LMFDB - Embedded newform 7800.2.a.bo.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.11753 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.11753 q^{7} +1.00000 q^{9} -3.21432 q^{11} +1.00000 q^{13} -6.05086 q^{17} -2.90321 q^{19} +4.11753 q^{21} -3.65878 q^{23} +1.00000 q^{27} -0.377784 q^{29} -1.40790 q^{31} -3.21432 q^{33} -10.0874 q^{37} +1.00000 q^{39} +1.33185 q^{41} -6.57628 q^{43} +8.11753 q^{47} +9.95407 q^{49} -6.05086 q^{51} -4.09679 q^{53} -2.90321 q^{57} -14.2192 q^{59} -2.57136 q^{61} +4.11753 q^{63} -7.02074 q^{67} -3.65878 q^{69} +7.19850 q^{71} -7.46520 q^{73} -13.2351 q^{77} +7.13828 q^{79} +1.00000 q^{81} -10.9748 q^{83} -0.377784 q^{87} -18.5303 q^{89} +4.11753 q^{91} -1.40790 q^{93} +12.2351 q^{97} -3.21432 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 2 q^{19} - q^{21} - 4 q^{23} + 3 q^{27} - q^{29} - 11 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 16 q^{41} + 11 q^{47} + 10 q^{49} - 5 q^{51} - 19 q^{53} - 2 q^{57} - 3 q^{59} - 21 q^{61} - q^{63} - q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - 13 q^{77} - 12 q^{79} + 3 q^{81} + 7 q^{83} - q^{87} - 16 q^{89} - q^{91} - 11 q^{93} + 10 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 - 5 * q^17 - 2 * q^19 - q^21 - 4 * q^23 + 3 * q^27 - q^29 - 11 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 16 * q^41 + 11 * q^47 + 10 * q^49 - 5 * q^51 - 19 * q^53 - 2 * q^57 - 3 * q^59 - 21 * q^61 - q^63 - q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - 13 * q^77 - 12 * q^79 + 3 * q^81 + 7 * q^83 - q^87 - 16 * q^89 - q^91 - 11 * q^93 + 10 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.11753	1.55628	0.778140	−	0.628090i	$-0.216163\pi$
$7$	4.11753	1.55628	0.778140	+	0.628090i	$0.216163\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.21432	−0.969154	−0.484577	−	0.874749i	$-0.661026\pi$
$11$	−3.21432	−0.969154	−0.484577	+	0.874749i	$0.661026\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.05086	−1.46755	−0.733774	−	0.679394i	$-0.762243\pi$
$17$	−6.05086	−1.46755	−0.733774	+	0.679394i	$0.762243\pi$
$18$	0	0
$19$	−2.90321	−0.666042	−0.333021	−	0.942919i	$-0.608068\pi$
$19$	−2.90321	−0.666042	−0.333021	+	0.942919i	$0.608068\pi$
$20$	0	0
$21$	4.11753	0.898519
$22$	0	0
$23$	−3.65878	−0.762909	−0.381454	−	0.924388i	$-0.624577\pi$
$23$	−3.65878	−0.762909	−0.381454	+	0.924388i	$0.624577\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.377784	−0.0701528	−0.0350764	−	0.999385i	$-0.511167\pi$
$29$	−0.377784	−0.0701528	−0.0350764	+	0.999385i	$0.511167\pi$
$30$	0	0
$31$	−1.40790	−0.252866	−0.126433	−	0.991975i	$-0.540353\pi$
$31$	−1.40790	−0.252866	−0.126433	+	0.991975i	$0.540353\pi$
$32$	0	0
$33$	−3.21432	−0.559541
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0874	−1.65836	−0.829181	−	0.558980i	$-0.811193\pi$
$37$	−10.0874	−1.65836	−0.829181	+	0.558980i	$0.811193\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.33185	0.208000	0.104000	−	0.994577i	$-0.466836\pi$
$41$	1.33185	0.208000	0.104000	+	0.994577i	$0.466836\pi$
$42$	0	0
$43$	−6.57628	−1.00287	−0.501437	−	0.865194i	$-0.667195\pi$
$43$	−6.57628	−1.00287	−0.501437	+	0.865194i	$0.667195\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.11753	1.18406	0.592032	−	0.805915i	$-0.298326\pi$
$47$	8.11753	1.18406	0.592032	+	0.805915i	$0.298326\pi$
$48$	0	0
$49$	9.95407	1.42201
$50$	0	0
$51$	−6.05086	−0.847289
$52$	0	0
$53$	−4.09679	−0.562737	−0.281369	−	0.959600i	$-0.590788\pi$
$53$	−4.09679	−0.562737	−0.281369	+	0.959600i	$0.590788\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.90321	−0.384540
$58$	0	0
$59$	−14.2192	−1.85119	−0.925594	−	0.378518i	$-0.876434\pi$
$59$	−14.2192	−1.85119	−0.925594	+	0.378518i	$0.876434\pi$
$60$	0	0
$61$	−2.57136	−0.329229	−0.164614	−	0.986358i	$-0.552638\pi$
$61$	−2.57136	−0.329229	−0.164614	+	0.986358i	$0.552638\pi$
$62$	0	0
$63$	4.11753	0.518760
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.02074	−0.857720	−0.428860	−	0.903371i	$-0.641085\pi$
$67$	−7.02074	−0.857720	−0.428860	+	0.903371i	$0.641085\pi$
$68$	0	0
$69$	−3.65878	−0.440465
$70$	0	0
$71$	7.19850	0.854305	0.427152	−	0.904180i	$-0.359517\pi$
$71$	7.19850	0.854305	0.427152	+	0.904180i	$0.359517\pi$
$72$	0	0
$73$	−7.46520	−0.873736	−0.436868	−	0.899526i	$-0.643912\pi$
$73$	−7.46520	−0.873736	−0.436868	+	0.899526i	$0.643912\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.2351	−1.50828
$78$	0	0
$79$	7.13828	0.803119	0.401559	−	0.915833i	$-0.368468\pi$
$79$	7.13828	0.803119	0.401559	+	0.915833i	$0.368468\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.9748	−1.20464	−0.602321	−	0.798254i	$-0.705757\pi$
$83$	−10.9748	−1.20464	−0.602321	+	0.798254i	$0.705757\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.377784	−0.0405027
$88$	0	0
$89$	−18.5303	−1.96421	−0.982107	−	0.188326i	$-0.939694\pi$
$89$	−18.5303	−1.96421	−0.982107	+	0.188326i	$0.939694\pi$
$90$	0	0
$91$	4.11753	0.431635
$92$	0	0
$93$	−1.40790	−0.145992
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.2351	1.24228	0.621141	−	0.783699i	$-0.286669\pi$
$97$	12.2351	1.24228	0.621141	+	0.783699i	$0.286669\pi$
$98$	0	0
$99$	−3.21432	−0.323051
$100$	0	0
$101$	−8.03657	−0.799668	−0.399834	−	0.916588i	$-0.630932\pi$
$101$	−8.03657	−0.799668	−0.399834	+	0.916588i	$0.630932\pi$
$102$	0	0
$103$	−4.57628	−0.450915	−0.225457	−	0.974253i	$-0.572388\pi$
$103$	−4.57628	−0.450915	−0.225457	+	0.974253i	$0.572388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.9906	−1.44920	−0.724600	−	0.689170i	$-0.757975\pi$
$107$	−14.9906	−1.44920	−0.724600	+	0.689170i	$0.757975\pi$
$108$	0	0
$109$	19.2716	1.84589	0.922944	−	0.384935i	$-0.125776\pi$
$109$	19.2716	1.84589	0.922944	+	0.384935i	$0.125776\pi$
$110$	0	0
$111$	−10.0874	−0.957456
$112$	0	0
$113$	6.04149	0.568335	0.284168	−	0.958775i	$-0.408283\pi$
$113$	6.04149	0.568335	0.284168	+	0.958775i	$0.408283\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−24.9146	−2.28392
$120$	0	0
$121$	−0.668149	−0.0607408
$122$	0	0
$123$	1.33185	0.120089
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.2716	1.71008	0.855040	−	0.518562i	$-0.173532\pi$
$127$	19.2716	1.71008	0.855040	+	0.518562i	$0.173532\pi$
$128$	0	0
$129$	−6.57628	−0.579009
$130$	0	0
$131$	−14.2494	−1.24497	−0.622486	−	0.782631i	$-0.713877\pi$
$131$	−14.2494	−1.24497	−0.622486	+	0.782631i	$0.713877\pi$
$132$	0	0
$133$	−11.9541	−1.03655
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.37778	0.459455	0.229728	−	0.973255i	$-0.426216\pi$
$137$	5.37778	0.459455	0.229728	+	0.973255i	$0.426216\pi$
$138$	0	0
$139$	7.24443	0.614465	0.307232	−	0.951635i	$-0.400597\pi$
$139$	7.24443	0.614465	0.307232	+	0.951635i	$0.400597\pi$
$140$	0	0
$141$	8.11753	0.683619
$142$	0	0
$143$	−3.21432	−0.268795
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.95407	0.820998
$148$	0	0
$149$	17.4795	1.43198	0.715988	−	0.698113i	$-0.245977\pi$
$149$	17.4795	1.43198	0.715988	+	0.698113i	$0.245977\pi$
$150$	0	0
$151$	−22.6795	−1.84563	−0.922817	−	0.385239i	$-0.874119\pi$
$151$	−22.6795	−1.84563	−0.922817	+	0.385239i	$0.874119\pi$
$152$	0	0
$153$	−6.05086	−0.489183
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.3225	1.06325	0.531625	−	0.846980i	$-0.321582\pi$
$157$	13.3225	1.06325	0.531625	+	0.846980i	$0.321582\pi$
$158$	0	0
$159$	−4.09679	−0.324896
$160$	0	0
$161$	−15.0651	−1.18730
$162$	0	0
$163$	16.0558	1.25759	0.628793	−	0.777573i	$-0.283549\pi$
$163$	16.0558	1.25759	0.628793	+	0.777573i	$0.283549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.84299	0.684291	0.342146	−	0.939647i	$-0.388846\pi$
$167$	8.84299	0.684291	0.342146	+	0.939647i	$0.388846\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.90321	−0.222014
$172$	0	0
$173$	−15.7699	−1.19896	−0.599480	−	0.800390i	$-0.704626\pi$
$173$	−15.7699	−1.19896	−0.599480	+	0.800390i	$0.704626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.2192	−1.06878
$178$	0	0
$179$	2.38271	0.178092	0.0890459	−	0.996028i	$-0.471618\pi$
$179$	2.38271	0.178092	0.0890459	+	0.996028i	$0.471618\pi$
$180$	0	0
$181$	−14.8207	−1.10161	−0.550807	−	0.834632i	$-0.685680\pi$
$181$	−14.8207	−1.10161	−0.550807	+	0.834632i	$0.685680\pi$
$182$	0	0
$183$	−2.57136	−0.190080
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19.4494	1.42228
$188$	0	0
$189$	4.11753	0.299506
$190$	0	0
$191$	−8.10171	−0.586219	−0.293110	−	0.956079i	$-0.594690\pi$
$191$	−8.10171	−0.586219	−0.293110	+	0.956079i	$0.594690\pi$
$192$	0	0
$193$	−7.65878	−0.551291	−0.275646	−	0.961259i	$-0.588892\pi$
$193$	−7.65878	−0.551291	−0.275646	+	0.961259i	$0.588892\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.5254	0.821153	0.410576	−	0.911826i	$-0.365328\pi$
$197$	11.5254	0.821153	0.410576	+	0.911826i	$0.365328\pi$
$198$	0	0
$199$	13.5210	0.958477	0.479238	−	0.877685i	$-0.340913\pi$
$199$	13.5210	0.958477	0.479238	+	0.877685i	$0.340913\pi$
$200$	0	0
$201$	−7.02074	−0.495205
$202$	0	0
$203$	−1.55554	−0.109177
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.65878	−0.254303
$208$	0	0
$209$	9.33185	0.645498
$210$	0	0
$211$	6.44293	0.443550	0.221775	−	0.975098i	$-0.428815\pi$
$211$	6.44293	0.443550	0.221775	+	0.975098i	$0.428815\pi$
$212$	0	0
$213$	7.19850	0.493233
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.79706	−0.393530
$218$	0	0
$219$	−7.46520	−0.504452
$220$	0	0
$221$	−6.05086	−0.407025
$222$	0	0
$223$	−2.51606	−0.168488	−0.0842439	−	0.996445i	$-0.526847\pi$
$223$	−2.51606	−0.168488	−0.0842439	+	0.996445i	$0.526847\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.22369	0.147591	0.0737957	−	0.997273i	$-0.476489\pi$
$227$	2.22369	0.147591	0.0737957	+	0.997273i	$0.476489\pi$
$228$	0	0
$229$	4.48886	0.296632	0.148316	−	0.988940i	$-0.452615\pi$
$229$	4.48886	0.296632	0.148316	+	0.988940i	$0.452615\pi$
$230$	0	0
$231$	−13.2351	−0.870803
$232$	0	0
$233$	10.8988	0.714002	0.357001	−	0.934104i	$-0.383799\pi$
$233$	10.8988	0.714002	0.357001	+	0.934104i	$0.383799\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.13828	0.463681
$238$	0	0
$239$	22.5877	1.46107	0.730537	−	0.682873i	$-0.239270\pi$
$239$	22.5877	1.46107	0.730537	+	0.682873i	$0.239270\pi$
$240$	0	0
$241$	−16.3412	−1.05263	−0.526315	−	0.850290i	$-0.676427\pi$
$241$	−16.3412	−1.05263	−0.526315	+	0.850290i	$0.676427\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.90321	−0.184727
$248$	0	0
$249$	−10.9748	−0.695500
$250$	0	0
$251$	10.6780	0.673989	0.336994	−	0.941507i	$-0.390590\pi$
$251$	10.6780	0.673989	0.336994	+	0.941507i	$0.390590\pi$
$252$	0	0
$253$	11.7605	0.739376
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.8622	0.677565	0.338783	−	0.940865i	$-0.389985\pi$
$257$	10.8622	0.677565	0.338783	+	0.940865i	$0.389985\pi$
$258$	0	0
$259$	−41.5353	−2.58088
$260$	0	0
$261$	−0.377784	−0.0233843
$262$	0	0
$263$	−16.2810	−1.00393	−0.501965	−	0.864888i	$-0.667389\pi$
$263$	−16.2810	−1.00393	−0.501965	+	0.864888i	$0.667389\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.5303	−1.13404
$268$	0	0
$269$	2.80642	0.171111	0.0855553	−	0.996333i	$-0.472734\pi$
$269$	2.80642	0.171111	0.0855553	+	0.996333i	$0.472734\pi$
$270$	0	0
$271$	−21.4637	−1.30383	−0.651913	−	0.758294i	$-0.726033\pi$
$271$	−21.4637	−1.30383	−0.651913	+	0.758294i	$0.726033\pi$
$272$	0	0
$273$	4.11753	0.249204
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.1941	1.87427	0.937134	−	0.348968i	$-0.113468\pi$
$277$	31.1941	1.87427	0.937134	+	0.348968i	$0.113468\pi$
$278$	0	0
$279$	−1.40790	−0.0842885
$280$	0	0
$281$	−17.3176	−1.03308	−0.516540	−	0.856263i	$-0.672780\pi$
$281$	−17.3176	−1.03308	−0.516540	+	0.856263i	$0.672780\pi$
$282$	0	0
$283$	21.5210	1.27929	0.639645	−	0.768671i	$-0.279081\pi$
$283$	21.5210	1.27929	0.639645	+	0.768671i	$0.279081\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.48394	0.323707
$288$	0	0
$289$	19.6128	1.15370
$290$	0	0
$291$	12.2351	0.717232
$292$	0	0
$293$	2.47013	0.144306	0.0721532	−	0.997394i	$-0.477013\pi$
$293$	2.47013	0.144306	0.0721532	+	0.997394i	$0.477013\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.21432	−0.186514
$298$	0	0
$299$	−3.65878	−0.211593
$300$	0	0
$301$	−27.0781	−1.56075
$302$	0	0
$303$	−8.03657	−0.461689
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.2810	0.929206	0.464603	−	0.885519i	$-0.346197\pi$
$307$	16.2810	0.929206	0.464603	+	0.885519i	$0.346197\pi$
$308$	0	0
$309$	−4.57628	−0.260336
$310$	0	0
$311$	11.4193	0.647527	0.323764	−	0.946138i	$-0.395052\pi$
$311$	11.4193	0.647527	0.323764	+	0.946138i	$0.395052\pi$
$312$	0	0
$313$	17.2208	0.973376	0.486688	−	0.873576i	$-0.338205\pi$
$313$	17.2208	0.973376	0.486688	+	0.873576i	$0.338205\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.00492	0.0564421	0.0282210	−	0.999602i	$-0.491016\pi$
$317$	1.00492	0.0564421	0.0282210	+	0.999602i	$0.491016\pi$
$318$	0	0
$319$	1.21432	0.0679889
$320$	0	0
$321$	−14.9906	−0.836695
$322$	0	0
$323$	17.5669	0.977449
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.2716	1.06572
$328$	0	0
$329$	33.4242	1.84274
$330$	0	0
$331$	−10.2494	−0.563355	−0.281678	−	0.959509i	$-0.590891\pi$
$331$	−10.2494	−0.563355	−0.281678	+	0.959509i	$0.590891\pi$
$332$	0	0
$333$	−10.0874	−0.552787
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.423717	0.0230814	0.0115407	−	0.999933i	$-0.496326\pi$
$337$	0.423717	0.0230814	0.0115407	+	0.999933i	$0.496326\pi$
$338$	0	0
$339$	6.04149	0.328129
$340$	0	0
$341$	4.52543	0.245066
$342$	0	0
$343$	12.1635	0.656765
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.4701	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$347$	−22.4701	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$348$	0	0
$349$	−6.38271	−0.341658	−0.170829	−	0.985301i	$-0.554645\pi$
$349$	−6.38271	−0.341658	−0.170829	+	0.985301i	$0.554645\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−11.7003	−0.622742	−0.311371	−	0.950288i	$-0.600788\pi$
$353$	−11.7003	−0.622742	−0.311371	+	0.950288i	$0.600788\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−24.9146	−1.31862
$358$	0	0
$359$	−4.99355	−0.263549	−0.131775	−	0.991280i	$-0.542068\pi$
$359$	−4.99355	−0.263549	−0.131775	+	0.991280i	$0.542068\pi$
$360$	0	0
$361$	−10.5714	−0.556387
$362$	0	0
$363$	−0.668149	−0.0350687
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.12399	−0.267470	−0.133735	−	0.991017i	$-0.542697\pi$
$367$	−5.12399	−0.267470	−0.133735	+	0.991017i	$0.542697\pi$
$368$	0	0
$369$	1.33185	0.0693334
$370$	0	0
$371$	−16.8687	−0.875777
$372$	0	0
$373$	−0.258721	−0.0133961	−0.00669804	−	0.999978i	$-0.502132\pi$
$373$	−0.258721	−0.0133961	−0.00669804	+	0.999978i	$0.502132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.377784	−0.0194569
$378$	0	0
$379$	18.0114	0.925182	0.462591	−	0.886572i	$-0.346920\pi$
$379$	18.0114	0.925182	0.462591	+	0.886572i	$0.346920\pi$
$380$	0	0
$381$	19.2716	0.987315
$382$	0	0
$383$	17.8336	0.911255	0.455628	−	0.890170i	$-0.349415\pi$
$383$	17.8336	0.911255	0.455628	+	0.890170i	$0.349415\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.57628	−0.334291
$388$	0	0
$389$	−8.32693	−0.422192	−0.211096	−	0.977465i	$-0.567703\pi$
$389$	−8.32693	−0.422192	−0.211096	+	0.977465i	$0.567703\pi$
$390$	0	0
$391$	22.1388	1.11960
$392$	0	0
$393$	−14.2494	−0.718785
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.2636	−0.816249	−0.408124	−	0.912926i	$-0.633817\pi$
$397$	−16.2636	−0.816249	−0.408124	+	0.912926i	$0.633817\pi$
$398$	0	0
$399$	−11.9541	−0.598452
$400$	0	0
$401$	−3.98571	−0.199037	−0.0995184	−	0.995036i	$-0.531730\pi$
$401$	−3.98571	−0.199037	−0.0995184	+	0.995036i	$0.531730\pi$
$402$	0	0
$403$	−1.40790	−0.0701323
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.4242	1.60721
$408$	0	0
$409$	−30.0687	−1.48680	−0.743400	−	0.668847i	$-0.766788\pi$
$409$	−30.0687	−1.48680	−0.743400	+	0.668847i	$0.766788\pi$
$410$	0	0
$411$	5.37778	0.265267
$412$	0	0
$413$	−58.5482	−2.88097
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.24443	0.354761
$418$	0	0
$419$	−30.9447	−1.51175	−0.755874	−	0.654717i	$-0.772788\pi$
$419$	−30.9447	−1.51175	−0.755874	+	0.654717i	$0.772788\pi$
$420$	0	0
$421$	23.2543	1.13334	0.566672	−	0.823943i	$-0.308231\pi$
$421$	23.2543	1.13334	0.566672	+	0.823943i	$0.308231\pi$
$422$	0	0
$423$	8.11753	0.394688
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.5877	−0.512373
$428$	0	0
$429$	−3.21432	−0.155189
$430$	0	0
$431$	34.9862	1.68523	0.842613	−	0.538520i	$-0.181016\pi$
$431$	34.9862	1.68523	0.842613	+	0.538520i	$0.181016\pi$
$432$	0	0
$433$	−35.0736	−1.68553	−0.842765	−	0.538282i	$-0.819074\pi$
$433$	−35.0736	−1.68553	−0.842765	+	0.538282i	$0.819074\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.6222	0.508129
$438$	0	0
$439$	17.6686	0.843277	0.421639	−	0.906764i	$-0.361455\pi$
$439$	17.6686	0.843277	0.421639	+	0.906764i	$0.361455\pi$
$440$	0	0
$441$	9.95407	0.474003
$442$	0	0
$443$	−13.2114	−0.627693	−0.313846	−	0.949474i	$-0.601618\pi$
$443$	−13.2114	−0.627693	−0.313846	+	0.949474i	$0.601618\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.4795	0.826752
$448$	0	0
$449$	−32.6450	−1.54061	−0.770306	−	0.637675i	$-0.779896\pi$
$449$	−32.6450	−1.54061	−0.770306	+	0.637675i	$0.779896\pi$
$450$	0	0
$451$	−4.28100	−0.201584
$452$	0	0
$453$	−22.6795	−1.06558
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.9037	−1.30528	−0.652640	−	0.757668i	$-0.726338\pi$
$457$	−27.9037	−1.30528	−0.652640	+	0.757668i	$0.726338\pi$
$458$	0	0
$459$	−6.05086	−0.282430
$460$	0	0
$461$	−25.5210	−1.18863	−0.594315	−	0.804232i	$-0.702577\pi$
$461$	−25.5210	−1.18863	−0.594315	+	0.804232i	$0.702577\pi$
$462$	0	0
$463$	0.0715987	0.00332747	0.00166374	−	0.999999i	$-0.499470\pi$
$463$	0.0715987	0.00332747	0.00166374	+	0.999999i	$0.499470\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.59364	0.120019	0.0600096	−	0.998198i	$-0.480887\pi$
$467$	2.59364	0.120019	0.0600096	+	0.998198i	$0.480887\pi$
$468$	0	0
$469$	−28.9081	−1.33485
$470$	0	0
$471$	13.3225	0.613868
$472$	0	0
$473$	21.1383	0.971939
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.09679	−0.187579
$478$	0	0
$479$	−14.1635	−0.647145	−0.323573	−	0.946203i	$-0.604884\pi$
$479$	−14.1635	−0.647145	−0.323573	+	0.946203i	$0.604884\pi$
$480$	0	0
$481$	−10.0874	−0.459947
$482$	0	0
$483$	−15.0651	−0.685488
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.9190	0.585417	0.292709	−	0.956202i	$-0.405443\pi$
$487$	12.9190	0.585417	0.292709	+	0.956202i	$0.405443\pi$
$488$	0	0
$489$	16.0558	0.726067
$490$	0	0
$491$	35.3274	1.59430	0.797152	−	0.603779i	$-0.206339\pi$
$491$	35.3274	1.59430	0.797152	+	0.603779i	$0.206339\pi$
$492$	0	0
$493$	2.28592	0.102953
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.6400	1.32954
$498$	0	0
$499$	−35.4449	−1.58673	−0.793367	−	0.608744i	$-0.791674\pi$
$499$	−35.4449	−1.58673	−0.793367	+	0.608744i	$0.791674\pi$
$500$	0	0
$501$	8.84299	0.395076
$502$	0	0
$503$	1.18865	0.0529995	0.0264997	−	0.999649i	$-0.491564\pi$
$503$	1.18865	0.0529995	0.0264997	+	0.999649i	$0.491564\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−10.0558	−0.445714	−0.222857	−	0.974851i	$-0.571538\pi$
$509$	−10.0558	−0.445714	−0.222857	+	0.974851i	$0.571538\pi$
$510$	0	0
$511$	−30.7382	−1.35978
$512$	0	0
$513$	−2.90321	−0.128180
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26.0923	−1.14754
$518$	0	0
$519$	−15.7699	−0.692220
$520$	0	0
$521$	−26.8573	−1.17664	−0.588319	−	0.808629i	$-0.700210\pi$
$521$	−26.8573	−1.17664	−0.588319	+	0.808629i	$0.700210\pi$
$522$	0	0
$523$	−14.3368	−0.626903	−0.313452	−	0.949604i	$-0.601485\pi$
$523$	−14.3368	−0.626903	−0.313452	+	0.949604i	$0.601485\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.51897	0.371092
$528$	0	0
$529$	−9.61332	−0.417971
$530$	0	0
$531$	−14.2192	−0.617063
$532$	0	0
$533$	1.33185	0.0576889
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.38271	0.102821
$538$	0	0
$539$	−31.9956	−1.37815
$540$	0	0
$541$	−18.9318	−0.813941	−0.406971	−	0.913441i	$-0.633415\pi$
$541$	−18.9318	−0.813941	−0.406971	+	0.913441i	$0.633415\pi$
$542$	0	0
$543$	−14.8207	−0.636018
$544$	0	0
$545$	0	0
$546$	0	0
$547$	45.7288	1.95522	0.977612	−	0.210415i	$-0.0674815\pi$
$547$	45.7288	1.95522	0.977612	+	0.210415i	$0.0674815\pi$
$548$	0	0
$549$	−2.57136	−0.109743
$550$	0	0
$551$	1.09679	0.0467247
$552$	0	0
$553$	29.3921	1.24988
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.3323	−1.87842	−0.939211	−	0.343342i	$-0.888441\pi$
$557$	−44.3323	−1.87842	−0.939211	+	0.343342i	$0.888441\pi$
$558$	0	0
$559$	−6.57628	−0.278147
$560$	0	0
$561$	19.4494	0.821154
$562$	0	0
$563$	−22.3511	−0.941985	−0.470993	−	0.882137i	$-0.656104\pi$
$563$	−22.3511	−0.941985	−0.470993	+	0.882137i	$0.656104\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.11753	0.172920
$568$	0	0
$569$	1.45584	0.0610318	0.0305159	−	0.999534i	$-0.490285\pi$
$569$	1.45584	0.0610318	0.0305159	+	0.999534i	$0.490285\pi$
$570$	0	0
$571$	7.12537	0.298187	0.149094	−	0.988823i	$-0.452364\pi$
$571$	7.12537	0.298187	0.149094	+	0.988823i	$0.452364\pi$
$572$	0	0
$573$	−8.10171	−0.338454
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.4197	1.76596	0.882979	−	0.469413i	$-0.155535\pi$
$577$	42.4197	1.76596	0.882979	+	0.469413i	$0.155535\pi$
$578$	0	0
$579$	−7.65878	−0.318288
$580$	0	0
$581$	−45.1891	−1.87476
$582$	0	0
$583$	13.1684	0.545379
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.59210	0.189536	0.0947682	−	0.995499i	$-0.469789\pi$
$587$	4.59210	0.189536	0.0947682	+	0.995499i	$0.469789\pi$
$588$	0	0
$589$	4.08742	0.168419
$590$	0	0
$591$	11.5254	0.474093
$592$	0	0
$593$	13.9081	0.571139	0.285569	−	0.958358i	$-0.407817\pi$
$593$	13.9081	0.571139	0.285569	+	0.958358i	$0.407817\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.5210	0.553377
$598$	0	0
$599$	−2.22077	−0.0907383	−0.0453692	−	0.998970i	$-0.514446\pi$
$599$	−2.22077	−0.0907383	−0.0453692	+	0.998970i	$0.514446\pi$
$600$	0	0
$601$	−3.54464	−0.144589	−0.0722944	−	0.997383i	$-0.523032\pi$
$601$	−3.54464	−0.144589	−0.0722944	+	0.997383i	$0.523032\pi$
$602$	0	0
$603$	−7.02074	−0.285907
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25.5714	1.03791	0.518955	−	0.854802i	$-0.326321\pi$
$607$	25.5714	1.03791	0.518955	+	0.854802i	$0.326321\pi$
$608$	0	0
$609$	−1.55554	−0.0630336
$610$	0	0
$611$	8.11753	0.328400
$612$	0	0
$613$	−28.5718	−1.15401	−0.577003	−	0.816742i	$-0.695778\pi$
$613$	−28.5718	−1.15401	−0.577003	+	0.816742i	$0.695778\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.67307	0.308906	0.154453	−	0.988000i	$-0.450638\pi$
$617$	7.67307	0.308906	0.154453	+	0.988000i	$0.450638\pi$
$618$	0	0
$619$	12.8716	0.517352	0.258676	−	0.965964i	$-0.416714\pi$
$619$	12.8716	0.517352	0.258676	+	0.965964i	$0.416714\pi$
$620$	0	0
$621$	−3.65878	−0.146822
$622$	0	0
$623$	−76.2993	−3.05687
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.33185	0.372678
$628$	0	0
$629$	61.0375	2.43373
$630$	0	0
$631$	−19.8938	−0.791961	−0.395981	−	0.918259i	$-0.629595\pi$
$631$	−19.8938	−0.791961	−0.395981	+	0.918259i	$0.629595\pi$
$632$	0	0
$633$	6.44293	0.256083
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.95407	0.394394
$638$	0	0
$639$	7.19850	0.284768
$640$	0	0
$641$	0.0508551	0.00200866	0.00100433	−	0.999999i	$-0.499680\pi$
$641$	0.0508551	0.00200866	0.00100433	+	0.999999i	$0.499680\pi$
$642$	0	0
$643$	34.5161	1.36118	0.680590	−	0.732664i	$-0.261723\pi$
$643$	34.5161	1.36118	0.680590	+	0.732664i	$0.261723\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.6321	1.44015	0.720077	−	0.693894i	$-0.244106\pi$
$647$	36.6321	1.44015	0.720077	+	0.693894i	$0.244106\pi$
$648$	0	0
$649$	45.7052	1.79409
$650$	0	0
$651$	−5.79706	−0.227205
$652$	0	0
$653$	−24.6860	−0.966037	−0.483018	−	0.875610i	$-0.660460\pi$
$653$	−24.6860	−0.966037	−0.483018	+	0.875610i	$0.660460\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.46520	−0.291245
$658$	0	0
$659$	−12.2623	−0.477670	−0.238835	−	0.971060i	$-0.576765\pi$
$659$	−12.2623	−0.477670	−0.238835	+	0.971060i	$0.576765\pi$
$660$	0	0
$661$	34.0973	1.32623	0.663115	−	0.748518i	$-0.269234\pi$
$661$	34.0973	1.32623	0.663115	+	0.748518i	$0.269234\pi$
$662$	0	0
$663$	−6.05086	−0.234996
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.38223	0.0535202
$668$	0	0
$669$	−2.51606	−0.0972765
$670$	0	0
$671$	8.26517	0.319074
$672$	0	0
$673$	−23.3926	−0.901717	−0.450858	−	0.892596i	$-0.648882\pi$
$673$	−23.3926	−0.901717	−0.450858	+	0.892596i	$0.648882\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.31756	0.358103	0.179051	−	0.983840i	$-0.442697\pi$
$677$	9.31756	0.358103	0.179051	+	0.983840i	$0.442697\pi$
$678$	0	0
$679$	50.3783	1.93334
$680$	0	0
$681$	2.22369	0.0852119
$682$	0	0
$683$	30.3669	1.16196	0.580978	−	0.813919i	$-0.302670\pi$
$683$	30.3669	1.16196	0.580978	+	0.813919i	$0.302670\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.48886	0.171261
$688$	0	0
$689$	−4.09679	−0.156075
$690$	0	0
$691$	−26.5462	−1.00986	−0.504932	−	0.863159i	$-0.668482\pi$
$691$	−26.5462	−1.00986	−0.504932	+	0.863159i	$0.668482\pi$
$692$	0	0
$693$	−13.2351	−0.502758
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.05884	−0.305250
$698$	0	0
$699$	10.8988	0.412229
$700$	0	0
$701$	−32.0923	−1.21211	−0.606056	−	0.795422i	$-0.707249\pi$
$701$	−32.0923	−1.21211	−0.606056	+	0.795422i	$0.707249\pi$
$702$	0	0
$703$	29.2859	1.10454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.0908	−1.24451
$708$	0	0
$709$	−29.3230	−1.10125	−0.550623	−	0.834754i	$-0.685610\pi$
$709$	−29.3230	−1.10125	−0.550623	+	0.834754i	$0.685610\pi$
$710$	0	0
$711$	7.13828	0.267706
$712$	0	0
$713$	5.15118	0.192913
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.5877	0.843552
$718$	0	0
$719$	−3.54416	−0.132175	−0.0660875	−	0.997814i	$-0.521052\pi$
$719$	−3.54416	−0.132175	−0.0660875	+	0.997814i	$0.521052\pi$
$720$	0	0
$721$	−18.8430	−0.701750
$722$	0	0
$723$	−16.3412	−0.607736
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20.2721	−0.751851	−0.375925	−	0.926650i	$-0.622675\pi$
$727$	−20.2721	−0.751851	−0.375925	+	0.926650i	$0.622675\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	39.7921	1.47177
$732$	0	0
$733$	33.5397	1.23882	0.619409	−	0.785069i	$-0.287372\pi$
$733$	33.5397	1.23882	0.619409	+	0.785069i	$0.287372\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.5669	0.831263
$738$	0	0
$739$	−49.0119	−1.80293	−0.901465	−	0.432852i	$-0.857507\pi$
$739$	−49.0119	−1.80293	−0.901465	+	0.432852i	$0.857507\pi$
$740$	0	0
$741$	−2.90321	−0.106652
$742$	0	0
$743$	35.6943	1.30950	0.654748	−	0.755847i	$-0.272775\pi$
$743$	35.6943	1.30950	0.654748	+	0.755847i	$0.272775\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.9748	−0.401547
$748$	0	0
$749$	−61.7244	−2.25536
$750$	0	0
$751$	−1.49378	−0.0545090	−0.0272545	−	0.999629i	$-0.508676\pi$
$751$	−1.49378	−0.0545090	−0.0272545	+	0.999629i	$0.508676\pi$
$752$	0	0
$753$	10.6780	0.389128
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.0830	−1.56588	−0.782939	−	0.622099i	$-0.786280\pi$
$757$	−43.0830	−1.56588	−0.782939	+	0.622099i	$0.786280\pi$
$758$	0	0
$759$	11.7605	0.426879
$760$	0	0
$761$	−5.57581	−0.202123	−0.101061	−	0.994880i	$-0.532224\pi$
$761$	−5.57581	−0.202123	−0.101061	+	0.994880i	$0.532224\pi$
$762$	0	0
$763$	79.3515	2.87272
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.2192	−0.513427
$768$	0	0
$769$	33.4608	1.20663	0.603313	−	0.797505i	$-0.293847\pi$
$769$	33.4608	1.20663	0.603313	+	0.797505i	$0.293847\pi$
$770$	0	0
$771$	10.8622	0.391193
$772$	0	0
$773$	−27.3778	−0.984710	−0.492355	−	0.870394i	$-0.663864\pi$
$773$	−27.3778	−0.984710	−0.492355	+	0.870394i	$0.663864\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−41.5353	−1.49007
$778$	0	0
$779$	−3.86665	−0.138537
$780$	0	0
$781$	−23.1383	−0.827953
$782$	0	0
$783$	−0.377784	−0.0135009
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.5288	1.08823	0.544117	−	0.839009i	$-0.316865\pi$
$787$	30.5288	1.08823	0.544117	+	0.839009i	$0.316865\pi$
$788$	0	0
$789$	−16.2810	−0.579619
$790$	0	0
$791$	24.8760	0.884489
$792$	0	0
$793$	−2.57136	−0.0913117
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.54909	−0.196559	−0.0982793	−	0.995159i	$-0.531334\pi$
$797$	−5.54909	−0.196559	−0.0982793	+	0.995159i	$0.531334\pi$
$798$	0	0
$799$	−49.1180	−1.73767
$800$	0	0
$801$	−18.5303	−0.654738
$802$	0	0
$803$	23.9956	0.846785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.80642	0.0987908
$808$	0	0
$809$	1.86665	0.0656278	0.0328139	−	0.999461i	$-0.489553\pi$
$809$	1.86665	0.0656278	0.0328139	+	0.999461i	$0.489553\pi$
$810$	0	0
$811$	0.721011	0.0253181	0.0126591	−	0.999920i	$-0.495970\pi$
$811$	0.721011	0.0253181	0.0126591	+	0.999920i	$0.495970\pi$
$812$	0	0
$813$	−21.4637	−0.752764
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.0923	0.667957
$818$	0	0
$819$	4.11753	0.143878
$820$	0	0
$821$	2.05884	0.0718540	0.0359270	−	0.999354i	$-0.488562\pi$
$821$	2.05884	0.0718540	0.0359270	+	0.999354i	$0.488562\pi$
$822$	0	0
$823$	−27.9210	−0.973266	−0.486633	−	0.873606i	$-0.661775\pi$
$823$	−27.9210	−0.973266	−0.486633	+	0.873606i	$0.661775\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.94761	0.172045	0.0860227	−	0.996293i	$-0.472584\pi$
$827$	4.94761	0.172045	0.0860227	+	0.996293i	$0.472584\pi$
$828$	0	0
$829$	−3.70519	−0.128687	−0.0643433	−	0.997928i	$-0.520495\pi$
$829$	−3.70519	−0.128687	−0.0643433	+	0.997928i	$0.520495\pi$
$830$	0	0
$831$	31.1941	1.08211
$832$	0	0
$833$	−60.2306	−2.08687
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.40790	−0.0486640
$838$	0	0
$839$	38.3956	1.32556	0.662782	−	0.748813i	$-0.269376\pi$
$839$	38.3956	1.32556	0.662782	+	0.748813i	$0.269376\pi$
$840$	0	0
$841$	−28.8573	−0.995079
$842$	0	0
$843$	−17.3176	−0.596448
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.75112	−0.0945297
$848$	0	0
$849$	21.5210	0.738598
$850$	0	0
$851$	36.9077	1.26518
$852$	0	0
$853$	41.5941	1.42416	0.712078	−	0.702101i	$-0.247754\pi$
$853$	41.5941	1.42416	0.712078	+	0.702101i	$0.247754\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.71456	0.126887	0.0634434	−	0.997985i	$-0.479792\pi$
$857$	3.71456	0.126887	0.0634434	+	0.997985i	$0.479792\pi$
$858$	0	0
$859$	−9.89384	−0.337574	−0.168787	−	0.985653i	$-0.553985\pi$
$859$	−9.89384	−0.337574	−0.168787	+	0.985653i	$0.553985\pi$
$860$	0	0
$861$	5.48394	0.186892
$862$	0	0
$863$	−25.7862	−0.877771	−0.438885	−	0.898543i	$-0.644627\pi$
$863$	−25.7862	−0.877771	−0.438885	+	0.898543i	$0.644627\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.6128	0.666087
$868$	0	0
$869$	−22.9447	−0.778346
$870$	0	0
$871$	−7.02074	−0.237889
$872$	0	0
$873$	12.2351	0.414094
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.5812	−1.67424	−0.837119	−	0.547021i	$-0.815762\pi$
$877$	−49.5812	−1.67424	−0.837119	+	0.547021i	$0.815762\pi$
$878$	0	0
$879$	2.47013	0.0833153
$880$	0	0
$881$	42.0593	1.41701	0.708507	−	0.705704i	$-0.249369\pi$
$881$	42.0593	1.41701	0.708507	+	0.705704i	$0.249369\pi$
$882$	0	0
$883$	−47.1753	−1.58758	−0.793788	−	0.608195i	$-0.791894\pi$
$883$	−47.1753	−1.58758	−0.793788	+	0.608195i	$0.791894\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0143	−1.00778	−0.503891	−	0.863767i	$-0.668099\pi$
$887$	−30.0143	−1.00778	−0.503891	+	0.863767i	$0.668099\pi$
$888$	0	0
$889$	79.3515	2.66137
$890$	0	0
$891$	−3.21432	−0.107684
$892$	0	0
$893$	−23.5669	−0.788637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.65878	−0.122163
$898$	0	0
$899$	0.531881	0.0177392
$900$	0	0
$901$	24.7891	0.825844
$902$	0	0
$903$	−27.0781	−0.901101
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.5941	1.11547	0.557737	−	0.830018i	$-0.311670\pi$
$907$	33.5941	1.11547	0.557737	+	0.830018i	$0.311670\pi$
$908$	0	0
$909$	−8.03657	−0.266556
$910$	0	0
$911$	−1.07805	−0.0357175	−0.0178587	−	0.999841i	$-0.505685\pi$
$911$	−1.07805	−0.0357175	−0.0178587	+	0.999841i	$0.505685\pi$
$912$	0	0
$913$	35.2766	1.16748
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−58.6722	−1.93753
$918$	0	0
$919$	−32.6133	−1.07581	−0.537907	−	0.843004i	$-0.680785\pi$
$919$	−32.6133	−1.07581	−0.537907	+	0.843004i	$0.680785\pi$
$920$	0	0
$921$	16.2810	0.536477
$922$	0	0
$923$	7.19850	0.236941
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.57628	−0.150305
$928$	0	0
$929$	2.35857	0.0773822	0.0386911	−	0.999251i	$-0.487681\pi$
$929$	2.35857	0.0773822	0.0386911	+	0.999251i	$0.487681\pi$
$930$	0	0
$931$	−28.8988	−0.947119
$932$	0	0
$933$	11.4193	0.373850
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.2859	−0.858724	−0.429362	−	0.903133i	$-0.641261\pi$
$937$	−26.2859	−0.858724	−0.429362	+	0.903133i	$0.641261\pi$
$938$	0	0
$939$	17.2208	0.561979
$940$	0	0
$941$	−7.07805	−0.230738	−0.115369	−	0.993323i	$-0.536805\pi$
$941$	−7.07805	−0.230738	−0.115369	+	0.993323i	$0.536805\pi$
$942$	0	0
$943$	−4.87295	−0.158685
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.3990	0.760365	0.380183	−	0.924911i	$-0.375861\pi$
$947$	23.3990	0.760365	0.380183	+	0.924911i	$0.375861\pi$
$948$	0	0
$949$	−7.46520	−0.242331
$950$	0	0
$951$	1.00492	0.0325868
$952$	0	0
$953$	44.6815	1.44738	0.723688	−	0.690127i	$-0.242445\pi$
$953$	44.6815	1.44738	0.723688	+	0.690127i	$0.242445\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.21432	0.0392534
$958$	0	0
$959$	22.1432	0.715041
$960$	0	0
$961$	−29.0178	−0.936059
$962$	0	0
$963$	−14.9906	−0.483066
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.5640	1.33661	0.668304	−	0.743888i	$-0.267021\pi$
$967$	41.5640	1.33661	0.668304	+	0.743888i	$0.267021\pi$
$968$	0	0
$969$	17.5669	0.564331
$970$	0	0
$971$	10.8988	0.349758	0.174879	−	0.984590i	$-0.444047\pi$
$971$	10.8988	0.349758	0.174879	+	0.984590i	$0.444047\pi$
$972$	0	0
$973$	29.8292	0.956279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.9688	1.31071	0.655355	−	0.755321i	$-0.272519\pi$
$977$	40.9688	1.31071	0.655355	+	0.755321i	$0.272519\pi$
$978$	0	0
$979$	59.5625	1.90362
$980$	0	0
$981$	19.2716	0.615296
$982$	0	0
$983$	47.5640	1.51706	0.758528	−	0.651640i	$-0.225919\pi$
$983$	47.5640	1.51706	0.758528	+	0.651640i	$0.225919\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33.4242	1.06390
$988$	0	0
$989$	24.0612	0.765101
$990$	0	0
$991$	−10.8845	−0.345757	−0.172878	−	0.984943i	$-0.555307\pi$
$991$	−10.8845	−0.345757	−0.172878	+	0.984943i	$0.555307\pi$
$992$	0	0
$993$	−10.2494	−0.325253
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.9684	−0.664075	−0.332037	−	0.943266i	$-0.607736\pi$
$997$	−20.9684	−0.664075	−0.332037	+	0.943266i	$0.607736\pi$
$998$	0	0
$999$	−10.0874	−0.319152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bo.1.3	yes	3
5.4	even	2		7800.2.a.bl.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bl.1.1	✓	3	5.4	even	2
7800.2.a.bo.1.3	yes	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.11753 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.11753 q^{7} +1.00000 q^{9} -3.21432 q^{11} +1.00000 q^{13} -6.05086 q^{17} -2.90321 q^{19} +4.11753 q^{21} -3.65878 q^{23} +1.00000 q^{27} -0.377784 q^{29} -1.40790 q^{31} -3.21432 q^{33} -10.0874 q^{37} +1.00000 q^{39} +1.33185 q^{41} -6.57628 q^{43} +8.11753 q^{47} +9.95407 q^{49} -6.05086 q^{51} -4.09679 q^{53} -2.90321 q^{57} -14.2192 q^{59} -2.57136 q^{61} +4.11753 q^{63} -7.02074 q^{67} -3.65878 q^{69} +7.19850 q^{71} -7.46520 q^{73} -13.2351 q^{77} +7.13828 q^{79} +1.00000 q^{81} -10.9748 q^{83} -0.377784 q^{87} -18.5303 q^{89} +4.11753 q^{91} -1.40790 q^{93} +12.2351 q^{97} -3.21432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} - 5 q^{17} - 2 q^{19} - q^{21} - 4 q^{23} + 3 q^{27} - q^{29} - 11 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 16 q^{41} + 11 q^{47} + 10 q^{49} - 5 q^{51} - 19 q^{53} - 2 q^{57} - 3 q^{59} - 21 q^{61} - q^{63} - q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - 13 q^{77} - 12 q^{79} + 3 q^{81} + 7 q^{83} - q^{87} - 16 q^{89} - q^{91} - 11 q^{93} + 10 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 - 5 * q^17 - 2 * q^19 - q^21 - 4 * q^23 + 3 * q^27 - q^29 - 11 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 16 * q^41 + 11 * q^47 + 10 * q^49 - 5 * q^51 - 19 * q^53 - 2 * q^57 - 3 * q^59 - 21 * q^61 - q^63 - q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - 13 * q^77 - 12 * q^79 + 3 * q^81 + 7 * q^83 - q^87 - 16 * q^89 - q^91 - 11 * q^93 + 10 * q^97 - 3 * q^99

Introduction

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