LMFDB - Embedded newform 7448.2.a.bx.1.14

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 11 x^{12} + 114 x^{11} - 10 x^{10} - 806 x^{9} + 523 x^{8} + 2586 x^{7} - 2226 x^{6} + \cdots + 8 $ x^14 - 6x^13 - 11x^12 + 114x^11 - 10x^10 - 806x^9 + 523x^8 + 2586x^7 - 2226x^6 - 3618x^5 + 3397x^4 + 1570x^3 - 1529x^2 + 8
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			1.14
Root			$3.32292$ 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)

$f(q)$	$=$	$q+3.32292 q^{3} +2.28248 q^{5} +8.04181 q^{9} +O(q^{10})$	$q+3.32292 q^{3} +2.28248 q^{5} +8.04181 q^{9} -2.44205 q^{11} -0.540933 q^{13} +7.58449 q^{15} +4.03920 q^{17} +1.00000 q^{19} +3.91812 q^{23} +0.209698 q^{25} +16.7535 q^{27} +5.85801 q^{29} -0.987329 q^{31} -8.11474 q^{33} -5.78204 q^{37} -1.79748 q^{39} +4.28939 q^{41} -6.87820 q^{43} +18.3552 q^{45} +1.82493 q^{47} +13.4219 q^{51} +7.01172 q^{53} -5.57392 q^{55} +3.32292 q^{57} +7.62454 q^{59} -14.7388 q^{61} -1.23467 q^{65} -13.5067 q^{67} +13.0196 q^{69} +11.0520 q^{71} -4.53613 q^{73} +0.696810 q^{75} +8.60932 q^{79} +31.5452 q^{81} -0.293270 q^{83} +9.21937 q^{85} +19.4657 q^{87} -15.6400 q^{89} -3.28082 q^{93} +2.28248 q^{95} -11.2659 q^{97} -19.6385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9	$ 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{19} - 4 q^{23} + 16 q^{25} + 36 q^{27} - 6 q^{29} + 16 q^{31} - 10 q^{33} + 6 q^{37} + 16 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{47} + 20 q^{51} - 6 q^{53} + 44 q^{55} + 6 q^{57} + 22 q^{59} + 10 q^{61} - 16 q^{65} + 4 q^{67} + 48 q^{69} + 6 q^{71} + 4 q^{73} + 64 q^{75} + 26 q^{79} + 30 q^{81} + 32 q^{83} - 8 q^{85} + 32 q^{87} - 54 q^{89} - 32 q^{93} + 2 q^{95} + 18 q^{97} - 38 q^{99}+O(q^{100}) $ 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^19 - 4 * q^23 + 16 * q^25 + 36 * q^27 - 6 * q^29 + 16 * q^31 - 10 * q^33 + 6 * q^37 + 16 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^47 + 20 * q^51 - 6 * q^53 + 44 * q^55 + 6 * q^57 + 22 * q^59 + 10 * q^61 - 16 * q^65 + 4 * q^67 + 48 * q^69 + 6 * q^71 + 4 * q^73 + 64 * q^75 + 26 * q^79 + 30 * q^81 + 32 * q^83 - 8 * q^85 + 32 * q^87 - 54 * q^89 - 32 * q^93 + 2 * q^95 + 18 * q^97 - 38 * 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
$3$	3.32292	1.91849	0.959245	−	0.282576i	$-0.0911890\pi$
$3$	3.32292	1.91849	0.959245	+	0.282576i	$0.0911890\pi$
$4$	0	0
$5$	2.28248	1.02075	0.510377	−	0.859951i	$-0.329506\pi$
$5$	2.28248	1.02075	0.510377	+	0.859951i	$0.329506\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.04181	2.68060
$10$	0	0
$11$	−2.44205	−0.736305	−0.368153	−	0.929765i	$-0.620010\pi$
$11$	−2.44205	−0.736305	−0.368153	+	0.929765i	$0.620010\pi$
$12$	0	0
$13$	−0.540933	−0.150028	−0.0750139	−	0.997182i	$-0.523900\pi$
$13$	−0.540933	−0.150028	−0.0750139	+	0.997182i	$0.523900\pi$
$14$	0	0
$15$	7.58449	1.95831
$16$	0	0
$17$	4.03920	0.979650	0.489825	−	0.871821i	$-0.337061\pi$
$17$	4.03920	0.979650	0.489825	+	0.871821i	$0.337061\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.91812	0.816984	0.408492	−	0.912762i	$-0.366055\pi$
$23$	3.91812	0.816984	0.408492	+	0.912762i	$0.366055\pi$
$24$	0	0
$25$	0.209698	0.0419396
$26$	0	0
$27$	16.7535	3.22422
$28$	0	0
$29$	5.85801	1.08780	0.543902	−	0.839149i	$-0.316946\pi$
$29$	5.85801	1.08780	0.543902	+	0.839149i	$0.316946\pi$
$30$	0	0
$31$	−0.987329	−0.177330	−0.0886648	−	0.996062i	$-0.528260\pi$
$31$	−0.987329	−0.177330	−0.0886648	+	0.996062i	$0.528260\pi$
$32$	0	0
$33$	−8.11474	−1.41259
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.78204	−0.950562	−0.475281	−	0.879834i	$-0.657654\pi$
$37$	−5.78204	−0.950562	−0.475281	+	0.879834i	$0.657654\pi$
$38$	0	0
$39$	−1.79748	−0.287827
$40$	0	0
$41$	4.28939	0.669891	0.334945	−	0.942238i	$-0.391282\pi$
$41$	4.28939	0.669891	0.334945	+	0.942238i	$0.391282\pi$
$42$	0	0
$43$	−6.87820	−1.04892	−0.524458	−	0.851436i	$-0.675732\pi$
$43$	−6.87820	−1.04892	−0.524458	+	0.851436i	$0.675732\pi$
$44$	0	0
$45$	18.3552	2.73624
$46$	0	0
$47$	1.82493	0.266193	0.133096	−	0.991103i	$-0.457508\pi$
$47$	1.82493	0.266193	0.133096	+	0.991103i	$0.457508\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.4219	1.87945
$52$	0	0
$53$	7.01172	0.963133	0.481567	−	0.876409i	$-0.340068\pi$
$53$	7.01172	0.963133	0.481567	+	0.876409i	$0.340068\pi$
$54$	0	0
$55$	−5.57392	−0.751587
$56$	0	0
$57$	3.32292	0.440132
$58$	0	0
$59$	7.62454	0.992631	0.496315	−	0.868142i	$-0.334686\pi$
$59$	7.62454	0.992631	0.496315	+	0.868142i	$0.334686\pi$
$60$	0	0
$61$	−14.7388	−1.88711	−0.943556	−	0.331213i	$-0.892542\pi$
$61$	−14.7388	−1.88711	−0.943556	+	0.331213i	$0.892542\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.23467	−0.153142
$66$	0	0
$67$	−13.5067	−1.65011	−0.825054	−	0.565054i	$-0.808855\pi$
$67$	−13.5067	−1.65011	−0.825054	+	0.565054i	$0.808855\pi$
$68$	0	0
$69$	13.0196	1.56738
$70$	0	0
$71$	11.0520	1.31164	0.655818	−	0.754919i	$-0.272324\pi$
$71$	11.0520	1.31164	0.655818	+	0.754919i	$0.272324\pi$
$72$	0	0
$73$	−4.53613	−0.530914	−0.265457	−	0.964123i	$-0.585523\pi$
$73$	−4.53613	−0.530914	−0.265457	+	0.964123i	$0.585523\pi$
$74$	0	0
$75$	0.696810	0.0804606
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.60932	0.968625	0.484312	−	0.874895i	$-0.339070\pi$
$79$	8.60932	0.968625	0.484312	+	0.874895i	$0.339070\pi$
$80$	0	0
$81$	31.5452	3.50503
$82$	0	0
$83$	−0.293270	−0.0321906	−0.0160953	−	0.999870i	$-0.505124\pi$
$83$	−0.293270	−0.0321906	−0.0160953	+	0.999870i	$0.505124\pi$
$84$	0	0
$85$	9.21937	0.999982
$86$	0	0
$87$	19.4657	2.08694
$88$	0	0
$89$	−15.6400	−1.65784	−0.828919	−	0.559369i	$-0.811044\pi$
$89$	−15.6400	−1.65784	−0.828919	+	0.559369i	$0.811044\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.28082	−0.340205
$94$	0	0
$95$	2.28248	0.234177
$96$	0	0
$97$	−11.2659	−1.14388	−0.571939	−	0.820296i	$-0.693809\pi$
$97$	−11.2659	−1.14388	−0.571939	+	0.820296i	$0.693809\pi$
$98$	0	0
$99$	−19.6385	−1.97374
$100$	0	0
$101$	−12.6626	−1.25997	−0.629987	−	0.776606i	$-0.716940\pi$
$101$	−12.6626	−1.25997	−0.629987	+	0.776606i	$0.716940\pi$
$102$	0	0
$103$	2.30674	0.227290	0.113645	−	0.993521i	$-0.463747\pi$
$103$	2.30674	0.227290	0.113645	+	0.993521i	$0.463747\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.30761	0.706453	0.353227	−	0.935538i	$-0.385084\pi$
$107$	7.30761	0.706453	0.353227	+	0.935538i	$0.385084\pi$
$108$	0	0
$109$	−7.61755	−0.729629	−0.364814	−	0.931080i	$-0.618868\pi$
$109$	−7.61755	−0.729629	−0.364814	+	0.931080i	$0.618868\pi$
$110$	0	0
$111$	−19.2133	−1.82364
$112$	0	0
$113$	−16.2801	−1.53150	−0.765751	−	0.643137i	$-0.777632\pi$
$113$	−16.2801	−1.53150	−0.765751	+	0.643137i	$0.777632\pi$
$114$	0	0
$115$	8.94301	0.833940
$116$	0	0
$117$	−4.35008	−0.402165
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.03640	−0.457854
$122$	0	0
$123$	14.2533	1.28518
$124$	0	0
$125$	−10.9338	−0.977944
$126$	0	0
$127$	12.4792	1.10735	0.553677	−	0.832732i	$-0.313224\pi$
$127$	12.4792	1.10735	0.553677	+	0.832732i	$0.313224\pi$
$128$	0	0
$129$	−22.8557	−2.01233
$130$	0	0
$131$	22.3980	1.95692	0.978460	−	0.206436i	$-0.0661864\pi$
$131$	22.3980	1.95692	0.978460	+	0.206436i	$0.0661864\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	38.2395	3.29114
$136$	0	0
$137$	16.7345	1.42972	0.714862	−	0.699265i	$-0.246489\pi$
$137$	16.7345	1.42972	0.714862	+	0.699265i	$0.246489\pi$
$138$	0	0
$139$	−3.90213	−0.330974	−0.165487	−	0.986212i	$-0.552920\pi$
$139$	−3.90213	−0.330974	−0.165487	+	0.986212i	$0.552920\pi$
$140$	0	0
$141$	6.06409	0.510688
$142$	0	0
$143$	1.32099	0.110466
$144$	0	0
$145$	13.3708	1.11038
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.2614	1.74180	0.870901	−	0.491459i	$-0.163536\pi$
$149$	21.2614	1.74180	0.870901	+	0.491459i	$0.163536\pi$
$150$	0	0
$151$	8.29128	0.674735	0.337367	−	0.941373i	$-0.390464\pi$
$151$	8.29128	0.674735	0.337367	+	0.941373i	$0.390464\pi$
$152$	0	0
$153$	32.4825	2.62605
$154$	0	0
$155$	−2.25355	−0.181010
$156$	0	0
$157$	4.23159	0.337718	0.168859	−	0.985640i	$-0.445992\pi$
$157$	4.23159	0.337718	0.168859	+	0.985640i	$0.445992\pi$
$158$	0	0
$159$	23.2994	1.84776
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.03166	−0.159132	−0.0795661	−	0.996830i	$-0.525353\pi$
$163$	−2.03166	−0.159132	−0.0795661	+	0.996830i	$0.525353\pi$
$164$	0	0
$165$	−18.5217	−1.44191
$166$	0	0
$167$	−6.25483	−0.484014	−0.242007	−	0.970275i	$-0.577806\pi$
$167$	−6.25483	−0.484014	−0.242007	+	0.970275i	$0.577806\pi$
$168$	0	0
$169$	−12.7074	−0.977492
$170$	0	0
$171$	8.04181	0.614972
$172$	0	0
$173$	14.2771	1.08547	0.542733	−	0.839905i	$-0.317389\pi$
$173$	14.2771	1.08547	0.542733	+	0.839905i	$0.317389\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.3357	1.90435
$178$	0	0
$179$	−7.64751	−0.571602	−0.285801	−	0.958289i	$-0.592260\pi$
$179$	−7.64751	−0.571602	−0.285801	+	0.958289i	$0.592260\pi$
$180$	0	0
$181$	2.74145	0.203770	0.101885	−	0.994796i	$-0.467513\pi$
$181$	2.74145	0.203770	0.101885	+	0.994796i	$0.467513\pi$
$182$	0	0
$183$	−48.9759	−3.62040
$184$	0	0
$185$	−13.1974	−0.970291
$186$	0	0
$187$	−9.86392	−0.721321
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.171653	0.0124203	0.00621017	−	0.999981i	$-0.498023\pi$
$191$	0.171653	0.0124203	0.00621017	+	0.999981i	$0.498023\pi$
$192$	0	0
$193$	−11.7899	−0.848656	−0.424328	−	0.905509i	$-0.639490\pi$
$193$	−11.7899	−0.848656	−0.424328	+	0.905509i	$0.639490\pi$
$194$	0	0
$195$	−4.10270	−0.293801
$196$	0	0
$197$	14.7004	1.04736	0.523680	−	0.851915i	$-0.324559\pi$
$197$	14.7004	1.04736	0.523680	+	0.851915i	$0.324559\pi$
$198$	0	0
$199$	3.50555	0.248502	0.124251	−	0.992251i	$-0.460347\pi$
$199$	3.50555	0.248502	0.124251	+	0.992251i	$0.460347\pi$
$200$	0	0
$201$	−44.8818	−3.16571
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.79044	0.683794
$206$	0	0
$207$	31.5088	2.19001
$208$	0	0
$209$	−2.44205	−0.168920
$210$	0	0
$211$	5.82075	0.400717	0.200359	−	0.979723i	$-0.435789\pi$
$211$	5.82075	0.400717	0.200359	+	0.979723i	$0.435789\pi$
$212$	0	0
$213$	36.7251	2.51636
$214$	0	0
$215$	−15.6993	−1.07069
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.0732	−1.01855
$220$	0	0
$221$	−2.18494	−0.146975
$222$	0	0
$223$	−16.1168	−1.07926	−0.539629	−	0.841903i	$-0.681436\pi$
$223$	−16.1168	−1.07926	−0.539629	+	0.841903i	$0.681436\pi$
$224$	0	0
$225$	1.68635	0.112423
$226$	0	0
$227$	−6.57054	−0.436102	−0.218051	−	0.975937i	$-0.569970\pi$
$227$	−6.57054	−0.436102	−0.218051	+	0.975937i	$0.569970\pi$
$228$	0	0
$229$	−25.7366	−1.70072	−0.850361	−	0.526200i	$-0.823616\pi$
$229$	−25.7366	−1.70072	−0.850361	+	0.526200i	$0.823616\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.4600	1.53692	0.768459	−	0.639899i	$-0.221024\pi$
$233$	23.4600	1.53692	0.768459	+	0.639899i	$0.221024\pi$
$234$	0	0
$235$	4.16535	0.271718
$236$	0	0
$237$	28.6081	1.85830
$238$	0	0
$239$	−18.0472	−1.16738	−0.583688	−	0.811978i	$-0.698391\pi$
$239$	−18.0472	−1.16738	−0.583688	+	0.811978i	$0.698391\pi$
$240$	0	0
$241$	−4.71077	−0.303447	−0.151724	−	0.988423i	$-0.548482\pi$
$241$	−4.71077	−0.303447	−0.151724	+	0.988423i	$0.548482\pi$
$242$	0	0
$243$	54.5618	3.50014
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.540933	−0.0344188
$248$	0	0
$249$	−0.974514	−0.0617573
$250$	0	0
$251$	12.4870	0.788175	0.394087	−	0.919073i	$-0.371061\pi$
$251$	12.4870	0.788175	0.394087	+	0.919073i	$0.371061\pi$
$252$	0	0
$253$	−9.56824	−0.601550
$254$	0	0
$255$	30.6353	1.91845
$256$	0	0
$257$	−14.7519	−0.920197	−0.460099	−	0.887868i	$-0.652186\pi$
$257$	−14.7519	−0.920197	−0.460099	+	0.887868i	$0.652186\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	47.1090	2.91597
$262$	0	0
$263$	−5.90282	−0.363983	−0.181992	−	0.983300i	$-0.558254\pi$
$263$	−5.90282	−0.363983	−0.181992	+	0.983300i	$0.558254\pi$
$264$	0	0
$265$	16.0041	0.983123
$266$	0	0
$267$	−51.9705	−3.18054
$268$	0	0
$269$	8.48262	0.517194	0.258597	−	0.965985i	$-0.416740\pi$
$269$	8.48262	0.517194	0.258597	+	0.965985i	$0.416740\pi$
$270$	0	0
$271$	15.2977	0.929270	0.464635	−	0.885502i	$-0.346186\pi$
$271$	15.2977	0.929270	0.464635	+	0.885502i	$0.346186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.512092	−0.0308803
$276$	0	0
$277$	25.4821	1.53107	0.765535	−	0.643395i	$-0.222475\pi$
$277$	25.4821	1.53107	0.765535	+	0.643395i	$0.222475\pi$
$278$	0	0
$279$	−7.93991	−0.475350
$280$	0	0
$281$	−15.7989	−0.942482	−0.471241	−	0.882004i	$-0.656194\pi$
$281$	−15.7989	−0.942482	−0.471241	+	0.882004i	$0.656194\pi$
$282$	0	0
$283$	33.3032	1.97967	0.989835	−	0.142223i	$-0.0454249\pi$
$283$	33.3032	1.97967	0.989835	+	0.142223i	$0.0454249\pi$
$284$	0	0
$285$	7.58449	0.449266
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−0.684874	−0.0402867
$290$	0	0
$291$	−37.4357	−2.19452
$292$	0	0
$293$	−2.16728	−0.126614	−0.0633068	−	0.997994i	$-0.520165\pi$
$293$	−2.16728	−0.126614	−0.0633068	+	0.997994i	$0.520165\pi$
$294$	0	0
$295$	17.4028	1.01323
$296$	0	0
$297$	−40.9129	−2.37401
$298$	0	0
$299$	−2.11944	−0.122570
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−42.0768	−2.41725
$304$	0	0
$305$	−33.6410	−1.92628
$306$	0	0
$307$	7.19059	0.410389	0.205194	−	0.978721i	$-0.434217\pi$
$307$	7.19059	0.410389	0.205194	+	0.978721i	$0.434217\pi$
$308$	0	0
$309$	7.66512	0.436054
$310$	0	0
$311$	−33.9587	−1.92562	−0.962812	−	0.270172i	$-0.912919\pi$
$311$	−33.9587	−1.92562	−0.962812	+	0.270172i	$0.912919\pi$
$312$	0	0
$313$	25.9839	1.46869	0.734347	−	0.678774i	$-0.237488\pi$
$313$	25.9839	1.46869	0.734347	+	0.678774i	$0.237488\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.0969	0.847927	0.423963	−	0.905679i	$-0.360639\pi$
$317$	15.0969	0.847927	0.423963	+	0.905679i	$0.360639\pi$
$318$	0	0
$319$	−14.3055	−0.800956
$320$	0	0
$321$	24.2826	1.35532
$322$	0	0
$323$	4.03920	0.224747
$324$	0	0
$325$	−0.113433	−0.00629211
$326$	0	0
$327$	−25.3125	−1.39979
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.2103	1.05589	0.527947	−	0.849278i	$-0.322962\pi$
$331$	19.2103	1.05589	0.527947	+	0.849278i	$0.322962\pi$
$332$	0	0
$333$	−46.4981	−2.54808
$334$	0	0
$335$	−30.8288	−1.68435
$336$	0	0
$337$	−19.5044	−1.06247	−0.531236	−	0.847224i	$-0.678272\pi$
$337$	−19.5044	−1.06247	−0.531236	+	0.847224i	$0.678272\pi$
$338$	0	0
$339$	−54.0975	−2.93817
$340$	0	0
$341$	2.41111	0.130569
$342$	0	0
$343$	0	0
$344$	0	0
$345$	29.7169	1.59991
$346$	0	0
$347$	−27.0691	−1.45314	−0.726572	−	0.687090i	$-0.758888\pi$
$347$	−27.0691	−1.45314	−0.726572	+	0.687090i	$0.758888\pi$
$348$	0	0
$349$	11.3187	0.605874	0.302937	−	0.953011i	$-0.402033\pi$
$349$	11.3187	0.605874	0.302937	+	0.953011i	$0.402033\pi$
$350$	0	0
$351$	−9.06254	−0.483723
$352$	0	0
$353$	−8.60330	−0.457907	−0.228954	−	0.973437i	$-0.573530\pi$
$353$	−8.60330	−0.457907	−0.228954	+	0.973437i	$0.573530\pi$
$354$	0	0
$355$	25.2260	1.33886
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.38351	−0.0730186	−0.0365093	−	0.999333i	$-0.511624\pi$
$359$	−1.38351	−0.0730186	−0.0365093	+	0.999333i	$0.511624\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−16.7356	−0.878389
$364$	0	0
$365$	−10.3536	−0.541933
$366$	0	0
$367$	21.4110	1.11764	0.558821	−	0.829288i	$-0.311254\pi$
$367$	21.4110	1.11764	0.558821	+	0.829288i	$0.311254\pi$
$368$	0	0
$369$	34.4945	1.79571
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.8218	−1.33700	−0.668502	−	0.743710i	$-0.733064\pi$
$373$	−25.8218	−1.33700	−0.668502	+	0.743710i	$0.733064\pi$
$374$	0	0
$375$	−36.3320	−1.87618
$376$	0	0
$377$	−3.16879	−0.163201
$378$	0	0
$379$	−3.95737	−0.203276	−0.101638	−	0.994821i	$-0.532408\pi$
$379$	−3.95737	−0.203276	−0.101638	+	0.994821i	$0.532408\pi$
$380$	0	0
$381$	41.4675	2.12445
$382$	0	0
$383$	3.44365	0.175962	0.0879811	−	0.996122i	$-0.471958\pi$
$383$	3.44365	0.175962	0.0879811	+	0.996122i	$0.471958\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−55.3132	−2.81173
$388$	0	0
$389$	−33.5042	−1.69873	−0.849366	−	0.527804i	$-0.823016\pi$
$389$	−33.5042	−1.69873	−0.849366	+	0.527804i	$0.823016\pi$
$390$	0	0
$391$	15.8261	0.800358
$392$	0	0
$393$	74.4267	3.75433
$394$	0	0
$395$	19.6506	0.988728
$396$	0	0
$397$	24.4899	1.22911	0.614556	−	0.788873i	$-0.289335\pi$
$397$	24.4899	1.22911	0.614556	+	0.788873i	$0.289335\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.3351	−0.566049	−0.283025	−	0.959113i	$-0.591338\pi$
$401$	−11.3351	−0.566049	−0.283025	+	0.959113i	$0.591338\pi$
$402$	0	0
$403$	0.534079	0.0266044
$404$	0	0
$405$	72.0013	3.57777
$406$	0	0
$407$	14.1200	0.699904
$408$	0	0
$409$	−20.7824	−1.02762	−0.513812	−	0.857903i	$-0.671767\pi$
$409$	−20.7824	−1.02762	−0.513812	+	0.857903i	$0.671767\pi$
$410$	0	0
$411$	55.6074	2.74291
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.669382	−0.0328587
$416$	0	0
$417$	−12.9665	−0.634971
$418$	0	0
$419$	−16.3476	−0.798634	−0.399317	−	0.916813i	$-0.630753\pi$
$419$	−16.3476	−0.798634	−0.399317	+	0.916813i	$0.630753\pi$
$420$	0	0
$421$	−31.2690	−1.52396	−0.761980	−	0.647601i	$-0.775772\pi$
$421$	−31.2690	−1.52396	−0.761980	+	0.647601i	$0.775772\pi$
$422$	0	0
$423$	14.6757	0.713557
$424$	0	0
$425$	0.847011	0.0410861
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.38953	0.211929
$430$	0	0
$431$	17.5009	0.842988	0.421494	−	0.906831i	$-0.361506\pi$
$431$	17.5009	0.842988	0.421494	+	0.906831i	$0.361506\pi$
$432$	0	0
$433$	−23.9411	−1.15053	−0.575267	−	0.817966i	$-0.695102\pi$
$433$	−23.9411	−1.15053	−0.575267	+	0.817966i	$0.695102\pi$
$434$	0	0
$435$	44.4300	2.13025
$436$	0	0
$437$	3.91812	0.187429
$438$	0	0
$439$	−6.09872	−0.291076	−0.145538	−	0.989353i	$-0.546491\pi$
$439$	−6.09872	−0.291076	−0.145538	+	0.989353i	$0.546491\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.8583	−0.991006	−0.495503	−	0.868606i	$-0.665016\pi$
$443$	−20.8583	−0.991006	−0.495503	+	0.868606i	$0.665016\pi$
$444$	0	0
$445$	−35.6979	−1.69225
$446$	0	0
$447$	70.6499	3.34163
$448$	0	0
$449$	9.39594	0.443422	0.221711	−	0.975112i	$-0.428836\pi$
$449$	9.39594	0.443422	0.221711	+	0.975112i	$0.428836\pi$
$450$	0	0
$451$	−10.4749	−0.493244
$452$	0	0
$453$	27.5513	1.29447
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.9169	−1.53979	−0.769893	−	0.638173i	$-0.779690\pi$
$457$	−32.9169	−1.53979	−0.769893	+	0.638173i	$0.779690\pi$
$458$	0	0
$459$	67.6708	3.15860
$460$	0	0
$461$	20.0099	0.931953	0.465976	−	0.884797i	$-0.345703\pi$
$461$	20.0099	0.931953	0.465976	+	0.884797i	$0.345703\pi$
$462$	0	0
$463$	−26.6221	−1.23724	−0.618618	−	0.785692i	$-0.712307\pi$
$463$	−26.6221	−1.23724	−0.618618	+	0.785692i	$0.712307\pi$
$464$	0	0
$465$	−7.48839	−0.347266
$466$	0	0
$467$	−10.2022	−0.472102	−0.236051	−	0.971741i	$-0.575853\pi$
$467$	−10.2022	−0.472102	−0.236051	+	0.971741i	$0.575853\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.0612	0.647908
$472$	0	0
$473$	16.7969	0.772322
$474$	0	0
$475$	0.209698	0.00962160
$476$	0	0
$477$	56.3869	2.58178
$478$	0	0
$479$	28.5157	1.30291	0.651457	−	0.758686i	$-0.274158\pi$
$479$	28.5157	1.30291	0.651457	+	0.758686i	$0.274158\pi$
$480$	0	0
$481$	3.12770	0.142611
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25.7141	−1.16762
$486$	0	0
$487$	−27.8697	−1.26290	−0.631449	−	0.775417i	$-0.717540\pi$
$487$	−27.8697	−1.26290	−0.631449	+	0.775417i	$0.717540\pi$
$488$	0	0
$489$	−6.75106	−0.305293
$490$	0	0
$491$	32.2557	1.45568	0.727839	−	0.685748i	$-0.240525\pi$
$491$	32.2557	1.45568	0.727839	+	0.685748i	$0.240525\pi$
$492$	0	0
$493$	23.6616	1.06567
$494$	0	0
$495$	−44.8244	−2.01471
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.76762	−0.347726	−0.173863	−	0.984770i	$-0.555625\pi$
$499$	−7.76762	−0.347726	−0.173863	+	0.984770i	$0.555625\pi$
$500$	0	0
$501$	−20.7843	−0.928575
$502$	0	0
$503$	−34.3592	−1.53200	−0.766001	−	0.642839i	$-0.777756\pi$
$503$	−34.3592	−1.53200	−0.766001	+	0.642839i	$0.777756\pi$
$504$	0	0
$505$	−28.9020	−1.28612
$506$	0	0
$507$	−42.2257	−1.87531
$508$	0	0
$509$	37.7513	1.67330	0.836648	−	0.547741i	$-0.184512\pi$
$509$	37.7513	1.67330	0.836648	+	0.547741i	$0.184512\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.7535	0.739686
$514$	0	0
$515$	5.26508	0.232007
$516$	0	0
$517$	−4.45656	−0.195999
$518$	0	0
$519$	47.4416	2.08246
$520$	0	0
$521$	−20.0248	−0.877302	−0.438651	−	0.898657i	$-0.644544\pi$
$521$	−20.0248	−0.877302	−0.438651	+	0.898657i	$0.644544\pi$
$522$	0	0
$523$	21.3354	0.932932	0.466466	−	0.884539i	$-0.345527\pi$
$523$	21.3354	0.932932	0.466466	+	0.884539i	$0.345527\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.98802	−0.173721
$528$	0	0
$529$	−7.64835	−0.332537
$530$	0	0
$531$	61.3151	2.66085
$532$	0	0
$533$	−2.32028	−0.100502
$534$	0	0
$535$	16.6794	0.721115
$536$	0	0
$537$	−25.4121	−1.09661
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.18189	0.0938070	0.0469035	−	0.998899i	$-0.485065\pi$
$541$	2.18189	0.0938070	0.0469035	+	0.998899i	$0.485065\pi$
$542$	0	0
$543$	9.10962	0.390931
$544$	0	0
$545$	−17.3869	−0.744772
$546$	0	0
$547$	16.4050	0.701428	0.350714	−	0.936483i	$-0.385939\pi$
$547$	16.4050	0.701428	0.350714	+	0.936483i	$0.385939\pi$
$548$	0	0
$549$	−118.527	−5.05860
$550$	0	0
$551$	5.85801	0.249559
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−43.8539	−1.86149
$556$	0	0
$557$	−19.6727	−0.833558	−0.416779	−	0.909008i	$-0.636841\pi$
$557$	−19.6727	−0.833558	−0.416779	+	0.909008i	$0.636841\pi$
$558$	0	0
$559$	3.72065	0.157367
$560$	0	0
$561$	−32.7770	−1.38385
$562$	0	0
$563$	−45.6249	−1.92286	−0.961430	−	0.275048i	$-0.911306\pi$
$563$	−45.6249	−1.92286	−0.961430	+	0.275048i	$0.911306\pi$
$564$	0	0
$565$	−37.1589	−1.56329
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.6091	−0.905901	−0.452950	−	0.891536i	$-0.649629\pi$
$569$	−21.6091	−0.905901	−0.452950	+	0.891536i	$0.649629\pi$
$570$	0	0
$571$	−18.9242	−0.791954	−0.395977	−	0.918260i	$-0.629594\pi$
$571$	−18.9242	−0.791954	−0.395977	+	0.918260i	$0.629594\pi$
$572$	0	0
$573$	0.570388	0.0238283
$574$	0	0
$575$	0.821621	0.0342640
$576$	0	0
$577$	31.2923	1.30271	0.651357	−	0.758771i	$-0.274200\pi$
$577$	31.2923	1.30271	0.651357	+	0.758771i	$0.274200\pi$
$578$	0	0
$579$	−39.1769	−1.62814
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−17.1230	−0.709160
$584$	0	0
$585$	−9.92896	−0.410512
$586$	0	0
$587$	−11.9024	−0.491265	−0.245632	−	0.969363i	$-0.578996\pi$
$587$	−11.9024	−0.491265	−0.245632	+	0.969363i	$0.578996\pi$
$588$	0	0
$589$	−0.987329	−0.0406822
$590$	0	0
$591$	48.8483	2.00935
$592$	0	0
$593$	0.879180	0.0361036	0.0180518	−	0.999837i	$-0.494254\pi$
$593$	0.879180	0.0361036	0.0180518	+	0.999837i	$0.494254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.6487	0.476748
$598$	0	0
$599$	−23.9852	−0.980008	−0.490004	−	0.871720i	$-0.663005\pi$
$599$	−23.9852	−0.980008	−0.490004	+	0.871720i	$0.663005\pi$
$600$	0	0
$601$	17.2344	0.703004	0.351502	−	0.936187i	$-0.385671\pi$
$601$	17.2344	0.703004	0.351502	+	0.936187i	$0.385671\pi$
$602$	0	0
$603$	−108.618	−4.42328
$604$	0	0
$605$	−11.4955	−0.467357
$606$	0	0
$607$	−20.7779	−0.843347	−0.421674	−	0.906748i	$-0.638557\pi$
$607$	−20.7779	−0.843347	−0.421674	+	0.906748i	$0.638557\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.987164	−0.0399364
$612$	0	0
$613$	34.1070	1.37757	0.688784	−	0.724966i	$-0.258145\pi$
$613$	34.1070	1.37757	0.688784	+	0.724966i	$0.258145\pi$
$614$	0	0
$615$	32.5329	1.31185
$616$	0	0
$617$	−12.8627	−0.517834	−0.258917	−	0.965900i	$-0.583366\pi$
$617$	−12.8627	−0.517834	−0.258917	+	0.965900i	$0.583366\pi$
$618$	0	0
$619$	−41.5579	−1.67035	−0.835176	−	0.549982i	$-0.814634\pi$
$619$	−41.5579	−1.67035	−0.835176	+	0.549982i	$0.814634\pi$
$620$	0	0
$621$	65.6423	2.63414
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.0045	−1.04018
$626$	0	0
$627$	−8.11474	−0.324071
$628$	0	0
$629$	−23.3548	−0.931218
$630$	0	0
$631$	−40.6384	−1.61779	−0.808895	−	0.587953i	$-0.799934\pi$
$631$	−40.6384	−1.61779	−0.808895	+	0.587953i	$0.799934\pi$
$632$	0	0
$633$	19.3419	0.768772
$634$	0	0
$635$	28.4836	1.13034
$636$	0	0
$637$	0	0
$638$	0	0
$639$	88.8784	3.51598
$640$	0	0
$641$	47.3527	1.87032	0.935159	−	0.354227i	$-0.115256\pi$
$641$	47.3527	1.87032	0.935159	+	0.354227i	$0.115256\pi$
$642$	0	0
$643$	−5.35267	−0.211089	−0.105544	−	0.994415i	$-0.533658\pi$
$643$	−5.35267	−0.211089	−0.105544	+	0.994415i	$0.533658\pi$
$644$	0	0
$645$	−52.1676	−2.05410
$646$	0	0
$647$	42.7370	1.68017	0.840083	−	0.542459i	$-0.182506\pi$
$647$	42.7370	1.68017	0.840083	+	0.542459i	$0.182506\pi$
$648$	0	0
$649$	−18.6195	−0.730879
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.05657	0.119613	0.0598064	−	0.998210i	$-0.480952\pi$
$653$	3.05657	0.119613	0.0598064	+	0.998210i	$0.480952\pi$
$654$	0	0
$655$	51.1229	1.99754
$656$	0	0
$657$	−36.4787	−1.42317
$658$	0	0
$659$	11.0483	0.430381	0.215190	−	0.976572i	$-0.430963\pi$
$659$	11.0483	0.430381	0.215190	+	0.976572i	$0.430963\pi$
$660$	0	0
$661$	−18.1767	−0.706993	−0.353496	−	0.935436i	$-0.615007\pi$
$661$	−18.1767	−0.706993	−0.353496	+	0.935436i	$0.615007\pi$
$662$	0	0
$663$	−7.26037	−0.281970
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22.9524	0.888719
$668$	0	0
$669$	−53.5547	−2.07055
$670$	0	0
$671$	35.9929	1.38949
$672$	0	0
$673$	−16.7660	−0.646283	−0.323141	−	0.946351i	$-0.604739\pi$
$673$	−16.7660	−0.646283	−0.323141	+	0.946351i	$0.604739\pi$
$674$	0	0
$675$	3.51318	0.135222
$676$	0	0
$677$	−0.159329	−0.00612350	−0.00306175	−	0.999995i	$-0.500975\pi$
$677$	−0.159329	−0.00612350	−0.00306175	+	0.999995i	$0.500975\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−21.8334	−0.836657
$682$	0	0
$683$	−24.6399	−0.942821	−0.471411	−	0.881914i	$-0.656255\pi$
$683$	−24.6399	−0.942821	−0.471411	+	0.881914i	$0.656255\pi$
$684$	0	0
$685$	38.1961	1.45940
$686$	0	0
$687$	−85.5207	−3.26282
$688$	0	0
$689$	−3.79287	−0.144497
$690$	0	0
$691$	42.4923	1.61648	0.808241	−	0.588852i	$-0.200420\pi$
$691$	42.4923	1.61648	0.808241	+	0.588852i	$0.200420\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.90651	−0.337843
$696$	0	0
$697$	17.3257	0.656258
$698$	0	0
$699$	77.9559	2.94856
$700$	0	0
$701$	17.8932	0.675815	0.337908	−	0.941179i	$-0.390281\pi$
$701$	17.8932	0.675815	0.337908	+	0.941179i	$0.390281\pi$
$702$	0	0
$703$	−5.78204	−0.218074
$704$	0	0
$705$	13.8411	0.521287
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26.6449	−1.00067	−0.500335	−	0.865832i	$-0.666790\pi$
$709$	−26.6449	−1.00067	−0.500335	+	0.865832i	$0.666790\pi$
$710$	0	0
$711$	69.2345	2.59650
$712$	0	0
$713$	−3.86847	−0.144875
$714$	0	0
$715$	3.01512	0.112759
$716$	0	0
$717$	−59.9695	−2.23960
$718$	0	0
$719$	45.8709	1.71070	0.855348	−	0.518053i	$-0.173343\pi$
$719$	45.8709	1.71070	0.855348	+	0.518053i	$0.173343\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−15.6535	−0.582161
$724$	0	0
$725$	1.22841	0.0456220
$726$	0	0
$727$	−15.0242	−0.557218	−0.278609	−	0.960405i	$-0.589873\pi$
$727$	−15.0242	−0.557218	−0.278609	+	0.960405i	$0.589873\pi$
$728$	0	0
$729$	86.6688	3.20996
$730$	0	0
$731$	−27.7824	−1.02757
$732$	0	0
$733$	−3.98158	−0.147063	−0.0735316	−	0.997293i	$-0.523427\pi$
$733$	−3.98158	−0.147063	−0.0735316	+	0.997293i	$0.523427\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.9841	1.21498
$738$	0	0
$739$	−50.8249	−1.86962	−0.934812	−	0.355144i	$-0.884432\pi$
$739$	−50.8249	−1.86962	−0.934812	+	0.355144i	$0.884432\pi$
$740$	0	0
$741$	−1.79748	−0.0660320
$742$	0	0
$743$	26.1138	0.958023	0.479011	−	0.877809i	$-0.340995\pi$
$743$	26.1138	0.958023	0.479011	+	0.877809i	$0.340995\pi$
$744$	0	0
$745$	48.5286	1.77795
$746$	0	0
$747$	−2.35842	−0.0862902
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.6796	−1.19250	−0.596248	−	0.802800i	$-0.703342\pi$
$751$	−32.6796	−1.19250	−0.596248	+	0.802800i	$0.703342\pi$
$752$	0	0
$753$	41.4934	1.51210
$754$	0	0
$755$	18.9246	0.688738
$756$	0	0
$757$	−18.2457	−0.663150	−0.331575	−	0.943429i	$-0.607580\pi$
$757$	−18.2457	−0.663150	−0.331575	+	0.943429i	$0.607580\pi$
$758$	0	0
$759$	−31.7945	−1.15407
$760$	0	0
$761$	−34.5400	−1.25207	−0.626036	−	0.779794i	$-0.715324\pi$
$761$	−34.5400	−1.25207	−0.626036	+	0.779794i	$0.715324\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	74.1404	2.68055
$766$	0	0
$767$	−4.12437	−0.148922
$768$	0	0
$769$	−20.9648	−0.756011	−0.378006	−	0.925803i	$-0.623390\pi$
$769$	−20.9648	−0.756011	−0.378006	+	0.925803i	$0.623390\pi$
$770$	0	0
$771$	−49.0194	−1.76539
$772$	0	0
$773$	7.08513	0.254834	0.127417	−	0.991849i	$-0.459331\pi$
$773$	7.08513	0.254834	0.127417	+	0.991849i	$0.459331\pi$
$774$	0	0
$775$	−0.207041	−0.00743712
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.28939	0.153684
$780$	0	0
$781$	−26.9896	−0.965765
$782$	0	0
$783$	98.1423	3.50732
$784$	0	0
$785$	9.65851	0.344727
$786$	0	0
$787$	5.66198	0.201828	0.100914	−	0.994895i	$-0.467823\pi$
$787$	5.66198	0.201828	0.100914	+	0.994895i	$0.467823\pi$
$788$	0	0
$789$	−19.6146	−0.698298
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.97272	0.283119
$794$	0	0
$795$	53.1803	1.88611
$796$	0	0
$797$	−39.3242	−1.39293	−0.696467	−	0.717589i	$-0.745246\pi$
$797$	−39.3242	−1.39293	−0.696467	+	0.717589i	$0.745246\pi$
$798$	0	0
$799$	7.37124	0.260776
$800$	0	0
$801$	−125.774	−4.44400
$802$	0	0
$803$	11.0775	0.390915
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.1871	0.992232
$808$	0	0
$809$	−19.8458	−0.697742	−0.348871	−	0.937171i	$-0.613435\pi$
$809$	−19.8458	−0.697742	−0.348871	+	0.937171i	$0.613435\pi$
$810$	0	0
$811$	−28.6083	−1.00457	−0.502286	−	0.864702i	$-0.667507\pi$
$811$	−28.6083	−1.00457	−0.502286	+	0.864702i	$0.667507\pi$
$812$	0	0
$813$	50.8331	1.78279
$814$	0	0
$815$	−4.63722	−0.162435
$816$	0	0
$817$	−6.87820	−0.240638
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.9740	1.95351	0.976753	−	0.214367i	$-0.0687687\pi$
$821$	55.9740	1.95351	0.976753	+	0.214367i	$0.0687687\pi$
$822$	0	0
$823$	51.5606	1.79729	0.898646	−	0.438676i	$-0.144552\pi$
$823$	51.5606	1.79729	0.898646	+	0.438676i	$0.144552\pi$
$824$	0	0
$825$	−1.70164	−0.0592436
$826$	0	0
$827$	−21.4455	−0.745735	−0.372867	−	0.927885i	$-0.621625\pi$
$827$	−21.4455	−0.745735	−0.372867	+	0.927885i	$0.621625\pi$
$828$	0	0
$829$	−20.3569	−0.707023	−0.353512	−	0.935430i	$-0.615013\pi$
$829$	−20.3569	−0.707023	−0.353512	+	0.935430i	$0.615013\pi$
$830$	0	0
$831$	84.6749	2.93734
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−14.2765	−0.494059
$836$	0	0
$837$	−16.5412	−0.571749
$838$	0	0
$839$	−39.6888	−1.37021	−0.685105	−	0.728444i	$-0.740244\pi$
$839$	−39.6888	−1.37021	−0.685105	+	0.728444i	$0.740244\pi$
$840$	0	0
$841$	5.31623	0.183318
$842$	0	0
$843$	−52.4985	−1.80814
$844$	0	0
$845$	−29.0043	−0.997779
$846$	0	0
$847$	0	0
$848$	0	0
$849$	110.664	3.79798
$850$	0	0
$851$	−22.6547	−0.776594
$852$	0	0
$853$	−39.6411	−1.35729	−0.678643	−	0.734468i	$-0.737432\pi$
$853$	−39.6411	−1.35729	−0.678643	+	0.734468i	$0.737432\pi$
$854$	0	0
$855$	18.3552	0.627736
$856$	0	0
$857$	3.27450	0.111855	0.0559275	−	0.998435i	$-0.482188\pi$
$857$	3.27450	0.111855	0.0559275	+	0.998435i	$0.482188\pi$
$858$	0	0
$859$	4.02589	0.137362	0.0686808	−	0.997639i	$-0.478121\pi$
$859$	4.02589	0.137362	0.0686808	+	0.997639i	$0.478121\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.3035	1.33791	0.668953	−	0.743304i	$-0.266743\pi$
$863$	39.3035	1.33791	0.668953	+	0.743304i	$0.266743\pi$
$864$	0	0
$865$	32.5871	1.10799
$866$	0	0
$867$	−2.27578	−0.0772897
$868$	0	0
$869$	−21.0244	−0.713204
$870$	0	0
$871$	7.30623	0.247562
$872$	0	0
$873$	−90.5982	−3.06628
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.3460	−0.349360	−0.174680	−	0.984625i	$-0.555889\pi$
$877$	−10.3460	−0.349360	−0.174680	+	0.984625i	$0.555889\pi$
$878$	0	0
$879$	−7.20169	−0.242907
$880$	0	0
$881$	−56.3514	−1.89853	−0.949264	−	0.314481i	$-0.898169\pi$
$881$	−56.3514	−1.89853	−0.949264	+	0.314481i	$0.898169\pi$
$882$	0	0
$883$	−17.5922	−0.592026	−0.296013	−	0.955184i	$-0.595657\pi$
$883$	−17.5922	−0.592026	−0.296013	+	0.955184i	$0.595657\pi$
$884$	0	0
$885$	57.8282	1.94388
$886$	0	0
$887$	35.9877	1.20835	0.604175	−	0.796852i	$-0.293503\pi$
$887$	35.9877	1.20835	0.604175	+	0.796852i	$0.293503\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−77.0350	−2.58077
$892$	0	0
$893$	1.82493	0.0610689
$894$	0	0
$895$	−17.4553	−0.583465
$896$	0	0
$897$	−7.04273	−0.235150
$898$	0	0
$899$	−5.78378	−0.192900
$900$	0	0
$901$	28.3217	0.943533
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.25729	0.208000
$906$	0	0
$907$	49.8102	1.65392	0.826961	−	0.562260i	$-0.190068\pi$
$907$	49.8102	1.65392	0.826961	+	0.562260i	$0.190068\pi$
$908$	0	0
$909$	−101.830	−3.37749
$910$	0	0
$911$	16.1757	0.535926	0.267963	−	0.963429i	$-0.413649\pi$
$911$	16.1757	0.535926	0.267963	+	0.963429i	$0.413649\pi$
$912$	0	0
$913$	0.716180	0.0237021
$914$	0	0
$915$	−111.786	−3.69554
$916$	0	0
$917$	0	0
$918$	0	0
$919$	56.5667	1.86596	0.932981	−	0.359926i	$-0.117198\pi$
$919$	56.5667	1.86596	0.932981	+	0.359926i	$0.117198\pi$
$920$	0	0
$921$	23.8938	0.787326
$922$	0	0
$923$	−5.97842	−0.196782
$924$	0	0
$925$	−1.21248	−0.0398662
$926$	0	0
$927$	18.5504	0.609274
$928$	0	0
$929$	7.19890	0.236188	0.118094	−	0.993002i	$-0.462322\pi$
$929$	7.19890	0.236188	0.118094	+	0.993002i	$0.462322\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−112.842	−3.69429
$934$	0	0
$935$	−22.5142	−0.736292
$936$	0	0
$937$	42.9203	1.40214	0.701072	−	0.713090i	$-0.252705\pi$
$937$	42.9203	1.40214	0.701072	+	0.713090i	$0.252705\pi$
$938$	0	0
$939$	86.3423	2.81768
$940$	0	0
$941$	45.4051	1.48016	0.740081	−	0.672517i	$-0.234787\pi$
$941$	45.4051	1.48016	0.740081	+	0.672517i	$0.234787\pi$
$942$	0	0
$943$	16.8064	0.547290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.9760	−0.389167	−0.194584	−	0.980886i	$-0.562336\pi$
$947$	−11.9760	−0.389167	−0.194584	+	0.980886i	$0.562336\pi$
$948$	0	0
$949$	2.45374	0.0796519
$950$	0	0
$951$	50.1658	1.62674
$952$	0	0
$953$	−16.2591	−0.526682	−0.263341	−	0.964703i	$-0.584824\pi$
$953$	−16.2591	−0.526682	−0.263341	+	0.964703i	$0.584824\pi$
$954$	0	0
$955$	0.391793	0.0126781
$956$	0	0
$957$	−47.5362	−1.53663
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0252	−0.968554
$962$	0	0
$963$	58.7664	1.89372
$964$	0	0
$965$	−26.9102	−0.866269
$966$	0	0
$967$	−7.56070	−0.243136	−0.121568	−	0.992583i	$-0.538792\pi$
$967$	−7.56070	−0.243136	−0.121568	+	0.992583i	$0.538792\pi$
$968$	0	0
$969$	13.4219	0.431175
$970$	0	0
$971$	32.0439	1.02834	0.514169	−	0.857689i	$-0.328101\pi$
$971$	32.0439	1.02834	0.514169	+	0.857689i	$0.328101\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.376927	−0.0120713
$976$	0	0
$977$	−31.7631	−1.01619	−0.508095	−	0.861301i	$-0.669650\pi$
$977$	−31.7631	−1.01619	−0.508095	+	0.861301i	$0.669650\pi$
$978$	0	0
$979$	38.1937	1.22067
$980$	0	0
$981$	−61.2589	−1.95585
$982$	0	0
$983$	23.2797	0.742507	0.371253	−	0.928532i	$-0.378928\pi$
$983$	23.2797	0.742507	0.371253	+	0.928532i	$0.378928\pi$
$984$	0	0
$985$	33.5533	1.06910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.9496	−0.856947
$990$	0	0
$991$	54.6609	1.73636	0.868181	−	0.496248i	$-0.165289\pi$
$991$	54.6609	1.73636	0.868181	+	0.496248i	$0.165289\pi$
$992$	0	0
$993$	63.8343	2.02572
$994$	0	0
$995$	8.00134	0.253659
$996$	0	0
$997$	−9.36301	−0.296529	−0.148265	−	0.988948i	$-0.547369\pi$
$997$	−9.36301	−0.296529	−0.148265	+	0.988948i	$0.547369\pi$
$998$	0	0
$999$	−96.8697	−3.06482

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bx.1.14	yes	14
7.6	odd	2		7448.2.a.bu.1.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bu.1.1	✓	14	7.6	odd	2
7448.2.a.bx.1.14	yes	14	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32292 q^{3} +2.28248 q^{5} +8.04181 q^{9} +O(q^{10})\)	\(q+3.32292 q^{3} +2.28248 q^{5} +8.04181 q^{9} -2.44205 q^{11} -0.540933 q^{13} +7.58449 q^{15} +4.03920 q^{17} +1.00000 q^{19} +3.91812 q^{23} +0.209698 q^{25} +16.7535 q^{27} +5.85801 q^{29} -0.987329 q^{31} -8.11474 q^{33} -5.78204 q^{37} -1.79748 q^{39} +4.28939 q^{41} -6.87820 q^{43} +18.3552 q^{45} +1.82493 q^{47} +13.4219 q^{51} +7.01172 q^{53} -5.57392 q^{55} +3.32292 q^{57} +7.62454 q^{59} -14.7388 q^{61} -1.23467 q^{65} -13.5067 q^{67} +13.0196 q^{69} +11.0520 q^{71} -4.53613 q^{73} +0.696810 q^{75} +8.60932 q^{79} +31.5452 q^{81} -0.293270 q^{83} +9.21937 q^{85} +19.4657 q^{87} -15.6400 q^{89} -3.28082 q^{93} +2.28248 q^{95} -11.2659 q^{97} -19.6385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9	\( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{19} - 4 q^{23} + 16 q^{25} + 36 q^{27} - 6 q^{29} + 16 q^{31} - 10 q^{33} + 6 q^{37} + 16 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{47} + 20 q^{51} - 6 q^{53} + 44 q^{55} + 6 q^{57} + 22 q^{59} + 10 q^{61} - 16 q^{65} + 4 q^{67} + 48 q^{69} + 6 q^{71} + 4 q^{73} + 64 q^{75} + 26 q^{79} + 30 q^{81} + 32 q^{83} - 8 q^{85} + 32 q^{87} - 54 q^{89} - 32 q^{93} + 2 q^{95} + 18 q^{97} - 38 q^{99}+O(q^{100}) \) 14 * q + 6 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^19 - 4 * q^23 + 16 * q^25 + 36 * q^27 - 6 * q^29 + 16 * q^31 - 10 * q^33 + 6 * q^37 + 16 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^47 + 20 * q^51 - 6 * q^53 + 44 * q^55 + 6 * q^57 + 22 * q^59 + 10 * q^61 - 16 * q^65 + 4 * q^67 + 48 * q^69 + 6 * q^71 + 4 * q^73 + 64 * q^75 + 26 * q^79 + 30 * q^81 + 32 * q^83 - 8 * q^85 + 32 * q^87 - 54 * q^89 - 32 * q^93 + 2 * q^95 + 18 * q^97 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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