LMFDB - Embedded newform 288.8.f.a.143.11

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			143.11
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-166.398 q^{5} +750.222i q^{7} +O(q^{10})$	$q-166.398 q^{5} +750.222i q^{7} +8691.94i q^{11} +9209.54i q^{13} +6845.42i q^{17} +12465.9 q^{19} +30540.3 q^{23} -50436.7 q^{25} +122115. q^{29} -247508. i q^{31} -124835. i q^{35} +270412. i q^{37} +465855. i q^{41} +191804. q^{43} -138358. q^{47} +260710. q^{49} -1.65255e6 q^{53} -1.44632e6i q^{55} -100518. i q^{59} +1.05236e6i q^{61} -1.53245e6i q^{65} -2.72917e6 q^{67} +5.83185e6 q^{71} -4.00385e6 q^{73} -6.52089e6 q^{77} +296509. i q^{79} +2.86831e6i q^{83} -1.13906e6i q^{85} +1.14123e7i q^{89} -6.90921e6 q^{91} -2.07429e6 q^{95} +1.12274e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−166.398	−0.595323	−0.297662	−	0.954671i	$-0.596207\pi$
$5$	−166.398	−0.595323	−0.297662	+	0.954671i	$0.596207\pi$
$6$	0	0
$7$	750.222i	0.826698i	0.910573	+	0.413349i	$0.135641\pi$
$7$	750.222i	0.826698i	−0.910573	+	0.413349i	$0.864359\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8691.94i	1.96899i	0.175427	+	0.984493i	$0.443870\pi$
$11$	8691.94i	1.96899i	−0.175427	+	0.984493i	$0.556130\pi$
$12$	0	0
$13$	9209.54i	1.16262i	0.813683	+	0.581308i	$0.197459\pi$
$13$	9209.54i	1.16262i	−0.813683	+	0.581308i	$0.802541\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6845.42i	0.337932i	0.985622	+	0.168966i	$0.0540427\pi$
$17$	6845.42i	0.337932i	−0.985622	+	0.168966i	$0.945957\pi$
$18$	0	0
$19$	12465.9	0.416951	0.208475	−	0.978028i	$-0.433150\pi$
$19$	12465.9	0.416951	0.208475	+	0.978028i	$0.433150\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30540.3	0.523391	0.261695	−	0.965151i	$-0.415718\pi$
$23$	30540.3	0.523391	0.261695	+	0.965151i	$0.415718\pi$
$24$	0	0
$25$	−50436.7	−0.645590
$26$	0	0
$27$	0	0
$28$	0	0
$29$	122115.	0.929768	0.464884	−	0.885372i	$-0.346096\pi$
$29$	122115.	0.929768	0.464884	+	0.885372i	$0.346096\pi$
$30$	0	0
$31$	− 247508.i	− 1.49218i	−0.665842	−	0.746092i	$-0.731928\pi$
$31$	− 247508.i	− 1.49218i	0.665842	−	0.746092i	$-0.268072\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 124835.i	− 0.492152i
$36$	0	0
$37$	270412.i	0.877648i	0.898573	+	0.438824i	$0.144605\pi$
$37$	270412.i	0.877648i	−0.898573	+	0.438824i	$0.855395\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	465855.i	1.05562i	0.849363	+	0.527810i	$0.176987\pi$
$41$	465855.i	1.05562i	−0.849363	+	0.527810i	$0.823013\pi$
$42$	0	0
$43$	191804.	0.367890	0.183945	−	0.982937i	$-0.441113\pi$
$43$	191804.	0.367890	0.183945	+	0.982937i	$0.441113\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−138358.	−0.194385	−0.0971923	−	0.995266i	$-0.530986\pi$
$47$	−138358.	−0.194385	−0.0971923	+	0.995266i	$0.530986\pi$
$48$	0	0
$49$	260710.	0.316571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.65255e6	−1.52471	−0.762356	−	0.647158i	$-0.775957\pi$
$53$	−1.65255e6	−1.52471	−0.762356	+	0.647158i	$0.775957\pi$
$54$	0	0
$55$	− 1.44632e6i	− 1.17218i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 100518.i	− 0.0637182i	−0.999492	−	0.0318591i	$-0.989857\pi$
$59$	− 100518.i	− 0.0637182i	0.999492	−	0.0318591i	$-0.0101428\pi$
$60$	0	0
$61$	1.05236e6i	0.593624i	0.954936	+	0.296812i	$0.0959234\pi$
$61$	1.05236e6i	0.593624i	−0.954936	+	0.296812i	$0.904077\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.53245e6i	− 0.692133i
$66$	0	0
$67$	−2.72917e6	−1.10858	−0.554292	−	0.832322i	$-0.687011\pi$
$67$	−2.72917e6	−1.10858	−0.554292	+	0.832322i	$0.687011\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.83185e6	1.93376	0.966879	−	0.255235i	$-0.0821528\pi$
$71$	5.83185e6	1.93376	0.966879	+	0.255235i	$0.0821528\pi$
$72$	0	0
$73$	−4.00385e6	−1.20461	−0.602307	−	0.798265i	$-0.705752\pi$
$73$	−4.00385e6	−1.20461	−0.602307	+	0.798265i	$0.705752\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.52089e6	−1.62776
$78$	0	0
$79$	296509.i	0.0676618i	0.999428	+	0.0338309i	$0.0107708\pi$
$79$	296509.i	0.0676618i	−0.999428	+	0.0338309i	$0.989229\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.86831e6i	0.550621i	0.961355	+	0.275311i	$0.0887807\pi$
$83$	2.86831e6i	0.550621i	−0.961355	+	0.275311i	$0.911219\pi$
$84$	0	0
$85$	− 1.13906e6i	− 0.201179i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.14123e7i	1.71596i	0.513679	+	0.857982i	$0.328282\pi$
$89$	1.14123e7i	1.71596i	−0.513679	+	0.857982i	$0.671718\pi$
$90$	0	0
$91$	−6.90921e6	−0.961132
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.07429e6	−0.248221
$96$	0	0
$97$	1.12274e7	1.24904	0.624521	−	0.781008i	$-0.285294\pi$
$97$	1.12274e7	1.24904	0.624521	+	0.781008i	$0.285294\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.15791e7	−1.11828	−0.559138	−	0.829074i	$-0.688868\pi$
$101$	−1.15791e7	−1.11828	−0.559138	+	0.829074i	$0.688868\pi$
$102$	0	0
$103$	− 1.80373e7i	− 1.62645i	−0.581947	−	0.813227i	$-0.697709\pi$
$103$	− 1.80373e7i	− 1.62645i	0.581947	−	0.813227i	$-0.302291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.04706e6i	0.398286i	0.979970	+	0.199143i	$0.0638158\pi$
$107$	5.04706e6i	0.398286i	−0.979970	+	0.199143i	$0.936184\pi$
$108$	0	0
$109$	− 1.66178e7i	− 1.22908i	−0.788884	−	0.614542i	$-0.789341\pi$
$109$	− 1.66178e7i	− 1.22908i	0.788884	−	0.614542i	$-0.210659\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.43084e6i	− 0.288876i	−0.989514	−	0.144438i	$-0.953863\pi$
$113$	− 4.43084e6i	− 0.288876i	0.989514	−	0.144438i	$-0.0461374\pi$
$114$	0	0
$115$	−5.08184e6	−0.311587
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.13559e6	−0.279367
$120$	0	0
$121$	−5.60627e7	−2.87690
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.13924e7	0.979658
$126$	0	0
$127$	− 2.77155e7i	− 1.20063i	−0.799762	−	0.600317i	$-0.795041\pi$
$127$	− 2.77155e7i	− 1.20063i	0.799762	−	0.600317i	$-0.204959\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.72039e7i	0.668616i	0.942464	+	0.334308i	$0.108503\pi$
$131$	1.72039e7i	0.668616i	−0.942464	+	0.334308i	$0.891497\pi$
$132$	0	0
$133$	9.35217e6i	0.344692i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.74030e7i	1.24275i	0.783513	+	0.621376i	$0.213426\pi$
$137$	3.74030e7i	1.24275i	−0.783513	+	0.621376i	$0.786574\pi$
$138$	0	0
$139$	−2.28530e7	−0.721758	−0.360879	−	0.932613i	$-0.617523\pi$
$139$	−2.28530e7	−0.721758	−0.360879	+	0.932613i	$0.617523\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00488e7	−2.28917
$144$	0	0
$145$	−2.03196e7	−0.553512
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.50798e7	−0.621116	−0.310558	−	0.950554i	$-0.600516\pi$
$149$	−2.50798e7	−0.621116	−0.310558	+	0.950554i	$0.600516\pi$
$150$	0	0
$151$	3.71651e7i	0.878448i	0.898377	+	0.439224i	$0.144747\pi$
$151$	3.71651e7i	0.878448i	−0.898377	+	0.439224i	$0.855253\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.11848e7i	0.888332i
$156$	0	0
$157$	− 7.74558e7i	− 1.59737i	−0.601750	−	0.798685i	$-0.705529\pi$
$157$	− 7.74558e7i	− 1.59737i	0.601750	−	0.798685i	$-0.294471\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.29120e7i	0.432686i
$162$	0	0
$163$	−7.06104e7	−1.27706	−0.638531	−	0.769596i	$-0.720458\pi$
$163$	−7.06104e7	−1.27706	−0.638531	+	0.769596i	$0.720458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.83606e7	−1.13579	−0.567895	−	0.823101i	$-0.692242\pi$
$167$	−6.83606e7	−1.13579	−0.567895	+	0.823101i	$0.692242\pi$
$168$	0	0
$169$	−2.20672e7	−0.351677
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.63994e7	0.681319	0.340660	−	0.940187i	$-0.389350\pi$
$173$	4.63994e7	0.681319	0.340660	+	0.940187i	$0.389350\pi$
$174$	0	0
$175$	− 3.78388e7i	− 0.533708i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.11843e7i	0.145755i	0.997341	+	0.0728777i	$0.0232183\pi$
$179$	1.11843e7i	0.145755i	−0.997341	+	0.0728777i	$0.976782\pi$
$180$	0	0
$181$	− 1.42553e8i	− 1.78690i	−0.449162	−	0.893450i	$-0.648277\pi$
$181$	− 1.42553e8i	− 1.78690i	0.449162	−	0.893450i	$-0.351723\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.49961e7i	− 0.522485i
$186$	0	0
$187$	−5.95000e7	−0.665382
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.46450e7	0.671303	0.335651	−	0.941986i	$-0.391044\pi$
$191$	6.46450e7	0.671303	0.335651	+	0.941986i	$0.391044\pi$
$192$	0	0
$193$	−1.07572e8	−1.07708	−0.538542	−	0.842599i	$-0.681025\pi$
$193$	−1.07572e8	−1.07708	−0.538542	+	0.842599i	$0.681025\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.48526e7	−0.697550	−0.348775	−	0.937207i	$-0.613402\pi$
$197$	−7.48526e7	−0.697550	−0.348775	+	0.937207i	$0.613402\pi$
$198$	0	0
$199$	3.50626e7i	0.315398i	0.987487	+	0.157699i	$0.0504075\pi$
$199$	3.50626e7i	0.315398i	−0.987487	+	0.157699i	$0.949592\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.16131e7i	0.768637i
$204$	0	0
$205$	− 7.75173e7i	− 0.628435i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.08353e8i	0.820970i
$210$	0	0
$211$	6.32841e7	0.463774	0.231887	−	0.972743i	$-0.425510\pi$
$211$	6.32841e7	0.463774	0.231887	+	0.972743i	$0.425510\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.19158e7	−0.219013
$216$	0	0
$217$	1.85686e8	1.23359
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.30432e7	−0.392885
$222$	0	0
$223$	− 1.91569e8i	− 1.15680i	−0.815753	−	0.578401i	$-0.803677\pi$
$223$	− 1.91569e8i	− 1.15680i	0.815753	−	0.578401i	$-0.196323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.07263e8i	0.608641i	0.952570	+	0.304320i	$0.0984293\pi$
$227$	1.07263e8i	0.608641i	−0.952570	+	0.304320i	$0.901571\pi$
$228$	0	0
$229$	2.78557e8i	1.53282i	0.642354	+	0.766408i	$0.277958\pi$
$229$	2.78557e8i	1.53282i	−0.642354	+	0.766408i	$0.722042\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.33025e6i	− 0.0379641i	−0.999820	−	0.0189820i	$-0.993957\pi$
$233$	− 7.33025e6i	− 0.0379641i	0.999820	−	0.0189820i	$-0.00604254\pi$
$234$	0	0
$235$	2.30225e7	0.115722
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.15951e7	0.339227	0.169614	−	0.985511i	$-0.445748\pi$
$239$	7.15951e7	0.339227	0.169614	+	0.985511i	$0.445748\pi$
$240$	0	0
$241$	−2.67697e8	−1.23192	−0.615961	−	0.787776i	$-0.711232\pi$
$241$	−2.67697e8	−1.23192	−0.615961	+	0.787776i	$0.711232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.33815e7	−0.188462
$246$	0	0
$247$	1.14805e8i	0.484754i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.02680e8i	− 1.60732i	−0.595090	−	0.803659i	$-0.702884\pi$
$251$	− 4.02680e8i	− 1.60732i	0.595090	−	0.803659i	$-0.297116\pi$
$252$	0	0
$253$	2.65454e8i	1.03055i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.64785e7i	0.354539i	0.984162	+	0.177270i	$0.0567265\pi$
$257$	9.64785e7i	0.354539i	−0.984162	+	0.177270i	$0.943274\pi$
$258$	0	0
$259$	−2.02869e8	−0.725550
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.51392e8	1.53006	0.765031	−	0.643993i	$-0.222724\pi$
$263$	4.51392e8	1.53006	0.765031	+	0.643993i	$0.222724\pi$
$264$	0	0
$265$	2.74980e8	0.907696
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.28565e8	0.402709	0.201354	−	0.979518i	$-0.435466\pi$
$269$	1.28565e8	0.402709	0.201354	+	0.979518i	$0.435466\pi$
$270$	0	0
$271$	− 2.72095e8i	− 0.830477i	−0.909713	−	0.415238i	$-0.863698\pi$
$271$	− 2.72095e8i	− 0.830477i	0.909713	−	0.415238i	$-0.136302\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.38393e8i	− 1.27116i
$276$	0	0
$277$	3.63332e8i	1.02713i	0.858051	+	0.513564i	$0.171675\pi$
$277$	3.63332e8i	1.02713i	−0.858051	+	0.513564i	$0.828325\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 4.12431e8i	− 1.10887i	−0.832229	−	0.554433i	$-0.812935\pi$
$281$	− 4.12431e8i	− 1.10887i	0.832229	−	0.554433i	$-0.187065\pi$
$282$	0	0
$283$	4.48777e8	1.17700	0.588502	−	0.808496i	$-0.299718\pi$
$283$	4.48777e8	1.17700	0.588502	+	0.808496i	$0.299718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.49495e8	−0.872678
$288$	0	0
$289$	3.63479e8	0.885802
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.75769e8	−0.640484	−0.320242	−	0.947336i	$-0.603764\pi$
$293$	−2.75769e8	−0.640484	−0.320242	+	0.947336i	$0.603764\pi$
$294$	0	0
$295$	1.67260e7i	0.0379329i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.81262e8i	0.608502i
$300$	0	0
$301$	1.43895e8i	0.304134i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.75111e8i	− 0.353398i
$306$	0	0
$307$	9.71621e8	1.91652	0.958258	−	0.285905i	$-0.0922941\pi$
$307$	9.71621e8	1.91652	0.958258	+	0.285905i	$0.0922941\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.54388e8	1.42211	0.711055	−	0.703136i	$-0.248218\pi$
$311$	7.54388e8	1.42211	0.711055	+	0.703136i	$0.248218\pi$
$312$	0	0
$313$	2.49396e8	0.459710	0.229855	−	0.973225i	$-0.426175\pi$
$313$	2.49396e8	0.459710	0.229855	+	0.973225i	$0.426175\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.59560e8	−0.810280	−0.405140	−	0.914255i	$-0.632777\pi$
$317$	−4.59560e8	−0.810280	−0.405140	+	0.914255i	$0.632777\pi$
$318$	0	0
$319$	1.06141e9i	1.83070i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.53341e7i	0.140901i
$324$	0	0
$325$	− 4.64499e8i	− 0.750574i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.03799e8i	− 0.160697i
$330$	0	0
$331$	−1.69556e8	−0.256989	−0.128495	−	0.991710i	$-0.541015\pi$
$331$	−1.69556e8	−0.256989	−0.128495	+	0.991710i	$0.541015\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.54128e8	0.659966
$336$	0	0
$337$	1.10636e9	1.57468	0.787341	−	0.616518i	$-0.211457\pi$
$337$	1.10636e9	1.57468	0.787341	+	0.616518i	$0.211457\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.15132e9	2.93809
$342$	0	0
$343$	8.13430e8i	1.08841i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.00799e7i	− 0.102889i	−0.998676	−	0.0514447i	$-0.983617\pi$
$347$	− 8.00799e7i	− 0.102889i	0.998676	−	0.0514447i	$-0.0163826\pi$
$348$	0	0
$349$	6.61204e8i	0.832619i	0.909223	+	0.416309i	$0.136677\pi$
$349$	6.61204e8i	0.832619i	−0.909223	+	0.416309i	$0.863323\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.07830e9i	− 1.30475i	−0.757895	−	0.652376i	$-0.773772\pi$
$353$	− 1.07830e9i	− 1.30475i	0.757895	−	0.652376i	$-0.226228\pi$
$354$	0	0
$355$	−9.70408e8	−1.15121
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.34950e8	−0.268006	−0.134003	−	0.990981i	$-0.542783\pi$
$359$	−2.34950e8	−0.268006	−0.134003	+	0.990981i	$0.542783\pi$
$360$	0	0
$361$	−7.38474e8	−0.826152
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.66232e8	0.717134
$366$	0	0
$367$	5.09803e8i	0.538358i	0.963090	+	0.269179i	$0.0867524\pi$
$367$	5.09803e8i	0.538358i	−0.963090	+	0.269179i	$0.913248\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.23978e9i	− 1.26048i
$372$	0	0
$373$	− 1.19562e9i	− 1.19293i	−0.802641	−	0.596463i	$-0.796572\pi$
$373$	− 1.19562e9i	− 1.19293i	0.802641	−	0.596463i	$-0.203428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.12462e9i	1.08096i
$378$	0	0
$379$	1.27016e9	1.19845	0.599226	−	0.800580i	$-0.295475\pi$
$379$	1.27016e9	1.19845	0.599226	+	0.800580i	$0.295475\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.06974e9	−0.972933	−0.486467	−	0.873699i	$-0.661714\pi$
$383$	−1.06974e9	−0.972933	−0.486467	+	0.873699i	$0.661714\pi$
$384$	0	0
$385$	1.08506e9	0.969041
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.09168e8	−0.352434	−0.176217	−	0.984351i	$-0.556386\pi$
$389$	−4.09168e8	−0.352434	−0.176217	+	0.984351i	$0.556386\pi$
$390$	0	0
$391$	2.09061e8i	0.176870i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.93385e7i	− 0.0402806i
$396$	0	0
$397$	4.20143e8i	0.337000i	0.985702	+	0.168500i	$0.0538923\pi$
$397$	4.20143e8i	0.337000i	−0.985702	+	0.168500i	$0.946108\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9.24601e8i	− 0.716060i	−0.933710	−	0.358030i	$-0.883449\pi$
$401$	− 9.24601e8i	− 0.716060i	0.933710	−	0.358030i	$-0.116551\pi$
$402$	0	0
$403$	2.27943e9	1.73484
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.35041e9	−1.72808
$408$	0	0
$409$	−1.39824e9	−1.01053	−0.505265	−	0.862964i	$-0.668605\pi$
$409$	−1.39824e9	−1.01053	−0.505265	+	0.862964i	$0.668605\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.54111e7	0.0526757
$414$	0	0
$415$	− 4.77281e8i	− 0.327798i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.83425e9i	1.21818i	0.793102	+	0.609088i	$0.208464\pi$
$419$	1.83425e9i	1.21818i	−0.793102	+	0.609088i	$0.791536\pi$
$420$	0	0
$421$	2.18531e9i	1.42733i	0.700485	+	0.713667i	$0.252967\pi$
$421$	2.18531e9i	1.42733i	−0.700485	+	0.713667i	$0.747033\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.45261e8i	− 0.218165i
$426$	0	0
$427$	−7.89506e8	−0.490747
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.35011e8	0.201553	0.100776	−	0.994909i	$-0.467867\pi$
$431$	3.35011e8	0.201553	0.100776	+	0.994909i	$0.467867\pi$
$432$	0	0
$433$	−8.65039e8	−0.512068	−0.256034	−	0.966668i	$-0.582416\pi$
$433$	−8.65039e8	−0.512068	−0.256034	+	0.966668i	$0.582416\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.80711e8	0.218228
$438$	0	0
$439$	− 1.73415e9i	− 0.978273i	−0.872207	−	0.489136i	$-0.837312\pi$
$439$	− 1.73415e9i	− 0.978273i	0.872207	−	0.489136i	$-0.162688\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.19800e8i	− 0.120120i	−0.998195	−	0.0600598i	$-0.980871\pi$
$443$	− 2.19800e8i	− 0.120120i	0.998195	−	0.0600598i	$-0.0191291\pi$
$444$	0	0
$445$	− 1.89898e9i	− 1.02155i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.50746e8i	0.235001i	0.993073	+	0.117501i	$0.0374882\pi$
$449$	4.50746e8i	0.235001i	−0.993073	+	0.117501i	$0.962512\pi$
$450$	0	0
$451$	−4.04919e9	−2.07850
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.14968e9	0.572185
$456$	0	0
$457$	2.75170e8	0.134863	0.0674317	−	0.997724i	$-0.478520\pi$
$457$	2.75170e8	0.134863	0.0674317	+	0.997724i	$0.478520\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.26285e8	−0.345266	−0.172633	−	0.984986i	$-0.555227\pi$
$461$	−7.26285e8	−0.345266	−0.172633	+	0.984986i	$0.555227\pi$
$462$	0	0
$463$	− 8.27651e8i	− 0.387537i	−0.981047	−	0.193769i	$-0.937929\pi$
$463$	− 8.27651e8i	− 0.387537i	0.981047	−	0.193769i	$-0.0620711\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.25212e9i	− 1.02325i	−0.859208	−	0.511626i	$-0.829043\pi$
$467$	− 2.25212e9i	− 1.02325i	0.859208	−	0.511626i	$-0.170957\pi$
$468$	0	0
$469$	− 2.04748e9i	− 0.916464i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.66715e9i	0.724369i
$474$	0	0
$475$	−6.28738e8	−0.269179
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.32761e9	1.38343	0.691717	−	0.722169i	$-0.256855\pi$
$479$	3.32761e9	1.38343	0.691717	+	0.722169i	$0.256855\pi$
$480$	0	0
$481$	−2.49038e9	−1.02037
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.86821e9	−0.743584
$486$	0	0
$487$	− 3.21388e8i	− 0.126089i	−0.998011	−	0.0630446i	$-0.979919\pi$
$487$	− 3.21388e8i	− 0.126089i	0.998011	−	0.0630446i	$-0.0200810\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.60400e9i	1.37404i	0.726639	+	0.687020i	$0.241081\pi$
$491$	3.60400e9i	1.37404i	−0.726639	+	0.687020i	$0.758919\pi$
$492$	0	0
$493$	8.35925e8i	0.314198i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.37518e9i	1.59863i
$498$	0	0
$499$	1.80781e8	0.0651331	0.0325666	−	0.999470i	$-0.489632\pi$
$499$	1.80781e8	0.0651331	0.0325666	+	0.999470i	$0.489632\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.24260e9	−0.785711	−0.392856	−	0.919600i	$-0.628513\pi$
$503$	−2.24260e9	−0.785711	−0.392856	+	0.919600i	$0.628513\pi$
$504$	0	0
$505$	1.92673e9	0.665736
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.67698e9	−1.57200	−0.786002	−	0.618224i	$-0.787852\pi$
$509$	−4.67698e9	−1.57200	−0.786002	+	0.618224i	$0.787852\pi$
$510$	0	0
$511$	− 3.00378e9i	− 0.995851i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.00137e9i	0.968266i
$516$	0	0
$517$	− 1.20260e9i	− 0.382741i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.70545e8i	− 0.269687i	−0.990867	−	0.134843i	$-0.956947\pi$
$521$	− 8.70545e8i	− 0.269687i	0.990867	−	0.134843i	$-0.0430531\pi$
$522$	0	0
$523$	−2.10726e9	−0.644114	−0.322057	−	0.946720i	$-0.604374\pi$
$523$	−2.10726e9	−0.644114	−0.322057	+	0.946720i	$0.604374\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.69429e9	0.504256
$528$	0	0
$529$	−2.47212e9	−0.726062
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.29031e9	−1.22728
$534$	0	0
$535$	− 8.39820e8i	− 0.237109i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.26607e9i	0.623323i
$540$	0	0
$541$	− 2.00110e9i	− 0.543347i	−0.962389	−	0.271674i	$-0.912423\pi$
$541$	− 2.00110e9i	− 0.543347i	0.962389	−	0.271674i	$-0.0875771\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.76517e9i	0.731703i
$546$	0	0
$547$	−6.96444e9	−1.81941	−0.909705	−	0.415254i	$-0.863693\pi$
$547$	−6.96444e9	−1.81941	−0.909705	+	0.415254i	$0.863693\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.52226e9	0.387668
$552$	0	0
$553$	−2.22448e8	−0.0559358
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.51765e8	0.0372115	0.0186058	−	0.999827i	$-0.494077\pi$
$557$	1.51765e8	0.0372115	0.0186058	+	0.999827i	$0.494077\pi$
$558$	0	0
$559$	1.76643e9i	0.427715i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.29891e9i	− 1.01526i	−0.861574	−	0.507632i	$-0.830521\pi$
$563$	− 4.29891e9i	− 1.01526i	0.861574	−	0.507632i	$-0.169479\pi$
$564$	0	0
$565$	7.37282e8i	0.171975i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5.91647e9i	− 1.34639i	−0.739467	−	0.673193i	$-0.764922\pi$
$569$	− 5.91647e9i	− 1.34639i	0.739467	−	0.673193i	$-0.235078\pi$
$570$	0	0
$571$	3.96765e9	0.891881	0.445941	−	0.895063i	$-0.352869\pi$
$571$	3.96765e9	0.891881	0.445941	+	0.895063i	$0.352869\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.54035e9	−0.337896
$576$	0	0
$577$	−6.30305e9	−1.36595	−0.682976	−	0.730441i	$-0.739315\pi$
$577$	−6.30305e9	−1.36595	−0.682976	+	0.730441i	$0.739315\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.15187e9	−0.455197
$582$	0	0
$583$	− 1.43638e10i	− 3.00213i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.41073e8i	0.0491944i	0.999697	+	0.0245972i	$0.00783032\pi$
$587$	2.41073e8i	0.0491944i	−0.999697	+	0.0245972i	$0.992170\pi$
$588$	0	0
$589$	− 3.08540e9i	− 0.622168i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.32868e9i	− 0.261655i	−0.991405	−	0.130828i	$-0.958237\pi$
$593$	− 1.32868e9i	− 0.261655i	0.991405	−	0.130828i	$-0.0417635\pi$
$594$	0	0
$595$	8.54551e8	0.166314
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.40415e9	−1.40761	−0.703803	−	0.710395i	$-0.748516\pi$
$599$	−7.40415e9	−1.40761	−0.703803	+	0.710395i	$0.748516\pi$
$600$	0	0
$601$	1.16287e7	0.00218509	0.00109254	−	0.999999i	$-0.499652\pi$
$601$	1.16287e7	0.00218509	0.00109254	+	0.999999i	$0.499652\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.32871e9	1.71269
$606$	0	0
$607$	7.47105e9i	1.35588i	0.735117	+	0.677940i	$0.237127\pi$
$607$	7.47105e9i	1.35588i	−0.735117	+	0.677940i	$0.762873\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.27422e9i	− 0.225995i
$612$	0	0
$613$	3.85770e9i	0.676420i	0.941071	+	0.338210i	$0.109821\pi$
$613$	3.85770e9i	0.676420i	−0.941071	+	0.338210i	$0.890179\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.73937e9i	1.15511i	0.816354	+	0.577553i	$0.195992\pi$
$617$	6.73937e9i	1.15511i	−0.816354	+	0.577553i	$0.804008\pi$
$618$	0	0
$619$	−2.08224e9	−0.352869	−0.176434	−	0.984312i	$-0.556456\pi$
$619$	−2.08224e9	−0.352869	−0.176434	+	0.984312i	$0.556456\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.56176e9	−1.41858
$624$	0	0
$625$	3.80718e8	0.0623769
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.85109e9	−0.296585
$630$	0	0
$631$	1.02675e10i	1.62690i	0.581633	+	0.813451i	$0.302414\pi$
$631$	1.02675e10i	1.62690i	−0.581633	+	0.813451i	$0.697586\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.61181e9i	0.714765i
$636$	0	0
$637$	2.40102e9i	0.368050i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.12515e9i	1.06854i	0.845314	+	0.534270i	$0.179414\pi$
$641$	7.12515e9i	1.06854i	−0.845314	+	0.534270i	$0.820586\pi$
$642$	0	0
$643$	4.07206e9	0.604054	0.302027	−	0.953299i	$-0.402337\pi$
$643$	4.07206e9	0.604054	0.302027	+	0.953299i	$0.402337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.52423e9	1.38250	0.691250	−	0.722616i	$-0.257060\pi$
$647$	9.52423e9	1.38250	0.691250	+	0.722616i	$0.257060\pi$
$648$	0	0
$649$	8.73700e8	0.125460
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.31382e9	1.30898	0.654488	−	0.756072i	$-0.272884\pi$
$653$	9.31382e9	1.30898	0.654488	+	0.756072i	$0.272884\pi$
$654$	0	0
$655$	− 2.86269e9i	− 0.398043i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.37116e9i	− 0.458860i	−0.973325	−	0.229430i	$-0.926314\pi$
$659$	− 3.37116e9i	− 0.458860i	0.973325	−	0.229430i	$-0.0736862\pi$
$660$	0	0
$661$	− 4.67132e8i	− 0.0629122i	−0.999505	−	0.0314561i	$-0.989986\pi$
$661$	− 4.67132e8i	− 0.0629122i	0.999505	−	0.0314561i	$-0.0100144\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.55618e9i	− 0.205203i
$666$	0	0
$667$	3.72942e9	0.486632
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.14708e9	−1.16884
$672$	0	0
$673$	7.88186e9	0.996727	0.498364	−	0.866968i	$-0.333935\pi$
$673$	7.88186e9	0.996727	0.498364	+	0.866968i	$0.333935\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.42433e10	1.76421	0.882104	−	0.471054i	$-0.156126\pi$
$677$	1.42433e10	1.76421	0.882104	+	0.471054i	$0.156126\pi$
$678$	0	0
$679$	8.42302e9i	1.03258i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.97439e9i	0.477308i	0.971105	+	0.238654i	$0.0767061\pi$
$683$	3.97439e9i	0.477308i	−0.971105	+	0.238654i	$0.923294\pi$
$684$	0	0
$685$	− 6.22378e9i	− 0.739839i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.52192e10i	− 1.77265i
$690$	0	0
$691$	−1.14648e9	−0.132189	−0.0660944	−	0.997813i	$-0.521054\pi$
$691$	−1.14648e9	−0.132189	−0.0660944	+	0.997813i	$0.521054\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.80269e9	0.429679
$696$	0	0
$697$	−3.18897e9	−0.356727
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.42915e9	−0.595277	−0.297638	−	0.954679i	$-0.596199\pi$
$701$	−5.42915e9	−0.595277	−0.297638	+	0.954679i	$0.596199\pi$
$702$	0	0
$703$	3.37092e9i	0.365936i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.68688e9i	− 0.924477i
$708$	0	0
$709$	− 2.46125e9i	− 0.259354i	−0.991556	−	0.129677i	$-0.958606\pi$
$709$	− 2.46125e9i	− 0.259354i	0.991556	−	0.129677i	$-0.0413941\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.55896e9i	− 0.780995i
$714$	0	0
$715$	1.33200e10	1.36280
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.87766e10	1.88393	0.941967	−	0.335705i	$-0.108975\pi$
$719$	1.87766e10	1.88393	0.941967	+	0.335705i	$0.108975\pi$
$720$	0	0
$721$	1.35320e10	1.34459
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.15906e9	−0.600249
$726$	0	0
$727$	2.64393e9i	0.255200i	0.991826	+	0.127600i	$0.0407273\pi$
$727$	2.64393e9i	0.255200i	−0.991826	+	0.127600i	$0.959273\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.31298e9i	0.124322i
$732$	0	0
$733$	− 1.00402e10i	− 0.941629i	−0.882232	−	0.470815i	$-0.843960\pi$
$733$	− 1.00402e10i	− 0.941629i	0.882232	−	0.470815i	$-0.156040\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.37218e10i	− 2.18279i
$738$	0	0
$739$	3.44469e8	0.0313975	0.0156987	−	0.999877i	$-0.495003\pi$
$739$	3.44469e8	0.0313975	0.0156987	+	0.999877i	$0.495003\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.52679e9	−0.583766	−0.291883	−	0.956454i	$-0.594282\pi$
$743$	−6.52679e9	−0.583766	−0.291883	+	0.956454i	$0.594282\pi$
$744$	0	0
$745$	4.17323e9	0.369765
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.78641e9	−0.329262
$750$	0	0
$751$	5.80249e9i	0.499890i	0.968260	+	0.249945i	$0.0804126\pi$
$751$	5.80249e9i	0.499890i	−0.968260	+	0.249945i	$0.919587\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.18420e9i	− 0.522961i
$756$	0	0
$757$	7.76287e9i	0.650409i	0.945644	+	0.325204i	$0.105433\pi$
$757$	7.76287e9i	0.650409i	−0.945644	+	0.325204i	$0.894567\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.04247e10i	1.68000i	0.542587	+	0.839999i	$0.317445\pi$
$761$	2.04247e10i	1.68000i	−0.542587	+	0.839999i	$0.682555\pi$
$762$	0	0
$763$	1.24671e10	1.01608
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.25728e8	0.0740798
$768$	0	0
$769$	1.69057e10	1.34058	0.670289	−	0.742100i	$-0.266170\pi$
$769$	1.69057e10	1.34058	0.670289	+	0.742100i	$0.266170\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.71619e9	0.756603	0.378301	−	0.925682i	$-0.376508\pi$
$773$	9.71619e9	0.756603	0.378301	+	0.925682i	$0.376508\pi$
$774$	0	0
$775$	1.24835e10i	0.963340i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.80729e9i	0.440141i
$780$	0	0
$781$	5.06901e10i	3.80754i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.28885e10i	0.950951i
$786$	0	0
$787$	−2.20410e10	−1.61183	−0.805915	−	0.592031i	$-0.798326\pi$
$787$	−2.20410e10	−1.61183	−0.805915	+	0.592031i	$0.798326\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.32411e9	0.238813
$792$	0	0
$793$	−9.69179e9	−0.690157
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.13976e10	−0.797460	−0.398730	−	0.917068i	$-0.630549\pi$
$797$	−1.13976e10	−0.797460	−0.398730	+	0.917068i	$0.630549\pi$
$798$	0	0
$799$	− 9.47119e8i	− 0.0656887i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 3.48012e10i	− 2.37187i
$804$	0	0
$805$	− 3.81251e9i	− 0.257588i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 3.32572e9i	− 0.220834i	−0.993885	−	0.110417i	$-0.964781\pi$
$809$	− 3.32572e9i	− 0.220834i	0.993885	−	0.110417i	$-0.0352186\pi$
$810$	0	0
$811$	1.93568e10	1.27427	0.637133	−	0.770754i	$-0.280120\pi$
$811$	1.93568e10	1.27427	0.637133	+	0.770754i	$0.280120\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.17494e10	0.760265
$816$	0	0
$817$	2.39100e9	0.153392
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.00753e9	0.0635417	0.0317709	−	0.999495i	$-0.489885\pi$
$821$	1.00753e9	0.0635417	0.0317709	+	0.999495i	$0.489885\pi$
$822$	0	0
$823$	6.91310e9i	0.432288i	0.976361	+	0.216144i	$0.0693481\pi$
$823$	6.91310e9i	0.432288i	−0.976361	+	0.216144i	$0.930652\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.99238e9i	0.122491i	0.998123	+	0.0612454i	$0.0195072\pi$
$827$	1.99238e9i	0.122491i	−0.998123	+	0.0612454i	$0.980493\pi$
$828$	0	0
$829$	− 9.21131e9i	− 0.561540i	−0.959775	−	0.280770i	$-0.909410\pi$
$829$	− 9.21131e9i	− 0.561540i	0.959775	−	0.280770i	$-0.0905898\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.78467e9i	0.106979i
$834$	0	0
$835$	1.13751e10	0.676162
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.95616e10	1.14351	0.571753	−	0.820426i	$-0.306264\pi$
$839$	1.95616e10	1.14351	0.571753	+	0.820426i	$0.306264\pi$
$840$	0	0
$841$	−2.33790e9	−0.135532
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.67194e9	0.209361
$846$	0	0
$847$	− 4.20595e10i	− 2.37833i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.25847e9i	0.459353i
$852$	0	0
$853$	2.07361e10i	1.14394i	0.820273	+	0.571972i	$0.193821\pi$
$853$	2.07361e10i	1.14394i	−0.820273	+	0.571972i	$0.806179\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.15860e10i	− 1.71420i	−0.515151	−	0.857099i	$-0.672264\pi$
$857$	− 3.15860e10i	− 1.71420i	0.515151	−	0.857099i	$-0.327736\pi$
$858$	0	0
$859$	−1.20261e10	−0.647362	−0.323681	−	0.946166i	$-0.604920\pi$
$859$	−1.20261e10	−0.647362	−0.323681	+	0.946166i	$0.604920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.93491e9	0.261362	0.130681	−	0.991424i	$-0.458284\pi$
$863$	4.93491e9	0.261362	0.130681	+	0.991424i	$0.458284\pi$
$864$	0	0
$865$	−7.72076e9	−0.405605
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.57724e9	−0.133225
$870$	0	0
$871$	− 2.51344e10i	− 1.28886i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.60491e10i	0.809881i
$876$	0	0
$877$	6.10257e9i	0.305502i	0.988265	+	0.152751i	$0.0488133\pi$
$877$	6.10257e9i	0.305502i	−0.988265	+	0.152751i	$0.951187\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.47462e9i	0.0726548i	0.999340	+	0.0363274i	$0.0115659\pi$
$881$	1.47462e9i	0.0726548i	−0.999340	+	0.0363274i	$0.988434\pi$
$882$	0	0
$883$	−1.11517e10	−0.545103	−0.272552	−	0.962141i	$-0.587868\pi$
$883$	−1.11517e10	−0.545103	−0.272552	+	0.962141i	$0.587868\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.52333e9	−0.410088	−0.205044	−	0.978753i	$-0.565734\pi$
$887$	−8.52333e9	−0.410088	−0.205044	+	0.978753i	$0.565734\pi$
$888$	0	0
$889$	2.07928e10	0.992561
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.72475e9	−0.0810489
$894$	0	0
$895$	− 1.86105e9i	− 0.0867716i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.02243e10i	− 1.38739i
$900$	0	0
$901$	− 1.13124e10i	− 0.515248i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.37205e10i	1.06378i
$906$	0	0
$907$	2.69804e10	1.20067	0.600334	−	0.799749i	$-0.295034\pi$
$907$	2.69804e10	1.20067	0.600334	+	0.799749i	$0.295034\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.36401e10	−0.597726	−0.298863	−	0.954296i	$-0.596607\pi$
$911$	−1.36401e10	−0.597726	−0.298863	+	0.954296i	$0.596607\pi$
$912$	0	0
$913$	−2.49312e10	−1.08416
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.29067e10	−0.552743
$918$	0	0
$919$	2.57488e10i	1.09434i	0.837020	+	0.547172i	$0.184295\pi$
$919$	2.57488e10i	1.09434i	−0.837020	+	0.547172i	$0.815705\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.37087e10i	2.24822i
$924$	0	0
$925$	− 1.36387e10i	− 0.566601i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.26697e10i	1.33687i	0.743769	+	0.668436i	$0.233036\pi$
$929$	3.26697e10i	1.33687i	−0.743769	+	0.668436i	$0.766964\pi$
$930$	0	0
$931$	3.24997e9	0.131994
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.90067e9	0.396118
$936$	0	0
$937$	1.41361e10	0.561358	0.280679	−	0.959802i	$-0.409440\pi$
$937$	1.41361e10	0.561358	0.280679	+	0.959802i	$0.409440\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.18935e10	−0.856548	−0.428274	−	0.903649i	$-0.640878\pi$
$941$	−2.18935e10	−0.856548	−0.428274	+	0.903649i	$0.640878\pi$
$942$	0	0
$943$	1.42274e10i	0.552501i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.93208e9i	− 0.150452i	−0.997167	−	0.0752259i	$-0.976032\pi$
$947$	− 3.93208e9i	− 0.150452i	0.997167	−	0.0752259i	$-0.0239678\pi$
$948$	0	0
$949$	− 3.68736e10i	− 1.40050i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.53127e9i	0.0947355i	0.998878	+	0.0473677i	$0.0150833\pi$
$953$	2.53127e9i	0.0947355i	−0.998878	+	0.0473677i	$0.984917\pi$
$954$	0	0
$955$	−1.07568e10	−0.399642
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.80606e10	−1.02738
$960$	0	0
$961$	−3.37474e10	−1.22662
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.78998e10	0.641213
$966$	0	0
$967$	1.74142e9i	0.0619314i	0.999520	+	0.0309657i	$0.00985826\pi$
$967$	1.74142e9i	0.0619314i	−0.999520	+	0.0309657i	$0.990142\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 5.71812e9i	− 0.200441i	−0.994965	−	0.100220i	$-0.968045\pi$
$971$	− 5.71812e9i	− 0.200441i	0.994965	−	0.100220i	$-0.0319548\pi$
$972$	0	0
$973$	− 1.71448e10i	− 0.596675i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.64895e10i	0.565688i	0.959166	+	0.282844i	$0.0912778\pi$
$977$	1.64895e10i	0.565688i	−0.959166	+	0.282844i	$0.908722\pi$
$978$	0	0
$979$	−9.91951e10	−3.37871
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.20432e10	−0.740182	−0.370091	−	0.928996i	$-0.620673\pi$
$983$	−2.20432e10	−0.740182	−0.370091	+	0.928996i	$0.620673\pi$
$984$	0	0
$985$	1.24553e10	0.415268
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.85775e9	0.192550
$990$	0	0
$991$	− 7.03533e9i	− 0.229629i	−0.993387	−	0.114814i	$-0.963373\pi$
$991$	− 7.03533e9i	− 0.229629i	0.993387	−	0.114814i	$-0.0366273\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.83434e9i	− 0.187764i
$996$	0	0
$997$	− 2.68981e10i	− 0.859584i	−0.902928	−	0.429792i	$-0.858587\pi$
$997$	− 2.68981e10i	− 0.859584i	0.902928	−	0.429792i	$-0.141413\pi$
$998$	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.143.11		28
3.2	odd	2	inner	288.8.f.a.143.17		28
4.3	odd	2		72.8.f.a.35.9	✓	28
8.3	odd	2	inner	288.8.f.a.143.18		28
8.5	even	2		72.8.f.a.35.19	yes	28
12.11	even	2		72.8.f.a.35.20	yes	28
24.5	odd	2		72.8.f.a.35.10	yes	28
24.11	even	2	inner	288.8.f.a.143.12		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.9	✓	28	4.3	odd	2
72.8.f.a.35.10	yes	28	24.5	odd	2
72.8.f.a.35.19	yes	28	8.5	even	2
72.8.f.a.35.20	yes	28	12.11	even	2
288.8.f.a.143.11		28	1.1	even	1	trivial
288.8.f.a.143.12		28	24.11	even	2	inner
288.8.f.a.143.17		28	3.2	odd	2	inner
288.8.f.a.143.18		28	8.3	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)

\(f(q)\)	\(=\)	\(q-166.398 q^{5} +750.222i q^{7} +O(q^{10})\)	\(q-166.398 q^{5} +750.222i q^{7} +8691.94i q^{11} +9209.54i q^{13} +6845.42i q^{17} +12465.9 q^{19} +30540.3 q^{23} -50436.7 q^{25} +122115. q^{29} -247508. i q^{31} -124835. i q^{35} +270412. i q^{37} +465855. i q^{41} +191804. q^{43} -138358. q^{47} +260710. q^{49} -1.65255e6 q^{53} -1.44632e6i q^{55} -100518. i q^{59} +1.05236e6i q^{61} -1.53245e6i q^{65} -2.72917e6 q^{67} +5.83185e6 q^{71} -4.00385e6 q^{73} -6.52089e6 q^{77} +296509. i q^{79} +2.86831e6i q^{83} -1.13906e6i q^{85} +1.14123e7i q^{89} -6.90921e6 q^{91} -2.07429e6 q^{95} +1.12274e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 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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