LMFDB - Embedded newform 1232.4.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 $ x^7 - 145x^5 - 10x^4 + 4790x^3 - 2452x^2 - 1496*x + 320
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-9.14775$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.14775 q^{3} +12.3118 q^{5} +7.00000 q^{7} +56.6814 q^{9} +11.0000 q^{11} +80.1862 q^{13} -112.625 q^{15} +68.2069 q^{17} +104.118 q^{19} -64.0343 q^{21} +184.098 q^{23} +26.5807 q^{25} -271.518 q^{27} +273.783 q^{29} -55.4648 q^{31} -100.625 q^{33} +86.1827 q^{35} -29.6327 q^{37} -733.523 q^{39} +188.501 q^{41} -65.0808 q^{43} +697.851 q^{45} +270.661 q^{47} +49.0000 q^{49} -623.940 q^{51} -711.859 q^{53} +135.430 q^{55} -952.441 q^{57} -464.547 q^{59} +415.489 q^{61} +396.770 q^{63} +987.237 q^{65} +78.8015 q^{67} -1684.08 q^{69} +660.432 q^{71} -1158.76 q^{73} -243.154 q^{75} +77.0000 q^{77} -1106.65 q^{79} +953.384 q^{81} -871.893 q^{83} +839.751 q^{85} -2504.50 q^{87} +64.4796 q^{89} +561.303 q^{91} +507.379 q^{93} +1281.88 q^{95} +557.755 q^{97} +623.495 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 6 q^{5} + 49 q^{7} + 101 q^{9} + 77 q^{11} + 88 q^{13} + 106 q^{15} + 134 q^{17} - 14 q^{19} - 42 q^{23} + 343 q^{25} + 30 q^{27} + 482 q^{29} - 50 q^{31} + 42 q^{35} + 152 q^{37} - 72 q^{39} + 234 q^{41}+ \cdots + 1111 q^{99}+O(q^{100}) $ 7 * q + 6 * q^5 + 49 * q^7 + 101 * q^9 + 77 * q^11 + 88 * q^13 + 106 * q^15 + 134 * q^17 - 14 * q^19 - 42 * q^23 + 343 * q^25 + 30 * q^27 + 482 * q^29 - 50 * q^31 + 42 * q^35 + 152 * q^37 - 72 * q^39 + 234 * q^41 - 472 * q^43 + 1384 * q^45 - 728 * q^47 + 343 * q^49 - 1572 * q^51 - 102 * q^53 + 66 * q^55 + 452 * q^57 - 1704 * q^59 + 656 * q^61 + 707 * q^63 + 2840 * q^65 - 1126 * q^67 - 350 * q^69 + 918 * q^71 + 1094 * q^73 - 274 * q^75 + 539 * q^77 - 672 * q^79 + 4503 * q^81 - 782 * q^83 + 1716 * q^85 - 3196 * q^87 - 464 * q^89 + 616 * q^91 + 1826 * q^93 + 1796 * q^95 + 3000 * q^97 + 1111 * q^99

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
$3$	−9.14775	−1.76049	−0.880243	−	0.474523i	$-0.842621\pi$
$3$	−9.14775	−1.76049	−0.880243	+	0.474523i	$0.842621\pi$
$4$	0	0
$5$	12.3118	1.10120	0.550601	−	0.834769i	$-0.314399\pi$
$5$	12.3118	1.10120	0.550601	+	0.834769i	$0.314399\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	56.6814	2.09931
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	80.1862	1.71074	0.855371	−	0.518016i	$-0.173329\pi$
$13$	80.1862	1.71074	0.855371	+	0.518016i	$0.173329\pi$
$14$	0	0
$15$	−112.625	−1.93865
$16$	0	0
$17$	68.2069	0.973095	0.486547	−	0.873654i	$-0.338256\pi$
$17$	68.2069	0.973095	0.486547	+	0.873654i	$0.338256\pi$
$18$	0	0
$19$	104.118	1.25717	0.628584	−	0.777742i	$-0.283635\pi$
$19$	104.118	1.25717	0.628584	+	0.777742i	$0.283635\pi$
$20$	0	0
$21$	−64.0343	−0.665401
$22$	0	0
$23$	184.098	1.66900	0.834501	−	0.551007i	$-0.185756\pi$
$23$	184.098	1.66900	0.834501	+	0.551007i	$0.185756\pi$
$24$	0	0
$25$	26.5807	0.212646
$26$	0	0
$27$	−271.518	−1.93532
$28$	0	0
$29$	273.783	1.75311	0.876556	−	0.481300i	$-0.159835\pi$
$29$	273.783	1.75311	0.876556	+	0.481300i	$0.159835\pi$
$30$	0	0
$31$	−55.4648	−0.321348	−0.160674	−	0.987008i	$-0.551367\pi$
$31$	−55.4648	−0.321348	−0.160674	+	0.987008i	$0.551367\pi$
$32$	0	0
$33$	−100.625	−0.530807
$34$	0	0
$35$	86.1827	0.416215
$36$	0	0
$37$	−29.6327	−0.131664	−0.0658321	−	0.997831i	$-0.520970\pi$
$37$	−29.6327	−0.131664	−0.0658321	+	0.997831i	$0.520970\pi$
$38$	0	0
$39$	−733.523	−3.01174
$40$	0	0
$41$	188.501	0.718022	0.359011	−	0.933333i	$-0.383114\pi$
$41$	188.501	0.718022	0.359011	+	0.933333i	$0.383114\pi$
$42$	0	0
$43$	−65.0808	−0.230807	−0.115404	−	0.993319i	$-0.536816\pi$
$43$	−65.0808	−0.230807	−0.115404	+	0.993319i	$0.536816\pi$
$44$	0	0
$45$	697.851	2.31177
$46$	0	0
$47$	270.661	0.839999	0.420000	−	0.907524i	$-0.362030\pi$
$47$	270.661	0.839999	0.420000	+	0.907524i	$0.362030\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−623.940	−1.71312
$52$	0	0
$53$	−711.859	−1.84493	−0.922466	−	0.386079i	$-0.873829\pi$
$53$	−711.859	−1.84493	−0.922466	+	0.386079i	$0.873829\pi$
$54$	0	0
$55$	135.430	0.332025
$56$	0	0
$57$	−952.441	−2.21323
$58$	0	0
$59$	−464.547	−1.02506	−0.512532	−	0.858668i	$-0.671293\pi$
$59$	−464.547	−1.02506	−0.512532	+	0.858668i	$0.671293\pi$
$60$	0	0
$61$	415.489	0.872097	0.436048	−	0.899923i	$-0.356378\pi$
$61$	415.489	0.872097	0.436048	+	0.899923i	$0.356378\pi$
$62$	0	0
$63$	396.770	0.793465
$64$	0	0
$65$	987.237	1.88387
$66$	0	0
$67$	78.8015	0.143689	0.0718443	−	0.997416i	$-0.477112\pi$
$67$	78.8015	0.143689	0.0718443	+	0.997416i	$0.477112\pi$
$68$	0	0
$69$	−1684.08	−2.93825
$70$	0	0
$71$	660.432	1.10393	0.551963	−	0.833868i	$-0.313879\pi$
$71$	660.432	1.10393	0.551963	+	0.833868i	$0.313879\pi$
$72$	0	0
$73$	−1158.76	−1.85784	−0.928922	−	0.370275i	$-0.879263\pi$
$73$	−1158.76	−1.85784	−0.928922	+	0.370275i	$0.879263\pi$
$74$	0	0
$75$	−243.154	−0.374360
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−1106.65	−1.57605	−0.788023	−	0.615645i	$-0.788895\pi$
$79$	−1106.65	−1.57605	−0.788023	+	0.615645i	$0.788895\pi$
$80$	0	0
$81$	953.384	1.30780
$82$	0	0
$83$	−871.893	−1.15304	−0.576522	−	0.817082i	$-0.695591\pi$
$83$	−871.893	−1.15304	−0.576522	+	0.817082i	$0.695591\pi$
$84$	0	0
$85$	839.751	1.07157
$86$	0	0
$87$	−2504.50	−3.08633
$88$	0	0
$89$	64.4796	0.0767958	0.0383979	−	0.999263i	$-0.487775\pi$
$89$	64.4796	0.0767958	0.0383979	+	0.999263i	$0.487775\pi$
$90$	0	0
$91$	561.303	0.646600
$92$	0	0
$93$	507.379	0.565728
$94$	0	0
$95$	1281.88	1.38440
$96$	0	0
$97$	557.755	0.583829	0.291915	−	0.956444i	$-0.405708\pi$
$97$	557.755	0.583829	0.291915	+	0.956444i	$0.405708\pi$
$98$	0	0
$99$	623.495	0.632966
$100$	0	0
$101$	−1407.37	−1.38652	−0.693261	−	0.720686i	$-0.743827\pi$
$101$	−1407.37	−1.38652	−0.693261	+	0.720686i	$0.743827\pi$
$102$	0	0
$103$	1606.59	1.53691	0.768457	−	0.639902i	$-0.221025\pi$
$103$	1606.59	1.53691	0.768457	+	0.639902i	$0.221025\pi$
$104$	0	0
$105$	−788.378	−0.732741
$106$	0	0
$107$	763.156	0.689505	0.344752	−	0.938694i	$-0.387963\pi$
$107$	763.156	0.689505	0.344752	+	0.938694i	$0.387963\pi$
$108$	0	0
$109$	−814.464	−0.715702	−0.357851	−	0.933779i	$-0.616490\pi$
$109$	−814.464	−0.715702	−0.357851	+	0.933779i	$0.616490\pi$
$110$	0	0
$111$	271.072	0.231793
$112$	0	0
$113$	794.810	0.661677	0.330838	−	0.943687i	$-0.392669\pi$
$113$	794.810	0.661677	0.330838	+	0.943687i	$0.392669\pi$
$114$	0	0
$115$	2266.58	1.83791
$116$	0	0
$117$	4545.06	3.59138
$118$	0	0
$119$	477.448	0.367795
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1724.36	−1.26407
$124$	0	0
$125$	−1211.72	−0.867036
$126$	0	0
$127$	329.002	0.229876	0.114938	−	0.993373i	$-0.463333\pi$
$127$	329.002	0.229876	0.114938	+	0.993373i	$0.463333\pi$
$128$	0	0
$129$	595.343	0.406333
$130$	0	0
$131$	−65.4279	−0.0436371	−0.0218186	−	0.999762i	$-0.506946\pi$
$131$	−65.4279	−0.0436371	−0.0218186	+	0.999762i	$0.506946\pi$
$132$	0	0
$133$	728.823	0.475165
$134$	0	0
$135$	−3342.88	−2.13118
$136$	0	0
$137$	2467.65	1.53887	0.769437	−	0.638723i	$-0.220537\pi$
$137$	2467.65	1.53887	0.769437	+	0.638723i	$0.220537\pi$
$138$	0	0
$139$	−42.2419	−0.0257763	−0.0128882	−	0.999917i	$-0.504103\pi$
$139$	−42.2419	−0.0257763	−0.0128882	+	0.999917i	$0.504103\pi$
$140$	0	0
$141$	−2475.94	−1.47881
$142$	0	0
$143$	882.048	0.515808
$144$	0	0
$145$	3370.77	1.93053
$146$	0	0
$147$	−448.240	−0.251498
$148$	0	0
$149$	1092.78	0.600832	0.300416	−	0.953808i	$-0.402875\pi$
$149$	1092.78	0.600832	0.300416	+	0.953808i	$0.402875\pi$
$150$	0	0
$151$	−3051.30	−1.64444	−0.822222	−	0.569167i	$-0.807266\pi$
$151$	−3051.30	−1.64444	−0.822222	+	0.569167i	$0.807266\pi$
$152$	0	0
$153$	3866.06	2.04283
$154$	0	0
$155$	−682.873	−0.353869
$156$	0	0
$157$	−12.0234	−0.00611191	−0.00305596	−	0.999995i	$-0.500973\pi$
$157$	−12.0234	−0.00611191	−0.00305596	+	0.999995i	$0.500973\pi$
$158$	0	0
$159$	6511.91	3.24798
$160$	0	0
$161$	1288.68	0.630823
$162$	0	0
$163$	−667.031	−0.320527	−0.160264	−	0.987074i	$-0.551234\pi$
$163$	−667.031	−0.320527	−0.160264	+	0.987074i	$0.551234\pi$
$164$	0	0
$165$	−1238.88	−0.584525
$166$	0	0
$167$	580.227	0.268858	0.134429	−	0.990923i	$-0.457080\pi$
$167$	580.227	0.268858	0.134429	+	0.990923i	$0.457080\pi$
$168$	0	0
$169$	4232.82	1.92664
$170$	0	0
$171$	5901.53	2.63919
$172$	0	0
$173$	−4354.49	−1.91367	−0.956837	−	0.290625i	$-0.906137\pi$
$173$	−4354.49	−1.91367	−0.956837	+	0.290625i	$0.906137\pi$
$174$	0	0
$175$	186.065	0.0803726
$176$	0	0
$177$	4249.56	1.80461
$178$	0	0
$179$	1804.21	0.753368	0.376684	−	0.926342i	$-0.377064\pi$
$179$	1804.21	0.753368	0.376684	+	0.926342i	$0.377064\pi$
$180$	0	0
$181$	−1370.66	−0.562876	−0.281438	−	0.959579i	$-0.590811\pi$
$181$	−1370.66	−0.562876	−0.281438	+	0.959579i	$0.590811\pi$
$182$	0	0
$183$	−3800.79	−1.53531
$184$	0	0
$185$	−364.832	−0.144989
$186$	0	0
$187$	750.276	0.293399
$188$	0	0
$189$	−1900.63	−0.731483
$190$	0	0
$191$	−3243.85	−1.22888	−0.614441	−	0.788963i	$-0.710618\pi$
$191$	−3243.85	−1.22888	−0.614441	+	0.788963i	$0.710618\pi$
$192$	0	0
$193$	3024.82	1.12814	0.564072	−	0.825726i	$-0.309234\pi$
$193$	3024.82	1.12814	0.564072	+	0.825726i	$0.309234\pi$
$194$	0	0
$195$	−9031.00	−3.31653
$196$	0	0
$197$	−3561.50	−1.28805	−0.644026	−	0.765004i	$-0.722737\pi$
$197$	−3561.50	−1.28805	−0.644026	+	0.765004i	$0.722737\pi$
$198$	0	0
$199$	4128.67	1.47072	0.735360	−	0.677676i	$-0.237013\pi$
$199$	4128.67	1.47072	0.735360	+	0.677676i	$0.237013\pi$
$200$	0	0
$201$	−720.857	−0.252962
$202$	0	0
$203$	1916.48	0.662614
$204$	0	0
$205$	2320.79	0.790687
$206$	0	0
$207$	10434.9	3.50375
$208$	0	0
$209$	1145.29	0.379050
$210$	0	0
$211$	741.318	0.241869	0.120935	−	0.992660i	$-0.461411\pi$
$211$	741.318	0.241869	0.120935	+	0.992660i	$0.461411\pi$
$212$	0	0
$213$	−6041.47	−1.94345
$214$	0	0
$215$	−801.262	−0.254166
$216$	0	0
$217$	−388.254	−0.121458
$218$	0	0
$219$	10600.1	3.27071
$220$	0	0
$221$	5469.25	1.66471
$222$	0	0
$223$	−1461.55	−0.438890	−0.219445	−	0.975625i	$-0.570425\pi$
$223$	−1461.55	−0.438890	−0.219445	+	0.975625i	$0.570425\pi$
$224$	0	0
$225$	1506.63	0.446410
$226$	0	0
$227$	3760.81	1.09962	0.549810	−	0.835289i	$-0.314700\pi$
$227$	3760.81	1.09962	0.549810	+	0.835289i	$0.314700\pi$
$228$	0	0
$229$	336.514	0.0971067	0.0485534	−	0.998821i	$-0.484539\pi$
$229$	336.514	0.0971067	0.0485534	+	0.998821i	$0.484539\pi$
$230$	0	0
$231$	−704.377	−0.200626
$232$	0	0
$233$	627.324	0.176383	0.0881917	−	0.996104i	$-0.471891\pi$
$233$	627.324	0.176383	0.0881917	+	0.996104i	$0.471891\pi$
$234$	0	0
$235$	3332.33	0.925009
$236$	0	0
$237$	10123.3	2.77461
$238$	0	0
$239$	1892.58	0.512220	0.256110	−	0.966648i	$-0.417559\pi$
$239$	1892.58	0.512220	0.256110	+	0.966648i	$0.417559\pi$
$240$	0	0
$241$	−1735.91	−0.463982	−0.231991	−	0.972718i	$-0.574524\pi$
$241$	−1735.91	−0.463982	−0.231991	+	0.972718i	$0.574524\pi$
$242$	0	0
$243$	−1390.33	−0.367036
$244$	0	0
$245$	603.279	0.157315
$246$	0	0
$247$	8348.78	2.15069
$248$	0	0
$249$	7975.86	2.02992
$250$	0	0
$251$	−2547.18	−0.640545	−0.320272	−	0.947326i	$-0.603774\pi$
$251$	−2547.18	−0.640545	−0.320272	+	0.947326i	$0.603774\pi$
$252$	0	0
$253$	2025.08	0.503223
$254$	0	0
$255$	−7681.83	−1.88649
$256$	0	0
$257$	−5298.56	−1.28605	−0.643025	−	0.765845i	$-0.722321\pi$
$257$	−5298.56	−1.28605	−0.643025	+	0.765845i	$0.722321\pi$
$258$	0	0
$259$	−207.429	−0.0497644
$260$	0	0
$261$	15518.4	3.68033
$262$	0	0
$263$	4866.37	1.14096	0.570482	−	0.821310i	$-0.306757\pi$
$263$	4866.37	1.14096	0.570482	+	0.821310i	$0.306757\pi$
$264$	0	0
$265$	−8764.28	−2.03164
$266$	0	0
$267$	−589.844	−0.135198
$268$	0	0
$269$	−7316.95	−1.65845	−0.829224	−	0.558916i	$-0.811217\pi$
$269$	−7316.95	−1.65845	−0.829224	+	0.558916i	$0.811217\pi$
$270$	0	0
$271$	−2149.75	−0.481875	−0.240937	−	0.970541i	$-0.577455\pi$
$271$	−2149.75	−0.481875	−0.240937	+	0.970541i	$0.577455\pi$
$272$	0	0
$273$	−5134.66	−1.13833
$274$	0	0
$275$	292.388	0.0641151
$276$	0	0
$277$	−6431.73	−1.39511	−0.697555	−	0.716531i	$-0.745729\pi$
$277$	−6431.73	−1.39511	−0.697555	+	0.716531i	$0.745729\pi$
$278$	0	0
$279$	−3143.83	−0.674609
$280$	0	0
$281$	−1252.60	−0.265921	−0.132961	−	0.991121i	$-0.542448\pi$
$281$	−1252.60	−0.265921	−0.132961	+	0.991121i	$0.542448\pi$
$282$	0	0
$283$	−6100.99	−1.28151	−0.640753	−	0.767747i	$-0.721378\pi$
$283$	−6100.99	−1.28151	−0.640753	+	0.767747i	$0.721378\pi$
$284$	0	0
$285$	−11726.3	−2.43721
$286$	0	0
$287$	1319.51	0.271387
$288$	0	0
$289$	−260.816	−0.0530869
$290$	0	0
$291$	−5102.20	−1.02782
$292$	0	0
$293$	7020.95	1.39989	0.699946	−	0.714196i	$-0.253207\pi$
$293$	7020.95	1.39989	0.699946	+	0.714196i	$0.253207\pi$
$294$	0	0
$295$	−5719.41	−1.12880
$296$	0	0
$297$	−2986.70	−0.583522
$298$	0	0
$299$	14762.1	2.85523
$300$	0	0
$301$	−455.565	−0.0872370
$302$	0	0
$303$	12874.3	2.44095
$304$	0	0
$305$	5115.42	0.960355
$306$	0	0
$307$	−1084.55	−0.201624	−0.100812	−	0.994905i	$-0.532144\pi$
$307$	−1084.55	−0.201624	−0.100812	+	0.994905i	$0.532144\pi$
$308$	0	0
$309$	−14696.7	−2.70572
$310$	0	0
$311$	−1013.41	−0.184775	−0.0923877	−	0.995723i	$-0.529450\pi$
$311$	−1013.41	−0.184775	−0.0923877	+	0.995723i	$0.529450\pi$
$312$	0	0
$313$	−7679.46	−1.38680	−0.693401	−	0.720552i	$-0.743888\pi$
$313$	−7679.46	−1.38680	−0.693401	+	0.720552i	$0.743888\pi$
$314$	0	0
$315$	4884.96	0.873765
$316$	0	0
$317$	6170.81	1.09334	0.546668	−	0.837350i	$-0.315896\pi$
$317$	6170.81	1.09334	0.546668	+	0.837350i	$0.315896\pi$
$318$	0	0
$319$	3011.61	0.528583
$320$	0	0
$321$	−6981.16	−1.21386
$322$	0	0
$323$	7101.53	1.22334
$324$	0	0
$325$	2131.41	0.363782
$326$	0	0
$327$	7450.52	1.25998
$328$	0	0
$329$	1894.63	0.317490
$330$	0	0
$331$	−1231.70	−0.204532	−0.102266	−	0.994757i	$-0.532609\pi$
$331$	−1231.70	−0.204532	−0.102266	+	0.994757i	$0.532609\pi$
$332$	0	0
$333$	−1679.62	−0.276404
$334$	0	0
$335$	970.190	0.158230
$336$	0	0
$337$	−877.044	−0.141767	−0.0708837	−	0.997485i	$-0.522582\pi$
$337$	−877.044	−0.141767	−0.0708837	+	0.997485i	$0.522582\pi$
$338$	0	0
$339$	−7270.73	−1.16487
$340$	0	0
$341$	−610.113	−0.0968900
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−20734.1	−3.23561
$346$	0	0
$347$	1722.78	0.266524	0.133262	−	0.991081i	$-0.457455\pi$
$347$	1722.78	0.266524	0.133262	+	0.991081i	$0.457455\pi$
$348$	0	0
$349$	−11582.3	−1.77647	−0.888236	−	0.459388i	$-0.848069\pi$
$349$	−11582.3	−1.77647	−0.888236	+	0.459388i	$0.848069\pi$
$350$	0	0
$351$	−21772.0	−3.31084
$352$	0	0
$353$	−3617.24	−0.545401	−0.272700	−	0.962099i	$-0.587917\pi$
$353$	−3617.24	−0.545401	−0.272700	+	0.962099i	$0.587917\pi$
$354$	0	0
$355$	8131.11	1.21565
$356$	0	0
$357$	−4367.58	−0.647498
$358$	0	0
$359$	−5092.73	−0.748702	−0.374351	−	0.927287i	$-0.622134\pi$
$359$	−5092.73	−0.748702	−0.374351	+	0.927287i	$0.622134\pi$
$360$	0	0
$361$	3981.45	0.580471
$362$	0	0
$363$	−1106.88	−0.160044
$364$	0	0
$365$	−14266.4	−2.04586
$366$	0	0
$367$	−7905.22	−1.12438	−0.562192	−	0.827007i	$-0.690042\pi$
$367$	−7905.22	−1.12438	−0.562192	+	0.827007i	$0.690042\pi$
$368$	0	0
$369$	10684.5	1.50735
$370$	0	0
$371$	−4983.01	−0.697318
$372$	0	0
$373$	−7760.82	−1.07732	−0.538660	−	0.842523i	$-0.681069\pi$
$373$	−7760.82	−1.07732	−0.538660	+	0.842523i	$0.681069\pi$
$374$	0	0
$375$	11084.5	1.52640
$376$	0	0
$377$	21953.6	2.99912
$378$	0	0
$379$	6800.27	0.921652	0.460826	−	0.887490i	$-0.347553\pi$
$379$	6800.27	0.921652	0.460826	+	0.887490i	$0.347553\pi$
$380$	0	0
$381$	−3009.63	−0.404693
$382$	0	0
$383$	−11602.9	−1.54800	−0.773998	−	0.633188i	$-0.781746\pi$
$383$	−11602.9	−1.54800	−0.773998	+	0.633188i	$0.781746\pi$
$384$	0	0
$385$	948.010	0.125494
$386$	0	0
$387$	−3688.87	−0.484537
$388$	0	0
$389$	−8729.14	−1.13775	−0.568875	−	0.822424i	$-0.692621\pi$
$389$	−8729.14	−1.13775	−0.568875	+	0.822424i	$0.692621\pi$
$390$	0	0
$391$	12556.7	1.62410
$392$	0	0
$393$	598.519	0.0768226
$394$	0	0
$395$	−13624.9	−1.73555
$396$	0	0
$397$	−6264.99	−0.792017	−0.396009	−	0.918247i	$-0.629605\pi$
$397$	−6264.99	−0.792017	−0.396009	+	0.918247i	$0.629605\pi$
$398$	0	0
$399$	−6667.09	−0.836521
$400$	0	0
$401$	13433.6	1.67293	0.836463	−	0.548023i	$-0.184619\pi$
$401$	13433.6	1.67293	0.836463	+	0.548023i	$0.184619\pi$
$402$	0	0
$403$	−4447.51	−0.549743
$404$	0	0
$405$	11737.9	1.44015
$406$	0	0
$407$	−325.959	−0.0396983
$408$	0	0
$409$	1416.39	0.171237	0.0856184	−	0.996328i	$-0.472713\pi$
$409$	1416.39	0.171237	0.0856184	+	0.996328i	$0.472713\pi$
$410$	0	0
$411$	−22573.5	−2.70917
$412$	0	0
$413$	−3251.83	−0.387438
$414$	0	0
$415$	−10734.6	−1.26973
$416$	0	0
$417$	386.419	0.0453789
$418$	0	0
$419$	−12604.4	−1.46960	−0.734801	−	0.678282i	$-0.762725\pi$
$419$	−12604.4	−1.46960	−0.734801	+	0.678282i	$0.762725\pi$
$420$	0	0
$421$	−1936.94	−0.224230	−0.112115	−	0.993695i	$-0.535762\pi$
$421$	−1936.94	−0.224230	−0.112115	+	0.993695i	$0.535762\pi$
$422$	0	0
$423$	15341.4	1.76342
$424$	0	0
$425$	1812.99	0.206925
$426$	0	0
$427$	2908.42	0.329622
$428$	0	0
$429$	−8068.76	−0.908073
$430$	0	0
$431$	9158.34	1.02353	0.511765	−	0.859125i	$-0.328992\pi$
$431$	9158.34	1.02353	0.511765	+	0.859125i	$0.328992\pi$
$432$	0	0
$433$	−831.437	−0.0922779	−0.0461390	−	0.998935i	$-0.514692\pi$
$433$	−831.437	−0.0922779	−0.0461390	+	0.998935i	$0.514692\pi$
$434$	0	0
$435$	−30834.9	−3.39867
$436$	0	0
$437$	19167.8	2.09822
$438$	0	0
$439$	6223.67	0.676627	0.338314	−	0.941033i	$-0.390144\pi$
$439$	6223.67	0.676627	0.338314	+	0.941033i	$0.390144\pi$
$440$	0	0
$441$	2777.39	0.299902
$442$	0	0
$443$	−7495.66	−0.803904	−0.401952	−	0.915661i	$-0.631668\pi$
$443$	−7495.66	−0.803904	−0.401952	+	0.915661i	$0.631668\pi$
$444$	0	0
$445$	793.861	0.0845677
$446$	0	0
$447$	−9996.47	−1.05776
$448$	0	0
$449$	8227.98	0.864816	0.432408	−	0.901678i	$-0.357664\pi$
$449$	8227.98	0.864816	0.432408	+	0.901678i	$0.357664\pi$
$450$	0	0
$451$	2073.51	0.216492
$452$	0	0
$453$	27912.5	2.89502
$454$	0	0
$455$	6910.66	0.712037
$456$	0	0
$457$	−1099.99	−0.112594	−0.0562969	−	0.998414i	$-0.517929\pi$
$457$	−1099.99	−0.112594	−0.0562969	+	0.998414i	$0.517929\pi$
$458$	0	0
$459$	−18519.4	−1.88325
$460$	0	0
$461$	−3290.30	−0.332417	−0.166209	−	0.986091i	$-0.553153\pi$
$461$	−3290.30	−0.332417	−0.166209	+	0.986091i	$0.553153\pi$
$462$	0	0
$463$	9480.79	0.951640	0.475820	−	0.879543i	$-0.342151\pi$
$463$	9480.79	0.951640	0.475820	+	0.879543i	$0.342151\pi$
$464$	0	0
$465$	6246.75	0.622981
$466$	0	0
$467$	−10021.6	−0.993033	−0.496516	−	0.868027i	$-0.665388\pi$
$467$	−10021.6	−0.993033	−0.496516	+	0.868027i	$0.665388\pi$
$468$	0	0
$469$	551.611	0.0543092
$470$	0	0
$471$	109.987	0.0107599
$472$	0	0
$473$	−715.888	−0.0695911
$474$	0	0
$475$	2767.52	0.267332
$476$	0	0
$477$	−40349.2	−3.87309
$478$	0	0
$479$	−4897.70	−0.467185	−0.233593	−	0.972335i	$-0.575048\pi$
$479$	−4897.70	−0.467185	−0.233593	+	0.972335i	$0.575048\pi$
$480$	0	0
$481$	−2376.13	−0.225244
$482$	0	0
$483$	−11788.6	−1.11056
$484$	0	0
$485$	6866.97	0.642914
$486$	0	0
$487$	13863.5	1.28997	0.644985	−	0.764196i	$-0.276864\pi$
$487$	13863.5	1.28997	0.644985	+	0.764196i	$0.276864\pi$
$488$	0	0
$489$	6101.84	0.564283
$490$	0	0
$491$	−6235.99	−0.573170	−0.286585	−	0.958055i	$-0.592520\pi$
$491$	−6235.99	−0.573170	−0.286585	+	0.958055i	$0.592520\pi$
$492$	0	0
$493$	18673.9	1.70594
$494$	0	0
$495$	7676.36	0.697024
$496$	0	0
$497$	4623.02	0.417245
$498$	0	0
$499$	12334.8	1.10658	0.553288	−	0.832990i	$-0.313373\pi$
$499$	12334.8	1.10658	0.553288	+	0.832990i	$0.313373\pi$
$500$	0	0
$501$	−5307.78	−0.473321
$502$	0	0
$503$	−10179.9	−0.902389	−0.451194	−	0.892426i	$-0.649002\pi$
$503$	−10179.9	−0.902389	−0.451194	+	0.892426i	$0.649002\pi$
$504$	0	0
$505$	−17327.3	−1.52684
$506$	0	0
$507$	−38720.8	−3.39182
$508$	0	0
$509$	10212.0	0.889267	0.444634	−	0.895713i	$-0.353334\pi$
$509$	10212.0	0.889267	0.444634	+	0.895713i	$0.353334\pi$
$510$	0	0
$511$	−8111.32	−0.702199
$512$	0	0
$513$	−28269.8	−2.43303
$514$	0	0
$515$	19780.0	1.69245
$516$	0	0
$517$	2977.27	0.253269
$518$	0	0
$519$	39833.8	3.36900
$520$	0	0
$521$	−8751.45	−0.735908	−0.367954	−	0.929844i	$-0.619942\pi$
$521$	−8751.45	−0.735908	−0.367954	+	0.929844i	$0.619942\pi$
$522$	0	0
$523$	16339.2	1.36608	0.683042	−	0.730379i	$-0.260657\pi$
$523$	16339.2	1.36608	0.683042	+	0.730379i	$0.260657\pi$
$524$	0	0
$525$	−1702.08	−0.141495
$526$	0	0
$527$	−3783.09	−0.312702
$528$	0	0
$529$	21725.0	1.78557
$530$	0	0
$531$	−26331.2	−2.15193
$532$	0	0
$533$	15115.2	1.22835
$534$	0	0
$535$	9395.83	0.759284
$536$	0	0
$537$	−16504.4	−1.32629
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−3508.22	−0.278799	−0.139399	−	0.990236i	$-0.544517\pi$
$541$	−3508.22	−0.278799	−0.139399	+	0.990236i	$0.544517\pi$
$542$	0	0
$543$	12538.5	0.990935
$544$	0	0
$545$	−10027.5	−0.788133
$546$	0	0
$547$	10613.7	0.829635	0.414817	−	0.909905i	$-0.363845\pi$
$547$	10613.7	0.829635	0.414817	+	0.909905i	$0.363845\pi$
$548$	0	0
$549$	23550.5	1.83080
$550$	0	0
$551$	28505.6	2.20396
$552$	0	0
$553$	−7746.54	−0.595690
$554$	0	0
$555$	3337.39	0.255251
$556$	0	0
$557$	19663.6	1.49583	0.747913	−	0.663797i	$-0.231056\pi$
$557$	19663.6	1.49583	0.747913	+	0.663797i	$0.231056\pi$
$558$	0	0
$559$	−5218.58	−0.394852
$560$	0	0
$561$	−6863.34	−0.516525
$562$	0	0
$563$	−16614.1	−1.24369	−0.621846	−	0.783139i	$-0.713617\pi$
$563$	−16614.1	−1.24369	−0.621846	+	0.783139i	$0.713617\pi$
$564$	0	0
$565$	9785.56	0.728640
$566$	0	0
$567$	6673.69	0.494301
$568$	0	0
$569$	1106.25	0.0815049	0.0407524	−	0.999169i	$-0.487025\pi$
$569$	1106.25	0.0815049	0.0407524	+	0.999169i	$0.487025\pi$
$570$	0	0
$571$	−13478.4	−0.987835	−0.493917	−	0.869509i	$-0.664436\pi$
$571$	−13478.4	−0.987835	−0.493917	+	0.869509i	$0.664436\pi$
$572$	0	0
$573$	29673.9	2.16343
$574$	0	0
$575$	4893.45	0.354906
$576$	0	0
$577$	−11069.4	−0.798656	−0.399328	−	0.916808i	$-0.630756\pi$
$577$	−11069.4	−0.798656	−0.399328	+	0.916808i	$0.630756\pi$
$578$	0	0
$579$	−27670.4	−1.98608
$580$	0	0
$581$	−6103.25	−0.435810
$582$	0	0
$583$	−7830.45	−0.556268
$584$	0	0
$585$	55958.0	3.95483
$586$	0	0
$587$	−3858.21	−0.271287	−0.135644	−	0.990758i	$-0.543310\pi$
$587$	−3858.21	−0.271287	−0.135644	+	0.990758i	$0.543310\pi$
$588$	0	0
$589$	−5774.86	−0.403988
$590$	0	0
$591$	32579.7	2.26760
$592$	0	0
$593$	9846.03	0.681835	0.340917	−	0.940093i	$-0.389262\pi$
$593$	9846.03	0.681835	0.340917	+	0.940093i	$0.389262\pi$
$594$	0	0
$595$	5878.26	0.405017
$596$	0	0
$597$	−37768.0	−2.58918
$598$	0	0
$599$	12370.3	0.843802	0.421901	−	0.906642i	$-0.361363\pi$
$599$	12370.3	0.843802	0.421901	+	0.906642i	$0.361363\pi$
$600$	0	0
$601$	13728.3	0.931765	0.465883	−	0.884847i	$-0.345737\pi$
$601$	13728.3	0.931765	0.465883	+	0.884847i	$0.345737\pi$
$602$	0	0
$603$	4466.58	0.301647
$604$	0	0
$605$	1489.73	0.100109
$606$	0	0
$607$	4689.25	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$607$	4689.25	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$608$	0	0
$609$	−17531.5	−1.16652
$610$	0	0
$611$	21703.3	1.43702
$612$	0	0
$613$	16666.9	1.09816	0.549079	−	0.835770i	$-0.314979\pi$
$613$	16666.9	1.09816	0.549079	+	0.835770i	$0.314979\pi$
$614$	0	0
$615$	−21230.0	−1.39199
$616$	0	0
$617$	16164.9	1.05474	0.527370	−	0.849636i	$-0.323178\pi$
$617$	16164.9	1.05474	0.527370	+	0.849636i	$0.323178\pi$
$618$	0	0
$619$	728.700	0.0473165	0.0236583	−	0.999720i	$-0.492469\pi$
$619$	728.700	0.0473165	0.0236583	+	0.999720i	$0.492469\pi$
$620$	0	0
$621$	−49985.9	−3.23006
$622$	0	0
$623$	451.357	0.0290261
$624$	0	0
$625$	−18241.1	−1.16743
$626$	0	0
$627$	−10476.9	−0.667313
$628$	0	0
$629$	−2021.15	−0.128122
$630$	0	0
$631$	−20704.7	−1.30625	−0.653124	−	0.757251i	$-0.726542\pi$
$631$	−20704.7	−1.30625	−0.653124	+	0.757251i	$0.726542\pi$
$632$	0	0
$633$	−6781.39	−0.425808
$634$	0	0
$635$	4050.61	0.253139
$636$	0	0
$637$	3929.12	0.244392
$638$	0	0
$639$	37434.2	2.31749
$640$	0	0
$641$	17599.4	1.08446	0.542228	−	0.840232i	$-0.317581\pi$
$641$	17599.4	1.08446	0.542228	+	0.840232i	$0.317581\pi$
$642$	0	0
$643$	12657.6	0.776310	0.388155	−	0.921594i	$-0.373112\pi$
$643$	12657.6	0.776310	0.388155	+	0.921594i	$0.373112\pi$
$644$	0	0
$645$	7329.75	0.447455
$646$	0	0
$647$	1456.90	0.0885263	0.0442631	−	0.999020i	$-0.485906\pi$
$647$	1456.90	0.0885263	0.0442631	+	0.999020i	$0.485906\pi$
$648$	0	0
$649$	−5110.01	−0.309069
$650$	0	0
$651$	3551.65	0.213825
$652$	0	0
$653$	3582.16	0.214672	0.107336	−	0.994223i	$-0.465768\pi$
$653$	3582.16	0.214672	0.107336	+	0.994223i	$0.465768\pi$
$654$	0	0
$655$	−805.536	−0.0480533
$656$	0	0
$657$	−65680.2	−3.90019
$658$	0	0
$659$	−30089.2	−1.77862	−0.889308	−	0.457309i	$-0.848813\pi$
$659$	−30089.2	−1.77862	−0.889308	+	0.457309i	$0.848813\pi$
$660$	0	0
$661$	27621.6	1.62535	0.812675	−	0.582717i	$-0.198011\pi$
$661$	27621.6	1.62535	0.812675	+	0.582717i	$0.198011\pi$
$662$	0	0
$663$	−50031.4	−2.93070
$664$	0	0
$665$	8973.13	0.523253
$666$	0	0
$667$	50402.8	2.92595
$668$	0	0
$669$	13369.9	0.772659
$670$	0	0
$671$	4570.38	0.262947
$672$	0	0
$673$	19934.5	1.14178	0.570891	−	0.821026i	$-0.306598\pi$
$673$	19934.5	1.14178	0.570891	+	0.821026i	$0.306598\pi$
$674$	0	0
$675$	−7217.15	−0.411538
$676$	0	0
$677$	18528.2	1.05184	0.525920	−	0.850534i	$-0.323721\pi$
$677$	18528.2	1.05184	0.525920	+	0.850534i	$0.323721\pi$
$678$	0	0
$679$	3904.28	0.220667
$680$	0	0
$681$	−34403.0	−1.93587
$682$	0	0
$683$	4577.85	0.256467	0.128233	−	0.991744i	$-0.459069\pi$
$683$	4577.85	0.256467	0.128233	+	0.991744i	$0.459069\pi$
$684$	0	0
$685$	30381.3	1.69461
$686$	0	0
$687$	−3078.34	−0.170955
$688$	0	0
$689$	−57081.2	−3.15620
$690$	0	0
$691$	−10369.2	−0.570858	−0.285429	−	0.958400i	$-0.592136\pi$
$691$	−10369.2	−0.570858	−0.285429	+	0.958400i	$0.592136\pi$
$692$	0	0
$693$	4364.47	0.239239
$694$	0	0
$695$	−520.074	−0.0283850
$696$	0	0
$697$	12857.1	0.698703
$698$	0	0
$699$	−5738.60	−0.310521
$700$	0	0
$701$	−29578.6	−1.59368	−0.796839	−	0.604191i	$-0.793496\pi$
$701$	−29578.6	−1.59368	−0.796839	+	0.604191i	$0.793496\pi$
$702$	0	0
$703$	−3085.28	−0.165524
$704$	0	0
$705$	−30483.3	−1.62847
$706$	0	0
$707$	−9851.61	−0.524056
$708$	0	0
$709$	−35837.9	−1.89834	−0.949168	−	0.314769i	$-0.898073\pi$
$709$	−35837.9	−1.89834	−0.949168	+	0.314769i	$0.898073\pi$
$710$	0	0
$711$	−62726.4	−3.30861
$712$	0	0
$713$	−10211.0	−0.536330
$714$	0	0
$715$	10859.6	0.568009
$716$	0	0
$717$	−17312.8	−0.901756
$718$	0	0
$719$	−24283.9	−1.25958	−0.629789	−	0.776766i	$-0.716859\pi$
$719$	−24283.9	−1.25958	−0.629789	+	0.776766i	$0.716859\pi$
$720$	0	0
$721$	11246.1	0.580899
$722$	0	0
$723$	15879.7	0.816834
$724$	0	0
$725$	7277.36	0.372792
$726$	0	0
$727$	18783.9	0.958259	0.479130	−	0.877744i	$-0.340952\pi$
$727$	18783.9	0.958259	0.479130	+	0.877744i	$0.340952\pi$
$728$	0	0
$729$	−13023.0	−0.661635
$730$	0	0
$731$	−4438.96	−0.224598
$732$	0	0
$733$	15115.4	0.761667	0.380833	−	0.924644i	$-0.375637\pi$
$733$	15115.4	0.761667	0.380833	+	0.924644i	$0.375637\pi$
$734$	0	0
$735$	−5518.65	−0.276950
$736$	0	0
$737$	866.817	0.0433238
$738$	0	0
$739$	−33698.9	−1.67745	−0.838724	−	0.544557i	$-0.816698\pi$
$739$	−33698.9	−1.67745	−0.838724	+	0.544557i	$0.816698\pi$
$740$	0	0
$741$	−76372.6	−3.78626
$742$	0	0
$743$	13689.2	0.675918	0.337959	−	0.941161i	$-0.390263\pi$
$743$	13689.2	0.675918	0.337959	+	0.941161i	$0.390263\pi$
$744$	0	0
$745$	13454.1	0.661637
$746$	0	0
$747$	−49420.1	−2.42060
$748$	0	0
$749$	5342.09	0.260608
$750$	0	0
$751$	−25644.2	−1.24603	−0.623016	−	0.782209i	$-0.714093\pi$
$751$	−25644.2	−1.24603	−0.623016	+	0.782209i	$0.714093\pi$
$752$	0	0
$753$	23301.0	1.12767
$754$	0	0
$755$	−37567.0	−1.81086
$756$	0	0
$757$	16529.6	0.793633	0.396816	−	0.917898i	$-0.370115\pi$
$757$	16529.6	0.793633	0.396816	+	0.917898i	$0.370115\pi$
$758$	0	0
$759$	−18524.9	−0.885917
$760$	0	0
$761$	−28422.9	−1.35392	−0.676958	−	0.736022i	$-0.736702\pi$
$761$	−28422.9	−1.35392	−0.676958	+	0.736022i	$0.736702\pi$
$762$	0	0
$763$	−5701.25	−0.270510
$764$	0	0
$765$	47598.3	2.24957
$766$	0	0
$767$	−37250.2	−1.75362
$768$	0	0
$769$	−2296.42	−0.107687	−0.0538433	−	0.998549i	$-0.517147\pi$
$769$	−2296.42	−0.107687	−0.0538433	+	0.998549i	$0.517147\pi$
$770$	0	0
$771$	48469.9	2.26407
$772$	0	0
$773$	30248.4	1.40745	0.703724	−	0.710473i	$-0.251519\pi$
$773$	30248.4	1.40745	0.703724	+	0.710473i	$0.251519\pi$
$774$	0	0
$775$	−1474.30	−0.0683333
$776$	0	0
$777$	1897.51	0.0876096
$778$	0	0
$779$	19626.2	0.902674
$780$	0	0
$781$	7264.75	0.332846
$782$	0	0
$783$	−74337.1	−3.39284
$784$	0	0
$785$	−148.030	−0.00673045
$786$	0	0
$787$	16251.6	0.736096	0.368048	−	0.929807i	$-0.380026\pi$
$787$	16251.6	0.736096	0.368048	+	0.929807i	$0.380026\pi$
$788$	0	0
$789$	−44516.4	−2.00865
$790$	0	0
$791$	5563.67	0.250090
$792$	0	0
$793$	33316.5	1.49193
$794$	0	0
$795$	80173.4	3.57668
$796$	0	0
$797$	13971.8	0.620962	0.310481	−	0.950580i	$-0.399510\pi$
$797$	13971.8	0.620962	0.310481	+	0.950580i	$0.399510\pi$
$798$	0	0
$799$	18461.0	0.817399
$800$	0	0
$801$	3654.80	0.161218
$802$	0	0
$803$	−12746.4	−0.560161
$804$	0	0
$805$	15866.0	0.694664
$806$	0	0
$807$	66933.7	2.91967
$808$	0	0
$809$	−27507.5	−1.19544	−0.597721	−	0.801704i	$-0.703927\pi$
$809$	−27507.5	−1.19544	−0.597721	+	0.801704i	$0.703927\pi$
$810$	0	0
$811$	−29240.5	−1.26606	−0.633028	−	0.774129i	$-0.718188\pi$
$811$	−29240.5	−1.26606	−0.633028	+	0.774129i	$0.718188\pi$
$812$	0	0
$813$	19665.4	0.848333
$814$	0	0
$815$	−8212.36	−0.352965
$816$	0	0
$817$	−6776.05	−0.290164
$818$	0	0
$819$	31815.5	1.35741
$820$	0	0
$821$	23841.8	1.01350	0.506751	−	0.862092i	$-0.330846\pi$
$821$	23841.8	1.01350	0.506751	+	0.862092i	$0.330846\pi$
$822$	0	0
$823$	12205.6	0.516961	0.258480	−	0.966016i	$-0.416778\pi$
$823$	12205.6	0.516961	0.258480	+	0.966016i	$0.416778\pi$
$824$	0	0
$825$	−2674.69	−0.112874
$826$	0	0
$827$	−39789.6	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$827$	−39789.6	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$828$	0	0
$829$	−18129.2	−0.759533	−0.379766	−	0.925082i	$-0.623996\pi$
$829$	−18129.2	−0.759533	−0.379766	+	0.925082i	$0.623996\pi$
$830$	0	0
$831$	58835.9	2.45607
$832$	0	0
$833$	3342.14	0.139014
$834$	0	0
$835$	7143.65	0.296067
$836$	0	0
$837$	15059.7	0.621911
$838$	0	0
$839$	5078.87	0.208989	0.104495	−	0.994525i	$-0.466677\pi$
$839$	5078.87	0.208989	0.104495	+	0.994525i	$0.466677\pi$
$840$	0	0
$841$	50568.2	2.07340
$842$	0	0
$843$	11458.5	0.468151
$844$	0	0
$845$	52113.7	2.12162
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	55810.4	2.25607
$850$	0	0
$851$	−5455.30	−0.219748
$852$	0	0
$853$	9616.04	0.385987	0.192994	−	0.981200i	$-0.438180\pi$
$853$	9616.04	0.385987	0.192994	+	0.981200i	$0.438180\pi$
$854$	0	0
$855$	72658.5	2.90628
$856$	0	0
$857$	11328.7	0.451552	0.225776	−	0.974179i	$-0.427508\pi$
$857$	11328.7	0.451552	0.225776	+	0.974179i	$0.427508\pi$
$858$	0	0
$859$	46045.4	1.82893	0.914464	−	0.404668i	$-0.132613\pi$
$859$	46045.4	1.82893	0.914464	+	0.404668i	$0.132613\pi$
$860$	0	0
$861$	−12070.5	−0.477773
$862$	0	0
$863$	−15695.8	−0.619111	−0.309555	−	0.950881i	$-0.600180\pi$
$863$	−15695.8	−0.619111	−0.309555	+	0.950881i	$0.600180\pi$
$864$	0	0
$865$	−53611.6	−2.10734
$866$	0	0
$867$	2385.88	0.0934588
$868$	0	0
$869$	−12173.1	−0.475196
$870$	0	0
$871$	6318.79	0.245814
$872$	0	0
$873$	31614.3	1.22564
$874$	0	0
$875$	−8482.04	−0.327709
$876$	0	0
$877$	−41186.1	−1.58581	−0.792906	−	0.609344i	$-0.791433\pi$
$877$	−41186.1	−1.58581	−0.792906	+	0.609344i	$0.791433\pi$
$878$	0	0
$879$	−64226.0	−2.46449
$880$	0	0
$881$	−31056.9	−1.18767	−0.593833	−	0.804588i	$-0.702386\pi$
$881$	−31056.9	−1.18767	−0.593833	+	0.804588i	$0.702386\pi$
$882$	0	0
$883$	−32020.9	−1.22037	−0.610187	−	0.792257i	$-0.708906\pi$
$883$	−32020.9	−1.22037	−0.610187	+	0.792257i	$0.708906\pi$
$884$	0	0
$885$	52319.8	1.98724
$886$	0	0
$887$	−17880.9	−0.676870	−0.338435	−	0.940990i	$-0.609897\pi$
$887$	−17880.9	−0.676870	−0.338435	+	0.940990i	$0.609897\pi$
$888$	0	0
$889$	2303.01	0.0868848
$890$	0	0
$891$	10487.2	0.394316
$892$	0	0
$893$	28180.5	1.05602
$894$	0	0
$895$	22213.1	0.829610
$896$	0	0
$897$	−135040.	−5.02659
$898$	0	0
$899$	−15185.3	−0.563358
$900$	0	0
$901$	−48553.7	−1.79529
$902$	0	0
$903$	4167.40	0.153580
$904$	0	0
$905$	−16875.3	−0.619840
$906$	0	0
$907$	28412.1	1.04014	0.520071	−	0.854123i	$-0.325905\pi$
$907$	28412.1	1.04014	0.520071	+	0.854123i	$0.325905\pi$
$908$	0	0
$909$	−79771.9	−2.91074
$910$	0	0
$911$	−52287.5	−1.90160	−0.950802	−	0.309800i	$-0.899738\pi$
$911$	−52287.5	−1.90160	−0.950802	+	0.309800i	$0.899738\pi$
$912$	0	0
$913$	−9590.82	−0.347656
$914$	0	0
$915$	−46794.6	−1.69069
$916$	0	0
$917$	−457.995	−0.0164933
$918$	0	0
$919$	7428.89	0.266655	0.133328	−	0.991072i	$-0.457434\pi$
$919$	7428.89	0.266655	0.133328	+	0.991072i	$0.457434\pi$
$920$	0	0
$921$	9921.21	0.354957
$922$	0	0
$923$	52957.5	1.88853
$924$	0	0
$925$	−787.658	−0.0279979
$926$	0	0
$927$	91063.8	3.22646
$928$	0	0
$929$	15872.5	0.560559	0.280280	−	0.959918i	$-0.409573\pi$
$929$	15872.5	0.560559	0.280280	+	0.959918i	$0.409573\pi$
$930$	0	0
$931$	5101.76	0.179595
$932$	0	0
$933$	9270.42	0.325295
$934$	0	0
$935$	9237.26	0.323092
$936$	0	0
$937$	21205.2	0.739321	0.369660	−	0.929167i	$-0.379474\pi$
$937$	21205.2	0.739321	0.369660	+	0.929167i	$0.379474\pi$
$938$	0	0
$939$	70249.8	2.44144
$940$	0	0
$941$	20982.6	0.726902	0.363451	−	0.931613i	$-0.381598\pi$
$941$	20982.6	0.726902	0.363451	+	0.931613i	$0.381598\pi$
$942$	0	0
$943$	34702.6	1.19838
$944$	0	0
$945$	−23400.2	−0.805511
$946$	0	0
$947$	−4468.49	−0.153333	−0.0766665	−	0.997057i	$-0.524428\pi$
$947$	−4468.49	−0.153333	−0.0766665	+	0.997057i	$0.524428\pi$
$948$	0	0
$949$	−92916.5	−3.17829
$950$	0	0
$951$	−56449.1	−1.92480
$952$	0	0
$953$	11117.3	0.377885	0.188943	−	0.981988i	$-0.439494\pi$
$953$	11117.3	0.377885	0.188943	+	0.981988i	$0.439494\pi$
$954$	0	0
$955$	−39937.6	−1.35325
$956$	0	0
$957$	−27549.5	−0.930563
$958$	0	0
$959$	17273.6	0.581639
$960$	0	0
$961$	−26714.7	−0.896736
$962$	0	0
$963$	43256.7	1.44749
$964$	0	0
$965$	37241.1	1.24231
$966$	0	0
$967$	11293.0	0.375550	0.187775	−	0.982212i	$-0.439872\pi$
$967$	11293.0	0.375550	0.187775	+	0.982212i	$0.439872\pi$
$968$	0	0
$969$	−64963.1	−2.15368
$970$	0	0
$971$	−30920.8	−1.02193	−0.510966	−	0.859601i	$-0.670712\pi$
$971$	−30920.8	−1.02193	−0.510966	+	0.859601i	$0.670712\pi$
$972$	0	0
$973$	−295.693	−0.00974254
$974$	0	0
$975$	−19497.6	−0.640433
$976$	0	0
$977$	−9639.96	−0.315670	−0.157835	−	0.987466i	$-0.550451\pi$
$977$	−9639.96	−0.315670	−0.157835	+	0.987466i	$0.550451\pi$
$978$	0	0
$979$	709.276	0.0231548
$980$	0	0
$981$	−46165.0	−1.50248
$982$	0	0
$983$	−26217.9	−0.850683	−0.425341	−	0.905033i	$-0.639846\pi$
$983$	−26217.9	−0.850683	−0.425341	+	0.905033i	$0.639846\pi$
$984$	0	0
$985$	−43848.5	−1.41840
$986$	0	0
$987$	−17331.6	−0.558937
$988$	0	0
$989$	−11981.2	−0.385218
$990$	0	0
$991$	14319.1	0.458991	0.229496	−	0.973310i	$-0.426292\pi$
$991$	14319.1	0.458991	0.229496	+	0.973310i	$0.426292\pi$
$992$	0	0
$993$	11267.3	0.360076
$994$	0	0
$995$	50831.4	1.61956
$996$	0	0
$997$	41490.0	1.31795	0.658977	−	0.752163i	$-0.270989\pi$
$997$	41490.0	1.31795	0.658977	+	0.752163i	$0.270989\pi$
$998$	0	0
$999$	8045.81	0.254813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bd.1.1		7
4.3	odd	2		616.4.a.j.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.j.1.7	✓	7	4.3	odd	2
1232.4.a.bd.1.1		7	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-9.14775 q^{3} +12.3118 q^{5} +7.00000 q^{7} +56.6814 q^{9} +11.0000 q^{11} +80.1862 q^{13} -112.625 q^{15} +68.2069 q^{17} +104.118 q^{19} -64.0343 q^{21} +184.098 q^{23} +26.5807 q^{25} -271.518 q^{27} +273.783 q^{29} -55.4648 q^{31} -100.625 q^{33} +86.1827 q^{35} -29.6327 q^{37} -733.523 q^{39} +188.501 q^{41} -65.0808 q^{43} +697.851 q^{45} +270.661 q^{47} +49.0000 q^{49} -623.940 q^{51} -711.859 q^{53} +135.430 q^{55} -952.441 q^{57} -464.547 q^{59} +415.489 q^{61} +396.770 q^{63} +987.237 q^{65} +78.8015 q^{67} -1684.08 q^{69} +660.432 q^{71} -1158.76 q^{73} -243.154 q^{75} +77.0000 q^{77} -1106.65 q^{79} +953.384 q^{81} -871.893 q^{83} +839.751 q^{85} -2504.50 q^{87} +64.4796 q^{89} +561.303 q^{91} +507.379 q^{93} +1281.88 q^{95} +557.755 q^{97} +623.495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 6 q^{5} + 49 q^{7} + 101 q^{9} + 77 q^{11} + 88 q^{13} + 106 q^{15} + 134 q^{17} - 14 q^{19} - 42 q^{23} + 343 q^{25} + 30 q^{27} + 482 q^{29} - 50 q^{31} + 42 q^{35} + 152 q^{37} - 72 q^{39} + 234 q^{41}+ \cdots + 1111 q^{99}+O(q^{100}) \) 7 * q + 6 * q^5 + 49 * q^7 + 101 * q^9 + 77 * q^11 + 88 * q^13 + 106 * q^15 + 134 * q^17 - 14 * q^19 - 42 * q^23 + 343 * q^25 + 30 * q^27 + 482 * q^29 - 50 * q^31 + 42 * q^35 + 152 * q^37 - 72 * q^39 + 234 * q^41 - 472 * q^43 + 1384 * q^45 - 728 * q^47 + 343 * q^49 - 1572 * q^51 - 102 * q^53 + 66 * q^55 + 452 * q^57 - 1704 * q^59 + 656 * q^61 + 707 * q^63 + 2840 * q^65 - 1126 * q^67 - 350 * q^69 + 918 * q^71 + 1094 * q^73 - 274 * q^75 + 539 * q^77 - 672 * q^79 + 4503 * q^81 - 782 * q^83 + 1716 * q^85 - 3196 * q^87 - 464 * q^89 + 616 * q^91 + 1826 * q^93 + 1796 * q^95 + 3000 * q^97 + 1111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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