LMFDB - Embedded newform 6039.2.a.k.1.6

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			1.6
Root			$1.98166$ 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)

$f(q)$	$=$	$q-1.98166 q^{2} +1.92698 q^{4} -2.60057 q^{5} -2.18138 q^{7} +0.144707 q^{8} +O(q^{10})$	\(q-1.98166 q^{2} +1.92698 q^{4} -2.60057 q^{5} -2.18138 q^{7} +0.144707 q^{8} +5.15344 q^{10} -1.00000 q^{11} +0.444418 q^{13} +4.32276 q^{14} -4.14071 q^{16} -6.05961 q^{17} -1.92163 q^{19} -5.01124 q^{20} +1.98166 q^{22} +9.17660 q^{23} +1.76296 q^{25} -0.880686 q^{26} -4.20348 q^{28} -10.5575 q^{29} +3.13863 q^{31} +7.91607 q^{32} +12.0081 q^{34} +5.67284 q^{35} -1.07395 q^{37} +3.80802 q^{38} -0.376321 q^{40} -1.30812 q^{41} -12.5238 q^{43} -1.92698 q^{44} -18.1849 q^{46} +1.08337 q^{47} -2.24156 q^{49} -3.49358 q^{50} +0.856384 q^{52} +4.16225 q^{53} +2.60057 q^{55} -0.315662 q^{56} +20.9213 q^{58} -8.24320 q^{59} -1.00000 q^{61} -6.21970 q^{62} -7.40554 q^{64} -1.15574 q^{65} -10.2859 q^{67} -11.6767 q^{68} -11.2416 q^{70} +5.20811 q^{71} -0.283380 q^{73} +2.12820 q^{74} -3.70294 q^{76} +2.18138 q^{77} +14.7634 q^{79} +10.7682 q^{80} +2.59225 q^{82} -15.2641 q^{83} +15.7584 q^{85} +24.8180 q^{86} -0.144707 q^{88} -16.3022 q^{89} -0.969447 q^{91} +17.6831 q^{92} -2.14687 q^{94} +4.99733 q^{95} +4.78591 q^{97} +4.44202 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.98166	−1.40125	−0.700623	−	0.713532i	$-0.747094\pi$
$2$	−1.98166	−1.40125	−0.700623	+	0.713532i	$0.747094\pi$
$3$	0	0
$3$	0	0
$4$	1.92698	0.963488
$5$	−2.60057	−1.16301	−0.581505	−	0.813543i	$-0.697536\pi$
$5$	−2.60057	−1.16301	−0.581505	+	0.813543i	$0.697536\pi$
$6$	0	0
$7$	−2.18138	−0.824486	−0.412243	−	0.911074i	$-0.635254\pi$
$7$	−2.18138	−0.824486	−0.412243	+	0.911074i	$0.635254\pi$
$8$	0.144707	0.0511617
$9$	0	0
$10$	5.15344	1.62966
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.444418	0.123259	0.0616297	−	0.998099i	$-0.480370\pi$
$13$	0.444418	0.123259	0.0616297	+	0.998099i	$0.480370\pi$
$14$	4.32276	1.15531
$15$	0	0
$16$	−4.14071	−1.03518
$17$	−6.05961	−1.46967	−0.734836	−	0.678245i	$-0.762741\pi$
$17$	−6.05961	−1.46967	−0.734836	+	0.678245i	$0.762741\pi$
$18$	0	0
$19$	−1.92163	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$19$	−1.92163	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$20$	−5.01124	−1.12055
$21$	0	0
$22$	1.98166	0.422491
$23$	9.17660	1.91345	0.956726	−	0.290990i	$-0.0939845\pi$
$23$	9.17660	1.91345	0.956726	+	0.290990i	$0.0939845\pi$
$24$	0	0
$25$	1.76296	0.352591
$26$	−0.880686	−0.172717
$27$	0	0
$28$	−4.20348	−0.794382
$29$	−10.5575	−1.96048	−0.980238	−	0.197822i	$-0.936613\pi$
$29$	−10.5575	−1.96048	−0.980238	+	0.197822i	$0.936613\pi$
$30$	0	0
$31$	3.13863	0.563715	0.281858	−	0.959456i	$-0.409049\pi$
$31$	3.13863	0.563715	0.281858	+	0.959456i	$0.409049\pi$
$32$	7.91607	1.39938
$33$	0	0
$34$	12.0081	2.05937
$35$	5.67284	0.958885
$36$	0	0
$37$	−1.07395	−0.176556	−0.0882780	−	0.996096i	$-0.528136\pi$
$37$	−1.07395	−0.176556	−0.0882780	+	0.996096i	$0.528136\pi$
$38$	3.80802	0.617742
$39$	0	0
$40$	−0.376321	−0.0595015
$41$	−1.30812	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$41$	−1.30812	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$42$	0	0
$43$	−12.5238	−1.90987	−0.954934	−	0.296817i	$-0.904075\pi$
$43$	−12.5238	−1.90987	−0.954934	+	0.296817i	$0.904075\pi$
$44$	−1.92698	−0.290503
$45$	0	0
$46$	−18.1849	−2.68122
$47$	1.08337	0.158025	0.0790127	−	0.996874i	$-0.474823\pi$
$47$	1.08337	0.158025	0.0790127	+	0.996874i	$0.474823\pi$
$48$	0	0
$49$	−2.24156	−0.320223
$50$	−3.49358	−0.494067
$51$	0	0
$52$	0.856384	0.118759
$53$	4.16225	0.571729	0.285864	−	0.958270i	$-0.407719\pi$
$53$	4.16225	0.571729	0.285864	+	0.958270i	$0.407719\pi$
$54$	0	0
$55$	2.60057	0.350661
$56$	−0.315662	−0.0421821
$57$	0	0
$58$	20.9213	2.74711
$59$	−8.24320	−1.07317	−0.536587	−	0.843845i	$-0.680287\pi$
$59$	−8.24320	−1.07317	−0.536587	+	0.843845i	$0.680287\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−6.21970	−0.789903
$63$	0	0
$64$	−7.40554	−0.925692
$65$	−1.15574	−0.143352
$66$	0	0
$67$	−10.2859	−1.25662	−0.628309	−	0.777964i	$-0.716253\pi$
$67$	−10.2859	−1.25662	−0.628309	+	0.777964i	$0.716253\pi$
$68$	−11.6767	−1.41601
$69$	0	0
$70$	−11.2416	−1.34363
$71$	5.20811	0.618089	0.309044	−	0.951048i	$-0.399991\pi$
$71$	5.20811	0.618089	0.309044	+	0.951048i	$0.399991\pi$
$72$	0	0
$73$	−0.283380	−0.0331671	−0.0165835	−	0.999862i	$-0.505279\pi$
$73$	−0.283380	−0.0331671	−0.0165835	+	0.999862i	$0.505279\pi$
$74$	2.12820	0.247398
$75$	0	0
$76$	−3.70294	−0.424756
$77$	2.18138	0.248592
$78$	0	0
$79$	14.7634	1.66101	0.830505	−	0.557012i	$-0.188052\pi$
$79$	14.7634	1.66101	0.830505	+	0.557012i	$0.188052\pi$
$80$	10.7682	1.20392
$81$	0	0
$82$	2.59225	0.286266
$83$	−15.2641	−1.67546	−0.837728	−	0.546088i	$-0.816116\pi$
$83$	−15.2641	−1.67546	−0.837728	+	0.546088i	$0.816116\pi$
$84$	0	0
$85$	15.7584	1.70924
$86$	24.8180	2.67619
$87$	0	0
$88$	−0.144707	−0.0154258
$89$	−16.3022	−1.72803	−0.864015	−	0.503466i	$-0.832058\pi$
$89$	−16.3022	−1.72803	−0.864015	+	0.503466i	$0.832058\pi$
$90$	0	0
$91$	−0.969447	−0.101626
$92$	17.6831	1.84359
$93$	0	0
$94$	−2.14687	−0.221432
$95$	4.99733	0.512716
$96$	0	0
$97$	4.78591	0.485935	0.242968	−	0.970034i	$-0.421879\pi$
$97$	4.78591	0.485935	0.242968	+	0.970034i	$0.421879\pi$
$98$	4.44202	0.448712
$99$	0	0
$100$	3.39718	0.339718
$101$	−3.67611	−0.365786	−0.182893	−	0.983133i	$-0.558546\pi$
$101$	−3.67611	−0.365786	−0.182893	+	0.983133i	$0.558546\pi$
$102$	0	0
$103$	8.87836	0.874811	0.437405	−	0.899264i	$-0.355898\pi$
$103$	8.87836	0.874811	0.437405	+	0.899264i	$0.355898\pi$
$104$	0.0643105	0.00630616
$105$	0	0
$106$	−8.24816	−0.801132
$107$	−3.89110	−0.376167	−0.188083	−	0.982153i	$-0.560228\pi$
$107$	−3.89110	−0.376167	−0.188083	+	0.982153i	$0.560228\pi$
$108$	0	0
$109$	−8.52962	−0.816990	−0.408495	−	0.912761i	$-0.633946\pi$
$109$	−8.52962	−0.816990	−0.408495	+	0.912761i	$0.633946\pi$
$110$	−5.15344	−0.491362
$111$	0	0
$112$	9.03249	0.853490
$113$	−9.75866	−0.918018	−0.459009	−	0.888432i	$-0.651795\pi$
$113$	−9.75866	−0.918018	−0.459009	+	0.888432i	$0.651795\pi$
$114$	0	0
$115$	−23.8644	−2.22536
$116$	−20.3440	−1.88890
$117$	0	0
$118$	16.3352	1.50378
$119$	13.2183	1.21172
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.98166	0.179411
$123$	0	0
$124$	6.04807	0.543133
$125$	8.41815	0.752942
$126$	0	0
$127$	−20.5051	−1.81953	−0.909765	−	0.415123i	$-0.863739\pi$
$127$	−20.5051	−1.81953	−0.909765	+	0.415123i	$0.863739\pi$
$128$	−1.15688	−0.102255
$129$	0	0
$130$	2.29028	0.200871
$131$	−13.2639	−1.15887	−0.579437	−	0.815017i	$-0.696727\pi$
$131$	−13.2639	−1.15887	−0.579437	+	0.815017i	$0.696727\pi$
$132$	0	0
$133$	4.19181	0.363476
$134$	20.3831	1.76083
$135$	0	0
$136$	−0.876869	−0.0751909
$137$	9.00446	0.769303	0.384652	−	0.923062i	$-0.374322\pi$
$137$	9.00446	0.769303	0.384652	+	0.923062i	$0.374322\pi$
$138$	0	0
$139$	15.4273	1.30852	0.654262	−	0.756268i	$-0.272979\pi$
$139$	15.4273	1.30852	0.654262	+	0.756268i	$0.272979\pi$
$140$	10.9314	0.923874
$141$	0	0
$142$	−10.3207	−0.866094
$143$	−0.444418	−0.0371641
$144$	0	0
$145$	27.4555	2.28005
$146$	0.561562	0.0464752
$147$	0	0
$148$	−2.06947	−0.170110
$149$	−9.29642	−0.761592	−0.380796	−	0.924659i	$-0.624350\pi$
$149$	−9.29642	−0.761592	−0.380796	+	0.924659i	$0.624350\pi$
$150$	0	0
$151$	−8.26936	−0.672951	−0.336475	−	0.941692i	$-0.609235\pi$
$151$	−8.26936	−0.672951	−0.336475	+	0.941692i	$0.609235\pi$
$152$	−0.278074	−0.0225547
$153$	0	0
$154$	−4.32276	−0.348338
$155$	−8.16223	−0.655606
$156$	0	0
$157$	−13.5257	−1.07946	−0.539732	−	0.841837i	$-0.681475\pi$
$157$	−13.5257	−1.07946	−0.539732	+	0.841837i	$0.681475\pi$
$158$	−29.2560	−2.32748
$159$	0	0
$160$	−20.5863	−1.62749
$161$	−20.0177	−1.57761
$162$	0	0
$163$	−12.9831	−1.01692	−0.508458	−	0.861087i	$-0.669784\pi$
$163$	−12.9831	−1.01692	−0.508458	+	0.861087i	$0.669784\pi$
$164$	−2.52072	−0.196835
$165$	0	0
$166$	30.2483	2.34772
$167$	21.3647	1.65325	0.826624	−	0.562755i	$-0.190258\pi$
$167$	21.3647	1.65325	0.826624	+	0.562755i	$0.190258\pi$
$168$	0	0
$169$	−12.8025	−0.984807
$170$	−31.2279	−2.39507
$171$	0	0
$172$	−24.1332	−1.84014
$173$	−13.8496	−1.05297	−0.526483	−	0.850186i	$-0.676490\pi$
$173$	−13.8496	−1.05297	−0.526483	+	0.850186i	$0.676490\pi$
$174$	0	0
$175$	−3.84569	−0.290707
$176$	4.14071	0.312118
$177$	0	0
$178$	32.3054	2.42139
$179$	1.03648	0.0774700	0.0387350	−	0.999250i	$-0.487667\pi$
$179$	1.03648	0.0774700	0.0387350	+	0.999250i	$0.487667\pi$
$180$	0	0
$181$	−13.2287	−0.983281	−0.491640	−	0.870798i	$-0.663603\pi$
$181$	−13.2287	−0.983281	−0.491640	+	0.870798i	$0.663603\pi$
$182$	1.92111	0.142402
$183$	0	0
$184$	1.32792	0.0978954
$185$	2.79288	0.205336
$186$	0	0
$187$	6.05961	0.443123
$188$	2.08762	0.152256
$189$	0	0
$190$	−9.90302	−0.718440
$191$	8.86840	0.641695	0.320848	−	0.947131i	$-0.396032\pi$
$191$	8.86840	0.641695	0.320848	+	0.947131i	$0.396032\pi$
$192$	0	0
$193$	3.17960	0.228872	0.114436	−	0.993431i	$-0.463494\pi$
$193$	3.17960	0.228872	0.114436	+	0.993431i	$0.463494\pi$
$194$	−9.48404	−0.680915
$195$	0	0
$196$	−4.31944	−0.308532
$197$	−17.0357	−1.21374	−0.606871	−	0.794801i	$-0.707575\pi$
$197$	−17.0357	−1.21374	−0.606871	+	0.794801i	$0.707575\pi$
$198$	0	0
$199$	27.9216	1.97931	0.989656	−	0.143458i	$-0.0458221\pi$
$199$	27.9216	1.97931	0.989656	+	0.143458i	$0.0458221\pi$
$200$	0.255112	0.0180392
$201$	0	0
$202$	7.28480	0.512557
$203$	23.0299	1.61638
$204$	0	0
$205$	3.40185	0.237596
$206$	−17.5939	−1.22582
$207$	0	0
$208$	−1.84021	−0.127596
$209$	1.92163	0.132922
$210$	0	0
$211$	14.8812	1.02446	0.512232	−	0.858847i	$-0.328819\pi$
$211$	14.8812	1.02446	0.512232	+	0.858847i	$0.328819\pi$
$212$	8.02056	0.550854
$213$	0	0
$214$	7.71084	0.527102
$215$	32.5691	2.22120
$216$	0	0
$217$	−6.84656	−0.464775
$218$	16.9028	1.14480
$219$	0	0
$220$	5.01124	0.337857
$221$	−2.69300	−0.181151
$222$	0	0
$223$	24.3925	1.63344	0.816722	−	0.577032i	$-0.195789\pi$
$223$	24.3925	1.63344	0.816722	+	0.577032i	$0.195789\pi$
$224$	−17.2680	−1.15377
$225$	0	0
$226$	19.3383	1.28637
$227$	4.57809	0.303859	0.151929	−	0.988391i	$-0.451451\pi$
$227$	4.57809	0.303859	0.151929	+	0.988391i	$0.451451\pi$
$228$	0	0
$229$	−16.3940	−1.08335	−0.541673	−	0.840589i	$-0.682209\pi$
$229$	−16.3940	−1.08335	−0.541673	+	0.840589i	$0.682209\pi$
$230$	47.2911	3.11828
$231$	0	0
$232$	−1.52774	−0.100301
$233$	−10.0688	−0.659626	−0.329813	−	0.944046i	$-0.606986\pi$
$233$	−10.0688	−0.659626	−0.329813	+	0.944046i	$0.606986\pi$
$234$	0	0
$235$	−2.81737	−0.183785
$236$	−15.8845	−1.03399
$237$	0	0
$238$	−26.1943	−1.69792
$239$	−8.26608	−0.534688	−0.267344	−	0.963601i	$-0.586146\pi$
$239$	−8.26608	−0.534688	−0.267344	+	0.963601i	$0.586146\pi$
$240$	0	0
$241$	1.52656	0.0983343	0.0491672	−	0.998791i	$-0.484343\pi$
$241$	1.52656	0.0983343	0.0491672	+	0.998791i	$0.484343\pi$
$242$	−1.98166	−0.127386
$243$	0	0
$244$	−1.92698	−0.123362
$245$	5.82934	0.372423
$246$	0	0
$247$	−0.854008	−0.0543392
$248$	0.454182	0.0288406
$249$	0	0
$250$	−16.6819	−1.05506
$251$	−21.1073	−1.33228	−0.666142	−	0.745825i	$-0.732055\pi$
$251$	−21.1073	−1.33228	−0.666142	+	0.745825i	$0.732055\pi$
$252$	0	0
$253$	−9.17660	−0.576928
$254$	40.6341	2.54961
$255$	0	0
$256$	17.1036	1.06898
$257$	−8.94976	−0.558271	−0.279135	−	0.960252i	$-0.590048\pi$
$257$	−8.94976	−0.558271	−0.279135	+	0.960252i	$0.590048\pi$
$258$	0	0
$259$	2.34269	0.145568
$260$	−2.22708	−0.138118
$261$	0	0
$262$	26.2846	1.62387
$263$	29.0355	1.79041	0.895204	−	0.445657i	$-0.147030\pi$
$263$	29.0355	1.79041	0.895204	+	0.445657i	$0.147030\pi$
$264$	0	0
$265$	−10.8242	−0.664926
$266$	−8.30675	−0.509320
$267$	0	0
$268$	−19.8206	−1.21074
$269$	0.271779	0.0165706	0.00828532	−	0.999966i	$-0.497363\pi$
$269$	0.271779	0.0165706	0.00828532	+	0.999966i	$0.497363\pi$
$270$	0	0
$271$	−9.91921	−0.602549	−0.301275	−	0.953537i	$-0.597412\pi$
$271$	−9.91921	−0.602549	−0.301275	+	0.953537i	$0.597412\pi$
$272$	25.0911	1.52137
$273$	0	0
$274$	−17.8438	−1.07798
$275$	−1.76296	−0.106310
$276$	0	0
$277$	−2.47418	−0.148659	−0.0743295	−	0.997234i	$-0.523682\pi$
$277$	−2.47418	−0.148659	−0.0743295	+	0.997234i	$0.523682\pi$
$278$	−30.5716	−1.83356
$279$	0	0
$280$	0.820900	0.0490581
$281$	12.2927	0.733321	0.366660	−	0.930355i	$-0.380501\pi$
$281$	12.2927	0.733321	0.366660	+	0.930355i	$0.380501\pi$
$282$	0	0
$283$	12.7133	0.755725	0.377862	−	0.925862i	$-0.376659\pi$
$283$	12.7133	0.755725	0.377862	+	0.925862i	$0.376659\pi$
$284$	10.0359	0.595521
$285$	0	0
$286$	0.880686	0.0520761
$287$	2.85351	0.168437
$288$	0	0
$289$	19.7189	1.15994
$290$	−54.4074	−3.19491
$291$	0	0
$292$	−0.546066	−0.0319561
$293$	10.1272	0.591637	0.295818	−	0.955244i	$-0.404408\pi$
$293$	10.1272	0.591637	0.295818	+	0.955244i	$0.404408\pi$
$294$	0	0
$295$	21.4370	1.24811
$296$	−0.155408	−0.00903290
$297$	0	0
$298$	18.4223	1.06718
$299$	4.07825	0.235851
$300$	0	0
$301$	27.3193	1.57466
$302$	16.3871	0.942969
$303$	0	0
$304$	7.95692	0.456361
$305$	2.60057	0.148908
$306$	0	0
$307$	−4.15427	−0.237097	−0.118548	−	0.992948i	$-0.537824\pi$
$307$	−4.15427	−0.237097	−0.118548	+	0.992948i	$0.537824\pi$
$308$	4.20348	0.239515
$309$	0	0
$310$	16.1748	0.918665
$311$	−10.2275	−0.579948	−0.289974	−	0.957035i	$-0.593647\pi$
$311$	−10.2275	−0.579948	−0.289974	+	0.957035i	$0.593647\pi$
$312$	0	0
$313$	−8.72686	−0.493271	−0.246636	−	0.969108i	$-0.579325\pi$
$313$	−8.72686	−0.493271	−0.246636	+	0.969108i	$0.579325\pi$
$314$	26.8033	1.51260
$315$	0	0
$316$	28.4487	1.60036
$317$	0.569591	0.0319914	0.0159957	−	0.999872i	$-0.494908\pi$
$317$	0.569591	0.0319914	0.0159957	+	0.999872i	$0.494908\pi$
$318$	0	0
$319$	10.5575	0.591106
$320$	19.2586	1.07659
$321$	0	0
$322$	39.6682	2.21062
$323$	11.6443	0.647908
$324$	0	0
$325$	0.783490	0.0434602
$326$	25.7281	1.42495
$327$	0	0
$328$	−0.189294	−0.0104520
$329$	−2.36324	−0.130290
$330$	0	0
$331$	−20.0001	−1.09930	−0.549652	−	0.835394i	$-0.685240\pi$
$331$	−20.0001	−1.09930	−0.549652	+	0.835394i	$0.685240\pi$
$332$	−29.4136	−1.61428
$333$	0	0
$334$	−42.3375	−2.31661
$335$	26.7491	1.46146
$336$	0	0
$337$	27.7992	1.51432	0.757161	−	0.653229i	$-0.226586\pi$
$337$	27.7992	1.51432	0.757161	+	0.653229i	$0.226586\pi$
$338$	25.3702	1.37996
$339$	0	0
$340$	30.3661	1.64684
$341$	−3.13863	−0.169966
$342$	0	0
$343$	20.1594	1.08851
$344$	−1.81229	−0.0977121
$345$	0	0
$346$	27.4452	1.47546
$347$	−14.0784	−0.755770	−0.377885	−	0.925853i	$-0.623349\pi$
$347$	−14.0784	−0.755770	−0.377885	+	0.925853i	$0.623349\pi$
$348$	0	0
$349$	26.9981	1.44517	0.722586	−	0.691281i	$-0.242953\pi$
$349$	26.9981	1.44517	0.722586	+	0.691281i	$0.242953\pi$
$350$	7.62084	0.407351
$351$	0	0
$352$	−7.91607	−0.421928
$353$	7.06670	0.376122	0.188061	−	0.982157i	$-0.439780\pi$
$353$	7.06670	0.376122	0.188061	+	0.982157i	$0.439780\pi$
$354$	0	0
$355$	−13.5440	−0.718843
$356$	−31.4140	−1.66494
$357$	0	0
$358$	−2.05395	−0.108554
$359$	1.11238	0.0587090	0.0293545	−	0.999569i	$-0.490655\pi$
$359$	1.11238	0.0587090	0.0293545	+	0.999569i	$0.490655\pi$
$360$	0	0
$361$	−15.3073	−0.805649
$362$	26.2148	1.37782
$363$	0	0
$364$	−1.86810	−0.0979151
$365$	0.736949	0.0385737
$366$	0	0
$367$	−15.6511	−0.816979	−0.408489	−	0.912763i	$-0.633944\pi$
$367$	−15.6511	−0.816979	−0.408489	+	0.912763i	$0.633944\pi$
$368$	−37.9977	−1.98076
$369$	0	0
$370$	−5.53453	−0.287727
$371$	−9.07946	−0.471382
$372$	0	0
$373$	−8.11997	−0.420436	−0.210218	−	0.977655i	$-0.567417\pi$
$373$	−8.11997	−0.420436	−0.210218	+	0.977655i	$0.567417\pi$
$374$	−12.0081	−0.620924
$375$	0	0
$376$	0.156771	0.00808484
$377$	−4.69194	−0.241647
$378$	0	0
$379$	35.5452	1.82583	0.912917	−	0.408145i	$-0.133824\pi$
$379$	35.5452	1.82583	0.912917	+	0.408145i	$0.133824\pi$
$380$	9.62975	0.493996
$381$	0	0
$382$	−17.5742	−0.899172
$383$	−25.4279	−1.29931	−0.649653	−	0.760231i	$-0.725086\pi$
$383$	−25.4279	−1.29931	−0.649653	+	0.760231i	$0.725086\pi$
$384$	0	0
$385$	−5.67284	−0.289115
$386$	−6.30088	−0.320706
$387$	0	0
$388$	9.22233	0.468193
$389$	2.40965	0.122174	0.0610872	−	0.998132i	$-0.480543\pi$
$389$	2.40965	0.122174	0.0610872	+	0.998132i	$0.480543\pi$
$390$	0	0
$391$	−55.6066	−2.81215
$392$	−0.324370	−0.0163832
$393$	0	0
$394$	33.7589	1.70075
$395$	−38.3932	−1.93177
$396$	0	0
$397$	8.86157	0.444750	0.222375	−	0.974961i	$-0.428619\pi$
$397$	8.86157	0.444750	0.222375	+	0.974961i	$0.428619\pi$
$398$	−55.3312	−2.77350
$399$	0	0
$400$	−7.29990	−0.364995
$401$	23.3978	1.16843	0.584214	−	0.811600i	$-0.301403\pi$
$401$	23.3978	1.16843	0.584214	+	0.811600i	$0.301403\pi$
$402$	0	0
$403$	1.39487	0.0694832
$404$	−7.08378	−0.352431
$405$	0	0
$406$	−45.6375	−2.26495
$407$	1.07395	0.0532337
$408$	0	0
$409$	−18.4508	−0.912336	−0.456168	−	0.889894i	$-0.650778\pi$
$409$	−18.4508	−0.912336	−0.456168	+	0.889894i	$0.650778\pi$
$410$	−6.74132	−0.332930
$411$	0	0
$412$	17.1084	0.842870
$413$	17.9816	0.884816
$414$	0	0
$415$	39.6954	1.94857
$416$	3.51805	0.172486
$417$	0	0
$418$	−3.80802	−0.186256
$419$	22.7893	1.11333	0.556665	−	0.830737i	$-0.312081\pi$
$419$	22.7893	1.11333	0.556665	+	0.830737i	$0.312081\pi$
$420$	0	0
$421$	16.9851	0.827802	0.413901	−	0.910322i	$-0.364166\pi$
$421$	16.9851	0.827802	0.413901	+	0.910322i	$0.364166\pi$
$422$	−29.4895	−1.43553
$423$	0	0
$424$	0.602307	0.0292506
$425$	−10.6828	−0.518194
$426$	0	0
$427$	2.18138	0.105565
$428$	−7.49806	−0.362432
$429$	0	0
$430$	−64.5409	−3.11244
$431$	26.8912	1.29531	0.647653	−	0.761936i	$-0.275751\pi$
$431$	26.8912	1.29531	0.647653	+	0.761936i	$0.275751\pi$
$432$	0	0
$433$	−4.94690	−0.237733	−0.118866	−	0.992910i	$-0.537926\pi$
$433$	−4.94690	−0.237733	−0.118866	+	0.992910i	$0.537926\pi$
$434$	13.5676	0.651264
$435$	0	0
$436$	−16.4364	−0.787160
$437$	−17.6340	−0.843550
$438$	0	0
$439$	−16.1331	−0.769989	−0.384995	−	0.922919i	$-0.625797\pi$
$439$	−16.1331	−0.769989	−0.384995	+	0.922919i	$0.625797\pi$
$440$	0.376321	0.0179404
$441$	0	0
$442$	5.33662	0.253837
$443$	−19.9786	−0.949213	−0.474607	−	0.880198i	$-0.657410\pi$
$443$	−19.9786	−0.949213	−0.474607	+	0.880198i	$0.657410\pi$
$444$	0	0
$445$	42.3950	2.00972
$446$	−48.3377	−2.28886
$447$	0	0
$448$	16.1543	0.763220
$449$	−3.49232	−0.164813	−0.0824064	−	0.996599i	$-0.526261\pi$
$449$	−3.49232	−0.164813	−0.0824064	+	0.996599i	$0.526261\pi$
$450$	0	0
$451$	1.30812	0.0615969
$452$	−18.8047	−0.884499
$453$	0	0
$454$	−9.07222	−0.425780
$455$	2.52111	0.118192
$456$	0	0
$457$	−8.64120	−0.404218	−0.202109	−	0.979363i	$-0.564780\pi$
$457$	−8.64120	−0.404218	−0.202109	+	0.979363i	$0.564780\pi$
$458$	32.4873	1.51803
$459$	0	0
$460$	−45.9861	−2.14411
$461$	−3.04671	−0.141899	−0.0709497	−	0.997480i	$-0.522603\pi$
$461$	−3.04671	−0.141899	−0.0709497	+	0.997480i	$0.522603\pi$
$462$	0	0
$463$	−8.03359	−0.373353	−0.186676	−	0.982421i	$-0.559772\pi$
$463$	−8.03359	−0.373353	−0.186676	+	0.982421i	$0.559772\pi$
$464$	43.7155	2.02944
$465$	0	0
$466$	19.9528	0.924298
$467$	−12.0195	−0.556195	−0.278098	−	0.960553i	$-0.589704\pi$
$467$	−12.0195	−0.556195	−0.278098	+	0.960553i	$0.589704\pi$
$468$	0	0
$469$	22.4374	1.03606
$470$	5.58307	0.257528
$471$	0	0
$472$	−1.19285	−0.0549054
$473$	12.5238	0.575847
$474$	0	0
$475$	−3.38775	−0.155441
$476$	25.4714	1.16748
$477$	0	0
$478$	16.3806	0.749230
$479$	21.3120	0.973768	0.486884	−	0.873467i	$-0.338133\pi$
$479$	21.3120	0.973768	0.486884	+	0.873467i	$0.338133\pi$
$480$	0	0
$481$	−0.477282	−0.0217622
$482$	−3.02512	−0.137791
$483$	0	0
$484$	1.92698	0.0875899
$485$	−12.4461	−0.565147
$486$	0	0
$487$	−37.5771	−1.70278	−0.851391	−	0.524532i	$-0.824240\pi$
$487$	−37.5771	−1.70278	−0.851391	+	0.524532i	$0.824240\pi$
$488$	−0.144707	−0.00655058
$489$	0	0
$490$	−11.5518	−0.521856
$491$	34.9908	1.57911	0.789557	−	0.613677i	$-0.210310\pi$
$491$	34.9908	1.57911	0.789557	+	0.613677i	$0.210310\pi$
$492$	0	0
$493$	63.9743	2.88126
$494$	1.69235	0.0761426
$495$	0	0
$496$	−12.9962	−0.583546
$497$	−11.3609	−0.509605
$498$	0	0
$499$	33.3039	1.49089	0.745445	−	0.666568i	$-0.232237\pi$
$499$	33.3039	1.49089	0.745445	+	0.666568i	$0.232237\pi$
$500$	16.2216	0.725451
$501$	0	0
$502$	41.8276	1.86686
$503$	−5.22511	−0.232976	−0.116488	−	0.993192i	$-0.537164\pi$
$503$	−5.22511	−0.232976	−0.116488	+	0.993192i	$0.537164\pi$
$504$	0	0
$505$	9.55997	0.425413
$506$	18.1849	0.808417
$507$	0	0
$508$	−39.5128	−1.75310
$509$	22.3134	0.989026	0.494513	−	0.869170i	$-0.335346\pi$
$509$	22.3134	0.989026	0.494513	+	0.869170i	$0.335346\pi$
$510$	0	0
$511$	0.618160	0.0273458
$512$	−31.5798	−1.39564
$513$	0	0
$514$	17.7354	0.782274
$515$	−23.0888	−1.01741
$516$	0	0
$517$	−1.08337	−0.0476465
$518$	−4.64242	−0.203976
$519$	0	0
$520$	−0.167244	−0.00733412
$521$	−6.24416	−0.273562	−0.136781	−	0.990601i	$-0.543676\pi$
$521$	−6.24416	−0.273562	−0.136781	+	0.990601i	$0.543676\pi$
$522$	0	0
$523$	−12.1521	−0.531372	−0.265686	−	0.964060i	$-0.585598\pi$
$523$	−12.1521	−0.531372	−0.265686	+	0.964060i	$0.585598\pi$
$524$	−25.5593	−1.11656
$525$	0	0
$526$	−57.5386	−2.50880
$527$	−19.0189	−0.828476
$528$	0	0
$529$	61.2099	2.66130
$530$	21.4499	0.931725
$531$	0	0
$532$	8.07753	0.350205
$533$	−0.581352	−0.0251812
$534$	0	0
$535$	10.1191	0.437486
$536$	−1.48844	−0.0642907
$537$	0	0
$538$	−0.538573	−0.0232195
$539$	2.24156	0.0965510
$540$	0	0
$541$	−16.4664	−0.707946	−0.353973	−	0.935256i	$-0.615170\pi$
$541$	−16.4664	−0.707946	−0.353973	+	0.935256i	$0.615170\pi$
$542$	19.6565	0.844319
$543$	0	0
$544$	−47.9683	−2.05663
$545$	22.1819	0.950167
$546$	0	0
$547$	5.93546	0.253782	0.126891	−	0.991917i	$-0.459500\pi$
$547$	5.93546	0.253782	0.126891	+	0.991917i	$0.459500\pi$
$548$	17.3514	0.741215
$549$	0	0
$550$	3.49358	0.148967
$551$	20.2876	0.864281
$552$	0	0
$553$	−32.2046	−1.36948
$554$	4.90298	0.208308
$555$	0	0
$556$	29.7280	1.26075
$557$	−11.3990	−0.482990	−0.241495	−	0.970402i	$-0.577638\pi$
$557$	−11.3990	−0.482990	−0.241495	+	0.970402i	$0.577638\pi$
$558$	0	0
$559$	−5.56583	−0.235409
$560$	−23.4896	−0.992617
$561$	0	0
$562$	−24.3599	−1.02756
$563$	26.2922	1.10808	0.554042	−	0.832489i	$-0.313085\pi$
$563$	26.2922	1.10808	0.554042	+	0.832489i	$0.313085\pi$
$564$	0	0
$565$	25.3781	1.06766
$566$	−25.1934	−1.05896
$567$	0	0
$568$	0.753650	0.0316225
$569$	40.8755	1.71359	0.856795	−	0.515658i	$-0.172452\pi$
$569$	40.8755	1.71359	0.856795	+	0.515658i	$0.172452\pi$
$570$	0	0
$571$	3.13722	0.131288	0.0656442	−	0.997843i	$-0.479090\pi$
$571$	3.13722	0.131288	0.0656442	+	0.997843i	$0.479090\pi$
$572$	−0.856384	−0.0358072
$573$	0	0
$574$	−5.65469	−0.236022
$575$	16.1779	0.674667
$576$	0	0
$577$	35.1099	1.46164	0.730822	−	0.682569i	$-0.239137\pi$
$577$	35.1099	1.46164	0.730822	+	0.682569i	$0.239137\pi$
$578$	−39.0762	−1.62535
$579$	0	0
$580$	52.9060	2.19680
$581$	33.2969	1.38139
$582$	0	0
$583$	−4.16225	−0.172383
$584$	−0.0410071	−0.00169688
$585$	0	0
$586$	−20.0686	−0.829028
$587$	11.7853	0.486430	0.243215	−	0.969972i	$-0.421798\pi$
$587$	11.7853	0.486430	0.243215	+	0.969972i	$0.421798\pi$
$588$	0	0
$589$	−6.03129	−0.248515
$590$	−42.4809	−1.74891
$591$	0	0
$592$	4.44691	0.182767
$593$	14.0578	0.577284	0.288642	−	0.957437i	$-0.406796\pi$
$593$	14.0578	0.577284	0.288642	+	0.957437i	$0.406796\pi$
$594$	0	0
$595$	−34.3752	−1.40925
$596$	−17.9140	−0.733785
$597$	0	0
$598$	−8.08170	−0.330485
$599$	23.2077	0.948243	0.474121	−	0.880460i	$-0.342766\pi$
$599$	23.2077	0.948243	0.474121	+	0.880460i	$0.342766\pi$
$600$	0	0
$601$	−9.72040	−0.396503	−0.198252	−	0.980151i	$-0.563526\pi$
$601$	−9.72040	−0.396503	−0.198252	+	0.980151i	$0.563526\pi$
$602$	−54.1376	−2.20648
$603$	0	0
$604$	−15.9349	−0.648380
$605$	−2.60057	−0.105728
$606$	0	0
$607$	−26.1875	−1.06292	−0.531459	−	0.847084i	$-0.678356\pi$
$607$	−26.1875	−1.06292	−0.531459	+	0.847084i	$0.678356\pi$
$608$	−15.2118	−0.616919
$609$	0	0
$610$	−5.15344	−0.208657
$611$	0.481468	0.0194781
$612$	0	0
$613$	−1.59718	−0.0645094	−0.0322547	−	0.999480i	$-0.510269\pi$
$613$	−1.59718	−0.0645094	−0.0322547	+	0.999480i	$0.510269\pi$
$614$	8.23234	0.332230
$615$	0	0
$616$	0.315662	0.0127184
$617$	4.95280	0.199392	0.0996962	−	0.995018i	$-0.468213\pi$
$617$	4.95280	0.199392	0.0996962	+	0.995018i	$0.468213\pi$
$618$	0	0
$619$	19.6334	0.789132	0.394566	−	0.918868i	$-0.370895\pi$
$619$	19.6334	0.789132	0.394566	+	0.918868i	$0.370895\pi$
$620$	−15.7284	−0.631669
$621$	0	0
$622$	20.2674	0.812649
$623$	35.5614	1.42474
$624$	0	0
$625$	−30.7068	−1.22827
$626$	17.2937	0.691194
$627$	0	0
$628$	−26.0636	−1.04005
$629$	6.50771	0.259479
$630$	0	0
$631$	15.1352	0.602521	0.301261	−	0.953542i	$-0.402593\pi$
$631$	15.1352	0.602521	0.301261	+	0.953542i	$0.402593\pi$
$632$	2.13636	0.0849800
$633$	0	0
$634$	−1.12874	−0.0448278
$635$	53.3248	2.11613
$636$	0	0
$637$	−0.996192	−0.0394706
$638$	−20.9213	−0.828284
$639$	0	0
$640$	3.00856	0.118924
$641$	40.4127	1.59621	0.798104	−	0.602520i	$-0.205837\pi$
$641$	40.4127	1.59621	0.798104	+	0.602520i	$0.205837\pi$
$642$	0	0
$643$	−16.3198	−0.643589	−0.321794	−	0.946810i	$-0.604286\pi$
$643$	−16.3198	−0.643589	−0.321794	+	0.946810i	$0.604286\pi$
$644$	−38.5736	−1.52001
$645$	0	0
$646$	−23.0751	−0.907878
$647$	22.2041	0.872933	0.436466	−	0.899721i	$-0.356230\pi$
$647$	22.2041	0.872933	0.436466	+	0.899721i	$0.356230\pi$
$648$	0	0
$649$	8.24320	0.323574
$650$	−1.55261	−0.0608984
$651$	0	0
$652$	−25.0182	−0.979787
$653$	−15.9029	−0.622327	−0.311163	−	0.950356i	$-0.600719\pi$
$653$	−15.9029	−0.622327	−0.311163	+	0.950356i	$0.600719\pi$
$654$	0	0
$655$	34.4937	1.34778
$656$	5.41655	0.211481
$657$	0	0
$658$	4.68314	0.182568
$659$	−2.35249	−0.0916402	−0.0458201	−	0.998950i	$-0.514590\pi$
$659$	−2.35249	−0.0916402	−0.0458201	+	0.998950i	$0.514590\pi$
$660$	0	0
$661$	−31.3644	−1.21994	−0.609968	−	0.792426i	$-0.708818\pi$
$661$	−31.3644	−1.21994	−0.609968	+	0.792426i	$0.708818\pi$
$662$	39.6333	1.54039
$663$	0	0
$664$	−2.20883	−0.0857191
$665$	−10.9011	−0.422727
$666$	0	0
$667$	−96.8818	−3.75128
$668$	41.1692	1.59289
$669$	0	0
$670$	−53.0076	−2.04786
$671$	1.00000	0.0386046
$672$	0	0
$673$	−33.8125	−1.30338	−0.651688	−	0.758487i	$-0.725939\pi$
$673$	−33.8125	−1.30338	−0.651688	+	0.758487i	$0.725939\pi$
$674$	−55.0886	−2.12194
$675$	0	0
$676$	−24.6701	−0.948850
$677$	−19.5191	−0.750180	−0.375090	−	0.926988i	$-0.622388\pi$
$677$	−19.5191	−0.750180	−0.375090	+	0.926988i	$0.622388\pi$
$678$	0	0
$679$	−10.4399	−0.400647
$680$	2.28036	0.0874477
$681$	0	0
$682$	6.21970	0.238165
$683$	34.3022	1.31254	0.656268	−	0.754528i	$-0.272134\pi$
$683$	34.3022	1.31254	0.656268	+	0.754528i	$0.272134\pi$
$684$	0	0
$685$	−23.4167	−0.894707
$686$	−39.9491	−1.52526
$687$	0	0
$688$	51.8577	1.97705
$689$	1.84978	0.0704710
$690$	0	0
$691$	−11.5937	−0.441046	−0.220523	−	0.975382i	$-0.570776\pi$
$691$	−11.5937	−0.441046	−0.220523	+	0.975382i	$0.570776\pi$
$692$	−26.6879	−1.01452
$693$	0	0
$694$	27.8987	1.05902
$695$	−40.1197	−1.52183
$696$	0	0
$697$	7.92670	0.300245
$698$	−53.5010	−2.02504
$699$	0	0
$700$	−7.41055	−0.280092
$701$	−6.50743	−0.245782	−0.122891	−	0.992420i	$-0.539217\pi$
$701$	−6.50743	−0.245782	−0.122891	+	0.992420i	$0.539217\pi$
$702$	0	0
$703$	2.06373	0.0778352
$704$	7.40554	0.279107
$705$	0	0
$706$	−14.0038	−0.527040
$707$	8.01900	0.301586
$708$	0	0
$709$	3.54453	0.133118	0.0665588	−	0.997783i	$-0.478798\pi$
$709$	3.54453	0.133118	0.0665588	+	0.997783i	$0.478798\pi$
$710$	26.8397	1.00728
$711$	0	0
$712$	−2.35904	−0.0884089
$713$	28.8020	1.07864
$714$	0	0
$715$	1.15574	0.0432222
$716$	1.99727	0.0746414
$717$	0	0
$718$	−2.20435	−0.0822657
$719$	40.7391	1.51931	0.759656	−	0.650325i	$-0.225367\pi$
$719$	40.7391	1.51931	0.759656	+	0.650325i	$0.225367\pi$
$720$	0	0
$721$	−19.3671	−0.721269
$722$	30.3339	1.12891
$723$	0	0
$724$	−25.4914	−0.947380
$725$	−18.6124	−0.691247
$726$	0	0
$727$	−34.3861	−1.27531	−0.637654	−	0.770323i	$-0.720095\pi$
$727$	−34.3861	−1.27531	−0.637654	+	0.770323i	$0.720095\pi$
$728$	−0.140286	−0.00519934
$729$	0	0
$730$	−1.46038	−0.0540512
$731$	75.8897	2.80688
$732$	0	0
$733$	24.0134	0.886956	0.443478	−	0.896285i	$-0.353744\pi$
$733$	24.0134	0.886956	0.443478	+	0.896285i	$0.353744\pi$
$734$	31.0151	1.14479
$735$	0	0
$736$	72.6426	2.67764
$737$	10.2859	0.378885
$738$	0	0
$739$	49.5917	1.82426	0.912131	−	0.409899i	$-0.134436\pi$
$739$	49.5917	1.82426	0.912131	+	0.409899i	$0.134436\pi$
$740$	5.38181	0.197839
$741$	0	0
$742$	17.9924	0.660522
$743$	13.9582	0.512075	0.256038	−	0.966667i	$-0.417583\pi$
$743$	13.9582	0.512075	0.256038	+	0.966667i	$0.417583\pi$
$744$	0	0
$745$	24.1760	0.885739
$746$	16.0910	0.589134
$747$	0	0
$748$	11.6767	0.426944
$749$	8.48799	0.310144
$750$	0	0
$751$	6.81173	0.248564	0.124282	−	0.992247i	$-0.460337\pi$
$751$	6.81173	0.248564	0.124282	+	0.992247i	$0.460337\pi$
$752$	−4.48592	−0.163584
$753$	0	0
$754$	9.29783	0.338607
$755$	21.5050	0.782648
$756$	0	0
$757$	21.4347	0.779057	0.389529	−	0.921014i	$-0.372638\pi$
$757$	21.4347	0.779057	0.389529	+	0.921014i	$0.372638\pi$
$758$	−70.4385	−2.55844
$759$	0	0
$760$	0.723149	0.0262314
$761$	−25.6533	−0.929931	−0.464965	−	0.885329i	$-0.653933\pi$
$761$	−25.6533	−0.929931	−0.464965	+	0.885329i	$0.653933\pi$
$762$	0	0
$763$	18.6064	0.673596
$764$	17.0892	0.618266
$765$	0	0
$766$	50.3895	1.82065
$767$	−3.66343	−0.132279
$768$	0	0
$769$	46.6407	1.68191	0.840953	−	0.541108i	$-0.181995\pi$
$769$	46.6407	1.68191	0.840953	+	0.541108i	$0.181995\pi$
$770$	11.2416	0.405121
$771$	0	0
$772$	6.12701	0.220516
$773$	25.4354	0.914849	0.457425	−	0.889248i	$-0.348772\pi$
$773$	25.4354	0.914849	0.457425	+	0.889248i	$0.348772\pi$
$774$	0	0
$775$	5.53328	0.198761
$776$	0.692555	0.0248613
$777$	0	0
$778$	−4.77512	−0.171196
$779$	2.51372	0.0900635
$780$	0	0
$781$	−5.20811	−0.186361
$782$	110.193	3.94051
$783$	0	0
$784$	9.28168	0.331488
$785$	35.1744	1.25543
$786$	0	0
$787$	3.14255	0.112020	0.0560099	−	0.998430i	$-0.482162\pi$
$787$	3.14255	0.112020	0.0560099	+	0.998430i	$0.482162\pi$
$788$	−32.8273	−1.16943
$789$	0	0
$790$	76.0822	2.70688
$791$	21.2874	0.756892
$792$	0	0
$793$	−0.444418	−0.0157818
$794$	−17.5606	−0.623203
$795$	0	0
$796$	53.8044	1.90705
$797$	24.9086	0.882309	0.441155	−	0.897431i	$-0.354569\pi$
$797$	24.9086	0.882309	0.441155	+	0.897431i	$0.354569\pi$
$798$	0	0
$799$	−6.56479	−0.232245
$800$	13.9557	0.493408
$801$	0	0
$802$	−46.3664	−1.63725
$803$	0.283380	0.0100003
$804$	0	0
$805$	52.0573	1.83478
$806$	−2.76415	−0.0973630
$807$	0	0
$808$	−0.531959	−0.0187142
$809$	7.32785	0.257634	0.128817	−	0.991668i	$-0.458882\pi$
$809$	7.32785	0.257634	0.128817	+	0.991668i	$0.458882\pi$
$810$	0	0
$811$	0.404969	0.0142204	0.00711019	−	0.999975i	$-0.497737\pi$
$811$	0.404969	0.0142204	0.00711019	+	0.999975i	$0.497737\pi$
$812$	44.3781	1.55737
$813$	0	0
$814$	−2.12820	−0.0745934
$815$	33.7635	1.18268
$816$	0	0
$817$	24.0662	0.841970
$818$	36.5633	1.27841
$819$	0	0
$820$	6.55530	0.228921
$821$	2.92173	0.101969	0.0509846	−	0.998699i	$-0.483764\pi$
$821$	2.92173	0.101969	0.0509846	+	0.998699i	$0.483764\pi$
$822$	0	0
$823$	33.2014	1.15733	0.578665	−	0.815566i	$-0.303574\pi$
$823$	33.2014	1.15733	0.578665	+	0.815566i	$0.303574\pi$
$824$	1.28476	0.0447568
$825$	0	0
$826$	−35.6334	−1.23984
$827$	−37.3203	−1.29775	−0.648877	−	0.760894i	$-0.724761\pi$
$827$	−37.3203	−1.29775	−0.648877	+	0.760894i	$0.724761\pi$
$828$	0	0
$829$	−18.7158	−0.650027	−0.325014	−	0.945709i	$-0.605369\pi$
$829$	−18.7158	−0.650027	−0.325014	+	0.945709i	$0.605369\pi$
$830$	−78.6628	−2.73043
$831$	0	0
$832$	−3.29116	−0.114100
$833$	13.5830	0.470623
$834$	0	0
$835$	−55.5603	−1.92274
$836$	3.70294	0.128069
$837$	0	0
$838$	−45.1606	−1.56005
$839$	0.337085	0.0116375	0.00581874	−	0.999983i	$-0.498148\pi$
$839$	0.337085	0.0116375	0.00581874	+	0.999983i	$0.498148\pi$
$840$	0	0
$841$	82.4605	2.84347
$842$	−33.6587	−1.15995
$843$	0	0
$844$	28.6758	0.987060
$845$	33.2938	1.14534
$846$	0	0
$847$	−2.18138	−0.0749532
$848$	−17.2347	−0.591841
$849$	0	0
$850$	21.1698	0.726117
$851$	−9.85519	−0.337832
$852$	0	0
$853$	−3.30841	−0.113278	−0.0566389	−	0.998395i	$-0.518038\pi$
$853$	−3.30841	−0.113278	−0.0566389	+	0.998395i	$0.518038\pi$
$854$	−4.32276	−0.147922
$855$	0	0
$856$	−0.563070	−0.0192453
$857$	−10.2920	−0.351568	−0.175784	−	0.984429i	$-0.556246\pi$
$857$	−10.2920	−0.351568	−0.175784	+	0.984429i	$0.556246\pi$
$858$	0	0
$859$	34.1456	1.16503	0.582517	−	0.812819i	$-0.302068\pi$
$859$	34.1456	1.16503	0.582517	+	0.812819i	$0.302068\pi$
$860$	62.7599	2.14010
$861$	0	0
$862$	−53.2893	−1.81504
$863$	40.4513	1.37698	0.688489	−	0.725247i	$-0.258274\pi$
$863$	40.4513	1.37698	0.688489	+	0.725247i	$0.258274\pi$
$864$	0	0
$865$	36.0168	1.22461
$866$	9.80307	0.333122
$867$	0	0
$868$	−13.1932	−0.447805
$869$	−14.7634	−0.500813
$870$	0	0
$871$	−4.57123	−0.154890
$872$	−1.23430	−0.0417986
$873$	0	0
$874$	34.9447	1.18202
$875$	−18.3632	−0.620790
$876$	0	0
$877$	52.6517	1.77792	0.888960	−	0.457984i	$-0.151428\pi$
$877$	52.6517	1.77792	0.888960	+	0.457984i	$0.151428\pi$
$878$	31.9703	1.07894
$879$	0	0
$880$	−10.7682	−0.362996
$881$	−3.36425	−0.113344	−0.0566722	−	0.998393i	$-0.518049\pi$
$881$	−3.36425	−0.113344	−0.0566722	+	0.998393i	$0.518049\pi$
$882$	0	0
$883$	−7.82221	−0.263238	−0.131619	−	0.991300i	$-0.542018\pi$
$883$	−7.82221	−0.263238	−0.131619	+	0.991300i	$0.542018\pi$
$884$	−5.18935	−0.174537
$885$	0	0
$886$	39.5909	1.33008
$887$	−26.2867	−0.882619	−0.441310	−	0.897355i	$-0.645486\pi$
$887$	−26.2867	−0.882619	−0.441310	+	0.897355i	$0.645486\pi$
$888$	0	0
$889$	44.7294	1.50018
$890$	−84.0125	−2.81611
$891$	0	0
$892$	47.0038	1.57380
$893$	−2.08183	−0.0696659
$894$	0	0
$895$	−2.69543	−0.0900983
$896$	2.52361	0.0843079
$897$	0	0
$898$	6.92059	0.230943
$899$	−33.1361	−1.10515
$900$	0	0
$901$	−25.2216	−0.840254
$902$	−2.59225	−0.0863124
$903$	0	0
$904$	−1.41215	−0.0469673
$905$	34.4021	1.14356
$906$	0	0
$907$	20.4355	0.678551	0.339275	−	0.940687i	$-0.389818\pi$
$907$	20.4355	0.678551	0.339275	+	0.940687i	$0.389818\pi$
$908$	8.82187	0.292764
$909$	0	0
$910$	−4.99599	−0.165615
$911$	8.55567	0.283462	0.141731	−	0.989905i	$-0.454733\pi$
$911$	8.55567	0.283462	0.141731	+	0.989905i	$0.454733\pi$
$912$	0	0
$913$	15.2641	0.505169
$914$	17.1239	0.566409
$915$	0	0
$916$	−31.5909	−1.04379
$917$	28.9337	0.955474
$918$	0	0
$919$	−48.6802	−1.60581	−0.802905	−	0.596106i	$-0.796714\pi$
$919$	−48.6802	−1.60581	−0.802905	+	0.596106i	$0.796714\pi$
$920$	−3.45334	−0.113853
$921$	0	0
$922$	6.03754	0.198836
$923$	2.31458	0.0761853
$924$	0	0
$925$	−1.89333	−0.0622522
$926$	15.9198	0.523159
$927$	0	0
$928$	−83.5738	−2.74345
$929$	−10.0819	−0.330778	−0.165389	−	0.986228i	$-0.552888\pi$
$929$	−10.0819	−0.330778	−0.165389	+	0.986228i	$0.552888\pi$
$930$	0	0
$931$	4.30746	0.141171
$932$	−19.4023	−0.635542
$933$	0	0
$934$	23.8185	0.779366
$935$	−15.7584	−0.515356
$936$	0	0
$937$	−4.03470	−0.131808	−0.0659039	−	0.997826i	$-0.520993\pi$
$937$	−4.03470	−0.131808	−0.0659039	+	0.997826i	$0.520993\pi$
$938$	−44.4633	−1.45178
$939$	0	0
$940$	−5.42901	−0.177075
$941$	14.9304	0.486716	0.243358	−	0.969937i	$-0.421751\pi$
$941$	14.9304	0.486716	0.243358	+	0.969937i	$0.421751\pi$
$942$	0	0
$943$	−12.0041	−0.390907
$944$	34.1328	1.11093
$945$	0	0
$946$	−24.8180	−0.806903
$947$	−18.6683	−0.606639	−0.303319	−	0.952889i	$-0.598095\pi$
$947$	−18.6683	−0.606639	−0.303319	+	0.952889i	$0.598095\pi$
$948$	0	0
$949$	−0.125939	−0.00408816
$950$	6.71338	0.217811
$951$	0	0
$952$	1.91279	0.0619938
$953$	−27.8528	−0.902239	−0.451120	−	0.892463i	$-0.648975\pi$
$953$	−27.8528	−0.902239	−0.451120	+	0.892463i	$0.648975\pi$
$954$	0	0
$955$	−23.0629	−0.746298
$956$	−15.9286	−0.515166
$957$	0	0
$958$	−42.2331	−1.36449
$959$	−19.6422	−0.634279
$960$	0	0
$961$	−21.1490	−0.682225
$962$	0.945811	0.0304942
$963$	0	0
$964$	2.94164	0.0947440
$965$	−8.26876	−0.266181
$966$	0	0
$967$	−46.3423	−1.49027	−0.745134	−	0.666915i	$-0.767614\pi$
$967$	−46.3423	−1.49027	−0.745134	+	0.666915i	$0.767614\pi$
$968$	0.144707	0.00465106
$969$	0	0
$970$	24.6639	0.791910
$971$	15.7597	0.505752	0.252876	−	0.967499i	$-0.418623\pi$
$971$	15.7597	0.505752	0.252876	+	0.967499i	$0.418623\pi$
$972$	0	0
$973$	−33.6528	−1.07886
$974$	74.4650	2.38601
$975$	0	0
$976$	4.14071	0.132541
$977$	10.4959	0.335792	0.167896	−	0.985805i	$-0.446303\pi$
$977$	10.4959	0.335792	0.167896	+	0.985805i	$0.446303\pi$
$978$	0	0
$979$	16.3022	0.521021
$980$	11.2330	0.358825
$981$	0	0
$982$	−69.3400	−2.21273
$983$	−36.0894	−1.15107	−0.575537	−	0.817776i	$-0.695207\pi$
$983$	−36.0894	−1.15107	−0.575537	+	0.817776i	$0.695207\pi$
$984$	0	0
$985$	44.3024	1.41159
$986$	−126.775	−4.03735
$987$	0	0
$988$	−1.64565	−0.0523552
$989$	−114.926	−3.65444
$990$	0	0
$991$	11.9025	0.378094	0.189047	−	0.981968i	$-0.439460\pi$
$991$	11.9025	0.378094	0.189047	+	0.981968i	$0.439460\pi$
$992$	24.8456	0.788850
$993$	0	0
$994$	22.5134	0.714082
$995$	−72.6122	−2.30196
$996$	0	0
$997$	−20.2428	−0.641097	−0.320549	−	0.947232i	$-0.603867\pi$
$997$	−20.2428	−0.641097	−0.320549	+	0.947232i	$0.603867\pi$
$998$	−65.9971	−2.08910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.6		19
3.2	odd	2		671.2.a.c.1.14	✓	19
33.32	even	2		7381.2.a.i.1.6		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.14	✓	19	3.2	odd	2
6039.2.a.k.1.6		19	1.1	even	1	trivial
7381.2.a.i.1.6		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)

\(f(q)\)	\(=\)	\(q-1.98166 q^{2} +1.92698 q^{4} -2.60057 q^{5} -2.18138 q^{7} +0.144707 q^{8} +O(q^{10})\)	\(q-1.98166 q^{2} +1.92698 q^{4} -2.60057 q^{5} -2.18138 q^{7} +0.144707 q^{8} +5.15344 q^{10} -1.00000 q^{11} +0.444418 q^{13} +4.32276 q^{14} -4.14071 q^{16} -6.05961 q^{17} -1.92163 q^{19} -5.01124 q^{20} +1.98166 q^{22} +9.17660 q^{23} +1.76296 q^{25} -0.880686 q^{26} -4.20348 q^{28} -10.5575 q^{29} +3.13863 q^{31} +7.91607 q^{32} +12.0081 q^{34} +5.67284 q^{35} -1.07395 q^{37} +3.80802 q^{38} -0.376321 q^{40} -1.30812 q^{41} -12.5238 q^{43} -1.92698 q^{44} -18.1849 q^{46} +1.08337 q^{47} -2.24156 q^{49} -3.49358 q^{50} +0.856384 q^{52} +4.16225 q^{53} +2.60057 q^{55} -0.315662 q^{56} +20.9213 q^{58} -8.24320 q^{59} -1.00000 q^{61} -6.21970 q^{62} -7.40554 q^{64} -1.15574 q^{65} -10.2859 q^{67} -11.6767 q^{68} -11.2416 q^{70} +5.20811 q^{71} -0.283380 q^{73} +2.12820 q^{74} -3.70294 q^{76} +2.18138 q^{77} +14.7634 q^{79} +10.7682 q^{80} +2.59225 q^{82} -15.2641 q^{83} +15.7584 q^{85} +24.8180 q^{86} -0.144707 q^{88} -16.3022 q^{89} -0.969447 q^{91} +17.6831 q^{92} -2.14687 q^{94} +4.99733 q^{95} +4.78591 q^{97} +4.44202 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 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

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