LMFDB - Embedded newform 1143.3.b.a.890.2

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			890.2
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.83

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} +O(q^{10})$	\(q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} -36.5274 q^{10} -14.1873i q^{11} +10.8150 q^{13} -43.7780i q^{14} +60.1947 q^{16} +9.53381i q^{17} -24.9634 q^{19} +103.477i q^{20} -54.8648 q^{22} +13.0513i q^{23} -64.2169 q^{25} -41.8237i q^{26} -124.016 q^{28} -18.2561i q^{29} -38.5312 q^{31} -125.197i q^{32} +36.8691 q^{34} -106.926i q^{35} -51.8983 q^{37} +96.5383i q^{38} +254.054 q^{40} +18.2318i q^{41} -31.3978 q^{43} +155.424i q^{44} +50.4719 q^{46} -4.56917i q^{47} +79.1505 q^{49} +248.339i q^{50} -118.480 q^{52} -31.5680i q^{53} -134.005 q^{55} +304.482i q^{56} -70.5998 q^{58} -62.5783i q^{59} +64.5712 q^{61} +149.008i q^{62} -243.380 q^{64} -102.153i q^{65} +74.3116 q^{67} -104.444i q^{68} -413.503 q^{70} +111.127i q^{71} -48.7312 q^{73} +200.701i q^{74} +273.478 q^{76} -160.605i q^{77} -63.2310 q^{79} -568.567i q^{80} +70.5059 q^{82} +51.0401i q^{83} +90.0514 q^{85} +121.421i q^{86} +381.593 q^{88} -19.4944i q^{89} +122.430 q^{91} -142.979i q^{92} -17.6698 q^{94} +235.791i q^{95} +72.1446 q^{97} -306.090i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.86719i	− 1.93359i	−0.255545	−	0.966797i	$-0.582255\pi$
$2$	− 3.86719i	− 1.93359i	0.255545	−	0.966797i	$-0.417745\pi$
$3$	0	0
$3$	0	0
$4$	−10.9551	−2.73879
$5$	− 9.44547i	− 1.88909i	−0.328376	−	0.944547i	$-0.606501\pi$
$5$	− 9.44547i	− 1.88909i	0.328376	−	0.944547i	$-0.393499\pi$
$6$	0	0
$7$	11.3204	1.61719	0.808597	−	0.588363i	$-0.200227\pi$
$7$	11.3204	1.61719	0.808597	+	0.588363i	$0.200227\pi$
$8$	26.8969i	3.36211i
$9$	0	0
$10$	−36.5274	−3.65274
$11$	− 14.1873i	− 1.28975i	−0.764288	−	0.644875i	$-0.776909\pi$
$11$	− 14.1873i	− 1.28975i	0.764288	−	0.644875i	$-0.223091\pi$
$12$	0	0
$13$	10.8150	0.831925	0.415962	−	0.909382i	$-0.363445\pi$
$13$	10.8150	0.831925	0.415962	+	0.909382i	$0.363445\pi$
$14$	− 43.7780i	− 3.12700i
$15$	0	0
$16$	60.1947	3.76217
$17$	9.53381i	0.560813i	0.959881	+	0.280406i	$0.0904692\pi$
$17$	9.53381i	0.560813i	−0.959881	+	0.280406i	$0.909531\pi$
$18$	0	0
$19$	−24.9634	−1.31387	−0.656933	−	0.753949i	$-0.728146\pi$
$19$	−24.9634	−1.31387	−0.656933	+	0.753949i	$0.728146\pi$
$20$	103.477i	5.17383i
$21$	0	0
$22$	−54.8648	−2.49385
$23$	13.0513i	0.567448i	0.958906	+	0.283724i	$0.0915700\pi$
$23$	13.0513i	0.567448i	−0.958906	+	0.283724i	$0.908430\pi$
$24$	0	0
$25$	−64.2169	−2.56868
$26$	− 41.8237i	− 1.60860i
$27$	0	0
$28$	−124.016	−4.42915
$29$	− 18.2561i	− 0.629521i	−0.949171	−	0.314761i	$-0.898076\pi$
$29$	− 18.2561i	− 0.629521i	0.949171	−	0.314761i	$-0.101924\pi$
$30$	0	0
$31$	−38.5312	−1.24294	−0.621472	−	0.783437i	$-0.713465\pi$
$31$	−38.5312	−1.24294	−0.621472	+	0.783437i	$0.713465\pi$
$32$	− 125.197i	− 3.91240i
$33$	0	0
$34$	36.8691	1.08438
$35$	− 106.926i	− 3.05503i
$36$	0	0
$37$	−51.8983	−1.40266	−0.701329	−	0.712838i	$-0.747410\pi$
$37$	−51.8983	−1.40266	−0.701329	+	0.712838i	$0.747410\pi$
$38$	96.5383i	2.54048i
$39$	0	0
$40$	254.054	6.35134
$41$	18.2318i	0.444679i	0.974969	+	0.222339i	$0.0713693\pi$
$41$	18.2318i	0.444679i	−0.974969	+	0.222339i	$0.928631\pi$
$42$	0	0
$43$	−31.3978	−0.730180	−0.365090	−	0.930972i	$-0.618962\pi$
$43$	−31.3978	−0.730180	−0.365090	+	0.930972i	$0.618962\pi$
$44$	155.424i	3.53235i
$45$	0	0
$46$	50.4719	1.09722
$47$	− 4.56917i	− 0.0972164i	−0.998818	−	0.0486082i	$-0.984521\pi$
$47$	− 4.56917i	− 0.0972164i	0.998818	−	0.0486082i	$-0.0154786\pi$
$48$	0	0
$49$	79.1505	1.61532
$50$	248.339i	4.96678i
$51$	0	0
$52$	−118.480	−2.27846
$53$	− 31.5680i	− 0.595623i	−0.954625	−	0.297812i	$-0.903743\pi$
$53$	− 31.5680i	− 0.595623i	0.954625	−	0.297812i	$-0.0962568\pi$
$54$	0	0
$55$	−134.005	−2.43646
$56$	304.482i	5.43718i
$57$	0	0
$58$	−70.5998	−1.21724
$59$	− 62.5783i	− 1.06065i	−0.847795	−	0.530325i	$-0.822070\pi$
$59$	− 62.5783i	− 1.06065i	0.847795	−	0.530325i	$-0.177930\pi$
$60$	0	0
$61$	64.5712	1.05854	0.529272	−	0.848452i	$-0.322465\pi$
$61$	64.5712	1.05854	0.529272	+	0.848452i	$0.322465\pi$
$62$	149.008i	2.40335i
$63$	0	0
$64$	−243.380	−3.80282
$65$	− 102.153i	− 1.57158i
$66$	0	0
$67$	74.3116	1.10913	0.554564	−	0.832141i	$-0.312885\pi$
$67$	74.3116	1.10913	0.554564	+	0.832141i	$0.312885\pi$
$68$	− 104.444i	− 1.53595i
$69$	0	0
$70$	−413.503	−5.90719
$71$	111.127i	1.56516i	0.622547	+	0.782582i	$0.286098\pi$
$71$	111.127i	1.56516i	−0.622547	+	0.782582i	$0.713902\pi$
$72$	0	0
$73$	−48.7312	−0.667550	−0.333775	−	0.942653i	$-0.608323\pi$
$73$	−48.7312	−0.667550	−0.333775	+	0.942653i	$0.608323\pi$
$74$	200.701i	2.71217i
$75$	0	0
$76$	273.478	3.59840
$77$	− 160.605i	− 2.08578i
$78$	0	0
$79$	−63.2310	−0.800393	−0.400196	−	0.916429i	$-0.631058\pi$
$79$	−63.2310	−0.800393	−0.400196	+	0.916429i	$0.631058\pi$
$80$	− 568.567i	− 7.10709i
$81$	0	0
$82$	70.5059	0.859829
$83$	51.0401i	0.614940i	0.951558	+	0.307470i	$0.0994824\pi$
$83$	51.0401i	0.614940i	−0.951558	+	0.307470i	$0.900518\pi$
$84$	0	0
$85$	90.0514	1.05943
$86$	121.421i	1.41187i
$87$	0	0
$88$	381.593	4.33628
$89$	− 19.4944i	− 0.219039i	−0.993985	−	0.109519i	$-0.965069\pi$
$89$	− 19.4944i	− 0.219039i	0.993985	−	0.109519i	$-0.0349312\pi$
$90$	0	0
$91$	122.430	1.34538
$92$	− 142.979i	− 1.55412i
$93$	0	0
$94$	−17.6698	−0.187977
$95$	235.791i	2.48202i
$96$	0	0
$97$	72.1446	0.743758	0.371879	−	0.928281i	$-0.378714\pi$
$97$	72.1446	0.743758	0.371879	+	0.928281i	$0.378714\pi$
$98$	− 306.090i	− 3.12337i
$99$	0	0
$100$	703.506	7.03506
$101$	− 130.320i	− 1.29030i	−0.764055	−	0.645151i	$-0.776794\pi$
$101$	− 130.320i	− 1.29030i	0.764055	−	0.645151i	$-0.223206\pi$
$102$	0	0
$103$	41.3239	0.401203	0.200602	−	0.979673i	$-0.435710\pi$
$103$	41.3239	0.401203	0.200602	+	0.979673i	$0.435710\pi$
$104$	290.890i	2.79702i
$105$	0	0
$106$	−122.080	−1.15169
$107$	− 169.058i	− 1.57999i	−0.613116	−	0.789993i	$-0.710084\pi$
$107$	− 169.058i	− 1.57999i	0.613116	−	0.789993i	$-0.289916\pi$
$108$	0	0
$109$	−67.9965	−0.623821	−0.311910	−	0.950111i	$-0.600969\pi$
$109$	−67.9965	−0.623821	−0.311910	+	0.950111i	$0.600969\pi$
$110$	518.224i	4.71113i
$111$	0	0
$112$	681.425	6.08415
$113$	44.6199i	0.394866i	0.980316	+	0.197433i	$0.0632605\pi$
$113$	44.6199i	0.394866i	−0.980316	+	0.197433i	$0.936739\pi$
$114$	0	0
$115$	123.276	1.07196
$116$	199.998i	1.72412i
$117$	0	0
$118$	−242.002	−2.05087
$119$	107.926i	0.906943i
$120$	0	0
$121$	−80.2783	−0.663457
$122$	− 249.709i	− 2.04679i
$123$	0	0
$124$	422.115	3.40416
$125$	370.422i	2.96338i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	440.411i	3.44071i
$129$	0	0
$130$	−395.045	−3.03881
$131$	− 157.351i	− 1.20115i	−0.799567	−	0.600576i	$-0.794938\pi$
$131$	− 157.351i	− 1.20115i	0.799567	−	0.600576i	$-0.205062\pi$
$132$	0	0
$133$	−282.595	−2.12477
$134$	− 287.377i	− 2.14460i
$135$	0	0
$136$	−256.430	−1.88551
$137$	− 112.882i	− 0.823958i	−0.911193	−	0.411979i	$-0.864838\pi$
$137$	− 112.882i	− 0.823958i	0.911193	−	0.411979i	$-0.135162\pi$
$138$	0	0
$139$	−98.1690	−0.706252	−0.353126	−	0.935576i	$-0.614881\pi$
$139$	−98.1690	−0.706252	−0.353126	+	0.935576i	$0.614881\pi$
$140$	1171.39i	8.36708i
$141$	0	0
$142$	429.748	3.02639
$143$	− 153.435i	− 1.07298i
$144$	0	0
$145$	−172.438	−1.18922
$146$	188.453i	1.29077i
$147$	0	0
$148$	568.554	3.84158
$149$	198.203i	1.33022i	0.746744	+	0.665112i	$0.231616\pi$
$149$	198.203i	1.33022i	−0.746744	+	0.665112i	$0.768384\pi$
$150$	0	0
$151$	272.334	1.80354	0.901768	−	0.432219i	$-0.142269\pi$
$151$	272.334	1.80354	0.901768	+	0.432219i	$0.142269\pi$
$152$	− 671.438i	− 4.41736i
$153$	0	0
$154$	−621.089	−4.03305
$155$	363.946i	2.34804i
$156$	0	0
$157$	−15.5821	−0.0992487	−0.0496244	−	0.998768i	$-0.515802\pi$
$157$	−15.5821	−0.0992487	−0.0496244	+	0.998768i	$0.515802\pi$
$158$	244.526i	1.54764i
$159$	0	0
$160$	−1182.54	−7.39088
$161$	147.746i	0.917674i
$162$	0	0
$163$	222.264	1.36358	0.681790	−	0.731548i	$-0.261202\pi$
$163$	222.264	1.36358	0.681790	+	0.731548i	$0.261202\pi$
$164$	− 199.732i	− 1.21788i
$165$	0	0
$166$	197.382	1.18905
$167$	0.247721i	0.00148336i	1.00000		0.000741679i	$0.000236084\pi$
$167$	0.247721i	0.00148336i	−1.00000		0.000741679i	$0.999764\pi$
$168$	0	0
$169$	−52.0354	−0.307901
$170$	− 348.246i	− 2.04850i
$171$	0	0
$172$	343.967	1.99981
$173$	− 177.674i	− 1.02702i	−0.858084	−	0.513509i	$-0.828345\pi$
$173$	− 177.674i	− 1.02702i	0.858084	−	0.513509i	$-0.171655\pi$
$174$	0	0
$175$	−726.958	−4.15405
$176$	− 853.997i	− 4.85226i
$177$	0	0
$178$	−75.3887	−0.423532
$179$	− 132.513i	− 0.740298i	−0.928972	−	0.370149i	$-0.879307\pi$
$179$	− 132.513i	− 0.740298i	0.928972	−	0.370149i	$-0.120693\pi$
$180$	0	0
$181$	16.4684	0.0909858	0.0454929	−	0.998965i	$-0.485514\pi$
$181$	16.4684	0.0909858	0.0454929	+	0.998965i	$0.485514\pi$
$182$	− 473.459i	− 2.60143i
$183$	0	0
$184$	−351.039	−1.90782
$185$	490.204i	2.64975i
$186$	0	0
$187$	135.259	0.723308
$188$	50.0559i	0.266255i
$189$	0	0
$190$	911.850	4.79921
$191$	− 89.1158i	− 0.466575i	−0.972408	−	0.233288i	$-0.925052\pi$
$191$	− 89.1158i	− 0.466575i	0.972408	−	0.233288i	$-0.0749483\pi$
$192$	0	0
$193$	−263.571	−1.36565	−0.682826	−	0.730581i	$-0.739249\pi$
$193$	−263.571	−1.36565	−0.682826	+	0.730581i	$0.739249\pi$
$194$	− 278.997i	− 1.43813i
$195$	0	0
$196$	−867.105	−4.42401
$197$	− 188.222i	− 0.955444i	−0.878511	−	0.477722i	$-0.841463\pi$
$197$	− 188.222i	− 0.955444i	0.878511	−	0.477722i	$-0.158537\pi$
$198$	0	0
$199$	87.7301	0.440855	0.220427	−	0.975403i	$-0.429255\pi$
$199$	87.7301	0.440855	0.220427	+	0.975403i	$0.429255\pi$
$200$	− 1727.23i	− 8.63617i
$201$	0	0
$202$	−503.974	−2.49492
$203$	− 206.666i	− 1.01806i
$204$	0	0
$205$	172.208	0.840040
$206$	− 159.807i	− 0.775764i
$207$	0	0
$208$	651.007	3.12984
$209$	354.163i	1.69456i
$210$	0	0
$211$	42.0224	0.199158	0.0995791	−	0.995030i	$-0.468250\pi$
$211$	42.0224	0.199158	0.0995791	+	0.995030i	$0.468250\pi$
$212$	345.833i	1.63129i
$213$	0	0
$214$	−653.781	−3.05505
$215$	296.567i	1.37938i
$216$	0	0
$217$	−436.187	−2.01008
$218$	262.955i	1.20622i
$219$	0	0
$220$	1468.05	6.67295
$221$	103.108i	0.466554i
$222$	0	0
$223$	167.396	0.750656	0.375328	−	0.926892i	$-0.377530\pi$
$223$	167.396	0.750656	0.375328	+	0.926892i	$0.377530\pi$
$224$	− 1417.27i	− 6.32710i
$225$	0	0
$226$	172.553	0.763511
$227$	− 374.151i	− 1.64824i	−0.566412	−	0.824122i	$-0.691669\pi$
$227$	− 374.151i	− 1.64824i	0.566412	−	0.824122i	$-0.308331\pi$
$228$	0	0
$229$	−144.506	−0.631030	−0.315515	−	0.948921i	$-0.602177\pi$
$229$	−144.506	−0.631030	−0.315515	+	0.948921i	$0.602177\pi$
$230$	− 476.731i	− 2.07274i
$231$	0	0
$232$	491.032	2.11652
$233$	219.708i	0.942955i	0.881878	+	0.471477i	$0.156279\pi$
$233$	219.708i	0.942955i	−0.881878	+	0.471477i	$0.843721\pi$
$234$	0	0
$235$	−43.1580	−0.183651
$236$	685.555i	2.90489i
$237$	0	0
$238$	417.371	1.75366
$239$	− 434.872i	− 1.81955i	−0.415103	−	0.909774i	$-0.636255\pi$
$239$	− 434.872i	− 1.81955i	0.415103	−	0.909774i	$-0.363745\pi$
$240$	0	0
$241$	198.766	0.824756	0.412378	−	0.911013i	$-0.364698\pi$
$241$	198.766	0.824756	0.412378	+	0.911013i	$0.364698\pi$
$242$	310.451i	1.28286i
$243$	0	0
$244$	−707.387	−2.89913
$245$	− 747.613i	− 3.05148i
$246$	0	0
$247$	−269.980	−1.09304
$248$	− 1036.37i	− 4.17891i
$249$	0	0
$250$	1432.49	5.72997
$251$	343.459i	1.36836i	0.729313	+	0.684180i	$0.239840\pi$
$251$	343.459i	1.36836i	−0.729313	+	0.684180i	$0.760160\pi$
$252$	0	0
$253$	185.162	0.731867
$254$	− 43.5810i	− 0.171579i
$255$	0	0
$256$	729.632	2.85013
$257$	261.700i	1.01829i	0.860681	+	0.509144i	$0.170038\pi$
$257$	261.700i	1.01829i	−0.860681	+	0.509144i	$0.829962\pi$
$258$	0	0
$259$	−587.508	−2.26837
$260$	1119.10i	4.30423i
$261$	0	0
$262$	−608.506	−2.32254
$263$	31.1538i	0.118456i	0.998244	+	0.0592278i	$0.0188638\pi$
$263$	31.1538i	0.118456i	−0.998244	+	0.0592278i	$0.981136\pi$
$264$	0	0
$265$	−298.175	−1.12519
$266$	1092.85i	4.10845i
$267$	0	0
$268$	−814.094	−3.03767
$269$	137.914i	0.512693i	0.966585	+	0.256346i	$0.0825188\pi$
$269$	137.914i	0.512693i	−0.966585	+	0.256346i	$0.917481\pi$
$270$	0	0
$271$	−194.468	−0.717593	−0.358797	−	0.933416i	$-0.616813\pi$
$271$	−194.468	−0.717593	−0.358797	+	0.933416i	$0.616813\pi$
$272$	573.885i	2.10987i
$273$	0	0
$274$	−436.537	−1.59320
$275$	911.062i	3.31295i
$276$	0	0
$277$	205.748	0.742773	0.371386	−	0.928478i	$-0.378883\pi$
$277$	205.748	0.742773	0.371386	+	0.928478i	$0.378883\pi$
$278$	379.638i	1.36560i
$279$	0	0
$280$	2875.98	10.2713
$281$	94.8530i	0.337555i	0.985654	+	0.168778i	$0.0539819\pi$
$281$	94.8530i	0.337555i	−0.985654	+	0.168778i	$0.946018\pi$
$282$	0	0
$283$	193.502	0.683754	0.341877	−	0.939745i	$-0.388937\pi$
$283$	193.502	0.683754	0.341877	+	0.939745i	$0.388937\pi$
$284$	− 1217.41i	− 4.28665i
$285$	0	0
$286$	−593.364	−2.07470
$287$	206.391i	0.719132i
$288$	0	0
$289$	198.106	0.685489
$290$	666.849i	2.29948i
$291$	0	0
$292$	533.857	1.82828
$293$	− 354.078i	− 1.20846i	−0.796811	−	0.604229i	$-0.793481\pi$
$293$	− 354.078i	− 1.20846i	0.796811	−	0.604229i	$-0.206519\pi$
$294$	0	0
$295$	−591.082	−2.00367
$296$	− 1395.90i	− 4.71589i
$297$	0	0
$298$	766.490	2.57211
$299$	141.150i	0.472074i
$300$	0	0
$301$	−355.434	−1.18084
$302$	− 1053.17i	− 3.48731i
$303$	0	0
$304$	−1502.67	−4.94298
$305$	− 609.905i	− 1.99969i
$306$	0	0
$307$	151.973	0.495026	0.247513	−	0.968885i	$-0.420387\pi$
$307$	151.973	0.495026	0.247513	+	0.968885i	$0.420387\pi$
$308$	1759.45i	5.71250i
$309$	0	0
$310$	1407.45	4.54015
$311$	− 63.8741i	− 0.205383i	−0.994713	−	0.102691i	$-0.967255\pi$
$311$	− 63.8741i	− 0.205383i	0.994713	−	0.102691i	$-0.0327454\pi$
$312$	0	0
$313$	−37.0453	−0.118356	−0.0591778	−	0.998247i	$-0.518848\pi$
$313$	−37.0453	−0.118356	−0.0591778	+	0.998247i	$0.518848\pi$
$314$	60.2587i	0.191907i
$315$	0	0
$316$	692.705	2.19211
$317$	− 399.581i	− 1.26051i	−0.776389	−	0.630254i	$-0.782951\pi$
$317$	− 399.581i	− 1.26051i	0.776389	−	0.630254i	$-0.217049\pi$
$318$	0	0
$319$	−259.004	−0.811926
$320$	2298.84i	7.18388i
$321$	0	0
$322$	571.360	1.77441
$323$	− 237.997i	− 0.736832i
$324$	0	0
$325$	−694.507	−2.13695
$326$	− 859.535i	− 2.63661i
$327$	0	0
$328$	−490.379	−1.49506
$329$	− 51.7246i	− 0.157218i
$330$	0	0
$331$	−37.3698	−0.112900	−0.0564498	−	0.998405i	$-0.517978\pi$
$331$	−37.3698	−0.112900	−0.0564498	+	0.998405i	$0.517978\pi$
$332$	− 559.151i	− 1.68419i
$333$	0	0
$334$	0.957984	0.00286821
$335$	− 701.908i	− 2.09525i
$336$	0	0
$337$	−131.462	−0.390094	−0.195047	−	0.980794i	$-0.562486\pi$
$337$	−131.462	−0.390094	−0.195047	+	0.980794i	$0.562486\pi$
$338$	201.231i	0.595357i
$339$	0	0
$340$	−986.526	−2.90155
$341$	546.653i	1.60309i
$342$	0	0
$343$	341.314	0.995085
$344$	− 844.501i	− 2.45495i
$345$	0	0
$346$	−687.099	−1.98583
$347$	− 362.839i	− 1.04564i	−0.852442	−	0.522822i	$-0.824879\pi$
$347$	− 362.839i	− 1.04564i	0.852442	−	0.522822i	$-0.175121\pi$
$348$	0	0
$349$	−473.272	−1.35608	−0.678040	−	0.735025i	$-0.737170\pi$
$349$	−473.272	−1.35608	−0.678040	+	0.735025i	$0.737170\pi$
$350$	2811.28i	8.03224i
$351$	0	0
$352$	−1776.20	−5.04602
$353$	264.341i	0.748841i	0.927259	+	0.374420i	$0.122158\pi$
$353$	264.341i	0.748841i	−0.927259	+	0.374420i	$0.877842\pi$
$354$	0	0
$355$	1049.64	2.95674
$356$	213.565i	0.599900i
$357$	0	0
$358$	−512.454	−1.43144
$359$	54.6728i	0.152292i	0.997097	+	0.0761459i	$0.0242615\pi$
$359$	54.6728i	0.152292i	−0.997097	+	0.0761459i	$0.975739\pi$
$360$	0	0
$361$	262.173	0.726242
$362$	− 63.6865i	− 0.175930i
$363$	0	0
$364$	−1341.24	−3.68472
$365$	460.289i	1.26107i
$366$	0	0
$367$	−340.491	−0.927769	−0.463885	−	0.885896i	$-0.653545\pi$
$367$	−340.491	−0.927769	−0.463885	+	0.885896i	$0.653545\pi$
$368$	785.620i	2.13484i
$369$	0	0
$370$	1895.71	5.12355
$371$	− 357.361i	− 0.963238i
$372$	0	0
$373$	168.009	0.450426	0.225213	−	0.974310i	$-0.427692\pi$
$373$	168.009	0.450426	0.225213	+	0.974310i	$0.427692\pi$
$374$	− 523.071i	− 1.39859i
$375$	0	0
$376$	122.896	0.326852
$377$	− 197.440i	− 0.523714i
$378$	0	0
$379$	29.6118	0.0781315	0.0390657	−	0.999237i	$-0.487562\pi$
$379$	29.6118	0.0781315	0.0390657	+	0.999237i	$0.487562\pi$
$380$	− 2583.13i	− 6.79771i
$381$	0	0
$382$	−344.628	−0.902167
$383$	− 670.916i	− 1.75174i	−0.482548	−	0.875870i	$-0.660289\pi$
$383$	− 670.916i	− 1.75174i	0.482548	−	0.875870i	$-0.339711\pi$
$384$	0	0
$385$	−1516.99	−3.94023
$386$	1019.28i	2.64062i
$387$	0	0
$388$	−790.354	−2.03700
$389$	651.906i	1.67585i	0.545785	+	0.837925i	$0.316232\pi$
$389$	651.906i	1.67585i	−0.545785	+	0.837925i	$0.683768\pi$
$390$	0	0
$391$	−124.429	−0.318232
$392$	2128.90i	5.43087i
$393$	0	0
$394$	−727.891	−1.84744
$395$	597.247i	1.51202i
$396$	0	0
$397$	−434.519	−1.09451	−0.547253	−	0.836967i	$-0.684326\pi$
$397$	−434.519	−1.09451	−0.547253	+	0.836967i	$0.684326\pi$
$398$	− 339.269i	− 0.852434i
$399$	0	0
$400$	−3865.52	−9.66379
$401$	− 181.485i	− 0.452581i	−0.974060	−	0.226290i	$-0.927340\pi$
$401$	− 181.485i	− 0.452581i	0.974060	−	0.226290i	$-0.0726598\pi$
$402$	0	0
$403$	−416.716	−1.03403
$404$	1427.68i	3.53386i
$405$	0	0
$406$	−799.215	−1.96851
$407$	736.295i	1.80908i
$408$	0	0
$409$	467.124	1.14211	0.571056	−	0.820911i	$-0.306534\pi$
$409$	467.124	1.14211	0.571056	+	0.820911i	$0.306534\pi$
$410$	− 665.962i	− 1.62430i
$411$	0	0
$412$	−452.710	−1.09881
$413$	− 708.409i	− 1.71528i
$414$	0	0
$415$	482.097	1.16168
$416$	− 1354.00i	− 3.25482i
$417$	0	0
$418$	1369.61	3.27659
$419$	457.963i	1.09299i	0.837462	+	0.546495i	$0.184038\pi$
$419$	457.963i	1.09299i	−0.837462	+	0.546495i	$0.815962\pi$
$420$	0	0
$421$	−656.507	−1.55940	−0.779699	−	0.626154i	$-0.784628\pi$
$421$	−656.507	−1.55940	−0.779699	+	0.626154i	$0.784628\pi$
$422$	− 162.508i	− 0.385091i
$423$	0	0
$424$	849.081	2.00255
$425$	− 612.232i	− 1.44055i
$426$	0	0
$427$	730.968	1.71187
$428$	1852.06i	4.32724i
$429$	0	0
$430$	1146.88	2.66716
$431$	− 295.176i	− 0.684864i	−0.939543	−	0.342432i	$-0.888749\pi$
$431$	− 295.176i	− 0.684864i	0.939543	−	0.342432i	$-0.111251\pi$
$432$	0	0
$433$	359.925	0.831236	0.415618	−	0.909539i	$-0.363565\pi$
$433$	359.925	0.831236	0.415618	+	0.909539i	$0.363565\pi$
$434$	1686.82i	3.88668i
$435$	0	0
$436$	744.912	1.70851
$437$	− 325.806i	− 0.745551i
$438$	0	0
$439$	550.931	1.25497	0.627484	−	0.778630i	$-0.284085\pi$
$439$	550.931	1.25497	0.627484	+	0.778630i	$0.284085\pi$
$440$	− 3604.32i	− 8.19165i
$441$	0	0
$442$	398.740	0.902126
$443$	487.185i	1.09974i	0.835250	+	0.549870i	$0.185323\pi$
$443$	487.185i	1.09974i	−0.835250	+	0.549870i	$0.814677\pi$
$444$	0	0
$445$	−184.134	−0.413785
$446$	− 647.353i	− 1.45146i
$447$	0	0
$448$	−2755.15	−6.14990
$449$	− 433.451i	− 0.965371i	−0.875794	−	0.482685i	$-0.839662\pi$
$449$	− 433.451i	− 0.965371i	0.875794	−	0.482685i	$-0.160338\pi$
$450$	0	0
$451$	258.660	0.573525
$452$	− 488.817i	− 1.08145i
$453$	0	0
$454$	−1446.91	−3.18704
$455$	− 1156.41i	− 2.54156i
$456$	0	0
$457$	544.259	1.19094	0.595469	−	0.803378i	$-0.296966\pi$
$457$	544.259	1.19094	0.595469	+	0.803378i	$0.296966\pi$
$458$	558.831i	1.22016i
$459$	0	0
$460$	−1350.50	−2.93588
$461$	− 275.026i	− 0.596586i	−0.954474	−	0.298293i	$-0.903583\pi$
$461$	− 275.026i	− 0.596586i	0.954474	−	0.298293i	$-0.0964172\pi$
$462$	0	0
$463$	221.762	0.478968	0.239484	−	0.970900i	$-0.423022\pi$
$463$	221.762	0.478968	0.239484	+	0.970900i	$0.423022\pi$
$464$	− 1098.92i	− 2.36836i
$465$	0	0
$466$	849.654	1.82329
$467$	− 387.031i	− 0.828761i	−0.910104	−	0.414380i	$-0.863998\pi$
$467$	− 387.031i	− 0.828761i	0.910104	−	0.414380i	$-0.136002\pi$
$468$	0	0
$469$	841.234	1.79368
$470$	166.900i	0.355106i
$471$	0	0
$472$	1683.16	3.56602
$473$	445.448i	0.941751i
$474$	0	0
$475$	1603.08	3.37490
$476$	− 1182.35i	− 2.48392i
$477$	0	0
$478$	−1681.73	−3.51827
$479$	50.1545i	0.104707i	0.998629	+	0.0523533i	$0.0166722\pi$
$479$	50.1545i	0.104707i	−0.998629	+	0.0523533i	$0.983328\pi$
$480$	0	0
$481$	−561.282	−1.16691
$482$	− 768.666i	− 1.59474i
$483$	0	0
$484$	879.461	1.81707
$485$	− 681.439i	− 1.40503i
$486$	0	0
$487$	425.468	0.873650	0.436825	−	0.899546i	$-0.356103\pi$
$487$	425.468	0.873650	0.436825	+	0.899546i	$0.356103\pi$
$488$	1736.76i	3.55894i
$489$	0	0
$490$	−2891.16	−5.90033
$491$	537.662i	1.09503i	0.836794	+	0.547517i	$0.184427\pi$
$491$	537.662i	1.09503i	−0.836794	+	0.547517i	$0.815573\pi$
$492$	0	0
$493$	174.050	0.353043
$494$	1044.06i	2.11349i
$495$	0	0
$496$	−2319.37	−4.67616
$497$	1257.99i	2.53117i
$498$	0	0
$499$	215.600	0.432064	0.216032	−	0.976386i	$-0.430688\pi$
$499$	215.600	0.432064	0.216032	+	0.976386i	$0.430688\pi$
$500$	− 4058.03i	− 8.11606i
$501$	0	0
$502$	1328.22	2.64585
$503$	231.627i	0.460491i	0.973133	+	0.230245i	$0.0739529\pi$
$503$	231.627i	0.460491i	−0.973133	+	0.230245i	$0.926047\pi$
$504$	0	0
$505$	−1230.94	−2.43750
$506$	− 716.058i	− 1.41513i
$507$	0	0
$508$	−123.458	−0.243028
$509$	− 663.901i	− 1.30432i	−0.758080	−	0.652162i	$-0.773862\pi$
$509$	− 663.901i	− 1.30432i	0.758080	−	0.652162i	$-0.226138\pi$
$510$	0	0
$511$	−551.654	−1.07956
$512$	− 1059.98i	− 2.07027i
$513$	0	0
$514$	1012.04	1.96896
$515$	− 390.324i	− 0.757911i
$516$	0	0
$517$	−64.8240	−0.125385
$518$	2272.00i	4.38611i
$519$	0	0
$520$	2747.59	5.28384
$521$	629.316i	1.20790i	0.797022	+	0.603950i	$0.206407\pi$
$521$	629.316i	1.20790i	−0.797022	+	0.603950i	$0.793593\pi$
$522$	0	0
$523$	464.271	0.887708	0.443854	−	0.896099i	$-0.353611\pi$
$523$	464.271	0.887708	0.443854	+	0.896099i	$0.353611\pi$
$524$	1723.80i	3.28970i
$525$	0	0
$526$	120.478	0.229045
$527$	− 367.350i	− 0.697058i
$528$	0	0
$529$	358.663	0.678002
$530$	1153.10i	2.17566i
$531$	0	0
$532$	3095.87	5.81931
$533$	197.178i	0.369939i
$534$	0	0
$535$	−1596.84	−2.98474
$536$	1998.75i	3.72901i
$537$	0	0
$538$	533.341	0.991340
$539$	− 1122.93i	− 2.08335i
$540$	0	0
$541$	−494.079	−0.913270	−0.456635	−	0.889654i	$-0.650945\pi$
$541$	−494.079	−0.913270	−0.456635	+	0.889654i	$0.650945\pi$
$542$	752.044i	1.38753i
$543$	0	0
$544$	1193.60	2.19412
$545$	642.259i	1.17846i
$546$	0	0
$547$	−673.432	−1.23114	−0.615568	−	0.788084i	$-0.711073\pi$
$547$	−673.432	−1.23114	−0.615568	+	0.788084i	$0.711073\pi$
$548$	1236.64i	2.25665i
$549$	0	0
$550$	3523.25	6.40591
$551$	455.736i	0.827106i
$552$	0	0
$553$	−715.798	−1.29439
$554$	− 795.667i	− 1.43622i
$555$	0	0
$556$	1075.46	1.93427
$557$	− 201.163i	− 0.361154i	−0.983561	−	0.180577i	$-0.942203\pi$
$557$	− 201.163i	− 0.361154i	0.983561	−	0.180577i	$-0.0577965\pi$
$558$	0	0
$559$	−339.567	−0.607455
$560$	− 6436.38i	− 11.4935i
$561$	0	0
$562$	366.814	0.652694
$563$	848.512i	1.50713i	0.657375	+	0.753563i	$0.271667\pi$
$563$	848.512i	1.50713i	−0.657375	+	0.753563i	$0.728333\pi$
$564$	0	0
$565$	421.456	0.745939
$566$	− 748.310i	− 1.32210i
$567$	0	0
$568$	−2988.96	−5.26225
$569$	1066.94i	1.87511i	0.347838	+	0.937555i	$0.386916\pi$
$569$	1066.94i	1.87511i	−0.347838	+	0.937555i	$0.613084\pi$
$570$	0	0
$571$	−998.217	−1.74819	−0.874095	−	0.485754i	$-0.838545\pi$
$571$	−998.217	−1.74819	−0.874095	+	0.485754i	$0.838545\pi$
$572$	1680.91i	2.93865i
$573$	0	0
$574$	798.152	1.39051
$575$	− 838.115i	− 1.45759i
$576$	0	0
$577$	−433.529	−0.751351	−0.375675	−	0.926751i	$-0.622589\pi$
$577$	−433.529	−0.751351	−0.375675	+	0.926751i	$0.622589\pi$
$578$	− 766.115i	− 1.32546i
$579$	0	0
$580$	1889.08	3.25703
$581$	577.792i	0.994478i
$582$	0	0
$583$	−447.864	−0.768206
$584$	− 1310.72i	− 2.24438i
$585$	0	0
$586$	−1369.29	−2.33667
$587$	− 643.448i	− 1.09616i	−0.836425	−	0.548081i	$-0.815358\pi$
$587$	− 643.448i	− 1.09616i	0.836425	−	0.548081i	$-0.184642\pi$
$588$	0	0
$589$	961.872	1.63306
$590$	2285.82i	3.87428i
$591$	0	0
$592$	−3124.00	−5.27703
$593$	− 398.873i	− 0.672635i	−0.941749	−	0.336318i	$-0.890818\pi$
$593$	− 398.873i	− 0.672635i	0.941749	−	0.336318i	$-0.109182\pi$
$594$	0	0
$595$	1019.41	1.71330
$596$	− 2171.35i	− 3.64320i
$597$	0	0
$598$	545.855	0.912800
$599$	− 1145.77i	− 1.91280i	−0.292055	−	0.956402i	$-0.594339\pi$
$599$	− 1145.77i	− 1.91280i	0.292055	−	0.956402i	$-0.405661\pi$
$600$	0	0
$601$	260.499	0.433443	0.216722	−	0.976233i	$-0.430464\pi$
$601$	260.499	0.433443	0.216722	+	0.976233i	$0.430464\pi$
$602$	1374.53i	2.28327i
$603$	0	0
$604$	−2983.46	−4.93950
$605$	758.267i	1.25333i
$606$	0	0
$607$	938.733	1.54651	0.773256	−	0.634094i	$-0.218627\pi$
$607$	938.733	1.54651	0.773256	+	0.634094i	$0.218627\pi$
$608$	3125.34i	5.14036i
$609$	0	0
$610$	−2358.62	−3.86659
$611$	− 49.4157i	− 0.0808767i
$612$	0	0
$613$	−608.571	−0.992774	−0.496387	−	0.868101i	$-0.665340\pi$
$613$	−608.571	−0.992774	−0.496387	+	0.868101i	$0.665340\pi$
$614$	− 587.708i	− 0.957179i
$615$	0	0
$616$	4319.77	7.01261
$617$	− 5.41005i	− 0.00876832i	−0.999990	−	0.00438416i	$-0.998604\pi$
$617$	− 5.41005i	− 0.00876832i	0.999990	−	0.00438416i	$-0.00139553\pi$
$618$	0	0
$619$	744.538	1.20281	0.601404	−	0.798945i	$-0.294608\pi$
$619$	744.538	1.20281	0.601404	+	0.798945i	$0.294608\pi$
$620$	− 3987.08i	− 6.43077i
$621$	0	0
$622$	−247.013	−0.397127
$623$	− 220.684i	− 0.354228i
$624$	0	0
$625$	1893.39	3.02942
$626$	143.261i	0.228852i
$627$	0	0
$628$	170.704	0.271821
$629$	− 494.789i	− 0.786628i
$630$	0	0
$631$	566.360	0.897560	0.448780	−	0.893642i	$-0.351859\pi$
$631$	566.360	0.897560	0.448780	+	0.893642i	$0.351859\pi$
$632$	− 1700.72i	− 2.69101i
$633$	0	0
$634$	−1545.26	−2.43731
$635$	− 106.445i	− 0.167630i
$636$	0	0
$637$	856.014	1.34382
$638$	1001.62i	1.56993i
$639$	0	0
$640$	4159.89	6.49983
$641$	1034.40i	1.61372i	0.590740	+	0.806862i	$0.298836\pi$
$641$	1034.40i	1.61372i	−0.590740	+	0.806862i	$0.701164\pi$
$642$	0	0
$643$	5.47434	0.00851375	0.00425688	−	0.999991i	$-0.498645\pi$
$643$	5.47434	0.00851375	0.00425688	+	0.999991i	$0.498645\pi$
$644$	− 1618.57i	− 2.51331i
$645$	0	0
$646$	−920.379	−1.42473
$647$	− 639.245i	− 0.988015i	−0.869458	−	0.494007i	$-0.835532\pi$
$647$	− 639.245i	− 0.988015i	0.869458	−	0.494007i	$-0.164468\pi$
$648$	0	0
$649$	−887.815	−1.36797
$650$	2685.79i	4.13198i
$651$	0	0
$652$	−2434.93	−3.73456
$653$	− 651.138i	− 0.997148i	−0.866847	−	0.498574i	$-0.833857\pi$
$653$	− 651.138i	− 0.997148i	0.866847	−	0.498574i	$-0.166143\pi$
$654$	0	0
$655$	−1486.25	−2.26909
$656$	1097.46i	1.67296i
$657$	0	0
$658$	−200.029	−0.303995
$659$	− 296.523i	− 0.449958i	−0.974364	−	0.224979i	$-0.927769\pi$
$659$	− 296.523i	− 0.449958i	0.974364	−	0.224979i	$-0.0722314\pi$
$660$	0	0
$661$	737.931	1.11639	0.558193	−	0.829711i	$-0.311495\pi$
$661$	737.931	1.11639	0.558193	+	0.829711i	$0.311495\pi$
$662$	144.516i	0.218302i
$663$	0	0
$664$	−1372.82	−2.06750
$665$	2669.24i	4.01390i
$666$	0	0
$667$	238.266	0.357221
$668$	− 2.71382i	− 0.00406260i
$669$	0	0
$670$	−2714.41	−4.05136
$671$	− 916.088i	− 1.36526i
$672$	0	0
$673$	−475.306	−0.706250	−0.353125	−	0.935576i	$-0.614881\pi$
$673$	−475.306	−0.706250	−0.353125	+	0.935576i	$0.614881\pi$
$674$	508.388i	0.754284i
$675$	0	0
$676$	570.055	0.843277
$677$	− 881.795i	− 1.30250i	−0.758862	−	0.651252i	$-0.774244\pi$
$677$	− 881.795i	− 1.30250i	0.758862	−	0.651252i	$-0.225756\pi$
$678$	0	0
$679$	816.702	1.20280
$680$	2422.10i	3.56191i
$681$	0	0
$682$	2114.01	3.09972
$683$	− 161.438i	− 0.236366i	−0.992992	−	0.118183i	$-0.962293\pi$
$683$	− 161.438i	− 0.236366i	0.992992	−	0.118183i	$-0.0377070\pi$
$684$	0	0
$685$	−1066.23	−1.55653
$686$	− 1319.93i	− 1.92409i
$687$	0	0
$688$	−1889.98	−2.74706
$689$	− 341.409i	− 0.495514i
$690$	0	0
$691$	366.744	0.530744	0.265372	−	0.964146i	$-0.414505\pi$
$691$	366.744	0.530744	0.265372	+	0.964146i	$0.414505\pi$
$692$	1946.44i	2.81278i
$693$	0	0
$694$	−1403.17	−2.02185
$695$	927.252i	1.33418i
$696$	0	0
$697$	−173.819	−0.249382
$698$	1830.23i	2.62211i
$699$	0	0
$700$	7963.94	11.3771
$701$	− 140.809i	− 0.200869i	−0.994944	−	0.100435i	$-0.967977\pi$
$701$	− 140.809i	− 0.200869i	0.994944	−	0.100435i	$-0.0320233\pi$
$702$	0	0
$703$	1295.56	1.84290
$704$	3452.90i	4.90469i
$705$	0	0
$706$	1022.26	1.44795
$707$	− 1475.27i	− 2.08667i
$708$	0	0
$709$	815.582	1.15033	0.575164	−	0.818038i	$-0.304938\pi$
$709$	815.582	1.15033	0.575164	+	0.818038i	$0.304938\pi$
$710$	− 4059.17i	− 5.71714i
$711$	0	0
$712$	524.340	0.736432
$713$	− 502.883i	− 0.705306i
$714$	0	0
$715$	−1449.27	−2.02695
$716$	1451.70i	2.02752i
$717$	0	0
$718$	211.430	0.294471
$719$	− 1073.20i	− 1.49263i	−0.665593	−	0.746315i	$-0.731821\pi$
$719$	− 1073.20i	− 1.49263i	0.665593	−	0.746315i	$-0.268179\pi$
$720$	0	0
$721$	467.802	0.648823
$722$	− 1013.87i	− 1.40426i
$723$	0	0
$724$	−180.414	−0.249191
$725$	1172.35i	1.61704i
$726$	0	0
$727$	−511.757	−0.703930	−0.351965	−	0.936013i	$-0.614486\pi$
$727$	−511.757	−0.703930	−0.351965	+	0.936013i	$0.614486\pi$
$728$	3292.98i	4.52332i
$729$	0	0
$730$	1780.02	2.43839
$731$	− 299.340i	− 0.409494i
$732$	0	0
$733$	630.537	0.860214	0.430107	−	0.902778i	$-0.358476\pi$
$733$	630.537	0.860214	0.430107	+	0.902778i	$0.358476\pi$
$734$	1316.74i	1.79393i
$735$	0	0
$736$	1633.98	2.22008
$737$	− 1054.28i	− 1.43050i
$738$	0	0
$739$	253.573	0.343130	0.171565	−	0.985173i	$-0.445118\pi$
$739$	253.573	0.343130	0.171565	+	0.985173i	$0.445118\pi$
$740$	− 5370.26i	− 7.25711i
$741$	0	0
$742$	−1381.98	−1.86251
$743$	1027.89i	1.38343i	0.722172	+	0.691714i	$0.243144\pi$
$743$	1027.89i	1.38343i	−0.722172	+	0.691714i	$0.756856\pi$
$744$	0	0
$745$	1872.12	2.51292
$746$	− 649.723i	− 0.870942i
$747$	0	0
$748$	−1481.78	−1.98099
$749$	− 1913.80i	− 2.55514i
$750$	0	0
$751$	−1089.29	−1.45045	−0.725224	−	0.688513i	$-0.758264\pi$
$751$	−1089.29	−1.45045	−0.725224	+	0.688513i	$0.758264\pi$
$752$	− 275.040i	− 0.365744i
$753$	0	0
$754$	−763.539	−1.01265
$755$	− 2572.32i	− 3.40705i
$756$	0	0
$757$	1244.71	1.64427	0.822136	−	0.569291i	$-0.192782\pi$
$757$	1244.71	1.64427	0.822136	+	0.569291i	$0.192782\pi$
$758$	− 114.514i	− 0.151075i
$759$	0	0
$760$	−6342.05	−8.34480
$761$	− 1341.43i	− 1.76272i	−0.472446	−	0.881359i	$-0.656629\pi$
$761$	− 1341.43i	− 1.76272i	0.472446	−	0.881359i	$-0.343371\pi$
$762$	0	0
$763$	−769.744	−1.00884
$764$	976.277i	1.27785i
$765$	0	0
$766$	−2594.56	−3.38715
$767$	− 676.786i	− 0.882380i
$768$	0	0
$769$	381.351	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$769$	381.351	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$770$	5866.48i	7.61880i
$771$	0	0
$772$	2887.46	3.74023
$773$	− 419.868i	− 0.543167i	−0.962415	−	0.271584i	$-0.912453\pi$
$773$	− 419.868i	− 0.543167i	0.962415	−	0.271584i	$-0.0875474\pi$
$774$	0	0
$775$	2474.36	3.19272
$776$	1940.46i	2.50060i
$777$	0	0
$778$	2521.04	3.24042
$779$	− 455.129i	− 0.584248i
$780$	0	0
$781$	1576.58	2.01867
$782$	481.190i	0.615332i
$783$	0	0
$784$	4764.44	6.07709
$785$	147.180i	0.187490i
$786$	0	0
$787$	−1063.32	−1.35110	−0.675551	−	0.737313i	$-0.736094\pi$
$787$	−1063.32	−1.35110	−0.675551	+	0.737313i	$0.736094\pi$
$788$	2062.00i	2.61676i
$789$	0	0
$790$	2309.67	2.92363
$791$	505.113i	0.638575i
$792$	0	0
$793$	698.338	0.880628
$794$	1680.37i	2.11633i
$795$	0	0
$796$	−961.096	−1.20741
$797$	324.161i	0.406727i	0.979103	+	0.203363i	$0.0651873\pi$
$797$	324.161i	0.406727i	−0.979103	+	0.203363i	$0.934813\pi$
$798$	0	0
$799$	43.5616	0.0545202
$800$	8039.74i	10.0497i
$801$	0	0
$802$	−701.836	−0.875108
$803$	691.362i	0.860973i
$804$	0	0
$805$	1395.53	1.73357
$806$	1611.52i	1.99940i
$807$	0	0
$808$	3505.21	4.33814
$809$	− 149.456i	− 0.184741i	−0.995725	−	0.0923707i	$-0.970556\pi$
$809$	− 149.456i	− 0.184741i	0.995725	−	0.0923707i	$-0.0294445\pi$
$810$	0	0
$811$	873.810	1.07745	0.538724	−	0.842483i	$-0.318907\pi$
$811$	873.810	1.07745	0.538724	+	0.842483i	$0.318907\pi$
$812$	2264.05i	2.78824i
$813$	0	0
$814$	2847.39	3.49803
$815$	− 2099.38i	− 2.57593i
$816$	0	0
$817$	783.796	0.959359
$818$	− 1806.46i	− 2.20838i
$819$	0	0
$820$	−1886.57	−2.30069
$821$	671.265i	0.817619i	0.912620	+	0.408809i	$0.134056\pi$
$821$	671.265i	0.817619i	−0.912620	+	0.408809i	$0.865944\pi$
$822$	0	0
$823$	−1205.43	−1.46467	−0.732337	−	0.680942i	$-0.761571\pi$
$823$	−1205.43	−1.46467	−0.732337	+	0.680942i	$0.761571\pi$
$824$	1111.48i	1.34889i
$825$	0	0
$826$	−2739.55	−3.31665
$827$	− 380.902i	− 0.460583i	−0.973122	−	0.230291i	$-0.926032\pi$
$827$	− 380.902i	− 0.460583i	0.973122	−	0.230291i	$-0.0739679\pi$
$828$	0	0
$829$	1532.03	1.84804	0.924020	−	0.382343i	$-0.124883\pi$
$829$	1532.03	1.84804	0.924020	+	0.382343i	$0.124883\pi$
$830$	− 1864.36i	− 2.24622i
$831$	0	0
$832$	−2632.16	−3.16366
$833$	754.606i	0.905889i
$834$	0	0
$835$	2.33984	0.00280220
$836$	− 3879.91i	− 4.64104i
$837$	0	0
$838$	1771.03	2.11340
$839$	270.488i	0.322394i	0.986922	+	0.161197i	$0.0515354\pi$
$839$	270.488i	0.322394i	−0.986922	+	0.161197i	$0.948465\pi$
$840$	0	0
$841$	507.714	0.603703
$842$	2538.83i	3.01524i
$843$	0	0
$844$	−460.361	−0.545452
$845$	491.498i	0.581655i
$846$	0	0
$847$	−908.779	−1.07294
$848$	− 1900.23i	− 2.24083i
$849$	0	0
$850$	−2367.62	−2.78543
$851$	− 677.342i	− 0.795936i
$852$	0	0
$853$	−929.016	−1.08912	−0.544558	−	0.838723i	$-0.683303\pi$
$853$	−929.016	−1.08912	−0.544558	+	0.838723i	$0.683303\pi$
$854$	− 2826.79i	− 3.31006i
$855$	0	0
$856$	4547.14	5.31208
$857$	207.368i	0.241969i	0.992654	+	0.120985i	$0.0386052\pi$
$857$	207.368i	0.241969i	−0.992654	+	0.120985i	$0.961395\pi$
$858$	0	0
$859$	−493.994	−0.575080	−0.287540	−	0.957769i	$-0.592837\pi$
$859$	−493.994	−0.575080	−0.287540	+	0.957769i	$0.592837\pi$
$860$	− 3248.93i	− 3.77783i
$861$	0	0
$862$	−1141.50	−1.32425
$863$	− 294.120i	− 0.340811i	−0.985374	−	0.170405i	$-0.945492\pi$
$863$	− 294.120i	− 0.340811i	0.985374	−	0.170405i	$-0.0545077\pi$
$864$	0	0
$865$	−1678.21	−1.94013
$866$	− 1391.90i	− 1.60727i
$867$	0	0
$868$	4778.50	5.50518
$869$	897.075i	1.03231i
$870$	0	0
$871$	803.681	0.922711
$872$	− 1828.89i	− 2.09735i
$873$	0	0
$874$	−1259.95	−1.44159
$875$	4193.31i	4.79236i
$876$	0	0
$877$	−1303.57	−1.48640	−0.743198	−	0.669071i	$-0.766692\pi$
$877$	−1303.57	−1.48640	−0.743198	+	0.669071i	$0.766692\pi$
$878$	− 2130.55i	− 2.42660i
$879$	0	0
$880$	−8066.41	−9.16637
$881$	− 1042.89i	− 1.18376i	−0.806026	−	0.591880i	$-0.798386\pi$
$881$	− 1042.89i	− 1.18376i	0.806026	−	0.591880i	$-0.201614\pi$
$882$	0	0
$883$	−895.269	−1.01390	−0.506948	−	0.861977i	$-0.669226\pi$
$883$	−895.269	−1.01390	−0.506948	+	0.861977i	$0.669226\pi$
$884$	− 1129.57i	− 1.27779i
$885$	0	0
$886$	1884.04	2.12645
$887$	− 490.009i	− 0.552434i	−0.961095	−	0.276217i	$-0.910919\pi$
$887$	− 490.009i	− 0.552434i	0.961095	−	0.276217i	$-0.0890808\pi$
$888$	0	0
$889$	127.574	0.143503
$890$	712.082i	0.800092i
$891$	0	0
$892$	−1833.85	−2.05589
$893$	114.062i	0.127729i
$894$	0	0
$895$	−1251.65	−1.39849
$896$	4985.61i	5.56430i
$897$	0	0
$898$	−1676.24	−1.86663
$899$	703.431i	0.782459i
$900$	0	0
$901$	300.964	0.334033
$902$	− 1000.29i	− 1.10896i
$903$	0	0
$904$	−1200.13	−1.32758
$905$	− 155.552i	− 0.171881i
$906$	0	0
$907$	−1592.09	−1.75534	−0.877668	−	0.479270i	$-0.840902\pi$
$907$	−1592.09	−1.75534	−0.877668	+	0.479270i	$0.840902\pi$
$908$	4098.88i	4.51419i
$909$	0	0
$910$	−4472.05	−4.91434
$911$	− 804.999i	− 0.883643i	−0.897103	−	0.441821i	$-0.854332\pi$
$911$	− 804.999i	− 0.883643i	0.897103	−	0.441821i	$-0.145668\pi$
$912$	0	0
$913$	724.118	0.793120
$914$	− 2104.75i	− 2.30279i
$915$	0	0
$916$	1583.08	1.72826
$917$	− 1781.27i	− 1.94250i
$918$	0	0
$919$	828.169	0.901164	0.450582	−	0.892735i	$-0.351217\pi$
$919$	828.169	0.901164	0.450582	+	0.892735i	$0.351217\pi$
$920$	3315.73i	3.60406i
$921$	0	0
$922$	−1063.58	−1.15355
$923$	1201.84i	1.30210i
$924$	0	0
$925$	3332.75	3.60297
$926$	− 857.597i	− 0.926131i
$927$	0	0
$928$	−2285.60	−2.46294
$929$	− 229.494i	− 0.247033i	−0.992342	−	0.123517i	$-0.960583\pi$
$929$	− 229.494i	− 0.247033i	0.992342	−	0.123517i	$-0.0394173\pi$
$930$	0	0
$931$	−1975.87	−2.12231
$932$	− 2406.94i	− 2.58255i
$933$	0	0
$934$	−1496.72	−1.60249
$935$	− 1277.58i	− 1.36640i
$936$	0	0
$937$	−1193.95	−1.27423	−0.637115	−	0.770769i	$-0.719872\pi$
$937$	−1193.95	−1.27423	−0.637115	+	0.770769i	$0.719872\pi$
$938$	− 3253.21i	− 3.46824i
$939$	0	0
$940$	472.802	0.502981
$941$	1658.41i	1.76239i	0.472752	+	0.881196i	$0.343261\pi$
$941$	1658.41i	1.76239i	−0.472752	+	0.881196i	$0.656739\pi$
$942$	0	0
$943$	−237.949	−0.252332
$944$	− 3766.88i	− 3.99034i
$945$	0	0
$946$	1722.63	1.82096
$947$	106.033i	0.111968i	0.998432	+	0.0559839i	$0.0178295\pi$
$947$	106.033i	0.111968i	−0.998432	+	0.0559839i	$0.982170\pi$
$948$	0	0
$949$	−527.028	−0.555351
$950$	− 6199.39i	− 6.52568i
$951$	0	0
$952$	−2902.88	−3.04924
$953$	− 1530.38i	− 1.60586i	−0.596073	−	0.802930i	$-0.703273\pi$
$953$	− 1530.38i	− 1.60586i	0.596073	−	0.802930i	$-0.296727\pi$
$954$	0	0
$955$	−841.741	−0.881404
$956$	4764.09i	4.98336i
$957$	0	0
$958$	193.957	0.202460
$959$	− 1277.87i	− 1.33250i
$960$	0	0
$961$	523.656	0.544907
$962$	2170.58i	2.25632i
$963$	0	0
$964$	−2177.51	−2.25883
$965$	2489.55i	2.57984i
$966$	0	0
$967$	886.692	0.916951	0.458475	−	0.888707i	$-0.348396\pi$
$967$	886.692	0.916951	0.458475	+	0.888707i	$0.348396\pi$
$968$	− 2159.24i	− 2.23062i
$969$	0	0
$970$	−2635.25	−2.71676
$971$	819.330i	0.843800i	0.906642	+	0.421900i	$0.138637\pi$
$971$	819.330i	0.843800i	−0.906642	+	0.421900i	$0.861363\pi$
$972$	0	0
$973$	−1111.31	−1.14215
$974$	− 1645.36i	− 1.68928i
$975$	0	0
$976$	3886.84	3.98242
$977$	841.090i	0.860890i	0.902617	+	0.430445i	$0.141643\pi$
$977$	841.090i	0.860890i	−0.902617	+	0.430445i	$0.858357\pi$
$978$	0	0
$979$	−276.573	−0.282505
$980$	8190.22i	8.35736i
$981$	0	0
$982$	2079.24	2.11735
$983$	1409.80i	1.43418i	0.696980	+	0.717091i	$0.254527\pi$
$983$	1409.80i	1.43418i	−0.696980	+	0.717091i	$0.745473\pi$
$984$	0	0
$985$	−1777.85	−1.80492
$986$	− 673.086i	− 0.682643i
$987$	0	0
$988$	2957.67	2.99360
$989$	− 409.782i	− 0.414340i
$990$	0	0
$991$	−1664.02	−1.67913	−0.839567	−	0.543256i	$-0.817191\pi$
$991$	−1664.02	−1.67913	−0.839567	+	0.543256i	$0.817191\pi$
$992$	4823.98i	4.86288i
$993$	0	0
$994$	4864.90	4.89426
$995$	− 828.652i	− 0.832816i
$996$	0	0
$997$	−193.171	−0.193752	−0.0968761	−	0.995296i	$-0.530885\pi$
$997$	−193.171	−0.193752	−0.0968761	+	0.995296i	$0.530885\pi$
$998$	− 833.765i	− 0.835436i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.2	✓	84
3.2	odd	2	inner	1143.3.b.a.890.83	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.2	✓	84	1.1	even	1	trivial
1143.3.b.a.890.83	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)

\(f(q)\)	\(=\)	\(q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} +O(q^{10})\)	\(q-3.86719i q^{2} -10.9551 q^{4} -9.44547i q^{5} +11.3204 q^{7} +26.8969i q^{8} -36.5274 q^{10} -14.1873i q^{11} +10.8150 q^{13} -43.7780i q^{14} +60.1947 q^{16} +9.53381i q^{17} -24.9634 q^{19} +103.477i q^{20} -54.8648 q^{22} +13.0513i q^{23} -64.2169 q^{25} -41.8237i q^{26} -124.016 q^{28} -18.2561i q^{29} -38.5312 q^{31} -125.197i q^{32} +36.8691 q^{34} -106.926i q^{35} -51.8983 q^{37} +96.5383i q^{38} +254.054 q^{40} +18.2318i q^{41} -31.3978 q^{43} +155.424i q^{44} +50.4719 q^{46} -4.56917i q^{47} +79.1505 q^{49} +248.339i q^{50} -118.480 q^{52} -31.5680i q^{53} -134.005 q^{55} +304.482i q^{56} -70.5998 q^{58} -62.5783i q^{59} +64.5712 q^{61} +149.008i q^{62} -243.380 q^{64} -102.153i q^{65} +74.3116 q^{67} -104.444i q^{68} -413.503 q^{70} +111.127i q^{71} -48.7312 q^{73} +200.701i q^{74} +273.478 q^{76} -160.605i q^{77} -63.2310 q^{79} -568.567i q^{80} +70.5059 q^{82} +51.0401i q^{83} +90.0514 q^{85} +121.421i q^{86} +381.593 q^{88} -19.4944i q^{89} +122.430 q^{91} -142.979i q^{92} -17.6698 q^{94} +235.791i q^{95} +72.1446 q^{97} -306.090i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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