LMFDB - Embedded newform 1143.3.b.a.890.71

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			890.71
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96858i q^{2} -4.81247 q^{4} -3.90126i q^{5} +11.9179 q^{7} -2.41189i q^{8} +O(q^{10})$	\(q+2.96858i q^{2} -4.81247 q^{4} -3.90126i q^{5} +11.9179 q^{7} -2.41189i q^{8} +11.5812 q^{10} -2.00981i q^{11} +10.4679 q^{13} +35.3793i q^{14} -12.0900 q^{16} -15.0361i q^{17} -12.2139 q^{19} +18.7747i q^{20} +5.96627 q^{22} -16.9967i q^{23} +9.78019 q^{25} +31.0748i q^{26} -57.3547 q^{28} -13.3323i q^{29} +9.79569 q^{31} -45.5377i q^{32} +44.6358 q^{34} -46.4949i q^{35} +23.1196 q^{37} -36.2579i q^{38} -9.40942 q^{40} -38.8506i q^{41} -8.44282 q^{43} +9.67214i q^{44} +50.4561 q^{46} -61.5275i q^{47} +93.0369 q^{49} +29.0333i q^{50} -50.3764 q^{52} +61.3578i q^{53} -7.84077 q^{55} -28.7448i q^{56} +39.5779 q^{58} -38.6575i q^{59} +49.1446 q^{61} +29.0793i q^{62} +86.8224 q^{64} -40.8379i q^{65} +8.24716 q^{67} +72.3606i q^{68} +138.024 q^{70} -20.5967i q^{71} -24.6261 q^{73} +68.6325i q^{74} +58.7790 q^{76} -23.9527i q^{77} +22.7212 q^{79} +47.1662i q^{80} +115.331 q^{82} +71.4711i q^{83} -58.6595 q^{85} -25.0632i q^{86} -4.84744 q^{88} +11.0072i q^{89} +124.755 q^{91} +81.7962i q^{92} +182.650 q^{94} +47.6495i q^{95} -93.8083 q^{97} +276.188i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.96858i	1.48429i	0.670239	+	0.742145i	$0.266192\pi$
$2$	2.96858i	1.48429i	−0.670239	+	0.742145i	$0.733808\pi$
$3$	0	0
$3$	0	0
$4$	−4.81247	−1.20312
$5$	− 3.90126i	− 0.780251i	−0.920762	−	0.390126i	$-0.872432\pi$
$5$	− 3.90126i	− 0.780251i	0.920762	−	0.390126i	$-0.127568\pi$
$6$	0	0
$7$	11.9179	1.70256	0.851280	−	0.524711i	$-0.175827\pi$
$7$	11.9179	1.70256	0.851280	+	0.524711i	$0.175827\pi$
$8$	− 2.41189i	− 0.301487i
$9$	0	0
$10$	11.5812	1.15812
$11$	− 2.00981i	− 0.182710i	−0.995818	−	0.0913548i	$-0.970880\pi$
$11$	− 2.00981i	− 0.182710i	0.995818	−	0.0913548i	$-0.0291197\pi$
$12$	0	0
$13$	10.4679	0.805222	0.402611	−	0.915371i	$-0.368103\pi$
$13$	10.4679	0.805222	0.402611	+	0.915371i	$0.368103\pi$
$14$	35.3793i	2.52709i
$15$	0	0
$16$	−12.0900	−0.755625
$17$	− 15.0361i	− 0.884474i	−0.896898	−	0.442237i	$-0.854185\pi$
$17$	− 15.0361i	− 0.884474i	0.896898	−	0.442237i	$-0.145815\pi$
$18$	0	0
$19$	−12.2139	−0.642836	−0.321418	−	0.946937i	$-0.604159\pi$
$19$	−12.2139	−0.642836	−0.321418	+	0.946937i	$0.604159\pi$
$20$	18.7747i	0.938735i
$21$	0	0
$22$	5.96627	0.271194
$23$	− 16.9967i	− 0.738987i	−0.929233	−	0.369494i	$-0.879531\pi$
$23$	− 16.9967i	− 0.738987i	0.929233	−	0.369494i	$-0.120469\pi$
$24$	0	0
$25$	9.78019	0.391208
$26$	31.0748i	1.19518i
$27$	0	0
$28$	−57.3547	−2.04838
$29$	− 13.3323i	− 0.459734i	−0.973222	−	0.229867i	$-0.926171\pi$
$29$	− 13.3323i	− 0.459734i	0.973222	−	0.229867i	$-0.0738291\pi$
$30$	0	0
$31$	9.79569	0.315990	0.157995	−	0.987440i	$-0.449497\pi$
$31$	9.79569	0.315990	0.157995	+	0.987440i	$0.449497\pi$
$32$	− 45.5377i	− 1.42305i
$33$	0	0
$34$	44.6358	1.31282
$35$	− 46.4949i	− 1.32843i
$36$	0	0
$37$	23.1196	0.624855	0.312428	−	0.949942i	$-0.398858\pi$
$37$	23.1196	0.624855	0.312428	+	0.949942i	$0.398858\pi$
$38$	− 36.2579i	− 0.954155i
$39$	0	0
$40$	−9.40942	−0.235235
$41$	− 38.8506i	− 0.947575i	−0.880639	−	0.473787i	$-0.842887\pi$
$41$	− 38.8506i	− 0.947575i	0.880639	−	0.473787i	$-0.157113\pi$
$42$	0	0
$43$	−8.44282	−0.196345	−0.0981724	−	0.995169i	$-0.531300\pi$
$43$	−8.44282	−0.196345	−0.0981724	+	0.995169i	$0.531300\pi$
$44$	9.67214i	0.219821i
$45$	0	0
$46$	50.4561	1.09687
$47$	− 61.5275i	− 1.30910i	−0.756020	−	0.654548i	$-0.772859\pi$
$47$	− 61.5275i	− 1.30910i	0.756020	−	0.654548i	$-0.227141\pi$
$48$	0	0
$49$	93.0369	1.89871
$50$	29.0333i	0.580666i
$51$	0	0
$52$	−50.3764	−0.968777
$53$	61.3578i	1.15769i	0.815436	+	0.578847i	$0.196497\pi$
$53$	61.3578i	1.15769i	−0.815436	+	0.578847i	$0.803503\pi$
$54$	0	0
$55$	−7.84077	−0.142559
$56$	− 28.7448i	− 0.513299i
$57$	0	0
$58$	39.5779	0.682378
$59$	− 38.6575i	− 0.655213i	−0.944814	−	0.327606i	$-0.893758\pi$
$59$	− 38.6575i	− 0.655213i	0.944814	−	0.327606i	$-0.106242\pi$
$60$	0	0
$61$	49.1446	0.805650	0.402825	−	0.915277i	$-0.368028\pi$
$61$	49.1446	0.805650	0.402825	+	0.915277i	$0.368028\pi$
$62$	29.0793i	0.469021i
$63$	0	0
$64$	86.8224	1.35660
$65$	− 40.8379i	− 0.628275i
$66$	0	0
$67$	8.24716	0.123092	0.0615460	−	0.998104i	$-0.480397\pi$
$67$	8.24716	0.123092	0.0615460	+	0.998104i	$0.480397\pi$
$68$	72.3606i	1.06413i
$69$	0	0
$70$	138.024	1.97177
$71$	− 20.5967i	− 0.290095i	−0.989425	−	0.145047i	$-0.953667\pi$
$71$	− 20.5967i	− 0.290095i	0.989425	−	0.145047i	$-0.0463334\pi$
$72$	0	0
$73$	−24.6261	−0.337344	−0.168672	−	0.985672i	$-0.553948\pi$
$73$	−24.6261	−0.337344	−0.168672	+	0.985672i	$0.553948\pi$
$74$	68.6325i	0.927466i
$75$	0	0
$76$	58.7790	0.773408
$77$	− 23.9527i	− 0.311074i
$78$	0	0
$79$	22.7212	0.287610	0.143805	−	0.989606i	$-0.454066\pi$
$79$	22.7212	0.287610	0.143805	+	0.989606i	$0.454066\pi$
$80$	47.1662i	0.589577i
$81$	0	0
$82$	115.331	1.40648
$83$	71.4711i	0.861097i	0.902567	+	0.430549i	$0.141680\pi$
$83$	71.4711i	0.861097i	−0.902567	+	0.430549i	$0.858320\pi$
$84$	0	0
$85$	−58.6595	−0.690112
$86$	− 25.0632i	− 0.291433i
$87$	0	0
$88$	−4.84744	−0.0550845
$89$	11.0072i	0.123676i	0.998086	+	0.0618381i	$0.0196962\pi$
$89$	11.0072i	0.123676i	−0.998086	+	0.0618381i	$0.980304\pi$
$90$	0	0
$91$	124.755	1.37094
$92$	81.7962i	0.889089i
$93$	0	0
$94$	182.650	1.94308
$95$	47.6495i	0.501574i
$96$	0	0
$97$	−93.8083	−0.967096	−0.483548	−	0.875318i	$-0.660652\pi$
$97$	−93.8083	−0.967096	−0.483548	+	0.875318i	$0.660652\pi$
$98$	276.188i	2.81824i
$99$	0	0
$100$	−47.0669	−0.470669
$101$	80.1451i	0.793516i	0.917923	+	0.396758i	$0.129865\pi$
$101$	80.1451i	0.793516i	−0.917923	+	0.396758i	$0.870135\pi$
$102$	0	0
$103$	106.830	1.03718	0.518591	−	0.855022i	$-0.326457\pi$
$103$	106.830	1.03718	0.518591	+	0.855022i	$0.326457\pi$
$104$	− 25.2474i	− 0.242764i
$105$	0	0
$106$	−182.146	−1.71836
$107$	184.771i	1.72683i	0.504494	+	0.863415i	$0.331679\pi$
$107$	184.771i	1.72683i	−0.504494	+	0.863415i	$0.668321\pi$
$108$	0	0
$109$	16.2143	0.148755	0.0743775	−	0.997230i	$-0.476303\pi$
$109$	16.2143	0.148755	0.0743775	+	0.997230i	$0.476303\pi$
$110$	− 23.2760i	− 0.211600i
$111$	0	0
$112$	−144.088	−1.28650
$113$	83.0951i	0.735354i	0.929953	+	0.367677i	$0.119847\pi$
$113$	83.0951i	0.735354i	−0.929953	+	0.367677i	$0.880153\pi$
$114$	0	0
$115$	−66.3085	−0.576596
$116$	64.1612i	0.553114i
$117$	0	0
$118$	114.758	0.972526
$119$	− 179.199i	− 1.50587i
$120$	0	0
$121$	116.961	0.966617
$122$	145.890i	1.19582i
$123$	0	0
$124$	−47.1415	−0.380173
$125$	− 135.686i	− 1.08549i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	75.5885i	0.590535i
$129$	0	0
$130$	121.231	0.932543
$131$	− 46.6377i	− 0.356013i	−0.984029	−	0.178006i	$-0.943035\pi$
$131$	− 46.6377i	− 0.356013i	0.984029	−	0.178006i	$-0.0569648\pi$
$132$	0	0
$133$	−145.564	−1.09447
$134$	24.4824i	0.182704i
$135$	0	0
$136$	−36.2654	−0.266657
$137$	2.14478i	0.0156553i	0.999969	+	0.00782765i	$0.00249164\pi$
$137$	2.14478i	0.0156553i	−0.999969	+	0.00782765i	$0.997508\pi$
$138$	0	0
$139$	68.4845	0.492694	0.246347	−	0.969182i	$-0.420770\pi$
$139$	68.4845	0.492694	0.246347	+	0.969182i	$0.420770\pi$
$140$	223.755i	1.59825i
$141$	0	0
$142$	61.1430	0.430585
$143$	− 21.0384i	− 0.147122i
$144$	0	0
$145$	−52.0126	−0.358708
$146$	− 73.1047i	− 0.500717i
$147$	0	0
$148$	−111.263	−0.751775
$149$	− 127.883i	− 0.858274i	−0.903239	−	0.429137i	$-0.858818\pi$
$149$	− 127.883i	− 0.858274i	0.903239	−	0.429137i	$-0.141182\pi$
$150$	0	0
$151$	42.5770	0.281967	0.140983	−	0.990012i	$-0.454974\pi$
$151$	42.5770	0.281967	0.140983	+	0.990012i	$0.454974\pi$
$152$	29.4586i	0.193806i
$153$	0	0
$154$	71.1056	0.461725
$155$	− 38.2155i	− 0.246552i
$156$	0	0
$157$	−185.690	−1.18274	−0.591370	−	0.806400i	$-0.701413\pi$
$157$	−185.690	−1.18274	−0.591370	+	0.806400i	$0.701413\pi$
$158$	67.4496i	0.426896i
$159$	0	0
$160$	−177.654	−1.11034
$161$	− 202.565i	− 1.25817i
$162$	0	0
$163$	218.551	1.34080	0.670402	−	0.741998i	$-0.266122\pi$
$163$	218.551	1.34080	0.670402	+	0.741998i	$0.266122\pi$
$164$	186.967i	1.14004i
$165$	0	0
$166$	−212.168	−1.27812
$167$	− 58.1575i	− 0.348248i	−0.984724	−	0.174124i	$-0.944291\pi$
$167$	− 58.1575i	− 0.348248i	0.984724	−	0.174124i	$-0.0557094\pi$
$168$	0	0
$169$	−59.4234	−0.351618
$170$	− 174.136i	− 1.02433i
$171$	0	0
$172$	40.6309	0.236226
$173$	287.885i	1.66407i	0.554720	+	0.832037i	$0.312825\pi$
$173$	287.885i	1.66407i	−0.554720	+	0.832037i	$0.687175\pi$
$174$	0	0
$175$	116.560	0.666055
$176$	24.2985i	0.138060i
$177$	0	0
$178$	−32.6757	−0.183571
$179$	− 25.5291i	− 0.142620i	−0.997454	−	0.0713102i	$-0.977282\pi$
$179$	− 25.5291i	− 0.142620i	0.997454	−	0.0713102i	$-0.0227180\pi$
$180$	0	0
$181$	−134.987	−0.745785	−0.372892	−	0.927875i	$-0.621634\pi$
$181$	−134.987	−0.745785	−0.372892	+	0.927875i	$0.621634\pi$
$182$	370.347i	2.03487i
$183$	0	0
$184$	−40.9942	−0.222795
$185$	− 90.1957i	− 0.487544i
$186$	0	0
$187$	−30.2196	−0.161602
$188$	296.100i	1.57500i
$189$	0	0
$190$	−141.451	−0.744481
$191$	188.564i	0.987244i	0.869677	+	0.493622i	$0.164327\pi$
$191$	188.564i	0.987244i	−0.869677	+	0.493622i	$0.835673\pi$
$192$	0	0
$193$	181.955	0.942773	0.471387	−	0.881927i	$-0.343754\pi$
$193$	181.955	0.942773	0.471387	+	0.881927i	$0.343754\pi$
$194$	− 278.478i	− 1.43545i
$195$	0	0
$196$	−447.738	−2.28438
$197$	− 77.3544i	− 0.392662i	−0.980538	−	0.196331i	$-0.937097\pi$
$197$	− 77.3544i	− 0.392662i	0.980538	−	0.196331i	$-0.0629027\pi$
$198$	0	0
$199$	−176.017	−0.884509	−0.442255	−	0.896890i	$-0.645821\pi$
$199$	−176.017	−0.884509	−0.442255	+	0.896890i	$0.645821\pi$
$200$	− 23.5888i	− 0.117944i
$201$	0	0
$202$	−237.917	−1.17781
$203$	− 158.893i	− 0.782724i
$204$	0	0
$205$	−151.566	−0.739347
$206$	317.133i	1.53948i
$207$	0	0
$208$	−126.557	−0.608445
$209$	24.5475i	0.117452i
$210$	0	0
$211$	243.707	1.15501	0.577504	−	0.816388i	$-0.304027\pi$
$211$	243.707	1.15501	0.577504	+	0.816388i	$0.304027\pi$
$212$	− 295.283i	− 1.39284i
$213$	0	0
$214$	−548.507	−2.56312
$215$	32.9376i	0.153198i
$216$	0	0
$217$	116.744	0.537992
$218$	48.1335i	0.220796i
$219$	0	0
$220$	37.7335	0.171516
$221$	− 157.396i	− 0.712198i
$222$	0	0
$223$	−351.699	−1.57713	−0.788564	−	0.614953i	$-0.789175\pi$
$223$	−351.699	−1.57713	−0.788564	+	0.614953i	$0.789175\pi$
$224$	− 542.715i	− 2.42283i
$225$	0	0
$226$	−246.674	−1.09148
$227$	199.659i	0.879556i	0.898107	+	0.439778i	$0.144943\pi$
$227$	199.659i	0.879556i	−0.898107	+	0.439778i	$0.855057\pi$
$228$	0	0
$229$	239.891	1.04756	0.523781	−	0.851853i	$-0.324521\pi$
$229$	239.891	1.04756	0.523781	+	0.851853i	$0.324521\pi$
$230$	− 196.842i	− 0.855836i
$231$	0	0
$232$	−32.1560	−0.138604
$233$	− 130.240i	− 0.558970i	−0.960150	−	0.279485i	$-0.909836\pi$
$233$	− 130.240i	− 0.558970i	0.960150	−	0.279485i	$-0.0901637\pi$
$234$	0	0
$235$	−240.035	−1.02142
$236$	186.038i	0.788298i
$237$	0	0
$238$	531.966	2.23515
$239$	− 424.397i	− 1.77572i	−0.460113	−	0.887861i	$-0.652191\pi$
$239$	− 424.397i	− 1.77572i	0.460113	−	0.887861i	$-0.347809\pi$
$240$	0	0
$241$	−330.531	−1.37150	−0.685749	−	0.727838i	$-0.740525\pi$
$241$	−330.531	−1.37150	−0.685749	+	0.727838i	$0.740525\pi$
$242$	347.207i	1.43474i
$243$	0	0
$244$	−236.507	−0.969292
$245$	− 362.961i	− 1.48147i
$246$	0	0
$247$	−127.854	−0.517626
$248$	− 23.6262i	− 0.0952668i
$249$	0	0
$250$	402.796	1.61119
$251$	− 186.099i	− 0.741429i	−0.928747	−	0.370714i	$-0.879113\pi$
$251$	− 186.099i	− 0.741429i	0.928747	−	0.370714i	$-0.120887\pi$
$252$	0	0
$253$	−34.1601	−0.135020
$254$	33.4542i	0.131709i
$255$	0	0
$256$	122.899	0.480074
$257$	81.5366i	0.317263i	0.987338	+	0.158632i	$0.0507082\pi$
$257$	81.5366i	0.317263i	−0.987338	+	0.158632i	$0.949292\pi$
$258$	0	0
$259$	275.538	1.06385
$260$	196.531i	0.755890i
$261$	0	0
$262$	138.448	0.528427
$263$	480.944i	1.82868i	0.404943	+	0.914342i	$0.367291\pi$
$263$	480.944i	1.82868i	−0.404943	+	0.914342i	$0.632709\pi$
$264$	0	0
$265$	239.373	0.903293
$266$	− 432.119i	− 1.62451i
$267$	0	0
$268$	−39.6893	−0.148094
$269$	454.168i	1.68836i	0.536063	+	0.844178i	$0.319911\pi$
$269$	454.168i	1.68836i	−0.536063	+	0.844178i	$0.680089\pi$
$270$	0	0
$271$	197.293	0.728018	0.364009	−	0.931395i	$-0.381408\pi$
$271$	197.293	0.728018	0.364009	+	0.931395i	$0.381408\pi$
$272$	181.786i	0.668330i
$273$	0	0
$274$	−6.36694	−0.0232370
$275$	− 19.6563i	− 0.0714774i
$276$	0	0
$277$	357.402	1.29026	0.645130	−	0.764073i	$-0.276803\pi$
$277$	357.402	1.29026	0.645130	+	0.764073i	$0.276803\pi$
$278$	203.302i	0.731302i
$279$	0	0
$280$	−112.141	−0.400503
$281$	236.862i	0.842925i	0.906846	+	0.421462i	$0.138483\pi$
$281$	236.862i	0.842925i	−0.906846	+	0.421462i	$0.861517\pi$
$282$	0	0
$283$	276.813	0.978138	0.489069	−	0.872245i	$-0.337337\pi$
$283$	276.813	0.978138	0.489069	+	0.872245i	$0.337337\pi$
$284$	99.1211i	0.349018i
$285$	0	0
$286$	62.4542	0.218371
$287$	− 463.018i	− 1.61330i
$288$	0	0
$289$	62.9170	0.217706
$290$	− 154.404i	− 0.532427i
$291$	0	0
$292$	118.513	0.405865
$293$	26.4288i	0.0902007i	0.998982	+	0.0451004i	$0.0143608\pi$
$293$	26.4288i	0.0902007i	−0.998982	+	0.0451004i	$0.985639\pi$
$294$	0	0
$295$	−150.813	−0.511231
$296$	− 55.7621i	− 0.188385i
$297$	0	0
$298$	379.631	1.27393
$299$	− 177.919i	− 0.595048i
$300$	0	0
$301$	−100.621	−0.334289
$302$	126.393i	0.418520i
$303$	0	0
$304$	147.666	0.485743
$305$	− 191.726i	− 0.628609i
$306$	0	0
$307$	−187.955	−0.612232	−0.306116	−	0.951994i	$-0.599030\pi$
$307$	−187.955	−0.612232	−0.306116	+	0.951994i	$0.599030\pi$
$308$	115.272i	0.374259i
$309$	0	0
$310$	113.446	0.365954
$311$	− 168.017i	− 0.540248i	−0.962825	−	0.270124i	$-0.912935\pi$
$311$	− 168.017i	− 0.540248i	0.962825	−	0.270124i	$-0.0870648\pi$
$312$	0	0
$313$	−498.185	−1.59164	−0.795822	−	0.605530i	$-0.792961\pi$
$313$	−498.185	−1.59164	−0.795822	+	0.605530i	$0.792961\pi$
$314$	− 551.237i	− 1.75553i
$315$	0	0
$316$	−109.345	−0.346028
$317$	− 170.371i	− 0.537447i	−0.963217	−	0.268724i	$-0.913398\pi$
$317$	− 170.371i	− 0.537447i	0.963217	−	0.268724i	$-0.0866019\pi$
$318$	0	0
$319$	−26.7953	−0.0839978
$320$	− 338.716i	− 1.05849i
$321$	0	0
$322$	601.332	1.86749
$323$	183.649i	0.568572i
$324$	0	0
$325$	102.378	0.315009
$326$	648.787i	1.99014i
$327$	0	0
$328$	−93.7034	−0.285681
$329$	− 733.281i	− 2.22882i
$330$	0	0
$331$	−91.5223	−0.276502	−0.138251	−	0.990397i	$-0.544148\pi$
$331$	−91.5223	−0.276502	−0.138251	+	0.990397i	$0.544148\pi$
$332$	− 343.953i	− 1.03600i
$333$	0	0
$334$	172.645	0.516902
$335$	− 32.1743i	− 0.0960427i
$336$	0	0
$337$	391.711	1.16235	0.581174	−	0.813779i	$-0.302594\pi$
$337$	391.711	1.16235	0.581174	+	0.813779i	$0.302594\pi$
$338$	− 176.403i	− 0.521903i
$339$	0	0
$340$	282.297	0.830286
$341$	− 19.6874i	− 0.0577344i
$342$	0	0
$343$	524.829	1.53011
$344$	20.3632i	0.0591953i
$345$	0	0
$346$	−854.609	−2.46997
$347$	266.022i	0.766633i	0.923617	+	0.383316i	$0.125218\pi$
$347$	266.022i	0.766633i	−0.923617	+	0.383316i	$0.874782\pi$
$348$	0	0
$349$	−140.116	−0.401479	−0.200739	−	0.979645i	$-0.564334\pi$
$349$	−140.116	−0.401479	−0.200739	+	0.979645i	$0.564334\pi$
$350$	346.017i	0.988619i
$351$	0	0
$352$	−91.5219	−0.260006
$353$	43.8632i	0.124258i	0.998068	+	0.0621292i	$0.0197891\pi$
$353$	43.8632i	0.124258i	−0.998068	+	0.0621292i	$0.980211\pi$
$354$	0	0
$355$	−80.3531	−0.226347
$356$	− 52.9717i	− 0.148797i
$357$	0	0
$358$	75.7851	0.211690
$359$	383.231i	1.06750i	0.845644	+	0.533748i	$0.179217\pi$
$359$	383.231i	1.06750i	−0.845644	+	0.533748i	$0.820783\pi$
$360$	0	0
$361$	−211.821	−0.586762
$362$	− 400.720i	− 1.10696i
$363$	0	0
$364$	−600.382	−1.64940
$365$	96.0729i	0.263213i
$366$	0	0
$367$	83.5467	0.227648	0.113824	−	0.993501i	$-0.463690\pi$
$367$	83.5467	0.227648	0.113824	+	0.993501i	$0.463690\pi$
$368$	205.490i	0.558397i
$369$	0	0
$370$	267.753	0.723657
$371$	731.258i	1.97105i
$372$	0	0
$373$	−254.080	−0.681181	−0.340590	−	0.940212i	$-0.610627\pi$
$373$	−254.080	−0.681181	−0.340590	+	0.940212i	$0.610627\pi$
$374$	− 89.7092i	− 0.239864i
$375$	0	0
$376$	−148.398	−0.394675
$377$	− 139.561i	− 0.370187i
$378$	0	0
$379$	628.995	1.65962	0.829809	−	0.558047i	$-0.188449\pi$
$379$	628.995	1.65962	0.829809	+	0.558047i	$0.188449\pi$
$380$	− 229.312i	− 0.603453i
$381$	0	0
$382$	−559.766	−1.46536
$383$	− 367.947i	− 0.960698i	−0.877077	−	0.480349i	$-0.840510\pi$
$383$	− 367.947i	− 0.960698i	0.877077	−	0.480349i	$-0.159490\pi$
$384$	0	0
$385$	−93.4457	−0.242716
$386$	540.149i	1.39935i
$387$	0	0
$388$	451.450	1.16353
$389$	− 530.476i	− 1.36369i	−0.731496	−	0.681845i	$-0.761178\pi$
$389$	− 530.476i	− 1.36369i	0.731496	−	0.681845i	$-0.238822\pi$
$390$	0	0
$391$	−255.563	−0.653615
$392$	− 224.395i	− 0.572437i
$393$	0	0
$394$	229.633	0.582824
$395$	− 88.6411i	− 0.224408i
$396$	0	0
$397$	−606.103	−1.52671	−0.763354	−	0.645981i	$-0.776449\pi$
$397$	−606.103	−1.52671	−0.763354	+	0.645981i	$0.776449\pi$
$398$	− 522.522i	− 1.31287i
$399$	0	0
$400$	−118.242	−0.295606
$401$	437.995i	1.09226i	0.837702	+	0.546128i	$0.183899\pi$
$401$	437.995i	1.09226i	−0.837702	+	0.546128i	$0.816101\pi$
$402$	0	0
$403$	102.540	0.254442
$404$	− 385.696i	− 0.954693i
$405$	0	0
$406$	471.687	1.16179
$407$	− 46.4660i	− 0.114167i
$408$	0	0
$409$	−470.257	−1.14977	−0.574886	−	0.818233i	$-0.694954\pi$
$409$	−470.257	−1.14977	−0.574886	+	0.818233i	$0.694954\pi$
$410$	− 449.936i	− 1.09741i
$411$	0	0
$412$	−514.115	−1.24785
$413$	− 460.718i	− 1.11554i
$414$	0	0
$415$	278.827	0.671872
$416$	− 476.683i	− 1.14587i
$417$	0	0
$418$	−72.8714	−0.174333
$419$	148.317i	0.353979i	0.984213	+	0.176990i	$0.0566359\pi$
$419$	148.317i	0.353979i	−0.984213	+	0.176990i	$0.943364\pi$
$420$	0	0
$421$	329.588	0.782868	0.391434	−	0.920206i	$-0.371979\pi$
$421$	329.588	0.782868	0.391434	+	0.920206i	$0.371979\pi$
$422$	723.463i	1.71437i
$423$	0	0
$424$	147.989	0.349030
$425$	− 147.056i	− 0.346013i
$426$	0	0
$427$	585.702	1.37167
$428$	− 889.205i	− 2.07758i
$429$	0	0
$430$	−97.7780	−0.227391
$431$	− 739.306i	− 1.71533i	−0.514211	−	0.857663i	$-0.671915\pi$
$431$	− 739.306i	− 1.71533i	0.514211	−	0.857663i	$-0.328085\pi$
$432$	0	0
$433$	−210.577	−0.486320	−0.243160	−	0.969986i	$-0.578184\pi$
$433$	−210.577	−0.486320	−0.243160	+	0.969986i	$0.578184\pi$
$434$	346.565i	0.798537i
$435$	0	0
$436$	−78.0309	−0.178970
$437$	207.596i	0.475047i
$438$	0	0
$439$	−795.985	−1.81318	−0.906589	−	0.422016i	$-0.861323\pi$
$439$	−795.985	−1.81318	−0.906589	+	0.422016i	$0.861323\pi$
$440$	18.9111i	0.0429798i
$441$	0	0
$442$	467.242	1.05711
$443$	− 186.660i	− 0.421355i	−0.977556	−	0.210678i	$-0.932433\pi$
$443$	− 186.660i	− 0.421355i	0.977556	−	0.210678i	$-0.0675671\pi$
$444$	0	0
$445$	42.9418	0.0964985
$446$	− 1044.05i	− 2.34091i
$447$	0	0
$448$	1034.74	2.30969
$449$	414.327i	0.922776i	0.887198	+	0.461388i	$0.152648\pi$
$449$	414.327i	0.922776i	−0.887198	+	0.461388i	$0.847352\pi$
$450$	0	0
$451$	−78.0821	−0.173131
$452$	− 399.893i	− 0.884718i
$453$	0	0
$454$	−592.704	−1.30552
$455$	− 486.703i	− 1.06968i
$456$	0	0
$457$	−672.941	−1.47252	−0.736259	−	0.676699i	$-0.763410\pi$
$457$	−672.941	−1.47252	−0.736259	+	0.676699i	$0.763410\pi$
$458$	712.137i	1.55488i
$459$	0	0
$460$	319.108	0.693713
$461$	− 876.511i	− 1.90133i	−0.310225	−	0.950663i	$-0.600404\pi$
$461$	− 876.511i	− 1.90133i	0.310225	−	0.950663i	$-0.399596\pi$
$462$	0	0
$463$	89.6318	0.193589	0.0967946	−	0.995304i	$-0.469141\pi$
$463$	89.6318	0.193589	0.0967946	+	0.995304i	$0.469141\pi$
$464$	161.187i	0.347386i
$465$	0	0
$466$	386.628	0.829673
$467$	− 95.3908i	− 0.204263i	−0.994771	−	0.102131i	$-0.967434\pi$
$467$	− 95.3908i	− 0.204263i	0.994771	−	0.102131i	$-0.0325662\pi$
$468$	0	0
$469$	98.2891	0.209572
$470$	− 712.563i	− 1.51609i
$471$	0	0
$472$	−93.2379	−0.197538
$473$	16.9684i	0.0358741i
$474$	0	0
$475$	−119.454	−0.251482
$476$	862.388i	1.81174i
$477$	0	0
$478$	1259.86	2.63569
$479$	− 748.279i	− 1.56217i	−0.624425	−	0.781085i	$-0.714667\pi$
$479$	− 748.279i	− 1.56217i	0.624425	−	0.781085i	$-0.285333\pi$
$480$	0	0
$481$	242.014	0.503147
$482$	− 981.208i	− 2.03570i
$483$	0	0
$484$	−562.870	−1.16295
$485$	365.970i	0.754578i
$486$	0	0
$487$	−674.647	−1.38531	−0.692656	−	0.721268i	$-0.743559\pi$
$487$	−674.647	−1.38531	−0.692656	+	0.721268i	$0.743559\pi$
$488$	− 118.532i	− 0.242893i
$489$	0	0
$490$	1077.48	2.19894
$491$	319.707i	0.651135i	0.945519	+	0.325568i	$0.105555\pi$
$491$	319.707i	0.651135i	−0.945519	+	0.325568i	$0.894445\pi$
$492$	0	0
$493$	−200.465	−0.406622
$494$	− 379.544i	− 0.768307i
$495$	0	0
$496$	−118.430	−0.238770
$497$	− 245.470i	− 0.493903i
$498$	0	0
$499$	−912.107	−1.82787	−0.913935	−	0.405862i	$-0.866972\pi$
$499$	−912.107	−1.82787	−0.913935	+	0.405862i	$0.866972\pi$
$500$	652.988i	1.30598i
$501$	0	0
$502$	552.449	1.10050
$503$	− 442.736i	− 0.880191i	−0.897951	−	0.440096i	$-0.854945\pi$
$503$	− 442.736i	− 0.880191i	0.897951	−	0.440096i	$-0.145055\pi$
$504$	0	0
$505$	312.667	0.619142
$506$	− 101.407i	− 0.200409i
$507$	0	0
$508$	−54.2338	−0.106759
$509$	− 240.686i	− 0.472860i	−0.971649	−	0.236430i	$-0.924023\pi$
$509$	− 240.686i	− 0.472860i	0.971649	−	0.236430i	$-0.0759774\pi$
$510$	0	0
$511$	−293.493	−0.574349
$512$	667.190i	1.30310i
$513$	0	0
$514$	−242.048	−0.470911
$515$	− 416.770i	− 0.809263i
$516$	0	0
$517$	−123.658	−0.239185
$518$	817.957i	1.57907i
$519$	0	0
$520$	−98.4967	−0.189417
$521$	416.011i	0.798486i	0.916845	+	0.399243i	$0.130727\pi$
$521$	416.011i	0.798486i	−0.916845	+	0.399243i	$0.869273\pi$
$522$	0	0
$523$	195.597	0.373991	0.186995	−	0.982361i	$-0.440125\pi$
$523$	195.597	0.373991	0.186995	+	0.982361i	$0.440125\pi$
$524$	224.443i	0.428326i
$525$	0	0
$526$	−1427.72	−2.71430
$527$	− 147.289i	− 0.279485i
$528$	0	0
$529$	240.112	0.453898
$530$	710.597i	1.34075i
$531$	0	0
$532$	700.524	1.31677
$533$	− 406.683i	− 0.763008i
$534$	0	0
$535$	720.839	1.34736
$536$	− 19.8913i	− 0.0371106i
$537$	0	0
$538$	−1348.23	−2.50601
$539$	− 186.986i	− 0.346913i
$540$	0	0
$541$	203.303	0.375792	0.187896	−	0.982189i	$-0.439833\pi$
$541$	203.303	0.375792	0.187896	+	0.982189i	$0.439833\pi$
$542$	585.680i	1.08059i
$543$	0	0
$544$	−684.707	−1.25865
$545$	− 63.2562i	− 0.116066i
$546$	0	0
$547$	−1006.26	−1.83960	−0.919801	−	0.392385i	$-0.871650\pi$
$547$	−1006.26	−1.83960	−0.919801	+	0.392385i	$0.871650\pi$
$548$	− 10.3217i	− 0.0188352i
$549$	0	0
$550$	58.3513	0.106093
$551$	162.839i	0.295533i
$552$	0	0
$553$	270.789	0.489673
$554$	1060.98i	1.91512i
$555$	0	0
$556$	−329.580	−0.592770
$557$	− 882.622i	− 1.58460i	−0.610132	−	0.792300i	$-0.708884\pi$
$557$	− 882.622i	− 1.58460i	0.610132	−	0.792300i	$-0.291116\pi$
$558$	0	0
$559$	−88.3785	−0.158101
$560$	562.123i	1.00379i
$561$	0	0
$562$	−703.143	−1.25114
$563$	− 15.7127i	− 0.0279089i	−0.999903	−	0.0139545i	$-0.995558\pi$
$563$	− 15.7127i	− 0.0279089i	0.999903	−	0.0139545i	$-0.00444199\pi$
$564$	0	0
$565$	324.175	0.573761
$566$	821.742i	1.45184i
$567$	0	0
$568$	−49.6771	−0.0874596
$569$	1032.00i	1.81371i	0.421447	+	0.906853i	$0.361522\pi$
$569$	1032.00i	1.81371i	−0.421447	+	0.906853i	$0.638478\pi$
$570$	0	0
$571$	−241.590	−0.423100	−0.211550	−	0.977367i	$-0.567851\pi$
$571$	−241.590	−0.423100	−0.211550	+	0.977367i	$0.567851\pi$
$572$	101.247i	0.177005i
$573$	0	0
$574$	1374.51	2.39461
$575$	− 166.231i	− 0.289097i
$576$	0	0
$577$	−661.808	−1.14698	−0.573490	−	0.819212i	$-0.694411\pi$
$577$	−661.808	−1.14698	−0.573490	+	0.819212i	$0.694411\pi$
$578$	186.774i	0.323139i
$579$	0	0
$580$	250.309	0.431568
$581$	851.787i	1.46607i
$582$	0	0
$583$	123.317	0.211522
$584$	59.3956i	0.101705i
$585$	0	0
$586$	−78.4561	−0.133884
$587$	− 315.909i	− 0.538175i	−0.963116	−	0.269088i	$-0.913278\pi$
$587$	− 315.909i	− 0.538175i	0.963116	−	0.269088i	$-0.0867221\pi$
$588$	0	0
$589$	−119.643	−0.203130
$590$	− 447.701i	− 0.758815i
$591$	0	0
$592$	−279.516	−0.472156
$593$	269.585i	0.454612i	0.973823	+	0.227306i	$0.0729918\pi$
$593$	269.585i	0.454612i	−0.973823	+	0.227306i	$0.927008\pi$
$594$	0	0
$595$	−699.100	−1.17496
$596$	615.433i	1.03261i
$597$	0	0
$598$	528.168	0.883225
$599$	− 136.849i	− 0.228462i	−0.993454	−	0.114231i	$-0.963560\pi$
$599$	− 136.849i	− 0.228462i	0.993454	−	0.114231i	$-0.0364404\pi$
$600$	0	0
$601$	306.108	0.509332	0.254666	−	0.967029i	$-0.418034\pi$
$601$	306.108	0.509332	0.254666	+	0.967029i	$0.418034\pi$
$602$	− 298.701i	− 0.496182i
$603$	0	0
$604$	−204.900	−0.339239
$605$	− 456.294i	− 0.754204i
$606$	0	0
$607$	−1101.53	−1.81470	−0.907352	−	0.420372i	$-0.861900\pi$
$607$	−1101.53	−1.81470	−0.907352	+	0.420372i	$0.861900\pi$
$608$	556.192i	0.914790i
$609$	0	0
$610$	569.154	0.933039
$611$	− 644.063i	− 1.05411i
$612$	0	0
$613$	−273.907	−0.446831	−0.223415	−	0.974723i	$-0.571721\pi$
$613$	−273.907	−0.446831	−0.223415	+	0.974723i	$0.571721\pi$
$614$	− 557.961i	− 0.908731i
$615$	0	0
$616$	−57.7714	−0.0937847
$617$	− 307.717i	− 0.498731i	−0.968409	−	0.249366i	$-0.919778\pi$
$617$	− 307.717i	− 0.498731i	0.968409	−	0.249366i	$-0.0802221\pi$
$618$	0	0
$619$	132.268	0.213680	0.106840	−	0.994276i	$-0.465927\pi$
$619$	132.268	0.213680	0.106840	+	0.994276i	$0.465927\pi$
$620$	183.911i	0.296631i
$621$	0	0
$622$	498.773	0.801886
$623$	131.183i	0.210566i
$624$	0	0
$625$	−284.843	−0.455749
$626$	− 1478.90i	− 2.36246i
$627$	0	0
$628$	893.629	1.42298
$629$	− 347.628i	− 0.552668i
$630$	0	0
$631$	1019.72	1.61604	0.808018	−	0.589157i	$-0.200540\pi$
$631$	1019.72	1.61604	0.808018	+	0.589157i	$0.200540\pi$
$632$	− 54.8010i	− 0.0867105i
$633$	0	0
$634$	505.760	0.797728
$635$	− 43.9649i	− 0.0692361i
$636$	0	0
$637$	973.900	1.52888
$638$	− 79.5440i	− 0.124677i
$639$	0	0
$640$	294.890	0.460766
$641$	176.132i	0.274777i	0.990517	+	0.137388i	$0.0438708\pi$
$641$	176.132i	0.274777i	−0.990517	+	0.137388i	$0.956129\pi$
$642$	0	0
$643$	1087.93	1.69196	0.845978	−	0.533217i	$-0.179017\pi$
$643$	1087.93	1.69196	0.845978	+	0.533217i	$0.179017\pi$
$644$	974.841i	1.51373i
$645$	0	0
$646$	−545.176	−0.843926
$647$	169.613i	0.262153i	0.991372	+	0.131076i	$0.0418433\pi$
$647$	169.613i	0.262153i	−0.991372	+	0.131076i	$0.958157\pi$
$648$	0	0
$649$	−77.6942	−0.119714
$650$	303.917i	0.467565i
$651$	0	0
$652$	−1051.77	−1.61315
$653$	581.712i	0.890830i	0.895324	+	0.445415i	$0.146944\pi$
$653$	581.712i	0.890830i	−0.895324	+	0.445415i	$0.853056\pi$
$654$	0	0
$655$	−181.946	−0.277780
$656$	469.703i	0.716011i
$657$	0	0
$658$	2176.80	3.30821
$659$	− 918.820i	− 1.39426i	−0.716942	−	0.697132i	$-0.754459\pi$
$659$	− 918.820i	− 1.39426i	0.716942	−	0.697132i	$-0.245541\pi$
$660$	0	0
$661$	−250.125	−0.378403	−0.189202	−	0.981938i	$-0.560590\pi$
$661$	−250.125	−0.378403	−0.189202	+	0.981938i	$0.560590\pi$
$662$	− 271.691i	− 0.410410i
$663$	0	0
$664$	172.381	0.259609
$665$	567.883i	0.853960i
$666$	0	0
$667$	−226.605	−0.339737
$668$	279.881i	0.418984i
$669$	0	0
$670$	95.5120	0.142555
$671$	− 98.7712i	− 0.147200i
$672$	0	0
$673$	876.265	1.30203	0.651014	−	0.759065i	$-0.274344\pi$
$673$	876.265	1.30203	0.651014	+	0.759065i	$0.274344\pi$
$674$	1162.83i	1.72526i
$675$	0	0
$676$	285.974	0.423038
$677$	814.452i	1.20303i	0.798861	+	0.601515i	$0.205436\pi$
$677$	814.452i	1.20303i	−0.798861	+	0.601515i	$0.794564\pi$
$678$	0	0
$679$	−1118.00	−1.64654
$680$	141.481i	0.208060i
$681$	0	0
$682$	58.4437	0.0856946
$683$	1331.38i	1.94931i	0.223722	+	0.974653i	$0.428179\pi$
$683$	1331.38i	1.94931i	−0.223722	+	0.974653i	$0.571821\pi$
$684$	0	0
$685$	8.36733	0.0122151
$686$	1558.00i	2.27113i
$687$	0	0
$688$	102.074	0.148363
$689$	642.287i	0.932201i
$690$	0	0
$691$	698.547	1.01092	0.505461	−	0.862850i	$-0.331323\pi$
$691$	698.547	1.01092	0.505461	+	0.862850i	$0.331323\pi$
$692$	− 1385.44i	− 2.00208i
$693$	0	0
$694$	−789.707	−1.13791
$695$	− 267.176i	− 0.384426i
$696$	0	0
$697$	−584.159	−0.838105
$698$	− 415.946i	− 0.595911i
$699$	0	0
$700$	−560.940	−0.801343
$701$	1006.22i	1.43540i	0.696351	+	0.717701i	$0.254806\pi$
$701$	1006.22i	1.43540i	−0.696351	+	0.717701i	$0.745194\pi$
$702$	0	0
$703$	−282.381	−0.401679
$704$	− 174.496i	− 0.247864i
$705$	0	0
$706$	−130.211	−0.184435
$707$	955.163i	1.35101i
$708$	0	0
$709$	−995.815	−1.40453	−0.702267	−	0.711913i	$-0.747829\pi$
$709$	−995.815	−1.40453	−0.702267	+	0.711913i	$0.747829\pi$
$710$	− 238.535i	− 0.335964i
$711$	0	0
$712$	26.5481	0.0372867
$713$	− 166.494i	− 0.233513i
$714$	0	0
$715$	−82.0763	−0.114792
$716$	122.858i	0.171589i
$717$	0	0
$718$	−1137.65	−1.58447
$719$	140.868i	0.195922i	0.995190	+	0.0979608i	$0.0312320\pi$
$719$	140.868i	0.195922i	−0.995190	+	0.0979608i	$0.968768\pi$
$720$	0	0
$721$	1273.19	1.76587
$722$	− 628.808i	− 0.870925i
$723$	0	0
$724$	649.621	0.897267
$725$	− 130.392i	− 0.179851i
$726$	0	0
$727$	−719.188	−0.989254	−0.494627	−	0.869105i	$-0.664695\pi$
$727$	−719.188	−0.989254	−0.494627	+	0.869105i	$0.664695\pi$
$728$	− 300.897i	− 0.413320i
$729$	0	0
$730$	−285.200	−0.390685
$731$	126.947i	0.173662i
$732$	0	0
$733$	1368.27	1.86668	0.933338	−	0.359000i	$-0.116882\pi$
$733$	1368.27	1.86668	0.933338	+	0.359000i	$0.116882\pi$
$734$	248.015i	0.337895i
$735$	0	0
$736$	−773.991	−1.05162
$737$	− 16.5752i	− 0.0224901i
$738$	0	0
$739$	−97.8408	−0.132396	−0.0661981	−	0.997806i	$-0.521087\pi$
$739$	−97.8408	−0.132396	−0.0661981	+	0.997806i	$0.521087\pi$
$740$	434.064i	0.586573i
$741$	0	0
$742$	−2170.80	−2.92560
$743$	− 393.158i	− 0.529149i	−0.964365	−	0.264575i	$-0.914768\pi$
$743$	− 393.158i	− 0.529149i	0.964365	−	0.264575i	$-0.0852315\pi$
$744$	0	0
$745$	−498.904	−0.669670
$746$	− 754.258i	− 1.01107i
$747$	0	0
$748$	145.431	0.194426
$749$	2202.09i	2.94003i
$750$	0	0
$751$	−1155.22	−1.53824	−0.769122	−	0.639102i	$-0.779306\pi$
$751$	−1155.22	−1.53824	−0.769122	+	0.639102i	$0.779306\pi$
$752$	743.868i	0.989186i
$753$	0	0
$754$	414.297	0.549466
$755$	− 166.104i	− 0.220005i
$756$	0	0
$757$	−1031.03	−1.36200	−0.680998	−	0.732285i	$-0.738454\pi$
$757$	−1031.03	−1.36200	−0.680998	+	0.732285i	$0.738454\pi$
$758$	1867.22i	2.46336i
$759$	0	0
$760$	114.926	0.151218
$761$	587.059i	0.771431i	0.922618	+	0.385715i	$0.126045\pi$
$761$	587.059i	0.771431i	−0.922618	+	0.385715i	$0.873955\pi$
$762$	0	0
$763$	193.241	0.253265
$764$	− 907.457i	− 1.18777i
$765$	0	0
$766$	1092.28	1.42596
$767$	− 404.663i	− 0.527592i
$768$	0	0
$769$	−803.230	−1.04451	−0.522256	−	0.852789i	$-0.674910\pi$
$769$	−803.230	−1.04451	−0.522256	+	0.852789i	$0.674910\pi$
$770$	− 277.401i	− 0.360261i
$771$	0	0
$772$	−875.655	−1.13427
$773$	653.000i	0.844761i	0.906419	+	0.422381i	$0.138805\pi$
$773$	653.000i	0.844761i	−0.906419	+	0.422381i	$0.861195\pi$
$774$	0	0
$775$	95.8037	0.123618
$776$	226.256i	0.291567i
$777$	0	0
$778$	1574.76	2.02411
$779$	474.516i	0.609135i
$780$	0	0
$781$	−41.3954	−0.0530031
$782$	− 758.661i	− 0.970154i
$783$	0	0
$784$	−1124.82	−1.43471
$785$	724.425i	0.922835i
$786$	0	0
$787$	−299.587	−0.380669	−0.190335	−	0.981719i	$-0.560957\pi$
$787$	−299.587	−0.380669	−0.190335	+	0.981719i	$0.560957\pi$
$788$	372.266i	0.472419i
$789$	0	0
$790$	263.138	0.333086
$791$	990.321i	1.25199i
$792$	0	0
$793$	514.440	0.648727
$794$	− 1799.27i	− 2.26608i
$795$	0	0
$796$	847.079	1.06417
$797$	698.510i	0.876424i	0.898872	+	0.438212i	$0.144388\pi$
$797$	698.510i	0.876424i	−0.898872	+	0.438212i	$0.855612\pi$
$798$	0	0
$799$	−925.132	−1.15786
$800$	− 445.367i	− 0.556709i
$801$	0	0
$802$	−1300.22	−1.62122
$803$	49.4938i	0.0616361i
$804$	0	0
$805$	−790.260	−0.981689
$806$	304.399i	0.377666i
$807$	0	0
$808$	193.301	0.239234
$809$	803.469i	0.993163i	0.867990	+	0.496582i	$0.165412\pi$
$809$	803.469i	0.993163i	−0.867990	+	0.496582i	$0.834588\pi$
$810$	0	0
$811$	−1104.38	−1.36175	−0.680874	−	0.732400i	$-0.738400\pi$
$811$	−1104.38	−1.36175	−0.680874	+	0.732400i	$0.738400\pi$
$812$	764.668i	0.941710i
$813$	0	0
$814$	137.938	0.169457
$815$	− 852.624i	− 1.04616i
$816$	0	0
$817$	103.120	0.126217
$818$	− 1396.00i	− 1.70660i
$819$	0	0
$820$	729.408	0.889521
$821$	1038.73i	1.26520i	0.774479	+	0.632600i	$0.218012\pi$
$821$	1038.73i	1.26520i	−0.774479	+	0.632600i	$0.781988\pi$
$822$	0	0
$823$	−273.226	−0.331988	−0.165994	−	0.986127i	$-0.553083\pi$
$823$	−273.226	−0.331988	−0.165994	+	0.986127i	$0.553083\pi$
$824$	− 257.662i	− 0.312697i
$825$	0	0
$826$	1367.68	1.65578
$827$	723.298i	0.874604i	0.899315	+	0.437302i	$0.144066\pi$
$827$	723.298i	0.874604i	−0.899315	+	0.437302i	$0.855934\pi$
$828$	0	0
$829$	−314.714	−0.379631	−0.189815	−	0.981820i	$-0.560789\pi$
$829$	−314.714	−0.379631	−0.189815	+	0.981820i	$0.560789\pi$
$830$	827.720i	0.997254i
$831$	0	0
$832$	908.846	1.09236
$833$	− 1398.91i	− 1.67936i
$834$	0	0
$835$	−226.887	−0.271721
$836$	− 118.134i	− 0.141309i
$837$	0	0
$838$	−440.292	−0.525408
$839$	− 418.313i	− 0.498585i	−0.968428	−	0.249293i	$-0.919802\pi$
$839$	− 418.313i	− 0.498585i	0.968428	−	0.249293i	$-0.0801981\pi$
$840$	0	0
$841$	663.250	0.788645
$842$	978.407i	1.16200i
$843$	0	0
$844$	−1172.83	−1.38961
$845$	231.826i	0.274350i
$846$	0	0
$847$	1393.93	1.64572
$848$	− 741.816i	− 0.874783i
$849$	0	0
$850$	436.546	0.513584
$851$	− 392.958i	− 0.461760i
$852$	0	0
$853$	1438.86	1.68682	0.843412	−	0.537267i	$-0.180543\pi$
$853$	1438.86	1.68682	0.843412	+	0.537267i	$0.180543\pi$
$854$	1738.70i	2.03595i
$855$	0	0
$856$	445.648	0.520616
$857$	1244.60i	1.45228i	0.687549	+	0.726138i	$0.258687\pi$
$857$	1244.60i	1.45228i	−0.687549	+	0.726138i	$0.741313\pi$
$858$	0	0
$859$	121.302	0.141214	0.0706068	−	0.997504i	$-0.477506\pi$
$859$	121.302	0.141214	0.0706068	+	0.997504i	$0.477506\pi$
$860$	− 158.511i	− 0.184316i
$861$	0	0
$862$	2194.69	2.54604
$863$	802.161i	0.929503i	0.885441	+	0.464752i	$0.153856\pi$
$863$	802.161i	0.929503i	−0.885441	+	0.464752i	$0.846144\pi$
$864$	0	0
$865$	1123.11	1.29840
$866$	− 625.114i	− 0.721841i
$867$	0	0
$868$	−561.829	−0.647268
$869$	− 45.6651i	− 0.0525491i
$870$	0	0
$871$	86.3303	0.0991163
$872$	− 39.1072i	− 0.0448477i
$873$	0	0
$874$	−616.265	−0.705108
$875$	− 1617.10i	− 1.84812i
$876$	0	0
$877$	181.374	0.206812	0.103406	−	0.994639i	$-0.467026\pi$
$877$	181.374	0.206812	0.103406	+	0.994639i	$0.467026\pi$
$878$	− 2362.95i	− 2.69128i
$879$	0	0
$880$	94.7949	0.107721
$881$	750.315i	0.851663i	0.904802	+	0.425832i	$0.140018\pi$
$881$	750.315i	0.851663i	−0.904802	+	0.425832i	$0.859982\pi$
$882$	0	0
$883$	1709.63	1.93616	0.968082	−	0.250635i	$-0.0806393\pi$
$883$	1709.63	1.93616	0.968082	+	0.250635i	$0.0806393\pi$
$884$	757.463i	0.856858i
$885$	0	0
$886$	554.117	0.625414
$887$	− 876.953i	− 0.988673i	−0.869271	−	0.494336i	$-0.835411\pi$
$887$	− 876.953i	− 0.988673i	0.869271	−	0.494336i	$-0.164589\pi$
$888$	0	0
$889$	134.308	0.151078
$890$	127.476i	0.143232i
$891$	0	0
$892$	1692.54	1.89747
$893$	751.490i	0.841535i
$894$	0	0
$895$	−99.5954	−0.111280
$896$	900.858i	1.00542i
$897$	0	0
$898$	−1229.96	−1.36967
$899$	− 130.599i	− 0.145271i
$900$	0	0
$901$	922.580	1.02395
$902$	− 231.793i	− 0.256977i
$903$	0	0
$904$	200.416	0.221700
$905$	526.619i	0.581900i
$906$	0	0
$907$	−604.074	−0.666013	−0.333007	−	0.942924i	$-0.608063\pi$
$907$	−604.074	−0.666013	−0.333007	+	0.942924i	$0.608063\pi$
$908$	− 960.854i	− 1.05821i
$909$	0	0
$910$	1444.82	1.58771
$911$	360.889i	0.396146i	0.980187	+	0.198073i	$0.0634683\pi$
$911$	360.889i	0.396146i	−0.980187	+	0.198073i	$0.936532\pi$
$912$	0	0
$913$	143.643	0.157331
$914$	− 1997.68i	− 2.18565i
$915$	0	0
$916$	−1154.47	−1.26034
$917$	− 555.825i	− 0.606134i
$918$	0	0
$919$	−267.944	−0.291561	−0.145780	−	0.989317i	$-0.546569\pi$
$919$	−267.944	−0.291561	−0.145780	+	0.989317i	$0.546569\pi$
$920$	159.929i	0.173836i
$921$	0	0
$922$	2602.00	2.82212
$923$	− 215.604i	− 0.233590i
$924$	0	0
$925$	226.114	0.244448
$926$	266.079i	0.287343i
$927$	0	0
$928$	−607.121	−0.654225
$929$	− 219.959i	− 0.236770i	−0.992968	−	0.118385i	$-0.962228\pi$
$929$	− 219.959i	− 0.236770i	0.992968	−	0.118385i	$-0.0377717\pi$
$930$	0	0
$931$	−1136.34	−1.22056
$932$	626.776i	0.672507i
$933$	0	0
$934$	283.175	0.303185
$935$	117.894i	0.126090i
$936$	0	0
$937$	−252.272	−0.269234	−0.134617	−	0.990898i	$-0.542980\pi$
$937$	−252.272	−0.269234	−0.134617	+	0.990898i	$0.542980\pi$
$938$	291.779i	0.311065i
$939$	0	0
$940$	1155.16	1.22889
$941$	− 600.523i	− 0.638176i	−0.947725	−	0.319088i	$-0.896623\pi$
$941$	− 600.523i	− 0.638176i	0.947725	−	0.319088i	$-0.103377\pi$
$942$	0	0
$943$	−660.331	−0.700245
$944$	467.369i	0.495095i
$945$	0	0
$946$	−50.3722	−0.0532476
$947$	− 38.7868i	− 0.0409575i	−0.999790	−	0.0204788i	$-0.993481\pi$
$947$	− 38.7868i	− 0.0409575i	0.999790	−	0.0204788i	$-0.00651905\pi$
$948$	0	0
$949$	−257.784	−0.271637
$950$	− 354.609i	− 0.373273i
$951$	0	0
$952$	−432.208	−0.454000
$953$	− 101.007i	− 0.105989i	−0.998595	−	0.0529944i	$-0.983123\pi$
$953$	− 101.007i	− 0.105989i	0.998595	−	0.0529944i	$-0.0168765\pi$
$954$	0	0
$955$	735.635	0.770299
$956$	2042.40i	2.13640i
$957$	0	0
$958$	2221.33	2.31871
$959$	25.5613i	0.0266541i
$960$	0	0
$961$	−865.044	−0.900150
$962$	718.437i	0.746816i
$963$	0	0
$964$	1590.67	1.65007
$965$	− 709.854i	− 0.735600i
$966$	0	0
$967$	−768.260	−0.794478	−0.397239	−	0.917715i	$-0.630032\pi$
$967$	−768.260	−0.794478	−0.397239	+	0.917715i	$0.630032\pi$
$968$	− 282.097i	− 0.291422i
$969$	0	0
$970$	−1086.41	−1.12001
$971$	688.506i	0.709069i	0.935043	+	0.354535i	$0.115361\pi$
$971$	688.506i	0.709069i	−0.935043	+	0.354535i	$0.884639\pi$
$972$	0	0
$973$	816.193	0.838842
$974$	− 2002.74i	− 2.05620i
$975$	0	0
$976$	−594.158	−0.608769
$977$	926.097i	0.947898i	0.880552	+	0.473949i	$0.157172\pi$
$977$	926.097i	0.947898i	−0.880552	+	0.473949i	$0.842828\pi$
$978$	0	0
$979$	22.1223	0.0225968
$980$	1746.74i	1.78239i
$981$	0	0
$982$	−949.077	−0.966474
$983$	− 1142.89i	− 1.16266i	−0.813669	−	0.581328i	$-0.802533\pi$
$983$	− 1142.89i	− 1.16266i	0.813669	−	0.581328i	$-0.197467\pi$
$984$	0	0
$985$	−301.779	−0.306375
$986$	− 595.096i	− 0.603546i
$987$	0	0
$988$	615.292	0.622765
$989$	143.500i	0.145096i
$990$	0	0
$991$	634.172	0.639931	0.319966	−	0.947429i	$-0.396329\pi$
$991$	634.172	0.639931	0.319966	+	0.947429i	$0.396329\pi$
$992$	− 446.073i	− 0.449671i
$993$	0	0
$994$	728.698	0.733096
$995$	686.689i	0.690140i
$996$	0	0
$997$	1132.63	1.13604	0.568020	−	0.823015i	$-0.307710\pi$
$997$	1132.63	1.13604	0.568020	+	0.823015i	$0.307710\pi$
$998$	− 2707.66i	− 2.71309i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.71	yes	84
3.2	odd	2	inner	1143.3.b.a.890.14	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.14	✓	84	3.2	odd	2	inner
1143.3.b.a.890.71	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)

\(f(q)\)	\(=\)	\(q+2.96858i q^{2} -4.81247 q^{4} -3.90126i q^{5} +11.9179 q^{7} -2.41189i q^{8} +O(q^{10})\)	\(q+2.96858i q^{2} -4.81247 q^{4} -3.90126i q^{5} +11.9179 q^{7} -2.41189i q^{8} +11.5812 q^{10} -2.00981i q^{11} +10.4679 q^{13} +35.3793i q^{14} -12.0900 q^{16} -15.0361i q^{17} -12.2139 q^{19} +18.7747i q^{20} +5.96627 q^{22} -16.9967i q^{23} +9.78019 q^{25} +31.0748i q^{26} -57.3547 q^{28} -13.3323i q^{29} +9.79569 q^{31} -45.5377i q^{32} +44.6358 q^{34} -46.4949i q^{35} +23.1196 q^{37} -36.2579i q^{38} -9.40942 q^{40} -38.8506i q^{41} -8.44282 q^{43} +9.67214i q^{44} +50.4561 q^{46} -61.5275i q^{47} +93.0369 q^{49} +29.0333i q^{50} -50.3764 q^{52} +61.3578i q^{53} -7.84077 q^{55} -28.7448i q^{56} +39.5779 q^{58} -38.6575i q^{59} +49.1446 q^{61} +29.0793i q^{62} +86.8224 q^{64} -40.8379i q^{65} +8.24716 q^{67} +72.3606i q^{68} +138.024 q^{70} -20.5967i q^{71} -24.6261 q^{73} +68.6325i q^{74} +58.7790 q^{76} -23.9527i q^{77} +22.7212 q^{79} +47.1662i q^{80} +115.331 q^{82} +71.4711i q^{83} -58.6595 q^{85} -25.0632i q^{86} -4.84744 q^{88} +11.0072i q^{89} +124.755 q^{91} +81.7962i q^{92} +182.650 q^{94} +47.6495i q^{95} -93.8083 q^{97} +276.188i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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

Related objects

Downloads

Learn more