LMFDB - Embedded newform 288.8.f.a.143.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(143,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("288.143");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.4
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-412.177 q^{5} -359.164i q^{7} +O(q^{10})$	$q-412.177 q^{5} -359.164i q^{7} -3819.02i q^{11} -14141.7i q^{13} +3331.48i q^{17} -34327.6 q^{19} +69918.3 q^{23} +91764.7 q^{25} +186332. q^{29} -189759. i q^{31} +148039. i q^{35} -190024. i q^{37} -159461. i q^{41} -757231. q^{43} +235539. q^{47} +694544. q^{49} -732030. q^{53} +1.57411e6i q^{55} -1.98198e6i q^{59} +1.37932e6i q^{61} +5.82890e6i q^{65} -2.28147e6 q^{67} -5.12011e6 q^{71} -521725. q^{73} -1.37165e6 q^{77} -3.85831e6i q^{79} -4.75271e6i q^{83} -1.37316e6i q^{85} +7.08773e6i q^{89} -5.07920e6 q^{91} +1.41490e7 q^{95} -1.34983e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−412.177	−1.47465	−0.737324	−	0.675539i	$-0.763911\pi$
$5$	−412.177	−1.47465	−0.737324	+	0.675539i	$0.763911\pi$
$6$	0	0
$7$	− 359.164i	− 0.395776i	−0.980225	−	0.197888i	$-0.936592\pi$
$7$	− 359.164i	− 0.395776i	0.980225	−	0.197888i	$-0.0634082\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3819.02i	− 0.865123i	−0.901604	−	0.432561i	$-0.857610\pi$
$11$	− 3819.02i	− 0.865123i	0.901604	−	0.432561i	$-0.142390\pi$
$12$	0	0
$13$	− 14141.7i	− 1.78526i	−0.450791	−	0.892630i	$-0.648858\pi$
$13$	− 14141.7i	− 1.78526i	0.450791	−	0.892630i	$-0.351142\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3331.48i	0.164462i	0.996613	+	0.0822312i	$0.0262046\pi$
$17$	3331.48i	0.164462i	−0.996613	+	0.0822312i	$0.973795\pi$
$18$	0	0
$19$	−34327.6	−1.14817	−0.574084	−	0.818796i	$-0.694642\pi$
$19$	−34327.6	−1.14817	−0.574084	+	0.818796i	$0.694642\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	69918.3	1.19824	0.599120	−	0.800659i	$-0.295517\pi$
$23$	69918.3	1.19824	0.599120	+	0.800659i	$0.295517\pi$
$24$	0	0
$25$	91764.7	1.17459
$26$	0	0
$27$	0	0
$28$	0	0
$29$	186332.	1.41872	0.709358	−	0.704848i	$-0.248985\pi$
$29$	186332.	1.41872	0.709358	+	0.704848i	$0.248985\pi$
$30$	0	0
$31$	− 189759.i	− 1.14403i	−0.820245	−	0.572013i	$-0.806163\pi$
$31$	− 189759.i	− 1.14403i	0.820245	−	0.572013i	$-0.193837\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	148039.i	0.583630i
$36$	0	0
$37$	− 190024.i	− 0.616739i	−0.951267	−	0.308369i	$-0.900217\pi$
$37$	− 190024.i	− 0.616739i	0.951267	−	0.308369i	$-0.0997833\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 159461.i	− 0.361335i	−0.983544	−	0.180668i	$-0.942174\pi$
$41$	− 159461.i	− 0.361335i	0.983544	−	0.180668i	$-0.0578258\pi$
$42$	0	0
$43$	−757231.	−1.45241	−0.726205	−	0.687478i	$-0.758718\pi$
$43$	−757231.	−1.45241	−0.726205	+	0.687478i	$0.758718\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	235539.	0.330918	0.165459	−	0.986217i	$-0.447089\pi$
$47$	235539.	0.330918	0.165459	+	0.986217i	$0.447089\pi$
$48$	0	0
$49$	694544.	0.843361
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−732030.	−0.675404	−0.337702	−	0.941253i	$-0.609650\pi$
$53$	−732030.	−0.675404	−0.337702	+	0.941253i	$0.609650\pi$
$54$	0	0
$55$	1.57411e6i	1.27575i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.98198e6i	− 1.25637i	−0.778064	−	0.628185i	$-0.783798\pi$
$59$	− 1.98198e6i	− 1.25637i	0.778064	−	0.628185i	$-0.216202\pi$
$60$	0	0
$61$	1.37932e6i	0.778055i	0.921226	+	0.389028i	$0.127189\pi$
$61$	1.37932e6i	0.778055i	−0.921226	+	0.389028i	$0.872811\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.82890e6i	2.63263i
$66$	0	0
$67$	−2.28147e6	−0.926731	−0.463366	−	0.886167i	$-0.653358\pi$
$67$	−2.28147e6	−0.926731	−0.463366	+	0.886167i	$0.653358\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.12011e6	−1.69776	−0.848878	−	0.528589i	$-0.822721\pi$
$71$	−5.12011e6	−1.69776	−0.848878	+	0.528589i	$0.822721\pi$
$72$	0	0
$73$	−521725.	−0.156968	−0.0784841	−	0.996915i	$-0.525008\pi$
$73$	−521725.	−0.156968	−0.0784841	+	0.996915i	$0.525008\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.37165e6	−0.342395
$78$	0	0
$79$	− 3.85831e6i	− 0.880446i	−0.897888	−	0.440223i	$-0.854899\pi$
$79$	− 3.85831e6i	− 0.880446i	0.897888	−	0.440223i	$-0.145101\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.75271e6i	− 0.912363i	−0.889887	−	0.456182i	$-0.849217\pi$
$83$	− 4.75271e6i	− 0.912363i	0.889887	−	0.456182i	$-0.150783\pi$
$84$	0	0
$85$	− 1.37316e6i	− 0.242524i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.08773e6i	1.06572i	0.846204	+	0.532859i	$0.178882\pi$
$89$	7.08773e6i	1.06572i	−0.846204	+	0.532859i	$0.821118\pi$
$90$	0	0
$91$	−5.07920e6	−0.706563
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.41490e7	1.69314
$96$	0	0
$97$	−1.34983e7	−1.50169	−0.750844	−	0.660480i	$-0.770353\pi$
$97$	−1.34983e7	−1.50169	−0.750844	+	0.660480i	$0.770353\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.25843e7	1.21535	0.607677	−	0.794184i	$-0.292101\pi$
$101$	1.25843e7	1.21535	0.607677	+	0.794184i	$0.292101\pi$
$102$	0	0
$103$	8.44045e6i	0.761089i	0.924763	+	0.380544i	$0.124263\pi$
$103$	8.44045e6i	0.761089i	−0.924763	+	0.380544i	$0.875737\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.12718e7i	0.889505i	0.895654	+	0.444752i	$0.146708\pi$
$107$	1.12718e7i	0.889505i	−0.895654	+	0.444752i	$0.853292\pi$
$108$	0	0
$109$	− 7.83959e6i	− 0.579830i	−0.957052	−	0.289915i	$-0.906373\pi$
$109$	− 7.83959e6i	− 0.579830i	0.957052	−	0.289915i	$-0.0936270\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.34305e7i	0.875623i	0.899067	+	0.437811i	$0.144246\pi$
$113$	1.34305e7i	0.875623i	−0.899067	+	0.437811i	$0.855754\pi$
$114$	0	0
$115$	−2.88187e7	−1.76698
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.19655e6	0.0650903
$120$	0	0
$121$	4.90224e6	0.251562
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.62197e6	−0.257456
$126$	0	0
$127$	− 2.93022e7i	− 1.26937i	−0.772771	−	0.634684i	$-0.781130\pi$
$127$	− 2.93022e7i	− 1.26937i	0.772771	−	0.634684i	$-0.218870\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.27923e7i	1.66309i	0.555457	+	0.831546i	$0.312544\pi$
$131$	4.27923e7i	1.66309i	−0.555457	+	0.831546i	$0.687456\pi$
$132$	0	0
$133$	1.23292e7i	0.454417i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.57159e6i	− 0.118670i	−0.998238	−	0.0593348i	$-0.981102\pi$
$137$	− 3.57159e6i	− 0.118670i	0.998238	−	0.0593348i	$-0.0188979\pi$
$138$	0	0
$139$	−4.65242e7	−1.46936	−0.734678	−	0.678417i	$-0.762667\pi$
$139$	−4.65242e7	−1.46936	−0.734678	+	0.678417i	$0.762667\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.40076e7	−1.54447
$144$	0	0
$145$	−7.68019e7	−2.09211
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.43823e7	−0.851496	−0.425748	−	0.904842i	$-0.639989\pi$
$149$	−3.43823e7	−0.851496	−0.425748	+	0.904842i	$0.639989\pi$
$150$	0	0
$151$	7.33709e7i	1.73422i	0.498116	+	0.867110i	$0.334025\pi$
$151$	7.33709e7i	1.73422i	−0.498116	+	0.867110i	$0.665975\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.82141e7i	1.68704i
$156$	0	0
$157$	3.78154e7i	0.779867i	0.920843	+	0.389933i	$0.127502\pi$
$157$	3.78154e7i	0.779867i	−0.920843	+	0.389933i	$0.872498\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.51121e7i	− 0.474234i
$162$	0	0
$163$	1.97142e7	0.356552	0.178276	−	0.983981i	$-0.442948\pi$
$163$	1.97142e7	0.356552	0.178276	+	0.983981i	$0.442948\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.21858e7	1.36549	0.682746	−	0.730656i	$-0.260786\pi$
$167$	8.21858e7	1.36549	0.682746	+	0.730656i	$0.260786\pi$
$168$	0	0
$169$	−1.37240e8	−2.18715
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.64838e7	−0.388883	−0.194441	−	0.980914i	$-0.562289\pi$
$173$	−2.64838e7	−0.388883	−0.194441	+	0.980914i	$0.562289\pi$
$174$	0	0
$175$	− 3.29586e7i	− 0.464874i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.25036e7i	1.20552i	0.797924	+	0.602758i	$0.205932\pi$
$179$	9.25036e7i	1.20552i	−0.797924	+	0.602758i	$0.794068\pi$
$180$	0	0
$181$	− 7.91658e7i	− 0.992344i	−0.868224	−	0.496172i	$-0.834738\pi$
$181$	− 7.91658e7i	− 0.992344i	0.868224	−	0.496172i	$-0.165262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.83233e7i	0.909473i
$186$	0	0
$187$	1.27230e7	0.142280
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.02924e8	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$191$	−1.02924e8	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$192$	0	0
$193$	−5.98055e7	−0.598812	−0.299406	−	0.954126i	$-0.596789\pi$
$193$	−5.98055e7	−0.598812	−0.299406	+	0.954126i	$0.596789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.68356e8	1.56890	0.784452	−	0.620190i	$-0.212945\pi$
$197$	1.68356e8	1.56890	0.784452	+	0.620190i	$0.212945\pi$
$198$	0	0
$199$	− 9.24295e7i	− 0.831428i	−0.909495	−	0.415714i	$-0.863532\pi$
$199$	− 9.24295e7i	− 0.831428i	0.909495	−	0.415714i	$-0.136468\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.69239e7i	− 0.561494i
$204$	0	0
$205$	6.57260e7i	0.532842i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.31098e8i	0.993306i
$210$	0	0
$211$	−1.21026e8	−0.886928	−0.443464	−	0.896292i	$-0.646251\pi$
$211$	−1.21026e8	−0.886928	−0.443464	+	0.896292i	$0.646251\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.12113e8	2.14179
$216$	0	0
$217$	−6.81544e7	−0.452778
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.71130e7	0.293608
$222$	0	0
$223$	− 6.34758e7i	− 0.383302i	−0.981463	−	0.191651i	$-0.938616\pi$
$223$	− 6.34758e7i	− 0.383302i	0.981463	−	0.191651i	$-0.0613842\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.62893e8i	− 0.924296i	−0.886803	−	0.462148i	$-0.847079\pi$
$227$	− 1.62893e8i	− 0.924296i	0.886803	−	0.462148i	$-0.152921\pi$
$228$	0	0
$229$	− 2.85285e7i	− 0.156984i	−0.996915	−	0.0784918i	$-0.974990\pi$
$229$	− 2.85285e7i	− 0.156984i	0.996915	−	0.0784918i	$-0.0250105\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.67347e7i	− 0.449207i	−0.974450	−	0.224604i	$-0.927891\pi$
$233$	− 8.67347e7i	− 0.449207i	0.974450	−	0.224604i	$-0.0721088\pi$
$234$	0	0
$235$	−9.70838e7	−0.487988
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.35437e8	−1.58935	−0.794673	−	0.607038i	$-0.792358\pi$
$239$	−3.35437e8	−1.58935	−0.794673	+	0.607038i	$0.792358\pi$
$240$	0	0
$241$	4.29134e8	1.97484	0.987422	−	0.158107i	$-0.0505391\pi$
$241$	4.29134e8	1.97484	0.987422	+	0.158107i	$0.0505391\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.86275e8	−1.24366
$246$	0	0
$247$	4.85452e8i	2.04978i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.57080e8i	0.626992i	0.949590	+	0.313496i	$0.101500\pi$
$251$	1.57080e8i	0.626992i	−0.949590	+	0.313496i	$0.898500\pi$
$252$	0	0
$253$	− 2.67020e8i	− 1.03662i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.69735e7i	− 0.172618i	−0.996268	−	0.0863092i	$-0.972493\pi$
$257$	− 4.69735e7i	− 0.172618i	0.996268	−	0.0863092i	$-0.0275073\pi$
$258$	0	0
$259$	−6.82496e7	−0.244090
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.00330e8	−0.340082	−0.170041	−	0.985437i	$-0.554390\pi$
$263$	−1.00330e8	−0.340082	−0.170041	+	0.985437i	$0.554390\pi$
$264$	0	0
$265$	3.01726e8	0.995983
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.70778e8	−0.848164	−0.424082	−	0.905624i	$-0.639403\pi$
$269$	−2.70778e8	−0.848164	−0.424082	+	0.905624i	$0.639403\pi$
$270$	0	0
$271$	− 6.20621e8i	− 1.89424i	−0.320885	−	0.947118i	$-0.603980\pi$
$271$	− 6.20621e8i	− 1.89424i	0.320885	−	0.947118i	$-0.396020\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.50452e8i	− 1.01616i
$276$	0	0
$277$	5.93506e8i	1.67782i	0.544268	+	0.838911i	$0.316807\pi$
$277$	5.93506e8i	1.67782i	−0.544268	+	0.838911i	$0.683193\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.49386e8i	0.670503i	0.942129	+	0.335251i	$0.108821\pi$
$281$	2.49386e8i	0.670503i	−0.942129	+	0.335251i	$0.891179\pi$
$282$	0	0
$283$	2.41236e8	0.632688	0.316344	−	0.948645i	$-0.397545\pi$
$283$	2.41236e8	0.632688	0.316344	+	0.948645i	$0.397545\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.72725e7	−0.143008
$288$	0	0
$289$	3.99240e8	0.972952
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.41368e8	1.02509	0.512547	−	0.858659i	$-0.328702\pi$
$293$	4.41368e8	1.02509	0.512547	+	0.858659i	$0.328702\pi$
$294$	0	0
$295$	8.16927e8i	1.85271i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.88767e8i	− 2.13917i
$300$	0	0
$301$	2.71970e8i	0.574829i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.68523e8i	− 1.14736i
$306$	0	0
$307$	4.02987e8	0.794889	0.397445	−	0.917626i	$-0.369897\pi$
$307$	4.02987e8	0.794889	0.397445	+	0.917626i	$0.369897\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.04195e8	0.196420	0.0982099	−	0.995166i	$-0.468688\pi$
$311$	1.04195e8	0.196420	0.0982099	+	0.995166i	$0.468688\pi$
$312$	0	0
$313$	6.28900e8	1.15925	0.579624	−	0.814884i	$-0.303200\pi$
$313$	6.28900e8	1.15925	0.579624	+	0.814884i	$0.303200\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.51759e8	0.972841	0.486421	−	0.873725i	$-0.338302\pi$
$317$	5.51759e8	0.972841	0.486421	+	0.873725i	$0.338302\pi$
$318$	0	0
$319$	− 7.11608e8i	− 1.22736i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.14362e8i	− 0.188830i
$324$	0	0
$325$	− 1.29771e9i	− 2.09694i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.45972e7i	− 0.130969i
$330$	0	0
$331$	8.03548e8	1.21791	0.608953	−	0.793206i	$-0.291590\pi$
$331$	8.03548e8	1.21791	0.608953	+	0.793206i	$0.291590\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.40371e8	1.36660
$336$	0	0
$337$	−8.74822e8	−1.24513	−0.622565	−	0.782568i	$-0.713910\pi$
$337$	−8.74822e8	−1.24513	−0.622565	+	0.782568i	$0.713910\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.24693e8	−0.989723
$342$	0	0
$343$	− 5.45242e8i	− 0.729558i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.16428e8i	0.406557i	0.979121	+	0.203279i	$0.0651597\pi$
$347$	3.16428e8i	0.406557i	−0.979121	+	0.203279i	$0.934840\pi$
$348$	0	0
$349$	− 4.41239e8i	− 0.555629i	−0.960635	−	0.277814i	$-0.910390\pi$
$349$	− 4.41239e8i	− 0.555629i	0.960635	−	0.277814i	$-0.0896101\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.79064e8i	− 0.458670i	−0.973348	−	0.229335i	$-0.926345\pi$
$353$	− 3.79064e8i	− 0.458670i	0.973348	−	0.229335i	$-0.0736552\pi$
$354$	0	0
$355$	2.11039e9	2.50359
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.43347e8	−0.505723	−0.252862	−	0.967502i	$-0.581372\pi$
$359$	−4.43347e8	−0.505723	−0.252862	+	0.967502i	$0.581372\pi$
$360$	0	0
$361$	2.84510e8	0.318289
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.15043e8	0.231473
$366$	0	0
$367$	2.76323e8i	0.291800i	0.989299	+	0.145900i	$0.0466078\pi$
$367$	2.76323e8i	0.291800i	−0.989299	+	0.145900i	$0.953392\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.62919e8i	0.267309i
$372$	0	0
$373$	1.23995e9i	1.23715i	0.785726	+	0.618575i	$0.212290\pi$
$373$	1.23995e9i	1.23715i	−0.785726	+	0.618575i	$0.787710\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.63507e9i	− 2.53278i
$378$	0	0
$379$	5.00434e8	0.472183	0.236091	−	0.971731i	$-0.424134\pi$
$379$	5.00434e8	0.472183	0.236091	+	0.971731i	$0.424134\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.63975e8	−0.421987	−0.210993	−	0.977487i	$-0.567670\pi$
$383$	−4.63975e8	−0.421987	−0.210993	+	0.977487i	$0.567670\pi$
$384$	0	0
$385$	5.65364e8	0.504912
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.74898e8	0.236781	0.118391	−	0.992967i	$-0.462226\pi$
$389$	2.74898e8	0.236781	0.118391	+	0.992967i	$0.462226\pi$
$390$	0	0
$391$	2.32932e8i	0.197065i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.59031e9i	1.29835i
$396$	0	0
$397$	1.28191e9i	1.02823i	0.857721	+	0.514116i	$0.171880\pi$
$397$	1.28191e9i	1.02823i	−0.857721	+	0.514116i	$0.828120\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.19451e8i	0.169954i	0.996383	+	0.0849772i	$0.0270817\pi$
$401$	2.19451e8i	0.169954i	−0.996383	+	0.0849772i	$0.972918\pi$
$402$	0	0
$403$	−2.68352e9	−2.04238
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.25704e8	−0.533555
$408$	0	0
$409$	−1.87644e8	−0.135614	−0.0678068	−	0.997698i	$-0.521600\pi$
$409$	−1.87644e8	−0.135614	−0.0678068	+	0.997698i	$0.521600\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.11856e8	−0.497241
$414$	0	0
$415$	1.95896e9i	1.34542i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 7.92192e8i	− 0.526116i	−0.964780	−	0.263058i	$-0.915269\pi$
$419$	− 7.92192e8i	− 0.526116i	0.964780	−	0.263058i	$-0.0847310\pi$
$420$	0	0
$421$	5.56076e7i	0.0363201i	0.999835	+	0.0181601i	$0.00578084\pi$
$421$	5.56076e7i	0.0363201i	−0.999835	+	0.0181601i	$0.994219\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.05713e8i	0.193176i
$426$	0	0
$427$	4.95402e8	0.307936
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.21789e8	−0.494413	−0.247206	−	0.968963i	$-0.579513\pi$
$431$	−8.21789e8	−0.494413	−0.247206	+	0.968963i	$0.579513\pi$
$432$	0	0
$433$	2.34831e8	0.139011	0.0695054	−	0.997582i	$-0.477858\pi$
$433$	2.34831e8	0.139011	0.0695054	+	0.997582i	$0.477858\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.40013e9	−1.37578
$438$	0	0
$439$	− 2.08919e9i	− 1.17856i	−0.807928	−	0.589281i	$-0.799411\pi$
$439$	− 2.08919e9i	− 1.17856i	0.807928	−	0.589281i	$-0.200589\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.20790e8i	0.229960i	0.993368	+	0.114980i	$0.0366804\pi$
$443$	4.20790e8i	0.229960i	−0.993368	+	0.114980i	$0.963320\pi$
$444$	0	0
$445$	− 2.92140e9i	− 1.57156i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.20517e9i	0.628329i	0.949369	+	0.314164i	$0.101724\pi$
$449$	1.20517e9i	0.628329i	−0.949369	+	0.314164i	$0.898276\pi$
$450$	0	0
$451$	−6.08984e8	−0.312599
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.09353e9	1.04193
$456$	0	0
$457$	1.17565e8	0.0576198	0.0288099	−	0.999585i	$-0.490828\pi$
$457$	1.17565e8	0.0576198	0.0288099	+	0.999585i	$0.490828\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.15104e9	0.547190	0.273595	−	0.961845i	$-0.411787\pi$
$461$	1.15104e9	0.547190	0.273595	+	0.961845i	$0.411787\pi$
$462$	0	0
$463$	2.00264e9i	0.937712i	0.883275	+	0.468856i	$0.155334\pi$
$463$	2.00264e9i	0.937712i	−0.883275	+	0.468856i	$0.844666\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.12001e9i	0.508876i	0.967089	+	0.254438i	$0.0818905\pi$
$467$	1.12001e9i	0.508876i	−0.967089	+	0.254438i	$0.918110\pi$
$468$	0	0
$469$	8.19423e8i	0.366778i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.89188e9i	1.25651i
$474$	0	0
$475$	−3.15006e9	−1.34862
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.11855e9	0.465031	0.232515	−	0.972593i	$-0.425304\pi$
$479$	1.11855e9	0.465031	0.232515	+	0.972593i	$0.425304\pi$
$480$	0	0
$481$	−2.68726e9	−1.10104
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.56371e9	2.21446
$486$	0	0
$487$	− 3.33703e9i	− 1.30921i	−0.755972	−	0.654604i	$-0.772835\pi$
$487$	− 3.33703e9i	− 1.30921i	0.755972	−	0.654604i	$-0.227165\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.90001e8i	0.186815i	0.995628	+	0.0934075i	$0.0297759\pi$
$491$	4.90001e8i	0.186815i	−0.995628	+	0.0934075i	$0.970224\pi$
$492$	0	0
$493$	6.20764e8i	0.233325i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.83896e9i	0.671931i
$498$	0	0
$499$	−1.65302e9	−0.595563	−0.297781	−	0.954634i	$-0.596247\pi$
$499$	−1.65302e9	−0.595563	−0.297781	+	0.954634i	$0.596247\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.28708e7	0.00450938	0.00225469	−	0.999997i	$-0.499282\pi$
$503$	1.28708e7	0.00450938	0.00225469	+	0.999997i	$0.499282\pi$
$504$	0	0
$505$	−5.18694e9	−1.79222
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.13852e8	−0.172713	−0.0863567	−	0.996264i	$-0.527522\pi$
$509$	−5.13852e8	−0.172713	−0.0863567	+	0.996264i	$0.527522\pi$
$510$	0	0
$511$	1.87385e8i	0.0621243i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.47896e9i	− 1.12234i
$516$	0	0
$517$	− 8.99530e8i	− 0.286285i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.90297e9i	1.51889i	0.650570	+	0.759446i	$0.274530\pi$
$521$	4.90297e9i	1.51889i	−0.650570	+	0.759446i	$0.725470\pi$
$522$	0	0
$523$	−2.66052e8	−0.0813224	−0.0406612	−	0.999173i	$-0.512946\pi$
$523$	−2.66052e8	−0.0813224	−0.0406612	+	0.999173i	$0.512946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.32178e8	0.188149
$528$	0	0
$529$	1.48375e9	0.435778
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.25505e9	−0.645077
$534$	0	0
$535$	− 4.64596e9i	− 1.31171i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.65248e9i	− 0.729611i
$540$	0	0
$541$	2.26607e9i	0.615295i	0.951500	+	0.307647i	$0.0995417\pi$
$541$	2.26607e9i	0.615295i	−0.951500	+	0.307647i	$0.900458\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.23130e9i	0.855045i
$546$	0	0
$547$	−6.06164e9	−1.58356	−0.791780	−	0.610806i	$-0.790845\pi$
$547$	−6.06164e9	−1.58356	−0.791780	+	0.610806i	$0.790845\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.39634e9	−1.62892
$552$	0	0
$553$	−1.38577e9	−0.348459
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.14513e9	−0.525968	−0.262984	−	0.964800i	$-0.584707\pi$
$557$	−2.14513e9	−0.525968	−0.262984	+	0.964800i	$0.584707\pi$
$558$	0	0
$559$	1.07086e10i	2.59293i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.66627e9i	0.865855i	0.901429	+	0.432927i	$0.142519\pi$
$563$	3.66627e9i	0.865855i	−0.901429	+	0.432927i	$0.857481\pi$
$564$	0	0
$565$	− 5.53573e9i	− 1.29124i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.28882e9i	0.975989i	0.872847	+	0.487994i	$0.162271\pi$
$569$	4.28882e9i	0.975989i	−0.872847	+	0.487994i	$0.837729\pi$
$570$	0	0
$571$	−2.79269e9	−0.627763	−0.313881	−	0.949462i	$-0.601629\pi$
$571$	−2.79269e9	−0.627763	−0.313881	+	0.949462i	$0.601629\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.41604e9	1.40744
$576$	0	0
$577$	−1.42684e9	−0.309214	−0.154607	−	0.987976i	$-0.549411\pi$
$577$	−1.42684e9	−0.309214	−0.154607	+	0.987976i	$0.549411\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.70700e9	−0.361092
$582$	0	0
$583$	2.79564e9i	0.584307i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.26290e9i	1.68616i	0.537787	+	0.843081i	$0.319260\pi$
$587$	8.26290e9i	1.68616i	−0.537787	+	0.843081i	$0.680740\pi$
$588$	0	0
$589$	6.51395e9i	1.31353i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 4.44985e9i	− 0.876302i	−0.898901	−	0.438151i	$-0.855634\pi$
$593$	− 4.44985e9i	− 0.876302i	0.898901	−	0.438151i	$-0.144366\pi$
$594$	0	0
$595$	−4.93190e8	−0.0959852
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.97431e9	−1.13578	−0.567890	−	0.823105i	$-0.692240\pi$
$599$	−5.97431e9	−1.13578	−0.567890	+	0.823105i	$0.692240\pi$
$600$	0	0
$601$	5.81457e9	1.09259	0.546295	−	0.837593i	$-0.316038\pi$
$601$	5.81457e9	1.09259	0.546295	+	0.837593i	$0.316038\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.02059e9	−0.370966
$606$	0	0
$607$	− 3.54687e9i	− 0.643703i	−0.946790	−	0.321851i	$-0.895695\pi$
$607$	− 3.54687e9i	− 0.643703i	0.946790	−	0.321851i	$-0.104305\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.33094e9i	− 0.590775i
$612$	0	0
$613$	8.92550e9i	1.56502i	0.622635	+	0.782512i	$0.286062\pi$
$613$	8.92550e9i	1.56502i	−0.622635	+	0.782512i	$0.713938\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.93844e9i	− 1.18923i	−0.804012	−	0.594613i	$-0.797305\pi$
$617$	− 6.93844e9i	− 1.18923i	0.804012	−	0.594613i	$-0.202695\pi$
$618$	0	0
$619$	1.41395e9	0.239617	0.119808	−	0.992797i	$-0.461772\pi$
$619$	1.41395e9	0.239617	0.119808	+	0.992797i	$0.461772\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.54566e9	0.421786
$624$	0	0
$625$	−4.85187e9	−0.794931
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.33060e8	0.101430
$630$	0	0
$631$	7.00062e8i	0.110926i	0.998461	+	0.0554631i	$0.0176635\pi$
$631$	7.00062e8i	0.110926i	−0.998461	+	0.0554631i	$0.982336\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.20777e10i	1.87187i
$636$	0	0
$637$	− 9.82207e9i	− 1.50562i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.19021e10i	− 1.78493i	−0.451117	−	0.892465i	$-0.648974\pi$
$641$	− 1.19021e10i	− 1.78493i	0.451117	−	0.892465i	$-0.351026\pi$
$642$	0	0
$643$	2.57955e9	0.382653	0.191326	−	0.981526i	$-0.438721\pi$
$643$	2.57955e9	0.382653	0.191326	+	0.981526i	$0.438721\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.01688e9	−0.728230	−0.364115	−	0.931354i	$-0.618629\pi$
$647$	−5.01688e9	−0.728230	−0.364115	+	0.931354i	$0.618629\pi$
$648$	0	0
$649$	−7.56924e9	−1.08692
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.90200e9	1.39164	0.695820	−	0.718216i	$-0.255041\pi$
$653$	9.90200e9	1.39164	0.695820	+	0.718216i	$0.255041\pi$
$654$	0	0
$655$	− 1.76380e10i	− 2.45247i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.56641e9i	− 0.485437i	−0.970097	−	0.242719i	$-0.921961\pi$
$659$	− 3.56641e9i	− 0.485437i	0.970097	−	0.242719i	$-0.0780391\pi$
$660$	0	0
$661$	− 1.03477e10i	− 1.39360i	−0.717265	−	0.696800i	$-0.754606\pi$
$661$	− 1.03477e10i	− 1.39360i	0.717265	−	0.696800i	$-0.245394\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.08182e9i	− 0.670106i
$666$	0	0
$667$	1.30281e10	1.69996
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.26765e9	0.673113
$672$	0	0
$673$	9.52836e9	1.20494	0.602470	−	0.798142i	$-0.294183\pi$
$673$	9.52836e9	1.20494	0.602470	+	0.798142i	$0.294183\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.90058e8	−0.0235411	−0.0117705	−	0.999931i	$-0.503747\pi$
$677$	−1.90058e8	−0.0235411	−0.0117705	+	0.999931i	$0.503747\pi$
$678$	0	0
$679$	4.84812e9i	0.594332i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.28062e9i	− 0.393988i	−0.980405	−	0.196994i	$-0.936882\pi$
$683$	− 3.28062e9i	− 0.393988i	0.980405	−	0.196994i	$-0.0631180\pi$
$684$	0	0
$685$	1.47213e9i	0.174996i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.03522e10i	1.20577i
$690$	0	0
$691$	−3.49476e9	−0.402944	−0.201472	−	0.979494i	$-0.564572\pi$
$691$	−3.49476e9	−0.402944	−0.201472	+	0.979494i	$0.564572\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.91762e10	2.16678
$696$	0	0
$697$	5.31241e8	0.0594260
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.22907e10	−1.34761	−0.673805	−	0.738909i	$-0.735341\pi$
$701$	−1.22907e10	−1.34761	−0.673805	+	0.738909i	$0.735341\pi$
$702$	0	0
$703$	6.52304e9i	0.708120i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.51981e9i	− 0.481008i
$708$	0	0
$709$	8.80574e9i	0.927906i	0.885860	+	0.463953i	$0.153569\pi$
$709$	8.80574e9i	0.927906i	−0.885860	+	0.463953i	$0.846431\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.32676e10i	− 1.37082i
$714$	0	0
$715$	2.22607e10	2.27755
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.43721e9	−0.645872	−0.322936	−	0.946421i	$-0.604670\pi$
$719$	−6.43721e9	−0.645872	−0.322936	+	0.946421i	$0.604670\pi$
$720$	0	0
$721$	3.03150e9	0.301221
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.70987e10	1.66641
$726$	0	0
$727$	1.15462e10i	1.11447i	0.830354	+	0.557236i	$0.188138\pi$
$727$	1.15462e10i	1.11447i	−0.830354	+	0.557236i	$0.811862\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.52270e9i	− 0.238867i
$732$	0	0
$733$	− 4.00798e9i	− 0.375890i	−0.982180	−	0.187945i	$-0.939817\pi$
$733$	− 4.00798e9i	− 0.375890i	0.982180	−	0.187945i	$-0.0601828\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.71300e9i	0.801736i
$738$	0	0
$739$	1.11260e10	1.01411	0.507055	−	0.861914i	$-0.330734\pi$
$739$	1.11260e10	1.01411	0.507055	+	0.861914i	$0.330734\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.41895e9	−0.305796	−0.152898	−	0.988242i	$-0.548861\pi$
$743$	−3.41895e9	−0.305796	−0.152898	+	0.988242i	$0.548861\pi$
$744$	0	0
$745$	1.41716e10	1.25566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.04841e9	0.352045
$750$	0	0
$751$	− 2.35287e9i	− 0.202702i	−0.994851	−	0.101351i	$-0.967683\pi$
$751$	− 2.35287e9i	− 0.202702i	0.994851	−	0.101351i	$-0.0323165\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 3.02418e10i	− 2.55737i
$756$	0	0
$757$	− 2.65364e9i	− 0.222334i	−0.993802	−	0.111167i	$-0.964541\pi$
$757$	− 2.65364e9i	− 0.222334i	0.993802	−	0.111167i	$-0.0354589\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 4.75292e9i	− 0.390944i	−0.980709	−	0.195472i	$-0.937376\pi$
$761$	− 4.75292e9i	− 0.390944i	0.980709	−	0.195472i	$-0.0626238\pi$
$762$	0	0
$763$	−2.81570e9	−0.229483
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.80287e10	−2.24295
$768$	0	0
$769$	−1.27901e10	−1.01422	−0.507108	−	0.861883i	$-0.669285\pi$
$769$	−1.27901e10	−1.01422	−0.507108	+	0.861883i	$0.669285\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.92520e10	−1.49916	−0.749579	−	0.661915i	$-0.769744\pi$
$773$	−1.92520e10	−1.49916	−0.749579	+	0.661915i	$0.769744\pi$
$774$	0	0
$775$	− 1.74131e10i	− 1.34376i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.47390e9i	0.414873i
$780$	0	0
$781$	1.95538e10i	1.46877i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.55866e10i	− 1.15003i
$786$	0	0
$787$	7.18441e9	0.525388	0.262694	−	0.964879i	$-0.415389\pi$
$787$	7.18441e9	0.525388	0.262694	+	0.964879i	$0.415389\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.82374e9	0.346550
$792$	0	0
$793$	1.95060e10	1.38903
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.37227e10	−1.65981	−0.829907	−	0.557901i	$-0.811607\pi$
$797$	−2.37227e10	−1.65981	−0.829907	+	0.557901i	$0.811607\pi$
$798$	0	0
$799$	7.84695e8i	0.0544236i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.99248e9i	0.135797i
$804$	0	0
$805$	1.03506e10i	0.699329i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.26128e9i	0.349359i	0.984625	+	0.174679i	$0.0558889\pi$
$809$	5.26128e9i	0.349359i	−0.984625	+	0.174679i	$0.944111\pi$
$810$	0	0
$811$	−6.25968e9	−0.412078	−0.206039	−	0.978544i	$-0.566057\pi$
$811$	−6.25968e9	−0.412078	−0.206039	+	0.978544i	$0.566057\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.12574e9	−0.525789
$816$	0	0
$817$	2.59939e10	1.66761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.64242e9	−0.292781	−0.146391	−	0.989227i	$-0.546766\pi$
$821$	−4.64242e9	−0.292781	−0.146391	+	0.989227i	$0.546766\pi$
$822$	0	0
$823$	1.47371e10i	0.921540i	0.887520	+	0.460770i	$0.152427\pi$
$823$	1.47371e10i	0.921540i	−0.887520	+	0.460770i	$0.847573\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.88827e9i	− 0.362008i	−0.983482	−	0.181004i	$-0.942065\pi$
$827$	− 5.88827e9i	− 0.362008i	0.983482	−	0.181004i	$-0.0579348\pi$
$828$	0	0
$829$	− 3.00855e10i	− 1.83407i	−0.398807	−	0.917035i	$-0.630576\pi$
$829$	− 3.00855e10i	− 1.83407i	0.398807	−	0.917035i	$-0.369424\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.31386e9i	0.138701i
$834$	0	0
$835$	−3.38751e10	−2.01362
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.18944e10	1.27987	0.639937	−	0.768428i	$-0.278961\pi$
$839$	2.18944e10	1.27987	0.639937	+	0.768428i	$0.278961\pi$
$840$	0	0
$841$	1.74699e10	1.01276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.65673e10	3.22528
$846$	0	0
$847$	− 1.76071e9i	− 0.0995623i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.32861e10i	− 0.739001i
$852$	0	0
$853$	− 1.74899e10i	− 0.964862i	−0.875934	−	0.482431i	$-0.839754\pi$
$853$	− 1.74899e10i	− 0.964862i	0.875934	−	0.482431i	$-0.160246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.86981e10i	1.01476i	0.861721	+	0.507382i	$0.169387\pi$
$857$	1.86981e10i	1.01476i	−0.861721	+	0.507382i	$0.830613\pi$
$858$	0	0
$859$	3.23322e10	1.74044	0.870221	−	0.492662i	$-0.163976\pi$
$859$	3.23322e10	1.74044	0.870221	+	0.492662i	$0.163976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.19116e9	0.0630859	0.0315429	−	0.999502i	$-0.489958\pi$
$863$	1.19116e9	0.0630859	0.0315429	+	0.999502i	$0.489958\pi$
$864$	0	0
$865$	1.09160e10	0.573465
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.47350e10	−0.761694
$870$	0	0
$871$	3.22640e10i	1.65445i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.01921e9i	0.101895i
$876$	0	0
$877$	2.08862e10i	1.04559i	0.852459	+	0.522794i	$0.175110\pi$
$877$	2.08862e10i	1.04559i	−0.852459	+	0.522794i	$0.824890\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.96861e9i	− 0.244804i	−0.992481	−	0.122402i	$-0.960940\pi$
$881$	− 4.96861e9i	− 0.244804i	0.992481	−	0.122402i	$-0.0390598\pi$
$882$	0	0
$883$	−2.57384e10	−1.25811	−0.629056	−	0.777360i	$-0.716558\pi$
$883$	−2.57384e10	−1.25811	−0.629056	+	0.777360i	$0.716558\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.14294e10	1.51218	0.756089	−	0.654468i	$-0.227108\pi$
$887$	3.14294e10	1.51218	0.756089	+	0.654468i	$0.227108\pi$
$888$	0	0
$889$	−1.05243e10	−0.502386
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.08549e9	−0.379950
$894$	0	0
$895$	− 3.81278e10i	− 1.77771i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.53582e10i	− 1.62305i
$900$	0	0
$901$	− 2.43875e9i	− 0.111079i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.26303e10i	1.46336i
$906$	0	0
$907$	−2.28352e10	−1.01620	−0.508099	−	0.861299i	$-0.669652\pi$
$907$	−2.28352e10	−1.01620	−0.508099	+	0.861299i	$0.669652\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.67176e10	−0.732589	−0.366295	−	0.930499i	$-0.619374\pi$
$911$	−1.67176e10	−0.732589	−0.366295	+	0.930499i	$0.619374\pi$
$912$	0	0
$913$	−1.81507e10	−0.789307
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.53694e10	0.658211
$918$	0	0
$919$	− 3.46269e10i	− 1.47167i	−0.677162	−	0.735834i	$-0.736790\pi$
$919$	− 3.46269e10i	− 1.47167i	0.677162	−	0.735834i	$-0.263210\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.24073e10i	3.03093i
$924$	0	0
$925$	− 1.74375e10i	− 0.724414i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.05754e10i	− 0.432753i	−0.976310	−	0.216377i	$-0.930576\pi$
$929$	− 1.05754e10i	− 0.432753i	0.976310	−	0.216377i	$-0.0694239\pi$
$930$	0	0
$931$	−2.38420e10	−0.968320
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.24413e9	−0.209813
$936$	0	0
$937$	−3.29022e10	−1.30658	−0.653290	−	0.757107i	$-0.726612\pi$
$937$	−3.29022e10	−1.30658	−0.653290	+	0.757107i	$0.726612\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.87420e9	0.151572	0.0757859	−	0.997124i	$-0.475853\pi$
$941$	3.87420e9	0.151572	0.0757859	+	0.997124i	$0.475853\pi$
$942$	0	0
$943$	− 1.11492e10i	− 0.432966i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.64475e10i	− 1.39458i	−0.716791	−	0.697288i	$-0.754390\pi$
$947$	− 3.64475e10i	− 1.39458i	0.716791	−	0.697288i	$-0.245610\pi$
$948$	0	0
$949$	7.37811e9i	0.280229i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.75252e10i	− 1.77868i	−0.457242	−	0.889342i	$-0.651163\pi$
$953$	− 4.75252e10i	− 1.77868i	0.457242	−	0.889342i	$-0.348837\pi$
$954$	0	0
$955$	4.24228e10	1.57611
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.28278e9	−0.0469666
$960$	0	0
$961$	−8.49574e9	−0.308794
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.46504e10	0.883037
$966$	0	0
$967$	5.17956e10i	1.84205i	0.389508	+	0.921023i	$0.372645\pi$
$967$	5.17956e10i	1.84205i	−0.389508	+	0.921023i	$0.627355\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.27726e10i	1.49933i	0.661816	+	0.749667i	$0.269786\pi$
$971$	4.27726e10i	1.49933i	−0.661816	+	0.749667i	$0.730214\pi$
$972$	0	0
$973$	1.67098e10i	0.581535i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.75207e10i	1.63024i	0.579291	+	0.815121i	$0.303330\pi$
$977$	4.75207e10i	1.63024i	−0.579291	+	0.815121i	$0.696670\pi$
$978$	0	0
$979$	2.70682e10	0.921977
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.39823e10	−1.14108	−0.570540	−	0.821270i	$-0.693266\pi$
$983$	−3.39823e10	−1.14108	−0.570540	+	0.821270i	$0.693266\pi$
$984$	0	0
$985$	−6.93923e10	−2.31358
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.29444e10	−1.74033
$990$	0	0
$991$	6.63551e9i	0.216579i	0.994119	+	0.108289i	$0.0345373\pi$
$991$	6.63551e9i	0.216579i	−0.994119	+	0.108289i	$0.965463\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.80973e10i	1.22606i
$996$	0	0
$997$	5.33996e10i	1.70650i	0.521506	+	0.853248i	$0.325370\pi$
$997$	5.33996e10i	1.70650i	−0.521506	+	0.853248i	$0.674630\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.f.a.143.4		28
3.2	odd	2	inner	288.8.f.a.143.26		28
4.3	odd	2		72.8.f.a.35.15	yes	28
8.3	odd	2	inner	288.8.f.a.143.25		28
8.5	even	2		72.8.f.a.35.13	✓	28
12.11	even	2		72.8.f.a.35.14	yes	28
24.5	odd	2		72.8.f.a.35.16	yes	28
24.11	even	2	inner	288.8.f.a.143.3		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.13	✓	28	8.5	even	2
72.8.f.a.35.14	yes	28	12.11	even	2
72.8.f.a.35.15	yes	28	4.3	odd	2
72.8.f.a.35.16	yes	28	24.5	odd	2
288.8.f.a.143.3		28	24.11	even	2	inner
288.8.f.a.143.4		28	1.1	even	1	trivial
288.8.f.a.143.25		28	8.3	odd	2	inner
288.8.f.a.143.26		28	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-412.177 q^{5} -359.164i q^{7} +O(q^{10})\)	\(q-412.177 q^{5} -359.164i q^{7} -3819.02i q^{11} -14141.7i q^{13} +3331.48i q^{17} -34327.6 q^{19} +69918.3 q^{23} +91764.7 q^{25} +186332. q^{29} -189759. i q^{31} +148039. i q^{35} -190024. i q^{37} -159461. i q^{41} -757231. q^{43} +235539. q^{47} +694544. q^{49} -732030. q^{53} +1.57411e6i q^{55} -1.98198e6i q^{59} +1.37932e6i q^{61} +5.82890e6i q^{65} -2.28147e6 q^{67} -5.12011e6 q^{71} -521725. q^{73} -1.37165e6 q^{77} -3.85831e6i q^{79} -4.75271e6i q^{83} -1.37316e6i q^{85} +7.08773e6i q^{89} -5.07920e6 q^{91} +1.41490e7 q^{95} -1.34983e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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