LMFDB - Embedded newform 18.26.a.e.1.1

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			1.1
Root			$163.829$ 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)

$f(q)$	$=$	\(q-4096.00 q^{2} +1.67772e7 q^{4} -8.78994e8 q^{5} +3.00388e10 q^{7} -6.87195e10 q^{8} +3.60036e12 q^{10} -5.73599e12 q^{11} -1.07343e13 q^{13} -1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -5.42738e15 q^{19} -1.47471e16 q^{20} +2.34946e16 q^{22} +1.04540e17 q^{23} +4.74607e17 q^{25} +4.39677e16 q^{26} +5.03967e17 q^{28} -3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +1.21845e19 q^{34} -2.64039e19 q^{35} -4.51028e19 q^{37} +2.22305e19 q^{38} +6.04040e19 q^{40} +7.56724e19 q^{41} -1.39036e20 q^{43} -9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} -4.38741e20 q^{49} -1.94399e21 q^{50} -1.80092e20 q^{52} +1.24315e21 q^{53} +5.04190e21 q^{55} -2.06425e21 q^{56} +1.26641e22 q^{58} +1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} +4.72237e21 q^{64} +9.43539e21 q^{65} -5.36922e22 q^{67} -4.99076e22 q^{68} +1.08150e23 q^{70} +2.27032e23 q^{71} +2.78528e23 q^{73} +1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} +6.36958e23 q^{79} -2.47415e23 q^{80} -3.09954e23 q^{82} -1.19662e24 q^{83} +2.61477e24 q^{85} +5.69491e23 q^{86} +3.94174e23 q^{88} +3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} +1.91418e24 q^{94} +4.77063e24 q^{95} +3.58465e24 q^{97} +1.79708e24 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.78994e8	−1.61013	−0.805065	−	0.593187i	$-0.797870\pi$
$5$	−8.78994e8	−1.61013	−0.805065	+	0.593187i	$0.797870\pi$
$6$	0	0
$7$	3.00388e10	0.820270	0.410135	−	0.912025i	$-0.365482\pi$
$7$	3.00388e10	0.820270	0.410135	+	0.912025i	$0.365482\pi$
$8$	−6.87195e10	−0.353553
$9$	0	0
$10$	3.60036e12	1.13853
$11$	−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	$0.520352\pi$
$14$	−1.23039e14	−0.580018
$15$	0	0
$16$	2.81475e14	0.250000
$17$	−2.97473e15	−1.23833	−0.619164	−	0.785262i	$-0.712528\pi$
$17$	−2.97473e15	−1.23833	−0.619164	+	0.785262i	$0.712528\pi$
$18$	0	0
$19$	−5.42738e15	−0.562561	−0.281281	−	0.959626i	$-0.590759\pi$
$19$	−5.42738e15	−0.562561	−0.281281	+	0.959626i	$0.590759\pi$
$20$	−1.47471e16	−0.805065
$21$	0	0
$22$	2.34946e16	0.389659
$23$	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
$26$	4.39677e16	0.0903581
$27$	0	0
$28$	5.03967e17	0.410135
$29$	−3.09182e18	−1.62270	−0.811352	−	0.584558i	$-0.801268\pi$
$29$	−3.09182e18	−1.62270	−0.811352	+	0.584558i	$0.801268\pi$
$30$	0	0
$31$	−4.26809e18	−0.973222	−0.486611	−	0.873619i	$-0.661767\pi$
$31$	−4.26809e18	−0.973222	−0.486611	+	0.873619i	$0.661767\pi$
$32$	−1.15292e18	−0.176777
$33$	0	0
$34$	1.21845e19	0.875630
$35$	−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	$0.690424\pi$
$38$	2.22305e19	0.397791
$39$	0	0
$40$	6.04040e19	0.569267
$41$	7.56724e19	0.523769	0.261884	−	0.965099i	$-0.415656\pi$
$41$	7.56724e19	0.523769	0.261884	+	0.965099i	$0.415656\pi$
$42$	0	0
$43$	−1.39036e20	−0.530605	−0.265302	−	0.964165i	$-0.585472\pi$
$43$	−1.39036e20	−0.530605	−0.265302	+	0.964165i	$0.585472\pi$
$44$	−9.62339e19	−0.275531
$45$	0	0
$46$	−4.28196e20	−0.703347
$47$	−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
$50$	−1.94399e21	−1.12608
$51$	0	0
$52$	−1.80092e20	−0.0638928
$53$	1.24315e21	0.347595	0.173797	−	0.984781i	$-0.444396\pi$
$53$	1.24315e21	0.347595	0.173797	+	0.984781i	$0.444396\pi$
$54$	0	0
$55$	5.04190e21	0.887280
$56$	−2.06425e21	−0.290009
$57$	0	0
$58$	1.26641e22	1.14743
$59$	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	$0.501819\pi$
$62$	1.74821e22	0.688172
$63$	0	0
$64$	4.72237e21	0.125000
$65$	9.43539e21	0.205752
$66$	0	0
$67$	−5.36922e22	−0.801634	−0.400817	−	0.916158i	$-0.631274\pi$
$67$	−5.36922e22	−0.801634	−0.400817	+	0.916158i	$0.631274\pi$
$68$	−4.99076e22	−0.619164
$69$	0	0
$70$	1.08150e23	0.933905
$71$	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	$0.247921\pi$
$74$	1.84741e23	0.796465
$75$	0	0
$76$	−9.10563e22	−0.281281
$77$	−1.72302e23	−0.452019
$78$	0	0
$79$	6.36958e23	1.21275	0.606376	−	0.795178i	$-0.292622\pi$
$79$	6.36958e23	1.21275	0.606376	+	0.795178i	$0.292622\pi$
$80$	−2.47415e23	−0.402532
$81$	0	0
$82$	−3.09954e23	−0.370360
$83$	−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
$86$	5.69491e23	0.375194
$87$	0	0
$88$	3.94174e23	0.194830
$89$	3.22134e24	1.38249	0.691244	−	0.722621i	$-0.257063\pi$
$89$	3.22134e24	1.38249	0.691244	+	0.722621i	$0.257063\pi$
$90$	0	0
$91$	−3.22445e23	−0.104819
$92$	1.75389e24	0.497341
$93$	0	0
$94$	1.91418e24	0.414843
$95$	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	$0.415525\pi$
$98$	1.79708e24	0.231335
$99$	0	0
$100$	7.96259e24	0.796259
$101$	3.40630e24	0.300792	0.150396	−	0.988626i	$-0.451945\pi$
$101$	3.40630e24	0.300792	0.150396	+	0.988626i	$0.451945\pi$
$102$	0	0
$103$	8.27075e24	0.571583	0.285792	−	0.958292i	$-0.407743\pi$
$103$	8.27075e24	0.571583	0.285792	+	0.958292i	$0.407743\pi$
$104$	7.37656e23	0.0451791
$105$	0	0
$106$	−5.09192e24	−0.245787
$107$	−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	$0.328797\pi$
$110$	−2.06516e25	−0.627402
$111$	0	0
$112$	8.45516e24	0.205067
$113$	8.06745e25	1.75088	0.875439	−	0.483329i	$-0.160572\pi$
$113$	8.06745e25	1.75088	0.875439	+	0.483329i	$0.160572\pi$
$114$	0	0
$115$	−9.18901e25	−1.60157
$116$	−5.18722e25	−0.811352
$117$	0	0
$118$	−5.42587e25	−0.685401
$119$	−8.93571e25	−1.01576
$120$	0	0
$121$	−7.54455e25	−0.696332
$122$	9.70685e23	0.00808315
$123$	0	0
$124$	−7.16066e25	−0.486611
$125$	−1.55216e26	−0.954030
$126$	0	0
$127$	−1.32973e26	−0.670221	−0.335111	−	0.942179i	$-0.608774\pi$
$127$	−1.32973e26	−0.670221	−0.335111	+	0.942179i	$0.608774\pi$
$128$	−1.93428e25	−0.0883883
$129$	0	0
$130$	−3.86474e25	−0.145488
$131$	−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
$134$	2.19923e26	0.566841
$135$	0	0
$136$	2.04422e26	0.437815
$137$	3.22963e26	0.631170	0.315585	−	0.948897i	$-0.397799\pi$
$137$	3.22963e26	0.631170	0.315585	+	0.948897i	$0.397799\pi$
$138$	0	0
$139$	−9.21581e25	−0.150262	−0.0751309	−	0.997174i	$-0.523937\pi$
$139$	−9.21581e25	−0.150262	−0.0751309	+	0.997174i	$0.523937\pi$
$140$	−4.42984e26	−0.660370
$141$	0	0
$142$	−9.29923e26	−1.16103
$143$	6.15719e25	0.0704177
$144$	0	0
$145$	2.71769e27	2.61276
$146$	−1.14085e27	−1.00651
$147$	0	0
$148$	−7.56699e26	−0.563186
$149$	7.12667e26	0.487594	0.243797	−	0.969826i	$-0.421607\pi$
$149$	7.12667e26	0.487594	0.243797	+	0.969826i	$0.421607\pi$
$150$	0	0
$151$	−3.06161e27	−1.77312	−0.886559	−	0.462616i	$-0.846911\pi$
$151$	−3.06161e27	−1.77312	−0.886559	+	0.462616i	$0.846911\pi$
$152$	3.72966e26	0.198895
$153$	0	0
$154$	7.05749e26	0.319626
$155$	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	$0.360219\pi$
$158$	−2.60898e27	−0.857546
$159$	0	0
$160$	1.01341e27	0.284633
$161$	3.14025e27	0.815908
$162$	0	0
$163$	6.63092e27	1.47648	0.738242	−	0.674536i	$-0.235656\pi$
$163$	6.63092e27	1.47648	0.738242	+	0.674536i	$0.235656\pi$
$164$	1.26957e27	0.261884
$165$	0	0
$166$	4.90135e27	0.868890
$167$	−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
$170$	−1.07101e28	−1.40988
$171$	0	0
$172$	−2.33263e27	−0.265302
$173$	8.68916e27	0.919182	0.459591	−	0.888131i	$-0.347996\pi$
$173$	8.68916e27	0.919182	0.459591	+	0.888131i	$0.347996\pi$
$174$	0	0
$175$	1.42566e28	1.30629
$176$	−1.61454e27	−0.137765
$177$	0	0
$178$	−1.31946e28	−0.977567
$179$	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	$0.266190\pi$
$182$	1.32074e27	0.0741180
$183$	0	0
$184$	−7.18394e27	−0.351674
$185$	3.96451e28	1.81360
$186$	0	0
$187$	1.70630e28	0.682394
$188$	−7.84046e27	−0.293338
$189$	0	0
$190$	−1.95405e28	−0.640495
$191$	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	$0.797337\pi$
$194$	−1.46827e28	−0.370924
$195$	0	0
$196$	−7.36085e27	−0.163579
$197$	−6.24204e28	−1.30166	−0.650832	−	0.759222i	$-0.725580\pi$
$197$	−6.24204e28	−1.30166	−0.650832	+	0.759222i	$0.725580\pi$
$198$	0	0
$199$	−3.24229e28	−0.595921	−0.297960	−	0.954578i	$-0.596306\pi$
$199$	−3.24229e28	−0.595921	−0.297960	+	0.954578i	$0.596306\pi$
$200$	−3.26148e28	−0.563040
$201$	0	0
$202$	−1.39522e28	−0.212692
$203$	−9.28745e28	−1.33106
$204$	0	0
$205$	−6.65156e28	−0.843335
$206$	−3.38770e28	−0.404171
$207$	0	0
$208$	−3.02144e27	−0.0319464
$209$	3.11314e28	0.310006
$210$	0	0
$211$	3.82197e28	0.337875	0.168938	−	0.985627i	$-0.445966\pi$
$211$	3.82197e28	0.337875	0.168938	+	0.985627i	$0.445966\pi$
$212$	2.08565e28	0.173797
$213$	0	0
$214$	6.47109e28	0.479519
$215$	1.22212e29	0.854342
$216$	0	0
$217$	−1.28208e29	−0.798305
$218$	−1.23236e29	−0.724487
$219$	0	0
$220$	8.45891e28	0.443640
$221$	3.19316e28	0.158241
$222$	0	0
$223$	3.11100e29	1.37749	0.688746	−	0.725003i	$-0.258161\pi$
$223$	3.11100e29	1.37749	0.688746	+	0.725003i	$0.258161\pi$
$224$	−3.46323e28	−0.145005
$225$	0	0
$226$	−3.30443e29	−1.23806
$227$	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	$0.700207\pi$
$230$	3.76382e29	1.13248
$231$	0	0
$232$	2.12468e29	0.573713
$233$	6.84781e28	0.175228	0.0876138	−	0.996155i	$-0.472076\pi$
$233$	6.84781e28	0.175228	0.0876138	+	0.996155i	$0.472076\pi$
$234$	0	0
$235$	4.10778e29	0.944625
$236$	2.22244e29	0.484652
$237$	0	0
$238$	3.66007e29	0.718253
$239$	−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	$0.163970\pi$
$242$	3.09025e29	0.492381
$243$	0	0
$244$	−3.97593e27	−0.00571565
$245$	3.85651e29	0.526766
$246$	0	0
$247$	5.82591e28	0.0718873
$248$	2.93301e29	0.344086
$249$	0	0
$250$	6.35766e29	0.674601
$251$	−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
$254$	5.44659e29	0.473918
$255$	0	0
$256$	7.92282e28	0.0625000
$257$	−6.75384e29	−0.507442	−0.253721	−	0.967277i	$-0.581655\pi$
$257$	−6.75384e29	−0.507442	−0.253721	+	0.967277i	$0.581655\pi$
$258$	0	0
$259$	−1.35483e30	−0.923929
$260$	1.58300e29	0.102876
$261$	0	0
$262$	2.76033e29	0.163003
$263$	−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
$266$	6.67778e29	0.326296
$267$	0	0
$268$	−9.00805e29	−0.400817
$269$	1.90129e30	0.807504	0.403752	−	0.914868i	$-0.367706\pi$
$269$	1.90129e30	0.807504	0.403752	+	0.914868i	$0.367706\pi$
$270$	0	0
$271$	−1.45800e30	−0.564471	−0.282236	−	0.959345i	$-0.591076\pi$
$271$	−1.45800e30	−0.564471	−0.282236	+	0.959345i	$0.591076\pi$
$272$	−8.37311e29	−0.309582
$273$	0	0
$274$	−1.32286e30	−0.446304
$275$	−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	$0.524913\pi$
$278$	3.77479e29	0.106251
$279$	0	0
$280$	1.81446e30	0.466952
$281$	3.64974e30	0.898323	0.449161	−	0.893451i	$-0.351723\pi$
$281$	3.64974e30	0.898323	0.449161	+	0.893451i	$0.351723\pi$
$282$	0	0
$283$	−1.43965e30	−0.324285	−0.162143	−	0.986767i	$-0.551840\pi$
$283$	−1.43965e30	−0.324285	−0.162143	+	0.986767i	$0.551840\pi$
$284$	3.80897e30	0.820971
$285$	0	0
$286$	−2.52199e29	−0.0497929
$287$	2.27311e30	0.429632
$288$	0	0
$289$	3.07837e30	0.533456
$290$	−1.11317e31	−1.84750
$291$	0	0
$292$	4.67292e30	0.711710
$293$	−1.24504e31	−1.81692	−0.908462	−	0.417967i	$-0.862743\pi$
$293$	−1.24504e31	−1.81692	−0.908462	+	0.417967i	$0.862743\pi$
$294$	0	0
$295$	−1.16438e31	−1.56070
$296$	3.09944e30	0.398233
$297$	0	0
$298$	−2.91909e30	−0.344781
$299$	−1.12217e30	−0.127106
$300$	0	0
$301$	−4.17647e30	−0.435239
$302$	1.25403e31	1.25378
$303$	0	0
$304$	−1.52767e30	−0.140640
$305$	2.08307e29	0.0184059
$306$	0	0
$307$	2.27193e31	1.84997	0.924986	−	0.380001i	$-0.124076\pi$
$307$	2.27193e31	1.84997	0.924986	+	0.380001i	$0.124076\pi$
$308$	−2.89075e30	−0.226009
$309$	0	0
$310$	−1.53666e31	−1.10805
$311$	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	$0.449043\pi$
$314$	−9.78780e30	−0.601263
$315$	0	0
$316$	1.06864e31	0.606376
$317$	8.79478e30	0.479716	0.239858	−	0.970808i	$-0.422899\pi$
$317$	8.79478e30	0.479716	0.239858	+	0.970808i	$0.422899\pi$
$318$	0	0
$319$	1.77347e31	0.894209
$320$	−4.15093e30	−0.201266
$321$	0	0
$322$	−1.28625e31	−0.576934
$323$	1.61450e31	0.696635
$324$	0	0
$325$	−5.09458e30	−0.203501
$326$	−2.71602e31	−1.04403
$327$	0	0
$328$	−5.20017e30	−0.185180
$329$	−1.40380e31	−0.481233
$330$	0	0
$331$	5.97122e31	1.89764	0.948818	−	0.315822i	$-0.102280\pi$
$331$	5.97122e31	1.89764	0.948818	+	0.315822i	$0.102280\pi$
$332$	−2.00759e31	−0.614398
$333$	0	0
$334$	1.66706e31	0.473283
$335$	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	$0.456049\pi$
$338$	2.84311e31	0.695560
$339$	0	0
$340$	4.38685e31	0.996934
$341$	2.44817e31	0.536305
$342$	0	0
$343$	−5.34633e31	−1.08863
$344$	9.55447e30	0.187597
$345$	0	0
$346$	−3.55908e31	−0.649960
$347$	−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	$0.357590\pi$
$350$	−5.83951e31	−0.923689
$351$	0	0
$352$	6.61315e30	0.0974148
$353$	5.67715e31	0.807135	0.403567	−	0.914950i	$-0.367770\pi$
$353$	5.67715e31	0.807135	0.403567	+	0.914950i	$0.367770\pi$
$354$	0	0
$355$	−1.99560e32	−2.64374
$356$	5.40451e31	0.691244
$357$	0	0
$358$	−1.76704e31	−0.210721
$359$	−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
$362$	−9.13280e31	−0.947866
$363$	0	0
$364$	−5.40974e30	−0.0524094
$365$	−2.44824e32	−2.29189
$366$	0	0
$367$	1.00500e32	0.878699	0.439349	−	0.898316i	$-0.355209\pi$
$367$	1.00500e32	0.878699	0.439349	+	0.898316i	$0.355209\pi$
$368$	2.94254e31	0.248671
$369$	0	0
$370$	−1.62386e32	−1.28241
$371$	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	$0.427678\pi$
$374$	−6.98901e31	−0.482526
$375$	0	0
$376$	3.21145e31	0.207421
$377$	3.31886e31	0.207358
$378$	0	0
$379$	−8.85755e31	−0.517993	−0.258996	−	0.965878i	$-0.583392\pi$
$379$	−8.85755e31	−0.517993	−0.258996	+	0.965878i	$0.583392\pi$
$380$	8.00379e31	0.452898
$381$	0	0
$382$	−7.83716e31	−0.415305
$383$	−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
$386$	2.44428e32	1.13713
$387$	0	0
$388$	6.01405e31	0.262283
$389$	−7.13043e31	−0.301124	−0.150562	−	0.988601i	$-0.548108\pi$
$389$	−7.13043e31	−0.301124	−0.150562	+	0.988601i	$0.548108\pi$
$390$	0	0
$391$	−3.10978e32	−1.23174
$392$	3.01500e31	0.115668
$393$	0	0
$394$	2.55674e32	0.920416
$395$	−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	$0.484390\pi$
$398$	1.32804e32	0.421380
$399$	0	0
$400$	1.33590e32	0.398129
$401$	5.00588e32	1.44603	0.723013	−	0.690835i	$-0.242757\pi$
$401$	5.00588e32	1.44603	0.723013	+	0.690835i	$0.242757\pi$
$402$	0	0
$403$	4.58150e31	0.124364
$404$	5.71482e31	0.150396
$405$	0	0
$406$	3.80414e32	0.941198
$407$	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	$0.601702\pi$
$410$	2.72448e32	0.596328
$411$	0	0
$412$	1.38760e32	0.285792
$413$	3.97916e32	0.795090
$414$	0	0
$415$	1.05182e33	1.97852
$416$	1.23758e31	0.0225895
$417$	0	0
$418$	−1.27514e32	−0.219207
$419$	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	$0.599775\pi$
$422$	−1.56548e32	−0.238914
$423$	0	0
$424$	−8.54283e31	−0.122893
$425$	−1.41183e33	−1.97206
$426$	0	0
$427$	−7.11870e30	−0.00937675
$428$	−2.65056e32	−0.339071
$429$	0	0
$430$	−5.00579e32	−0.604111
$431$	−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	$0.331497\pi$
$434$	5.25140e32	0.564487
$435$	0	0
$436$	5.04774e32	0.512290
$437$	−5.67378e32	−0.559570
$438$	0	0
$439$	6.07601e32	0.565995	0.282997	−	0.959121i	$-0.408671\pi$
$439$	6.07601e32	0.565995	0.282997	+	0.959121i	$0.408671\pi$
$440$	−3.46477e32	−0.313701
$441$	0	0
$442$	−1.30792e32	−0.111893
$443$	−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
$446$	−1.27427e33	−0.974034
$447$	0	0
$448$	1.41854e32	0.102534
$449$	1.21579e33	0.854630	0.427315	−	0.904103i	$-0.359460\pi$
$449$	1.21579e33	0.854630	0.427315	+	0.904103i	$0.359460\pi$
$450$	0	0
$451$	−4.34056e32	−0.288629
$452$	1.35349e33	0.875439
$453$	0	0
$454$	−4.32422e32	−0.264673
$455$	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	$0.550409\pi$
$458$	1.51684e33	0.831997
$459$	0	0
$460$	−1.54166e33	−0.800784
$461$	3.78592e32	0.191386	0.0956930	−	0.995411i	$-0.469493\pi$
$461$	3.78592e32	0.191386	0.0956930	+	0.995411i	$0.469493\pi$
$462$	0	0
$463$	2.77039e33	1.32672	0.663359	−	0.748301i	$-0.269130\pi$
$463$	2.77039e33	1.32672	0.663359	+	0.748301i	$0.269130\pi$
$464$	−8.70270e32	−0.405676
$465$	0	0
$466$	−2.80486e32	−0.123905
$467$	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
$470$	−1.68255e33	−0.667951
$471$	0	0
$472$	−9.10311e32	−0.342700
$473$	7.97508e32	0.292396
$474$	0	0
$475$	−2.57587e33	−0.895889
$476$	−1.49916e33	−0.507881
$477$	0	0
$478$	1.58832e33	0.510609
$479$	−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
$482$	−4.24851e33	−1.23069
$483$	0	0
$484$	−1.26577e33	−0.348166
$485$	−3.15089e33	−0.844620
$486$	0	0
$487$	3.71393e32	0.0945632	0.0472816	−	0.998882i	$-0.484944\pi$
$487$	3.71393e32	0.0945632	0.0472816	+	0.998882i	$0.484944\pi$
$488$	1.62854e31	0.00404157
$489$	0	0
$490$	−1.57963e33	−0.372480
$491$	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
$494$	−2.38629e32	−0.0508320
$495$	0	0
$496$	−1.20136e33	−0.243306
$497$	6.81976e33	1.34684
$498$	0	0
$499$	−2.39117e33	−0.449111	−0.224556	−	0.974461i	$-0.572093\pi$
$499$	−2.39117e33	−0.449111	−0.224556	+	0.974461i	$0.572093\pi$
$500$	−2.60410e33	−0.477015
$501$	0	0
$502$	6.06166e33	1.05632
$503$	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
$506$	2.45613e33	0.387587
$507$	0	0
$508$	−2.23092e33	−0.335111
$509$	3.62528e32	0.0531337	0.0265668	−	0.999647i	$-0.491543\pi$
$509$	3.62528e32	0.0531337	0.0265668	+	0.999647i	$0.491543\pi$
$510$	0	0
$511$	8.36664e33	1.16759
$512$	−3.24519e32	−0.0441942
$513$	0	0
$514$	2.76637e33	0.358816
$515$	−7.26994e33	−0.920323
$516$	0	0
$517$	2.68059e33	0.323295
$518$	5.54939e33	0.653316
$519$	0	0
$520$	−6.48395e32	−0.0727442
$521$	−8.23573e33	−0.902050	−0.451025	−	0.892511i	$-0.648942\pi$
$521$	−8.23573e33	−0.902050	−0.451025	+	0.892511i	$0.648942\pi$
$522$	0	0
$523$	−1.20443e34	−1.25750	−0.628752	−	0.777606i	$-0.716434\pi$
$523$	−1.20443e34	−1.25750	−0.628752	+	0.777606i	$0.716434\pi$
$524$	−1.13063e33	−0.115260
$525$	0	0
$526$	5.82051e33	0.565770
$527$	1.26964e34	1.20517
$528$	0	0
$529$	−1.17149e32	−0.0106057
$530$	4.47577e33	0.395749
$531$	0	0
$532$	−2.73522e33	−0.230726
$533$	−8.12291e32	−0.0669301
$534$	0	0
$535$	1.38868e34	1.09190
$536$	3.68970e33	0.283420
$537$	0	0
$538$	−7.78769e33	−0.570992
$539$	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	$0.352909\pi$
$542$	5.97197e33	0.399141
$543$	0	0
$544$	3.42963e33	0.218907
$545$	−2.64462e34	−1.64971
$546$	0	0
$547$	−1.78334e34	−1.06266	−0.531328	−	0.847166i	$-0.678307\pi$
$547$	−1.78334e34	−1.06266	−0.531328	+	0.847166i	$0.678307\pi$
$548$	5.41843e33	0.315585
$549$	0	0
$550$	1.11507e34	0.620539
$551$	1.67805e34	0.912871
$552$	0	0
$553$	1.91334e34	0.994784
$554$	2.17534e33	0.110574
$555$	0	0
$556$	−1.54616e33	−0.0751309
$557$	−8.73339e33	−0.414947	−0.207474	−	0.978241i	$-0.566524\pi$
$557$	−8.73339e33	−0.414947	−0.207474	+	0.978241i	$0.566524\pi$
$558$	0	0
$559$	1.49245e33	0.0678037
$560$	−7.43204e33	−0.330185
$561$	0	0
$562$	−1.49493e34	−0.635210
$563$	−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
$566$	5.89682e33	0.229304
$567$	0	0
$568$	−1.56015e34	−0.580514
$569$	−3.49591e34	−1.27250	−0.636248	−	0.771484i	$-0.719515\pi$
$569$	−3.49591e34	−1.27250	−0.636248	+	0.771484i	$0.719515\pi$
$570$	0	0
$571$	−3.60524e34	−1.25598	−0.627989	−	0.778222i	$-0.716122\pi$
$571$	−3.60524e34	−1.25598	−0.627989	+	0.778222i	$0.716122\pi$
$572$	1.03301e33	0.0352089
$573$	0	0
$574$	−9.31065e33	−0.303795
$575$	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	$0.738719\pi$
$578$	−1.26090e34	−0.377210
$579$	0	0
$580$	4.55953e34	1.30638
$581$	−3.59450e34	−1.00794
$582$	0	0
$583$	−7.13067e33	−0.191546
$584$	−1.91403e34	−0.503255
$585$	0	0
$586$	5.09967e34	1.28476
$587$	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
$590$	4.76931e34	1.10358
$591$	0	0
$592$	−1.26953e34	−0.281593
$593$	4.88078e33	0.106000	0.0530000	−	0.998595i	$-0.483122\pi$
$593$	4.88078e33	0.106000	0.0530000	+	0.998595i	$0.483122\pi$
$594$	0	0
$595$	7.85444e34	1.63551
$596$	1.19566e34	0.243797
$597$	0	0
$598$	4.59639e33	0.0898777
$599$	−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	$0.591106\pi$
$602$	1.71068e34	0.307760
$603$	0	0
$604$	−5.13652e34	−0.886559
$605$	6.63161e34	1.12118
$606$	0	0
$607$	1.03737e35	1.68297	0.841484	−	0.540282i	$-0.181682\pi$
$607$	1.03737e35	1.68297	0.841484	+	0.540282i	$0.181682\pi$
$608$	6.25734e33	0.0994477
$609$	0	0
$610$	−8.53226e32	−0.0130149
$611$	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	$0.434290\pi$
$614$	−9.30582e34	−1.30813
$615$	0	0
$616$	1.18405e34	0.159813
$617$	−5.36336e34	−0.709370	−0.354685	−	0.934986i	$-0.615412\pi$
$617$	−5.36336e34	−0.709370	−0.354685	+	0.934986i	$0.615412\pi$
$618$	0	0
$619$	6.60909e34	0.839477	0.419738	−	0.907645i	$-0.362122\pi$
$619$	6.60909e34	0.839477	0.419738	+	0.907645i	$0.362122\pi$
$620$	6.29418e34	0.783507
$621$	0	0
$622$	−7.55041e34	−0.902798
$623$	9.67651e34	1.13401
$624$	0	0
$625$	−5.00979e33	−0.0564053
$626$	−2.04264e34	−0.225431
$627$	0	0
$628$	4.00908e34	0.425157
$629$	1.34168e35	1.39482
$630$	0	0
$631$	8.62816e34	0.862087	0.431043	−	0.902331i	$-0.358146\pi$
$631$	8.62816e34	0.862087	0.431043	+	0.902331i	$0.358146\pi$
$632$	−4.37714e34	−0.428773
$633$	0	0
$634$	−3.60234e34	−0.339210
$635$	1.16883e35	1.07914
$636$	0	0
$637$	4.70958e33	0.0418061
$638$	−7.26412e34	−0.632302
$639$	0	0
$640$	1.70022e34	0.142317
$641$	−1.77093e35	−1.45370	−0.726850	−	0.686796i	$-0.759017\pi$
$641$	−1.77093e35	−1.45370	−0.726850	+	0.686796i	$0.759017\pi$
$642$	0	0
$643$	−7.98617e34	−0.630524	−0.315262	−	0.949005i	$-0.602092\pi$
$643$	−7.98617e34	−0.630524	−0.315262	+	0.949005i	$0.602092\pi$
$644$	5.26847e34	0.407954
$645$	0	0
$646$	−6.61298e34	−0.492596
$647$	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
$650$	2.08674e34	0.143897
$651$	0	0
$652$	1.11248e35	0.738242
$653$	−2.41184e35	−1.57013	−0.785063	−	0.619416i	$-0.787369\pi$
$653$	−2.41184e35	−1.57013	−0.785063	+	0.619416i	$0.787369\pi$
$654$	0	0
$655$	5.92361e34	0.371168
$656$	2.12999e34	0.130942
$657$	0	0
$658$	5.74995e34	0.340283
$659$	−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	$0.580072\pi$
$662$	−2.44581e35	−1.34183
$663$	0	0
$664$	8.22310e34	0.434445
$665$	1.43304e35	0.742998
$666$	0	0
$667$	−3.23219e35	−1.61408
$668$	−6.82828e34	−0.334662
$669$	0	0
$670$	−1.93311e35	−0.912687
$671$	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	$0.320297\pi$
$674$	−4.44120e34	−0.194649
$675$	0	0
$676$	−1.16454e35	−0.491835
$677$	3.79216e34	0.157227	0.0786137	−	0.996905i	$-0.474951\pi$
$677$	3.79216e34	0.157227	0.0786137	+	0.996905i	$0.474951\pi$
$678$	0	0
$679$	1.07679e35	0.430286
$680$	−1.79685e35	−0.704939
$681$	0	0
$682$	−1.00277e35	−0.379225
$683$	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
$686$	2.18986e35	0.769776
$687$	0	0
$688$	−3.91351e34	−0.132651
$689$	−1.33443e34	−0.0444177
$690$	0	0
$691$	8.34540e34	0.267899	0.133950	−	0.990988i	$-0.457234\pi$
$691$	8.34540e34	0.267899	0.133950	+	0.990988i	$0.457234\pi$
$692$	1.45780e35	0.459591
$693$	0	0
$694$	2.94991e34	0.0897047
$695$	8.10064e34	0.241941
$696$	0	0
$697$	−2.25105e35	−0.648597
$698$	−2.16178e35	−0.611814
$699$	0	0
$700$	2.39186e35	0.653147
$701$	−1.50008e35	−0.402383	−0.201191	−	0.979552i	$-0.564481\pi$
$701$	−1.50008e35	−0.402383	−0.201191	+	0.979552i	$0.564481\pi$
$702$	0	0
$703$	2.44790e35	0.633653
$704$	−2.70874e34	−0.0688827
$705$	0	0
$706$	−2.32536e35	−0.570731
$707$	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	$0.579373\pi$
$710$	8.17397e35	1.86941
$711$	0	0
$712$	−2.21369e35	−0.488783
$713$	−4.46186e35	−0.968048
$714$	0	0
$715$	−5.41213e34	−0.113382
$716$	7.23778e34	0.149002
$717$	0	0
$718$	8.96718e34	0.178279
$719$	−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
$722$	2.60588e35	0.483325
$723$	0	0
$724$	3.74080e35	0.670243
$725$	−1.46740e36	−2.58419
$726$	0	0
$727$	2.40795e35	0.409701	0.204850	−	0.978793i	$-0.434329\pi$
$727$	2.40795e35	0.409701	0.204850	+	0.978793i	$0.434329\pi$
$728$	2.21583e34	0.0370590
$729$	0	0
$730$	1.00280e36	1.62061
$731$	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	$0.583850\pi$
$734$	−4.11648e35	−0.621334
$735$	0	0
$736$	−1.20526e35	−0.175837
$737$	3.07978e35	0.441749
$738$	0	0
$739$	1.23366e36	1.71057	0.855284	−	0.518159i	$-0.173383\pi$
$739$	1.23366e36	1.71057	0.855284	+	0.518159i	$0.173383\pi$
$740$	6.65134e35	0.906802
$741$	0	0
$742$	−1.52955e35	−0.201611
$743$	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
$746$	−2.58479e35	−0.318559
$747$	0	0
$748$	2.86270e35	0.341197
$749$	−4.74570e35	−0.556259
$750$	0	0
$751$	−4.41291e35	−0.500294	−0.250147	−	0.968208i	$-0.580479\pi$
$751$	−4.41291e35	−0.500294	−0.250147	+	0.968208i	$0.580479\pi$
$752$	−1.31541e35	−0.146669
$753$	0	0
$754$	−1.35940e35	−0.146625
$755$	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	$0.470019\pi$
$758$	3.62805e35	0.366276
$759$	0	0
$760$	−3.27835e35	−0.320247
$761$	−1.88027e36	−1.80681	−0.903406	−	0.428787i	$-0.858941\pi$
$761$	−1.88027e36	−1.80681	−0.903406	+	0.428787i	$0.858941\pi$
$762$	0	0
$763$	9.03772e35	0.840431
$764$	3.21010e35	0.293665
$765$	0	0
$766$	3.72046e35	0.329410
$767$	−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	$0.317839\pi$
$770$	−6.20349e35	−0.514639
$771$	0	0
$772$	−1.00118e36	−0.804071
$773$	1.04808e36	0.828226	0.414113	−	0.910225i	$-0.364092\pi$
$773$	1.04808e36	0.828226	0.414113	+	0.910225i	$0.364092\pi$
$774$	0	0
$775$	−2.02566e36	−1.54987
$776$	−2.46335e35	−0.185462
$777$	0	0
$778$	2.92063e35	0.212927
$779$	−4.10703e35	−0.294652
$780$	0	0
$781$	−1.30225e36	−0.904811
$782$	1.27377e36	0.870974
$783$	0	0
$784$	−1.23495e35	−0.0817894
$785$	−2.10044e36	−1.36912
$786$	0	0
$787$	−8.93146e35	−0.563947	−0.281974	−	0.959422i	$-0.590989\pi$
$787$	−8.93146e35	−0.563947	−0.281974	+	0.959422i	$0.590989\pi$
$788$	−1.04724e36	−0.650832
$789$	0	0
$790$	2.29328e36	1.38076
$791$	2.42336e36	1.43619
$792$	0	0
$793$	2.54386e33	0.00146076
$794$	−1.22645e35	−0.0693257
$795$	0	0
$796$	−5.43966e35	−0.297960
$797$	1.02840e36	0.554540	0.277270	−	0.960792i	$-0.410570\pi$
$797$	1.02840e36	0.554540	0.277270	+	0.960792i	$0.410570\pi$
$798$	0	0
$799$	1.39017e36	0.726498
$800$	−5.47185e35	−0.281520
$801$	0	0
$802$	−2.05041e36	−1.02249
$803$	−1.59763e36	−0.784392
$804$	0	0
$805$	−2.76026e36	−1.31372
$806$	−1.87658e35	−0.0879385
$807$	0	0
$808$	−2.34079e35	−0.106346
$809$	−1.33897e36	−0.598981	−0.299490	−	0.954099i	$-0.596817\pi$
$809$	−1.33897e36	−0.598981	−0.299490	+	0.954099i	$0.596817\pi$
$810$	0	0
$811$	1.29485e36	0.561643	0.280821	−	0.959760i	$-0.409393\pi$
$811$	1.29485e36	0.561643	0.280821	+	0.959760i	$0.409393\pi$
$812$	−1.55818e36	−0.665528
$813$	0	0
$814$	−1.05967e36	−0.438901
$815$	−5.82854e36	−2.37733
$816$	0	0
$817$	7.54600e35	0.298498
$818$	1.14024e36	0.444201
$819$	0	0
$820$	−1.11595e36	−0.421668
$821$	−3.91664e34	−0.0145755	−0.00728777	−	0.999973i	$-0.502320\pi$
$821$	−3.91664e34	−0.0145755	−0.00728777	+	0.999973i	$0.502320\pi$
$822$	0	0
$823$	−3.40400e36	−1.22883	−0.614414	−	0.788984i	$-0.710608\pi$
$823$	−3.40400e36	−1.22883	−0.614414	+	0.788984i	$0.710608\pi$
$824$	−5.68362e35	−0.202085
$825$	0	0
$826$	−1.62987e36	−0.562213
$827$	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	$0.456819\pi$
$830$	−4.30826e36	−1.39902
$831$	0	0
$832$	−5.06913e34	−0.0159732
$833$	1.30513e36	0.405128
$834$	0	0
$835$	3.57748e36	1.07770
$836$	5.22298e35	0.155003
$837$	0	0
$838$	−2.60845e36	−0.751332
$839$	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
$842$	1.60676e36	0.436063
$843$	0	0
$844$	6.41220e35	0.168938
$845$	6.10126e36	1.58384
$846$	0	0
$847$	−2.26629e36	−0.571180
$848$	3.49914e35	0.0868987
$849$	0	0
$850$	5.78284e36	1.39446
$851$	−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	$0.464843\pi$
$854$	2.91582e34	0.00663036
$855$	0	0
$856$	1.08567e36	0.239759
$857$	5.39118e36	1.17334	0.586669	−	0.809827i	$-0.300439\pi$
$857$	5.39118e36	1.17334	0.586669	+	0.809827i	$0.300439\pi$
$858$	0	0
$859$	5.48992e36	1.16052	0.580258	−	0.814432i	$-0.302952\pi$
$859$	5.48992e36	1.16052	0.580258	+	0.814432i	$0.302952\pi$
$860$	2.05037e36	0.427171
$861$	0	0
$862$	1.00016e36	0.202407
$863$	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
$866$	−3.73912e36	−0.714161
$867$	0	0
$868$	−2.15097e36	−0.399152
$869$	−3.65359e36	−0.668301
$870$	0	0
$871$	5.76349e35	0.102437
$872$	−2.06755e36	−0.362244
$873$	0	0
$874$	2.32398e36	0.395676
$875$	−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	$0.446679\pi$
$878$	−2.48874e36	−0.400219
$879$	0	0
$880$	1.41917e36	0.221820
$881$	1.76163e36	0.271466	0.135733	−	0.990745i	$-0.456661\pi$
$881$	1.76163e36	0.271466	0.135733	+	0.990745i	$0.456661\pi$
$882$	0	0
$883$	1.25672e37	1.88248	0.941240	−	0.337738i	$-0.109662\pi$
$883$	1.25672e37	1.88248	0.941240	+	0.337738i	$0.109662\pi$
$884$	5.35724e35	0.0791203
$885$	0	0
$886$	3.71216e36	0.532973
$887$	−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
$890$	1.15980e37	1.57401
$891$	0	0
$892$	5.21939e36	0.688746
$893$	2.53636e36	0.330041
$894$	0	0
$895$	−3.79203e36	−0.479826
$896$	−5.81034e35	−0.0725023
$897$	0	0
$898$	−4.97986e36	−0.604315
$899$	1.31962e37	1.57925
$900$	0	0
$901$	−3.69802e36	−0.430436
$902$	1.77789e36	0.204091
$903$	0	0
$904$	−5.54391e36	−0.619029
$905$	−1.95988e37	−2.15835
$906$	0	0
$907$	−1.59523e37	−1.70896	−0.854482	−	0.519481i	$-0.826125\pi$
$907$	−1.59523e37	−1.70896	−0.854482	+	0.519481i	$0.826125\pi$
$908$	1.77120e36	0.187152
$909$	0	0
$910$	−1.16092e36	−0.119340
$911$	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
$914$	2.29183e36	0.223026
$915$	0	0
$916$	−6.21298e36	−0.588310
$917$	−2.02434e36	−0.189089
$918$	0	0
$919$	−7.70373e35	−0.0700256	−0.0350128	−	0.999387i	$-0.511147\pi$
$919$	−7.70373e35	−0.0700256	−0.0350128	+	0.999387i	$0.511147\pi$
$920$	6.31464e36	0.566240
$921$	0	0
$922$	−1.55071e36	−0.135330
$923$	−2.43703e36	−0.209817
$924$	0	0
$925$	−2.14061e37	−1.79377
$926$	−1.13475e37	−0.938132
$927$	0	0
$928$	3.56463e36	0.286856
$929$	−1.66420e37	−1.32132	−0.660662	−	0.750684i	$-0.729724\pi$
$929$	−1.66420e37	−1.32132	−0.660662	+	0.750684i	$0.729724\pi$
$930$	0	0
$931$	2.38121e36	0.184046
$932$	1.14887e36	0.0876138
$933$	0	0
$934$	−9.49954e36	−0.705288
$935$	−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	$0.387993\pi$
$938$	6.60622e36	0.464962
$939$	0	0
$940$	6.89172e36	0.472312
$941$	1.33808e37	0.904923	0.452461	−	0.891784i	$-0.350546\pi$
$941$	1.33808e37	0.904923	0.452461	+	0.891784i	$0.350546\pi$
$942$	0	0
$943$	7.91080e36	0.520984
$944$	3.72863e36	0.242326
$945$	0	0
$946$	−3.26659e36	−0.206755
$947$	−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
$950$	1.05508e37	0.633489
$951$	0	0
$952$	6.14058e36	0.359126
$953$	9.10378e36	0.525485	0.262742	−	0.964866i	$-0.415373\pi$
$953$	9.10378e36	0.525485	0.262742	+	0.964866i	$0.415373\pi$
$954$	0	0
$955$	−1.68184e37	−0.945677
$956$	−6.50575e36	−0.361055
$957$	0	0
$958$	9.19044e36	0.496898
$959$	9.70142e36	0.517729
$960$	0	0
$961$	−1.01623e36	−0.0528386
$962$	−1.98307e36	−0.101777
$963$	0	0
$964$	1.74019e37	0.870230
$965$	5.24538e37	2.58932
$966$	0	0
$967$	−1.34794e36	−0.0648394	−0.0324197	−	0.999474i	$-0.510321\pi$
$967$	−1.34794e36	−0.0648394	−0.0324197	+	0.999474i	$0.510321\pi$
$968$	5.18457e36	0.246190
$969$	0	0
$970$	1.29060e37	0.597236
$971$	−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
$974$	−1.52123e36	−0.0668663
$975$	0	0
$976$	−6.67050e34	−0.00285782
$977$	−1.16914e37	−0.494522	−0.247261	−	0.968949i	$-0.579530\pi$
$977$	−1.16914e37	−0.494522	−0.247261	+	0.968949i	$0.579530\pi$
$978$	0	0
$979$	−1.84776e37	−0.761835
$980$	6.47014e36	0.263383
$981$	0	0
$982$	−1.08752e37	−0.431563
$983$	−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
$986$	−3.76722e37	−1.42089
$987$	0	0
$988$	9.77426e35	0.0359436
$989$	−1.45348e37	−0.527784
$990$	0	0
$991$	−3.79186e37	−1.34255	−0.671276	−	0.741207i	$-0.734254\pi$
$991$	−3.79186e37	−1.34255	−0.671276	+	0.741207i	$0.734254\pi$
$992$	4.92077e36	0.172043
$993$	0	0
$994$	−2.79338e37	−0.952356
$995$	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	$0.159106\pi$
$998$	9.79424e36	0.317570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.26.a.e.1.1		2
3.2	odd	2		2.26.a.b.1.1	✓	2
12.11	even	2		16.26.a.c.1.2		2
15.2	even	4		50.26.b.e.49.4		4
15.8	even	4		50.26.b.e.49.1		4
15.14	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	3.2	odd	2
16.26.a.c.1.2		2	12.11	even	2
18.26.a.e.1.1		2	1.1	even	1	trivial
50.26.a.c.1.2		2	15.14	odd	2
50.26.b.e.49.1		4	15.8	even	4
50.26.b.e.49.4		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)

\(f(q)\)	\(=\)	\(q-4096.00 q^{2} +1.67772e7 q^{4} -8.78994e8 q^{5} +3.00388e10 q^{7} -6.87195e10 q^{8} +3.60036e12 q^{10} -5.73599e12 q^{11} -1.07343e13 q^{13} -1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -5.42738e15 q^{19} -1.47471e16 q^{20} +2.34946e16 q^{22} +1.04540e17 q^{23} +4.74607e17 q^{25} +4.39677e16 q^{26} +5.03967e17 q^{28} -3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +1.21845e19 q^{34} -2.64039e19 q^{35} -4.51028e19 q^{37} +2.22305e19 q^{38} +6.04040e19 q^{40} +7.56724e19 q^{41} -1.39036e20 q^{43} -9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} -4.38741e20 q^{49} -1.94399e21 q^{50} -1.80092e20 q^{52} +1.24315e21 q^{53} +5.04190e21 q^{55} -2.06425e21 q^{56} +1.26641e22 q^{58} +1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} +4.72237e21 q^{64} +9.43539e21 q^{65} -5.36922e22 q^{67} -4.99076e22 q^{68} +1.08150e23 q^{70} +2.27032e23 q^{71} +2.78528e23 q^{73} +1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} +6.36958e23 q^{79} -2.47415e23 q^{80} -3.09954e23 q^{82} -1.19662e24 q^{83} +2.61477e24 q^{85} +5.69491e23 q^{86} +3.94174e23 q^{88} +3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} +1.91418e24 q^{94} +4.77063e24 q^{95} +3.58465e24 q^{97} +1.79708e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 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

Modular forms

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Database

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