00571565	−	0.999984i	$-0.501819\pi$
$61$	−2.36984e20	−0.0114313	−0.00571565	+	0.999984i	$0.501819\pi$
$62$	0	0
$63$	−3.15912e22	−1.01814
$64$	0	0
$65$	−9.43539e21	−0.205752
$66$	0	0
$67$	5.36922e22	0.801634	0.400817	−	0.916158i	$-0.368726\pi$
$67$	5.36922e22	0.801634	0.400817	+	0.916158i	$0.368726\pi$
$68$	0	0
$69$	1.44059e23	1.48911
$70$	0	0
$71$	2.27032e23	1.64194	0.820971	−	0.570970i	$-0.193433\pi$
$71$	2.27032e23	1.64194	0.820971	+	0.570970i	$0.193433\pi$
$72$	0	0
$73$	2.78528e23	1.42342	0.711710	−	0.702473i	$-0.247921\pi$
$73$	2.78528e23	1.42342	0.711710	+	0.702473i	$0.247921\pi$
$74$	0	0
$75$	6.54024e23	2.38412
$76$	0	0
$77$	1.72302e23	0.452019
$78$	0	0
$79$	−6.36958e23	−1.21275	−0.606376	−	0.795178i	$-0.707378\pi$
$79$	−6.36958e23	−1.21275	−0.606376	+	0.795178i	$0.707378\pi$
$80$	0	0
$81$	−5.02942e23	−0.700576
$82$	0	0
$83$	−1.19662e24	−1.22880	−0.614398	−	0.788996i	$-0.710601\pi$
$83$	−1.19662e24	−1.22880	−0.614398	+	0.788996i	$0.710601\pi$
$84$	0	0
$85$	2.61477e24	1.99387
$86$	0	0
$87$	4.26063e24	2.42931
$88$	0	0
$89$	−3.22134e24	−1.38249	−0.691244	−	0.722621i	$-0.742937\pi$
$89$	−3.22134e24	−1.38249	−0.691244	+	0.722621i	$0.742937\pi$
$90$	0	0
$91$	3.22445e23	0.104819
$92$	0	0
$93$	5.88156e24	1.45699
$94$	0	0
$95$	4.77063e24	0.905797
$96$	0	0
$97$	3.58465e24	0.524566	0.262283	−	0.964991i	$-0.415525\pi$
$97$	3.58465e24	0.524566	0.262283	+	0.964991i	$0.415525\pi$
$98$	0	0
$99$	−6.03243e24	−0.683994
$100$	0	0
$101$	−3.40630e24	−0.300792	−0.150396	−	0.988626i	$-0.548055\pi$
$101$	−3.40630e24	−0.300792	−0.150396	+	0.988626i	$0.548055\pi$
$102$	0	0
$103$	−8.27075e24	−0.571583	−0.285792	−	0.958292i	$-0.592257\pi$
$103$	−8.27075e24	−0.571583	−0.285792	+	0.958292i	$0.592257\pi$
$104$	0	0
$105$	−3.63854e25	−1.97725
$106$	0	0
$107$	−1.57986e25	−0.678142	−0.339071	−	0.940761i	$-0.610113\pi$
$107$	−1.57986e25	−0.678142	−0.339071	+	0.940761i	$0.610113\pi$
$108$	0	0
$109$	3.00869e25	1.02458	0.512290	−	0.858813i	$-0.328797\pi$
$109$	3.00869e25	1.02458	0.512290	+	0.858813i	$0.328797\pi$
$110$	0	0
$111$	−6.21530e25	−1.68626
$112$	0	0
$113$	−8.06745e25	−1.75088	−0.875439	−	0.483329i	$-0.839428\pi$
$113$	−8.06745e25	−1.75088	−0.875439	+	0.483329i	$0.839428\pi$
$114$	0	0
$115$	9.18901e25	1.60157
$116$	0	0
$117$	−1.12891e25	−0.158612
$118$	0	0
$119$	−8.93571e25	−1.01576
$120$	0	0
$121$	−7.54455e25	−0.696332
$122$	0	0
$123$	−1.04279e26	−0.784121
$124$	0	0
$125$	1.55216e26	0.954030
$126$	0	0
$127$	1.32973e26	0.670221	0.335111	−	0.942179i	$-0.391226\pi$
$127$	1.32973e26	0.670221	0.335111	+	0.942179i	$0.391226\pi$
$128$	0	0
$129$	1.91596e26	0.794355
$130$	0	0
$131$	−6.73908e25	−0.230520	−0.115260	−	0.993335i	$-0.536770\pi$
$131$	−6.73908e25	−0.230520	−0.115260	+	0.993335i	$0.536770\pi$
$132$	0	0
$133$	−1.63032e26	−0.461452
$134$	0	0
$135$	2.47577e26	0.581484
$136$	0	0
$137$	−3.22963e26	−0.631170	−0.315585	−	0.948897i	$-0.602201\pi$
$137$	−3.22963e26	−0.631170	−0.315585	+	0.948897i	$0.602201\pi$
$138$	0	0
$139$	9.21581e25	0.150262	0.0751309	−	0.997174i	$-0.476063\pi$
$139$	9.21581e25	0.150262	0.0751309	+	0.997174i	$0.476063\pi$
$140$	0	0
$141$	−6.43992e26	−0.878298
$142$	0	0
$143$	6.15719e25	0.0704177
$144$	0	0
$145$	2.71769e27	2.61276
$146$	0	0
$147$	−6.04599e26	−0.489779
$148$	0	0
$149$	−7.12667e26	−0.487594	−0.243797	−	0.969826i	$-0.578393\pi$
$149$	−7.12667e26	−0.487594	−0.243797	+	0.969826i	$0.578393\pi$
$150$	0	0
$151$	3.06161e27	1.77312	0.886559	−	0.462616i	$-0.153089\pi$
$151$	3.06161e27	1.77312	0.886559	+	0.462616i	$0.153089\pi$
$152$	0	0
$153$	3.12846e27	1.53705
$154$	0	0
$155$	3.75162e27	1.56701
$156$	0	0
$157$	2.38960e27	0.850315	0.425157	−	0.905119i	$-0.360219\pi$
$157$	2.38960e27	0.850315	0.425157	+	0.905119i	$0.360219\pi$
$158$	0	0
$159$	−1.71309e27	−0.520375
$160$	0	0
$161$	−3.14025e27	−0.815908
$162$	0	0
$163$	−6.63092e27	−1.47648	−0.738242	−	0.674536i	$-0.764344\pi$
$163$	−6.63092e27	−1.47648	−0.738242	+	0.674536i	$0.764344\pi$
$164$	0	0
$165$	−6.94790e27	−1.32832
$166$	0	0
$167$	−4.06997e27	−0.669323	−0.334662	−	0.942338i	$-0.608622\pi$
$167$	−4.06997e27	−0.669323	−0.334662	+	0.942338i	$0.608622\pi$
$168$	0	0
$169$	−6.94118e27	−0.983671
$170$	0	0
$171$	5.70787e27	0.698269
$172$	0	0
$173$	−8.68916e27	−0.919182	−0.459591	−	0.888131i	$-0.652004\pi$
$173$	−8.68916e27	−0.919182	−0.459591	+	0.888131i	$0.652004\pi$
$174$	0	0
$175$	−1.42566e28	−1.30629
$176$	0	0
$177$	1.82545e28	1.45112
$178$	0	0
$179$	4.31406e27	0.298005	0.149002	−	0.988837i	$-0.452394\pi$
$179$	4.31406e27	0.298005	0.149002	+	0.988837i	$0.452394\pi$
$180$	0	0
$181$	2.22969e28	1.34049	0.670243	−	0.742142i	$-0.266190\pi$
$181$	2.22969e28	1.34049	0.670243	+	0.742142i	$0.266190\pi$
$182$	0	0
$183$	−3.26571e26	−0.0171135
$184$	0	0
$185$	−3.96451e28	−1.81360
$186$	0	0
$187$	−1.70630e28	−0.682394
$188$	0	0
$189$	−8.46070e27	−0.296233
$190$	0	0
$191$	1.91337e28	0.587329	0.293665	−	0.955908i	$-0.405125\pi$
$191$	1.91337e28	0.587329	0.293665	+	0.955908i	$0.405125\pi$
$192$	0	0
$193$	−5.96748e28	−1.60814	−0.804071	−	0.594533i	$-0.797337\pi$
$193$	−5.96748e28	−1.60814	−0.804071	+	0.594533i	$0.797337\pi$
$194$	0	0
$195$	−1.30023e28	−0.308025
$196$	0	0
$197$	6.24204e28	1.30166	0.650832	−	0.759222i	$-0.274420\pi$
$197$	6.24204e28	1.30166	0.650832	+	0.759222i	$0.274420\pi$
$198$	0	0
$199$	3.24229e28	0.595921	0.297960	−	0.954578i	$-0.403694\pi$
$199$	3.24229e28	0.595921	0.297960	+	0.954578i	$0.403694\pi$
$200$	0	0
$201$	7.39895e28	1.20011
$202$	0	0
$203$	−9.28745e28	−1.33106
$204$	0	0
$205$	−6.65156e28	−0.843335
$206$	0	0
$207$	1.09943e29	1.23463
$208$	0	0
$209$	−3.11314e28	−0.310006
$210$	0	0
$211$	−3.82197e28	−0.337875	−0.168938	−	0.985627i	$-0.554034\pi$
$211$	−3.82197e28	−0.337875	−0.168938	+	0.985627i	$0.554034\pi$
$212$	0	0
$213$	3.12857e29	2.45811
$214$	0	0
$215$	1.22212e29	0.854342
$216$	0	0
$217$	−1.28208e29	−0.798305
$218$	0	0
$219$	3.83820e29	2.13097
$220$	0	0
$221$	−3.19316e28	−0.158241
$222$	0	0
$223$	−3.11100e29	−1.37749	−0.688746	−	0.725003i	$-0.741839\pi$
$223$	−3.11100e29	−1.37749	−0.688746	+	0.725003i	$0.741839\pi$
$224$	0	0
$225$	4.99135e29	1.97668
$226$	0	0
$227$	1.05572e29	0.374304	0.187152	−	0.982331i	$-0.440074\pi$
$227$	1.05572e29	0.374304	0.187152	+	0.982331i	$0.440074\pi$
$228$	0	0
$229$	−3.70322e29	−1.17662	−0.588310	−	0.808635i	$-0.700207\pi$
$229$	−3.70322e29	−1.17662	−0.588310	+	0.808635i	$0.700207\pi$
$230$	0	0
$231$	2.37438e29	0.676706
$232$	0	0
$233$	−6.84781e28	−0.175228	−0.0876138	−	0.996155i	$-0.527924\pi$
$233$	−6.84781e28	−0.175228	−0.0876138	+	0.996155i	$0.527924\pi$
$234$	0	0
$235$	−4.10778e29	−0.944625
$236$	0	0
$237$	−8.77748e29	−1.81558
$238$	0	0
$239$	−3.87773e29	−0.722111	−0.361055	−	0.932544i	$-0.617583\pi$
$239$	−3.87773e29	−0.722111	−0.361055	+	0.932544i	$0.617583\pi$
$240$	0	0
$241$	1.03723e30	1.74046	0.870230	−	0.492646i	$-0.163970\pi$
$241$	1.03723e30	1.74046	0.870230	+	0.492646i	$0.163970\pi$
$242$	0	0
$243$	−9.31717e29	−1.40996
$244$	0	0
$245$	−3.85651e29	−0.526766
$246$	0	0
$247$	−5.82591e28	−0.0718873
$248$	0	0
$249$	−1.64898e30	−1.83960
$250$	0	0
$251$	−1.47990e30	−1.49386	−0.746930	−	0.664903i	$-0.768473\pi$
$251$	−1.47990e30	−1.49386	−0.746930	+	0.664903i	$0.768473\pi$
$252$	0	0
$253$	−5.99641e29	−0.548131
$254$	0	0
$255$	3.60323e30	2.98497
$256$	0	0
$257$	6.75384e29	0.507442	0.253721	−	0.967277i	$-0.418345\pi$
$257$	6.75384e29	0.507442	0.253721	+	0.967277i	$0.418345\pi$
$258$	0	0
$259$	1.35483e30	0.923929
$260$	0	0
$261$	3.25161e30	2.01415
$262$	0	0
$263$	−1.42102e30	−0.800119	−0.400059	−	0.916489i	$-0.631011\pi$
$263$	−1.42102e30	−0.800119	−0.400059	+	0.916489i	$0.631011\pi$
$264$	0	0
$265$	−1.09272e30	−0.559673
$266$	0	0
$267$	−4.43911e30	−2.06969
$268$	0	0
$269$	−1.90129e30	−0.807504	−0.403752	−	0.914868i	$-0.632294\pi$
$269$	−1.90129e30	−0.807504	−0.403752	+	0.914868i	$0.632294\pi$
$270$	0	0
$271$	1.45800e30	0.564471	0.282236	−	0.959345i	$-0.408924\pi$
$271$	1.45800e30	0.564471	0.282236	+	0.959345i	$0.408924\pi$
$272$	0	0
$273$	4.44340e29	0.156921
$274$	0	0
$275$	−2.72234e30	−0.877575
$276$	0	0
$277$	−5.31088e29	−0.156376	−0.0781879	−	0.996939i	$-0.524913\pi$
$277$	−5.31088e29	−0.156376	−0.0781879	+	0.996939i	$0.524913\pi$
$278$	0	0
$279$	4.48867e30	1.20799
$280$	0	0
$281$	−3.64974e30	−0.898323	−0.449161	−	0.893451i	$-0.648277\pi$
$281$	−3.64974e30	−0.898323	−0.449161	+	0.893451i	$0.648277\pi$
$282$	0	0
$283$	1.43965e30	0.324285	0.162143	−	0.986767i	$-0.448160\pi$
$283$	1.43965e30	0.324285	0.162143	+	0.986767i	$0.448160\pi$
$284$	0	0
$285$	6.57408e30	1.35604
$286$	0	0
$287$	2.27311e30	0.429632
$288$	0	0
$289$	3.07837e30	0.533456
$290$	0	0
$291$	4.93976e30	0.785315
$292$	0	0
$293$	1.24504e31	1.81692	0.908462	−	0.417967i	$-0.137257\pi$
$293$	1.24504e31	1.81692	0.908462	+	0.417967i	$0.137257\pi$
$294$	0	0
$295$	1.16438e31	1.56070
$296$	0	0
$297$	−1.61559e30	−0.199011
$298$	0	0
$299$	−1.12217e30	−0.127106
$300$	0	0
$301$	−4.17647e30	−0.435239
$302$	0	0
$303$	−4.69399e30	−0.450307
$304$	0	0
$305$	−2.08307e29	−0.0184059
$306$	0	0
$307$	−2.27193e31	−1.84997	−0.924986	−	0.380001i	$-0.875924\pi$
$307$	−2.27193e31	−1.84997	−0.924986	+	0.380001i	$0.875924\pi$
$308$	0	0
$309$	−1.13974e31	−0.855703
$310$	0	0
$311$	1.84336e31	1.27675	0.638374	−	0.769726i	$-0.279607\pi$
$311$	1.84336e31	1.27675	0.638374	+	0.769726i	$0.279607\pi$
$312$	0	0
$313$	4.98693e30	0.318808	0.159404	−	0.987213i	$-0.449043\pi$
$313$	4.98693e30	0.318808	0.159404	+	0.987213i	$0.449043\pi$
$314$	0	0
$315$	−2.77685e31	−1.63934
$316$	0	0
$317$	−8.79478e30	−0.479716	−0.239858	−	0.970808i	$-0.577101\pi$
$317$	−8.79478e30	−0.479716	−0.239858	+	0.970808i	$0.577101\pi$
$318$	0	0
$319$	−1.77347e31	−0.894209
$320$	0	0
$321$	−2.17709e31	−1.01523
$322$	0	0
$323$	1.61450e31	0.696635
$324$	0	0
$325$	−5.09458e30	−0.203501
$326$	0	0
$327$	4.14606e31	1.53387
$328$	0	0
$329$	1.40380e31	0.481233
$330$	0	0
$331$	−5.97122e31	−1.89764	−0.948818	−	0.315822i	$-0.897720\pi$
$331$	−5.97122e31	−1.89764	−0.948818	+	0.315822i	$0.897720\pi$
$332$	0	0
$333$	−4.74337e31	−1.39809
$334$	0	0
$335$	4.71951e31	1.29073
$336$	0	0
$337$	1.08428e31	0.275275	0.137638	−	0.990483i	$-0.456049\pi$
$337$	1.08428e31	0.275275	0.137638	+	0.990483i	$0.456049\pi$
$338$	0	0
$339$	−1.11172e32	−2.62119
$340$	0	0
$341$	−2.44817e31	−0.536305
$342$	0	0
$343$	5.34633e31	1.08863
$344$	0	0
$345$	1.26627e32	2.39767
$346$	0	0
$347$	−7.20193e30	−0.126862	−0.0634308	−	0.997986i	$-0.520204\pi$
$347$	−7.20193e30	−0.126862	−0.0634308	+	0.997986i	$0.520204\pi$
$348$	0	0
$349$	5.27779e31	0.865235	0.432618	−	0.901577i	$-0.357590\pi$
$349$	5.27779e31	0.865235	0.432618	+	0.901577i	$0.357590\pi$
$350$	0	0
$351$	−3.02342e30	−0.0461487
$352$	0	0
$353$	−5.67715e31	−0.807135	−0.403567	−	0.914950i	$-0.632230\pi$
$353$	−5.67715e31	−0.807135	−0.403567	+	0.914950i	$0.632230\pi$
$354$	0	0
$355$	1.99560e32	2.64374
$356$	0	0
$357$	−1.23137e32	−1.52067
$358$	0	0
$359$	−2.18925e31	−0.252125	−0.126062	−	0.992022i	$-0.540234\pi$
$359$	−2.18925e31	−0.252125	−0.126062	+	0.992022i	$0.540234\pi$
$360$	0	0
$361$	−6.36201e31	−0.683525
$362$	0	0
$363$	−1.03966e32	−1.04246
$364$	0	0
$365$	2.44824e32	2.29189
$366$	0	0
$367$	−1.00500e32	−0.878699	−0.439349	−	0.898316i	$-0.644791\pi$
$367$	−1.00500e32	−0.878699	−0.439349	+	0.898316i	$0.644791\pi$
$368$	0	0
$369$	−7.95833e31	−0.650118
$370$	0	0
$371$	3.73425e31	0.285122
$372$	0	0
$373$	6.31052e31	0.450510	0.225255	−	0.974300i	$-0.427678\pi$
$373$	6.31052e31	0.450510	0.225255	+	0.974300i	$0.427678\pi$
$374$	0	0
$375$	2.13893e32	1.42825
$376$	0	0
$377$	−3.31886e31	−0.207358
$378$	0	0
$379$	8.85755e31	0.517993	0.258996	−	0.965878i	$-0.416608\pi$
$379$	8.85755e31	0.517993	0.258996	+	0.965878i	$0.416608\pi$
$380$	0	0
$381$	1.83241e32	1.00337
$382$	0	0
$383$	−9.08315e31	−0.465856	−0.232928	−	0.972494i	$-0.574831\pi$
$383$	−9.08315e31	−0.465856	−0.232928	+	0.972494i	$0.574831\pi$
$384$	0	0
$385$	1.51453e32	0.727809
$386$	0	0
$387$	1.46221e32	0.658603
$388$	0	0
$389$	7.13043e31	0.301124	0.150562	−	0.988601i	$-0.451892\pi$
$389$	7.13043e31	0.301124	0.150562	+	0.988601i	$0.451892\pi$
$390$	0	0
$391$	3.10978e32	1.23174
$392$	0	0
$393$	−9.28666e31	−0.345106
$394$	0	0
$395$	−5.59883e32	−1.95269
$396$	0	0
$397$	2.99427e31	0.0980413	0.0490206	−	0.998798i	$-0.484390\pi$
$397$	2.99427e31	0.0980413	0.0490206	+	0.998798i	$0.484390\pi$
$398$	0	0
$399$	−2.24663e32	−0.690828
$400$	0	0
$401$	−5.00588e32	−1.44603	−0.723013	−	0.690835i	$-0.757243\pi$
$401$	−5.00588e32	−1.44603	−0.723013	+	0.690835i	$0.757243\pi$
$402$	0	0
$403$	−4.58150e31	−0.124364
$404$	0	0
$405$	−4.42083e32	−1.12802
$406$	0	0
$407$	2.58709e32	0.620700
$408$	0	0
$409$	−2.78379e32	−0.628195	−0.314097	−	0.949391i	$-0.601702\pi$
$409$	−2.78379e32	−0.628195	−0.314097	+	0.949391i	$0.601702\pi$
$410$	0	0
$411$	−4.45054e32	−0.944908
$412$	0	0
$413$	−3.97916e32	−0.795090
$414$	0	0
$415$	−1.05182e33	−1.97852
$416$	0	0
$417$	1.26997e32	0.224953
$418$	0	0
$419$	6.36828e32	1.06254	0.531272	−	0.847201i	$-0.321714\pi$
$419$	6.36828e32	1.06254	0.531272	+	0.847201i	$0.321714\pi$
$420$	0	0
$421$	−3.92275e32	−0.616686	−0.308343	−	0.951275i	$-0.599775\pi$
$421$	−3.92275e32	−0.616686	−0.308343	+	0.951275i	$0.599775\pi$
$422$	0	0
$423$	−4.91480e32	−0.728201
$424$	0	0
$425$	1.41183e33	1.97206
$426$	0	0
$427$	7.11870e30	0.00937675
$428$	0	0
$429$	8.48480e31	0.105421
$430$	0	0
$431$	−2.44178e32	−0.286247	−0.143124	−	0.989705i	$-0.545715\pi$
$431$	−2.44178e32	−0.286247	−0.143124	+	0.989705i	$0.545715\pi$
$432$	0	0
$433$	9.12872e32	1.00998	0.504988	−	0.863126i	$-0.331497\pi$
$433$	9.12872e32	1.00998	0.504988	+	0.863126i	$0.331497\pi$
$434$	0	0
$435$	3.74506e33	3.91150
$436$	0	0
$437$	5.67378e32	0.559570
$438$	0	0
$439$	−6.07601e32	−0.565995	−0.282997	−	0.959121i	$-0.591329\pi$
$439$	−6.07601e32	−0.565995	−0.282997	+	0.959121i	$0.591329\pi$
$440$	0	0
$441$	−4.61416e32	−0.406078
$442$	0	0
$443$	−9.06289e32	−0.753737	−0.376869	−	0.926267i	$-0.622999\pi$
$443$	−9.06289e32	−0.753737	−0.376869	+	0.926267i	$0.622999\pi$
$444$	0	0
$445$	−2.83154e33	−2.22598
$446$	0	0
$447$	−9.82078e32	−0.729965
$448$	0	0
$449$	−1.21579e33	−0.854630	−0.427315	−	0.904103i	$-0.640540\pi$
$449$	−1.21579e33	−0.854630	−0.427315	+	0.904103i	$0.640540\pi$
$450$	0	0
$451$	4.34056e32	0.288629
$452$	0	0
$453$	4.21899e33	2.65449
$454$	0	0
$455$	2.83428e32	0.168772
$456$	0	0
$457$	−5.59530e32	−0.315407	−0.157703	−	0.987487i	$-0.550409\pi$
$457$	−5.59530e32	−0.315407	−0.157703	+	0.987487i	$0.550409\pi$
$458$	0	0
$459$	8.37859e32	0.447211
$460$	0	0
$461$	−3.78592e32	−0.191386	−0.0956930	−	0.995411i	$-0.530507\pi$
$461$	−3.78592e32	−0.191386	−0.0956930	+	0.995411i	$0.530507\pi$
$462$	0	0
$463$	−2.77039e33	−1.32672	−0.663359	−	0.748301i	$-0.730870\pi$
$463$	−2.77039e33	−1.32672	−0.663359	+	0.748301i	$0.730870\pi$
$464$	0	0
$465$	5.16985e33	2.34594
$466$	0	0
$467$	2.31922e33	0.997428	0.498714	−	0.866767i	$-0.333806\pi$
$467$	2.31922e33	0.997428	0.498714	+	0.866767i	$0.333806\pi$
$468$	0	0
$469$	−1.61285e33	−0.657556
$470$	0	0
$471$	3.29294e33	1.27298
$472$	0	0
$473$	−7.97508e32	−0.292396
$474$	0	0
$475$	2.57587e33	0.895889
$476$	0	0
$477$	−1.30739e33	−0.431446
$478$	0	0
$479$	−2.24376e33	−0.702720	−0.351360	−	0.936240i	$-0.614281\pi$
$479$	−2.24376e33	−0.702720	−0.351360	+	0.936240i	$0.614281\pi$
$480$	0	0
$481$	4.84147e32	0.143934
$482$	0	0
$483$	−4.32737e33	−1.22148
$484$	0	0
$485$	3.15089e33	0.844620
$486$	0	0
$487$	−3.71393e32	−0.0945632	−0.0472816	−	0.998882i	$-0.515056\pi$
$487$	−3.71393e32	−0.0945632	−0.0472816	+	0.998882i	$0.515056\pi$
$488$	0	0
$489$	−9.13761e33	−2.21041
$490$	0	0
$491$	2.65508e33	0.610323	0.305161	−	0.952301i	$-0.401290\pi$
$491$	2.65508e33	0.610323	0.305161	+	0.952301i	$0.401290\pi$
$492$	0	0
$493$	9.19732e33	2.00944
$494$	0	0
$495$	−5.30247e33	−1.10132
$496$	0	0
$497$	−6.81976e33	−1.34684
$498$	0	0
$499$	2.39117e33	0.449111	0.224556	−	0.974461i	$-0.427907\pi$
$499$	2.39117e33	0.449111	0.224556	+	0.974461i	$0.427907\pi$
$500$	0	0
$501$	−5.60855e33	−1.00203
$502$	0	0
$503$	6.34146e33	1.07793	0.538964	−	0.842329i	$-0.318816\pi$
$503$	6.34146e33	1.07793	0.538964	+	0.842329i	$0.318816\pi$
$504$	0	0
$505$	−2.99412e33	−0.484314
$506$	0	0
$507$	−9.56517e33	−1.47263
$508$	0	0
$509$	−3.62528e32	−0.0531337	−0.0265668	−	0.999647i	$-0.508457\pi$
$509$	−3.62528e32	−0.0531337	−0.0265668	+	0.999647i	$0.508457\pi$
$510$	0	0
$511$	−8.36664e33	−1.16759
$512$	0	0
$513$	1.52867e33	0.203164
$514$	0	0
$515$	−7.26994e33	−0.920323
$516$	0	0
$517$	2.68059e33	0.323295
$518$	0	0
$519$	−1.19739e34	−1.37608
$520$	0	0
$521$	8.23573e33	0.902050	0.451025	−	0.892511i	$-0.351058\pi$
$521$	8.23573e33	0.902050	0.451025	+	0.892511i	$0.351058\pi$
$522$	0	0
$523$	1.20443e34	1.25750	0.628752	−	0.777606i	$-0.283566\pi$
$523$	1.20443e34	1.25750	0.628752	+	0.777606i	$0.283566\pi$
$524$	0	0
$525$	−1.96461e34	−1.95562
$526$	0	0
$527$	1.26964e34	1.20517
$528$	0	0
$529$	−1.17149e32	−0.0106057
$530$	0	0
$531$	1.39314e34	1.20313
$532$	0	0
$533$	8.12291e32	0.0669301
$534$	0	0
$535$	−1.38868e34	−1.09190
$536$	0	0
$537$	5.94490e33	0.446135
$538$	0	0
$539$	2.51661e33	0.180284
$540$	0	0
$541$	1.30366e34	0.891657	0.445829	−	0.895118i	$-0.352909\pi$
$541$	1.30366e34	0.891657	0.445829	+	0.895118i	$0.352909\pi$
$542$	0	0
$543$	3.07258e34	2.00681
$544$	0	0
$545$	2.64462e34	1.64971
$546$	0	0
$547$	1.78334e34	1.06266	0.531328	−	0.847166i	$-0.321693\pi$
$547$	1.78334e34	1.06266	0.531328	+	0.847166i	$0.321693\pi$
$548$	0	0
$549$	−2.49231e32	−0.0141889
$550$	0	0
$551$	1.67805e34	0.912871
$552$	0	0
$553$	1.91334e34	0.994784
$554$	0	0
$555$	−5.46321e34	−2.71510
$556$	0	0
$557$	8.73339e33	0.414947	0.207474	−	0.978241i	$-0.433476\pi$
$557$	8.73339e33	0.414947	0.207474	+	0.978241i	$0.433476\pi$
$558$	0	0
$559$	−1.49245e33	−0.0678037
$560$	0	0
$561$	−2.35134e34	−1.02160
$562$	0	0
$563$	−3.82416e34	−1.58921	−0.794606	−	0.607126i	$-0.792322\pi$
$563$	−3.82416e34	−1.58921	−0.794606	+	0.607126i	$0.792322\pi$
$564$	0	0
$565$	−7.09124e34	−2.81914
$566$	0	0
$567$	1.51078e34	0.574661
$568$	0	0
$569$	3.49591e34	1.27250	0.636248	−	0.771484i	$-0.280485\pi$
$569$	3.49591e34	1.27250	0.636248	+	0.771484i	$0.280485\pi$
$570$	0	0
$571$	3.60524e34	1.25598	0.627989	−	0.778222i	$-0.283878\pi$
$571$	3.60524e34	1.25598	0.627989	+	0.778222i	$0.283878\pi$
$572$	0	0
$573$	2.63668e34	0.879276
$574$	0	0
$575$	4.96155e34	1.58405
$576$	0	0
$577$	−4.45926e34	−1.36322	−0.681609	−	0.731717i	$-0.738719\pi$
$577$	−4.45926e34	−1.36322	−0.681609	+	0.731717i	$0.738719\pi$
$578$	0	0
$579$	−8.22337e34	−2.40751
$580$	0	0
$581$	3.59450e34	1.00794
$582$	0	0
$583$	7.13067e33	0.191546
$584$	0	0
$585$	−9.92303e33	−0.255385
$586$	0	0
$587$	2.50958e34	0.618905	0.309452	−	0.950915i	$-0.399854\pi$
$587$	2.50958e34	0.618905	0.309452	+	0.950915i	$0.399854\pi$
$588$	0	0
$589$	2.31645e34	0.547497
$590$	0	0
$591$	8.60173e34	1.94869
$592$	0	0
$593$	−4.88078e33	−0.106000	−0.0530000	−	0.998595i	$-0.516878\pi$
$593$	−4.88078e33	−0.106000	−0.0530000	+	0.998595i	$0.516878\pi$
$594$	0	0
$595$	−7.85444e34	−1.63551
$596$	0	0
$597$	4.46798e34	0.892138
$598$	0	0
$599$	−4.61838e34	−0.884412	−0.442206	−	0.896914i	$-0.645804\pi$
$599$	−4.61838e34	−0.884412	−0.442206	+	0.896914i	$0.645804\pi$
$600$	0	0
$601$	−3.07405e34	−0.564651	−0.282325	−	0.959319i	$-0.591106\pi$
$601$	−3.07405e34	−0.564651	−0.282325	+	0.959319i	$0.591106\pi$
$602$	0	0
$603$	5.64671e34	0.995013
$604$	0	0
$605$	−6.63161e34	−1.12118
$606$	0	0
$607$	−1.03737e35	−1.68297	−0.841484	−	0.540282i	$-0.818318\pi$
$607$	−1.03737e35	−1.68297	−0.841484	+	0.540282i	$0.818318\pi$
$608$	0	0
$609$	−1.27984e35	−1.99269
$610$	0	0
$611$	5.01644e33	0.0749688
$612$	0	0
$613$	2.85744e34	0.409941	0.204970	−	0.978768i	$-0.434290\pi$
$613$	2.85744e34	0.409941	0.204970	+	0.978768i	$0.434290\pi$
$614$	0	0
$615$	−9.16606e34	−1.26254
$616$	0	0
$617$	5.36336e34	0.709370	0.354685	−	0.934986i	$-0.384588\pi$
$617$	5.36336e34	0.709370	0.354685	+	0.934986i	$0.384588\pi$
$618$	0	0
$619$	−6.60909e34	−0.839477	−0.419738	−	0.907645i	$-0.637878\pi$
$619$	−6.60909e34	−0.839477	−0.419738	+	0.907645i	$0.637878\pi$
$620$	0	0
$621$	2.94447e34	0.359221
$622$	0	0
$623$	9.67651e34	1.13401
$624$	0	0
$625$	−5.00979e33	−0.0564053
$626$	0	0
$627$	−4.29000e34	−0.464102
$628$	0	0
$629$	−1.34168e35	−1.39482
$630$	0	0
$631$	−8.62816e34	−0.862087	−0.431043	−	0.902331i	$-0.641854\pi$
$631$	−8.62816e34	−0.862087	−0.431043	+	0.902331i	$0.641854\pi$
$632$	0	0
$633$	−5.26679e34	−0.505824
$634$	0	0
$635$	1.16883e35	1.07914
$636$	0	0
$637$	4.70958e33	0.0418061
$638$	0	0
$639$	2.38765e35	2.03803
$640$	0	0
$641$	1.77093e35	1.45370	0.726850	−	0.686796i	$-0.240983\pi$
$641$	1.77093e35	1.45370	0.726850	+	0.686796i	$0.240983\pi$
$642$	0	0
$643$	7.98617e34	0.630524	0.315262	−	0.949005i	$-0.397908\pi$
$643$	7.98617e34	0.630524	0.315262	+	0.949005i	$0.397908\pi$
$644$	0	0
$645$	1.68412e35	1.27901
$646$	0	0
$647$	2.16671e35	1.58306	0.791530	−	0.611130i	$-0.209285\pi$
$647$	2.16671e35	1.58306	0.791530	+	0.611130i	$0.209285\pi$
$648$	0	0
$649$	−7.59833e34	−0.534145
$650$	0	0
$651$	−1.76675e35	−1.19512
$652$	0	0
$653$	2.41184e35	1.57013	0.785063	−	0.619416i	$-0.212631\pi$
$653$	2.41184e35	1.57013	0.785063	+	0.619416i	$0.212631\pi$
$654$	0	0
$655$	−5.92361e34	−0.371168
$656$	0	0
$657$	2.92923e35	1.76679
$658$	0	0
$659$	−2.18788e35	−1.27045	−0.635223	−	0.772329i	$-0.719092\pi$
$659$	−2.18788e35	−1.27045	−0.635223	+	0.772329i	$0.719092\pi$
$660$	0	0
$661$	−8.90411e34	−0.497820	−0.248910	−	0.968527i	$-0.580072\pi$
$661$	−8.90411e34	−0.497820	−0.248910	+	0.968527i	$0.580072\pi$
$662$	0	0
$663$	−4.40028e34	−0.236898
$664$	0	0
$665$	−1.43304e35	−0.742998
$666$	0	0
$667$	3.23219e35	1.61408
$668$	0	0
$669$	−4.28705e35	−2.06221
$670$	0	0
$671$	1.35934e33	0.00629934
$672$	0	0
$673$	2.39665e35	1.07008	0.535040	−	0.844827i	$-0.320297\pi$
$673$	2.39665e35	1.07008	0.535040	+	0.844827i	$0.320297\pi$
$674$	0	0
$675$	1.33678e35	0.575124
$676$	0	0
$677$	−3.79216e34	−0.157227	−0.0786137	−	0.996905i	$-0.525049\pi$
$677$	−3.79216e34	−0.157227	−0.0786137	+	0.996905i	$0.525049\pi$
$678$	0	0
$679$	−1.07679e35	−0.430286
$680$	0	0
$681$	1.45481e35	0.560361
$682$	0	0
$683$	3.21208e35	1.19269	0.596345	−	0.802729i	$-0.296619\pi$
$683$	3.21208e35	1.19269	0.596345	+	0.802729i	$0.296619\pi$
$684$	0	0
$685$	−2.83883e35	−1.01627
$686$	0	0
$687$	−5.10316e35	−1.76149
$688$	0	0
$689$	1.33443e34	0.0444177
$690$	0	0
$691$	−8.34540e34	−0.267899	−0.133950	−	0.990988i	$-0.542766\pi$
$691$	−8.34540e34	−0.267899	−0.133950	+	0.990988i	$0.542766\pi$
$692$	0	0
$693$	1.81207e35	0.561060
$694$	0	0
$695$	8.10064e34	0.241941
$696$	0	0
$697$	−2.25105e35	−0.648597
$698$	0	0
$699$	−9.43649e34	−0.262329
$700$	0	0
$701$	1.50008e35	0.402383	0.201191	−	0.979552i	$-0.435519\pi$
$701$	1.50008e35	0.402383	0.201191	+	0.979552i	$0.435519\pi$
$702$	0	0
$703$	−2.44790e35	−0.633653
$704$	0	0
$705$	−5.66066e35	−1.41417
$706$	0	0
$707$	1.02321e35	0.246730
$708$	0	0
$709$	−2.12042e35	−0.493564	−0.246782	−	0.969071i	$-0.579373\pi$
$709$	−2.12042e35	−0.493564	−0.246782	+	0.969071i	$0.579373\pi$
$710$	0	0
$711$	−6.69877e35	−1.50531
$712$	0	0
$713$	4.46186e35	0.968048
$714$	0	0
$715$	5.41213e34	0.113382
$716$	0	0
$717$	−5.34363e35	−1.08105
$718$	0	0
$719$	−3.27911e35	−0.640686	−0.320343	−	0.947302i	$-0.603798\pi$
$719$	−3.27911e35	−0.640686	−0.320343	+	0.947302i	$0.603798\pi$
$720$	0	0
$721$	2.48443e35	0.468853
$722$	0	0
$723$	1.42934e36	2.60560
$724$	0	0
$725$	1.46740e36	2.58419
$726$	0	0
$727$	−2.40795e35	−0.409701	−0.204850	−	0.978793i	$-0.565671\pi$
$727$	−2.40795e35	−0.409701	−0.204850	+	0.978793i	$0.565671\pi$
$728$	0	0
$729$	−8.57797e35	−1.41023
$730$	0	0
$731$	4.13594e35	0.657063
$732$	0	0
$733$	−3.39196e35	−0.520776	−0.260388	−	0.965504i	$-0.583850\pi$
$733$	−3.39196e35	−0.520776	−0.260388	+	0.965504i	$0.583850\pi$
$734$	0	0
$735$	−5.31439e35	−0.788608
$736$	0	0
$737$	−3.07978e35	−0.441749
$738$	0	0
$739$	−1.23366e36	−1.71057	−0.855284	−	0.518159i	$-0.826617\pi$
$739$	−1.23366e36	−1.71057	−0.855284	+	0.518159i	$0.826617\pi$
$740$	0	0
$741$	−8.02829e34	−0.107621
$742$	0	0
$743$	6.24478e35	0.809389	0.404694	−	0.914452i	$-0.367378\pi$
$743$	6.24478e35	0.809389	0.404694	+	0.914452i	$0.367378\pi$
$744$	0	0
$745$	−6.26430e35	−0.785090
$746$	0	0
$747$	−1.25846e36	−1.52522
$748$	0	0
$749$	4.74570e35	0.556259
$750$	0	0
$751$	4.41291e35	0.500294	0.250147	−	0.968208i	$-0.419521\pi$
$751$	4.41291e35	0.500294	0.250147	+	0.968208i	$0.419521\pi$
$752$	0	0
$753$	−2.03934e36	−2.23642
$754$	0	0
$755$	2.69113e36	2.85495
$756$	0	0
$757$	1.83266e35	0.188098	0.0940489	−	0.995568i	$-0.470019\pi$
$757$	1.83266e35	0.188098	0.0940489	+	0.995568i	$0.470019\pi$
$758$	0	0
$759$	−8.26323e35	−0.820593
$760$	0	0
$761$	1.88027e36	1.80681	0.903406	−	0.428787i	$-0.141059\pi$
$761$	1.88027e36	1.80681	0.903406	+	0.428787i	$0.141059\pi$
$762$	0	0
$763$	−9.03772e35	−0.840431
$764$	0	0
$765$	2.74990e36	2.47485
$766$	0	0
$767$	−1.42195e35	−0.123863
$768$	0	0
$769$	1.28453e36	1.08309	0.541546	−	0.840671i	$-0.317839\pi$
$769$	1.28453e36	1.08309	0.541546	+	0.840671i	$0.317839\pi$
$770$	0	0
$771$	9.30700e35	0.759679
$772$	0	0
$773$	−1.04808e36	−0.828226	−0.414113	−	0.910225i	$-0.635908\pi$
$773$	−1.04808e36	−0.828226	−0.414113	+	0.910225i	$0.635908\pi$
$774$	0	0
$775$	2.02566e36	1.54987
$776$	0	0
$777$	1.86700e36	1.38319
$778$	0	0
$779$	−4.10703e35	−0.294652
$780$	0	0
$781$	−1.30225e36	−0.904811
$782$	0	0
$783$	8.70840e35	0.586025
$784$	0	0
$785$	2.10044e36	1.36912
$786$	0	0
$787$	8.93146e35	0.563947	0.281974	−	0.959422i	$-0.409011\pi$
$787$	8.93146e35	0.563947	0.281974	+	0.959422i	$0.409011\pi$
$788$	0	0
$789$	−1.95822e36	−1.19784
$790$	0	0
$791$	2.42336e36	1.43619
$792$	0	0
$793$	2.54386e33	0.00146076
$794$	0	0
$795$	−1.50580e36	−0.837872
$796$	0	0
$797$	−1.02840e36	−0.554540	−0.277270	−	0.960792i	$-0.589430\pi$
$797$	−1.02840e36	−0.554540	−0.277270	+	0.960792i	$0.589430\pi$
$798$	0	0
$799$	−1.39017e36	−0.726498
$800$	0	0
$801$	−3.38782e36	−1.71599
$802$	0	0
$803$	−1.59763e36	−0.784392
$804$	0	0
$805$	−2.76026e36	−1.31372
$806$	0	0
$807$	−2.62004e36	−1.20889
$808$	0	0
$809$	1.33897e36	0.598981	0.299490	−	0.954099i	$-0.403183\pi$
$809$	1.33897e36	0.598981	0.299490	+	0.954099i	$0.403183\pi$
$810$	0	0
$811$	−1.29485e36	−0.561643	−0.280821	−	0.959760i	$-0.590607\pi$
$811$	−1.29485e36	−0.561643	−0.280821	+	0.959760i	$0.590607\pi$
$812$	0	0
$813$	2.00917e36	0.845055
$814$	0	0
$815$	−5.82854e36	−2.37733
$816$	0	0
$817$	7.54600e35	0.298498
$818$	0	0
$819$	3.39110e35	0.130104
$820$	0	0
$821$	3.91664e34	0.0145755	0.00728777	−	0.999973i	$-0.497680\pi$
$821$	3.91664e34	0.0145755	0.00728777	+	0.999973i	$0.497680\pi$
$822$	0	0
$823$	3.40400e36	1.22883	0.614414	−	0.788984i	$-0.289392\pi$
$823$	3.40400e36	1.22883	0.614414	+	0.788984i	$0.289392\pi$
$824$	0	0
$825$	−3.75147e36	−1.31379
$826$	0	0
$827$	5.02186e36	1.70626	0.853132	−	0.521696i	$-0.174700\pi$
$827$	5.02186e36	1.70626	0.853132	+	0.521696i	$0.174700\pi$
$828$	0	0
$829$	8.20485e35	0.270482	0.135241	−	0.990813i	$-0.456819\pi$
$829$	8.20485e35	0.270482	0.135241	+	0.990813i	$0.456819\pi$
$830$	0	0
$831$	−7.31856e35	−0.234106
$832$	0	0
$833$	−1.30513e36	−0.405128
$834$	0	0
$835$	−3.57748e36	−1.07770
$836$	0	0
$837$	1.20215e36	0.351471
$838$	0	0
$839$	5.38664e36	1.52860	0.764300	−	0.644861i	$-0.223085\pi$
$839$	5.38664e36	1.52860	0.764300	+	0.644861i	$0.223085\pi$
$840$	0	0
$841$	5.92900e36	1.63317
$842$	0	0
$843$	−5.02945e36	−1.34486
$844$	0	0
$845$	−6.10126e36	−1.58384
$846$	0	0
$847$	2.26629e36	0.571180
$848$	0	0
$849$	1.98389e36	0.485479
$850$	0	0
$851$	−4.71505e36	−1.12038
$852$	0	0
$853$	9.55369e35	0.220449	0.110225	−	0.993907i	$-0.464843\pi$
$853$	9.55369e35	0.220449	0.110225	+	0.993907i	$0.464843\pi$
$854$	0	0
$855$	5.01718e36	1.12430
$856$	0	0
$857$	−5.39118e36	−1.17334	−0.586669	−	0.809827i	$-0.699561\pi$
$857$	−5.39118e36	−1.17334	−0.586669	+	0.809827i	$0.699561\pi$
$858$	0	0
$859$	−5.48992e36	−1.16052	−0.580258	−	0.814432i	$-0.697048\pi$
$859$	−5.48992e36	−1.16052	−0.580258	+	0.814432i	$0.697048\pi$
$860$	0	0
$861$	3.13241e36	0.643190
$862$	0	0
$863$	2.89510e36	0.577469	0.288735	−	0.957409i	$-0.406766\pi$
$863$	2.89510e36	0.577469	0.288735	+	0.957409i	$0.406766\pi$
$864$	0	0
$865$	−7.63772e36	−1.48000
$866$	0	0
$867$	4.24209e36	0.798623
$868$	0	0
$869$	3.65359e36	0.668301
$870$	0	0
$871$	−5.76349e35	−0.102437
$872$	0	0
$873$	3.76991e36	0.651108
$874$	0	0
$875$	−4.66251e36	−0.782562
$876$	0	0
$877$	2.04429e36	0.333463	0.166731	−	0.986002i	$-0.446679\pi$
$877$	2.04429e36	0.333463	0.166731	+	0.986002i	$0.446679\pi$
$878$	0	0
$879$	1.71570e37	2.72007
$880$	0	0
$881$	−1.76163e36	−0.271466	−0.135733	−	0.990745i	$-0.543339\pi$
$881$	−1.76163e36	−0.271466	−0.135733	+	0.990745i	$0.543339\pi$
$882$	0	0
$883$	−1.25672e37	−1.88248	−0.941240	−	0.337738i	$-0.890338\pi$
$883$	−1.25672e37	−1.88248	−0.941240	+	0.337738i	$0.890338\pi$
$884$	0	0
$885$	1.60456e37	2.33649
$886$	0	0
$887$	−8.81003e36	−1.24719	−0.623595	−	0.781748i	$-0.714328\pi$
$887$	−8.81003e36	−1.24719	−0.623595	+	0.781748i	$0.714328\pi$
$888$	0	0
$889$	−3.99436e36	−0.549762
$890$	0	0
$891$	2.88487e36	0.386060
$892$	0	0
$893$	−2.53636e36	−0.330041
$894$	0	0
$895$	3.79203e36	0.479826
$896$	0	0
$897$	−1.54638e36	−0.190288
$898$	0	0
$899$	1.31962e37	1.57925
$900$	0	0
$901$	−3.69802e36	−0.430436
$902$	0	0
$903$	−5.75530e36	−0.651585
$904$	0	0
$905$	1.95988e37	2.15835
$906$	0	0
$907$	1.59523e37	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$907$	1.59523e37	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$908$	0	0
$909$	−3.58234e36	−0.373352
$910$	0	0
$911$	1.71775e36	0.174172	0.0870862	−	0.996201i	$-0.472244\pi$
$911$	1.71775e36	0.174172	0.0870862	+	0.996201i	$0.472244\pi$
$912$	0	0
$913$	6.86379e36	0.677142
$914$	0	0
$915$	−2.87054e35	−0.0275550
$916$	0	0
$917$	2.02434e36	0.189089
$918$	0	0
$919$	7.70373e35	0.0700256	0.0350128	−	0.999387i	$-0.488853\pi$
$919$	7.70373e35	0.0700256	0.0350128	+	0.999387i	$0.488853\pi$
$920$	0	0
$921$	−3.13079e37	−2.76955
$922$	0	0
$923$	−2.43703e36	−0.209817
$924$	0	0
$925$	−2.14061e37	−1.79377
$926$	0	0
$927$	−8.69819e36	−0.709467
$928$	0	0
$929$	1.66420e37	1.32132	0.660662	−	0.750684i	$-0.270276\pi$
$929$	1.66420e37	1.32132	0.660662	+	0.750684i	$0.270276\pi$
$930$	0	0
$931$	−2.38121e36	−0.184046
$932$	0	0
$933$	2.54021e37	1.91139
$934$	0	0
$935$	−1.49983e37	−1.09874
$936$	0	0
$937$	9.66427e36	0.689325	0.344663	−	0.938727i	$-0.387993\pi$
$937$	9.66427e36	0.689325	0.344663	+	0.938727i	$0.387993\pi$
$938$	0	0
$939$	6.87214e36	0.477279
$940$	0	0
$941$	−1.33808e37	−0.904923	−0.452461	−	0.891784i	$-0.649454\pi$
$941$	−1.33808e37	−0.904923	−0.452461	+	0.891784i	$0.649454\pi$
$942$	0	0
$943$	−7.91080e36	−0.520984
$944$	0	0
$945$	−7.43690e36	−0.476974
$946$	0	0
$947$	−2.63078e37	−1.64328	−0.821638	−	0.570010i	$-0.806939\pi$
$947$	−2.63078e37	−1.64328	−0.821638	+	0.570010i	$0.806939\pi$
$948$	0	0
$949$	−2.98981e36	−0.181893
$950$	0	0
$951$	−1.21195e37	−0.718170
$952$	0	0
$953$	−9.10378e36	−0.525485	−0.262742	−	0.964866i	$-0.584627\pi$
$953$	−9.10378e36	−0.525485	−0.262742	+	0.964866i	$0.584627\pi$
$954$	0	0
$955$	1.68184e37	0.945677
$956$	0	0
$957$	−2.44389e37	−1.33870
$958$	0	0
$959$	9.70142e36	0.517729
$960$	0	0
$961$	−1.01623e36	−0.0528386
$962$	0	0
$963$	−1.66151e37	−0.841731
$964$	0	0
$965$	−5.24538e37	−2.58932
$966$	0	0
$967$	1.34794e36	0.0648394	0.0324197	−	0.999474i	$-0.489679\pi$
$967$	1.34794e36	0.0648394	0.0324197	+	0.999474i	$0.489679\pi$
$968$	0	0
$969$	2.22483e37	1.04291
$970$	0	0
$971$	−3.82504e36	−0.174741	−0.0873707	−	0.996176i	$-0.527846\pi$
$971$	−3.82504e36	−0.174741	−0.0873707	+	0.996176i	$0.527846\pi$
$972$	0	0
$973$	−2.76831e36	−0.123255
$974$	0	0
$975$	−7.02049e36	−0.304656
$976$	0	0
$977$	1.16914e37	0.494522	0.247261	−	0.968949i	$-0.420470\pi$
$977$	1.16914e37	0.494522	0.247261	+	0.968949i	$0.420470\pi$
$978$	0	0
$979$	1.84776e37	0.761835
$980$	0	0
$981$	3.16418e37	1.27174
$982$	0	0
$983$	−1.38138e37	−0.541244	−0.270622	−	0.962686i	$-0.587229\pi$
$983$	−1.38138e37	−0.541244	−0.270622	+	0.962686i	$0.587229\pi$
$984$	0	0
$985$	5.48672e37	2.09585
$986$	0	0
$987$	1.93447e37	0.720441
$988$	0	0
$989$	1.45348e37	0.527784
$990$	0	0
$991$	3.79186e37	1.34255	0.671276	−	0.741207i	$-0.265746\pi$
$991$	3.79186e37	1.34255	0.671276	+	0.741207i	$0.265746\pi$
$992$	0	0
$993$	−8.22853e37	−2.84090
$994$	0	0
$995$	2.84995e37	0.959510
$996$	0	0
$997$	5.34619e37	1.75531	0.877657	−	0.479289i	$-0.159106\pi$
$997$	5.34619e37	1.75531	0.877657	+	0.479289i	$0.159106\pi$
$998$	0	0
$999$	−1.27036e37	−0.406779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.26.a.c.1.2		2
4.3	odd	2		2.26.a.b.1.1	✓	2
12.11	even	2		18.26.a.e.1.1		2
20.3	even	4		50.26.b.e.49.1		4
20.7	even	4		50.26.b.e.49.4		4
20.19	odd	2		50.26.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.26.a.b.1.1	✓	2	4.3	odd	2
16.26.a.c.1.2		2	1.1	even	1	trivial
18.26.a.e.1.1		2	12.11	even	2
50.26.a.c.1.2		2	20.19	odd	2
50.26.b.e.49.1		4	20.3	even	4
50.26.b.e.49.4		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)

\(f(q)\)	\(=\)	\(q+1.37803e6 q^{3} +8.78994e8 q^{5} -3.00388e10 q^{7} +1.05168e12 q^{9} -5.73599e12 q^{11} -1.07343e13 q^{13} +1.21128e15 q^{15} +2.97473e15 q^{17} +5.42738e15 q^{19} -4.13944e16 q^{21} +1.04540e17 q^{23} +4.74607e17 q^{25} +2.81659e17 q^{27} +3.09182e18 q^{29} +4.26809e18 q^{31} -7.90437e18 q^{33} -2.64039e19 q^{35} -4.51028e19 q^{37} -1.47922e19 q^{39} -7.56724e19 q^{41} +1.39036e20 q^{43} +9.24421e20 q^{45} -4.67328e20 q^{47} -4.38741e20 q^{49} +4.09927e21 q^{51} -1.24315e21 q^{53} -5.04190e21 q^{55} +7.47909e21 q^{57} +1.32468e22 q^{59} -2.36984e20 q^{61} -3.15912e22 q^{63} -9.43539e21 q^{65} +5.36922e22 q^{67} +1.44059e23 q^{69} +2.27032e23 q^{71} +2.78528e23 q^{73} +6.54024e23 q^{75} +1.72302e23 q^{77} -6.36958e23 q^{79} -5.02942e23 q^{81} -1.19662e24 q^{83} +2.61477e24 q^{85} +4.26063e24 q^{87} -3.22134e24 q^{89} +3.22445e23 q^{91} +5.88156e24 q^{93} +4.77063e24 q^{95} +3.58465e24 q^{97} -6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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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