LMFDB - Embedded newform 1143.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.a (trivial)

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:	yes
Analytic conductor:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 $ x^9 - 2x^8 - 14x^7 + 26x^6 + 59x^5 - 99x^4 - 66x^3 + 102x^2 - 24x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.55353$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55353 q^{2} +4.52051 q^{4} -2.44899 q^{5} +3.89705 q^{7} +6.43620 q^{8} +O(q^{10})$	\(q+2.55353 q^{2} +4.52051 q^{4} -2.44899 q^{5} +3.89705 q^{7} +6.43620 q^{8} -6.25358 q^{10} -3.82323 q^{11} +4.41381 q^{13} +9.95122 q^{14} +7.39400 q^{16} +7.77710 q^{17} -5.77710 q^{19} -11.0707 q^{20} -9.76272 q^{22} +4.71830 q^{23} +0.997574 q^{25} +11.2708 q^{26} +17.6166 q^{28} +1.05007 q^{29} -0.879109 q^{31} +6.00840 q^{32} +19.8590 q^{34} -9.54384 q^{35} +2.79920 q^{37} -14.7520 q^{38} -15.7622 q^{40} -0.154217 q^{41} -12.0189 q^{43} -17.2829 q^{44} +12.0483 q^{46} -5.65025 q^{47} +8.18697 q^{49} +2.54734 q^{50} +19.9527 q^{52} +5.76838 q^{53} +9.36306 q^{55} +25.0822 q^{56} +2.68139 q^{58} -4.10330 q^{59} +4.94156 q^{61} -2.24483 q^{62} +0.554626 q^{64} -10.8094 q^{65} -11.4692 q^{67} +35.1565 q^{68} -24.3705 q^{70} -14.7531 q^{71} -12.0563 q^{73} +7.14784 q^{74} -26.1154 q^{76} -14.8993 q^{77} +6.49805 q^{79} -18.1079 q^{80} -0.393799 q^{82} +14.3278 q^{83} -19.0461 q^{85} -30.6907 q^{86} -24.6071 q^{88} -2.55754 q^{89} +17.2008 q^{91} +21.3292 q^{92} -14.4281 q^{94} +14.1481 q^{95} -2.23970 q^{97} +20.9057 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8	$ 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8} - 4 q^{10} - 8 q^{11} + 14 q^{13} - 4 q^{14} + 32 q^{16} + 6 q^{17} + 12 q^{19} + 28 q^{20} - 18 q^{22} + 4 q^{23} + 21 q^{25} + 14 q^{26} + 8 q^{29} + 4 q^{31} + 29 q^{32} - 3 q^{34} - 6 q^{35} + 22 q^{37} + 7 q^{38} + 2 q^{41} + 6 q^{43} - 17 q^{44} - 10 q^{46} + 2 q^{47} + 23 q^{49} + 20 q^{50} - 9 q^{52} + 12 q^{53} - 22 q^{55} - 18 q^{56} - 28 q^{58} + 6 q^{59} + 2 q^{61} + 15 q^{62} + 24 q^{64} - 4 q^{65} + 18 q^{67} + 24 q^{68} - 72 q^{70} - 24 q^{71} + 14 q^{73} - 3 q^{74} + 4 q^{76} + 18 q^{77} + 12 q^{79} + 86 q^{80} + 4 q^{82} + 20 q^{83} - 24 q^{85} - 16 q^{86} - 55 q^{88} + 30 q^{89} + 14 q^{91} + 46 q^{92} - 66 q^{94} + 32 q^{95} + 12 q^{97} + 62 q^{98}+O(q^{100}) $ 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8 - 4 * q^10 - 8 * q^11 + 14 * q^13 - 4 * q^14 + 32 * q^16 + 6 * q^17 + 12 * q^19 + 28 * q^20 - 18 * q^22 + 4 * q^23 + 21 * q^25 + 14 * q^26 + 8 * q^29 + 4 * q^31 + 29 * q^32 - 3 * q^34 - 6 * q^35 + 22 * q^37 + 7 * q^38 + 2 * q^41 + 6 * q^43 - 17 * q^44 - 10 * q^46 + 2 * q^47 + 23 * q^49 + 20 * q^50 - 9 * q^52 + 12 * q^53 - 22 * q^55 - 18 * q^56 - 28 * q^58 + 6 * q^59 + 2 * q^61 + 15 * q^62 + 24 * q^64 - 4 * q^65 + 18 * q^67 + 24 * q^68 - 72 * q^70 - 24 * q^71 + 14 * q^73 - 3 * q^74 + 4 * q^76 + 18 * q^77 + 12 * q^79 + 86 * q^80 + 4 * q^82 + 20 * q^83 - 24 * q^85 - 16 * q^86 - 55 * q^88 + 30 * q^89 + 14 * q^91 + 46 * q^92 - 66 * q^94 + 32 * q^95 + 12 * q^97 + 62 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.55353	1.80562	0.902809	−	0.430042i	$-0.141501\pi$
$2$	2.55353	1.80562	0.902809	+	0.430042i	$0.141501\pi$
$3$	0	0
$3$	0	0
$4$	4.52051	2.26026
$5$	−2.44899	−1.09522	−0.547612	−	0.836733i	$-0.684463\pi$
$5$	−2.44899	−1.09522	−0.547612	+	0.836733i	$0.684463\pi$
$6$	0	0
$7$	3.89705	1.47294	0.736472	−	0.676468i	$-0.236490\pi$
$7$	3.89705	1.47294	0.736472	+	0.676468i	$0.236490\pi$
$8$	6.43620	2.27554
$9$	0	0
$10$	−6.25358	−1.97756
$11$	−3.82323	−1.15275	−0.576373	−	0.817187i	$-0.695533\pi$
$11$	−3.82323	−1.15275	−0.576373	+	0.817187i	$0.695533\pi$
$12$	0	0
$13$	4.41381	1.22417	0.612085	−	0.790792i	$-0.290331\pi$
$13$	4.41381	1.22417	0.612085	+	0.790792i	$0.290331\pi$
$14$	9.95122	2.65958
$15$	0	0
$16$	7.39400	1.84850
$17$	7.77710	1.88622	0.943112	−	0.332476i	$-0.107884\pi$
$17$	7.77710	1.88622	0.943112	+	0.332476i	$0.107884\pi$
$18$	0	0
$19$	−5.77710	−1.32536	−0.662679	−	0.748904i	$-0.730580\pi$
$19$	−5.77710	−1.32536	−0.662679	+	0.748904i	$0.730580\pi$
$20$	−11.0707	−2.47549
$21$	0	0
$22$	−9.76272	−2.08142
$23$	4.71830	0.983835	0.491917	−	0.870642i	$-0.336296\pi$
$23$	4.71830	0.983835	0.491917	+	0.870642i	$0.336296\pi$
$24$	0	0
$25$	0.997574	0.199515
$26$	11.2708	2.21038
$27$	0	0
$28$	17.6166	3.32923
$29$	1.05007	0.194993	0.0974966	−	0.995236i	$-0.468916\pi$
$29$	1.05007	0.194993	0.0974966	+	0.995236i	$0.468916\pi$
$30$	0	0
$31$	−0.879109	−0.157893	−0.0789463	−	0.996879i	$-0.525156\pi$
$31$	−0.879109	−0.157893	−0.0789463	+	0.996879i	$0.525156\pi$
$32$	6.00840	1.06215
$33$	0	0
$34$	19.8590	3.40580
$35$	−9.54384	−1.61320
$36$	0	0
$37$	2.79920	0.460186	0.230093	−	0.973169i	$-0.426097\pi$
$37$	2.79920	0.460186	0.230093	+	0.973169i	$0.426097\pi$
$38$	−14.7520	−2.39309
$39$	0	0
$40$	−15.7622	−2.49223
$41$	−0.154217	−0.0240847	−0.0120424	−	0.999927i	$-0.503833\pi$
$41$	−0.154217	−0.0240847	−0.0120424	+	0.999927i	$0.503833\pi$
$42$	0	0
$43$	−12.0189	−1.83287	−0.916434	−	0.400185i	$-0.868946\pi$
$43$	−12.0189	−1.83287	−0.916434	+	0.400185i	$0.868946\pi$
$44$	−17.2829	−2.60550
$45$	0	0
$46$	12.0483	1.77643
$47$	−5.65025	−0.824174	−0.412087	−	0.911145i	$-0.635200\pi$
$47$	−5.65025	−0.824174	−0.412087	+	0.911145i	$0.635200\pi$
$48$	0	0
$49$	8.18697	1.16957
$50$	2.54734	0.360248
$51$	0	0
$52$	19.9527	2.76694
$53$	5.76838	0.792347	0.396174	−	0.918176i	$-0.370338\pi$
$53$	5.76838	0.792347	0.396174	+	0.918176i	$0.370338\pi$
$54$	0	0
$55$	9.36306	1.26252
$56$	25.0822	3.35175
$57$	0	0
$58$	2.68139	0.352083
$59$	−4.10330	−0.534205	−0.267102	−	0.963668i	$-0.586066\pi$
$59$	−4.10330	−0.534205	−0.267102	+	0.963668i	$0.586066\pi$
$60$	0	0
$61$	4.94156	0.632701	0.316351	−	0.948642i	$-0.397542\pi$
$61$	4.94156	0.632701	0.316351	+	0.948642i	$0.397542\pi$
$62$	−2.24483	−0.285094
$63$	0	0
$64$	0.554626	0.0693283
$65$	−10.8094	−1.34074
$66$	0	0
$67$	−11.4692	−1.40118	−0.700592	−	0.713562i	$-0.747080\pi$
$67$	−11.4692	−1.40118	−0.700592	+	0.713562i	$0.747080\pi$
$68$	35.1565	4.26335
$69$	0	0
$70$	−24.3705	−2.91283
$71$	−14.7531	−1.75087	−0.875433	−	0.483339i	$-0.839424\pi$
$71$	−14.7531	−1.75087	−0.875433	+	0.483339i	$0.839424\pi$
$72$	0	0
$73$	−12.0563	−1.41108	−0.705542	−	0.708668i	$-0.749296\pi$
$73$	−12.0563	−1.41108	−0.705542	+	0.708668i	$0.749296\pi$
$74$	7.14784	0.830919
$75$	0	0
$76$	−26.1154	−2.99565
$77$	−14.8993	−1.69793
$78$	0	0
$79$	6.49805	0.731088	0.365544	−	0.930794i	$-0.380883\pi$
$79$	6.49805	0.731088	0.365544	+	0.930794i	$0.380883\pi$
$80$	−18.1079	−2.02452
$81$	0	0
$82$	−0.393799	−0.0434878
$83$	14.3278	1.57268	0.786338	−	0.617796i	$-0.211974\pi$
$83$	14.3278	1.57268	0.786338	+	0.617796i	$0.211974\pi$
$84$	0	0
$85$	−19.0461	−2.06584
$86$	−30.6907	−3.30946
$87$	0	0
$88$	−24.6071	−2.62312
$89$	−2.55754	−0.271098	−0.135549	−	0.990771i	$-0.543280\pi$
$89$	−2.55754	−0.271098	−0.135549	+	0.990771i	$0.543280\pi$
$90$	0	0
$91$	17.2008	1.80314
$92$	21.3292	2.22372
$93$	0	0
$94$	−14.4281	−1.48814
$95$	14.1481	1.45156
$96$	0	0
$97$	−2.23970	−0.227408	−0.113704	−	0.993515i	$-0.536271\pi$
$97$	−2.23970	−0.227408	−0.113704	+	0.993515i	$0.536271\pi$
$98$	20.9057	2.11179
$99$	0	0
$100$	4.50955	0.450955
$101$	6.82762	0.679374	0.339687	−	0.940539i	$-0.389679\pi$
$101$	6.82762	0.679374	0.339687	+	0.940539i	$0.389679\pi$
$102$	0	0
$103$	12.4296	1.22472	0.612361	−	0.790578i	$-0.290220\pi$
$103$	12.4296	1.22472	0.612361	+	0.790578i	$0.290220\pi$
$104$	28.4082	2.78565
$105$	0	0
$106$	14.7297	1.43068
$107$	−15.4762	−1.49614	−0.748070	−	0.663620i	$-0.769019\pi$
$107$	−15.4762	−1.49614	−0.748070	+	0.663620i	$0.769019\pi$
$108$	0	0
$109$	−11.3499	−1.08713	−0.543563	−	0.839368i	$-0.682925\pi$
$109$	−11.3499	−1.08713	−0.543563	+	0.839368i	$0.682925\pi$
$110$	23.9089	2.27962
$111$	0	0
$112$	28.8148	2.72274
$113$	−2.22852	−0.209642	−0.104821	−	0.994491i	$-0.533427\pi$
$113$	−2.22852	−0.209642	−0.104821	+	0.994491i	$0.533427\pi$
$114$	0	0
$115$	−11.5551	−1.07752
$116$	4.74686	0.440735
$117$	0	0
$118$	−10.4779	−0.964569
$119$	30.3077	2.77830
$120$	0	0
$121$	3.61707	0.328825
$122$	12.6184	1.14242
$123$	0	0
$124$	−3.97402	−0.356878
$125$	9.80192	0.876710
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.6006	−0.936965
$129$	0	0
$130$	−27.6021	−2.42087
$131$	0.0851636	0.00744078	0.00372039	−	0.999993i	$-0.498816\pi$
$131$	0.0851636	0.00744078	0.00372039	+	0.999993i	$0.498816\pi$
$132$	0	0
$133$	−22.5136	−1.95218
$134$	−29.2869	−2.53000
$135$	0	0
$136$	50.0550	4.29218
$137$	−0.378848	−0.0323672	−0.0161836	−	0.999869i	$-0.505152\pi$
$137$	−0.378848	−0.0323672	−0.0161836	+	0.999869i	$0.505152\pi$
$138$	0	0
$139$	−3.02979	−0.256983	−0.128492	−	0.991711i	$-0.541014\pi$
$139$	−3.02979	−0.256983	−0.128492	+	0.991711i	$0.541014\pi$
$140$	−43.1431	−3.64625
$141$	0	0
$142$	−37.6724	−3.16140
$143$	−16.8750	−1.41116
$144$	0	0
$145$	−2.57162	−0.213561
$146$	−30.7861	−2.54788
$147$	0	0
$148$	12.6538	1.04014
$149$	−7.96111	−0.652200	−0.326100	−	0.945335i	$-0.605735\pi$
$149$	−7.96111	−0.652200	−0.326100	+	0.945335i	$0.605735\pi$
$150$	0	0
$151$	20.9642	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$151$	20.9642	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$152$	−37.1826	−3.01590
$153$	0	0
$154$	−38.0458	−3.06582
$155$	2.15293	0.172928
$156$	0	0
$157$	3.36423	0.268495	0.134247	−	0.990948i	$-0.457138\pi$
$157$	3.36423	0.268495	0.134247	+	0.990948i	$0.457138\pi$
$158$	16.5930	1.32007
$159$	0	0
$160$	−14.7145	−1.16329
$161$	18.3875	1.44913
$162$	0	0
$163$	4.23778	0.331928	0.165964	−	0.986132i	$-0.446926\pi$
$163$	4.23778	0.331928	0.165964	+	0.986132i	$0.446926\pi$
$164$	−0.697142	−0.0544376
$165$	0	0
$166$	36.5864	2.83965
$167$	−12.5341	−0.969918	−0.484959	−	0.874537i	$-0.661165\pi$
$167$	−12.5341	−0.969918	−0.484959	+	0.874537i	$0.661165\pi$
$168$	0	0
$169$	6.48173	0.498594
$170$	−48.6347	−3.73011
$171$	0	0
$172$	−54.3317	−4.14275
$173$	9.15129	0.695760	0.347880	−	0.937539i	$-0.386902\pi$
$173$	9.15129	0.695760	0.347880	+	0.937539i	$0.386902\pi$
$174$	0	0
$175$	3.88759	0.293874
$176$	−28.2690	−2.13085
$177$	0	0
$178$	−6.53074	−0.489500
$179$	−17.5670	−1.31302	−0.656511	−	0.754317i	$-0.727968\pi$
$179$	−17.5670	−1.31302	−0.656511	+	0.754317i	$0.727968\pi$
$180$	0	0
$181$	6.77582	0.503643	0.251822	−	0.967774i	$-0.418970\pi$
$181$	6.77582	0.503643	0.251822	+	0.967774i	$0.418970\pi$
$182$	43.9228	3.25578
$183$	0	0
$184$	30.3680	2.23876
$185$	−6.85522	−0.504006
$186$	0	0
$187$	−29.7336	−2.17434
$188$	−25.5420	−1.86284
$189$	0	0
$190$	36.1275	2.62097
$191$	−1.65216	−0.119546	−0.0597730	−	0.998212i	$-0.519038\pi$
$191$	−1.65216	−0.119546	−0.0597730	+	0.998212i	$0.519038\pi$
$192$	0	0
$193$	−22.2545	−1.60192	−0.800959	−	0.598720i	$-0.795676\pi$
$193$	−22.2545	−1.60192	−0.800959	+	0.598720i	$0.795676\pi$
$194$	−5.71915	−0.410611
$195$	0	0
$196$	37.0093	2.64352
$197$	−14.8955	−1.06126	−0.530630	−	0.847603i	$-0.678045\pi$
$197$	−14.8955	−1.06126	−0.530630	+	0.847603i	$0.678045\pi$
$198$	0	0
$199$	−7.89145	−0.559410	−0.279705	−	0.960086i	$-0.590237\pi$
$199$	−7.89145	−0.559410	−0.279705	+	0.960086i	$0.590237\pi$
$200$	6.42059	0.454004
$201$	0	0
$202$	17.4345	1.22669
$203$	4.09217	0.287214
$204$	0	0
$205$	0.377678	0.0263781
$206$	31.7393	2.21138
$207$	0	0
$208$	32.6357	2.26288
$209$	22.0872	1.52780
$210$	0	0
$211$	10.3296	0.711117	0.355558	−	0.934654i	$-0.384291\pi$
$211$	10.3296	0.711117	0.355558	+	0.934654i	$0.384291\pi$
$212$	26.0760	1.79091
$213$	0	0
$214$	−39.5189	−2.70146
$215$	29.4343	2.00740
$216$	0	0
$217$	−3.42593	−0.232567
$218$	−28.9824	−1.96293
$219$	0	0
$220$	42.3258	2.85361
$221$	34.3266	2.30906
$222$	0	0
$223$	9.21433	0.617037	0.308518	−	0.951218i	$-0.400167\pi$
$223$	9.21433	0.617037	0.308518	+	0.951218i	$0.400167\pi$
$224$	23.4150	1.56448
$225$	0	0
$226$	−5.69060	−0.378533
$227$	6.19616	0.411254	0.205627	−	0.978630i	$-0.434077\pi$
$227$	6.19616	0.411254	0.205627	+	0.978630i	$0.434077\pi$
$228$	0	0
$229$	10.3639	0.684864	0.342432	−	0.939543i	$-0.388749\pi$
$229$	10.3639	0.684864	0.342432	+	0.939543i	$0.388749\pi$
$230$	−29.5063	−1.94559
$231$	0	0
$232$	6.75847	0.443715
$233$	7.72395	0.506012	0.253006	−	0.967465i	$-0.418581\pi$
$233$	7.72395	0.506012	0.253006	+	0.967465i	$0.418581\pi$
$234$	0	0
$235$	13.8374	0.902654
$236$	−18.5490	−1.20744
$237$	0	0
$238$	77.3916	5.01655
$239$	12.5030	0.808753	0.404377	−	0.914593i	$-0.367488\pi$
$239$	12.5030	0.808753	0.404377	+	0.914593i	$0.367488\pi$
$240$	0	0
$241$	−1.33181	−0.0857894	−0.0428947	−	0.999080i	$-0.513658\pi$
$241$	−1.33181	−0.0857894	−0.0428947	+	0.999080i	$0.513658\pi$
$242$	9.23629	0.593731
$243$	0	0
$244$	22.3384	1.43007
$245$	−20.0498	−1.28094
$246$	0	0
$247$	−25.4990	−1.62246
$248$	−5.65812	−0.359291
$249$	0	0
$250$	25.0295	1.58300
$251$	−5.84067	−0.368660	−0.184330	−	0.982864i	$-0.559012\pi$
$251$	−5.84067	−0.368660	−0.184330	+	0.982864i	$0.559012\pi$
$252$	0	0
$253$	−18.0392	−1.13411
$254$	−2.55353	−0.160223
$255$	0	0
$256$	−28.1781	−1.76113
$257$	25.3594	1.58188	0.790938	−	0.611896i	$-0.209593\pi$
$257$	25.3594	1.58188	0.790938	+	0.611896i	$0.209593\pi$
$258$	0	0
$259$	10.9086	0.677828
$260$	−48.8640	−3.03042
$261$	0	0
$262$	0.217468	0.0134352
$263$	21.5029	1.32593	0.662963	−	0.748652i	$-0.269299\pi$
$263$	21.5029	1.32593	0.662963	+	0.748652i	$0.269299\pi$
$264$	0	0
$265$	−14.1267	−0.867798
$266$	−57.4892	−3.52489
$267$	0	0
$268$	−51.8466	−3.16703
$269$	−12.4571	−0.759522	−0.379761	−	0.925085i	$-0.623994\pi$
$269$	−12.4571	−0.759522	−0.379761	+	0.925085i	$0.623994\pi$
$270$	0	0
$271$	24.6211	1.49563	0.747814	−	0.663909i	$-0.231104\pi$
$271$	24.6211	1.49563	0.747814	+	0.663909i	$0.231104\pi$
$272$	57.5039	3.48669
$273$	0	0
$274$	−0.967399	−0.0584427
$275$	−3.81395	−0.229990
$276$	0	0
$277$	8.66539	0.520653	0.260326	−	0.965521i	$-0.416170\pi$
$277$	8.66539	0.520653	0.260326	+	0.965521i	$0.416170\pi$
$278$	−7.73665	−0.464014
$279$	0	0
$280$	−61.4261	−3.67091
$281$	9.51624	0.567691	0.283846	−	0.958870i	$-0.408390\pi$
$281$	9.51624	0.567691	0.283846	+	0.958870i	$0.408390\pi$
$282$	0	0
$283$	3.00430	0.178587	0.0892934	−	0.996005i	$-0.471539\pi$
$283$	3.00430	0.178587	0.0892934	+	0.996005i	$0.471539\pi$
$284$	−66.6914	−3.95741
$285$	0	0
$286$	−43.0908	−2.54801
$287$	−0.600992	−0.0354755
$288$	0	0
$289$	43.4833	2.55784
$290$	−6.56670	−0.385610
$291$	0	0
$292$	−54.5007	−3.18941
$293$	−12.7522	−0.744993	−0.372497	−	0.928034i	$-0.621498\pi$
$293$	−12.7522	−0.744993	−0.372497	+	0.928034i	$0.621498\pi$
$294$	0	0
$295$	10.0490	0.585073
$296$	18.0162	1.04717
$297$	0	0
$298$	−20.3289	−1.17762
$299$	20.8257	1.20438
$300$	0	0
$301$	−46.8383	−2.69971
$302$	53.5328	3.08047
$303$	0	0
$304$	−42.7159	−2.44992
$305$	−12.1018	−0.692949
$306$	0	0
$307$	−23.4460	−1.33813	−0.669067	−	0.743202i	$-0.733306\pi$
$307$	−23.4460	−1.33813	−0.669067	+	0.743202i	$0.733306\pi$
$308$	−67.3524	−3.83776
$309$	0	0
$310$	5.49758	0.312242
$311$	11.9555	0.677933	0.338967	−	0.940798i	$-0.389923\pi$
$311$	11.9555	0.677933	0.338967	+	0.940798i	$0.389923\pi$
$312$	0	0
$313$	23.0054	1.30034	0.650172	−	0.759787i	$-0.274697\pi$
$313$	23.0054	1.30034	0.650172	+	0.759787i	$0.274697\pi$
$314$	8.59066	0.484799
$315$	0	0
$316$	29.3745	1.65245
$317$	9.24711	0.519369	0.259685	−	0.965693i	$-0.416381\pi$
$317$	9.24711	0.519369	0.259685	+	0.965693i	$0.416381\pi$
$318$	0	0
$319$	−4.01466	−0.224778
$320$	−1.35828	−0.0759300
$321$	0	0
$322$	46.9529	2.61658
$323$	−44.9291	−2.49992
$324$	0	0
$325$	4.40310	0.244240
$326$	10.8213	0.599335
$327$	0	0
$328$	−0.992574	−0.0548057
$329$	−22.0193	−1.21396
$330$	0	0
$331$	31.5913	1.73641	0.868206	−	0.496204i	$-0.165273\pi$
$331$	31.5913	1.73641	0.868206	+	0.496204i	$0.165273\pi$
$332$	64.7688	3.55465
$333$	0	0
$334$	−32.0062	−1.75130
$335$	28.0880	1.53461
$336$	0	0
$337$	−15.7777	−0.859465	−0.429733	−	0.902956i	$-0.641392\pi$
$337$	−15.7777	−0.859465	−0.429733	+	0.902956i	$0.641392\pi$
$338$	16.5513	0.900271
$339$	0	0
$340$	−86.0980	−4.66932
$341$	3.36103	0.182010
$342$	0	0
$343$	4.62567	0.249763
$344$	−77.3562	−4.17077
$345$	0	0
$346$	23.3681	1.25628
$347$	−21.3471	−1.14597	−0.572987	−	0.819564i	$-0.694215\pi$
$347$	−21.3471	−1.14597	−0.572987	+	0.819564i	$0.694215\pi$
$348$	0	0
$349$	−2.19003	−0.117230	−0.0586148	−	0.998281i	$-0.518668\pi$
$349$	−2.19003	−0.117230	−0.0586148	+	0.998281i	$0.518668\pi$
$350$	9.92708	0.530625
$351$	0	0
$352$	−22.9715	−1.22438
$353$	0.825057	0.0439133	0.0219567	−	0.999759i	$-0.493010\pi$
$353$	0.825057	0.0439133	0.0219567	+	0.999759i	$0.493010\pi$
$354$	0	0
$355$	36.1302	1.91759
$356$	−11.5614	−0.612751
$357$	0	0
$358$	−44.8579	−2.37082
$359$	22.3795	1.18115	0.590574	−	0.806984i	$-0.298902\pi$
$359$	22.3795	1.18115	0.590574	+	0.806984i	$0.298902\pi$
$360$	0	0
$361$	14.3749	0.756572
$362$	17.3023	0.909387
$363$	0	0
$364$	77.7565	4.07555
$365$	29.5258	1.54545
$366$	0	0
$367$	29.0135	1.51449	0.757246	−	0.653130i	$-0.226545\pi$
$367$	29.0135	1.51449	0.757246	+	0.653130i	$0.226545\pi$
$368$	34.8872	1.81862
$369$	0	0
$370$	−17.5050	−0.910043
$371$	22.4796	1.16708
$372$	0	0
$373$	2.32666	0.120470	0.0602348	−	0.998184i	$-0.480815\pi$
$373$	2.32666	0.120470	0.0602348	+	0.998184i	$0.480815\pi$
$374$	−75.9257	−3.92602
$375$	0	0
$376$	−36.3661	−1.87544
$377$	4.63481	0.238705
$378$	0	0
$379$	14.5711	0.748466	0.374233	−	0.927335i	$-0.377906\pi$
$379$	14.5711	0.748466	0.374233	+	0.927335i	$0.377906\pi$
$380$	63.9566	3.28090
$381$	0	0
$382$	−4.21884	−0.215854
$383$	−9.95839	−0.508850	−0.254425	−	0.967093i	$-0.581886\pi$
$383$	−9.95839	−0.508850	−0.254425	+	0.967093i	$0.581886\pi$
$384$	0	0
$385$	36.4883	1.85962
$386$	−56.8276	−2.89245
$387$	0	0
$388$	−10.1246	−0.513999
$389$	30.7848	1.56085	0.780426	−	0.625248i	$-0.215002\pi$
$389$	30.7848	1.56085	0.780426	+	0.625248i	$0.215002\pi$
$390$	0	0
$391$	36.6947	1.85573
$392$	52.6930	2.66140
$393$	0	0
$394$	−38.0361	−1.91623
$395$	−15.9137	−0.800705
$396$	0	0
$397$	−21.9196	−1.10011	−0.550057	−	0.835127i	$-0.685394\pi$
$397$	−21.9196	−1.10011	−0.550057	+	0.835127i	$0.685394\pi$
$398$	−20.1511	−1.01008
$399$	0	0
$400$	7.37607	0.368803
$401$	2.34919	0.117313	0.0586566	−	0.998278i	$-0.481318\pi$
$401$	2.34919	0.117313	0.0586566	+	0.998278i	$0.481318\pi$
$402$	0	0
$403$	−3.88022	−0.193288
$404$	30.8643	1.53556
$405$	0	0
$406$	10.4495	0.518599
$407$	−10.7020	−0.530477
$408$	0	0
$409$	−38.2148	−1.88960	−0.944800	−	0.327648i	$-0.893744\pi$
$409$	−38.2148	−1.88960	−0.944800	+	0.327648i	$0.893744\pi$
$410$	0.964411	0.0476288
$411$	0	0
$412$	56.1880	2.76819
$413$	−15.9908	−0.786854
$414$	0	0
$415$	−35.0886	−1.72243
$416$	26.5200	1.30025
$417$	0	0
$418$	56.4002	2.75862
$419$	−34.8133	−1.70074	−0.850371	−	0.526184i	$-0.823622\pi$
$419$	−34.8133	−1.70074	−0.850371	+	0.526184i	$0.823622\pi$
$420$	0	0
$421$	1.27040	0.0619155	0.0309577	−	0.999521i	$-0.490144\pi$
$421$	1.27040	0.0619155	0.0309577	+	0.999521i	$0.490144\pi$
$422$	26.3769	1.28401
$423$	0	0
$424$	37.1264	1.80302
$425$	7.75823	0.376330
$426$	0	0
$427$	19.2575	0.931934
$428$	−69.9603	−3.38166
$429$	0	0
$430$	75.1613	3.62460
$431$	−19.7115	−0.949472	−0.474736	−	0.880128i	$-0.657456\pi$
$431$	−19.7115	−0.949472	−0.474736	+	0.880128i	$0.657456\pi$
$432$	0	0
$433$	26.3026	1.26402	0.632011	−	0.774960i	$-0.282230\pi$
$433$	26.3026	1.26402	0.632011	+	0.774960i	$0.282230\pi$
$434$	−8.74821	−0.419928
$435$	0	0
$436$	−51.3075	−2.45718
$437$	−27.2581	−1.30393
$438$	0	0
$439$	−6.21716	−0.296729	−0.148364	−	0.988933i	$-0.547401\pi$
$439$	−6.21716	−0.296729	−0.148364	+	0.988933i	$0.547401\pi$
$440$	60.2626	2.87290
$441$	0	0
$442$	87.6541	4.16928
$443$	−17.0352	−0.809365	−0.404682	−	0.914457i	$-0.632618\pi$
$443$	−17.0352	−0.809365	−0.404682	+	0.914457i	$0.632618\pi$
$444$	0	0
$445$	6.26339	0.296913
$446$	23.5291	1.11413
$447$	0	0
$448$	2.16140	0.102117
$449$	31.8463	1.50292	0.751461	−	0.659778i	$-0.229350\pi$
$449$	31.8463	1.50292	0.751461	+	0.659778i	$0.229350\pi$
$450$	0	0
$451$	0.589608	0.0277636
$452$	−10.0741	−0.473844
$453$	0	0
$454$	15.8221	0.742567
$455$	−42.1247	−1.97484
$456$	0	0
$457$	35.9612	1.68220	0.841098	−	0.540883i	$-0.181910\pi$
$457$	35.9612	1.68220	0.841098	+	0.540883i	$0.181910\pi$
$458$	26.4644	1.23660
$459$	0	0
$460$	−52.2350	−2.43547
$461$	19.4444	0.905617	0.452809	−	0.891608i	$-0.350422\pi$
$461$	19.4444	0.905617	0.452809	+	0.891608i	$0.350422\pi$
$462$	0	0
$463$	−26.9817	−1.25395	−0.626973	−	0.779041i	$-0.715706\pi$
$463$	−26.9817	−1.25395	−0.626973	+	0.779041i	$0.715706\pi$
$464$	7.76423	0.360445
$465$	0	0
$466$	19.7233	0.913665
$467$	34.8369	1.61206	0.806029	−	0.591876i	$-0.201613\pi$
$467$	34.8369	1.61206	0.806029	+	0.591876i	$0.201613\pi$
$468$	0	0
$469$	−44.6959	−2.06387
$470$	35.3343	1.62985
$471$	0	0
$472$	−26.4097	−1.21560
$473$	45.9511	2.11283
$474$	0	0
$475$	−5.76308	−0.264428
$476$	137.006	6.27968
$477$	0	0
$478$	31.9268	1.46030
$479$	6.86253	0.313557	0.156779	−	0.987634i	$-0.449889\pi$
$479$	6.86253	0.313557	0.156779	+	0.987634i	$0.449889\pi$
$480$	0	0
$481$	12.3551	0.563346
$482$	−3.40082	−0.154903
$483$	0	0
$484$	16.3510	0.743228
$485$	5.48502	0.249062
$486$	0	0
$487$	−16.9608	−0.768565	−0.384283	−	0.923216i	$-0.625551\pi$
$487$	−16.9608	−0.768565	−0.384283	+	0.923216i	$0.625551\pi$
$488$	31.8048	1.43974
$489$	0	0
$490$	−51.1979	−2.31288
$491$	1.90424	0.0859372	0.0429686	−	0.999076i	$-0.486318\pi$
$491$	1.90424	0.0859372	0.0429686	+	0.999076i	$0.486318\pi$
$492$	0	0
$493$	8.16650	0.367801
$494$	−65.1125	−2.92955
$495$	0	0
$496$	−6.50014	−0.291865
$497$	−57.4934	−2.57893
$498$	0	0
$499$	−36.4821	−1.63316	−0.816582	−	0.577229i	$-0.804134\pi$
$499$	−36.4821	−1.63316	−0.816582	+	0.577229i	$0.804134\pi$
$500$	44.3097	1.98159
$501$	0	0
$502$	−14.9143	−0.665659
$503$	−36.6800	−1.63548	−0.817741	−	0.575586i	$-0.804774\pi$
$503$	−36.6800	−1.63548	−0.817741	+	0.575586i	$0.804774\pi$
$504$	0	0
$505$	−16.7208	−0.744066
$506$	−46.0635	−2.04777
$507$	0	0
$508$	−4.52051	−0.200565
$509$	31.3055	1.38759	0.693797	−	0.720171i	$-0.255937\pi$
$509$	31.3055	1.38759	0.693797	+	0.720171i	$0.255937\pi$
$510$	0	0
$511$	−46.9840	−2.07845
$512$	−50.7524	−2.24296
$513$	0	0
$514$	64.7560	2.85627
$515$	−30.4400	−1.34134
$516$	0	0
$517$	21.6022	0.950063
$518$	27.8555	1.22390
$519$	0	0
$520$	−69.5715	−3.05091
$521$	4.68368	0.205196	0.102598	−	0.994723i	$-0.467284\pi$
$521$	4.68368	0.205196	0.102598	+	0.994723i	$0.467284\pi$
$522$	0	0
$523$	9.51110	0.415891	0.207946	−	0.978140i	$-0.433322\pi$
$523$	9.51110	0.415891	0.207946	+	0.978140i	$0.433322\pi$
$524$	0.384983	0.0168181
$525$	0	0
$526$	54.9083	2.39412
$527$	−6.83692	−0.297821
$528$	0	0
$529$	−0.737598	−0.0320695
$530$	−36.0730	−1.56691
$531$	0	0
$532$	−101.773	−4.41242
$533$	−0.680686	−0.0294838
$534$	0	0
$535$	37.9011	1.63861
$536$	−73.8179	−3.18845
$537$	0	0
$538$	−31.8095	−1.37141
$539$	−31.3006	−1.34821
$540$	0	0
$541$	12.9800	0.558055	0.279028	−	0.960283i	$-0.409988\pi$
$541$	12.9800	0.558055	0.279028	+	0.960283i	$0.409988\pi$
$542$	62.8708	2.70053
$543$	0	0
$544$	46.7279	2.00344
$545$	27.7959	1.19065
$546$	0	0
$547$	−17.8217	−0.762002	−0.381001	−	0.924575i	$-0.624421\pi$
$547$	−17.8217	−0.762002	−0.381001	+	0.924575i	$0.624421\pi$
$548$	−1.71259	−0.0731581
$549$	0	0
$550$	−9.73904	−0.415274
$551$	−6.06636	−0.258436
$552$	0	0
$553$	25.3232	1.07685
$554$	22.1273	0.940100
$555$	0	0
$556$	−13.6962	−0.580848
$557$	−7.06935	−0.299538	−0.149769	−	0.988721i	$-0.547853\pi$
$557$	−7.06935	−0.299538	−0.149769	+	0.988721i	$0.547853\pi$
$558$	0	0
$559$	−53.0493	−2.24374
$560$	−70.5672	−2.98201
$561$	0	0
$562$	24.3000	1.02503
$563$	29.7344	1.25316	0.626579	−	0.779358i	$-0.284455\pi$
$563$	29.7344	1.25316	0.626579	+	0.779358i	$0.284455\pi$
$564$	0	0
$565$	5.45764	0.229605
$566$	7.67156	0.322460
$567$	0	0
$568$	−94.9537	−3.98417
$569$	−19.0302	−0.797786	−0.398893	−	0.916997i	$-0.630606\pi$
$569$	−19.0302	−0.797786	−0.398893	+	0.916997i	$0.630606\pi$
$570$	0	0
$571$	31.1583	1.30393	0.651966	−	0.758248i	$-0.273944\pi$
$571$	31.1583	1.30393	0.651966	+	0.758248i	$0.273944\pi$
$572$	−76.2837	−3.18958
$573$	0	0
$574$	−1.53465	−0.0640551
$575$	4.70686	0.196290
$576$	0	0
$577$	−5.26050	−0.218998	−0.109499	−	0.993987i	$-0.534925\pi$
$577$	−5.26050	−0.218998	−0.109499	+	0.993987i	$0.534925\pi$
$578$	111.036	4.61848
$579$	0	0
$580$	−11.6250	−0.482703
$581$	55.8360	2.31647
$582$	0	0
$583$	−22.0538	−0.913376
$584$	−77.5968	−3.21098
$585$	0	0
$586$	−32.5632	−1.34517
$587$	−17.7687	−0.733391	−0.366696	−	0.930341i	$-0.619511\pi$
$587$	−17.7687	−0.733391	−0.366696	+	0.930341i	$0.619511\pi$
$588$	0	0
$589$	5.07870	0.209264
$590$	25.6603	1.05642
$591$	0	0
$592$	20.6973	0.850653
$593$	11.9800	0.491961	0.245981	−	0.969275i	$-0.420890\pi$
$593$	11.9800	0.491961	0.245981	+	0.969275i	$0.420890\pi$
$594$	0	0
$595$	−74.2234	−3.04286
$596$	−35.9883	−1.47414
$597$	0	0
$598$	53.1791	2.17465
$599$	37.2111	1.52041	0.760203	−	0.649686i	$-0.225100\pi$
$599$	37.2111	1.52041	0.760203	+	0.649686i	$0.225100\pi$
$600$	0	0
$601$	−25.5961	−1.04409	−0.522044	−	0.852919i	$-0.674830\pi$
$601$	−25.5961	−1.04409	−0.522044	+	0.852919i	$0.674830\pi$
$602$	−119.603	−4.87465
$603$	0	0
$604$	94.7691	3.85610
$605$	−8.85818	−0.360136
$606$	0	0
$607$	−4.02451	−0.163350	−0.0816748	−	0.996659i	$-0.526027\pi$
$607$	−4.02451	−0.163350	−0.0816748	+	0.996659i	$0.526027\pi$
$608$	−34.7111	−1.40772
$609$	0	0
$610$	−30.9024	−1.25120
$611$	−24.9391	−1.00893
$612$	0	0
$613$	24.6810	0.996857	0.498429	−	0.866931i	$-0.333911\pi$
$613$	24.6810	0.996857	0.498429	+	0.866931i	$0.333911\pi$
$614$	−59.8701	−2.41616
$615$	0	0
$616$	−95.8948	−3.86371
$617$	−23.2530	−0.936132	−0.468066	−	0.883693i	$-0.655049\pi$
$617$	−23.2530	−0.936132	−0.468066	+	0.883693i	$0.655049\pi$
$618$	0	0
$619$	3.14807	0.126532	0.0632658	−	0.997997i	$-0.479848\pi$
$619$	3.14807	0.126532	0.0632658	+	0.997997i	$0.479848\pi$
$620$	9.73236	0.390861
$621$	0	0
$622$	30.5287	1.22409
$623$	−9.96684	−0.399313
$624$	0	0
$625$	−28.9927	−1.15971
$626$	58.7450	2.34792
$627$	0	0
$628$	15.2080	0.606867
$629$	21.7697	0.868013
$630$	0	0
$631$	−14.8108	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$631$	−14.8108	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$632$	41.8228	1.66362
$633$	0	0
$634$	23.6128	0.937782
$635$	2.44899	0.0971854
$636$	0	0
$637$	36.1357	1.43175
$638$	−10.2515	−0.405863
$639$	0	0
$640$	25.9607	1.02619
$641$	−4.47852	−0.176891	−0.0884454	−	0.996081i	$-0.528190\pi$
$641$	−4.47852	−0.176891	−0.0884454	+	0.996081i	$0.528190\pi$
$642$	0	0
$643$	30.8163	1.21528	0.607639	−	0.794213i	$-0.292117\pi$
$643$	30.8163	1.21528	0.607639	+	0.794213i	$0.292117\pi$
$644$	83.1207	3.27541
$645$	0	0
$646$	−114.728	−4.51390
$647$	4.58862	0.180397	0.0901987	−	0.995924i	$-0.471250\pi$
$647$	4.58862	0.180397	0.0901987	+	0.995924i	$0.471250\pi$
$648$	0	0
$649$	15.6879	0.615802
$650$	11.2435	0.441005
$651$	0	0
$652$	19.1569	0.750243
$653$	−1.41093	−0.0552139	−0.0276069	−	0.999619i	$-0.508789\pi$
$653$	−1.41093	−0.0552139	−0.0276069	+	0.999619i	$0.508789\pi$
$654$	0	0
$655$	−0.208565	−0.00814932
$656$	−1.14028	−0.0445206
$657$	0	0
$658$	−56.2269	−2.19195
$659$	−37.3998	−1.45689	−0.728444	−	0.685105i	$-0.759756\pi$
$659$	−37.3998	−1.45689	−0.728444	+	0.685105i	$0.759756\pi$
$660$	0	0
$661$	−27.1575	−1.05630	−0.528152	−	0.849150i	$-0.677115\pi$
$661$	−27.1575	−1.05630	−0.528152	+	0.849150i	$0.677115\pi$
$662$	80.6692	3.13530
$663$	0	0
$664$	92.2164	3.57869
$665$	55.1357	2.13807
$666$	0	0
$667$	4.95455	0.191841
$668$	−56.6605	−2.19226
$669$	0	0
$670$	71.7234	2.77092
$671$	−18.8927	−0.729344
$672$	0	0
$673$	−0.583317	−0.0224852	−0.0112426	−	0.999937i	$-0.503579\pi$
$673$	−0.583317	−0.0224852	−0.0112426	+	0.999937i	$0.503579\pi$
$674$	−40.2888	−1.55187
$675$	0	0
$676$	29.3007	1.12695
$677$	6.45528	0.248097	0.124048	−	0.992276i	$-0.460412\pi$
$677$	6.45528	0.248097	0.124048	+	0.992276i	$0.460412\pi$
$678$	0	0
$679$	−8.72823	−0.334959
$680$	−122.584	−4.70089
$681$	0	0
$682$	8.58250	0.328641
$683$	−4.93815	−0.188953	−0.0944765	−	0.995527i	$-0.530118\pi$
$683$	−4.93815	−0.188953	−0.0944765	+	0.995527i	$0.530118\pi$
$684$	0	0
$685$	0.927797	0.0354493
$686$	11.8118	0.450976
$687$	0	0
$688$	−88.8680	−3.38806
$689$	25.4605	0.969969
$690$	0	0
$691$	23.1086	0.879091	0.439546	−	0.898220i	$-0.355139\pi$
$691$	23.1086	0.879091	0.439546	+	0.898220i	$0.355139\pi$
$692$	41.3685	1.57260
$693$	0	0
$694$	−54.5105	−2.06919
$695$	7.41994	0.281454
$696$	0	0
$697$	−1.19936	−0.0454291
$698$	−5.59231	−0.211672
$699$	0	0
$700$	17.5739	0.664231
$701$	16.0358	0.605664	0.302832	−	0.953044i	$-0.402068\pi$
$701$	16.0358	0.605664	0.302832	+	0.953044i	$0.402068\pi$
$702$	0	0
$703$	−16.1713	−0.609910
$704$	−2.12046	−0.0799179
$705$	0	0
$706$	2.10681	0.0792907
$707$	26.6076	1.00068
$708$	0	0
$709$	−13.3502	−0.501376	−0.250688	−	0.968068i	$-0.580657\pi$
$709$	−13.3502	−0.501376	−0.250688	+	0.968068i	$0.580657\pi$
$710$	92.2594	3.46243
$711$	0	0
$712$	−16.4608	−0.616895
$713$	−4.14791	−0.155340
$714$	0	0
$715$	41.3268	1.54553
$716$	−79.4120	−2.96776
$717$	0	0
$718$	57.1468	2.13270
$719$	−34.9098	−1.30192	−0.650959	−	0.759113i	$-0.725633\pi$
$719$	−34.9098	−1.30192	−0.650959	+	0.759113i	$0.725633\pi$
$720$	0	0
$721$	48.4386	1.80395
$722$	36.7066	1.36608
$723$	0	0
$724$	30.6302	1.13836
$725$	1.04752	0.0389040
$726$	0	0
$727$	51.1378	1.89659	0.948297	−	0.317383i	$-0.102804\pi$
$727$	51.1378	1.89659	0.948297	+	0.317383i	$0.102804\pi$
$728$	110.708	4.10311
$729$	0	0
$730$	75.3951	2.79050
$731$	−93.4723	−3.45720
$732$	0	0
$733$	0.278165	0.0102743	0.00513713	−	0.999987i	$-0.498365\pi$
$733$	0.278165	0.0102743	0.00513713	+	0.999987i	$0.498365\pi$
$734$	74.0868	2.73459
$735$	0	0
$736$	28.3495	1.04498
$737$	43.8493	1.61521
$738$	0	0
$739$	−7.90691	−0.290860	−0.145430	−	0.989369i	$-0.546457\pi$
$739$	−7.90691	−0.290860	−0.145430	+	0.989369i	$0.546457\pi$
$740$	−30.9891	−1.13918
$741$	0	0
$742$	57.4024	2.10731
$743$	−20.1569	−0.739487	−0.369743	−	0.929134i	$-0.620554\pi$
$743$	−20.1569	−0.739487	−0.369743	+	0.929134i	$0.620554\pi$
$744$	0	0
$745$	19.4967	0.714305
$746$	5.94118	0.217522
$747$	0	0
$748$	−134.411	−4.91456
$749$	−60.3114	−2.20373
$750$	0	0
$751$	18.2248	0.665031	0.332515	−	0.943098i	$-0.392103\pi$
$751$	18.2248	0.665031	0.332515	+	0.943098i	$0.392103\pi$
$752$	−41.7780	−1.52349
$753$	0	0
$754$	11.8351	0.431010
$755$	−51.3413	−1.86850
$756$	0	0
$757$	17.3018	0.628843	0.314422	−	0.949283i	$-0.398189\pi$
$757$	17.3018	0.628843	0.314422	+	0.949283i	$0.398189\pi$
$758$	37.2077	1.35144
$759$	0	0
$760$	91.0599	3.30309
$761$	2.44843	0.0887556	0.0443778	−	0.999015i	$-0.485869\pi$
$761$	2.44843	0.0887556	0.0443778	+	0.999015i	$0.485869\pi$
$762$	0	0
$763$	−44.2312	−1.60128
$764$	−7.46860	−0.270205
$765$	0	0
$766$	−25.4290	−0.918788
$767$	−18.1112	−0.653958
$768$	0	0
$769$	31.3994	1.13229	0.566146	−	0.824305i	$-0.308434\pi$
$769$	31.3994	1.13229	0.566146	+	0.824305i	$0.308434\pi$
$770$	93.1739	3.35775
$771$	0	0
$772$	−100.602	−3.62074
$773$	−7.09070	−0.255035	−0.127517	−	0.991836i	$-0.540701\pi$
$773$	−7.09070	−0.255035	−0.127517	+	0.991836i	$0.540701\pi$
$774$	0	0
$775$	−0.876977	−0.0315019
$776$	−14.4152	−0.517475
$777$	0	0
$778$	78.6100	2.81830
$779$	0.890929	0.0319208
$780$	0	0
$781$	56.4043	2.01830
$782$	93.7010	3.35074
$783$	0	0
$784$	60.5345	2.16195
$785$	−8.23898	−0.294062
$786$	0	0
$787$	11.6900	0.416702	0.208351	−	0.978054i	$-0.433190\pi$
$787$	11.6900	0.416702	0.208351	+	0.978054i	$0.433190\pi$
$788$	−67.3353	−2.39872
$789$	0	0
$790$	−40.6361	−1.44577
$791$	−8.68465	−0.308791
$792$	0	0
$793$	21.8111	0.774535
$794$	−55.9724	−1.98638
$795$	0	0
$796$	−35.6734	−1.26441
$797$	46.9377	1.66262	0.831310	−	0.555809i	$-0.187591\pi$
$797$	46.9377	1.66262	0.831310	+	0.555809i	$0.187591\pi$
$798$	0	0
$799$	−43.9425	−1.55458
$800$	5.99383	0.211914
$801$	0	0
$802$	5.99874	0.211823
$803$	46.0940	1.62662
$804$	0	0
$805$	−45.0308	−1.58713
$806$	−9.90826	−0.349004
$807$	0	0
$808$	43.9439	1.54594
$809$	23.0704	0.811113	0.405556	−	0.914070i	$-0.367078\pi$
$809$	23.0704	0.811113	0.405556	+	0.914070i	$0.367078\pi$
$810$	0	0
$811$	−5.97647	−0.209862	−0.104931	−	0.994480i	$-0.533462\pi$
$811$	−5.97647	−0.209862	−0.104931	+	0.994480i	$0.533462\pi$
$812$	18.4987	0.649178
$813$	0	0
$814$	−27.3278	−0.957839
$815$	−10.3783	−0.363536
$816$	0	0
$817$	69.4345	2.42921
$818$	−97.5826	−3.41190
$819$	0	0
$820$	1.70730	0.0596214
$821$	−39.5214	−1.37931	−0.689653	−	0.724140i	$-0.742237\pi$
$821$	−39.5214	−1.37931	−0.689653	+	0.724140i	$0.742237\pi$
$822$	0	0
$823$	−41.1393	−1.43403	−0.717013	−	0.697060i	$-0.754491\pi$
$823$	−41.1393	−1.43403	−0.717013	+	0.697060i	$0.754491\pi$
$824$	79.9992	2.78690
$825$	0	0
$826$	−40.8329	−1.42076
$827$	−38.8786	−1.35194	−0.675971	−	0.736928i	$-0.736275\pi$
$827$	−38.8786	−1.35194	−0.675971	+	0.736928i	$0.736275\pi$
$828$	0	0
$829$	5.18917	0.180227	0.0901136	−	0.995931i	$-0.471277\pi$
$829$	5.18917	0.180227	0.0901136	+	0.995931i	$0.471277\pi$
$830$	−89.5998	−3.11006
$831$	0	0
$832$	2.44801	0.0848696
$833$	63.6709	2.20606
$834$	0	0
$835$	30.6959	1.06228
$836$	99.8453	3.45322
$837$	0	0
$838$	−88.8968	−3.07089
$839$	11.6267	0.401399	0.200699	−	0.979653i	$-0.435679\pi$
$839$	11.6267	0.401399	0.200699	+	0.979653i	$0.435679\pi$
$840$	0	0
$841$	−27.8974	−0.961978
$842$	3.24400	0.111796
$843$	0	0
$844$	46.6949	1.60731
$845$	−15.8737	−0.546072
$846$	0	0
$847$	14.0959	0.484340
$848$	42.6514	1.46465
$849$	0	0
$850$	19.8109	0.679507
$851$	13.2075	0.452747
$852$	0	0
$853$	−2.04861	−0.0701432	−0.0350716	−	0.999385i	$-0.511166\pi$
$853$	−2.04861	−0.0701432	−0.0350716	+	0.999385i	$0.511166\pi$
$854$	49.1745	1.68272
$855$	0	0
$856$	−99.6078	−3.40453
$857$	−3.08215	−0.105284	−0.0526421	−	0.998613i	$-0.516764\pi$
$857$	−3.08215	−0.105284	−0.0526421	+	0.998613i	$0.516764\pi$
$858$	0	0
$859$	−49.3371	−1.68336	−0.841680	−	0.539976i	$-0.818433\pi$
$859$	−49.3371	−1.68336	−0.841680	+	0.539976i	$0.818433\pi$
$860$	133.058	4.53724
$861$	0	0
$862$	−50.3340	−1.71438
$863$	7.72911	0.263102	0.131551	−	0.991309i	$-0.458004\pi$
$863$	7.72911	0.263102	0.131551	+	0.991309i	$0.458004\pi$
$864$	0	0
$865$	−22.4115	−0.762013
$866$	67.1644	2.28234
$867$	0	0
$868$	−15.4870	−0.525661
$869$	−24.8435	−0.842759
$870$	0	0
$871$	−50.6228	−1.71529
$872$	−73.0504	−2.47380
$873$	0	0
$874$	−69.6044	−2.35440
$875$	38.1985	1.29135
$876$	0	0
$877$	29.1830	0.985439	0.492720	−	0.870188i	$-0.336003\pi$
$877$	29.1830	0.985439	0.492720	+	0.870188i	$0.336003\pi$
$878$	−15.8757	−0.535779
$879$	0	0
$880$	69.2305	2.33376
$881$	17.3481	0.584473	0.292237	−	0.956346i	$-0.405601\pi$
$881$	17.3481	0.584473	0.292237	+	0.956346i	$0.405601\pi$
$882$	0	0
$883$	2.60337	0.0876104	0.0438052	−	0.999040i	$-0.486052\pi$
$883$	2.60337	0.0876104	0.0438052	+	0.999040i	$0.486052\pi$
$884$	155.174	5.21907
$885$	0	0
$886$	−43.4998	−1.46140
$887$	51.0504	1.71411	0.857053	−	0.515229i	$-0.172293\pi$
$887$	51.0504	1.71411	0.857053	+	0.515229i	$0.172293\pi$
$888$	0	0
$889$	−3.89705	−0.130703
$890$	15.9938	0.536112
$891$	0	0
$892$	41.6535	1.39466
$893$	32.6420	1.09232
$894$	0	0
$895$	43.0216	1.43805
$896$	−41.3108	−1.38010
$897$	0	0
$898$	81.3206	2.71370
$899$	−0.923127	−0.0307880
$900$	0	0
$901$	44.8612	1.49454
$902$	1.50558	0.0501304
$903$	0	0
$904$	−14.3432	−0.477048
$905$	−16.5940	−0.551602
$906$	0	0
$907$	−14.0974	−0.468098	−0.234049	−	0.972225i	$-0.575198\pi$
$907$	−14.0974	−0.468098	−0.234049	+	0.972225i	$0.575198\pi$
$908$	28.0098	0.929539
$909$	0	0
$910$	−107.567	−3.56580
$911$	−46.1625	−1.52943	−0.764716	−	0.644368i	$-0.777121\pi$
$911$	−46.1625	−1.52943	−0.764716	+	0.644368i	$0.777121\pi$
$912$	0	0
$913$	−54.7783	−1.81290
$914$	91.8281	3.03740
$915$	0	0
$916$	46.8500	1.54797
$917$	0.331887	0.0109599
$918$	0	0
$919$	25.0206	0.825354	0.412677	−	0.910877i	$-0.364594\pi$
$919$	25.0206	0.825354	0.412677	+	0.910877i	$0.364594\pi$
$920$	−74.3710	−2.45194
$921$	0	0
$922$	49.6519	1.63520
$923$	−65.1172	−2.14336
$924$	0	0
$925$	2.79241	0.0918139
$926$	−68.8985	−2.26415
$927$	0	0
$928$	6.30925	0.207111
$929$	21.4236	0.702884	0.351442	−	0.936210i	$-0.385691\pi$
$929$	21.4236	0.702884	0.351442	+	0.936210i	$0.385691\pi$
$930$	0	0
$931$	−47.2969	−1.55009
$932$	34.9162	1.14372
$933$	0	0
$934$	88.9569	2.91076
$935$	72.8175	2.38139
$936$	0	0
$937$	15.0693	0.492293	0.246147	−	0.969233i	$-0.420835\pi$
$937$	15.0693	0.492293	0.246147	+	0.969233i	$0.420835\pi$
$938$	−114.132	−3.72655
$939$	0	0
$940$	62.5523	2.04023
$941$	−27.3929	−0.892983	−0.446492	−	0.894788i	$-0.647327\pi$
$941$	−27.3929	−0.892983	−0.446492	+	0.894788i	$0.647327\pi$
$942$	0	0
$943$	−0.727645	−0.0236954
$944$	−30.3398	−0.987478
$945$	0	0
$946$	117.337	3.81497
$947$	32.9494	1.07071	0.535356	−	0.844627i	$-0.320177\pi$
$947$	32.9494	1.07071	0.535356	+	0.844627i	$0.320177\pi$
$948$	0	0
$949$	−53.2143	−1.72741
$950$	−14.7162	−0.477457
$951$	0	0
$952$	195.067	6.32214
$953$	−35.6548	−1.15497	−0.577487	−	0.816400i	$-0.695966\pi$
$953$	−35.6548	−1.15497	−0.577487	+	0.816400i	$0.695966\pi$
$954$	0	0
$955$	4.04613	0.130930
$956$	56.5201	1.82799
$957$	0	0
$958$	17.5237	0.566164
$959$	−1.47639	−0.0476751
$960$	0	0
$961$	−30.2272	−0.975070
$962$	31.5492	1.01719
$963$	0	0
$964$	−6.02046	−0.193906
$965$	54.5013	1.75446
$966$	0	0
$967$	49.3265	1.58623	0.793117	−	0.609069i	$-0.208457\pi$
$967$	49.3265	1.58623	0.793117	+	0.609069i	$0.208457\pi$
$968$	23.2802	0.748254
$969$	0	0
$970$	14.0062	0.449711
$971$	18.4196	0.591113	0.295556	−	0.955325i	$-0.404495\pi$
$971$	18.4196	0.591113	0.295556	+	0.955325i	$0.404495\pi$
$972$	0	0
$973$	−11.8072	−0.378522
$974$	−43.3098	−1.38773
$975$	0	0
$976$	36.5379	1.16955
$977$	−28.6577	−0.916840	−0.458420	−	0.888736i	$-0.651584\pi$
$977$	−28.6577	−0.916840	−0.458420	+	0.888736i	$0.651584\pi$
$978$	0	0
$979$	9.77804	0.312508
$980$	−90.6355	−2.89525
$981$	0	0
$982$	4.86254	0.155170
$983$	−11.3897	−0.363275	−0.181638	−	0.983366i	$-0.558140\pi$
$983$	−11.3897	−0.363275	−0.181638	+	0.983366i	$0.558140\pi$
$984$	0	0
$985$	36.4790	1.16232
$986$	20.8534	0.664108
$987$	0	0
$988$	−115.269	−3.66718
$989$	−56.7089	−1.80324
$990$	0	0
$991$	−22.5603	−0.716650	−0.358325	−	0.933597i	$-0.616652\pi$
$991$	−22.5603	−0.716650	−0.358325	+	0.933597i	$0.616652\pi$
$992$	−5.28204	−0.167705
$993$	0	0
$994$	−146.811	−4.65656
$995$	19.3261	0.612679
$996$	0	0
$997$	−30.4430	−0.964139	−0.482069	−	0.876133i	$-0.660115\pi$
$997$	−30.4430	−0.964139	−0.482069	+	0.876133i	$0.660115\pi$
$998$	−93.1582	−2.94887
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.j.1.8		9
3.2	odd	2		381.2.a.e.1.2	✓	9
12.11	even	2		6096.2.a.bk.1.8		9
15.14	odd	2		9525.2.a.p.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.e.1.2	✓	9	3.2	odd	2
1143.2.a.j.1.8		9	1.1	even	1	trivial
6096.2.a.bk.1.8		9	12.11	even	2
9525.2.a.p.1.8		9	15.14	odd	2

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55353 q^{2} +4.52051 q^{4} -2.44899 q^{5} +3.89705 q^{7} +6.43620 q^{8} +O(q^{10})\)	\(q+2.55353 q^{2} +4.52051 q^{4} -2.44899 q^{5} +3.89705 q^{7} +6.43620 q^{8} -6.25358 q^{10} -3.82323 q^{11} +4.41381 q^{13} +9.95122 q^{14} +7.39400 q^{16} +7.77710 q^{17} -5.77710 q^{19} -11.0707 q^{20} -9.76272 q^{22} +4.71830 q^{23} +0.997574 q^{25} +11.2708 q^{26} +17.6166 q^{28} +1.05007 q^{29} -0.879109 q^{31} +6.00840 q^{32} +19.8590 q^{34} -9.54384 q^{35} +2.79920 q^{37} -14.7520 q^{38} -15.7622 q^{40} -0.154217 q^{41} -12.0189 q^{43} -17.2829 q^{44} +12.0483 q^{46} -5.65025 q^{47} +8.18697 q^{49} +2.54734 q^{50} +19.9527 q^{52} +5.76838 q^{53} +9.36306 q^{55} +25.0822 q^{56} +2.68139 q^{58} -4.10330 q^{59} +4.94156 q^{61} -2.24483 q^{62} +0.554626 q^{64} -10.8094 q^{65} -11.4692 q^{67} +35.1565 q^{68} -24.3705 q^{70} -14.7531 q^{71} -12.0563 q^{73} +7.14784 q^{74} -26.1154 q^{76} -14.8993 q^{77} +6.49805 q^{79} -18.1079 q^{80} -0.393799 q^{82} +14.3278 q^{83} -19.0461 q^{85} -30.6907 q^{86} -24.6071 q^{88} -2.55754 q^{89} +17.2008 q^{91} +21.3292 q^{92} -14.4281 q^{94} +14.1481 q^{95} -2.23970 q^{97} +20.9057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8	\( 9 q + 2 q^{2} + 14 q^{4} + 4 q^{5} + 10 q^{7} + 6 q^{8} - 4 q^{10} - 8 q^{11} + 14 q^{13} - 4 q^{14} + 32 q^{16} + 6 q^{17} + 12 q^{19} + 28 q^{20} - 18 q^{22} + 4 q^{23} + 21 q^{25} + 14 q^{26} + 8 q^{29} + 4 q^{31} + 29 q^{32} - 3 q^{34} - 6 q^{35} + 22 q^{37} + 7 q^{38} + 2 q^{41} + 6 q^{43} - 17 q^{44} - 10 q^{46} + 2 q^{47} + 23 q^{49} + 20 q^{50} - 9 q^{52} + 12 q^{53} - 22 q^{55} - 18 q^{56} - 28 q^{58} + 6 q^{59} + 2 q^{61} + 15 q^{62} + 24 q^{64} - 4 q^{65} + 18 q^{67} + 24 q^{68} - 72 q^{70} - 24 q^{71} + 14 q^{73} - 3 q^{74} + 4 q^{76} + 18 q^{77} + 12 q^{79} + 86 q^{80} + 4 q^{82} + 20 q^{83} - 24 q^{85} - 16 q^{86} - 55 q^{88} + 30 q^{89} + 14 q^{91} + 46 q^{92} - 66 q^{94} + 32 q^{95} + 12 q^{97} + 62 q^{98}+O(q^{100}) \) 9 * q + 2 * q^2 + 14 * q^4 + 4 * q^5 + 10 * q^7 + 6 * q^8 - 4 * q^10 - 8 * q^11 + 14 * q^13 - 4 * q^14 + 32 * q^16 + 6 * q^17 + 12 * q^19 + 28 * q^20 - 18 * q^22 + 4 * q^23 + 21 * q^25 + 14 * q^26 + 8 * q^29 + 4 * q^31 + 29 * q^32 - 3 * q^34 - 6 * q^35 + 22 * q^37 + 7 * q^38 + 2 * q^41 + 6 * q^43 - 17 * q^44 - 10 * q^46 + 2 * q^47 + 23 * q^49 + 20 * q^50 - 9 * q^52 + 12 * q^53 - 22 * q^55 - 18 * q^56 - 28 * q^58 + 6 * q^59 + 2 * q^61 + 15 * q^62 + 24 * q^64 - 4 * q^65 + 18 * q^67 + 24 * q^68 - 72 * q^70 - 24 * q^71 + 14 * q^73 - 3 * q^74 + 4 * q^76 + 18 * q^77 + 12 * q^79 + 86 * q^80 + 4 * q^82 + 20 * q^83 - 24 * q^85 - 16 * q^86 - 55 * q^88 + 30 * q^89 + 14 * q^91 + 46 * q^92 - 66 * q^94 + 32 * q^95 + 12 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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