LMFDB - Embedded newform 1143.2.a.i.1.4

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 15x^{4} + 17x^{3} - 28x^{2} - 11x + 15 $ x^7 - 2x^6 - 8x^5 + 15x^4 + 17x^3 - 28x^2 - 11x + 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.818322$ 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)

$f(q)$	$=$	$q-0.818322 q^{2} -1.33035 q^{4} -2.74338 q^{5} -0.135055 q^{7} +2.72530 q^{8} +O(q^{10})$	\(q-0.818322 q^{2} -1.33035 q^{4} -2.74338 q^{5} -0.135055 q^{7} +2.72530 q^{8} +2.24497 q^{10} +0.266231 q^{11} +3.28557 q^{13} +0.110519 q^{14} +0.430525 q^{16} +4.65292 q^{17} -3.29662 q^{19} +3.64965 q^{20} -0.217863 q^{22} +1.80234 q^{23} +2.52613 q^{25} -2.68866 q^{26} +0.179671 q^{28} -3.76272 q^{29} -2.53611 q^{31} -5.80291 q^{32} -3.80759 q^{34} +0.370508 q^{35} +0.0550810 q^{37} +2.69770 q^{38} -7.47653 q^{40} -7.78083 q^{41} +3.55180 q^{43} -0.354180 q^{44} -1.47489 q^{46} +6.62310 q^{47} -6.98176 q^{49} -2.06718 q^{50} -4.37096 q^{52} -13.7361 q^{53} -0.730373 q^{55} -0.368066 q^{56} +3.07912 q^{58} -13.5354 q^{59} +9.30061 q^{61} +2.07535 q^{62} +3.88760 q^{64} -9.01357 q^{65} +6.45752 q^{67} -6.19001 q^{68} -0.303195 q^{70} -2.95281 q^{71} -11.8165 q^{73} -0.0450740 q^{74} +4.38566 q^{76} -0.0359559 q^{77} -11.4993 q^{79} -1.18109 q^{80} +6.36722 q^{82} +0.0301401 q^{83} -12.7647 q^{85} -2.90652 q^{86} +0.725560 q^{88} -2.02198 q^{89} -0.443734 q^{91} -2.39774 q^{92} -5.41983 q^{94} +9.04388 q^{95} +14.6568 q^{97} +5.71333 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * 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 $

$2$	−0.818322	−0.578641	−0.289321	−	0.957232i	$-0.593429\pi$
$2$	−0.818322	−0.578641	−0.289321	+	0.957232i	$0.593429\pi$
$3$	0	0
$3$	0	0
$4$	−1.33035	−0.665174
$5$	−2.74338	−1.22688	−0.613438	−	0.789743i	$-0.710214\pi$
$5$	−2.74338	−1.22688	−0.613438	+	0.789743i	$0.710214\pi$
$6$	0	0
$7$	−0.135055	−0.0510461	−0.0255231	−	0.999674i	$-0.508125\pi$
$7$	−0.135055	−0.0510461	−0.0255231	+	0.999674i	$0.508125\pi$
$8$	2.72530	0.963539
$9$	0	0
$10$	2.24497	0.709921
$11$	0.266231	0.0802717	0.0401359	−	0.999194i	$-0.487221\pi$
$11$	0.266231	0.0802717	0.0401359	+	0.999194i	$0.487221\pi$
$12$	0	0
$13$	3.28557	0.911254	0.455627	−	0.890171i	$-0.349415\pi$
$13$	3.28557	0.911254	0.455627	+	0.890171i	$0.349415\pi$
$14$	0.110519	0.0295374
$15$	0	0
$16$	0.430525	0.107631
$17$	4.65292	1.12850	0.564250	−	0.825604i	$-0.309165\pi$
$17$	4.65292	1.12850	0.564250	+	0.825604i	$0.309165\pi$
$18$	0	0
$19$	−3.29662	−0.756297	−0.378148	−	0.925745i	$-0.623439\pi$
$19$	−3.29662	−0.756297	−0.378148	+	0.925745i	$0.623439\pi$
$20$	3.64965	0.816087
$21$	0	0
$22$	−0.217863	−0.0464485
$23$	1.80234	0.375814	0.187907	−	0.982187i	$-0.439830\pi$
$23$	1.80234	0.375814	0.187907	+	0.982187i	$0.439830\pi$
$24$	0	0
$25$	2.52613	0.505225
$26$	−2.68866	−0.527289
$27$	0	0
$28$	0.179671	0.0339546
$29$	−3.76272	−0.698720	−0.349360	−	0.936989i	$-0.613601\pi$
$29$	−3.76272	−0.698720	−0.349360	+	0.936989i	$0.613601\pi$
$30$	0	0
$31$	−2.53611	−0.455498	−0.227749	−	0.973720i	$-0.573137\pi$
$31$	−2.53611	−0.455498	−0.227749	+	0.973720i	$0.573137\pi$
$32$	−5.80291	−1.02582
$33$	0	0
$34$	−3.80759	−0.652996
$35$	0.370508	0.0626273
$36$	0	0
$37$	0.0550810	0.00905525	0.00452763	−	0.999990i	$-0.498559\pi$
$37$	0.0550810	0.00905525	0.00452763	+	0.999990i	$0.498559\pi$
$38$	2.69770	0.437624
$39$	0	0
$40$	−7.47653	−1.18214
$41$	−7.78083	−1.21516	−0.607580	−	0.794258i	$-0.707860\pi$
$41$	−7.78083	−1.21516	−0.607580	+	0.794258i	$0.707860\pi$
$42$	0	0
$43$	3.55180	0.541645	0.270823	−	0.962629i	$-0.412704\pi$
$43$	3.55180	0.541645	0.270823	+	0.962629i	$0.412704\pi$
$44$	−0.354180	−0.0533947
$45$	0	0
$46$	−1.47489	−0.217461
$47$	6.62310	0.966078	0.483039	−	0.875599i	$-0.339533\pi$
$47$	6.62310	0.966078	0.483039	+	0.875599i	$0.339533\pi$
$48$	0	0
$49$	−6.98176	−0.997394
$50$	−2.06718	−0.292344
$51$	0	0
$52$	−4.37096	−0.606143
$53$	−13.7361	−1.88680	−0.943398	−	0.331664i	$-0.892390\pi$
$53$	−13.7361	−1.88680	−0.943398	+	0.331664i	$0.892390\pi$
$54$	0	0
$55$	−0.730373	−0.0984835
$56$	−0.368066	−0.0491849
$57$	0	0
$58$	3.07912	0.404308
$59$	−13.5354	−1.76216	−0.881082	−	0.472963i	$-0.843185\pi$
$59$	−13.5354	−1.76216	−0.881082	+	0.472963i	$0.843185\pi$
$60$	0	0
$61$	9.30061	1.19082	0.595411	−	0.803421i	$-0.296989\pi$
$61$	9.30061	1.19082	0.595411	+	0.803421i	$0.296989\pi$
$62$	2.07535	0.263570
$63$	0	0
$64$	3.88760	0.485950
$65$	−9.01357	−1.11800
$66$	0	0
$67$	6.45752	0.788911	0.394456	−	0.918915i	$-0.370933\pi$
$67$	6.45752	0.788911	0.394456	+	0.918915i	$0.370933\pi$
$68$	−6.19001	−0.750649
$69$	0	0
$70$	−0.303195	−0.0362387
$71$	−2.95281	−0.350434	−0.175217	−	0.984530i	$-0.556063\pi$
$71$	−2.95281	−0.350434	−0.175217	+	0.984530i	$0.556063\pi$
$72$	0	0
$73$	−11.8165	−1.38302	−0.691509	−	0.722368i	$-0.743054\pi$
$73$	−11.8165	−1.38302	−0.691509	+	0.722368i	$0.743054\pi$
$74$	−0.0450740	−0.00523974
$75$	0	0
$76$	4.38566	0.503069
$77$	−0.0359559	−0.00409756
$78$	0	0
$79$	−11.4993	−1.29378	−0.646888	−	0.762585i	$-0.723930\pi$
$79$	−11.4993	−1.29378	−0.646888	+	0.762585i	$0.723930\pi$
$80$	−1.18109	−0.132050
$81$	0	0
$82$	6.36722	0.703142
$83$	0.0301401	0.00330830	0.00165415	−	0.999999i	$-0.499473\pi$
$83$	0.0301401	0.00330830	0.00165415	+	0.999999i	$0.499473\pi$
$84$	0	0
$85$	−12.7647	−1.38453
$86$	−2.90652	−0.313418
$87$	0	0
$88$	0.725560	0.0773449
$89$	−2.02198	−0.214330	−0.107165	−	0.994241i	$-0.534177\pi$
$89$	−2.02198	−0.214330	−0.107165	+	0.994241i	$0.534177\pi$
$90$	0	0
$91$	−0.443734	−0.0465160
$92$	−2.39774	−0.249982
$93$	0	0
$94$	−5.41983	−0.559013
$95$	9.04388	0.927882
$96$	0	0
$97$	14.6568	1.48817	0.744084	−	0.668086i	$-0.232886\pi$
$97$	14.6568	1.48817	0.744084	+	0.668086i	$0.232886\pi$
$98$	5.71333	0.577133
$99$	0	0
$100$	−3.36063	−0.336063
$101$	1.54470	0.153703	0.0768516	−	0.997043i	$-0.475513\pi$
$101$	1.54470	0.153703	0.0768516	+	0.997043i	$0.475513\pi$
$102$	0	0
$103$	−14.2915	−1.40819	−0.704093	−	0.710108i	$-0.748646\pi$
$103$	−14.2915	−1.40819	−0.704093	+	0.710108i	$0.748646\pi$
$104$	8.95417	0.878028
$105$	0	0
$106$	11.2405	1.09178
$107$	19.6847	1.90299	0.951494	−	0.307667i	$-0.0995482\pi$
$107$	19.6847	1.90299	0.951494	+	0.307667i	$0.0995482\pi$
$108$	0	0
$109$	3.84703	0.368479	0.184239	−	0.982881i	$-0.441018\pi$
$109$	3.84703	0.368479	0.184239	+	0.982881i	$0.441018\pi$
$110$	0.597681	0.0569866
$111$	0	0
$112$	−0.0581447	−0.00549416
$113$	−11.2267	−1.05612	−0.528058	−	0.849208i	$-0.677080\pi$
$113$	−11.2267	−1.05612	−0.528058	+	0.849208i	$0.677080\pi$
$114$	0	0
$115$	−4.94450	−0.461077
$116$	5.00573	0.464770
$117$	0	0
$118$	11.0764	1.01966
$119$	−0.628402	−0.0576055
$120$	0	0
$121$	−10.9291	−0.993556
$122$	−7.61090	−0.689059
$123$	0	0
$124$	3.37391	0.302986
$125$	6.78677	0.607027
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.42450	0.744628
$129$	0	0
$130$	7.37601	0.646919
$131$	−5.32947	−0.465638	−0.232819	−	0.972520i	$-0.574795\pi$
$131$	−5.32947	−0.465638	−0.232819	+	0.972520i	$0.574795\pi$
$132$	0	0
$133$	0.445226	0.0386060
$134$	−5.28433	−0.456496
$135$	0	0
$136$	12.6806	1.08735
$137$	−7.84540	−0.670278	−0.335139	−	0.942169i	$-0.608783\pi$
$137$	−7.84540	−0.670278	−0.335139	+	0.942169i	$0.608783\pi$
$138$	0	0
$139$	−12.4749	−1.05810	−0.529052	−	0.848590i	$-0.677452\pi$
$139$	−12.4749	−1.05810	−0.529052	+	0.848590i	$0.677452\pi$
$140$	−0.492905	−0.0416580
$141$	0	0
$142$	2.41635	0.202775
$143$	0.874722	0.0731480
$144$	0	0
$145$	10.3226	0.857243
$146$	9.66971	0.800271
$147$	0	0
$148$	−0.0732769	−0.00602332
$149$	17.3556	1.42183	0.710913	−	0.703279i	$-0.248282\pi$
$149$	17.3556	1.42183	0.710913	+	0.703279i	$0.248282\pi$
$150$	0	0
$151$	−21.4400	−1.74476	−0.872380	−	0.488828i	$-0.837425\pi$
$151$	−21.4400	−1.74476	−0.872380	+	0.488828i	$0.837425\pi$
$152$	−8.98428	−0.728721
$153$	0	0
$154$	0.0294236	0.00237102
$155$	6.95750	0.558840
$156$	0	0
$157$	−18.0373	−1.43953	−0.719765	−	0.694217i	$-0.755751\pi$
$157$	−18.0373	−1.43953	−0.719765	+	0.694217i	$0.755751\pi$
$158$	9.41017	0.748633
$159$	0	0
$160$	15.9196	1.25855
$161$	−0.243415	−0.0191838
$162$	0	0
$163$	11.3953	0.892552	0.446276	−	0.894895i	$-0.352750\pi$
$163$	11.3953	0.892552	0.446276	+	0.894895i	$0.352750\pi$
$164$	10.3512	0.808294
$165$	0	0
$166$	−0.0246643	−0.00191432
$167$	−7.23205	−0.559633	−0.279816	−	0.960054i	$-0.590274\pi$
$167$	−7.23205	−0.559633	−0.279816	+	0.960054i	$0.590274\pi$
$168$	0	0
$169$	−2.20501	−0.169616
$170$	10.4457	0.801146
$171$	0	0
$172$	−4.72514	−0.360288
$173$	−8.65578	−0.658087	−0.329043	−	0.944315i	$-0.606726\pi$
$173$	−8.65578	−0.658087	−0.329043	+	0.944315i	$0.606726\pi$
$174$	0	0
$175$	−0.341167	−0.0257898
$176$	0.114619	0.00863975
$177$	0	0
$178$	1.65464	0.124020
$179$	−8.38070	−0.626403	−0.313201	−	0.949687i	$-0.601402\pi$
$179$	−8.38070	−0.626403	−0.313201	+	0.949687i	$0.601402\pi$
$180$	0	0
$181$	19.8253	1.47360	0.736802	−	0.676108i	$-0.236335\pi$
$181$	19.8253	1.47360	0.736802	+	0.676108i	$0.236335\pi$
$182$	0.363118	0.0269161
$183$	0	0
$184$	4.91191	0.362111
$185$	−0.151108	−0.0111097
$186$	0	0
$187$	1.23875	0.0905866
$188$	−8.81103	−0.642611
$189$	0	0
$190$	−7.40081	−0.536911
$191$	−2.78774	−0.201714	−0.100857	−	0.994901i	$-0.532158\pi$
$191$	−2.78774	−0.201714	−0.100857	+	0.994901i	$0.532158\pi$
$192$	0	0
$193$	5.12334	0.368786	0.184393	−	0.982853i	$-0.440968\pi$
$193$	5.12334	0.368786	0.184393	+	0.982853i	$0.440968\pi$
$194$	−11.9940	−0.861116
$195$	0	0
$196$	9.28818	0.663441
$197$	−14.0543	−1.00133	−0.500665	−	0.865641i	$-0.666911\pi$
$197$	−14.0543	−1.00133	−0.500665	+	0.865641i	$0.666911\pi$
$198$	0	0
$199$	−24.5793	−1.74238	−0.871191	−	0.490944i	$-0.836652\pi$
$199$	−24.5793	−1.74238	−0.871191	+	0.490944i	$0.836652\pi$
$200$	6.88445	0.486804
$201$	0	0
$202$	−1.26406	−0.0889390
$203$	0.508175	0.0356669
$204$	0	0
$205$	21.3458	1.49085
$206$	11.6951	0.814834
$207$	0	0
$208$	1.41452	0.0980794
$209$	−0.877664	−0.0607093
$210$	0	0
$211$	−11.8022	−0.812494	−0.406247	−	0.913763i	$-0.633163\pi$
$211$	−11.8022	−0.812494	−0.406247	+	0.913763i	$0.633163\pi$
$212$	18.2738	1.25505
$213$	0	0
$214$	−16.1084	−1.10115
$215$	−9.74394	−0.664532
$216$	0	0
$217$	0.342515	0.0232514
$218$	−3.14811	−0.213217
$219$	0	0
$220$	0.971651	0.0655087
$221$	15.2875	1.02835
$222$	0	0
$223$	20.6060	1.37988	0.689940	−	0.723867i	$-0.257637\pi$
$223$	20.6060	1.37988	0.689940	+	0.723867i	$0.257637\pi$
$224$	0.783713	0.0523640
$225$	0	0
$226$	9.18704	0.611113
$227$	−9.78820	−0.649665	−0.324833	−	0.945772i	$-0.605308\pi$
$227$	−9.78820	−0.649665	−0.324833	+	0.945772i	$0.605308\pi$
$228$	0	0
$229$	5.51529	0.364460	0.182230	−	0.983256i	$-0.441668\pi$
$229$	5.51529	0.364460	0.182230	+	0.983256i	$0.441668\pi$
$230$	4.04619	0.266798
$231$	0	0
$232$	−10.2545	−0.673243
$233$	−16.5524	−1.08438	−0.542192	−	0.840255i	$-0.682405\pi$
$233$	−16.5524	−1.08438	−0.542192	+	0.840255i	$0.682405\pi$
$234$	0	0
$235$	−18.1697	−1.18526
$236$	18.0069	1.17215
$237$	0	0
$238$	0.514235	0.0333329
$239$	18.7871	1.21524	0.607618	−	0.794229i	$-0.292125\pi$
$239$	18.7871	1.21524	0.607618	+	0.794229i	$0.292125\pi$
$240$	0	0
$241$	−17.9210	−1.15439	−0.577196	−	0.816605i	$-0.695853\pi$
$241$	−17.9210	−1.15439	−0.577196	+	0.816605i	$0.695853\pi$
$242$	8.94354	0.574913
$243$	0	0
$244$	−12.3731	−0.792104
$245$	19.1536	1.22368
$246$	0	0
$247$	−10.8313	−0.689178
$248$	−6.91165	−0.438890
$249$	0	0
$250$	−5.55377	−0.351251
$251$	8.22074	0.518888	0.259444	−	0.965758i	$-0.416461\pi$
$251$	8.22074	0.518888	0.259444	+	0.965758i	$0.416461\pi$
$252$	0	0
$253$	0.479839	0.0301672
$254$	−0.818322	−0.0513461
$255$	0	0
$256$	−14.6692	−0.916822
$257$	−18.7527	−1.16976	−0.584880	−	0.811120i	$-0.698859\pi$
$257$	−18.7527	−1.16976	−0.584880	+	0.811120i	$0.698859\pi$
$258$	0	0
$259$	−0.00743898	−0.000462235 0
$260$	11.9912	0.743662
$261$	0	0
$262$	4.36122	0.269437
$263$	−6.97068	−0.429831	−0.214915	−	0.976633i	$-0.568948\pi$
$263$	−6.97068	−0.429831	−0.214915	+	0.976633i	$0.568948\pi$
$264$	0	0
$265$	37.6833	2.31486
$266$	−0.364339	−0.0223390
$267$	0	0
$268$	−8.59075	−0.524763
$269$	0.735286	0.0448312	0.0224156	−	0.999749i	$-0.492864\pi$
$269$	0.735286	0.0448312	0.0224156	+	0.999749i	$0.492864\pi$
$270$	0	0
$271$	20.6629	1.25518	0.627591	−	0.778543i	$-0.284041\pi$
$271$	20.6629	1.25518	0.627591	+	0.778543i	$0.284041\pi$
$272$	2.00320	0.121462
$273$	0	0
$274$	6.42007	0.387850
$275$	0.672534	0.0405553
$276$	0	0
$277$	−0.175839	−0.0105652	−0.00528258	−	0.999986i	$-0.501682\pi$
$277$	−0.175839	−0.0105652	−0.00528258	+	0.999986i	$0.501682\pi$
$278$	10.2085	0.612262
$279$	0	0
$280$	1.00974	0.0603438
$281$	20.2767	1.20961	0.604804	−	0.796374i	$-0.293251\pi$
$281$	20.2767	1.20961	0.604804	+	0.796374i	$0.293251\pi$
$282$	0	0
$283$	−16.4628	−0.978610	−0.489305	−	0.872113i	$-0.662749\pi$
$283$	−16.4628	−0.978610	−0.489305	+	0.872113i	$0.662749\pi$
$284$	3.92826	0.233100
$285$	0	0
$286$	−0.715805	−0.0423264
$287$	1.05084	0.0620292
$288$	0	0
$289$	4.64969	0.273511
$290$	−8.44719	−0.496036
$291$	0	0
$292$	15.7201	0.919948
$293$	25.7168	1.50239	0.751196	−	0.660080i	$-0.229477\pi$
$293$	25.7168	1.50239	0.751196	+	0.660080i	$0.229477\pi$
$294$	0	0
$295$	37.1329	2.16196
$296$	0.150112	0.00872508
$297$	0	0
$298$	−14.2025	−0.822728
$299$	5.92172	0.342462
$300$	0	0
$301$	−0.479690	−0.0276489
$302$	17.5448	1.00959
$303$	0	0
$304$	−1.41928	−0.0814011
$305$	−25.5151	−1.46099
$306$	0	0
$307$	23.8011	1.35840	0.679201	−	0.733952i	$-0.262326\pi$
$307$	23.8011	1.35840	0.679201	+	0.733952i	$0.262326\pi$
$308$	0.0478339	0.00272559
$309$	0	0
$310$	−5.69348	−0.323368
$311$	7.55647	0.428488	0.214244	−	0.976780i	$-0.431271\pi$
$311$	7.55647	0.428488	0.214244	+	0.976780i	$0.431271\pi$
$312$	0	0
$313$	−0.126520	−0.00715132	−0.00357566	−	0.999994i	$-0.501138\pi$
$313$	−0.126520	−0.00715132	−0.00357566	+	0.999994i	$0.501138\pi$
$314$	14.7603	0.832972
$315$	0	0
$316$	15.2981	0.860587
$317$	−0.143760	−0.00807437	−0.00403718	−	0.999992i	$-0.501285\pi$
$317$	−0.143760	−0.00807437	−0.00403718	+	0.999992i	$0.501285\pi$
$318$	0	0
$319$	−1.00175	−0.0560874
$320$	−10.6651	−0.596200
$321$	0	0
$322$	0.199192	0.0111006
$323$	−15.3389	−0.853480
$324$	0	0
$325$	8.29977	0.460388
$326$	−9.32507	−0.516468
$327$	0	0
$328$	−21.2051	−1.17085
$329$	−0.894485	−0.0493145
$330$	0	0
$331$	10.1633	0.558626	0.279313	−	0.960200i	$-0.409893\pi$
$331$	10.1633	0.558626	0.279313	+	0.960200i	$0.409893\pi$
$332$	−0.0400968	−0.00220060
$333$	0	0
$334$	5.91815	0.323827
$335$	−17.7154	−0.967896
$336$	0	0
$337$	−2.83802	−0.154597	−0.0772983	−	0.997008i	$-0.524629\pi$
$337$	−2.83802	−0.154597	−0.0772983	+	0.997008i	$0.524629\pi$
$338$	1.80441	0.0981468
$339$	0	0
$340$	16.9815	0.920953
$341$	−0.675191	−0.0365636
$342$	0	0
$343$	1.88831	0.101959
$344$	9.67973	0.521896
$345$	0	0
$346$	7.08322	0.380796
$347$	5.51387	0.296000	0.148000	−	0.988987i	$-0.452716\pi$
$347$	5.51387	0.296000	0.148000	+	0.988987i	$0.452716\pi$
$348$	0	0
$349$	−23.3543	−1.25013	−0.625065	−	0.780573i	$-0.714927\pi$
$349$	−23.3543	−1.25013	−0.625065	+	0.780573i	$0.714927\pi$
$350$	0.279184	0.0149230
$351$	0	0
$352$	−1.54491	−0.0823442
$353$	2.00940	0.106949	0.0534747	−	0.998569i	$-0.482970\pi$
$353$	2.00940	0.106949	0.0534747	+	0.998569i	$0.482970\pi$
$354$	0	0
$355$	8.10067	0.429939
$356$	2.68994	0.142567
$357$	0	0
$358$	6.85811	0.362463
$359$	−28.7692	−1.51838	−0.759189	−	0.650870i	$-0.774404\pi$
$359$	−28.7692	−1.51838	−0.759189	+	0.650870i	$0.774404\pi$
$360$	0	0
$361$	−8.13229	−0.428015
$362$	−16.2235	−0.852688
$363$	0	0
$364$	0.590321	0.0309412
$365$	32.4172	1.69679
$366$	0	0
$367$	35.8154	1.86955	0.934774	−	0.355243i	$-0.115602\pi$
$367$	35.8154	1.86955	0.934774	+	0.355243i	$0.115602\pi$
$368$	0.775952	0.0404493
$369$	0	0
$370$	0.123655	0.00642852
$371$	1.85513	0.0963136
$372$	0	0
$373$	−2.54824	−0.131943	−0.0659714	−	0.997822i	$-0.521015\pi$
$373$	−2.54824	−0.131943	−0.0659714	+	0.997822i	$0.521015\pi$
$374$	−1.01370	−0.0524172
$375$	0	0
$376$	18.0499	0.930854
$377$	−12.3627	−0.636711
$378$	0	0
$379$	28.6007	1.46912	0.734561	−	0.678543i	$-0.237388\pi$
$379$	28.6007	1.46912	0.734561	+	0.678543i	$0.237388\pi$
$380$	−12.0315	−0.617204
$381$	0	0
$382$	2.28127	0.116720
$383$	12.0698	0.616736	0.308368	−	0.951267i	$-0.400217\pi$
$383$	12.0698	0.616736	0.308368	+	0.951267i	$0.400217\pi$
$384$	0	0
$385$	0.0986408	0.00502720
$386$	−4.19254	−0.213395
$387$	0	0
$388$	−19.4986	−0.989892
$389$	22.4001	1.13573	0.567865	−	0.823122i	$-0.307770\pi$
$389$	22.4001	1.13573	0.567865	+	0.823122i	$0.307770\pi$
$390$	0	0
$391$	8.38614	0.424106
$392$	−19.0274	−0.961028
$393$	0	0
$394$	11.5010	0.579410
$395$	31.5470	1.58730
$396$	0	0
$397$	−31.3195	−1.57188	−0.785940	−	0.618303i	$-0.787820\pi$
$397$	−31.3195	−1.57188	−0.785940	+	0.618303i	$0.787820\pi$
$398$	20.1138	1.00821
$399$	0	0
$400$	1.08756	0.0543780
$401$	−4.25162	−0.212316	−0.106158	−	0.994349i	$-0.533855\pi$
$401$	−4.25162	−0.212316	−0.106158	+	0.994349i	$0.533855\pi$
$402$	0	0
$403$	−8.33257	−0.415075
$404$	−2.05499	−0.102239
$405$	0	0
$406$	−0.415851	−0.0206384
$407$	0.0146643	0.000726881 0
$408$	0	0
$409$	−25.0374	−1.23802	−0.619010	−	0.785383i	$-0.712466\pi$
$409$	−25.0374	−1.23802	−0.619010	+	0.785383i	$0.712466\pi$
$410$	−17.4677	−0.862668
$411$	0	0
$412$	19.0127	0.936689
$413$	1.82803	0.0899517
$414$	0	0
$415$	−0.0826856	−0.00405888
$416$	−19.0659	−0.934781
$417$	0	0
$418$	0.718212	0.0351289
$419$	−19.2538	−0.940612	−0.470306	−	0.882503i	$-0.655856\pi$
$419$	−19.2538	−0.940612	−0.470306	+	0.882503i	$0.655856\pi$
$420$	0	0
$421$	−5.07775	−0.247474	−0.123737	−	0.992315i	$-0.539488\pi$
$421$	−5.07775	−0.247474	−0.123737	+	0.992315i	$0.539488\pi$
$422$	9.65797	0.470142
$423$	0	0
$424$	−37.4349	−1.81800
$425$	11.7539	0.570146
$426$	0	0
$427$	−1.25610	−0.0607868
$428$	−26.1875	−1.26582
$429$	0	0
$430$	7.97369	0.384525
$431$	23.7235	1.14272	0.571360	−	0.820699i	$-0.306416\pi$
$431$	23.7235	1.14272	0.571360	+	0.820699i	$0.306416\pi$
$432$	0	0
$433$	−1.24058	−0.0596187	−0.0298093	−	0.999556i	$-0.509490\pi$
$433$	−1.24058	−0.0596187	−0.0298093	+	0.999556i	$0.509490\pi$
$434$	−0.280288	−0.0134542
$435$	0	0
$436$	−5.11790	−0.245103
$437$	−5.94163	−0.284227
$438$	0	0
$439$	16.1814	0.772294	0.386147	−	0.922437i	$-0.373806\pi$
$439$	16.1814	0.772294	0.386147	+	0.922437i	$0.373806\pi$
$440$	−1.99048	−0.0948926
$441$	0	0
$442$	−12.5101	−0.595046
$443$	17.0999	0.812441	0.406220	−	0.913775i	$-0.366847\pi$
$443$	17.0999	0.812441	0.406220	+	0.913775i	$0.366847\pi$
$444$	0	0
$445$	5.54707	0.262956
$446$	−16.8624	−0.798455
$447$	0	0
$448$	−0.525041	−0.0248058
$449$	−36.9220	−1.74246	−0.871228	−	0.490879i	$-0.836676\pi$
$449$	−36.9220	−1.74246	−0.871228	+	0.490879i	$0.836676\pi$
$450$	0	0
$451$	−2.07150	−0.0975431
$452$	14.9354	0.702502
$453$	0	0
$454$	8.00990	0.375923
$455$	1.21733	0.0570693
$456$	0	0
$457$	−13.2331	−0.619017	−0.309508	−	0.950897i	$-0.600164\pi$
$457$	−13.2331	−0.619017	−0.309508	+	0.950897i	$0.600164\pi$
$458$	−4.51328	−0.210892
$459$	0	0
$460$	6.57791	0.306696
$461$	−17.0587	−0.794503	−0.397251	−	0.917710i	$-0.630036\pi$
$461$	−17.0587	−0.794503	−0.397251	+	0.917710i	$0.630036\pi$
$462$	0	0
$463$	−18.1318	−0.842657	−0.421328	−	0.906908i	$-0.638436\pi$
$463$	−18.1318	−0.842657	−0.421328	+	0.906908i	$0.638436\pi$
$464$	−1.61994	−0.0752040
$465$	0	0
$466$	13.5452	0.627469
$467$	−32.6217	−1.50955	−0.754776	−	0.655982i	$-0.772255\pi$
$467$	−32.6217	−1.50955	−0.754776	+	0.655982i	$0.772255\pi$
$468$	0	0
$469$	−0.872122	−0.0402708
$470$	14.8686	0.685839
$471$	0	0
$472$	−36.8881	−1.69791
$473$	0.945601	0.0434788
$474$	0	0
$475$	−8.32768	−0.382100
$476$	0.835994	0.0383177
$477$	0	0
$478$	−15.3739	−0.703186
$479$	−30.5570	−1.39619	−0.698093	−	0.716007i	$-0.745968\pi$
$479$	−30.5570	−1.39619	−0.698093	+	0.716007i	$0.745968\pi$
$480$	0	0
$481$	0.180973	0.00825164
$482$	14.6651	0.667979
$483$	0	0
$484$	14.5395	0.660888
$485$	−40.2090	−1.82580
$486$	0	0
$487$	20.5767	0.932419	0.466209	−	0.884674i	$-0.345619\pi$
$487$	20.5767	0.932419	0.466209	+	0.884674i	$0.345619\pi$
$488$	25.3470	1.14740
$489$	0	0
$490$	−15.6738	−0.708071
$491$	26.3733	1.19021	0.595105	−	0.803648i	$-0.297111\pi$
$491$	26.3733	1.19021	0.595105	+	0.803648i	$0.297111\pi$
$492$	0	0
$493$	−17.5076	−0.788505
$494$	8.86349	0.398787
$495$	0	0
$496$	−1.09186	−0.0490258
$497$	0.398792	0.0178883
$498$	0	0
$499$	5.20889	0.233182	0.116591	−	0.993180i	$-0.462803\pi$
$499$	5.20889	0.233182	0.116591	+	0.993180i	$0.462803\pi$
$500$	−9.02877	−0.403779
$501$	0	0
$502$	−6.72721	−0.300250
$503$	9.38865	0.418619	0.209310	−	0.977849i	$-0.432878\pi$
$503$	9.38865	0.418619	0.209310	+	0.977849i	$0.432878\pi$
$504$	0	0
$505$	−4.23769	−0.188575
$506$	−0.392663	−0.0174560
$507$	0	0
$508$	−1.33035	−0.0590247
$509$	31.6611	1.40335	0.701676	−	0.712496i	$-0.252435\pi$
$509$	31.6611	1.40335	0.701676	+	0.712496i	$0.252435\pi$
$510$	0	0
$511$	1.59588	0.0705977
$512$	−4.84491	−0.214117
$513$	0	0
$514$	15.3457	0.676872
$515$	39.2071	1.72767
$516$	0	0
$517$	1.76328	0.0775488
$518$	0.00608748	0.000267468 0
$519$	0	0
$520$	−24.5647	−1.07723
$521$	−6.22207	−0.272594	−0.136297	−	0.990668i	$-0.543520\pi$
$521$	−6.22207	−0.272594	−0.136297	+	0.990668i	$0.543520\pi$
$522$	0	0
$523$	−38.3006	−1.67477	−0.837385	−	0.546614i	$-0.815916\pi$
$523$	−38.3006	−1.67477	−0.837385	+	0.546614i	$0.815916\pi$
$524$	7.09005	0.309730
$525$	0	0
$526$	5.70426	0.248718
$527$	−11.8003	−0.514030
$528$	0	0
$529$	−19.7516	−0.858764
$530$	−30.8370	−1.33948
$531$	0	0
$532$	−0.592306	−0.0256797
$533$	−25.5645	−1.10732
$534$	0	0
$535$	−54.0025	−2.33473
$536$	17.5987	0.760146
$537$	0	0
$538$	−0.601701	−0.0259412
$539$	−1.85876	−0.0800626
$540$	0	0
$541$	−41.0790	−1.76612	−0.883061	−	0.469258i	$-0.844522\pi$
$541$	−41.0790	−1.76612	−0.883061	+	0.469258i	$0.844522\pi$
$542$	−16.9089	−0.726300
$543$	0	0
$544$	−27.0005	−1.15764
$545$	−10.5539	−0.452078
$546$	0	0
$547$	8.06670	0.344907	0.172454	−	0.985018i	$-0.444830\pi$
$547$	8.06670	0.344907	0.172454	+	0.985018i	$0.444830\pi$
$548$	10.4371	0.445852
$549$	0	0
$550$	−0.550349	−0.0234670
$551$	12.4043	0.528439
$552$	0	0
$553$	1.55305	0.0660423
$554$	0.143893	0.00611344
$555$	0	0
$556$	16.5959	0.703823
$557$	3.87559	0.164214	0.0821071	−	0.996624i	$-0.473835\pi$
$557$	3.87559	0.164214	0.0821071	+	0.996624i	$0.473835\pi$
$558$	0	0
$559$	11.6697	0.493576
$560$	0.159513	0.00674065
$561$	0	0
$562$	−16.5929	−0.699930
$563$	−16.5843	−0.698945	−0.349472	−	0.936947i	$-0.613639\pi$
$563$	−16.5843	−0.698945	−0.349472	+	0.936947i	$0.613639\pi$
$564$	0	0
$565$	30.7990	1.29572
$566$	13.4718	0.566264
$567$	0	0
$568$	−8.04728	−0.337657
$569$	−39.5356	−1.65742	−0.828710	−	0.559678i	$-0.810925\pi$
$569$	−39.5356	−1.65742	−0.828710	+	0.559678i	$0.810925\pi$
$570$	0	0
$571$	17.7137	0.741293	0.370647	−	0.928774i	$-0.379136\pi$
$571$	17.7137	0.741293	0.370647	+	0.928774i	$0.379136\pi$
$572$	−1.16369	−0.0486561
$573$	0	0
$574$	−0.859927	−0.0358927
$575$	4.55293	0.189871
$576$	0	0
$577$	40.1099	1.66980	0.834898	−	0.550405i	$-0.185527\pi$
$577$	40.1099	1.66980	0.834898	+	0.550405i	$0.185527\pi$
$578$	−3.80495	−0.158265
$579$	0	0
$580$	−13.7326	−0.570216
$581$	−0.00407057	−0.000168876 0
$582$	0	0
$583$	−3.65697	−0.151456
$584$	−32.2035	−1.33259
$585$	0	0
$586$	−21.0446	−0.869345
$587$	−24.7808	−1.02281	−0.511406	−	0.859339i	$-0.670875\pi$
$587$	−24.7808	−1.02281	−0.511406	+	0.859339i	$0.670875\pi$
$588$	0	0
$589$	8.36059	0.344492
$590$	−30.3866	−1.25100
$591$	0	0
$592$	0.0237137	0.000974628 0
$593$	−34.3299	−1.40976	−0.704880	−	0.709327i	$-0.748999\pi$
$593$	−34.3299	−1.40976	−0.704880	+	0.709327i	$0.748999\pi$
$594$	0	0
$595$	1.72394	0.0706748
$596$	−23.0890	−0.945763
$597$	0	0
$598$	−4.84587	−0.198162
$599$	45.8613	1.87384	0.936921	−	0.349541i	$-0.113663\pi$
$599$	45.8613	1.87384	0.936921	+	0.349541i	$0.113663\pi$
$600$	0	0
$601$	11.0494	0.450715	0.225358	−	0.974276i	$-0.427645\pi$
$601$	11.0494	0.450715	0.225358	+	0.974276i	$0.427645\pi$
$602$	0.392541	0.0159988
$603$	0	0
$604$	28.5226	1.16057
$605$	29.9827	1.21897
$606$	0	0
$607$	4.91310	0.199416	0.0997082	−	0.995017i	$-0.468209\pi$
$607$	4.91310	0.199416	0.0997082	+	0.995017i	$0.468209\pi$
$608$	19.1300	0.775823
$609$	0	0
$610$	20.8796	0.845390
$611$	21.7607	0.880343
$612$	0	0
$613$	−30.9734	−1.25101	−0.625503	−	0.780222i	$-0.715106\pi$
$613$	−30.9734	−1.25101	−0.625503	+	0.780222i	$0.715106\pi$
$614$	−19.4770	−0.786027
$615$	0	0
$616$	−0.0979907	−0.00394816
$617$	12.3335	0.496530	0.248265	−	0.968692i	$-0.420140\pi$
$617$	12.3335	0.496530	0.248265	+	0.968692i	$0.420140\pi$
$618$	0	0
$619$	4.48347	0.180206	0.0901030	−	0.995932i	$-0.471280\pi$
$619$	4.48347	0.180206	0.0901030	+	0.995932i	$0.471280\pi$
$620$	−9.25590	−0.371726
$621$	0	0
$622$	−6.18363	−0.247941
$623$	0.273080	0.0109407
$624$	0	0
$625$	−31.2493	−1.24997
$626$	0.103534	0.00413805
$627$	0	0
$628$	23.9959	0.957539
$629$	0.256287	0.0102188
$630$	0	0
$631$	47.1030	1.87514	0.937570	−	0.347798i	$-0.113070\pi$
$631$	47.1030	1.87514	0.937570	+	0.347798i	$0.113070\pi$
$632$	−31.3391	−1.24660
$633$	0	0
$634$	0.117642	0.00467216
$635$	−2.74338	−0.108868
$636$	0	0
$637$	−22.9391	−0.908880
$638$	0.819758	0.0324545
$639$	0	0
$640$	−23.1116	−0.913566
$641$	25.8930	1.02271	0.511357	−	0.859369i	$-0.329143\pi$
$641$	25.8930	1.02271	0.511357	+	0.859369i	$0.329143\pi$
$642$	0	0
$643$	22.4912	0.886965	0.443483	−	0.896283i	$-0.353743\pi$
$643$	22.4912	0.886965	0.443483	+	0.896283i	$0.353743\pi$
$644$	0.323827	0.0127606
$645$	0	0
$646$	12.5522	0.493859
$647$	−29.3863	−1.15530	−0.577648	−	0.816286i	$-0.696029\pi$
$647$	−29.3863	−1.15530	−0.577648	+	0.816286i	$0.696029\pi$
$648$	0	0
$649$	−3.60356	−0.141452
$650$	−6.79189	−0.266400
$651$	0	0
$652$	−15.1598	−0.593703
$653$	−15.8265	−0.619338	−0.309669	−	0.950844i	$-0.600218\pi$
$653$	−15.8265	−0.619338	−0.309669	+	0.950844i	$0.600218\pi$
$654$	0	0
$655$	14.6207	0.571280
$656$	−3.34984	−0.130789
$657$	0	0
$658$	0.731977	0.0285354
$659$	−39.9760	−1.55724	−0.778622	−	0.627493i	$-0.784081\pi$
$659$	−39.9760	−1.55724	−0.778622	+	0.627493i	$0.784081\pi$
$660$	0	0
$661$	12.3341	0.479741	0.239870	−	0.970805i	$-0.422895\pi$
$661$	12.3341	0.479741	0.239870	+	0.970805i	$0.422895\pi$
$662$	−8.31686	−0.323244
$663$	0	0
$664$	0.0821406	0.00318768
$665$	−1.22142	−0.0473648
$666$	0	0
$667$	−6.78170	−0.262588
$668$	9.62115	0.372253
$669$	0	0
$670$	14.4969	0.560065
$671$	2.47611	0.0955893
$672$	0	0
$673$	12.2289	0.471390	0.235695	−	0.971827i	$-0.424263\pi$
$673$	12.2289	0.471390	0.235695	+	0.971827i	$0.424263\pi$
$674$	2.32241	0.0894560
$675$	0	0
$676$	2.93343	0.112824
$677$	−23.6394	−0.908534	−0.454267	−	0.890865i	$-0.650099\pi$
$677$	−23.6394	−0.908534	−0.454267	+	0.890865i	$0.650099\pi$
$678$	0	0
$679$	−1.97947	−0.0759652
$680$	−34.7877	−1.33405
$681$	0	0
$682$	0.552524	0.0211572
$683$	34.0115	1.30142	0.650708	−	0.759328i	$-0.274472\pi$
$683$	34.0115	1.30142	0.650708	+	0.759328i	$0.274472\pi$
$684$	0	0
$685$	21.5229	0.822348
$686$	−1.54525	−0.0589978
$687$	0	0
$688$	1.52914	0.0582979
$689$	−45.1309	−1.71935
$690$	0	0
$691$	−45.9774	−1.74906	−0.874531	−	0.484971i	$-0.838830\pi$
$691$	−45.9774	−1.74906	−0.874531	+	0.484971i	$0.838830\pi$
$692$	11.5152	0.437743
$693$	0	0
$694$	−4.51213	−0.171278
$695$	34.2232	1.29816
$696$	0	0
$697$	−36.2036	−1.37131
$698$	19.1114	0.723376
$699$	0	0
$700$	0.453871	0.0171547
$701$	−19.5014	−0.736557	−0.368279	−	0.929715i	$-0.620053\pi$
$701$	−19.5014	−0.736557	−0.368279	+	0.929715i	$0.620053\pi$
$702$	0	0
$703$	−0.181581	−0.00684846
$704$	1.03500	0.0390080
$705$	0	0
$706$	−1.64434	−0.0618854
$707$	−0.208620	−0.00784595
$708$	0	0
$709$	16.1356	0.605984	0.302992	−	0.952993i	$-0.402015\pi$
$709$	16.1356	0.605984	0.302992	+	0.952993i	$0.402015\pi$
$710$	−6.62896	−0.248780
$711$	0	0
$712$	−5.51051	−0.206515
$713$	−4.57093	−0.171182
$714$	0	0
$715$	−2.39969	−0.0897435
$716$	11.1493	0.416667
$717$	0	0
$718$	23.5425	0.878597
$719$	16.7165	0.623419	0.311710	−	0.950177i	$-0.399098\pi$
$719$	16.7165	0.623419	0.311710	+	0.950177i	$0.399098\pi$
$720$	0	0
$721$	1.93015	0.0718824
$722$	6.65484	0.247667
$723$	0	0
$724$	−26.3746	−0.980204
$725$	−9.50511	−0.353011
$726$	0	0
$727$	−22.7271	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$727$	−22.7271	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$728$	−1.20931	−0.0448199
$729$	0	0
$730$	−26.5277	−0.981834
$731$	16.5263	0.611246
$732$	0	0
$733$	26.4585	0.977266	0.488633	−	0.872490i	$-0.337496\pi$
$733$	26.4585	0.977266	0.488633	+	0.872490i	$0.337496\pi$
$734$	−29.3085	−1.08180
$735$	0	0
$736$	−10.4588	−0.385517
$737$	1.71919	0.0633273
$738$	0	0
$739$	50.7663	1.86747	0.933733	−	0.357969i	$-0.116531\pi$
$739$	50.7663	1.86747	0.933733	+	0.357969i	$0.116531\pi$
$740$	0.201026	0.00738987
$741$	0	0
$742$	−1.51809	−0.0557310
$743$	10.2193	0.374910	0.187455	−	0.982273i	$-0.439976\pi$
$743$	10.2193	0.374910	0.187455	+	0.982273i	$0.439976\pi$
$744$	0	0
$745$	−47.6130	−1.74441
$746$	2.08528	0.0763475
$747$	0	0
$748$	−1.64797	−0.0602559
$749$	−2.65852	−0.0971402
$750$	0	0
$751$	41.4093	1.51105	0.755524	−	0.655121i	$-0.227382\pi$
$751$	41.4093	1.51105	0.755524	+	0.655121i	$0.227382\pi$
$752$	2.85141	0.103980
$753$	0	0
$754$	10.1167	0.368427
$755$	58.8180	2.14061
$756$	0	0
$757$	23.1449	0.841214	0.420607	−	0.907243i	$-0.361817\pi$
$757$	23.1449	0.841214	0.420607	+	0.907243i	$0.361817\pi$
$758$	−23.4046	−0.850094
$759$	0	0
$760$	24.6473	0.894050
$761$	39.4782	1.43108	0.715541	−	0.698571i	$-0.246180\pi$
$761$	39.4782	1.43108	0.715541	+	0.698571i	$0.246180\pi$
$762$	0	0
$763$	−0.519562	−0.0188094
$764$	3.70867	0.134175
$765$	0	0
$766$	−9.87696	−0.356869
$767$	−44.4717	−1.60578
$768$	0	0
$769$	−4.81634	−0.173682	−0.0868408	−	0.996222i	$-0.527677\pi$
$769$	−4.81634	−0.173682	−0.0868408	+	0.996222i	$0.527677\pi$
$770$	−0.0807199	−0.00290894
$771$	0	0
$772$	−6.81582	−0.245307
$773$	10.4872	0.377198	0.188599	−	0.982054i	$-0.439605\pi$
$773$	10.4872	0.377198	0.188599	+	0.982054i	$0.439605\pi$
$774$	0	0
$775$	−6.40653	−0.230129
$776$	39.9441	1.43391
$777$	0	0
$778$	−18.3305	−0.657180
$779$	25.6504	0.919022
$780$	0	0
$781$	−0.786130	−0.0281299
$782$	−6.86257	−0.245405
$783$	0	0
$784$	−3.00582	−0.107351
$785$	49.4831	1.76613
$786$	0	0
$787$	−4.39250	−0.156576	−0.0782879	−	0.996931i	$-0.524945\pi$
$787$	−4.39250	−0.156576	−0.0782879	+	0.996931i	$0.524945\pi$
$788$	18.6972	0.666058
$789$	0	0
$790$	−25.8157	−0.918480
$791$	1.51622	0.0539107
$792$	0	0
$793$	30.5579	1.08514
$794$	25.6294	0.909554
$795$	0	0
$796$	32.6991	1.15899
$797$	−39.2121	−1.38896	−0.694481	−	0.719511i	$-0.744366\pi$
$797$	−39.2121	−1.38896	−0.694481	+	0.719511i	$0.744366\pi$
$798$	0	0
$799$	30.8168	1.09022
$800$	−14.6589	−0.518269
$801$	0	0
$802$	3.47920	0.122855
$803$	−3.14592	−0.111017
$804$	0	0
$805$	0.667781	0.0235362
$806$	6.81873	0.240179
$807$	0	0
$808$	4.20976	0.148099
$809$	0.316252	0.0111188	0.00555942	−	0.999985i	$-0.498230\pi$
$809$	0.316252	0.0111188	0.00555942	+	0.999985i	$0.498230\pi$
$810$	0	0
$811$	−50.8916	−1.78705	−0.893524	−	0.449016i	$-0.851774\pi$
$811$	−50.8916	−1.78705	−0.893524	+	0.449016i	$0.851774\pi$
$812$	−0.676051	−0.0237247
$813$	0	0
$814$	−0.0120001	−0.000420603 0
$815$	−31.2617	−1.09505
$816$	0	0
$817$	−11.7090	−0.409644
$818$	20.4887	0.716369
$819$	0	0
$820$	−28.3973	−0.991676
$821$	15.4178	0.538085	0.269043	−	0.963128i	$-0.413293\pi$
$821$	15.4178	0.538085	0.269043	+	0.963128i	$0.413293\pi$
$822$	0	0
$823$	−48.8660	−1.70336	−0.851680	−	0.524062i	$-0.824416\pi$
$823$	−48.8660	−1.70336	−0.851680	+	0.524062i	$0.824416\pi$
$824$	−38.9487	−1.35684
$825$	0	0
$826$	−1.49592	−0.0520497
$827$	42.1609	1.46608	0.733039	−	0.680186i	$-0.238101\pi$
$827$	42.1609	1.46608	0.733039	+	0.680186i	$0.238101\pi$
$828$	0	0
$829$	28.4114	0.986769	0.493385	−	0.869811i	$-0.335760\pi$
$829$	28.4114	0.986769	0.493385	+	0.869811i	$0.335760\pi$
$830$	0.0676635	0.00234863
$831$	0	0
$832$	12.7730	0.442824
$833$	−32.4856	−1.12556
$834$	0	0
$835$	19.8402	0.686600
$836$	1.16760	0.0403822
$837$	0	0
$838$	15.7558	0.544277
$839$	10.9648	0.378546	0.189273	−	0.981924i	$-0.439387\pi$
$839$	10.9648	0.378546	0.189273	+	0.981924i	$0.439387\pi$
$840$	0	0
$841$	−14.8419	−0.511791
$842$	4.15524	0.143199
$843$	0	0
$844$	15.7010	0.540450
$845$	6.04917	0.208098
$846$	0	0
$847$	1.47604	0.0507172
$848$	−5.91372	−0.203078
$849$	0	0
$850$	−9.61845	−0.329910
$851$	0.0992745	0.00340309
$852$	0	0
$853$	−8.12843	−0.278312	−0.139156	−	0.990270i	$-0.544439\pi$
$853$	−8.12843	−0.278312	−0.139156	+	0.990270i	$0.544439\pi$
$854$	1.02789	0.0351738
$855$	0	0
$856$	53.6466	1.83360
$857$	−4.51225	−0.154135	−0.0770677	−	0.997026i	$-0.524556\pi$
$857$	−4.51225	−0.154135	−0.0770677	+	0.997026i	$0.524556\pi$
$858$	0	0
$859$	26.8887	0.917431	0.458716	−	0.888583i	$-0.348310\pi$
$859$	26.8887	0.917431	0.458716	+	0.888583i	$0.348310\pi$
$860$	12.9628	0.442029
$861$	0	0
$862$	−19.4135	−0.661225
$863$	47.8004	1.62715	0.813573	−	0.581463i	$-0.197519\pi$
$863$	47.8004	1.62715	0.813573	+	0.581463i	$0.197519\pi$
$864$	0	0
$865$	23.7461	0.807391
$866$	1.01520	0.0344978
$867$	0	0
$868$	−0.455664	−0.0154662
$869$	−3.06148	−0.103854
$870$	0	0
$871$	21.2166	0.718898
$872$	10.4843	0.355044
$873$	0	0
$874$	4.86217	0.164465
$875$	−0.916590	−0.0309864
$876$	0	0
$877$	−3.96696	−0.133955	−0.0669774	−	0.997754i	$-0.521336\pi$
$877$	−3.96696	−0.133955	−0.0669774	+	0.997754i	$0.521336\pi$
$878$	−13.2416	−0.446881
$879$	0	0
$880$	−0.314444	−0.0105999
$881$	−0.114398	−0.00385416	−0.00192708	−	0.999998i	$-0.500613\pi$
$881$	−0.114398	−0.00385416	−0.00192708	+	0.999998i	$0.500613\pi$
$882$	0	0
$883$	12.1141	0.407672	0.203836	−	0.979005i	$-0.434659\pi$
$883$	12.1141	0.407672	0.203836	+	0.979005i	$0.434659\pi$
$884$	−20.3377	−0.684032
$885$	0	0
$886$	−13.9932	−0.470112
$887$	−11.0340	−0.370484	−0.185242	−	0.982693i	$-0.559307\pi$
$887$	−11.0340	−0.370484	−0.185242	+	0.982693i	$0.559307\pi$
$888$	0	0
$889$	−0.135055	−0.00452961
$890$	−4.53929	−0.152157
$891$	0	0
$892$	−27.4132	−0.917861
$893$	−21.8338	−0.730642
$894$	0	0
$895$	22.9914	0.768519
$896$	−1.13777	−0.0380104
$897$	0	0
$898$	30.2141	1.00826
$899$	9.54266	0.318266
$900$	0	0
$901$	−63.9129	−2.12925
$902$	1.69515	0.0564424
$903$	0	0
$904$	−30.5960	−1.01761
$905$	−54.3884	−1.80793
$906$	0	0
$907$	44.1494	1.46596	0.732979	−	0.680251i	$-0.238129\pi$
$907$	44.1494	1.46596	0.732979	+	0.680251i	$0.238129\pi$
$908$	13.0217	0.432141
$909$	0	0
$910$	−0.996169	−0.0330227
$911$	−3.45436	−0.114448	−0.0572241	−	0.998361i	$-0.518225\pi$
$911$	−3.45436	−0.114448	−0.0572241	+	0.998361i	$0.518225\pi$
$912$	0	0
$913$	0.00802423	0.000265563 0
$914$	10.8289	0.358189
$915$	0	0
$916$	−7.33726	−0.242430
$917$	0.719773	0.0237690
$918$	0	0
$919$	6.38517	0.210627	0.105314	−	0.994439i	$-0.466415\pi$
$919$	6.38517	0.210627	0.105314	+	0.994439i	$0.466415\pi$
$920$	−13.4752	−0.444265
$921$	0	0
$922$	13.9595	0.459732
$923$	−9.70167	−0.319334
$924$	0	0
$925$	0.139141	0.00457494
$926$	14.8377	0.487596
$927$	0	0
$928$	21.8347	0.716760
$929$	−5.06890	−0.166305	−0.0831525	−	0.996537i	$-0.526499\pi$
$929$	−5.06890	−0.166305	−0.0831525	+	0.996537i	$0.526499\pi$
$930$	0	0
$931$	23.0162	0.754326
$932$	22.0205	0.721304
$933$	0	0
$934$	26.6951	0.873489
$935$	−3.39837	−0.111139
$936$	0	0
$937$	−16.2537	−0.530984	−0.265492	−	0.964113i	$-0.585534\pi$
$937$	−16.2537	−0.530984	−0.265492	+	0.964113i	$0.585534\pi$
$938$	0.713677	0.0233024
$939$	0	0
$940$	24.1720	0.788404
$941$	37.5175	1.22304	0.611518	−	0.791230i	$-0.290559\pi$
$941$	37.5175	1.22304	0.611518	+	0.791230i	$0.290559\pi$
$942$	0	0
$943$	−14.0237	−0.456674
$944$	−5.82735	−0.189664
$945$	0	0
$946$	−0.773807	−0.0251586
$947$	15.0972	0.490592	0.245296	−	0.969448i	$-0.421115\pi$
$947$	15.0972	0.490592	0.245296	+	0.969448i	$0.421115\pi$
$948$	0	0
$949$	−38.8240	−1.26028
$950$	6.81472	0.221099
$951$	0	0
$952$	−1.71258	−0.0555051
$953$	35.0151	1.13425	0.567125	−	0.823632i	$-0.308056\pi$
$953$	35.0151	1.13425	0.567125	+	0.823632i	$0.308056\pi$
$954$	0	0
$955$	7.64782	0.247478
$956$	−24.9934	−0.808344
$957$	0	0
$958$	25.0055	0.807891
$959$	1.05956	0.0342151
$960$	0	0
$961$	−24.5682	−0.792521
$962$	−0.148094	−0.00477474
$963$	0	0
$964$	23.8412	0.767872
$965$	−14.0553	−0.452455
$966$	0	0
$967$	24.4686	0.786856	0.393428	−	0.919355i	$-0.371289\pi$
$967$	24.4686	0.786856	0.393428	+	0.919355i	$0.371289\pi$
$968$	−29.7851	−0.957330
$969$	0	0
$970$	32.9040	1.05648
$971$	−18.9699	−0.608773	−0.304387	−	0.952549i	$-0.598451\pi$
$971$	−18.9699	−0.608773	−0.304387	+	0.952549i	$0.598451\pi$
$972$	0	0
$973$	1.68480	0.0540121
$974$	−16.8384	−0.539536
$975$	0	0
$976$	4.00415	0.128170
$977$	−38.3716	−1.22762	−0.613809	−	0.789455i	$-0.710363\pi$
$977$	−38.3716	−1.22762	−0.613809	+	0.789455i	$0.710363\pi$
$978$	0	0
$979$	−0.538316	−0.0172046
$980$	−25.4810	−0.813960
$981$	0	0
$982$	−21.5818	−0.688704
$983$	−15.5044	−0.494514	−0.247257	−	0.968950i	$-0.579529\pi$
$983$	−15.5044	−0.494514	−0.247257	+	0.968950i	$0.579529\pi$
$984$	0	0
$985$	38.5563	1.22851
$986$	14.3269	0.456261
$987$	0	0
$988$	14.4094	0.458424
$989$	6.40156	0.203558
$990$	0	0
$991$	13.0862	0.415698	0.207849	−	0.978161i	$-0.433354\pi$
$991$	13.0862	0.415698	0.207849	+	0.978161i	$0.433354\pi$
$992$	14.7168	0.467259
$993$	0	0
$994$	−0.326341	−0.0103509
$995$	67.4304	2.13769
$996$	0	0
$997$	−3.78357	−0.119827	−0.0599135	−	0.998204i	$-0.519082\pi$
$997$	−3.78357	−0.119827	−0.0599135	+	0.998204i	$0.519082\pi$
$998$	−4.26255	−0.134929
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.i.1.4		7
3.2	odd	2		127.2.a.b.1.4	✓	7
12.11	even	2		2032.2.a.p.1.4		7
15.14	odd	2		3175.2.a.j.1.4		7
21.20	even	2		6223.2.a.h.1.4		7
24.5	odd	2		8128.2.a.bi.1.4		7
24.11	even	2		8128.2.a.bj.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.2.a.b.1.4	✓	7	3.2	odd	2
1143.2.a.i.1.4		7	1.1	even	1	trivial
2032.2.a.p.1.4		7	12.11	even	2
3175.2.a.j.1.4		7	15.14	odd	2
6223.2.a.h.1.4		7	21.20	even	2
8128.2.a.bi.1.4		7	24.5	odd	2
8128.2.a.bj.1.4		7	24.11	even	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)

\(f(q)\)	\(=\)	\(q-0.818322 q^{2} -1.33035 q^{4} -2.74338 q^{5} -0.135055 q^{7} +2.72530 q^{8} +O(q^{10})\)	\(q-0.818322 q^{2} -1.33035 q^{4} -2.74338 q^{5} -0.135055 q^{7} +2.72530 q^{8} +2.24497 q^{10} +0.266231 q^{11} +3.28557 q^{13} +0.110519 q^{14} +0.430525 q^{16} +4.65292 q^{17} -3.29662 q^{19} +3.64965 q^{20} -0.217863 q^{22} +1.80234 q^{23} +2.52613 q^{25} -2.68866 q^{26} +0.179671 q^{28} -3.76272 q^{29} -2.53611 q^{31} -5.80291 q^{32} -3.80759 q^{34} +0.370508 q^{35} +0.0550810 q^{37} +2.69770 q^{38} -7.47653 q^{40} -7.78083 q^{41} +3.55180 q^{43} -0.354180 q^{44} -1.47489 q^{46} +6.62310 q^{47} -6.98176 q^{49} -2.06718 q^{50} -4.37096 q^{52} -13.7361 q^{53} -0.730373 q^{55} -0.368066 q^{56} +3.07912 q^{58} -13.5354 q^{59} +9.30061 q^{61} +2.07535 q^{62} +3.88760 q^{64} -9.01357 q^{65} +6.45752 q^{67} -6.19001 q^{68} -0.303195 q^{70} -2.95281 q^{71} -11.8165 q^{73} -0.0450740 q^{74} +4.38566 q^{76} -0.0359559 q^{77} -11.4993 q^{79} -1.18109 q^{80} +6.36722 q^{82} +0.0301401 q^{83} -12.7647 q^{85} -2.90652 q^{86} +0.725560 q^{88} -2.02198 q^{89} -0.443734 q^{91} -2.39774 q^{92} -5.41983 q^{94} +9.04388 q^{95} +14.6568 q^{97} +5.71333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * q^98

Introduction

L-functions

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