LMFDB - Embedded newform 7448.2.a.bo.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 $ x^7 - x^6 - 14x^5 + 13x^4 + 50x^3 - 53x^2 - 25*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.824576$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.824576 q^{3} +1.46996 q^{5} -2.32007 q^{9} +O(q^{10})$	$q-0.824576 q^{3} +1.46996 q^{5} -2.32007 q^{9} -0.500020 q^{11} -0.566709 q^{13} -1.21209 q^{15} +4.55510 q^{17} -1.00000 q^{19} +5.64733 q^{23} -2.83922 q^{25} +4.38680 q^{27} -7.45835 q^{29} -1.28341 q^{31} +0.412305 q^{33} -6.49950 q^{37} +0.467295 q^{39} +4.07796 q^{41} +1.50379 q^{43} -3.41041 q^{45} -11.8111 q^{47} -3.75603 q^{51} +12.8260 q^{53} -0.735009 q^{55} +0.824576 q^{57} -10.8733 q^{59} +7.81111 q^{61} -0.833039 q^{65} +8.38629 q^{67} -4.65665 q^{69} +1.43972 q^{71} -16.8196 q^{73} +2.34115 q^{75} +2.79455 q^{79} +3.34297 q^{81} +14.8707 q^{83} +6.69581 q^{85} +6.14998 q^{87} -13.4355 q^{89} +1.05827 q^{93} -1.46996 q^{95} +7.05323 q^{97} +1.16008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) $ 7 * q - q^3 - q^5 + 8 * q^9	$ 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) $ 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * 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
$3$	−0.824576	−0.476069	−0.238035	−	0.971257i	$-0.576503\pi$
$3$	−0.824576	−0.476069	−0.238035	+	0.971257i	$0.576503\pi$
$4$	0	0
$5$	1.46996	0.657385	0.328693	−	0.944437i	$-0.393392\pi$
$5$	1.46996	0.657385	0.328693	+	0.944437i	$0.393392\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.32007	−0.773358
$10$	0	0
$11$	−0.500020	−0.150762	−0.0753809	−	0.997155i	$-0.524017\pi$
$11$	−0.500020	−0.150762	−0.0753809	+	0.997155i	$0.524017\pi$
$12$	0	0
$13$	−0.566709	−0.157177	−0.0785885	−	0.996907i	$-0.525041\pi$
$13$	−0.566709	−0.157177	−0.0785885	+	0.996907i	$0.525041\pi$
$14$	0	0
$15$	−1.21209	−0.312961
$16$	0	0
$17$	4.55510	1.10478	0.552388	−	0.833587i	$-0.313717\pi$
$17$	4.55510	1.10478	0.552388	+	0.833587i	$0.313717\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.64733	1.17755	0.588775	−	0.808297i	$-0.299610\pi$
$23$	5.64733	1.17755	0.588775	+	0.808297i	$0.299610\pi$
$24$	0	0
$25$	−2.83922	−0.567845
$26$	0	0
$27$	4.38680	0.844241
$28$	0	0
$29$	−7.45835	−1.38498	−0.692491	−	0.721427i	$-0.743487\pi$
$29$	−7.45835	−1.38498	−0.692491	+	0.721427i	$0.743487\pi$
$30$	0	0
$31$	−1.28341	−0.230507	−0.115253	−	0.993336i	$-0.536768\pi$
$31$	−1.28341	−0.230507	−0.115253	+	0.993336i	$0.536768\pi$
$32$	0	0
$33$	0.412305	0.0717730
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.49950	−1.06851	−0.534256	−	0.845323i	$-0.679408\pi$
$37$	−6.49950	−1.06851	−0.534256	+	0.845323i	$0.679408\pi$
$38$	0	0
$39$	0.467295	0.0748271
$40$	0	0
$41$	4.07796	0.636871	0.318435	−	0.947945i	$-0.396843\pi$
$41$	4.07796	0.636871	0.318435	+	0.947945i	$0.396843\pi$
$42$	0	0
$43$	1.50379	0.229326	0.114663	−	0.993404i	$-0.463421\pi$
$43$	1.50379	0.229326	0.114663	+	0.993404i	$0.463421\pi$
$44$	0	0
$45$	−3.41041	−0.508394
$46$	0	0
$47$	−11.8111	−1.72283	−0.861414	−	0.507904i	$-0.830421\pi$
$47$	−11.8111	−1.72283	−0.861414	+	0.507904i	$0.830421\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.75603	−0.525949
$52$	0	0
$53$	12.8260	1.76179	0.880897	−	0.473309i	$-0.156940\pi$
$53$	12.8260	1.76179	0.880897	+	0.473309i	$0.156940\pi$
$54$	0	0
$55$	−0.735009	−0.0991086
$56$	0	0
$57$	0.824576	0.109218
$58$	0	0
$59$	−10.8733	−1.41559	−0.707793	−	0.706420i	$-0.750309\pi$
$59$	−10.8733	−1.41559	−0.707793	+	0.706420i	$0.750309\pi$
$60$	0	0
$61$	7.81111	1.00011	0.500055	−	0.865994i	$-0.333313\pi$
$61$	7.81111	1.00011	0.500055	+	0.865994i	$0.333313\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.833039	−0.103326
$66$	0	0
$67$	8.38629	1.02455	0.512274	−	0.858822i	$-0.328803\pi$
$67$	8.38629	1.02455	0.512274	+	0.858822i	$0.328803\pi$
$68$	0	0
$69$	−4.65665	−0.560595
$70$	0	0
$71$	1.43972	0.170864	0.0854318	−	0.996344i	$-0.472773\pi$
$71$	1.43972	0.170864	0.0854318	+	0.996344i	$0.472773\pi$
$72$	0	0
$73$	−16.8196	−1.96859	−0.984294	−	0.176539i	$-0.943510\pi$
$73$	−16.8196	−1.96859	−0.984294	+	0.176539i	$0.943510\pi$
$74$	0	0
$75$	2.34115	0.270333
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.79455	0.314412	0.157206	−	0.987566i	$-0.449751\pi$
$79$	2.79455	0.314412	0.157206	+	0.987566i	$0.449751\pi$
$80$	0	0
$81$	3.34297	0.371441
$82$	0	0
$83$	14.8707	1.63227	0.816136	−	0.577860i	$-0.196112\pi$
$83$	14.8707	1.63227	0.816136	+	0.577860i	$0.196112\pi$
$84$	0	0
$85$	6.69581	0.726263
$86$	0	0
$87$	6.14998	0.659347
$88$	0	0
$89$	−13.4355	−1.42416	−0.712079	−	0.702100i	$-0.752246\pi$
$89$	−13.4355	−1.42416	−0.712079	+	0.702100i	$0.752246\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.05827	0.109737
$94$	0	0
$95$	−1.46996	−0.150815
$96$	0	0
$97$	7.05323	0.716147	0.358073	−	0.933693i	$-0.383434\pi$
$97$	7.05323	0.716147	0.358073	+	0.933693i	$0.383434\pi$
$98$	0	0
$99$	1.16008	0.116593
$100$	0	0
$101$	−4.80886	−0.478500	−0.239250	−	0.970958i	$-0.576902\pi$
$101$	−4.80886	−0.478500	−0.239250	+	0.970958i	$0.576902\pi$
$102$	0	0
$103$	4.84327	0.477221	0.238611	−	0.971115i	$-0.423308\pi$
$103$	4.84327	0.477221	0.238611	+	0.971115i	$0.423308\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.1277	−1.26911	−0.634554	−	0.772879i	$-0.718816\pi$
$107$	−13.1277	−1.26911	−0.634554	+	0.772879i	$0.718816\pi$
$108$	0	0
$109$	−1.43218	−0.137178	−0.0685892	−	0.997645i	$-0.521850\pi$
$109$	−1.43218	−0.137178	−0.0685892	+	0.997645i	$0.521850\pi$
$110$	0	0
$111$	5.35933	0.508685
$112$	0	0
$113$	−4.26248	−0.400981	−0.200490	−	0.979696i	$-0.564253\pi$
$113$	−4.26248	−0.400981	−0.200490	+	0.979696i	$0.564253\pi$
$114$	0	0
$115$	8.30135	0.774104
$116$	0	0
$117$	1.31481	0.121554
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.7500	−0.977271
$122$	0	0
$123$	−3.36259	−0.303194
$124$	0	0
$125$	−11.5233	−1.03068
$126$	0	0
$127$	5.32388	0.472418	0.236209	−	0.971702i	$-0.424095\pi$
$127$	5.32388	0.472418	0.236209	+	0.971702i	$0.424095\pi$
$128$	0	0
$129$	−1.23999	−0.109175
$130$	0	0
$131$	−19.2686	−1.68350	−0.841752	−	0.539864i	$-0.818476\pi$
$131$	−19.2686	−1.68350	−0.841752	+	0.539864i	$0.818476\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.44842	0.554992
$136$	0	0
$137$	−11.0396	−0.943177	−0.471589	−	0.881819i	$-0.656319\pi$
$137$	−11.0396	−0.943177	−0.471589	+	0.881819i	$0.656319\pi$
$138$	0	0
$139$	−5.09816	−0.432421	−0.216210	−	0.976347i	$-0.569370\pi$
$139$	−5.09816	−0.432421	−0.216210	+	0.976347i	$0.569370\pi$
$140$	0	0
$141$	9.73916	0.820185
$142$	0	0
$143$	0.283366	0.0236963
$144$	0	0
$145$	−10.9635	−0.910466
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.5039	−1.67975	−0.839873	−	0.542783i	$-0.817371\pi$
$149$	−20.5039	−1.67975	−0.839873	+	0.542783i	$0.817371\pi$
$150$	0	0
$151$	23.8383	1.93993	0.969967	−	0.243238i	$-0.0782096\pi$
$151$	23.8383	1.93993	0.969967	+	0.243238i	$0.0782096\pi$
$152$	0	0
$153$	−10.5682	−0.854387
$154$	0	0
$155$	−1.88655	−0.151532
$156$	0	0
$157$	0.363216	0.0289878	0.0144939	−	0.999895i	$-0.495386\pi$
$157$	0.363216	0.0289878	0.0144939	+	0.999895i	$0.495386\pi$
$158$	0	0
$159$	−10.5760	−0.838735
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.0550	1.64916	0.824578	−	0.565748i	$-0.191413\pi$
$163$	21.0550	1.64916	0.824578	+	0.565748i	$0.191413\pi$
$164$	0	0
$165$	0.606071	0.0471825
$166$	0	0
$167$	3.35996	0.260001	0.130001	−	0.991514i	$-0.458502\pi$
$167$	3.35996	0.260001	0.130001	+	0.991514i	$0.458502\pi$
$168$	0	0
$169$	−12.6788	−0.975295
$170$	0	0
$171$	2.32007	0.177421
$172$	0	0
$173$	−12.2803	−0.933651	−0.466825	−	0.884350i	$-0.654602\pi$
$173$	−12.2803	−0.933651	−0.466825	+	0.884350i	$0.654602\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.96588	0.673917
$178$	0	0
$179$	10.6624	0.796943	0.398472	−	0.917181i	$-0.369541\pi$
$179$	10.6624	0.796943	0.398472	+	0.917181i	$0.369541\pi$
$180$	0	0
$181$	−1.41679	−0.105309	−0.0526544	−	0.998613i	$-0.516768\pi$
$181$	−1.41679	−0.105309	−0.0526544	+	0.998613i	$0.516768\pi$
$182$	0	0
$183$	−6.44085	−0.476122
$184$	0	0
$185$	−9.55399	−0.702424
$186$	0	0
$187$	−2.27765	−0.166558
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.4037	−1.40400	−0.702001	−	0.712176i	$-0.747710\pi$
$191$	−19.4037	−1.40400	−0.702001	+	0.712176i	$0.747710\pi$
$192$	0	0
$193$	−4.60376	−0.331386	−0.165693	−	0.986177i	$-0.552986\pi$
$193$	−4.60376	−0.331386	−0.165693	+	0.986177i	$0.552986\pi$
$194$	0	0
$195$	0.686904	0.0491902
$196$	0	0
$197$	2.81443	0.200520	0.100260	−	0.994961i	$-0.468033\pi$
$197$	2.81443	0.200520	0.100260	+	0.994961i	$0.468033\pi$
$198$	0	0
$199$	−11.7736	−0.834610	−0.417305	−	0.908766i	$-0.637025\pi$
$199$	−11.7736	−0.834610	−0.417305	+	0.908766i	$0.637025\pi$
$200$	0	0
$201$	−6.91513	−0.487756
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.99443	0.418669
$206$	0	0
$207$	−13.1022	−0.910669
$208$	0	0
$209$	0.500020	0.0345871
$210$	0	0
$211$	7.24807	0.498978	0.249489	−	0.968378i	$-0.419737\pi$
$211$	7.24807	0.498978	0.249489	+	0.968378i	$0.419737\pi$
$212$	0	0
$213$	−1.18716	−0.0813429
$214$	0	0
$215$	2.21051	0.150755
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13.8690	0.937183
$220$	0	0
$221$	−2.58142	−0.173645
$222$	0	0
$223$	−13.1011	−0.877317	−0.438659	−	0.898654i	$-0.644546\pi$
$223$	−13.1011	−0.877317	−0.438659	+	0.898654i	$0.644546\pi$
$224$	0	0
$225$	6.58721	0.439147
$226$	0	0
$227$	0.467908	0.0310561	0.0155281	−	0.999879i	$-0.495057\pi$
$227$	0.467908	0.0310561	0.0155281	+	0.999879i	$0.495057\pi$
$228$	0	0
$229$	−18.9033	−1.24917	−0.624583	−	0.780959i	$-0.714731\pi$
$229$	−18.9033	−1.24917	−0.624583	+	0.780959i	$0.714731\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.8170	1.10172	0.550859	−	0.834599i	$-0.314300\pi$
$233$	16.8170	1.10172	0.550859	+	0.834599i	$0.314300\pi$
$234$	0	0
$235$	−17.3618	−1.13256
$236$	0	0
$237$	−2.30432	−0.149682
$238$	0	0
$239$	2.84661	0.184132	0.0920660	−	0.995753i	$-0.470653\pi$
$239$	2.84661	0.184132	0.0920660	+	0.995753i	$0.470653\pi$
$240$	0	0
$241$	−26.1732	−1.68597	−0.842983	−	0.537940i	$-0.819203\pi$
$241$	−26.1732	−1.68597	−0.842983	+	0.537940i	$0.819203\pi$
$242$	0	0
$243$	−15.9169	−1.02107
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.566709	0.0360589
$248$	0	0
$249$	−12.2620	−0.777074
$250$	0	0
$251$	22.7254	1.43441	0.717207	−	0.696860i	$-0.245420\pi$
$251$	22.7254	1.43441	0.717207	+	0.696860i	$0.245420\pi$
$252$	0	0
$253$	−2.82378	−0.177530
$254$	0	0
$255$	−5.52120	−0.345751
$256$	0	0
$257$	10.0510	0.626965	0.313482	−	0.949594i	$-0.398504\pi$
$257$	10.0510	0.626965	0.313482	+	0.949594i	$0.398504\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	17.3039	1.07109
$262$	0	0
$263$	−12.4111	−0.765300	−0.382650	−	0.923893i	$-0.624988\pi$
$263$	−12.4111	−0.765300	−0.382650	+	0.923893i	$0.624988\pi$
$264$	0	0
$265$	18.8538	1.15818
$266$	0	0
$267$	11.0786	0.677997
$268$	0	0
$269$	−9.25058	−0.564018	−0.282009	−	0.959412i	$-0.591001\pi$
$269$	−9.25058	−0.564018	−0.282009	+	0.959412i	$0.591001\pi$
$270$	0	0
$271$	−31.6248	−1.92107	−0.960535	−	0.278160i	$-0.910275\pi$
$271$	−31.6248	−1.92107	−0.960535	+	0.278160i	$0.910275\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.41967	0.0856093
$276$	0	0
$277$	13.1740	0.791551	0.395776	−	0.918347i	$-0.370476\pi$
$277$	13.1740	0.791551	0.395776	+	0.918347i	$0.370476\pi$
$278$	0	0
$279$	2.97760	0.178264
$280$	0	0
$281$	19.7332	1.17718	0.588591	−	0.808431i	$-0.299683\pi$
$281$	19.7332	1.17718	0.588591	+	0.808431i	$0.299683\pi$
$282$	0	0
$283$	−8.32769	−0.495030	−0.247515	−	0.968884i	$-0.579614\pi$
$283$	−8.32769	−0.495030	−0.247515	+	0.968884i	$0.579614\pi$
$284$	0	0
$285$	1.21209	0.0717981
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.74898	0.220528
$290$	0	0
$291$	−5.81592	−0.340935
$292$	0	0
$293$	14.3314	0.837252	0.418626	−	0.908159i	$-0.362512\pi$
$293$	14.3314	0.837252	0.418626	+	0.908159i	$0.362512\pi$
$294$	0	0
$295$	−15.9833	−0.930585
$296$	0	0
$297$	−2.19349	−0.127279
$298$	0	0
$299$	−3.20040	−0.185084
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.96527	0.227799
$304$	0	0
$305$	11.4820	0.657458
$306$	0	0
$307$	2.76323	0.157706	0.0788529	−	0.996886i	$-0.474874\pi$
$307$	2.76323	0.157706	0.0788529	+	0.996886i	$0.474874\pi$
$308$	0	0
$309$	−3.99364	−0.227190
$310$	0	0
$311$	4.41422	0.250307	0.125154	−	0.992137i	$-0.460058\pi$
$311$	4.41422	0.250307	0.125154	+	0.992137i	$0.460058\pi$
$312$	0	0
$313$	25.4639	1.43931	0.719653	−	0.694334i	$-0.244301\pi$
$313$	25.4639	1.43931	0.719653	+	0.694334i	$0.244301\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.6542	−1.49705	−0.748525	−	0.663106i	$-0.769238\pi$
$317$	−26.6542	−1.49705	−0.748525	+	0.663106i	$0.769238\pi$
$318$	0	0
$319$	3.72933	0.208802
$320$	0	0
$321$	10.8248	0.604183
$322$	0	0
$323$	−4.55510	−0.253453
$324$	0	0
$325$	1.60901	0.0892521
$326$	0	0
$327$	1.18094	0.0653064
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.46537	0.135509	0.0677546	−	0.997702i	$-0.478417\pi$
$331$	2.46537	0.135509	0.0677546	+	0.997702i	$0.478417\pi$
$332$	0	0
$333$	15.0793	0.826342
$334$	0	0
$335$	12.3275	0.673523
$336$	0	0
$337$	−11.9893	−0.653099	−0.326549	−	0.945180i	$-0.605886\pi$
$337$	−11.9893	−0.653099	−0.326549	+	0.945180i	$0.605886\pi$
$338$	0	0
$339$	3.51474	0.190894
$340$	0	0
$341$	0.641730	0.0347516
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.84509	−0.368527
$346$	0	0
$347$	5.04848	0.271017	0.135508	−	0.990776i	$-0.456733\pi$
$347$	5.04848	0.271017	0.135508	+	0.990776i	$0.456733\pi$
$348$	0	0
$349$	−21.4869	−1.15017	−0.575083	−	0.818095i	$-0.695030\pi$
$349$	−21.4869	−1.15017	−0.575083	+	0.818095i	$0.695030\pi$
$350$	0	0
$351$	−2.48604	−0.132695
$352$	0	0
$353$	−33.8358	−1.80090	−0.900449	−	0.434962i	$-0.856762\pi$
$353$	−33.8358	−1.80090	−0.900449	+	0.434962i	$0.856762\pi$
$354$	0	0
$355$	2.11633	0.112323
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3540	0.599239	0.299620	−	0.954059i	$-0.403140\pi$
$359$	11.3540	0.599239	0.299620	+	0.954059i	$0.403140\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.86417	0.465248
$364$	0	0
$365$	−24.7241	−1.29412
$366$	0	0
$367$	5.21186	0.272057	0.136028	−	0.990705i	$-0.456566\pi$
$367$	5.21186	0.272057	0.136028	+	0.990705i	$0.456566\pi$
$368$	0	0
$369$	−9.46118	−0.492529
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.16058	−0.111871	−0.0559353	−	0.998434i	$-0.517814\pi$
$373$	−2.16058	−0.111871	−0.0559353	+	0.998434i	$0.517814\pi$
$374$	0	0
$375$	9.50186	0.490674
$376$	0	0
$377$	4.22672	0.217687
$378$	0	0
$379$	32.7737	1.68347	0.841737	−	0.539888i	$-0.181533\pi$
$379$	32.7737	1.68347	0.841737	+	0.539888i	$0.181533\pi$
$380$	0	0
$381$	−4.38995	−0.224904
$382$	0	0
$383$	8.96618	0.458150	0.229075	−	0.973409i	$-0.426430\pi$
$383$	8.96618	0.458150	0.229075	+	0.973409i	$0.426430\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.48890	−0.177351
$388$	0	0
$389$	−17.5899	−0.891845	−0.445923	−	0.895072i	$-0.647124\pi$
$389$	−17.5899	−0.891845	−0.445923	+	0.895072i	$0.647124\pi$
$390$	0	0
$391$	25.7242	1.30093
$392$	0	0
$393$	15.8884	0.801464
$394$	0	0
$395$	4.10788	0.206690
$396$	0	0
$397$	−25.0146	−1.25545	−0.627723	−	0.778437i	$-0.716013\pi$
$397$	−25.0146	−1.25545	−0.627723	+	0.778437i	$0.716013\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.7358	−0.935621	−0.467810	−	0.883829i	$-0.654957\pi$
$401$	−18.7358	−0.935621	−0.467810	+	0.883829i	$0.654957\pi$
$402$	0	0
$403$	0.727319	0.0362303
$404$	0	0
$405$	4.91403	0.244180
$406$	0	0
$407$	3.24988	0.161091
$408$	0	0
$409$	−26.8066	−1.32550	−0.662750	−	0.748841i	$-0.730611\pi$
$409$	−26.8066	−1.32550	−0.662750	+	0.748841i	$0.730611\pi$
$410$	0	0
$411$	9.10299	0.449017
$412$	0	0
$413$	0	0
$414$	0	0
$415$	21.8593	1.07303
$416$	0	0
$417$	4.20382	0.205862
$418$	0	0
$419$	−17.4413	−0.852065	−0.426033	−	0.904708i	$-0.640089\pi$
$419$	−17.4413	−0.852065	−0.426033	+	0.904708i	$0.640089\pi$
$420$	0	0
$421$	−5.06222	−0.246717	−0.123359	−	0.992362i	$-0.539367\pi$
$421$	−5.06222	−0.246717	−0.123359	+	0.992362i	$0.539367\pi$
$422$	0	0
$423$	27.4027	1.33236
$424$	0	0
$425$	−12.9330	−0.627341
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.233657	−0.0112811
$430$	0	0
$431$	−37.5380	−1.80814	−0.904071	−	0.427382i	$-0.859436\pi$
$431$	−37.5380	−1.80814	−0.904071	+	0.427382i	$0.859436\pi$
$432$	0	0
$433$	−3.92429	−0.188590	−0.0942948	−	0.995544i	$-0.530060\pi$
$433$	−3.92429	−0.188590	−0.0942948	+	0.995544i	$0.530060\pi$
$434$	0	0
$435$	9.04021	0.433445
$436$	0	0
$437$	−5.64733	−0.270149
$438$	0	0
$439$	−9.66387	−0.461231	−0.230616	−	0.973045i	$-0.574074\pi$
$439$	−9.66387	−0.461231	−0.230616	+	0.973045i	$0.574074\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.50149	−0.0713380	−0.0356690	−	0.999364i	$-0.511356\pi$
$443$	−1.50149	−0.0713380	−0.0356690	+	0.999364i	$0.511356\pi$
$444$	0	0
$445$	−19.7496	−0.936220
$446$	0	0
$447$	16.9070	0.799675
$448$	0	0
$449$	22.8025	1.07612	0.538058	−	0.842908i	$-0.319158\pi$
$449$	22.8025	1.07612	0.538058	+	0.842908i	$0.319158\pi$
$450$	0	0
$451$	−2.03906	−0.0960158
$452$	0	0
$453$	−19.6565	−0.923542
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.4100	1.23541	0.617704	−	0.786411i	$-0.288063\pi$
$457$	26.4100	1.23541	0.617704	+	0.786411i	$0.288063\pi$
$458$	0	0
$459$	19.9824	0.932696
$460$	0	0
$461$	−31.0967	−1.44832	−0.724160	−	0.689632i	$-0.757772\pi$
$461$	−31.0967	−1.44832	−0.724160	+	0.689632i	$0.757772\pi$
$462$	0	0
$463$	19.8323	0.921687	0.460844	−	0.887481i	$-0.347547\pi$
$463$	19.8323	0.921687	0.460844	+	0.887481i	$0.347547\pi$
$464$	0	0
$465$	1.55561	0.0721396
$466$	0	0
$467$	−30.0413	−1.39015	−0.695073	−	0.718939i	$-0.744628\pi$
$467$	−30.0413	−1.39015	−0.695073	+	0.718939i	$0.744628\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.299499	−0.0138002
$472$	0	0
$473$	−0.751926	−0.0345736
$474$	0	0
$475$	2.83922	0.130272
$476$	0	0
$477$	−29.7574	−1.36250
$478$	0	0
$479$	3.26282	0.149082	0.0745411	−	0.997218i	$-0.476251\pi$
$479$	3.26282	0.149082	0.0745411	+	0.997218i	$0.476251\pi$
$480$	0	0
$481$	3.68333	0.167945
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.3679	0.470784
$486$	0	0
$487$	−15.2659	−0.691765	−0.345883	−	0.938278i	$-0.612421\pi$
$487$	−15.2659	−0.691765	−0.345883	+	0.938278i	$0.612421\pi$
$488$	0	0
$489$	−17.3615	−0.785112
$490$	0	0
$491$	−4.58118	−0.206746	−0.103373	−	0.994643i	$-0.532964\pi$
$491$	−4.58118	−0.206746	−0.103373	+	0.994643i	$0.532964\pi$
$492$	0	0
$493$	−33.9736	−1.53009
$494$	0	0
$495$	1.70528	0.0766465
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−25.8383	−1.15668	−0.578340	−	0.815796i	$-0.696299\pi$
$499$	−25.8383	−1.15668	−0.578340	+	0.815796i	$0.696299\pi$
$500$	0	0
$501$	−2.77054	−0.123778
$502$	0	0
$503$	44.1618	1.96908	0.984539	−	0.175166i	$-0.0560462\pi$
$503$	44.1618	1.96908	0.984539	+	0.175166i	$0.0560462\pi$
$504$	0	0
$505$	−7.06883	−0.314559
$506$	0	0
$507$	10.4547	0.464308
$508$	0	0
$509$	24.5022	1.08604	0.543020	−	0.839720i	$-0.317281\pi$
$509$	24.5022	1.08604	0.543020	+	0.839720i	$0.317281\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.38680	−0.193682
$514$	0	0
$515$	7.11940	0.313718
$516$	0	0
$517$	5.90580	0.259737
$518$	0	0
$519$	10.1260	0.444482
$520$	0	0
$521$	−14.8210	−0.649320	−0.324660	−	0.945831i	$-0.605250\pi$
$521$	−14.8210	−0.649320	−0.324660	+	0.945831i	$0.605250\pi$
$522$	0	0
$523$	−38.4495	−1.68128	−0.840639	−	0.541595i	$-0.817821\pi$
$523$	−38.4495	−1.68128	−0.840639	+	0.541595i	$0.817821\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.84605	−0.254658
$528$	0	0
$529$	8.89238	0.386625
$530$	0	0
$531$	25.2269	1.09476
$532$	0	0
$533$	−2.31102	−0.100101
$534$	0	0
$535$	−19.2972	−0.834292
$536$	0	0
$537$	−8.79194	−0.379400
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13.4048	−0.576319	−0.288159	−	0.957582i	$-0.593043\pi$
$541$	−13.4048	−0.576319	−0.288159	+	0.957582i	$0.593043\pi$
$542$	0	0
$543$	1.16825	0.0501343
$544$	0	0
$545$	−2.10525	−0.0901791
$546$	0	0
$547$	−21.0090	−0.898279	−0.449139	−	0.893462i	$-0.648269\pi$
$547$	−21.0090	−0.898279	−0.449139	+	0.893462i	$0.648269\pi$
$548$	0	0
$549$	−18.1224	−0.773444
$550$	0	0
$551$	7.45835	0.317737
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.87799	0.334402
$556$	0	0
$557$	3.15955	0.133875	0.0669373	−	0.997757i	$-0.478677\pi$
$557$	3.15955	0.133875	0.0669373	+	0.997757i	$0.478677\pi$
$558$	0	0
$559$	−0.852212	−0.0360447
$560$	0	0
$561$	1.87809	0.0792931
$562$	0	0
$563$	33.6928	1.41998	0.709990	−	0.704211i	$-0.248699\pi$
$563$	33.6928	1.41998	0.709990	+	0.704211i	$0.248699\pi$
$564$	0	0
$565$	−6.26567	−0.263599
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.9420	1.33908	0.669539	−	0.742777i	$-0.266492\pi$
$569$	31.9420	1.33908	0.669539	+	0.742777i	$0.266492\pi$
$570$	0	0
$571$	13.6664	0.571922	0.285961	−	0.958241i	$-0.407687\pi$
$571$	13.6664	0.571922	0.285961	+	0.958241i	$0.407687\pi$
$572$	0	0
$573$	15.9998	0.668402
$574$	0	0
$575$	−16.0340	−0.668666
$576$	0	0
$577$	−0.650281	−0.0270716	−0.0135358	−	0.999908i	$-0.504309\pi$
$577$	−0.650281	−0.0270716	−0.0135358	+	0.999908i	$0.504309\pi$
$578$	0	0
$579$	3.79615	0.157762
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.41329	−0.265611
$584$	0	0
$585$	1.93271	0.0799079
$586$	0	0
$587$	−15.3982	−0.635550	−0.317775	−	0.948166i	$-0.602936\pi$
$587$	−15.3982	−0.635550	−0.317775	+	0.948166i	$0.602936\pi$
$588$	0	0
$589$	1.28341	0.0528819
$590$	0	0
$591$	−2.32071	−0.0954612
$592$	0	0
$593$	−31.3015	−1.28540	−0.642699	−	0.766119i	$-0.722185\pi$
$593$	−31.3015	−1.28540	−0.642699	+	0.766119i	$0.722185\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.70825	0.397332
$598$	0	0
$599$	−30.7464	−1.25627	−0.628133	−	0.778106i	$-0.716180\pi$
$599$	−30.7464	−1.25627	−0.628133	+	0.778106i	$0.716180\pi$
$600$	0	0
$601$	−7.53574	−0.307389	−0.153695	−	0.988118i	$-0.549117\pi$
$601$	−7.53574	−0.307389	−0.153695	+	0.988118i	$0.549117\pi$
$602$	0	0
$603$	−19.4568	−0.792343
$604$	0	0
$605$	−15.8020	−0.642443
$606$	0	0
$607$	−33.6774	−1.36692	−0.683461	−	0.729987i	$-0.739526\pi$
$607$	−33.6774	−1.36692	−0.683461	+	0.729987i	$0.739526\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.69347	0.270789
$612$	0	0
$613$	−7.38873	−0.298428	−0.149214	−	0.988805i	$-0.547674\pi$
$613$	−7.38873	−0.298428	−0.149214	+	0.988805i	$0.547674\pi$
$614$	0	0
$615$	−4.94286	−0.199316
$616$	0	0
$617$	−8.16793	−0.328829	−0.164414	−	0.986391i	$-0.552573\pi$
$617$	−8.16793	−0.328829	−0.164414	+	0.986391i	$0.552573\pi$
$618$	0	0
$619$	−7.77403	−0.312465	−0.156232	−	0.987720i	$-0.549935\pi$
$619$	−7.77403	−0.312465	−0.156232	+	0.987720i	$0.549935\pi$
$620$	0	0
$621$	24.7738	0.994136
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2.74270	−0.109708
$626$	0	0
$627$	−0.412305	−0.0164659
$628$	0	0
$629$	−29.6059	−1.18046
$630$	0	0
$631$	−38.9982	−1.55250	−0.776248	−	0.630428i	$-0.782879\pi$
$631$	−38.9982	−1.55250	−0.776248	+	0.630428i	$0.782879\pi$
$632$	0	0
$633$	−5.97658	−0.237548
$634$	0	0
$635$	7.82589	0.310561
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.34026	−0.132139
$640$	0	0
$641$	−25.6900	−1.01469	−0.507347	−	0.861742i	$-0.669374\pi$
$641$	−25.6900	−1.01469	−0.507347	+	0.861742i	$0.669374\pi$
$642$	0	0
$643$	15.9712	0.629843	0.314921	−	0.949118i	$-0.398022\pi$
$643$	15.9712	0.629843	0.314921	+	0.949118i	$0.398022\pi$
$644$	0	0
$645$	−1.82273	−0.0717700
$646$	0	0
$647$	−9.37120	−0.368420	−0.184210	−	0.982887i	$-0.558973\pi$
$647$	−9.37120	−0.368420	−0.184210	+	0.982887i	$0.558973\pi$
$648$	0	0
$649$	5.43688	0.213416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.83567	0.189234	0.0946172	−	0.995514i	$-0.469837\pi$
$653$	4.83567	0.189234	0.0946172	+	0.995514i	$0.469837\pi$
$654$	0	0
$655$	−28.3240	−1.10671
$656$	0	0
$657$	39.0228	1.52242
$658$	0	0
$659$	−13.1087	−0.510644	−0.255322	−	0.966856i	$-0.582181\pi$
$659$	−13.1087	−0.510644	−0.255322	+	0.966856i	$0.582181\pi$
$660$	0	0
$661$	8.08722	0.314556	0.157278	−	0.987554i	$-0.449728\pi$
$661$	8.08722	0.314556	0.157278	+	0.987554i	$0.449728\pi$
$662$	0	0
$663$	2.12858	0.0826671
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.1198	−1.63089
$668$	0	0
$669$	10.8029	0.417664
$670$	0	0
$671$	−3.90572	−0.150778
$672$	0	0
$673$	−18.9893	−0.731982	−0.365991	−	0.930618i	$-0.619270\pi$
$673$	−18.9893	−0.731982	−0.365991	+	0.930618i	$0.619270\pi$
$674$	0	0
$675$	−12.4551	−0.479398
$676$	0	0
$677$	38.7895	1.49080	0.745401	−	0.666616i	$-0.232258\pi$
$677$	38.7895	1.49080	0.745401	+	0.666616i	$0.232258\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.385826	−0.0147849
$682$	0	0
$683$	12.7177	0.486627	0.243314	−	0.969948i	$-0.421766\pi$
$683$	12.7177	0.486627	0.243314	+	0.969948i	$0.421766\pi$
$684$	0	0
$685$	−16.2278	−0.620031
$686$	0	0
$687$	15.5872	0.594689
$688$	0	0
$689$	−7.26864	−0.276913
$690$	0	0
$691$	20.8065	0.791518	0.395759	−	0.918354i	$-0.370482\pi$
$691$	20.8065	0.791518	0.395759	+	0.918354i	$0.370482\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.49409	−0.284267
$696$	0	0
$697$	18.5755	0.703599
$698$	0	0
$699$	−13.8669	−0.524493
$700$	0	0
$701$	−27.1326	−1.02478	−0.512392	−	0.858752i	$-0.671240\pi$
$701$	−27.1326	−1.02478	−0.512392	+	0.858752i	$0.671240\pi$
$702$	0	0
$703$	6.49950	0.245133
$704$	0	0
$705$	14.3162	0.539178
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.0838	0.866929	0.433464	−	0.901171i	$-0.357291\pi$
$709$	23.0838	0.866929	0.433464	+	0.901171i	$0.357291\pi$
$710$	0	0
$711$	−6.48358	−0.243153
$712$	0	0
$713$	−7.24783	−0.271433
$714$	0	0
$715$	0.416537	0.0155776
$716$	0	0
$717$	−2.34725	−0.0876595
$718$	0	0
$719$	15.7757	0.588334	0.294167	−	0.955754i	$-0.404958\pi$
$719$	15.7757	0.588334	0.294167	+	0.955754i	$0.404958\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.5818	0.802636
$724$	0	0
$725$	21.1759	0.786454
$726$	0	0
$727$	−10.9589	−0.406443	−0.203222	−	0.979133i	$-0.565141\pi$
$727$	−10.9589	−0.406443	−0.203222	+	0.979133i	$0.565141\pi$
$728$	0	0
$729$	3.09581	0.114660
$730$	0	0
$731$	6.84992	0.253353
$732$	0	0
$733$	14.1398	0.522264	0.261132	−	0.965303i	$-0.415904\pi$
$733$	14.1398	0.522264	0.261132	+	0.965303i	$0.415904\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.19331	−0.154463
$738$	0	0
$739$	−8.18284	−0.301011	−0.150505	−	0.988609i	$-0.548090\pi$
$739$	−8.18284	−0.301011	−0.150505	+	0.988609i	$0.548090\pi$
$740$	0	0
$741$	−0.467295	−0.0171665
$742$	0	0
$743$	50.3161	1.84592	0.922958	−	0.384899i	$-0.125764\pi$
$743$	50.3161	1.84592	0.922958	+	0.384899i	$0.125764\pi$
$744$	0	0
$745$	−30.1399	−1.10424
$746$	0	0
$747$	−34.5011	−1.26233
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.58232	−0.313173	−0.156587	−	0.987664i	$-0.550049\pi$
$751$	−8.58232	−0.313173	−0.156587	+	0.987664i	$0.550049\pi$
$752$	0	0
$753$	−18.7388	−0.682880
$754$	0	0
$755$	35.0413	1.27528
$756$	0	0
$757$	−6.77542	−0.246257	−0.123128	−	0.992391i	$-0.539293\pi$
$757$	−6.77542	−0.246257	−0.123128	+	0.992391i	$0.539293\pi$
$758$	0	0
$759$	2.32842	0.0845164
$760$	0	0
$761$	−33.9843	−1.23193	−0.615964	−	0.787774i	$-0.711234\pi$
$761$	−33.9843	−1.23193	−0.615964	+	0.787774i	$0.711234\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−15.5348	−0.561661
$766$	0	0
$767$	6.16201	0.222497
$768$	0	0
$769$	−35.4639	−1.27886	−0.639430	−	0.768849i	$-0.720830\pi$
$769$	−35.4639	−1.27886	−0.639430	+	0.768849i	$0.720830\pi$
$770$	0	0
$771$	−8.28782	−0.298479
$772$	0	0
$773$	45.0018	1.61860	0.809302	−	0.587393i	$-0.199846\pi$
$773$	45.0018	1.61860	0.809302	+	0.587393i	$0.199846\pi$
$774$	0	0
$775$	3.64388	0.130892
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.07796	−0.146108
$780$	0	0
$781$	−0.719891	−0.0257597
$782$	0	0
$783$	−32.7183	−1.16926
$784$	0	0
$785$	0.533912	0.0190561
$786$	0	0
$787$	47.3048	1.68623	0.843117	−	0.537730i	$-0.180718\pi$
$787$	47.3048	1.68623	0.843117	+	0.537730i	$0.180718\pi$
$788$	0	0
$789$	10.2339	0.364335
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.42663	−0.157194
$794$	0	0
$795$	−15.5464	−0.551372
$796$	0	0
$797$	−24.3861	−0.863799	−0.431900	−	0.901922i	$-0.642157\pi$
$797$	−24.3861	−0.863799	−0.431900	+	0.901922i	$0.642157\pi$
$798$	0	0
$799$	−53.8009	−1.90334
$800$	0	0
$801$	31.1713	1.10138
$802$	0	0
$803$	8.41015	0.296788
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.62781	0.268512
$808$	0	0
$809$	36.0429	1.26720	0.633601	−	0.773660i	$-0.281576\pi$
$809$	36.0429	1.26720	0.633601	+	0.773660i	$0.281576\pi$
$810$	0	0
$811$	−10.3020	−0.361751	−0.180876	−	0.983506i	$-0.557893\pi$
$811$	−10.3020	−0.361751	−0.180876	+	0.983506i	$0.557893\pi$
$812$	0	0
$813$	26.0770	0.914562
$814$	0	0
$815$	30.9500	1.08413
$816$	0	0
$817$	−1.50379	−0.0526109
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.5181	0.471785	0.235892	−	0.971779i	$-0.424199\pi$
$821$	13.5181	0.471785	0.235892	+	0.971779i	$0.424199\pi$
$822$	0	0
$823$	49.7542	1.73432	0.867162	−	0.498026i	$-0.165942\pi$
$823$	49.7542	1.73432	0.867162	+	0.498026i	$0.165942\pi$
$824$	0	0
$825$	−1.17063	−0.0407559
$826$	0	0
$827$	41.7602	1.45214	0.726072	−	0.687618i	$-0.241344\pi$
$827$	41.7602	1.45214	0.726072	+	0.687618i	$0.241344\pi$
$828$	0	0
$829$	1.80429	0.0626654	0.0313327	−	0.999509i	$-0.490025\pi$
$829$	1.80429	0.0626654	0.0313327	+	0.999509i	$0.490025\pi$
$830$	0	0
$831$	−10.8630	−0.376833
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.93899	0.170921
$836$	0	0
$837$	−5.63006	−0.194603
$838$	0	0
$839$	36.3125	1.25365	0.626824	−	0.779161i	$-0.284354\pi$
$839$	36.3125	1.25365	0.626824	+	0.779161i	$0.284354\pi$
$840$	0	0
$841$	26.6270	0.918173
$842$	0	0
$843$	−16.2715	−0.560420
$844$	0	0
$845$	−18.6374	−0.641145
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.86681	0.235668
$850$	0	0
$851$	−36.7049	−1.25823
$852$	0	0
$853$	−16.4708	−0.563950	−0.281975	−	0.959422i	$-0.590990\pi$
$853$	−16.4708	−0.563950	−0.281975	+	0.959422i	$0.590990\pi$
$854$	0	0
$855$	3.41041	0.116634
$856$	0	0
$857$	−25.6099	−0.874818	−0.437409	−	0.899263i	$-0.644104\pi$
$857$	−25.6099	−0.874818	−0.437409	+	0.899263i	$0.644104\pi$
$858$	0	0
$859$	26.7008	0.911020	0.455510	−	0.890231i	$-0.349457\pi$
$859$	26.7008	0.911020	0.455510	+	0.890231i	$0.349457\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.2997	−1.03141	−0.515706	−	0.856765i	$-0.672470\pi$
$863$	−30.2997	−1.03141	−0.515706	+	0.856765i	$0.672470\pi$
$864$	0	0
$865$	−18.0515	−0.613768
$866$	0	0
$867$	−3.09131	−0.104987
$868$	0	0
$869$	−1.39733	−0.0474013
$870$	0	0
$871$	−4.75259	−0.161035
$872$	0	0
$873$	−16.3640	−0.553838
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.3151	−0.449620	−0.224810	−	0.974403i	$-0.572176\pi$
$877$	−13.3151	−0.449620	−0.224810	+	0.974403i	$0.572176\pi$
$878$	0	0
$879$	−11.8174	−0.398590
$880$	0	0
$881$	11.5234	0.388234	0.194117	−	0.980978i	$-0.437816\pi$
$881$	11.5234	0.388234	0.194117	+	0.980978i	$0.437816\pi$
$882$	0	0
$883$	−45.5807	−1.53391	−0.766956	−	0.641700i	$-0.778229\pi$
$883$	−45.5807	−1.53391	−0.766956	+	0.641700i	$0.778229\pi$
$884$	0	0
$885$	13.1795	0.443023
$886$	0	0
$887$	−35.2945	−1.18507	−0.592536	−	0.805544i	$-0.701873\pi$
$887$	−35.2945	−1.18507	−0.592536	+	0.805544i	$0.701873\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.67155	−0.0559992
$892$	0	0
$893$	11.8111	0.395244
$894$	0	0
$895$	15.6732	0.523899
$896$	0	0
$897$	2.63897	0.0881127
$898$	0	0
$899$	9.57210	0.319248
$900$	0	0
$901$	58.4240	1.94639
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.08262	−0.0692285
$906$	0	0
$907$	15.9735	0.530392	0.265196	−	0.964195i	$-0.414563\pi$
$907$	15.9735	0.530392	0.265196	+	0.964195i	$0.414563\pi$
$908$	0	0
$909$	11.1569	0.370052
$910$	0	0
$911$	−43.6031	−1.44464	−0.722318	−	0.691561i	$-0.756923\pi$
$911$	−43.6031	−1.44464	−0.722318	+	0.691561i	$0.756923\pi$
$912$	0	0
$913$	−7.43565	−0.246084
$914$	0	0
$915$	−9.46779	−0.312995
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.1000	−0.531090	−0.265545	−	0.964098i	$-0.585552\pi$
$919$	−16.1000	−0.531090	−0.265545	+	0.964098i	$0.585552\pi$
$920$	0	0
$921$	−2.27849	−0.0750788
$922$	0	0
$923$	−0.815905	−0.0268558
$924$	0	0
$925$	18.4535	0.606748
$926$	0	0
$927$	−11.2367	−0.369063
$928$	0	0
$929$	−25.9577	−0.851644	−0.425822	−	0.904807i	$-0.640015\pi$
$929$	−25.9577	−0.851644	−0.425822	+	0.904807i	$0.640015\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.63986	−0.119164
$934$	0	0
$935$	−3.34804	−0.109493
$936$	0	0
$937$	−60.1590	−1.96531	−0.982654	−	0.185447i	$-0.940627\pi$
$937$	−60.1590	−1.96531	−0.982654	+	0.185447i	$0.940627\pi$
$938$	0	0
$939$	−20.9969	−0.685209
$940$	0	0
$941$	52.4822	1.71087	0.855435	−	0.517910i	$-0.173290\pi$
$941$	52.4822	1.71087	0.855435	+	0.517910i	$0.173290\pi$
$942$	0	0
$943$	23.0296	0.749947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.8937	−1.39386	−0.696929	−	0.717140i	$-0.745451\pi$
$947$	−42.8937	−1.39386	−0.696929	+	0.717140i	$0.745451\pi$
$948$	0	0
$949$	9.53184	0.309416
$950$	0	0
$951$	21.9784	0.712699
$952$	0	0
$953$	25.2939	0.819349	0.409675	−	0.912232i	$-0.365642\pi$
$953$	25.2939	0.819349	0.409675	+	0.912232i	$0.365642\pi$
$954$	0	0
$955$	−28.5226	−0.922970
$956$	0	0
$957$	−3.07511	−0.0994043
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.3529	−0.946867
$962$	0	0
$963$	30.4574	0.981475
$964$	0	0
$965$	−6.76733	−0.217848
$966$	0	0
$967$	15.7996	0.508081	0.254041	−	0.967194i	$-0.418240\pi$
$967$	15.7996	0.508081	0.254041	+	0.967194i	$0.418240\pi$
$968$	0	0
$969$	3.75603	0.120661
$970$	0	0
$971$	−43.6937	−1.40220	−0.701099	−	0.713064i	$-0.747307\pi$
$971$	−43.6937	−1.40220	−0.701099	+	0.713064i	$0.747307\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.32675	−0.0424901
$976$	0	0
$977$	−14.8417	−0.474830	−0.237415	−	0.971408i	$-0.576300\pi$
$977$	−14.8417	−0.474830	−0.237415	+	0.971408i	$0.576300\pi$
$978$	0	0
$979$	6.71801	0.214709
$980$	0	0
$981$	3.32278	0.106088
$982$	0	0
$983$	43.8607	1.39894	0.699470	−	0.714662i	$-0.253420\pi$
$983$	43.8607	1.39894	0.699470	+	0.714662i	$0.253420\pi$
$984$	0	0
$985$	4.13709	0.131819
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.49240	0.270043
$990$	0	0
$991$	19.8815	0.631558	0.315779	−	0.948833i	$-0.397734\pi$
$991$	19.8815	0.631558	0.315779	+	0.948833i	$0.397734\pi$
$992$	0	0
$993$	−2.03289	−0.0645117
$994$	0	0
$995$	−17.3067	−0.548661
$996$	0	0
$997$	−12.2770	−0.388816	−0.194408	−	0.980921i	$-0.562279\pi$
$997$	−12.2770	−0.388816	−0.194408	+	0.980921i	$0.562279\pi$
$998$	0	0
$999$	−28.5120	−0.902081

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bo.1.4	✓	7
7.6	odd	2		7448.2.a.bp.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.4	✓	7	1.1	even	1	trivial
7448.2.a.bp.1.4	yes	7	7.6	odd	2

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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.824576 q^{3} +1.46996 q^{5} -2.32007 q^{9} +O(q^{10})\)	\(q-0.824576 q^{3} +1.46996 q^{5} -2.32007 q^{9} -0.500020 q^{11} -0.566709 q^{13} -1.21209 q^{15} +4.55510 q^{17} -1.00000 q^{19} +5.64733 q^{23} -2.83922 q^{25} +4.38680 q^{27} -7.45835 q^{29} -1.28341 q^{31} +0.412305 q^{33} -6.49950 q^{37} +0.467295 q^{39} +4.07796 q^{41} +1.50379 q^{43} -3.41041 q^{45} -11.8111 q^{47} -3.75603 q^{51} +12.8260 q^{53} -0.735009 q^{55} +0.824576 q^{57} -10.8733 q^{59} +7.81111 q^{61} -0.833039 q^{65} +8.38629 q^{67} -4.65665 q^{69} +1.43972 q^{71} -16.8196 q^{73} +2.34115 q^{75} +2.79455 q^{79} +3.34297 q^{81} +14.8707 q^{83} +6.69581 q^{85} +6.14998 q^{87} -13.4355 q^{89} +1.05827 q^{93} -1.46996 q^{95} +7.05323 q^{97} +1.16008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) \) 7 * q - q^3 - q^5 + 8 * q^9	\( 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) \) 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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