LMFDB - Embedded newform 1143.3.b.a.890.79

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			890.79
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.62368i q^{2} -9.13107 q^{4} +3.89223i q^{5} -8.96960 q^{7} -18.5934i q^{8} +O(q^{10})$	\(q+3.62368i q^{2} -9.13107 q^{4} +3.89223i q^{5} -8.96960 q^{7} -18.5934i q^{8} -14.1042 q^{10} +3.39451i q^{11} +4.51211 q^{13} -32.5030i q^{14} +30.8521 q^{16} -0.588323i q^{17} -29.7256 q^{19} -35.5402i q^{20} -12.3006 q^{22} -25.8399i q^{23} +9.85055 q^{25} +16.3504i q^{26} +81.9021 q^{28} -30.9303i q^{29} +11.9611 q^{31} +37.4249i q^{32} +2.13190 q^{34} -34.9117i q^{35} +5.42650 q^{37} -107.716i q^{38} +72.3696 q^{40} +40.4950i q^{41} -32.6461 q^{43} -30.9955i q^{44} +93.6356 q^{46} +41.3299i q^{47} +31.4538 q^{49} +35.6953i q^{50} -41.2004 q^{52} +16.8242i q^{53} -13.2122 q^{55} +166.775i q^{56} +112.081 q^{58} -60.6872i q^{59} +75.7718 q^{61} +43.3430i q^{62} -12.2073 q^{64} +17.5622i q^{65} +33.9228 q^{67} +5.37202i q^{68} +126.509 q^{70} -30.0600i q^{71} -0.689938 q^{73} +19.6639i q^{74} +271.427 q^{76} -30.4474i q^{77} -72.2732 q^{79} +120.084i q^{80} -146.741 q^{82} +21.0132i q^{83} +2.28989 q^{85} -118.299i q^{86} +63.1153 q^{88} +139.493i q^{89} -40.4718 q^{91} +235.946i q^{92} -149.766 q^{94} -115.699i q^{95} -32.4602 q^{97} +113.978i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.62368i	1.81184i	0.423448	+	0.905920i	$0.360820\pi$
$2$	3.62368i	1.81184i	−0.423448	+	0.905920i	$0.639180\pi$
$3$	0	0
$3$	0	0
$4$	−9.13107	−2.28277
$5$	3.89223i	0.778446i	0.921144	+	0.389223i	$0.127256\pi$
$5$	3.89223i	0.778446i	−0.921144	+	0.389223i	$0.872744\pi$
$6$	0	0
$7$	−8.96960	−1.28137	−0.640686	−	0.767803i	$-0.721350\pi$
$7$	−8.96960	−1.28137	−0.640686	+	0.767803i	$0.721350\pi$
$8$	− 18.5934i	− 2.32417i
$9$	0	0
$10$	−14.1042	−1.41042
$11$	3.39451i	0.308592i	0.988025	+	0.154296i	$0.0493109\pi$
$11$	3.39451i	0.308592i	−0.988025	+	0.154296i	$0.950689\pi$
$12$	0	0
$13$	4.51211	0.347085	0.173543	−	0.984826i	$-0.444479\pi$
$13$	4.51211	0.347085	0.173543	+	0.984826i	$0.444479\pi$
$14$	− 32.5030i	− 2.32164i
$15$	0	0
$16$	30.8521	1.92826
$17$	− 0.588323i	− 0.0346073i	−0.999850	−	0.0173036i	$-0.994492\pi$
$17$	− 0.588323i	− 0.0346073i	0.999850	−	0.0173036i	$-0.00550819\pi$
$18$	0	0
$19$	−29.7256	−1.56451	−0.782253	−	0.622960i	$-0.785930\pi$
$19$	−29.7256	−1.56451	−0.782253	+	0.622960i	$0.785930\pi$
$20$	− 35.5402i	− 1.77701i
$21$	0	0
$22$	−12.3006	−0.559119
$23$	− 25.8399i	− 1.12347i	−0.827316	−	0.561737i	$-0.810133\pi$
$23$	− 25.8399i	− 1.12347i	0.827316	−	0.561737i	$-0.189867\pi$
$24$	0	0
$25$	9.85055	0.394022
$26$	16.3504i	0.628863i
$27$	0	0
$28$	81.9021	2.92507
$29$	− 30.9303i	− 1.06656i	−0.845938	−	0.533281i	$-0.820959\pi$
$29$	− 30.9303i	− 1.06656i	0.845938	−	0.533281i	$-0.179041\pi$
$30$	0	0
$31$	11.9611	0.385840	0.192920	−	0.981214i	$-0.438204\pi$
$31$	11.9611	0.385840	0.192920	+	0.981214i	$0.438204\pi$
$32$	37.4249i	1.16953i
$33$	0	0
$34$	2.13190	0.0627029
$35$	− 34.9117i	− 0.997479i
$36$	0	0
$37$	5.42650	0.146662	0.0733311	−	0.997308i	$-0.476637\pi$
$37$	5.42650	0.146662	0.0733311	+	0.997308i	$0.476637\pi$
$38$	− 107.716i	− 2.83464i
$39$	0	0
$40$	72.3696	1.80924
$41$	40.4950i	0.987682i	0.869552	+	0.493841i	$0.164407\pi$
$41$	40.4950i	0.987682i	−0.869552	+	0.493841i	$0.835593\pi$
$42$	0	0
$43$	−32.6461	−0.759211	−0.379606	−	0.925148i	$-0.623940\pi$
$43$	−32.6461	−0.759211	−0.379606	+	0.925148i	$0.623940\pi$
$44$	− 30.9955i	− 0.704443i
$45$	0	0
$46$	93.6356	2.03556
$47$	41.3299i	0.879359i	0.898155	+	0.439680i	$0.144908\pi$
$47$	41.3299i	0.879359i	−0.898155	+	0.439680i	$0.855092\pi$
$48$	0	0
$49$	31.4538	0.641914
$50$	35.6953i	0.713905i
$51$	0	0
$52$	−41.2004	−0.792315
$53$	16.8242i	0.317437i	0.987324	+	0.158719i	$0.0507363\pi$
$53$	16.8242i	0.317437i	−0.987324	+	0.158719i	$0.949264\pi$
$54$	0	0
$55$	−13.2122	−0.240222
$56$	166.775i	2.97813i
$57$	0	0
$58$	112.081	1.93244
$59$	− 60.6872i	− 1.02860i	−0.857611	−	0.514298i	$-0.828052\pi$
$59$	− 60.6872i	− 1.02860i	0.857611	−	0.514298i	$-0.171948\pi$
$60$	0	0
$61$	75.7718	1.24216	0.621080	−	0.783747i	$-0.286694\pi$
$61$	75.7718	1.24216	0.621080	+	0.783747i	$0.286694\pi$
$62$	43.3430i	0.699081i
$63$	0	0
$64$	−12.2073	−0.190739
$65$	17.5622i	0.270187i
$66$	0	0
$67$	33.9228	0.506311	0.253155	−	0.967426i	$-0.418532\pi$
$67$	33.9228	0.506311	0.253155	+	0.967426i	$0.418532\pi$
$68$	5.37202i	0.0790003i
$69$	0	0
$70$	126.509	1.80727
$71$	− 30.0600i	− 0.423380i	−0.977337	−	0.211690i	$-0.932103\pi$
$71$	− 30.0600i	− 0.423380i	0.977337	−	0.211690i	$-0.0678967\pi$
$72$	0	0
$73$	−0.689938	−0.00945120	−0.00472560	−	0.999989i	$-0.501504\pi$
$73$	−0.689938	−0.00945120	−0.00472560	+	0.999989i	$0.501504\pi$
$74$	19.6639i	0.265728i
$75$	0	0
$76$	271.427	3.57140
$77$	− 30.4474i	− 0.395421i
$78$	0	0
$79$	−72.2732	−0.914850	−0.457425	−	0.889248i	$-0.651228\pi$
$79$	−72.2732	−0.914850	−0.457425	+	0.889248i	$0.651228\pi$
$80$	120.084i	1.50104i
$81$	0	0
$82$	−146.741	−1.78952
$83$	21.0132i	0.253171i	0.991956	+	0.126585i	$0.0404018\pi$
$83$	21.0132i	0.253171i	−0.991956	+	0.126585i	$0.959598\pi$
$84$	0	0
$85$	2.28989	0.0269399
$86$	− 118.299i	− 1.37557i
$87$	0	0
$88$	63.1153	0.717219
$89$	139.493i	1.56733i	0.621183	+	0.783666i	$0.286653\pi$
$89$	139.493i	1.56733i	−0.621183	+	0.783666i	$0.713347\pi$
$90$	0	0
$91$	−40.4718	−0.444745
$92$	235.946i	2.56463i
$93$	0	0
$94$	−149.766	−1.59326
$95$	− 115.699i	− 1.21788i
$96$	0	0
$97$	−32.4602	−0.334641	−0.167321	−	0.985903i	$-0.553512\pi$
$97$	−32.4602	−0.334641	−0.167321	+	0.985903i	$0.553512\pi$
$98$	113.978i	1.16305i
$99$	0	0
$100$	−89.9461	−0.899461
$101$	− 13.2198i	− 0.130889i	−0.997856	−	0.0654447i	$-0.979153\pi$
$101$	− 13.2198i	− 0.130889i	0.997856	−	0.0654447i	$-0.0208466\pi$
$102$	0	0
$103$	106.910	1.03796	0.518981	−	0.854786i	$-0.326311\pi$
$103$	106.910	1.03796	0.518981	+	0.854786i	$0.326311\pi$
$104$	− 83.8952i	− 0.806685i
$105$	0	0
$106$	−60.9655	−0.575146
$107$	9.87832i	0.0923207i	0.998934	+	0.0461604i	$0.0146985\pi$
$107$	9.87832i	0.0923207i	−0.998934	+	0.0461604i	$0.985301\pi$
$108$	0	0
$109$	7.38727	0.0677731	0.0338866	−	0.999426i	$-0.489212\pi$
$109$	7.38727	0.0677731	0.0338866	+	0.999426i	$0.489212\pi$
$110$	− 47.8768i	− 0.435244i
$111$	0	0
$112$	−276.731	−2.47082
$113$	− 145.173i	− 1.28472i	−0.766404	−	0.642359i	$-0.777956\pi$
$113$	− 145.173i	− 1.28472i	0.766404	−	0.642359i	$-0.222044\pi$
$114$	0	0
$115$	100.575	0.874564
$116$	282.426i	2.43471i
$117$	0	0
$118$	219.911	1.86365
$119$	5.27703i	0.0443448i
$120$	0	0
$121$	109.477	0.904771
$122$	274.573i	2.25060i
$123$	0	0
$124$	−109.217	−0.880784
$125$	135.646i	1.08517i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	105.464i	0.823938i
$129$	0	0
$130$	−63.6397	−0.489536
$131$	− 127.633i	− 0.974301i	−0.873318	−	0.487151i	$-0.838036\pi$
$131$	− 127.633i	− 0.974301i	0.873318	−	0.487151i	$-0.161964\pi$
$132$	0	0
$133$	266.627	2.00471
$134$	122.925i	0.917354i
$135$	0	0
$136$	−10.9389	−0.0804332
$137$	− 172.488i	− 1.25903i	−0.776987	−	0.629517i	$-0.783253\pi$
$137$	− 172.488i	− 1.25903i	0.776987	−	0.629517i	$-0.216747\pi$
$138$	0	0
$139$	243.875	1.75450	0.877249	−	0.480035i	$-0.159376\pi$
$139$	243.875	1.75450	0.877249	+	0.480035i	$0.159376\pi$
$140$	318.782i	2.27701i
$141$	0	0
$142$	108.928	0.767097
$143$	15.3164i	0.107108i
$144$	0	0
$145$	120.388	0.830260
$146$	− 2.50011i	− 0.0171241i
$147$	0	0
$148$	−49.5497	−0.334796
$149$	− 100.923i	− 0.677337i	−0.940906	−	0.338668i	$-0.890024\pi$
$149$	− 100.923i	− 0.677337i	0.940906	−	0.338668i	$-0.109976\pi$
$150$	0	0
$151$	−261.415	−1.73122	−0.865611	−	0.500717i	$-0.833070\pi$
$151$	−261.415	−1.73122	−0.865611	+	0.500717i	$0.833070\pi$
$152$	552.699i	3.63618i
$153$	0	0
$154$	110.332	0.716439
$155$	46.5551i	0.300356i
$156$	0	0
$157$	−120.944	−0.770345	−0.385172	−	0.922845i	$-0.625858\pi$
$157$	−120.944	−0.770345	−0.385172	+	0.922845i	$0.625858\pi$
$158$	− 261.895i	− 1.65756i
$159$	0	0
$160$	−145.666	−0.910414
$161$	231.774i	1.43959i
$162$	0	0
$163$	−75.6333	−0.464008	−0.232004	−	0.972715i	$-0.574528\pi$
$163$	−75.6333	−0.464008	−0.232004	+	0.972715i	$0.574528\pi$
$164$	− 369.762i	− 2.25465i
$165$	0	0
$166$	−76.1450	−0.458705
$167$	− 141.550i	− 0.847606i	−0.905754	−	0.423803i	$-0.860695\pi$
$167$	− 141.550i	− 0.847606i	0.905754	−	0.423803i	$-0.139305\pi$
$168$	0	0
$169$	−148.641	−0.879532
$170$	8.29783i	0.0488108i
$171$	0	0
$172$	298.094	1.73310
$173$	− 131.050i	− 0.757512i	−0.925497	−	0.378756i	$-0.876352\pi$
$173$	− 131.050i	− 0.757512i	0.925497	−	0.378756i	$-0.123648\pi$
$174$	0	0
$175$	−88.3555	−0.504889
$176$	104.728i	0.595044i
$177$	0	0
$178$	−505.477	−2.83976
$179$	− 179.416i	− 1.00233i	−0.865353	−	0.501163i	$-0.832906\pi$
$179$	− 179.416i	− 1.00233i	0.865353	−	0.501163i	$-0.167094\pi$
$180$	0	0
$181$	339.063	1.87327	0.936637	−	0.350301i	$-0.113921\pi$
$181$	339.063	1.87327	0.936637	+	0.350301i	$0.113921\pi$
$182$	− 146.657i	− 0.805808i
$183$	0	0
$184$	−480.451	−2.61114
$185$	21.1212i	0.114169i
$186$	0	0
$187$	1.99707	0.0106795
$188$	− 377.386i	− 2.00737i
$189$	0	0
$190$	419.256	2.20661
$191$	− 314.303i	− 1.64557i	−0.568354	−	0.822784i	$-0.692420\pi$
$191$	− 314.303i	− 1.64557i	0.568354	−	0.822784i	$-0.307580\pi$
$192$	0	0
$193$	−151.480	−0.784872	−0.392436	−	0.919779i	$-0.628367\pi$
$193$	−151.480	−0.784872	−0.392436	+	0.919779i	$0.628367\pi$
$194$	− 117.625i	− 0.606317i
$195$	0	0
$196$	−287.207	−1.46534
$197$	13.8619i	0.0703650i	0.999381	+	0.0351825i	$0.0112013\pi$
$197$	13.8619i	0.0703650i	−0.999381	+	0.0351825i	$0.988799\pi$
$198$	0	0
$199$	−351.838	−1.76803	−0.884015	−	0.467459i	$-0.845169\pi$
$199$	−351.838	−1.76803	−0.884015	+	0.467459i	$0.845169\pi$
$200$	− 183.155i	− 0.915774i
$201$	0	0
$202$	47.9045	0.237151
$203$	277.432i	1.36666i
$204$	0	0
$205$	−157.616	−0.768857
$206$	387.408i	1.88062i
$207$	0	0
$208$	139.208	0.669270
$209$	− 100.904i	− 0.482794i
$210$	0	0
$211$	−193.186	−0.915574	−0.457787	−	0.889062i	$-0.651358\pi$
$211$	−193.186	−0.915574	−0.457787	+	0.889062i	$0.651358\pi$
$212$	− 153.623i	− 0.724635i
$213$	0	0
$214$	−35.7959	−0.167270
$215$	− 127.066i	− 0.591005i
$216$	0	0
$217$	−107.286	−0.494405
$218$	26.7691i	0.122794i
$219$	0	0
$220$	120.642	0.548371
$221$	− 2.65458i	− 0.0120117i
$222$	0	0
$223$	312.551	1.40158	0.700788	−	0.713370i	$-0.252832\pi$
$223$	312.551	1.40158	0.700788	+	0.713370i	$0.252832\pi$
$224$	− 335.686i	− 1.49860i
$225$	0	0
$226$	526.061	2.32770
$227$	− 111.603i	− 0.491644i	−0.969315	−	0.245822i	$-0.920942\pi$
$227$	− 111.603i	− 0.491644i	0.969315	−	0.245822i	$-0.0790579\pi$
$228$	0	0
$229$	123.668	0.540034	0.270017	−	0.962855i	$-0.412971\pi$
$229$	123.668	0.540034	0.270017	+	0.962855i	$0.412971\pi$
$230$	364.451i	1.58457i
$231$	0	0
$232$	−575.098	−2.47887
$233$	327.222i	1.40439i	0.711986	+	0.702194i	$0.247796\pi$
$233$	327.222i	1.40439i	−0.711986	+	0.702194i	$0.752204\pi$
$234$	0	0
$235$	−160.865	−0.684534
$236$	554.139i	2.34805i
$237$	0	0
$238$	−19.1223	−0.0803457
$239$	− 16.7075i	− 0.0699059i	−0.999389	−	0.0349529i	$-0.988872\pi$
$239$	− 16.7075i	− 0.0699059i	0.999389	−	0.0349529i	$-0.0111281\pi$
$240$	0	0
$241$	84.7568	0.351688	0.175844	−	0.984418i	$-0.443735\pi$
$241$	84.7568	0.351688	0.175844	+	0.984418i	$0.443735\pi$
$242$	396.711i	1.63930i
$243$	0	0
$244$	−691.877	−2.83556
$245$	122.425i	0.499695i
$246$	0	0
$247$	−134.125	−0.543017
$248$	− 222.396i	− 0.896758i
$249$	0	0
$250$	−491.539	−1.96616
$251$	− 132.843i	− 0.529255i	−0.964351	−	0.264628i	$-0.914751\pi$
$251$	− 132.843i	− 0.529255i	0.964351	−	0.264628i	$-0.0852491\pi$
$252$	0	0
$253$	87.7138	0.346695
$254$	40.8368i	0.160775i
$255$	0	0
$256$	−430.998	−1.68358
$257$	− 285.101i	− 1.10934i	−0.832070	−	0.554671i	$-0.812844\pi$
$257$	− 285.101i	− 1.10934i	0.832070	−	0.554671i	$-0.187156\pi$
$258$	0	0
$259$	−48.6735	−0.187929
$260$	− 160.361i	− 0.616774i
$261$	0	0
$262$	462.503	1.76528
$263$	− 299.719i	− 1.13962i	−0.821777	−	0.569809i	$-0.807017\pi$
$263$	− 299.719i	− 1.13962i	0.821777	−	0.569809i	$-0.192983\pi$
$264$	0	0
$265$	−65.4836	−0.247108
$266$	966.172i	3.63222i
$267$	0	0
$268$	−309.751	−1.15579
$269$	− 263.792i	− 0.980641i	−0.871542	−	0.490320i	$-0.836880\pi$
$269$	− 263.792i	− 0.980641i	0.871542	−	0.490320i	$-0.163120\pi$
$270$	0	0
$271$	−61.5576	−0.227150	−0.113575	−	0.993529i	$-0.536230\pi$
$271$	−61.5576	−0.227150	−0.113575	+	0.993529i	$0.536230\pi$
$272$	− 18.1510i	− 0.0667317i
$273$	0	0
$274$	625.041	2.28117
$275$	33.4378i	0.121592i
$276$	0	0
$277$	−494.241	−1.78426	−0.892132	−	0.451774i	$-0.850791\pi$
$277$	−494.241	−1.78426	−0.892132	+	0.451774i	$0.850791\pi$
$278$	883.726i	3.17887i
$279$	0	0
$280$	−649.127	−2.31831
$281$	− 329.072i	− 1.17107i	−0.810646	−	0.585537i	$-0.800884\pi$
$281$	− 329.072i	− 1.17107i	0.810646	−	0.585537i	$-0.199116\pi$
$282$	0	0
$283$	−66.0556	−0.233412	−0.116706	−	0.993167i	$-0.537234\pi$
$283$	−66.0556	−0.233412	−0.116706	+	0.993167i	$0.537234\pi$
$284$	274.480i	0.966478i
$285$	0	0
$286$	−55.5017	−0.194062
$287$	− 363.224i	− 1.26559i
$288$	0	0
$289$	288.654	0.998802
$290$	436.247i	1.50430i
$291$	0	0
$292$	6.29987	0.0215749
$293$	− 369.480i	− 1.26102i	−0.776180	−	0.630512i	$-0.782845\pi$
$293$	− 369.480i	− 1.26102i	0.776180	−	0.630512i	$-0.217155\pi$
$294$	0	0
$295$	236.209	0.800707
$296$	− 100.897i	− 0.340868i
$297$	0	0
$298$	365.714	1.22723
$299$	− 116.592i	− 0.389941i
$300$	0	0
$301$	292.822	0.972832
$302$	− 947.283i	− 3.13670i
$303$	0	0
$304$	−917.099	−3.01677
$305$	294.921i	0.966955i
$306$	0	0
$307$	162.715	0.530016	0.265008	−	0.964246i	$-0.414625\pi$
$307$	162.715	0.530016	0.265008	+	0.964246i	$0.414625\pi$
$308$	278.017i	0.902653i
$309$	0	0
$310$	−168.701	−0.544197
$311$	308.574i	0.992201i	0.868265	+	0.496100i	$0.165235\pi$
$311$	308.574i	0.992201i	−0.868265	+	0.496100i	$0.834765\pi$
$312$	0	0
$313$	136.494	0.436083	0.218041	−	0.975940i	$-0.430033\pi$
$313$	136.494	0.436083	0.218041	+	0.975940i	$0.430033\pi$
$314$	− 438.263i	− 1.39574i
$315$	0	0
$316$	659.931	2.08839
$317$	145.648i	0.459457i	0.973255	+	0.229729i	$0.0737839\pi$
$317$	145.648i	0.459457i	−0.973255	+	0.229729i	$0.926216\pi$
$318$	0	0
$319$	104.993	0.329132
$320$	− 47.5137i	− 0.148480i
$321$	0	0
$322$	−839.874	−2.60830
$323$	17.4883i	0.0541433i
$324$	0	0
$325$	44.4468	0.136759
$326$	− 274.071i	− 0.840708i
$327$	0	0
$328$	752.937	2.29554
$329$	− 370.713i	− 1.12679i
$330$	0	0
$331$	−133.660	−0.403806	−0.201903	−	0.979406i	$-0.564713\pi$
$331$	−133.660	−0.403806	−0.201903	+	0.979406i	$0.564713\pi$
$332$	− 191.873i	− 0.577930i
$333$	0	0
$334$	512.933	1.53573
$335$	132.035i	0.394135i
$336$	0	0
$337$	43.6985	0.129669	0.0648346	−	0.997896i	$-0.479348\pi$
$337$	43.6985	0.129669	0.0648346	+	0.997896i	$0.479348\pi$
$338$	− 538.627i	− 1.59357i
$339$	0	0
$340$	−20.9091	−0.0614975
$341$	40.6019i	0.119067i
$342$	0	0
$343$	157.383	0.458842
$344$	607.000i	1.76454i
$345$	0	0
$346$	474.882	1.37249
$347$	− 142.865i	− 0.411715i	−0.978582	−	0.205858i	$-0.934002\pi$
$347$	− 142.865i	− 0.411715i	0.978582	−	0.205858i	$-0.0659984\pi$
$348$	0	0
$349$	404.034	1.15769	0.578845	−	0.815438i	$-0.303504\pi$
$349$	404.034	1.15769	0.578845	+	0.815438i	$0.303504\pi$
$350$	− 320.172i	− 0.914778i
$351$	0	0
$352$	−127.039	−0.360906
$353$	− 212.275i	− 0.601345i	−0.953727	−	0.300672i	$-0.902789\pi$
$353$	− 212.275i	− 0.601345i	0.953727	−	0.300672i	$-0.0972111\pi$
$354$	0	0
$355$	117.000	0.329579
$356$	− 1273.72i	− 3.57785i
$357$	0	0
$358$	650.147	1.81605
$359$	107.708i	0.300024i	0.988684	+	0.150012i	$0.0479312\pi$
$359$	107.708i	0.300024i	−0.988684	+	0.150012i	$0.952069\pi$
$360$	0	0
$361$	522.613	1.44768
$362$	1228.66i	3.39408i
$363$	0	0
$364$	369.551	1.01525
$365$	− 2.68540i	− 0.00735725i
$366$	0	0
$367$	60.3564	0.164459	0.0822295	−	0.996613i	$-0.473796\pi$
$367$	60.3564	0.164459	0.0822295	+	0.996613i	$0.473796\pi$
$368$	− 797.216i	− 2.16635i
$369$	0	0
$370$	−76.5364	−0.206855
$371$	− 150.906i	− 0.406755i
$372$	0	0
$373$	−288.400	−0.773191	−0.386596	−	0.922249i	$-0.626349\pi$
$373$	−288.400	−0.773191	−0.386596	+	0.922249i	$0.626349\pi$
$374$	7.23674i	0.0193496i
$375$	0	0
$376$	768.462	2.04378
$377$	− 139.561i	− 0.370188i
$378$	0	0
$379$	369.425	0.974735	0.487368	−	0.873197i	$-0.337957\pi$
$379$	369.425	0.974735	0.487368	+	0.873197i	$0.337957\pi$
$380$	1056.45i	2.78014i
$381$	0	0
$382$	1138.94	2.98151
$383$	− 209.087i	− 0.545918i	−0.962026	−	0.272959i	$-0.911998\pi$
$383$	− 209.087i	− 0.545918i	0.962026	−	0.272959i	$-0.0880023\pi$
$384$	0	0
$385$	118.508	0.307814
$386$	− 548.916i	− 1.42206i
$387$	0	0
$388$	296.396	0.763908
$389$	− 686.324i	− 1.76433i	−0.470942	−	0.882164i	$-0.656086\pi$
$389$	− 686.324i	− 1.76433i	0.470942	−	0.882164i	$-0.343914\pi$
$390$	0	0
$391$	−15.2022	−0.0388804
$392$	− 584.831i	− 1.49192i
$393$	0	0
$394$	−50.2312	−0.127490
$395$	− 281.304i	− 0.712161i
$396$	0	0
$397$	178.613	0.449906	0.224953	−	0.974370i	$-0.427777\pi$
$397$	178.613	0.449906	0.224953	+	0.974370i	$0.427777\pi$
$398$	− 1274.95i	− 3.20339i
$399$	0	0
$400$	303.911	0.759776
$401$	− 26.1163i	− 0.0651279i	−0.999470	−	0.0325640i	$-0.989633\pi$
$401$	− 26.1163i	− 0.0651279i	0.999470	−	0.0325640i	$-0.0103673\pi$
$402$	0	0
$403$	53.9696	0.133919
$404$	120.711i	0.298790i
$405$	0	0
$406$	−1005.33	−2.47617
$407$	18.4203i	0.0452587i
$408$	0	0
$409$	194.587	0.475763	0.237882	−	0.971294i	$-0.423547\pi$
$409$	194.587	0.475763	0.237882	+	0.971294i	$0.423547\pi$
$410$	− 571.149i	− 1.39305i
$411$	0	0
$412$	−976.203	−2.36943
$413$	544.340i	1.31801i
$414$	0	0
$415$	−81.7880	−0.197080
$416$	168.865i	0.405926i
$417$	0	0
$418$	365.644	0.874745
$419$	138.890i	0.331481i	0.986169	+	0.165740i	$0.0530014\pi$
$419$	138.890i	0.331481i	−0.986169	+	0.165740i	$0.946999\pi$
$420$	0	0
$421$	582.377	1.38332	0.691659	−	0.722224i	$-0.256880\pi$
$421$	582.377	1.38332	0.691659	+	0.722224i	$0.256880\pi$
$422$	− 700.045i	− 1.65887i
$423$	0	0
$424$	312.818	0.737778
$425$	− 5.79531i	− 0.0136360i
$426$	0	0
$427$	−679.643	−1.59167
$428$	− 90.1996i	− 0.210747i
$429$	0	0
$430$	460.447	1.07081
$431$	489.076i	1.13475i	0.823461	+	0.567373i	$0.192040\pi$
$431$	489.076i	1.13475i	−0.823461	+	0.567373i	$0.807960\pi$
$432$	0	0
$433$	−684.180	−1.58009	−0.790046	−	0.613047i	$-0.789944\pi$
$433$	−684.180	−1.58009	−0.790046	+	0.613047i	$0.789944\pi$
$434$	− 388.770i	− 0.895783i
$435$	0	0
$436$	−67.4537	−0.154710
$437$	768.107i	1.75768i
$438$	0	0
$439$	−350.343	−0.798048	−0.399024	−	0.916941i	$-0.630651\pi$
$439$	−350.343	−0.798048	−0.399024	+	0.916941i	$0.630651\pi$
$440$	245.659i	0.558316i
$441$	0	0
$442$	9.61935	0.0217632
$443$	419.851i	0.947745i	0.880593	+	0.473873i	$0.157144\pi$
$443$	419.851i	0.947745i	−0.880593	+	0.473873i	$0.842856\pi$
$444$	0	0
$445$	−542.937	−1.22008
$446$	1132.59i	2.53943i
$447$	0	0
$448$	109.495	0.244408
$449$	− 338.737i	− 0.754426i	−0.926126	−	0.377213i	$-0.876882\pi$
$449$	− 338.737i	− 0.754426i	0.926126	−	0.377213i	$-0.123118\pi$
$450$	0	0
$451$	−137.460	−0.304790
$452$	1325.58i	2.93271i
$453$	0	0
$454$	404.415	0.890781
$455$	− 157.526i	− 0.346210i
$456$	0	0
$457$	401.316	0.878153	0.439076	−	0.898450i	$-0.355306\pi$
$457$	401.316	0.878153	0.439076	+	0.898450i	$0.355306\pi$
$458$	448.133i	0.978456i
$459$	0	0
$460$	−918.355	−1.99642
$461$	187.767i	0.407303i	0.979043	+	0.203651i	$0.0652809\pi$
$461$	187.767i	0.407303i	−0.979043	+	0.203651i	$0.934719\pi$
$462$	0	0
$463$	489.118	1.05641	0.528205	−	0.849117i	$-0.322865\pi$
$463$	489.118	1.05641	0.528205	+	0.849117i	$0.322865\pi$
$464$	− 954.265i	− 2.05661i
$465$	0	0
$466$	−1185.75	−2.54453
$467$	371.837i	0.796225i	0.917337	+	0.398112i	$0.130335\pi$
$467$	371.837i	0.796225i	−0.917337	+	0.398112i	$0.869665\pi$
$468$	0	0
$469$	−304.274	−0.648772
$470$	− 582.925i	− 1.24027i
$471$	0	0
$472$	−1128.38	−2.39063
$473$	− 110.817i	− 0.234286i
$474$	0	0
$475$	−292.814	−0.616450
$476$	− 48.1849i	− 0.101229i
$477$	0	0
$478$	60.5427	0.126658
$479$	− 733.351i	− 1.53100i	−0.643433	−	0.765502i	$-0.722491\pi$
$479$	− 733.351i	− 1.53100i	0.643433	−	0.765502i	$-0.277509\pi$
$480$	0	0
$481$	24.4850	0.0509043
$482$	307.132i	0.637203i
$483$	0	0
$484$	−999.645	−2.06538
$485$	− 126.343i	− 0.260500i
$486$	0	0
$487$	248.594	0.510459	0.255229	−	0.966881i	$-0.417849\pi$
$487$	248.594	0.510459	0.255229	+	0.966881i	$0.417849\pi$
$488$	− 1408.85i	− 2.88699i
$489$	0	0
$490$	−443.630	−0.905368
$491$	− 324.038i	− 0.659955i	−0.943989	−	0.329977i	$-0.892959\pi$
$491$	− 324.038i	− 0.659955i	0.943989	−	0.329977i	$-0.107041\pi$
$492$	0	0
$493$	−18.1970	−0.0369108
$494$	− 486.027i	− 0.983861i
$495$	0	0
$496$	369.024	0.744000
$497$	269.626i	0.542507i
$498$	0	0
$499$	−682.389	−1.36751	−0.683756	−	0.729710i	$-0.739655\pi$
$499$	−682.389	−1.36751	−0.683756	+	0.729710i	$0.739655\pi$
$500$	− 1238.60i	− 2.47719i
$501$	0	0
$502$	481.381	0.958927
$503$	− 778.156i	− 1.54703i	−0.633779	−	0.773515i	$-0.718497\pi$
$503$	− 778.156i	− 1.54703i	0.633779	−	0.773515i	$-0.281503\pi$
$504$	0	0
$505$	51.4546	0.101890
$506$	317.847i	0.628156i
$507$	0	0
$508$	−102.902	−0.202563
$509$	147.442i	0.289669i	0.989456	+	0.144835i	$0.0462650\pi$
$509$	147.442i	0.289669i	−0.989456	+	0.144835i	$0.953735\pi$
$510$	0	0
$511$	6.18847	0.0121105
$512$	− 1139.94i	− 2.22645i
$513$	0	0
$514$	1033.11	2.00995
$515$	416.119i	0.807997i
$516$	0	0
$517$	−140.295	−0.271363
$518$	− 176.377i	− 0.340497i
$519$	0	0
$520$	326.540	0.627961
$521$	− 361.413i	− 0.693690i	−0.937922	−	0.346845i	$-0.887253\pi$
$521$	− 361.413i	− 0.693690i	0.937922	−	0.346845i	$-0.112747\pi$
$522$	0	0
$523$	−19.7777	−0.0378159	−0.0189079	−	0.999821i	$-0.506019\pi$
$523$	−19.7777	−0.0378159	−0.0189079	+	0.999821i	$0.506019\pi$
$524$	1165.43i	2.22410i
$525$	0	0
$526$	1086.09	2.06480
$527$	− 7.03697i	− 0.0133529i
$528$	0	0
$529$	−138.700	−0.262194
$530$	− 237.292i	− 0.447720i
$531$	0	0
$532$	−2434.59	−4.57630
$533$	182.718i	0.342810i
$534$	0	0
$535$	−38.4487	−0.0718667
$536$	− 630.739i	− 1.17675i
$537$	0	0
$538$	955.899	1.77676
$539$	106.770i	0.198089i
$540$	0	0
$541$	−663.746	−1.22689	−0.613444	−	0.789739i	$-0.710216\pi$
$541$	−663.746	−1.22689	−0.613444	+	0.789739i	$0.710216\pi$
$542$	− 223.065i	− 0.411559i
$543$	0	0
$544$	22.0179	0.0404741
$545$	28.7529i	0.0527577i
$546$	0	0
$547$	171.872	0.314209	0.157105	−	0.987582i	$-0.449784\pi$
$547$	171.872	0.314209	0.157105	+	0.987582i	$0.449784\pi$
$548$	1575.00i	2.87408i
$549$	0	0
$550$	−121.168	−0.220305
$551$	919.422i	1.66864i
$552$	0	0
$553$	648.262	1.17226
$554$	− 1790.97i	− 3.23280i
$555$	0	0
$556$	−2226.84	−4.00511
$557$	200.437i	0.359852i	0.983680	+	0.179926i	$0.0575858\pi$
$557$	200.437i	0.359852i	−0.983680	+	0.179926i	$0.942414\pi$
$558$	0	0
$559$	−147.303	−0.263511
$560$	− 1077.10i	− 1.92340i
$561$	0	0
$562$	1192.45	2.12180
$563$	753.744i	1.33880i	0.742902	+	0.669400i	$0.233449\pi$
$563$	753.744i	1.33880i	−0.742902	+	0.669400i	$0.766551\pi$
$564$	0	0
$565$	565.047	1.00008
$566$	− 239.364i	− 0.422905i
$567$	0	0
$568$	−558.916	−0.984007
$569$	85.1626i	0.149671i	0.997196	+	0.0748354i	$0.0238431\pi$
$569$	85.1626i	0.149671i	−0.997196	+	0.0748354i	$0.976157\pi$
$570$	0	0
$571$	−826.091	−1.44674	−0.723372	−	0.690458i	$-0.757409\pi$
$571$	−826.091	−1.44674	−0.723372	+	0.690458i	$0.757409\pi$
$572$	− 139.855i	− 0.244502i
$573$	0	0
$574$	1316.21	2.29304
$575$	− 254.537i	− 0.442674i
$576$	0	0
$577$	253.424	0.439209	0.219605	−	0.975589i	$-0.429523\pi$
$577$	253.424	0.439209	0.219605	+	0.975589i	$0.429523\pi$
$578$	1045.99i	1.80967i
$579$	0	0
$580$	−1099.27	−1.89529
$581$	− 188.480i	− 0.324406i
$582$	0	0
$583$	−57.1098	−0.0979585
$584$	12.8283i	0.0219662i
$585$	0	0
$586$	1338.88	2.28477
$587$	− 385.659i	− 0.657000i	−0.944504	−	0.328500i	$-0.893457\pi$
$587$	− 385.659i	− 0.657000i	0.944504	−	0.328500i	$-0.106543\pi$
$588$	0	0
$589$	−355.550	−0.603650
$590$	855.945i	1.45075i
$591$	0	0
$592$	167.419	0.282803
$593$	809.736i	1.36549i	0.730656	+	0.682745i	$0.239214\pi$
$593$	809.736i	1.36549i	−0.730656	+	0.682745i	$0.760786\pi$
$594$	0	0
$595$	−20.5394	−0.0345200
$596$	921.537i	1.54620i
$597$	0	0
$598$	422.494	0.706511
$599$	− 29.2794i	− 0.0488805i	−0.999701	−	0.0244402i	$-0.992220\pi$
$599$	− 29.2794i	− 0.0488805i	0.999701	−	0.0244402i	$-0.00778034\pi$
$600$	0	0
$601$	−934.241	−1.55448	−0.777239	−	0.629206i	$-0.783380\pi$
$601$	−934.241	−1.55448	−0.777239	+	0.629206i	$0.783380\pi$
$602$	1061.09i	1.76262i
$603$	0	0
$604$	2386.99	3.95198
$605$	426.111i	0.704315i
$606$	0	0
$607$	−531.512	−0.875638	−0.437819	−	0.899063i	$-0.644249\pi$
$607$	−531.512	−0.875638	−0.437819	+	0.899063i	$0.644249\pi$
$608$	− 1112.48i	− 1.82973i
$609$	0	0
$610$	−1068.70	−1.75197
$611$	186.485i	0.305213i
$612$	0	0
$613$	−259.694	−0.423645	−0.211822	−	0.977308i	$-0.567940\pi$
$613$	−259.694	−0.423645	−0.211822	+	0.977308i	$0.567940\pi$
$614$	589.627i	0.960305i
$615$	0	0
$616$	−566.119	−0.919025
$617$	− 702.234i	− 1.13814i	−0.822288	−	0.569071i	$-0.807303\pi$
$617$	− 702.234i	− 1.13814i	0.822288	−	0.569071i	$-0.192697\pi$
$618$	0	0
$619$	−1151.24	−1.85985	−0.929923	−	0.367755i	$-0.880127\pi$
$619$	−1151.24	−1.85985	−0.929923	+	0.367755i	$0.880127\pi$
$620$	− 425.098i	− 0.685642i
$621$	0	0
$622$	−1118.18	−1.79771
$623$	− 1251.19i	− 2.00833i
$624$	0	0
$625$	−281.703	−0.450725
$626$	494.611i	0.790113i
$627$	0	0
$628$	1104.35	1.75852
$629$	− 3.19254i	− 0.00507558i
$630$	0	0
$631$	−873.923	−1.38498	−0.692490	−	0.721427i	$-0.743486\pi$
$631$	−873.923	−1.38498	−0.692490	+	0.721427i	$0.743486\pi$
$632$	1343.80i	2.12627i
$633$	0	0
$634$	−527.782	−0.832464
$635$	43.8632i	0.0690759i
$636$	0	0
$637$	141.923	0.222799
$638$	380.461i	0.596335i
$639$	0	0
$640$	−410.490	−0.641391
$641$	614.060i	0.957973i	0.877822	+	0.478986i	$0.158996\pi$
$641$	614.060i	0.957973i	−0.877822	+	0.478986i	$0.841004\pi$
$642$	0	0
$643$	299.770	0.466205	0.233103	−	0.972452i	$-0.425112\pi$
$643$	299.770	0.466205	0.233103	+	0.972452i	$0.425112\pi$
$644$	− 2116.34i	− 3.28624i
$645$	0	0
$646$	−63.3720	−0.0980990
$647$	976.042i	1.50857i	0.656549	+	0.754283i	$0.272015\pi$
$647$	976.042i	1.50857i	−0.656549	+	0.754283i	$0.727985\pi$
$648$	0	0
$649$	206.003	0.317416
$650$	161.061i	0.247786i
$651$	0	0
$652$	690.613	1.05922
$653$	− 1196.90i	− 1.83293i	−0.400116	−	0.916465i	$-0.631030\pi$
$653$	− 1196.90i	− 1.83293i	0.400116	−	0.916465i	$-0.368970\pi$
$654$	0	0
$655$	496.779	0.758441
$656$	1249.36i	1.90451i
$657$	0	0
$658$	1343.34	2.04156
$659$	− 225.322i	− 0.341916i	−0.985278	−	0.170958i	$-0.945314\pi$
$659$	− 225.322i	− 0.341916i	0.985278	−	0.170958i	$-0.0546862\pi$
$660$	0	0
$661$	−1119.76	−1.69405	−0.847023	−	0.531556i	$-0.821608\pi$
$661$	−1119.76	−1.69405	−0.847023	+	0.531556i	$0.821608\pi$
$662$	− 484.340i	− 0.731632i
$663$	0	0
$664$	390.705	0.588412
$665$	1037.77i	1.56056i
$666$	0	0
$667$	−799.235	−1.19825
$668$	1292.50i	1.93489i
$669$	0	0
$670$	−478.454	−0.714110
$671$	257.208i	0.383320i
$672$	0	0
$673$	526.066	0.781673	0.390836	−	0.920460i	$-0.372186\pi$
$673$	526.066	0.781673	0.390836	+	0.920460i	$0.372186\pi$
$674$	158.349i	0.234940i
$675$	0	0
$676$	1357.25	2.00777
$677$	− 277.685i	− 0.410170i	−0.978744	−	0.205085i	$-0.934253\pi$
$677$	− 277.685i	− 0.410170i	0.978744	−	0.205085i	$-0.0657471\pi$
$678$	0	0
$679$	291.155	0.428800
$680$	− 42.5767i	− 0.0626129i
$681$	0	0
$682$	−147.128	−0.215731
$683$	− 1275.87i	− 1.86804i	−0.357227	−	0.934018i	$-0.616278\pi$
$683$	− 1275.87i	− 1.86804i	0.357227	−	0.934018i	$-0.383722\pi$
$684$	0	0
$685$	671.362	0.980090
$686$	570.305i	0.831348i
$687$	0	0
$688$	−1007.20	−1.46396
$689$	75.9125i	0.110178i
$690$	0	0
$691$	−1181.70	−1.71013	−0.855067	−	0.518518i	$-0.826484\pi$
$691$	−1181.70	−1.71013	−0.855067	+	0.518518i	$0.826484\pi$
$692$	1196.62i	1.72922i
$693$	0	0
$694$	517.698	0.745963
$695$	949.218i	1.36578i
$696$	0	0
$697$	23.8241	0.0341810
$698$	1464.09i	2.09755i
$699$	0	0
$700$	806.780	1.15254
$701$	1082.89i	1.54478i	0.635148	+	0.772390i	$0.280939\pi$
$701$	1082.89i	1.54478i	−0.635148	+	0.772390i	$0.719061\pi$
$702$	0	0
$703$	−161.306	−0.229454
$704$	− 41.4378i	− 0.0588606i
$705$	0	0
$706$	769.216	1.08954
$707$	118.577i	0.167718i
$708$	0	0
$709$	−208.077	−0.293480	−0.146740	−	0.989175i	$-0.546878\pi$
$709$	−208.077	−0.293480	−0.146740	+	0.989175i	$0.546878\pi$
$710$	423.972i	0.597144i
$711$	0	0
$712$	2593.63	3.64275
$713$	− 309.072i	− 0.433482i
$714$	0	0
$715$	−59.6149	−0.0833775
$716$	1638.26i	2.28808i
$717$	0	0
$718$	−390.301	−0.543595
$719$	− 1011.36i	− 1.40662i	−0.710884	−	0.703309i	$-0.751705\pi$
$719$	− 1011.36i	− 1.40662i	0.710884	−	0.703309i	$-0.248295\pi$
$720$	0	0
$721$	−958.941	−1.33002
$722$	1893.78i	2.62297i
$723$	0	0
$724$	−3096.00	−4.27625
$725$	− 304.680i	− 0.420249i
$726$	0	0
$727$	774.643	1.06553	0.532767	−	0.846262i	$-0.321152\pi$
$727$	774.643	1.06553	0.532767	+	0.846262i	$0.321152\pi$
$728$	752.507i	1.03366i
$729$	0	0
$730$	9.73102	0.0133302
$731$	19.2065i	0.0262742i
$732$	0	0
$733$	−1018.11	−1.38896	−0.694480	−	0.719512i	$-0.744366\pi$
$733$	−1018.11	−1.38896	−0.694480	+	0.719512i	$0.744366\pi$
$734$	218.713i	0.297973i
$735$	0	0
$736$	967.055	1.31393
$737$	115.151i	0.156243i
$738$	0	0
$739$	−857.914	−1.16091	−0.580456	−	0.814292i	$-0.697126\pi$
$739$	−857.914	−1.16091	−0.580456	+	0.814292i	$0.697126\pi$
$740$	− 192.859i	− 0.260620i
$741$	0	0
$742$	546.836	0.736976
$743$	1216.92i	1.63784i	0.573906	+	0.818921i	$0.305428\pi$
$743$	1216.92i	1.63784i	−0.573906	+	0.818921i	$0.694572\pi$
$744$	0	0
$745$	392.816	0.527270
$746$	− 1045.07i	− 1.40090i
$747$	0	0
$748$	−18.2354	−0.0243788
$749$	− 88.6046i	− 0.118297i
$750$	0	0
$751$	−50.4386	−0.0671619	−0.0335809	−	0.999436i	$-0.510691\pi$
$751$	−50.4386	−0.0671619	−0.0335809	+	0.999436i	$0.510691\pi$
$752$	1275.12i	1.69563i
$753$	0	0
$754$	505.724	0.670721
$755$	− 1017.49i	− 1.34766i
$756$	0	0
$757$	1139.84	1.50573	0.752864	−	0.658176i	$-0.228672\pi$
$757$	1139.84	1.50573	0.752864	+	0.658176i	$0.228672\pi$
$758$	1338.68i	1.76607i
$759$	0	0
$760$	−2151.23	−2.83057
$761$	672.607i	0.883847i	0.897053	+	0.441923i	$0.145704\pi$
$761$	672.607i	0.883847i	−0.897053	+	0.441923i	$0.854296\pi$
$762$	0	0
$763$	−66.2609	−0.0868426
$764$	2869.93i	3.75645i
$765$	0	0
$766$	757.663	0.989116
$767$	− 273.827i	− 0.357011i
$768$	0	0
$769$	−742.811	−0.965944	−0.482972	−	0.875636i	$-0.660443\pi$
$769$	−742.811	−0.965944	−0.482972	+	0.875636i	$0.660443\pi$
$770$	429.436i	0.557709i
$771$	0	0
$772$	1383.18	1.79168
$773$	1222.52i	1.58152i	0.612124	+	0.790762i	$0.290315\pi$
$773$	1222.52i	1.58152i	−0.612124	+	0.790762i	$0.709685\pi$
$774$	0	0
$775$	117.823	0.152030
$776$	603.544i	0.777763i
$777$	0	0
$778$	2487.02	3.19668
$779$	− 1203.74i	− 1.54523i
$780$	0	0
$781$	102.039	0.130652
$782$	− 55.0880i	− 0.0704450i
$783$	0	0
$784$	970.416	1.23778
$785$	− 470.742i	− 0.599672i
$786$	0	0
$787$	−337.342	−0.428643	−0.214321	−	0.976763i	$-0.568754\pi$
$787$	−337.342	−0.428643	−0.214321	+	0.976763i	$0.568754\pi$
$788$	− 126.574i	− 0.160627i
$789$	0	0
$790$	1019.36	1.29032
$791$	1302.14i	1.64620i
$792$	0	0
$793$	341.891	0.431136
$794$	647.235i	0.815158i
$795$	0	0
$796$	3212.66	4.03600
$797$	582.790i	0.731230i	0.930766	+	0.365615i	$0.119141\pi$
$797$	582.790i	0.731230i	−0.930766	+	0.365615i	$0.880859\pi$
$798$	0	0
$799$	24.3153	0.0304322
$800$	368.656i	0.460820i
$801$	0	0
$802$	94.6371	0.118001
$803$	− 2.34200i	− 0.00291656i
$804$	0	0
$805$	−902.116	−1.12064
$806$	195.568i	0.242641i
$807$	0	0
$808$	−245.801	−0.304209
$809$	850.348i	1.05111i	0.850759	+	0.525555i	$0.176142\pi$
$809$	850.348i	1.05111i	−0.850759	+	0.525555i	$0.823858\pi$
$810$	0	0
$811$	513.085	0.632658	0.316329	−	0.948650i	$-0.397550\pi$
$811$	513.085	0.632658	0.316329	+	0.948650i	$0.397550\pi$
$812$	− 2533.25i	− 3.11977i
$813$	0	0
$814$	−66.7493	−0.0820016
$815$	− 294.382i	− 0.361205i
$816$	0	0
$817$	970.425	1.18779
$818$	705.122i	0.862007i
$819$	0	0
$820$	1439.20	1.75512
$821$	507.426i	0.618058i	0.951053	+	0.309029i	$0.100004\pi$
$821$	507.426i	0.618058i	−0.951053	+	0.309029i	$0.899996\pi$
$822$	0	0
$823$	−715.645	−0.869557	−0.434779	−	0.900537i	$-0.643173\pi$
$823$	−715.645	−0.869557	−0.434779	+	0.900537i	$0.643173\pi$
$824$	− 1987.82i	− 2.41240i
$825$	0	0
$826$	−1972.52	−2.38803
$827$	1010.84i	1.22230i	0.791516	+	0.611149i	$0.209292\pi$
$827$	1010.84i	1.22230i	−0.791516	+	0.611149i	$0.790708\pi$
$828$	0	0
$829$	−497.757	−0.600431	−0.300215	−	0.953871i	$-0.597059\pi$
$829$	−497.757	−0.600431	−0.300215	+	0.953871i	$0.597059\pi$
$830$	− 296.374i	− 0.357077i
$831$	0	0
$832$	−55.0808	−0.0662028
$833$	− 18.5050i	− 0.0222149i
$834$	0	0
$835$	550.946	0.659815
$836$	921.360i	1.10211i
$837$	0	0
$838$	−503.295	−0.600591
$839$	− 918.153i	− 1.09434i	−0.837021	−	0.547171i	$-0.815705\pi$
$839$	− 918.153i	− 1.09434i	0.837021	−	0.547171i	$-0.184295\pi$
$840$	0	0
$841$	−115.682	−0.137553
$842$	2110.35i	2.50635i
$843$	0	0
$844$	1764.00	2.09004
$845$	− 578.544i	− 0.684668i
$846$	0	0
$847$	−981.968	−1.15935
$848$	519.062i	0.612101i
$849$	0	0
$850$	21.0004	0.0247063
$851$	− 140.220i	− 0.164771i
$852$	0	0
$853$	1103.31	1.29344	0.646720	−	0.762727i	$-0.276140\pi$
$853$	1103.31	1.29344	0.646720	+	0.762727i	$0.276140\pi$
$854$	− 2462.81i	− 2.88385i
$855$	0	0
$856$	183.671	0.214569
$857$	− 785.899i	− 0.917036i	−0.888685	−	0.458518i	$-0.848381\pi$
$857$	− 785.899i	− 0.917036i	0.888685	−	0.458518i	$-0.151619\pi$
$858$	0	0
$859$	−395.687	−0.460637	−0.230319	−	0.973115i	$-0.573977\pi$
$859$	−395.687	−0.460637	−0.230319	+	0.973115i	$0.573977\pi$
$860$	1160.25i	1.34913i
$861$	0	0
$862$	−1772.26	−2.05598
$863$	896.554i	1.03888i	0.854507	+	0.519440i	$0.173860\pi$
$863$	896.554i	1.03888i	−0.854507	+	0.519440i	$0.826140\pi$
$864$	0	0
$865$	510.075	0.589682
$866$	− 2479.25i	− 2.86288i
$867$	0	0
$868$	979.635	1.12861
$869$	− 245.332i	− 0.282315i
$870$	0	0
$871$	153.063	0.175733
$872$	− 137.354i	− 0.157516i
$873$	0	0
$874$	−2783.38	−3.18464
$875$	− 1216.69i	− 1.39051i
$876$	0	0
$877$	−1390.20	−1.58518	−0.792590	−	0.609755i	$-0.791268\pi$
$877$	−1390.20	−1.58518	−0.792590	+	0.609755i	$0.791268\pi$
$878$	− 1269.53i	− 1.44594i
$879$	0	0
$880$	−407.625	−0.463210
$881$	− 1509.07i	− 1.71290i	−0.516228	−	0.856451i	$-0.672664\pi$
$881$	− 1509.07i	− 1.71290i	0.516228	−	0.856451i	$-0.327336\pi$
$882$	0	0
$883$	−409.196	−0.463416	−0.231708	−	0.972785i	$-0.574431\pi$
$883$	−409.196	−0.463416	−0.231708	+	0.972785i	$0.574431\pi$
$884$	24.2391i	0.0274198i
$885$	0	0
$886$	−1521.41	−1.71716
$887$	− 201.476i	− 0.227143i	−0.993530	−	0.113572i	$-0.963771\pi$
$887$	− 201.476i	− 0.227143i	0.993530	−	0.113572i	$-0.0362292\pi$
$888$	0	0
$889$	−101.082	−0.113703
$890$	− 1967.43i	− 2.21060i
$891$	0	0
$892$	−2853.93	−3.19947
$893$	− 1228.56i	− 1.37576i
$894$	0	0
$895$	698.329	0.780256
$896$	− 945.971i	− 1.05577i
$897$	0	0
$898$	1227.48	1.36690
$899$	− 369.959i	− 0.411522i
$900$	0	0
$901$	9.89806	0.0109856
$902$	− 498.113i	− 0.552232i
$903$	0	0
$904$	−2699.25	−2.98590
$905$	1319.71i	1.45824i
$906$	0	0
$907$	1740.01	1.91843	0.959213	−	0.282683i	$-0.0912244\pi$
$907$	1740.01	1.91843	0.959213	+	0.282683i	$0.0912244\pi$
$908$	1019.06i	1.12231i
$909$	0	0
$910$	570.823	0.627278
$911$	1651.26i	1.81258i	0.422662	+	0.906288i	$0.361096\pi$
$911$	1651.26i	1.81258i	−0.422662	+	0.906288i	$0.638904\pi$
$912$	0	0
$913$	−71.3294	−0.0781264
$914$	1454.24i	1.59107i
$915$	0	0
$916$	−1129.22	−1.23277
$917$	1144.82i	1.24844i
$918$	0	0
$919$	−157.497	−0.171378	−0.0856892	−	0.996322i	$-0.527309\pi$
$919$	−157.497	−0.171378	−0.0856892	+	0.996322i	$0.527309\pi$
$920$	− 1870.02i	− 2.03263i
$921$	0	0
$922$	−680.406	−0.737968
$923$	− 135.634i	− 0.146949i
$924$	0	0
$925$	53.4540	0.0577881
$926$	1772.41i	1.91405i
$927$	0	0
$928$	1157.56	1.24737
$929$	− 262.416i	− 0.282472i	−0.989976	−	0.141236i	$-0.954892\pi$
$929$	− 262.416i	− 0.282472i	0.989976	−	0.141236i	$-0.0451076\pi$
$930$	0	0
$931$	−934.983	−1.00428
$932$	− 2987.89i	− 3.20589i
$933$	0	0
$934$	−1347.42	−1.44263
$935$	7.77305i	0.00831342i
$936$	0	0
$937$	−47.0928	−0.0502591	−0.0251295	−	0.999684i	$-0.508000\pi$
$937$	−47.0928	−0.0502591	−0.0251295	+	0.999684i	$0.508000\pi$
$938$	− 1102.59i	− 1.17547i
$939$	0	0
$940$	1468.87	1.56263
$941$	362.486i	0.385213i	0.981276	+	0.192607i	$0.0616941\pi$
$941$	362.486i	0.385213i	−0.981276	+	0.192607i	$0.938306\pi$
$942$	0	0
$943$	1046.39	1.10963
$944$	− 1872.33i	− 1.98340i
$945$	0	0
$946$	401.567	0.424489
$947$	− 775.594i	− 0.819001i	−0.912310	−	0.409500i	$-0.865703\pi$
$947$	− 775.594i	− 0.819001i	0.912310	−	0.409500i	$-0.134297\pi$
$948$	0	0
$949$	−3.11307	−0.00328037
$950$	− 1061.06i	− 1.11691i
$951$	0	0
$952$	98.1177	0.103065
$953$	− 1296.53i	− 1.36047i	−0.732994	−	0.680235i	$-0.761878\pi$
$953$	− 1296.53i	− 1.36047i	0.732994	−	0.680235i	$-0.238122\pi$
$954$	0	0
$955$	1223.34	1.28099
$956$	152.557i	0.159579i
$957$	0	0
$958$	2657.43	2.77394
$959$	1547.15i	1.61329i
$960$	0	0
$961$	−817.933	−0.851127
$962$	88.7257i	0.0922304i
$963$	0	0
$964$	−773.920	−0.802822
$965$	− 589.596i	− 0.610980i
$966$	0	0
$967$	1033.76	1.06904	0.534521	−	0.845155i	$-0.320492\pi$
$967$	1033.76	1.06904	0.534521	+	0.845155i	$0.320492\pi$
$968$	− 2035.55i	− 2.10284i
$969$	0	0
$970$	457.825	0.471985
$971$	− 1439.64i	− 1.48263i	−0.671155	−	0.741317i	$-0.734201\pi$
$971$	− 1439.64i	− 1.48263i	0.671155	−	0.741317i	$-0.265799\pi$
$972$	0	0
$973$	−2187.46	−2.24816
$974$	900.824i	0.924870i
$975$	0	0
$976$	2337.72	2.39521
$977$	− 1126.35i	− 1.15287i	−0.817144	−	0.576434i	$-0.804444\pi$
$977$	− 1126.35i	− 1.15287i	0.817144	−	0.576434i	$-0.195556\pi$
$978$	0	0
$979$	−473.509	−0.483666
$980$	− 1117.87i	− 1.14069i
$981$	0	0
$982$	1174.21	1.19573
$983$	2.00328i	0.00203793i	0.999999	+	0.00101896i	$0.000324346\pi$
$983$	2.00328i	0.00203793i	−0.999999	+	0.00101896i	$0.999676\pi$
$984$	0	0
$985$	−53.9537	−0.0547754
$986$	− 65.9402i	− 0.0668764i
$987$	0	0
$988$	1224.71	1.23958
$989$	843.571i	0.852954i
$990$	0	0
$991$	835.250	0.842836	0.421418	−	0.906867i	$-0.361533\pi$
$991$	835.250	0.842836	0.421418	+	0.906867i	$0.361533\pi$
$992$	447.641i	0.451251i
$993$	0	0
$994$	−977.039	−0.982937
$995$	− 1369.43i	− 1.37631i
$996$	0	0
$997$	878.694	0.881338	0.440669	−	0.897670i	$-0.354741\pi$
$997$	878.694	0.881338	0.440669	+	0.897670i	$0.354741\pi$
$998$	− 2472.76i	− 2.47771i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.79	yes	84
3.2	odd	2	inner	1143.3.b.a.890.6	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.6	✓	84	3.2	odd	2	inner
1143.3.b.a.890.79	yes	84	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.62368i q^{2} -9.13107 q^{4} +3.89223i q^{5} -8.96960 q^{7} -18.5934i q^{8} +O(q^{10})\)	\(q+3.62368i q^{2} -9.13107 q^{4} +3.89223i q^{5} -8.96960 q^{7} -18.5934i q^{8} -14.1042 q^{10} +3.39451i q^{11} +4.51211 q^{13} -32.5030i q^{14} +30.8521 q^{16} -0.588323i q^{17} -29.7256 q^{19} -35.5402i q^{20} -12.3006 q^{22} -25.8399i q^{23} +9.85055 q^{25} +16.3504i q^{26} +81.9021 q^{28} -30.9303i q^{29} +11.9611 q^{31} +37.4249i q^{32} +2.13190 q^{34} -34.9117i q^{35} +5.42650 q^{37} -107.716i q^{38} +72.3696 q^{40} +40.4950i q^{41} -32.6461 q^{43} -30.9955i q^{44} +93.6356 q^{46} +41.3299i q^{47} +31.4538 q^{49} +35.6953i q^{50} -41.2004 q^{52} +16.8242i q^{53} -13.2122 q^{55} +166.775i q^{56} +112.081 q^{58} -60.6872i q^{59} +75.7718 q^{61} +43.3430i q^{62} -12.2073 q^{64} +17.5622i q^{65} +33.9228 q^{67} +5.37202i q^{68} +126.509 q^{70} -30.0600i q^{71} -0.689938 q^{73} +19.6639i q^{74} +271.427 q^{76} -30.4474i q^{77} -72.2732 q^{79} +120.084i q^{80} -146.741 q^{82} +21.0132i q^{83} +2.28989 q^{85} -118.299i q^{86} +63.1153 q^{88} +139.493i q^{89} -40.4718 q^{91} +235.946i q^{92} -149.766 q^{94} -115.699i q^{95} -32.4602 q^{97} +113.978i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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