LMFDB - Embedded newform 7448.2.a.bw.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:	$0$
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} - 2 x^{13} - 27 x^{12} + 46 x^{11} + 286 x^{10} - 386 x^{9} - 1525 x^{8} + 1414 x^{7} + \cdots + 392 $ x^14 - 2x^13 - 27x^12 + 46x^11 + 286x^10 - 386x^9 - 1525x^8 + 1414x^7 + 4390x^6 - 1910x^5 - 6603x^4 - 610x^3 + 4031x^2 + 2352*x + 392
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			$-1.85150$ 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)

$f(q)$	$=$	$q-1.85150 q^{3} -4.22552 q^{5} +0.428065 q^{9} +O(q^{10})$	$q-1.85150 q^{3} -4.22552 q^{5} +0.428065 q^{9} -5.16503 q^{11} +5.14176 q^{13} +7.82356 q^{15} +4.34942 q^{17} -1.00000 q^{19} -4.98189 q^{23} +12.8550 q^{25} +4.76195 q^{27} -9.92138 q^{29} -2.62473 q^{31} +9.56306 q^{33} -5.25681 q^{37} -9.51999 q^{39} +3.66743 q^{41} -7.10832 q^{43} -1.80880 q^{45} -11.5360 q^{47} -8.05296 q^{51} +1.39745 q^{53} +21.8249 q^{55} +1.85150 q^{57} -2.64929 q^{59} -11.5426 q^{61} -21.7266 q^{65} -7.39763 q^{67} +9.22399 q^{69} +5.38073 q^{71} -5.36421 q^{73} -23.8011 q^{75} -4.37152 q^{79} -10.1010 q^{81} -16.0383 q^{83} -18.3786 q^{85} +18.3695 q^{87} +16.2368 q^{89} +4.85971 q^{93} +4.22552 q^{95} -12.8732 q^{97} -2.21096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * 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
$3$	−1.85150	−1.06897	−0.534483	−	0.845179i	$-0.679494\pi$
$3$	−1.85150	−1.06897	−0.534483	+	0.845179i	$0.679494\pi$
$4$	0	0
$5$	−4.22552	−1.88971	−0.944855	−	0.327490i	$-0.893797\pi$
$5$	−4.22552	−1.88971	−0.944855	+	0.327490i	$0.893797\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.428065	0.142688
$10$	0	0
$11$	−5.16503	−1.55731	−0.778657	−	0.627450i	$-0.784099\pi$
$11$	−5.16503	−1.55731	−0.778657	+	0.627450i	$0.784099\pi$
$12$	0	0
$13$	5.14176	1.42607	0.713034	−	0.701129i	$-0.247320\pi$
$13$	5.14176	1.42607	0.713034	+	0.701129i	$0.247320\pi$
$14$	0	0
$15$	7.82356	2.02004
$16$	0	0
$17$	4.34942	1.05489	0.527445	−	0.849589i	$-0.323150\pi$
$17$	4.34942	1.05489	0.527445	+	0.849589i	$0.323150\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.98189	−1.03880	−0.519398	−	0.854532i	$-0.673844\pi$
$23$	−4.98189	−1.03880	−0.519398	+	0.854532i	$0.673844\pi$
$24$	0	0
$25$	12.8550	2.57100
$26$	0	0
$27$	4.76195	0.916437
$28$	0	0
$29$	−9.92138	−1.84235	−0.921177	−	0.389143i	$-0.872771\pi$
$29$	−9.92138	−1.84235	−0.921177	+	0.389143i	$0.872771\pi$
$30$	0	0
$31$	−2.62473	−0.471416	−0.235708	−	0.971824i	$-0.575741\pi$
$31$	−2.62473	−0.471416	−0.235708	+	0.971824i	$0.575741\pi$
$32$	0	0
$33$	9.56306	1.66472
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.25681	−0.864214	−0.432107	−	0.901822i	$-0.642230\pi$
$37$	−5.25681	−0.864214	−0.432107	+	0.901822i	$0.642230\pi$
$38$	0	0
$39$	−9.51999	−1.52442
$40$	0	0
$41$	3.66743	0.572757	0.286378	−	0.958117i	$-0.407549\pi$
$41$	3.66743	0.572757	0.286378	+	0.958117i	$0.407549\pi$
$42$	0	0
$43$	−7.10832	−1.08401	−0.542004	−	0.840376i	$-0.682334\pi$
$43$	−7.10832	−1.08401	−0.542004	+	0.840376i	$0.682334\pi$
$44$	0	0
$45$	−1.80880	−0.269639
$46$	0	0
$47$	−11.5360	−1.68269	−0.841346	−	0.540497i	$-0.818236\pi$
$47$	−11.5360	−1.68269	−0.841346	+	0.540497i	$0.818236\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.05296	−1.12764
$52$	0	0
$53$	1.39745	0.191955	0.0959774	−	0.995384i	$-0.469402\pi$
$53$	1.39745	0.191955	0.0959774	+	0.995384i	$0.469402\pi$
$54$	0	0
$55$	21.8249	2.94287
$56$	0	0
$57$	1.85150	0.245238
$58$	0	0
$59$	−2.64929	−0.344908	−0.172454	−	0.985018i	$-0.555170\pi$
$59$	−2.64929	−0.344908	−0.172454	+	0.985018i	$0.555170\pi$
$60$	0	0
$61$	−11.5426	−1.47788	−0.738941	−	0.673770i	$-0.764674\pi$
$61$	−11.5426	−1.47788	−0.738941	+	0.673770i	$0.764674\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21.7266	−2.69485
$66$	0	0
$67$	−7.39763	−0.903764	−0.451882	−	0.892078i	$-0.649247\pi$
$67$	−7.39763	−0.903764	−0.451882	+	0.892078i	$0.649247\pi$
$68$	0	0
$69$	9.22399	1.11044
$70$	0	0
$71$	5.38073	0.638575	0.319288	−	0.947658i	$-0.396556\pi$
$71$	5.38073	0.638575	0.319288	+	0.947658i	$0.396556\pi$
$72$	0	0
$73$	−5.36421	−0.627833	−0.313916	−	0.949451i	$-0.601641\pi$
$73$	−5.36421	−0.627833	−0.313916	+	0.949451i	$0.601641\pi$
$74$	0	0
$75$	−23.8011	−2.74831
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.37152	−0.491834	−0.245917	−	0.969291i	$-0.579089\pi$
$79$	−4.37152	−0.491834	−0.245917	+	0.969291i	$0.579089\pi$
$80$	0	0
$81$	−10.1010	−1.12233
$82$	0	0
$83$	−16.0383	−1.76043	−0.880216	−	0.474573i	$-0.842603\pi$
$83$	−16.0383	−1.76043	−0.880216	+	0.474573i	$0.842603\pi$
$84$	0	0
$85$	−18.3786	−1.99343
$86$	0	0
$87$	18.3695	1.96941
$88$	0	0
$89$	16.2368	1.72110	0.860549	−	0.509367i	$-0.170120\pi$
$89$	16.2368	1.72110	0.860549	+	0.509367i	$0.170120\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.85971	0.503928
$94$	0	0
$95$	4.22552	0.433529
$96$	0	0
$97$	−12.8732	−1.30708	−0.653540	−	0.756892i	$-0.726717\pi$
$97$	−12.8732	−1.30708	−0.653540	+	0.756892i	$0.726717\pi$
$98$	0	0
$99$	−2.21096	−0.222210
$100$	0	0
$101$	−7.17688	−0.714127	−0.357063	−	0.934080i	$-0.616222\pi$
$101$	−7.17688	−0.714127	−0.357063	+	0.934080i	$0.616222\pi$
$102$	0	0
$103$	7.64911	0.753689	0.376845	−	0.926277i	$-0.377009\pi$
$103$	7.64911	0.753689	0.376845	+	0.926277i	$0.377009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.26166	0.508664	0.254332	−	0.967117i	$-0.418144\pi$
$107$	5.26166	0.508664	0.254332	+	0.967117i	$0.418144\pi$
$108$	0	0
$109$	−9.40490	−0.900826	−0.450413	−	0.892820i	$-0.648723\pi$
$109$	−9.40490	−0.900826	−0.450413	+	0.892820i	$0.648723\pi$
$110$	0	0
$111$	9.73300	0.923815
$112$	0	0
$113$	−5.70651	−0.536824	−0.268412	−	0.963304i	$-0.586499\pi$
$113$	−5.70651	−0.536824	−0.268412	+	0.963304i	$0.586499\pi$
$114$	0	0
$115$	21.0511	1.96302
$116$	0	0
$117$	2.20101	0.203483
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.6775	1.42523
$122$	0	0
$123$	−6.79026	−0.612257
$124$	0	0
$125$	−33.1915	−2.96874
$126$	0	0
$127$	−15.8275	−1.40447	−0.702233	−	0.711947i	$-0.747813\pi$
$127$	−15.8275	−1.40447	−0.702233	+	0.711947i	$0.747813\pi$
$128$	0	0
$129$	13.1611	1.15877
$130$	0	0
$131$	−4.81240	−0.420461	−0.210231	−	0.977652i	$-0.567421\pi$
$131$	−4.81240	−0.420461	−0.210231	+	0.977652i	$0.567421\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−20.1217	−1.73180
$136$	0	0
$137$	−11.9654	−1.02227	−0.511137	−	0.859499i	$-0.670776\pi$
$137$	−11.9654	−1.02227	−0.511137	+	0.859499i	$0.670776\pi$
$138$	0	0
$139$	9.37805	0.795436	0.397718	−	0.917508i	$-0.369802\pi$
$139$	9.37805	0.795436	0.397718	+	0.917508i	$0.369802\pi$
$140$	0	0
$141$	21.3589	1.79874
$142$	0	0
$143$	−26.5573	−2.22084
$144$	0	0
$145$	41.9230	3.48151
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.5780	−1.76774	−0.883869	−	0.467735i	$-0.845070\pi$
$149$	−21.5780	−1.76774	−0.883869	+	0.467735i	$0.845070\pi$
$150$	0	0
$151$	−0.623954	−0.0507767	−0.0253883	−	0.999678i	$-0.508082\pi$
$151$	−0.623954	−0.0507767	−0.0253883	+	0.999678i	$0.508082\pi$
$152$	0	0
$153$	1.86183	0.150520
$154$	0	0
$155$	11.0909	0.890840
$156$	0	0
$157$	19.0614	1.52127	0.760633	−	0.649182i	$-0.224889\pi$
$157$	19.0614	1.52127	0.760633	+	0.649182i	$0.224889\pi$
$158$	0	0
$159$	−2.58739	−0.205193
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.93264	0.777985	0.388992	−	0.921241i	$-0.372823\pi$
$163$	9.93264	0.777985	0.388992	+	0.921241i	$0.372823\pi$
$164$	0	0
$165$	−40.4089	−3.14583
$166$	0	0
$167$	−3.87941	−0.300198	−0.150099	−	0.988671i	$-0.547959\pi$
$167$	−3.87941	−0.300198	−0.150099	+	0.988671i	$0.547959\pi$
$168$	0	0
$169$	13.4377	1.03367
$170$	0	0
$171$	−0.428065	−0.0327349
$172$	0	0
$173$	5.80054	0.441007	0.220504	−	0.975386i	$-0.429230\pi$
$173$	5.80054	0.441007	0.220504	+	0.975386i	$0.429230\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.90517	0.368695
$178$	0	0
$179$	20.0802	1.50086	0.750431	−	0.660949i	$-0.229846\pi$
$179$	20.0802	1.50086	0.750431	+	0.660949i	$0.229846\pi$
$180$	0	0
$181$	22.9622	1.70676	0.853382	−	0.521286i	$-0.174547\pi$
$181$	22.9622	1.70676	0.853382	+	0.521286i	$0.174547\pi$
$182$	0	0
$183$	21.3712	1.57981
$184$	0	0
$185$	22.2127	1.63311
$186$	0	0
$187$	−22.4649	−1.64279
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.8648	−1.00322	−0.501611	−	0.865093i	$-0.667259\pi$
$191$	−13.8648	−1.00322	−0.501611	+	0.865093i	$0.667259\pi$
$192$	0	0
$193$	−8.85169	−0.637159	−0.318579	−	0.947896i	$-0.603206\pi$
$193$	−8.85169	−0.637159	−0.318579	+	0.947896i	$0.603206\pi$
$194$	0	0
$195$	40.2269	2.88071
$196$	0	0
$197$	5.68820	0.405268	0.202634	−	0.979255i	$-0.435050\pi$
$197$	5.68820	0.405268	0.202634	+	0.979255i	$0.435050\pi$
$198$	0	0
$199$	−5.80886	−0.411779	−0.205890	−	0.978575i	$-0.566009\pi$
$199$	−5.80886	−0.411779	−0.205890	+	0.978575i	$0.566009\pi$
$200$	0	0
$201$	13.6967	0.966093
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−15.4968	−1.08234
$206$	0	0
$207$	−2.13257	−0.148224
$208$	0	0
$209$	5.16503	0.357272
$210$	0	0
$211$	6.79631	0.467877	0.233938	−	0.972251i	$-0.424839\pi$
$211$	6.79631	0.467877	0.233938	+	0.972251i	$0.424839\pi$
$212$	0	0
$213$	−9.96244	−0.682615
$214$	0	0
$215$	30.0363	2.04846
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.93185	0.671132
$220$	0	0
$221$	22.3637	1.50434
$222$	0	0
$223$	5.63115	0.377090	0.188545	−	0.982065i	$-0.439623\pi$
$223$	5.63115	0.377090	0.188545	+	0.982065i	$0.439623\pi$
$224$	0	0
$225$	5.50277	0.366852
$226$	0	0
$227$	−5.45894	−0.362323	−0.181161	−	0.983453i	$-0.557986\pi$
$227$	−5.45894	−0.362323	−0.181161	+	0.983453i	$0.557986\pi$
$228$	0	0
$229$	1.46101	0.0965462	0.0482731	−	0.998834i	$-0.484628\pi$
$229$	1.46101	0.0965462	0.0482731	+	0.998834i	$0.484628\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.5231	−0.885928	−0.442964	−	0.896539i	$-0.646073\pi$
$233$	−13.5231	−0.885928	−0.442964	+	0.896539i	$0.646073\pi$
$234$	0	0
$235$	48.7454	3.17980
$236$	0	0
$237$	8.09388	0.525754
$238$	0	0
$239$	3.58654	0.231994	0.115997	−	0.993250i	$-0.462994\pi$
$239$	3.58654	0.231994	0.115997	+	0.993250i	$0.462994\pi$
$240$	0	0
$241$	18.0246	1.16107	0.580534	−	0.814236i	$-0.302844\pi$
$241$	18.0246	1.16107	0.580534	+	0.814236i	$0.302844\pi$
$242$	0	0
$243$	4.41611	0.283294
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.14176	−0.327163
$248$	0	0
$249$	29.6950	1.88184
$250$	0	0
$251$	−20.0749	−1.26712	−0.633559	−	0.773695i	$-0.718407\pi$
$251$	−20.0749	−1.26712	−0.633559	+	0.773695i	$0.718407\pi$
$252$	0	0
$253$	25.7316	1.61773
$254$	0	0
$255$	34.0280	2.13091
$256$	0	0
$257$	3.95471	0.246688	0.123344	−	0.992364i	$-0.460638\pi$
$257$	3.95471	0.246688	0.123344	+	0.992364i	$0.460638\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.24699	−0.262882
$262$	0	0
$263$	−11.6175	−0.716367	−0.358183	−	0.933651i	$-0.616604\pi$
$263$	−11.6175	−0.716367	−0.358183	+	0.933651i	$0.616604\pi$
$264$	0	0
$265$	−5.90496	−0.362739
$266$	0	0
$267$	−30.0625	−1.83980
$268$	0	0
$269$	−26.5790	−1.62055	−0.810274	−	0.586052i	$-0.800682\pi$
$269$	−26.5790	−1.62055	−0.810274	+	0.586052i	$0.800682\pi$
$270$	0	0
$271$	3.88968	0.236281	0.118141	−	0.992997i	$-0.462307\pi$
$271$	3.88968	0.236281	0.118141	+	0.992997i	$0.462307\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−66.3965	−4.00386
$276$	0	0
$277$	17.2658	1.03740	0.518701	−	0.854956i	$-0.326416\pi$
$277$	17.2658	1.03740	0.518701	+	0.854956i	$0.326416\pi$
$278$	0	0
$279$	−1.12356	−0.0672655
$280$	0	0
$281$	7.48638	0.446600	0.223300	−	0.974750i	$-0.428317\pi$
$281$	7.48638	0.446600	0.223300	+	0.974750i	$0.428317\pi$
$282$	0	0
$283$	−25.1138	−1.49286	−0.746430	−	0.665464i	$-0.768234\pi$
$283$	−25.1138	−1.49286	−0.746430	+	0.665464i	$0.768234\pi$
$284$	0	0
$285$	−7.82356	−0.463428
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.91745	0.112791
$290$	0	0
$291$	23.8349	1.39722
$292$	0	0
$293$	1.75016	0.102245	0.0511227	−	0.998692i	$-0.483720\pi$
$293$	1.75016	0.102245	0.0511227	+	0.998692i	$0.483720\pi$
$294$	0	0
$295$	11.1946	0.651777
$296$	0	0
$297$	−24.5956	−1.42718
$298$	0	0
$299$	−25.6157	−1.48139
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.2880	0.763377
$304$	0	0
$305$	48.7736	2.79277
$306$	0	0
$307$	21.5528	1.23009	0.615043	−	0.788494i	$-0.289139\pi$
$307$	21.5528	1.23009	0.615043	+	0.788494i	$0.289139\pi$
$308$	0	0
$309$	−14.1623	−0.805668
$310$	0	0
$311$	−24.9198	−1.41307	−0.706537	−	0.707676i	$-0.749743\pi$
$311$	−24.9198	−1.41307	−0.706537	+	0.707676i	$0.749743\pi$
$312$	0	0
$313$	−1.78219	−0.100735	−0.0503677	−	0.998731i	$-0.516039\pi$
$313$	−1.78219	−0.100735	−0.0503677	+	0.998731i	$0.516039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.9338	0.838766	0.419383	−	0.907809i	$-0.362247\pi$
$317$	14.9338	0.838766	0.419383	+	0.907809i	$0.362247\pi$
$318$	0	0
$319$	51.2442	2.86912
$320$	0	0
$321$	−9.74199	−0.543745
$322$	0	0
$323$	−4.34942	−0.242008
$324$	0	0
$325$	66.0974	3.66642
$326$	0	0
$327$	17.4132	0.962952
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.10459	−0.225609	−0.112804	−	0.993617i	$-0.535983\pi$
$331$	−4.10459	−0.225609	−0.112804	+	0.993617i	$0.535983\pi$
$332$	0	0
$333$	−2.25025	−0.123313
$334$	0	0
$335$	31.2588	1.70785
$336$	0	0
$337$	23.4740	1.27871	0.639355	−	0.768912i	$-0.279201\pi$
$337$	23.4740	1.27871	0.639355	+	0.768912i	$0.279201\pi$
$338$	0	0
$339$	10.5656	0.573846
$340$	0	0
$341$	13.5568	0.734143
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−38.9761	−2.09840
$346$	0	0
$347$	−31.3867	−1.68493	−0.842464	−	0.538752i	$-0.818896\pi$
$347$	−31.3867	−1.68493	−0.842464	+	0.538752i	$0.818896\pi$
$348$	0	0
$349$	−18.1985	−0.974141	−0.487071	−	0.873363i	$-0.661935\pi$
$349$	−18.1985	−0.974141	−0.487071	+	0.873363i	$0.661935\pi$
$350$	0	0
$351$	24.4848	1.30690
$352$	0	0
$353$	24.0173	1.27831	0.639157	−	0.769076i	$-0.279283\pi$
$353$	24.0173	1.27831	0.639157	+	0.769076i	$0.279283\pi$
$354$	0	0
$355$	−22.7364	−1.20672
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.7703	1.14899	0.574497	−	0.818507i	$-0.305198\pi$
$359$	21.7703	1.14899	0.574497	+	0.818507i	$0.305198\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−29.0269	−1.52352
$364$	0	0
$365$	22.6666	1.18642
$366$	0	0
$367$	−12.3697	−0.645692	−0.322846	−	0.946452i	$-0.604640\pi$
$367$	−12.3697	−0.645692	−0.322846	+	0.946452i	$0.604640\pi$
$368$	0	0
$369$	1.56990	0.0817256
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.1353	−0.576562	−0.288281	−	0.957546i	$-0.593084\pi$
$373$	−11.1353	−0.576562	−0.288281	+	0.957546i	$0.593084\pi$
$374$	0	0
$375$	61.4542	3.17348
$376$	0	0
$377$	−51.0134	−2.62732
$378$	0	0
$379$	27.6099	1.41822	0.709112	−	0.705096i	$-0.249096\pi$
$379$	27.6099	1.41822	0.709112	+	0.705096i	$0.249096\pi$
$380$	0	0
$381$	29.3047	1.50133
$382$	0	0
$383$	−20.1811	−1.03121	−0.515603	−	0.856827i	$-0.672432\pi$
$383$	−20.1811	−1.03121	−0.515603	+	0.856827i	$0.672432\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.04282	−0.154675
$388$	0	0
$389$	9.88684	0.501283	0.250641	−	0.968080i	$-0.419359\pi$
$389$	9.88684	0.501283	0.250641	+	0.968080i	$0.419359\pi$
$390$	0	0
$391$	−21.6683	−1.09581
$392$	0	0
$393$	8.91017	0.449459
$394$	0	0
$395$	18.4719	0.929423
$396$	0	0
$397$	0.237270	0.0119082	0.00595412	−	0.999982i	$-0.498105\pi$
$397$	0.237270	0.0119082	0.00595412	+	0.999982i	$0.498105\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.41247	0.0705355	0.0352678	−	0.999378i	$-0.488772\pi$
$401$	1.41247	0.0705355	0.0352678	+	0.999378i	$0.488772\pi$
$402$	0	0
$403$	−13.4958	−0.672272
$404$	0	0
$405$	42.6818	2.12087
$406$	0	0
$407$	27.1516	1.34585
$408$	0	0
$409$	−25.0406	−1.23818	−0.619088	−	0.785322i	$-0.712498\pi$
$409$	−25.0406	−1.23818	−0.619088	+	0.785322i	$0.712498\pi$
$410$	0	0
$411$	22.1540	1.09278
$412$	0	0
$413$	0	0
$414$	0	0
$415$	67.7702	3.32671
$416$	0	0
$417$	−17.3635	−0.850294
$418$	0	0
$419$	−3.87107	−0.189114	−0.0945570	−	0.995519i	$-0.530143\pi$
$419$	−3.87107	−0.189114	−0.0945570	+	0.995519i	$0.530143\pi$
$420$	0	0
$421$	5.86378	0.285783	0.142892	−	0.989738i	$-0.454360\pi$
$421$	5.86378	0.285783	0.142892	+	0.989738i	$0.454360\pi$
$422$	0	0
$423$	−4.93813	−0.240100
$424$	0	0
$425$	55.9118	2.71212
$426$	0	0
$427$	0	0
$428$	0	0
$429$	49.1710	2.37400
$430$	0	0
$431$	−31.5901	−1.52164	−0.760820	−	0.648963i	$-0.775203\pi$
$431$	−31.5901	−1.52164	−0.760820	+	0.648963i	$0.775203\pi$
$432$	0	0
$433$	0.390316	0.0187574	0.00937869	−	0.999956i	$-0.497015\pi$
$433$	0.390316	0.0187574	0.00937869	+	0.999956i	$0.497015\pi$
$434$	0	0
$435$	−77.6205	−3.72162
$436$	0	0
$437$	4.98189	0.238316
$438$	0	0
$439$	31.7294	1.51436	0.757180	−	0.653206i	$-0.226576\pi$
$439$	31.7294	1.51436	0.757180	+	0.653206i	$0.226576\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.02627	−0.0487597	−0.0243799	−	0.999703i	$-0.507761\pi$
$443$	−1.02627	−0.0487597	−0.0243799	+	0.999703i	$0.507761\pi$
$444$	0	0
$445$	−68.6090	−3.25238
$446$	0	0
$447$	39.9517	1.88965
$448$	0	0
$449$	−4.84004	−0.228416	−0.114208	−	0.993457i	$-0.536433\pi$
$449$	−4.84004	−0.228416	−0.114208	+	0.993457i	$0.536433\pi$
$450$	0	0
$451$	−18.9424	−0.891962
$452$	0	0
$453$	1.15525	0.0542785
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.7400	−0.595953	−0.297977	−	0.954573i	$-0.596312\pi$
$457$	−12.7400	−0.595953	−0.297977	+	0.954573i	$0.596312\pi$
$458$	0	0
$459$	20.7117	0.966740
$460$	0	0
$461$	2.23355	0.104027	0.0520135	−	0.998646i	$-0.483436\pi$
$461$	2.23355	0.104027	0.0520135	+	0.998646i	$0.483436\pi$
$462$	0	0
$463$	−36.0746	−1.67653	−0.838264	−	0.545265i	$-0.816429\pi$
$463$	−36.0746	−1.67653	−0.838264	+	0.545265i	$0.816429\pi$
$464$	0	0
$465$	−20.5348	−0.952277
$466$	0	0
$467$	−28.9597	−1.34009	−0.670047	−	0.742318i	$-0.733726\pi$
$467$	−28.9597	−1.34009	−0.670047	+	0.742318i	$0.733726\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−35.2923	−1.62618
$472$	0	0
$473$	36.7147	1.68814
$474$	0	0
$475$	−12.8550	−0.589828
$476$	0	0
$477$	0.598200	0.0273897
$478$	0	0
$479$	17.6376	0.805884	0.402942	−	0.915225i	$-0.367988\pi$
$479$	17.6376	0.805884	0.402942	+	0.915225i	$0.367988\pi$
$480$	0	0
$481$	−27.0293	−1.23243
$482$	0	0
$483$	0	0
$484$	0	0
$485$	54.3961	2.47000
$486$	0	0
$487$	−2.31369	−0.104843	−0.0524217	−	0.998625i	$-0.516694\pi$
$487$	−2.31369	−0.104843	−0.0524217	+	0.998625i	$0.516694\pi$
$488$	0	0
$489$	−18.3903	−0.831639
$490$	0	0
$491$	−9.76074	−0.440496	−0.220248	−	0.975444i	$-0.570687\pi$
$491$	−9.76074	−0.440496	−0.220248	+	0.975444i	$0.570687\pi$
$492$	0	0
$493$	−43.1522	−1.94348
$494$	0	0
$495$	9.34247	0.419913
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−40.1194	−1.79599	−0.897995	−	0.440005i	$-0.854977\pi$
$499$	−40.1194	−1.79599	−0.897995	+	0.440005i	$0.854977\pi$
$500$	0	0
$501$	7.18275	0.320902
$502$	0	0
$503$	−1.05275	−0.0469397	−0.0234698	−	0.999725i	$-0.507471\pi$
$503$	−1.05275	−0.0469397	−0.0234698	+	0.999725i	$0.507471\pi$
$504$	0	0
$505$	30.3261	1.34949
$506$	0	0
$507$	−24.8800	−1.10496
$508$	0	0
$509$	23.4107	1.03766	0.518831	−	0.854877i	$-0.326367\pi$
$509$	23.4107	1.03766	0.518831	+	0.854877i	$0.326367\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.76195	−0.210245
$514$	0	0
$515$	−32.3215	−1.42425
$516$	0	0
$517$	59.5835	2.62048
$518$	0	0
$519$	−10.7397	−0.471422
$520$	0	0
$521$	13.9590	0.611555	0.305778	−	0.952103i	$-0.401084\pi$
$521$	13.9590	0.611555	0.305778	+	0.952103i	$0.401084\pi$
$522$	0	0
$523$	33.9104	1.48280	0.741399	−	0.671065i	$-0.234163\pi$
$523$	33.9104	1.48280	0.741399	+	0.671065i	$0.234163\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.4161	−0.497292
$528$	0	0
$529$	1.81923	0.0790971
$530$	0	0
$531$	−1.13407	−0.0492144
$532$	0	0
$533$	18.8571	0.816790
$534$	0	0
$535$	−22.2333	−0.961228
$536$	0	0
$537$	−37.1785	−1.60437
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.72469	−0.289117	−0.144558	−	0.989496i	$-0.546176\pi$
$541$	−6.72469	−0.289117	−0.144558	+	0.989496i	$0.546176\pi$
$542$	0	0
$543$	−42.5145	−1.82447
$544$	0	0
$545$	39.7406	1.70230
$546$	0	0
$547$	28.8045	1.23159	0.615795	−	0.787906i	$-0.288835\pi$
$547$	28.8045	1.23159	0.615795	+	0.787906i	$0.288835\pi$
$548$	0	0
$549$	−4.94099	−0.210876
$550$	0	0
$551$	9.92138	0.422665
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−41.1270	−1.74574
$556$	0	0
$557$	−4.38136	−0.185644	−0.0928222	−	0.995683i	$-0.529589\pi$
$557$	−4.38136	−0.185644	−0.0928222	+	0.995683i	$0.529589\pi$
$558$	0	0
$559$	−36.5493	−1.54587
$560$	0	0
$561$	41.5938	1.75609
$562$	0	0
$563$	5.62218	0.236947	0.118473	−	0.992957i	$-0.462200\pi$
$563$	5.62218	0.236947	0.118473	+	0.992957i	$0.462200\pi$
$564$	0	0
$565$	24.1130	1.01444
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.77998	0.242309	0.121155	−	0.992634i	$-0.461340\pi$
$569$	5.77998	0.242309	0.121155	+	0.992634i	$0.461340\pi$
$570$	0	0
$571$	26.8779	1.12481	0.562403	−	0.826863i	$-0.309877\pi$
$571$	26.8779	1.12481	0.562403	+	0.826863i	$0.309877\pi$
$572$	0	0
$573$	25.6707	1.07241
$574$	0	0
$575$	−64.0423	−2.67075
$576$	0	0
$577$	16.7211	0.696109	0.348054	−	0.937474i	$-0.386842\pi$
$577$	16.7211	0.696109	0.348054	+	0.937474i	$0.386842\pi$
$578$	0	0
$579$	16.3889	0.681101
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.21788	−0.298934
$584$	0	0
$585$	−9.30039	−0.384524
$586$	0	0
$587$	−28.3078	−1.16839	−0.584193	−	0.811615i	$-0.698589\pi$
$587$	−28.3078	−1.16839	−0.584193	+	0.811615i	$0.698589\pi$
$588$	0	0
$589$	2.62473	0.108150
$590$	0	0
$591$	−10.5317	−0.433217
$592$	0	0
$593$	37.4570	1.53817	0.769087	−	0.639144i	$-0.220711\pi$
$593$	37.4570	1.53817	0.769087	+	0.639144i	$0.220711\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.7551	0.440178
$598$	0	0
$599$	40.4889	1.65433	0.827166	−	0.561957i	$-0.189951\pi$
$599$	40.4889	1.65433	0.827166	+	0.561957i	$0.189951\pi$
$600$	0	0
$601$	10.2505	0.418125	0.209062	−	0.977902i	$-0.432959\pi$
$601$	10.2505	0.418125	0.209062	+	0.977902i	$0.432959\pi$
$602$	0	0
$603$	−3.16666	−0.128956
$604$	0	0
$605$	−66.2455	−2.69326
$606$	0	0
$607$	−0.676671	−0.0274652	−0.0137326	−	0.999906i	$-0.504371\pi$
$607$	−0.676671	−0.0274652	−0.0137326	+	0.999906i	$0.504371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.3151	−2.39963
$612$	0	0
$613$	28.9931	1.17102	0.585511	−	0.810665i	$-0.300894\pi$
$613$	28.9931	1.17102	0.585511	+	0.810665i	$0.300894\pi$
$614$	0	0
$615$	28.6924	1.15699
$616$	0	0
$617$	−23.6529	−0.952228	−0.476114	−	0.879383i	$-0.657955\pi$
$617$	−23.6529	−0.952228	−0.476114	+	0.879383i	$0.657955\pi$
$618$	0	0
$619$	−21.0744	−0.847054	−0.423527	−	0.905884i	$-0.639208\pi$
$619$	−21.0744	−0.847054	−0.423527	+	0.905884i	$0.639208\pi$
$620$	0	0
$621$	−23.7235	−0.951991
$622$	0	0
$623$	0	0
$624$	0	0
$625$	75.9762	3.03905
$626$	0	0
$627$	−9.56306	−0.381912
$628$	0	0
$629$	−22.8641	−0.911650
$630$	0	0
$631$	27.2647	1.08539	0.542695	−	0.839930i	$-0.317404\pi$
$631$	27.2647	1.08539	0.542695	+	0.839930i	$0.317404\pi$
$632$	0	0
$633$	−12.5834	−0.500145
$634$	0	0
$635$	66.8795	2.65403
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.30330	0.0911171
$640$	0	0
$641$	33.3904	1.31884	0.659421	−	0.751774i	$-0.270802\pi$
$641$	33.3904	1.31884	0.659421	+	0.751774i	$0.270802\pi$
$642$	0	0
$643$	−12.6713	−0.499707	−0.249854	−	0.968284i	$-0.580383\pi$
$643$	−12.6713	−0.499707	−0.249854	+	0.968284i	$0.580383\pi$
$644$	0	0
$645$	−55.6124	−2.18974
$646$	0	0
$647$	−20.2334	−0.795457	−0.397729	−	0.917503i	$-0.630201\pi$
$647$	−20.2334	−0.795457	−0.397729	+	0.917503i	$0.630201\pi$
$648$	0	0
$649$	13.6837	0.537131
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.3576	0.405324	0.202662	−	0.979249i	$-0.435041\pi$
$653$	10.3576	0.405324	0.202662	+	0.979249i	$0.435041\pi$
$654$	0	0
$655$	20.3349	0.794549
$656$	0	0
$657$	−2.29623	−0.0895844
$658$	0	0
$659$	−5.45001	−0.212302	−0.106151	−	0.994350i	$-0.533853\pi$
$659$	−5.45001	−0.212302	−0.106151	+	0.994350i	$0.533853\pi$
$660$	0	0
$661$	−20.7326	−0.806403	−0.403202	−	0.915111i	$-0.632103\pi$
$661$	−20.7326	−0.806403	−0.403202	+	0.915111i	$0.632103\pi$
$662$	0	0
$663$	−41.4064	−1.60809
$664$	0	0
$665$	0	0
$666$	0	0
$667$	49.4272	1.91383
$668$	0	0
$669$	−10.4261	−0.403096
$670$	0	0
$671$	59.6179	2.30153
$672$	0	0
$673$	16.2718	0.627233	0.313617	−	0.949550i	$-0.398459\pi$
$673$	16.2718	0.627233	0.313617	+	0.949550i	$0.398459\pi$
$674$	0	0
$675$	61.2149	2.35616
$676$	0	0
$677$	40.4623	1.55509	0.777546	−	0.628826i	$-0.216464\pi$
$677$	40.4623	1.55509	0.777546	+	0.628826i	$0.216464\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.1073	0.387311
$682$	0	0
$683$	14.6544	0.560734	0.280367	−	0.959893i	$-0.409544\pi$
$683$	14.6544	0.560734	0.280367	+	0.959893i	$0.409544\pi$
$684$	0	0
$685$	50.5601	1.93180
$686$	0	0
$687$	−2.70506	−0.103205
$688$	0	0
$689$	7.18537	0.273741
$690$	0	0
$691$	19.8723	0.755978	0.377989	−	0.925810i	$-0.376616\pi$
$691$	19.8723	0.755978	0.377989	+	0.925810i	$0.376616\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−39.6271	−1.50314
$696$	0	0
$697$	15.9512	0.604195
$698$	0	0
$699$	25.0381	0.947027
$700$	0	0
$701$	−15.8350	−0.598081	−0.299040	−	0.954240i	$-0.596667\pi$
$701$	−15.8350	−0.598081	−0.299040	+	0.954240i	$0.596667\pi$
$702$	0	0
$703$	5.25681	0.198264
$704$	0	0
$705$	−90.2522	−3.39910
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26.9803	−1.01327	−0.506633	−	0.862162i	$-0.669110\pi$
$709$	−26.9803	−1.01327	−0.506633	+	0.862162i	$0.669110\pi$
$710$	0	0
$711$	−1.87129	−0.0701789
$712$	0	0
$713$	13.0761	0.489705
$714$	0	0
$715$	112.219	4.19673
$716$	0	0
$717$	−6.64048	−0.247993
$718$	0	0
$719$	−0.797404	−0.0297382	−0.0148691	−	0.999889i	$-0.504733\pi$
$719$	−0.797404	−0.0297382	−0.0148691	+	0.999889i	$0.504733\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−33.3726	−1.24114
$724$	0	0
$725$	−127.539	−4.73670
$726$	0	0
$727$	45.4087	1.68412	0.842058	−	0.539388i	$-0.181344\pi$
$727$	45.4087	1.68412	0.842058	+	0.539388i	$0.181344\pi$
$728$	0	0
$729$	22.1264	0.819497
$730$	0	0
$731$	−30.9171	−1.14351
$732$	0	0
$733$	−5.65213	−0.208766	−0.104383	−	0.994537i	$-0.533287\pi$
$733$	−5.65213	−0.208766	−0.104383	+	0.994537i	$0.533287\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.2089	1.40744
$738$	0	0
$739$	−24.0449	−0.884507	−0.442253	−	0.896890i	$-0.645821\pi$
$739$	−24.0449	−0.884507	−0.442253	+	0.896890i	$0.645821\pi$
$740$	0	0
$741$	9.51999	0.349726
$742$	0	0
$743$	−23.3836	−0.857862	−0.428931	−	0.903337i	$-0.641110\pi$
$743$	−23.3836	−0.857862	−0.428931	+	0.903337i	$0.641110\pi$
$744$	0	0
$745$	91.1782	3.34051
$746$	0	0
$747$	−6.86543	−0.251193
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.5434	0.603679	0.301839	−	0.953359i	$-0.402399\pi$
$751$	16.5434	0.603679	0.301839	+	0.953359i	$0.402399\pi$
$752$	0	0
$753$	37.1688	1.35451
$754$	0	0
$755$	2.63653	0.0959532
$756$	0	0
$757$	9.67569	0.351669	0.175834	−	0.984420i	$-0.443738\pi$
$757$	9.67569	0.351669	0.175834	+	0.984420i	$0.443738\pi$
$758$	0	0
$759$	−47.6421	−1.72930
$760$	0	0
$761$	7.09326	0.257130	0.128565	−	0.991701i	$-0.458963\pi$
$761$	7.09326	0.257130	0.128565	+	0.991701i	$0.458963\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−7.86721	−0.284440
$766$	0	0
$767$	−13.6220	−0.491863
$768$	0	0
$769$	14.8444	0.535304	0.267652	−	0.963516i	$-0.413752\pi$
$769$	14.8444	0.535304	0.267652	+	0.963516i	$0.413752\pi$
$770$	0	0
$771$	−7.32215	−0.263701
$772$	0	0
$773$	45.3307	1.63043	0.815216	−	0.579157i	$-0.196618\pi$
$773$	45.3307	1.63043	0.815216	+	0.579157i	$0.196618\pi$
$774$	0	0
$775$	−33.7410	−1.21201
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.66743	−0.131399
$780$	0	0
$781$	−27.7916	−0.994462
$782$	0	0
$783$	−47.2451	−1.68840
$784$	0	0
$785$	−80.5443	−2.87475
$786$	0	0
$787$	−7.69870	−0.274429	−0.137214	−	0.990541i	$-0.543815\pi$
$787$	−7.69870	−0.274429	−0.137214	+	0.990541i	$0.543815\pi$
$788$	0	0
$789$	21.5099	0.765772
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−59.3494	−2.10756
$794$	0	0
$795$	10.9331	0.387756
$796$	0	0
$797$	17.5731	0.622472	0.311236	−	0.950333i	$-0.399257\pi$
$797$	17.5731	0.622472	0.311236	+	0.950333i	$0.399257\pi$
$798$	0	0
$799$	−50.1747	−1.77505
$800$	0	0
$801$	6.95040	0.245580
$802$	0	0
$803$	27.7063	0.977733
$804$	0	0
$805$	0	0
$806$	0	0
$807$	49.2110	1.73231
$808$	0	0
$809$	−41.8181	−1.47024	−0.735122	−	0.677935i	$-0.762875\pi$
$809$	−41.8181	−1.47024	−0.735122	+	0.677935i	$0.762875\pi$
$810$	0	0
$811$	14.8364	0.520976	0.260488	−	0.965477i	$-0.416117\pi$
$811$	14.8364	0.520976	0.260488	+	0.965477i	$0.416117\pi$
$812$	0	0
$813$	−7.20175	−0.252577
$814$	0	0
$815$	−41.9706	−1.47016
$816$	0	0
$817$	7.10832	0.248689
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.0222	−1.25718	−0.628591	−	0.777736i	$-0.716368\pi$
$821$	−36.0222	−1.25718	−0.628591	+	0.777736i	$0.716368\pi$
$822$	0	0
$823$	−30.8190	−1.07428	−0.537142	−	0.843492i	$-0.680496\pi$
$823$	−30.8190	−1.07428	−0.537142	+	0.843492i	$0.680496\pi$
$824$	0	0
$825$	122.933	4.27999
$826$	0	0
$827$	−16.8333	−0.585350	−0.292675	−	0.956212i	$-0.594545\pi$
$827$	−16.8333	−0.585350	−0.292675	+	0.956212i	$0.594545\pi$
$828$	0	0
$829$	12.4429	0.432160	0.216080	−	0.976376i	$-0.430673\pi$
$829$	12.4429	0.432160	0.216080	+	0.976376i	$0.430673\pi$
$830$	0	0
$831$	−31.9677	−1.10895
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.3925	0.567287
$836$	0	0
$837$	−12.4988	−0.432023
$838$	0	0
$839$	−35.9073	−1.23966	−0.619829	−	0.784737i	$-0.712798\pi$
$839$	−35.9073	−1.23966	−0.619829	+	0.784737i	$0.712798\pi$
$840$	0	0
$841$	69.4338	2.39427
$842$	0	0
$843$	−13.8611	−0.477400
$844$	0	0
$845$	−56.7813	−1.95334
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.4983	1.59582
$850$	0	0
$851$	26.1888	0.897742
$852$	0	0
$853$	−37.7399	−1.29219	−0.646095	−	0.763257i	$-0.723599\pi$
$853$	−37.7399	−1.29219	−0.646095	+	0.763257i	$0.723599\pi$
$854$	0	0
$855$	1.80880	0.0618595
$856$	0	0
$857$	−43.9099	−1.49994	−0.749968	−	0.661475i	$-0.769931\pi$
$857$	−43.9099	−1.49994	−0.749968	+	0.661475i	$0.769931\pi$
$858$	0	0
$859$	−2.52855	−0.0862730	−0.0431365	−	0.999069i	$-0.513735\pi$
$859$	−2.52855	−0.0862730	−0.0431365	+	0.999069i	$0.513735\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.65674	0.328719	0.164360	−	0.986400i	$-0.447444\pi$
$863$	9.65674	0.328719	0.164360	+	0.986400i	$0.447444\pi$
$864$	0	0
$865$	−24.5103	−0.833375
$866$	0	0
$867$	−3.55016	−0.120570
$868$	0	0
$869$	22.5790	0.765940
$870$	0	0
$871$	−38.0369	−1.28883
$872$	0	0
$873$	−5.51058	−0.186505
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.1078	1.28681	0.643405	−	0.765526i	$-0.277521\pi$
$877$	38.1078	1.28681	0.643405	+	0.765526i	$0.277521\pi$
$878$	0	0
$879$	−3.24043	−0.109297
$880$	0	0
$881$	17.3623	0.584952	0.292476	−	0.956273i	$-0.405521\pi$
$881$	17.3623	0.584952	0.292476	+	0.956273i	$0.405521\pi$
$882$	0	0
$883$	−28.1085	−0.945925	−0.472963	−	0.881083i	$-0.656815\pi$
$883$	−28.1085	−0.945925	−0.472963	+	0.881083i	$0.656815\pi$
$884$	0	0
$885$	−20.7269	−0.696727
$886$	0	0
$887$	39.5290	1.32725	0.663627	−	0.748064i	$-0.269016\pi$
$887$	39.5290	1.32725	0.663627	+	0.748064i	$0.269016\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	52.1717	1.74782
$892$	0	0
$893$	11.5360	0.386036
$894$	0	0
$895$	−84.8491	−2.83619
$896$	0	0
$897$	47.4275	1.58356
$898$	0	0
$899$	26.0410	0.868516
$900$	0	0
$901$	6.07811	0.202491
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−97.0271	−3.22529
$906$	0	0
$907$	−49.7230	−1.65103	−0.825513	−	0.564383i	$-0.809114\pi$
$907$	−49.7230	−1.65103	−0.825513	+	0.564383i	$0.809114\pi$
$908$	0	0
$909$	−3.07217	−0.101897
$910$	0	0
$911$	30.0376	0.995191	0.497595	−	0.867409i	$-0.334216\pi$
$911$	30.0376	0.995191	0.497595	+	0.867409i	$0.334216\pi$
$912$	0	0
$913$	82.8383	2.74155
$914$	0	0
$915$	−90.3044	−2.98537
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44.4099	1.46495	0.732474	−	0.680795i	$-0.238365\pi$
$919$	44.4099	1.46495	0.732474	+	0.680795i	$0.238365\pi$
$920$	0	0
$921$	−39.9051	−1.31492
$922$	0	0
$923$	27.6664	0.910652
$924$	0	0
$925$	−67.5763	−2.22190
$926$	0	0
$927$	3.27431	0.107543
$928$	0	0
$929$	51.5358	1.69083	0.845417	−	0.534107i	$-0.179352\pi$
$929$	51.5358	1.69083	0.845417	+	0.534107i	$0.179352\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	46.1391	1.51053
$934$	0	0
$935$	94.9257	3.10440
$936$	0	0
$937$	31.4862	1.02861	0.514305	−	0.857607i	$-0.328050\pi$
$937$	31.4862	1.02861	0.514305	+	0.857607i	$0.328050\pi$
$938$	0	0
$939$	3.29973	0.107683
$940$	0	0
$941$	−33.0699	−1.07805	−0.539024	−	0.842290i	$-0.681207\pi$
$941$	−33.0699	−1.07805	−0.539024	+	0.842290i	$0.681207\pi$
$942$	0	0
$943$	−18.2707	−0.594977
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.2485	−0.982944	−0.491472	−	0.870893i	$-0.663541\pi$
$947$	−30.2485	−0.982944	−0.491472	+	0.870893i	$0.663541\pi$
$948$	0	0
$949$	−27.5815	−0.895333
$950$	0	0
$951$	−27.6500	−0.896612
$952$	0	0
$953$	19.1311	0.619716	0.309858	−	0.950783i	$-0.399718\pi$
$953$	19.1311	0.619716	0.309858	+	0.950783i	$0.399718\pi$
$954$	0	0
$955$	58.5860	1.89580
$956$	0	0
$957$	−94.8788	−3.06700
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.1108	−0.777767
$962$	0	0
$963$	2.25233	0.0725804
$964$	0	0
$965$	37.4030	1.20404
$966$	0	0
$967$	3.75264	0.120677	0.0603384	−	0.998178i	$-0.480782\pi$
$967$	3.75264	0.120677	0.0603384	+	0.998178i	$0.480782\pi$
$968$	0	0
$969$	8.05296	0.258698
$970$	0	0
$971$	−31.4099	−1.00799	−0.503997	−	0.863706i	$-0.668138\pi$
$971$	−31.4099	−1.00799	−0.503997	+	0.863706i	$0.668138\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−122.380	−3.91928
$976$	0	0
$977$	32.8995	1.05255	0.526273	−	0.850315i	$-0.323589\pi$
$977$	32.8995	1.05255	0.526273	+	0.850315i	$0.323589\pi$
$978$	0	0
$979$	−83.8636	−2.68029
$980$	0	0
$981$	−4.02590	−0.128537
$982$	0	0
$983$	−32.4943	−1.03641	−0.518204	−	0.855257i	$-0.673399\pi$
$983$	−32.4943	−1.03641	−0.518204	+	0.855257i	$0.673399\pi$
$984$	0	0
$985$	−24.0356	−0.765838
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.4129	1.12606
$990$	0	0
$991$	37.2383	1.18291	0.591456	−	0.806337i	$-0.298553\pi$
$991$	37.2383	1.18291	0.591456	+	0.806337i	$0.298553\pi$
$992$	0	0
$993$	7.59967	0.241168
$994$	0	0
$995$	24.5455	0.778143
$996$	0	0
$997$	−21.1919	−0.671155	−0.335577	−	0.942013i	$-0.608931\pi$
$997$	−21.1919	−0.671155	−0.335577	+	0.942013i	$0.608931\pi$
$998$	0	0
$999$	−25.0326	−0.791998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bw.1.4	yes	14
7.6	odd	2		7448.2.a.bv.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bv.1.11	✓	14	7.6	odd	2
7448.2.a.bw.1.4	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)

\(f(q)\)	\(=\)	\(q-1.85150 q^{3} -4.22552 q^{5} +0.428065 q^{9} +O(q^{10})\)	\(q-1.85150 q^{3} -4.22552 q^{5} +0.428065 q^{9} -5.16503 q^{11} +5.14176 q^{13} +7.82356 q^{15} +4.34942 q^{17} -1.00000 q^{19} -4.98189 q^{23} +12.8550 q^{25} +4.76195 q^{27} -9.92138 q^{29} -2.62473 q^{31} +9.56306 q^{33} -5.25681 q^{37} -9.51999 q^{39} +3.66743 q^{41} -7.10832 q^{43} -1.80880 q^{45} -11.5360 q^{47} -8.05296 q^{51} +1.39745 q^{53} +21.8249 q^{55} +1.85150 q^{57} -2.64929 q^{59} -11.5426 q^{61} -21.7266 q^{65} -7.39763 q^{67} +9.22399 q^{69} +5.38073 q^{71} -5.36421 q^{73} -23.8011 q^{75} -4.37152 q^{79} -10.1010 q^{81} -16.0383 q^{83} -18.3786 q^{85} +18.3695 q^{87} +16.2368 q^{89} +4.85971 q^{93} +4.22552 q^{95} -12.8732 q^{97} -2.21096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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