LMFDB - Embedded newform 1161.4.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1161,4,Mod(1,1161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1161, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1161.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1161 = 3^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1161.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:	$68.5012175167$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	1161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.17639 q^{2} +9.44227 q^{4} -10.0927 q^{5} -15.5108 q^{7} +6.02347 q^{8} +O(q^{10})$	\(q+4.17639 q^{2} +9.44227 q^{4} -10.0927 q^{5} -15.5108 q^{7} +6.02347 q^{8} -42.1510 q^{10} -31.6635 q^{11} +9.63159 q^{13} -64.7792 q^{14} -50.3817 q^{16} +128.277 q^{17} +129.242 q^{19} -95.2979 q^{20} -132.239 q^{22} -100.964 q^{23} -23.1376 q^{25} +40.2253 q^{26} -146.457 q^{28} +281.950 q^{29} +186.047 q^{31} -258.602 q^{32} +535.734 q^{34} +156.546 q^{35} -30.2943 q^{37} +539.765 q^{38} -60.7931 q^{40} +118.551 q^{41} -43.0000 q^{43} -298.975 q^{44} -421.664 q^{46} +386.321 q^{47} -102.416 q^{49} -96.6318 q^{50} +90.9440 q^{52} -49.4232 q^{53} +319.570 q^{55} -93.4288 q^{56} +1177.53 q^{58} +287.325 q^{59} +306.393 q^{61} +777.006 q^{62} -676.969 q^{64} -97.2086 q^{65} +406.767 q^{67} +1211.22 q^{68} +653.796 q^{70} +1191.55 q^{71} +358.827 q^{73} -126.521 q^{74} +1220.34 q^{76} +491.125 q^{77} -1065.27 q^{79} +508.487 q^{80} +495.115 q^{82} -1307.45 q^{83} -1294.66 q^{85} -179.585 q^{86} -190.724 q^{88} +808.245 q^{89} -149.393 q^{91} -953.325 q^{92} +1613.43 q^{94} -1304.40 q^{95} +1083.78 q^{97} -427.728 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + q^{2} + 87 q^{4} + 3 q^{5} + 3 q^{7} + 69 q^{8}+O(q^{10}) $ 21 * q + q^2 + 87 * q^4 + 3 * q^5 + 3 * q^7 + 69 * q^8	$ 21 q + q^{2} + 87 q^{4} + 3 q^{5} + 3 q^{7} + 69 q^{8} - 9 q^{10} + 129 q^{11} + 54 q^{13} + 184 q^{14} + 111 q^{16} + 68 q^{17} + 78 q^{19} + 123 q^{20} - 51 q^{22} + 586 q^{23} + 402 q^{25} + 125 q^{26} - 6 q^{28} + 556 q^{29} - 111 q^{31} + 758 q^{32} - 420 q^{34} + 1409 q^{35} + 330 q^{37} + 1067 q^{38} + 180 q^{40} + 678 q^{41} - 903 q^{43} + 1510 q^{44} - 1062 q^{46} + 1580 q^{47} + 1218 q^{49} + 2054 q^{50} + 717 q^{52} + 1069 q^{53} - 2259 q^{55} + 2823 q^{56} + 666 q^{58} + 2248 q^{59} - 162 q^{61} + 3211 q^{62} - 2025 q^{64} + 910 q^{65} - 216 q^{67} + 1868 q^{68} + 3636 q^{70} + 3088 q^{71} - 1605 q^{73} + 2516 q^{74} + 1263 q^{76} + 891 q^{77} + 2094 q^{79} + 5251 q^{80} - 966 q^{82} + 1993 q^{83} - 114 q^{85} - 43 q^{86} + 846 q^{88} + 2494 q^{89} - 2382 q^{91} + 8034 q^{92} - 1062 q^{94} + 4798 q^{95} + 2565 q^{97} + 3103 q^{98}+O(q^{100}) $ 21 * q + q^2 + 87 * q^4 + 3 * q^5 + 3 * q^7 + 69 * q^8 - 9 * q^10 + 129 * q^11 + 54 * q^13 + 184 * q^14 + 111 * q^16 + 68 * q^17 + 78 * q^19 + 123 * q^20 - 51 * q^22 + 586 * q^23 + 402 * q^25 + 125 * q^26 - 6 * q^28 + 556 * q^29 - 111 * q^31 + 758 * q^32 - 420 * q^34 + 1409 * q^35 + 330 * q^37 + 1067 * q^38 + 180 * q^40 + 678 * q^41 - 903 * q^43 + 1510 * q^44 - 1062 * q^46 + 1580 * q^47 + 1218 * q^49 + 2054 * q^50 + 717 * q^52 + 1069 * q^53 - 2259 * q^55 + 2823 * q^56 + 666 * q^58 + 2248 * q^59 - 162 * q^61 + 3211 * q^62 - 2025 * q^64 + 910 * q^65 - 216 * q^67 + 1868 * q^68 + 3636 * q^70 + 3088 * q^71 - 1605 * q^73 + 2516 * q^74 + 1263 * q^76 + 891 * q^77 + 2094 * q^79 + 5251 * q^80 - 966 * q^82 + 1993 * q^83 - 114 * q^85 - 43 * q^86 + 846 * q^88 + 2494 * q^89 - 2382 * q^91 + 8034 * q^92 - 1062 * q^94 + 4798 * q^95 + 2565 * q^97 + 3103 * q^98

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 $

$2$	4.17639	1.47658	0.738289	−	0.674484i	$-0.235634\pi$
$2$	4.17639	1.47658	0.738289	+	0.674484i	$0.235634\pi$
$3$	0	0
$3$	0	0
$4$	9.44227	1.18028
$5$	−10.0927	−0.902718	−0.451359	−	0.892343i	$-0.649061\pi$
$5$	−10.0927	−0.902718	−0.451359	+	0.892343i	$0.649061\pi$
$6$	0	0
$7$	−15.5108	−0.837504	−0.418752	−	0.908101i	$-0.637532\pi$
$7$	−15.5108	−0.837504	−0.418752	+	0.908101i	$0.637532\pi$
$8$	6.02347	0.266202
$9$	0	0
$10$	−42.1510	−1.33293
$11$	−31.6635	−0.867900	−0.433950	−	0.900937i	$-0.642880\pi$
$11$	−31.6635	−0.867900	−0.433950	+	0.900937i	$0.642880\pi$
$12$	0	0
$13$	9.63159	0.205486	0.102743	−	0.994708i	$-0.467238\pi$
$13$	9.63159	0.205486	0.102743	+	0.994708i	$0.467238\pi$
$14$	−64.7792	−1.23664
$15$	0	0
$16$	−50.3817	−0.787215
$17$	128.277	1.83010	0.915049	−	0.403342i	$-0.132152\pi$
$17$	128.277	1.83010	0.915049	+	0.403342i	$0.132152\pi$
$18$	0	0
$19$	129.242	1.56053	0.780266	−	0.625448i	$-0.215084\pi$
$19$	129.242	1.56053	0.780266	+	0.625448i	$0.215084\pi$
$20$	−95.2979	−1.06546
$21$	0	0
$22$	−132.239	−1.28152
$23$	−100.964	−0.915320	−0.457660	−	0.889127i	$-0.651312\pi$
$23$	−100.964	−0.915320	−0.457660	+	0.889127i	$0.651312\pi$
$24$	0	0
$25$	−23.1376	−0.185101
$26$	40.2253	0.303417
$27$	0	0
$28$	−146.457	−0.988492
$29$	281.950	1.80541	0.902703	−	0.430264i	$-0.141579\pi$
$29$	281.950	1.80541	0.902703	+	0.430264i	$0.141579\pi$
$30$	0	0
$31$	186.047	1.07790	0.538952	−	0.842336i	$-0.318820\pi$
$31$	186.047	1.07790	0.538952	+	0.842336i	$0.318820\pi$
$32$	−258.602	−1.42859
$33$	0	0
$34$	535.734	2.70228
$35$	156.546	0.756029
$36$	0	0
$37$	−30.2943	−0.134604	−0.0673021	−	0.997733i	$-0.521439\pi$
$37$	−30.2943	−0.134604	−0.0673021	+	0.997733i	$0.521439\pi$
$38$	539.765	2.30425
$39$	0	0
$40$	−60.7931	−0.240306
$41$	118.551	0.451574	0.225787	−	0.974177i	$-0.427505\pi$
$41$	118.551	0.451574	0.225787	+	0.974177i	$0.427505\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−298.975	−1.02437
$45$	0	0
$46$	−421.664	−1.35154
$47$	386.321	1.19895	0.599476	−	0.800393i	$-0.295376\pi$
$47$	386.321	1.19895	0.599476	+	0.800393i	$0.295376\pi$
$48$	0	0
$49$	−102.416	−0.298588
$50$	−96.6318	−0.273316
$51$	0	0
$52$	90.9440	0.242532
$53$	−49.4232	−0.128091	−0.0640453	−	0.997947i	$-0.520400\pi$
$53$	−49.4232	−0.128091	−0.0640453	+	0.997947i	$0.520400\pi$
$54$	0	0
$55$	319.570	0.783468
$56$	−93.4288	−0.222946
$57$	0	0
$58$	1177.53	2.66582
$59$	287.325	0.634009	0.317004	−	0.948424i	$-0.397323\pi$
$59$	287.325	0.634009	0.317004	+	0.948424i	$0.397323\pi$
$60$	0	0
$61$	306.393	0.643108	0.321554	−	0.946891i	$-0.395795\pi$
$61$	306.393	0.643108	0.321554	+	0.946891i	$0.395795\pi$
$62$	777.006	1.59161
$63$	0	0
$64$	−676.969	−1.32221
$65$	−97.2086	−0.185496
$66$	0	0
$67$	406.767	0.741710	0.370855	−	0.928691i	$-0.379065\pi$
$67$	406.767	0.741710	0.370855	+	0.928691i	$0.379065\pi$
$68$	1211.22	2.16003
$69$	0	0
$70$	653.796	1.11634
$71$	1191.55	1.99170	0.995852	−	0.0909928i	$-0.0290040\pi$
$71$	1191.55	1.99170	0.995852	+	0.0909928i	$0.0290040\pi$
$72$	0	0
$73$	358.827	0.575309	0.287655	−	0.957734i	$-0.407124\pi$
$73$	358.827	0.575309	0.287655	+	0.957734i	$0.407124\pi$
$74$	−126.521	−0.198754
$75$	0	0
$76$	1220.34	1.84187
$77$	491.125	0.726869
$78$	0	0
$79$	−1065.27	−1.51711	−0.758556	−	0.651608i	$-0.774095\pi$
$79$	−1065.27	−1.51711	−0.758556	+	0.651608i	$0.774095\pi$
$80$	508.487	0.710632
$81$	0	0
$82$	495.115	0.666784
$83$	−1307.45	−1.72905	−0.864523	−	0.502593i	$-0.832379\pi$
$83$	−1307.45	−1.72905	−0.864523	+	0.502593i	$0.832379\pi$
$84$	0	0
$85$	−1294.66	−1.65206
$86$	−179.585	−0.225176
$87$	0	0
$88$	−190.724	−0.231037
$89$	808.245	0.962627	0.481313	−	0.876549i	$-0.340160\pi$
$89$	808.245	0.962627	0.481313	+	0.876549i	$0.340160\pi$
$90$	0	0
$91$	−149.393	−0.172095
$92$	−953.325	−1.08034
$93$	0	0
$94$	1613.43	1.77035
$95$	−1304.40	−1.40872
$96$	0	0
$97$	1083.78	1.13444	0.567222	−	0.823565i	$-0.308018\pi$
$97$	1083.78	1.13444	0.567222	+	0.823565i	$0.308018\pi$
$98$	−427.728	−0.440888
$99$	0	0
$100$	−218.471	−0.218471
$101$	−185.482	−0.182735	−0.0913673	−	0.995817i	$-0.529124\pi$
$101$	−185.482	−0.182735	−0.0913673	+	0.995817i	$0.529124\pi$
$102$	0	0
$103$	1147.19	1.09744	0.548719	−	0.836007i	$-0.315116\pi$
$103$	1147.19	1.09744	0.548719	+	0.836007i	$0.315116\pi$
$104$	58.0156	0.0547009
$105$	0	0
$106$	−206.411	−0.189136
$107$	−2172.77	−1.96308	−0.981539	−	0.191263i	$-0.938742\pi$
$107$	−2172.77	−1.96308	−0.981539	+	0.191263i	$0.938742\pi$
$108$	0	0
$109$	−1121.51	−0.985514	−0.492757	−	0.870167i	$-0.664011\pi$
$109$	−1121.51	−0.985514	−0.492757	+	0.870167i	$0.664011\pi$
$110$	1334.65	1.15685
$111$	0	0
$112$	781.460	0.659295
$113$	−869.624	−0.723959	−0.361980	−	0.932186i	$-0.617899\pi$
$113$	−869.624	−0.723959	−0.361980	+	0.932186i	$0.617899\pi$
$114$	0	0
$115$	1018.99	0.826275
$116$	2662.25	2.13089
$117$	0	0
$118$	1199.98	0.936163
$119$	−1989.67	−1.53271
$120$	0	0
$121$	−328.425	−0.246750
$122$	1279.62	0.949599
$123$	0	0
$124$	1756.71	1.27223
$125$	1495.11	1.06981
$126$	0	0
$127$	651.419	0.455150	0.227575	−	0.973761i	$-0.426920\pi$
$127$	651.419	0.455150	0.227575	+	0.973761i	$0.426920\pi$
$128$	−758.475	−0.523753
$129$	0	0
$130$	−405.981	−0.273899
$131$	1609.32	1.07334	0.536669	−	0.843793i	$-0.319682\pi$
$131$	1609.32	1.07334	0.536669	+	0.843793i	$0.319682\pi$
$132$	0	0
$133$	−2004.64	−1.30695
$134$	1698.82	1.09519
$135$	0	0
$136$	772.671	0.487177
$137$	−33.3763	−0.0208141	−0.0104070	−	0.999946i	$-0.503313\pi$
$137$	−33.3763	−0.0208141	−0.0104070	+	0.999946i	$0.503313\pi$
$138$	0	0
$139$	933.296	0.569505	0.284752	−	0.958601i	$-0.408089\pi$
$139$	933.296	0.569505	0.284752	+	0.958601i	$0.408089\pi$
$140$	1478.14	0.892329
$141$	0	0
$142$	4976.38	2.94091
$143$	−304.969	−0.178341
$144$	0	0
$145$	−2845.63	−1.62977
$146$	1498.60	0.849489
$147$	0	0
$148$	−286.047	−0.158871
$149$	−1792.88	−0.985759	−0.492880	−	0.870098i	$-0.664056\pi$
$149$	−1792.88	−0.985759	−0.492880	+	0.870098i	$0.664056\pi$
$150$	0	0
$151$	2462.06	1.32688	0.663442	−	0.748227i	$-0.269095\pi$
$151$	2462.06	1.32688	0.663442	+	0.748227i	$0.269095\pi$
$152$	778.485	0.415417
$153$	0	0
$154$	2051.13	1.07328
$155$	−1877.72	−0.973044
$156$	0	0
$157$	−2954.53	−1.50189	−0.750945	−	0.660364i	$-0.770402\pi$
$157$	−2954.53	−1.50189	−0.750945	+	0.660364i	$0.770402\pi$
$158$	−4448.97	−2.24013
$159$	0	0
$160$	2609.99	1.28961
$161$	1566.02	0.766584
$162$	0	0
$163$	2215.36	1.06454	0.532271	−	0.846574i	$-0.321339\pi$
$163$	2215.36	1.06454	0.532271	+	0.846574i	$0.321339\pi$
$164$	1119.39	0.532985
$165$	0	0
$166$	−5460.41	−2.55307
$167$	3179.56	1.47330	0.736652	−	0.676273i	$-0.236406\pi$
$167$	3179.56	1.47330	0.736652	+	0.676273i	$0.236406\pi$
$168$	0	0
$169$	−2104.23	−0.957775
$170$	−5407.00	−2.43940
$171$	0	0
$172$	−406.017	−0.179992
$173$	−2650.84	−1.16497	−0.582485	−	0.812841i	$-0.697920\pi$
$173$	−2650.84	−1.16497	−0.582485	+	0.812841i	$0.697920\pi$
$174$	0	0
$175$	358.883	0.155023
$176$	1595.26	0.683223
$177$	0	0
$178$	3375.55	1.42139
$179$	3005.71	1.25507	0.627535	−	0.778588i	$-0.284064\pi$
$179$	3005.71	1.25507	0.627535	+	0.778588i	$0.284064\pi$
$180$	0	0
$181$	−668.298	−0.274443	−0.137222	−	0.990540i	$-0.543817\pi$
$181$	−668.298	−0.274443	−0.137222	+	0.990540i	$0.543817\pi$
$182$	−623.926	−0.254112
$183$	0	0
$184$	−608.151	−0.243660
$185$	305.751	0.121510
$186$	0	0
$187$	−4061.69	−1.58834
$188$	3647.75	1.41510
$189$	0	0
$190$	−5447.68	−2.08009
$191$	5033.53	1.90688	0.953439	−	0.301586i	$-0.0975162\pi$
$191$	5033.53	1.90688	0.953439	+	0.301586i	$0.0975162\pi$
$192$	0	0
$193$	−3617.71	−1.34927	−0.674634	−	0.738152i	$-0.735699\pi$
$193$	−3617.71	−1.34927	−0.674634	+	0.738152i	$0.735699\pi$
$194$	4526.29	1.67510
$195$	0	0
$196$	−967.035	−0.352418
$197$	4735.77	1.71274	0.856369	−	0.516364i	$-0.172715\pi$
$197$	4735.77	1.71274	0.856369	+	0.516364i	$0.172715\pi$
$198$	0	0
$199$	−145.044	−0.0516679	−0.0258340	−	0.999666i	$-0.508224\pi$
$199$	−145.044	−0.0516679	−0.0258340	+	0.999666i	$0.508224\pi$
$200$	−139.369	−0.0492743
$201$	0	0
$202$	−774.648	−0.269822
$203$	−4373.26	−1.51203
$204$	0	0
$205$	−1196.50	−0.407644
$206$	4791.12	1.62045
$207$	0	0
$208$	−485.256	−0.161762
$209$	−4092.24	−1.35439
$210$	0	0
$211$	723.090	0.235922	0.117961	−	0.993018i	$-0.462364\pi$
$211$	723.090	0.235922	0.117961	+	0.993018i	$0.462364\pi$
$212$	−466.667	−0.151183
$213$	0	0
$214$	−9074.33	−2.89864
$215$	433.986	0.137663
$216$	0	0
$217$	−2885.74	−0.902749
$218$	−4683.86	−1.45519
$219$	0	0
$220$	3017.46	0.924714
$221$	1235.51	0.376060
$222$	0	0
$223$	−5721.92	−1.71824	−0.859122	−	0.511771i	$-0.828990\pi$
$223$	−5721.92	−1.71824	−0.859122	+	0.511771i	$0.828990\pi$
$224$	4011.12	1.19645
$225$	0	0
$226$	−3631.89	−1.06898
$227$	4611.86	1.34846	0.674228	−	0.738523i	$-0.264476\pi$
$227$	4611.86	1.34846	0.674228	+	0.738523i	$0.264476\pi$
$228$	0	0
$229$	2946.42	0.850240	0.425120	−	0.905137i	$-0.360232\pi$
$229$	2946.42	0.850240	0.425120	+	0.905137i	$0.360232\pi$
$230$	4255.72	1.22006
$231$	0	0
$232$	1698.32	0.480604
$233$	1574.02	0.442566	0.221283	−	0.975210i	$-0.428976\pi$
$233$	1574.02	0.442566	0.221283	+	0.975210i	$0.428976\pi$
$234$	0	0
$235$	−3899.02	−1.08232
$236$	2713.00	0.748310
$237$	0	0
$238$	−8309.66	−2.26317
$239$	2887.44	0.781477	0.390739	−	0.920502i	$-0.372220\pi$
$239$	2887.44	0.781477	0.390739	+	0.920502i	$0.372220\pi$
$240$	0	0
$241$	2236.69	0.597832	0.298916	−	0.954279i	$-0.403375\pi$
$241$	2236.69	0.597832	0.298916	+	0.954279i	$0.403375\pi$
$242$	−1371.63	−0.364346
$243$	0	0
$244$	2893.04	0.759049
$245$	1033.65	0.269540
$246$	0	0
$247$	1244.80	0.320668
$248$	1120.65	0.286941
$249$	0	0
$250$	6244.16	1.57966
$251$	6507.21	1.63638	0.818190	−	0.574948i	$-0.194978\pi$
$251$	6507.21	1.63638	0.818190	+	0.574948i	$0.194978\pi$
$252$	0	0
$253$	3196.86	0.794406
$254$	2720.58	0.672065
$255$	0	0
$256$	2248.06	0.548843
$257$	2085.45	0.506173	0.253087	−	0.967444i	$-0.418554\pi$
$257$	2085.45	0.506173	0.253087	+	0.967444i	$0.418554\pi$
$258$	0	0
$259$	469.889	0.112731
$260$	−917.870	−0.218938
$261$	0	0
$262$	6721.17	1.58487
$263$	−7267.24	−1.70387	−0.851934	−	0.523648i	$-0.824571\pi$
$263$	−7267.24	−1.70387	−0.851934	+	0.523648i	$0.824571\pi$
$264$	0	0
$265$	498.813	0.115630
$266$	−8372.18	−1.92982
$267$	0	0
$268$	3840.81	0.875428
$269$	−190.159	−0.0431011	−0.0215505	−	0.999768i	$-0.506860\pi$
$269$	−190.159	−0.0431011	−0.0215505	+	0.999768i	$0.506860\pi$
$270$	0	0
$271$	−6655.02	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$271$	−6655.02	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$272$	−6462.80	−1.44068
$273$	0	0
$274$	−139.392	−0.0307336
$275$	732.617	0.160649
$276$	0	0
$277$	−6655.59	−1.44367	−0.721833	−	0.692067i	$-0.756700\pi$
$277$	−6655.59	−1.44367	−0.721833	+	0.692067i	$0.756700\pi$
$278$	3897.81	0.840918
$279$	0	0
$280$	942.948	0.201257
$281$	−1211.81	−0.257261	−0.128631	−	0.991693i	$-0.541058\pi$
$281$	−1211.81	−0.257261	−0.128631	+	0.991693i	$0.541058\pi$
$282$	0	0
$283$	838.483	0.176122	0.0880612	−	0.996115i	$-0.471933\pi$
$283$	838.483	0.176122	0.0880612	+	0.996115i	$0.471933\pi$
$284$	11250.9	2.35077
$285$	0	0
$286$	−1273.67	−0.263335
$287$	−1838.82	−0.378195
$288$	0	0
$289$	11541.9	2.34926
$290$	−11884.5	−2.40649
$291$	0	0
$292$	3388.14	0.679028
$293$	1955.40	0.389883	0.194942	−	0.980815i	$-0.437548\pi$
$293$	1955.40	0.389883	0.194942	+	0.980815i	$0.437548\pi$
$294$	0	0
$295$	−2899.88	−0.572331
$296$	−182.477	−0.0358320
$297$	0	0
$298$	−7487.76	−1.45555
$299$	−972.439	−0.188086
$300$	0	0
$301$	666.964	0.127718
$302$	10282.5	1.95925
$303$	0	0
$304$	−6511.43	−1.22847
$305$	−3092.33	−0.580545
$306$	0	0
$307$	−2160.95	−0.401733	−0.200867	−	0.979619i	$-0.564376\pi$
$307$	−2160.95	−0.401733	−0.200867	+	0.979619i	$0.564376\pi$
$308$	4637.34	0.857911
$309$	0	0
$310$	−7842.08	−1.43677
$311$	5814.29	1.06012	0.530061	−	0.847960i	$-0.322169\pi$
$311$	5814.29	1.06012	0.530061	+	0.847960i	$0.322169\pi$
$312$	0	0
$313$	−1285.35	−0.232115	−0.116058	−	0.993242i	$-0.537026\pi$
$313$	−1285.35	−0.232115	−0.116058	+	0.993242i	$0.537026\pi$
$314$	−12339.3	−2.21766
$315$	0	0
$316$	−10058.5	−1.79062
$317$	321.395	0.0569442	0.0284721	−	0.999595i	$-0.490936\pi$
$317$	321.395	0.0569442	0.0284721	+	0.999595i	$0.490936\pi$
$318$	0	0
$319$	−8927.51	−1.56691
$320$	6832.44	1.19358
$321$	0	0
$322$	6540.33	1.13192
$323$	16578.7	2.85593
$324$	0	0
$325$	−222.852	−0.0380357
$326$	9252.21	1.57188
$327$	0	0
$328$	714.088	0.120210
$329$	−5992.15	−1.00413
$330$	0	0
$331$	672.505	0.111674	0.0558371	−	0.998440i	$-0.482217\pi$
$331$	672.505	0.111674	0.0558371	+	0.998440i	$0.482217\pi$
$332$	−12345.3	−2.04076
$333$	0	0
$334$	13279.1	2.17545
$335$	−4105.38	−0.669554
$336$	0	0
$337$	−8206.88	−1.32658	−0.663290	−	0.748363i	$-0.730840\pi$
$337$	−8206.88	−1.32658	−0.663290	+	0.748363i	$0.730840\pi$
$338$	−8788.10	−1.41423
$339$	0	0
$340$	−12224.5	−1.94990
$341$	−5890.90	−0.935513
$342$	0	0
$343$	6908.74	1.08757
$344$	−259.009	−0.0405955
$345$	0	0
$346$	−11071.0	−1.72017
$347$	−4338.04	−0.671119	−0.335559	−	0.942019i	$-0.608925\pi$
$347$	−4338.04	−0.671119	−0.335559	+	0.942019i	$0.608925\pi$
$348$	0	0
$349$	3803.24	0.583332	0.291666	−	0.956520i	$-0.405790\pi$
$349$	3803.24	0.583332	0.291666	+	0.956520i	$0.405790\pi$
$350$	1498.83	0.228903
$351$	0	0
$352$	8188.23	1.23987
$353$	6041.29	0.910894	0.455447	−	0.890263i	$-0.349479\pi$
$353$	6041.29	0.910894	0.455447	+	0.890263i	$0.349479\pi$
$354$	0	0
$355$	−12025.9	−1.79795
$356$	7631.67	1.13617
$357$	0	0
$358$	12553.0	1.85321
$359$	−3321.58	−0.488318	−0.244159	−	0.969735i	$-0.578512\pi$
$359$	−3321.58	−0.488318	−0.244159	+	0.969735i	$0.578512\pi$
$360$	0	0
$361$	9844.45	1.43526
$362$	−2791.08	−0.405237
$363$	0	0
$364$	−1410.61	−0.203121
$365$	−3621.53	−0.519342
$366$	0	0
$367$	9199.50	1.30847	0.654237	−	0.756289i	$-0.272990\pi$
$367$	9199.50	1.30847	0.654237	+	0.756289i	$0.272990\pi$
$368$	5086.72	0.720553
$369$	0	0
$370$	1276.94	0.179418
$371$	766.593	0.107276
$372$	0	0
$373$	1331.32	0.184808	0.0924038	−	0.995722i	$-0.470545\pi$
$373$	1331.32	0.184808	0.0924038	+	0.995722i	$0.470545\pi$
$374$	−16963.2	−2.34531
$375$	0	0
$376$	2327.00	0.319164
$377$	2715.62	0.370986
$378$	0	0
$379$	−9002.23	−1.22009	−0.610044	−	0.792367i	$-0.708848\pi$
$379$	−9002.23	−1.22009	−0.610044	+	0.792367i	$0.708848\pi$
$380$	−12316.5	−1.66269
$381$	0	0
$382$	21022.0	2.81565
$383$	6403.08	0.854262	0.427131	−	0.904190i	$-0.359524\pi$
$383$	6403.08	0.854262	0.427131	+	0.904190i	$0.359524\pi$
$384$	0	0
$385$	−4956.77	−0.656158
$386$	−15109.0	−1.99230
$387$	0	0
$388$	10233.3	1.33897
$389$	1133.99	0.147804	0.0739018	−	0.997266i	$-0.476455\pi$
$389$	1133.99	0.147804	0.0739018	+	0.997266i	$0.476455\pi$
$390$	0	0
$391$	−12951.3	−1.67513
$392$	−616.897	−0.0794847
$393$	0	0
$394$	19778.4	2.52899
$395$	10751.4	1.36952
$396$	0	0
$397$	5945.81	0.751667	0.375833	−	0.926687i	$-0.377356\pi$
$397$	5945.81	0.751667	0.375833	+	0.926687i	$0.377356\pi$
$398$	−605.762	−0.0762917
$399$	0	0
$400$	1165.71	0.145714
$401$	7735.31	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$401$	7735.31	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$402$	0	0
$403$	1791.93	0.221495
$404$	−1751.37	−0.215679
$405$	0	0
$406$	−18264.5	−2.23264
$407$	959.223	0.116823
$408$	0	0
$409$	−7186.25	−0.868795	−0.434398	−	0.900721i	$-0.643039\pi$
$409$	−7186.25	−0.868795	−0.434398	+	0.900721i	$0.643039\pi$
$410$	−4997.04	−0.601918
$411$	0	0
$412$	10832.1	1.29529
$413$	−4456.63	−0.530985
$414$	0	0
$415$	13195.6	1.56084
$416$	−2490.74	−0.293555
$417$	0	0
$418$	−17090.8	−1.99986
$419$	14289.5	1.66607	0.833037	−	0.553217i	$-0.186600\pi$
$419$	14289.5	1.66607	0.833037	+	0.553217i	$0.186600\pi$
$420$	0	0
$421$	2982.48	0.345267	0.172633	−	0.984986i	$-0.444772\pi$
$421$	2982.48	0.345267	0.172633	+	0.984986i	$0.444772\pi$
$422$	3019.91	0.348357
$423$	0	0
$424$	−297.699	−0.0340980
$425$	−2968.02	−0.338753
$426$	0	0
$427$	−4752.39	−0.538605
$428$	−20515.8	−2.31699
$429$	0	0
$430$	1812.50	0.203270
$431$	−5917.81	−0.661371	−0.330685	−	0.943741i	$-0.607280\pi$
$431$	−5917.81	−0.661371	−0.330685	+	0.943741i	$0.607280\pi$
$432$	0	0
$433$	−3785.46	−0.420133	−0.210067	−	0.977687i	$-0.567368\pi$
$433$	−3785.46	−0.420133	−0.210067	+	0.977687i	$0.567368\pi$
$434$	−12052.0	−1.33298
$435$	0	0
$436$	−10589.6	−1.16319
$437$	−13048.7	−1.42839
$438$	0	0
$439$	−11034.5	−1.19966	−0.599829	−	0.800128i	$-0.704765\pi$
$439$	−11034.5	−1.19966	−0.599829	+	0.800128i	$0.704765\pi$
$440$	1924.92	0.208561
$441$	0	0
$442$	5159.97	0.555282
$443$	−5346.24	−0.573380	−0.286690	−	0.958023i	$-0.592555\pi$
$443$	−5346.24	−0.573380	−0.286690	+	0.958023i	$0.592555\pi$
$444$	0	0
$445$	−8157.37	−0.868980
$446$	−23897.0	−2.53712
$447$	0	0
$448$	10500.3	1.10735
$449$	−10990.6	−1.15518	−0.577592	−	0.816326i	$-0.696008\pi$
$449$	−10990.6	−1.15518	−0.577592	+	0.816326i	$0.696008\pi$
$450$	0	0
$451$	−3753.73	−0.391921
$452$	−8211.23	−0.854477
$453$	0	0
$454$	19260.9	1.99110
$455$	1507.78	0.155354
$456$	0	0
$457$	−7379.33	−0.755340	−0.377670	−	0.925940i	$-0.623275\pi$
$457$	−7379.33	−0.755340	−0.377670	+	0.925940i	$0.623275\pi$
$458$	12305.4	1.25545
$459$	0	0
$460$	9621.61	0.975239
$461$	13922.6	1.40659	0.703296	−	0.710897i	$-0.251711\pi$
$461$	13922.6	1.40659	0.703296	+	0.710897i	$0.251711\pi$
$462$	0	0
$463$	6298.10	0.632176	0.316088	−	0.948730i	$-0.397630\pi$
$463$	6298.10	0.632176	0.316088	+	0.948730i	$0.397630\pi$
$464$	−14205.1	−1.42124
$465$	0	0
$466$	6573.75	0.653483
$467$	10801.6	1.07032	0.535159	−	0.844752i	$-0.320252\pi$
$467$	10801.6	1.07032	0.535159	+	0.844752i	$0.320252\pi$
$468$	0	0
$469$	−6309.28	−0.621185
$470$	−16283.9	−1.59812
$471$	0	0
$472$	1730.69	0.168775
$473$	1361.53	0.132353
$474$	0	0
$475$	−2990.35	−0.288856
$476$	−18787.0	−1.80904
$477$	0	0
$478$	12059.1	1.15391
$479$	−8630.57	−0.823259	−0.411629	−	0.911351i	$-0.635040\pi$
$479$	−8630.57	−0.823259	−0.411629	+	0.911351i	$0.635040\pi$
$480$	0	0
$481$	−291.782	−0.0276593
$482$	9341.28	0.882746
$483$	0	0
$484$	−3101.07	−0.291235
$485$	−10938.2	−1.02408
$486$	0	0
$487$	−13855.0	−1.28918	−0.644591	−	0.764527i	$-0.722972\pi$
$487$	−13855.0	−1.28918	−0.644591	+	0.764527i	$0.722972\pi$
$488$	1845.55	0.171197
$489$	0	0
$490$	4316.92	0.397997
$491$	−14749.5	−1.35567	−0.677835	−	0.735214i	$-0.737082\pi$
$491$	−14749.5	−1.35567	−0.677835	+	0.735214i	$0.737082\pi$
$492$	0	0
$493$	36167.6	3.30407
$494$	5198.79	0.473491
$495$	0	0
$496$	−9373.37	−0.848542
$497$	−18481.9	−1.66806
$498$	0	0
$499$	−11013.6	−0.988053	−0.494027	−	0.869447i	$-0.664475\pi$
$499$	−11013.6	−0.988053	−0.494027	+	0.869447i	$0.664475\pi$
$500$	14117.2	1.26268
$501$	0	0
$502$	27176.7	2.41624
$503$	1140.04	0.101058	0.0505288	−	0.998723i	$-0.483909\pi$
$503$	1140.04	0.101058	0.0505288	+	0.998723i	$0.483909\pi$
$504$	0	0
$505$	1872.02	0.164958
$506$	13351.3	1.17300
$507$	0	0
$508$	6150.87	0.537206
$509$	4900.78	0.426765	0.213382	−	0.976969i	$-0.431552\pi$
$509$	4900.78	0.426765	0.213382	+	0.976969i	$0.431552\pi$
$510$	0	0
$511$	−5565.70	−0.481824
$512$	15456.6	1.33416
$513$	0	0
$514$	8709.64	0.747404
$515$	−11578.2	−0.990676
$516$	0	0
$517$	−12232.3	−1.04057
$518$	1962.44	0.166457
$519$	0	0
$520$	−585.534	−0.0493795
$521$	5561.51	0.467666	0.233833	−	0.972277i	$-0.424873\pi$
$521$	5561.51	0.467666	0.233833	+	0.972277i	$0.424873\pi$
$522$	0	0
$523$	−10554.3	−0.882423	−0.441212	−	0.897403i	$-0.645451\pi$
$523$	−10554.3	−0.882423	−0.441212	+	0.897403i	$0.645451\pi$
$524$	15195.7	1.26684
$525$	0	0
$526$	−30350.9	−2.51590
$527$	23865.5	1.97267
$528$	0	0
$529$	−1973.36	−0.162190
$530$	2083.24	0.170736
$531$	0	0
$532$	−18928.4	−1.54257
$533$	1141.83	0.0927922
$534$	0	0
$535$	21929.1	1.77210
$536$	2450.15	0.197445
$537$	0	0
$538$	−794.178	−0.0636421
$539$	3242.83	0.259144
$540$	0	0
$541$	13816.9	1.09803	0.549014	−	0.835813i	$-0.315003\pi$
$541$	13816.9	1.09803	0.549014	+	0.835813i	$0.315003\pi$
$542$	−27794.0	−2.20268
$543$	0	0
$544$	−33172.6	−2.61445
$545$	11319.0	0.889641
$546$	0	0
$547$	15208.0	1.18875	0.594377	−	0.804187i	$-0.297399\pi$
$547$	15208.0	1.18875	0.594377	+	0.804187i	$0.297399\pi$
$548$	−315.147	−0.0245665
$549$	0	0
$550$	3059.70	0.237211
$551$	36439.7	2.81739
$552$	0	0
$553$	16523.1	1.27059
$554$	−27796.3	−2.13169
$555$	0	0
$556$	8812.43	0.672177
$557$	17731.5	1.34885	0.674425	−	0.738343i	$-0.264392\pi$
$557$	17731.5	1.34885	0.674425	+	0.738343i	$0.264392\pi$
$558$	0	0
$559$	−414.158	−0.0313364
$560$	−7887.04	−0.595157
$561$	0	0
$562$	−5060.99	−0.379866
$563$	−10029.6	−0.750794	−0.375397	−	0.926864i	$-0.622494\pi$
$563$	−10029.6	−0.750794	−0.375397	+	0.926864i	$0.622494\pi$
$564$	0	0
$565$	8776.85	0.653531
$566$	3501.83	0.260058
$567$	0	0
$568$	7177.27	0.530196
$569$	−6679.40	−0.492118	−0.246059	−	0.969255i	$-0.579136\pi$
$569$	−6679.40	−0.492118	−0.246059	+	0.969255i	$0.579136\pi$
$570$	0	0
$571$	15686.4	1.14966	0.574828	−	0.818274i	$-0.305069\pi$
$571$	15686.4	1.14966	0.574828	+	0.818274i	$0.305069\pi$
$572$	−2879.60	−0.210493
$573$	0	0
$574$	−7679.62	−0.558434
$575$	2336.06	0.169427
$576$	0	0
$577$	5413.56	0.390588	0.195294	−	0.980745i	$-0.437434\pi$
$577$	5413.56	0.390588	0.195294	+	0.980745i	$0.437434\pi$
$578$	48203.6	3.46887
$579$	0	0
$580$	−26869.2	−1.92359
$581$	20279.5	1.44808
$582$	0	0
$583$	1564.91	0.111170
$584$	2161.39	0.153149
$585$	0	0
$586$	8166.53	0.575693
$587$	13114.7	0.922150	0.461075	−	0.887361i	$-0.347464\pi$
$587$	13114.7	0.922150	0.461075	+	0.887361i	$0.347464\pi$
$588$	0	0
$589$	24045.1	1.68210
$590$	−12111.0	−0.845091
$591$	0	0
$592$	1526.28	0.105962
$593$	−24835.0	−1.71981	−0.859907	−	0.510451i	$-0.829479\pi$
$593$	−24835.0	−1.71981	−0.859907	+	0.510451i	$0.829479\pi$
$594$	0	0
$595$	20081.1	1.38361
$596$	−16928.8	−1.16348
$597$	0	0
$598$	−4061.29	−0.277723
$599$	6090.86	0.415468	0.207734	−	0.978185i	$-0.433391\pi$
$599$	6090.86	0.415468	0.207734	+	0.978185i	$0.433391\pi$
$600$	0	0
$601$	12147.7	0.824484	0.412242	−	0.911074i	$-0.364746\pi$
$601$	12147.7	0.824484	0.412242	+	0.911074i	$0.364746\pi$
$602$	2785.50	0.188586
$603$	0	0
$604$	23247.4	1.56610
$605$	3314.69	0.222746
$606$	0	0
$607$	14032.4	0.938314	0.469157	−	0.883115i	$-0.344558\pi$
$607$	14032.4	0.938314	0.469157	+	0.883115i	$0.344558\pi$
$608$	−33422.2	−2.22935
$609$	0	0
$610$	−12914.8	−0.857220
$611$	3720.89	0.246368
$612$	0	0
$613$	28027.3	1.84668	0.923339	−	0.383986i	$-0.125449\pi$
$613$	28027.3	1.84668	0.923339	+	0.383986i	$0.125449\pi$
$614$	−9024.99	−0.593190
$615$	0	0
$616$	2958.28	0.193494
$617$	−3382.42	−0.220699	−0.110349	−	0.993893i	$-0.535197\pi$
$617$	−3382.42	−0.220699	−0.110349	+	0.993893i	$0.535197\pi$
$618$	0	0
$619$	−16285.9	−1.05749	−0.528745	−	0.848781i	$-0.677337\pi$
$619$	−16285.9	−1.05749	−0.528745	+	0.848781i	$0.677337\pi$
$620$	−17729.9	−1.14847
$621$	0	0
$622$	24282.7	1.56535
$623$	−12536.5	−0.806204
$624$	0	0
$625$	−12197.4	−0.780637
$626$	−5368.11	−0.342736
$627$	0	0
$628$	−27897.4	−1.77266
$629$	−3886.05	−0.246339
$630$	0	0
$631$	11924.9	0.752335	0.376168	−	0.926552i	$-0.377242\pi$
$631$	11924.9	0.752335	0.376168	+	0.926552i	$0.377242\pi$
$632$	−6416.60	−0.403859
$633$	0	0
$634$	1342.27	0.0840826
$635$	−6574.57	−0.410872
$636$	0	0
$637$	−986.424	−0.0613556
$638$	−37284.8	−2.31367
$639$	0	0
$640$	7655.05	0.472801
$641$	−8137.40	−0.501416	−0.250708	−	0.968063i	$-0.580663\pi$
$641$	−8137.40	−0.501416	−0.250708	+	0.968063i	$0.580663\pi$
$642$	0	0
$643$	806.867	0.0494864	0.0247432	−	0.999694i	$-0.492123\pi$
$643$	806.867	0.0494864	0.0247432	+	0.999694i	$0.492123\pi$
$644$	14786.8	0.904786
$645$	0	0
$646$	69239.2	4.21700
$647$	−9989.06	−0.606972	−0.303486	−	0.952836i	$-0.598150\pi$
$647$	−9989.06	−0.606972	−0.303486	+	0.952836i	$0.598150\pi$
$648$	0	0
$649$	−9097.70	−0.550256
$650$	−930.717	−0.0561627
$651$	0	0
$652$	20918.0	1.25646
$653$	−20378.4	−1.22124	−0.610620	−	0.791924i	$-0.709080\pi$
$653$	−20378.4	−1.22124	−0.610620	+	0.791924i	$0.709080\pi$
$654$	0	0
$655$	−16242.4	−0.968921
$656$	−5972.80	−0.355486
$657$	0	0
$658$	−25025.6	−1.48267
$659$	−14624.9	−0.864501	−0.432251	−	0.901753i	$-0.642280\pi$
$659$	−14624.9	−0.864501	−0.432251	+	0.901753i	$0.642280\pi$
$660$	0	0
$661$	−15941.2	−0.938033	−0.469017	−	0.883189i	$-0.655392\pi$
$661$	−15941.2	−0.938033	−0.469017	+	0.883189i	$0.655392\pi$
$662$	2808.64	0.164896
$663$	0	0
$664$	−7875.37	−0.460276
$665$	20232.2	1.17981
$666$	0	0
$667$	−28466.7	−1.65252
$668$	30022.2	1.73892
$669$	0	0
$670$	−17145.7	−0.988650
$671$	−9701.45	−0.558153
$672$	0	0
$673$	−1702.62	−0.0975201	−0.0487601	−	0.998811i	$-0.515527\pi$
$673$	−1702.62	−0.0975201	−0.0487601	+	0.998811i	$0.515527\pi$
$674$	−34275.2	−1.95880
$675$	0	0
$676$	−19868.7	−1.13045
$677$	21877.0	1.24195	0.620975	−	0.783831i	$-0.286737\pi$
$677$	21877.0	1.24195	0.620975	+	0.783831i	$0.286737\pi$
$678$	0	0
$679$	−16810.3	−0.950101
$680$	−7798.33	−0.439783
$681$	0	0
$682$	−24602.7	−1.38136
$683$	10536.1	0.590268	0.295134	−	0.955456i	$-0.404636\pi$
$683$	10536.1	0.590268	0.295134	+	0.955456i	$0.404636\pi$
$684$	0	0
$685$	336.856	0.0187892
$686$	28853.6	1.60588
$687$	0	0
$688$	2166.41	0.120049
$689$	−476.024	−0.0263209
$690$	0	0
$691$	−17238.3	−0.949024	−0.474512	−	0.880249i	$-0.657375\pi$
$691$	−17238.3	−0.949024	−0.474512	+	0.880249i	$0.657375\pi$
$692$	−25030.0	−1.37500
$693$	0	0
$694$	−18117.4	−0.990960
$695$	−9419.47	−0.514102
$696$	0	0
$697$	15207.3	0.826425
$698$	15883.8	0.861336
$699$	0	0
$700$	3388.66	0.182971
$701$	27913.4	1.50396	0.751979	−	0.659187i	$-0.229099\pi$
$701$	27913.4	1.50396	0.751979	+	0.659187i	$0.229099\pi$
$702$	0	0
$703$	−3915.29	−0.210054
$704$	21435.2	1.14754
$705$	0	0
$706$	25230.8	1.34501
$707$	2876.98	0.153041
$708$	0	0
$709$	−794.124	−0.0420648	−0.0210324	−	0.999779i	$-0.506695\pi$
$709$	−794.124	−0.0420648	−0.0210324	+	0.999779i	$0.506695\pi$
$710$	−50225.1	−2.65481
$711$	0	0
$712$	4868.44	0.256254
$713$	−18784.0	−0.986628
$714$	0	0
$715$	3077.96	0.160992
$716$	28380.8	1.48134
$717$	0	0
$718$	−13872.2	−0.721040
$719$	23826.4	1.23585	0.617923	−	0.786239i	$-0.287974\pi$
$719$	23826.4	1.23585	0.617923	+	0.786239i	$0.287974\pi$
$720$	0	0
$721$	−17793.8	−0.919108
$722$	41114.3	2.11927
$723$	0	0
$724$	−6310.25	−0.323921
$725$	−6523.65	−0.334182
$726$	0	0
$727$	−9038.11	−0.461080	−0.230540	−	0.973063i	$-0.574049\pi$
$727$	−9038.11	−0.461080	−0.230540	+	0.973063i	$0.574049\pi$
$728$	−899.868	−0.0458122
$729$	0	0
$730$	−15125.0	−0.766849
$731$	−5515.90	−0.279087
$732$	0	0
$733$	−9282.63	−0.467751	−0.233876	−	0.972267i	$-0.575141\pi$
$733$	−9282.63	−0.467751	−0.233876	+	0.972267i	$0.575141\pi$
$734$	38420.7	1.93206
$735$	0	0
$736$	26109.4	1.30761
$737$	−12879.7	−0.643730
$738$	0	0
$739$	−27923.0	−1.38994	−0.694969	−	0.719039i	$-0.744582\pi$
$739$	−27923.0	−1.38994	−0.694969	+	0.719039i	$0.744582\pi$
$740$	2886.98	0.143416
$741$	0	0
$742$	3201.59	0.158402
$743$	−10656.5	−0.526178	−0.263089	−	0.964772i	$-0.584741\pi$
$743$	−10656.5	−0.526178	−0.263089	+	0.964772i	$0.584741\pi$
$744$	0	0
$745$	18094.9	0.889862
$746$	5560.12	0.272883
$747$	0	0
$748$	−38351.5	−1.87469
$749$	33701.3	1.64408
$750$	0	0
$751$	10607.6	0.515417	0.257708	−	0.966223i	$-0.417033\pi$
$751$	10607.6	0.515417	0.257708	+	0.966223i	$0.417033\pi$
$752$	−19463.5	−0.943833
$753$	0	0
$754$	11341.5	0.547790
$755$	−24848.8	−1.19780
$756$	0	0
$757$	−31400.3	−1.50761	−0.753806	−	0.657097i	$-0.771784\pi$
$757$	−31400.3	−1.50761	−0.753806	+	0.657097i	$0.771784\pi$
$758$	−37596.9	−1.80156
$759$	0	0
$760$	−7857.01	−0.375005
$761$	−30161.8	−1.43675	−0.718373	−	0.695658i	$-0.755113\pi$
$761$	−30161.8	−1.43675	−0.718373	+	0.695658i	$0.755113\pi$
$762$	0	0
$763$	17395.5	0.825372
$764$	47528.0	2.25066
$765$	0	0
$766$	26741.8	1.26138
$767$	2767.39	0.130280
$768$	0	0
$769$	−13107.1	−0.614636	−0.307318	−	0.951607i	$-0.599431\pi$
$769$	−13107.1	−0.614636	−0.307318	+	0.951607i	$0.599431\pi$
$770$	−20701.4	−0.968868
$771$	0	0
$772$	−34159.4	−1.59252
$773$	−27874.7	−1.29700	−0.648502	−	0.761213i	$-0.724604\pi$
$773$	−27874.7	−1.29700	−0.648502	+	0.761213i	$0.724604\pi$
$774$	0	0
$775$	−4304.69	−0.199521
$776$	6528.11	0.301992
$777$	0	0
$778$	4735.99	0.218244
$779$	15321.7	0.704695
$780$	0	0
$781$	−37728.6	−1.72860
$782$	−54089.6	−2.47345
$783$	0	0
$784$	5159.87	0.235052
$785$	29819.1	1.35578
$786$	0	0
$787$	−9865.07	−0.446826	−0.223413	−	0.974724i	$-0.571720\pi$
$787$	−9865.07	−0.446826	−0.223413	+	0.974724i	$0.571720\pi$
$788$	44716.4	2.02152
$789$	0	0
$790$	44902.1	2.02221
$791$	13488.6	0.606318
$792$	0	0
$793$	2951.05	0.132150
$794$	24832.1	1.10989
$795$	0	0
$796$	−1369.55	−0.0609828
$797$	−22731.7	−1.01029	−0.505144	−	0.863035i	$-0.668560\pi$
$797$	−22731.7	−1.01029	−0.505144	+	0.863035i	$0.668560\pi$
$798$	0	0
$799$	49556.0	2.19420
$800$	5983.43	0.264433
$801$	0	0
$802$	32305.7	1.42239
$803$	−11361.7	−0.499311
$804$	0	0
$805$	−15805.4	−0.692009
$806$	7483.80	0.327054
$807$	0	0
$808$	−1117.25	−0.0486444
$809$	27981.1	1.21602	0.608012	−	0.793928i	$-0.291967\pi$
$809$	27981.1	1.21602	0.608012	+	0.793928i	$0.291967\pi$
$810$	0	0
$811$	−708.550	−0.0306789	−0.0153394	−	0.999882i	$-0.504883\pi$
$811$	−708.550	−0.0306789	−0.0153394	+	0.999882i	$0.504883\pi$
$812$	−41293.5	−1.78463
$813$	0	0
$814$	4006.09	0.172498
$815$	−22358.9	−0.960981
$816$	0	0
$817$	−5557.40	−0.237979
$818$	−30012.6	−1.28284
$819$	0	0
$820$	−11297.6	−0.481135
$821$	23071.5	0.980755	0.490378	−	0.871510i	$-0.336859\pi$
$821$	23071.5	0.980755	0.490378	+	0.871510i	$0.336859\pi$
$822$	0	0
$823$	1629.38	0.0690117	0.0345059	−	0.999404i	$-0.489014\pi$
$823$	1629.38	0.0690117	0.0345059	+	0.999404i	$0.489014\pi$
$824$	6910.07	0.292141
$825$	0	0
$826$	−18612.7	−0.784040
$827$	30783.9	1.29439	0.647196	−	0.762324i	$-0.275942\pi$
$827$	30783.9	1.29439	0.647196	+	0.762324i	$0.275942\pi$
$828$	0	0
$829$	−19420.8	−0.813644	−0.406822	−	0.913507i	$-0.633363\pi$
$829$	−19420.8	−0.813644	−0.406822	+	0.913507i	$0.633363\pi$
$830$	55110.2	2.30470
$831$	0	0
$832$	−6520.28	−0.271695
$833$	−13137.5	−0.546445
$834$	0	0
$835$	−32090.3	−1.32998
$836$	−38640.1	−1.59856
$837$	0	0
$838$	59678.4	2.46009
$839$	3935.30	0.161933	0.0809664	−	0.996717i	$-0.474199\pi$
$839$	3935.30	0.161933	0.0809664	+	0.996717i	$0.474199\pi$
$840$	0	0
$841$	55106.8	2.25949
$842$	12456.0	0.509813
$843$	0	0
$844$	6827.61	0.278455
$845$	21237.4	0.864601
$846$	0	0
$847$	5094.13	0.206654
$848$	2490.03	0.100835
$849$	0	0
$850$	−12395.6	−0.500195
$851$	3058.62	0.123206
$852$	0	0
$853$	−28482.8	−1.14330	−0.571649	−	0.820498i	$-0.693696\pi$
$853$	−28482.8	−1.14330	−0.571649	+	0.820498i	$0.693696\pi$
$854$	−19847.9	−0.795292
$855$	0	0
$856$	−13087.6	−0.522576
$857$	−4698.11	−0.187263	−0.0936314	−	0.995607i	$-0.529848\pi$
$857$	−4698.11	−0.187263	−0.0936314	+	0.995607i	$0.529848\pi$
$858$	0	0
$859$	14606.9	0.580189	0.290094	−	0.956998i	$-0.406313\pi$
$859$	14606.9	0.580189	0.290094	+	0.956998i	$0.406313\pi$
$860$	4097.81	0.162482
$861$	0	0
$862$	−24715.1	−0.976565
$863$	14804.4	0.583948	0.291974	−	0.956426i	$-0.405688\pi$
$863$	14804.4	0.583948	0.291974	+	0.956426i	$0.405688\pi$
$864$	0	0
$865$	26754.1	1.05164
$866$	−15809.6	−0.620359
$867$	0	0
$868$	−27247.9	−1.06550
$869$	33730.0	1.31670
$870$	0	0
$871$	3917.82	0.152411
$872$	−6755.38	−0.262346
$873$	0	0
$874$	−54496.6	−2.10912
$875$	−23190.3	−0.895971
$876$	0	0
$877$	−20260.7	−0.780108	−0.390054	−	0.920792i	$-0.627544\pi$
$877$	−20260.7	−0.780108	−0.390054	+	0.920792i	$0.627544\pi$
$878$	−46084.6	−1.77139
$879$	0	0
$880$	−16100.5	−0.616758
$881$	−20626.8	−0.788803	−0.394401	−	0.918938i	$-0.629048\pi$
$881$	−20626.8	−0.788803	−0.394401	+	0.918938i	$0.629048\pi$
$882$	0	0
$883$	−21447.7	−0.817410	−0.408705	−	0.912667i	$-0.634020\pi$
$883$	−21447.7	−0.817410	−0.408705	+	0.912667i	$0.634020\pi$
$884$	11666.0	0.443857
$885$	0	0
$886$	−22328.0	−0.846640
$887$	−41042.9	−1.55365	−0.776824	−	0.629718i	$-0.783170\pi$
$887$	−41042.9	−1.55365	−0.776824	+	0.629718i	$0.783170\pi$
$888$	0	0
$889$	−10104.0	−0.381190
$890$	−34068.4	−1.28312
$891$	0	0
$892$	−54027.9	−2.02801
$893$	49928.9	1.87100
$894$	0	0
$895$	−30335.7	−1.13297
$896$	11764.5	0.438645
$897$	0	0
$898$	−45901.0	−1.70572
$899$	52456.0	1.94606
$900$	0	0
$901$	−6339.85	−0.234418
$902$	−15677.1	−0.578702
$903$	0	0
$904$	−5238.16	−0.192720
$905$	6744.93	0.247745
$906$	0	0
$907$	−1183.40	−0.0433231	−0.0216615	−	0.999765i	$-0.506896\pi$
$907$	−1183.40	−0.0433231	−0.0216615	+	0.999765i	$0.506896\pi$
$908$	43546.4	1.59156
$909$	0	0
$910$	6297.09	0.229392
$911$	48148.3	1.75107	0.875535	−	0.483154i	$-0.160509\pi$
$911$	48148.3	1.75107	0.875535	+	0.483154i	$0.160509\pi$
$912$	0	0
$913$	41398.3	1.50064
$914$	−30819.0	−1.11532
$915$	0	0
$916$	27820.9	1.00352
$917$	−24961.9	−0.898925
$918$	0	0
$919$	6038.40	0.216745	0.108372	−	0.994110i	$-0.465436\pi$
$919$	6038.40	0.216745	0.108372	+	0.994110i	$0.465436\pi$
$920$	6137.88	0.219957
$921$	0	0
$922$	58146.1	2.07694
$923$	11476.5	0.409268
$924$	0	0
$925$	700.938	0.0249154
$926$	26303.4	0.933458
$927$	0	0
$928$	−72912.7	−2.57918
$929$	−38992.8	−1.37709	−0.688543	−	0.725195i	$-0.741750\pi$
$929$	−38992.8	−1.37709	−0.688543	+	0.725195i	$0.741750\pi$
$930$	0	0
$931$	−13236.4	−0.465955
$932$	14862.4	0.522353
$933$	0	0
$934$	45111.7	1.58041
$935$	40993.3	1.43382
$936$	0	0
$937$	48142.4	1.67849	0.839245	−	0.543753i	$-0.182997\pi$
$937$	48142.4	1.67849	0.839245	+	0.543753i	$0.182997\pi$
$938$	−26350.1	−0.917228
$939$	0	0
$940$	−36815.6	−1.27744
$941$	−19733.8	−0.683637	−0.341818	−	0.939766i	$-0.611043\pi$
$941$	−19733.8	−0.683637	−0.341818	+	0.939766i	$0.611043\pi$
$942$	0	0
$943$	−11969.3	−0.413334
$944$	−14475.9	−0.499101
$945$	0	0
$946$	5686.28	0.195430
$947$	42705.3	1.46540	0.732702	−	0.680550i	$-0.238259\pi$
$947$	42705.3	1.46540	0.732702	+	0.680550i	$0.238259\pi$
$948$	0	0
$949$	3456.08	0.118218
$950$	−12488.9	−0.426518
$951$	0	0
$952$	−11984.7	−0.408012
$953$	8305.61	0.282314	0.141157	−	0.989987i	$-0.454918\pi$
$953$	8305.61	0.282314	0.141157	+	0.989987i	$0.454918\pi$
$954$	0	0
$955$	−50801.9	−1.72137
$956$	27264.0	0.922364
$957$	0	0
$958$	−36044.7	−1.21561
$959$	517.692	0.0174318
$960$	0	0
$961$	4822.52	0.161878
$962$	−1218.60	−0.0408411
$963$	0	0
$964$	21119.4	0.705611
$965$	36512.5	1.21801
$966$	0	0
$967$	8502.39	0.282749	0.141375	−	0.989956i	$-0.454848\pi$
$967$	8502.39	0.282749	0.141375	+	0.989956i	$0.454848\pi$
$968$	−1978.26	−0.0656856
$969$	0	0
$970$	−45682.4	−1.51214
$971$	−28522.6	−0.942672	−0.471336	−	0.881954i	$-0.656228\pi$
$971$	−28522.6	−0.942672	−0.471336	+	0.881954i	$0.656228\pi$
$972$	0	0
$973$	−14476.2	−0.476962
$974$	−57864.1	−1.90358
$975$	0	0
$976$	−15436.6	−0.506264
$977$	4979.17	0.163048	0.0815240	−	0.996671i	$-0.474021\pi$
$977$	4979.17	0.163048	0.0815240	+	0.996671i	$0.474021\pi$
$978$	0	0
$979$	−25591.8	−0.835463
$980$	9759.98	0.318134
$981$	0	0
$982$	−61599.6	−2.00175
$983$	51380.1	1.66711	0.833556	−	0.552435i	$-0.186301\pi$
$983$	51380.1	1.66711	0.833556	+	0.552435i	$0.186301\pi$
$984$	0	0
$985$	−47796.6	−1.54612
$986$	151050.	4.87872
$987$	0	0
$988$	11753.8	0.378479
$989$	4341.43	0.139585
$990$	0	0
$991$	53340.5	1.70980	0.854902	−	0.518790i	$-0.173617\pi$
$991$	53340.5	1.70980	0.854902	+	0.518790i	$0.173617\pi$
$992$	−48112.1	−1.53988
$993$	0	0
$994$	−77187.6	−2.46302
$995$	1463.89	0.0466415
$996$	0	0
$997$	16138.6	0.512652	0.256326	−	0.966590i	$-0.417488\pi$
$997$	16138.6	0.512652	0.256326	+	0.966590i	$0.417488\pi$
$998$	−45997.3	−1.45894
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1161.4.a.e.1.18	yes	21
3.2	odd	2		1161.4.a.d.1.4	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1161.4.a.d.1.4	✓	21	3.2	odd	2
1161.4.a.e.1.18	yes	21	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:	\( N \)	\(=\)	\( 1161 = 3^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1161.a (trivial)

\(f(q)\)	\(=\)	\(q+4.17639 q^{2} +9.44227 q^{4} -10.0927 q^{5} -15.5108 q^{7} +6.02347 q^{8} +O(q^{10})\)	\(q+4.17639 q^{2} +9.44227 q^{4} -10.0927 q^{5} -15.5108 q^{7} +6.02347 q^{8} -42.1510 q^{10} -31.6635 q^{11} +9.63159 q^{13} -64.7792 q^{14} -50.3817 q^{16} +128.277 q^{17} +129.242 q^{19} -95.2979 q^{20} -132.239 q^{22} -100.964 q^{23} -23.1376 q^{25} +40.2253 q^{26} -146.457 q^{28} +281.950 q^{29} +186.047 q^{31} -258.602 q^{32} +535.734 q^{34} +156.546 q^{35} -30.2943 q^{37} +539.765 q^{38} -60.7931 q^{40} +118.551 q^{41} -43.0000 q^{43} -298.975 q^{44} -421.664 q^{46} +386.321 q^{47} -102.416 q^{49} -96.6318 q^{50} +90.9440 q^{52} -49.4232 q^{53} +319.570 q^{55} -93.4288 q^{56} +1177.53 q^{58} +287.325 q^{59} +306.393 q^{61} +777.006 q^{62} -676.969 q^{64} -97.2086 q^{65} +406.767 q^{67} +1211.22 q^{68} +653.796 q^{70} +1191.55 q^{71} +358.827 q^{73} -126.521 q^{74} +1220.34 q^{76} +491.125 q^{77} -1065.27 q^{79} +508.487 q^{80} +495.115 q^{82} -1307.45 q^{83} -1294.66 q^{85} -179.585 q^{86} -190.724 q^{88} +808.245 q^{89} -149.393 q^{91} -953.325 q^{92} +1613.43 q^{94} -1304.40 q^{95} +1083.78 q^{97} -427.728 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + q^{2} + 87 q^{4} + 3 q^{5} + 3 q^{7} + 69 q^{8}+O(q^{10}) \) 21 * q + q^2 + 87 * q^4 + 3 * q^5 + 3 * q^7 + 69 * q^8	\( 21 q + q^{2} + 87 q^{4} + 3 q^{5} + 3 q^{7} + 69 q^{8} - 9 q^{10} + 129 q^{11} + 54 q^{13} + 184 q^{14} + 111 q^{16} + 68 q^{17} + 78 q^{19} + 123 q^{20} - 51 q^{22} + 586 q^{23} + 402 q^{25} + 125 q^{26} - 6 q^{28} + 556 q^{29} - 111 q^{31} + 758 q^{32} - 420 q^{34} + 1409 q^{35} + 330 q^{37} + 1067 q^{38} + 180 q^{40} + 678 q^{41} - 903 q^{43} + 1510 q^{44} - 1062 q^{46} + 1580 q^{47} + 1218 q^{49} + 2054 q^{50} + 717 q^{52} + 1069 q^{53} - 2259 q^{55} + 2823 q^{56} + 666 q^{58} + 2248 q^{59} - 162 q^{61} + 3211 q^{62} - 2025 q^{64} + 910 q^{65} - 216 q^{67} + 1868 q^{68} + 3636 q^{70} + 3088 q^{71} - 1605 q^{73} + 2516 q^{74} + 1263 q^{76} + 891 q^{77} + 2094 q^{79} + 5251 q^{80} - 966 q^{82} + 1993 q^{83} - 114 q^{85} - 43 q^{86} + 846 q^{88} + 2494 q^{89} - 2382 q^{91} + 8034 q^{92} - 1062 q^{94} + 4798 q^{95} + 2565 q^{97} + 3103 q^{98}+O(q^{100}) \) 21 * q + q^2 + 87 * q^4 + 3 * q^5 + 3 * q^7 + 69 * q^8 - 9 * q^10 + 129 * q^11 + 54 * q^13 + 184 * q^14 + 111 * q^16 + 68 * q^17 + 78 * q^19 + 123 * q^20 - 51 * q^22 + 586 * q^23 + 402 * q^25 + 125 * q^26 - 6 * q^28 + 556 * q^29 - 111 * q^31 + 758 * q^32 - 420 * q^34 + 1409 * q^35 + 330 * q^37 + 1067 * q^38 + 180 * q^40 + 678 * q^41 - 903 * q^43 + 1510 * q^44 - 1062 * q^46 + 1580 * q^47 + 1218 * q^49 + 2054 * q^50 + 717 * q^52 + 1069 * q^53 - 2259 * q^55 + 2823 * q^56 + 666 * q^58 + 2248 * q^59 - 162 * q^61 + 3211 * q^62 - 2025 * q^64 + 910 * q^65 - 216 * q^67 + 1868 * q^68 + 3636 * q^70 + 3088 * q^71 - 1605 * q^73 + 2516 * q^74 + 1263 * q^76 + 891 * q^77 + 2094 * q^79 + 5251 * q^80 - 966 * q^82 + 1993 * q^83 - 114 * q^85 - 43 * q^86 + 846 * q^88 + 2494 * q^89 - 2382 * q^91 + 8034 * q^92 - 1062 * q^94 + 4798 * q^95 + 2565 * q^97 + 3103 * q^98

Introduction

L-functions

Modular forms

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Database

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