LMFDB - Embedded newform 6039.2.a.k.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-0.556474$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.556474 q^{2} -1.69034 q^{4} -0.874867 q^{5} +4.50273 q^{7} -2.05358 q^{8} +O(q^{10})$	\(q+0.556474 q^{2} -1.69034 q^{4} -0.874867 q^{5} +4.50273 q^{7} -2.05358 q^{8} -0.486841 q^{10} -1.00000 q^{11} +1.21259 q^{13} +2.50565 q^{14} +2.23791 q^{16} +6.11300 q^{17} -3.45103 q^{19} +1.47882 q^{20} -0.556474 q^{22} +1.78258 q^{23} -4.23461 q^{25} +0.674772 q^{26} -7.61114 q^{28} -2.94546 q^{29} +9.52559 q^{31} +5.35249 q^{32} +3.40172 q^{34} -3.93930 q^{35} -1.86424 q^{37} -1.92041 q^{38} +1.79661 q^{40} -6.97280 q^{41} +7.84183 q^{43} +1.69034 q^{44} +0.991960 q^{46} -12.1891 q^{47} +13.2746 q^{49} -2.35645 q^{50} -2.04968 q^{52} +10.8203 q^{53} +0.874867 q^{55} -9.24670 q^{56} -1.63907 q^{58} +10.0621 q^{59} -1.00000 q^{61} +5.30074 q^{62} -1.49731 q^{64} -1.06085 q^{65} -10.1945 q^{67} -10.3330 q^{68} -2.19211 q^{70} +12.0311 q^{71} -8.81240 q^{73} -1.03740 q^{74} +5.83340 q^{76} -4.50273 q^{77} +13.9690 q^{79} -1.95788 q^{80} -3.88018 q^{82} -1.87604 q^{83} -5.34806 q^{85} +4.36377 q^{86} +2.05358 q^{88} +5.70475 q^{89} +5.45995 q^{91} -3.01316 q^{92} -6.78292 q^{94} +3.01919 q^{95} -13.1986 q^{97} +7.38697 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	0.556474	0.393486	0.196743	−	0.980455i	$-0.436963\pi$
$2$	0.556474	0.393486	0.196743	+	0.980455i	$0.436963\pi$
$3$	0	0
$3$	0	0
$4$	−1.69034	−0.845169
$5$	−0.874867	−0.391253	−0.195626	−	0.980679i	$-0.562674\pi$
$5$	−0.874867	−0.391253	−0.195626	+	0.980679i	$0.562674\pi$
$6$	0	0
$7$	4.50273	1.70187	0.850937	−	0.525268i	$-0.176035\pi$
$7$	4.50273	1.70187	0.850937	+	0.525268i	$0.176035\pi$
$8$	−2.05358	−0.726049
$9$	0	0
$10$	−0.486841	−0.153953
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.21259	0.336311	0.168155	−	0.985761i	$-0.446219\pi$
$13$	1.21259	0.336311	0.168155	+	0.985761i	$0.446219\pi$
$14$	2.50565	0.669664
$15$	0	0
$16$	2.23791	0.559478
$17$	6.11300	1.48262	0.741310	−	0.671163i	$-0.234205\pi$
$17$	6.11300	1.48262	0.741310	+	0.671163i	$0.234205\pi$
$18$	0	0
$19$	−3.45103	−0.791721	−0.395860	−	0.918311i	$-0.629554\pi$
$19$	−3.45103	−0.791721	−0.395860	+	0.918311i	$0.629554\pi$
$20$	1.47882	0.330674
$21$	0	0
$22$	−0.556474	−0.118641
$23$	1.78258	0.371694	0.185847	−	0.982579i	$-0.440497\pi$
$23$	1.78258	0.371694	0.185847	+	0.982579i	$0.440497\pi$
$24$	0	0
$25$	−4.23461	−0.846921
$26$	0.674772	0.132334
$27$	0	0
$28$	−7.61114	−1.43837
$29$	−2.94546	−0.546958	−0.273479	−	0.961878i	$-0.588174\pi$
$29$	−2.94546	−0.546958	−0.273479	+	0.961878i	$0.588174\pi$
$30$	0	0
$31$	9.52559	1.71085	0.855423	−	0.517930i	$-0.173297\pi$
$31$	9.52559	1.71085	0.855423	+	0.517930i	$0.173297\pi$
$32$	5.35249	0.946196
$33$	0	0
$34$	3.40172	0.583391
$35$	−3.93930	−0.665862
$36$	0	0
$37$	−1.86424	−0.306479	−0.153239	−	0.988189i	$-0.548971\pi$
$37$	−1.86424	−0.306479	−0.153239	+	0.988189i	$0.548971\pi$
$38$	−1.92041	−0.311531
$39$	0	0
$40$	1.79661	0.284068
$41$	−6.97280	−1.08897	−0.544485	−	0.838771i	$-0.683275\pi$
$41$	−6.97280	−1.08897	−0.544485	+	0.838771i	$0.683275\pi$
$42$	0	0
$43$	7.84183	1.19587	0.597934	−	0.801545i	$-0.295989\pi$
$43$	7.84183	1.19587	0.597934	+	0.801545i	$0.295989\pi$
$44$	1.69034	0.254828
$45$	0	0
$46$	0.991960	0.146256
$47$	−12.1891	−1.77796	−0.888982	−	0.457942i	$-0.848587\pi$
$47$	−12.1891	−1.77796	−0.888982	+	0.457942i	$0.848587\pi$
$48$	0	0
$49$	13.2746	1.89637
$50$	−2.35645	−0.333252
$51$	0	0
$52$	−2.04968	−0.284239
$53$	10.8203	1.48628	0.743141	−	0.669135i	$-0.233335\pi$
$53$	10.8203	1.48628	0.743141	+	0.669135i	$0.233335\pi$
$54$	0	0
$55$	0.874867	0.117967
$56$	−9.24670	−1.23564
$57$	0	0
$58$	−1.63907	−0.215220
$59$	10.0621	1.30998	0.654988	−	0.755639i	$-0.272674\pi$
$59$	10.0621	1.30998	0.654988	+	0.755639i	$0.272674\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	5.30074	0.673195
$63$	0	0
$64$	−1.49731	−0.187163
$65$	−1.06085	−0.131582
$66$	0	0
$67$	−10.1945	−1.24546	−0.622730	−	0.782437i	$-0.713977\pi$
$67$	−10.1945	−1.24546	−0.622730	+	0.782437i	$0.713977\pi$
$68$	−10.3330	−1.25306
$69$	0	0
$70$	−2.19211	−0.262008
$71$	12.0311	1.42783	0.713916	−	0.700231i	$-0.246920\pi$
$71$	12.0311	1.42783	0.713916	+	0.700231i	$0.246920\pi$
$72$	0	0
$73$	−8.81240	−1.03141	−0.515707	−	0.856765i	$-0.672471\pi$
$73$	−8.81240	−1.03141	−0.515707	+	0.856765i	$0.672471\pi$
$74$	−1.03740	−0.120595
$75$	0	0
$76$	5.83340	0.669137
$77$	−4.50273	−0.513134
$78$	0	0
$79$	13.9690	1.57163	0.785816	−	0.618460i	$-0.212243\pi$
$79$	13.9690	1.57163	0.785816	+	0.618460i	$0.212243\pi$
$80$	−1.95788	−0.218897
$81$	0	0
$82$	−3.88018	−0.428494
$83$	−1.87604	−0.205923	−0.102961	−	0.994685i	$-0.532832\pi$
$83$	−1.87604	−0.205923	−0.102961	+	0.994685i	$0.532832\pi$
$84$	0	0
$85$	−5.34806	−0.580079
$86$	4.36377	0.470558
$87$	0	0
$88$	2.05358	0.218912
$89$	5.70475	0.604703	0.302351	−	0.953197i	$-0.402228\pi$
$89$	5.70475	0.604703	0.302351	+	0.953197i	$0.402228\pi$
$90$	0	0
$91$	5.45995	0.572358
$92$	−3.01316	−0.314144
$93$	0	0
$94$	−6.78292	−0.699605
$95$	3.01919	0.309763
$96$	0	0
$97$	−13.1986	−1.34011	−0.670057	−	0.742310i	$-0.733730\pi$
$97$	−13.1986	−1.34011	−0.670057	+	0.742310i	$0.733730\pi$
$98$	7.38697	0.746197
$99$	0	0
$100$	7.15791	0.715791
$101$	−0.948657	−0.0943949	−0.0471974	−	0.998886i	$-0.515029\pi$
$101$	−0.948657	−0.0943949	−0.0471974	+	0.998886i	$0.515029\pi$
$102$	0	0
$103$	0.859347	0.0846739	0.0423370	−	0.999103i	$-0.486520\pi$
$103$	0.859347	0.0846739	0.0423370	+	0.999103i	$0.486520\pi$
$104$	−2.49014	−0.244178
$105$	0	0
$106$	6.02121	0.584832
$107$	−16.0349	−1.55015	−0.775076	−	0.631869i	$-0.782288\pi$
$107$	−16.0349	−1.55015	−0.775076	+	0.631869i	$0.782288\pi$
$108$	0	0
$109$	−10.5285	−1.00845	−0.504224	−	0.863573i	$-0.668221\pi$
$109$	−10.5285	−1.00845	−0.504224	+	0.863573i	$0.668221\pi$
$110$	0.486841	0.0464184
$111$	0	0
$112$	10.0767	0.952161
$113$	0.0682609	0.00642144	0.00321072	−	0.999995i	$-0.498978\pi$
$113$	0.0682609	0.00642144	0.00321072	+	0.999995i	$0.498978\pi$
$114$	0	0
$115$	−1.55952	−0.145426
$116$	4.97882	0.462271
$117$	0	0
$118$	5.59930	0.515457
$119$	27.5252	2.52323
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.556474	−0.0503808
$123$	0	0
$124$	−16.1015	−1.44595
$125$	8.07906	0.722613
$126$	0	0
$127$	−0.618699	−0.0549006	−0.0274503	−	0.999623i	$-0.508739\pi$
$127$	−0.618699	−0.0549006	−0.0274503	+	0.999623i	$0.508739\pi$
$128$	−11.5382	−1.01984
$129$	0	0
$130$	−0.590336	−0.0517759
$131$	−2.74292	−0.239650	−0.119825	−	0.992795i	$-0.538233\pi$
$131$	−2.74292	−0.239650	−0.119825	+	0.992795i	$0.538233\pi$
$132$	0	0
$133$	−15.5391	−1.34741
$134$	−5.67299	−0.490071
$135$	0	0
$136$	−12.5535	−1.07645
$137$	9.11318	0.778591	0.389296	−	0.921113i	$-0.372718\pi$
$137$	9.11318	0.778591	0.389296	+	0.921113i	$0.372718\pi$
$138$	0	0
$139$	−13.0007	−1.10270	−0.551352	−	0.834272i	$-0.685888\pi$
$139$	−13.0007	−1.10270	−0.551352	+	0.834272i	$0.685888\pi$
$140$	6.65874	0.562766
$141$	0	0
$142$	6.69501	0.561832
$143$	−1.21259	−0.101402
$144$	0	0
$145$	2.57688	0.213999
$146$	−4.90387	−0.405847
$147$	0	0
$148$	3.15119	0.259026
$149$	12.0242	0.985062	0.492531	−	0.870295i	$-0.336072\pi$
$149$	12.0242	0.985062	0.492531	+	0.870295i	$0.336072\pi$
$150$	0	0
$151$	19.4699	1.58444	0.792218	−	0.610239i	$-0.208926\pi$
$151$	19.4699	1.58444	0.792218	+	0.610239i	$0.208926\pi$
$152$	7.08695	0.574828
$153$	0	0
$154$	−2.50565	−0.201911
$155$	−8.33363	−0.669373
$156$	0	0
$157$	11.2579	0.898476	0.449238	−	0.893412i	$-0.351695\pi$
$157$	11.2579	0.898476	0.449238	+	0.893412i	$0.351695\pi$
$158$	7.77337	0.618416
$159$	0	0
$160$	−4.68272	−0.370201
$161$	8.02649	0.632576
$162$	0	0
$163$	10.2276	0.801091	0.400546	−	0.916277i	$-0.368821\pi$
$163$	10.2276	0.801091	0.400546	+	0.916277i	$0.368821\pi$
$164$	11.7864	0.920362
$165$	0	0
$166$	−1.04397	−0.0810277
$167$	15.6162	1.20842	0.604210	−	0.796825i	$-0.293489\pi$
$167$	15.6162	1.20842	0.604210	+	0.796825i	$0.293489\pi$
$168$	0	0
$169$	−11.5296	−0.886895
$170$	−2.97606	−0.228253
$171$	0	0
$172$	−13.2553	−1.01071
$173$	10.4072	0.791247	0.395623	−	0.918413i	$-0.370529\pi$
$173$	10.4072	0.791247	0.395623	+	0.918413i	$0.370529\pi$
$174$	0	0
$175$	−19.0673	−1.44135
$176$	−2.23791	−0.168689
$177$	0	0
$178$	3.17455	0.237942
$179$	5.31572	0.397316	0.198658	−	0.980069i	$-0.436342\pi$
$179$	5.31572	0.397316	0.198658	+	0.980069i	$0.436342\pi$
$180$	0	0
$181$	9.67871	0.719413	0.359707	−	0.933065i	$-0.382877\pi$
$181$	9.67871	0.719413	0.359707	+	0.933065i	$0.382877\pi$
$182$	3.03832	0.225215
$183$	0	0
$184$	−3.66067	−0.269868
$185$	1.63096	0.119911
$186$	0	0
$187$	−6.11300	−0.447027
$188$	20.6037	1.50268
$189$	0	0
$190$	1.68010	0.121887
$191$	21.8469	1.58079	0.790393	−	0.612600i	$-0.209876\pi$
$191$	21.8469	1.58079	0.790393	+	0.612600i	$0.209876\pi$
$192$	0	0
$193$	20.7290	1.49210	0.746052	−	0.665888i	$-0.231947\pi$
$193$	20.7290	1.49210	0.746052	+	0.665888i	$0.231947\pi$
$194$	−7.34467	−0.527317
$195$	0	0
$196$	−22.4386	−1.60276
$197$	8.74690	0.623191	0.311595	−	0.950215i	$-0.399137\pi$
$197$	8.74690	0.623191	0.311595	+	0.950215i	$0.399137\pi$
$198$	0	0
$199$	22.5954	1.60175	0.800873	−	0.598834i	$-0.204369\pi$
$199$	22.5954	1.60175	0.800873	+	0.598834i	$0.204369\pi$
$200$	8.69608	0.614906
$201$	0	0
$202$	−0.527903	−0.0371431
$203$	−13.2626	−0.930853
$204$	0	0
$205$	6.10028	0.426062
$206$	0.478204	0.0333180
$207$	0	0
$208$	2.71366	0.188159
$209$	3.45103	0.238713
$210$	0	0
$211$	7.08528	0.487771	0.243885	−	0.969804i	$-0.421578\pi$
$211$	7.08528	0.487771	0.243885	+	0.969804i	$0.421578\pi$
$212$	−18.2899	−1.25616
$213$	0	0
$214$	−8.92299	−0.609963
$215$	−6.86056	−0.467886
$216$	0	0
$217$	42.8912	2.91164
$218$	−5.85883	−0.396810
$219$	0	0
$220$	−1.47882	−0.0997021
$221$	7.41254	0.498621
$222$	0	0
$223$	−10.1032	−0.676561	−0.338281	−	0.941045i	$-0.609845\pi$
$223$	−10.1032	−0.676561	−0.338281	+	0.941045i	$0.609845\pi$
$224$	24.1008	1.61031
$225$	0	0
$226$	0.0379854	0.00252675
$227$	−6.61082	−0.438776	−0.219388	−	0.975638i	$-0.570406\pi$
$227$	−6.61082	−0.438776	−0.219388	+	0.975638i	$0.570406\pi$
$228$	0	0
$229$	−14.1361	−0.934140	−0.467070	−	0.884220i	$-0.654690\pi$
$229$	−14.1361	−0.934140	−0.467070	+	0.884220i	$0.654690\pi$
$230$	−0.867833	−0.0572232
$231$	0	0
$232$	6.04872	0.397118
$233$	−0.866200	−0.0567466	−0.0283733	−	0.999597i	$-0.509033\pi$
$233$	−0.866200	−0.0567466	−0.0283733	+	0.999597i	$0.509033\pi$
$234$	0	0
$235$	10.6639	0.695633
$236$	−17.0084	−1.10715
$237$	0	0
$238$	15.3171	0.992857
$239$	−1.61805	−0.104663	−0.0523315	−	0.998630i	$-0.516665\pi$
$239$	−1.61805	−0.104663	−0.0523315	+	0.998630i	$0.516665\pi$
$240$	0	0
$241$	23.8643	1.53724	0.768619	−	0.639707i	$-0.220944\pi$
$241$	23.8643	1.53724	0.768619	+	0.639707i	$0.220944\pi$
$242$	0.556474	0.0357715
$243$	0	0
$244$	1.69034	0.108213
$245$	−11.6135	−0.741961
$246$	0	0
$247$	−4.18467	−0.266264
$248$	−19.5615	−1.24216
$249$	0	0
$250$	4.49578	0.284338
$251$	6.36741	0.401907	0.200954	−	0.979601i	$-0.435596\pi$
$251$	6.36741	0.401907	0.200954	+	0.979601i	$0.435596\pi$
$252$	0	0
$253$	−1.78258	−0.112070
$254$	−0.344290	−0.0216026
$255$	0	0
$256$	−3.42609	−0.214130
$257$	14.0913	0.878991	0.439496	−	0.898245i	$-0.355157\pi$
$257$	14.0913	0.878991	0.439496	+	0.898245i	$0.355157\pi$
$258$	0	0
$259$	−8.39416	−0.521588
$260$	1.79320	0.111209
$261$	0	0
$262$	−1.52636	−0.0942988
$263$	17.5337	1.08117	0.540586	−	0.841289i	$-0.318202\pi$
$263$	17.5337	1.08117	0.540586	+	0.841289i	$0.318202\pi$
$264$	0	0
$265$	−9.46632	−0.581512
$266$	−8.64708	−0.530187
$267$	0	0
$268$	17.2322	1.05262
$269$	11.9094	0.726127	0.363064	−	0.931764i	$-0.381731\pi$
$269$	11.9094	0.726127	0.363064	+	0.931764i	$0.381731\pi$
$270$	0	0
$271$	5.31183	0.322670	0.161335	−	0.986900i	$-0.448420\pi$
$271$	5.31183	0.322670	0.161335	+	0.986900i	$0.448420\pi$
$272$	13.6804	0.829494
$273$	0	0
$274$	5.07124	0.306365
$275$	4.23461	0.255356
$276$	0	0
$277$	−2.59032	−0.155637	−0.0778185	−	0.996968i	$-0.524795\pi$
$277$	−2.59032	−0.155637	−0.0778185	+	0.996968i	$0.524795\pi$
$278$	−7.23455	−0.433899
$279$	0	0
$280$	8.08964	0.483448
$281$	9.47583	0.565281	0.282640	−	0.959226i	$-0.408790\pi$
$281$	9.47583	0.565281	0.282640	+	0.959226i	$0.408790\pi$
$282$	0	0
$283$	−11.9329	−0.709338	−0.354669	−	0.934992i	$-0.615406\pi$
$283$	−11.9329	−0.709338	−0.354669	+	0.934992i	$0.615406\pi$
$284$	−20.3367	−1.20676
$285$	0	0
$286$	−0.674772	−0.0399001
$287$	−31.3967	−1.85329
$288$	0	0
$289$	20.3688	1.19816
$290$	1.43397	0.0842055
$291$	0	0
$292$	14.8959	0.871719
$293$	−18.7030	−1.09264	−0.546322	−	0.837575i	$-0.683972\pi$
$293$	−18.7030	−1.09264	−0.546322	+	0.837575i	$0.683972\pi$
$294$	0	0
$295$	−8.80301	−0.512531
$296$	3.82835	0.222518
$297$	0	0
$298$	6.69116	0.387609
$299$	2.16153	0.125005
$300$	0	0
$301$	35.3097	2.03522
$302$	10.8345	0.623454
$303$	0	0
$304$	−7.72311	−0.442951
$305$	0.874867	0.0500948
$306$	0	0
$307$	18.2224	1.04001	0.520004	−	0.854164i	$-0.325931\pi$
$307$	18.2224	1.04001	0.520004	+	0.854164i	$0.325931\pi$
$308$	7.61114	0.433685
$309$	0	0
$310$	−4.63744	−0.263389
$311$	0.0331052	0.00187722	0.000938611	−	1.00000i	$-0.499701\pi$
$311$	0.0331052	0.00187722	0.000938611		1.00000i	$0.499701\pi$
$312$	0	0
$313$	22.8589	1.29206	0.646031	−	0.763311i	$-0.276428\pi$
$313$	22.8589	1.29206	0.646031	+	0.763311i	$0.276428\pi$
$314$	6.26471	0.353538
$315$	0	0
$316$	−23.6123	−1.32829
$317$	5.00300	0.280996	0.140498	−	0.990081i	$-0.455130\pi$
$317$	5.00300	0.280996	0.140498	+	0.990081i	$0.455130\pi$
$318$	0	0
$319$	2.94546	0.164914
$320$	1.30994	0.0732282
$321$	0	0
$322$	4.46653	0.248910
$323$	−21.0961	−1.17382
$324$	0	0
$325$	−5.13482	−0.284829
$326$	5.69142	0.315218
$327$	0	0
$328$	14.3192	0.790644
$329$	−54.8843	−3.02587
$330$	0	0
$331$	−1.03641	−0.0569664	−0.0284832	−	0.999594i	$-0.509068\pi$
$331$	−1.03641	−0.0569664	−0.0284832	+	0.999594i	$0.509068\pi$
$332$	3.17115	0.174039
$333$	0	0
$334$	8.69002	0.475497
$335$	8.91886	0.487290
$336$	0	0
$337$	−32.5078	−1.77081	−0.885407	−	0.464816i	$-0.846120\pi$
$337$	−32.5078	−1.77081	−0.885407	+	0.464816i	$0.846120\pi$
$338$	−6.41594	−0.348981
$339$	0	0
$340$	9.04003	0.490264
$341$	−9.52559	−0.515840
$342$	0	0
$343$	28.2529	1.52551
$344$	−16.1038	−0.868258
$345$	0	0
$346$	5.79135	0.311345
$347$	−1.02130	−0.0548262	−0.0274131	−	0.999624i	$-0.508727\pi$
$347$	−1.02130	−0.0548262	−0.0274131	+	0.999624i	$0.508727\pi$
$348$	0	0
$349$	4.68103	0.250570	0.125285	−	0.992121i	$-0.460016\pi$
$349$	4.68103	0.250570	0.125285	+	0.992121i	$0.460016\pi$
$350$	−10.6105	−0.567153
$351$	0	0
$352$	−5.35249	−0.285289
$353$	7.82918	0.416705	0.208353	−	0.978054i	$-0.433190\pi$
$353$	7.82918	0.416705	0.208353	+	0.978054i	$0.433190\pi$
$354$	0	0
$355$	−10.5256	−0.558643
$356$	−9.64296	−0.511076
$357$	0	0
$358$	2.95806	0.156338
$359$	−10.2262	−0.539717	−0.269859	−	0.962900i	$-0.586977\pi$
$359$	−10.2262	−0.539717	−0.269859	+	0.962900i	$0.586977\pi$
$360$	0	0
$361$	−7.09039	−0.373179
$362$	5.38595	0.283079
$363$	0	0
$364$	−9.22916	−0.483739
$365$	7.70969	0.403543
$366$	0	0
$367$	30.9579	1.61599	0.807995	−	0.589189i	$-0.200553\pi$
$367$	30.9579	1.61599	0.807995	+	0.589189i	$0.200553\pi$
$368$	3.98926	0.207955
$369$	0	0
$370$	0.907586	0.0471832
$371$	48.7209	2.52946
$372$	0	0
$373$	26.4894	1.37157	0.685784	−	0.727805i	$-0.259460\pi$
$373$	26.4894	1.37157	0.685784	+	0.727805i	$0.259460\pi$
$374$	−3.40172	−0.175899
$375$	0	0
$376$	25.0313	1.29089
$377$	−3.57162	−0.183948
$378$	0	0
$379$	−4.68172	−0.240484	−0.120242	−	0.992745i	$-0.538367\pi$
$379$	−4.68172	−0.240484	−0.120242	+	0.992745i	$0.538367\pi$
$380$	−5.10345	−0.261802
$381$	0	0
$382$	12.1572	0.622018
$383$	−18.2494	−0.932502	−0.466251	−	0.884652i	$-0.654396\pi$
$383$	−18.2494	−0.932502	−0.466251	+	0.884652i	$0.654396\pi$
$384$	0	0
$385$	3.93930	0.200765
$386$	11.5351	0.587122
$387$	0	0
$388$	22.3101	1.13262
$389$	−32.3196	−1.63867	−0.819334	−	0.573317i	$-0.805657\pi$
$389$	−32.3196	−1.63867	−0.819334	+	0.573317i	$0.805657\pi$
$390$	0	0
$391$	10.8969	0.551081
$392$	−27.2604	−1.37686
$393$	0	0
$394$	4.86742	0.245217
$395$	−12.2210	−0.614905
$396$	0	0
$397$	−32.5665	−1.63446	−0.817232	−	0.576309i	$-0.804493\pi$
$397$	−32.5665	−1.63446	−0.817232	+	0.576309i	$0.804493\pi$
$398$	12.5737	0.630265
$399$	0	0
$400$	−9.47668	−0.473834
$401$	−31.9546	−1.59573	−0.797867	−	0.602833i	$-0.794039\pi$
$401$	−31.9546	−1.59573	−0.797867	+	0.602833i	$0.794039\pi$
$402$	0	0
$403$	11.5506	0.575376
$404$	1.60355	0.0797796
$405$	0	0
$406$	−7.38029	−0.366278
$407$	1.86424	0.0924068
$408$	0	0
$409$	−27.6360	−1.36651	−0.683255	−	0.730180i	$-0.739436\pi$
$409$	−27.6360	−1.36651	−0.683255	+	0.730180i	$0.739436\pi$
$410$	3.39464	0.167650
$411$	0	0
$412$	−1.45259	−0.0715638
$413$	45.3070	2.22941
$414$	0	0
$415$	1.64129	0.0805677
$416$	6.49035	0.318216
$417$	0	0
$418$	1.92041	0.0939302
$419$	−21.9679	−1.07320	−0.536601	−	0.843836i	$-0.680292\pi$
$419$	−21.9679	−1.07320	−0.536601	+	0.843836i	$0.680292\pi$
$420$	0	0
$421$	15.1470	0.738218	0.369109	−	0.929386i	$-0.379663\pi$
$421$	15.1470	0.738218	0.369109	+	0.929386i	$0.379663\pi$
$422$	3.94277	0.191931
$423$	0	0
$424$	−22.2203	−1.07911
$425$	−25.8861	−1.25566
$426$	0	0
$427$	−4.50273	−0.217903
$428$	27.1044	1.31014
$429$	0	0
$430$	−3.81772	−0.184107
$431$	26.8606	1.29383	0.646916	−	0.762561i	$-0.276059\pi$
$431$	26.8606	1.29383	0.646916	+	0.762561i	$0.276059\pi$
$432$	0	0
$433$	14.8367	0.713009	0.356504	−	0.934294i	$-0.383969\pi$
$433$	14.8367	0.713009	0.356504	+	0.934294i	$0.383969\pi$
$434$	23.8678	1.14569
$435$	0	0
$436$	17.7967	0.852308
$437$	−6.15174	−0.294278
$438$	0	0
$439$	−5.65532	−0.269914	−0.134957	−	0.990851i	$-0.543090\pi$
$439$	−5.65532	−0.269914	−0.134957	+	0.990851i	$0.543090\pi$
$440$	−1.79661	−0.0856498
$441$	0	0
$442$	4.12488	0.196201
$443$	11.7795	0.559659	0.279829	−	0.960050i	$-0.409722\pi$
$443$	11.7795	0.559659	0.279829	+	0.960050i	$0.409722\pi$
$444$	0	0
$445$	−4.99090	−0.236592
$446$	−5.62217	−0.266218
$447$	0	0
$448$	−6.74197	−0.318528
$449$	−27.1242	−1.28007	−0.640036	−	0.768345i	$-0.721080\pi$
$449$	−27.1242	−1.28007	−0.640036	+	0.768345i	$0.721080\pi$
$450$	0	0
$451$	6.97280	0.328337
$452$	−0.115384	−0.00542720
$453$	0	0
$454$	−3.67875	−0.172652
$455$	−4.77673	−0.223937
$456$	0	0
$457$	24.7028	1.15555	0.577773	−	0.816197i	$-0.303922\pi$
$457$	24.7028	1.15555	0.577773	+	0.816197i	$0.303922\pi$
$458$	−7.86637	−0.367571
$459$	0	0
$460$	2.63612	0.122910
$461$	14.8449	0.691397	0.345699	−	0.938346i	$-0.387642\pi$
$461$	14.8449	0.691397	0.345699	+	0.938346i	$0.387642\pi$
$462$	0	0
$463$	−19.2182	−0.893146	−0.446573	−	0.894747i	$-0.647356\pi$
$463$	−19.2182	−0.893146	−0.446573	+	0.894747i	$0.647356\pi$
$464$	−6.59168	−0.306011
$465$	0	0
$466$	−0.482017	−0.0223290
$467$	10.9690	0.507585	0.253792	−	0.967259i	$-0.418322\pi$
$467$	10.9690	0.507585	0.253792	+	0.967259i	$0.418322\pi$
$468$	0	0
$469$	−45.9033	−2.11962
$470$	5.93415	0.273722
$471$	0	0
$472$	−20.6633	−0.951106
$473$	−7.84183	−0.360568
$474$	0	0
$475$	14.6138	0.670525
$476$	−46.5269	−2.13256
$477$	0	0
$478$	−0.900403	−0.0411835
$479$	−0.883373	−0.0403624	−0.0201812	−	0.999796i	$-0.506424\pi$
$479$	−0.883373	−0.0403624	−0.0201812	+	0.999796i	$0.506424\pi$
$480$	0	0
$481$	−2.26055	−0.103072
$482$	13.2799	0.604882
$483$	0	0
$484$	−1.69034	−0.0768335
$485$	11.5470	0.524323
$486$	0	0
$487$	−3.96107	−0.179493	−0.0897467	−	0.995965i	$-0.528606\pi$
$487$	−3.96107	−0.179493	−0.0897467	+	0.995965i	$0.528606\pi$
$488$	2.05358	0.0929610
$489$	0	0
$490$	−6.46262	−0.291951
$491$	2.92664	0.132077	0.0660387	−	0.997817i	$-0.478964\pi$
$491$	2.92664	0.132077	0.0660387	+	0.997817i	$0.478964\pi$
$492$	0	0
$493$	−18.0056	−0.810931
$494$	−2.32866	−0.104771
$495$	0	0
$496$	21.3174	0.957182
$497$	54.1730	2.42999
$498$	0	0
$499$	−4.64531	−0.207953	−0.103976	−	0.994580i	$-0.533157\pi$
$499$	−4.64531	−0.207953	−0.103976	+	0.994580i	$0.533157\pi$
$500$	−13.6563	−0.610730
$501$	0	0
$502$	3.54329	0.158145
$503$	16.2825	0.726001	0.363000	−	0.931789i	$-0.381752\pi$
$503$	16.2825	0.726001	0.363000	+	0.931789i	$0.381752\pi$
$504$	0	0
$505$	0.829949	0.0369322
$506$	−0.991960	−0.0440980
$507$	0	0
$508$	1.04581	0.0464003
$509$	18.9031	0.837867	0.418934	−	0.908017i	$-0.362404\pi$
$509$	18.9031	0.837867	0.418934	+	0.908017i	$0.362404\pi$
$510$	0	0
$511$	−39.6799	−1.75534
$512$	21.1699	0.935584
$513$	0	0
$514$	7.84144	0.345871
$515$	−0.751814	−0.0331289
$516$	0	0
$517$	12.1891	0.536076
$518$	−4.67113	−0.205238
$519$	0	0
$520$	2.17854	0.0955353
$521$	16.9237	0.741439	0.370719	−	0.928745i	$-0.379111\pi$
$521$	16.9237	0.741439	0.370719	+	0.928745i	$0.379111\pi$
$522$	0	0
$523$	25.2740	1.10515	0.552577	−	0.833462i	$-0.313644\pi$
$523$	25.2740	1.10515	0.552577	+	0.833462i	$0.313644\pi$
$524$	4.63645	0.202544
$525$	0	0
$526$	9.75703	0.425427
$527$	58.2299	2.53654
$528$	0	0
$529$	−19.8224	−0.861844
$530$	−5.26776	−0.228817
$531$	0	0
$532$	26.2663	1.13879
$533$	−8.45512	−0.366232
$534$	0	0
$535$	14.0284	0.606501
$536$	20.9352	0.904265
$537$	0	0
$538$	6.62725	0.285721
$539$	−13.2746	−0.571778
$540$	0	0
$541$	12.8723	0.553423	0.276711	−	0.960953i	$-0.410755\pi$
$541$	12.8723	0.553423	0.276711	+	0.960953i	$0.410755\pi$
$542$	2.95589	0.126966
$543$	0	0
$544$	32.7198	1.40285
$545$	9.21104	0.394558
$546$	0	0
$547$	12.6284	0.539953	0.269977	−	0.962867i	$-0.412984\pi$
$547$	12.6284	0.539953	0.269977	+	0.962867i	$0.412984\pi$
$548$	−15.4043	−0.658041
$549$	0	0
$550$	2.35645	0.100479
$551$	10.1649	0.433038
$552$	0	0
$553$	62.8986	2.67472
$554$	−1.44144	−0.0612410
$555$	0	0
$556$	21.9756	0.931971
$557$	42.4320	1.79790	0.898950	−	0.438051i	$-0.144331\pi$
$557$	42.4320	1.79790	0.898950	+	0.438051i	$0.144331\pi$
$558$	0	0
$559$	9.50889	0.402183
$560$	−8.81580	−0.372536
$561$	0	0
$562$	5.27305	0.222430
$563$	−39.6342	−1.67038	−0.835191	−	0.549960i	$-0.814643\pi$
$563$	−39.6342	−1.67038	−0.835191	+	0.549960i	$0.814643\pi$
$564$	0	0
$565$	−0.0597192	−0.00251241
$566$	−6.64035	−0.279115
$567$	0	0
$568$	−24.7068	−1.03668
$569$	−28.4337	−1.19200	−0.596002	−	0.802983i	$-0.703245\pi$
$569$	−28.4337	−1.19200	−0.596002	+	0.802983i	$0.703245\pi$
$570$	0	0
$571$	6.08101	0.254482	0.127241	−	0.991872i	$-0.459388\pi$
$571$	6.08101	0.254482	0.127241	+	0.991872i	$0.459388\pi$
$572$	2.04968	0.0857014
$573$	0	0
$574$	−17.4714	−0.729243
$575$	−7.54853	−0.314796
$576$	0	0
$577$	45.8340	1.90809	0.954047	−	0.299657i	$-0.0968724\pi$
$577$	45.8340	1.90809	0.954047	+	0.299657i	$0.0968724\pi$
$578$	11.3347	0.471460
$579$	0	0
$580$	−4.35580	−0.180865
$581$	−8.44733	−0.350454
$582$	0	0
$583$	−10.8203	−0.448131
$584$	18.0969	0.748857
$585$	0	0
$586$	−10.4077	−0.429940
$587$	25.9912	1.07277	0.536387	−	0.843972i	$-0.319789\pi$
$587$	25.9912	1.07277	0.536387	+	0.843972i	$0.319789\pi$
$588$	0	0
$589$	−32.8731	−1.35451
$590$	−4.89865	−0.201674
$591$	0	0
$592$	−4.17200	−0.171468
$593$	−31.7176	−1.30248	−0.651242	−	0.758870i	$-0.725752\pi$
$593$	−31.7176	−1.30248	−0.651242	+	0.758870i	$0.725752\pi$
$594$	0	0
$595$	−24.0809	−0.987221
$596$	−20.3250	−0.832544
$597$	0	0
$598$	1.20284	0.0491876
$599$	−19.0067	−0.776595	−0.388297	−	0.921534i	$-0.626937\pi$
$599$	−19.0067	−0.776595	−0.388297	+	0.921534i	$0.626937\pi$
$600$	0	0
$601$	2.77378	0.113145	0.0565725	−	0.998398i	$-0.481983\pi$
$601$	2.77378	0.113145	0.0565725	+	0.998398i	$0.481983\pi$
$602$	19.6489	0.800829
$603$	0	0
$604$	−32.9106	−1.33911
$605$	−0.874867	−0.0355684
$606$	0	0
$607$	−29.2284	−1.18634	−0.593172	−	0.805076i	$-0.702125\pi$
$607$	−29.2284	−1.18634	−0.593172	+	0.805076i	$0.702125\pi$
$608$	−18.4716	−0.749123
$609$	0	0
$610$	0.486841	0.0197116
$611$	−14.7803	−0.597949
$612$	0	0
$613$	22.7281	0.917979	0.458989	−	0.888442i	$-0.348212\pi$
$613$	22.7281	0.917979	0.458989	+	0.888442i	$0.348212\pi$
$614$	10.1403	0.409229
$615$	0	0
$616$	9.24670	0.372560
$617$	−33.6346	−1.35408	−0.677038	−	0.735948i	$-0.736737\pi$
$617$	−33.6346	−1.35408	−0.677038	+	0.735948i	$0.736737\pi$
$618$	0	0
$619$	21.9944	0.884028	0.442014	−	0.897008i	$-0.354264\pi$
$619$	21.9944	0.884028	0.442014	+	0.897008i	$0.354264\pi$
$620$	14.0866	0.565733
$621$	0	0
$622$	0.0184222	0.000738661 0
$623$	25.6870	1.02913
$624$	0	0
$625$	14.1049	0.564197
$626$	12.7204	0.508408
$627$	0	0
$628$	−19.0296	−0.759364
$629$	−11.3961	−0.454391
$630$	0	0
$631$	13.2436	0.527218	0.263609	−	0.964630i	$-0.415087\pi$
$631$	13.2436	0.527218	0.263609	+	0.964630i	$0.415087\pi$
$632$	−28.6863	−1.14108
$633$	0	0
$634$	2.78404	0.110568
$635$	0.541279	0.0214800
$636$	0	0
$637$	16.0966	0.637771
$638$	1.63907	0.0648914
$639$	0	0
$640$	10.0944	0.399016
$641$	−34.9376	−1.37995	−0.689976	−	0.723832i	$-0.742379\pi$
$641$	−34.9376	−1.37995	−0.689976	+	0.723832i	$0.742379\pi$
$642$	0	0
$643$	−19.7526	−0.778965	−0.389483	−	0.921034i	$-0.627346\pi$
$643$	−19.7526	−0.778965	−0.389483	+	0.921034i	$0.627346\pi$
$644$	−13.5675	−0.534633
$645$	0	0
$646$	−11.7394	−0.461882
$647$	11.1966	0.440185	0.220092	−	0.975479i	$-0.429364\pi$
$647$	11.1966	0.440185	0.220092	+	0.975479i	$0.429364\pi$
$648$	0	0
$649$	−10.0621	−0.394972
$650$	−2.85739	−0.112076
$651$	0	0
$652$	−17.2882	−0.677057
$653$	−24.2801	−0.950152	−0.475076	−	0.879945i	$-0.657579\pi$
$653$	−24.2801	−0.950152	−0.475076	+	0.879945i	$0.657579\pi$
$654$	0	0
$655$	2.39969	0.0937635
$656$	−15.6045	−0.609255
$657$	0	0
$658$	−30.5417	−1.19064
$659$	38.4569	1.49807	0.749034	−	0.662531i	$-0.230518\pi$
$659$	38.4569	1.49807	0.749034	+	0.662531i	$0.230518\pi$
$660$	0	0
$661$	22.3228	0.868257	0.434129	−	0.900851i	$-0.357056\pi$
$661$	22.3228	0.868257	0.434129	+	0.900851i	$0.357056\pi$
$662$	−0.576737	−0.0224155
$663$	0	0
$664$	3.85260	0.149510
$665$	13.5946	0.527177
$666$	0	0
$667$	−5.25052	−0.203301
$668$	−26.3967	−1.02132
$669$	0	0
$670$	4.96311	0.191742
$671$	1.00000	0.0386046
$672$	0	0
$673$	−15.3279	−0.590846	−0.295423	−	0.955367i	$-0.595461\pi$
$673$	−15.3279	−0.590846	−0.295423	+	0.955367i	$0.595461\pi$
$674$	−18.0898	−0.696791
$675$	0	0
$676$	19.4890	0.749576
$677$	−36.4164	−1.39960	−0.699799	−	0.714340i	$-0.746727\pi$
$677$	−36.4164	−1.39960	−0.699799	+	0.714340i	$0.746727\pi$
$678$	0	0
$679$	−59.4298	−2.28070
$680$	10.9827	0.421165
$681$	0	0
$682$	−5.30074	−0.202976
$683$	−20.0282	−0.766356	−0.383178	−	0.923674i	$-0.625170\pi$
$683$	−20.0282	−0.766356	−0.383178	+	0.923674i	$0.625170\pi$
$684$	0	0
$685$	−7.97282	−0.304626
$686$	15.7220	0.600269
$687$	0	0
$688$	17.5493	0.669062
$689$	13.1205	0.499853
$690$	0	0
$691$	0.630961	0.0240029	0.0120014	−	0.999928i	$-0.496180\pi$
$691$	0.630961	0.0240029	0.0120014	+	0.999928i	$0.496180\pi$
$692$	−17.5917	−0.668737
$693$	0	0
$694$	−0.568326	−0.0215733
$695$	11.3739	0.431436
$696$	0	0
$697$	−42.6247	−1.61453
$698$	2.60487	0.0985957
$699$	0	0
$700$	32.2302	1.21819
$701$	−35.0510	−1.32386	−0.661929	−	0.749567i	$-0.730262\pi$
$701$	−35.0510	−1.32386	−0.661929	+	0.749567i	$0.730262\pi$
$702$	0	0
$703$	6.43354	0.242645
$704$	1.49731	0.0564319
$705$	0	0
$706$	4.35673	0.163968
$707$	−4.27155	−0.160648
$708$	0	0
$709$	−8.54746	−0.321007	−0.160503	−	0.987035i	$-0.551312\pi$
$709$	−8.54746	−0.321007	−0.160503	+	0.987035i	$0.551312\pi$
$710$	−5.85724	−0.219818
$711$	0	0
$712$	−11.7151	−0.439044
$713$	16.9801	0.635911
$714$	0	0
$715$	1.06085	0.0396736
$716$	−8.98536	−0.335799
$717$	0	0
$718$	−5.69060	−0.212371
$719$	−14.9763	−0.558522	−0.279261	−	0.960215i	$-0.590089\pi$
$719$	−14.9763	−0.558522	−0.279261	+	0.960215i	$0.590089\pi$
$720$	0	0
$721$	3.86941	0.144104
$722$	−3.94562	−0.146841
$723$	0	0
$724$	−16.3603	−0.608025
$725$	12.4729	0.463230
$726$	0	0
$727$	9.53447	0.353614	0.176807	−	0.984246i	$-0.443423\pi$
$727$	9.53447	0.353614	0.176807	+	0.984246i	$0.443423\pi$
$728$	−11.2124	−0.415560
$729$	0	0
$730$	4.29024	0.158789
$731$	47.9371	1.77302
$732$	0	0
$733$	30.4593	1.12504	0.562521	−	0.826783i	$-0.309832\pi$
$733$	30.4593	1.12504	0.562521	+	0.826783i	$0.309832\pi$
$734$	17.2273	0.635870
$735$	0	0
$736$	9.54125	0.351695
$737$	10.1945	0.375520
$738$	0	0
$739$	−12.4841	−0.459237	−0.229618	−	0.973281i	$-0.573748\pi$
$739$	−12.4841	−0.459237	−0.229618	+	0.973281i	$0.573748\pi$
$740$	−2.75687	−0.101345
$741$	0	0
$742$	27.1119	0.995309
$743$	26.1928	0.960922	0.480461	−	0.877016i	$-0.340469\pi$
$743$	26.1928	0.960922	0.480461	+	0.877016i	$0.340469\pi$
$744$	0	0
$745$	−10.5196	−0.385408
$746$	14.7406	0.539693
$747$	0	0
$748$	10.3330	0.377813
$749$	−72.2008	−2.63816
$750$	0	0
$751$	−41.7868	−1.52482	−0.762412	−	0.647092i	$-0.775985\pi$
$751$	−41.7868	−1.52482	−0.762412	+	0.647092i	$0.775985\pi$
$752$	−27.2782	−0.994733
$753$	0	0
$754$	−1.98751	−0.0723809
$755$	−17.0336	−0.619914
$756$	0	0
$757$	12.4416	0.452197	0.226099	−	0.974104i	$-0.427403\pi$
$757$	12.4416	0.452197	0.226099	+	0.974104i	$0.427403\pi$
$758$	−2.60525	−0.0946271
$759$	0	0
$760$	−6.20014	−0.224903
$761$	0.915749	0.0331959	0.0165979	−	0.999862i	$-0.494716\pi$
$761$	0.915749	0.0331959	0.0165979	+	0.999862i	$0.494716\pi$
$762$	0	0
$763$	−47.4070	−1.71625
$764$	−36.9286	−1.33603
$765$	0	0
$766$	−10.1553	−0.366927
$767$	12.2012	0.440559
$768$	0	0
$769$	24.8887	0.897510	0.448755	−	0.893655i	$-0.351868\pi$
$769$	24.8887	0.897510	0.448755	+	0.893655i	$0.351868\pi$
$770$	2.19211	0.0789983
$771$	0	0
$772$	−35.0389	−1.26108
$773$	−15.3156	−0.550863	−0.275432	−	0.961321i	$-0.588821\pi$
$773$	−15.3156	−0.550863	−0.275432	+	0.961321i	$0.588821\pi$
$774$	0	0
$775$	−40.3371	−1.44895
$776$	27.1043	0.972988
$777$	0	0
$778$	−17.9850	−0.644793
$779$	24.0634	0.862159
$780$	0	0
$781$	−12.0311	−0.430508
$782$	6.06385	0.216843
$783$	0	0
$784$	29.7074	1.06098
$785$	−9.84915	−0.351531
$786$	0	0
$787$	−42.7248	−1.52298	−0.761488	−	0.648180i	$-0.775531\pi$
$787$	−42.7248	−1.52298	−0.761488	+	0.648180i	$0.775531\pi$
$788$	−14.7852	−0.526701
$789$	0	0
$790$	−6.80066	−0.241957
$791$	0.307361	0.0109285
$792$	0	0
$793$	−1.21259	−0.0430602
$794$	−18.1224	−0.643139
$795$	0	0
$796$	−38.1939	−1.35375
$797$	−51.8936	−1.83817	−0.919083	−	0.394065i	$-0.871068\pi$
$797$	−51.8936	−1.83817	−0.919083	+	0.394065i	$0.871068\pi$
$798$	0	0
$799$	−74.5120	−2.63605
$800$	−22.6657	−0.801353
$801$	0	0
$802$	−17.7819	−0.627900
$803$	8.81240	0.310983
$804$	0	0
$805$	−7.02211	−0.247497
$806$	6.42760	0.226403
$807$	0	0
$808$	1.94814	0.0685353
$809$	−35.8080	−1.25894	−0.629471	−	0.777024i	$-0.716728\pi$
$809$	−35.8080	−1.25894	−0.629471	+	0.777024i	$0.716728\pi$
$810$	0	0
$811$	52.0749	1.82860	0.914299	−	0.405041i	$-0.132743\pi$
$811$	52.0749	1.82860	0.914299	+	0.405041i	$0.132743\pi$
$812$	22.4183	0.786728
$813$	0	0
$814$	1.03740	0.0363608
$815$	−8.94784	−0.313429
$816$	0	0
$817$	−27.0624	−0.946793
$818$	−15.3787	−0.537703
$819$	0	0
$820$	−10.3115	−0.360094
$821$	−19.7668	−0.689865	−0.344932	−	0.938628i	$-0.612098\pi$
$821$	−19.7668	−0.689865	−0.344932	+	0.938628i	$0.612098\pi$
$822$	0	0
$823$	−3.36122	−0.117165	−0.0585824	−	0.998283i	$-0.518658\pi$
$823$	−3.36122	−0.117165	−0.0585824	+	0.998283i	$0.518658\pi$
$824$	−1.76473	−0.0614774
$825$	0	0
$826$	25.2122	0.877243
$827$	0.864402	0.0300582	0.0150291	−	0.999887i	$-0.495216\pi$
$827$	0.864402	0.0300582	0.0150291	+	0.999887i	$0.495216\pi$
$828$	0	0
$829$	−10.8633	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$829$	−10.8633	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$830$	0.913334	0.0317023
$831$	0	0
$832$	−1.81561	−0.0629451
$833$	81.1477	2.81160
$834$	0	0
$835$	−13.6621	−0.472797
$836$	−5.83340	−0.201752
$837$	0	0
$838$	−12.2246	−0.422291
$839$	−3.05864	−0.105596	−0.0527981	−	0.998605i	$-0.516814\pi$
$839$	−3.05864	−0.105596	−0.0527981	+	0.998605i	$0.516814\pi$
$840$	0	0
$841$	−20.3243	−0.700837
$842$	8.42889	0.290479
$843$	0	0
$844$	−11.9765	−0.412248
$845$	10.0869	0.347000
$846$	0	0
$847$	4.50273	0.154716
$848$	24.2149	0.831543
$849$	0	0
$850$	−14.4050	−0.494086
$851$	−3.32315	−0.113916
$852$	0	0
$853$	−16.6079	−0.568645	−0.284323	−	0.958729i	$-0.591769\pi$
$853$	−16.6079	−0.568645	−0.284323	+	0.958729i	$0.591769\pi$
$854$	−2.50565	−0.0857417
$855$	0	0
$856$	32.9289	1.12548
$857$	39.9964	1.36625	0.683126	−	0.730300i	$-0.260620\pi$
$857$	39.9964	1.36625	0.683126	+	0.730300i	$0.260620\pi$
$858$	0	0
$859$	1.61392	0.0550662	0.0275331	−	0.999621i	$-0.491235\pi$
$859$	1.61392	0.0550662	0.0275331	+	0.999621i	$0.491235\pi$
$860$	11.5967	0.395443
$861$	0	0
$862$	14.9472	0.509105
$863$	23.8576	0.812121	0.406060	−	0.913846i	$-0.366902\pi$
$863$	23.8576	0.812121	0.406060	+	0.913846i	$0.366902\pi$
$864$	0	0
$865$	−9.10494	−0.309577
$866$	8.25626	0.280559
$867$	0	0
$868$	−72.5006	−2.46083
$869$	−13.9690	−0.473865
$870$	0	0
$871$	−12.3617	−0.418862
$872$	21.6211	0.732182
$873$	0	0
$874$	−3.42328	−0.115794
$875$	36.3778	1.22980
$876$	0	0
$877$	−5.51352	−0.186178	−0.0930892	−	0.995658i	$-0.529674\pi$
$877$	−5.51352	−0.186178	−0.0930892	+	0.995658i	$0.529674\pi$
$878$	−3.14704	−0.106207
$879$	0	0
$880$	1.95788	0.0660000
$881$	35.5918	1.19912	0.599559	−	0.800330i	$-0.295343\pi$
$881$	35.5918	1.19912	0.599559	+	0.800330i	$0.295343\pi$
$882$	0	0
$883$	−32.3481	−1.08860	−0.544300	−	0.838891i	$-0.683204\pi$
$883$	−32.3481	−1.08860	−0.544300	+	0.838891i	$0.683204\pi$
$884$	−12.5297	−0.421419
$885$	0	0
$886$	6.55495	0.220218
$887$	−30.3059	−1.01757	−0.508786	−	0.860893i	$-0.669905\pi$
$887$	−30.3059	−1.01757	−0.508786	+	0.860893i	$0.669905\pi$
$888$	0	0
$889$	−2.78584	−0.0934339
$890$	−2.77731	−0.0930955
$891$	0	0
$892$	17.0778	0.571808
$893$	42.0650	1.40765
$894$	0	0
$895$	−4.65055	−0.155451
$896$	−51.9534	−1.73564
$897$	0	0
$898$	−15.0939	−0.503691
$899$	−28.0572	−0.935761
$900$	0	0
$901$	66.1445	2.20359
$902$	3.88018	0.129196
$903$	0	0
$904$	−0.140179	−0.00466228
$905$	−8.46759	−0.281472
$906$	0	0
$907$	−30.7190	−1.02001	−0.510005	−	0.860172i	$-0.670356\pi$
$907$	−30.7190	−1.02001	−0.510005	+	0.860172i	$0.670356\pi$
$908$	11.1745	0.370839
$909$	0	0
$910$	−2.65813	−0.0881160
$911$	27.8528	0.922805	0.461403	−	0.887191i	$-0.347346\pi$
$911$	27.8528	0.922805	0.461403	+	0.887191i	$0.347346\pi$
$912$	0	0
$913$	1.87604	0.0620880
$914$	13.7464	0.454692
$915$	0	0
$916$	23.8948	0.789506
$917$	−12.3506	−0.407853
$918$	0	0
$919$	27.8547	0.918842	0.459421	−	0.888219i	$-0.348057\pi$
$919$	27.8547	0.918842	0.459421	+	0.888219i	$0.348057\pi$
$920$	3.20260	0.105586
$921$	0	0
$922$	8.26081	0.272055
$923$	14.5888	0.480195
$924$	0	0
$925$	7.89431	0.259563
$926$	−10.6944	−0.351441
$927$	0	0
$928$	−15.7655	−0.517529
$929$	−39.3291	−1.29035	−0.645174	−	0.764036i	$-0.723215\pi$
$929$	−39.3291	−1.29035	−0.645174	+	0.764036i	$0.723215\pi$
$930$	0	0
$931$	−45.8111	−1.50140
$932$	1.46417	0.0479605
$933$	0	0
$934$	6.10396	0.199728
$935$	5.34806	0.174900
$936$	0	0
$937$	13.6631	0.446355	0.223177	−	0.974778i	$-0.428357\pi$
$937$	13.6631	0.446355	0.223177	+	0.974778i	$0.428357\pi$
$938$	−25.5440	−0.834040
$939$	0	0
$940$	−18.0255	−0.587927
$941$	22.7565	0.741841	0.370920	−	0.928665i	$-0.379042\pi$
$941$	22.7565	0.741841	0.370920	+	0.928665i	$0.379042\pi$
$942$	0	0
$943$	−12.4296	−0.404763
$944$	22.5181	0.732903
$945$	0	0
$946$	−4.36377	−0.141878
$947$	−39.2683	−1.27605	−0.638024	−	0.770017i	$-0.720248\pi$
$947$	−39.2683	−1.27605	−0.638024	+	0.770017i	$0.720248\pi$
$948$	0	0
$949$	−10.6858	−0.346876
$950$	8.13217	0.263842
$951$	0	0
$952$	−56.5251	−1.83199
$953$	−9.50916	−0.308032	−0.154016	−	0.988068i	$-0.549221\pi$
$953$	−9.50916	−0.308032	−0.154016	+	0.988068i	$0.549221\pi$
$954$	0	0
$955$	−19.1131	−0.618487
$956$	2.73505	0.0884579
$957$	0	0
$958$	−0.491574	−0.0158820
$959$	41.0342	1.32506
$960$	0	0
$961$	59.7369	1.92700
$962$	−1.25793	−0.0405574
$963$	0	0
$964$	−40.3388	−1.29922
$965$	−18.1351	−0.583789
$966$	0	0
$967$	21.7852	0.700566	0.350283	−	0.936644i	$-0.386085\pi$
$967$	21.7852	0.700566	0.350283	+	0.936644i	$0.386085\pi$
$968$	−2.05358	−0.0660044
$969$	0	0
$970$	6.42561	0.206314
$971$	−36.0043	−1.15543	−0.577717	−	0.816237i	$-0.696056\pi$
$971$	−36.0043	−1.15543	−0.577717	+	0.816237i	$0.696056\pi$
$972$	0	0
$973$	−58.5387	−1.87666
$974$	−2.20423	−0.0706282
$975$	0	0
$976$	−2.23791	−0.0716339
$977$	−1.76560	−0.0564867	−0.0282433	−	0.999601i	$-0.508991\pi$
$977$	−1.76560	−0.0564867	−0.0282433	+	0.999601i	$0.508991\pi$
$978$	0	0
$979$	−5.70475	−0.182325
$980$	19.6308	0.627082
$981$	0	0
$982$	1.62860	0.0519707
$983$	−27.6600	−0.882215	−0.441108	−	0.897454i	$-0.645414\pi$
$983$	−27.6600	−0.882215	−0.441108	+	0.897454i	$0.645414\pi$
$984$	0	0
$985$	−7.65238	−0.243825
$986$	−10.0196	−0.319090
$987$	0	0
$988$	7.07350	0.225038
$989$	13.9787	0.444497
$990$	0	0
$991$	−7.58233	−0.240861	−0.120430	−	0.992722i	$-0.538427\pi$
$991$	−7.58233	−0.240861	−0.120430	+	0.992722i	$0.538427\pi$
$992$	50.9856	1.61880
$993$	0	0
$994$	30.1458	0.956168
$995$	−19.7680	−0.626687
$996$	0	0
$997$	−40.7474	−1.29048	−0.645241	−	0.763979i	$-0.723243\pi$
$997$	−40.7474	−1.29048	−0.645241	+	0.763979i	$0.723243\pi$
$998$	−2.58499	−0.0818266
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.13		19
3.2	odd	2		671.2.a.c.1.7	✓	19
33.32	even	2		7381.2.a.i.1.13		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.7	✓	19	3.2	odd	2
6039.2.a.k.1.13		19	1.1	even	1	trivial
7381.2.a.i.1.13		19	33.32	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.556474 q^{2} -1.69034 q^{4} -0.874867 q^{5} +4.50273 q^{7} -2.05358 q^{8} +O(q^{10})\)	\(q+0.556474 q^{2} -1.69034 q^{4} -0.874867 q^{5} +4.50273 q^{7} -2.05358 q^{8} -0.486841 q^{10} -1.00000 q^{11} +1.21259 q^{13} +2.50565 q^{14} +2.23791 q^{16} +6.11300 q^{17} -3.45103 q^{19} +1.47882 q^{20} -0.556474 q^{22} +1.78258 q^{23} -4.23461 q^{25} +0.674772 q^{26} -7.61114 q^{28} -2.94546 q^{29} +9.52559 q^{31} +5.35249 q^{32} +3.40172 q^{34} -3.93930 q^{35} -1.86424 q^{37} -1.92041 q^{38} +1.79661 q^{40} -6.97280 q^{41} +7.84183 q^{43} +1.69034 q^{44} +0.991960 q^{46} -12.1891 q^{47} +13.2746 q^{49} -2.35645 q^{50} -2.04968 q^{52} +10.8203 q^{53} +0.874867 q^{55} -9.24670 q^{56} -1.63907 q^{58} +10.0621 q^{59} -1.00000 q^{61} +5.30074 q^{62} -1.49731 q^{64} -1.06085 q^{65} -10.1945 q^{67} -10.3330 q^{68} -2.19211 q^{70} +12.0311 q^{71} -8.81240 q^{73} -1.03740 q^{74} +5.83340 q^{76} -4.50273 q^{77} +13.9690 q^{79} -1.95788 q^{80} -3.88018 q^{82} -1.87604 q^{83} -5.34806 q^{85} +4.36377 q^{86} +2.05358 q^{88} +5.70475 q^{89} +5.45995 q^{91} -3.01316 q^{92} -6.78292 q^{94} +3.01919 q^{95} -13.1986 q^{97} +7.38697 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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