LMFDB - Embedded newform 1143.3.b.a.890.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.19
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.66

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} +O(q^{10})$	\(q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} -18.4878 q^{10} +1.28153i q^{11} -14.4150 q^{13} +8.84618i q^{14} -19.9864 q^{16} +10.3912i q^{17} -3.18314 q^{19} +15.8230i q^{20} +3.16948 q^{22} +24.2785i q^{23} -30.8796 q^{25} +35.6512i q^{26} +7.57108 q^{28} -17.6691i q^{29} +32.6017 q^{31} +30.7993i q^{32} +25.6995 q^{34} +26.7377i q^{35} -49.9752 q^{37} +7.87253i q^{38} -34.8180 q^{40} +10.5982i q^{41} +38.4537 q^{43} -2.71262i q^{44} +60.0455 q^{46} +10.6464i q^{47} -36.2064 q^{49} +76.3715i q^{50} +30.5124 q^{52} -63.3509i q^{53} +9.57977 q^{55} +16.6599i q^{56} -43.6992 q^{58} +1.85123i q^{59} +18.8709 q^{61} -80.6304i q^{62} -3.77284 q^{64} +107.756i q^{65} -40.7181 q^{67} -21.9951i q^{68} +66.1275 q^{70} -5.52989i q^{71} -27.0768 q^{73} +123.599i q^{74} +6.73777 q^{76} -4.58380i q^{77} +40.6548 q^{79} +149.404i q^{80} +26.2115 q^{82} -45.3514i q^{83} +77.6770 q^{85} -95.1037i q^{86} +5.96905 q^{88} -77.0121i q^{89} +51.5599 q^{91} -51.3905i q^{92} +26.3305 q^{94} +23.7948i q^{95} -112.360 q^{97} +89.5455i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.47320i	− 1.23660i	−0.785943	−	0.618300i	$-0.787822\pi$
$2$	− 2.47320i	− 1.23660i	0.785943	−	0.618300i	$-0.212178\pi$
$3$	0	0
$3$	0	0
$4$	−2.11671	−0.529177
$5$	− 7.47527i	− 1.49505i	−0.664232	−	0.747527i	$-0.731241\pi$
$5$	− 7.47527i	− 1.49505i	0.664232	−	0.747527i	$-0.268759\pi$
$6$	0	0
$7$	−3.57682	−0.510974	−0.255487	−	0.966813i	$-0.582236\pi$
$7$	−3.57682	−0.510974	−0.255487	+	0.966813i	$0.582236\pi$
$8$	− 4.65775i	− 0.582219i
$9$	0	0
$10$	−18.4878	−1.84878
$11$	1.28153i	0.116503i	0.998302	+	0.0582513i	$0.0185525\pi$
$11$	1.28153i	0.116503i	−0.998302	+	0.0582513i	$0.981448\pi$
$12$	0	0
$13$	−14.4150	−1.10885	−0.554424	−	0.832235i	$-0.687061\pi$
$13$	−14.4150	−1.10885	−0.554424	+	0.832235i	$0.687061\pi$
$14$	8.84618i	0.631870i
$15$	0	0
$16$	−19.9864	−1.24915
$17$	10.3912i	0.611247i	0.952152	+	0.305623i	$0.0988648\pi$
$17$	10.3912i	0.611247i	−0.952152	+	0.305623i	$0.901135\pi$
$18$	0	0
$19$	−3.18314	−0.167534	−0.0837668	−	0.996485i	$-0.526695\pi$
$19$	−3.18314	−0.167534	−0.0837668	+	0.996485i	$0.526695\pi$
$20$	15.8230i	0.791148i
$21$	0	0
$22$	3.16948	0.144067
$23$	24.2785i	1.05559i	0.849373	+	0.527794i	$0.176981\pi$
$23$	24.2785i	1.05559i	−0.849373	+	0.527794i	$0.823019\pi$
$24$	0	0
$25$	−30.8796	−1.23519
$26$	35.6512i	1.37120i
$27$	0	0
$28$	7.57108	0.270396
$29$	− 17.6691i	− 0.609280i	−0.952468	−	0.304640i	$-0.901464\pi$
$29$	− 17.6691i	− 0.609280i	0.952468	−	0.304640i	$-0.0985361\pi$
$30$	0	0
$31$	32.6017	1.05167	0.525833	−	0.850588i	$-0.323754\pi$
$31$	32.6017	1.05167	0.525833	+	0.850588i	$0.323754\pi$
$32$	30.7993i	0.962477i
$33$	0	0
$34$	25.6995	0.755867
$35$	26.7377i	0.763933i
$36$	0	0
$37$	−49.9752	−1.35068	−0.675340	−	0.737506i	$-0.736003\pi$
$37$	−49.9752	−1.35068	−0.675340	+	0.737506i	$0.736003\pi$
$38$	7.87253i	0.207172i
$39$	0	0
$40$	−34.8180	−0.870449
$41$	10.5982i	0.258493i	0.991613	+	0.129246i	$0.0412558\pi$
$41$	10.5982i	0.258493i	−0.991613	+	0.129246i	$0.958744\pi$
$42$	0	0
$43$	38.4537	0.894273	0.447136	−	0.894466i	$-0.352444\pi$
$43$	38.4537	0.894273	0.447136	+	0.894466i	$0.352444\pi$
$44$	− 2.71262i	− 0.0616505i
$45$	0	0
$46$	60.0455	1.30534
$47$	10.6464i	0.226518i	0.993565	+	0.113259i	$0.0361290\pi$
$47$	10.6464i	0.226518i	−0.993565	+	0.113259i	$0.963871\pi$
$48$	0	0
$49$	−36.2064	−0.738906
$50$	76.3715i	1.52743i
$51$	0	0
$52$	30.5124	0.586777
$53$	− 63.3509i	− 1.19530i	−0.801757	−	0.597650i	$-0.796101\pi$
$53$	− 63.3509i	− 1.19530i	0.801757	−	0.597650i	$-0.203899\pi$
$54$	0	0
$55$	9.57977	0.174178
$56$	16.6599i	0.297499i
$57$	0	0
$58$	−43.6992	−0.753435
$59$	1.85123i	0.0313768i	0.999877	+	0.0156884i	$0.00499398\pi$
$59$	1.85123i	0.0313768i	−0.999877	+	0.0156884i	$0.995006\pi$
$60$	0	0
$61$	18.8709	0.309358	0.154679	−	0.987965i	$-0.450566\pi$
$61$	18.8709	0.309358	0.154679	+	0.987965i	$0.450566\pi$
$62$	− 80.6304i	− 1.30049i
$63$	0	0
$64$	−3.77284	−0.0589506
$65$	107.756i	1.65779i
$66$	0	0
$67$	−40.7181	−0.607733	−0.303867	−	0.952715i	$-0.598278\pi$
$67$	−40.7181	−0.607733	−0.303867	+	0.952715i	$0.598278\pi$
$68$	− 21.9951i	− 0.323458i
$69$	0	0
$70$	66.1275	0.944679
$71$	− 5.52989i	− 0.0778857i	−0.999241	−	0.0389429i	$-0.987601\pi$
$71$	− 5.52989i	− 0.0778857i	0.999241	−	0.0389429i	$-0.0123990\pi$
$72$	0	0
$73$	−27.0768	−0.370915	−0.185458	−	0.982652i	$-0.559377\pi$
$73$	−27.0768	−0.370915	−0.185458	+	0.982652i	$0.559377\pi$
$74$	123.599i	1.67025i
$75$	0	0
$76$	6.73777	0.0886549
$77$	− 4.58380i	− 0.0595298i
$78$	0	0
$79$	40.6548	0.514618	0.257309	−	0.966329i	$-0.417164\pi$
$79$	40.6548	0.514618	0.257309	+	0.966329i	$0.417164\pi$
$80$	149.404i	1.86754i
$81$	0	0
$82$	26.2115	0.319652
$83$	− 45.3514i	− 0.546403i	−0.961957	−	0.273201i	$-0.911917\pi$
$83$	− 45.3514i	− 0.546403i	0.961957	−	0.273201i	$-0.0880826\pi$
$84$	0	0
$85$	77.6770	0.913847
$86$	− 95.1037i	− 1.10586i
$87$	0	0
$88$	5.96905	0.0678301
$89$	− 77.0121i	− 0.865305i	−0.901561	−	0.432652i	$-0.857578\pi$
$89$	− 77.0121i	− 0.865305i	0.901561	−	0.432652i	$-0.142422\pi$
$90$	0	0
$91$	51.5599	0.566592
$92$	− 51.3905i	− 0.558593i
$93$	0	0
$94$	26.3305	0.280112
$95$	23.7948i	0.250472i
$96$	0	0
$97$	−112.360	−1.15835	−0.579174	−	0.815204i	$-0.696625\pi$
$97$	−112.360	−1.15835	−0.579174	+	0.815204i	$0.696625\pi$
$98$	89.5455i	0.913730i
$99$	0	0
$100$	65.3632	0.653632
$101$	85.3813i	0.845359i	0.906279	+	0.422680i	$0.138910\pi$
$101$	85.3813i	0.845359i	−0.906279	+	0.422680i	$0.861090\pi$
$102$	0	0
$103$	32.7075	0.317549	0.158774	−	0.987315i	$-0.449246\pi$
$103$	32.7075	0.317549	0.158774	+	0.987315i	$0.449246\pi$
$104$	67.1416i	0.645592i
$105$	0	0
$106$	−156.679	−1.47811
$107$	200.409i	1.87298i	0.350694	+	0.936490i	$0.385946\pi$
$107$	200.409i	1.87298i	−0.350694	+	0.936490i	$0.614054\pi$
$108$	0	0
$109$	−38.9278	−0.357136	−0.178568	−	0.983928i	$-0.557146\pi$
$109$	−38.9278	−0.357136	−0.178568	+	0.983928i	$0.557146\pi$
$110$	− 23.6927i	− 0.215388i
$111$	0	0
$112$	71.4876	0.638282
$113$	− 24.1549i	− 0.213760i	−0.994272	−	0.106880i	$-0.965914\pi$
$113$	− 24.1549i	− 0.213760i	0.994272	−	0.106880i	$-0.0340860\pi$
$114$	0	0
$115$	181.488	1.57816
$116$	37.4004i	0.322417i
$117$	0	0
$118$	4.57846	0.0388005
$119$	− 37.1674i	− 0.312331i
$120$	0	0
$121$	119.358	0.986427
$122$	− 46.6714i	− 0.382552i
$123$	0	0
$124$	−69.0082	−0.556518
$125$	43.9519i	0.351615i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	132.528i	1.03538i
$129$	0	0
$130$	266.502	2.05002
$131$	− 26.7909i	− 0.204511i	−0.994758	−	0.102255i	$-0.967394\pi$
$131$	− 26.7909i	− 0.204511i	0.994758	−	0.102255i	$-0.0326059\pi$
$132$	0	0
$133$	11.3855	0.0856053
$134$	100.704i	0.751522i
$135$	0	0
$136$	48.3996	0.355880
$137$	205.936i	1.50318i	0.659629	+	0.751592i	$0.270714\pi$
$137$	205.936i	1.50318i	−0.659629	+	0.751592i	$0.729286\pi$
$138$	0	0
$139$	−159.715	−1.14903	−0.574513	−	0.818495i	$-0.694809\pi$
$139$	−159.715	−1.14903	−0.574513	+	0.818495i	$0.694809\pi$
$140$	− 56.5959i	− 0.404256i
$141$	0	0
$142$	−13.6765	−0.0963134
$143$	− 18.4733i	− 0.129184i
$144$	0	0
$145$	−132.081	−0.910906
$146$	66.9663i	0.458674i
$147$	0	0
$148$	105.783	0.714749
$149$	43.3666i	0.291051i	0.989355	+	0.145526i	$0.0464873\pi$
$149$	43.3666i	0.291051i	−0.989355	+	0.145526i	$0.953513\pi$
$150$	0	0
$151$	−227.463	−1.50638	−0.753189	−	0.657804i	$-0.771486\pi$
$151$	−227.463	−1.50638	−0.753189	+	0.657804i	$0.771486\pi$
$152$	14.8263i	0.0975412i
$153$	0	0
$154$	−11.3366	−0.0736145
$155$	− 243.706i	− 1.57230i
$156$	0	0
$157$	59.8855	0.381437	0.190718	−	0.981645i	$-0.438918\pi$
$157$	59.8855	0.381437	0.190718	+	0.981645i	$0.438918\pi$
$158$	− 100.547i	− 0.636376i
$159$	0	0
$160$	230.233	1.43895
$161$	− 86.8398i	− 0.539377i
$162$	0	0
$163$	143.932	0.883017	0.441508	−	0.897257i	$-0.354444\pi$
$163$	143.932	0.883017	0.441508	+	0.897257i	$0.354444\pi$
$164$	− 22.4333i	− 0.136788i
$165$	0	0
$166$	−112.163	−0.675681
$167$	− 18.5191i	− 0.110893i	−0.998462	−	0.0554463i	$-0.982342\pi$
$167$	− 18.5191i	− 0.110893i	0.998462	−	0.0554463i	$-0.0176582\pi$
$168$	0	0
$169$	38.7927	0.229543
$170$	− 192.111i	− 1.13006i
$171$	0	0
$172$	−81.3954	−0.473229
$173$	8.77175i	0.0507038i	0.999679	+	0.0253519i	$0.00807062\pi$
$173$	8.77175i	0.0507038i	−0.999679	+	0.0253519i	$0.991929\pi$
$174$	0	0
$175$	110.451	0.631148
$176$	− 25.6131i	− 0.145529i
$177$	0	0
$178$	−190.466	−1.07004
$179$	− 82.8360i	− 0.462771i	−0.972862	−	0.231385i	$-0.925674\pi$
$179$	− 82.8360i	− 0.462771i	0.972862	−	0.231385i	$-0.0743258\pi$
$180$	0	0
$181$	219.331	1.21178	0.605888	−	0.795550i	$-0.292818\pi$
$181$	219.331	1.21178	0.605888	+	0.795550i	$0.292818\pi$
$182$	− 127.518i	− 0.700647i
$183$	0	0
$184$	113.083	0.614583
$185$	373.578i	2.01934i
$186$	0	0
$187$	−13.3166	−0.0712119
$188$	− 22.5352i	− 0.119868i
$189$	0	0
$190$	58.8493	0.309733
$191$	− 44.5096i	− 0.233034i	−0.993189	−	0.116517i	$-0.962827\pi$
$191$	− 44.5096i	− 0.233034i	0.993189	−	0.116517i	$-0.0371730\pi$
$192$	0	0
$193$	−73.5363	−0.381017	−0.190508	−	0.981686i	$-0.561014\pi$
$193$	−73.5363	−0.381017	−0.190508	+	0.981686i	$0.561014\pi$
$194$	277.888i	1.43241i
$195$	0	0
$196$	76.6384	0.391012
$197$	− 267.608i	− 1.35842i	−0.733945	−	0.679209i	$-0.762323\pi$
$197$	− 267.608i	− 1.35842i	0.733945	−	0.679209i	$-0.237677\pi$
$198$	0	0
$199$	−173.678	−0.872753	−0.436376	−	0.899764i	$-0.643738\pi$
$199$	−173.678	−0.872753	−0.436376	+	0.899764i	$0.643738\pi$
$200$	143.830i	0.719149i
$201$	0	0
$202$	211.165	1.04537
$203$	63.1992i	0.311326i
$204$	0	0
$205$	79.2244	0.386461
$206$	− 80.8921i	− 0.392680i
$207$	0	0
$208$	288.104	1.38512
$209$	− 4.07928i	− 0.0195181i
$210$	0	0
$211$	−384.984	−1.82457	−0.912284	−	0.409559i	$-0.865683\pi$
$211$	−384.984	−1.82457	−0.912284	+	0.409559i	$0.865683\pi$
$212$	134.095i	0.632526i
$213$	0	0
$214$	495.651	2.31613
$215$	− 287.452i	− 1.33699i
$216$	0	0
$217$	−116.610	−0.537374
$218$	96.2762i	0.441634i
$219$	0	0
$220$	−20.2776	−0.0921709
$221$	− 149.789i	− 0.677779i
$222$	0	0
$223$	−391.016	−1.75344	−0.876719	−	0.481004i	$-0.840272\pi$
$223$	−391.016	−1.75344	−0.876719	+	0.481004i	$0.840272\pi$
$224$	− 110.163i	− 0.491801i
$225$	0	0
$226$	−59.7398	−0.264335
$227$	118.447i	0.521792i	0.965367	+	0.260896i	$0.0840180\pi$
$227$	118.447i	0.521792i	−0.965367	+	0.260896i	$0.915982\pi$
$228$	0	0
$229$	−13.0828	−0.0571299	−0.0285650	−	0.999592i	$-0.509094\pi$
$229$	−13.0828	−0.0571299	−0.0285650	+	0.999592i	$0.509094\pi$
$230$	− 448.857i	− 1.95155i
$231$	0	0
$232$	−82.2984	−0.354734
$233$	31.3816i	0.134685i	0.997730	+	0.0673425i	$0.0214520\pi$
$233$	31.3816i	0.134685i	−0.997730	+	0.0673425i	$0.978548\pi$
$234$	0	0
$235$	79.5844	0.338657
$236$	− 3.91852i	− 0.0166039i
$237$	0	0
$238$	−91.9223	−0.386228
$239$	− 151.225i	− 0.632741i	−0.948636	−	0.316370i	$-0.897536\pi$
$239$	− 151.225i	− 0.632741i	0.948636	−	0.316370i	$-0.102464\pi$
$240$	0	0
$241$	−138.477	−0.574593	−0.287297	−	0.957842i	$-0.592757\pi$
$241$	−138.477	−0.574593	−0.287297	+	0.957842i	$0.592757\pi$
$242$	− 295.195i	− 1.21981i
$243$	0	0
$244$	−39.9441	−0.163705
$245$	270.652i	1.10470i
$246$	0	0
$247$	45.8850	0.185769
$248$	− 151.850i	− 0.612300i
$249$	0	0
$250$	108.702	0.434807
$251$	− 243.313i	− 0.969376i	−0.874687	−	0.484688i	$-0.838933\pi$
$251$	− 243.313i	− 0.969376i	0.874687	−	0.484688i	$-0.161067\pi$
$252$	0	0
$253$	−31.1136	−0.122979
$254$	− 27.8715i	− 0.109730i
$255$	0	0
$256$	312.677	1.22139
$257$	− 158.311i	− 0.615996i	−0.951387	−	0.307998i	$-0.900341\pi$
$257$	− 158.311i	− 0.615996i	0.951387	−	0.307998i	$-0.0996590\pi$
$258$	0	0
$259$	178.752	0.690163
$260$	− 228.088i	− 0.877263i
$261$	0	0
$262$	−66.2593	−0.252898
$263$	− 85.3123i	− 0.324381i	−0.986759	−	0.162191i	$-0.948144\pi$
$263$	− 85.3123i	− 0.324381i	0.986759	−	0.162191i	$-0.0518559\pi$
$264$	0	0
$265$	−473.565	−1.78704
$266$	− 28.1586i	− 0.105859i
$267$	0	0
$268$	86.1884	0.321598
$269$	− 118.069i	− 0.438920i	−0.975622	−	0.219460i	$-0.929570\pi$
$269$	− 118.069i	− 0.438920i	0.975622	−	0.219460i	$-0.0704295\pi$
$270$	0	0
$271$	334.758	1.23527	0.617634	−	0.786466i	$-0.288091\pi$
$271$	334.758	1.23527	0.617634	+	0.786466i	$0.288091\pi$
$272$	− 207.682i	− 0.763538i
$273$	0	0
$274$	509.321	1.85884
$275$	− 39.5732i	− 0.143902i
$276$	0	0
$277$	−30.6223	−0.110550	−0.0552750	−	0.998471i	$-0.517604\pi$
$277$	−30.6223	−0.110550	−0.0552750	+	0.998471i	$0.517604\pi$
$278$	395.006i	1.42089i
$279$	0	0
$280$	124.537	0.444777
$281$	− 4.97564i	− 0.0177069i	−0.999961	−	0.00885345i	$-0.997182\pi$
$281$	− 4.97564i	− 0.0177069i	0.999961	−	0.00885345i	$-0.00281818\pi$
$282$	0	0
$283$	−230.905	−0.815918	−0.407959	−	0.913000i	$-0.633759\pi$
$283$	−230.905	−0.815918	−0.407959	+	0.913000i	$0.633759\pi$
$284$	11.7052i	0.0412153i
$285$	0	0
$286$	−45.6880	−0.159748
$287$	− 37.9078i	− 0.132083i
$288$	0	0
$289$	181.023	0.626377
$290$	326.663i	1.12643i
$291$	0	0
$292$	57.3137	0.196280
$293$	420.765i	1.43606i	0.696014	+	0.718028i	$0.254955\pi$
$293$	420.765i	1.43606i	−0.696014	+	0.718028i	$0.745045\pi$
$294$	0	0
$295$	13.8384	0.0469100
$296$	232.772i	0.786392i
$297$	0	0
$298$	107.254	0.359913
$299$	− 349.975i	− 1.17049i
$300$	0	0
$301$	−137.542	−0.456950
$302$	562.562i	1.86279i
$303$	0	0
$304$	63.6194	0.209274
$305$	− 141.065i	− 0.462507i
$306$	0	0
$307$	176.646	0.575393	0.287697	−	0.957722i	$-0.407111\pi$
$307$	176.646	0.575393	0.287697	+	0.957722i	$0.407111\pi$
$308$	9.70256i	0.0315018i
$309$	0	0
$310$	−602.734	−1.94430
$311$	10.4719i	0.0336717i	0.999858	+	0.0168359i	$0.00535927\pi$
$311$	10.4719i	0.0336717i	−0.999858	+	0.0168359i	$0.994641\pi$
$312$	0	0
$313$	−569.849	−1.82060	−0.910302	−	0.413946i	$-0.864150\pi$
$313$	−569.849	−1.82060	−0.910302	+	0.413946i	$0.864150\pi$
$314$	− 148.109i	− 0.471684i
$315$	0	0
$316$	−86.0543	−0.272324
$317$	− 68.0365i	− 0.214626i	−0.994225	−	0.107313i	$-0.965775\pi$
$317$	− 68.0365i	− 0.214626i	0.994225	−	0.107313i	$-0.0342247\pi$
$318$	0	0
$319$	22.6435	0.0709827
$320$	28.2030i	0.0881343i
$321$	0	0
$322$	−214.772	−0.666994
$323$	− 33.0766i	− 0.102404i
$324$	0	0
$325$	445.131	1.36963
$326$	− 355.972i	− 1.09194i
$327$	0	0
$328$	49.3638	0.150499
$329$	− 38.0801i	− 0.115745i
$330$	0	0
$331$	80.7152	0.243853	0.121926	−	0.992539i	$-0.461093\pi$
$331$	80.7152	0.243853	0.121926	+	0.992539i	$0.461093\pi$
$332$	95.9958i	0.289144i
$333$	0	0
$334$	−45.8013	−0.137130
$335$	304.379i	0.908594i
$336$	0	0
$337$	−652.457	−1.93607	−0.968037	−	0.250806i	$-0.919304\pi$
$337$	−652.457	−1.93607	−0.968037	+	0.250806i	$0.919304\pi$
$338$	− 95.9421i	− 0.283852i
$339$	0	0
$340$	−164.420	−0.483587
$341$	41.7800i	0.122522i
$342$	0	0
$343$	304.768	0.888535
$344$	− 179.108i	− 0.520663i
$345$	0	0
$346$	21.6943	0.0627002
$347$	− 409.886i	− 1.18123i	−0.806955	−	0.590613i	$-0.798886\pi$
$347$	− 409.886i	− 1.18123i	0.806955	−	0.590613i	$-0.201114\pi$
$348$	0	0
$349$	−415.021	−1.18917	−0.594586	−	0.804032i	$-0.702684\pi$
$349$	−415.021	−1.18917	−0.594586	+	0.804032i	$0.702684\pi$
$350$	− 273.167i	− 0.780476i
$351$	0	0
$352$	−39.4702	−0.112131
$353$	145.012i	0.410799i	0.978678	+	0.205400i	$0.0658494\pi$
$353$	145.012i	0.410799i	−0.978678	+	0.205400i	$0.934151\pi$
$354$	0	0
$355$	−41.3374	−0.116443
$356$	163.012i	0.457899i
$357$	0	0
$358$	−204.870	−0.572262
$359$	− 188.574i	− 0.525277i	−0.964894	−	0.262638i	$-0.915407\pi$
$359$	− 188.574i	− 0.525277i	0.964894	−	0.262638i	$-0.0845926\pi$
$360$	0	0
$361$	−350.868	−0.971933
$362$	− 542.450i	− 1.49848i
$363$	0	0
$364$	−109.137	−0.299828
$365$	202.406i	0.554538i
$366$	0	0
$367$	−693.878	−1.89068	−0.945338	−	0.326093i	$-0.894268\pi$
$367$	−693.878	−1.89068	−0.945338	+	0.326093i	$0.894268\pi$
$368$	− 485.239i	− 1.31859i
$369$	0	0
$370$	923.932	2.49711
$371$	226.595i	0.610767i
$372$	0	0
$373$	−698.882	−1.87368	−0.936839	−	0.349760i	$-0.886263\pi$
$373$	−698.882	−1.87368	−0.936839	+	0.349760i	$0.886263\pi$
$374$	32.9346i	0.0880605i
$375$	0	0
$376$	49.5881	0.131883
$377$	254.701i	0.675598i
$378$	0	0
$379$	−117.384	−0.309721	−0.154861	−	0.987936i	$-0.549493\pi$
$379$	−117.384	−0.309721	−0.154861	+	0.987936i	$0.549493\pi$
$380$	− 50.3667i	− 0.132544i
$381$	0	0
$382$	−110.081	−0.288170
$383$	565.781i	1.47724i	0.674124	+	0.738618i	$0.264521\pi$
$383$	565.781i	1.47724i	−0.674124	+	0.738618i	$0.735479\pi$
$384$	0	0
$385$	−34.2651	−0.0890003
$386$	181.870i	0.471165i
$387$	0	0
$388$	237.833	0.612971
$389$	− 293.432i	− 0.754325i	−0.926147	−	0.377163i	$-0.876900\pi$
$389$	− 293.432i	− 0.754325i	0.926147	−	0.377163i	$-0.123100\pi$
$390$	0	0
$391$	−252.283	−0.645224
$392$	168.640i	0.430205i
$393$	0	0
$394$	−661.848	−1.67982
$395$	− 303.905i	− 0.769381i
$396$	0	0
$397$	−156.374	−0.393890	−0.196945	−	0.980415i	$-0.563102\pi$
$397$	−156.374	−0.393890	−0.196945	+	0.980415i	$0.563102\pi$
$398$	429.539i	1.07924i
$399$	0	0
$400$	617.172	1.54293
$401$	547.372i	1.36502i	0.730878	+	0.682508i	$0.239111\pi$
$401$	547.372i	1.36502i	−0.730878	+	0.682508i	$0.760889\pi$
$402$	0	0
$403$	−469.953	−1.16614
$404$	− 180.727i	− 0.447345i
$405$	0	0
$406$	156.304	0.384985
$407$	− 64.0447i	− 0.157358i
$408$	0	0
$409$	−256.309	−0.626674	−0.313337	−	0.949642i	$-0.601447\pi$
$409$	−256.309	−0.626674	−0.313337	+	0.949642i	$0.601447\pi$
$410$	− 195.938i	− 0.477897i
$411$	0	0
$412$	−69.2323	−0.168039
$413$	− 6.62152i	− 0.0160327i
$414$	0	0
$415$	−339.014	−0.816902
$416$	− 443.972i	− 1.06724i
$417$	0	0
$418$	−10.0889	−0.0241361
$419$	− 6.95844i	− 0.0166072i	−0.999966	−	0.00830362i	$-0.997357\pi$
$419$	− 6.95844i	− 0.0166072i	0.999966	−	0.00830362i	$-0.00264316\pi$
$420$	0	0
$421$	164.509	0.390757	0.195378	−	0.980728i	$-0.437407\pi$
$421$	164.509	0.390757	0.195378	+	0.980728i	$0.437407\pi$
$422$	952.141i	2.25626i
$423$	0	0
$424$	−295.073	−0.695927
$425$	− 320.876i	− 0.755003i
$426$	0	0
$427$	−67.4976	−0.158074
$428$	− 424.207i	− 0.991138i
$429$	0	0
$430$	−710.926	−1.65332
$431$	− 436.670i	− 1.01316i	−0.862194	−	0.506578i	$-0.830910\pi$
$431$	− 436.670i	− 1.01316i	0.862194	−	0.506578i	$-0.169090\pi$
$432$	0	0
$433$	64.1886	0.148242	0.0741208	−	0.997249i	$-0.476385\pi$
$433$	64.1886	0.148242	0.0741208	+	0.997249i	$0.476385\pi$
$434$	288.400i	0.664516i
$435$	0	0
$436$	82.3989	0.188988
$437$	− 77.2818i	− 0.176846i
$438$	0	0
$439$	−258.306	−0.588397	−0.294199	−	0.955744i	$-0.595053\pi$
$439$	−258.306	−0.588397	−0.294199	+	0.955744i	$0.595053\pi$
$440$	− 44.6202i	− 0.101410i
$441$	0	0
$442$	−370.459	−0.838141
$443$	276.205i	0.623487i	0.950166	+	0.311744i	$0.100913\pi$
$443$	276.205i	0.623487i	−0.950166	+	0.311744i	$0.899087\pi$
$444$	0	0
$445$	−575.686	−1.29368
$446$	967.061i	2.16830i
$447$	0	0
$448$	13.4948	0.0301222
$449$	− 555.627i	− 1.23748i	−0.785597	−	0.618738i	$-0.787644\pi$
$449$	− 555.627i	− 1.23748i	0.785597	−	0.618738i	$-0.212356\pi$
$450$	0	0
$451$	−13.5819	−0.0301151
$452$	51.1288i	0.113117i
$453$	0	0
$454$	292.943	0.645248
$455$	− 385.424i	− 0.847086i
$456$	0	0
$457$	67.3557	0.147387	0.0736934	−	0.997281i	$-0.476521\pi$
$457$	67.3557	0.147387	0.0736934	+	0.997281i	$0.476521\pi$
$458$	32.3562i	0.0706468i
$459$	0	0
$460$	−384.158	−0.835126
$461$	799.216i	1.73366i	0.498605	+	0.866829i	$0.333846\pi$
$461$	799.216i	1.73366i	−0.498605	+	0.866829i	$0.666154\pi$
$462$	0	0
$463$	−469.674	−1.01441	−0.507207	−	0.861824i	$-0.669322\pi$
$463$	−469.674	−1.01441	−0.507207	+	0.861824i	$0.669322\pi$
$464$	353.142i	0.761081i
$465$	0	0
$466$	77.6130	0.166551
$467$	− 584.567i	− 1.25175i	−0.779923	−	0.625875i	$-0.784742\pi$
$467$	− 584.567i	− 1.25175i	0.779923	−	0.625875i	$-0.215258\pi$
$468$	0	0
$469$	145.641	0.310536
$470$	− 196.828i	− 0.418783i
$471$	0	0
$472$	8.62258	0.0182682
$473$	49.2796i	0.104185i
$474$	0	0
$475$	98.2941	0.206935
$476$	78.6726i	0.165278i
$477$	0	0
$478$	−374.010	−0.782447
$479$	− 442.649i	− 0.924111i	−0.886851	−	0.462055i	$-0.847112\pi$
$479$	− 442.649i	− 0.924111i	0.886851	−	0.462055i	$-0.152888\pi$
$480$	0	0
$481$	720.393	1.49770
$482$	342.481i	0.710541i
$483$	0	0
$484$	−252.645	−0.521995
$485$	839.919i	1.73179i
$486$	0	0
$487$	730.873	1.50077	0.750383	−	0.661004i	$-0.229869\pi$
$487$	730.873	1.50077	0.750383	+	0.661004i	$0.229869\pi$
$488$	− 87.8958i	− 0.180114i
$489$	0	0
$490$	669.377	1.36608
$491$	− 275.276i	− 0.560644i	−0.959906	−	0.280322i	$-0.909559\pi$
$491$	− 275.276i	− 0.560644i	0.959906	−	0.280322i	$-0.0904412\pi$
$492$	0	0
$493$	183.603	0.372420
$494$	− 113.483i	− 0.229722i
$495$	0	0
$496$	−651.589	−1.31369
$497$	19.7794i	0.0397976i
$498$	0	0
$499$	−185.214	−0.371170	−0.185585	−	0.982628i	$-0.559418\pi$
$499$	−185.214	−0.371170	−0.185585	+	0.982628i	$0.559418\pi$
$500$	− 93.0333i	− 0.186067i
$501$	0	0
$502$	−601.762	−1.19873
$503$	− 634.099i	− 1.26064i	−0.776337	−	0.630318i	$-0.782925\pi$
$503$	− 634.099i	− 1.26064i	0.776337	−	0.630318i	$-0.217075\pi$
$504$	0	0
$505$	638.248	1.26386
$506$	76.9501i	0.152075i
$507$	0	0
$508$	−23.8541	−0.0469569
$509$	492.444i	0.967473i	0.875214	+	0.483736i	$0.160721\pi$
$509$	492.444i	0.967473i	−0.875214	+	0.483736i	$0.839279\pi$
$510$	0	0
$511$	96.8488	0.189528
$512$	− 243.199i	− 0.474999i
$513$	0	0
$514$	−391.534	−0.761740
$515$	− 244.497i	− 0.474752i
$516$	0	0
$517$	−13.6436	−0.0263900
$518$	− 442.089i	− 0.853454i
$519$	0	0
$520$	501.901	0.965195
$521$	− 861.089i	− 1.65276i	−0.563111	−	0.826381i	$-0.690396\pi$
$521$	− 861.089i	− 1.65276i	0.563111	−	0.826381i	$-0.309604\pi$
$522$	0	0
$523$	58.5594	0.111968	0.0559842	−	0.998432i	$-0.482170\pi$
$523$	58.5594	0.111968	0.0559842	+	0.998432i	$0.482170\pi$
$524$	56.7086i	0.108223i
$525$	0	0
$526$	−210.994	−0.401129
$527$	338.770i	0.642828i
$528$	0	0
$529$	−60.4458	−0.114264
$530$	1171.22i	2.20985i
$531$	0	0
$532$	−24.0998	−0.0453003
$533$	− 152.773i	− 0.286629i
$534$	0	0
$535$	1498.11	2.80021
$536$	189.655i	0.353834i
$537$	0	0
$538$	−292.009	−0.542768
$539$	− 46.3995i	− 0.0860845i
$540$	0	0
$541$	794.836	1.46920	0.734599	−	0.678501i	$-0.237370\pi$
$541$	794.836	1.46920	0.734599	+	0.678501i	$0.237370\pi$
$542$	− 827.922i	− 1.52753i
$543$	0	0
$544$	−320.041	−0.588311
$545$	290.996i	0.533938i
$546$	0	0
$547$	−164.077	−0.299957	−0.149979	−	0.988689i	$-0.547921\pi$
$547$	−164.077	−0.299957	−0.149979	+	0.988689i	$0.547921\pi$
$548$	− 435.907i	− 0.795450i
$549$	0	0
$550$	−97.8723	−0.177950
$551$	56.2432i	0.102075i
$552$	0	0
$553$	−145.415	−0.262956
$554$	75.7351i	0.136706i
$555$	0	0
$556$	338.069	0.608039
$557$	− 806.590i	− 1.44810i	−0.689749	−	0.724048i	$-0.742279\pi$
$557$	− 806.590i	− 1.44810i	0.689749	−	0.724048i	$-0.257721\pi$
$558$	0	0
$559$	−554.311	−0.991612
$560$	− 534.389i	− 0.954266i
$561$	0	0
$562$	−12.3057	−0.0218963
$563$	− 136.258i	− 0.242022i	−0.992651	−	0.121011i	$-0.961386\pi$
$563$	− 136.258i	− 0.242022i	0.992651	−	0.121011i	$-0.0386136\pi$
$564$	0	0
$565$	−180.564	−0.319582
$566$	571.073i	1.00896i
$567$	0	0
$568$	−25.7568	−0.0453465
$569$	− 438.619i	− 0.770860i	−0.922737	−	0.385430i	$-0.874053\pi$
$569$	− 438.619i	− 0.770860i	0.922737	−	0.385430i	$-0.125947\pi$
$570$	0	0
$571$	279.001	0.488618	0.244309	−	0.969697i	$-0.421439\pi$
$571$	279.001	0.488618	0.244309	+	0.969697i	$0.421439\pi$
$572$	39.1025i	0.0683610i
$573$	0	0
$574$	−93.7536	−0.163334
$575$	− 749.711i	− 1.30385i
$576$	0	0
$577$	−193.520	−0.335391	−0.167695	−	0.985839i	$-0.553633\pi$
$577$	−193.520	−0.335391	−0.167695	+	0.985839i	$0.553633\pi$
$578$	− 447.706i	− 0.774578i
$579$	0	0
$580$	279.578	0.482031
$581$	162.214i	0.279198i
$582$	0	0
$583$	81.1861	0.139256
$584$	126.117i	0.215954i
$585$	0	0
$586$	1040.63	1.77583
$587$	159.092i	0.271025i	0.990776	+	0.135512i	$0.0432681\pi$
$587$	159.092i	0.271025i	−0.990776	+	0.135512i	$0.956732\pi$
$588$	0	0
$589$	−103.776	−0.176189
$590$	− 34.2252i	− 0.0580089i
$591$	0	0
$592$	998.823	1.68720
$593$	502.148i	0.846793i	0.905945	+	0.423396i	$0.139162\pi$
$593$	502.148i	0.846793i	−0.905945	+	0.423396i	$0.860838\pi$
$594$	0	0
$595$	−277.836	−0.466952
$596$	− 91.7945i	− 0.154018i
$597$	0	0
$598$	−865.558	−1.44742
$599$	− 190.124i	− 0.317403i	−0.987327	−	0.158701i	$-0.949269\pi$
$599$	− 190.124i	− 0.317403i	0.987327	−	0.158701i	$-0.0507307\pi$
$600$	0	0
$601$	199.419	0.331811	0.165906	−	0.986142i	$-0.446945\pi$
$601$	199.419	0.331811	0.165906	+	0.986142i	$0.446945\pi$
$602$	340.169i	0.565064i
$603$	0	0
$604$	481.473	0.797141
$605$	− 892.231i	− 1.47476i
$606$	0	0
$607$	748.425	1.23299	0.616495	−	0.787359i	$-0.288552\pi$
$607$	748.425	1.23299	0.616495	+	0.787359i	$0.288552\pi$
$608$	− 98.0383i	− 0.161247i
$609$	0	0
$610$	−348.881	−0.571936
$611$	− 153.467i	− 0.251174i
$612$	0	0
$613$	458.107	0.747319	0.373660	−	0.927566i	$-0.378103\pi$
$613$	458.107	0.747319	0.373660	+	0.927566i	$0.378103\pi$
$614$	− 436.880i	− 0.711530i
$615$	0	0
$616$	−21.3502	−0.0346594
$617$	783.814i	1.27036i	0.772363	+	0.635182i	$0.219075\pi$
$617$	783.814i	1.27036i	−0.772363	+	0.635182i	$0.780925\pi$
$618$	0	0
$619$	1123.39	1.81484	0.907421	−	0.420223i	$-0.138048\pi$
$619$	1123.39	1.81484	0.907421	+	0.420223i	$0.138048\pi$
$620$	515.855i	0.832024i
$621$	0	0
$622$	25.8991	0.0416384
$623$	275.458i	0.442148i
$624$	0	0
$625$	−443.439	−0.709502
$626$	1409.35i	2.25136i
$627$	0	0
$628$	−126.760	−0.201848
$629$	− 519.302i	− 0.825599i
$630$	0	0
$631$	367.570	0.582520	0.291260	−	0.956644i	$-0.405925\pi$
$631$	367.570	0.582520	0.291260	+	0.956644i	$0.405925\pi$
$632$	− 189.360i	− 0.299620i
$633$	0	0
$634$	−168.268	−0.265407
$635$	− 84.2420i	− 0.132665i
$636$	0	0
$637$	521.916	0.819334
$638$	− 56.0018i	− 0.0877771i
$639$	0	0
$640$	990.683	1.54794
$641$	− 165.990i	− 0.258955i	−0.991582	−	0.129477i	$-0.958670\pi$
$641$	− 165.990i	− 0.258955i	0.991582	−	0.129477i	$-0.0413300\pi$
$642$	0	0
$643$	−225.814	−0.351188	−0.175594	−	0.984463i	$-0.556185\pi$
$643$	−225.814	−0.351188	−0.175594	+	0.984463i	$0.556185\pi$
$644$	183.814i	0.285426i
$645$	0	0
$646$	−81.8050	−0.126633
$647$	− 727.421i	− 1.12430i	−0.827036	−	0.562149i	$-0.809975\pi$
$647$	− 727.421i	− 1.12430i	0.827036	−	0.562149i	$-0.190025\pi$
$648$	0	0
$649$	−2.37241	−0.00365548
$650$	− 1100.90i	− 1.69369i
$651$	0	0
$652$	−304.662	−0.467272
$653$	− 445.966i	− 0.682949i	−0.939891	−	0.341475i	$-0.889074\pi$
$653$	− 445.966i	− 0.682949i	0.939891	−	0.341475i	$-0.110926\pi$
$654$	0	0
$655$	−200.269	−0.305755
$656$	− 211.820i	− 0.322896i
$657$	0	0
$658$	−94.1796	−0.143130
$659$	− 1167.70i	− 1.77192i	−0.463760	−	0.885961i	$-0.653500\pi$
$659$	− 1167.70i	− 1.77192i	0.463760	−	0.885961i	$-0.346500\pi$
$660$	0	0
$661$	1018.80	1.54130	0.770652	−	0.637256i	$-0.219931\pi$
$661$	1018.80	1.54130	0.770652	+	0.637256i	$0.219931\pi$
$662$	− 199.625i	− 0.301548i
$663$	0	0
$664$	−211.236	−0.318126
$665$	− 85.1097i	− 0.127984i
$666$	0	0
$667$	428.980	0.643148
$668$	39.1995i	0.0586818i
$669$	0	0
$670$	752.789	1.12357
$671$	24.1836i	0.0360411i
$672$	0	0
$673$	547.102	0.812930	0.406465	−	0.913666i	$-0.366761\pi$
$673$	547.102	0.812930	0.406465	+	0.913666i	$0.366761\pi$
$674$	1613.66i	2.39415i
$675$	0	0
$676$	−82.1129	−0.121469
$677$	619.136i	0.914529i	0.889331	+	0.457264i	$0.151171\pi$
$677$	619.136i	0.914529i	−0.889331	+	0.457264i	$0.848829\pi$
$678$	0	0
$679$	401.890	0.591885
$680$	− 361.800i	− 0.532059i
$681$	0	0
$682$	103.330	0.151510
$683$	− 667.806i	− 0.977754i	−0.872353	−	0.488877i	$-0.837407\pi$
$683$	− 667.806i	− 0.977754i	0.872353	−	0.488877i	$-0.162593\pi$
$684$	0	0
$685$	1539.43	2.24734
$686$	− 753.751i	− 1.09876i
$687$	0	0
$688$	−768.551	−1.11708
$689$	913.205i	1.32541i
$690$	0	0
$691$	624.648	0.903977	0.451988	−	0.892024i	$-0.350715\pi$
$691$	624.648	0.903977	0.451988	+	0.892024i	$0.350715\pi$
$692$	− 18.5672i	− 0.0268313i
$693$	0	0
$694$	−1013.73	−1.46070
$695$	1193.91i	1.71786i
$696$	0	0
$697$	−110.128	−0.158003
$698$	1026.43i	1.47053i
$699$	0	0
$700$	−233.792	−0.333989
$701$	− 714.318i	− 1.01900i	−0.860471	−	0.509499i	$-0.829831\pi$
$701$	− 714.318i	− 1.01900i	0.860471	−	0.509499i	$-0.170169\pi$
$702$	0	0
$703$	159.078	0.226284
$704$	− 4.83500i	− 0.00686790i
$705$	0	0
$706$	358.644	0.507994
$707$	− 305.393i	− 0.431956i
$708$	0	0
$709$	483.391	0.681792	0.340896	−	0.940101i	$-0.389270\pi$
$709$	483.391	0.681792	0.340896	+	0.940101i	$0.389270\pi$
$710$	102.236i	0.143994i
$711$	0	0
$712$	−358.703	−0.503797
$713$	791.519i	1.11013i
$714$	0	0
$715$	−138.093	−0.193137
$716$	175.340i	0.244888i
$717$	0	0
$718$	−466.382	−0.649556
$719$	− 1146.34i	− 1.59436i	−0.603745	−	0.797178i	$-0.706325\pi$
$719$	− 1146.34i	− 1.59436i	0.603745	−	0.797178i	$-0.293675\pi$
$720$	0	0
$721$	−116.989	−0.162259
$722$	867.765i	1.20189i
$723$	0	0
$724$	−464.260	−0.641244
$725$	545.616i	0.752574i
$726$	0	0
$727$	−396.069	−0.544799	−0.272400	−	0.962184i	$-0.587817\pi$
$727$	−396.069	−0.544799	−0.272400	+	0.962184i	$0.587817\pi$
$728$	− 240.153i	− 0.329881i
$729$	0	0
$730$	500.591	0.685742
$731$	399.580i	0.546621i
$732$	0	0
$733$	1143.30	1.55975	0.779876	−	0.625934i	$-0.215282\pi$
$733$	1143.30	1.55975	0.779876	+	0.625934i	$0.215282\pi$
$734$	1716.10i	2.33801i
$735$	0	0
$736$	−747.760	−1.01598
$737$	− 52.1815i	− 0.0708025i
$738$	0	0
$739$	−940.492	−1.27265	−0.636327	−	0.771419i	$-0.719547\pi$
$739$	−940.492	−1.27265	−0.636327	+	0.771419i	$0.719547\pi$
$740$	− 790.756i	− 1.06859i
$741$	0	0
$742$	560.414	0.755274
$743$	1295.75i	1.74394i	0.489555	+	0.871972i	$0.337159\pi$
$743$	1295.75i	1.74394i	−0.489555	+	0.871972i	$0.662841\pi$
$744$	0	0
$745$	324.177	0.435137
$746$	1728.47i	2.31699i
$747$	0	0
$748$	28.1874	0.0376837
$749$	− 716.826i	− 0.957044i
$750$	0	0
$751$	−245.217	−0.326521	−0.163260	−	0.986583i	$-0.552201\pi$
$751$	−245.217	−0.326521	−0.163260	+	0.986583i	$0.552201\pi$
$752$	− 212.782i	− 0.282955i
$753$	0	0
$754$	629.925	0.835444
$755$	1700.35i	2.25212i
$756$	0	0
$757$	−210.908	−0.278610	−0.139305	−	0.990250i	$-0.544487\pi$
$757$	−210.908	−0.278610	−0.139305	+	0.990250i	$0.544487\pi$
$758$	290.315i	0.383001i
$759$	0	0
$760$	110.830	0.145829
$761$	237.328i	0.311863i	0.987768	+	0.155932i	$0.0498379\pi$
$761$	237.328i	0.311863i	−0.987768	+	0.155932i	$0.950162\pi$
$762$	0	0
$763$	139.238	0.182487
$764$	94.2138i	0.123316i
$765$	0	0
$766$	1399.29	1.82675
$767$	− 26.6855i	− 0.0347921i
$768$	0	0
$769$	−231.536	−0.301087	−0.150543	−	0.988603i	$-0.548102\pi$
$769$	−231.536	−0.301087	−0.150543	+	0.988603i	$0.548102\pi$
$770$	84.7444i	0.110058i
$771$	0	0
$772$	155.655	0.201625
$773$	1365.28i	1.76621i	0.469171	+	0.883107i	$0.344553\pi$
$773$	1365.28i	1.76621i	−0.469171	+	0.883107i	$0.655447\pi$
$774$	0	0
$775$	−1006.73	−1.29900
$776$	523.344i	0.674412i
$777$	0	0
$778$	−725.717	−0.932798
$779$	− 33.7355i	− 0.0433062i
$780$	0	0
$781$	7.08671	0.00907389
$782$	623.945i	0.797884i
$783$	0	0
$784$	723.634	0.923003
$785$	− 447.660i	− 0.570268i
$786$	0	0
$787$	690.289	0.877114	0.438557	−	0.898703i	$-0.355490\pi$
$787$	690.289	0.877114	0.438557	+	0.898703i	$0.355490\pi$
$788$	566.449i	0.718844i
$789$	0	0
$790$	−751.618	−0.951416
$791$	86.3975i	0.109226i
$792$	0	0
$793$	−272.024	−0.343031
$794$	386.744i	0.487083i
$795$	0	0
$796$	367.625	0.461841
$797$	− 877.407i	− 1.10089i	−0.834873	−	0.550443i	$-0.814459\pi$
$797$	− 877.407i	− 1.10089i	0.834873	−	0.550443i	$-0.185541\pi$
$798$	0	0
$799$	−110.628	−0.138459
$800$	− 951.070i	− 1.18884i
$801$	0	0
$802$	1353.76	1.68798
$803$	− 34.6997i	− 0.0432126i
$804$	0	0
$805$	−649.151	−0.806398
$806$	1162.29i	1.44204i
$807$	0	0
$808$	397.685	0.492184
$809$	− 271.708i	− 0.335857i	−0.985799	−	0.167928i	$-0.946292\pi$
$809$	− 271.708i	− 0.335857i	0.985799	−	0.167928i	$-0.0537077\pi$
$810$	0	0
$811$	590.185	0.727725	0.363862	−	0.931453i	$-0.381458\pi$
$811$	590.185	0.727725	0.363862	+	0.931453i	$0.381458\pi$
$812$	− 133.774i	− 0.164747i
$813$	0	0
$814$	−158.395	−0.194589
$815$	− 1075.93i	− 1.32016i
$816$	0	0
$817$	−122.404	−0.149821
$818$	633.904i	0.774944i
$819$	0	0
$820$	−167.695	−0.204506
$821$	− 1133.04i	− 1.38007i	−0.723774	−	0.690037i	$-0.757594\pi$
$821$	− 1133.04i	− 1.38007i	0.723774	−	0.690037i	$-0.242406\pi$
$822$	0	0
$823$	−839.497	−1.02004	−0.510022	−	0.860161i	$-0.670363\pi$
$823$	−839.497	−1.02004	−0.510022	+	0.860161i	$0.670363\pi$
$824$	− 152.343i	− 0.184883i
$825$	0	0
$826$	−16.3763	−0.0198261
$827$	− 104.942i	− 0.126895i	−0.997985	−	0.0634477i	$-0.979790\pi$
$827$	− 104.942i	− 0.126895i	0.997985	−	0.0634477i	$-0.0202096\pi$
$828$	0	0
$829$	−361.576	−0.436159	−0.218079	−	0.975931i	$-0.569979\pi$
$829$	−361.576	−0.436159	−0.218079	+	0.975931i	$0.569979\pi$
$830$	838.449i	1.01018i
$831$	0	0
$832$	54.3855	0.0653672
$833$	− 376.228i	− 0.451654i
$834$	0	0
$835$	−138.435	−0.165790
$836$	8.63465i	0.0103285i
$837$	0	0
$838$	−17.2096	−0.0205365
$839$	1033.20i	1.23146i	0.787957	+	0.615730i	$0.211139\pi$
$839$	1033.20i	1.23146i	−0.787957	+	0.615730i	$0.788861\pi$
$840$	0	0
$841$	528.802	0.628778
$842$	− 406.862i	− 0.483209i
$843$	0	0
$844$	814.898	0.965519
$845$	− 289.986i	− 0.343179i
$846$	0	0
$847$	−426.921	−0.504038
$848$	1266.16i	1.49311i
$849$	0	0
$850$	−793.591	−0.933636
$851$	− 1213.32i	− 1.42576i
$852$	0	0
$853$	1162.20	1.36249	0.681245	−	0.732056i	$-0.261439\pi$
$853$	1162.20	1.36249	0.681245	+	0.732056i	$0.261439\pi$
$854$	166.935i	0.195474i
$855$	0	0
$856$	933.455	1.09048
$857$	172.029i	0.200735i	0.994950	+	0.100367i	$0.0320018\pi$
$857$	172.029i	0.200735i	−0.994950	+	0.100367i	$0.967998\pi$
$858$	0	0
$859$	−1205.40	−1.40326	−0.701631	−	0.712540i	$-0.747545\pi$
$859$	−1205.40	−1.40326	−0.701631	+	0.712540i	$0.747545\pi$
$860$	608.452i	0.707503i
$861$	0	0
$862$	−1079.97	−1.25287
$863$	1547.43i	1.79308i	0.442965	+	0.896539i	$0.353927\pi$
$863$	1547.43i	1.79308i	−0.442965	+	0.896539i	$0.646073\pi$
$864$	0	0
$865$	65.5712	0.0758049
$866$	− 158.751i	− 0.183315i
$867$	0	0
$868$	246.830	0.284366
$869$	52.1003i	0.0599543i
$870$	0	0
$871$	586.952	0.673883
$872$	181.316i	0.207931i
$873$	0	0
$874$	−191.133	−0.218688
$875$	− 157.208i	− 0.179666i
$876$	0	0
$877$	−1072.48	−1.22289	−0.611445	−	0.791287i	$-0.709412\pi$
$877$	−1072.48	−1.22289	−0.611445	+	0.791287i	$0.709412\pi$
$878$	638.843i	0.727611i
$879$	0	0
$880$	−191.465	−0.217574
$881$	− 363.670i	− 0.412793i	−0.978468	−	0.206396i	$-0.933826\pi$
$881$	− 363.670i	− 0.412793i	0.978468	−	0.206396i	$-0.0661736\pi$
$882$	0	0
$883$	−910.225	−1.03083	−0.515416	−	0.856940i	$-0.672363\pi$
$883$	−910.225	−1.03083	−0.515416	+	0.856940i	$0.672363\pi$
$884$	317.060i	0.358665i
$885$	0	0
$886$	683.109	0.771004
$887$	− 425.744i	− 0.479982i	−0.970775	−	0.239991i	$-0.922855\pi$
$887$	− 425.744i	− 0.479982i	0.970775	−	0.239991i	$-0.0771446\pi$
$888$	0	0
$889$	−40.3087	−0.0453416
$890$	1423.79i	1.59976i
$891$	0	0
$892$	827.668	0.927879
$893$	− 33.8888i	− 0.0379494i
$894$	0	0
$895$	−619.221	−0.691867
$896$	− 474.029i	− 0.529050i
$897$	0	0
$898$	−1374.17	−1.53026
$899$	− 576.042i	− 0.640759i
$900$	0	0
$901$	658.292	0.730624
$902$	33.5907i	0.0372403i
$903$	0	0
$904$	−112.507	−0.124455
$905$	− 1639.56i	− 1.81167i
$906$	0	0
$907$	−735.774	−0.811218	−0.405609	−	0.914047i	$-0.632940\pi$
$907$	−735.774	−0.811218	−0.405609	+	0.914047i	$0.632940\pi$
$908$	− 250.717i	− 0.276121i
$909$	0	0
$910$	−953.230	−1.04751
$911$	− 791.514i	− 0.868841i	−0.900710	−	0.434420i	$-0.856953\pi$
$911$	− 791.514i	− 0.868841i	0.900710	−	0.434420i	$-0.143047\pi$
$912$	0	0
$913$	58.1192	0.0636574
$914$	− 166.584i	− 0.182258i
$915$	0	0
$916$	27.6924	0.0302319
$917$	95.8263i	0.104500i
$918$	0	0
$919$	−264.051	−0.287324	−0.143662	−	0.989627i	$-0.545888\pi$
$919$	−264.051	−0.287324	−0.143662	+	0.989627i	$0.545888\pi$
$920$	− 845.328i	− 0.918835i
$921$	0	0
$922$	1976.62	2.14384
$923$	79.7134i	0.0863634i
$924$	0	0
$925$	1543.22	1.66834
$926$	1161.60i	1.25442i
$927$	0	0
$928$	544.196	0.586418
$929$	− 197.077i	− 0.212139i	−0.994359	−	0.106070i	$-0.966173\pi$
$929$	− 197.077i	− 0.212139i	0.994359	−	0.106070i	$-0.0338266\pi$
$930$	0	0
$931$	115.250	0.123791
$932$	− 66.4257i	− 0.0712722i
$933$	0	0
$934$	−1445.75	−1.54791
$935$	99.5453i	0.106466i
$936$	0	0
$937$	−1280.61	−1.36671	−0.683354	−	0.730087i	$-0.739479\pi$
$937$	−1280.61	−1.36671	−0.683354	+	0.730087i	$0.739479\pi$
$938$	− 360.200i	− 0.384008i
$939$	0	0
$940$	−168.457	−0.179209
$941$	− 571.596i	− 0.607434i	−0.952762	−	0.303717i	$-0.901772\pi$
$941$	− 571.596i	− 0.607434i	0.952762	−	0.303717i	$-0.0982278\pi$
$942$	0	0
$943$	−257.309	−0.272862
$944$	− 36.9994i	− 0.0391943i
$945$	0	0
$946$	121.878	0.128835
$947$	− 135.984i	− 0.143594i	−0.997419	−	0.0717971i	$-0.977127\pi$
$947$	− 135.984i	− 0.143594i	0.997419	−	0.0717971i	$-0.0228734\pi$
$948$	0	0
$949$	390.313	0.411289
$950$	− 243.101i	− 0.255896i
$951$	0	0
$952$	−173.117	−0.181845
$953$	− 360.322i	− 0.378092i	−0.981968	−	0.189046i	$-0.939460\pi$
$953$	− 360.322i	− 0.378092i	0.981968	−	0.189046i	$-0.0605395\pi$
$954$	0	0
$955$	−332.721	−0.348399
$956$	320.099i	0.334832i
$957$	0	0
$958$	−1094.76	−1.14275
$959$	− 736.596i	− 0.768088i
$960$	0	0
$961$	101.868	0.106002
$962$	− 1781.68i	− 1.85205i
$963$	0	0
$964$	293.115	0.304062
$965$	549.703i	0.569641i
$966$	0	0
$967$	1222.22	1.26393	0.631963	−	0.774999i	$-0.282250\pi$
$967$	1222.22	1.26393	0.631963	+	0.774999i	$0.282250\pi$
$968$	− 555.939i	− 0.574317i
$969$	0	0
$970$	2077.29	2.14153
$971$	286.990i	0.295561i	0.989020	+	0.147781i	$0.0472130\pi$
$971$	286.990i	0.295561i	−0.989020	+	0.147781i	$0.952787\pi$
$972$	0	0
$973$	571.270	0.587123
$974$	− 1807.59i	− 1.85585i
$975$	0	0
$976$	−377.160	−0.386435
$977$	1714.15i	1.75451i	0.480028	+	0.877253i	$0.340627\pi$
$977$	1714.15i	1.75451i	−0.480028	+	0.877253i	$0.659373\pi$
$978$	0	0
$979$	98.6933	0.100810
$980$	− 572.892i	− 0.584584i
$981$	0	0
$982$	−680.812	−0.693292
$983$	807.305i	0.821267i	0.911801	+	0.410633i	$0.134692\pi$
$983$	807.305i	0.821267i	−0.911801	+	0.410633i	$0.865308\pi$
$984$	0	0
$985$	−2000.44	−2.03091
$986$	− 454.087i	− 0.460535i
$987$	0	0
$988$	−97.1251	−0.0983048
$989$	933.599i	0.943983i
$990$	0	0
$991$	167.284	0.168804	0.0844019	−	0.996432i	$-0.473102\pi$
$991$	167.284	0.168804	0.0844019	+	0.996432i	$0.473102\pi$
$992$	1004.11i	1.01220i
$993$	0	0
$994$	48.9183	0.0492136
$995$	1298.29i	1.30481i
$996$	0	0
$997$	1733.30	1.73851	0.869257	−	0.494360i	$-0.164598\pi$
$997$	1733.30	1.73851	0.869257	+	0.494360i	$0.164598\pi$
$998$	458.071i	0.458989i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.19	✓	84
3.2	odd	2	inner	1143.3.b.a.890.66	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.19	✓	84	1.1	even	1	trivial
1143.3.b.a.890.66	yes	84	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} +O(q^{10})\)	\(q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} -18.4878 q^{10} +1.28153i q^{11} -14.4150 q^{13} +8.84618i q^{14} -19.9864 q^{16} +10.3912i q^{17} -3.18314 q^{19} +15.8230i q^{20} +3.16948 q^{22} +24.2785i q^{23} -30.8796 q^{25} +35.6512i q^{26} +7.57108 q^{28} -17.6691i q^{29} +32.6017 q^{31} +30.7993i q^{32} +25.6995 q^{34} +26.7377i q^{35} -49.9752 q^{37} +7.87253i q^{38} -34.8180 q^{40} +10.5982i q^{41} +38.4537 q^{43} -2.71262i q^{44} +60.0455 q^{46} +10.6464i q^{47} -36.2064 q^{49} +76.3715i q^{50} +30.5124 q^{52} -63.3509i q^{53} +9.57977 q^{55} +16.6599i q^{56} -43.6992 q^{58} +1.85123i q^{59} +18.8709 q^{61} -80.6304i q^{62} -3.77284 q^{64} +107.756i q^{65} -40.7181 q^{67} -21.9951i q^{68} +66.1275 q^{70} -5.52989i q^{71} -27.0768 q^{73} +123.599i q^{74} +6.73777 q^{76} -4.58380i q^{77} +40.6548 q^{79} +149.404i q^{80} +26.2115 q^{82} -45.3514i q^{83} +77.6770 q^{85} -95.1037i q^{86} +5.96905 q^{88} -77.0121i q^{89} +51.5599 q^{91} -51.3905i q^{92} +26.3305 q^{94} +23.7948i q^{95} -112.360 q^{97} +89.5455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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