LMFDB - Embedded newform 1143.3.b.a.890.1

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			890.1
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.84

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +O(q^{10})$	\(q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +7.89464 q^{10} -7.97333i q^{11} -22.6045 q^{13} +22.1957i q^{14} +62.7235 q^{16} +24.8287i q^{17} +15.9522 q^{19} -22.5449i q^{20} -30.9785 q^{22} -13.4635i q^{23} +20.8712 q^{25} +87.8245i q^{26} +63.3848 q^{28} +22.5281i q^{29} -1.55815 q^{31} -133.430i q^{32} +96.4661 q^{34} -11.6081i q^{35} +3.12185 q^{37} -61.9785i q^{38} -56.0144 q^{40} +4.15270i q^{41} +9.89437 q^{43} +88.4661i q^{44} -52.3091 q^{46} -35.6762i q^{47} -16.3640 q^{49} -81.0900i q^{50} +250.803 q^{52} -98.5702i q^{53} +16.2014 q^{55} -157.484i q^{56} +87.5273 q^{58} +70.4093i q^{59} +37.5028 q^{61} +6.05382i q^{62} -267.515 q^{64} -45.9312i q^{65} +102.001 q^{67} -275.481i q^{68} -45.1004 q^{70} -140.711i q^{71} -37.1066 q^{73} -12.1292i q^{74} -176.994 q^{76} +45.5500i q^{77} -69.2564 q^{79} +127.451i q^{80} +16.1343 q^{82} -82.8101i q^{83} -50.4507 q^{85} -38.4422i q^{86} +219.800 q^{88} +148.742i q^{89} +129.135 q^{91} +149.381i q^{92} -138.611 q^{94} +32.4140i q^{95} +25.7301 q^{97} +63.5784i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.88526i	− 1.94263i	−0.237796	−	0.971315i	$-0.576425\pi$
$2$	− 3.88526i	− 1.94263i	0.237796	−	0.971315i	$-0.423575\pi$
$3$	0	0
$3$	0	0
$4$	−11.0952	−2.77381
$5$	2.03195i	0.406389i	0.979138	+	0.203195i	$0.0651324\pi$
$5$	2.03195i	0.406389i	−0.979138	+	0.203195i	$0.934868\pi$
$6$	0	0
$7$	−5.71279	−0.816113	−0.408057	−	0.912957i	$-0.633793\pi$
$7$	−5.71279	−0.816113	−0.408057	+	0.912957i	$0.633793\pi$
$8$	27.5669i	3.44586i
$9$	0	0
$10$	7.89464	0.789464
$11$	− 7.97333i	− 0.724848i	−0.932013	−	0.362424i	$-0.881949\pi$
$11$	− 7.97333i	− 0.724848i	0.932013	−	0.362424i	$-0.118051\pi$
$12$	0	0
$13$	−22.6045	−1.73881	−0.869405	−	0.494100i	$-0.835498\pi$
$13$	−22.6045	−1.73881	−0.869405	+	0.494100i	$0.835498\pi$
$14$	22.1957i	1.58541i
$15$	0	0
$16$	62.7235	3.92022
$17$	24.8287i	1.46051i	0.683172	+	0.730257i	$0.260600\pi$
$17$	24.8287i	1.46051i	−0.683172	+	0.730257i	$0.739400\pi$
$18$	0	0
$19$	15.9522	0.839590	0.419795	−	0.907619i	$-0.362102\pi$
$19$	15.9522	0.839590	0.419795	+	0.907619i	$0.362102\pi$
$20$	− 22.5449i	− 1.12725i
$21$	0	0
$22$	−30.9785	−1.40811
$23$	− 13.4635i	− 0.585369i	−0.956209	−	0.292684i	$-0.905452\pi$
$23$	− 13.4635i	− 0.585369i	0.956209	−	0.292684i	$-0.0945485\pi$
$24$	0	0
$25$	20.8712	0.834848
$26$	87.8245i	3.37787i
$27$	0	0
$28$	63.3848	2.26374
$29$	22.5281i	0.776829i	0.921485	+	0.388415i	$0.126977\pi$
$29$	22.5281i	0.776829i	−0.921485	+	0.388415i	$0.873023\pi$
$30$	0	0
$31$	−1.55815	−0.0502629	−0.0251315	−	0.999684i	$-0.508000\pi$
$31$	−1.55815	−0.0502629	−0.0251315	+	0.999684i	$0.508000\pi$
$32$	− 133.430i	− 4.16968i
$33$	0	0
$34$	96.4661	2.83724
$35$	− 11.6081i	− 0.331660i
$36$	0	0
$37$	3.12185	0.0843744	0.0421872	−	0.999110i	$-0.486567\pi$
$37$	3.12185	0.0843744	0.0421872	+	0.999110i	$0.486567\pi$
$38$	− 61.9785i	− 1.63101i
$39$	0	0
$40$	−56.0144	−1.40036
$41$	4.15270i	0.101285i	0.998717	+	0.0506427i	$0.0161270\pi$
$41$	4.15270i	0.101285i	−0.998717	+	0.0506427i	$0.983873\pi$
$42$	0	0
$43$	9.89437	0.230102	0.115051	−	0.993360i	$-0.463297\pi$
$43$	9.89437	0.230102	0.115051	+	0.993360i	$0.463297\pi$
$44$	88.4661i	2.01059i
$45$	0	0
$46$	−52.3091	−1.13715
$47$	− 35.6762i	− 0.759067i	−0.925178	−	0.379534i	$-0.876084\pi$
$47$	− 35.6762i	− 0.759067i	0.925178	−	0.379534i	$-0.123916\pi$
$48$	0	0
$49$	−16.3640	−0.333959
$50$	− 81.0900i	− 1.62180i
$51$	0	0
$52$	250.803	4.82313
$53$	− 98.5702i	− 1.85982i	−0.367793	−	0.929908i	$-0.619886\pi$
$53$	− 98.5702i	− 1.85982i	0.367793	−	0.929908i	$-0.380114\pi$
$54$	0	0
$55$	16.2014	0.294571
$56$	− 157.484i	− 2.81221i
$57$	0	0
$58$	87.5273	1.50909
$59$	70.4093i	1.19338i	0.802472	+	0.596689i	$0.203517\pi$
$59$	70.4093i	1.19338i	−0.802472	+	0.596689i	$0.796483\pi$
$60$	0	0
$61$	37.5028	0.614801	0.307400	−	0.951580i	$-0.400541\pi$
$61$	37.5028	0.614801	0.307400	+	0.951580i	$0.400541\pi$
$62$	6.05382i	0.0976423i
$63$	0	0
$64$	−267.515	−4.17992
$65$	− 45.9312i	− 0.706634i
$66$	0	0
$67$	102.001	1.52240	0.761198	−	0.648519i	$-0.224611\pi$
$67$	102.001	1.52240	0.761198	+	0.648519i	$0.224611\pi$
$68$	− 275.481i	− 4.05119i
$69$	0	0
$70$	−45.1004	−0.644292
$71$	− 140.711i	− 1.98185i	−0.134410	−	0.990926i	$-0.542914\pi$
$71$	− 140.711i	− 1.98185i	0.134410	−	0.990926i	$-0.457086\pi$
$72$	0	0
$73$	−37.1066	−0.508310	−0.254155	−	0.967163i	$-0.581797\pi$
$73$	−37.1066	−0.508310	−0.254155	+	0.967163i	$0.581797\pi$
$74$	− 12.1292i	− 0.163908i
$75$	0	0
$76$	−176.994	−2.32886
$77$	45.5500i	0.591558i
$78$	0	0
$79$	−69.2564	−0.876663	−0.438331	−	0.898813i	$-0.644430\pi$
$79$	−69.2564	−0.876663	−0.438331	+	0.898813i	$0.644430\pi$
$80$	127.451i	1.59313i
$81$	0	0
$82$	16.1343	0.196760
$83$	− 82.8101i	− 0.997712i	−0.866685	−	0.498856i	$-0.833754\pi$
$83$	− 82.8101i	− 0.997712i	0.866685	−	0.498856i	$-0.166246\pi$
$84$	0	0
$85$	−50.4507	−0.593537
$86$	− 38.4422i	− 0.447002i
$87$	0	0
$88$	219.800	2.49773
$89$	148.742i	1.67125i	0.549298	+	0.835627i	$0.314895\pi$
$89$	148.742i	1.67125i	−0.549298	+	0.835627i	$0.685105\pi$
$90$	0	0
$91$	129.135	1.41907
$92$	149.381i	1.62370i
$93$	0	0
$94$	−138.611	−1.47459
$95$	32.4140i	0.341200i
$96$	0	0
$97$	25.7301	0.265258	0.132629	−	0.991166i	$-0.457658\pi$
$97$	25.7301	0.265258	0.132629	+	0.991166i	$0.457658\pi$
$98$	63.5784i	0.648760i
$99$	0	0
$100$	−231.571	−2.31571
$101$	80.0970i	0.793040i	0.918026	+	0.396520i	$0.129782\pi$
$101$	80.0970i	0.793040i	−0.918026	+	0.396520i	$0.870218\pi$
$102$	0	0
$103$	89.2829	0.866824	0.433412	−	0.901196i	$-0.357309\pi$
$103$	89.2829	0.866824	0.433412	+	0.901196i	$0.357309\pi$
$104$	− 623.137i	− 5.99170i
$105$	0	0
$106$	−382.971	−3.61293
$107$	− 2.37443i	− 0.0221910i	−0.999938	−	0.0110955i	$-0.996468\pi$
$107$	− 2.37443i	− 0.0221910i	0.999938	−	0.0110955i	$-0.00353188\pi$
$108$	0	0
$109$	2.17902	0.0199910	0.00999551	−	0.999950i	$-0.496818\pi$
$109$	2.17902	0.0199910	0.00999551	+	0.999950i	$0.496818\pi$
$110$	− 62.9466i	− 0.572242i
$111$	0	0
$112$	−358.326	−3.19934
$113$	− 116.498i	− 1.03096i	−0.856902	−	0.515480i	$-0.827614\pi$
$113$	− 116.498i	− 1.03096i	0.856902	−	0.515480i	$-0.172386\pi$
$114$	0	0
$115$	27.3571	0.237887
$116$	− 249.954i	− 2.15478i
$117$	0	0
$118$	273.559	2.31829
$119$	− 141.841i	− 1.19195i
$120$	0	0
$121$	57.4260	0.474595
$122$	− 145.708i	− 1.19433i
$123$	0	0
$124$	17.2881	0.139420
$125$	93.2078i	0.745662i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	505.646i	3.95036i
$129$	0	0
$130$	−178.455	−1.37273
$131$	− 78.9935i	− 0.603004i	−0.953466	−	0.301502i	$-0.902512\pi$
$131$	− 78.9935i	− 0.603004i	0.953466	−	0.301502i	$-0.0974879\pi$
$132$	0	0
$133$	−91.1316	−0.685200
$134$	− 396.299i	− 2.95745i
$135$	0	0
$136$	−684.451	−5.03273
$137$	87.1459i	0.636102i	0.948074	+	0.318051i	$0.103028\pi$
$137$	87.1459i	0.636102i	−0.948074	+	0.318051i	$0.896972\pi$
$138$	0	0
$139$	232.830	1.67504	0.837519	−	0.546408i	$-0.184005\pi$
$139$	232.830	1.67504	0.837519	+	0.546408i	$0.184005\pi$
$140$	128.795i	0.919961i
$141$	0	0
$142$	−546.701	−3.85000
$143$	180.233i	1.26037i
$144$	0	0
$145$	−45.7758	−0.315695
$146$	144.169i	0.987459i
$147$	0	0
$148$	−34.6377	−0.234039
$149$	285.179i	1.91395i	0.290164	+	0.956977i	$0.406290\pi$
$149$	285.179i	1.91395i	−0.290164	+	0.956977i	$0.593710\pi$
$150$	0	0
$151$	123.029	0.814759	0.407380	−	0.913259i	$-0.366443\pi$
$151$	123.029	0.814759	0.407380	+	0.913259i	$0.366443\pi$
$152$	439.753i	2.89311i
$153$	0	0
$154$	176.974	1.14918
$155$	− 3.16608i	− 0.0204263i
$156$	0	0
$157$	209.967	1.33737	0.668685	−	0.743546i	$-0.266858\pi$
$157$	209.967	1.33737	0.668685	+	0.743546i	$0.266858\pi$
$158$	269.079i	1.70303i
$159$	0	0
$160$	271.122	1.69451
$161$	76.9140i	0.477727i
$162$	0	0
$163$	222.316	1.36390	0.681952	−	0.731397i	$-0.261131\pi$
$163$	222.316	1.36390	0.681952	+	0.731397i	$0.261131\pi$
$164$	− 46.0753i	− 0.280947i
$165$	0	0
$166$	−321.739	−1.93819
$167$	172.780i	1.03461i	0.855801	+	0.517306i	$0.173065\pi$
$167$	172.780i	1.03461i	−0.855801	+	0.517306i	$0.826935\pi$
$168$	0	0
$169$	341.965	2.02346
$170$	196.014i	1.15302i
$171$	0	0
$172$	−109.780	−0.638259
$173$	− 252.077i	− 1.45709i	−0.684996	−	0.728547i	$-0.740196\pi$
$173$	− 252.077i	− 1.45709i	0.684996	−	0.728547i	$-0.259804\pi$
$174$	0	0
$175$	−119.233	−0.681330
$176$	− 500.115i	− 2.84156i
$177$	0	0
$178$	577.899	3.24663
$179$	− 26.4932i	− 0.148007i	−0.997258	−	0.0740034i	$-0.976422\pi$
$179$	− 26.4932i	− 0.148007i	0.997258	−	0.0740034i	$-0.0235776\pi$
$180$	0	0
$181$	−63.2787	−0.349606	−0.174803	−	0.984603i	$-0.555929\pi$
$181$	−63.2787	−0.349606	−0.174803	+	0.984603i	$0.555929\pi$
$182$	− 501.723i	− 2.75672i
$183$	0	0
$184$	371.146	2.01710
$185$	6.34344i	0.0342888i
$186$	0	0
$187$	197.968	1.05865
$188$	395.836i	2.10551i
$189$	0	0
$190$	125.937	0.662826
$191$	− 143.322i	− 0.750379i	−0.926948	−	0.375189i	$-0.877578\pi$
$191$	− 143.322i	− 0.750379i	0.926948	−	0.375189i	$-0.122422\pi$
$192$	0	0
$193$	364.106	1.88656	0.943279	−	0.332002i	$-0.107724\pi$
$193$	364.106	1.88656	0.943279	+	0.332002i	$0.107724\pi$
$194$	− 99.9680i	− 0.515299i
$195$	0	0
$196$	181.563	0.926340
$197$	236.066i	1.19830i	0.800635	+	0.599152i	$0.204495\pi$
$197$	236.066i	1.19830i	−0.800635	+	0.599152i	$0.795505\pi$
$198$	0	0
$199$	278.694	1.40047	0.700236	−	0.713912i	$-0.253078\pi$
$199$	278.694	1.40047	0.700236	+	0.713912i	$0.253078\pi$
$200$	575.354i	2.87677i
$201$	0	0
$202$	311.198	1.54058
$203$	− 128.698i	− 0.633981i
$204$	0	0
$205$	−8.43807	−0.0411613
$206$	− 346.887i	− 1.68392i
$207$	0	0
$208$	−1417.84	−6.81652
$209$	− 127.192i	− 0.608575i
$210$	0	0
$211$	−386.586	−1.83216	−0.916080	−	0.400996i	$-0.868664\pi$
$211$	−386.586	−1.83216	−0.916080	+	0.400996i	$0.868664\pi$
$212$	1093.66i	5.15878i
$213$	0	0
$214$	−9.22530	−0.0431089
$215$	20.1048i	0.0935108i
$216$	0	0
$217$	8.90139	0.0410202
$218$	− 8.46606i	− 0.0388352i
$219$	0	0
$220$	−179.758	−0.817083
$221$	− 561.242i	− 2.53956i
$222$	0	0
$223$	7.23993	0.0324660	0.0162330	−	0.999868i	$-0.494833\pi$
$223$	7.23993	0.0324660	0.0162330	+	0.999868i	$0.494833\pi$
$224$	762.256i	3.40293i
$225$	0	0
$226$	−452.627	−2.00277
$227$	298.177i	1.31356i	0.754084	+	0.656778i	$0.228081\pi$
$227$	298.177i	1.31356i	−0.754084	+	0.656778i	$0.771919\pi$
$228$	0	0
$229$	−156.233	−0.682241	−0.341121	−	0.940019i	$-0.610806\pi$
$229$	−156.233	−0.682241	−0.341121	+	0.940019i	$0.610806\pi$
$230$	− 106.289i	− 0.462127i
$231$	0	0
$232$	−621.028	−2.67684
$233$	101.771i	0.436784i	0.975861	+	0.218392i	$0.0700811\pi$
$233$	101.771i	0.436784i	−0.975861	+	0.218392i	$0.929919\pi$
$234$	0	0
$235$	72.4920	0.308477
$236$	− 781.209i	− 3.31021i
$237$	0	0
$238$	−551.091	−2.31551
$239$	122.338i	0.511875i	0.966693	+	0.255937i	$0.0823841\pi$
$239$	122.338i	0.511875i	−0.966693	+	0.255937i	$0.917616\pi$
$240$	0	0
$241$	0.637189	0.00264394	0.00132197	−	0.999999i	$-0.499579\pi$
$241$	0.637189	0.00264394	0.00132197	+	0.999999i	$0.499579\pi$
$242$	− 223.115i	− 0.921962i
$243$	0	0
$244$	−416.103	−1.70534
$245$	− 33.2508i	− 0.135717i
$246$	0	0
$247$	−360.592	−1.45989
$248$	− 42.9534i	− 0.173199i
$249$	0	0
$250$	362.137	1.44855
$251$	135.183i	0.538577i	0.963060	+	0.269289i	$0.0867886\pi$
$251$	135.183i	0.538577i	−0.963060	+	0.269289i	$0.913211\pi$
$252$	0	0
$253$	−107.349	−0.424303
$254$	43.7847i	0.172381i
$255$	0	0
$256$	894.507	3.49417
$257$	− 109.808i	− 0.427267i	−0.976914	−	0.213633i	$-0.931470\pi$
$257$	− 109.808i	− 0.427267i	0.976914	−	0.213633i	$-0.0685298\pi$
$258$	0	0
$259$	−17.8345	−0.0688591
$260$	509.618i	1.96007i
$261$	0	0
$262$	−306.910	−1.17141
$263$	− 266.392i	− 1.01290i	−0.862270	−	0.506448i	$-0.830958\pi$
$263$	− 266.392i	− 1.01290i	0.862270	−	0.506448i	$-0.169042\pi$
$264$	0	0
$265$	200.289	0.755809
$266$	354.070i	1.33109i
$267$	0	0
$268$	−1131.72	−4.22284
$269$	79.5758i	0.295821i	0.989001	+	0.147910i	$0.0472547\pi$
$269$	79.5758i	0.295821i	−0.989001	+	0.147910i	$0.952745\pi$
$270$	0	0
$271$	193.955	0.715700	0.357850	−	0.933779i	$-0.383510\pi$
$271$	193.955	0.715700	0.357850	+	0.933779i	$0.383510\pi$
$272$	1557.35i	5.72554i
$273$	0	0
$274$	338.585	1.23571
$275$	− 166.413i	− 0.605138i
$276$	0	0
$277$	118.474	0.427702	0.213851	−	0.976866i	$-0.431399\pi$
$277$	118.474	0.427702	0.213851	+	0.976866i	$0.431399\pi$
$278$	− 904.607i	− 3.25398i
$279$	0	0
$280$	319.999	1.14285
$281$	− 57.3404i	− 0.204058i	−0.994781	−	0.102029i	$-0.967467\pi$
$281$	− 57.3404i	− 0.204058i	0.994781	−	0.102029i	$-0.0325335\pi$
$282$	0	0
$283$	−179.085	−0.632810	−0.316405	−	0.948624i	$-0.602476\pi$
$283$	−179.085	−0.632810	−0.316405	+	0.948624i	$0.602476\pi$
$284$	1561.23i	5.49728i
$285$	0	0
$286$	700.254	2.44844
$287$	− 23.7235i	− 0.0826604i
$288$	0	0
$289$	−327.467	−1.13310
$290$	177.851i	0.613279i
$291$	0	0
$292$	411.707	1.40996
$293$	− 423.884i	− 1.44670i	−0.690481	−	0.723351i	$-0.742601\pi$
$293$	− 423.884i	− 1.44670i	0.690481	−	0.723351i	$-0.257399\pi$
$294$	0	0
$295$	−143.068	−0.484976
$296$	86.0597i	0.290742i
$297$	0	0
$298$	1108.00	3.71810
$299$	304.336i	1.01784i
$300$	0	0
$301$	−56.5245	−0.187789
$302$	− 477.998i	− 1.58278i
$303$	0	0
$304$	1000.58	3.29138
$305$	76.2037i	0.249848i
$306$	0	0
$307$	−474.504	−1.54561	−0.772807	−	0.634641i	$-0.781148\pi$
$307$	−474.504	−1.54561	−0.772807	+	0.634641i	$0.781148\pi$
$308$	− 505.388i	− 1.64087i
$309$	0	0
$310$	−12.3010	−0.0396808
$311$	206.584i	0.664257i	0.943234	+	0.332128i	$0.107767\pi$
$311$	206.584i	0.664257i	−0.943234	+	0.332128i	$0.892233\pi$
$312$	0	0
$313$	112.876	0.360626	0.180313	−	0.983609i	$-0.442289\pi$
$313$	112.876	0.360626	0.180313	+	0.983609i	$0.442289\pi$
$314$	− 815.776i	− 2.59801i
$315$	0	0
$316$	768.416	2.43170
$317$	− 434.770i	− 1.37151i	−0.727831	−	0.685756i	$-0.759472\pi$
$317$	− 434.770i	− 1.37151i	0.727831	−	0.685756i	$-0.240528\pi$
$318$	0	0
$319$	179.624	0.563084
$320$	− 543.576i	− 1.69867i
$321$	0	0
$322$	298.831	0.928047
$323$	396.073i	1.22623i
$324$	0	0
$325$	−471.784	−1.45164
$326$	− 863.757i	− 2.64956i
$327$	0	0
$328$	−114.477	−0.349016
$329$	203.810i	0.619485i
$330$	0	0
$331$	261.919	0.791295	0.395647	−	0.918402i	$-0.370520\pi$
$331$	261.919	0.791295	0.395647	+	0.918402i	$0.370520\pi$
$332$	918.799i	2.76747i
$333$	0	0
$334$	671.296	2.00987
$335$	207.260i	0.618685i
$336$	0	0
$337$	364.685	1.08215	0.541076	−	0.840974i	$-0.318017\pi$
$337$	364.685	1.08215	0.541076	+	0.840974i	$0.318017\pi$
$338$	− 1328.62i	− 3.93084i
$339$	0	0
$340$	559.763	1.64636
$341$	12.4237i	0.0364330i
$342$	0	0
$343$	373.411	1.08866
$344$	272.757i	0.792898i
$345$	0	0
$346$	−979.385	−2.83059
$347$	149.497i	0.430828i	0.976523	+	0.215414i	$0.0691101\pi$
$347$	149.497i	0.430828i	−0.976523	+	0.215414i	$0.930890\pi$
$348$	0	0
$349$	−463.218	−1.32727	−0.663636	−	0.748056i	$-0.730988\pi$
$349$	−463.218	−1.32727	−0.663636	+	0.748056i	$0.730988\pi$
$350$	463.250i	1.32357i
$351$	0	0
$352$	−1063.88	−3.02238
$353$	419.892i	1.18950i	0.803912	+	0.594748i	$0.202748\pi$
$353$	419.892i	1.18950i	−0.803912	+	0.594748i	$0.797252\pi$
$354$	0	0
$355$	285.918	0.805403
$356$	− 1650.32i	− 4.63574i
$357$	0	0
$358$	−102.933	−0.287523
$359$	78.8625i	0.219673i	0.993950	+	0.109836i	$0.0350327\pi$
$359$	78.8625i	0.219673i	−0.993950	+	0.109836i	$0.964967\pi$
$360$	0	0
$361$	−106.527	−0.295089
$362$	245.854i	0.679155i
$363$	0	0
$364$	−1432.78	−3.93622
$365$	− 75.3987i	− 0.206572i
$366$	0	0
$367$	584.688	1.59316	0.796578	−	0.604536i	$-0.206642\pi$
$367$	584.688	1.59316	0.796578	+	0.604536i	$0.206642\pi$
$368$	− 844.476i	− 2.29477i
$369$	0	0
$370$	24.6459	0.0666105
$371$	563.111i	1.51782i
$372$	0	0
$373$	24.0162	0.0643866	0.0321933	−	0.999482i	$-0.489751\pi$
$373$	24.0162	0.0643866	0.0321933	+	0.999482i	$0.489751\pi$
$374$	− 769.157i	− 2.05657i
$375$	0	0
$376$	983.480	2.61564
$377$	− 509.236i	− 1.35076i
$378$	0	0
$379$	609.893	1.60922	0.804609	−	0.593806i	$-0.202375\pi$
$379$	609.893	1.60922	0.804609	+	0.593806i	$0.202375\pi$
$380$	− 359.642i	− 0.946425i
$381$	0	0
$382$	−556.845	−1.45771
$383$	− 545.175i	− 1.42343i	−0.702467	−	0.711716i	$-0.747918\pi$
$383$	− 545.175i	− 1.42343i	0.702467	−	0.711716i	$-0.252082\pi$
$384$	0	0
$385$	−92.5551	−0.240403
$386$	− 1414.64i	− 3.66488i
$387$	0	0
$388$	−285.481	−0.735777
$389$	106.310i	0.273291i	0.990620	+	0.136646i	$0.0436322\pi$
$389$	106.310i	0.273291i	−0.990620	+	0.136646i	$0.956368\pi$
$390$	0	0
$391$	334.281	0.854939
$392$	− 451.105i	− 1.15078i
$393$	0	0
$394$	917.177	2.32786
$395$	− 140.725i	− 0.356266i
$396$	0	0
$397$	710.850	1.79056	0.895278	−	0.445509i	$-0.146977\pi$
$397$	710.850	1.79056	0.895278	+	0.445509i	$0.146977\pi$
$398$	− 1082.80i	− 2.72060i
$399$	0	0
$400$	1309.11	3.27279
$401$	208.003i	0.518711i	0.965782	+	0.259356i	$0.0835102\pi$
$401$	208.003i	0.518711i	−0.965782	+	0.259356i	$0.916490\pi$
$402$	0	0
$403$	35.2213	0.0873977
$404$	− 888.696i	− 2.19974i
$405$	0	0
$406$	−500.025	−1.23159
$407$	− 24.8916i	− 0.0611587i
$408$	0	0
$409$	−655.639	−1.60303	−0.801515	−	0.597974i	$-0.795972\pi$
$409$	−655.639	−1.60303	−0.801515	+	0.597974i	$0.795972\pi$
$410$	32.7841i	0.0799612i
$411$	0	0
$412$	−990.616	−2.40441
$413$	− 402.234i	− 0.973932i
$414$	0	0
$415$	168.266	0.405460
$416$	3016.12i	7.25028i
$417$	0	0
$418$	−494.175	−1.18224
$419$	69.6645i	0.166264i	0.996539	+	0.0831318i	$0.0264923\pi$
$419$	69.6645i	0.166264i	−0.996539	+	0.0831318i	$0.973508\pi$
$420$	0	0
$421$	280.499	0.666268	0.333134	−	0.942880i	$-0.391894\pi$
$421$	280.499	0.666268	0.333134	+	0.942880i	$0.391894\pi$
$422$	1501.99i	3.55921i
$423$	0	0
$424$	2717.27	6.40866
$425$	518.206i	1.21931i
$426$	0	0
$427$	−214.246	−0.501747
$428$	26.3449i	0.0615536i
$429$	0	0
$430$	78.1125	0.181657
$431$	− 507.000i	− 1.17633i	−0.808740	−	0.588167i	$-0.799850\pi$
$431$	− 507.000i	− 1.17633i	0.808740	−	0.588167i	$-0.200150\pi$
$432$	0	0
$433$	221.867	0.512396	0.256198	−	0.966624i	$-0.417530\pi$
$433$	221.867	0.512396	0.256198	+	0.966624i	$0.417530\pi$
$434$	− 34.5842i	− 0.0796872i
$435$	0	0
$436$	−24.1768	−0.0554513
$437$	− 214.772i	− 0.491470i
$438$	0	0
$439$	−243.190	−0.553964	−0.276982	−	0.960875i	$-0.589334\pi$
$439$	−243.190	−0.553964	−0.276982	+	0.960875i	$0.589334\pi$
$440$	446.621i	1.01505i
$441$	0	0
$442$	−2180.57	−4.93342
$443$	− 317.504i	− 0.716714i	−0.933585	−	0.358357i	$-0.883337\pi$
$443$	− 317.504i	− 0.716714i	0.933585	−	0.358357i	$-0.116663\pi$
$444$	0	0
$445$	−302.235	−0.679179
$446$	− 28.1290i	− 0.0630695i
$447$	0	0
$448$	1528.26	3.41129
$449$	− 706.223i	− 1.57288i	−0.617666	−	0.786440i	$-0.711922\pi$
$449$	− 706.223i	− 1.57288i	0.617666	−	0.786440i	$-0.288078\pi$
$450$	0	0
$451$	33.1109	0.0734166
$452$	1292.58i	2.85969i
$453$	0	0
$454$	1158.50	2.55175
$455$	262.395i	0.576693i
$456$	0	0
$457$	657.374	1.43846	0.719228	−	0.694774i	$-0.244496\pi$
$457$	657.374	1.43846	0.719228	+	0.694774i	$0.244496\pi$
$458$	607.007i	1.32534i
$459$	0	0
$460$	−303.533	−0.659855
$461$	− 558.636i	− 1.21179i	−0.795544	−	0.605896i	$-0.792815\pi$
$461$	− 558.636i	− 1.21179i	0.795544	−	0.605896i	$-0.207185\pi$
$462$	0	0
$463$	248.378	0.536454	0.268227	−	0.963356i	$-0.413562\pi$
$463$	248.378	0.536454	0.268227	+	0.963356i	$0.413562\pi$
$464$	1413.04i	3.04534i
$465$	0	0
$466$	395.405	0.848510
$467$	449.563i	0.962662i	0.876539	+	0.481331i	$0.159847\pi$
$467$	449.563i	0.962662i	−0.876539	+	0.481331i	$0.840153\pi$
$468$	0	0
$469$	−582.708	−1.24245
$470$	− 281.650i	− 0.599256i
$471$	0	0
$472$	−1940.97	−4.11221
$473$	− 78.8911i	− 0.166789i
$474$	0	0
$475$	332.942	0.700930
$476$	1573.77i	3.30623i
$477$	0	0
$478$	475.315	0.994383
$479$	581.495i	1.21398i	0.794711	+	0.606988i	$0.207622\pi$
$479$	581.495i	1.21398i	−0.794711	+	0.606988i	$0.792378\pi$
$480$	0	0
$481$	−70.5680	−0.146711
$482$	− 2.47564i	− 0.00513619i
$483$	0	0
$484$	−637.155	−1.31644
$485$	52.2821i	0.107798i
$486$	0	0
$487$	117.988	0.242276	0.121138	−	0.992636i	$-0.461346\pi$
$487$	117.988	0.242276	0.121138	+	0.992636i	$0.461346\pi$
$488$	1033.84i	2.11852i
$489$	0	0
$490$	−129.188	−0.263649
$491$	536.142i	1.09194i	0.837805	+	0.545969i	$0.183838\pi$
$491$	536.142i	1.09194i	−0.837805	+	0.545969i	$0.816162\pi$
$492$	0	0
$493$	−559.343	−1.13457
$494$	1400.99i	2.83602i
$495$	0	0
$496$	−97.7327	−0.197042
$497$	803.855i	1.61741i
$498$	0	0
$499$	162.152	0.324954	0.162477	−	0.986712i	$-0.448052\pi$
$499$	162.152	0.324954	0.162477	+	0.986712i	$0.448052\pi$
$500$	− 1034.16i	− 2.06833i
$501$	0	0
$502$	525.221	1.04626
$503$	80.4958i	0.160031i	0.996794	+	0.0800157i	$0.0254970\pi$
$503$	80.4958i	0.160031i	−0.996794	+	0.0800157i	$0.974503\pi$
$504$	0	0
$505$	−162.753	−0.322283
$506$	417.078i	0.824265i
$507$	0	0
$508$	125.037	0.246136
$509$	− 662.979i	− 1.30251i	−0.758858	−	0.651256i	$-0.774242\pi$
$509$	− 662.979i	− 1.30251i	0.758858	−	0.651256i	$-0.225758\pi$
$510$	0	0
$511$	211.983	0.414839
$512$	− 1452.81i	− 2.83752i
$513$	0	0
$514$	−426.631	−0.830021
$515$	181.418i	0.352268i
$516$	0	0
$517$	−284.458	−0.550209
$518$	69.2917i	0.133768i
$519$	0	0
$520$	1266.18	2.43496
$521$	619.817i	1.18967i	0.803848	+	0.594834i	$0.202782\pi$
$521$	619.817i	1.18967i	−0.803848	+	0.594834i	$0.797218\pi$
$522$	0	0
$523$	114.109	0.218182	0.109091	−	0.994032i	$-0.465206\pi$
$523$	114.109	0.218182	0.109091	+	0.994032i	$0.465206\pi$
$524$	876.452i	1.67262i
$525$	0	0
$526$	−1035.00	−1.96768
$527$	− 38.6869i	− 0.0734098i
$528$	0	0
$529$	347.735	0.657344
$530$	− 778.176i	− 1.46826i
$531$	0	0
$532$	1011.13	1.90062
$533$	− 93.8700i	− 0.176116i
$534$	0	0
$535$	4.82472	0.00901817
$536$	2811.84i	5.24596i
$537$	0	0
$538$	309.173	0.574670
$539$	130.476i	0.242070i
$540$	0	0
$541$	18.1841	0.0336120	0.0168060	−	0.999859i	$-0.494650\pi$
$541$	18.1841	0.0336120	0.0168060	+	0.999859i	$0.494650\pi$
$542$	− 753.565i	− 1.39034i
$543$	0	0
$544$	3312.89	6.08987
$545$	4.42765i	0.00812413i
$546$	0	0
$547$	−946.157	−1.72972	−0.864860	−	0.502013i	$-0.832593\pi$
$547$	−946.157	−1.72972	−0.864860	+	0.502013i	$0.832593\pi$
$548$	− 966.906i	− 1.76443i
$549$	0	0
$550$	−646.558	−1.17556
$551$	359.372i	0.652218i
$552$	0	0
$553$	395.647	0.715456
$554$	− 460.301i	− 0.830868i
$555$	0	0
$556$	−2583.31	−4.64624
$557$	226.354i	0.406380i	0.979139	+	0.203190i	$0.0651309\pi$
$557$	226.354i	0.406380i	−0.979139	+	0.203190i	$0.934869\pi$
$558$	0	0
$559$	−223.658	−0.400103
$560$	− 728.100i	− 1.30018i
$561$	0	0
$562$	−222.782	−0.396410
$563$	813.544i	1.44502i	0.691363	+	0.722508i	$0.257011\pi$
$563$	813.544i	1.44502i	−0.691363	+	0.722508i	$0.742989\pi$
$564$	0	0
$565$	236.719	0.418971
$566$	695.793i	1.22932i
$567$	0	0
$568$	3878.98	6.82918
$569$	− 672.821i	− 1.18246i	−0.806502	−	0.591231i	$-0.798642\pi$
$569$	− 672.821i	− 1.18246i	0.806502	−	0.591231i	$-0.201358\pi$
$570$	0	0
$571$	335.830	0.588144	0.294072	−	0.955783i	$-0.404989\pi$
$571$	335.830	0.588144	0.294072	+	0.955783i	$0.404989\pi$
$572$	− 1999.73i	− 3.49604i
$573$	0	0
$574$	−92.1721	−0.160579
$575$	− 280.999i	− 0.488694i
$576$	0	0
$577$	−124.392	−0.215584	−0.107792	−	0.994173i	$-0.534378\pi$
$577$	−124.392	−0.215584	−0.107792	+	0.994173i	$0.534378\pi$
$578$	1272.29i	2.20120i
$579$	0	0
$580$	507.894	0.875679
$581$	473.077i	0.814246i
$582$	0	0
$583$	−785.933	−1.34808
$584$	− 1022.91i	− 1.75157i
$585$	0	0
$586$	−1646.90	−2.81041
$587$	170.171i	0.289899i	0.989439	+	0.144950i	$0.0463020\pi$
$587$	170.171i	0.289899i	−0.989439	+	0.144950i	$0.953698\pi$
$588$	0	0
$589$	−24.8560	−0.0422003
$590$	555.856i	0.942129i
$591$	0	0
$592$	195.814	0.330766
$593$	− 233.420i	− 0.393625i	−0.980441	−	0.196812i	$-0.936941\pi$
$593$	− 233.420i	− 0.393625i	0.980441	−	0.196812i	$-0.0630590\pi$
$594$	0	0
$595$	288.214	0.484394
$596$	− 3164.13i	− 5.30895i
$597$	0	0
$598$	1182.42	1.97730
$599$	876.407i	1.46312i	0.681779	+	0.731559i	$0.261207\pi$
$599$	876.407i	1.46312i	−0.681779	+	0.731559i	$0.738793\pi$
$600$	0	0
$601$	−821.070	−1.36617	−0.683087	−	0.730337i	$-0.739363\pi$
$601$	−821.070	−1.36617	−0.683087	+	0.730337i	$0.739363\pi$
$602$	219.612i	0.364805i
$603$	0	0
$604$	−1365.03	−2.25999
$605$	116.686i	0.192870i
$606$	0	0
$607$	−243.151	−0.400579	−0.200289	−	0.979737i	$-0.564188\pi$
$607$	−243.151	−0.400579	−0.200289	+	0.979737i	$0.564188\pi$
$608$	− 2128.50i	− 3.50082i
$609$	0	0
$610$	296.071	0.485363
$611$	806.443i	1.31987i
$612$	0	0
$613$	−496.958	−0.810698	−0.405349	−	0.914162i	$-0.632850\pi$
$613$	−496.958	−0.810698	−0.405349	+	0.914162i	$0.632850\pi$
$614$	1843.57i	3.00256i
$615$	0	0
$616$	−1255.67	−2.03843
$617$	− 866.834i	− 1.40492i	−0.711724	−	0.702459i	$-0.752086\pi$
$617$	− 866.834i	− 1.40492i	0.711724	−	0.702459i	$-0.247914\pi$
$618$	0	0
$619$	−766.155	−1.23773	−0.618866	−	0.785497i	$-0.712407\pi$
$619$	−766.155	−1.23773	−0.618866	+	0.785497i	$0.712407\pi$
$620$	35.1284i	0.0566588i
$621$	0	0
$622$	802.632	1.29041
$623$	− 849.729i	− 1.36393i
$624$	0	0
$625$	332.387	0.531819
$626$	− 438.553i	− 0.700564i
$627$	0	0
$628$	−2329.64	−3.70961
$629$	77.5117i	0.123230i
$630$	0	0
$631$	−878.466	−1.39218	−0.696090	−	0.717954i	$-0.745079\pi$
$631$	−878.466	−1.39218	−0.696090	+	0.717954i	$0.745079\pi$
$632$	− 1909.18i	− 3.02086i
$633$	0	0
$634$	−1689.19	−2.66434
$635$	− 22.8989i	− 0.0360612i
$636$	0	0
$637$	369.901	0.580692
$638$	− 697.885i	− 1.09386i
$639$	0	0
$640$	−1027.45	−1.60538
$641$	901.514i	1.40642i	0.710983	+	0.703209i	$0.248250\pi$
$641$	901.514i	1.40642i	−0.710983	+	0.703209i	$0.751750\pi$
$642$	0	0
$643$	−96.0218	−0.149334	−0.0746671	−	0.997209i	$-0.523789\pi$
$643$	−96.0218	−0.149334	−0.0746671	+	0.997209i	$0.523789\pi$
$644$	− 853.380i	− 1.32512i
$645$	0	0
$646$	1538.85	2.38212
$647$	932.456i	1.44120i	0.693351	+	0.720600i	$0.256134\pi$
$647$	932.456i	1.44120i	−0.693351	+	0.720600i	$0.743866\pi$
$648$	0	0
$649$	561.397	0.865018
$650$	1833.00i	2.82000i
$651$	0	0
$652$	−2466.66	−3.78321
$653$	75.4491i	0.115542i	0.998330	+	0.0577711i	$0.0183994\pi$
$653$	75.4491i	0.115542i	−0.998330	+	0.0577711i	$0.981601\pi$
$654$	0	0
$655$	160.510	0.245054
$656$	260.472i	0.397061i
$657$	0	0
$658$	791.857	1.20343
$659$	383.689i	0.582229i	0.956688	+	0.291114i	$0.0940260\pi$
$659$	383.689i	0.582229i	−0.956688	+	0.291114i	$0.905974\pi$
$660$	0	0
$661$	134.541	0.203542	0.101771	−	0.994808i	$-0.467549\pi$
$661$	134.541	0.203542	0.101771	+	0.994808i	$0.467549\pi$
$662$	− 1017.62i	− 1.53719i
$663$	0	0
$664$	2282.82	3.43798
$665$	− 185.175i	− 0.278458i
$666$	0	0
$667$	303.306	0.454731
$668$	− 1917.04i	− 2.86982i
$669$	0	0
$670$	805.257	1.20188
$671$	− 299.023i	− 0.445637i
$672$	0	0
$673$	8.81088	0.0130920	0.00654598	−	0.999979i	$-0.497916\pi$
$673$	8.81088	0.0130920	0.00654598	+	0.999979i	$0.497916\pi$
$674$	− 1416.90i	− 2.10222i
$675$	0	0
$676$	−3794.19	−5.61270
$677$	− 1222.18i	− 1.80529i	−0.430388	−	0.902644i	$-0.641623\pi$
$677$	− 1222.18i	− 1.80529i	0.430388	−	0.902644i	$-0.358377\pi$
$678$	0	0
$679$	−146.990	−0.216481
$680$	− 1390.77i	− 2.04525i
$681$	0	0
$682$	48.2691	0.0707759
$683$	− 650.368i	− 0.952222i	−0.879385	−	0.476111i	$-0.842046\pi$
$683$	− 650.368i	− 0.952222i	0.879385	−	0.476111i	$-0.157954\pi$
$684$	0	0
$685$	−177.076	−0.258505
$686$	− 1450.80i	− 2.11487i
$687$	0	0
$688$	620.610	0.902049
$689$	2228.13i	3.23387i
$690$	0	0
$691$	709.417	1.02665	0.513326	−	0.858194i	$-0.328413\pi$
$691$	709.417	1.02665	0.513326	+	0.858194i	$0.328413\pi$
$692$	2796.86i	4.04170i
$693$	0	0
$694$	580.836	0.836939
$695$	473.099i	0.680718i
$696$	0	0
$697$	−103.106	−0.147929
$698$	1799.72i	2.57840i
$699$	0	0
$700$	1322.92	1.88988
$701$	850.556i	1.21335i	0.794951	+	0.606673i	$0.207496\pi$
$701$	850.556i	1.21335i	−0.794951	+	0.606673i	$0.792504\pi$
$702$	0	0
$703$	49.8005	0.0708399
$704$	2132.98i	3.02981i
$705$	0	0
$706$	1631.39	2.31075
$707$	− 457.578i	− 0.647210i
$708$	0	0
$709$	−1199.05	−1.69118	−0.845591	−	0.533831i	$-0.820752\pi$
$709$	−1199.05	−1.69118	−0.845591	+	0.533831i	$0.820752\pi$
$710$	− 1110.87i	− 1.56460i
$711$	0	0
$712$	−4100.34	−5.75890
$713$	20.9781i	0.0294223i
$714$	0	0
$715$	−366.225	−0.512202
$716$	293.949i	0.410543i
$717$	0	0
$718$	306.401	0.426743
$719$	− 255.580i	− 0.355466i	−0.984079	−	0.177733i	$-0.943124\pi$
$719$	− 255.580i	− 0.355466i	0.984079	−	0.177733i	$-0.0568763\pi$
$720$	0	0
$721$	−510.055	−0.707427
$722$	413.885i	0.573248i
$723$	0	0
$724$	702.093	0.969741
$725$	470.187i	0.648534i
$726$	0	0
$727$	−481.234	−0.661945	−0.330972	−	0.943640i	$-0.607377\pi$
$727$	−481.234	−0.661945	−0.330972	+	0.943640i	$0.607377\pi$
$728$	3559.85i	4.88990i
$729$	0	0
$730$	−292.944	−0.401293
$731$	245.665i	0.336067i
$732$	0	0
$733$	−245.297	−0.334648	−0.167324	−	0.985902i	$-0.553513\pi$
$733$	−245.297	−0.334648	−0.167324	+	0.985902i	$0.553513\pi$
$734$	− 2271.67i	− 3.09491i
$735$	0	0
$736$	−1796.43	−2.44080
$737$	− 813.284i	− 1.10351i
$738$	0	0
$739$	−761.537	−1.03050	−0.515248	−	0.857041i	$-0.672300\pi$
$739$	−761.537	−1.03050	−0.515248	+	0.857041i	$0.672300\pi$
$740$	− 70.3820i	− 0.0951108i
$741$	0	0
$742$	2187.83	2.94856
$743$	− 1378.21i	− 1.85493i	−0.373912	−	0.927464i	$-0.621984\pi$
$743$	− 1378.21i	− 1.85493i	0.373912	−	0.927464i	$-0.378016\pi$
$744$	0	0
$745$	−579.469	−0.777810
$746$	− 93.3092i	− 0.125079i
$747$	0	0
$748$	−2196.50	−2.93650
$749$	13.5647i	0.0181103i
$750$	0	0
$751$	−85.4470	−0.113778	−0.0568888	−	0.998381i	$-0.518118\pi$
$751$	−85.4470	−0.113778	−0.0568888	+	0.998381i	$0.518118\pi$
$752$	− 2237.73i	− 2.97571i
$753$	0	0
$754$	−1978.52	−2.62403
$755$	249.988i	0.331109i
$756$	0	0
$757$	89.9427	0.118815	0.0594074	−	0.998234i	$-0.481079\pi$
$757$	89.9427	0.118815	0.0594074	+	0.998234i	$0.481079\pi$
$758$	− 2369.59i	− 3.12611i
$759$	0	0
$760$	−893.554	−1.17573
$761$	1194.58i	1.56975i	0.619653	+	0.784876i	$0.287273\pi$
$761$	1194.58i	1.56975i	−0.619653	+	0.784876i	$0.712727\pi$
$762$	0	0
$763$	−12.4483	−0.0163149
$764$	1590.20i	2.08141i
$765$	0	0
$766$	−2118.15	−2.76520
$767$	− 1591.57i	− 2.07506i
$768$	0	0
$769$	1240.46	1.61308	0.806542	−	0.591177i	$-0.201337\pi$
$769$	1240.46	1.61308	0.806542	+	0.591177i	$0.201337\pi$
$770$	359.601i	0.467014i
$771$	0	0
$772$	−4039.84	−5.23295
$773$	1146.78i	1.48354i	0.670655	+	0.741770i	$0.266013\pi$
$773$	1146.78i	1.48354i	−0.670655	+	0.741770i	$0.733987\pi$
$774$	0	0
$775$	−32.5205	−0.0419619
$776$	709.297i	0.914043i
$777$	0	0
$778$	413.044	0.530904
$779$	66.2448i	0.0850383i
$780$	0	0
$781$	−1121.94	−1.43654
$782$	− 1298.77i	− 1.66083i
$783$	0	0
$784$	−1026.41	−1.30919
$785$	426.642i	0.543492i
$786$	0	0
$787$	1153.33	1.46548	0.732740	−	0.680509i	$-0.238241\pi$
$787$	1153.33	1.46548	0.732740	+	0.680509i	$0.238241\pi$
$788$	− 2619.21i	− 3.32387i
$789$	0	0
$790$	−546.754	−0.692094
$791$	665.532i	0.841380i
$792$	0	0
$793$	−847.734	−1.06902
$794$	− 2761.84i	− 3.47839i
$795$	0	0
$796$	−3092.18	−3.88464
$797$	934.090i	1.17201i	0.810308	+	0.586004i	$0.199300\pi$
$797$	934.090i	1.17201i	−0.810308	+	0.586004i	$0.800700\pi$
$798$	0	0
$799$	885.794	1.10863
$800$	− 2784.84i	− 3.48104i
$801$	0	0
$802$	808.147	1.00766
$803$	295.864i	0.368448i
$804$	0	0
$805$	−156.285	−0.194143
$806$	− 136.844i	− 0.169781i
$807$	0	0
$808$	−2208.03	−2.73270
$809$	− 10.3335i	− 0.0127732i	−0.999980	−	0.00638658i	$-0.997967\pi$
$809$	− 10.3335i	− 0.0127732i	0.999980	−	0.00638658i	$-0.00203293\pi$
$810$	0	0
$811$	1442.35	1.77848	0.889240	−	0.457440i	$-0.151234\pi$
$811$	1442.35	1.77848	0.889240	+	0.457440i	$0.151234\pi$
$812$	1427.94i	1.75854i
$813$	0	0
$814$	−96.7102	−0.118809
$815$	451.735i	0.554276i
$816$	0	0
$817$	157.837	0.193191
$818$	2547.33i	3.11410i
$819$	0	0
$820$	93.6225	0.114174
$821$	− 1546.21i	− 1.88333i	−0.336552	−	0.941665i	$-0.609261\pi$
$821$	− 1546.21i	− 1.88333i	0.336552	−	0.941665i	$-0.390739\pi$
$822$	0	0
$823$	1582.02	1.92226	0.961129	−	0.276100i	$-0.0890421\pi$
$823$	1582.02	1.92226	0.961129	+	0.276100i	$0.0890421\pi$
$824$	2461.25i	2.98695i
$825$	0	0
$826$	−1562.78	−1.89199
$827$	− 58.0629i	− 0.0702091i	−0.999384	−	0.0351045i	$-0.988824\pi$
$827$	− 58.0629i	− 0.0702091i	0.999384	−	0.0351045i	$-0.0111764\pi$
$828$	0	0
$829$	−279.841	−0.337565	−0.168783	−	0.985653i	$-0.553984\pi$
$829$	−279.841	−0.337565	−0.168783	+	0.985653i	$0.553984\pi$
$830$	− 653.756i	− 0.787658i
$831$	0	0
$832$	6047.05	7.26809
$833$	− 406.298i	− 0.487753i
$834$	0	0
$835$	−351.080	−0.420455
$836$	1411.23i	1.68807i
$837$	0	0
$838$	270.665	0.322989
$839$	611.537i	0.728888i	0.931225	+	0.364444i	$0.118741\pi$
$839$	611.537i	0.728888i	−0.931225	+	0.364444i	$0.881259\pi$
$840$	0	0
$841$	333.487	0.396536
$842$	− 1089.81i	− 1.29431i
$843$	0	0
$844$	4289.26	5.08207
$845$	694.855i	0.822313i
$846$	0	0
$847$	−328.063	−0.387323
$848$	− 6182.67i	− 7.29088i
$849$	0	0
$850$	2013.36	2.36866
$851$	− 42.0310i	− 0.0493901i
$852$	0	0
$853$	−780.661	−0.915195	−0.457597	−	0.889159i	$-0.651290\pi$
$853$	−780.661	−0.915195	−0.457597	+	0.889159i	$0.651290\pi$
$854$	832.401i	0.974708i
$855$	0	0
$856$	65.4558	0.0764670
$857$	− 91.5738i	− 0.106854i	−0.998572	−	0.0534269i	$-0.982986\pi$
$857$	− 91.5738i	− 0.106854i	0.998572	−	0.0534269i	$-0.0170144\pi$
$858$	0	0
$859$	−976.219	−1.13646	−0.568230	−	0.822870i	$-0.692372\pi$
$859$	−976.219	−1.13646	−0.568230	+	0.822870i	$0.692372\pi$
$860$	− 223.068i	− 0.259381i
$861$	0	0
$862$	−1969.83	−2.28518
$863$	− 514.017i	− 0.595616i	−0.954626	−	0.297808i	$-0.903744\pi$
$863$	− 514.017i	− 0.595616i	0.954626	−	0.297808i	$-0.0962555\pi$
$864$	0	0
$865$	512.207	0.592147
$866$	− 862.012i	− 0.995395i
$867$	0	0
$868$	−98.7631	−0.113782
$869$	552.204i	0.635448i
$870$	0	0
$871$	−2305.68	−2.64716
$872$	60.0688i	0.0688862i
$873$	0	0
$874$	−834.446	−0.954743
$875$	− 532.477i	− 0.608545i
$876$	0	0
$877$	609.520	0.695006	0.347503	−	0.937679i	$-0.387030\pi$
$877$	609.520	0.695006	0.347503	+	0.937679i	$0.387030\pi$
$878$	944.858i	1.07615i
$879$	0	0
$880$	1016.21	1.15478
$881$	− 981.132i	− 1.11366i	−0.830628	−	0.556828i	$-0.812018\pi$
$881$	− 981.132i	− 1.11366i	0.830628	−	0.556828i	$-0.187982\pi$
$882$	0	0
$883$	549.991	0.622867	0.311433	−	0.950268i	$-0.399191\pi$
$883$	549.991	0.622867	0.311433	+	0.950268i	$0.399191\pi$
$884$	6227.12i	7.04426i
$885$	0	0
$886$	−1233.59	−1.39231
$887$	− 381.280i	− 0.429853i	−0.976630	−	0.214927i	$-0.931049\pi$
$887$	− 381.280i	− 0.429853i	0.976630	−	0.214927i	$-0.0689512\pi$
$888$	0	0
$889$	64.3799	0.0724183
$890$	1174.26i	1.31939i
$891$	0	0
$892$	−80.3288	−0.0900547
$893$	− 569.114i	− 0.637305i
$894$	0	0
$895$	53.8328	0.0601484
$896$	− 2888.65i	− 3.22394i
$897$	0	0
$898$	−2743.86	−3.05553
$899$	− 35.1021i	− 0.0390457i
$900$	0	0
$901$	2447.38	2.71629
$902$	− 128.644i	− 0.142621i
$903$	0	0
$904$	3211.50	3.55254
$905$	− 128.579i	− 0.142076i
$906$	0	0
$907$	−701.779	−0.773737	−0.386869	−	0.922135i	$-0.626443\pi$
$907$	−701.779	−0.773737	−0.386869	+	0.922135i	$0.626443\pi$
$908$	− 3308.35i	− 3.64356i
$909$	0	0
$910$	1019.47	1.12030
$911$	514.496i	0.564760i	0.959303	+	0.282380i	$0.0911239\pi$
$911$	514.496i	0.564760i	−0.959303	+	0.282380i	$0.908876\pi$
$912$	0	0
$913$	−660.273	−0.723190
$914$	− 2554.07i	− 2.79439i
$915$	0	0
$916$	1733.45	1.89241
$917$	451.273i	0.492119i
$918$	0	0
$919$	753.168	0.819551	0.409776	−	0.912186i	$-0.365607\pi$
$919$	753.168	0.819551	0.409776	+	0.912186i	$0.365607\pi$
$920$	754.149i	0.819727i
$921$	0	0
$922$	−2170.45	−2.35406
$923$	3180.72i	3.44606i
$924$	0	0
$925$	65.1568	0.0704398
$926$	− 965.015i	− 1.04213i
$927$	0	0
$928$	3005.91	3.23913
$929$	646.826i	0.696261i	0.937446	+	0.348130i	$0.113183\pi$
$929$	646.826i	0.696261i	−0.937446	+	0.348130i	$0.886817\pi$
$930$	0	0
$931$	−261.042	−0.280389
$932$	− 1129.17i	− 1.21156i
$933$	0	0
$934$	1746.67	1.87010
$935$	402.260i	0.430225i
$936$	0	0
$937$	1059.11	1.13032	0.565160	−	0.824981i	$-0.308814\pi$
$937$	1059.11	1.13032	0.565160	+	0.824981i	$0.308814\pi$
$938$	2263.97i	2.41362i
$939$	0	0
$940$	−804.317	−0.855656
$941$	66.0163i	0.0701555i	0.999385	+	0.0350777i	$0.0111679\pi$
$941$	66.0163i	0.0701555i	−0.999385	+	0.0350777i	$0.988832\pi$
$942$	0	0
$943$	55.9098	0.0592893
$944$	4416.32i	4.67830i
$945$	0	0
$946$	−306.512	−0.324009
$947$	− 1136.85i	− 1.20048i	−0.799821	−	0.600238i	$-0.795073\pi$
$947$	− 1136.85i	− 1.20048i	0.799821	−	0.600238i	$-0.204927\pi$
$948$	0	0
$949$	838.778	0.883855
$950$	− 1293.57i	− 1.36165i
$951$	0	0
$952$	3910.13	4.10728
$953$	35.9360i	0.0377083i	0.999822	+	0.0188542i	$0.00600182\pi$
$953$	35.9360i	0.0377083i	−0.999822	+	0.0188542i	$0.993998\pi$
$954$	0	0
$955$	291.223	0.304946
$956$	− 1357.37i	− 1.41984i
$957$	0	0
$958$	2259.26	2.35831
$959$	− 497.847i	− 0.519131i
$960$	0	0
$961$	−958.572	−0.997474
$962$	274.175i	0.285005i
$963$	0	0
$964$	−7.06977	−0.00733378
$965$	739.843i	0.766676i
$966$	0	0
$967$	−773.061	−0.799443	−0.399721	−	0.916637i	$-0.630893\pi$
$967$	−773.061	−0.799443	−0.399721	+	0.916637i	$0.630893\pi$
$968$	1583.05i	1.63539i
$969$	0	0
$970$	203.129	0.209412
$971$	745.333i	0.767593i	0.923418	+	0.383796i	$0.125384\pi$
$971$	745.333i	0.767593i	−0.923418	+	0.383796i	$0.874616\pi$
$972$	0	0
$973$	−1330.11	−1.36702
$974$	− 458.415i	− 0.470652i
$975$	0	0
$976$	2352.31	2.41015
$977$	187.891i	0.192314i	0.995366	+	0.0961572i	$0.0306551\pi$
$977$	187.891i	0.192314i	−0.995366	+	0.0961572i	$0.969345\pi$
$978$	0	0
$979$	1185.97	1.21141
$980$	368.926i	0.376455i
$981$	0	0
$982$	2083.05	2.12123
$983$	970.266i	0.987046i	0.869733	+	0.493523i	$0.164291\pi$
$983$	970.266i	0.987046i	−0.869733	+	0.493523i	$0.835709\pi$
$984$	0	0
$985$	−479.673	−0.486978
$986$	2173.19i	2.20405i
$987$	0	0
$988$	4000.86	4.04945
$989$	− 133.213i	− 0.134694i
$990$	0	0
$991$	654.491	0.660435	0.330218	−	0.943905i	$-0.392878\pi$
$991$	654.491	0.660435	0.330218	+	0.943905i	$0.392878\pi$
$992$	207.904i	0.209580i
$993$	0	0
$994$	3123.19	3.14204
$995$	566.291i	0.569137i
$996$	0	0
$997$	1089.23	1.09251	0.546253	−	0.837620i	$-0.316054\pi$
$997$	1089.23	1.09251	0.546253	+	0.837620i	$0.316054\pi$
$998$	− 630.003i	− 0.631266i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.1	✓	84
3.2	odd	2	inner	1143.3.b.a.890.84	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.1	✓	84	1.1	even	1	trivial
1143.3.b.a.890.84	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)

\(f(q)\)	\(=\)	\(q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +O(q^{10})\)	\(q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +7.89464 q^{10} -7.97333i q^{11} -22.6045 q^{13} +22.1957i q^{14} +62.7235 q^{16} +24.8287i q^{17} +15.9522 q^{19} -22.5449i q^{20} -30.9785 q^{22} -13.4635i q^{23} +20.8712 q^{25} +87.8245i q^{26} +63.3848 q^{28} +22.5281i q^{29} -1.55815 q^{31} -133.430i q^{32} +96.4661 q^{34} -11.6081i q^{35} +3.12185 q^{37} -61.9785i q^{38} -56.0144 q^{40} +4.15270i q^{41} +9.89437 q^{43} +88.4661i q^{44} -52.3091 q^{46} -35.6762i q^{47} -16.3640 q^{49} -81.0900i q^{50} +250.803 q^{52} -98.5702i q^{53} +16.2014 q^{55} -157.484i q^{56} +87.5273 q^{58} +70.4093i q^{59} +37.5028 q^{61} +6.05382i q^{62} -267.515 q^{64} -45.9312i q^{65} +102.001 q^{67} -275.481i q^{68} -45.1004 q^{70} -140.711i q^{71} -37.1066 q^{73} -12.1292i q^{74} -176.994 q^{76} +45.5500i q^{77} -69.2564 q^{79} +127.451i q^{80} +16.1343 q^{82} -82.8101i q^{83} -50.4507 q^{85} -38.4422i q^{86} +219.800 q^{88} +148.742i q^{89} +129.135 q^{91} +149.381i q^{92} -138.611 q^{94} +32.4140i q^{95} +25.7301 q^{97} +63.5784i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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