LMFDB - Embedded newform 48.22.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.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:	$134.149125258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 162 $ x^2 - x - 162
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-12.2377$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} -5.97585e10 q^{11} +7.38499e11 q^{13} +1.25642e12 q^{15} -8.35876e12 q^{17} -4.19061e13 q^{19} +3.73234e13 q^{21} -4.48926e13 q^{23} -2.41012e13 q^{25} -2.05891e14 q^{27} -2.76669e15 q^{29} -8.36452e15 q^{31} +3.52868e15 q^{33} +1.34490e16 q^{35} -1.77675e16 q^{37} -4.36077e16 q^{39} +1.45253e17 q^{41} -1.24744e17 q^{43} -7.41904e16 q^{45} -4.28566e17 q^{47} -1.59026e17 q^{49} +4.93577e17 q^{51} -4.77017e17 q^{53} +1.27152e18 q^{55} +2.47452e18 q^{57} -1.61959e18 q^{59} -3.76882e18 q^{61} -2.20391e18 q^{63} -1.57135e19 q^{65} +2.81797e18 q^{67} +2.65086e18 q^{69} -1.00228e19 q^{71} -1.72739e19 q^{73} +1.42315e18 q^{75} +3.77719e19 q^{77} +3.28276e19 q^{79} +1.21577e19 q^{81} -3.05240e17 q^{83} +1.77854e20 q^{85} +1.63370e20 q^{87} +2.34593e20 q^{89} -4.66787e20 q^{91} +4.93917e20 q^{93} +8.91662e20 q^{95} -5.92086e20 q^{97} -2.08365e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10}) $ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9	$ 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100}) $ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	0	0
$5$	−2.12776e7	−0.974400	−0.487200	−	0.873290i	$-0.661982\pi$
$5$	−2.12776e7	−0.974400	−0.487200	+	0.873290i	$0.661982\pi$
$6$	0	0
$7$	−6.32076e8	−0.845745	−0.422873	−	0.906189i	$-0.638978\pi$
$7$	−6.32076e8	−0.845745	−0.422873	+	0.906189i	$0.638978\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−5.97585e10	−0.694666	−0.347333	−	0.937742i	$-0.612913\pi$
$11$	−5.97585e10	−0.694666	−0.347333	+	0.937742i	$0.612913\pi$
$12$	0	0
$13$	7.38499e11	1.48575	0.742874	−	0.669432i	$-0.233462\pi$
$13$	7.38499e11	1.48575	0.742874	+	0.669432i	$0.233462\pi$
$14$	0	0
$15$	1.25642e12	0.562570
$16$	0	0
$17$	−8.35876e12	−1.00561	−0.502804	−	0.864401i	$-0.667698\pi$
$17$	−8.35876e12	−1.00561	−0.502804	+	0.864401i	$0.667698\pi$
$18$	0	0
$19$	−4.19061e13	−1.56807	−0.784034	−	0.620718i	$-0.786841\pi$
$19$	−4.19061e13	−1.56807	−0.784034	+	0.620718i	$0.786841\pi$
$20$	0	0
$21$	3.73234e13	0.488291
$22$	0	0
$23$	−4.48926e13	−0.225961	−0.112980	−	0.993597i	$-0.536040\pi$
$23$	−4.48926e13	−0.225961	−0.112980	+	0.993597i	$0.536040\pi$
$24$	0	0
$25$	−2.41012e13	−0.0505438
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	−2.76669e15	−1.22119	−0.610593	−	0.791945i	$-0.709069\pi$
$29$	−2.76669e15	−1.22119	−0.610593	+	0.791945i	$0.709069\pi$
$30$	0	0
$31$	−8.36452e15	−1.83292	−0.916459	−	0.400128i	$-0.868966\pi$
$31$	−8.36452e15	−1.83292	−0.916459	+	0.400128i	$0.868966\pi$
$32$	0	0
$33$	3.52868e15	0.401066
$34$	0	0
$35$	1.34490e16	0.824095
$36$	0	0
$37$	−1.77675e16	−0.607447	−0.303724	−	0.952760i	$-0.598230\pi$
$37$	−1.77675e16	−0.607447	−0.303724	+	0.952760i	$0.598230\pi$
$38$	0	0
$39$	−4.36077e16	−0.857797
$40$	0	0
$41$	1.45253e17	1.69003	0.845013	−	0.534745i	$-0.179592\pi$
$41$	1.45253e17	1.69003	0.845013	+	0.534745i	$0.179592\pi$
$42$	0	0
$43$	−1.24744e17	−0.880239	−0.440119	−	0.897939i	$-0.645064\pi$
$43$	−1.24744e17	−0.880239	−0.440119	+	0.897939i	$0.645064\pi$
$44$	0	0
$45$	−7.41904e16	−0.324800
$46$	0	0
$47$	−4.28566e17	−1.18847	−0.594237	−	0.804290i	$-0.702546\pi$
$47$	−4.28566e17	−1.18847	−0.594237	+	0.804290i	$0.702546\pi$
$48$	0	0
$49$	−1.59026e17	−0.284715
$50$	0	0
$51$	4.93577e17	0.580588
$52$	0	0
$53$	−4.77017e17	−0.374660	−0.187330	−	0.982297i	$-0.559983\pi$
$53$	−4.77017e17	−0.374660	−0.187330	+	0.982297i	$0.559983\pi$
$54$	0	0
$55$	1.27152e18	0.676883
$56$	0	0
$57$	2.47452e18	0.905324
$58$	0	0
$59$	−1.61959e18	−0.412533	−0.206267	−	0.978496i	$-0.566131\pi$
$59$	−1.61959e18	−0.412533	−0.206267	+	0.978496i	$0.566131\pi$
$60$	0	0
$61$	−3.76882e18	−0.676461	−0.338230	−	0.941063i	$-0.609828\pi$
$61$	−3.76882e18	−0.676461	−0.338230	+	0.941063i	$0.609828\pi$
$62$	0	0
$63$	−2.20391e18	−0.281915
$64$	0	0
$65$	−1.57135e19	−1.44771
$66$	0	0
$67$	2.81797e18	0.188865	0.0944324	−	0.995531i	$-0.469896\pi$
$67$	2.81797e18	0.188865	0.0944324	+	0.995531i	$0.469896\pi$
$68$	0	0
$69$	2.65086e18	0.130458
$70$	0	0
$71$	−1.00228e19	−0.365407	−0.182704	−	0.983168i	$-0.558485\pi$
$71$	−1.00228e19	−0.365407	−0.182704	+	0.983168i	$0.558485\pi$
$72$	0	0
$73$	−1.72739e19	−0.470435	−0.235218	−	0.971943i	$-0.575580\pi$
$73$	−1.72739e19	−0.470435	−0.235218	+	0.971943i	$0.575580\pi$
$74$	0	0
$75$	1.42315e18	0.0291815
$76$	0	0
$77$	3.77719e19	0.587511
$78$	0	0
$79$	3.28276e19	0.390080	0.195040	−	0.980795i	$-0.437516\pi$
$79$	3.28276e19	0.390080	0.195040	+	0.980795i	$0.437516\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−3.05240e17	−0.00215934	−0.00107967	−	0.999999i	$-0.500344\pi$
$83$	−3.05240e17	−0.00215934	−0.00107967	+	0.999999i	$0.500344\pi$
$84$	0	0
$85$	1.77854e20	0.979864
$86$	0	0
$87$	1.63370e20	0.705052
$88$	0	0
$89$	2.34593e20	0.797480	0.398740	−	0.917064i	$-0.369448\pi$
$89$	2.34593e20	0.797480	0.398740	+	0.917064i	$0.369448\pi$
$90$	0	0
$91$	−4.66787e20	−1.25656
$92$	0	0
$93$	4.93917e20	1.05824
$94$	0	0
$95$	8.91662e20	1.52793
$96$	0	0
$97$	−5.92086e20	−0.815233	−0.407616	−	0.913153i	$-0.633640\pi$
$97$	−5.92086e20	−0.815233	−0.407616	+	0.913153i	$0.633640\pi$
$98$	0	0
$99$	−2.08365e20	−0.231555
$100$	0	0
$101$	−1.66229e21	−1.49739	−0.748693	−	0.662917i	$-0.769318\pi$
$101$	−1.66229e21	−1.49739	−0.748693	+	0.662917i	$0.769318\pi$
$102$	0	0
$103$	−1.17919e21	−0.864552	−0.432276	−	0.901741i	$-0.642289\pi$
$103$	−1.17919e21	−0.864552	−0.432276	+	0.901741i	$0.642289\pi$
$104$	0	0
$105$	−7.94153e20	−0.475791
$106$	0	0
$107$	8.69160e20	0.427140	0.213570	−	0.976928i	$-0.431491\pi$
$107$	8.69160e20	0.427140	0.213570	+	0.976928i	$0.431491\pi$
$108$	0	0
$109$	4.12021e20	0.166702	0.0833510	−	0.996520i	$-0.473438\pi$
$109$	4.12021e20	0.166702	0.0833510	+	0.996520i	$0.473438\pi$
$110$	0	0
$111$	1.04915e21	0.350710
$112$	0	0
$113$	2.45943e21	0.681569	0.340785	−	0.940141i	$-0.389307\pi$
$113$	2.45943e21	0.681569	0.340785	+	0.940141i	$0.389307\pi$
$114$	0	0
$115$	9.55207e20	0.220176
$116$	0	0
$117$	2.57499e21	0.495249
$118$	0	0
$119$	5.28337e21	0.850487
$120$	0	0
$121$	−3.82917e21	−0.517439
$122$	0	0
$123$	−8.57702e21	−0.975737
$124$	0	0
$125$	1.06588e22	1.02365
$126$	0	0
$127$	−5.53783e21	−0.450195	−0.225097	−	0.974336i	$-0.572270\pi$
$127$	−5.53783e21	−0.450195	−0.225097	+	0.974336i	$0.572270\pi$
$128$	0	0
$129$	7.36600e21	0.508206
$130$	0	0
$131$	2.54630e22	1.49472	0.747362	−	0.664417i	$-0.231320\pi$
$131$	2.54630e22	1.49472	0.747362	+	0.664417i	$0.231320\pi$
$132$	0	0
$133$	2.64878e22	1.32619
$134$	0	0
$135$	4.38087e21	0.187523
$136$	0	0
$137$	−4.82088e22	−1.76832	−0.884158	−	0.467188i	$-0.845267\pi$
$137$	−4.82088e22	−1.76832	−0.884158	+	0.467188i	$0.845267\pi$
$138$	0	0
$139$	2.18194e22	0.687363	0.343682	−	0.939086i	$-0.388326\pi$
$139$	2.18194e22	0.687363	0.343682	+	0.939086i	$0.388326\pi$
$140$	0	0
$141$	2.53064e22	0.686166
$142$	0	0
$143$	−4.41316e22	−1.03210
$144$	0	0
$145$	5.88685e22	1.18992
$146$	0	0
$147$	9.39035e21	0.164380
$148$	0	0
$149$	4.84140e22	0.735386	0.367693	−	0.929947i	$-0.380148\pi$
$149$	4.84140e22	0.735386	0.367693	+	0.929947i	$0.380148\pi$
$150$	0	0
$151$	1.22948e23	1.62354	0.811770	−	0.583977i	$-0.198504\pi$
$151$	1.22948e23	1.62354	0.811770	+	0.583977i	$0.198504\pi$
$152$	0	0
$153$	−2.91452e22	−0.335202
$154$	0	0
$155$	1.77977e23	1.78600
$156$	0	0
$157$	5.89828e22	0.517344	0.258672	−	0.965965i	$-0.416715\pi$
$157$	5.89828e22	0.517344	0.258672	+	0.965965i	$0.416715\pi$
$158$	0	0
$159$	2.81674e22	0.216310
$160$	0	0
$161$	2.83755e22	0.191105
$162$	0	0
$163$	3.18244e22	0.188274	0.0941369	−	0.995559i	$-0.469991\pi$
$163$	3.18244e22	0.188274	0.0941369	+	0.995559i	$0.469991\pi$
$164$	0	0
$165$	−7.50818e22	−0.390799
$166$	0	0
$167$	3.62822e22	0.166407	0.0832033	−	0.996533i	$-0.473485\pi$
$167$	3.62822e22	0.166407	0.0832033	+	0.996533i	$0.473485\pi$
$168$	0	0
$169$	2.98317e23	1.20744
$170$	0	0
$171$	−1.46118e23	−0.522689
$172$	0	0
$173$	3.21510e23	1.01791	0.508956	−	0.860792i	$-0.330031\pi$
$173$	3.21510e23	1.01791	0.508956	+	0.860792i	$0.330031\pi$
$174$	0	0
$175$	1.52338e22	0.0427472
$176$	0	0
$177$	9.56351e22	0.238176
$178$	0	0
$179$	1.80418e23	0.399321	0.199661	−	0.979865i	$-0.436016\pi$
$179$	1.80418e23	0.399321	0.199661	+	0.979865i	$0.436016\pi$
$180$	0	0
$181$	−6.36646e21	−0.0125393	−0.00626965	−	0.999980i	$-0.501996\pi$
$181$	−6.36646e21	−0.0125393	−0.00626965	+	0.999980i	$0.501996\pi$
$182$	0	0
$183$	2.22545e23	0.390555
$184$	0	0
$185$	3.78050e23	0.591897
$186$	0	0
$187$	4.99507e23	0.698561
$188$	0	0
$189$	1.30139e23	0.162764
$190$	0	0
$191$	−8.17882e23	−0.915883	−0.457942	−	0.888982i	$-0.651413\pi$
$191$	−8.17882e23	−0.915883	−0.457942	+	0.888982i	$0.651413\pi$
$192$	0	0
$193$	−1.80064e23	−0.180748	−0.0903742	−	0.995908i	$-0.528806\pi$
$193$	−1.80064e23	−0.180748	−0.0903742	+	0.995908i	$0.528806\pi$
$194$	0	0
$195$	9.27866e23	0.835837
$196$	0	0
$197$	1.65216e24	1.33708	0.668541	−	0.743675i	$-0.266919\pi$
$197$	1.65216e24	1.33708	0.668541	+	0.743675i	$0.266919\pi$
$198$	0	0
$199$	1.52537e24	1.11024	0.555120	−	0.831770i	$-0.312672\pi$
$199$	1.52537e24	1.11024	0.555120	+	0.831770i	$0.312672\pi$
$200$	0	0
$201$	−1.66398e23	−0.109041
$202$	0	0
$203$	1.74876e24	1.03281
$204$	0	0
$205$	−3.09063e24	−1.64676
$206$	0	0
$207$	−1.56531e23	−0.0753202
$208$	0	0
$209$	2.50425e24	1.08928
$210$	0	0
$211$	−1.35748e24	−0.534277	−0.267138	−	0.963658i	$-0.586078\pi$
$211$	−1.35748e24	−0.534277	−0.267138	+	0.963658i	$0.586078\pi$
$212$	0	0
$213$	5.91838e23	0.210968
$214$	0	0
$215$	2.65425e24	0.857705
$216$	0	0
$217$	5.28701e24	1.55018
$218$	0	0
$219$	1.02001e24	0.271606
$220$	0	0
$221$	−6.17294e24	−1.49408
$222$	0	0
$223$	−4.73192e24	−1.04192	−0.520962	−	0.853580i	$-0.674427\pi$
$223$	−4.73192e24	−1.04192	−0.520962	+	0.853580i	$0.674427\pi$
$224$	0	0
$225$	−8.40356e22	−0.0168479
$226$	0	0
$227$	−8.00226e24	−1.46198	−0.730989	−	0.682389i	$-0.760941\pi$
$227$	−8.00226e24	−1.46198	−0.730989	+	0.682389i	$0.760941\pi$
$228$	0	0
$229$	2.72111e24	0.453391	0.226696	−	0.973966i	$-0.427208\pi$
$229$	2.72111e24	0.453391	0.226696	+	0.973966i	$0.427208\pi$
$230$	0	0
$231$	−2.23039e24	−0.339200
$232$	0	0
$233$	−6.60115e24	−0.917028	−0.458514	−	0.888687i	$-0.651618\pi$
$233$	−6.60115e24	−0.917028	−0.458514	+	0.888687i	$0.651618\pi$
$234$	0	0
$235$	9.11885e24	1.15805
$236$	0	0
$237$	−1.93843e24	−0.225213
$238$	0	0
$239$	−9.92224e24	−1.05544	−0.527718	−	0.849420i	$-0.676952\pi$
$239$	−9.92224e24	−1.05544	−0.527718	+	0.849420i	$0.676952\pi$
$240$	0	0
$241$	−1.54177e25	−1.50259	−0.751295	−	0.659966i	$-0.770571\pi$
$241$	−1.54177e25	−1.50259	−0.751295	+	0.659966i	$0.770571\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	3.38370e24	0.277426
$246$	0	0
$247$	−3.09477e25	−2.32975
$248$	0	0
$249$	1.80241e22	0.00124670
$250$	0	0
$251$	1.00009e25	0.636014	0.318007	−	0.948088i	$-0.396986\pi$
$251$	1.00009e25	0.636014	0.318007	+	0.948088i	$0.396986\pi$
$252$	0	0
$253$	2.68271e24	0.156967
$254$	0	0
$255$	−1.05021e25	−0.565725
$256$	0	0
$257$	−3.06596e25	−1.52149	−0.760744	−	0.649052i	$-0.775166\pi$
$257$	−3.06596e25	−1.52149	−0.760744	+	0.649052i	$0.775166\pi$
$258$	0	0
$259$	1.12304e25	0.513746
$260$	0	0
$261$	−9.64685e24	−0.407062
$262$	0	0
$263$	2.33455e25	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$263$	2.33455e25	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$264$	0	0
$265$	1.01498e25	0.365069
$266$	0	0
$267$	−1.38525e25	−0.460425
$268$	0	0
$269$	2.29737e25	0.706043	0.353022	−	0.935615i	$-0.385154\pi$
$269$	2.29737e25	0.706043	0.353022	+	0.935615i	$0.385154\pi$
$270$	0	0
$271$	4.02085e25	1.14325	0.571624	−	0.820516i	$-0.306314\pi$
$271$	4.02085e25	1.14325	0.571624	+	0.820516i	$0.306314\pi$
$272$	0	0
$273$	2.75633e25	0.725477
$274$	0	0
$275$	1.44025e24	0.0351111
$276$	0	0
$277$	−4.48722e25	−1.01377	−0.506886	−	0.862013i	$-0.669203\pi$
$277$	−4.48722e25	−1.01377	−0.506886	+	0.862013i	$0.669203\pi$
$278$	0	0
$279$	−2.91653e25	−0.610973
$280$	0	0
$281$	−6.06489e25	−1.17871	−0.589355	−	0.807874i	$-0.700618\pi$
$281$	−6.06489e25	−1.17871	−0.589355	+	0.807874i	$0.700618\pi$
$282$	0	0
$283$	−1.04229e26	−1.88032	−0.940158	−	0.340739i	$-0.889323\pi$
$283$	−1.04229e26	−1.88032	−0.940158	+	0.340739i	$0.889323\pi$
$284$	0	0
$285$	−5.26517e25	−0.882148
$286$	0	0
$287$	−9.18107e25	−1.42933
$288$	0	0
$289$	7.76980e23	0.0112456
$290$	0	0
$291$	3.49621e25	0.470675
$292$	0	0
$293$	6.24634e25	0.782556	0.391278	−	0.920273i	$-0.372033\pi$
$293$	6.24634e25	0.782556	0.391278	+	0.920273i	$0.372033\pi$
$294$	0	0
$295$	3.44610e25	0.401972
$296$	0	0
$297$	1.23037e25	0.133689
$298$	0	0
$299$	−3.31532e25	−0.335720
$300$	0	0
$301$	7.88476e25	0.744458
$302$	0	0
$303$	9.81568e25	0.864516
$304$	0	0
$305$	8.01915e25	0.659144
$306$	0	0
$307$	−7.14411e25	−0.548271	−0.274135	−	0.961691i	$-0.588392\pi$
$307$	−7.14411e25	−0.548271	−0.274135	+	0.961691i	$0.588392\pi$
$308$	0	0
$309$	6.96297e25	0.499150
$310$	0	0
$311$	6.01915e25	0.403228	0.201614	−	0.979465i	$-0.435381\pi$
$311$	6.01915e25	0.403228	0.201614	+	0.979465i	$0.435381\pi$
$312$	0	0
$313$	−1.69049e26	−1.05876	−0.529379	−	0.848385i	$-0.677575\pi$
$313$	−1.69049e26	−1.05876	−0.529379	+	0.848385i	$0.677575\pi$
$314$	0	0
$315$	4.68939e25	0.274698
$316$	0	0
$317$	−5.71691e25	−0.313357	−0.156678	−	0.987650i	$-0.550079\pi$
$317$	−5.71691e25	−0.313357	−0.156678	+	0.987650i	$0.550079\pi$
$318$	0	0
$319$	1.65333e26	0.848316
$320$	0	0
$321$	−5.13230e25	−0.246609
$322$	0	0
$323$	3.50283e26	1.57686
$324$	0	0
$325$	−1.77987e25	−0.0750954
$326$	0	0
$327$	−2.43294e25	−0.0962454
$328$	0	0
$329$	2.70886e26	1.00515
$330$	0	0
$331$	−6.67957e25	−0.232571	−0.116285	−	0.993216i	$-0.537099\pi$
$331$	−6.67957e25	−0.232571	−0.116285	+	0.993216i	$0.537099\pi$
$332$	0	0
$333$	−6.19515e25	−0.202482
$334$	0	0
$335$	−5.99597e25	−0.184030
$336$	0	0
$337$	3.55365e26	1.02461	0.512307	−	0.858802i	$-0.328791\pi$
$337$	3.55365e26	1.02461	0.512307	+	0.858802i	$0.328791\pi$
$338$	0	0
$339$	−1.45227e26	−0.393504
$340$	0	0
$341$	4.99851e26	1.27327
$342$	0	0
$343$	4.53560e26	1.08654
$344$	0	0
$345$	−5.64040e25	−0.127119
$346$	0	0
$347$	3.94238e25	0.0836179	0.0418090	−	0.999126i	$-0.486688\pi$
$347$	3.94238e25	0.0836179	0.0418090	+	0.999126i	$0.486688\pi$
$348$	0	0
$349$	−5.84299e26	−1.16672	−0.583362	−	0.812212i	$-0.698263\pi$
$349$	−5.84299e26	−1.16672	−0.583362	+	0.812212i	$0.698263\pi$
$350$	0	0
$351$	−1.52050e26	−0.285932
$352$	0	0
$353$	4.96062e26	0.878823	0.439412	−	0.898286i	$-0.355187\pi$
$353$	4.96062e26	0.878823	0.439412	+	0.898286i	$0.355187\pi$
$354$	0	0
$355$	2.13262e26	0.356053
$356$	0	0
$357$	−3.11978e26	−0.491029
$358$	0	0
$359$	3.35136e26	0.497426	0.248713	−	0.968577i	$-0.419992\pi$
$359$	3.35136e26	0.497426	0.248713	+	0.968577i	$0.419992\pi$
$360$	0	0
$361$	1.04192e27	1.45884
$362$	0	0
$363$	2.26109e26	0.298743
$364$	0	0
$365$	3.67547e26	0.458392
$366$	0	0
$367$	−9.70198e25	−0.114253	−0.0571264	−	0.998367i	$-0.518194\pi$
$367$	−9.70198e25	−0.114253	−0.0571264	+	0.998367i	$0.518194\pi$
$368$	0	0
$369$	5.06465e26	0.563342
$370$	0	0
$371$	3.01511e26	0.316867
$372$	0	0
$373$	1.03252e27	1.02554	0.512771	−	0.858525i	$-0.328619\pi$
$373$	1.03252e27	1.02554	0.512771	+	0.858525i	$0.328619\pi$
$374$	0	0
$375$	−6.29389e26	−0.591005
$376$	0	0
$377$	−2.04320e27	−1.81437
$378$	0	0
$379$	−2.20881e26	−0.185544	−0.0927721	−	0.995687i	$-0.529573\pi$
$379$	−2.20881e26	−0.185544	−0.0927721	+	0.995687i	$0.529573\pi$
$380$	0	0
$381$	3.27003e26	0.259920
$382$	0	0
$383$	1.59547e27	1.20033	0.600165	−	0.799876i	$-0.295101\pi$
$383$	1.59547e27	1.20033	0.600165	+	0.799876i	$0.295101\pi$
$384$	0	0
$385$	−8.03695e26	−0.572471
$386$	0	0
$387$	−4.34955e26	−0.293413
$388$	0	0
$389$	−3.60474e26	−0.230358	−0.115179	−	0.993345i	$-0.536744\pi$
$389$	−3.60474e26	−0.230358	−0.115179	+	0.993345i	$0.536744\pi$
$390$	0	0
$391$	3.75247e26	0.227228
$392$	0	0
$393$	−1.50357e27	−0.862980
$394$	0	0
$395$	−6.98492e26	−0.380095
$396$	0	0
$397$	−1.47191e27	−0.759595	−0.379798	−	0.925070i	$-0.624006\pi$
$397$	−1.47191e27	−0.759595	−0.379798	+	0.925070i	$0.624006\pi$
$398$	0	0
$399$	−1.56408e27	−0.765674
$400$	0	0
$401$	1.00401e27	0.466360	0.233180	−	0.972434i	$-0.425087\pi$
$401$	1.00401e27	0.466360	0.233180	+	0.972434i	$0.425087\pi$
$402$	0	0
$403$	−6.17719e27	−2.72325
$404$	0	0
$405$	−2.58686e26	−0.108267
$406$	0	0
$407$	1.06176e27	0.421973
$408$	0	0
$409$	−4.41838e27	−1.66789	−0.833947	−	0.551845i	$-0.813924\pi$
$409$	−4.41838e27	−1.66789	−0.833947	+	0.551845i	$0.813924\pi$
$410$	0	0
$411$	2.84668e27	1.02094
$412$	0	0
$413$	1.02370e27	0.348898
$414$	0	0
$415$	6.49477e24	0.00210406
$416$	0	0
$417$	−1.28841e27	−0.396849
$418$	0	0
$419$	2.36639e27	0.693169	0.346584	−	0.938019i	$-0.387341\pi$
$419$	2.36639e27	0.693169	0.346584	+	0.938019i	$0.387341\pi$
$420$	0	0
$421$	2.72067e27	0.758078	0.379039	−	0.925381i	$-0.376255\pi$
$421$	2.72067e27	0.758078	0.379039	+	0.925381i	$0.376255\pi$
$422$	0	0
$423$	−1.49432e27	−0.396158
$424$	0	0
$425$	2.01456e26	0.0508272
$426$	0	0
$427$	2.38218e27	0.572114
$428$	0	0
$429$	2.60593e27	0.595883
$430$	0	0
$431$	−8.87397e27	−1.93244	−0.966222	−	0.257713i	$-0.917031\pi$
$431$	−8.87397e27	−1.93244	−0.966222	+	0.257713i	$0.917031\pi$
$432$	0	0
$433$	−3.39994e27	−0.705258	−0.352629	−	0.935763i	$-0.614712\pi$
$433$	−3.39994e27	−0.705258	−0.352629	+	0.935763i	$0.614712\pi$
$434$	0	0
$435$	−3.47613e27	−0.687003
$436$	0	0
$437$	1.88128e27	0.354322
$438$	0	0
$439$	−2.29516e27	−0.412037	−0.206018	−	0.978548i	$-0.566051\pi$
$439$	−2.29516e27	−0.412037	−0.206018	+	0.978548i	$0.566051\pi$
$440$	0	0
$441$	−5.54491e26	−0.0949050
$442$	0	0
$443$	5.65265e27	0.922599	0.461299	−	0.887245i	$-0.347383\pi$
$443$	5.65265e27	0.922599	0.461299	+	0.887245i	$0.347383\pi$
$444$	0	0
$445$	−4.99157e27	−0.777065
$446$	0	0
$447$	−2.85880e27	−0.424576
$448$	0	0
$449$	1.23324e28	1.74768	0.873839	−	0.486216i	$-0.161623\pi$
$449$	1.23324e28	1.74768	0.873839	+	0.486216i	$0.161623\pi$
$450$	0	0
$451$	−8.68008e27	−1.17400
$452$	0	0
$453$	−7.25995e27	−0.937351
$454$	0	0
$455$	9.93211e27	1.22440
$456$	0	0
$457$	−7.62646e27	−0.897848	−0.448924	−	0.893570i	$-0.648193\pi$
$457$	−7.62646e27	−0.897848	−0.448924	+	0.893570i	$0.648193\pi$
$458$	0	0
$459$	1.72100e27	0.193529
$460$	0	0
$461$	−9.04851e26	−0.0972114	−0.0486057	−	0.998818i	$-0.515478\pi$
$461$	−9.04851e26	−0.0972114	−0.0486057	+	0.998818i	$0.515478\pi$
$462$	0	0
$463$	1.19705e27	0.122889	0.0614443	−	0.998111i	$-0.480429\pi$
$463$	1.19705e27	0.122889	0.0614443	+	0.998111i	$0.480429\pi$
$464$	0	0
$465$	−1.05094e28	−1.03115
$466$	0	0
$467$	1.16951e28	1.09693	0.548464	−	0.836174i	$-0.315213\pi$
$467$	1.16951e28	1.09693	0.548464	+	0.836174i	$0.315213\pi$
$468$	0	0
$469$	−1.78117e27	−0.159732
$470$	0	0
$471$	−3.48288e27	−0.298689
$472$	0	0
$473$	7.45451e27	0.611472
$474$	0	0
$475$	1.00999e27	0.0792562
$476$	0	0
$477$	−1.66326e27	−0.124887
$478$	0	0
$479$	−2.04856e28	−1.47206	−0.736029	−	0.676950i	$-0.763301\pi$
$479$	−2.04856e28	−1.47206	−0.736029	+	0.676950i	$0.763301\pi$
$480$	0	0
$481$	−1.31213e28	−0.902513
$482$	0	0
$483$	−1.67555e27	−0.110335
$484$	0	0
$485$	1.25982e28	0.794363
$486$	0	0
$487$	3.75693e27	0.226871	0.113436	−	0.993545i	$-0.463814\pi$
$487$	3.75693e27	0.226871	0.113436	+	0.993545i	$0.463814\pi$
$488$	0	0
$489$	−1.87920e27	−0.108700
$490$	0	0
$491$	1.53173e28	0.848841	0.424420	−	0.905465i	$-0.360478\pi$
$491$	1.53173e28	0.848841	0.424420	+	0.905465i	$0.360478\pi$
$492$	0	0
$493$	2.31261e28	1.22803
$494$	0	0
$495$	4.43350e27	0.225628
$496$	0	0
$497$	6.33518e27	0.309042
$498$	0	0
$499$	−3.84836e27	−0.179978	−0.0899890	−	0.995943i	$-0.528683\pi$
$499$	−3.84836e27	−0.179978	−0.0899890	+	0.995943i	$0.528683\pi$
$500$	0	0
$501$	−2.14243e27	−0.0960749
$502$	0	0
$503$	2.25177e28	0.968411	0.484205	−	0.874954i	$-0.339109\pi$
$503$	2.25177e28	0.968411	0.484205	+	0.874954i	$0.339109\pi$
$504$	0	0
$505$	3.53696e28	1.45905
$506$	0	0
$507$	−1.76153e28	−0.697119
$508$	0	0
$509$	2.09122e27	0.0794078	0.0397039	−	0.999211i	$-0.487359\pi$
$509$	2.09122e27	0.0794078	0.0397039	+	0.999211i	$0.487359\pi$
$510$	0	0
$511$	1.09184e28	0.397869
$512$	0	0
$513$	8.62810e27	0.301775
$514$	0	0
$515$	2.50902e28	0.842420
$516$	0	0
$517$	2.56104e28	0.825594
$518$	0	0
$519$	−1.89849e28	−0.587692
$520$	0	0
$521$	−3.76486e27	−0.111932	−0.0559659	−	0.998433i	$-0.517824\pi$
$521$	−3.76486e27	−0.111932	−0.0559659	+	0.998433i	$0.517824\pi$
$522$	0	0
$523$	1.74944e28	0.499611	0.249806	−	0.968296i	$-0.419633\pi$
$523$	1.74944e28	0.499611	0.249806	+	0.968296i	$0.419633\pi$
$524$	0	0
$525$	−8.99539e26	−0.0246801
$526$	0	0
$527$	6.99170e28	1.84320
$528$	0	0
$529$	−3.74562e28	−0.948942
$530$	0	0
$531$	−5.64716e27	−0.137511
$532$	0	0
$533$	1.07269e29	2.51095
$534$	0	0
$535$	−1.84936e28	−0.416205
$536$	0	0
$537$	−1.06535e28	−0.230548
$538$	0	0
$539$	9.50317e27	0.197782
$540$	0	0
$541$	−6.99960e28	−1.40121	−0.700603	−	0.713552i	$-0.747085\pi$
$541$	−6.99960e28	−1.40121	−0.700603	+	0.713552i	$0.747085\pi$
$542$	0	0
$543$	3.75933e26	0.00723956
$544$	0	0
$545$	−8.76680e27	−0.162434
$546$	0	0
$547$	−3.00896e28	−0.536476	−0.268238	−	0.963353i	$-0.586441\pi$
$547$	−3.00896e28	−0.536476	−0.268238	+	0.963353i	$0.586441\pi$
$548$	0	0
$549$	−1.31411e28	−0.225487
$550$	0	0
$551$	1.15941e29	1.91490
$552$	0	0
$553$	−2.07495e28	−0.329909
$554$	0	0
$555$	−2.23235e28	−0.341732
$556$	0	0
$557$	1.17263e29	1.72855	0.864277	−	0.503017i	$-0.167777\pi$
$557$	1.17263e29	1.72855	0.864277	+	0.503017i	$0.167777\pi$
$558$	0	0
$559$	−9.21233e28	−1.30781
$560$	0	0
$561$	−2.94954e28	−0.403315
$562$	0	0
$563$	−6.46986e28	−0.852230	−0.426115	−	0.904669i	$-0.640118\pi$
$563$	−6.46986e28	−0.852230	−0.426115	+	0.904669i	$0.640118\pi$
$564$	0	0
$565$	−5.23307e28	−0.664121
$566$	0	0
$567$	−7.68456e27	−0.0939717
$568$	0	0
$569$	−1.35009e29	−1.59105	−0.795527	−	0.605918i	$-0.792806\pi$
$569$	−1.35009e29	−1.59105	−0.795527	+	0.605918i	$0.792806\pi$
$570$	0	0
$571$	−2.55750e28	−0.290494	−0.145247	−	0.989395i	$-0.546398\pi$
$571$	−2.55750e28	−0.290494	−0.145247	+	0.989395i	$0.546398\pi$
$572$	0	0
$573$	4.82951e28	0.528786
$574$	0	0
$575$	1.08197e27	0.0114209
$576$	0	0
$577$	−1.52580e29	−1.55293	−0.776465	−	0.630160i	$-0.782989\pi$
$577$	−1.52580e29	−1.55293	−0.776465	+	0.630160i	$0.782989\pi$
$578$	0	0
$579$	1.06326e28	0.104355
$580$	0	0
$581$	1.92935e26	0.00182625
$582$	0	0
$583$	2.85058e28	0.260264
$584$	0	0
$585$	−5.47896e28	−0.482571
$586$	0	0
$587$	1.08485e29	0.921868	0.460934	−	0.887434i	$-0.347514\pi$
$587$	1.08485e29	0.921868	0.460934	+	0.887434i	$0.347514\pi$
$588$	0	0
$589$	3.50525e29	2.87414
$590$	0	0
$591$	−9.75587e28	−0.771965
$592$	0	0
$593$	8.82363e28	0.673865	0.336933	−	0.941529i	$-0.390611\pi$
$593$	8.82363e28	0.673865	0.336933	+	0.941529i	$0.390611\pi$
$594$	0	0
$595$	−1.12417e29	−0.828715
$596$	0	0
$597$	−9.00714e28	−0.640998
$598$	0	0
$599$	−9.23342e28	−0.634426	−0.317213	−	0.948354i	$-0.602747\pi$
$599$	−9.23342e28	−0.634426	−0.317213	+	0.948354i	$0.602747\pi$
$600$	0	0
$601$	−7.30699e28	−0.484793	−0.242397	−	0.970177i	$-0.577934\pi$
$601$	−7.30699e28	−0.484793	−0.242397	+	0.970177i	$0.577934\pi$
$602$	0	0
$603$	9.82566e27	0.0629550
$604$	0	0
$605$	8.14756e28	0.504192
$606$	0	0
$607$	−8.07544e28	−0.482709	−0.241354	−	0.970437i	$-0.577592\pi$
$607$	−8.07544e28	−0.482709	−0.241354	+	0.970437i	$0.577592\pi$
$608$	0	0
$609$	−1.03262e29	−0.596294
$610$	0	0
$611$	−3.16496e29	−1.76577
$612$	0	0
$613$	−2.21326e29	−1.19316	−0.596578	−	0.802555i	$-0.703473\pi$
$613$	−2.21326e29	−1.19316	−0.596578	+	0.802555i	$0.703473\pi$
$614$	0	0
$615$	1.82498e29	0.950759
$616$	0	0
$617$	−2.16026e29	−1.08771	−0.543854	−	0.839180i	$-0.683035\pi$
$617$	−2.16026e29	−1.08771	−0.543854	+	0.839180i	$0.683035\pi$
$618$	0	0
$619$	2.09908e29	1.02159	0.510795	−	0.859702i	$-0.329351\pi$
$619$	2.09908e29	1.02159	0.510795	+	0.859702i	$0.329351\pi$
$620$	0	0
$621$	9.24299e27	0.0434861
$622$	0	0
$623$	−1.48280e29	−0.674465
$624$	0	0
$625$	−2.15300e29	−0.946901
$626$	0	0
$627$	−1.47873e29	−0.628899
$628$	0	0
$629$	1.48514e29	0.610853
$630$	0	0
$631$	−4.81094e28	−0.191391	−0.0956955	−	0.995411i	$-0.530508\pi$
$631$	−4.81094e28	−0.191391	−0.0956955	+	0.995411i	$0.530508\pi$
$632$	0	0
$633$	8.01576e28	0.308465
$634$	0	0
$635$	1.17832e29	0.438670
$636$	0	0
$637$	−1.17441e29	−0.423014
$638$	0	0
$639$	−3.49474e28	−0.121802
$640$	0	0
$641$	−5.04867e28	−0.170282	−0.0851408	−	0.996369i	$-0.527134\pi$
$641$	−5.04867e28	−0.170282	−0.0851408	+	0.996369i	$0.527134\pi$
$642$	0	0
$643$	3.93930e29	1.28589	0.642945	−	0.765912i	$-0.277712\pi$
$643$	3.93930e29	1.28589	0.642945	+	0.765912i	$0.277712\pi$
$644$	0	0
$645$	−1.56731e29	−0.495196
$646$	0	0
$647$	1.38475e29	0.423523	0.211762	−	0.977321i	$-0.432080\pi$
$647$	1.38475e29	0.423523	0.211762	+	0.977321i	$0.432080\pi$
$648$	0	0
$649$	9.67842e28	0.286573
$650$	0	0
$651$	−3.12193e29	−0.894998
$652$	0	0
$653$	1.16655e29	0.323830	0.161915	−	0.986805i	$-0.448233\pi$
$653$	1.16655e29	0.323830	0.161915	+	0.986805i	$0.448233\pi$
$654$	0	0
$655$	−5.41792e29	−1.45646
$656$	0	0
$657$	−6.02303e28	−0.156812
$658$	0	0
$659$	3.36819e29	0.849377	0.424688	−	0.905340i	$-0.360384\pi$
$659$	3.36819e29	0.849377	0.424688	+	0.905340i	$0.360384\pi$
$660$	0	0
$661$	4.31449e29	1.05394	0.526968	−	0.849885i	$-0.323329\pi$
$661$	4.31449e29	1.05394	0.526968	+	0.849885i	$0.323329\pi$
$662$	0	0
$663$	3.64506e29	0.862606
$664$	0	0
$665$	−5.63598e29	−1.29224
$666$	0	0
$667$	1.24204e29	0.275940
$668$	0	0
$669$	2.79415e29	0.601555
$670$	0	0
$671$	2.25219e29	0.469915
$672$	0	0
$673$	−1.19710e29	−0.242088	−0.121044	−	0.992647i	$-0.538624\pi$
$673$	−1.19710e29	−0.242088	−0.121044	+	0.992647i	$0.538624\pi$
$674$	0	0
$675$	4.96222e27	0.00972717
$676$	0	0
$677$	7.71503e29	1.46608	0.733038	−	0.680188i	$-0.238102\pi$
$677$	7.71503e29	1.46608	0.733038	+	0.680188i	$0.238102\pi$
$678$	0	0
$679$	3.74243e29	0.689479
$680$	0	0
$681$	4.72525e29	0.844074
$682$	0	0
$683$	−6.40488e29	−1.10941	−0.554707	−	0.832046i	$-0.687170\pi$
$683$	−6.40488e29	−1.10941	−0.554707	+	0.832046i	$0.687170\pi$
$684$	0	0
$685$	1.02577e30	1.72305
$686$	0	0
$687$	−1.60679e29	−0.261766
$688$	0	0
$689$	−3.52277e29	−0.556650
$690$	0	0
$691$	−1.07309e30	−1.64482	−0.822408	−	0.568898i	$-0.807370\pi$
$691$	−1.07309e30	−1.64482	−0.822408	+	0.568898i	$0.807370\pi$
$692$	0	0
$693$	1.31702e29	0.195837
$694$	0	0
$695$	−4.64263e29	−0.669767
$696$	0	0
$697$	−1.21413e30	−1.69950
$698$	0	0
$699$	3.89791e29	0.529446
$700$	0	0
$701$	−7.18115e29	−0.946574	−0.473287	−	0.880908i	$-0.656933\pi$
$701$	−7.18115e29	−0.946574	−0.473287	+	0.880908i	$0.656933\pi$
$702$	0	0
$703$	7.44568e29	0.952518
$704$	0	0
$705$	−5.38459e29	−0.668601
$706$	0	0
$707$	1.05070e30	1.26641
$708$	0	0
$709$	4.24360e29	0.496534	0.248267	−	0.968692i	$-0.420139\pi$
$709$	4.24360e29	0.496534	0.248267	+	0.968692i	$0.420139\pi$
$710$	0	0
$711$	1.14463e29	0.130027
$712$	0	0
$713$	3.75505e29	0.414167
$714$	0	0
$715$	9.39014e29	1.00568
$716$	0	0
$717$	5.85898e29	0.609356
$718$	0	0
$719$	5.73272e29	0.579038	0.289519	−	0.957172i	$-0.406505\pi$
$719$	5.73272e29	0.579038	0.289519	+	0.957172i	$0.406505\pi$
$720$	0	0
$721$	7.45334e29	0.731191
$722$	0	0
$723$	9.10400e29	0.867521
$724$	0	0
$725$	6.66805e28	0.0617234
$726$	0	0
$727$	−1.42987e30	−1.28583	−0.642916	−	0.765937i	$-0.722275\pi$
$727$	−1.42987e30	−1.28583	−0.642916	+	0.765937i	$0.722275\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.04270e30	0.885175
$732$	0	0
$733$	1.99143e30	1.64276	0.821380	−	0.570381i	$-0.193205\pi$
$733$	1.99143e30	1.64276	0.821380	+	0.570381i	$0.193205\pi$
$734$	0	0
$735$	−1.99804e29	−0.160172
$736$	0	0
$737$	−1.68398e29	−0.131198
$738$	0	0
$739$	2.11124e30	1.59872	0.799359	−	0.600854i	$-0.205173\pi$
$739$	2.11124e30	1.59872	0.799359	+	0.600854i	$0.205173\pi$
$740$	0	0
$741$	1.82743e30	1.34508
$742$	0	0
$743$	−1.11538e30	−0.798067	−0.399033	−	0.916936i	$-0.630654\pi$
$743$	−1.11538e30	−0.798067	−0.399033	+	0.916936i	$0.630654\pi$
$744$	0	0
$745$	−1.03013e30	−0.716561
$746$	0	0
$747$	−1.06431e27	−0.000719781 0
$748$	0	0
$749$	−5.49375e29	−0.361251
$750$	0	0
$751$	2.28371e30	1.46023	0.730116	−	0.683323i	$-0.239466\pi$
$751$	2.28371e30	1.46023	0.730116	+	0.683323i	$0.239466\pi$
$752$	0	0
$753$	−5.90546e29	−0.367203
$754$	0	0
$755$	−2.61603e30	−1.58198
$756$	0	0
$757$	9.73411e29	0.572519	0.286259	−	0.958152i	$-0.407588\pi$
$757$	9.73411e29	0.572519	0.286259	+	0.958152i	$0.407588\pi$
$758$	0	0
$759$	−1.58412e29	−0.0906251
$760$	0	0
$761$	−3.06985e30	−1.70836	−0.854179	−	0.519979i	$-0.825940\pi$
$761$	−3.06985e30	−1.70836	−0.854179	+	0.519979i	$0.825940\pi$
$762$	0	0
$763$	−2.60428e29	−0.140987
$764$	0	0
$765$	6.20140e29	0.326621
$766$	0	0
$767$	−1.19607e30	−0.612920
$768$	0	0
$769$	−7.51816e29	−0.374874	−0.187437	−	0.982277i	$-0.560018\pi$
$769$	−7.51816e29	−0.374874	−0.187437	+	0.982277i	$0.560018\pi$
$770$	0	0
$771$	1.81042e30	0.878432
$772$	0	0
$773$	1.23771e30	0.584434	0.292217	−	0.956352i	$-0.405607\pi$
$773$	1.23771e30	0.584434	0.292217	+	0.956352i	$0.405607\pi$
$774$	0	0
$775$	2.01595e29	0.0926427
$776$	0	0
$777$	−6.63144e29	−0.296611
$778$	0	0
$779$	−6.08698e30	−2.65008
$780$	0	0
$781$	5.98949e29	0.253836
$782$	0	0
$783$	5.69637e29	0.235017
$784$	0	0
$785$	−1.25501e30	−0.504100
$786$	0	0
$787$	−2.14352e30	−0.838288	−0.419144	−	0.907920i	$-0.637670\pi$
$787$	−2.14352e30	−0.838288	−0.419144	+	0.907920i	$0.637670\pi$
$788$	0	0
$789$	−1.37853e30	−0.524936
$790$	0	0
$791$	−1.55454e30	−0.576434
$792$	0	0
$793$	−2.78327e30	−1.00505
$794$	0	0
$795$	−5.99334e29	−0.210773
$796$	0	0
$797$	3.01280e30	1.03195	0.515974	−	0.856605i	$-0.327430\pi$
$797$	3.01280e30	1.03195	0.515974	+	0.856605i	$0.327430\pi$
$798$	0	0
$799$	3.58228e30	1.19514
$800$	0	0
$801$	8.17975e29	0.265827
$802$	0	0
$803$	1.03226e30	0.326796
$804$	0	0
$805$	−6.03763e29	−0.186213
$806$	0	0
$807$	−1.35657e30	−0.407634
$808$	0	0
$809$	−5.76883e30	−1.68900	−0.844498	−	0.535559i	$-0.820101\pi$
$809$	−5.76883e30	−1.68900	−0.844498	+	0.535559i	$0.820101\pi$
$810$	0	0
$811$	−3.98728e30	−1.13752	−0.568759	−	0.822504i	$-0.692576\pi$
$811$	−3.98728e30	−1.13752	−0.568759	+	0.822504i	$0.692576\pi$
$812$	0	0
$813$	−2.37427e30	−0.660055
$814$	0	0
$815$	−6.77146e29	−0.183454
$816$	0	0
$817$	5.22754e30	1.38027
$818$	0	0
$819$	−1.62759e30	−0.418855
$820$	0	0
$821$	−4.09615e30	−1.02748	−0.513739	−	0.857946i	$-0.671740\pi$
$821$	−4.09615e30	−1.02748	−0.513739	+	0.857946i	$0.671740\pi$
$822$	0	0
$823$	−1.43384e30	−0.350592	−0.175296	−	0.984516i	$-0.556088\pi$
$823$	−1.43384e30	−0.350592	−0.175296	+	0.984516i	$0.556088\pi$
$824$	0	0
$825$	−8.50453e28	−0.0202714
$826$	0	0
$827$	3.13142e30	0.727668	0.363834	−	0.931464i	$-0.381468\pi$
$827$	3.13142e30	0.727668	0.363834	+	0.931464i	$0.381468\pi$
$828$	0	0
$829$	−4.44329e30	−1.00666	−0.503329	−	0.864095i	$-0.667892\pi$
$829$	−4.44329e30	−1.00666	−0.503329	+	0.864095i	$0.667892\pi$
$830$	0	0
$831$	2.64966e30	0.585301
$832$	0	0
$833$	1.32926e30	0.286311
$834$	0	0
$835$	−7.71998e29	−0.162147
$836$	0	0
$837$	1.72218e30	0.352745
$838$	0	0
$839$	9.30057e29	0.185784	0.0928922	−	0.995676i	$-0.470389\pi$
$839$	9.30057e29	0.185784	0.0928922	+	0.995676i	$0.470389\pi$
$840$	0	0
$841$	2.52173e30	0.491293
$842$	0	0
$843$	3.58126e30	0.680528
$844$	0	0
$845$	−6.34746e30	−1.17653
$846$	0	0
$847$	2.42033e30	0.437621
$848$	0	0
$849$	6.15462e30	1.08560
$850$	0	0
$851$	7.97630e29	0.137259
$852$	0	0
$853$	−2.27471e30	−0.381911	−0.190955	−	0.981599i	$-0.561159\pi$
$853$	−2.27471e30	−0.381911	−0.190955	+	0.981599i	$0.561159\pi$
$854$	0	0
$855$	3.10903e30	0.509309
$856$	0	0
$857$	3.05003e30	0.487534	0.243767	−	0.969834i	$-0.421617\pi$
$857$	3.05003e30	0.487534	0.243767	+	0.969834i	$0.421617\pi$
$858$	0	0
$859$	−1.34073e30	−0.209128	−0.104564	−	0.994518i	$-0.533345\pi$
$859$	−1.34073e30	−0.209128	−0.104564	+	0.994518i	$0.533345\pi$
$860$	0	0
$861$	5.42133e30	0.825225
$862$	0	0
$863$	−1.14916e31	−1.70713	−0.853566	−	0.520985i	$-0.825565\pi$
$863$	−1.14916e31	−1.70713	−0.853566	+	0.520985i	$0.825565\pi$
$864$	0	0
$865$	−6.84096e30	−0.991855
$866$	0	0
$867$	−4.58799e28	−0.00649265
$868$	0	0
$869$	−1.96173e30	−0.270976
$870$	0	0
$871$	2.08107e30	0.280606
$872$	0	0
$873$	−2.06448e30	−0.271744
$874$	0	0
$875$	−6.73714e30	−0.865747
$876$	0	0
$877$	9.06320e30	1.13707	0.568533	−	0.822660i	$-0.307511\pi$
$877$	9.06320e30	1.13707	0.568533	+	0.822660i	$0.307511\pi$
$878$	0	0
$879$	−3.68840e30	−0.451809
$880$	0	0
$881$	−6.50767e30	−0.778356	−0.389178	−	0.921163i	$-0.627241\pi$
$881$	−6.50767e30	−0.778356	−0.389178	+	0.921163i	$0.627241\pi$
$882$	0	0
$883$	−5.90742e30	−0.689939	−0.344969	−	0.938614i	$-0.612111\pi$
$883$	−5.90742e30	−0.689939	−0.344969	+	0.938614i	$0.612111\pi$
$884$	0	0
$885$	−2.03489e30	−0.232079
$886$	0	0
$887$	3.42312e30	0.381263	0.190632	−	0.981662i	$-0.438946\pi$
$887$	3.42312e30	0.381263	0.190632	+	0.981662i	$0.438946\pi$
$888$	0	0
$889$	3.50032e30	0.380750
$890$	0	0
$891$	−7.26524e29	−0.0771852
$892$	0	0
$893$	1.79595e31	1.86361
$894$	0	0
$895$	−3.83886e30	−0.389099
$896$	0	0
$897$	1.95766e30	0.193828
$898$	0	0
$899$	2.31420e31	2.23833
$900$	0	0
$901$	3.98728e30	0.376761
$902$	0	0
$903$	−4.65587e30	−0.429813
$904$	0	0
$905$	1.35463e29	0.0122183
$906$	0	0
$907$	1.16251e31	1.02452	0.512262	−	0.858829i	$-0.328808\pi$
$907$	1.16251e31	1.02452	0.512262	+	0.858829i	$0.328808\pi$
$908$	0	0
$909$	−5.79606e30	−0.499128
$910$	0	0
$911$	1.34510e31	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$911$	1.34510e31	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$912$	0	0
$913$	1.82407e28	0.00150002
$914$	0	0
$915$	−4.73523e30	−0.380557
$916$	0	0
$917$	−1.60945e31	−1.26416
$918$	0	0
$919$	8.55562e30	0.656808	0.328404	−	0.944537i	$-0.393489\pi$
$919$	8.55562e30	0.656808	0.328404	+	0.944537i	$0.393489\pi$
$920$	0	0
$921$	4.21853e30	0.316544
$922$	0	0
$923$	−7.40185e30	−0.542903
$924$	0	0
$925$	4.28218e29	0.0307027
$926$	0	0
$927$	−4.11157e30	−0.288184
$928$	0	0
$929$	9.93712e30	0.680919	0.340460	−	0.940259i	$-0.389417\pi$
$929$	9.93712e30	0.680919	0.340460	+	0.940259i	$0.389417\pi$
$930$	0	0
$931$	6.66418e30	0.446452
$932$	0	0
$933$	−3.55425e30	−0.232804
$934$	0	0
$935$	−1.06283e31	−0.680679
$936$	0	0
$937$	−2.14272e30	−0.134183	−0.0670917	−	0.997747i	$-0.521372\pi$
$937$	−2.14272e30	−0.134183	−0.0670917	+	0.997747i	$0.521372\pi$
$938$	0	0
$939$	9.98217e30	0.611275
$940$	0	0
$941$	1.69148e31	1.01292	0.506459	−	0.862264i	$-0.330954\pi$
$941$	1.69148e31	1.01292	0.506459	+	0.862264i	$0.330954\pi$
$942$	0	0
$943$	−6.52077e30	−0.381879
$944$	0	0
$945$	−2.76904e30	−0.158597
$946$	0	0
$947$	1.96530e31	1.10092	0.550459	−	0.834862i	$-0.314453\pi$
$947$	1.96530e31	1.10092	0.550459	+	0.834862i	$0.314453\pi$
$948$	0	0
$949$	−1.27568e31	−0.698948
$950$	0	0
$951$	3.37578e30	0.180917
$952$	0	0
$953$	2.39001e31	1.25292	0.626462	−	0.779452i	$-0.284502\pi$
$953$	2.39001e31	1.25292	0.626462	+	0.779452i	$0.284502\pi$
$954$	0	0
$955$	1.74026e31	0.892437
$956$	0	0
$957$	−9.76276e30	−0.489776
$958$	0	0
$959$	3.04716e31	1.49555
$960$	0	0
$961$	4.91397e31	2.35959
$962$	0	0
$963$	3.03057e30	0.142380
$964$	0	0
$965$	3.83132e30	0.176121
$966$	0	0
$967$	−1.63072e31	−0.733504	−0.366752	−	0.930319i	$-0.619530\pi$
$967$	−1.63072e31	−0.733504	−0.366752	+	0.930319i	$0.619530\pi$
$968$	0	0
$969$	−2.06839e31	−0.910401
$970$	0	0
$971$	−3.09343e31	−1.33241	−0.666207	−	0.745767i	$-0.732083\pi$
$971$	−3.09343e31	−1.33241	−0.666207	+	0.745767i	$0.732083\pi$
$972$	0	0
$973$	−1.37915e31	−0.581334
$974$	0	0
$975$	1.05100e30	0.0433563
$976$	0	0
$977$	1.98044e31	0.799592	0.399796	−	0.916604i	$-0.369081\pi$
$977$	1.98044e31	0.799592	0.399796	+	0.916604i	$0.369081\pi$
$978$	0	0
$979$	−1.40189e31	−0.553982
$980$	0	0
$981$	1.43663e30	0.0555673
$982$	0	0
$983$	4.34284e30	0.164423	0.0822114	−	0.996615i	$-0.473802\pi$
$983$	4.34284e30	0.164423	0.0822114	+	0.996615i	$0.473802\pi$
$984$	0	0
$985$	−3.51541e31	−1.30285
$986$	0	0
$987$	−1.59955e31	−0.580322
$988$	0	0
$989$	5.60008e30	0.198899
$990$	0	0
$991$	−4.70795e30	−0.163704	−0.0818518	−	0.996645i	$-0.526083\pi$
$991$	−4.70795e30	−0.163704	−0.0818518	+	0.996645i	$0.526083\pi$
$992$	0	0
$993$	3.94422e30	0.134275
$994$	0	0
$995$	−3.24561e31	−1.08182
$996$	0	0
$997$	−1.58934e31	−0.518700	−0.259350	−	0.965783i	$-0.583508\pi$
$997$	−1.58934e31	−0.518700	−0.259350	+	0.965783i	$0.583508\pi$
$998$	0	0
$999$	3.65817e30	0.116903

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.g.1.1		2
4.3	odd	2		3.22.a.c.1.1	✓	2
12.11	even	2		9.22.a.e.1.2		2
20.3	even	4		75.22.b.d.49.3		4
20.7	even	4		75.22.b.d.49.2		4
20.19	odd	2		75.22.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.c.1.1	✓	2	4.3	odd	2
9.22.a.e.1.2		2	12.11	even	2
48.22.a.g.1.1		2	1.1	even	1	trivial
75.22.a.d.1.2		2	20.19	odd	2
75.22.b.d.49.2		4	20.7	even	4
75.22.b.d.49.3		4	20.3	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q-59049.0 q^{3} -2.12776e7 q^{5} -6.32076e8 q^{7} +3.48678e9 q^{9} -5.97585e10 q^{11} +7.38499e11 q^{13} +1.25642e12 q^{15} -8.35876e12 q^{17} -4.19061e13 q^{19} +3.73234e13 q^{21} -4.48926e13 q^{23} -2.41012e13 q^{25} -2.05891e14 q^{27} -2.76669e15 q^{29} -8.36452e15 q^{31} +3.52868e15 q^{33} +1.34490e16 q^{35} -1.77675e16 q^{37} -4.36077e16 q^{39} +1.45253e17 q^{41} -1.24744e17 q^{43} -7.41904e16 q^{45} -4.28566e17 q^{47} -1.59026e17 q^{49} +4.93577e17 q^{51} -4.77017e17 q^{53} +1.27152e18 q^{55} +2.47452e18 q^{57} -1.61959e18 q^{59} -3.76882e18 q^{61} -2.20391e18 q^{63} -1.57135e19 q^{65} +2.81797e18 q^{67} +2.65086e18 q^{69} -1.00228e19 q^{71} -1.72739e19 q^{73} +1.42315e18 q^{75} +3.77719e19 q^{77} +3.28276e19 q^{79} +1.21577e19 q^{81} -3.05240e17 q^{83} +1.77854e20 q^{85} +1.63370e20 q^{87} +2.34593e20 q^{89} -4.66787e20 q^{91} +4.93917e20 q^{93} +8.91662e20 q^{95} -5.92086e20 q^{97} -2.08365e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10}) \) 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9	\( 2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100}) \) 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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