LMFDB - Embedded newform 1143.3.b.a.890.68

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			890.68
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56806i q^{2} -2.59492 q^{4} -0.662170i q^{5} -12.8177 q^{7} +3.60832i q^{8} +O(q^{10})$	\(q+2.56806i q^{2} -2.59492 q^{4} -0.662170i q^{5} -12.8177 q^{7} +3.60832i q^{8} +1.70049 q^{10} +20.5391i q^{11} +4.92172 q^{13} -32.9165i q^{14} -19.6461 q^{16} +3.69398i q^{17} -1.02559 q^{19} +1.71828i q^{20} -52.7457 q^{22} -26.1310i q^{23} +24.5615 q^{25} +12.6393i q^{26} +33.2609 q^{28} +27.6705i q^{29} -29.6107 q^{31} -36.0190i q^{32} -9.48635 q^{34} +8.48748i q^{35} -2.50356 q^{37} -2.63378i q^{38} +2.38932 q^{40} -69.8927i q^{41} -66.7086 q^{43} -53.2974i q^{44} +67.1060 q^{46} +81.6550i q^{47} +115.293 q^{49} +63.0754i q^{50} -12.7715 q^{52} -81.3765i q^{53} +13.6004 q^{55} -46.2502i q^{56} -71.0595 q^{58} -66.1371i q^{59} +5.87636 q^{61} -76.0419i q^{62} +13.9145 q^{64} -3.25902i q^{65} +60.9345 q^{67} -9.58559i q^{68} -21.7963 q^{70} -115.840i q^{71} -104.353 q^{73} -6.42930i q^{74} +2.66133 q^{76} -263.264i q^{77} +15.1002 q^{79} +13.0090i q^{80} +179.489 q^{82} -0.533557i q^{83} +2.44604 q^{85} -171.312i q^{86} -74.1117 q^{88} -76.1149i q^{89} -63.0850 q^{91} +67.8080i q^{92} -209.695 q^{94} +0.679115i q^{95} +89.0106 q^{97} +296.078i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.56806i	1.28403i	0.766692	+	0.642015i	$0.221901\pi$
$2$	2.56806i	1.28403i	−0.766692	+	0.642015i	$0.778099\pi$
$3$	0	0
$3$	0	0
$4$	−2.59492	−0.648731
$5$	− 0.662170i	− 0.132434i	−0.997805	−	0.0662170i	$-0.978907\pi$
$5$	− 0.662170i	− 0.132434i	0.997805	−	0.0662170i	$-0.0210930\pi$
$6$	0	0
$7$	−12.8177	−1.83110	−0.915548	−	0.402210i	$-0.868242\pi$
$7$	−12.8177	−1.83110	−0.915548	+	0.402210i	$0.868242\pi$
$8$	3.60832i	0.451040i
$9$	0	0
$10$	1.70049	0.170049
$11$	20.5391i	1.86719i	0.358326	+	0.933597i	$0.383348\pi$
$11$	20.5391i	1.86719i	−0.358326	+	0.933597i	$0.616652\pi$
$12$	0	0
$13$	4.92172	0.378594	0.189297	−	0.981920i	$-0.439379\pi$
$13$	4.92172	0.378594	0.189297	+	0.981920i	$0.439379\pi$
$14$	− 32.9165i	− 2.35118i
$15$	0	0
$16$	−19.6461	−1.22788
$17$	3.69398i	0.217293i	0.994080	+	0.108646i	$0.0346516\pi$
$17$	3.69398i	0.217293i	−0.994080	+	0.108646i	$0.965348\pi$
$18$	0	0
$19$	−1.02559	−0.0539784	−0.0269892	−	0.999636i	$-0.508592\pi$
$19$	−1.02559	−0.0539784	−0.0269892	+	0.999636i	$0.508592\pi$
$20$	1.71828i	0.0859140i
$21$	0	0
$22$	−52.7457	−2.39753
$23$	− 26.1310i	− 1.13613i	−0.822983	−	0.568066i	$-0.807692\pi$
$23$	− 26.1310i	− 1.13613i	0.822983	−	0.568066i	$-0.192308\pi$
$24$	0	0
$25$	24.5615	0.982461
$26$	12.6393i	0.486126i
$27$	0	0
$28$	33.2609	1.18789
$29$	27.6705i	0.954156i	0.878861	+	0.477078i	$0.158304\pi$
$29$	27.6705i	0.954156i	−0.878861	+	0.477078i	$0.841696\pi$
$30$	0	0
$31$	−29.6107	−0.955183	−0.477591	−	0.878582i	$-0.658490\pi$
$31$	−29.6107	−0.955183	−0.477591	+	0.878582i	$0.658490\pi$
$32$	− 36.0190i	− 1.12559i
$33$	0	0
$34$	−9.48635	−0.279010
$35$	8.48748i	0.242499i
$36$	0	0
$37$	−2.50356	−0.0676639	−0.0338319	−	0.999428i	$-0.510771\pi$
$37$	−2.50356	−0.0676639	−0.0338319	+	0.999428i	$0.510771\pi$
$38$	− 2.63378i	− 0.0693099i
$39$	0	0
$40$	2.38932	0.0597331
$41$	− 69.8927i	− 1.70470i	−0.522972	−	0.852350i	$-0.675177\pi$
$41$	− 69.8927i	− 1.70470i	0.522972	−	0.852350i	$-0.324823\pi$
$42$	0	0
$43$	−66.7086	−1.55136	−0.775682	−	0.631124i	$-0.782594\pi$
$43$	−66.7086	−1.55136	−0.775682	+	0.631124i	$0.782594\pi$
$44$	− 53.2974i	− 1.21131i
$45$	0	0
$46$	67.1060	1.45883
$47$	81.6550i	1.73734i	0.495390	+	0.868671i	$0.335025\pi$
$47$	81.6550i	1.73734i	−0.495390	+	0.868671i	$0.664975\pi$
$48$	0	0
$49$	115.293	2.35291
$50$	63.0754i	1.26151i
$51$	0	0
$52$	−12.7715	−0.245606
$53$	− 81.3765i	− 1.53541i	−0.640806	−	0.767703i	$-0.721400\pi$
$53$	− 81.3765i	− 1.53541i	0.640806	−	0.767703i	$-0.278600\pi$
$54$	0	0
$55$	13.6004	0.247280
$56$	− 46.2502i	− 0.825897i
$57$	0	0
$58$	−71.0595	−1.22516
$59$	− 66.1371i	− 1.12097i	−0.828165	−	0.560484i	$-0.810615\pi$
$59$	− 66.1371i	− 1.12097i	0.828165	−	0.560484i	$-0.189385\pi$
$60$	0	0
$61$	5.87636	0.0963337	0.0481669	−	0.998839i	$-0.484662\pi$
$61$	5.87636	0.0963337	0.0481669	+	0.998839i	$0.484662\pi$
$62$	− 76.0419i	− 1.22648i
$63$	0	0
$64$	13.9145	0.217414
$65$	− 3.25902i	− 0.0501387i
$66$	0	0
$67$	60.9345	0.909470	0.454735	−	0.890627i	$-0.349734\pi$
$67$	60.9345	0.909470	0.454735	+	0.890627i	$0.349734\pi$
$68$	− 9.58559i	− 0.140965i
$69$	0	0
$70$	−21.7963	−0.311376
$71$	− 115.840i	− 1.63155i	−0.578372	−	0.815773i	$-0.696312\pi$
$71$	− 115.840i	− 1.63155i	0.578372	−	0.815773i	$-0.303688\pi$
$72$	0	0
$73$	−104.353	−1.42950	−0.714750	−	0.699380i	$-0.753459\pi$
$73$	−104.353	−1.42950	−0.714750	+	0.699380i	$0.753459\pi$
$74$	− 6.42930i	− 0.0868824i
$75$	0	0
$76$	2.66133	0.0350175
$77$	− 263.264i	− 3.41901i
$78$	0	0
$79$	15.1002	0.191142	0.0955710	−	0.995423i	$-0.469532\pi$
$79$	15.1002	0.191142	0.0955710	+	0.995423i	$0.469532\pi$
$80$	13.0090i	0.162613i
$81$	0	0
$82$	179.489	2.18888
$83$	− 0.533557i	− 0.00642839i	−0.999995	−	0.00321420i	$-0.998977\pi$
$83$	− 0.533557i	− 0.00642839i	0.999995	−	0.00321420i	$-0.00102311\pi$
$84$	0	0
$85$	2.44604	0.0287770
$86$	− 171.312i	− 1.99200i
$87$	0	0
$88$	−74.1117	−0.842179
$89$	− 76.1149i	− 0.855223i	−0.903963	−	0.427612i	$-0.859355\pi$
$89$	− 76.1149i	− 0.855223i	0.903963	−	0.427612i	$-0.140645\pi$
$90$	0	0
$91$	−63.0850	−0.693242
$92$	67.8080i	0.737043i
$93$	0	0
$94$	−209.695	−2.23080
$95$	0.679115i	0.00714858i
$96$	0	0
$97$	89.0106	0.917635	0.458817	−	0.888531i	$-0.348273\pi$
$97$	89.0106	0.917635	0.458817	+	0.888531i	$0.348273\pi$
$98$	296.078i	3.02120i
$99$	0	0
$100$	−63.7353	−0.637353
$101$	50.7640i	0.502614i	0.967907	+	0.251307i	$0.0808603\pi$
$101$	50.7640i	0.502614i	−0.967907	+	0.251307i	$0.919140\pi$
$102$	0	0
$103$	−202.537	−1.96638	−0.983191	−	0.182581i	$-0.941555\pi$
$103$	−202.537	−1.96638	−0.983191	+	0.182581i	$0.941555\pi$
$104$	17.7591i	0.170761i
$105$	0	0
$106$	208.980	1.97151
$107$	6.15381i	0.0575122i	0.999586	+	0.0287561i	$0.00915461\pi$
$107$	6.15381i	0.0575122i	−0.999586	+	0.0287561i	$0.990845\pi$
$108$	0	0
$109$	42.1393	0.386599	0.193300	−	0.981140i	$-0.438081\pi$
$109$	42.1393	0.386599	0.193300	+	0.981140i	$0.438081\pi$
$110$	34.9266i	0.317515i
$111$	0	0
$112$	251.817	2.24836
$113$	48.6558i	0.430583i	0.976550	+	0.215291i	$0.0690701\pi$
$113$	48.6558i	0.430583i	−0.976550	+	0.215291i	$0.930930\pi$
$114$	0	0
$115$	−17.3032	−0.150462
$116$	− 71.8028i	− 0.618990i
$117$	0	0
$118$	169.844	1.43936
$119$	− 47.3482i	− 0.397884i
$120$	0	0
$121$	−300.856	−2.48641
$122$	15.0908i	0.123695i
$123$	0	0
$124$	76.8374	0.619656
$125$	− 32.8182i	− 0.262545i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	− 108.343i	− 0.846426i
$129$	0	0
$130$	8.36935	0.0643796
$131$	98.0583i	0.748536i	0.927321	+	0.374268i	$0.122106\pi$
$131$	98.0583i	0.748536i	−0.927321	+	0.374268i	$0.877894\pi$
$132$	0	0
$133$	13.1457	0.0988397
$134$	156.483i	1.16779i
$135$	0	0
$136$	−13.3291	−0.0980078
$137$	− 135.480i	− 0.988904i	−0.869205	−	0.494452i	$-0.835369\pi$
$137$	− 135.480i	− 0.988904i	0.869205	−	0.494452i	$-0.164631\pi$
$138$	0	0
$139$	18.1978	0.130920	0.0654599	−	0.997855i	$-0.479149\pi$
$139$	18.1978	0.130920	0.0654599	+	0.997855i	$0.479149\pi$
$140$	− 22.0244i	− 0.157317i
$141$	0	0
$142$	297.483	2.09495
$143$	101.088i	0.706908i
$144$	0	0
$145$	18.3226	0.126363
$146$	− 267.986i	− 1.83552i
$147$	0	0
$148$	6.49655	0.0438956
$149$	133.336i	0.894869i	0.894317	+	0.447435i	$0.147662\pi$
$149$	133.336i	0.894869i	−0.894317	+	0.447435i	$0.852338\pi$
$150$	0	0
$151$	184.374	1.22102	0.610511	−	0.792008i	$-0.290964\pi$
$151$	184.374	1.22102	0.610511	+	0.792008i	$0.290964\pi$
$152$	− 3.70066i	− 0.0243464i
$153$	0	0
$154$	676.076	4.39011
$155$	19.6073i	0.126499i
$156$	0	0
$157$	138.955	0.885065	0.442532	−	0.896752i	$-0.354080\pi$
$157$	138.955	0.885065	0.442532	+	0.896752i	$0.354080\pi$
$158$	38.7783i	0.245432i
$159$	0	0
$160$	−23.8507	−0.149067
$161$	334.939i	2.08036i
$162$	0	0
$163$	−299.670	−1.83847	−0.919233	−	0.393715i	$-0.871190\pi$
$163$	−299.670	−1.83847	−0.919233	+	0.393715i	$0.871190\pi$
$164$	181.366i	1.10589i
$165$	0	0
$166$	1.37020	0.00825424
$167$	− 105.420i	− 0.631257i	−0.948883	−	0.315628i	$-0.897785\pi$
$167$	− 105.420i	− 0.631257i	0.948883	−	0.315628i	$-0.102215\pi$
$168$	0	0
$169$	−144.777	−0.856667
$170$	6.28158i	0.0369505i
$171$	0	0
$172$	173.104	1.00642
$173$	110.157i	0.636744i	0.947966	+	0.318372i	$0.103136\pi$
$173$	110.157i	0.636744i	−0.947966	+	0.318372i	$0.896864\pi$
$174$	0	0
$175$	−314.822	−1.79898
$176$	− 403.513i	− 2.29269i
$177$	0	0
$178$	195.467	1.09813
$179$	205.760i	1.14950i	0.818329	+	0.574750i	$0.194901\pi$
$179$	205.760i	1.14950i	−0.818329	+	0.574750i	$0.805099\pi$
$180$	0	0
$181$	43.1373	0.238328	0.119164	−	0.992875i	$-0.461979\pi$
$181$	43.1373	0.238328	0.119164	+	0.992875i	$0.461979\pi$
$182$	− 162.006i	− 0.890142i
$183$	0	0
$184$	94.2891	0.512441
$185$	1.65779i	0.00896100i
$186$	0	0
$187$	−75.8711	−0.405728
$188$	− 211.889i	− 1.12707i
$189$	0	0
$190$	−1.74401	−0.00917899
$191$	64.0595i	0.335390i	0.985839	+	0.167695i	$0.0536324\pi$
$191$	64.0595i	0.335390i	−0.985839	+	0.167695i	$0.946368\pi$
$192$	0	0
$193$	191.411	0.991766	0.495883	−	0.868389i	$-0.334845\pi$
$193$	191.411	0.991766	0.495883	+	0.868389i	$0.334845\pi$
$194$	228.584i	1.17827i
$195$	0	0
$196$	−299.175	−1.52640
$197$	− 254.592i	− 1.29234i	−0.763192	−	0.646172i	$-0.776369\pi$
$197$	− 254.592i	− 1.29234i	0.763192	−	0.646172i	$-0.223631\pi$
$198$	0	0
$199$	−49.1976	−0.247224	−0.123612	−	0.992331i	$-0.539448\pi$
$199$	−49.1976	−0.247224	−0.123612	+	0.992331i	$0.539448\pi$
$200$	88.6259i	0.443129i
$201$	0	0
$202$	−130.365	−0.645371
$203$	− 354.671i	− 1.74715i
$204$	0	0
$205$	−46.2809	−0.225760
$206$	− 520.128i	− 2.52489i
$207$	0	0
$208$	−96.6925	−0.464868
$209$	− 21.0647i	− 0.100788i
$210$	0	0
$211$	164.180	0.778105	0.389053	−	0.921216i	$-0.372802\pi$
$211$	164.180	0.778105	0.389053	+	0.921216i	$0.372802\pi$
$212$	211.166i	0.996065i
$213$	0	0
$214$	−15.8033	−0.0738473
$215$	44.1725i	0.205453i
$216$	0	0
$217$	379.540	1.74903
$218$	108.216i	0.496405i
$219$	0	0
$220$	−35.2920	−0.160418
$221$	18.1807i	0.0822658i
$222$	0	0
$223$	−173.368	−0.777434	−0.388717	−	0.921357i	$-0.627082\pi$
$223$	−173.368	−0.777434	−0.388717	+	0.921357i	$0.627082\pi$
$224$	461.679i	2.06107i
$225$	0	0
$226$	−124.951	−0.552881
$227$	394.569i	1.73819i	0.494645	+	0.869095i	$0.335298\pi$
$227$	394.569i	1.73819i	−0.494645	+	0.869095i	$0.664702\pi$
$228$	0	0
$229$	86.8349	0.379191	0.189596	−	0.981862i	$-0.439282\pi$
$229$	86.8349	0.379191	0.189596	+	0.981862i	$0.439282\pi$
$230$	− 44.4356i	− 0.193198i
$231$	0	0
$232$	−99.8441	−0.430362
$233$	344.348i	1.47789i	0.673767	+	0.738944i	$0.264675\pi$
$233$	344.348i	1.47789i	−0.673767	+	0.738944i	$0.735325\pi$
$234$	0	0
$235$	54.0695	0.230083
$236$	171.621i	0.727206i
$237$	0	0
$238$	121.593	0.510895
$239$	− 98.9845i	− 0.414161i	−0.978324	−	0.207081i	$-0.933604\pi$
$239$	− 98.9845i	− 0.414161i	0.978324	−	0.207081i	$-0.0663962\pi$
$240$	0	0
$241$	−343.314	−1.42454	−0.712269	−	0.701906i	$-0.752333\pi$
$241$	−343.314	−1.42454	−0.712269	+	0.701906i	$0.752333\pi$
$242$	− 772.615i	− 3.19262i
$243$	0	0
$244$	−15.2487	−0.0624946
$245$	− 76.3433i	− 0.311605i
$246$	0	0
$247$	−5.04767	−0.0204359
$248$	− 106.845i	− 0.430826i
$249$	0	0
$250$	84.2790	0.337116
$251$	211.652i	0.843236i	0.906774	+	0.421618i	$0.138538\pi$
$251$	211.652i	0.843236i	−0.906774	+	0.421618i	$0.861462\pi$
$252$	0	0
$253$	536.708	2.12138
$254$	28.9405i	0.113939i
$255$	0	0
$256$	333.888	1.30425
$257$	348.091i	1.35444i	0.735781	+	0.677219i	$0.236815\pi$
$257$	348.091i	1.35444i	−0.735781	+	0.677219i	$0.763185\pi$
$258$	0	0
$259$	32.0898	0.123899
$260$	8.45690i	0.0325265i
$261$	0	0
$262$	−251.819	−0.961142
$263$	− 290.421i	− 1.10426i	−0.833757	−	0.552131i	$-0.813815\pi$
$263$	− 290.421i	− 1.10426i	0.833757	−	0.552131i	$-0.186185\pi$
$264$	0	0
$265$	−53.8851	−0.203340
$266$	33.7589i	0.126913i
$267$	0	0
$268$	−158.120	−0.590001
$269$	351.648i	1.30724i	0.756822	+	0.653621i	$0.226751\pi$
$269$	351.648i	1.30724i	−0.756822	+	0.653621i	$0.773249\pi$
$270$	0	0
$271$	−256.884	−0.947912	−0.473956	−	0.880548i	$-0.657174\pi$
$271$	−256.884	−0.947912	−0.473956	+	0.880548i	$0.657174\pi$
$272$	− 72.5722i	− 0.266809i
$273$	0	0
$274$	347.920	1.26978
$275$	504.472i	1.83445i
$276$	0	0
$277$	−162.433	−0.586399	−0.293200	−	0.956051i	$-0.594720\pi$
$277$	−162.433	−0.586399	−0.293200	+	0.956051i	$0.594720\pi$
$278$	46.7331i	0.168105i
$279$	0	0
$280$	−30.6255	−0.109377
$281$	102.926i	0.366284i	0.983086	+	0.183142i	$0.0586268\pi$
$281$	102.926i	0.366284i	−0.983086	+	0.183142i	$0.941373\pi$
$282$	0	0
$283$	−463.379	−1.63738	−0.818691	−	0.574235i	$-0.805300\pi$
$283$	−463.379	−1.63738	−0.818691	+	0.574235i	$0.805300\pi$
$284$	300.595i	1.05843i
$285$	0	0
$286$	−259.599	−0.907691
$287$	895.861i	3.12147i
$288$	0	0
$289$	275.355	0.952784
$290$	47.0535i	0.162253i
$291$	0	0
$292$	270.789	0.927360
$293$	303.314i	1.03520i	0.855623	+	0.517600i	$0.173175\pi$
$293$	303.314i	1.03520i	−0.855623	+	0.517600i	$0.826825\pi$
$294$	0	0
$295$	−43.7940	−0.148454
$296$	− 9.03366i	− 0.0305191i
$297$	0	0
$298$	−342.413	−1.14904
$299$	− 128.610i	− 0.430132i
$300$	0	0
$301$	855.049	2.84069
$302$	473.484i	1.56783i
$303$	0	0
$304$	20.1488	0.0662790
$305$	− 3.89115i	− 0.0127579i
$306$	0	0
$307$	−282.642	−0.920659	−0.460330	−	0.887748i	$-0.652269\pi$
$307$	−282.642	−0.920659	−0.460330	+	0.887748i	$0.652269\pi$
$308$	683.149i	2.21802i
$309$	0	0
$310$	−50.3527	−0.162428
$311$	− 292.688i	− 0.941119i	−0.882368	−	0.470559i	$-0.844052\pi$
$311$	− 292.688i	− 0.941119i	0.882368	−	0.470559i	$-0.155948\pi$
$312$	0	0
$313$	−189.990	−0.606998	−0.303499	−	0.952832i	$-0.598155\pi$
$313$	−189.990	−0.606998	−0.303499	+	0.952832i	$0.598155\pi$
$314$	356.845i	1.13645i
$315$	0	0
$316$	−39.1839	−0.124000
$317$	− 427.306i	− 1.34797i	−0.738745	−	0.673985i	$-0.764581\pi$
$317$	− 427.306i	− 1.34797i	0.738745	−	0.673985i	$-0.235419\pi$
$318$	0	0
$319$	−568.328	−1.78159
$320$	− 9.21379i	− 0.0287931i
$321$	0	0
$322$	−860.142	−2.67125
$323$	− 3.78851i	− 0.0117291i
$324$	0	0
$325$	120.885	0.371954
$326$	− 769.570i	− 2.36064i
$327$	0	0
$328$	252.195	0.768888
$329$	− 1046.63i	− 3.18124i
$330$	0	0
$331$	−92.8700	−0.280574	−0.140287	−	0.990111i	$-0.544803\pi$
$331$	−92.8700	−0.280574	−0.140287	+	0.990111i	$0.544803\pi$
$332$	1.38454i	0.00417030i
$333$	0	0
$334$	270.724	0.810552
$335$	− 40.3490i	− 0.120445i
$336$	0	0
$337$	−199.485	−0.591944	−0.295972	−	0.955197i	$-0.595644\pi$
$337$	−199.485	−0.591944	−0.295972	+	0.955197i	$0.595644\pi$
$338$	− 371.795i	− 1.09998i
$339$	0	0
$340$	−6.34729	−0.0186685
$341$	− 608.177i	− 1.78351i
$342$	0	0
$343$	−849.716	−2.47731
$344$	− 240.706i	− 0.699727i
$345$	0	0
$346$	−282.889	−0.817597
$347$	12.9677i	0.0373709i	0.999825	+	0.0186855i	$0.00594812\pi$
$347$	12.9677i	0.0373709i	−0.999825	+	0.0186855i	$0.994052\pi$
$348$	0	0
$349$	−421.158	−1.20676	−0.603378	−	0.797455i	$-0.706179\pi$
$349$	−421.158	−1.20676	−0.603378	+	0.797455i	$0.706179\pi$
$350$	− 808.480i	− 2.30994i
$351$	0	0
$352$	739.798	2.10170
$353$	− 668.166i	− 1.89282i	−0.322965	−	0.946411i	$-0.604680\pi$
$353$	− 668.166i	− 1.89282i	0.322965	−	0.946411i	$-0.395320\pi$
$354$	0	0
$355$	−76.7057	−0.216072
$356$	197.512i	0.554810i
$357$	0	0
$358$	−528.405	−1.47599
$359$	− 368.277i	− 1.02584i	−0.858436	−	0.512920i	$-0.828564\pi$
$359$	− 368.277i	− 1.02584i	0.858436	−	0.512920i	$-0.171436\pi$
$360$	0	0
$361$	−359.948	−0.997086
$362$	110.779i	0.306020i
$363$	0	0
$364$	163.701	0.449727
$365$	69.0997i	0.189314i
$366$	0	0
$367$	−19.2091	−0.0523409	−0.0261705	−	0.999657i	$-0.508331\pi$
$367$	−19.2091	−0.0523409	−0.0261705	+	0.999657i	$0.508331\pi$
$368$	513.372i	1.39503i
$369$	0	0
$370$	−4.25729	−0.0115062
$371$	1043.06i	2.81147i
$372$	0	0
$373$	−79.8145	−0.213980	−0.106990	−	0.994260i	$-0.534121\pi$
$373$	−79.8145	−0.213980	−0.106990	+	0.994260i	$0.534121\pi$
$374$	− 194.841i	− 0.520966i
$375$	0	0
$376$	−294.638	−0.783610
$377$	136.187i	0.361238i
$378$	0	0
$379$	116.514	0.307424	0.153712	−	0.988116i	$-0.450877\pi$
$379$	116.514	0.307424	0.153712	+	0.988116i	$0.450877\pi$
$380$	− 1.76225i	− 0.00463751i
$381$	0	0
$382$	−164.508	−0.430650
$383$	− 409.174i	− 1.06834i	−0.845377	−	0.534170i	$-0.820624\pi$
$383$	− 409.174i	− 1.06834i	0.845377	−	0.534170i	$-0.179376\pi$
$384$	0	0
$385$	−174.325	−0.452793
$386$	491.554i	1.27346i
$387$	0	0
$388$	−230.976	−0.595298
$389$	390.255i	1.00323i	0.865092	+	0.501613i	$0.167260\pi$
$389$	390.255i	1.00323i	−0.865092	+	0.501613i	$0.832740\pi$
$390$	0	0
$391$	96.5274	0.246873
$392$	416.013i	1.06126i
$393$	0	0
$394$	653.806	1.65941
$395$	− 9.99892i	− 0.0253137i
$396$	0	0
$397$	−564.041	−1.42076	−0.710379	−	0.703819i	$-0.751476\pi$
$397$	−564.041	−1.42076	−0.710379	+	0.703819i	$0.751476\pi$
$398$	− 126.342i	− 0.317443i
$399$	0	0
$400$	−482.537	−1.20634
$401$	− 383.854i	− 0.957242i	−0.878022	−	0.478621i	$-0.841137\pi$
$401$	− 383.854i	− 0.957242i	0.878022	−	0.478621i	$-0.158863\pi$
$402$	0	0
$403$	−145.735	−0.361626
$404$	− 131.729i	− 0.326061i
$405$	0	0
$406$	910.817	2.24339
$407$	− 51.4210i	− 0.126342i
$408$	0	0
$409$	−134.307	−0.328378	−0.164189	−	0.986429i	$-0.552501\pi$
$409$	−134.307	−0.328378	−0.164189	+	0.986429i	$0.552501\pi$
$410$	− 118.852i	− 0.289883i
$411$	0	0
$412$	525.569	1.27565
$413$	847.724i	2.05260i
$414$	0	0
$415$	−0.353305	−0.000851338 0
$416$	− 177.275i	− 0.426143i
$417$	0	0
$418$	54.0955	0.129415
$419$	568.735i	1.35736i	0.734433	+	0.678682i	$0.237448\pi$
$419$	568.735i	1.35736i	−0.734433	+	0.678682i	$0.762552\pi$
$420$	0	0
$421$	14.6978	0.0349117	0.0174559	−	0.999848i	$-0.494443\pi$
$421$	14.6978	0.0349117	0.0174559	+	0.999848i	$0.494443\pi$
$422$	421.624i	0.999110i
$423$	0	0
$424$	293.632	0.692529
$425$	90.7298i	0.213482i
$426$	0	0
$427$	−75.3212	−0.176396
$428$	− 15.9686i	− 0.0373099i
$429$	0	0
$430$	−113.437	−0.263808
$431$	− 416.758i	− 0.966955i	−0.875357	−	0.483478i	$-0.839373\pi$
$431$	− 416.758i	− 0.966955i	0.875357	−	0.483478i	$-0.160627\pi$
$432$	0	0
$433$	−714.420	−1.64993	−0.824966	−	0.565183i	$-0.808806\pi$
$433$	−714.420	−1.64993	−0.824966	+	0.565183i	$0.808806\pi$
$434$	974.680i	2.24581i
$435$	0	0
$436$	−109.348	−0.250799
$437$	26.7997i	0.0613266i
$438$	0	0
$439$	205.791	0.468771	0.234386	−	0.972144i	$-0.424692\pi$
$439$	205.791	0.468771	0.234386	+	0.972144i	$0.424692\pi$
$440$	49.0746i	0.111533i
$441$	0	0
$442$	−46.6892	−0.105632
$443$	− 303.789i	− 0.685755i	−0.939380	−	0.342877i	$-0.888598\pi$
$443$	− 303.789i	− 0.685755i	0.939380	−	0.342877i	$-0.111402\pi$
$444$	0	0
$445$	−50.4010	−0.113261
$446$	− 445.219i	− 0.998248i
$447$	0	0
$448$	−178.352	−0.398107
$449$	81.4465i	0.181395i	0.995878	+	0.0906976i	$0.0289097\pi$
$449$	81.4465i	0.181395i	−0.995878	+	0.0906976i	$0.971090\pi$
$450$	0	0
$451$	1435.53	3.18300
$452$	− 126.258i	− 0.279332i
$453$	0	0
$454$	−1013.28	−2.23189
$455$	41.7730i	0.0918088i
$456$	0	0
$457$	864.984	1.89274	0.946372	−	0.323079i	$-0.104718\pi$
$457$	864.984	1.89274	0.946372	+	0.323079i	$0.104718\pi$
$458$	222.997i	0.486893i
$459$	0	0
$460$	44.9004	0.0976096
$461$	− 126.563i	− 0.274540i	−0.990534	−	0.137270i	$-0.956167\pi$
$461$	− 126.563i	− 0.274540i	0.990534	−	0.137270i	$-0.0438328\pi$
$462$	0	0
$463$	27.5640	0.0595334	0.0297667	−	0.999557i	$-0.490524\pi$
$463$	27.5640	0.0595334	0.0297667	+	0.999557i	$0.490524\pi$
$464$	− 543.617i	− 1.17159i
$465$	0	0
$466$	−884.305	−1.89765
$467$	− 193.863i	− 0.415124i	−0.978222	−	0.207562i	$-0.933447\pi$
$467$	− 193.863i	− 0.415124i	0.978222	−	0.207562i	$-0.0665529\pi$
$468$	0	0
$469$	−781.038	−1.66533
$470$	138.854i	0.295433i
$471$	0	0
$472$	238.644	0.505601
$473$	− 1370.14i	− 2.89670i
$474$	0	0
$475$	−25.1901	−0.0530317
$476$	122.865i	0.258120i
$477$	0	0
$478$	254.198	0.531795
$479$	102.074i	0.213099i	0.994307	+	0.106549i	$0.0339802\pi$
$479$	102.074i	0.213099i	−0.994307	+	0.106549i	$0.966020\pi$
$480$	0	0
$481$	−12.3218	−0.0256171
$482$	− 881.650i	− 1.82915i
$483$	0	0
$484$	780.697	1.61301
$485$	− 58.9402i	− 0.121526i
$486$	0	0
$487$	−658.072	−1.35128	−0.675639	−	0.737233i	$-0.736132\pi$
$487$	−658.072	−1.35128	−0.675639	+	0.737233i	$0.736132\pi$
$488$	21.2038i	0.0434504i
$489$	0	0
$490$	196.054	0.400110
$491$	99.1405i	0.201915i	0.994891	+	0.100958i	$0.0321907\pi$
$491$	99.1405i	0.201915i	−0.994891	+	0.100958i	$0.967809\pi$
$492$	0	0
$493$	−102.214	−0.207331
$494$	− 12.9627i	− 0.0262403i
$495$	0	0
$496$	581.733	1.17285
$497$	1484.80i	2.98752i
$498$	0	0
$499$	378.592	0.758701	0.379351	−	0.925253i	$-0.376147\pi$
$499$	378.592	0.758701	0.379351	+	0.925253i	$0.376147\pi$
$500$	85.1606i	0.170321i
$501$	0	0
$502$	−543.535	−1.08274
$503$	− 509.788i	− 1.01349i	−0.862095	−	0.506747i	$-0.830848\pi$
$503$	− 509.788i	− 1.01349i	0.862095	−	0.506747i	$-0.169152\pi$
$504$	0	0
$505$	33.6144	0.0665632
$506$	1378.30i	2.72391i
$507$	0	0
$508$	−29.2433	−0.0575655
$509$	75.7404i	0.148802i	0.997228	+	0.0744012i	$0.0237045\pi$
$509$	75.7404i	0.148802i	−0.997228	+	0.0744012i	$0.976295\pi$
$510$	0	0
$511$	1337.57	2.61755
$512$	424.074i	0.828269i
$513$	0	0
$514$	−893.917	−1.73914
$515$	134.114i	0.260416i
$516$	0	0
$517$	−1677.12	−3.24395
$518$	82.4086i	0.159090i
$519$	0	0
$520$	11.7596	0.0226146
$521$	− 16.3142i	− 0.0313132i	−0.999877	−	0.0156566i	$-0.995016\pi$
$521$	− 16.3142i	− 0.0313132i	0.999877	−	0.0156566i	$-0.00498386\pi$
$522$	0	0
$523$	430.771	0.823654	0.411827	−	0.911262i	$-0.364891\pi$
$523$	430.771	0.823654	0.411827	+	0.911262i	$0.364891\pi$
$524$	− 254.454i	− 0.485599i
$525$	0	0
$526$	745.818	1.41790
$527$	− 109.381i	− 0.207554i
$528$	0	0
$529$	−153.830	−0.290794
$530$	− 138.380i	− 0.261094i
$531$	0	0
$532$	−34.1120	−0.0641203
$533$	− 343.992i	− 0.645389i
$534$	0	0
$535$	4.07487	0.00761657
$536$	219.871i	0.410207i
$537$	0	0
$538$	−903.052	−1.67854
$539$	2368.01i	4.39334i
$540$	0	0
$541$	200.420	0.370461	0.185231	−	0.982695i	$-0.440697\pi$
$541$	200.420	0.370461	0.185231	+	0.982695i	$0.440697\pi$
$542$	− 659.694i	− 1.21715i
$543$	0	0
$544$	133.053	0.244583
$545$	− 27.9034i	− 0.0511989i
$546$	0	0
$547$	−842.613	−1.54043	−0.770213	−	0.637787i	$-0.779850\pi$
$547$	−842.613	−1.54043	−0.770213	+	0.637787i	$0.779850\pi$
$548$	351.560i	0.641532i
$549$	0	0
$550$	−1295.51	−2.35548
$551$	− 28.3786i	− 0.0515038i
$552$	0	0
$553$	−193.550	−0.349999
$554$	− 417.136i	− 0.752954i
$555$	0	0
$556$	−47.2220	−0.0849317
$557$	− 730.584i	− 1.31164i	−0.754917	−	0.655821i	$-0.772323\pi$
$557$	− 730.584i	− 1.31164i	0.754917	−	0.655821i	$-0.227677\pi$
$558$	0	0
$559$	−328.321	−0.587337
$560$	− 166.746i	− 0.297760i
$561$	0	0
$562$	−264.320	−0.470320
$563$	933.078i	1.65733i	0.559744	+	0.828666i	$0.310900\pi$
$563$	933.078i	1.65733i	−0.559744	+	0.828666i	$0.689100\pi$
$564$	0	0
$565$	32.2184	0.0570238
$566$	− 1189.98i	− 2.10245i
$567$	0	0
$568$	417.987	0.735893
$569$	− 327.272i	− 0.575170i	−0.957755	−	0.287585i	$-0.907148\pi$
$569$	− 327.272i	− 0.575170i	0.957755	−	0.287585i	$-0.0928523\pi$
$570$	0	0
$571$	654.425	1.14610	0.573051	−	0.819519i	$-0.305760\pi$
$571$	654.425	1.14610	0.573051	+	0.819519i	$0.305760\pi$
$572$	− 262.315i	− 0.458593i
$573$	0	0
$574$	−2300.62	−4.00806
$575$	− 641.818i	− 1.11620i
$576$	0	0
$577$	220.648	0.382405	0.191203	−	0.981551i	$-0.438761\pi$
$577$	220.648	0.382405	0.191203	+	0.981551i	$0.438761\pi$
$578$	707.126i	1.22340i
$579$	0	0
$580$	−47.5457	−0.0819754
$581$	6.83895i	0.0117710i
$582$	0	0
$583$	1671.40	2.86690
$584$	− 376.541i	− 0.644761i
$585$	0	0
$586$	−778.927	−1.32923
$587$	641.542i	1.09292i	0.837486	+	0.546458i	$0.184024\pi$
$587$	641.542i	1.09292i	−0.837486	+	0.546458i	$0.815976\pi$
$588$	0	0
$589$	30.3684	0.0515593
$590$	− 112.466i	− 0.190620i
$591$	0	0
$592$	49.1852	0.0830831
$593$	798.467i	1.34649i	0.739421	+	0.673244i	$0.235099\pi$
$593$	798.467i	1.34649i	−0.739421	+	0.673244i	$0.764901\pi$
$594$	0	0
$595$	−31.3526	−0.0526934
$596$	− 345.995i	− 0.580529i
$597$	0	0
$598$	330.277	0.552302
$599$	486.587i	0.812332i	0.913799	+	0.406166i	$0.133135\pi$
$599$	486.587i	0.812332i	−0.913799	+	0.406166i	$0.866865\pi$
$600$	0	0
$601$	−564.769	−0.939715	−0.469858	−	0.882742i	$-0.655695\pi$
$601$	−564.769	−0.939715	−0.469858	+	0.882742i	$0.655695\pi$
$602$	2195.82i	3.64753i
$603$	0	0
$604$	−478.437	−0.792114
$605$	199.218i	0.329286i
$606$	0	0
$607$	−543.288	−0.895038	−0.447519	−	0.894274i	$-0.647692\pi$
$607$	−543.288	−0.895038	−0.447519	+	0.894274i	$0.647692\pi$
$608$	36.9407i	0.0607577i
$609$	0	0
$610$	9.99270	0.0163815
$611$	401.883i	0.657747i
$612$	0	0
$613$	562.465	0.917561	0.458781	−	0.888550i	$-0.348286\pi$
$613$	562.465	0.917561	0.458781	+	0.888550i	$0.348286\pi$
$614$	− 725.842i	− 1.18215i
$615$	0	0
$616$	949.940	1.54211
$617$	− 394.884i	− 0.640006i	−0.947416	−	0.320003i	$-0.896316\pi$
$617$	− 394.884i	− 0.640006i	0.947416	−	0.320003i	$-0.103684\pi$
$618$	0	0
$619$	502.096	0.811141	0.405570	−	0.914064i	$-0.367073\pi$
$619$	502.096	0.811141	0.405570	+	0.914064i	$0.367073\pi$
$620$	− 50.8794i	− 0.0820636i
$621$	0	0
$622$	751.639	1.20842
$623$	975.615i	1.56600i
$624$	0	0
$625$	592.307	0.947691
$626$	− 487.906i	− 0.779403i
$627$	0	0
$628$	−360.578	−0.574169
$629$	− 9.24811i	− 0.0147029i
$630$	0	0
$631$	22.7911	0.0361191	0.0180595	−	0.999837i	$-0.494251\pi$
$631$	22.7911	0.0361191	0.0180595	+	0.999837i	$0.494251\pi$
$632$	54.4864i	0.0862127i
$633$	0	0
$634$	1097.35	1.73083
$635$	− 7.46228i	− 0.0117516i
$636$	0	0
$637$	567.438	0.890797
$638$	− 1459.50i	− 2.28762i
$639$	0	0
$640$	−71.7412	−0.112096
$641$	293.293i	0.457556i	0.973479	+	0.228778i	$0.0734729\pi$
$641$	293.293i	0.457556i	−0.973479	+	0.228778i	$0.926527\pi$
$642$	0	0
$643$	−627.634	−0.976103	−0.488051	−	0.872815i	$-0.662292\pi$
$643$	−627.634	−0.976103	−0.488051	+	0.872815i	$0.662292\pi$
$644$	− 869.140i	− 1.34960i
$645$	0	0
$646$	9.72911	0.0150605
$647$	− 953.959i	− 1.47443i	−0.675656	−	0.737217i	$-0.736140\pi$
$647$	− 953.959i	− 1.47443i	0.675656	−	0.737217i	$-0.263860\pi$
$648$	0	0
$649$	1358.40	2.09306
$650$	310.440i	0.477600i
$651$	0	0
$652$	777.620	1.19267
$653$	− 1017.63i	− 1.55839i	−0.626783	−	0.779194i	$-0.715629\pi$
$653$	− 1017.63i	− 1.55839i	0.626783	−	0.779194i	$-0.284371\pi$
$654$	0	0
$655$	64.9313	0.0991317
$656$	1373.12i	2.09317i
$657$	0	0
$658$	2687.80	4.08480
$659$	961.378i	1.45884i	0.684064	+	0.729422i	$0.260211\pi$
$659$	961.378i	1.45884i	−0.684064	+	0.729422i	$0.739789\pi$
$660$	0	0
$661$	−325.601	−0.492589	−0.246294	−	0.969195i	$-0.579213\pi$
$661$	−325.601	−0.492589	−0.246294	+	0.969195i	$0.579213\pi$
$662$	− 238.496i	− 0.360265i
$663$	0	0
$664$	1.92524	0.00289946
$665$	− 8.70468i	− 0.0130897i
$666$	0	0
$667$	723.059	1.08405
$668$	273.556i	0.409516i
$669$	0	0
$670$	103.619	0.154655
$671$	120.695i	0.179874i
$672$	0	0
$673$	467.782	0.695069	0.347535	−	0.937667i	$-0.387019\pi$
$673$	467.782	0.695069	0.347535	+	0.937667i	$0.387019\pi$
$674$	− 512.290i	− 0.760074i
$675$	0	0
$676$	375.684	0.555746
$677$	− 846.092i	− 1.24977i	−0.780718	−	0.624884i	$-0.785146\pi$
$677$	− 846.092i	− 1.24977i	0.780718	−	0.624884i	$-0.214854\pi$
$678$	0	0
$679$	−1140.91	−1.68028
$680$	8.82611i	0.0129796i
$681$	0	0
$682$	1561.83	2.29008
$683$	758.834i	1.11103i	0.831506	+	0.555515i	$0.187479\pi$
$683$	758.834i	1.11103i	−0.831506	+	0.555515i	$0.812521\pi$
$684$	0	0
$685$	−89.7107	−0.130965
$686$	− 2182.12i	− 3.18093i
$687$	0	0
$688$	1310.56	1.90489
$689$	− 400.512i	− 0.581295i
$690$	0	0
$691$	−27.0426	−0.0391354	−0.0195677	−	0.999809i	$-0.506229\pi$
$691$	−27.0426	−0.0391354	−0.0195677	+	0.999809i	$0.506229\pi$
$692$	− 285.848i	− 0.413075i
$693$	0	0
$694$	−33.3018	−0.0479854
$695$	− 12.0501i	− 0.0173382i
$696$	0	0
$697$	258.182	0.370419
$698$	− 1081.56i	− 1.54951i
$699$	0	0
$700$	816.938	1.16705
$701$	1088.82i	1.55324i	0.629972	+	0.776618i	$0.283066\pi$
$701$	1088.82i	1.55324i	−0.629972	+	0.776618i	$0.716934\pi$
$702$	0	0
$703$	2.56763	0.00365239
$704$	285.792i	0.405955i
$705$	0	0
$706$	1715.89	2.43044
$707$	− 650.676i	− 0.920334i
$708$	0	0
$709$	−1141.14	−1.60950	−0.804750	−	0.593614i	$-0.797701\pi$
$709$	−1141.14	−1.60950	−0.804750	+	0.593614i	$0.797701\pi$
$710$	− 196.985i	− 0.277443i
$711$	0	0
$712$	274.647	0.385740
$713$	773.757i	1.08521i
$714$	0	0
$715$	66.9374	0.0936187
$716$	− 533.932i	− 0.745716i
$717$	0	0
$718$	945.757	1.31721
$719$	− 358.180i	− 0.498164i	−0.968482	−	0.249082i	$-0.919871\pi$
$719$	− 358.180i	− 0.498164i	0.968482	−	0.249082i	$-0.0801288\pi$
$720$	0	0
$721$	2596.06	3.60063
$722$	− 924.368i	− 1.28029i
$723$	0	0
$724$	−111.938	−0.154610
$725$	679.630i	0.937421i
$726$	0	0
$727$	500.703	0.688725	0.344362	−	0.938837i	$-0.388095\pi$
$727$	500.703	0.688725	0.344362	+	0.938837i	$0.388095\pi$
$728$	− 227.631i	− 0.312680i
$729$	0	0
$730$	−177.452	−0.243085
$731$	− 246.420i	− 0.337100i
$732$	0	0
$733$	174.957	0.238686	0.119343	−	0.992853i	$-0.461921\pi$
$733$	174.957	0.238686	0.119343	+	0.992853i	$0.461921\pi$
$734$	− 49.3301i	− 0.0672073i
$735$	0	0
$736$	−941.212	−1.27882
$737$	1251.54i	1.69816i
$738$	0	0
$739$	−147.666	−0.199819	−0.0999095	−	0.994997i	$-0.531855\pi$
$739$	−147.666	−0.199819	−0.0999095	+	0.994997i	$0.531855\pi$
$740$	− 4.30182i	− 0.00581328i
$741$	0	0
$742$	−2678.63	−3.61001
$743$	− 127.109i	− 0.171075i	−0.996335	−	0.0855375i	$-0.972739\pi$
$743$	− 127.109i	− 0.171075i	0.996335	−	0.0855375i	$-0.0272607\pi$
$744$	0	0
$745$	88.2908	0.118511
$746$	− 204.968i	− 0.274756i
$747$	0	0
$748$	196.880	0.263208
$749$	− 78.8774i	− 0.105310i
$750$	0	0
$751$	−42.3505	−0.0563922	−0.0281961	−	0.999602i	$-0.508976\pi$
$751$	−42.3505	−0.0563922	−0.0281961	+	0.999602i	$0.508976\pi$
$752$	− 1604.20i	− 2.13325i
$753$	0	0
$754$	−349.735	−0.463839
$755$	− 122.087i	− 0.161705i
$756$	0	0
$757$	192.852	0.254758	0.127379	−	0.991854i	$-0.459343\pi$
$757$	192.852	0.254758	0.127379	+	0.991854i	$0.459343\pi$
$758$	299.214i	0.394742i
$759$	0	0
$760$	−2.45047	−0.00322430
$761$	1239.77i	1.62914i	0.580068	+	0.814568i	$0.303026\pi$
$761$	1239.77i	1.62914i	−0.580068	+	0.814568i	$0.696974\pi$
$762$	0	0
$763$	−540.128	−0.707900
$764$	− 166.229i	− 0.217578i
$765$	0	0
$766$	1050.78	1.37178
$767$	− 325.508i	− 0.424392i
$768$	0	0
$769$	−286.471	−0.372524	−0.186262	−	0.982500i	$-0.559637\pi$
$769$	−286.471	−0.372524	−0.186262	+	0.982500i	$0.559637\pi$
$770$	− 447.678i	− 0.581400i
$771$	0	0
$772$	−496.696	−0.643389
$773$	1184.33i	1.53212i	0.642767	+	0.766062i	$0.277786\pi$
$773$	1184.33i	1.53212i	−0.642767	+	0.766062i	$0.722214\pi$
$774$	0	0
$775$	−727.283	−0.938430
$776$	321.179i	0.413890i
$777$	0	0
$778$	−1002.20	−1.28817
$779$	71.6813i	0.0920170i
$780$	0	0
$781$	2379.25	3.04641
$782$	247.888i	0.316992i
$783$	0	0
$784$	−2265.05	−2.88909
$785$	− 92.0120i	− 0.117213i
$786$	0	0
$787$	884.950	1.12446	0.562230	−	0.826981i	$-0.309944\pi$
$787$	884.950	1.12446	0.562230	+	0.826981i	$0.309944\pi$
$788$	660.646i	0.838383i
$789$	0	0
$790$	25.6778	0.0325036
$791$	− 623.654i	− 0.788438i
$792$	0	0
$793$	28.9218	0.0364714
$794$	− 1448.49i	− 1.82429i
$795$	0	0
$796$	127.664	0.160382
$797$	− 264.634i	− 0.332037i	−0.986123	−	0.166019i	$-0.946909\pi$
$797$	− 264.634i	− 0.332037i	0.986123	−	0.166019i	$-0.0530912\pi$
$798$	0	0
$799$	−301.632	−0.377512
$800$	− 884.681i	− 1.10585i
$801$	0	0
$802$	985.760	1.22913
$803$	− 2143.33i	− 2.66915i
$804$	0	0
$805$	221.786	0.275511
$806$	− 374.257i	− 0.464339i
$807$	0	0
$808$	−183.173	−0.226699
$809$	57.1698i	0.0706673i	0.999376	+	0.0353336i	$0.0112494\pi$
$809$	57.1698i	0.0706673i	−0.999376	+	0.0353336i	$0.988751\pi$
$810$	0	0
$811$	−1433.72	−1.76784	−0.883918	−	0.467641i	$-0.845104\pi$
$811$	−1433.72	−1.76784	−0.883918	+	0.467641i	$0.845104\pi$
$812$	920.345i	1.13343i
$813$	0	0
$814$	132.052	0.162226
$815$	198.432i	0.243475i
$816$	0	0
$817$	68.4157	0.0837402
$818$	− 344.907i	− 0.421647i
$819$	0	0
$820$	120.095	0.146458
$821$	− 582.135i	− 0.709056i	−0.935045	−	0.354528i	$-0.884642\pi$
$821$	− 582.135i	− 0.709056i	0.935045	−	0.354528i	$-0.115358\pi$
$822$	0	0
$823$	−380.700	−0.462576	−0.231288	−	0.972885i	$-0.574294\pi$
$823$	−380.700	−0.462576	−0.231288	+	0.972885i	$0.574294\pi$
$824$	− 730.819i	− 0.886917i
$825$	0	0
$826$	−2177.00	−2.63560
$827$	66.1038i	0.0799321i	0.999201	+	0.0399660i	$0.0127250\pi$
$827$	66.1038i	0.0799321i	−0.999201	+	0.0399660i	$0.987275\pi$
$828$	0	0
$829$	−1179.40	−1.42268	−0.711339	−	0.702849i	$-0.751911\pi$
$829$	−1179.40	−1.42268	−0.711339	+	0.702849i	$0.751911\pi$
$830$	− 0.907309i	− 0.00109314i
$831$	0	0
$832$	68.4834	0.0823118
$833$	425.888i	0.511271i
$834$	0	0
$835$	−69.8059	−0.0835999
$836$	54.6614i	0.0653844i
$837$	0	0
$838$	−1460.55	−1.74289
$839$	− 189.991i	− 0.226449i	−0.993569	−	0.113225i	$-0.963882\pi$
$839$	− 189.991i	− 0.226449i	0.993569	−	0.113225i	$-0.0361179\pi$
$840$	0	0
$841$	75.3427	0.0895870
$842$	37.7449i	0.0448277i
$843$	0	0
$844$	−426.035	−0.504781
$845$	95.8668i	0.113452i
$846$	0	0
$847$	3856.27	4.55286
$848$	1598.73i	1.88529i
$849$	0	0
$850$	−232.999	−0.274117
$851$	65.4207i	0.0768750i
$852$	0	0
$853$	−830.684	−0.973838	−0.486919	−	0.873447i	$-0.661879\pi$
$853$	−830.684	−0.973838	−0.486919	+	0.873447i	$0.661879\pi$
$854$	− 193.429i	− 0.226498i
$855$	0	0
$856$	−22.2049	−0.0259403
$857$	− 157.459i	− 0.183732i	−0.995771	−	0.0918662i	$-0.970717\pi$
$857$	− 157.459i	− 0.183732i	0.995771	−	0.0918662i	$-0.0292832\pi$
$858$	0	0
$859$	1278.39	1.48823	0.744117	−	0.668049i	$-0.232870\pi$
$859$	1278.39	1.48823	0.744117	+	0.668049i	$0.232870\pi$
$860$	− 114.624i	− 0.133284i
$861$	0	0
$862$	1070.26	1.24160
$863$	− 1164.50i	− 1.34937i	−0.738107	−	0.674683i	$-0.764280\pi$
$863$	− 1164.50i	− 1.34937i	0.738107	−	0.674683i	$-0.235720\pi$
$864$	0	0
$865$	72.9424	0.0843265
$866$	− 1834.67i	− 2.11856i
$867$	0	0
$868$	−984.876	−1.13465
$869$	310.145i	0.356899i
$870$	0	0
$871$	299.903	0.344320
$872$	152.052i	0.174372i
$873$	0	0
$874$	−68.8232	−0.0787451
$875$	420.652i	0.480746i
$876$	0	0
$877$	−625.815	−0.713586	−0.356793	−	0.934183i	$-0.616130\pi$
$877$	−625.815	−0.713586	−0.356793	+	0.934183i	$0.616130\pi$
$878$	528.482i	0.601916i
$879$	0	0
$880$	−267.194	−0.303630
$881$	− 762.102i	− 0.865042i	−0.901624	−	0.432521i	$-0.857624\pi$
$881$	− 762.102i	− 0.865042i	0.901624	−	0.432521i	$-0.142376\pi$
$882$	0	0
$883$	−877.735	−0.994038	−0.497019	−	0.867740i	$-0.665572\pi$
$883$	−877.735	−0.994038	−0.497019	+	0.867740i	$0.665572\pi$
$884$	− 47.1776i	− 0.0533683i
$885$	0	0
$886$	780.149	0.880529
$887$	699.683i	0.788820i	0.918935	+	0.394410i	$0.129051\pi$
$887$	699.683i	0.788820i	−0.918935	+	0.394410i	$0.870949\pi$
$888$	0	0
$889$	−144.448	−0.162483
$890$	− 129.433i	− 0.145430i
$891$	0	0
$892$	449.876	0.504345
$893$	− 83.7446i	− 0.0937790i
$894$	0	0
$895$	136.248	0.152233
$896$	1388.70i	1.54989i
$897$	0	0
$898$	−209.159	−0.232917
$899$	− 819.342i	− 0.911393i
$900$	0	0
$901$	300.603	0.333633
$902$	3686.54i	4.08707i
$903$	0	0
$904$	−175.566	−0.194210
$905$	− 28.5642i	− 0.0315627i
$906$	0	0
$907$	−1077.34	−1.18780	−0.593901	−	0.804538i	$-0.702413\pi$
$907$	−1077.34	−1.18780	−0.593901	+	0.804538i	$0.702413\pi$
$908$	− 1023.88i	− 1.12762i
$909$	0	0
$910$	−107.275	−0.117885
$911$	581.178i	0.637956i	0.947762	+	0.318978i	$0.103340\pi$
$911$	581.178i	0.637956i	−0.947762	+	0.318978i	$0.896660\pi$
$912$	0	0
$913$	10.9588	0.0120031
$914$	2221.33i	2.43034i
$915$	0	0
$916$	−225.330	−0.245993
$917$	− 1256.88i	− 1.37064i
$918$	0	0
$919$	721.090	0.784647	0.392323	−	0.919827i	$-0.371671\pi$
$919$	721.090	0.784647	0.392323	+	0.919827i	$0.371671\pi$
$920$	− 62.4354i	− 0.0678646i
$921$	0	0
$922$	325.021	0.352517
$923$	− 570.131i	− 0.617694i
$924$	0	0
$925$	−61.4913	−0.0664771
$926$	70.7858i	0.0764426i
$927$	0	0
$928$	996.663	1.07399
$929$	− 553.878i	− 0.596209i	−0.954533	−	0.298105i	$-0.903646\pi$
$929$	− 553.878i	− 0.596209i	0.954533	−	0.298105i	$-0.0963544\pi$
$930$	0	0
$931$	−118.243	−0.127006
$932$	− 893.556i	− 0.958751i
$933$	0	0
$934$	497.852	0.533032
$935$	50.2396i	0.0537322i
$936$	0	0
$937$	−1271.69	−1.35720	−0.678598	−	0.734510i	$-0.737412\pi$
$937$	−1271.69	−1.35720	−0.678598	+	0.734510i	$0.737412\pi$
$938$	− 2005.75i	− 2.13833i
$939$	0	0
$940$	−140.306	−0.149262
$941$	− 318.512i	− 0.338483i	−0.985575	−	0.169241i	$-0.945868\pi$
$941$	− 318.512i	− 0.338483i	0.985575	−	0.169241i	$-0.0541317\pi$
$942$	0	0
$943$	−1826.37	−1.93676
$944$	1299.33i	1.37641i
$945$	0	0
$946$	3518.59	3.71944
$947$	− 784.960i	− 0.828891i	−0.910074	−	0.414446i	$-0.863975\pi$
$947$	− 784.960i	− 0.828891i	0.910074	−	0.414446i	$-0.136025\pi$
$948$	0	0
$949$	−513.599	−0.541200
$950$	− 64.6896i	− 0.0680943i
$951$	0	0
$952$	170.847	0.179462
$953$	1016.76i	1.06690i	0.845831	+	0.533452i	$0.179105\pi$
$953$	1016.76i	1.06690i	−0.845831	+	0.533452i	$0.820895\pi$
$954$	0	0
$955$	42.4183	0.0444170
$956$	256.857i	0.268679i
$957$	0	0
$958$	−262.133	−0.273625
$959$	1736.53i	1.81078i
$960$	0	0
$961$	−84.2084	−0.0876258
$962$	− 31.6432i	− 0.0328931i
$963$	0	0
$964$	890.873	0.924142
$965$	− 126.747i	− 0.131344i
$966$	0	0
$967$	−1823.75	−1.88598	−0.942992	−	0.332816i	$-0.892001\pi$
$967$	−1823.75	−1.88598	−0.942992	+	0.332816i	$0.892001\pi$
$968$	− 1085.58i	− 1.12147i
$969$	0	0
$970$	151.362	0.156043
$971$	1079.74i	1.11199i	0.831187	+	0.555993i	$0.187662\pi$
$971$	1079.74i	1.11199i	−0.831187	+	0.555993i	$0.812338\pi$
$972$	0	0
$973$	−233.254	−0.239727
$974$	− 1689.97i	− 1.73508i
$975$	0	0
$976$	−115.447	−0.118286
$977$	− 630.788i	− 0.645637i	−0.946461	−	0.322819i	$-0.895370\pi$
$977$	− 630.788i	− 0.645637i	0.946461	−	0.322819i	$-0.104630\pi$
$978$	0	0
$979$	1563.33	1.59687
$980$	198.105i	0.202148i
$981$	0	0
$982$	−254.599	−0.259265
$983$	− 1211.78i	− 1.23274i	−0.787458	−	0.616369i	$-0.788603\pi$
$983$	− 1211.78i	− 1.23274i	0.787458	−	0.616369i	$-0.211397\pi$
$984$	0	0
$985$	−168.583	−0.171150
$986$	− 262.492i	− 0.266219i
$987$	0	0
$988$	13.0983	0.0132574
$989$	1743.16i	1.76255i
$990$	0	0
$991$	453.179	0.457295	0.228647	−	0.973509i	$-0.426570\pi$
$991$	453.179	0.457295	0.228647	+	0.973509i	$0.426570\pi$
$992$	1066.55i	1.07515i
$993$	0	0
$994$	−3813.04	−3.83606
$995$	32.5772i	0.0327409i
$996$	0	0
$997$	−1127.93	−1.13133	−0.565663	−	0.824637i	$-0.691380\pi$
$997$	−1127.93	−1.13133	−0.565663	+	0.824637i	$0.691380\pi$
$998$	972.246i	0.974195i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.68	yes	84
3.2	odd	2	inner	1143.3.b.a.890.17	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.17	✓	84	3.2	odd	2	inner
1143.3.b.a.890.68	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)

\(f(q)\)	\(=\)	\(q+2.56806i q^{2} -2.59492 q^{4} -0.662170i q^{5} -12.8177 q^{7} +3.60832i q^{8} +O(q^{10})\)	\(q+2.56806i q^{2} -2.59492 q^{4} -0.662170i q^{5} -12.8177 q^{7} +3.60832i q^{8} +1.70049 q^{10} +20.5391i q^{11} +4.92172 q^{13} -32.9165i q^{14} -19.6461 q^{16} +3.69398i q^{17} -1.02559 q^{19} +1.71828i q^{20} -52.7457 q^{22} -26.1310i q^{23} +24.5615 q^{25} +12.6393i q^{26} +33.2609 q^{28} +27.6705i q^{29} -29.6107 q^{31} -36.0190i q^{32} -9.48635 q^{34} +8.48748i q^{35} -2.50356 q^{37} -2.63378i q^{38} +2.38932 q^{40} -69.8927i q^{41} -66.7086 q^{43} -53.2974i q^{44} +67.1060 q^{46} +81.6550i q^{47} +115.293 q^{49} +63.0754i q^{50} -12.7715 q^{52} -81.3765i q^{53} +13.6004 q^{55} -46.2502i q^{56} -71.0595 q^{58} -66.1371i q^{59} +5.87636 q^{61} -76.0419i q^{62} +13.9145 q^{64} -3.25902i q^{65} +60.9345 q^{67} -9.58559i q^{68} -21.7963 q^{70} -115.840i q^{71} -104.353 q^{73} -6.42930i q^{74} +2.66133 q^{76} -263.264i q^{77} +15.1002 q^{79} +13.0090i q^{80} +179.489 q^{82} -0.533557i q^{83} +2.44604 q^{85} -171.312i q^{86} -74.1117 q^{88} -76.1149i q^{89} -63.0850 q^{91} +67.8080i q^{92} -209.695 q^{94} +0.679115i q^{95} +89.0106 q^{97} +296.078i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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