LMFDB - Embedded newform 1143.3.b.a.890.82

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			890.82
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.83653i q^{2} -10.7190 q^{4} -7.45843i q^{5} +0.844054 q^{7} -25.7776i q^{8} +O(q^{10})$	\(q+3.83653i q^{2} -10.7190 q^{4} -7.45843i q^{5} +0.844054 q^{7} -25.7776i q^{8} +28.6145 q^{10} -12.8528i q^{11} -4.71633 q^{13} +3.23824i q^{14} +56.0208 q^{16} -5.69167i q^{17} -12.0832 q^{19} +79.9469i q^{20} +49.3100 q^{22} +8.89247i q^{23} -30.6282 q^{25} -18.0944i q^{26} -9.04740 q^{28} +27.8216i q^{29} -26.6004 q^{31} +111.815i q^{32} +21.8363 q^{34} -6.29532i q^{35} -8.01864 q^{37} -46.3577i q^{38} -192.261 q^{40} +74.7809i q^{41} -6.24122 q^{43} +137.769i q^{44} -34.1163 q^{46} +31.6811i q^{47} -48.2876 q^{49} -117.506i q^{50} +50.5543 q^{52} -23.4701i q^{53} -95.8615 q^{55} -21.7577i q^{56} -106.739 q^{58} +82.7889i q^{59} -81.7300 q^{61} -102.053i q^{62} -204.900 q^{64} +35.1764i q^{65} +94.4999 q^{67} +61.0090i q^{68} +24.1522 q^{70} -137.250i q^{71} +104.478 q^{73} -30.7638i q^{74} +129.520 q^{76} -10.8484i q^{77} +44.4068 q^{79} -417.828i q^{80} -286.899 q^{82} +68.1282i q^{83} -42.4509 q^{85} -23.9447i q^{86} -331.314 q^{88} -38.7934i q^{89} -3.98084 q^{91} -95.3183i q^{92} -121.546 q^{94} +90.1219i q^{95} +101.020 q^{97} -185.257i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.83653i	1.91827i	0.282954	+	0.959133i	$0.408686\pi$
$2$	3.83653i	1.91827i	−0.282954	+	0.959133i	$0.591314\pi$
$3$	0	0
$3$	0	0
$4$	−10.7190	−2.67975
$5$	− 7.45843i	− 1.49169i	−0.666121	−	0.745843i	$-0.732047\pi$
$5$	− 7.45843i	− 1.49169i	0.666121	−	0.745843i	$-0.267953\pi$
$6$	0	0
$7$	0.844054	0.120579	0.0602895	−	0.998181i	$-0.480798\pi$
$7$	0.844054	0.120579	0.0602895	+	0.998181i	$0.480798\pi$
$8$	− 25.7776i	− 3.22220i
$9$	0	0
$10$	28.6145	2.86145
$11$	− 12.8528i	− 1.16843i	−0.811598	−	0.584216i	$-0.801402\pi$
$11$	− 12.8528i	− 1.16843i	0.811598	−	0.584216i	$-0.198598\pi$
$12$	0	0
$13$	−4.71633	−0.362795	−0.181397	−	0.983410i	$-0.558062\pi$
$13$	−4.71633	−0.362795	−0.181397	+	0.983410i	$0.558062\pi$
$14$	3.23824i	0.231303i
$15$	0	0
$16$	56.0208	3.50130
$17$	− 5.69167i	− 0.334804i	−0.985889	−	0.167402i	$-0.946462\pi$
$17$	− 5.69167i	− 0.334804i	0.985889	−	0.167402i	$-0.0535378\pi$
$18$	0	0
$19$	−12.0832	−0.635959	−0.317979	−	0.948098i	$-0.603004\pi$
$19$	−12.0832	−0.635959	−0.317979	+	0.948098i	$0.603004\pi$
$20$	79.9469i	3.99734i
$21$	0	0
$22$	49.3100	2.24137
$23$	8.89247i	0.386629i	0.981137	+	0.193315i	$0.0619238\pi$
$23$	8.89247i	0.386629i	−0.981137	+	0.193315i	$0.938076\pi$
$24$	0	0
$25$	−30.6282	−1.22513
$26$	− 18.0944i	− 0.695937i
$27$	0	0
$28$	−9.04740	−0.323122
$29$	27.8216i	0.959366i	0.877442	+	0.479683i	$0.159248\pi$
$29$	27.8216i	0.959366i	−0.877442	+	0.479683i	$0.840752\pi$
$30$	0	0
$31$	−26.6004	−0.858078	−0.429039	−	0.903286i	$-0.641148\pi$
$31$	−26.6004	−0.858078	−0.429039	+	0.903286i	$0.641148\pi$
$32$	111.815i	3.49423i
$33$	0	0
$34$	21.8363	0.642244
$35$	− 6.29532i	− 0.179866i
$36$	0	0
$37$	−8.01864	−0.216720	−0.108360	−	0.994112i	$-0.534560\pi$
$37$	−8.01864	−0.216720	−0.108360	+	0.994112i	$0.534560\pi$
$38$	− 46.3577i	− 1.21994i
$39$	0	0
$40$	−192.261	−4.80652
$41$	74.7809i	1.82392i	0.410274	+	0.911962i	$0.365433\pi$
$41$	74.7809i	1.82392i	−0.410274	+	0.911962i	$0.634567\pi$
$42$	0	0
$43$	−6.24122	−0.145145	−0.0725723	−	0.997363i	$-0.523121\pi$
$43$	−6.24122	−0.145145	−0.0725723	+	0.997363i	$0.523121\pi$
$44$	137.769i	3.13111i
$45$	0	0
$46$	−34.1163	−0.741658
$47$	31.6811i	0.674066i	0.941493	+	0.337033i	$0.109423\pi$
$47$	31.6811i	0.674066i	−0.941493	+	0.337033i	$0.890577\pi$
$48$	0	0
$49$	−48.2876	−0.985461
$50$	− 117.506i	− 2.35012i
$51$	0	0
$52$	50.5543	0.972198
$53$	− 23.4701i	− 0.442833i	−0.975179	−	0.221416i	$-0.928932\pi$
$53$	− 23.4701i	− 0.442833i	0.975179	−	0.221416i	$-0.0710680\pi$
$54$	0	0
$55$	−95.8615	−1.74294
$56$	− 21.7577i	− 0.388531i
$57$	0	0
$58$	−106.739	−1.84032
$59$	82.7889i	1.40320i	0.712570	+	0.701601i	$0.247531\pi$
$59$	82.7889i	1.40320i	−0.712570	+	0.701601i	$0.752469\pi$
$60$	0	0
$61$	−81.7300	−1.33984	−0.669918	−	0.742435i	$-0.733671\pi$
$61$	−81.7300	−1.33984	−0.669918	+	0.742435i	$0.733671\pi$
$62$	− 102.053i	− 1.64602i
$63$	0	0
$64$	−204.900	−3.20156
$65$	35.1764i	0.541176i
$66$	0	0
$67$	94.4999	1.41045	0.705223	−	0.708985i	$-0.250847\pi$
$67$	94.4999	1.41045	0.705223	+	0.708985i	$0.250847\pi$
$68$	61.0090i	0.897191i
$69$	0	0
$70$	24.1522	0.345031
$71$	− 137.250i	− 1.93310i	−0.256483	−	0.966549i	$-0.582564\pi$
$71$	− 137.250i	− 1.93310i	0.256483	−	0.966549i	$-0.417436\pi$
$72$	0	0
$73$	104.478	1.43121	0.715605	−	0.698505i	$-0.246151\pi$
$73$	104.478	1.43121	0.715605	+	0.698505i	$0.246151\pi$
$74$	− 30.7638i	− 0.415727i
$75$	0	0
$76$	129.520	1.70421
$77$	− 10.8484i	− 0.140889i
$78$	0	0
$79$	44.4068	0.562112	0.281056	−	0.959691i	$-0.409315\pi$
$79$	44.4068	0.562112	0.281056	+	0.959691i	$0.409315\pi$
$80$	− 417.828i	− 5.22284i
$81$	0	0
$82$	−286.899	−3.49877
$83$	68.1282i	0.820822i	0.911901	+	0.410411i	$0.134615\pi$
$83$	68.1282i	0.820822i	−0.911901	+	0.410411i	$0.865385\pi$
$84$	0	0
$85$	−42.4509	−0.499423
$86$	− 23.9447i	− 0.278426i
$87$	0	0
$88$	−331.314	−3.76493
$89$	− 38.7934i	− 0.435881i	−0.975962	−	0.217941i	$-0.930066\pi$
$89$	− 38.7934i	− 0.435881i	0.975962	−	0.217941i	$-0.0699339\pi$
$90$	0	0
$91$	−3.98084	−0.0437454
$92$	− 95.3183i	− 1.03607i
$93$	0	0
$94$	−121.546	−1.29304
$95$	90.1219i	0.948651i
$96$	0	0
$97$	101.020	1.04144	0.520722	−	0.853726i	$-0.325663\pi$
$97$	101.020	1.04144	0.520722	+	0.853726i	$0.325663\pi$
$98$	− 185.257i	− 1.89038i
$99$	0	0
$100$	328.304	3.28304
$101$	66.5209i	0.658622i	0.944221	+	0.329311i	$0.106817\pi$
$101$	66.5209i	0.658622i	−0.944221	+	0.329311i	$0.893183\pi$
$102$	0	0
$103$	−156.271	−1.51719	−0.758595	−	0.651562i	$-0.774114\pi$
$103$	−156.271	−1.51719	−0.758595	+	0.651562i	$0.774114\pi$
$104$	121.576i	1.16900i
$105$	0	0
$106$	90.0440	0.849471
$107$	133.785i	1.25033i	0.780493	+	0.625164i	$0.214968\pi$
$107$	133.785i	1.25033i	−0.780493	+	0.625164i	$0.785032\pi$
$108$	0	0
$109$	−134.500	−1.23395	−0.616974	−	0.786983i	$-0.711642\pi$
$109$	−134.500	−1.23395	−0.616974	+	0.786983i	$0.711642\pi$
$110$	− 367.776i	− 3.34342i
$111$	0	0
$112$	47.2846	0.422184
$113$	137.454i	1.21641i	0.793781	+	0.608203i	$0.208109\pi$
$113$	137.454i	1.21641i	−0.793781	+	0.608203i	$0.791891\pi$
$114$	0	0
$115$	66.3239	0.576729
$116$	− 298.220i	− 2.57086i
$117$	0	0
$118$	−317.622	−2.69171
$119$	− 4.80408i	− 0.0403704i
$120$	0	0
$121$	−44.1934	−0.365235
$122$	− 313.560i	− 2.57016i
$123$	0	0
$124$	285.130	2.29943
$125$	41.9777i	0.335822i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	− 338.843i	− 2.64721i
$129$	0	0
$130$	−134.956	−1.03812
$131$	− 193.561i	− 1.47756i	−0.673946	−	0.738781i	$-0.735402\pi$
$131$	− 193.561i	− 1.47756i	0.673946	−	0.738781i	$-0.264598\pi$
$132$	0	0
$133$	−10.1989	−0.0766834
$134$	362.552i	2.70561i
$135$	0	0
$136$	−146.718	−1.07881
$137$	− 117.539i	− 0.857947i	−0.903317	−	0.428973i	$-0.858875\pi$
$137$	− 117.539i	− 0.857947i	0.903317	−	0.428973i	$-0.141125\pi$
$138$	0	0
$139$	−139.824	−1.00593	−0.502964	−	0.864307i	$-0.667757\pi$
$139$	−139.824	−1.00593	−0.502964	+	0.864307i	$0.667757\pi$
$140$	67.4795i	0.481996i
$141$	0	0
$142$	526.564	3.70820
$143$	60.6179i	0.423901i
$144$	0	0
$145$	207.506	1.43107
$146$	400.835i	2.74544i
$147$	0	0
$148$	85.9518	0.580755
$149$	− 6.83029i	− 0.0458409i	−0.999737	−	0.0229204i	$-0.992704\pi$
$149$	− 6.83029i	− 0.0458409i	0.999737	−	0.0229204i	$-0.00729644\pi$
$150$	0	0
$151$	142.106	0.941096	0.470548	−	0.882374i	$-0.344056\pi$
$151$	142.106	0.941096	0.470548	+	0.882374i	$0.344056\pi$
$152$	311.477i	2.04919i
$153$	0	0
$154$	41.6203	0.270262
$155$	198.397i	1.27998i
$156$	0	0
$157$	129.637	0.825717	0.412858	−	0.910795i	$-0.364530\pi$
$157$	129.637	0.825717	0.412858	+	0.910795i	$0.364530\pi$
$158$	170.368i	1.07828i
$159$	0	0
$160$	833.966	5.21229
$161$	7.50572i	0.0466194i
$162$	0	0
$163$	−28.9706	−0.177734	−0.0888670	−	0.996044i	$-0.528325\pi$
$163$	−28.9706	−0.177734	−0.0888670	+	0.996044i	$0.528325\pi$
$164$	− 801.576i	− 4.88766i
$165$	0	0
$166$	−261.376	−1.57456
$167$	139.833i	0.837324i	0.908142	+	0.418662i	$0.137501\pi$
$167$	139.833i	0.837324i	−0.908142	+	0.418662i	$0.862499\pi$
$168$	0	0
$169$	−146.756	−0.868380
$170$	− 162.864i	− 0.958026i
$171$	0	0
$172$	66.8996	0.388951
$173$	152.490i	0.881443i	0.897644	+	0.440721i	$0.145277\pi$
$173$	152.490i	0.881443i	−0.897644	+	0.440721i	$0.854723\pi$
$174$	0	0
$175$	−25.8519	−0.147725
$176$	− 720.022i	− 4.09104i
$177$	0	0
$178$	148.832	0.836137
$179$	301.532i	1.68454i	0.539058	+	0.842268i	$0.318780\pi$
$179$	301.532i	1.68454i	−0.539058	+	0.842268i	$0.681220\pi$
$180$	0	0
$181$	−256.150	−1.41519	−0.707597	−	0.706617i	$-0.750221\pi$
$181$	−256.150	−1.41519	−0.707597	+	0.706617i	$0.750221\pi$
$182$	− 15.2726i	− 0.0839154i
$183$	0	0
$184$	229.227	1.24580
$185$	59.8065i	0.323278i
$186$	0	0
$187$	−73.1537	−0.391196
$188$	− 339.589i	− 1.80633i
$189$	0	0
$190$	−345.756	−1.81977
$191$	186.541i	0.976656i	0.872660	+	0.488328i	$0.162393\pi$
$191$	186.541i	0.976656i	−0.872660	+	0.488328i	$0.837607\pi$
$192$	0	0
$193$	−2.32954	−0.0120701	−0.00603507	−	0.999982i	$-0.501921\pi$
$193$	−2.32954	−0.0120701	−0.00603507	+	0.999982i	$0.501921\pi$
$194$	387.567i	1.99777i
$195$	0	0
$196$	517.594	2.64079
$197$	220.708i	1.12035i	0.828376	+	0.560173i	$0.189265\pi$
$197$	220.708i	1.12035i	−0.828376	+	0.560173i	$0.810735\pi$
$198$	0	0
$199$	−266.198	−1.33768	−0.668840	−	0.743406i	$-0.733209\pi$
$199$	−266.198	−1.33768	−0.668840	+	0.743406i	$0.733209\pi$
$200$	789.523i	3.94762i
$201$	0	0
$202$	−255.210	−1.26341
$203$	23.4829i	0.115679i
$204$	0	0
$205$	557.748	2.72072
$206$	− 599.537i	− 2.91038i
$207$	0	0
$208$	−264.213	−1.27025
$209$	155.303i	0.743075i
$210$	0	0
$211$	−196.024	−0.929022	−0.464511	−	0.885567i	$-0.653770\pi$
$211$	−196.024	−0.929022	−0.464511	+	0.885567i	$0.653770\pi$
$212$	251.576i	1.18668i
$213$	0	0
$214$	−513.271	−2.39846
$215$	46.5497i	0.216510i
$216$	0	0
$217$	−22.4522	−0.103466
$218$	− 516.015i	− 2.36704i
$219$	0	0
$220$	1027.54	4.67063
$221$	26.8438i	0.121465i
$222$	0	0
$223$	152.579	0.684211	0.342105	−	0.939662i	$-0.388860\pi$
$223$	152.579	0.684211	0.342105	+	0.939662i	$0.388860\pi$
$224$	94.3780i	0.421331i
$225$	0	0
$226$	−527.347	−2.33339
$227$	171.350i	0.754846i	0.926041	+	0.377423i	$0.123190\pi$
$227$	171.350i	0.754846i	−0.926041	+	0.377423i	$0.876810\pi$
$228$	0	0
$229$	267.113	1.16643	0.583217	−	0.812316i	$-0.301794\pi$
$229$	267.113	1.16643	0.583217	+	0.812316i	$0.301794\pi$
$230$	254.454i	1.10632i
$231$	0	0
$232$	717.175	3.09127
$233$	− 321.626i	− 1.38037i	−0.723633	−	0.690185i	$-0.757529\pi$
$233$	− 321.626i	− 1.38037i	0.723633	−	0.690185i	$-0.242471\pi$
$234$	0	0
$235$	236.291	1.00550
$236$	− 887.413i	− 3.76023i
$237$	0	0
$238$	18.4310	0.0774412
$239$	− 387.078i	− 1.61957i	−0.586725	−	0.809786i	$-0.699583\pi$
$239$	− 387.078i	− 1.61957i	0.586725	−	0.809786i	$-0.300417\pi$
$240$	0	0
$241$	−217.992	−0.904533	−0.452266	−	0.891883i	$-0.649384\pi$
$241$	−217.992	−0.904533	−0.452266	+	0.891883i	$0.649384\pi$
$242$	− 169.550i	− 0.700618i
$243$	0	0
$244$	876.063	3.59042
$245$	360.150i	1.47000i
$246$	0	0
$247$	56.9885	0.230722
$248$	685.696i	2.76490i
$249$	0	0
$250$	−161.049	−0.644196
$251$	− 341.604i	− 1.36097i	−0.732762	−	0.680485i	$-0.761769\pi$
$251$	− 341.604i	− 1.36097i	0.732762	−	0.680485i	$-0.238231\pi$
$252$	0	0
$253$	114.293	0.451750
$254$	− 43.2355i	− 0.170219i
$255$	0	0
$256$	480.385	1.87651
$257$	− 472.336i	− 1.83789i	−0.394391	−	0.918943i	$-0.629045\pi$
$257$	− 472.336i	− 1.83789i	0.394391	−	0.918943i	$-0.370955\pi$
$258$	0	0
$259$	−6.76816	−0.0261319
$260$	− 377.056i	− 1.45021i
$261$	0	0
$262$	742.602	2.83436
$263$	146.715i	0.557851i	0.960313	+	0.278926i	$0.0899783\pi$
$263$	146.715i	0.557851i	−0.960313	+	0.278926i	$0.910022\pi$
$264$	0	0
$265$	−175.050	−0.660568
$266$	− 39.1284i	− 0.147099i
$267$	0	0
$268$	−1012.94	−3.77964
$269$	− 16.6908i	− 0.0620474i	−0.999519	−	0.0310237i	$-0.990123\pi$
$269$	− 16.6908i	− 0.0620474i	0.999519	−	0.0310237i	$-0.00987674\pi$
$270$	0	0
$271$	−120.702	−0.445396	−0.222698	−	0.974888i	$-0.571486\pi$
$271$	−120.702	−0.445396	−0.222698	+	0.974888i	$0.571486\pi$
$272$	− 318.852i	− 1.17225i
$273$	0	0
$274$	450.941	1.64577
$275$	393.657i	1.43148i
$276$	0	0
$277$	−488.270	−1.76271	−0.881355	−	0.472456i	$-0.843368\pi$
$277$	−488.270	−1.76271	−0.881355	+	0.472456i	$0.843368\pi$
$278$	− 536.440i	− 1.92964i
$279$	0	0
$280$	−162.278	−0.579566
$281$	− 329.738i	− 1.17345i	−0.809788	−	0.586723i	$-0.800418\pi$
$281$	− 329.738i	− 1.17345i	0.809788	−	0.586723i	$-0.199582\pi$
$282$	0	0
$283$	−452.880	−1.60028	−0.800141	−	0.599812i	$-0.795242\pi$
$283$	−452.880	−1.60028	−0.800141	+	0.599812i	$0.795242\pi$
$284$	1471.18i	5.18021i
$285$	0	0
$286$	−232.562	−0.813155
$287$	63.1191i	0.219927i
$288$	0	0
$289$	256.605	0.887906
$290$	796.102i	2.74518i
$291$	0	0
$292$	−1119.90	−3.83528
$293$	− 95.4461i	− 0.325755i	−0.986646	−	0.162877i	$-0.947923\pi$
$293$	− 95.4461i	− 0.325755i	0.986646	−	0.162877i	$-0.0520775\pi$
$294$	0	0
$295$	617.475	2.09314
$296$	206.702i	0.698316i
$297$	0	0
$298$	26.2046	0.0879351
$299$	− 41.9398i	− 0.140267i
$300$	0	0
$301$	−5.26793	−0.0175014
$302$	545.193i	1.80527i
$303$	0	0
$304$	−676.912	−2.22668
$305$	609.578i	1.99862i
$306$	0	0
$307$	−274.354	−0.893663	−0.446831	−	0.894618i	$-0.647448\pi$
$307$	−274.354	−0.893663	−0.446831	+	0.894618i	$0.647448\pi$
$308$	116.284i	0.377546i
$309$	0	0
$310$	−761.158	−2.45535
$311$	244.443i	0.785989i	0.919541	+	0.392995i	$0.128561\pi$
$311$	244.443i	0.785989i	−0.919541	+	0.392995i	$0.871439\pi$
$312$	0	0
$313$	−200.326	−0.640021	−0.320010	−	0.947414i	$-0.603686\pi$
$313$	−200.326	−0.640021	−0.320010	+	0.947414i	$0.603686\pi$
$314$	497.359i	1.58394i
$315$	0	0
$316$	−475.997	−1.50632
$317$	− 490.938i	− 1.54870i	−0.632758	−	0.774350i	$-0.718077\pi$
$317$	− 490.938i	− 1.54870i	0.632758	−	0.774350i	$-0.281923\pi$
$318$	0	0
$319$	357.584	1.12095
$320$	1528.23i	4.77572i
$321$	0	0
$322$	−28.7960	−0.0894284
$323$	68.7737i	0.212922i
$324$	0	0
$325$	144.453	0.444470
$326$	− 111.147i	− 0.340941i
$327$	0	0
$328$	1927.68	5.87706
$329$	26.7405i	0.0812783i
$330$	0	0
$331$	63.0543	0.190496	0.0952482	−	0.995454i	$-0.469636\pi$
$331$	63.0543	0.190496	0.0952482	+	0.995454i	$0.469636\pi$
$332$	− 730.266i	− 2.19960i
$333$	0	0
$334$	−536.474	−1.60621
$335$	− 704.821i	− 2.10394i
$336$	0	0
$337$	−64.9287	−0.192667	−0.0963333	−	0.995349i	$-0.530711\pi$
$337$	−64.9287	−0.192667	−0.0963333	+	0.995349i	$0.530711\pi$
$338$	− 563.035i	− 1.66578i
$339$	0	0
$340$	455.031	1.33833
$341$	341.889i	1.00261i
$342$	0	0
$343$	−82.1159	−0.239405
$344$	160.884i	0.467686i
$345$	0	0
$346$	−585.032	−1.69084
$347$	157.767i	0.454659i	0.973818	+	0.227330i	$0.0729995\pi$
$347$	157.767i	0.454659i	−0.973818	+	0.227330i	$0.927000\pi$
$348$	0	0
$349$	348.778	0.999365	0.499682	−	0.866209i	$-0.333450\pi$
$349$	348.778	0.999365	0.499682	+	0.866209i	$0.333450\pi$
$350$	− 99.1816i	− 0.283376i
$351$	0	0
$352$	1437.13	4.08277
$353$	− 361.738i	− 1.02475i	−0.858761	−	0.512377i	$-0.828765\pi$
$353$	− 361.738i	− 1.02475i	0.858761	−	0.512377i	$-0.171235\pi$
$354$	0	0
$355$	−1023.67	−2.88358
$356$	415.827i	1.16805i
$357$	0	0
$358$	−1156.84	−3.23139
$359$	440.805i	1.22787i	0.789357	+	0.613934i	$0.210414\pi$
$359$	440.805i	1.22787i	−0.789357	+	0.613934i	$0.789586\pi$
$360$	0	0
$361$	−214.996	−0.595556
$362$	− 982.728i	− 2.71472i
$363$	0	0
$364$	42.6705	0.117227
$365$	− 779.245i	− 2.13492i
$366$	0	0
$367$	−21.4041	−0.0583219	−0.0291609	−	0.999575i	$-0.509284\pi$
$367$	−21.4041	−0.0583219	−0.0291609	+	0.999575i	$0.509284\pi$
$368$	498.163i	1.35371i
$369$	0	0
$370$	−229.450	−0.620134
$371$	− 19.8101i	− 0.0533964i
$372$	0	0
$373$	−573.207	−1.53675	−0.768374	−	0.640001i	$-0.778934\pi$
$373$	−573.207	−1.53675	−0.768374	+	0.640001i	$0.778934\pi$
$374$	− 280.657i	− 0.750419i
$375$	0	0
$376$	816.664	2.17198
$377$	− 131.216i	− 0.348053i
$378$	0	0
$379$	−469.913	−1.23988	−0.619938	−	0.784651i	$-0.712842\pi$
$379$	−469.913	−1.23988	−0.619938	+	0.784651i	$0.712842\pi$
$380$	− 966.016i	− 2.54215i
$381$	0	0
$382$	−715.672	−1.87349
$383$	116.747i	0.304821i	0.988317	+	0.152411i	$0.0487036\pi$
$383$	116.747i	0.304821i	−0.988317	+	0.152411i	$0.951296\pi$
$384$	0	0
$385$	−80.9122	−0.210162
$386$	− 8.93735i	− 0.0231538i
$387$	0	0
$388$	−1082.83	−2.79081
$389$	357.982i	0.920261i	0.887851	+	0.460130i	$0.152197\pi$
$389$	357.982i	0.920261i	−0.887851	+	0.460130i	$0.847803\pi$
$390$	0	0
$391$	50.6130	0.129445
$392$	1244.74i	3.17536i
$393$	0	0
$394$	−846.755	−2.14912
$395$	− 331.206i	− 0.838495i
$396$	0	0
$397$	327.680	0.825389	0.412695	−	0.910869i	$-0.364588\pi$
$397$	327.680	0.825389	0.412695	+	0.910869i	$0.364588\pi$
$398$	− 1021.28i	− 2.56603i
$399$	0	0
$400$	−1715.82	−4.28955
$401$	− 326.702i	− 0.814717i	−0.913268	−	0.407359i	$-0.866450\pi$
$401$	− 326.702i	− 0.814717i	0.913268	−	0.407359i	$-0.133550\pi$
$402$	0	0
$403$	125.456	0.311306
$404$	− 713.037i	− 1.76494i
$405$	0	0
$406$	−90.0931	−0.221904
$407$	103.062i	0.253223i
$408$	0	0
$409$	431.498	1.05501	0.527504	−	0.849553i	$-0.323128\pi$
$409$	431.498	1.05501	0.527504	+	0.849553i	$0.323128\pi$
$410$	2139.82i	5.21907i
$411$	0	0
$412$	1675.06	4.06569
$413$	69.8783i	0.169197i
$414$	0	0
$415$	508.130	1.22441
$416$	− 527.357i	− 1.26769i
$417$	0	0
$418$	−595.824	−1.42542
$419$	440.104i	1.05037i	0.850989	+	0.525184i	$0.176004\pi$
$419$	440.104i	1.05037i	−0.850989	+	0.525184i	$0.823996\pi$
$420$	0	0
$421$	−192.973	−0.458369	−0.229184	−	0.973383i	$-0.573606\pi$
$421$	−192.973	−0.458369	−0.229184	+	0.973383i	$0.573606\pi$
$422$	− 752.051i	− 1.78211i
$423$	0	0
$424$	−605.005	−1.42690
$425$	174.326i	0.410178i
$426$	0	0
$427$	−68.9845	−0.161556
$428$	− 1434.04i	− 3.35057i
$429$	0	0
$430$	−178.590	−0.415325
$431$	186.224i	0.432074i	0.976385	+	0.216037i	$0.0693132\pi$
$431$	186.224i	0.432074i	−0.976385	+	0.216037i	$0.930687\pi$
$432$	0	0
$433$	835.214	1.92890	0.964450	−	0.264266i	$-0.0851296\pi$
$433$	835.214	1.92890	0.964450	+	0.264266i	$0.0851296\pi$
$434$	− 86.1385i	− 0.198476i
$435$	0	0
$436$	1441.71	3.30667
$437$	− 107.450i	− 0.245880i
$438$	0	0
$439$	65.1154	0.148327	0.0741633	−	0.997246i	$-0.476371\pi$
$439$	65.1154	0.148327	0.0741633	+	0.997246i	$0.476371\pi$
$440$	2471.08i	5.61610i
$441$	0	0
$442$	−102.987	−0.233003
$443$	146.097i	0.329790i	0.986311	+	0.164895i	$0.0527286\pi$
$443$	146.097i	0.329790i	−0.986311	+	0.164895i	$0.947271\pi$
$444$	0	0
$445$	−289.338	−0.650198
$446$	585.375i	1.31250i
$447$	0	0
$448$	−172.946	−0.386041
$449$	415.030i	0.924342i	0.886791	+	0.462171i	$0.152929\pi$
$449$	415.030i	0.924342i	−0.886791	+	0.462171i	$0.847071\pi$
$450$	0	0
$451$	961.141	2.13113
$452$	− 1473.37i	− 3.25966i
$453$	0	0
$454$	−657.391	−1.44800
$455$	29.6908i	0.0652545i
$456$	0	0
$457$	−600.629	−1.31429	−0.657143	−	0.753766i	$-0.728235\pi$
$457$	−600.629	−1.31429	−0.657143	+	0.753766i	$0.728235\pi$
$458$	1024.79i	2.23753i
$459$	0	0
$460$	−710.925	−1.54549
$461$	623.725i	1.35298i	0.736451	+	0.676491i	$0.236500\pi$
$461$	623.725i	1.35298i	−0.736451	+	0.676491i	$0.763500\pi$
$462$	0	0
$463$	−242.413	−0.523570	−0.261785	−	0.965126i	$-0.584311\pi$
$463$	−242.413	−0.523570	−0.261785	+	0.965126i	$0.584311\pi$
$464$	1558.59i	3.35903i
$465$	0	0
$466$	1233.93	2.64792
$467$	− 394.124i	− 0.843948i	−0.906608	−	0.421974i	$-0.861337\pi$
$467$	− 394.124i	− 0.843948i	0.906608	−	0.421974i	$-0.138663\pi$
$468$	0	0
$469$	79.7630	0.170070
$470$	906.540i	1.92881i
$471$	0	0
$472$	2134.10	4.52140
$473$	80.2169i	0.169592i
$474$	0	0
$475$	370.088	0.779132
$476$	51.4948i	0.108182i
$477$	0	0
$478$	1485.04	3.10677
$479$	560.196i	1.16951i	0.811210	+	0.584755i	$0.198809\pi$
$479$	560.196i	1.16951i	−0.811210	+	0.584755i	$0.801191\pi$
$480$	0	0
$481$	37.8186	0.0786249
$482$	− 836.335i	− 1.73514i
$483$	0	0
$484$	473.709	0.978738
$485$	− 753.451i	− 1.55351i
$486$	0	0
$487$	473.772	0.972837	0.486418	−	0.873726i	$-0.338303\pi$
$487$	473.772	0.972837	0.486418	+	0.873726i	$0.338303\pi$
$488$	2106.81i	4.31723i
$489$	0	0
$490$	−1381.73	−2.81985
$491$	− 599.215i	− 1.22040i	−0.792249	−	0.610198i	$-0.791090\pi$
$491$	− 599.215i	− 1.22040i	0.792249	−	0.610198i	$-0.208910\pi$
$492$	0	0
$493$	158.351	0.321200
$494$	218.638i	0.442587i
$495$	0	0
$496$	−1490.18	−3.00439
$497$	− 115.846i	− 0.233091i
$498$	0	0
$499$	−799.002	−1.60121	−0.800603	−	0.599195i	$-0.795487\pi$
$499$	−799.002	−1.60121	−0.800603	+	0.599195i	$0.795487\pi$
$500$	− 449.959i	− 0.899918i
$501$	0	0
$502$	1310.57	2.61071
$503$	613.592i	1.21986i	0.792454	+	0.609932i	$0.208803\pi$
$503$	613.592i	1.21986i	−0.792454	+	0.609932i	$0.791197\pi$
$504$	0	0
$505$	496.141	0.982458
$506$	438.488i	0.866577i
$507$	0	0
$508$	120.797	0.237789
$509$	178.490i	0.350667i	0.984509	+	0.175334i	$0.0561004\pi$
$509$	178.490i	0.350667i	−0.984509	+	0.175334i	$0.943900\pi$
$510$	0	0
$511$	88.1854	0.172574
$512$	487.642i	0.952425i
$513$	0	0
$514$	1812.13	3.52555
$515$	1165.53i	2.26317i
$516$	0	0
$517$	407.190	0.787601
$518$	− 25.9663i	− 0.0501280i
$519$	0	0
$520$	906.765	1.74378
$521$	824.524i	1.58258i	0.611441	+	0.791290i	$0.290590\pi$
$521$	824.524i	1.58258i	−0.611441	+	0.791290i	$0.709410\pi$
$522$	0	0
$523$	−35.4452	−0.0677729	−0.0338864	−	0.999426i	$-0.510788\pi$
$523$	−35.4452	−0.0677729	−0.0338864	+	0.999426i	$0.510788\pi$
$524$	2074.77i	3.95949i
$525$	0	0
$526$	−562.877	−1.07011
$527$	151.401i	0.287288i
$528$	0	0
$529$	449.924	0.850518
$530$	− 671.587i	− 1.26714i
$531$	0	0
$532$	109.322	0.205492
$533$	− 352.691i	− 0.661710i
$534$	0	0
$535$	997.828	1.86510
$536$	− 2435.98i	− 4.54475i
$537$	0	0
$538$	64.0347	0.119024
$539$	620.629i	1.15144i
$540$	0	0
$541$	277.739	0.513381	0.256691	−	0.966494i	$-0.417368\pi$
$541$	277.739	0.513381	0.256691	+	0.966494i	$0.417368\pi$
$542$	− 463.078i	− 0.854388i
$543$	0	0
$544$	636.415	1.16988
$545$	1003.16i	1.84066i
$546$	0	0
$547$	−746.686	−1.36506	−0.682529	−	0.730859i	$-0.739120\pi$
$547$	−746.686	−1.36506	−0.682529	+	0.730859i	$0.739120\pi$
$548$	1259.90i	2.29908i
$549$	0	0
$550$	−1510.28	−2.74596
$551$	− 336.175i	− 0.610117i
$552$	0	0
$553$	37.4818	0.0677790
$554$	− 1873.27i	− 3.38135i
$555$	0	0
$556$	1498.77	2.69563
$557$	− 702.305i	− 1.26087i	−0.776241	−	0.630436i	$-0.782876\pi$
$557$	− 702.305i	− 1.26087i	0.776241	−	0.630436i	$-0.217124\pi$
$558$	0	0
$559$	29.4357	0.0526577
$560$	− 352.669i	− 0.629766i
$561$	0	0
$562$	1265.05	2.25098
$563$	403.982i	0.717553i	0.933424	+	0.358776i	$0.116806\pi$
$563$	403.982i	0.717553i	−0.933424	+	0.358776i	$0.883194\pi$
$564$	0	0
$565$	1025.19	1.81450
$566$	− 1737.49i	− 3.06977i
$567$	0	0
$568$	−3537.98	−6.22884
$569$	− 597.989i	− 1.05095i	−0.850810	−	0.525474i	$-0.823888\pi$
$569$	− 597.989i	− 1.05095i	0.850810	−	0.525474i	$-0.176112\pi$
$570$	0	0
$571$	498.679	0.873343	0.436672	−	0.899621i	$-0.356157\pi$
$571$	498.679	0.873343	0.436672	+	0.899621i	$0.356157\pi$
$572$	− 649.762i	− 1.13595i
$573$	0	0
$574$	−242.159	−0.421879
$575$	− 272.361i	− 0.473671i
$576$	0	0
$577$	704.908	1.22168	0.610839	−	0.791755i	$-0.290832\pi$
$577$	704.908	1.22168	0.610839	+	0.791755i	$0.290832\pi$
$578$	984.473i	1.70324i
$579$	0	0
$580$	−2224.25	−3.83492
$581$	57.5039i	0.0989740i
$582$	0	0
$583$	−301.656	−0.517420
$584$	− 2693.21i	− 4.61165i
$585$	0	0
$586$	366.182	0.624884
$587$	685.573i	1.16793i	0.811780	+	0.583964i	$0.198499\pi$
$587$	685.573i	1.16793i	−0.811780	+	0.583964i	$0.801501\pi$
$588$	0	0
$589$	321.419	0.545702
$590$	2368.97i	4.01520i
$591$	0	0
$592$	−449.211	−0.758802
$593$	− 265.529i	− 0.447773i	−0.974615	−	0.223886i	$-0.928126\pi$
$593$	− 265.529i	− 0.447773i	0.974615	−	0.223886i	$-0.0718745\pi$
$594$	0	0
$595$	−35.8309	−0.0602200
$596$	73.2139i	0.122842i
$597$	0	0
$598$	160.904	0.269069
$599$	− 667.031i	− 1.11357i	−0.830655	−	0.556787i	$-0.812034\pi$
$599$	− 667.031i	− 1.11357i	0.830655	−	0.556787i	$-0.187966\pi$
$600$	0	0
$601$	136.316	0.226816	0.113408	−	0.993549i	$-0.463823\pi$
$601$	136.316	0.226816	0.113408	+	0.993549i	$0.463823\pi$
$602$	− 20.2106i	− 0.0335724i
$603$	0	0
$604$	−1523.23	−2.52190
$605$	329.614i	0.544816i
$606$	0	0
$607$	−883.875	−1.45614	−0.728068	−	0.685504i	$-0.759582\pi$
$607$	−883.875	−1.45614	−0.728068	+	0.685504i	$0.759582\pi$
$608$	− 1351.09i	− 2.22218i
$609$	0	0
$610$	−2338.67	−3.83388
$611$	− 149.419i	− 0.244548i
$612$	0	0
$613$	−343.773	−0.560805	−0.280403	−	0.959883i	$-0.590468\pi$
$613$	−343.773	−0.560805	−0.280403	+	0.959883i	$0.590468\pi$
$614$	− 1052.57i	− 1.71428i
$615$	0	0
$616$	−279.647	−0.453972
$617$	− 556.940i	− 0.902657i	−0.892358	−	0.451329i	$-0.850950\pi$
$617$	− 556.940i	− 0.902657i	0.892358	−	0.451329i	$-0.149050\pi$
$618$	0	0
$619$	600.216	0.969654	0.484827	−	0.874610i	$-0.338882\pi$
$619$	600.216	0.969654	0.484827	+	0.874610i	$0.338882\pi$
$620$	− 2126.62i	− 3.43003i
$621$	0	0
$622$	−937.812	−1.50774
$623$	− 32.7437i	− 0.0525582i
$624$	0	0
$625$	−452.617	−0.724188
$626$	− 768.559i	− 1.22773i
$627$	0	0
$628$	−1389.58	−2.21271
$629$	45.6395i	0.0725588i
$630$	0	0
$631$	−293.725	−0.465491	−0.232746	−	0.972538i	$-0.574771\pi$
$631$	−293.725	−0.465491	−0.232746	+	0.972538i	$0.574771\pi$
$632$	− 1144.70i	− 1.81124i
$633$	0	0
$634$	1883.50	2.97082
$635$	84.0523i	0.132366i
$636$	0	0
$637$	227.740	0.357520
$638$	1371.88i	2.15029i
$639$	0	0
$640$	−2527.24	−3.94881
$641$	− 520.355i	− 0.811786i	−0.913921	−	0.405893i	$-0.866961\pi$
$641$	− 520.355i	− 0.811786i	0.913921	−	0.405893i	$-0.133039\pi$
$642$	0	0
$643$	786.529	1.22322	0.611609	−	0.791160i	$-0.290523\pi$
$643$	786.529	1.22322	0.611609	+	0.791160i	$0.290523\pi$
$644$	− 80.4538i	− 0.124928i
$645$	0	0
$646$	−263.853	−0.408441
$647$	628.680i	0.971685i	0.874046	+	0.485842i	$0.161487\pi$
$647$	628.680i	0.971685i	−0.874046	+	0.485842i	$0.838513\pi$
$648$	0	0
$649$	1064.07	1.63955
$650$	554.198i	0.852613i
$651$	0	0
$652$	310.536	0.476282
$653$	− 703.238i	− 1.07693i	−0.842646	−	0.538467i	$-0.819004\pi$
$653$	− 703.238i	− 1.07693i	0.842646	−	0.538467i	$-0.180996\pi$
$654$	0	0
$655$	−1443.66	−2.20406
$656$	4189.29i	6.38611i
$657$	0	0
$658$	−102.591	−0.155913
$659$	930.433i	1.41189i	0.708268	+	0.705943i	$0.249477\pi$
$659$	930.433i	1.41189i	−0.708268	+	0.705943i	$0.750523\pi$
$660$	0	0
$661$	−590.748	−0.893718	−0.446859	−	0.894604i	$-0.647458\pi$
$661$	−590.748	−0.893718	−0.446859	+	0.894604i	$0.647458\pi$
$662$	241.910i	0.365423i
$663$	0	0
$664$	1756.18	2.64486
$665$	76.0677i	0.114388i
$666$	0	0
$667$	−247.403	−0.370919
$668$	− 1498.87i	− 2.24382i
$669$	0	0
$670$	2704.07	4.03593
$671$	1050.46i	1.56551i
$672$	0	0
$673$	−272.854	−0.405429	−0.202714	−	0.979238i	$-0.564976\pi$
$673$	−272.854	−0.405429	−0.202714	+	0.979238i	$0.564976\pi$
$674$	− 249.101i	− 0.369586i
$675$	0	0
$676$	1573.08	2.32704
$677$	− 1232.99i	− 1.82126i	−0.413224	−	0.910629i	$-0.635597\pi$
$677$	− 1232.99i	− 1.82126i	0.413224	−	0.910629i	$-0.364403\pi$
$678$	0	0
$679$	85.2663	0.125576
$680$	1094.29i	1.60924i
$681$	0	0
$682$	−1311.67	−1.92327
$683$	− 1079.42i	− 1.58041i	−0.612840	−	0.790207i	$-0.709973\pi$
$683$	− 1079.42i	− 1.58041i	0.612840	−	0.790207i	$-0.290027\pi$
$684$	0	0
$685$	−876.655	−1.27979
$686$	− 315.041i	− 0.459243i
$687$	0	0
$688$	−349.638	−0.508195
$689$	110.693i	0.160657i
$690$	0	0
$691$	521.237	0.754323	0.377161	−	0.926148i	$-0.376900\pi$
$691$	521.237	0.754323	0.377161	+	0.926148i	$0.376900\pi$
$692$	− 1634.54i	− 2.36204i
$693$	0	0
$694$	−605.278	−0.872158
$695$	1042.87i	1.50053i
$696$	0	0
$697$	425.628	0.610657
$698$	1338.10i	1.91705i
$699$	0	0
$700$	277.106	0.395866
$701$	240.666i	0.343319i	0.985156	+	0.171659i	$0.0549128\pi$
$701$	240.666i	0.343319i	−0.985156	+	0.171659i	$0.945087\pi$
$702$	0	0
$703$	96.8910	0.137825
$704$	2633.53i	3.74080i
$705$	0	0
$706$	1387.82	1.96575
$707$	56.1472i	0.0794161i
$708$	0	0
$709$	205.759	0.290210	0.145105	−	0.989416i	$-0.453648\pi$
$709$	205.759	0.290210	0.145105	+	0.989416i	$0.453648\pi$
$710$	− 3927.34i	− 5.53147i
$711$	0	0
$712$	−1000.00	−1.40450
$713$	− 236.543i	− 0.331758i
$714$	0	0
$715$	452.114	0.632328
$716$	− 3232.12i	− 4.51413i
$717$	0	0
$718$	−1691.16	−2.35538
$719$	450.051i	0.625941i	0.949763	+	0.312970i	$0.101324\pi$
$719$	450.051i	0.625941i	−0.949763	+	0.312970i	$0.898676\pi$
$720$	0	0
$721$	−131.901	−0.182941
$722$	− 824.839i	− 1.14244i
$723$	0	0
$724$	2745.67	3.79236
$725$	− 852.127i	− 1.17535i
$726$	0	0
$727$	16.4297	0.0225993	0.0112996	−	0.999936i	$-0.496403\pi$
$727$	16.4297	0.0225993	0.0112996	+	0.999936i	$0.496403\pi$
$728$	102.617i	0.140957i
$729$	0	0
$730$	2989.60	4.09534
$731$	35.5230i	0.0485950i
$732$	0	0
$733$	−556.901	−0.759756	−0.379878	−	0.925037i	$-0.624034\pi$
$733$	−556.901	−0.759756	−0.379878	+	0.925037i	$0.624034\pi$
$734$	− 82.1177i	− 0.111877i
$735$	0	0
$736$	−994.313	−1.35097
$737$	− 1214.58i	− 1.64801i
$738$	0	0
$739$	−850.227	−1.15051	−0.575255	−	0.817974i	$-0.695097\pi$
$739$	−850.227	−1.15051	−0.575255	+	0.817974i	$0.695097\pi$
$740$	− 641.065i	− 0.866305i
$741$	0	0
$742$	76.0019	0.102428
$743$	63.6337i	0.0856443i	0.999083	+	0.0428221i	$0.0136349\pi$
$743$	63.6337i	0.0856443i	−0.999083	+	0.0428221i	$0.986365\pi$
$744$	0	0
$745$	−50.9433	−0.0683802
$746$	− 2199.13i	− 2.94789i
$747$	0	0
$748$	784.134	1.04831
$749$	112.922i	0.150764i
$750$	0	0
$751$	1198.70	1.59614	0.798072	−	0.602562i	$-0.205853\pi$
$751$	1198.70	1.59614	0.798072	+	0.602562i	$0.205853\pi$
$752$	1774.80i	2.36011i
$753$	0	0
$754$	503.414	0.667658
$755$	− 1059.88i	− 1.40382i
$756$	0	0
$757$	−426.213	−0.563028	−0.281514	−	0.959557i	$-0.590837\pi$
$757$	−426.213	−0.563028	−0.281514	+	0.959557i	$0.590837\pi$
$758$	− 1802.84i	− 2.37841i
$759$	0	0
$760$	2323.13	3.05675
$761$	− 885.119i	− 1.16310i	−0.813511	−	0.581550i	$-0.802447\pi$
$761$	− 885.119i	− 1.16310i	0.813511	−	0.581550i	$-0.197553\pi$
$762$	0	0
$763$	−113.526	−0.148788
$764$	− 1999.53i	− 2.61719i
$765$	0	0
$766$	−447.902	−0.584729
$767$	− 390.460i	− 0.509074i
$768$	0	0
$769$	489.713	0.636818	0.318409	−	0.947953i	$-0.396851\pi$
$769$	489.713	0.636818	0.318409	+	0.947953i	$0.396851\pi$
$770$	− 310.422i	− 0.403146i
$771$	0	0
$772$	24.9703	0.0323449
$773$	478.239i	0.618679i	0.950952	+	0.309340i	$0.100108\pi$
$773$	478.239i	0.618679i	−0.950952	+	0.309340i	$0.899892\pi$
$774$	0	0
$775$	814.723	1.05126
$776$	− 2604.06i	− 3.35575i
$777$	0	0
$778$	−1373.41	−1.76531
$779$	− 903.594i	− 1.15994i
$780$	0	0
$781$	−1764.04	−2.25869
$782$	194.179i	0.248310i
$783$	0	0
$784$	−2705.11	−3.45039
$785$	− 966.893i	− 1.23171i
$786$	0	0
$787$	−636.491	−0.808757	−0.404378	−	0.914592i	$-0.632512\pi$
$787$	−636.491	−0.808757	−0.404378	+	0.914592i	$0.632512\pi$
$788$	− 2365.77i	− 3.00225i
$789$	0	0
$790$	1270.68	1.60846
$791$	116.019i	0.146673i
$792$	0	0
$793$	385.466	0.486085
$794$	1257.15i	1.58332i
$795$	0	0
$796$	2853.38	3.58465
$797$	382.445i	0.479855i	0.970791	+	0.239928i	$0.0771237\pi$
$797$	382.445i	0.479855i	−0.970791	+	0.239928i	$0.922876\pi$
$798$	0	0
$799$	180.318	0.225680
$800$	− 3424.70i	− 4.28088i
$801$	0	0
$802$	1253.40	1.56285
$803$	− 1342.84i	− 1.67227i
$804$	0	0
$805$	55.9809	0.0695415
$806$	481.317i	0.597168i
$807$	0	0
$808$	1714.75	2.12222
$809$	1031.19i	1.27465i	0.770597	+	0.637323i	$0.219958\pi$
$809$	1031.19i	1.27465i	−0.770597	+	0.637323i	$0.780042\pi$
$810$	0	0
$811$	407.070	0.501936	0.250968	−	0.967995i	$-0.419251\pi$
$811$	407.070	0.501936	0.250968	+	0.967995i	$0.419251\pi$
$812$	− 251.713i	− 0.309992i
$813$	0	0
$814$	−395.400	−0.485749
$815$	216.076i	0.265123i
$816$	0	0
$817$	75.4140	0.0923061
$818$	1655.46i	2.02379i
$819$	0	0
$820$	−5978.50	−7.29085
$821$	446.485i	0.543830i	0.962321	+	0.271915i	$0.0876570\pi$
$821$	446.485i	0.543830i	−0.962321	+	0.271915i	$0.912343\pi$
$822$	0	0
$823$	706.495	0.858438	0.429219	−	0.903200i	$-0.358789\pi$
$823$	706.495	0.858438	0.429219	+	0.903200i	$0.358789\pi$
$824$	4028.29i	4.88870i
$825$	0	0
$826$	−268.090	−0.324565
$827$	1052.40i	1.27255i	0.771461	+	0.636276i	$0.219526\pi$
$827$	1052.40i	1.27255i	−0.771461	+	0.636276i	$0.780474\pi$
$828$	0	0
$829$	−1352.99	−1.63208	−0.816040	−	0.577995i	$-0.803835\pi$
$829$	−1352.99	−1.63208	−0.816040	+	0.577995i	$0.803835\pi$
$830$	1949.46i	2.34874i
$831$	0	0
$832$	966.374	1.16151
$833$	274.837i	0.329936i
$834$	0	0
$835$	1042.94	1.24902
$836$	− 1664.69i	− 1.99125i
$837$	0	0
$838$	−1688.47	−2.01489
$839$	− 1124.03i	− 1.33973i	−0.742485	−	0.669863i	$-0.766353\pi$
$839$	− 1124.03i	− 1.33973i	0.742485	−	0.669863i	$-0.233647\pi$
$840$	0	0
$841$	66.9580	0.0796172
$842$	− 740.348i	− 0.879274i
$843$	0	0
$844$	2101.18	2.48954
$845$	1094.57i	1.29535i
$846$	0	0
$847$	−37.3016	−0.0440397
$848$	− 1314.82i	− 1.55049i
$849$	0	0
$850$	−668.807	−0.786831
$851$	− 71.3055i	− 0.0837903i
$852$	0	0
$853$	−224.933	−0.263697	−0.131848	−	0.991270i	$-0.542091\pi$
$853$	−224.933	−0.263697	−0.131848	+	0.991270i	$0.542091\pi$
$854$	− 264.661i	− 0.309908i
$855$	0	0
$856$	3448.67	4.02882
$857$	971.825i	1.13398i	0.823723	+	0.566992i	$0.191893\pi$
$857$	971.825i	1.13398i	−0.823723	+	0.566992i	$0.808107\pi$
$858$	0	0
$859$	871.022	1.01399	0.506997	−	0.861948i	$-0.330755\pi$
$859$	871.022	1.01399	0.506997	+	0.861948i	$0.330755\pi$
$860$	− 498.966i	− 0.580193i
$861$	0	0
$862$	−714.454	−0.828833
$863$	− 1554.76i	− 1.80157i	−0.434262	−	0.900786i	$-0.642991\pi$
$863$	− 1554.76i	− 1.80157i	0.434262	−	0.900786i	$-0.357009\pi$
$864$	0	0
$865$	1137.33	1.31484
$866$	3204.33i	3.70014i
$867$	0	0
$868$	240.665	0.277263
$869$	− 570.751i	− 0.656790i
$870$	0	0
$871$	−445.693	−0.511702
$872$	3467.10i	3.97603i
$873$	0	0
$874$	412.234	0.471664
$875$	35.4315i	0.0404931i
$876$	0	0
$877$	−1171.39	−1.33568	−0.667839	−	0.744306i	$-0.732780\pi$
$877$	−1171.39	−1.33568	−0.667839	+	0.744306i	$0.732780\pi$
$878$	249.817i	0.284530i
$879$	0	0
$880$	−5370.24	−6.10254
$881$	212.080i	0.240727i	0.992730	+	0.120363i	$0.0384060\pi$
$881$	212.080i	0.240727i	−0.992730	+	0.120363i	$0.961594\pi$
$882$	0	0
$883$	−534.932	−0.605812	−0.302906	−	0.953020i	$-0.597957\pi$
$883$	−534.932	−0.605812	−0.302906	+	0.953020i	$0.597957\pi$
$884$	− 287.738i	− 0.325496i
$885$	0	0
$886$	−560.507	−0.632626
$887$	− 649.299i	− 0.732017i	−0.930611	−	0.366009i	$-0.880724\pi$
$887$	− 649.299i	− 0.732017i	0.930611	−	0.366009i	$-0.119276\pi$
$888$	0	0
$889$	−9.51200	−0.0106997
$890$	− 1110.06i	− 1.24725i
$891$	0	0
$892$	−1635.49	−1.83351
$893$	− 382.810i	− 0.428678i
$894$	0	0
$895$	2248.96	2.51280
$896$	− 286.002i	− 0.319199i
$897$	0	0
$898$	−1592.28	−1.77314
$899$	− 740.066i	− 0.823210i
$900$	0	0
$901$	−133.584	−0.148262
$902$	3687.45i	4.08808i
$903$	0	0
$904$	3543.24	3.91951
$905$	1910.48i	2.11102i
$906$	0	0
$907$	960.314	1.05878	0.529390	−	0.848378i	$-0.322421\pi$
$907$	960.314	1.05878	0.529390	+	0.848378i	$0.322421\pi$
$908$	− 1836.70i	− 2.02280i
$909$	0	0
$910$	−113.910	−0.125176
$911$	− 19.1928i	− 0.0210679i	−0.999945	−	0.0105339i	$-0.996647\pi$
$911$	− 19.1928i	− 0.0210679i	0.999945	−	0.0105339i	$-0.00335312\pi$
$912$	0	0
$913$	875.636	0.959075
$914$	− 2304.33i	− 2.52115i
$915$	0	0
$916$	−2863.19	−3.12575
$917$	− 163.376i	− 0.178163i
$918$	0	0
$919$	−222.712	−0.242341	−0.121171	−	0.992632i	$-0.538665\pi$
$919$	−222.712	−0.242341	−0.121171	+	0.992632i	$0.538665\pi$
$920$	− 1709.67i	− 1.85834i
$921$	0	0
$922$	−2392.94	−2.59538
$923$	647.316i	0.701317i
$924$	0	0
$925$	245.597	0.265510
$926$	− 930.025i	− 1.00435i
$927$	0	0
$928$	−3110.88	−3.35224
$929$	− 1217.46i	− 1.31051i	−0.755408	−	0.655254i	$-0.772562\pi$
$929$	− 1217.46i	− 1.31051i	0.755408	−	0.655254i	$-0.227438\pi$
$930$	0	0
$931$	583.469	0.626713
$932$	3447.51i	3.69904i
$933$	0	0
$934$	1512.07	1.61892
$935$	545.612i	0.583542i
$936$	0	0
$937$	−676.327	−0.721800	−0.360900	−	0.932605i	$-0.617530\pi$
$937$	−676.327	−0.721800	−0.360900	+	0.932605i	$0.617530\pi$
$938$	306.013i	0.326240i
$939$	0	0
$940$	−2532.81	−2.69447
$941$	− 367.884i	− 0.390951i	−0.980709	−	0.195475i	$-0.937375\pi$
$941$	− 367.884i	− 0.390951i	0.980709	−	0.195475i	$-0.0626249\pi$
$942$	0	0
$943$	−664.987	−0.705182
$944$	4637.90i	4.91303i
$945$	0	0
$946$	−307.755	−0.325322
$947$	1836.82i	1.93962i	0.243862	+	0.969810i	$0.421585\pi$
$947$	1836.82i	1.93962i	−0.243862	+	0.969810i	$0.578415\pi$
$948$	0	0
$949$	−492.755	−0.519236
$950$	1419.85i	1.49458i
$951$	0	0
$952$	−123.838	−0.130082
$953$	− 340.367i	− 0.357154i	−0.983926	−	0.178577i	$-0.942851\pi$
$953$	− 340.367i	− 0.357154i	0.983926	−	0.178577i	$-0.0571493\pi$
$954$	0	0
$955$	1391.31	1.45686
$956$	4149.09i	4.34005i
$957$	0	0
$958$	−2149.21	−2.24343
$959$	− 99.2090i	− 0.103450i
$960$	0	0
$961$	−253.418	−0.263703
$962$	145.092i	0.150823i
$963$	0	0
$964$	2336.66	2.42392
$965$	17.3747i	0.0180049i
$966$	0	0
$967$	242.367	0.250638	0.125319	−	0.992116i	$-0.460005\pi$
$967$	242.367	0.250638	0.125319	+	0.992116i	$0.460005\pi$
$968$	1139.20i	1.17686i
$969$	0	0
$970$	2890.64	2.98004
$971$	− 210.675i	− 0.216967i	−0.994098	−	0.108483i	$-0.965401\pi$
$971$	− 210.675i	− 0.216967i	0.994098	−	0.108483i	$-0.0345995\pi$
$972$	0	0
$973$	−118.019	−0.121294
$974$	1817.64i	1.86616i
$975$	0	0
$976$	−4578.58	−4.69117
$977$	− 45.8883i	− 0.0469686i	−0.999724	−	0.0234843i	$-0.992524\pi$
$977$	− 45.8883i	− 0.0469686i	0.999724	−	0.0234843i	$-0.00747596\pi$
$978$	0	0
$979$	−498.603	−0.509298
$980$	− 3860.44i	− 3.93923i
$981$	0	0
$982$	2298.91	2.34105
$983$	− 146.995i	− 0.149538i	−0.997201	−	0.0747688i	$-0.976178\pi$
$983$	− 146.995i	− 0.149538i	0.997201	−	0.0747688i	$-0.0238219\pi$
$984$	0	0
$985$	1646.14	1.67121
$986$	607.521i	0.616147i
$987$	0	0
$988$	−610.859	−0.618278
$989$	− 55.4999i	− 0.0561172i
$990$	0	0
$991$	1099.39	1.10938	0.554689	−	0.832058i	$-0.312837\pi$
$991$	1099.39	1.10938	0.554689	+	0.832058i	$0.312837\pi$
$992$	− 2974.33i	− 2.99832i
$993$	0	0
$994$	444.448	0.447131
$995$	1985.42i	1.99540i
$996$	0	0
$997$	−1142.73	−1.14617	−0.573085	−	0.819496i	$-0.694254\pi$
$997$	−1142.73	−1.14617	−0.573085	+	0.819496i	$0.694254\pi$
$998$	− 3065.40i	− 3.07154i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.82	yes	84
3.2	odd	2	inner	1143.3.b.a.890.3	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.3	✓	84	3.2	odd	2	inner
1143.3.b.a.890.82	yes	84	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.83653i q^{2} -10.7190 q^{4} -7.45843i q^{5} +0.844054 q^{7} -25.7776i q^{8} +O(q^{10})\)	\(q+3.83653i q^{2} -10.7190 q^{4} -7.45843i q^{5} +0.844054 q^{7} -25.7776i q^{8} +28.6145 q^{10} -12.8528i q^{11} -4.71633 q^{13} +3.23824i q^{14} +56.0208 q^{16} -5.69167i q^{17} -12.0832 q^{19} +79.9469i q^{20} +49.3100 q^{22} +8.89247i q^{23} -30.6282 q^{25} -18.0944i q^{26} -9.04740 q^{28} +27.8216i q^{29} -26.6004 q^{31} +111.815i q^{32} +21.8363 q^{34} -6.29532i q^{35} -8.01864 q^{37} -46.3577i q^{38} -192.261 q^{40} +74.7809i q^{41} -6.24122 q^{43} +137.769i q^{44} -34.1163 q^{46} +31.6811i q^{47} -48.2876 q^{49} -117.506i q^{50} +50.5543 q^{52} -23.4701i q^{53} -95.8615 q^{55} -21.7577i q^{56} -106.739 q^{58} +82.7889i q^{59} -81.7300 q^{61} -102.053i q^{62} -204.900 q^{64} +35.1764i q^{65} +94.4999 q^{67} +61.0090i q^{68} +24.1522 q^{70} -137.250i q^{71} +104.478 q^{73} -30.7638i q^{74} +129.520 q^{76} -10.8484i q^{77} +44.4068 q^{79} -417.828i q^{80} -286.899 q^{82} +68.1282i q^{83} -42.4509 q^{85} -23.9447i q^{86} -331.314 q^{88} -38.7934i q^{89} -3.98084 q^{91} -95.3183i q^{92} -121.546 q^{94} +90.1219i q^{95} +101.020 q^{97} -185.257i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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