LMFDB - Embedded newform 2349.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2349,2,Mod(1,2349)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2349.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2349, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2349 = 3^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2349.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-1,0,5,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.7568594348$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 13x^{9} + 11x^{8} + 59x^{7} - 41x^{6} - 110x^{5} + 61x^{4} + 76x^{3} - 28x^{2} - 13x - 1 $ x^11 - x^10 - 13x^9 + 11x^8 + 59x^7 - 41x^6 - 110x^5 + 61x^4 + 76x^3 - 28x^2 - 13*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 261)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.52818$ of defining polynomial
Character	$\chi$	$=$	2349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.52818 q^{2} +0.335346 q^{4} +0.151216 q^{5} -0.0579426 q^{7} +2.54390 q^{8} -0.231086 q^{10} -1.08157 q^{11} -2.45294 q^{13} +0.0885469 q^{14} -4.55824 q^{16} +1.83487 q^{17} +2.89537 q^{19} +0.0507097 q^{20} +1.65284 q^{22} +0.781359 q^{23} -4.97713 q^{25} +3.74855 q^{26} -0.0194308 q^{28} +1.00000 q^{29} -1.20902 q^{31} +1.87803 q^{32} -2.80403 q^{34} -0.00876184 q^{35} -1.28794 q^{37} -4.42466 q^{38} +0.384678 q^{40} +4.37194 q^{41} -7.44418 q^{43} -0.362701 q^{44} -1.19406 q^{46} +5.87634 q^{47} -6.99664 q^{49} +7.60598 q^{50} -0.822584 q^{52} +0.0547103 q^{53} -0.163551 q^{55} -0.147400 q^{56} -1.52818 q^{58} +7.23633 q^{59} -4.04570 q^{61} +1.84760 q^{62} +6.24650 q^{64} -0.370924 q^{65} -4.33877 q^{67} +0.615318 q^{68} +0.0133897 q^{70} -14.9063 q^{71} -3.30924 q^{73} +1.96822 q^{74} +0.970952 q^{76} +0.0626690 q^{77} +3.21824 q^{79} -0.689278 q^{80} -6.68112 q^{82} +8.10784 q^{83} +0.277462 q^{85} +11.3761 q^{86} -2.75141 q^{88} +3.75415 q^{89} +0.142130 q^{91} +0.262026 q^{92} -8.98013 q^{94} +0.437827 q^{95} +15.5691 q^{97} +10.6922 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - q^{2} + 5 q^{4} + q^{5} - 7 q^{7} - 3 q^{8} - 10 q^{10} - 3 q^{11} - 7 q^{13} + 10 q^{14} - 7 q^{16} - q^{17} - 28 q^{19} + 4 q^{20} - 13 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} - 16 q^{28} + 11 q^{29}+ \cdots - 43 q^{98}+O(q^{100}) $ 11 * q - q^2 + 5 * q^4 + q^5 - 7 * q^7 - 3 * q^8 - 10 * q^10 - 3 * q^11 - 7 * q^13 + 10 * q^14 - 7 * q^16 - q^17 - 28 * q^19 + 4 * q^20 - 13 * q^22 - 4 * q^23 - 4 * q^25 + 6 * q^26 - 16 * q^28 + 11 * q^29 - 27 * q^31 - 7 * q^32 - 19 * q^34 + 11 * q^35 - 14 * q^37 + 5 * q^38 - 24 * q^40 + 9 * q^41 - 25 * q^43 - 23 * q^44 - 10 * q^46 - 6 * q^47 - 12 * q^49 - 41 * q^50 - 20 * q^52 + 6 * q^53 - 35 * q^55 + 10 * q^56 - q^58 - 25 * q^61 + 6 * q^62 - 25 * q^64 - 5 * q^65 - 29 * q^67 + 50 * q^68 - 7 * q^70 + q^71 - 13 * q^73 - 14 * q^74 - 38 * q^76 - 9 * q^77 - 27 * q^79 + 37 * q^80 - 17 * q^82 + 12 * q^83 - 7 * q^85 - q^86 + 5 * q^88 - 23 * q^89 - 43 * q^91 - 19 * q^92 + 10 * q^94 - 8 * q^95 - 36 * q^97 - 43 * q^98

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 $

$2$	−1.52818	−1.08059	−0.540295	−	0.841476i	$-0.681687\pi$
$2$	−1.52818	−1.08059	−0.540295	+	0.841476i	$0.681687\pi$
$3$	0	0
$3$	0	0
$4$	0.335346	0.167673
$5$	0.151216	0.0676258	0.0338129	−	0.999428i	$-0.489235\pi$
$5$	0.151216	0.0676258	0.0338129	+	0.999428i	$0.489235\pi$
$6$	0	0
$7$	−0.0579426	−0.0219002	−0.0109501	−	0.999940i	$-0.503486\pi$
$7$	−0.0579426	−0.0219002	−0.0109501	+	0.999940i	$0.503486\pi$
$8$	2.54390	0.899404
$9$	0	0
$10$	−0.231086	−0.0730757
$11$	−1.08157	−0.326106	−0.163053	−	0.986617i	$-0.552134\pi$
$11$	−1.08157	−0.326106	−0.163053	+	0.986617i	$0.552134\pi$
$12$	0	0
$13$	−2.45294	−0.680324	−0.340162	−	0.940367i	$-0.610482\pi$
$13$	−2.45294	−0.680324	−0.340162	+	0.940367i	$0.610482\pi$
$14$	0.0885469	0.0236652
$15$	0	0
$16$	−4.55824	−1.13956
$17$	1.83487	0.445022	0.222511	−	0.974930i	$-0.428575\pi$
$17$	1.83487	0.445022	0.222511	+	0.974930i	$0.428575\pi$
$18$	0	0
$19$	2.89537	0.664244	0.332122	−	0.943236i	$-0.392235\pi$
$19$	2.89537	0.664244	0.332122	+	0.943236i	$0.392235\pi$
$20$	0.0507097	0.0113390
$21$	0	0
$22$	1.65284	0.352387
$23$	0.781359	0.162925	0.0814623	−	0.996676i	$-0.474041\pi$
$23$	0.781359	0.162925	0.0814623	+	0.996676i	$0.474041\pi$
$24$	0	0
$25$	−4.97713	−0.995427
$26$	3.74855	0.735150
$27$	0	0
$28$	−0.0194308	−0.00367208
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.20902	−0.217146	−0.108573	−	0.994088i	$-0.534628\pi$
$31$	−1.20902	−0.217146	−0.108573	+	0.994088i	$0.534628\pi$
$32$	1.87803	0.331991
$33$	0	0
$34$	−2.80403	−0.480886
$35$	−0.00876184	−0.00148102
$36$	0	0
$37$	−1.28794	−0.211737	−0.105868	−	0.994380i	$-0.533762\pi$
$37$	−1.28794	−0.211737	−0.105868	+	0.994380i	$0.533762\pi$
$38$	−4.42466	−0.717775
$39$	0	0
$40$	0.384678	0.0608229
$41$	4.37194	0.682782	0.341391	−	0.939921i	$-0.389102\pi$
$41$	4.37194	0.682782	0.341391	+	0.939921i	$0.389102\pi$
$42$	0	0
$43$	−7.44418	−1.13523	−0.567613	−	0.823295i	$-0.692133\pi$
$43$	−7.44418	−1.13523	−0.567613	+	0.823295i	$0.692133\pi$
$44$	−0.362701	−0.0546792
$45$	0	0
$46$	−1.19406	−0.176054
$47$	5.87634	0.857153	0.428576	−	0.903506i	$-0.359015\pi$
$47$	5.87634	0.857153	0.428576	+	0.903506i	$0.359015\pi$
$48$	0	0
$49$	−6.99664	−0.999520
$50$	7.60598	1.07565
$51$	0	0
$52$	−0.822584	−0.114072
$53$	0.0547103	0.00751503	0.00375752	−	0.999993i	$-0.498804\pi$
$53$	0.0547103	0.00751503	0.00375752	+	0.999993i	$0.498804\pi$
$54$	0	0
$55$	−0.163551	−0.0220532
$56$	−0.147400	−0.0196971
$57$	0	0
$58$	−1.52818	−0.200660
$59$	7.23633	0.942089	0.471045	−	0.882109i	$-0.343877\pi$
$59$	7.23633	0.942089	0.471045	+	0.882109i	$0.343877\pi$
$60$	0	0
$61$	−4.04570	−0.517998	−0.258999	−	0.965878i	$-0.583393\pi$
$61$	−4.04570	−0.517998	−0.258999	+	0.965878i	$0.583393\pi$
$62$	1.84760	0.234646
$63$	0	0
$64$	6.24650	0.780812
$65$	−0.370924	−0.0460074
$66$	0	0
$67$	−4.33877	−0.530065	−0.265032	−	0.964239i	$-0.585383\pi$
$67$	−4.33877	−0.530065	−0.265032	+	0.964239i	$0.585383\pi$
$68$	0.615318	0.0746183
$69$	0	0
$70$	0.0133897	0.00160038
$71$	−14.9063	−1.76905	−0.884527	−	0.466489i	$-0.845519\pi$
$71$	−14.9063	−1.76905	−0.884527	+	0.466489i	$0.845519\pi$
$72$	0	0
$73$	−3.30924	−0.387317	−0.193658	−	0.981069i	$-0.562035\pi$
$73$	−3.30924	−0.387317	−0.193658	+	0.981069i	$0.562035\pi$
$74$	1.96822	0.228800
$75$	0	0
$76$	0.970952	0.111376
$77$	0.0626690	0.00714180
$78$	0	0
$79$	3.21824	0.362080	0.181040	−	0.983476i	$-0.442054\pi$
$79$	3.21824	0.362080	0.181040	+	0.983476i	$0.442054\pi$
$80$	−0.689278	−0.0770636
$81$	0	0
$82$	−6.68112	−0.737807
$83$	8.10784	0.889952	0.444976	−	0.895543i	$-0.353212\pi$
$83$	8.10784	0.889952	0.444976	+	0.895543i	$0.353212\pi$
$84$	0	0
$85$	0.277462	0.0300950
$86$	11.3761	1.22671
$87$	0	0
$88$	−2.75141	−0.293301
$89$	3.75415	0.397939	0.198970	−	0.980006i	$-0.436241\pi$
$89$	3.75415	0.397939	0.198970	+	0.980006i	$0.436241\pi$
$90$	0	0
$91$	0.142130	0.0148992
$92$	0.262026	0.0273181
$93$	0	0
$94$	−8.98013	−0.926230
$95$	0.437827	0.0449201
$96$	0	0
$97$	15.5691	1.58080	0.790402	−	0.612589i	$-0.209872\pi$
$97$	15.5691	1.58080	0.790402	+	0.612589i	$0.209872\pi$
$98$	10.6922	1.08007
$99$	0	0
$100$	−1.66906	−0.166906
$101$	11.9917	1.19322	0.596609	−	0.802532i	$-0.296514\pi$
$101$	11.9917	1.19322	0.596609	+	0.802532i	$0.296514\pi$
$102$	0	0
$103$	10.9416	1.07810	0.539052	−	0.842272i	$-0.318782\pi$
$103$	10.9416	1.07810	0.539052	+	0.842272i	$0.318782\pi$
$104$	−6.24003	−0.611886
$105$	0	0
$106$	−0.0836074	−0.00812066
$107$	−3.86363	−0.373511	−0.186756	−	0.982406i	$-0.559797\pi$
$107$	−3.86363	−0.373511	−0.186756	+	0.982406i	$0.559797\pi$
$108$	0	0
$109$	−15.3103	−1.46646	−0.733230	−	0.679980i	$-0.761988\pi$
$109$	−15.3103	−1.46646	−0.733230	+	0.679980i	$0.761988\pi$
$110$	0.249936	0.0238304
$111$	0	0
$112$	0.264116	0.0249566
$113$	−9.17086	−0.862721	−0.431361	−	0.902180i	$-0.641966\pi$
$113$	−9.17086	−0.862721	−0.431361	+	0.902180i	$0.641966\pi$
$114$	0	0
$115$	0.118154	0.0110179
$116$	0.335346	0.0311361
$117$	0	0
$118$	−11.0584	−1.01801
$119$	−0.106317	−0.00974609
$120$	0	0
$121$	−9.83020	−0.893655
$122$	6.18257	0.559743
$123$	0	0
$124$	−0.405440	−0.0364096
$125$	−1.50870	−0.134942
$126$	0	0
$127$	−12.5497	−1.11360	−0.556801	−	0.830646i	$-0.687972\pi$
$127$	−12.5497	−1.11360	−0.556801	+	0.830646i	$0.687972\pi$
$128$	−13.3019	−1.17573
$129$	0	0
$130$	0.566840	0.0497152
$131$	−18.4477	−1.61178	−0.805891	−	0.592064i	$-0.798313\pi$
$131$	−18.4477	−1.61178	−0.805891	+	0.592064i	$0.798313\pi$
$132$	0	0
$133$	−0.167765	−0.0145471
$134$	6.63044	0.572782
$135$	0	0
$136$	4.66773	0.400255
$137$	−17.1951	−1.46908	−0.734540	−	0.678565i	$-0.762602\pi$
$137$	−17.1951	−1.46908	−0.734540	+	0.678565i	$0.762602\pi$
$138$	0	0
$139$	1.63401	0.138595	0.0692976	−	0.997596i	$-0.477924\pi$
$139$	1.63401	0.138595	0.0692976	+	0.997596i	$0.477924\pi$
$140$	−0.00293825	−0.000248327 0
$141$	0	0
$142$	22.7796	1.91162
$143$	2.65303	0.221858
$144$	0	0
$145$	0.151216	0.0125578
$146$	5.05712	0.418530
$147$	0	0
$148$	−0.431907	−0.0355025
$149$	−5.09418	−0.417331	−0.208666	−	0.977987i	$-0.566912\pi$
$149$	−5.09418	−0.417331	−0.208666	+	0.977987i	$0.566912\pi$
$150$	0	0
$151$	−17.6243	−1.43425	−0.717124	−	0.696945i	$-0.754542\pi$
$151$	−17.6243	−1.43425	−0.717124	+	0.696945i	$0.754542\pi$
$152$	7.36553	0.597424
$153$	0	0
$154$	−0.0957698	−0.00771735
$155$	−0.182823	−0.0146847
$156$	0	0
$157$	1.47416	0.117651	0.0588256	−	0.998268i	$-0.481264\pi$
$157$	1.47416	0.117651	0.0588256	+	0.998268i	$0.481264\pi$
$158$	−4.91806	−0.391260
$159$	0	0
$160$	0.283988	0.0224512
$161$	−0.0452739	−0.00356809
$162$	0	0
$163$	−14.2857	−1.11894	−0.559469	−	0.828851i	$-0.688995\pi$
$163$	−14.2857	−1.11894	−0.559469	+	0.828851i	$0.688995\pi$
$164$	1.46611	0.114484
$165$	0	0
$166$	−12.3903	−0.961672
$167$	−14.3390	−1.10958	−0.554791	−	0.831990i	$-0.687202\pi$
$167$	−14.3390	−1.10958	−0.554791	+	0.831990i	$0.687202\pi$
$168$	0	0
$169$	−6.98308	−0.537160
$170$	−0.424013	−0.0325203
$171$	0	0
$172$	−2.49638	−0.190347
$173$	7.05344	0.536263	0.268132	−	0.963382i	$-0.413594\pi$
$173$	7.05344	0.536263	0.268132	+	0.963382i	$0.413594\pi$
$174$	0	0
$175$	0.288388	0.0218001
$176$	4.93006	0.371617
$177$	0	0
$178$	−5.73703	−0.430009
$179$	−11.7051	−0.874878	−0.437439	−	0.899248i	$-0.644114\pi$
$179$	−11.7051	−0.874878	−0.437439	+	0.899248i	$0.644114\pi$
$180$	0	0
$181$	−7.15646	−0.531936	−0.265968	−	0.963982i	$-0.585691\pi$
$181$	−7.15646	−0.531936	−0.265968	+	0.963982i	$0.585691\pi$
$182$	−0.217200	−0.0161000
$183$	0	0
$184$	1.98770	0.146535
$185$	−0.194758	−0.0143189
$186$	0	0
$187$	−1.98455	−0.145124
$188$	1.97061	0.143721
$189$	0	0
$190$	−0.669080	−0.0485401
$191$	22.8547	1.65371	0.826853	−	0.562418i	$-0.190129\pi$
$191$	22.8547	1.65371	0.826853	+	0.562418i	$0.190129\pi$
$192$	0	0
$193$	0.567703	0.0408641	0.0204321	−	0.999791i	$-0.493496\pi$
$193$	0.567703	0.0408641	0.0204321	+	0.999791i	$0.493496\pi$
$194$	−23.7925	−1.70820
$195$	0	0
$196$	−2.34630	−0.167593
$197$	−7.85018	−0.559302	−0.279651	−	0.960102i	$-0.590219\pi$
$197$	−7.85018	−0.559302	−0.279651	+	0.960102i	$0.590219\pi$
$198$	0	0
$199$	15.9248	1.12888	0.564441	−	0.825473i	$-0.309092\pi$
$199$	15.9248	1.12888	0.564441	+	0.825473i	$0.309092\pi$
$200$	−12.6613	−0.895290
$201$	0	0
$202$	−18.3255	−1.28938
$203$	−0.0579426	−0.00406677
$204$	0	0
$205$	0.661107	0.0461737
$206$	−16.7207	−1.16499
$207$	0	0
$208$	11.1811	0.775269
$209$	−3.13155	−0.216614
$210$	0	0
$211$	20.1127	1.38462	0.692308	−	0.721602i	$-0.256594\pi$
$211$	20.1127	1.38462	0.692308	+	0.721602i	$0.256594\pi$
$212$	0.0183469	0.00126007
$213$	0	0
$214$	5.90434	0.403612
$215$	−1.12568	−0.0767706
$216$	0	0
$217$	0.0700537	0.00475555
$218$	23.3970	1.58464
$219$	0	0
$220$	−0.0548461	−0.00369773
$221$	−4.50084	−0.302759
$222$	0	0
$223$	−24.4543	−1.63758	−0.818790	−	0.574093i	$-0.805355\pi$
$223$	−24.4543	−1.63758	−0.818790	+	0.574093i	$0.805355\pi$
$224$	−0.108818	−0.00727069
$225$	0	0
$226$	14.0148	0.932248
$227$	−28.4313	−1.88705	−0.943527	−	0.331295i	$-0.892514\pi$
$227$	−28.4313	−1.88705	−0.943527	+	0.331295i	$0.892514\pi$
$228$	0	0
$229$	−14.3532	−0.948485	−0.474242	−	0.880394i	$-0.657278\pi$
$229$	−14.3532	−0.948485	−0.474242	+	0.880394i	$0.657278\pi$
$230$	−0.180561	−0.0119058
$231$	0	0
$232$	2.54390	0.167015
$233$	3.54955	0.232539	0.116269	−	0.993218i	$-0.462906\pi$
$233$	3.54955	0.232539	0.116269	+	0.993218i	$0.462906\pi$
$234$	0	0
$235$	0.888597	0.0579657
$236$	2.42667	0.157963
$237$	0	0
$238$	0.162472	0.0105315
$239$	−13.9450	−0.902029	−0.451015	−	0.892517i	$-0.648938\pi$
$239$	−13.9450	−0.902029	−0.451015	+	0.892517i	$0.648938\pi$
$240$	0	0
$241$	21.1250	1.36078	0.680389	−	0.732851i	$-0.261811\pi$
$241$	21.1250	1.36078	0.680389	+	0.732851i	$0.261811\pi$
$242$	15.0224	0.965674
$243$	0	0
$244$	−1.35671	−0.0868543
$245$	−1.05800	−0.0675934
$246$	0	0
$247$	−7.10218	−0.451901
$248$	−3.07562	−0.195302
$249$	0	0
$250$	2.30557	0.145817
$251$	−6.98888	−0.441134	−0.220567	−	0.975372i	$-0.570791\pi$
$251$	−6.98888	−0.441134	−0.220567	+	0.975372i	$0.570791\pi$
$252$	0	0
$253$	−0.845095	−0.0531307
$254$	19.1782	1.20335
$255$	0	0
$256$	7.83468	0.489668
$257$	1.63268	0.101844	0.0509218	−	0.998703i	$-0.483784\pi$
$257$	1.63268	0.101844	0.0509218	+	0.998703i	$0.483784\pi$
$258$	0	0
$259$	0.0746268	0.00463708
$260$	−0.124388	−0.00771421
$261$	0	0
$262$	28.1915	1.74167
$263$	−16.5193	−1.01862	−0.509312	−	0.860582i	$-0.670100\pi$
$263$	−16.5193	−1.01862	−0.509312	+	0.860582i	$0.670100\pi$
$264$	0	0
$265$	0.00827306	0.000508210 0
$266$	0.256376	0.0157194
$267$	0	0
$268$	−1.45499	−0.0888776
$269$	−4.33854	−0.264525	−0.132263	−	0.991215i	$-0.542224\pi$
$269$	−4.33854	−0.264525	−0.132263	+	0.991215i	$0.542224\pi$
$270$	0	0
$271$	−18.7332	−1.13796	−0.568980	−	0.822351i	$-0.692662\pi$
$271$	−18.7332	−1.13796	−0.568980	+	0.822351i	$0.692662\pi$
$272$	−8.36379	−0.507129
$273$	0	0
$274$	26.2773	1.58747
$275$	5.38313	0.324615
$276$	0	0
$277$	12.4377	0.747308	0.373654	−	0.927568i	$-0.378105\pi$
$277$	12.4377	0.747308	0.373654	+	0.927568i	$0.378105\pi$
$278$	−2.49707	−0.149765
$279$	0	0
$280$	−0.0222892	−0.00133204
$281$	0.145092	0.00865548	0.00432774	−	0.999991i	$-0.498622\pi$
$281$	0.145092	0.00865548	0.00432774	+	0.999991i	$0.498622\pi$
$282$	0	0
$283$	8.11009	0.482095	0.241047	−	0.970513i	$-0.422509\pi$
$283$	8.11009	0.482095	0.241047	+	0.970513i	$0.422509\pi$
$284$	−4.99877	−0.296623
$285$	0	0
$286$	−4.05432	−0.239737
$287$	−0.253321	−0.0149531
$288$	0	0
$289$	−13.6332	−0.801955
$290$	−0.231086	−0.0135698
$291$	0	0
$292$	−1.10974	−0.0649426
$293$	−18.9313	−1.10598	−0.552990	−	0.833188i	$-0.686513\pi$
$293$	−18.9313	−1.10598	−0.552990	+	0.833188i	$0.686513\pi$
$294$	0	0
$295$	1.09425	0.0637096
$296$	−3.27640	−0.190437
$297$	0	0
$298$	7.78484	0.450964
$299$	−1.91663	−0.110841
$300$	0	0
$301$	0.431335	0.0248617
$302$	26.9332	1.54983
$303$	0	0
$304$	−13.1978	−0.756945
$305$	−0.611774	−0.0350301
$306$	0	0
$307$	−19.9124	−1.13646	−0.568230	−	0.822870i	$-0.692372\pi$
$307$	−19.9124	−1.13646	−0.568230	+	0.822870i	$0.692372\pi$
$308$	0.0210158	0.00119749
$309$	0	0
$310$	0.279387	0.0158681
$311$	27.7656	1.57444	0.787222	−	0.616670i	$-0.211519\pi$
$311$	27.7656	1.57444	0.787222	+	0.616670i	$0.211519\pi$
$312$	0	0
$313$	−9.31087	−0.526281	−0.263141	−	0.964757i	$-0.584758\pi$
$313$	−9.31087	−0.526281	−0.263141	+	0.964757i	$0.584758\pi$
$314$	−2.25279	−0.127133
$315$	0	0
$316$	1.07922	0.0607110
$317$	−12.2762	−0.689498	−0.344749	−	0.938695i	$-0.612036\pi$
$317$	−12.2762	−0.689498	−0.344749	+	0.938695i	$0.612036\pi$
$318$	0	0
$319$	−1.08157	−0.0605564
$320$	0.944570	0.0528031
$321$	0	0
$322$	0.0691869	0.00385563
$323$	5.31265	0.295604
$324$	0	0
$325$	12.2086	0.677212
$326$	21.8311	1.20911
$327$	0	0
$328$	11.1218	0.614096
$329$	−0.340490	−0.0187718
$330$	0	0
$331$	−21.7765	−1.19694	−0.598472	−	0.801143i	$-0.704225\pi$
$331$	−21.7765	−1.19694	−0.598472	+	0.801143i	$0.704225\pi$
$332$	2.71893	0.149221
$333$	0	0
$334$	21.9126	1.19900
$335$	−0.656091	−0.0358461
$336$	0	0
$337$	−11.9304	−0.649892	−0.324946	−	0.945733i	$-0.605346\pi$
$337$	−11.9304	−0.649892	−0.324946	+	0.945733i	$0.605346\pi$
$338$	10.6714	0.580449
$339$	0	0
$340$	0.0930459	0.00504612
$341$	1.30764	0.0708127
$342$	0	0
$343$	0.811001	0.0437900
$344$	−18.9372	−1.02103
$345$	0	0
$346$	−10.7790	−0.579480
$347$	−12.1588	−0.652716	−0.326358	−	0.945246i	$-0.605822\pi$
$347$	−12.1588	−0.652716	−0.326358	+	0.945246i	$0.605822\pi$
$348$	0	0
$349$	10.1326	0.542388	0.271194	−	0.962525i	$-0.412581\pi$
$349$	10.1326	0.542388	0.271194	+	0.962525i	$0.412581\pi$
$350$	−0.440710	−0.0235569
$351$	0	0
$352$	−2.03122	−0.108264
$353$	−12.3668	−0.658218	−0.329109	−	0.944292i	$-0.606748\pi$
$353$	−12.3668	−0.658218	−0.329109	+	0.944292i	$0.606748\pi$
$354$	0	0
$355$	−2.25407	−0.119634
$356$	1.25894	0.0667237
$357$	0	0
$358$	17.8875	0.945384
$359$	−19.6499	−1.03708	−0.518541	−	0.855053i	$-0.673525\pi$
$359$	−19.6499	−1.03708	−0.518541	+	0.855053i	$0.673525\pi$
$360$	0	0
$361$	−10.6168	−0.558779
$362$	10.9364	0.574804
$363$	0	0
$364$	0.0476627	0.00249820
$365$	−0.500409	−0.0261926
$366$	0	0
$367$	9.86294	0.514841	0.257421	−	0.966299i	$-0.417127\pi$
$367$	9.86294	0.514841	0.257421	+	0.966299i	$0.417127\pi$
$368$	−3.56162	−0.185662
$369$	0	0
$370$	0.297625	0.0154728
$371$	−0.00317005	−0.000164581 0
$372$	0	0
$373$	20.8855	1.08141	0.540705	−	0.841212i	$-0.318157\pi$
$373$	20.8855	1.08141	0.540705	+	0.841212i	$0.318157\pi$
$374$	3.03275	0.156820
$375$	0	0
$376$	14.9488	0.770926
$377$	−2.45294	−0.126333
$378$	0	0
$379$	−29.3384	−1.50701	−0.753506	−	0.657442i	$-0.771639\pi$
$379$	−29.3384	−1.50701	−0.753506	+	0.657442i	$0.771639\pi$
$380$	0.146823	0.00753188
$381$	0	0
$382$	−34.9261	−1.78698
$383$	32.6387	1.66776	0.833879	−	0.551947i	$-0.186115\pi$
$383$	32.6387	1.66776	0.833879	+	0.551947i	$0.186115\pi$
$384$	0	0
$385$	0.00947656	0.000482970 0
$386$	−0.867554	−0.0441573
$387$	0	0
$388$	5.22104	0.265058
$389$	−33.8652	−1.71704	−0.858518	−	0.512784i	$-0.828614\pi$
$389$	−33.8652	−1.71704	−0.858518	+	0.512784i	$0.828614\pi$
$390$	0	0
$391$	1.43369	0.0725051
$392$	−17.7987	−0.898972
$393$	0	0
$394$	11.9965	0.604376
$395$	0.486649	0.0244860
$396$	0	0
$397$	30.3217	1.52180	0.760902	−	0.648867i	$-0.224757\pi$
$397$	30.3217	1.52180	0.760902	+	0.648867i	$0.224757\pi$
$398$	−24.3361	−1.21986
$399$	0	0
$400$	22.6869	1.13435
$401$	34.6591	1.73079	0.865397	−	0.501087i	$-0.167066\pi$
$401$	34.6591	1.73079	0.865397	+	0.501087i	$0.167066\pi$
$402$	0	0
$403$	2.96565	0.147730
$404$	4.02137	0.200071
$405$	0	0
$406$	0.0885469	0.00439451
$407$	1.39300	0.0690486
$408$	0	0
$409$	−13.3446	−0.659849	−0.329925	−	0.944007i	$-0.607023\pi$
$409$	−13.3446	−0.659849	−0.329925	+	0.944007i	$0.607023\pi$
$410$	−1.01029	−0.0498948
$411$	0	0
$412$	3.66921	0.180769
$413$	−0.419291	−0.0206320
$414$	0	0
$415$	1.22604	0.0601837
$416$	−4.60669	−0.225862
$417$	0	0
$418$	4.78559	0.234071
$419$	−4.41552	−0.215712	−0.107856	−	0.994167i	$-0.534399\pi$
$419$	−4.41552	−0.215712	−0.107856	+	0.994167i	$0.534399\pi$
$420$	0	0
$421$	17.1362	0.835168	0.417584	−	0.908638i	$-0.362877\pi$
$421$	17.1362	0.835168	0.417584	+	0.908638i	$0.362877\pi$
$422$	−30.7359	−1.49620
$423$	0	0
$424$	0.139177	0.00675905
$425$	−9.13241	−0.442987
$426$	0	0
$427$	0.234418	0.0113443
$428$	−1.29565	−0.0626278
$429$	0	0
$430$	1.72024	0.0829575
$431$	−12.9834	−0.625388	−0.312694	−	0.949854i	$-0.601231\pi$
$431$	−12.9834	−0.625388	−0.312694	+	0.949854i	$0.601231\pi$
$432$	0	0
$433$	24.6959	1.18681	0.593404	−	0.804905i	$-0.297784\pi$
$433$	24.6959	1.18681	0.593404	+	0.804905i	$0.297784\pi$
$434$	−0.107055	−0.00513880
$435$	0	0
$436$	−5.13425	−0.245886
$437$	2.26233	0.108222
$438$	0	0
$439$	−35.0650	−1.67356	−0.836781	−	0.547537i	$-0.815565\pi$
$439$	−35.0650	−1.67356	−0.836781	+	0.547537i	$0.815565\pi$
$440$	−0.416057	−0.0198347
$441$	0	0
$442$	6.87811	0.327158
$443$	−3.44567	−0.163709	−0.0818544	−	0.996644i	$-0.526084\pi$
$443$	−3.44567	−0.163709	−0.0818544	+	0.996644i	$0.526084\pi$
$444$	0	0
$445$	0.567687	0.0269110
$446$	37.3706	1.76955
$447$	0	0
$448$	−0.361938	−0.0171000
$449$	2.91179	0.137416	0.0687079	−	0.997637i	$-0.478112\pi$
$449$	2.91179	0.137416	0.0687079	+	0.997637i	$0.478112\pi$
$450$	0	0
$451$	−4.72856	−0.222659
$452$	−3.07541	−0.144655
$453$	0	0
$454$	43.4483	2.03913
$455$	0.0214923	0.00100757
$456$	0	0
$457$	28.9796	1.35561	0.677805	−	0.735241i	$-0.262931\pi$
$457$	28.9796	1.35561	0.677805	+	0.735241i	$0.262931\pi$
$458$	21.9343	1.02492
$459$	0	0
$460$	0.0396224	0.00184741
$461$	21.7458	1.01280	0.506402	−	0.862297i	$-0.330975\pi$
$461$	21.7458	1.01280	0.506402	+	0.862297i	$0.330975\pi$
$462$	0	0
$463$	20.8839	0.970556	0.485278	−	0.874360i	$-0.338718\pi$
$463$	20.8839	0.970556	0.485278	+	0.874360i	$0.338718\pi$
$464$	−4.55824	−0.211611
$465$	0	0
$466$	−5.42436	−0.251279
$467$	18.3950	0.851221	0.425610	−	0.904906i	$-0.360059\pi$
$467$	18.3950	0.851221	0.425610	+	0.904906i	$0.360059\pi$
$468$	0	0
$469$	0.251399	0.0116085
$470$	−1.35794	−0.0626371
$471$	0	0
$472$	18.4085	0.847319
$473$	8.05141	0.370204
$474$	0	0
$475$	−14.4107	−0.661207
$476$	−0.0356531	−0.00163416
$477$	0	0
$478$	21.3106	0.974723
$479$	9.30916	0.425347	0.212673	−	0.977123i	$-0.431783\pi$
$479$	9.30916	0.425347	0.212673	+	0.977123i	$0.431783\pi$
$480$	0	0
$481$	3.15925	0.144049
$482$	−32.2828	−1.47044
$483$	0	0
$484$	−3.29652	−0.149842
$485$	2.35430	0.106903
$486$	0	0
$487$	20.2090	0.915755	0.457877	−	0.889015i	$-0.348610\pi$
$487$	20.2090	0.915755	0.457877	+	0.889015i	$0.348610\pi$
$488$	−10.2918	−0.465889
$489$	0	0
$490$	1.61682	0.0730407
$491$	9.51156	0.429251	0.214625	−	0.976696i	$-0.431147\pi$
$491$	9.51156	0.429251	0.214625	+	0.976696i	$0.431147\pi$
$492$	0	0
$493$	1.83487	0.0826386
$494$	10.8534	0.488320
$495$	0	0
$496$	5.51099	0.247451
$497$	0.863710	0.0387427
$498$	0	0
$499$	−4.58145	−0.205094	−0.102547	−	0.994728i	$-0.532699\pi$
$499$	−4.58145	−0.205094	−0.102547	+	0.994728i	$0.532699\pi$
$500$	−0.505937	−0.0226262
$501$	0	0
$502$	10.6803	0.476685
$503$	−8.85873	−0.394991	−0.197496	−	0.980304i	$-0.563281\pi$
$503$	−8.85873	−0.394991	−0.197496	+	0.980304i	$0.563281\pi$
$504$	0	0
$505$	1.81334	0.0806924
$506$	1.29146	0.0574124
$507$	0	0
$508$	−4.20848	−0.186721
$509$	−27.7440	−1.22973	−0.614865	−	0.788632i	$-0.710790\pi$
$509$	−27.7440	−1.22973	−0.614865	+	0.788632i	$0.710790\pi$
$510$	0	0
$511$	0.191746	0.00848233
$512$	14.6309	0.646599
$513$	0	0
$514$	−2.49503	−0.110051
$515$	1.65454	0.0729077
$516$	0	0
$517$	−6.35568	−0.279523
$518$	−0.114043	−0.00501078
$519$	0	0
$520$	−0.943592	−0.0413793
$521$	25.4524	1.11509	0.557546	−	0.830146i	$-0.311743\pi$
$521$	25.4524	1.11509	0.557546	+	0.830146i	$0.311743\pi$
$522$	0	0
$523$	25.8717	1.13129	0.565646	−	0.824648i	$-0.308627\pi$
$523$	25.8717	1.13129	0.565646	+	0.824648i	$0.308627\pi$
$524$	−6.18636	−0.270252
$525$	0	0
$526$	25.2445	1.10071
$527$	−2.21840	−0.0966349
$528$	0	0
$529$	−22.3895	−0.973456
$530$	−0.0126428	−0.000549167 0
$531$	0	0
$532$	−0.0562595	−0.00243916
$533$	−10.7241	−0.464513
$534$	0	0
$535$	−0.584243	−0.0252590
$536$	−11.0374	−0.476742
$537$	0	0
$538$	6.63008	0.285843
$539$	7.56737	0.325950
$540$	0	0
$541$	4.29297	0.184569	0.0922845	−	0.995733i	$-0.470583\pi$
$541$	4.29297	0.184569	0.0922845	+	0.995733i	$0.470583\pi$
$542$	28.6278	1.22967
$543$	0	0
$544$	3.44594	0.147744
$545$	−2.31516	−0.0991706
$546$	0	0
$547$	−18.5239	−0.792026	−0.396013	−	0.918245i	$-0.629606\pi$
$547$	−18.5239	−0.792026	−0.396013	+	0.918245i	$0.629606\pi$
$548$	−5.76632	−0.246325
$549$	0	0
$550$	−8.22641	−0.350775
$551$	2.89537	0.123347
$552$	0	0
$553$	−0.186473	−0.00792964
$554$	−19.0071	−0.807532
$555$	0	0
$556$	0.547960	0.0232387
$557$	25.5627	1.08313	0.541564	−	0.840660i	$-0.317833\pi$
$557$	25.5627	1.08313	0.541564	+	0.840660i	$0.317833\pi$
$558$	0	0
$559$	18.2601	0.772321
$560$	0.0399385	0.00168771
$561$	0	0
$562$	−0.221728	−0.00935302
$563$	37.7859	1.59248	0.796242	−	0.604978i	$-0.206818\pi$
$563$	37.7859	1.59248	0.796242	+	0.604978i	$0.206818\pi$
$564$	0	0
$565$	−1.38678	−0.0583422
$566$	−12.3937	−0.520946
$567$	0	0
$568$	−37.9201	−1.59109
$569$	37.5277	1.57324	0.786622	−	0.617435i	$-0.211828\pi$
$569$	37.5277	1.57324	0.786622	+	0.617435i	$0.211828\pi$
$570$	0	0
$571$	−10.1411	−0.424392	−0.212196	−	0.977227i	$-0.568061\pi$
$571$	−10.1411	−0.424392	−0.212196	+	0.977227i	$0.568061\pi$
$572$	0.889684	0.0371996
$573$	0	0
$574$	0.387122	0.0161581
$575$	−3.88893	−0.162179
$576$	0	0
$577$	42.3619	1.76355	0.881775	−	0.471670i	$-0.156349\pi$
$577$	42.3619	1.76355	0.881775	+	0.471670i	$0.156349\pi$
$578$	20.8341	0.866584
$579$	0	0
$580$	0.0507097	0.00210560
$581$	−0.469789	−0.0194901
$582$	0	0
$583$	−0.0591731	−0.00245070
$584$	−8.41836	−0.348354
$585$	0	0
$586$	28.9305	1.19511
$587$	41.3222	1.70555	0.852775	−	0.522279i	$-0.174918\pi$
$587$	41.3222	1.70555	0.852775	+	0.522279i	$0.174918\pi$
$588$	0	0
$589$	−3.50056	−0.144238
$590$	−1.67221	−0.0688439
$591$	0	0
$592$	5.87075	0.241286
$593$	27.3325	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$593$	27.3325	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$594$	0	0
$595$	−0.0160769	−0.000659088 0
$596$	−1.70831	−0.0699752
$597$	0	0
$598$	2.92896	0.119774
$599$	16.5875	0.677748	0.338874	−	0.940832i	$-0.389954\pi$
$599$	16.5875	0.677748	0.338874	+	0.940832i	$0.389954\pi$
$600$	0	0
$601$	−29.9840	−1.22307	−0.611536	−	0.791217i	$-0.709448\pi$
$601$	−29.9840	−1.22307	−0.611536	+	0.791217i	$0.709448\pi$
$602$	−0.659159	−0.0268653
$603$	0	0
$604$	−5.91025	−0.240485
$605$	−1.48648	−0.0604341
$606$	0	0
$607$	−32.6427	−1.32493	−0.662464	−	0.749094i	$-0.730489\pi$
$607$	−32.6427	−1.32493	−0.662464	+	0.749094i	$0.730489\pi$
$608$	5.43759	0.220523
$609$	0	0
$610$	0.934903	0.0378531
$611$	−14.4143	−0.583141
$612$	0	0
$613$	25.4682	1.02865	0.514325	−	0.857596i	$-0.328043\pi$
$613$	25.4682	1.02865	0.514325	+	0.857596i	$0.328043\pi$
$614$	30.4298	1.22805
$615$	0	0
$616$	0.159424	0.00642336
$617$	−5.58811	−0.224969	−0.112484	−	0.993653i	$-0.535881\pi$
$617$	−5.58811	−0.224969	−0.112484	+	0.993653i	$0.535881\pi$
$618$	0	0
$619$	35.9924	1.44666	0.723329	−	0.690504i	$-0.242611\pi$
$619$	35.9924	1.44666	0.723329	+	0.690504i	$0.242611\pi$
$620$	−0.0613090	−0.00246223
$621$	0	0
$622$	−42.4310	−1.70133
$623$	−0.217525	−0.00871496
$624$	0	0
$625$	24.6575	0.986301
$626$	14.2287	0.568694
$627$	0	0
$628$	0.494355	0.0197269
$629$	−2.36322	−0.0942276
$630$	0	0
$631$	−1.71845	−0.0684105	−0.0342053	−	0.999415i	$-0.510890\pi$
$631$	−1.71845	−0.0684105	−0.0342053	+	0.999415i	$0.510890\pi$
$632$	8.18686	0.325656
$633$	0	0
$634$	18.7602	0.745064
$635$	−1.89771	−0.0753083
$636$	0	0
$637$	17.1624	0.679997
$638$	1.65284	0.0654366
$639$	0	0
$640$	−2.01145	−0.0795096
$641$	−4.82736	−0.190669	−0.0953347	−	0.995445i	$-0.530392\pi$
$641$	−4.82736	−0.190669	−0.0953347	+	0.995445i	$0.530392\pi$
$642$	0	0
$643$	−33.0164	−1.30204	−0.651020	−	0.759061i	$-0.725659\pi$
$643$	−33.0164	−1.30204	−0.651020	+	0.759061i	$0.725659\pi$
$644$	−0.0151824	−0.000598272 0
$645$	0	0
$646$	−8.11870	−0.319426
$647$	7.25011	0.285031	0.142516	−	0.989793i	$-0.454481\pi$
$647$	7.25011	0.285031	0.142516	+	0.989793i	$0.454481\pi$
$648$	0	0
$649$	−7.82660	−0.307221
$650$	−18.6570	−0.731788
$651$	0	0
$652$	−4.79064	−0.187616
$653$	−40.2602	−1.57550	−0.787751	−	0.615994i	$-0.788755\pi$
$653$	−40.2602	−1.57550	−0.787751	+	0.615994i	$0.788755\pi$
$654$	0	0
$655$	−2.78958	−0.108998
$656$	−19.9283	−0.778070
$657$	0	0
$658$	0.520332	0.0202847
$659$	−34.4719	−1.34284	−0.671418	−	0.741079i	$-0.734314\pi$
$659$	−34.4719	−1.34284	−0.671418	+	0.741079i	$0.734314\pi$
$660$	0	0
$661$	−31.9494	−1.24269	−0.621344	−	0.783538i	$-0.713413\pi$
$661$	−31.9494	−1.24269	−0.621344	+	0.783538i	$0.713413\pi$
$662$	33.2785	1.29341
$663$	0	0
$664$	20.6255	0.800426
$665$	−0.0253688	−0.000983760 0
$666$	0	0
$667$	0.781359	0.0302543
$668$	−4.80852	−0.186047
$669$	0	0
$670$	1.00263	0.0387349
$671$	4.37571	0.168922
$672$	0	0
$673$	22.8571	0.881077	0.440538	−	0.897734i	$-0.354788\pi$
$673$	22.8571	0.881077	0.440538	+	0.897734i	$0.354788\pi$
$674$	18.2319	0.702267
$675$	0	0
$676$	−2.34175	−0.0900672
$677$	42.6273	1.63830	0.819150	−	0.573580i	$-0.194446\pi$
$677$	42.6273	1.63830	0.819150	+	0.573580i	$0.194446\pi$
$678$	0	0
$679$	−0.902114	−0.0346200
$680$	0.705835	0.0270676
$681$	0	0
$682$	−1.99831	−0.0765194
$683$	41.5204	1.58873	0.794367	−	0.607438i	$-0.207803\pi$
$683$	41.5204	1.58873	0.794367	+	0.607438i	$0.207803\pi$
$684$	0	0
$685$	−2.60018	−0.0993478
$686$	−1.23936	−0.0473190
$687$	0	0
$688$	33.9323	1.29366
$689$	−0.134201	−0.00511266
$690$	0	0
$691$	−16.2283	−0.617353	−0.308676	−	0.951167i	$-0.599886\pi$
$691$	−16.2283	−0.617353	−0.308676	+	0.951167i	$0.599886\pi$
$692$	2.36534	0.0899169
$693$	0	0
$694$	18.5808	0.705318
$695$	0.247089	0.00937262
$696$	0	0
$697$	8.02196	0.303853
$698$	−15.4845	−0.586099
$699$	0	0
$700$	0.0967098	0.00365529
$701$	48.6772	1.83851	0.919257	−	0.393658i	$-0.128791\pi$
$701$	48.6772	1.83851	0.919257	+	0.393658i	$0.128791\pi$
$702$	0	0
$703$	−3.72908	−0.140645
$704$	−6.75604	−0.254628
$705$	0	0
$706$	18.8987	0.711263
$707$	−0.694830	−0.0261318
$708$	0	0
$709$	51.1081	1.91941	0.959703	−	0.281017i	$-0.0906717\pi$
$709$	51.1081	1.91941	0.959703	+	0.281017i	$0.0906717\pi$
$710$	3.44464	0.129275
$711$	0	0
$712$	9.55017	0.357908
$713$	−0.944677	−0.0353784
$714$	0	0
$715$	0.401181	0.0150033
$716$	−3.92525	−0.146693
$717$	0	0
$718$	30.0287	1.12066
$719$	−33.1064	−1.23466	−0.617330	−	0.786704i	$-0.711786\pi$
$719$	−33.1064	−1.23466	−0.617330	+	0.786704i	$0.711786\pi$
$720$	0	0
$721$	−0.633983	−0.0236107
$722$	16.2244	0.603811
$723$	0	0
$724$	−2.39989	−0.0891913
$725$	−4.97713	−0.184846
$726$	0	0
$727$	−15.0301	−0.557435	−0.278717	−	0.960373i	$-0.589909\pi$
$727$	−15.0301	−0.557435	−0.278717	+	0.960373i	$0.589909\pi$
$728$	0.361564	0.0134004
$729$	0	0
$730$	0.764717	0.0283035
$731$	−13.6591	−0.505201
$732$	0	0
$733$	37.0618	1.36891	0.684454	−	0.729056i	$-0.260040\pi$
$733$	37.0618	1.36891	0.684454	+	0.729056i	$0.260040\pi$
$734$	−15.0724	−0.556332
$735$	0	0
$736$	1.46741	0.0540895
$737$	4.69269	0.172857
$738$	0	0
$739$	−32.1984	−1.18444	−0.592219	−	0.805777i	$-0.701748\pi$
$739$	−32.1984	−1.18444	−0.592219	+	0.805777i	$0.701748\pi$
$740$	−0.0653112	−0.00240089
$741$	0	0
$742$	0.00484442	0.000177844 0
$743$	8.93641	0.327845	0.163923	−	0.986473i	$-0.447585\pi$
$743$	8.93641	0.327845	0.163923	+	0.986473i	$0.447585\pi$
$744$	0	0
$745$	−0.770321	−0.0282224
$746$	−31.9169	−1.16856
$747$	0	0
$748$	−0.665510	−0.0243335
$749$	0.223869	0.00817999
$750$	0	0
$751$	−0.773158	−0.0282129	−0.0141065	−	0.999900i	$-0.504490\pi$
$751$	−0.773158	−0.0282129	−0.0141065	+	0.999900i	$0.504490\pi$
$752$	−26.7858	−0.976776
$753$	0	0
$754$	3.74855	0.136514
$755$	−2.66508	−0.0969922
$756$	0	0
$757$	−36.5342	−1.32786	−0.663929	−	0.747796i	$-0.731112\pi$
$757$	−36.5342	−1.32786	−0.663929	+	0.747796i	$0.731112\pi$
$758$	44.8344	1.62846
$759$	0	0
$760$	1.11379	0.0404013
$761$	−6.50231	−0.235709	−0.117854	−	0.993031i	$-0.537602\pi$
$761$	−6.50231	−0.235709	−0.117854	+	0.993031i	$0.537602\pi$
$762$	0	0
$763$	0.887118	0.0321158
$764$	7.66423	0.277282
$765$	0	0
$766$	−49.8779	−1.80216
$767$	−17.7503	−0.640926
$768$	0	0
$769$	−36.1435	−1.30337	−0.651683	−	0.758491i	$-0.725937\pi$
$769$	−36.1435	−1.30337	−0.651683	+	0.758491i	$0.725937\pi$
$770$	−0.0144819	−0.000521892 0
$771$	0	0
$772$	0.190377	0.00685181
$773$	3.21379	0.115592	0.0577959	−	0.998328i	$-0.481593\pi$
$773$	3.21379	0.115592	0.0577959	+	0.998328i	$0.481593\pi$
$774$	0	0
$775$	6.01745	0.216153
$776$	39.6062	1.42178
$777$	0	0
$778$	51.7523	1.85541
$779$	12.6584	0.453534
$780$	0	0
$781$	16.1222	0.576899
$782$	−2.19095	−0.0783482
$783$	0	0
$784$	31.8923	1.13901
$785$	0.222917	0.00795625
$786$	0	0
$787$	−29.1577	−1.03936	−0.519680	−	0.854361i	$-0.673949\pi$
$787$	−29.1577	−1.03936	−0.519680	+	0.854361i	$0.673949\pi$
$788$	−2.63253	−0.0937799
$789$	0	0
$790$	−0.743689	−0.0264593
$791$	0.531383	0.0188938
$792$	0	0
$793$	9.92386	0.352407
$794$	−46.3371	−1.64444
$795$	0	0
$796$	5.34034	0.189283
$797$	−16.3600	−0.579500	−0.289750	−	0.957102i	$-0.593572\pi$
$797$	−16.3600	−0.579500	−0.289750	+	0.957102i	$0.593572\pi$
$798$	0	0
$799$	10.7823	0.381452
$800$	−9.34719	−0.330473
$801$	0	0
$802$	−52.9655	−1.87028
$803$	3.57918	0.126306
$804$	0	0
$805$	−0.00684614	−0.000241295 0
$806$	−4.53206	−0.159635
$807$	0	0
$808$	30.5057	1.07319
$809$	−0.968702	−0.0340577	−0.0170289	−	0.999855i	$-0.505421\pi$
$809$	−0.968702	−0.0340577	−0.0170289	+	0.999855i	$0.505421\pi$
$810$	0	0
$811$	−32.8852	−1.15476	−0.577378	−	0.816477i	$-0.695924\pi$
$811$	−32.8852	−1.15476	−0.577378	+	0.816477i	$0.695924\pi$
$812$	−0.0194308	−0.000681888 0
$813$	0	0
$814$	−2.12877	−0.0746132
$815$	−2.16022	−0.0756691
$816$	0	0
$817$	−21.5537	−0.754068
$818$	20.3930	0.713026
$819$	0	0
$820$	0.221700	0.00774208
$821$	13.8449	0.483192	0.241596	−	0.970377i	$-0.422329\pi$
$821$	13.8449	0.483192	0.241596	+	0.970377i	$0.422329\pi$
$822$	0	0
$823$	19.8751	0.692804	0.346402	−	0.938086i	$-0.387403\pi$
$823$	19.8751	0.692804	0.346402	+	0.938086i	$0.387403\pi$
$824$	27.8342	0.969651
$825$	0	0
$826$	0.640754	0.0222947
$827$	−24.5929	−0.855181	−0.427590	−	0.903973i	$-0.640637\pi$
$827$	−24.5929	−0.855181	−0.427590	+	0.903973i	$0.640637\pi$
$828$	0	0
$829$	−44.4191	−1.54274	−0.771370	−	0.636387i	$-0.780428\pi$
$829$	−44.4191	−1.54274	−0.771370	+	0.636387i	$0.780428\pi$
$830$	−1.87361	−0.0650339
$831$	0	0
$832$	−15.3223	−0.531205
$833$	−12.8380	−0.444809
$834$	0	0
$835$	−2.16828	−0.0750364
$836$	−1.05015	−0.0363203
$837$	0	0
$838$	6.74773	0.233096
$839$	18.9679	0.654844	0.327422	−	0.944878i	$-0.393820\pi$
$839$	18.9679	0.654844	0.327422	+	0.944878i	$0.393820\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−26.1873	−0.902474
$843$	0	0
$844$	6.74472	0.232163
$845$	−1.05595	−0.0363259
$846$	0	0
$847$	0.569587	0.0195712
$848$	−0.249382	−0.00856382
$849$	0	0
$850$	13.9560	0.478687
$851$	−1.00635	−0.0344971
$852$	0	0
$853$	−6.18209	−0.211671	−0.105835	−	0.994384i	$-0.533752\pi$
$853$	−6.18209	−0.211671	−0.105835	+	0.994384i	$0.533752\pi$
$854$	−0.358234	−0.0122585
$855$	0	0
$856$	−9.82869	−0.335938
$857$	−26.6845	−0.911526	−0.455763	−	0.890101i	$-0.650634\pi$
$857$	−26.6845	−0.911526	−0.455763	+	0.890101i	$0.650634\pi$
$858$	0	0
$859$	−43.7240	−1.49184	−0.745922	−	0.666033i	$-0.767991\pi$
$859$	−43.7240	−1.49184	−0.745922	+	0.666033i	$0.767991\pi$
$860$	−0.377492	−0.0128724
$861$	0	0
$862$	19.8410	0.675788
$863$	32.7025	1.11320	0.556602	−	0.830779i	$-0.312105\pi$
$863$	32.7025	1.11320	0.556602	+	0.830779i	$0.312105\pi$
$864$	0	0
$865$	1.06659	0.0362652
$866$	−37.7398	−1.28245
$867$	0	0
$868$	0.0234922	0.000797378 0
$869$	−3.48075	−0.118076
$870$	0	0
$871$	10.6427	0.360616
$872$	−38.9478	−1.31894
$873$	0	0
$874$	−3.45725	−0.116943
$875$	0.0874180	0.00295527
$876$	0	0
$877$	−23.1085	−0.780317	−0.390159	−	0.920748i	$-0.627580\pi$
$877$	−23.1085	−0.780317	−0.390159	+	0.920748i	$0.627580\pi$
$878$	53.5858	1.80843
$879$	0	0
$880$	0.745503	0.0251309
$881$	−42.3370	−1.42637	−0.713184	−	0.700976i	$-0.752748\pi$
$881$	−42.3370	−1.42637	−0.713184	+	0.700976i	$0.752748\pi$
$882$	0	0
$883$	−40.9401	−1.37774	−0.688872	−	0.724883i	$-0.741894\pi$
$883$	−40.9401	−1.37774	−0.688872	+	0.724883i	$0.741894\pi$
$884$	−1.50934	−0.0507646
$885$	0	0
$886$	5.26562	0.176902
$887$	−11.9422	−0.400980	−0.200490	−	0.979696i	$-0.564253\pi$
$887$	−11.9422	−0.400980	−0.200490	+	0.979696i	$0.564253\pi$
$888$	0	0
$889$	0.727160	0.0243882
$890$	−0.867531	−0.0290797
$891$	0	0
$892$	−8.20065	−0.274578
$893$	17.0142	0.569359
$894$	0	0
$895$	−1.76999	−0.0591643
$896$	0.770744	0.0257487
$897$	0	0
$898$	−4.44975	−0.148490
$899$	−1.20902	−0.0403230
$900$	0	0
$901$	0.100386	0.00334436
$902$	7.22611	0.240603
$903$	0	0
$904$	−23.3297	−0.775935
$905$	−1.08217	−0.0359726
$906$	0	0
$907$	16.9795	0.563795	0.281898	−	0.959444i	$-0.409036\pi$
$907$	16.9795	0.563795	0.281898	+	0.959444i	$0.409036\pi$
$908$	−9.53434	−0.316408
$909$	0	0
$910$	−0.0328442	−0.00108877
$911$	−29.0868	−0.963687	−0.481844	−	0.876257i	$-0.660033\pi$
$911$	−29.0868	−0.963687	−0.481844	+	0.876257i	$0.660033\pi$
$912$	0	0
$913$	−8.76921	−0.290219
$914$	−44.2862	−1.46486
$915$	0	0
$916$	−4.81328	−0.159035
$917$	1.06891	0.0352984
$918$	0	0
$919$	56.7951	1.87350	0.936749	−	0.350002i	$-0.113819\pi$
$919$	56.7951	1.87350	0.936749	+	0.350002i	$0.113819\pi$
$920$	0.300571	0.00990954
$921$	0	0
$922$	−33.2316	−1.09443
$923$	36.5643	1.20353
$924$	0	0
$925$	6.41027	0.210768
$926$	−31.9144	−1.04877
$927$	0	0
$928$	1.87803	0.0616493
$929$	5.22657	0.171478	0.0857391	−	0.996318i	$-0.472675\pi$
$929$	5.22657	0.171478	0.0857391	+	0.996318i	$0.472675\pi$
$930$	0	0
$931$	−20.2579	−0.663926
$932$	1.19033	0.0389905
$933$	0	0
$934$	−28.1110	−0.919820
$935$	−0.300095	−0.00981416
$936$	0	0
$937$	24.4385	0.798372	0.399186	−	0.916870i	$-0.369293\pi$
$937$	24.4385	0.798372	0.399186	+	0.916870i	$0.369293\pi$
$938$	−0.384184	−0.0125441
$939$	0	0
$940$	0.297987	0.00971928
$941$	21.9511	0.715586	0.357793	−	0.933801i	$-0.383529\pi$
$941$	21.9511	0.715586	0.357793	+	0.933801i	$0.383529\pi$
$942$	0	0
$943$	3.41605	0.111242
$944$	−32.9849	−1.07357
$945$	0	0
$946$	−12.3040	−0.400039
$947$	−2.29429	−0.0745543	−0.0372771	−	0.999305i	$-0.511868\pi$
$947$	−2.29429	−0.0745543	−0.0372771	+	0.999305i	$0.511868\pi$
$948$	0	0
$949$	8.11737	0.263501
$950$	22.0221	0.714493
$951$	0	0
$952$	−0.270460	−0.00876567
$953$	−3.38896	−0.109779	−0.0548896	−	0.998492i	$-0.517481\pi$
$953$	−3.38896	−0.109779	−0.0548896	+	0.998492i	$0.517481\pi$
$954$	0	0
$955$	3.45599	0.111833
$956$	−4.67641	−0.151246
$957$	0	0
$958$	−14.2261	−0.459625
$959$	0.996331	0.0321732
$960$	0	0
$961$	−29.5383	−0.952848
$962$	−4.82792	−0.155658
$963$	0	0
$964$	7.08417	0.228166
$965$	0.0858457	0.00276347
$966$	0	0
$967$	22.3252	0.717929	0.358965	−	0.933351i	$-0.383130\pi$
$967$	22.3252	0.717929	0.358965	+	0.933351i	$0.383130\pi$
$968$	−25.0070	−0.803756
$969$	0	0
$970$	−3.59780	−0.115518
$971$	25.9936	0.834174	0.417087	−	0.908867i	$-0.363051\pi$
$971$	25.9936	0.834174	0.417087	+	0.908867i	$0.363051\pi$
$972$	0	0
$973$	−0.0946790	−0.00303527
$974$	−30.8830	−0.989555
$975$	0	0
$976$	18.4412	0.590289
$977$	−5.59628	−0.179041	−0.0895204	−	0.995985i	$-0.528533\pi$
$977$	−5.59628	−0.179041	−0.0895204	+	0.995985i	$0.528533\pi$
$978$	0	0
$979$	−4.06038	−0.129770
$980$	−0.354797	−0.0113336
$981$	0	0
$982$	−14.5354	−0.463844
$983$	47.1635	1.50428	0.752141	−	0.659002i	$-0.229021\pi$
$983$	47.1635	1.50428	0.752141	+	0.659002i	$0.229021\pi$
$984$	0	0
$985$	−1.18707	−0.0378233
$986$	−2.80403	−0.0892984
$987$	0	0
$988$	−2.38169	−0.0757716
$989$	−5.81657	−0.184956
$990$	0	0
$991$	21.3455	0.678061	0.339030	−	0.940775i	$-0.389901\pi$
$991$	21.3455	0.678061	0.339030	+	0.940775i	$0.389901\pi$
$992$	−2.27057	−0.0720907
$993$	0	0
$994$	−1.31991	−0.0418649
$995$	2.40809	0.0763416
$996$	0	0
$997$	−38.4855	−1.21885	−0.609424	−	0.792844i	$-0.708599\pi$
$997$	−38.4855	−1.21885	−0.609424	+	0.792844i	$0.708599\pi$
$998$	7.00129	0.221622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2349.2.a.e.1.3		11
3.2	odd	2		2349.2.a.f.1.9		11
9.2	odd	6		261.2.e.a.175.3	yes	22
9.4	even	3		783.2.e.a.262.9		22
9.5	odd	6		261.2.e.a.88.3	✓	22
9.7	even	3		783.2.e.a.523.9		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
261.2.e.a.88.3	✓	22	9.5	odd	6
261.2.e.a.175.3	yes	22	9.2	odd	6
783.2.e.a.262.9		22	9.4	even	3
783.2.e.a.523.9		22	9.7	even	3
2349.2.a.e.1.3		11	1.1	even	1	trivial
2349.2.a.f.1.9		11	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2349 = 3^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2349.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52818 q^{2} +0.335346 q^{4} +0.151216 q^{5} -0.0579426 q^{7} +2.54390 q^{8} -0.231086 q^{10} -1.08157 q^{11} -2.45294 q^{13} +0.0885469 q^{14} -4.55824 q^{16} +1.83487 q^{17} +2.89537 q^{19} +0.0507097 q^{20} +1.65284 q^{22} +0.781359 q^{23} -4.97713 q^{25} +3.74855 q^{26} -0.0194308 q^{28} +1.00000 q^{29} -1.20902 q^{31} +1.87803 q^{32} -2.80403 q^{34} -0.00876184 q^{35} -1.28794 q^{37} -4.42466 q^{38} +0.384678 q^{40} +4.37194 q^{41} -7.44418 q^{43} -0.362701 q^{44} -1.19406 q^{46} +5.87634 q^{47} -6.99664 q^{49} +7.60598 q^{50} -0.822584 q^{52} +0.0547103 q^{53} -0.163551 q^{55} -0.147400 q^{56} -1.52818 q^{58} +7.23633 q^{59} -4.04570 q^{61} +1.84760 q^{62} +6.24650 q^{64} -0.370924 q^{65} -4.33877 q^{67} +0.615318 q^{68} +0.0133897 q^{70} -14.9063 q^{71} -3.30924 q^{73} +1.96822 q^{74} +0.970952 q^{76} +0.0626690 q^{77} +3.21824 q^{79} -0.689278 q^{80} -6.68112 q^{82} +8.10784 q^{83} +0.277462 q^{85} +11.3761 q^{86} -2.75141 q^{88} +3.75415 q^{89} +0.142130 q^{91} +0.262026 q^{92} -8.98013 q^{94} +0.437827 q^{95} +15.5691 q^{97} +10.6922 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - q^{2} + 5 q^{4} + q^{5} - 7 q^{7} - 3 q^{8} - 10 q^{10} - 3 q^{11} - 7 q^{13} + 10 q^{14} - 7 q^{16} - q^{17} - 28 q^{19} + 4 q^{20} - 13 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} - 16 q^{28} + 11 q^{29}+ \cdots - 43 q^{98}+O(q^{100}) \) 11 * q - q^2 + 5 * q^4 + q^5 - 7 * q^7 - 3 * q^8 - 10 * q^10 - 3 * q^11 - 7 * q^13 + 10 * q^14 - 7 * q^16 - q^17 - 28 * q^19 + 4 * q^20 - 13 * q^22 - 4 * q^23 - 4 * q^25 + 6 * q^26 - 16 * q^28 + 11 * q^29 - 27 * q^31 - 7 * q^32 - 19 * q^34 + 11 * q^35 - 14 * q^37 + 5 * q^38 - 24 * q^40 + 9 * q^41 - 25 * q^43 - 23 * q^44 - 10 * q^46 - 6 * q^47 - 12 * q^49 - 41 * q^50 - 20 * q^52 + 6 * q^53 - 35 * q^55 + 10 * q^56 - q^58 - 25 * q^61 + 6 * q^62 - 25 * q^64 - 5 * q^65 - 29 * q^67 + 50 * q^68 - 7 * q^70 + q^71 - 13 * q^73 - 14 * q^74 - 38 * q^76 - 9 * q^77 - 27 * q^79 + 37 * q^80 - 17 * q^82 + 12 * q^83 - 7 * q^85 - q^86 + 5 * q^88 - 23 * q^89 - 43 * q^91 - 19 * q^92 + 10 * q^94 - 8 * q^95 - 36 * q^97 - 43 * q^98

Introduction

L-functions

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