LMFDB - Embedded newform 1143.3.b.a.890.7

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			890.7
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.78

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} +O(q^{10})$	\(q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} -25.8324 q^{10} +15.5657i q^{11} -9.78351 q^{13} -16.8833i q^{14} +26.9194 q^{16} +23.5044i q^{17} +25.7089 q^{19} +63.7749i q^{20} +55.7976 q^{22} -5.95651i q^{23} -26.9320 q^{25} +35.0705i q^{26} -41.6813 q^{28} +53.9788i q^{29} +10.8286 q^{31} -26.9577i q^{32} +84.2553 q^{34} -33.9412i q^{35} -62.1323 q^{37} -92.1576i q^{38} +125.281 q^{40} +33.5546i q^{41} +77.7688 q^{43} -137.753i q^{44} -21.3520 q^{46} +59.3622i q^{47} -26.8171 q^{49} +96.5421i q^{50} +86.5819 q^{52} +88.7794i q^{53} +112.172 q^{55} +81.8801i q^{56} +193.496 q^{58} -14.4209i q^{59} -78.6267 q^{61} -38.8169i q^{62} +11.0435 q^{64} +70.5038i q^{65} -18.8207 q^{67} -208.009i q^{68} -121.668 q^{70} -75.8330i q^{71} -45.6238 q^{73} +222.723i q^{74} -227.518 q^{76} +73.3124i q^{77} -56.3446 q^{79} -193.991i q^{80} +120.282 q^{82} +128.210i q^{83} +169.382 q^{85} -278.775i q^{86} -270.606 q^{88} -70.1992i q^{89} -46.0791 q^{91} +52.7137i q^{92} +212.793 q^{94} -185.268i q^{95} -4.59545 q^{97} +96.1301i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.58466i	− 1.79233i	−0.443722	−	0.896164i	$-0.646342\pi$
$2$	− 3.58466i	− 1.79233i	0.443722	−	0.896164i	$-0.353658\pi$
$3$	0	0
$3$	0	0
$4$	−8.84977	−2.21244
$5$	− 7.20639i	− 1.44128i	−0.693311	−	0.720639i	$-0.743849\pi$
$5$	− 7.20639i	− 1.44128i	0.693311	−	0.720639i	$-0.256151\pi$
$6$	0	0
$7$	4.70987	0.672839	0.336420	−	0.941712i	$-0.390784\pi$
$7$	4.70987	0.672839	0.336420	+	0.941712i	$0.390784\pi$
$8$	17.3848i	2.17310i
$9$	0	0
$10$	−25.8324	−2.58324
$11$	15.5657i	1.41506i	0.706683	+	0.707531i	$0.250191\pi$
$11$	15.5657i	1.41506i	−0.706683	+	0.707531i	$0.749809\pi$
$12$	0	0
$13$	−9.78351	−0.752578	−0.376289	−	0.926502i	$-0.622800\pi$
$13$	−9.78351	−0.752578	−0.376289	+	0.926502i	$0.622800\pi$
$14$	− 16.8833i	− 1.20595i
$15$	0	0
$16$	26.9194	1.68246
$17$	23.5044i	1.38261i	0.722562	+	0.691306i	$0.242964\pi$
$17$	23.5044i	1.38261i	−0.722562	+	0.691306i	$0.757036\pi$
$18$	0	0
$19$	25.7089	1.35310	0.676550	−	0.736397i	$-0.263474\pi$
$19$	25.7089	1.35310	0.676550	+	0.736397i	$0.263474\pi$
$20$	63.7749i	3.18874i
$21$	0	0
$22$	55.7976	2.53626
$23$	− 5.95651i	− 0.258979i	−0.991581	−	0.129489i	$-0.958666\pi$
$23$	− 5.95651i	− 0.258979i	0.991581	−	0.129489i	$-0.0413338\pi$
$24$	0	0
$25$	−26.9320	−1.07728
$26$	35.0705i	1.34887i
$27$	0	0
$28$	−41.6813	−1.48862
$29$	53.9788i	1.86134i	0.365861	+	0.930669i	$0.380774\pi$
$29$	53.9788i	1.86134i	−0.365861	+	0.930669i	$0.619226\pi$
$30$	0	0
$31$	10.8286	0.349310	0.174655	−	0.984630i	$-0.444119\pi$
$31$	10.8286	0.349310	0.174655	+	0.984630i	$0.444119\pi$
$32$	− 26.9577i	− 0.842427i
$33$	0	0
$34$	84.2553	2.47810
$35$	− 33.9412i	− 0.969748i
$36$	0	0
$37$	−62.1323	−1.67925	−0.839625	−	0.543166i	$-0.817225\pi$
$37$	−62.1323	−1.67925	−0.839625	+	0.543166i	$0.817225\pi$
$38$	− 92.1576i	− 2.42520i
$39$	0	0
$40$	125.281	3.13204
$41$	33.5546i	0.818405i	0.912444	+	0.409202i	$0.134193\pi$
$41$	33.5546i	0.818405i	−0.912444	+	0.409202i	$0.865807\pi$
$42$	0	0
$43$	77.7688	1.80858	0.904288	−	0.426922i	$-0.140402\pi$
$43$	77.7688	1.80858	0.904288	+	0.426922i	$0.140402\pi$
$44$	− 137.753i	− 3.13074i
$45$	0	0
$46$	−21.3520	−0.464175
$47$	59.3622i	1.26303i	0.775365	+	0.631513i	$0.217566\pi$
$47$	59.3622i	1.26303i	−0.775365	+	0.631513i	$0.782434\pi$
$48$	0	0
$49$	−26.8171	−0.547287
$50$	96.5421i	1.93084i
$51$	0	0
$52$	86.5819	1.66504
$53$	88.7794i	1.67508i	0.546374	+	0.837542i	$0.316008\pi$
$53$	88.7794i	1.67508i	−0.546374	+	0.837542i	$0.683992\pi$
$54$	0	0
$55$	112.172	2.03950
$56$	81.8801i	1.46214i
$57$	0	0
$58$	193.496	3.33613
$59$	− 14.4209i	− 0.244421i	−0.992504	−	0.122211i	$-0.961002\pi$
$59$	− 14.4209i	− 0.244421i	0.992504	−	0.122211i	$-0.0389983\pi$
$60$	0	0
$61$	−78.6267	−1.28896	−0.644481	−	0.764620i	$-0.722927\pi$
$61$	−78.6267	−1.28896	−0.644481	+	0.764620i	$0.722927\pi$
$62$	− 38.8169i	− 0.626079i
$63$	0	0
$64$	11.0435	0.172555
$65$	70.5038i	1.08467i
$66$	0	0
$67$	−18.8207	−0.280906	−0.140453	−	0.990087i	$-0.544856\pi$
$67$	−18.8207	−0.280906	−0.140453	+	0.990087i	$0.544856\pi$
$68$	− 208.009i	− 3.05895i
$69$	0	0
$70$	−121.668	−1.73811
$71$	− 75.8330i	− 1.06807i	−0.845462	−	0.534035i	$-0.820675\pi$
$71$	− 75.8330i	− 1.06807i	0.845462	−	0.534035i	$-0.179325\pi$
$72$	0	0
$73$	−45.6238	−0.624984	−0.312492	−	0.949920i	$-0.601164\pi$
$73$	−45.6238	−0.624984	−0.312492	+	0.949920i	$0.601164\pi$
$74$	222.723i	3.00977i
$75$	0	0
$76$	−227.518	−2.99366
$77$	73.3124i	0.952109i
$78$	0	0
$79$	−56.3446	−0.713222	−0.356611	−	0.934253i	$-0.616068\pi$
$79$	−56.3446	−0.713222	−0.356611	+	0.934253i	$0.616068\pi$
$80$	− 193.991i	− 2.42489i
$81$	0	0
$82$	120.282	1.46685
$83$	128.210i	1.54470i	0.635199	+	0.772348i	$0.280918\pi$
$83$	128.210i	1.54470i	−0.635199	+	0.772348i	$0.719082\pi$
$84$	0	0
$85$	169.382	1.99273
$86$	− 278.775i	− 3.24156i
$87$	0	0
$88$	−270.606	−3.07506
$89$	− 70.1992i	− 0.788756i	−0.918948	−	0.394378i	$-0.870960\pi$
$89$	− 70.1992i	− 0.788756i	0.918948	−	0.394378i	$-0.129040\pi$
$90$	0	0
$91$	−46.0791	−0.506364
$92$	52.7137i	0.572975i
$93$	0	0
$94$	212.793	2.26376
$95$	− 185.268i	− 1.95019i
$96$	0	0
$97$	−4.59545	−0.0473758	−0.0236879	−	0.999719i	$-0.507541\pi$
$97$	−4.59545	−0.0473758	−0.0236879	+	0.999719i	$0.507541\pi$
$98$	96.1301i	0.980919i
$99$	0	0
$100$	238.342	2.38342
$101$	− 119.961i	− 1.18774i	−0.804562	−	0.593868i	$-0.797600\pi$
$101$	− 119.961i	− 1.18774i	0.804562	−	0.593868i	$-0.202400\pi$
$102$	0	0
$103$	−158.473	−1.53858	−0.769289	−	0.638901i	$-0.779389\pi$
$103$	−158.473	−1.53858	−0.769289	+	0.638901i	$0.779389\pi$
$104$	− 170.084i	− 1.63542i
$105$	0	0
$106$	318.244	3.00230
$107$	− 158.986i	− 1.48585i	−0.669372	−	0.742927i	$-0.733437\pi$
$107$	− 158.986i	− 1.48585i	0.669372	−	0.742927i	$-0.266563\pi$
$108$	0	0
$109$	58.4012	0.535790	0.267895	−	0.963448i	$-0.413672\pi$
$109$	58.4012	0.535790	0.267895	+	0.963448i	$0.413672\pi$
$110$	− 402.099i	− 3.65545i
$111$	0	0
$112$	126.787	1.13203
$113$	− 80.9446i	− 0.716324i	−0.933659	−	0.358162i	$-0.883404\pi$
$113$	− 80.9446i	− 0.716324i	0.933659	−	0.358162i	$-0.116596\pi$
$114$	0	0
$115$	−42.9249	−0.373260
$116$	− 477.700i	− 4.11811i
$117$	0	0
$118$	−51.6938	−0.438083
$119$	110.703i	0.930276i
$120$	0	0
$121$	−121.290	−1.00240
$122$	281.850i	2.31025i
$123$	0	0
$124$	−95.8308	−0.772829
$125$	13.9230i	0.111384i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	− 147.418i	− 1.15170i
$129$	0	0
$130$	252.732	1.94409
$131$	17.1308i	0.130770i	0.997860	+	0.0653848i	$0.0208275\pi$
$131$	17.1308i	0.130770i	−0.997860	+	0.0653848i	$0.979173\pi$
$132$	0	0
$133$	121.086	0.910419
$134$	67.4659i	0.503477i
$135$	0	0
$136$	−408.619	−3.00455
$137$	− 28.8275i	− 0.210420i	−0.994450	−	0.105210i	$-0.966449\pi$
$137$	− 28.8275i	− 0.210420i	0.994450	−	0.105210i	$-0.0335515\pi$
$138$	0	0
$139$	25.6315	0.184399	0.0921997	−	0.995741i	$-0.470610\pi$
$139$	25.6315	0.184399	0.0921997	+	0.995741i	$0.470610\pi$
$140$	300.372i	2.14551i
$141$	0	0
$142$	−271.836	−1.91433
$143$	− 152.287i	− 1.06494i
$144$	0	0
$145$	388.992	2.68271
$146$	163.546i	1.12018i
$147$	0	0
$148$	549.856	3.71525
$149$	60.3128i	0.404784i	0.979305	+	0.202392i	$0.0648715\pi$
$149$	60.3128i	0.404784i	−0.979305	+	0.202392i	$0.935128\pi$
$150$	0	0
$151$	31.2208	0.206760	0.103380	−	0.994642i	$-0.467034\pi$
$151$	31.2208	0.206760	0.103380	+	0.994642i	$0.467034\pi$
$152$	446.943i	2.94042i
$153$	0	0
$154$	262.800	1.70649
$155$	− 78.0352i	− 0.503453i
$156$	0	0
$157$	−58.7863	−0.374435	−0.187218	−	0.982318i	$-0.559947\pi$
$157$	−58.7863	−0.374435	−0.187218	+	0.982318i	$0.559947\pi$
$158$	201.976i	1.27833i
$159$	0	0
$160$	−194.267	−1.21417
$161$	− 28.0544i	− 0.174251i
$162$	0	0
$163$	−97.1285	−0.595881	−0.297940	−	0.954585i	$-0.596300\pi$
$163$	−97.1285	−0.595881	−0.297940	+	0.954585i	$0.596300\pi$
$164$	− 296.950i	− 1.81067i
$165$	0	0
$166$	459.588	2.76860
$167$	147.196i	0.881415i	0.897651	+	0.440707i	$0.145272\pi$
$167$	147.196i	0.881415i	−0.897651	+	0.440707i	$0.854728\pi$
$168$	0	0
$169$	−73.2829	−0.433627
$170$	− 607.176i	− 3.57162i
$171$	0	0
$172$	−688.236	−4.00137
$173$	− 37.9424i	− 0.219320i	−0.993969	−	0.109660i	$-0.965024\pi$
$173$	− 37.9424i	− 0.219320i	0.993969	−	0.109660i	$-0.0349763\pi$
$174$	0	0
$175$	−126.846	−0.724837
$176$	419.018i	2.38079i
$177$	0	0
$178$	−251.640	−1.41371
$179$	220.858i	1.23384i	0.787024	+	0.616922i	$0.211621\pi$
$179$	220.858i	1.23384i	−0.787024	+	0.616922i	$0.788379\pi$
$180$	0	0
$181$	136.292	0.752996	0.376498	−	0.926417i	$-0.377128\pi$
$181$	136.292	0.752996	0.376498	+	0.926417i	$0.377128\pi$
$182$	165.178i	0.907571i
$183$	0	0
$184$	103.553	0.562785
$185$	447.749i	2.42027i
$186$	0	0
$187$	−365.862	−1.95648
$188$	− 525.342i	− 2.79437i
$189$	0	0
$190$	−664.124	−3.49539
$191$	202.812i	1.06185i	0.847420	+	0.530923i	$0.178155\pi$
$191$	202.812i	1.06185i	−0.847420	+	0.530923i	$0.821845\pi$
$192$	0	0
$193$	193.960	1.00497	0.502486	−	0.864585i	$-0.332419\pi$
$193$	193.960	1.00497	0.502486	+	0.864585i	$0.332419\pi$
$194$	16.4731i	0.0849130i
$195$	0	0
$196$	237.325	1.21084
$197$	266.715i	1.35388i	0.736036	+	0.676942i	$0.236695\pi$
$197$	266.715i	1.35388i	−0.736036	+	0.676942i	$0.763305\pi$
$198$	0	0
$199$	160.272	0.805388	0.402694	−	0.915335i	$-0.368074\pi$
$199$	160.272	0.805388	0.402694	+	0.915335i	$0.368074\pi$
$200$	− 468.207i	− 2.34104i
$201$	0	0
$202$	−430.020	−2.12881
$203$	254.233i	1.25238i
$204$	0	0
$205$	241.807	1.17955
$206$	568.073i	2.75764i
$207$	0	0
$208$	−263.366	−1.26618
$209$	400.176i	1.91472i
$210$	0	0
$211$	−210.891	−0.999483	−0.499741	−	0.866175i	$-0.666572\pi$
$211$	−210.891	−0.999483	−0.499741	+	0.866175i	$0.666572\pi$
$212$	− 785.677i	− 3.70603i
$213$	0	0
$214$	−569.912	−2.66314
$215$	− 560.432i	− 2.60666i
$216$	0	0
$217$	51.0014	0.235030
$218$	− 209.348i	− 0.960313i
$219$	0	0
$220$	−992.699	−4.51227
$221$	− 229.956i	− 1.04052i
$222$	0	0
$223$	392.407	1.75967	0.879835	−	0.475279i	$-0.157653\pi$
$223$	392.407	1.75967	0.879835	+	0.475279i	$0.157653\pi$
$224$	− 126.967i	− 0.566818i
$225$	0	0
$226$	−290.159	−1.28389
$227$	− 22.0499i	− 0.0971360i	−0.998820	−	0.0485680i	$-0.984534\pi$
$227$	− 22.0499i	− 0.0971360i	0.998820	−	0.0485680i	$-0.0154658\pi$
$228$	0	0
$229$	−263.211	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$229$	−263.211	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$230$	153.871i	0.669005i
$231$	0	0
$232$	−938.410	−4.04487
$233$	129.088i	0.554025i	0.960866	+	0.277013i	$0.0893444\pi$
$233$	129.088i	0.554025i	−0.960866	+	0.277013i	$0.910656\pi$
$234$	0	0
$235$	427.787	1.82037
$236$	127.621i	0.540768i
$237$	0	0
$238$	396.832	1.66736
$239$	289.053i	1.20943i	0.796443	+	0.604713i	$0.206712\pi$
$239$	289.053i	1.20943i	−0.796443	+	0.604713i	$0.793288\pi$
$240$	0	0
$241$	−104.175	−0.432260	−0.216130	−	0.976365i	$-0.569344\pi$
$241$	−104.175	−0.432260	−0.216130	+	0.976365i	$0.569344\pi$
$242$	434.784i	1.79663i
$243$	0	0
$244$	695.829	2.85176
$245$	193.254i	0.788793i
$246$	0	0
$247$	−251.523	−1.01831
$248$	188.253i	0.759085i
$249$	0	0
$250$	49.9091	0.199636
$251$	− 440.383i	− 1.75451i	−0.480023	−	0.877256i	$-0.659372\pi$
$251$	− 440.383i	− 1.75451i	0.480023	−	0.877256i	$-0.340628\pi$
$252$	0	0
$253$	92.7170	0.366470
$254$	− 40.3970i	− 0.159043i
$255$	0	0
$256$	−484.269	−1.89167
$257$	− 104.246i	− 0.405628i	−0.979217	−	0.202814i	$-0.934991\pi$
$257$	− 104.246i	− 0.405628i	0.979217	−	0.202814i	$-0.0650087\pi$
$258$	0	0
$259$	−292.635	−1.12987
$260$	− 623.942i	− 2.39978i
$261$	0	0
$262$	61.4082	0.234382
$263$	106.847i	0.406262i	0.979152	+	0.203131i	$0.0651118\pi$
$263$	106.847i	0.406262i	−0.979152	+	0.203131i	$0.934888\pi$
$264$	0	0
$265$	639.779	2.41426
$266$	− 434.051i	− 1.63177i
$267$	0	0
$268$	166.559	0.621490
$269$	280.299i	1.04200i	0.853556	+	0.521001i	$0.174441\pi$
$269$	280.299i	1.04200i	−0.853556	+	0.521001i	$0.825559\pi$
$270$	0	0
$271$	25.5309	0.0942100	0.0471050	−	0.998890i	$-0.485000\pi$
$271$	25.5309	0.0942100	0.0471050	+	0.998890i	$0.485000\pi$
$272$	632.724i	2.32619i
$273$	0	0
$274$	−103.337	−0.377142
$275$	− 419.215i	− 1.52442i
$276$	0	0
$277$	281.436	1.01602	0.508008	−	0.861352i	$-0.330382\pi$
$277$	281.436	1.01602	0.508008	+	0.861352i	$0.330382\pi$
$278$	− 91.8802i	− 0.330504i
$279$	0	0
$280$	590.060	2.10736
$281$	521.602i	1.85623i	0.372289	+	0.928117i	$0.378573\pi$
$281$	521.602i	1.85623i	−0.372289	+	0.928117i	$0.621427\pi$
$282$	0	0
$283$	219.479	0.775543	0.387771	−	0.921756i	$-0.373245\pi$
$283$	219.479	0.775543	0.387771	+	0.921756i	$0.373245\pi$
$284$	671.105i	2.36305i
$285$	0	0
$286$	−545.897	−1.90873
$287$	158.038i	0.550655i
$288$	0	0
$289$	−263.457	−0.911616
$290$	− 1394.40i	− 4.80829i
$291$	0	0
$292$	403.760	1.38274
$293$	− 122.921i	− 0.419524i	−0.977752	−	0.209762i	$-0.932731\pi$
$293$	− 122.921i	− 0.419524i	0.977752	−	0.209762i	$-0.0672689\pi$
$294$	0	0
$295$	−103.922	−0.352279
$296$	− 1080.16i	− 3.64917i
$297$	0	0
$298$	216.201	0.725506
$299$	58.2755i	0.194901i
$300$	0	0
$301$	366.281	1.21688
$302$	− 111.916i	− 0.370582i
$303$	0	0
$304$	692.068	2.27654
$305$	566.615i	1.85775i
$306$	0	0
$307$	358.046	1.16627	0.583136	−	0.812375i	$-0.301825\pi$
$307$	358.046	1.16627	0.583136	+	0.812375i	$0.301825\pi$
$308$	− 648.798i	− 2.10649i
$309$	0	0
$310$	−279.730	−0.902353
$311$	− 101.682i	− 0.326953i	−0.986547	−	0.163476i	$-0.947729\pi$
$311$	− 101.682i	− 0.326953i	0.986547	−	0.163476i	$-0.0522707\pi$
$312$	0	0
$313$	−113.540	−0.362747	−0.181374	−	0.983414i	$-0.558054\pi$
$313$	−113.540	−0.362747	−0.181374	+	0.983414i	$0.558054\pi$
$314$	210.729i	0.671111i
$315$	0	0
$316$	498.637	1.57796
$317$	− 563.141i	− 1.77647i	−0.459389	−	0.888235i	$-0.651932\pi$
$317$	− 563.141i	− 1.77647i	0.459389	−	0.888235i	$-0.348068\pi$
$318$	0	0
$319$	−840.217	−2.63391
$320$	− 79.5838i	− 0.248699i
$321$	0	0
$322$	−100.565	−0.312315
$323$	604.272i	1.87081i
$324$	0	0
$325$	263.490	0.810738
$326$	348.173i	1.06801i
$327$	0	0
$328$	−583.339	−1.77847
$329$	279.589i	0.849814i
$330$	0	0
$331$	248.907	0.751985	0.375993	−	0.926623i	$-0.377302\pi$
$331$	248.907	0.751985	0.375993	+	0.926623i	$0.377302\pi$
$332$	− 1134.63i	− 3.41755i
$333$	0	0
$334$	527.648	1.57979
$335$	135.630i	0.404864i
$336$	0	0
$337$	81.4574	0.241713	0.120857	−	0.992670i	$-0.461436\pi$
$337$	81.4574	0.241713	0.120857	+	0.992670i	$0.461436\pi$
$338$	262.694i	0.777201i
$339$	0	0
$340$	−1498.99	−4.40880
$341$	168.555i	0.494295i
$342$	0	0
$343$	−357.089	−1.04108
$344$	1351.99i	3.93021i
$345$	0	0
$346$	−136.011	−0.393094
$347$	94.2754i	0.271687i	0.990730	+	0.135844i	$0.0433745\pi$
$347$	94.2754i	0.271687i	−0.990730	+	0.135844i	$0.956626\pi$
$348$	0	0
$349$	−544.452	−1.56003	−0.780017	−	0.625759i	$-0.784789\pi$
$349$	−544.452	−1.56003	−0.780017	+	0.625759i	$0.784789\pi$
$350$	454.701i	1.29915i
$351$	0	0
$352$	419.614	1.19209
$353$	− 420.122i	− 1.19015i	−0.803671	−	0.595073i	$-0.797123\pi$
$353$	− 420.122i	− 1.19015i	0.803671	−	0.595073i	$-0.202877\pi$
$354$	0	0
$355$	−546.482	−1.53939
$356$	621.247i	1.74508i
$357$	0	0
$358$	791.701	2.21145
$359$	381.254i	1.06199i	0.847375	+	0.530995i	$0.178182\pi$
$359$	381.254i	1.06199i	−0.847375	+	0.530995i	$0.821818\pi$
$360$	0	0
$361$	299.947	0.830879
$362$	− 488.561i	− 1.34962i
$363$	0	0
$364$	407.790	1.12030
$365$	328.783i	0.900775i
$366$	0	0
$367$	9.78670	0.0266668	0.0133334	−	0.999911i	$-0.495756\pi$
$367$	9.78670	0.0266668	0.0133334	+	0.999911i	$0.495756\pi$
$368$	− 160.345i	− 0.435721i
$369$	0	0
$370$	1605.03	4.33791
$371$	418.140i	1.12706i
$372$	0	0
$373$	254.878	0.683319	0.341659	−	0.939824i	$-0.389011\pi$
$373$	254.878	0.683319	0.341659	+	0.939824i	$0.389011\pi$
$374$	1311.49i	3.50666i
$375$	0	0
$376$	−1032.00	−2.74468
$377$	− 528.102i	− 1.40080i
$378$	0	0
$379$	−99.9256	−0.263656	−0.131828	−	0.991273i	$-0.542085\pi$
$379$	−99.9256	−0.263656	−0.131828	+	0.991273i	$0.542085\pi$
$380$	1639.58i	4.31469i
$381$	0	0
$382$	727.013	1.90318
$383$	− 10.5938i	− 0.0276600i	−0.999904	−	0.0138300i	$-0.995598\pi$
$383$	− 10.5938i	− 0.0276600i	0.999904	−	0.0138300i	$-0.00440236\pi$
$384$	0	0
$385$	528.317	1.37225
$386$	− 695.279i	− 1.80124i
$387$	0	0
$388$	40.6687	0.104816
$389$	− 62.8874i	− 0.161664i	−0.996728	−	0.0808322i	$-0.974242\pi$
$389$	− 62.8874i	− 0.161664i	0.996728	−	0.0808322i	$-0.0257578\pi$
$390$	0	0
$391$	140.004	0.358067
$392$	− 466.209i	− 1.18931i
$393$	0	0
$394$	956.082	2.42661
$395$	406.041i	1.02795i
$396$	0	0
$397$	461.795	1.16321	0.581605	−	0.813471i	$-0.302425\pi$
$397$	461.795	1.16321	0.581605	+	0.813471i	$0.302425\pi$
$398$	− 574.521i	− 1.44352i
$399$	0	0
$400$	−724.994	−1.81248
$401$	496.739i	1.23875i	0.785095	+	0.619376i	$0.212614\pi$
$401$	496.739i	1.23875i	−0.785095	+	0.619376i	$0.787386\pi$
$402$	0	0
$403$	−105.942	−0.262883
$404$	1061.63i	2.62780i
$405$	0	0
$406$	911.340	2.24468
$407$	− 967.130i	− 2.37624i
$408$	0	0
$409$	364.666	0.891605	0.445802	−	0.895131i	$-0.352918\pi$
$409$	364.666	0.891605	0.445802	+	0.895131i	$0.352918\pi$
$410$	− 866.797i	− 2.11414i
$411$	0	0
$412$	1402.45	3.40401
$413$	− 67.9204i	− 0.164456i
$414$	0	0
$415$	923.930	2.22634
$416$	263.741i	0.633992i
$417$	0	0
$418$	1434.50	3.43181
$419$	430.068i	1.02642i	0.858264	+	0.513208i	$0.171543\pi$
$419$	430.068i	1.02642i	−0.858264	+	0.513208i	$0.828457\pi$
$420$	0	0
$421$	−522.669	−1.24149	−0.620747	−	0.784011i	$-0.713171\pi$
$421$	−522.669	−1.24149	−0.620747	+	0.784011i	$0.713171\pi$
$422$	755.971i	1.79140i
$423$	0	0
$424$	−1543.41	−3.64012
$425$	− 633.021i	− 1.48946i
$426$	0	0
$427$	−370.322	−0.867265
$428$	1406.99i	3.28737i
$429$	0	0
$430$	−2008.96	−4.67199
$431$	110.538i	0.256469i	0.991744	+	0.128235i	$0.0409310\pi$
$431$	110.538i	0.256469i	−0.991744	+	0.128235i	$0.959069\pi$
$432$	0	0
$433$	−73.8530	−0.170561	−0.0852806	−	0.996357i	$-0.527179\pi$
$433$	−73.8530	−0.170561	−0.0852806	+	0.996357i	$0.527179\pi$
$434$	− 182.823i	− 0.421250i
$435$	0	0
$436$	−516.837	−1.18541
$437$	− 153.135i	− 0.350424i
$438$	0	0
$439$	330.242	0.752260	0.376130	−	0.926567i	$-0.377255\pi$
$439$	330.242	0.752260	0.376130	+	0.926567i	$0.377255\pi$
$440$	1950.09i	4.43202i
$441$	0	0
$442$	−824.312	−1.86496
$443$	425.440i	0.960360i	0.877170	+	0.480180i	$0.159429\pi$
$443$	425.440i	0.960360i	−0.877170	+	0.480180i	$0.840571\pi$
$444$	0	0
$445$	−505.883	−1.13682
$446$	− 1406.64i	− 3.15391i
$447$	0	0
$448$	52.0135	0.116102
$449$	− 149.718i	− 0.333447i	−0.986004	−	0.166724i	$-0.946681\pi$
$449$	− 149.718i	− 0.333447i	0.986004	−	0.166724i	$-0.0533187\pi$
$450$	0	0
$451$	−522.300	−1.15809
$452$	716.341i	1.58483i
$453$	0	0
$454$	−79.0412	−0.174100
$455$	332.064i	0.729811i
$456$	0	0
$457$	−424.824	−0.929593	−0.464797	−	0.885418i	$-0.653873\pi$
$457$	−424.824	−0.929593	−0.464797	+	0.885418i	$0.653873\pi$
$458$	943.520i	2.06009i
$459$	0	0
$460$	379.876	0.825816
$461$	− 28.8882i	− 0.0626642i	−0.999509	−	0.0313321i	$-0.990025\pi$
$461$	− 28.8882i	− 0.0626642i	0.999509	−	0.0313321i	$-0.00997495\pi$
$462$	0	0
$463$	514.451	1.11113	0.555563	−	0.831474i	$-0.312503\pi$
$463$	514.451	1.11113	0.555563	+	0.831474i	$0.312503\pi$
$464$	1453.08i	3.13163i
$465$	0	0
$466$	462.736	0.992995
$467$	424.921i	0.909894i	0.890518	+	0.454947i	$0.150342\pi$
$467$	424.921i	0.909894i	−0.890518	+	0.454947i	$0.849658\pi$
$468$	0	0
$469$	−88.6433	−0.189005
$470$	− 1533.47i	− 3.26270i
$471$	0	0
$472$	250.703	0.531151
$473$	1210.52i	2.55925i
$474$	0	0
$475$	−692.393	−1.45767
$476$	− 979.695i	− 2.05818i
$477$	0	0
$478$	1036.16	2.16769
$479$	− 400.532i	− 0.836183i	−0.908405	−	0.418091i	$-0.862699\pi$
$479$	− 400.532i	− 0.836183i	0.908405	−	0.418091i	$-0.137301\pi$
$480$	0	0
$481$	607.872	1.26377
$482$	373.431i	0.774753i
$483$	0	0
$484$	1073.39	2.21775
$485$	33.1166i	0.0682817i
$486$	0	0
$487$	−823.321	−1.69060	−0.845299	−	0.534293i	$-0.820578\pi$
$487$	−823.321	−1.69060	−0.845299	+	0.534293i	$0.820578\pi$
$488$	− 1366.91i	− 2.80104i
$489$	0	0
$490$	692.751	1.41378
$491$	− 258.140i	− 0.525744i	−0.964831	−	0.262872i	$-0.915330\pi$
$491$	− 258.140i	− 0.525744i	0.964831	−	0.262872i	$-0.0846698\pi$
$492$	0	0
$493$	−1268.74	−2.57351
$494$	901.625i	1.82515i
$495$	0	0
$496$	291.500	0.587701
$497$	− 357.164i	− 0.718640i
$498$	0	0
$499$	−49.8261	−0.0998520	−0.0499260	−	0.998753i	$-0.515899\pi$
$499$	−49.8261	−0.0998520	−0.0499260	+	0.998753i	$0.515899\pi$
$500$	− 123.215i	− 0.246430i
$501$	0	0
$502$	−1578.62	−3.14466
$503$	− 879.672i	− 1.74885i	−0.485160	−	0.874426i	$-0.661238\pi$
$503$	− 879.672i	− 1.74885i	0.485160	−	0.874426i	$-0.338762\pi$
$504$	0	0
$505$	−864.488	−1.71186
$506$	− 332.359i	− 0.656836i
$507$	0	0
$508$	−99.7319	−0.196323
$509$	− 116.561i	− 0.229001i	−0.993423	−	0.114500i	$-0.963473\pi$
$509$	− 116.561i	− 0.229001i	0.993423	−	0.114500i	$-0.0365267\pi$
$510$	0	0
$511$	−214.882	−0.420513
$512$	1146.27i	2.23880i
$513$	0	0
$514$	−373.688	−0.727019
$515$	1142.02i	2.21752i
$516$	0	0
$517$	−924.013	−1.78726
$518$	1049.00i	2.02509i
$519$	0	0
$520$	−1225.69	−2.35710
$521$	− 909.374i	− 1.74544i	−0.488222	−	0.872719i	$-0.662354\pi$
$521$	− 909.374i	− 1.74544i	0.488222	−	0.872719i	$-0.337646\pi$
$522$	0	0
$523$	942.453	1.80201	0.901007	−	0.433804i	$-0.142829\pi$
$523$	942.453	1.80201	0.901007	+	0.433804i	$0.142829\pi$
$524$	− 151.604i	− 0.289320i
$525$	0	0
$526$	383.010	0.728156
$527$	254.520i	0.482960i
$528$	0	0
$529$	493.520	0.932930
$530$	− 2293.39i	− 4.32715i
$531$	0	0
$532$	−1071.58	−2.01425
$533$	− 328.282i	− 0.615913i
$534$	0	0
$535$	−1145.72	−2.14153
$536$	− 327.194i	− 0.610437i
$537$	0	0
$538$	1004.77	1.86761
$539$	− 417.426i	− 0.774445i
$540$	0	0
$541$	697.261	1.28884	0.644419	−	0.764673i	$-0.277099\pi$
$541$	697.261	1.28884	0.644419	+	0.764673i	$0.277099\pi$
$542$	− 91.5196i	− 0.168855i
$543$	0	0
$544$	633.624	1.16475
$545$	− 420.861i	− 0.772223i
$546$	0	0
$547$	951.659	1.73978	0.869890	−	0.493246i	$-0.164190\pi$
$547$	951.659	1.73978	0.869890	+	0.493246i	$0.164190\pi$
$548$	255.117i	0.465542i
$549$	0	0
$550$	−1502.74	−2.73226
$551$	1387.74i	2.51858i
$552$	0	0
$553$	−265.376	−0.479884
$554$	− 1008.85i	− 1.82103i
$555$	0	0
$556$	−226.833	−0.407973
$557$	− 840.713i	− 1.50936i	−0.656093	−	0.754680i	$-0.727792\pi$
$557$	− 840.713i	− 1.50936i	0.656093	−	0.754680i	$-0.272208\pi$
$558$	0	0
$559$	−760.852	−1.36109
$560$	− 913.676i	− 1.63156i
$561$	0	0
$562$	1869.76	3.32698
$563$	− 838.241i	− 1.48888i	−0.667688	−	0.744442i	$-0.732716\pi$
$563$	− 838.241i	− 1.48888i	0.667688	−	0.744442i	$-0.267284\pi$
$564$	0	0
$565$	−583.318	−1.03242
$566$	− 786.756i	− 1.39003i
$567$	0	0
$568$	1318.34	2.32102
$569$	940.180i	1.65234i	0.563423	+	0.826168i	$0.309484\pi$
$569$	940.180i	1.65234i	−0.563423	+	0.826168i	$0.690516\pi$
$570$	0	0
$571$	612.707	1.07304	0.536521	−	0.843887i	$-0.319738\pi$
$571$	612.707	1.07304	0.536521	+	0.843887i	$0.319738\pi$
$572$	1347.70i	2.35613i
$573$	0	0
$574$	566.512	0.986954
$575$	160.421i	0.278993i
$576$	0	0
$577$	−68.6703	−0.119013	−0.0595064	−	0.998228i	$-0.518953\pi$
$577$	−68.6703	−0.119013	−0.0595064	+	0.998228i	$0.518953\pi$
$578$	944.404i	1.63392i
$579$	0	0
$580$	−3442.49	−5.93533
$581$	603.852i	1.03933i
$582$	0	0
$583$	−1381.91	−2.37035
$584$	− 793.160i	− 1.35815i
$585$	0	0
$586$	−440.628	−0.751925
$587$	613.626i	1.04536i	0.852529	+	0.522680i	$0.175068\pi$
$587$	613.626i	1.04536i	−0.852529	+	0.522680i	$0.824932\pi$
$588$	0	0
$589$	278.392	0.472652
$590$	372.526i	0.631400i
$591$	0	0
$592$	−1672.56	−2.82527
$593$	− 10.5482i	− 0.0177878i	−0.999960	−	0.00889392i	$-0.997169\pi$
$593$	− 10.5482i	− 0.0177878i	0.999960	−	0.00889392i	$-0.00283106\pi$
$594$	0	0
$595$	797.767	1.34079
$596$	− 533.755i	− 0.895562i
$597$	0	0
$598$	208.898	0.349328
$599$	12.8106i	0.0213866i	0.999943	+	0.0106933i	$0.00340385\pi$
$599$	12.8106i	0.0213866i	−0.999943	+	0.0106933i	$0.996596\pi$
$600$	0	0
$601$	1099.50	1.82946	0.914728	−	0.404071i	$-0.132405\pi$
$601$	1099.50	1.82946	0.914728	+	0.404071i	$0.132405\pi$
$602$	− 1312.99i	− 2.18105i
$603$	0	0
$604$	−276.297	−0.457445
$605$	874.064i	1.44473i
$606$	0	0
$607$	169.097	0.278579	0.139289	−	0.990252i	$-0.455518\pi$
$607$	169.097	0.278579	0.139289	+	0.990252i	$0.455518\pi$
$608$	− 693.052i	− 1.13989i
$609$	0	0
$610$	2031.12	3.32970
$611$	− 580.771i	− 0.950526i
$612$	0	0
$613$	−52.3964	−0.0854754	−0.0427377	−	0.999086i	$-0.513608\pi$
$613$	−52.3964	−0.0854754	−0.0427377	+	0.999086i	$0.513608\pi$
$614$	− 1283.47i	− 2.09034i
$615$	0	0
$616$	−1274.52	−2.06902
$617$	425.530i	0.689677i	0.938662	+	0.344838i	$0.112066\pi$
$617$	425.530i	0.689677i	−0.938662	+	0.344838i	$0.887934\pi$
$618$	0	0
$619$	167.798	0.271079	0.135539	−	0.990772i	$-0.456723\pi$
$619$	167.798	0.271079	0.135539	+	0.990772i	$0.456723\pi$
$620$	690.594i	1.11386i
$621$	0	0
$622$	−364.496	−0.586007
$623$	− 330.630i	− 0.530706i
$624$	0	0
$625$	−572.966	−0.916746
$626$	407.002i	0.650163i
$627$	0	0
$628$	520.246	0.828416
$629$	− 1460.38i	− 2.32175i
$630$	0	0
$631$	−930.073	−1.47397	−0.736983	−	0.675911i	$-0.763750\pi$
$631$	−930.073	−1.47397	−0.736983	+	0.675911i	$0.763750\pi$
$632$	− 979.538i	− 1.54990i
$633$	0	0
$634$	−2018.67	−3.18402
$635$	− 81.2119i	− 0.127893i
$636$	0	0
$637$	262.365	0.411876
$638$	3011.89i	4.72083i
$639$	0	0
$640$	−1062.35	−1.65992
$641$	23.8249i	0.0371683i	0.999827	+	0.0185841i	$0.00591585\pi$
$641$	23.8249i	0.0371683i	−0.999827	+	0.0185841i	$0.994084\pi$
$642$	0	0
$643$	762.785	1.18629	0.593146	−	0.805095i	$-0.297886\pi$
$643$	762.785	1.18629	0.593146	+	0.805095i	$0.297886\pi$
$644$	248.275i	0.385520i
$645$	0	0
$646$	2166.11	3.35311
$647$	− 1067.19i	− 1.64944i	−0.565539	−	0.824722i	$-0.691332\pi$
$647$	− 1067.19i	− 1.64944i	0.565539	−	0.824722i	$-0.308668\pi$
$648$	0	0
$649$	224.470	0.345871
$650$	− 944.521i	− 1.45311i
$651$	0	0
$652$	859.565	1.31835
$653$	− 437.847i	− 0.670516i	−0.942126	−	0.335258i	$-0.891176\pi$
$653$	− 437.847i	− 0.670516i	0.942126	−	0.335258i	$-0.108824\pi$
$654$	0	0
$655$	123.451	0.188475
$656$	903.269i	1.37693i
$657$	0	0
$658$	1002.23	1.52315
$659$	− 528.030i	− 0.801260i	−0.916240	−	0.400630i	$-0.868791\pi$
$659$	− 528.030i	− 0.801260i	0.916240	−	0.400630i	$-0.131209\pi$
$660$	0	0
$661$	−1149.03	−1.73831	−0.869157	−	0.494536i	$-0.835338\pi$
$661$	−1149.03	−1.73831	−0.869157	+	0.494536i	$0.835338\pi$
$662$	− 892.247i	− 1.34780i
$663$	0	0
$664$	−2228.90	−3.35678
$665$	− 872.590i	− 1.31217i
$666$	0	0
$667$	321.525	0.482047
$668$	− 1302.65i	− 1.95008i
$669$	0	0
$670$	486.185	0.725650
$671$	− 1223.88i	− 1.82396i
$672$	0	0
$673$	932.678	1.38585	0.692926	−	0.721009i	$-0.256321\pi$
$673$	932.678	1.38585	0.692926	+	0.721009i	$0.256321\pi$
$674$	− 291.997i	− 0.433230i
$675$	0	0
$676$	648.537	0.959374
$677$	− 216.494i	− 0.319785i	−0.987134	−	0.159892i	$-0.948885\pi$
$677$	− 216.494i	− 0.319785i	0.987134	−	0.159892i	$-0.0511147\pi$
$678$	0	0
$679$	−21.6440	−0.0318763
$680$	2944.67i	4.33039i
$681$	0	0
$682$	604.211	0.885940
$683$	230.195i	0.337036i	0.985699	+	0.168518i	$0.0538981\pi$
$683$	230.195i	0.337036i	−0.985699	+	0.168518i	$0.946102\pi$
$684$	0	0
$685$	−207.742	−0.303273
$686$	1280.04i	1.86595i
$687$	0	0
$688$	2093.49	3.04286
$689$	− 868.574i	− 1.26063i
$690$	0	0
$691$	−13.7096	−0.0198402	−0.00992011	−	0.999951i	$-0.503158\pi$
$691$	−13.7096	−0.0198402	−0.00992011	+	0.999951i	$0.503158\pi$
$692$	335.782i	0.485234i
$693$	0	0
$694$	337.945	0.486953
$695$	− 184.711i	− 0.265771i
$696$	0	0
$697$	−788.681	−1.13154
$698$	1951.67i	2.79609i
$699$	0	0
$700$	1122.56	1.60366
$701$	− 310.382i	− 0.442770i	−0.975186	−	0.221385i	$-0.928942\pi$
$701$	− 310.382i	− 0.442770i	0.975186	−	0.221385i	$-0.0710578\pi$
$702$	0	0
$703$	−1597.35	−2.27219
$704$	171.900i	0.244175i
$705$	0	0
$706$	−1505.99	−2.13313
$707$	− 565.003i	− 0.799155i
$708$	0	0
$709$	−1053.71	−1.48619	−0.743093	−	0.669188i	$-0.766642\pi$
$709$	−1053.71	−1.48619	−0.743093	+	0.669188i	$0.766642\pi$
$710$	1958.95i	2.75909i
$711$	0	0
$712$	1220.40	1.71404
$713$	− 64.5007i	− 0.0904638i
$714$	0	0
$715$	−1097.44	−1.53488
$716$	− 1954.54i	− 2.72981i
$717$	0	0
$718$	1366.67	1.90343
$719$	1131.63i	1.57390i	0.617017	+	0.786950i	$0.288341\pi$
$719$	1131.63i	1.57390i	−0.617017	+	0.786950i	$0.711659\pi$
$720$	0	0
$721$	−746.390	−1.03522
$722$	− 1075.21i	− 1.48921i
$723$	0	0
$724$	−1206.16	−1.66596
$725$	− 1453.76i	− 2.00519i
$726$	0	0
$727$	70.7147	0.0972692	0.0486346	−	0.998817i	$-0.484513\pi$
$727$	70.7147	0.0972692	0.0486346	+	0.998817i	$0.484513\pi$
$728$	− 801.075i	− 1.10038i
$729$	0	0
$730$	1178.57	1.61448
$731$	1827.91i	2.50056i
$732$	0	0
$733$	980.186	1.33722	0.668612	−	0.743611i	$-0.266889\pi$
$733$	980.186	1.33722	0.668612	+	0.743611i	$0.266889\pi$
$734$	− 35.0820i	− 0.0477956i
$735$	0	0
$736$	−160.574	−0.218171
$737$	− 292.957i	− 0.397500i
$738$	0	0
$739$	−483.337	−0.654042	−0.327021	−	0.945017i	$-0.606045\pi$
$739$	−483.337	−0.654042	−0.327021	+	0.945017i	$0.606045\pi$
$740$	− 3962.48i	− 5.35470i
$741$	0	0
$742$	1498.89	2.02007
$743$	− 530.012i	− 0.713341i	−0.934230	−	0.356671i	$-0.883912\pi$
$743$	− 530.012i	− 0.713341i	0.934230	−	0.356671i	$-0.116088\pi$
$744$	0	0
$745$	434.638	0.583406
$746$	− 913.650i	− 1.22473i
$747$	0	0
$748$	3237.79	4.32860
$749$	− 748.806i	− 0.999741i
$750$	0	0
$751$	263.871	0.351360	0.175680	−	0.984447i	$-0.443788\pi$
$751$	263.871	0.351360	0.175680	+	0.984447i	$0.443788\pi$
$752$	1597.99i	2.12499i
$753$	0	0
$754$	−1893.07	−2.51070
$755$	− 224.989i	− 0.297999i
$756$	0	0
$757$	−1216.09	−1.60646	−0.803232	−	0.595667i	$-0.796888\pi$
$757$	−1216.09	−1.60646	−0.803232	+	0.595667i	$0.796888\pi$
$758$	358.199i	0.472558i
$759$	0	0
$760$	3220.85	4.23796
$761$	599.778i	0.788145i	0.919079	+	0.394073i	$0.128934\pi$
$761$	599.778i	0.788145i	−0.919079	+	0.394073i	$0.871066\pi$
$762$	0	0
$763$	275.062	0.360501
$764$	− 1794.84i	− 2.34927i
$765$	0	0
$766$	−37.9750	−0.0495758
$767$	141.087i	0.183946i
$768$	0	0
$769$	378.397	0.492063	0.246032	−	0.969262i	$-0.420873\pi$
$769$	378.397	0.492063	0.246032	+	0.969262i	$0.420873\pi$
$770$	− 1893.84i	− 2.45953i
$771$	0	0
$772$	−1716.50	−2.22344
$773$	− 868.664i	− 1.12376i	−0.827220	−	0.561879i	$-0.810079\pi$
$773$	− 868.664i	− 1.12376i	0.827220	−	0.561879i	$-0.189921\pi$
$774$	0	0
$775$	−291.637	−0.376305
$776$	− 79.8909i	− 0.102952i
$777$	0	0
$778$	−225.430	−0.289756
$779$	862.652i	1.10738i
$780$	0	0
$781$	1180.39	1.51139
$782$	− 501.867i	− 0.641774i
$783$	0	0
$784$	−721.899	−0.920790
$785$	423.637i	0.539665i
$786$	0	0
$787$	56.4537	0.0717328	0.0358664	−	0.999357i	$-0.488581\pi$
$787$	56.4537	0.0717328	0.0358664	+	0.999357i	$0.488581\pi$
$788$	− 2360.37i	− 2.99539i
$789$	0	0
$790$	1455.52	1.84243
$791$	− 381.239i	− 0.481971i
$792$	0	0
$793$	769.245	0.970045
$794$	− 1655.38i	− 2.08486i
$795$	0	0
$796$	−1418.37	−1.78187
$797$	457.708i	0.574289i	0.957887	+	0.287145i	$0.0927060\pi$
$797$	457.708i	0.574289i	−0.957887	+	0.287145i	$0.907294\pi$
$798$	0	0
$799$	−1395.27	−1.74628
$800$	726.025i	0.907531i
$801$	0	0
$802$	1780.64	2.22025
$803$	− 710.165i	− 0.884390i
$804$	0	0
$805$	−202.171	−0.251144
$806$	379.765i	0.471173i
$807$	0	0
$808$	2085.50	2.58107
$809$	1039.81i	1.28530i	0.766159	+	0.642651i	$0.222166\pi$
$809$	1039.81i	1.28530i	−0.766159	+	0.642651i	$0.777834\pi$
$810$	0	0
$811$	−855.629	−1.05503	−0.527515	−	0.849546i	$-0.676876\pi$
$811$	−855.629	−1.05503	−0.527515	+	0.849546i	$0.676876\pi$
$812$	− 2249.91i	− 2.77082i
$813$	0	0
$814$	−3466.83	−4.25901
$815$	699.946i	0.858829i
$816$	0	0
$817$	1999.35	2.44719
$818$	− 1307.20i	− 1.59805i
$819$	0	0
$820$	−2139.94	−2.60968
$821$	− 701.585i	− 0.854549i	−0.904122	−	0.427275i	$-0.859474\pi$
$821$	− 701.585i	− 0.854549i	0.904122	−	0.427275i	$-0.140526\pi$
$822$	0	0
$823$	−507.346	−0.616460	−0.308230	−	0.951312i	$-0.599737\pi$
$823$	−507.346	−0.616460	−0.308230	+	0.951312i	$0.599737\pi$
$824$	− 2755.03i	− 3.34348i
$825$	0	0
$826$	−243.471	−0.294760
$827$	− 1126.11i	− 1.36168i	−0.732434	−	0.680838i	$-0.761616\pi$
$827$	− 1126.11i	− 1.36168i	0.732434	−	0.680838i	$-0.238384\pi$
$828$	0	0
$829$	−1160.25	−1.39957	−0.699786	−	0.714352i	$-0.746721\pi$
$829$	−1160.25	−1.39957	−0.699786	+	0.714352i	$0.746721\pi$
$830$	− 3311.97i	− 3.99033i
$831$	0	0
$832$	−108.044	−0.129861
$833$	− 630.320i	− 0.756686i
$834$	0	0
$835$	1060.75	1.27036
$836$	− 3541.47i	− 4.23621i
$837$	0	0
$838$	1541.65	1.83967
$839$	607.665i	0.724272i	0.932125	+	0.362136i	$0.117952\pi$
$839$	607.665i	0.724272i	−0.932125	+	0.362136i	$0.882048\pi$
$840$	0	0
$841$	−2072.71	−2.46458
$842$	1873.59i	2.22517i
$843$	0	0
$844$	1866.34	2.21130
$845$	528.105i	0.624976i
$846$	0	0
$847$	−571.261	−0.674453
$848$	2389.89i	2.81826i
$849$	0	0
$850$	−2269.17	−2.66961
$851$	370.091i	0.434890i
$852$	0	0
$853$	−273.450	−0.320574	−0.160287	−	0.987070i	$-0.551242\pi$
$853$	−273.450	−0.320574	−0.160287	+	0.987070i	$0.551242\pi$
$854$	1327.48i	1.55442i
$855$	0	0
$856$	2763.94	3.22890
$857$	1326.64i	1.54800i	0.633184	+	0.774001i	$0.281747\pi$
$857$	1326.64i	1.54800i	−0.633184	+	0.774001i	$0.718253\pi$
$858$	0	0
$859$	251.982	0.293343	0.146672	−	0.989185i	$-0.453144\pi$
$859$	251.982	0.293343	0.146672	+	0.989185i	$0.453144\pi$
$860$	4959.70i	5.76709i
$861$	0	0
$862$	396.242	0.459677
$863$	− 456.453i	− 0.528915i	−0.964397	−	0.264457i	$-0.914807\pi$
$863$	− 456.453i	− 0.528915i	0.964397	−	0.264457i	$-0.0851929\pi$
$864$	0	0
$865$	−273.428	−0.316101
$866$	264.738i	0.305702i
$867$	0	0
$868$	−451.351	−0.519990
$869$	− 877.041i	− 1.00925i
$870$	0	0
$871$	184.133	0.211404
$872$	1015.29i	1.16432i
$873$	0	0
$874$	−548.937	−0.628075
$875$	65.5755i	0.0749434i
$876$	0	0
$877$	537.509	0.612896	0.306448	−	0.951887i	$-0.400860\pi$
$877$	537.509	0.612896	0.306448	+	0.951887i	$0.400860\pi$
$878$	− 1183.81i	− 1.34830i
$879$	0	0
$880$	3019.61	3.43137
$881$	1228.12i	1.39401i	0.717067	+	0.697004i	$0.245484\pi$
$881$	1228.12i	1.39401i	−0.717067	+	0.697004i	$0.754516\pi$
$882$	0	0
$883$	−57.2818	−0.0648718	−0.0324359	−	0.999474i	$-0.510326\pi$
$883$	−57.2818	−0.0648718	−0.0324359	+	0.999474i	$0.510326\pi$
$884$	2035.06i	2.30210i
$885$	0	0
$886$	1525.06	1.72128
$887$	1335.11i	1.50520i	0.658481	+	0.752598i	$0.271199\pi$
$887$	1335.11i	1.50520i	−0.658481	+	0.752598i	$0.728801\pi$
$888$	0	0
$889$	53.0776	0.0597048
$890$	1813.42i	2.03755i
$891$	0	0
$892$	−3472.71	−3.89317
$893$	1526.14i	1.70900i
$894$	0	0
$895$	1591.59	1.77831
$896$	− 694.320i	− 0.774910i
$897$	0	0
$898$	−536.687	−0.597647
$899$	584.516i	0.650184i
$900$	0	0
$901$	−2086.71	−2.31599
$902$	1872.27i	2.07568i
$903$	0	0
$904$	1407.20	1.55664
$905$	− 982.175i	− 1.08528i
$906$	0	0
$907$	−1726.30	−1.90331	−0.951654	−	0.307173i	$-0.900617\pi$
$907$	−1726.30	−1.90331	−0.951654	+	0.307173i	$0.900617\pi$
$908$	195.136i	0.214908i
$909$	0	0
$910$	1190.34	1.30806
$911$	1710.32i	1.87741i	0.344721	+	0.938705i	$0.387974\pi$
$911$	1710.32i	1.87741i	−0.344721	+	0.938705i	$0.612026\pi$
$912$	0	0
$913$	−1995.67	−2.18584
$914$	1522.85i	1.66614i
$915$	0	0
$916$	2329.35	2.54296
$917$	80.6840i	0.0879870i
$918$	0	0
$919$	−307.462	−0.334561	−0.167280	−	0.985909i	$-0.553499\pi$
$919$	−307.462	−0.334561	−0.167280	+	0.985909i	$0.553499\pi$
$920$	− 746.240i	− 0.811130i
$921$	0	0
$922$	−103.554	−0.112315
$923$	741.913i	0.803807i
$924$	0	0
$925$	1673.35	1.80902
$926$	− 1844.13i	− 1.99150i
$927$	0	0
$928$	1455.14	1.56804
$929$	1510.66i	1.62612i	0.582180	+	0.813060i	$0.302200\pi$
$929$	1510.66i	1.62612i	−0.582180	+	0.813060i	$0.697800\pi$
$930$	0	0
$931$	−689.438	−0.740535
$932$	− 1142.40i	− 1.22575i
$933$	0	0
$934$	1523.20	1.63083
$935$	2636.54i	2.81983i
$936$	0	0
$937$	−452.031	−0.482423	−0.241212	−	0.970473i	$-0.577545\pi$
$937$	−452.031	−0.482423	−0.241212	+	0.970473i	$0.577545\pi$
$938$	317.756i	0.338759i
$939$	0	0
$940$	−3785.82	−4.02747
$941$	− 706.234i	− 0.750515i	−0.926921	−	0.375257i	$-0.877554\pi$
$941$	− 706.234i	− 0.750515i	0.926921	−	0.375257i	$-0.122446\pi$
$942$	0	0
$943$	199.868	0.211949
$944$	− 388.200i	− 0.411229i
$945$	0	0
$946$	4339.31	4.58701
$947$	− 663.862i	− 0.701016i	−0.936560	−	0.350508i	$-0.886009\pi$
$947$	− 663.862i	− 0.701016i	0.936560	−	0.350508i	$-0.113991\pi$
$948$	0	0
$949$	446.361	0.470349
$950$	2481.99i	2.61262i
$951$	0	0
$952$	−1924.54	−2.02158
$953$	59.7064i	0.0626510i	0.999509	+	0.0313255i	$0.00997285\pi$
$953$	59.7064i	0.0626510i	−0.999509	+	0.0313255i	$0.990027\pi$
$954$	0	0
$955$	1461.55	1.53041
$956$	− 2558.05i	− 2.67579i
$957$	0	0
$958$	−1435.77	−1.49871
$959$	− 135.774i	− 0.141579i
$960$	0	0
$961$	−843.741	−0.877982
$962$	− 2179.01i	− 2.26509i
$963$	0	0
$964$	921.923	0.956351
$965$	− 1397.75i	− 1.44844i
$966$	0	0
$967$	1441.18	1.49036	0.745182	−	0.666861i	$-0.232363\pi$
$967$	1441.18	1.49036	0.745182	+	0.666861i	$0.232363\pi$
$968$	− 2108.60i	− 2.17831i
$969$	0	0
$970$	118.712	0.122383
$971$	− 942.274i	− 0.970416i	−0.874399	−	0.485208i	$-0.838744\pi$
$971$	− 942.274i	− 0.970416i	0.874399	−	0.485208i	$-0.161256\pi$
$972$	0	0
$973$	120.721	0.124071
$974$	2951.33i	3.03011i
$975$	0	0
$976$	−2116.58	−2.16863
$977$	1642.03i	1.68068i	0.542058	+	0.840341i	$0.317645\pi$
$977$	1642.03i	1.68068i	−0.542058	+	0.840341i	$0.682355\pi$
$978$	0	0
$979$	1092.70	1.11614
$980$	− 1710.26i	− 1.74516i
$981$	0	0
$982$	−925.345	−0.942307
$983$	704.163i	0.716341i	0.933656	+	0.358170i	$0.116599\pi$
$983$	704.163i	0.716341i	−0.933656	+	0.358170i	$0.883401\pi$
$984$	0	0
$985$	1922.05	1.95132
$986$	4548.00i	4.61258i
$987$	0	0
$988$	2225.92	2.25296
$989$	− 463.230i	− 0.468383i
$990$	0	0
$991$	−1524.19	−1.53803	−0.769015	−	0.639230i	$-0.779253\pi$
$991$	−1524.19	−1.53803	−0.769015	+	0.639230i	$0.779253\pi$
$992$	− 291.914i	− 0.294268i
$993$	0	0
$994$	−1280.31	−1.28804
$995$	− 1154.98i	− 1.16079i
$996$	0	0
$997$	−165.044	−0.165540	−0.0827702	−	0.996569i	$-0.526377\pi$
$997$	−165.044	−0.165540	−0.0827702	+	0.996569i	$0.526377\pi$
$998$	178.610i	0.178968i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.7	✓	84
3.2	odd	2	inner	1143.3.b.a.890.78	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.7	✓	84	1.1	even	1	trivial
1143.3.b.a.890.78	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)

\(f(q)\)	\(=\)	\(q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} +O(q^{10})\)	\(q-3.58466i q^{2} -8.84977 q^{4} -7.20639i q^{5} +4.70987 q^{7} +17.3848i q^{8} -25.8324 q^{10} +15.5657i q^{11} -9.78351 q^{13} -16.8833i q^{14} +26.9194 q^{16} +23.5044i q^{17} +25.7089 q^{19} +63.7749i q^{20} +55.7976 q^{22} -5.95651i q^{23} -26.9320 q^{25} +35.0705i q^{26} -41.6813 q^{28} +53.9788i q^{29} +10.8286 q^{31} -26.9577i q^{32} +84.2553 q^{34} -33.9412i q^{35} -62.1323 q^{37} -92.1576i q^{38} +125.281 q^{40} +33.5546i q^{41} +77.7688 q^{43} -137.753i q^{44} -21.3520 q^{46} +59.3622i q^{47} -26.8171 q^{49} +96.5421i q^{50} +86.5819 q^{52} +88.7794i q^{53} +112.172 q^{55} +81.8801i q^{56} +193.496 q^{58} -14.4209i q^{59} -78.6267 q^{61} -38.8169i q^{62} +11.0435 q^{64} +70.5038i q^{65} -18.8207 q^{67} -208.009i q^{68} -121.668 q^{70} -75.8330i q^{71} -45.6238 q^{73} +222.723i q^{74} -227.518 q^{76} +73.3124i q^{77} -56.3446 q^{79} -193.991i q^{80} +120.282 q^{82} +128.210i q^{83} +169.382 q^{85} -278.775i q^{86} -270.606 q^{88} -70.1992i q^{89} -46.0791 q^{91} +52.7137i q^{92} +212.793 q^{94} -185.268i q^{95} -4.59545 q^{97} +96.1301i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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