LMFDB - Embedded newform 7232.2.a.bn.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7232,2,Mod(1,7232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7232 = 2^{6} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,-10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.7478107418$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 43 x^{16} + 760 x^{14} - 7095 x^{12} + 37240 x^{10} - 107142 x^{8} + 149152 x^{6} - 72200 x^{4} + \cdots - 800 $ x^18 - 43x^16 + 760x^14 - 7095x^12 + 37240x^10 - 107142x^8 + 149152x^6 - 72200x^4 + 13504x^2 - 800
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 3616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.40641$ of defining polynomial
Character	$\chi$	$=$	7232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.40641 q^{3} -4.21710 q^{5} +3.22264 q^{7} +2.79083 q^{9} -4.08286 q^{11} +1.84698 q^{13} -10.1481 q^{15} +5.67522 q^{17} +4.09027 q^{19} +7.75501 q^{21} -6.11509 q^{23} +12.7839 q^{25} -0.503343 q^{27} -3.67363 q^{29} +8.84028 q^{31} -9.82506 q^{33} -13.5902 q^{35} -11.9721 q^{37} +4.44461 q^{39} +4.78001 q^{41} +4.73093 q^{43} -11.7692 q^{45} -6.11041 q^{47} +3.38540 q^{49} +13.6569 q^{51} +0.898949 q^{53} +17.2178 q^{55} +9.84288 q^{57} -1.68397 q^{59} +14.3393 q^{61} +8.99385 q^{63} -7.78891 q^{65} +11.9139 q^{67} -14.7154 q^{69} +8.45105 q^{71} -6.29847 q^{73} +30.7634 q^{75} -13.1576 q^{77} -4.73671 q^{79} -9.58375 q^{81} +2.28784 q^{83} -23.9329 q^{85} -8.84028 q^{87} -4.43267 q^{89} +5.95216 q^{91} +21.2734 q^{93} -17.2491 q^{95} +14.7164 q^{97} -11.3946 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) $ 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

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
$3$	2.40641	1.38934	0.694672	−	0.719326i	$-0.255549\pi$
$3$	2.40641	1.38934	0.694672	+	0.719326i	$0.255549\pi$
$4$	0	0
$5$	−4.21710	−1.88594	−0.942972	−	0.332873i	$-0.891982\pi$
$5$	−4.21710	−1.88594	−0.942972	+	0.332873i	$0.891982\pi$
$6$	0	0
$7$	3.22264	1.21804	0.609021	−	0.793154i	$-0.291562\pi$
$7$	3.22264	1.21804	0.609021	+	0.793154i	$0.291562\pi$
$8$	0	0
$9$	2.79083	0.930278
$10$	0	0
$11$	−4.08286	−1.23103	−0.615514	−	0.788126i	$-0.711052\pi$
$11$	−4.08286	−1.23103	−0.615514	+	0.788126i	$0.711052\pi$
$12$	0	0
$13$	1.84698	0.512261	0.256130	−	0.966642i	$-0.417552\pi$
$13$	1.84698	0.512261	0.256130	+	0.966642i	$0.417552\pi$
$14$	0	0
$15$	−10.1481	−2.62022
$16$	0	0
$17$	5.67522	1.37644	0.688221	−	0.725501i	$-0.258392\pi$
$17$	5.67522	1.37644	0.688221	+	0.725501i	$0.258392\pi$
$18$	0	0
$19$	4.09027	0.938372	0.469186	−	0.883099i	$-0.344547\pi$
$19$	4.09027	0.938372	0.469186	+	0.883099i	$0.344547\pi$
$20$	0	0
$21$	7.75501	1.69228
$22$	0	0
$23$	−6.11509	−1.27508	−0.637542	−	0.770416i	$-0.720049\pi$
$23$	−6.11509	−1.27508	−0.637542	+	0.770416i	$0.720049\pi$
$24$	0	0
$25$	12.7839	2.55678
$26$	0	0
$27$	−0.503343	−0.0968684
$28$	0	0
$29$	−3.67363	−0.682176	−0.341088	−	0.940031i	$-0.610795\pi$
$29$	−3.67363	−0.682176	−0.341088	+	0.940031i	$0.610795\pi$
$30$	0	0
$31$	8.84028	1.58776	0.793881	−	0.608073i	$-0.208057\pi$
$31$	8.84028	1.58776	0.793881	+	0.608073i	$0.208057\pi$
$32$	0	0
$33$	−9.82506	−1.71032
$34$	0	0
$35$	−13.5902	−2.29716
$36$	0	0
$37$	−11.9721	−1.96820	−0.984101	−	0.177611i	$-0.943163\pi$
$37$	−11.9721	−1.96820	−0.984101	+	0.177611i	$0.943163\pi$
$38$	0	0
$39$	4.44461	0.711707
$40$	0	0
$41$	4.78001	0.746513	0.373256	−	0.927728i	$-0.378241\pi$
$41$	4.78001	0.746513	0.373256	+	0.927728i	$0.378241\pi$
$42$	0	0
$43$	4.73093	0.721460	0.360730	−	0.932670i	$-0.382528\pi$
$43$	4.73093	0.721460	0.360730	+	0.932670i	$0.382528\pi$
$44$	0	0
$45$	−11.7692	−1.75445
$46$	0	0
$47$	−6.11041	−0.891295	−0.445647	−	0.895209i	$-0.647026\pi$
$47$	−6.11041	−0.891295	−0.445647	+	0.895209i	$0.647026\pi$
$48$	0	0
$49$	3.38540	0.483628
$50$	0	0
$51$	13.6569	1.91235
$52$	0	0
$53$	0.898949	0.123480	0.0617401	−	0.998092i	$-0.480335\pi$
$53$	0.898949	0.123480	0.0617401	+	0.998092i	$0.480335\pi$
$54$	0	0
$55$	17.2178	2.32165
$56$	0	0
$57$	9.84288	1.30372
$58$	0	0
$59$	−1.68397	−0.219234	−0.109617	−	0.993974i	$-0.534962\pi$
$59$	−1.68397	−0.219234	−0.109617	+	0.993974i	$0.534962\pi$
$60$	0	0
$61$	14.3393	1.83595	0.917977	−	0.396633i	$-0.129821\pi$
$61$	14.3393	1.83595	0.917977	+	0.396633i	$0.129821\pi$
$62$	0	0
$63$	8.99385	1.13312
$64$	0	0
$65$	−7.78891	−0.966095
$66$	0	0
$67$	11.9139	1.45552	0.727758	−	0.685834i	$-0.240562\pi$
$67$	11.9139	1.45552	0.727758	+	0.685834i	$0.240562\pi$
$68$	0	0
$69$	−14.7154	−1.77153
$70$	0	0
$71$	8.45105	1.00296	0.501478	−	0.865171i	$-0.332790\pi$
$71$	8.45105	1.00296	0.501478	+	0.865171i	$0.332790\pi$
$72$	0	0
$73$	−6.29847	−0.737180	−0.368590	−	0.929592i	$-0.620159\pi$
$73$	−6.29847	−0.737180	−0.368590	+	0.929592i	$0.620159\pi$
$74$	0	0
$75$	30.7634	3.55225
$76$	0	0
$77$	−13.1576	−1.49945
$78$	0	0
$79$	−4.73671	−0.532922	−0.266461	−	0.963846i	$-0.585854\pi$
$79$	−4.73671	−0.532922	−0.266461	+	0.963846i	$0.585854\pi$
$80$	0	0
$81$	−9.58375	−1.06486
$82$	0	0
$83$	2.28784	0.251123	0.125561	−	0.992086i	$-0.459927\pi$
$83$	2.28784	0.251123	0.125561	+	0.992086i	$0.459927\pi$
$84$	0	0
$85$	−23.9329	−2.59589
$86$	0	0
$87$	−8.84028	−0.947778
$88$	0	0
$89$	−4.43267	−0.469862	−0.234931	−	0.972012i	$-0.575486\pi$
$89$	−4.43267	−0.469862	−0.234931	+	0.972012i	$0.575486\pi$
$90$	0	0
$91$	5.95216	0.623956
$92$	0	0
$93$	21.2734	2.20595
$94$	0	0
$95$	−17.2491	−1.76972
$96$	0	0
$97$	14.7164	1.49423	0.747114	−	0.664695i	$-0.231439\pi$
$97$	14.7164	1.49423	0.747114	+	0.664695i	$0.231439\pi$
$98$	0	0
$99$	−11.3946	−1.14520
$100$	0	0
$101$	8.16162	0.812112	0.406056	−	0.913848i	$-0.366904\pi$
$101$	8.16162	0.812112	0.406056	+	0.913848i	$0.366904\pi$
$102$	0	0
$103$	0.579459	0.0570958	0.0285479	−	0.999592i	$-0.490912\pi$
$103$	0.579459	0.0570958	0.0285479	+	0.999592i	$0.490912\pi$
$104$	0	0
$105$	−32.7036	−3.19155
$106$	0	0
$107$	17.6236	1.70373	0.851867	−	0.523758i	$-0.175470\pi$
$107$	17.6236	1.70373	0.851867	+	0.523758i	$0.175470\pi$
$108$	0	0
$109$	−8.83689	−0.846420	−0.423210	−	0.906032i	$-0.639097\pi$
$109$	−8.83689	−0.846420	−0.423210	+	0.906032i	$0.639097\pi$
$110$	0	0
$111$	−28.8098	−2.73451
$112$	0	0
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	0	0
$115$	25.7879	2.40474
$116$	0	0
$117$	5.15462	0.476545
$118$	0	0
$119$	18.2892	1.67657
$120$	0	0
$121$	5.66975	0.515432
$122$	0	0
$123$	11.5027	1.03716
$124$	0	0
$125$	−32.8255	−2.93600
$126$	0	0
$127$	8.06072	0.715274	0.357637	−	0.933861i	$-0.383583\pi$
$127$	8.06072	0.715274	0.357637	+	0.933861i	$0.383583\pi$
$128$	0	0
$129$	11.3846	1.00236
$130$	0	0
$131$	−5.46852	−0.477787	−0.238894	−	0.971046i	$-0.576785\pi$
$131$	−5.46852	−0.477787	−0.238894	+	0.971046i	$0.576785\pi$
$132$	0	0
$133$	13.1815	1.14298
$134$	0	0
$135$	2.12265	0.182688
$136$	0	0
$137$	−6.18480	−0.528403	−0.264202	−	0.964467i	$-0.585108\pi$
$137$	−6.18480	−0.528403	−0.264202	+	0.964467i	$0.585108\pi$
$138$	0	0
$139$	−6.96213	−0.590520	−0.295260	−	0.955417i	$-0.595406\pi$
$139$	−6.96213	−0.590520	−0.295260	+	0.955417i	$0.595406\pi$
$140$	0	0
$141$	−14.7042	−1.23832
$142$	0	0
$143$	−7.54098	−0.630608
$144$	0	0
$145$	15.4921	1.28655
$146$	0	0
$147$	8.14667	0.671926
$148$	0	0
$149$	19.7956	1.62172	0.810858	−	0.585243i	$-0.199001\pi$
$149$	19.7956	1.62172	0.810858	+	0.585243i	$0.199001\pi$
$150$	0	0
$151$	18.5020	1.50567	0.752837	−	0.658207i	$-0.228685\pi$
$151$	18.5020	1.50567	0.752837	+	0.658207i	$0.228685\pi$
$152$	0	0
$153$	15.8386	1.28047
$154$	0	0
$155$	−37.2803	−2.99443
$156$	0	0
$157$	−2.38935	−0.190691	−0.0953454	−	0.995444i	$-0.530396\pi$
$157$	−2.38935	−0.190691	−0.0953454	+	0.995444i	$0.530396\pi$
$158$	0	0
$159$	2.16325	0.171556
$160$	0	0
$161$	−19.7067	−1.55311
$162$	0	0
$163$	2.03808	0.159634	0.0798172	−	0.996810i	$-0.474566\pi$
$163$	2.03808	0.159634	0.0798172	+	0.996810i	$0.474566\pi$
$164$	0	0
$165$	41.4332	3.22557
$166$	0	0
$167$	9.43837	0.730363	0.365182	−	0.930936i	$-0.381007\pi$
$167$	9.43837	0.730363	0.365182	+	0.930936i	$0.381007\pi$
$168$	0	0
$169$	−9.58865	−0.737589
$170$	0	0
$171$	11.4153	0.872946
$172$	0	0
$173$	13.8683	1.05439	0.527195	−	0.849744i	$-0.323244\pi$
$173$	13.8683	1.05439	0.527195	+	0.849744i	$0.323244\pi$
$174$	0	0
$175$	41.1979	3.11427
$176$	0	0
$177$	−4.05233	−0.304592
$178$	0	0
$179$	5.67339	0.424049	0.212025	−	0.977264i	$-0.431994\pi$
$179$	5.67339	0.424049	0.212025	+	0.977264i	$0.431994\pi$
$180$	0	0
$181$	16.1549	1.20078	0.600391	−	0.799706i	$-0.295012\pi$
$181$	16.1549	1.20078	0.600391	+	0.799706i	$0.295012\pi$
$182$	0	0
$183$	34.5062	2.55077
$184$	0	0
$185$	50.4875	3.71192
$186$	0	0
$187$	−23.1711	−1.69444
$188$	0	0
$189$	−1.62209	−0.117990
$190$	0	0
$191$	1.19109	0.0861841	0.0430920	−	0.999071i	$-0.486279\pi$
$191$	1.19109	0.0861841	0.0430920	+	0.999071i	$0.486279\pi$
$192$	0	0
$193$	19.4781	1.40206	0.701032	−	0.713130i	$-0.252723\pi$
$193$	19.4781	1.40206	0.701032	+	0.713130i	$0.252723\pi$
$194$	0	0
$195$	−18.7433	−1.34224
$196$	0	0
$197$	22.9253	1.63336	0.816679	−	0.577092i	$-0.195813\pi$
$197$	22.9253	1.63336	0.816679	+	0.577092i	$0.195813\pi$
$198$	0	0
$199$	17.4729	1.23862	0.619311	−	0.785146i	$-0.287412\pi$
$199$	17.4729	1.23862	0.619311	+	0.785146i	$0.287412\pi$
$200$	0	0
$201$	28.6698	2.02221
$202$	0	0
$203$	−11.8388	−0.830920
$204$	0	0
$205$	−20.1578	−1.40788
$206$	0	0
$207$	−17.0662	−1.18618
$208$	0	0
$209$	−16.7000	−1.15516
$210$	0	0
$211$	−3.24015	−0.223061	−0.111530	−	0.993761i	$-0.535575\pi$
$211$	−3.24015	−0.223061	−0.111530	+	0.993761i	$0.535575\pi$
$212$	0	0
$213$	20.3367	1.39345
$214$	0	0
$215$	−19.9508	−1.36063
$216$	0	0
$217$	28.4890	1.93396
$218$	0	0
$219$	−15.1567	−1.02420
$220$	0	0
$221$	10.4820	0.705098
$222$	0	0
$223$	−27.3711	−1.83290	−0.916452	−	0.400145i	$-0.868960\pi$
$223$	−27.3711	−1.83290	−0.916452	+	0.400145i	$0.868960\pi$
$224$	0	0
$225$	35.6778	2.37852
$226$	0	0
$227$	−14.4995	−0.962364	−0.481182	−	0.876621i	$-0.659792\pi$
$227$	−14.4995	−0.962364	−0.481182	+	0.876621i	$0.659792\pi$
$228$	0	0
$229$	−9.51830	−0.628987	−0.314494	−	0.949260i	$-0.601835\pi$
$229$	−9.51830	−0.628987	−0.314494	+	0.949260i	$0.601835\pi$
$230$	0	0
$231$	−31.6626	−2.08325
$232$	0	0
$233$	12.8068	0.839004	0.419502	−	0.907755i	$-0.362205\pi$
$233$	12.8068	0.839004	0.419502	+	0.907755i	$0.362205\pi$
$234$	0	0
$235$	25.7682	1.68093
$236$	0	0
$237$	−11.3985	−0.740412
$238$	0	0
$239$	13.2257	0.855497	0.427748	−	0.903898i	$-0.359307\pi$
$239$	13.2257	0.855497	0.427748	+	0.903898i	$0.359307\pi$
$240$	0	0
$241$	−6.22900	−0.401245	−0.200623	−	0.979669i	$-0.564297\pi$
$241$	−6.22900	−0.401245	−0.200623	+	0.979669i	$0.564297\pi$
$242$	0	0
$243$	−21.5525	−1.38259
$244$	0	0
$245$	−14.2766	−0.912096
$246$	0	0
$247$	7.55466	0.480691
$248$	0	0
$249$	5.50549	0.348896
$250$	0	0
$251$	15.0165	0.947833	0.473916	−	0.880570i	$-0.342840\pi$
$251$	15.0165	0.947833	0.473916	+	0.880570i	$0.342840\pi$
$252$	0	0
$253$	24.9671	1.56967
$254$	0	0
$255$	−57.5926	−3.60659
$256$	0	0
$257$	−18.2547	−1.13869	−0.569347	−	0.822097i	$-0.692804\pi$
$257$	−18.2547	−1.13869	−0.569347	+	0.822097i	$0.692804\pi$
$258$	0	0
$259$	−38.5818	−2.39735
$260$	0	0
$261$	−10.2525	−0.634613
$262$	0	0
$263$	29.1786	1.79923	0.899614	−	0.436686i	$-0.143848\pi$
$263$	29.1786	1.79923	0.899614	+	0.436686i	$0.143848\pi$
$264$	0	0
$265$	−3.79096	−0.232877
$266$	0	0
$267$	−10.6668	−0.652801
$268$	0	0
$269$	−9.20086	−0.560986	−0.280493	−	0.959856i	$-0.590498\pi$
$269$	−9.20086	−0.560986	−0.280493	+	0.959856i	$0.590498\pi$
$270$	0	0
$271$	−18.5052	−1.12411	−0.562055	−	0.827100i	$-0.689989\pi$
$271$	−18.5052	−1.12411	−0.562055	+	0.827100i	$0.689989\pi$
$272$	0	0
$273$	14.3234	0.866889
$274$	0	0
$275$	−52.1949	−3.14747
$276$	0	0
$277$	−2.14328	−0.128777	−0.0643885	−	0.997925i	$-0.520510\pi$
$277$	−2.14328	−0.128777	−0.0643885	+	0.997925i	$0.520510\pi$
$278$	0	0
$279$	24.6718	1.47706
$280$	0	0
$281$	−28.2585	−1.68576	−0.842882	−	0.538099i	$-0.819143\pi$
$281$	−28.2585	−1.68576	−0.842882	+	0.538099i	$0.819143\pi$
$282$	0	0
$283$	1.06295	0.0631860	0.0315930	−	0.999501i	$-0.489942\pi$
$283$	1.06295	0.0631860	0.0315930	+	0.999501i	$0.489942\pi$
$284$	0	0
$285$	−41.5084	−2.45874
$286$	0	0
$287$	15.4043	0.909285
$288$	0	0
$289$	15.2081	0.894594
$290$	0	0
$291$	35.4139	2.07600
$292$	0	0
$293$	−2.82846	−0.165241	−0.0826203	−	0.996581i	$-0.526329\pi$
$293$	−2.82846	−0.165241	−0.0826203	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	7.10147	0.413463
$296$	0	0
$297$	2.05508	0.119248
$298$	0	0
$299$	−11.2945	−0.653176
$300$	0	0
$301$	15.2461	0.878770
$302$	0	0
$303$	19.6402	1.12830
$304$	0	0
$305$	−60.4701	−3.46251
$306$	0	0
$307$	12.3863	0.706922	0.353461	−	0.935449i	$-0.385005\pi$
$307$	12.3863	0.706922	0.353461	+	0.935449i	$0.385005\pi$
$308$	0	0
$309$	1.39442	0.0793257
$310$	0	0
$311$	34.2618	1.94281	0.971403	−	0.237436i	$-0.0763071\pi$
$311$	34.2618	1.94281	0.971403	+	0.237436i	$0.0763071\pi$
$312$	0	0
$313$	12.0262	0.679759	0.339880	−	0.940469i	$-0.389614\pi$
$313$	12.0262	0.679759	0.339880	+	0.940469i	$0.389614\pi$
$314$	0	0
$315$	−37.9279	−2.13700
$316$	0	0
$317$	15.6528	0.879148	0.439574	−	0.898206i	$-0.355129\pi$
$317$	15.6528	0.879148	0.439574	+	0.898206i	$0.355129\pi$
$318$	0	0
$319$	14.9989	0.839779
$320$	0	0
$321$	42.4096	2.36707
$322$	0	0
$323$	23.2132	1.29161
$324$	0	0
$325$	23.6117	1.30974
$326$	0	0
$327$	−21.2652	−1.17597
$328$	0	0
$329$	−19.6916	−1.08564
$330$	0	0
$331$	−10.2599	−0.563937	−0.281968	−	0.959424i	$-0.590987\pi$
$331$	−10.2599	−0.563937	−0.281968	+	0.959424i	$0.590987\pi$
$332$	0	0
$333$	−33.4121	−1.83097
$334$	0	0
$335$	−50.2422	−2.74502
$336$	0	0
$337$	−2.05514	−0.111951	−0.0559753	−	0.998432i	$-0.517827\pi$
$337$	−2.05514	−0.111951	−0.0559753	+	0.998432i	$0.517827\pi$
$338$	0	0
$339$	−2.40641	−0.130699
$340$	0	0
$341$	−36.0936	−1.95458
$342$	0	0
$343$	−11.6486	−0.628963
$344$	0	0
$345$	62.0564	3.34101
$346$	0	0
$347$	19.6174	1.05312	0.526559	−	0.850138i	$-0.323482\pi$
$347$	19.6174	1.05312	0.526559	+	0.850138i	$0.323482\pi$
$348$	0	0
$349$	−3.99826	−0.214022	−0.107011	−	0.994258i	$-0.534128\pi$
$349$	−3.99826	−0.214022	−0.107011	+	0.994258i	$0.534128\pi$
$350$	0	0
$351$	−0.929666	−0.0496219
$352$	0	0
$353$	24.4500	1.30134	0.650671	−	0.759360i	$-0.274488\pi$
$353$	24.4500	1.30134	0.650671	+	0.759360i	$0.274488\pi$
$354$	0	0
$355$	−35.6389	−1.89152
$356$	0	0
$357$	44.0114	2.32933
$358$	0	0
$359$	−11.4392	−0.603737	−0.301869	−	0.953350i	$-0.597610\pi$
$359$	−11.4392	−0.603737	−0.301869	+	0.953350i	$0.597610\pi$
$360$	0	0
$361$	−2.26971	−0.119458
$362$	0	0
$363$	13.6438	0.716113
$364$	0	0
$365$	26.5613	1.39028
$366$	0	0
$367$	−4.06455	−0.212168	−0.106084	−	0.994357i	$-0.533831\pi$
$367$	−4.06455	−0.212168	−0.106084	+	0.994357i	$0.533831\pi$
$368$	0	0
$369$	13.3402	0.694464
$370$	0	0
$371$	2.89699	0.150404
$372$	0	0
$373$	30.0409	1.55546	0.777728	−	0.628601i	$-0.216372\pi$
$373$	30.0409	1.55546	0.777728	+	0.628601i	$0.216372\pi$
$374$	0	0
$375$	−78.9918	−4.07912
$376$	0	0
$377$	−6.78514	−0.349452
$378$	0	0
$379$	−31.5651	−1.62139	−0.810695	−	0.585469i	$-0.800910\pi$
$379$	−31.5651	−1.62139	−0.810695	+	0.585469i	$0.800910\pi$
$380$	0	0
$381$	19.3974	0.993761
$382$	0	0
$383$	9.16822	0.468474	0.234237	−	0.972180i	$-0.424741\pi$
$383$	9.16822	0.468474	0.234237	+	0.972180i	$0.424741\pi$
$384$	0	0
$385$	55.4868	2.82787
$386$	0	0
$387$	13.2032	0.671158
$388$	0	0
$389$	−2.52717	−0.128133	−0.0640664	−	0.997946i	$-0.520407\pi$
$389$	−2.52717	−0.128133	−0.0640664	+	0.997946i	$0.520407\pi$
$390$	0	0
$391$	−34.7045	−1.75508
$392$	0	0
$393$	−13.1595	−0.663811
$394$	0	0
$395$	19.9752	1.00506
$396$	0	0
$397$	−11.4012	−0.572212	−0.286106	−	0.958198i	$-0.592361\pi$
$397$	−11.4012	−0.572212	−0.286106	+	0.958198i	$0.592361\pi$
$398$	0	0
$399$	31.7200	1.58799
$400$	0	0
$401$	37.2948	1.86241	0.931207	−	0.364491i	$-0.118757\pi$
$401$	37.2948	1.86241	0.931207	+	0.364491i	$0.118757\pi$
$402$	0	0
$403$	16.3279	0.813348
$404$	0	0
$405$	40.4156	2.00827
$406$	0	0
$407$	48.8804	2.42291
$408$	0	0
$409$	1.16461	0.0575863	0.0287932	−	0.999585i	$-0.490834\pi$
$409$	1.16461	0.0575863	0.0287932	+	0.999585i	$0.490834\pi$
$410$	0	0
$411$	−14.8832	−0.734134
$412$	0	0
$413$	−5.42683	−0.267037
$414$	0	0
$415$	−9.64804	−0.473603
$416$	0	0
$417$	−16.7538	−0.820435
$418$	0	0
$419$	−0.570557	−0.0278735	−0.0139368	−	0.999903i	$-0.504436\pi$
$419$	−0.570557	−0.0278735	−0.0139368	+	0.999903i	$0.504436\pi$
$420$	0	0
$421$	−38.6626	−1.88430	−0.942151	−	0.335189i	$-0.891200\pi$
$421$	−38.6626	−1.88430	−0.942151	+	0.335189i	$0.891200\pi$
$422$	0	0
$423$	−17.0531	−0.829152
$424$	0	0
$425$	72.5515	3.51926
$426$	0	0
$427$	46.2103	2.23627
$428$	0	0
$429$	−18.1467	−0.876132
$430$	0	0
$431$	9.66471	0.465532	0.232766	−	0.972533i	$-0.425222\pi$
$431$	9.66471	0.465532	0.232766	+	0.972533i	$0.425222\pi$
$432$	0	0
$433$	−0.362198	−0.0174061	−0.00870307	−	0.999962i	$-0.502770\pi$
$433$	−0.362198	−0.0174061	−0.00870307	+	0.999962i	$0.502770\pi$
$434$	0	0
$435$	37.2803	1.78746
$436$	0	0
$437$	−25.0123	−1.19650
$438$	0	0
$439$	26.7847	1.27837	0.639183	−	0.769055i	$-0.279273\pi$
$439$	26.7847	1.27837	0.639183	+	0.769055i	$0.279273\pi$
$440$	0	0
$441$	9.44808	0.449909
$442$	0	0
$443$	9.61014	0.456591	0.228296	−	0.973592i	$-0.426685\pi$
$443$	9.61014	0.456591	0.228296	+	0.973592i	$0.426685\pi$
$444$	0	0
$445$	18.6930	0.886134
$446$	0	0
$447$	47.6364	2.25312
$448$	0	0
$449$	6.77080	0.319534	0.159767	−	0.987155i	$-0.448926\pi$
$449$	6.77080	0.319534	0.159767	+	0.987155i	$0.448926\pi$
$450$	0	0
$451$	−19.5161	−0.918979
$452$	0	0
$453$	44.5236	2.09190
$454$	0	0
$455$	−25.1008	−1.17675
$456$	0	0
$457$	7.34575	0.343620	0.171810	−	0.985130i	$-0.445039\pi$
$457$	7.34575	0.343620	0.171810	+	0.985130i	$0.445039\pi$
$458$	0	0
$459$	−2.85658	−0.133334
$460$	0	0
$461$	25.6056	1.19257	0.596287	−	0.802772i	$-0.296642\pi$
$461$	25.6056	1.19257	0.596287	+	0.802772i	$0.296642\pi$
$462$	0	0
$463$	−1.82649	−0.0848840	−0.0424420	−	0.999099i	$-0.513514\pi$
$463$	−1.82649	−0.0848840	−0.0424420	+	0.999099i	$0.513514\pi$
$464$	0	0
$465$	−89.7120	−4.16029
$466$	0	0
$467$	−12.1542	−0.562430	−0.281215	−	0.959645i	$-0.590737\pi$
$467$	−12.1542	−0.562430	−0.281215	+	0.959645i	$0.590737\pi$
$468$	0	0
$469$	38.3943	1.77288
$470$	0	0
$471$	−5.74976	−0.264935
$472$	0	0
$473$	−19.3157	−0.888139
$474$	0	0
$475$	52.2896	2.39921
$476$	0	0
$477$	2.50882	0.114871
$478$	0	0
$479$	−24.8130	−1.13373	−0.566867	−	0.823809i	$-0.691845\pi$
$479$	−24.8130	−1.13373	−0.566867	+	0.823809i	$0.691845\pi$
$480$	0	0
$481$	−22.1123	−1.00823
$482$	0	0
$483$	−47.4225	−2.15780
$484$	0	0
$485$	−62.0607	−2.81803
$486$	0	0
$487$	−11.6570	−0.528228	−0.264114	−	0.964491i	$-0.585080\pi$
$487$	−11.6570	−0.528228	−0.264114	+	0.964491i	$0.585080\pi$
$488$	0	0
$489$	4.90445	0.221787
$490$	0	0
$491$	−17.8948	−0.807581	−0.403790	−	0.914852i	$-0.632307\pi$
$491$	−17.8948	−0.807581	−0.403790	+	0.914852i	$0.632307\pi$
$492$	0	0
$493$	−20.8487	−0.938977
$494$	0	0
$495$	48.0521	2.15978
$496$	0	0
$497$	27.2347	1.22164
$498$	0	0
$499$	24.9134	1.11528	0.557638	−	0.830084i	$-0.311708\pi$
$499$	24.9134	1.11528	0.557638	+	0.830084i	$0.311708\pi$
$500$	0	0
$501$	22.7126	1.01473
$502$	0	0
$503$	−9.24362	−0.412153	−0.206076	−	0.978536i	$-0.566070\pi$
$503$	−9.24362	−0.412153	−0.206076	+	0.978536i	$0.566070\pi$
$504$	0	0
$505$	−34.4183	−1.53160
$506$	0	0
$507$	−23.0743	−1.02476
$508$	0	0
$509$	−12.7041	−0.563100	−0.281550	−	0.959546i	$-0.590849\pi$
$509$	−12.7041	−0.563100	−0.281550	+	0.959546i	$0.590849\pi$
$510$	0	0
$511$	−20.2977	−0.897917
$512$	0	0
$513$	−2.05881	−0.0908985
$514$	0	0
$515$	−2.44364	−0.107679
$516$	0	0
$517$	24.9480	1.09721
$518$	0	0
$519$	33.3730	1.46491
$520$	0	0
$521$	6.38394	0.279685	0.139843	−	0.990174i	$-0.455340\pi$
$521$	6.38394	0.279685	0.139843	+	0.990174i	$0.455340\pi$
$522$	0	0
$523$	−29.8510	−1.30529	−0.652647	−	0.757662i	$-0.726342\pi$
$523$	−29.8510	−1.30529	−0.652647	+	0.757662i	$0.726342\pi$
$524$	0	0
$525$	99.1393	4.32679
$526$	0	0
$527$	50.1705	2.18546
$528$	0	0
$529$	14.3943	0.625839
$530$	0	0
$531$	−4.69968	−0.203949
$532$	0	0
$533$	8.82861	0.382409
$534$	0	0
$535$	−74.3203	−3.21315
$536$	0	0
$537$	13.6525	0.589150
$538$	0	0
$539$	−13.8221	−0.595361
$540$	0	0
$541$	−19.0989	−0.821125	−0.410563	−	0.911832i	$-0.634668\pi$
$541$	−19.0989	−0.821125	−0.410563	+	0.911832i	$0.634668\pi$
$542$	0	0
$543$	38.8753	1.66830
$544$	0	0
$545$	37.2660	1.59630
$546$	0	0
$547$	−24.8167	−1.06109	−0.530543	−	0.847658i	$-0.678012\pi$
$547$	−24.8167	−1.06109	−0.530543	+	0.847658i	$0.678012\pi$
$548$	0	0
$549$	40.0185	1.70795
$550$	0	0
$551$	−15.0261	−0.640135
$552$	0	0
$553$	−15.2647	−0.649122
$554$	0	0
$555$	121.494	5.15713
$556$	0	0
$557$	−42.1720	−1.78688	−0.893442	−	0.449179i	$-0.851717\pi$
$557$	−42.1720	−1.78688	−0.893442	+	0.449179i	$0.851717\pi$
$558$	0	0
$559$	8.73795	0.369576
$560$	0	0
$561$	−55.7594	−2.35416
$562$	0	0
$563$	21.7866	0.918197	0.459098	−	0.888385i	$-0.348173\pi$
$563$	21.7866	0.918197	0.459098	+	0.888385i	$0.348173\pi$
$564$	0	0
$565$	4.21710	0.177415
$566$	0	0
$567$	−30.8850	−1.29705
$568$	0	0
$569$	−14.8381	−0.622047	−0.311024	−	0.950402i	$-0.600672\pi$
$569$	−14.8381	−0.622047	−0.311024	+	0.950402i	$0.600672\pi$
$570$	0	0
$571$	−3.32147	−0.138999	−0.0694996	−	0.997582i	$-0.522140\pi$
$571$	−3.32147	−0.138999	−0.0694996	+	0.997582i	$0.522140\pi$
$572$	0	0
$573$	2.86625	0.119739
$574$	0	0
$575$	−78.1747	−3.26011
$576$	0	0
$577$	−20.9502	−0.872168	−0.436084	−	0.899906i	$-0.643635\pi$
$577$	−20.9502	−0.872168	−0.436084	+	0.899906i	$0.643635\pi$
$578$	0	0
$579$	46.8724	1.94795
$580$	0	0
$581$	7.37288	0.305878
$582$	0	0
$583$	−3.67029	−0.152008
$584$	0	0
$585$	−21.7375	−0.898737
$586$	0	0
$587$	−40.5165	−1.67229	−0.836147	−	0.548506i	$-0.815197\pi$
$587$	−40.5165	−1.67229	−0.836147	+	0.548506i	$0.815197\pi$
$588$	0	0
$589$	36.1591	1.48991
$590$	0	0
$591$	55.1677	2.26930
$592$	0	0
$593$	−12.4878	−0.512814	−0.256407	−	0.966569i	$-0.582539\pi$
$593$	−12.4878	−0.512814	−0.256407	+	0.966569i	$0.582539\pi$
$594$	0	0
$595$	−77.1272	−3.16191
$596$	0	0
$597$	42.0471	1.72087
$598$	0	0
$599$	3.52440	0.144003	0.0720016	−	0.997405i	$-0.477061\pi$
$599$	3.52440	0.144003	0.0720016	+	0.997405i	$0.477061\pi$
$600$	0	0
$601$	−9.14227	−0.372921	−0.186461	−	0.982462i	$-0.559702\pi$
$601$	−9.14227	−0.372921	−0.186461	+	0.982462i	$0.559702\pi$
$602$	0	0
$603$	33.2498	1.35403
$604$	0	0
$605$	−23.9099	−0.972076
$606$	0	0
$607$	−14.6463	−0.594473	−0.297237	−	0.954804i	$-0.596065\pi$
$607$	−14.6463	−0.594473	−0.297237	+	0.954804i	$0.596065\pi$
$608$	0	0
$609$	−28.4890	−1.15443
$610$	0	0
$611$	−11.2858	−0.456576
$612$	0	0
$613$	5.53656	0.223619	0.111810	−	0.993730i	$-0.464335\pi$
$613$	5.53656	0.223619	0.111810	+	0.993730i	$0.464335\pi$
$614$	0	0
$615$	−48.5080	−1.95603
$616$	0	0
$617$	−2.78797	−0.112239	−0.0561197	−	0.998424i	$-0.517873\pi$
$617$	−2.78797	−0.112239	−0.0561197	+	0.998424i	$0.517873\pi$
$618$	0	0
$619$	27.9864	1.12487	0.562434	−	0.826842i	$-0.309865\pi$
$619$	27.9864	1.12487	0.562434	+	0.826842i	$0.309865\pi$
$620$	0	0
$621$	3.07799	0.123515
$622$	0	0
$623$	−14.2849	−0.572312
$624$	0	0
$625$	74.5088	2.98035
$626$	0	0
$627$	−40.1871	−1.60492
$628$	0	0
$629$	−67.9443	−2.70912
$630$	0	0
$631$	17.6827	0.703937	0.351968	−	0.936012i	$-0.385512\pi$
$631$	17.6827	0.703937	0.351968	+	0.936012i	$0.385512\pi$
$632$	0	0
$633$	−7.79714	−0.309908
$634$	0	0
$635$	−33.9929	−1.34897
$636$	0	0
$637$	6.25277	0.247744
$638$	0	0
$639$	23.5855	0.933027
$640$	0	0
$641$	2.69744	0.106542	0.0532711	−	0.998580i	$-0.483035\pi$
$641$	2.69744	0.106542	0.0532711	+	0.998580i	$0.483035\pi$
$642$	0	0
$643$	14.9538	0.589721	0.294860	−	0.955540i	$-0.404727\pi$
$643$	14.9538	0.589721	0.294860	+	0.955540i	$0.404727\pi$
$644$	0	0
$645$	−48.0099	−1.89039
$646$	0	0
$647$	−23.1195	−0.908921	−0.454461	−	0.890767i	$-0.650168\pi$
$647$	−23.1195	−0.908921	−0.454461	+	0.890767i	$0.650168\pi$
$648$	0	0
$649$	6.87542	0.269884
$650$	0	0
$651$	68.5564	2.68694
$652$	0	0
$653$	16.4346	0.643136	0.321568	−	0.946886i	$-0.395790\pi$
$653$	16.4346	0.643136	0.321568	+	0.946886i	$0.395790\pi$
$654$	0	0
$655$	23.0613	0.901079
$656$	0	0
$657$	−17.5780	−0.685782
$658$	0	0
$659$	31.2449	1.21713	0.608565	−	0.793504i	$-0.291746\pi$
$659$	31.2449	1.21713	0.608565	+	0.793504i	$0.291746\pi$
$660$	0	0
$661$	−35.2963	−1.37287	−0.686434	−	0.727193i	$-0.740825\pi$
$661$	−35.2963	−1.37287	−0.686434	+	0.727193i	$0.740825\pi$
$662$	0	0
$663$	25.2241	0.979624
$664$	0	0
$665$	−55.5875	−2.15559
$666$	0	0
$667$	22.4646	0.869832
$668$	0	0
$669$	−65.8662	−2.54653
$670$	0	0
$671$	−58.5452	−2.26011
$672$	0	0
$673$	−44.7092	−1.72341	−0.861707	−	0.507407i	$-0.830604\pi$
$673$	−44.7092	−1.72341	−0.861707	+	0.507407i	$0.830604\pi$
$674$	0	0
$675$	−6.43469	−0.247671
$676$	0	0
$677$	−11.5552	−0.444101	−0.222051	−	0.975035i	$-0.571275\pi$
$677$	−11.5552	−0.444101	−0.222051	+	0.975035i	$0.571275\pi$
$678$	0	0
$679$	47.4258	1.82003
$680$	0	0
$681$	−34.8918	−1.33705
$682$	0	0
$683$	−25.8795	−0.990253	−0.495127	−	0.868821i	$-0.664878\pi$
$683$	−25.8795	−0.990253	−0.495127	+	0.868821i	$0.664878\pi$
$684$	0	0
$685$	26.0819	0.996539
$686$	0	0
$687$	−22.9050	−0.873880
$688$	0	0
$689$	1.66034	0.0632541
$690$	0	0
$691$	−10.9614	−0.416991	−0.208496	−	0.978023i	$-0.566857\pi$
$691$	−10.9614	−0.416991	−0.208496	+	0.978023i	$0.566857\pi$
$692$	0	0
$693$	−36.7206	−1.39490
$694$	0	0
$695$	29.3600	1.11369
$696$	0	0
$697$	27.1276	1.02753
$698$	0	0
$699$	30.8186	1.16566
$700$	0	0
$701$	−23.5186	−0.888285	−0.444142	−	0.895956i	$-0.646492\pi$
$701$	−23.5186	−0.888285	−0.444142	+	0.895956i	$0.646492\pi$
$702$	0	0
$703$	−48.9691	−1.84690
$704$	0	0
$705$	62.0090	2.33539
$706$	0	0
$707$	26.3020	0.989187
$708$	0	0
$709$	18.3779	0.690198	0.345099	−	0.938566i	$-0.387845\pi$
$709$	18.3779	0.690198	0.345099	+	0.938566i	$0.387845\pi$
$710$	0	0
$711$	−13.2194	−0.495765
$712$	0	0
$713$	−54.0591	−2.02453
$714$	0	0
$715$	31.8010	1.18929
$716$	0	0
$717$	31.8264	1.18858
$718$	0	0
$719$	0.0257786	0.000961380 0	0.000480690	−	1.00000i	$-0.499847\pi$
$719$	0.0257786	0.000961380 0	0.000480690		1.00000i	$0.499847\pi$
$720$	0	0
$721$	1.86739	0.0695451
$722$	0	0
$723$	−14.9896	−0.557468
$724$	0	0
$725$	−46.9634	−1.74418
$726$	0	0
$727$	−50.2243	−1.86272	−0.931359	−	0.364103i	$-0.881375\pi$
$727$	−50.2243	−1.86272	−0.931359	+	0.364103i	$0.881375\pi$
$728$	0	0
$729$	−23.1129	−0.856033
$730$	0	0
$731$	26.8491	0.993049
$732$	0	0
$733$	8.67622	0.320464	0.160232	−	0.987079i	$-0.448776\pi$
$733$	8.67622	0.320464	0.160232	+	0.987079i	$0.448776\pi$
$734$	0	0
$735$	−34.3553	−1.26722
$736$	0	0
$737$	−48.6429	−1.79178
$738$	0	0
$739$	44.7510	1.64619	0.823096	−	0.567903i	$-0.192245\pi$
$739$	44.7510	1.64619	0.823096	+	0.567903i	$0.192245\pi$
$740$	0	0
$741$	18.1796	0.667846
$742$	0	0
$743$	−28.0886	−1.03047	−0.515236	−	0.857048i	$-0.672296\pi$
$743$	−28.0886	−1.03047	−0.515236	+	0.857048i	$0.672296\pi$
$744$	0	0
$745$	−83.4799	−3.05847
$746$	0	0
$747$	6.38497	0.233614
$748$	0	0
$749$	56.7944	2.07522
$750$	0	0
$751$	−49.5578	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$751$	−49.5578	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$752$	0	0
$753$	36.1359	1.31687
$754$	0	0
$755$	−78.0249	−2.83962
$756$	0	0
$757$	23.2082	0.843515	0.421757	−	0.906709i	$-0.361413\pi$
$757$	23.2082	0.843515	0.421757	+	0.906709i	$0.361413\pi$
$758$	0	0
$759$	60.0811	2.18081
$760$	0	0
$761$	4.84815	0.175745	0.0878726	−	0.996132i	$-0.471993\pi$
$761$	4.84815	0.175745	0.0878726	+	0.996132i	$0.471993\pi$
$762$	0	0
$763$	−28.4781	−1.03098
$764$	0	0
$765$	−66.7929	−2.41490
$766$	0	0
$767$	−3.11027	−0.112305
$768$	0	0
$769$	−0.566722	−0.0204365	−0.0102183	−	0.999948i	$-0.503253\pi$
$769$	−0.566722	−0.0204365	−0.0102183	+	0.999948i	$0.503253\pi$
$770$	0	0
$771$	−43.9283	−1.58204
$772$	0	0
$773$	−6.14638	−0.221070	−0.110535	−	0.993872i	$-0.535256\pi$
$773$	−6.14638	−0.221070	−0.110535	+	0.993872i	$0.535256\pi$
$774$	0	0
$775$	113.013	4.05956
$776$	0	0
$777$	−92.8437	−3.33075
$778$	0	0
$779$	19.5515	0.700507
$780$	0	0
$781$	−34.5045	−1.23467
$782$	0	0
$783$	1.84910	0.0660813
$784$	0	0
$785$	10.0761	0.359632
$786$	0	0
$787$	−8.53758	−0.304332	−0.152166	−	0.988355i	$-0.548625\pi$
$787$	−8.53758	−0.304332	−0.152166	+	0.988355i	$0.548625\pi$
$788$	0	0
$789$	70.2158	2.49975
$790$	0	0
$791$	−3.22264	−0.114584
$792$	0	0
$793$	26.4844	0.940488
$794$	0	0
$795$	−9.12262	−0.323546
$796$	0	0
$797$	14.1869	0.502527	0.251263	−	0.967919i	$-0.419154\pi$
$797$	14.1869	0.502527	0.251263	+	0.967919i	$0.419154\pi$
$798$	0	0
$799$	−34.6779	−1.22682
$800$	0	0
$801$	−12.3708	−0.437102
$802$	0	0
$803$	25.7158	0.907490
$804$	0	0
$805$	83.1051	2.92907
$806$	0	0
$807$	−22.1411	−0.779403
$808$	0	0
$809$	−33.8689	−1.19077	−0.595384	−	0.803441i	$-0.703000\pi$
$809$	−33.8689	−1.19077	−0.595384	+	0.803441i	$0.703000\pi$
$810$	0	0
$811$	−31.7865	−1.11618	−0.558088	−	0.829782i	$-0.688465\pi$
$811$	−31.7865	−1.11618	−0.558088	+	0.829782i	$0.688465\pi$
$812$	0	0
$813$	−44.5311	−1.56178
$814$	0	0
$815$	−8.59476	−0.301061
$816$	0	0
$817$	19.3508	0.676998
$818$	0	0
$819$	16.6115	0.580452
$820$	0	0
$821$	40.1354	1.40073	0.700367	−	0.713783i	$-0.253020\pi$
$821$	40.1354	1.40073	0.700367	+	0.713783i	$0.253020\pi$
$822$	0	0
$823$	−52.6075	−1.83378	−0.916892	−	0.399136i	$-0.869310\pi$
$823$	−52.6075	−1.83378	−0.916892	+	0.399136i	$0.869310\pi$
$824$	0	0
$825$	−125.603	−4.37292
$826$	0	0
$827$	−0.342687	−0.0119164	−0.00595819	−	0.999982i	$-0.501897\pi$
$827$	−0.342687	−0.0119164	−0.00595819	+	0.999982i	$0.501897\pi$
$828$	0	0
$829$	46.6832	1.62137	0.810687	−	0.585480i	$-0.199094\pi$
$829$	46.6832	1.62137	0.810687	+	0.585480i	$0.199094\pi$
$830$	0	0
$831$	−5.15761	−0.178916
$832$	0	0
$833$	19.2129	0.665687
$834$	0	0
$835$	−39.8025	−1.37742
$836$	0	0
$837$	−4.44969	−0.153804
$838$	0	0
$839$	32.3054	1.11531	0.557654	−	0.830074i	$-0.311702\pi$
$839$	32.3054	1.11531	0.557654	+	0.830074i	$0.311702\pi$
$840$	0	0
$841$	−15.5044	−0.534635
$842$	0	0
$843$	−68.0018	−2.34211
$844$	0	0
$845$	40.4363	1.39105
$846$	0	0
$847$	18.2716	0.627819
$848$	0	0
$849$	2.55791	0.0877872
$850$	0	0
$851$	73.2105	2.50962
$852$	0	0
$853$	6.50867	0.222853	0.111426	−	0.993773i	$-0.464458\pi$
$853$	6.50867	0.222853	0.111426	+	0.993773i	$0.464458\pi$
$854$	0	0
$855$	−48.1392	−1.64633
$856$	0	0
$857$	−7.30729	−0.249612	−0.124806	−	0.992181i	$-0.539831\pi$
$857$	−7.30729	−0.249612	−0.124806	+	0.992181i	$0.539831\pi$
$858$	0	0
$859$	−37.3444	−1.27417	−0.637087	−	0.770792i	$-0.719861\pi$
$859$	−37.3444	−1.27417	−0.637087	+	0.770792i	$0.719861\pi$
$860$	0	0
$861$	37.0690	1.26331
$862$	0	0
$863$	−1.32465	−0.0450915	−0.0225457	−	0.999746i	$-0.507177\pi$
$863$	−1.32465	−0.0450915	−0.0225457	+	0.999746i	$0.507177\pi$
$864$	0	0
$865$	−58.4841	−1.98852
$866$	0	0
$867$	36.5970	1.24290
$868$	0	0
$869$	19.3393	0.656042
$870$	0	0
$871$	22.0048	0.745604
$872$	0	0
$873$	41.0711	1.39005
$874$	0	0
$875$	−105.785	−3.57618
$876$	0	0
$877$	36.9740	1.24852	0.624262	−	0.781215i	$-0.285400\pi$
$877$	36.9740	1.24852	0.624262	+	0.781215i	$0.285400\pi$
$878$	0	0
$879$	−6.80646	−0.229576
$880$	0	0
$881$	41.1447	1.38620	0.693100	−	0.720842i	$-0.256245\pi$
$881$	41.1447	1.38620	0.693100	+	0.720842i	$0.256245\pi$
$882$	0	0
$883$	−0.110300	−0.00371191	−0.00185595	−	0.999998i	$-0.500591\pi$
$883$	−0.110300	−0.00371191	−0.00185595	+	0.999998i	$0.500591\pi$
$884$	0	0
$885$	17.0891	0.574443
$886$	0	0
$887$	3.60289	0.120973	0.0604866	−	0.998169i	$-0.480735\pi$
$887$	3.60289	0.120973	0.0604866	+	0.998169i	$0.480735\pi$
$888$	0	0
$889$	25.9768	0.871234
$890$	0	0
$891$	39.1291	1.31087
$892$	0	0
$893$	−24.9932	−0.836366
$894$	0	0
$895$	−23.9252	−0.799733
$896$	0	0
$897$	−27.1792	−0.907486
$898$	0	0
$899$	−32.4759	−1.08313
$900$	0	0
$901$	5.10173	0.169963
$902$	0	0
$903$	36.6884	1.22091
$904$	0	0
$905$	−68.1267	−2.26461
$906$	0	0
$907$	−1.99421	−0.0662165	−0.0331083	−	0.999452i	$-0.510541\pi$
$907$	−1.99421	−0.0662165	−0.0331083	+	0.999452i	$0.510541\pi$
$908$	0	0
$909$	22.7777	0.755489
$910$	0	0
$911$	18.7165	0.620105	0.310053	−	0.950719i	$-0.399653\pi$
$911$	18.7165	0.620105	0.310053	+	0.950719i	$0.399653\pi$
$912$	0	0
$913$	−9.34092	−0.309139
$914$	0	0
$915$	−145.516	−4.81061
$916$	0	0
$917$	−17.6231	−0.581965
$918$	0	0
$919$	−25.7155	−0.848276	−0.424138	−	0.905597i	$-0.639423\pi$
$919$	−25.7155	−0.848276	−0.424138	+	0.905597i	$0.639423\pi$
$920$	0	0
$921$	29.8065	0.982158
$922$	0	0
$923$	15.6089	0.513775
$924$	0	0
$925$	−153.050	−5.03226
$926$	0	0
$927$	1.61717	0.0531150
$928$	0	0
$929$	32.1464	1.05469	0.527345	−	0.849651i	$-0.323188\pi$
$929$	32.1464	1.05469	0.527345	+	0.849651i	$0.323188\pi$
$930$	0	0
$931$	13.8472	0.453823
$932$	0	0
$933$	82.4480	2.69923
$934$	0	0
$935$	97.7149	3.19562
$936$	0	0
$937$	−20.6830	−0.675683	−0.337841	−	0.941203i	$-0.609697\pi$
$937$	−20.6830	−0.675683	−0.337841	+	0.941203i	$0.609697\pi$
$938$	0	0
$939$	28.9400	0.944420
$940$	0	0
$941$	−21.8443	−0.712105	−0.356052	−	0.934466i	$-0.615877\pi$
$941$	−21.8443	−0.712105	−0.356052	+	0.934466i	$0.615877\pi$
$942$	0	0
$943$	−29.2302	−0.951867
$944$	0	0
$945$	6.84052	0.222522
$946$	0	0
$947$	−29.6038	−0.961995	−0.480997	−	0.876722i	$-0.659725\pi$
$947$	−29.6038	−0.961995	−0.480997	+	0.876722i	$0.659725\pi$
$948$	0	0
$949$	−11.6332	−0.377629
$950$	0	0
$951$	37.6671	1.22144
$952$	0	0
$953$	−26.0399	−0.843516	−0.421758	−	0.906708i	$-0.638587\pi$
$953$	−26.0399	−0.843516	−0.421758	+	0.906708i	$0.638587\pi$
$954$	0	0
$955$	−5.02293	−0.162538
$956$	0	0
$957$	36.0936	1.16674
$958$	0	0
$959$	−19.9314	−0.643618
$960$	0	0
$961$	47.1506	1.52099
$962$	0	0
$963$	49.1844	1.58495
$964$	0	0
$965$	−82.1410	−2.64421
$966$	0	0
$967$	24.2513	0.779870	0.389935	−	0.920842i	$-0.372497\pi$
$967$	24.2513	0.779870	0.389935	+	0.920842i	$0.372497\pi$
$968$	0	0
$969$	55.8605	1.79450
$970$	0	0
$971$	32.2887	1.03619	0.518096	−	0.855322i	$-0.326641\pi$
$971$	32.2887	1.03619	0.518096	+	0.855322i	$0.326641\pi$
$972$	0	0
$973$	−22.4364	−0.719278
$974$	0	0
$975$	56.8195	1.81968
$976$	0	0
$977$	47.1989	1.51003	0.755014	−	0.655709i	$-0.227630\pi$
$977$	47.1989	1.51003	0.755014	+	0.655709i	$0.227630\pi$
$978$	0	0
$979$	18.0980	0.578414
$980$	0	0
$981$	−24.6623	−0.787406
$982$	0	0
$983$	−25.7368	−0.820878	−0.410439	−	0.911888i	$-0.634624\pi$
$983$	−25.7368	−0.820878	−0.410439	+	0.911888i	$0.634624\pi$
$984$	0	0
$985$	−96.6781	−3.08042
$986$	0	0
$987$	−47.3863	−1.50832
$988$	0	0
$989$	−28.9301	−0.919922
$990$	0	0
$991$	21.2202	0.674082	0.337041	−	0.941490i	$-0.390574\pi$
$991$	21.2202	0.674082	0.337041	+	0.941490i	$0.390574\pi$
$992$	0	0
$993$	−24.6897	−0.783502
$994$	0	0
$995$	−73.6850	−2.33597
$996$	0	0
$997$	3.15458	0.0999065	0.0499532	−	0.998752i	$-0.484093\pi$
$997$	3.15458	0.0999065	0.0499532	+	0.998752i	$0.484093\pi$
$998$	0	0
$999$	6.02607	0.190656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7232.2.a.bn.1.14		18
4.3	odd	2	inner	7232.2.a.bn.1.5		18
8.3	odd	2		3616.2.a.j.1.14	yes	18
8.5	even	2		3616.2.a.j.1.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3616.2.a.j.1.5	✓	18	8.5	even	2
3616.2.a.j.1.14	yes	18	8.3	odd	2
7232.2.a.bn.1.5		18	4.3	odd	2	inner
7232.2.a.bn.1.14		18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7232 = 2^{6} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7232.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40641 q^{3} -4.21710 q^{5} +3.22264 q^{7} +2.79083 q^{9} -4.08286 q^{11} +1.84698 q^{13} -10.1481 q^{15} +5.67522 q^{17} +4.09027 q^{19} +7.75501 q^{21} -6.11509 q^{23} +12.7839 q^{25} -0.503343 q^{27} -3.67363 q^{29} +8.84028 q^{31} -9.82506 q^{33} -13.5902 q^{35} -11.9721 q^{37} +4.44461 q^{39} +4.78001 q^{41} +4.73093 q^{43} -11.7692 q^{45} -6.11041 q^{47} +3.38540 q^{49} +13.6569 q^{51} +0.898949 q^{53} +17.2178 q^{55} +9.84288 q^{57} -1.68397 q^{59} +14.3393 q^{61} +8.99385 q^{63} -7.78891 q^{65} +11.9139 q^{67} -14.7154 q^{69} +8.45105 q^{71} -6.29847 q^{73} +30.7634 q^{75} -13.1576 q^{77} -4.73671 q^{79} -9.58375 q^{81} +2.28784 q^{83} -23.9329 q^{85} -8.84028 q^{87} -4.43267 q^{89} +5.95216 q^{91} +21.2734 q^{93} -17.2491 q^{95} +14.7164 q^{97} -11.3946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) \) 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

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Groups

Database

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