LMFDB - Embedded newform 6039.2.a.h.1.3

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} + \cdots - 35 $ x^13 - 4x^12 - 11x^11 + 57x^10 + 28x^9 - 290x^8 + 51x^7 + 644x^6 - 259x^5 - 640x^4 + 274x^3 + 256x^2 - 74x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.25716$ 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)

$f(q)$	$=$	$q-2.25716 q^{2} +3.09478 q^{4} -1.69786 q^{5} +0.963803 q^{7} -2.47110 q^{8} +O(q^{10})$	\(q-2.25716 q^{2} +3.09478 q^{4} -1.69786 q^{5} +0.963803 q^{7} -2.47110 q^{8} +3.83236 q^{10} -1.00000 q^{11} -4.15967 q^{13} -2.17546 q^{14} -0.611895 q^{16} -6.69466 q^{17} +1.25294 q^{19} -5.25452 q^{20} +2.25716 q^{22} -0.411188 q^{23} -2.11726 q^{25} +9.38905 q^{26} +2.98276 q^{28} +3.52892 q^{29} +3.52150 q^{31} +6.32334 q^{32} +15.1109 q^{34} -1.63641 q^{35} +11.0031 q^{37} -2.82809 q^{38} +4.19559 q^{40} +8.91329 q^{41} +6.77048 q^{43} -3.09478 q^{44} +0.928119 q^{46} +5.52308 q^{47} -6.07108 q^{49} +4.77899 q^{50} -12.8733 q^{52} +0.321548 q^{53} +1.69786 q^{55} -2.38165 q^{56} -7.96534 q^{58} +10.3468 q^{59} +1.00000 q^{61} -7.94859 q^{62} -13.0490 q^{64} +7.06255 q^{65} -12.8483 q^{67} -20.7185 q^{68} +3.69364 q^{70} -9.11055 q^{71} +15.1546 q^{73} -24.8358 q^{74} +3.87757 q^{76} -0.963803 q^{77} -6.47197 q^{79} +1.03891 q^{80} -20.1187 q^{82} +4.87774 q^{83} +11.3666 q^{85} -15.2821 q^{86} +2.47110 q^{88} -0.431277 q^{89} -4.00910 q^{91} -1.27254 q^{92} -12.4665 q^{94} -2.12732 q^{95} -4.74378 q^{97} +13.7034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.25716	−1.59605	−0.798027	−	0.602621i	$-0.794123\pi$
$2$	−2.25716	−1.59605	−0.798027	+	0.602621i	$0.794123\pi$
$3$	0	0
$3$	0	0
$4$	3.09478	1.54739
$5$	−1.69786	−0.759308	−0.379654	−	0.925129i	$-0.623957\pi$
$5$	−1.69786	−0.759308	−0.379654	+	0.925129i	$0.623957\pi$
$6$	0	0
$7$	0.963803	0.364283	0.182142	−	0.983272i	$-0.441697\pi$
$7$	0.963803	0.364283	0.182142	+	0.983272i	$0.441697\pi$
$8$	−2.47110	−0.873665
$9$	0	0
$10$	3.83236	1.21190
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.15967	−1.15368	−0.576842	−	0.816856i	$-0.695715\pi$
$13$	−4.15967	−1.15368	−0.576842	+	0.816856i	$0.695715\pi$
$14$	−2.17546	−0.581416
$15$	0	0
$16$	−0.611895	−0.152974
$17$	−6.69466	−1.62369	−0.811847	−	0.583870i	$-0.801538\pi$
$17$	−6.69466	−1.62369	−0.811847	+	0.583870i	$0.801538\pi$
$18$	0	0
$19$	1.25294	0.287444	0.143722	−	0.989618i	$-0.454093\pi$
$19$	1.25294	0.287444	0.143722	+	0.989618i	$0.454093\pi$
$20$	−5.25452	−1.17495
$21$	0	0
$22$	2.25716	0.481229
$23$	−0.411188	−0.0857387	−0.0428694	−	0.999081i	$-0.513650\pi$
$23$	−0.411188	−0.0857387	−0.0428694	+	0.999081i	$0.513650\pi$
$24$	0	0
$25$	−2.11726	−0.423451
$26$	9.38905	1.84134
$27$	0	0
$28$	2.98276	0.563688
$29$	3.52892	0.655304	0.327652	−	0.944799i	$-0.393743\pi$
$29$	3.52892	0.655304	0.327652	+	0.944799i	$0.393743\pi$
$30$	0	0
$31$	3.52150	0.632480	0.316240	−	0.948679i	$-0.397580\pi$
$31$	3.52150	0.632480	0.316240	+	0.948679i	$0.397580\pi$
$32$	6.32334	1.11782
$33$	0	0
$34$	15.1109	2.59151
$35$	−1.63641	−0.276603
$36$	0	0
$37$	11.0031	1.80890	0.904449	−	0.426582i	$-0.140282\pi$
$37$	11.0031	1.80890	0.904449	+	0.426582i	$0.140282\pi$
$38$	−2.82809	−0.458776
$39$	0	0
$40$	4.19559	0.663381
$41$	8.91329	1.39202	0.696011	−	0.718031i	$-0.254956\pi$
$41$	8.91329	1.39202	0.696011	+	0.718031i	$0.254956\pi$
$42$	0	0
$43$	6.77048	1.03249	0.516244	−	0.856442i	$-0.327330\pi$
$43$	6.77048	1.03249	0.516244	+	0.856442i	$0.327330\pi$
$44$	−3.09478	−0.466556
$45$	0	0
$46$	0.928119	0.136844
$47$	5.52308	0.805623	0.402812	−	0.915283i	$-0.368033\pi$
$47$	5.52308	0.805623	0.402812	+	0.915283i	$0.368033\pi$
$48$	0	0
$49$	−6.07108	−0.867298
$50$	4.77899	0.675851
$51$	0	0
$52$	−12.8733	−1.78520
$53$	0.321548	0.0441680	0.0220840	−	0.999756i	$-0.492970\pi$
$53$	0.321548	0.0441680	0.0220840	+	0.999756i	$0.492970\pi$
$54$	0	0
$55$	1.69786	0.228940
$56$	−2.38165	−0.318261
$57$	0	0
$58$	−7.96534	−1.04590
$59$	10.3468	1.34704	0.673519	−	0.739170i	$-0.264782\pi$
$59$	10.3468	1.34704	0.673519	+	0.739170i	$0.264782\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−7.94859	−1.00947
$63$	0	0
$64$	−13.0490	−1.63113
$65$	7.06255	0.876002
$66$	0	0
$67$	−12.8483	−1.56967	−0.784835	−	0.619705i	$-0.787252\pi$
$67$	−12.8483	−1.56967	−0.784835	+	0.619705i	$0.787252\pi$
$68$	−20.7185	−2.51249
$69$	0	0
$70$	3.69364	0.441474
$71$	−9.11055	−1.08122	−0.540612	−	0.841272i	$-0.681807\pi$
$71$	−9.11055	−1.08122	−0.540612	+	0.841272i	$0.681807\pi$
$72$	0	0
$73$	15.1546	1.77371	0.886854	−	0.462050i	$-0.152886\pi$
$73$	15.1546	1.77371	0.886854	+	0.462050i	$0.152886\pi$
$74$	−24.8358	−2.88710
$75$	0	0
$76$	3.87757	0.444788
$77$	−0.963803	−0.109836
$78$	0	0
$79$	−6.47197	−0.728153	−0.364077	−	0.931369i	$-0.618615\pi$
$79$	−6.47197	−0.728153	−0.364077	+	0.931369i	$0.618615\pi$
$80$	1.03891	0.116154
$81$	0	0
$82$	−20.1187	−2.22174
$83$	4.87774	0.535401	0.267701	−	0.963502i	$-0.413736\pi$
$83$	4.87774	0.535401	0.267701	+	0.963502i	$0.413736\pi$
$84$	0	0
$85$	11.3666	1.23288
$86$	−15.2821	−1.64791
$87$	0	0
$88$	2.47110	0.263420
$89$	−0.431277	−0.0457152	−0.0228576	−	0.999739i	$-0.507276\pi$
$89$	−0.431277	−0.0457152	−0.0228576	+	0.999739i	$0.507276\pi$
$90$	0	0
$91$	−4.00910	−0.420268
$92$	−1.27254	−0.132671
$93$	0	0
$94$	−12.4665	−1.28582
$95$	−2.12732	−0.218259
$96$	0	0
$97$	−4.74378	−0.481658	−0.240829	−	0.970568i	$-0.577419\pi$
$97$	−4.74378	−0.481658	−0.240829	+	0.970568i	$0.577419\pi$
$98$	13.7034	1.38425
$99$	0	0
$100$	−6.55244	−0.655244
$101$	−15.9306	−1.58516	−0.792579	−	0.609770i	$-0.791262\pi$
$101$	−15.9306	−1.58516	−0.792579	+	0.609770i	$0.791262\pi$
$102$	0	0
$103$	−12.1342	−1.19562	−0.597808	−	0.801639i	$-0.703961\pi$
$103$	−12.1342	−1.19562	−0.597808	+	0.801639i	$0.703961\pi$
$104$	10.2789	1.00793
$105$	0	0
$106$	−0.725786	−0.0704946
$107$	13.5134	1.30639	0.653197	−	0.757188i	$-0.273428\pi$
$107$	13.5134	1.30639	0.653197	+	0.757188i	$0.273428\pi$
$108$	0	0
$109$	−18.4480	−1.76700	−0.883498	−	0.468436i	$-0.844818\pi$
$109$	−18.4480	−1.76700	−0.883498	+	0.468436i	$0.844818\pi$
$110$	−3.83236	−0.365401
$111$	0	0
$112$	−0.589746	−0.0557258
$113$	9.25623	0.870753	0.435377	−	0.900248i	$-0.356615\pi$
$113$	9.25623	0.870753	0.435377	+	0.900248i	$0.356615\pi$
$114$	0	0
$115$	0.698142	0.0651021
$116$	10.9212	1.01401
$117$	0	0
$118$	−23.3544	−2.14995
$119$	−6.45234	−0.591485
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.25716	−0.204354
$123$	0	0
$124$	10.8983	0.978693
$125$	12.0841	1.08084
$126$	0	0
$127$	11.8074	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$127$	11.8074	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$128$	16.8071	1.48555
$129$	0	0
$130$	−15.9413	−1.39815
$131$	−13.8749	−1.21226	−0.606130	−	0.795366i	$-0.707279\pi$
$131$	−13.8749	−1.21226	−0.606130	+	0.795366i	$0.707279\pi$
$132$	0	0
$133$	1.20759	0.104711
$134$	29.0007	2.50528
$135$	0	0
$136$	16.5432	1.41856
$137$	−1.79062	−0.152983	−0.0764914	−	0.997070i	$-0.524372\pi$
$137$	−1.79062	−0.152983	−0.0764914	+	0.997070i	$0.524372\pi$
$138$	0	0
$139$	−1.55700	−0.132063	−0.0660314	−	0.997818i	$-0.521034\pi$
$139$	−1.55700	−0.132063	−0.0660314	+	0.997818i	$0.521034\pi$
$140$	−5.06432	−0.428013
$141$	0	0
$142$	20.5640	1.72569
$143$	4.15967	0.347849
$144$	0	0
$145$	−5.99162	−0.497577
$146$	−34.2063	−2.83093
$147$	0	0
$148$	34.0522	2.79907
$149$	−8.75151	−0.716952	−0.358476	−	0.933539i	$-0.616703\pi$
$149$	−8.75151	−0.716952	−0.358476	+	0.933539i	$0.616703\pi$
$150$	0	0
$151$	17.7750	1.44651	0.723255	−	0.690581i	$-0.242645\pi$
$151$	17.7750	1.44651	0.723255	+	0.690581i	$0.242645\pi$
$152$	−3.09614	−0.251130
$153$	0	0
$154$	2.17546	0.175304
$155$	−5.97903	−0.480247
$156$	0	0
$157$	−1.98295	−0.158256	−0.0791282	−	0.996864i	$-0.525214\pi$
$157$	−1.98295	−0.158256	−0.0791282	+	0.996864i	$0.525214\pi$
$158$	14.6083	1.16217
$159$	0	0
$160$	−10.7362	−0.848769
$161$	−0.396305	−0.0312332
$162$	0	0
$163$	11.8205	0.925857	0.462928	−	0.886396i	$-0.346799\pi$
$163$	11.8205	0.925857	0.462928	+	0.886396i	$0.346799\pi$
$164$	27.5847	2.15400
$165$	0	0
$166$	−11.0098	−0.854530
$167$	23.0437	1.78318	0.891588	−	0.452848i	$-0.149592\pi$
$167$	23.0437	1.78318	0.891588	+	0.452848i	$0.149592\pi$
$168$	0	0
$169$	4.30284	0.330988
$170$	−25.6563	−1.96775
$171$	0	0
$172$	20.9531	1.59766
$173$	20.3192	1.54484	0.772420	−	0.635112i	$-0.219046\pi$
$173$	20.3192	1.54484	0.772420	+	0.635112i	$0.219046\pi$
$174$	0	0
$175$	−2.04062	−0.154256
$176$	0.611895	0.0461233
$177$	0	0
$178$	0.973462	0.0729640
$179$	−7.03429	−0.525768	−0.262884	−	0.964827i	$-0.584674\pi$
$179$	−7.03429	−0.525768	−0.262884	+	0.964827i	$0.584674\pi$
$180$	0	0
$181$	14.0497	1.04431	0.522155	−	0.852851i	$-0.325128\pi$
$181$	14.0497	1.04431	0.522155	+	0.852851i	$0.325128\pi$
$182$	9.04919	0.670771
$183$	0	0
$184$	1.01609	0.0749069
$185$	−18.6818	−1.37351
$186$	0	0
$187$	6.69466	0.489562
$188$	17.0927	1.24661
$189$	0	0
$190$	4.80171	0.348353
$191$	−13.7269	−0.993244	−0.496622	−	0.867967i	$-0.665426\pi$
$191$	−13.7269	−0.993244	−0.496622	+	0.867967i	$0.665426\pi$
$192$	0	0
$193$	−11.8848	−0.855484	−0.427742	−	0.903901i	$-0.640691\pi$
$193$	−11.8848	−0.855484	−0.427742	+	0.903901i	$0.640691\pi$
$194$	10.7075	0.768752
$195$	0	0
$196$	−18.7887	−1.34205
$197$	−16.0080	−1.14052	−0.570260	−	0.821464i	$-0.693157\pi$
$197$	−16.0080	−1.14052	−0.570260	+	0.821464i	$0.693157\pi$
$198$	0	0
$199$	−10.9760	−0.778068	−0.389034	−	0.921223i	$-0.627191\pi$
$199$	−10.9760	−0.778068	−0.389034	+	0.921223i	$0.627191\pi$
$200$	5.23195	0.369954
$201$	0	0
$202$	35.9580	2.53000
$203$	3.40118	0.238716
$204$	0	0
$205$	−15.1336	−1.05697
$206$	27.3888	1.90827
$207$	0	0
$208$	2.54528	0.176483
$209$	−1.25294	−0.0866676
$210$	0	0
$211$	−15.8142	−1.08869	−0.544347	−	0.838860i	$-0.683223\pi$
$211$	−15.8142	−1.08869	−0.544347	+	0.838860i	$0.683223\pi$
$212$	0.995120	0.0683452
$213$	0	0
$214$	−30.5020	−2.08507
$215$	−11.4953	−0.783976
$216$	0	0
$217$	3.39403	0.230402
$218$	41.6401	2.82022
$219$	0	0
$220$	5.25452	0.354260
$221$	27.8476	1.87323
$222$	0	0
$223$	−14.5392	−0.973616	−0.486808	−	0.873509i	$-0.661839\pi$
$223$	−14.5392	−0.973616	−0.486808	+	0.873509i	$0.661839\pi$
$224$	6.09445	0.407203
$225$	0	0
$226$	−20.8928	−1.38977
$227$	−25.8655	−1.71675	−0.858375	−	0.513022i	$-0.828526\pi$
$227$	−25.8655	−1.71675	−0.858375	+	0.513022i	$0.828526\pi$
$228$	0	0
$229$	−17.4634	−1.15402	−0.577009	−	0.816738i	$-0.695780\pi$
$229$	−17.4634	−1.15402	−0.577009	+	0.816738i	$0.695780\pi$
$230$	−1.57582	−0.103907
$231$	0	0
$232$	−8.72030	−0.572516
$233$	−8.04346	−0.526945	−0.263472	−	0.964667i	$-0.584868\pi$
$233$	−8.04346	−0.526945	−0.263472	+	0.964667i	$0.584868\pi$
$234$	0	0
$235$	−9.37743	−0.611716
$236$	32.0210	2.08439
$237$	0	0
$238$	14.5640	0.944042
$239$	3.42277	0.221401	0.110700	−	0.993854i	$-0.464691\pi$
$239$	3.42277	0.221401	0.110700	+	0.993854i	$0.464691\pi$
$240$	0	0
$241$	16.4937	1.06245	0.531226	−	0.847230i	$-0.321732\pi$
$241$	16.4937	1.06245	0.531226	+	0.847230i	$0.321732\pi$
$242$	−2.25716	−0.145096
$243$	0	0
$244$	3.09478	0.198123
$245$	10.3079	0.658546
$246$	0	0
$247$	−5.21181	−0.331620
$248$	−8.70196	−0.552575
$249$	0	0
$250$	−27.2759	−1.72508
$251$	−28.8276	−1.81958	−0.909791	−	0.415067i	$-0.863758\pi$
$251$	−28.8276	−1.81958	−0.909791	+	0.415067i	$0.863758\pi$
$252$	0	0
$253$	0.411188	0.0258512
$254$	−26.6511	−1.67224
$255$	0	0
$256$	−11.8382	−0.739889
$257$	19.9412	1.24390	0.621949	−	0.783058i	$-0.286341\pi$
$257$	19.9412	1.24390	0.621949	+	0.783058i	$0.286341\pi$
$258$	0	0
$259$	10.6048	0.658951
$260$	21.8571	1.35552
$261$	0	0
$262$	31.3180	1.93483
$263$	−11.9859	−0.739085	−0.369543	−	0.929214i	$-0.620486\pi$
$263$	−11.9859	−0.739085	−0.369543	+	0.929214i	$0.620486\pi$
$264$	0	0
$265$	−0.545945	−0.0335371
$266$	−2.72572	−0.167125
$267$	0	0
$268$	−39.7627	−2.42889
$269$	10.0911	0.615266	0.307633	−	0.951505i	$-0.400463\pi$
$269$	10.0911	0.615266	0.307633	+	0.951505i	$0.400463\pi$
$270$	0	0
$271$	5.76608	0.350264	0.175132	−	0.984545i	$-0.443965\pi$
$271$	5.76608	0.350264	0.175132	+	0.984545i	$0.443965\pi$
$272$	4.09643	0.248383
$273$	0	0
$274$	4.04171	0.244169
$275$	2.11726	0.127675
$276$	0	0
$277$	−10.3414	−0.621357	−0.310678	−	0.950515i	$-0.600556\pi$
$277$	−10.3414	−0.621357	−0.310678	+	0.950515i	$0.600556\pi$
$278$	3.51439	0.210779
$279$	0	0
$280$	4.04372	0.241659
$281$	−24.5170	−1.46256	−0.731280	−	0.682077i	$-0.761077\pi$
$281$	−24.5170	−1.46256	−0.731280	+	0.682077i	$0.761077\pi$
$282$	0	0
$283$	−21.7457	−1.29265	−0.646324	−	0.763063i	$-0.723694\pi$
$283$	−21.7457	−1.29265	−0.646324	+	0.763063i	$0.723694\pi$
$284$	−28.1952	−1.67308
$285$	0	0
$286$	−9.38905	−0.555186
$287$	8.59066	0.507091
$288$	0	0
$289$	27.8185	1.63638
$290$	13.5241	0.794161
$291$	0	0
$292$	46.9001	2.74462
$293$	−14.3684	−0.839409	−0.419704	−	0.907661i	$-0.637866\pi$
$293$	−14.3684	−0.839409	−0.419704	+	0.907661i	$0.637866\pi$
$294$	0	0
$295$	−17.5674	−1.02282
$296$	−27.1897	−1.58037
$297$	0	0
$298$	19.7536	1.14429
$299$	1.71041	0.0989154
$300$	0	0
$301$	6.52540	0.376118
$302$	−40.1211	−2.30871
$303$	0	0
$304$	−0.766667	−0.0439714
$305$	−1.69786	−0.0972194
$306$	0	0
$307$	9.09697	0.519192	0.259596	−	0.965717i	$-0.416411\pi$
$307$	9.09697	0.519192	0.259596	+	0.965717i	$0.416411\pi$
$308$	−2.98276	−0.169958
$309$	0	0
$310$	13.4956	0.766500
$311$	−24.8297	−1.40796	−0.703982	−	0.710218i	$-0.748596\pi$
$311$	−24.8297	−1.40796	−0.703982	+	0.710218i	$0.748596\pi$
$312$	0	0
$313$	10.2747	0.580761	0.290381	−	0.956911i	$-0.406218\pi$
$313$	10.2747	0.580761	0.290381	+	0.956911i	$0.406218\pi$
$314$	4.47583	0.252586
$315$	0	0
$316$	−20.0293	−1.12674
$317$	29.8704	1.67769	0.838844	−	0.544373i	$-0.183232\pi$
$317$	29.8704	1.67769	0.838844	+	0.544373i	$0.183232\pi$
$318$	0	0
$319$	−3.52892	−0.197581
$320$	22.1555	1.23853
$321$	0	0
$322$	0.894524	0.0498499
$323$	−8.38801	−0.466721
$324$	0	0
$325$	8.80708	0.488529
$326$	−26.6809	−1.47772
$327$	0	0
$328$	−22.0256	−1.21616
$329$	5.32316	0.293475
$330$	0	0
$331$	8.73092	0.479895	0.239947	−	0.970786i	$-0.422870\pi$
$331$	8.73092	0.479895	0.239947	+	0.970786i	$0.422870\pi$
$332$	15.0955	0.828475
$333$	0	0
$334$	−52.0134	−2.84604
$335$	21.8147	1.19186
$336$	0	0
$337$	−20.8462	−1.13556	−0.567781	−	0.823179i	$-0.692198\pi$
$337$	−20.8462	−1.13556	−0.567781	+	0.823179i	$0.692198\pi$
$338$	−9.71221	−0.528275
$339$	0	0
$340$	35.1772	1.90775
$341$	−3.52150	−0.190700
$342$	0	0
$343$	−12.5980	−0.680225
$344$	−16.7305	−0.902048
$345$	0	0
$346$	−45.8637	−2.46565
$347$	−0.840518	−0.0451214	−0.0225607	−	0.999745i	$-0.507182\pi$
$347$	−0.840518	−0.0451214	−0.0225607	+	0.999745i	$0.507182\pi$
$348$	0	0
$349$	17.6570	0.945156	0.472578	−	0.881289i	$-0.343324\pi$
$349$	17.6570	0.945156	0.472578	+	0.881289i	$0.343324\pi$
$350$	4.60601	0.246201
$351$	0	0
$352$	−6.32334	−0.337035
$353$	4.14349	0.220536	0.110268	−	0.993902i	$-0.464829\pi$
$353$	4.14349	0.220536	0.110268	+	0.993902i	$0.464829\pi$
$354$	0	0
$355$	15.4685	0.820982
$356$	−1.33471	−0.0707393
$357$	0	0
$358$	15.8775	0.839154
$359$	29.2977	1.54628	0.773138	−	0.634238i	$-0.218686\pi$
$359$	29.2977	1.54628	0.773138	+	0.634238i	$0.218686\pi$
$360$	0	0
$361$	−17.4301	−0.917376
$362$	−31.7125	−1.66677
$363$	0	0
$364$	−12.4073	−0.650319
$365$	−25.7304	−1.34679
$366$	0	0
$367$	−36.8166	−1.92181	−0.960906	−	0.276876i	$-0.910701\pi$
$367$	−36.8166	−1.92181	−0.960906	+	0.276876i	$0.910701\pi$
$368$	0.251604	0.0131158
$369$	0	0
$370$	42.1678	2.19220
$371$	0.309909	0.0160897
$372$	0	0
$373$	10.4299	0.540039	0.270020	−	0.962855i	$-0.412970\pi$
$373$	10.4299	0.540039	0.270020	+	0.962855i	$0.412970\pi$
$374$	−15.1109	−0.781368
$375$	0	0
$376$	−13.6481	−0.703845
$377$	−14.6791	−0.756014
$378$	0	0
$379$	−23.3596	−1.19990	−0.599951	−	0.800037i	$-0.704813\pi$
$379$	−23.3596	−1.19990	−0.599951	+	0.800037i	$0.704813\pi$
$380$	−6.58359	−0.337731
$381$	0	0
$382$	30.9838	1.58527
$383$	−13.9500	−0.712810	−0.356405	−	0.934331i	$-0.615998\pi$
$383$	−13.9500	−0.712810	−0.356405	+	0.934331i	$0.615998\pi$
$384$	0	0
$385$	1.63641	0.0833990
$386$	26.8258	1.36540
$387$	0	0
$388$	−14.6810	−0.745312
$389$	36.5993	1.85566	0.927829	−	0.373006i	$-0.121673\pi$
$389$	36.5993	1.85566	0.927829	+	0.373006i	$0.121673\pi$
$390$	0	0
$391$	2.75277	0.139213
$392$	15.0022	0.757727
$393$	0	0
$394$	36.1325	1.82033
$395$	10.9885	0.552893
$396$	0	0
$397$	−0.968472	−0.0486062	−0.0243031	−	0.999705i	$-0.507737\pi$
$397$	−0.968472	−0.0486062	−0.0243031	+	0.999705i	$0.507737\pi$
$398$	24.7746	1.24184
$399$	0	0
$400$	1.29554	0.0647769
$401$	−2.44043	−0.121869	−0.0609347	−	0.998142i	$-0.519408\pi$
$401$	−2.44043	−0.121869	−0.0609347	+	0.998142i	$0.519408\pi$
$402$	0	0
$403$	−14.6483	−0.729682
$404$	−49.3018	−2.45286
$405$	0	0
$406$	−7.67702	−0.381004
$407$	−11.0031	−0.545403
$408$	0	0
$409$	15.7514	0.778858	0.389429	−	0.921056i	$-0.372672\pi$
$409$	15.7514	0.778858	0.389429	+	0.921056i	$0.372672\pi$
$410$	34.1589	1.68699
$411$	0	0
$412$	−37.5526	−1.85008
$413$	9.97227	0.490703
$414$	0	0
$415$	−8.28174	−0.406535
$416$	−26.3030	−1.28961
$417$	0	0
$418$	2.82809	0.138326
$419$	−39.5394	−1.93163	−0.965814	−	0.259236i	$-0.916529\pi$
$419$	−39.5394	−1.93163	−0.965814	+	0.259236i	$0.916529\pi$
$420$	0	0
$421$	−17.8106	−0.868034	−0.434017	−	0.900905i	$-0.642904\pi$
$421$	−17.8106	−0.868034	−0.434017	+	0.900905i	$0.642904\pi$
$422$	35.6952	1.73762
$423$	0	0
$424$	−0.794576	−0.0385880
$425$	14.1743	0.687555
$426$	0	0
$427$	0.963803	0.0466417
$428$	41.8211	2.02150
$429$	0	0
$430$	25.9469	1.25127
$431$	33.0900	1.59389	0.796945	−	0.604052i	$-0.206448\pi$
$431$	33.0900	1.59389	0.796945	+	0.604052i	$0.206448\pi$
$432$	0	0
$433$	13.6141	0.654250	0.327125	−	0.944981i	$-0.393920\pi$
$433$	13.6141	0.654250	0.327125	+	0.944981i	$0.393920\pi$
$434$	−7.66088	−0.367734
$435$	0	0
$436$	−57.0924	−2.73423
$437$	−0.515194	−0.0246451
$438$	0	0
$439$	15.0351	0.717587	0.358794	−	0.933417i	$-0.383188\pi$
$439$	15.0351	0.717587	0.358794	+	0.933417i	$0.383188\pi$
$440$	−4.19559	−0.200017
$441$	0	0
$442$	−62.8565	−2.98978
$443$	−25.7815	−1.22492	−0.612458	−	0.790503i	$-0.709819\pi$
$443$	−25.7815	−1.22492	−0.612458	+	0.790503i	$0.709819\pi$
$444$	0	0
$445$	0.732249	0.0347120
$446$	32.8173	1.55394
$447$	0	0
$448$	−12.5767	−0.594192
$449$	−17.0888	−0.806471	−0.403236	−	0.915096i	$-0.632115\pi$
$449$	−17.0888	−0.806471	−0.403236	+	0.915096i	$0.632115\pi$
$450$	0	0
$451$	−8.91329	−0.419711
$452$	28.6460	1.34739
$453$	0	0
$454$	58.3825	2.74003
$455$	6.80691	0.319113
$456$	0	0
$457$	33.7234	1.57751	0.788756	−	0.614706i	$-0.210725\pi$
$457$	33.7234	1.57751	0.788756	+	0.614706i	$0.210725\pi$
$458$	39.4178	1.84187
$459$	0	0
$460$	2.16060	0.100738
$461$	1.19356	0.0555898	0.0277949	−	0.999614i	$-0.491151\pi$
$461$	1.19356	0.0555898	0.0277949	+	0.999614i	$0.491151\pi$
$462$	0	0
$463$	−29.0640	−1.35072	−0.675360	−	0.737488i	$-0.736012\pi$
$463$	−29.0640	−1.35072	−0.675360	+	0.737488i	$0.736012\pi$
$464$	−2.15933	−0.100244
$465$	0	0
$466$	18.1554	0.841032
$467$	−28.8315	−1.33416	−0.667082	−	0.744984i	$-0.732457\pi$
$467$	−28.8315	−1.33416	−0.667082	+	0.744984i	$0.732457\pi$
$468$	0	0
$469$	−12.3832	−0.571804
$470$	21.1664	0.976333
$471$	0	0
$472$	−25.5679	−1.17686
$473$	−6.77048	−0.311307
$474$	0	0
$475$	−2.65279	−0.121719
$476$	−19.9686	−0.915258
$477$	0	0
$478$	−7.72575	−0.353368
$479$	−7.10069	−0.324439	−0.162219	−	0.986755i	$-0.551865\pi$
$479$	−7.10069	−0.324439	−0.162219	+	0.986755i	$0.551865\pi$
$480$	0	0
$481$	−45.7692	−2.08690
$482$	−37.2289	−1.69573
$483$	0	0
$484$	3.09478	0.140672
$485$	8.05429	0.365727
$486$	0	0
$487$	29.0146	1.31478	0.657390	−	0.753551i	$-0.271661\pi$
$487$	29.0146	1.31478	0.657390	+	0.753551i	$0.271661\pi$
$488$	−2.47110	−0.111861
$489$	0	0
$490$	−23.2665	−1.05108
$491$	−31.7488	−1.43280	−0.716402	−	0.697688i	$-0.754212\pi$
$491$	−31.7488	−1.43280	−0.716402	+	0.697688i	$0.754212\pi$
$492$	0	0
$493$	−23.6249	−1.06401
$494$	11.7639	0.529283
$495$	0	0
$496$	−2.15479	−0.0967527
$497$	−8.78078	−0.393872
$498$	0	0
$499$	13.6833	0.612551	0.306275	−	0.951943i	$-0.400917\pi$
$499$	13.6833	0.612551	0.306275	+	0.951943i	$0.400917\pi$
$500$	37.3977	1.67248
$501$	0	0
$502$	65.0686	2.90415
$503$	−32.0779	−1.43028	−0.715141	−	0.698980i	$-0.753637\pi$
$503$	−32.0779	−1.43028	−0.715141	+	0.698980i	$0.753637\pi$
$504$	0	0
$505$	27.0481	1.20362
$506$	−0.928119	−0.0412599
$507$	0	0
$508$	36.5412	1.62125
$509$	−34.7846	−1.54180	−0.770901	−	0.636956i	$-0.780193\pi$
$509$	−34.7846	−1.54180	−0.770901	+	0.636956i	$0.780193\pi$
$510$	0	0
$511$	14.6060	0.646132
$512$	−6.89332	−0.304645
$513$	0	0
$514$	−45.0105	−1.98533
$515$	20.6022	0.907841
$516$	0	0
$517$	−5.52308	−0.242905
$518$	−23.9368	−1.05172
$519$	0	0
$520$	−17.4523	−0.765332
$521$	31.1784	1.36595	0.682976	−	0.730441i	$-0.260685\pi$
$521$	31.1784	1.36595	0.682976	+	0.730441i	$0.260685\pi$
$522$	0	0
$523$	15.7009	0.686552	0.343276	−	0.939235i	$-0.388463\pi$
$523$	15.7009	0.686552	0.343276	+	0.939235i	$0.388463\pi$
$524$	−42.9399	−1.87584
$525$	0	0
$526$	27.0542	1.17962
$527$	−23.5752	−1.02695
$528$	0	0
$529$	−22.8309	−0.992649
$530$	1.23229	0.0535271
$531$	0	0
$532$	3.73722	0.162029
$533$	−37.0763	−1.60595
$534$	0	0
$535$	−22.9440	−0.991955
$536$	31.7494	1.37136
$537$	0	0
$538$	−22.7773	−0.981999
$539$	6.07108	0.261500
$540$	0	0
$541$	0.934541	0.0401791	0.0200895	−	0.999798i	$-0.493605\pi$
$541$	0.934541	0.0401791	0.0200895	+	0.999798i	$0.493605\pi$
$542$	−13.0150	−0.559041
$543$	0	0
$544$	−42.3326	−1.81500
$545$	31.3222	1.34169
$546$	0	0
$547$	3.15556	0.134922	0.0674610	−	0.997722i	$-0.478510\pi$
$547$	3.15556	0.134922	0.0674610	+	0.997722i	$0.478510\pi$
$548$	−5.54157	−0.236724
$549$	0	0
$550$	−4.77899	−0.203777
$551$	4.42152	0.188363
$552$	0	0
$553$	−6.23770	−0.265254
$554$	23.3423	0.991720
$555$	0	0
$556$	−4.81856	−0.204353
$557$	32.0032	1.35602	0.678011	−	0.735052i	$-0.262842\pi$
$557$	32.0032	1.35602	0.678011	+	0.735052i	$0.262842\pi$
$558$	0	0
$559$	−28.1629	−1.19117
$560$	1.00131	0.0423130
$561$	0	0
$562$	55.3388	2.33433
$563$	10.4892	0.442067	0.221033	−	0.975266i	$-0.429057\pi$
$563$	10.4892	0.442067	0.221033	+	0.975266i	$0.429057\pi$
$564$	0	0
$565$	−15.7158	−0.661170
$566$	49.0836	2.06314
$567$	0	0
$568$	22.5131	0.944627
$569$	27.2937	1.14421	0.572105	−	0.820180i	$-0.306127\pi$
$569$	27.2937	1.14421	0.572105	+	0.820180i	$0.306127\pi$
$570$	0	0
$571$	39.7605	1.66392	0.831962	−	0.554832i	$-0.187218\pi$
$571$	39.7605	1.66392	0.831962	+	0.554832i	$0.187218\pi$
$572$	12.8733	0.538258
$573$	0	0
$574$	−19.3905	−0.809344
$575$	0.870591	0.0363062
$576$	0	0
$577$	−27.2686	−1.13520	−0.567602	−	0.823303i	$-0.692129\pi$
$577$	−27.2686	−1.13520	−0.567602	+	0.823303i	$0.692129\pi$
$578$	−62.7909	−2.61176
$579$	0	0
$580$	−18.5428	−0.769946
$581$	4.70118	0.195038
$582$	0	0
$583$	−0.321548	−0.0133172
$584$	−37.4484	−1.54963
$585$	0	0
$586$	32.4317	1.33974
$587$	11.4261	0.471604	0.235802	−	0.971801i	$-0.424228\pi$
$587$	11.4261	0.471604	0.235802	+	0.971801i	$0.424228\pi$
$588$	0	0
$589$	4.41222	0.181803
$590$	39.6526	1.63247
$591$	0	0
$592$	−6.73274	−0.276714
$593$	−31.3763	−1.28847	−0.644235	−	0.764827i	$-0.722824\pi$
$593$	−31.3763	−1.28847	−0.644235	+	0.764827i	$0.722824\pi$
$594$	0	0
$595$	10.9552	0.449119
$596$	−27.0840	−1.10940
$597$	0	0
$598$	−3.86067	−0.157874
$599$	−25.6067	−1.04626	−0.523130	−	0.852253i	$-0.675236\pi$
$599$	−25.6067	−1.04626	−0.523130	+	0.852253i	$0.675236\pi$
$600$	0	0
$601$	−27.1881	−1.10902	−0.554512	−	0.832176i	$-0.687095\pi$
$601$	−27.1881	−1.10902	−0.554512	+	0.832176i	$0.687095\pi$
$602$	−14.7289	−0.600305
$603$	0	0
$604$	55.0098	2.23832
$605$	−1.69786	−0.0690280
$606$	0	0
$607$	13.0462	0.529530	0.264765	−	0.964313i	$-0.414706\pi$
$607$	13.0462	0.529530	0.264765	+	0.964313i	$0.414706\pi$
$608$	7.92276	0.321310
$609$	0	0
$610$	3.83236	0.155168
$611$	−22.9742	−0.929435
$612$	0	0
$613$	−12.8209	−0.517833	−0.258916	−	0.965900i	$-0.583365\pi$
$613$	−12.8209	−0.517833	−0.258916	+	0.965900i	$0.583365\pi$
$614$	−20.5333	−0.828658
$615$	0	0
$616$	2.38165	0.0959594
$617$	−22.0526	−0.887803	−0.443901	−	0.896076i	$-0.646406\pi$
$617$	−22.0526	−0.887803	−0.443901	+	0.896076i	$0.646406\pi$
$618$	0	0
$619$	41.3944	1.66378	0.831890	−	0.554941i	$-0.187259\pi$
$619$	41.3944	1.66378	0.831890	+	0.554941i	$0.187259\pi$
$620$	−18.5038	−0.743129
$621$	0	0
$622$	56.0447	2.24719
$623$	−0.415666	−0.0166533
$624$	0	0
$625$	−9.93095	−0.397238
$626$	−23.1917	−0.926927
$627$	0	0
$628$	−6.13678	−0.244884
$629$	−73.6620	−2.93710
$630$	0	0
$631$	33.2300	1.32287	0.661433	−	0.750004i	$-0.269948\pi$
$631$	33.2300	1.32287	0.661433	+	0.750004i	$0.269948\pi$
$632$	15.9929	0.636162
$633$	0	0
$634$	−67.4223	−2.67768
$635$	−20.0473	−0.795552
$636$	0	0
$637$	25.2537	1.00059
$638$	7.96534	0.315351
$639$	0	0
$640$	−28.5361	−1.12799
$641$	15.6738	0.619077	0.309538	−	0.950887i	$-0.399825\pi$
$641$	15.6738	0.619077	0.309538	+	0.950887i	$0.399825\pi$
$642$	0	0
$643$	−3.64712	−0.143828	−0.0719142	−	0.997411i	$-0.522911\pi$
$643$	−3.64712	−0.143828	−0.0719142	+	0.997411i	$0.522911\pi$
$644$	−1.22648	−0.0483299
$645$	0	0
$646$	18.9331	0.744913
$647$	22.3324	0.877976	0.438988	−	0.898493i	$-0.355337\pi$
$647$	22.3324	0.877976	0.438988	+	0.898493i	$0.355337\pi$
$648$	0	0
$649$	−10.3468	−0.406147
$650$	−19.8790	−0.779719
$651$	0	0
$652$	36.5820	1.43266
$653$	−28.0945	−1.09942	−0.549711	−	0.835355i	$-0.685262\pi$
$653$	−28.0945	−1.09942	−0.549711	+	0.835355i	$0.685262\pi$
$654$	0	0
$655$	23.5578	0.920479
$656$	−5.45400	−0.212943
$657$	0	0
$658$	−12.0152	−0.468402
$659$	5.05621	0.196962	0.0984810	−	0.995139i	$-0.468602\pi$
$659$	5.05621	0.196962	0.0984810	+	0.995139i	$0.468602\pi$
$660$	0	0
$661$	−20.4642	−0.795967	−0.397983	−	0.917393i	$-0.630290\pi$
$661$	−20.4642	−0.795967	−0.397983	+	0.917393i	$0.630290\pi$
$662$	−19.7071	−0.765938
$663$	0	0
$664$	−12.0534	−0.467761
$665$	−2.05032	−0.0795080
$666$	0	0
$667$	−1.45105	−0.0561849
$668$	71.3152	2.75927
$669$	0	0
$670$	−49.2392	−1.90228
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	25.6964	0.990524	0.495262	−	0.868744i	$-0.335072\pi$
$673$	25.6964	0.990524	0.495262	+	0.868744i	$0.335072\pi$
$674$	47.0532	1.81242
$675$	0	0
$676$	13.3164	0.512167
$677$	−24.7114	−0.949737	−0.474869	−	0.880057i	$-0.657504\pi$
$677$	−24.7114	−0.949737	−0.474869	+	0.880057i	$0.657504\pi$
$678$	0	0
$679$	−4.57207	−0.175460
$680$	−28.0881	−1.07713
$681$	0	0
$682$	7.94859	0.304367
$683$	7.66587	0.293327	0.146663	−	0.989186i	$-0.453147\pi$
$683$	7.66587	0.293327	0.146663	+	0.989186i	$0.453147\pi$
$684$	0	0
$685$	3.04023	0.116161
$686$	28.4356	1.08568
$687$	0	0
$688$	−4.14282	−0.157943
$689$	−1.33753	−0.0509560
$690$	0	0
$691$	−3.16847	−0.120534	−0.0602672	−	0.998182i	$-0.519195\pi$
$691$	−3.16847	−0.120534	−0.0602672	+	0.998182i	$0.519195\pi$
$692$	62.8834	2.39047
$693$	0	0
$694$	1.89718	0.0720162
$695$	2.64357	0.100276
$696$	0	0
$697$	−59.6715	−2.26022
$698$	−39.8546	−1.50852
$699$	0	0
$700$	−6.31526	−0.238695
$701$	−18.3145	−0.691728	−0.345864	−	0.938285i	$-0.612414\pi$
$701$	−18.3145	−0.691728	−0.345864	+	0.938285i	$0.612414\pi$
$702$	0	0
$703$	13.7862	0.519957
$704$	13.0490	0.491803
$705$	0	0
$706$	−9.35253	−0.351987
$707$	−15.3540	−0.577446
$708$	0	0
$709$	−39.4547	−1.48175	−0.740876	−	0.671642i	$-0.765589\pi$
$709$	−39.4547	−1.48175	−0.740876	+	0.671642i	$0.765589\pi$
$710$	−34.9149	−1.31033
$711$	0	0
$712$	1.06573	0.0399398
$713$	−1.44800	−0.0542280
$714$	0	0
$715$	−7.06255	−0.264125
$716$	−21.7696	−0.813568
$717$	0	0
$718$	−66.1298	−2.46794
$719$	−27.9814	−1.04353	−0.521765	−	0.853089i	$-0.674726\pi$
$719$	−27.9814	−1.04353	−0.521765	+	0.853089i	$0.674726\pi$
$720$	0	0
$721$	−11.6950	−0.435543
$722$	39.3427	1.46418
$723$	0	0
$724$	43.4809	1.61595
$725$	−7.47162	−0.277489
$726$	0	0
$727$	−13.8256	−0.512764	−0.256382	−	0.966576i	$-0.582530\pi$
$727$	−13.8256	−0.512764	−0.256382	+	0.966576i	$0.582530\pi$
$728$	9.90688	0.367173
$729$	0	0
$730$	58.0777	2.14955
$731$	−45.3261	−1.67644
$732$	0	0
$733$	31.5671	1.16596	0.582978	−	0.812488i	$-0.301887\pi$
$733$	31.5671	1.16596	0.582978	+	0.812488i	$0.301887\pi$
$734$	83.1011	3.06732
$735$	0	0
$736$	−2.60008	−0.0958404
$737$	12.8483	0.473273
$738$	0	0
$739$	25.6387	0.943135	0.471567	−	0.881830i	$-0.343688\pi$
$739$	25.6387	0.943135	0.471567	+	0.881830i	$0.343688\pi$
$740$	−57.8160	−2.12536
$741$	0	0
$742$	−0.699515	−0.0256800
$743$	−10.2003	−0.374212	−0.187106	−	0.982340i	$-0.559911\pi$
$743$	−10.2003	−0.374212	−0.187106	+	0.982340i	$0.559911\pi$
$744$	0	0
$745$	14.8589	0.544387
$746$	−23.5419	−0.861932
$747$	0	0
$748$	20.7185	0.757544
$749$	13.0243	0.475897
$750$	0	0
$751$	−50.0262	−1.82548	−0.912742	−	0.408538i	$-0.866039\pi$
$751$	−50.0262	−1.82548	−0.912742	+	0.408538i	$0.866039\pi$
$752$	−3.37954	−0.123239
$753$	0	0
$754$	33.1332	1.20664
$755$	−30.1796	−1.09835
$756$	0	0
$757$	−3.80748	−0.138385	−0.0691926	−	0.997603i	$-0.522042\pi$
$757$	−3.80748	−0.138385	−0.0691926	+	0.997603i	$0.522042\pi$
$758$	52.7264	1.91511
$759$	0	0
$760$	5.25682	0.190685
$761$	−28.4714	−1.03209	−0.516044	−	0.856562i	$-0.672596\pi$
$761$	−28.4714	−1.03209	−0.516044	+	0.856562i	$0.672596\pi$
$762$	0	0
$763$	−17.7802	−0.643687
$764$	−42.4818	−1.53694
$765$	0	0
$766$	31.4874	1.13768
$767$	−43.0392	−1.55406
$768$	0	0
$769$	21.6675	0.781350	0.390675	−	0.920529i	$-0.372242\pi$
$769$	21.6675	0.781350	0.390675	+	0.920529i	$0.372242\pi$
$770$	−3.69364	−0.133109
$771$	0	0
$772$	−36.7807	−1.32377
$773$	18.7695	0.675093	0.337546	−	0.941309i	$-0.390403\pi$
$773$	18.7695	0.675093	0.337546	+	0.941309i	$0.390403\pi$
$774$	0	0
$775$	−7.45591	−0.267824
$776$	11.7223	0.420807
$777$	0	0
$778$	−82.6105	−2.96173
$779$	11.1678	0.400129
$780$	0	0
$781$	9.11055	0.326001
$782$	−6.21345	−0.222192
$783$	0	0
$784$	3.71486	0.132674
$785$	3.36677	0.120165
$786$	0	0
$787$	10.7123	0.381851	0.190926	−	0.981605i	$-0.438851\pi$
$787$	10.7123	0.381851	0.190926	+	0.981605i	$0.438851\pi$
$788$	−49.5411	−1.76483
$789$	0	0
$790$	−24.8029	−0.882447
$791$	8.92119	0.317201
$792$	0	0
$793$	−4.15967	−0.147714
$794$	2.18600	0.0775782
$795$	0	0
$796$	−33.9683	−1.20397
$797$	−51.8932	−1.83815	−0.919076	−	0.394081i	$-0.871063\pi$
$797$	−51.8932	−1.83815	−0.919076	+	0.394081i	$0.871063\pi$
$798$	0	0
$799$	−36.9751	−1.30809
$800$	−13.3881	−0.473342
$801$	0	0
$802$	5.50845	0.194510
$803$	−15.1546	−0.534793
$804$	0	0
$805$	0.672872	0.0237156
$806$	33.0635	1.16461
$807$	0	0
$808$	39.3661	1.38490
$809$	−1.20097	−0.0422239	−0.0211120	−	0.999777i	$-0.506721\pi$
$809$	−1.20097	−0.0422239	−0.0211120	+	0.999777i	$0.506721\pi$
$810$	0	0
$811$	−11.2842	−0.396241	−0.198121	−	0.980178i	$-0.563484\pi$
$811$	−11.2842	−0.396241	−0.198121	+	0.980178i	$0.563484\pi$
$812$	10.5259	0.369387
$813$	0	0
$814$	24.8358	0.870493
$815$	−20.0697	−0.703011
$816$	0	0
$817$	8.48300	0.296783
$818$	−35.5535	−1.24310
$819$	0	0
$820$	−46.8351	−1.63555
$821$	24.1209	0.841824	0.420912	−	0.907102i	$-0.361710\pi$
$821$	24.1209	0.841824	0.420912	+	0.907102i	$0.361710\pi$
$822$	0	0
$823$	−4.05215	−0.141249	−0.0706246	−	0.997503i	$-0.522499\pi$
$823$	−4.05215	−0.141249	−0.0706246	+	0.997503i	$0.522499\pi$
$824$	29.9847	1.04457
$825$	0	0
$826$	−22.5090	−0.783189
$827$	−43.4022	−1.50924	−0.754621	−	0.656161i	$-0.772179\pi$
$827$	−43.4022	−1.50924	−0.754621	+	0.656161i	$0.772179\pi$
$828$	0	0
$829$	−26.5208	−0.921106	−0.460553	−	0.887632i	$-0.652349\pi$
$829$	−26.5208	−0.921106	−0.460553	+	0.887632i	$0.652349\pi$
$830$	18.6932	0.648851
$831$	0	0
$832$	54.2796	1.88181
$833$	40.6439	1.40823
$834$	0	0
$835$	−39.1251	−1.35398
$836$	−3.87757	−0.134109
$837$	0	0
$838$	89.2469	3.08298
$839$	8.93322	0.308409	0.154205	−	0.988039i	$-0.450719\pi$
$839$	8.93322	0.308409	0.154205	+	0.988039i	$0.450719\pi$
$840$	0	0
$841$	−16.5467	−0.570577
$842$	40.2014	1.38543
$843$	0	0
$844$	−48.9415	−1.68463
$845$	−7.30564	−0.251322
$846$	0	0
$847$	0.963803	0.0331167
$848$	−0.196754	−0.00675654
$849$	0	0
$850$	−31.9937	−1.09738
$851$	−4.52435	−0.155093
$852$	0	0
$853$	34.3410	1.17581	0.587906	−	0.808929i	$-0.299952\pi$
$853$	34.3410	1.17581	0.587906	+	0.808929i	$0.299952\pi$
$854$	−2.17546	−0.0744427
$855$	0	0
$856$	−33.3930	−1.14135
$857$	−12.0616	−0.412018	−0.206009	−	0.978550i	$-0.566048\pi$
$857$	−12.0616	−0.412018	−0.206009	+	0.978550i	$0.566048\pi$
$858$	0	0
$859$	−1.74396	−0.0595032	−0.0297516	−	0.999557i	$-0.509472\pi$
$859$	−1.74396	−0.0595032	−0.0297516	+	0.999557i	$0.509472\pi$
$860$	−35.5756	−1.21312
$861$	0	0
$862$	−74.6895	−2.54393
$863$	28.4493	0.968424	0.484212	−	0.874951i	$-0.339106\pi$
$863$	28.4493	0.968424	0.484212	+	0.874951i	$0.339106\pi$
$864$	0	0
$865$	−34.4992	−1.17301
$866$	−30.7291	−1.04422
$867$	0	0
$868$	10.5038	0.356521
$869$	6.47197	0.219546
$870$	0	0
$871$	53.4446	1.81090
$872$	45.5867	1.54376
$873$	0	0
$874$	1.16288	0.0393349
$875$	11.6467	0.393731
$876$	0	0
$877$	9.88665	0.333848	0.166924	−	0.985970i	$-0.446616\pi$
$877$	9.88665	0.333848	0.166924	+	0.985970i	$0.446616\pi$
$878$	−33.9367	−1.14531
$879$	0	0
$880$	−1.03891	−0.0350218
$881$	24.3630	0.820812	0.410406	−	0.911903i	$-0.365387\pi$
$881$	24.3630	0.820812	0.410406	+	0.911903i	$0.365387\pi$
$882$	0	0
$883$	45.7119	1.53833	0.769164	−	0.639052i	$-0.220673\pi$
$883$	45.7119	1.53833	0.769164	+	0.639052i	$0.220673\pi$
$884$	86.1822	2.89862
$885$	0	0
$886$	58.1930	1.95503
$887$	39.8349	1.33752	0.668762	−	0.743476i	$-0.266824\pi$
$887$	39.8349	1.33752	0.668762	+	0.743476i	$0.266824\pi$
$888$	0	0
$889$	11.3800	0.381672
$890$	−1.65281	−0.0554022
$891$	0	0
$892$	−44.9956	−1.50656
$893$	6.92008	0.231572
$894$	0	0
$895$	11.9433	0.399220
$896$	16.1987	0.541160
$897$	0	0
$898$	38.5722	1.28717
$899$	12.4271	0.414466
$900$	0	0
$901$	−2.15266	−0.0717154
$902$	20.1187	0.669881
$903$	0	0
$904$	−22.8730	−0.760746
$905$	−23.8546	−0.792952
$906$	0	0
$907$	20.1360	0.668604	0.334302	−	0.942466i	$-0.391499\pi$
$907$	20.1360	0.668604	0.334302	+	0.942466i	$0.391499\pi$
$908$	−80.0479	−2.65648
$909$	0	0
$910$	−15.3643	−0.509322
$911$	−42.6508	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$911$	−42.6508	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$912$	0	0
$913$	−4.87774	−0.161430
$914$	−76.1191	−2.51780
$915$	0	0
$916$	−54.0455	−1.78571
$917$	−13.3727	−0.441606
$918$	0	0
$919$	−41.9945	−1.38527	−0.692636	−	0.721288i	$-0.743551\pi$
$919$	−41.9945	−1.38527	−0.692636	+	0.721288i	$0.743551\pi$
$920$	−1.72518	−0.0568774
$921$	0	0
$922$	−2.69407	−0.0887244
$923$	37.8969	1.24739
$924$	0	0
$925$	−23.2964	−0.765980
$926$	65.6022	2.15582
$927$	0	0
$928$	22.3145	0.732511
$929$	−39.5385	−1.29722	−0.648609	−	0.761122i	$-0.724649\pi$
$929$	−39.5385	−1.29722	−0.648609	+	0.761122i	$0.724649\pi$
$930$	0	0
$931$	−7.60670	−0.249300
$932$	−24.8927	−0.815389
$933$	0	0
$934$	65.0775	2.12940
$935$	−11.3666	−0.371729
$936$	0	0
$937$	21.5016	0.702428	0.351214	−	0.936295i	$-0.385769\pi$
$937$	21.5016	0.702428	0.351214	+	0.936295i	$0.385769\pi$
$938$	27.9509	0.912631
$939$	0	0
$940$	−29.0211	−0.946564
$941$	−39.2600	−1.27984	−0.639920	−	0.768441i	$-0.721033\pi$
$941$	−39.2600	−1.27984	−0.639920	+	0.768441i	$0.721033\pi$
$942$	0	0
$943$	−3.66504	−0.119350
$944$	−6.33115	−0.206061
$945$	0	0
$946$	15.2821	0.496863
$947$	−10.4430	−0.339351	−0.169675	−	0.985500i	$-0.554272\pi$
$947$	−10.4430	−0.339351	−0.169675	+	0.985500i	$0.554272\pi$
$948$	0	0
$949$	−63.0380	−2.04630
$950$	5.98779	0.194269
$951$	0	0
$952$	15.9444	0.516759
$953$	60.5732	1.96216	0.981080	−	0.193605i	$-0.0620180\pi$
$953$	60.5732	1.96216	0.981080	+	0.193605i	$0.0620180\pi$
$954$	0	0
$955$	23.3064	0.754178
$956$	10.5927	0.342593
$957$	0	0
$958$	16.0274	0.517822
$959$	−1.72580	−0.0557291
$960$	0	0
$961$	−18.5991	−0.599969
$962$	103.309	3.33080
$963$	0	0
$964$	51.0443	1.64403
$965$	20.1787	0.649576
$966$	0	0
$967$	−3.40980	−0.109652	−0.0548259	−	0.998496i	$-0.517460\pi$
$967$	−3.40980	−0.109652	−0.0548259	+	0.998496i	$0.517460\pi$
$968$	−2.47110	−0.0794241
$969$	0	0
$970$	−18.1798	−0.583720
$971$	8.09632	0.259823	0.129912	−	0.991526i	$-0.458531\pi$
$971$	8.09632	0.259823	0.129912	+	0.991526i	$0.458531\pi$
$972$	0	0
$973$	−1.50064	−0.0481083
$974$	−65.4908	−2.09846
$975$	0	0
$976$	−0.611895	−0.0195863
$977$	−21.4406	−0.685947	−0.342973	−	0.939345i	$-0.611434\pi$
$977$	−21.4406	−0.685947	−0.342973	+	0.939345i	$0.611434\pi$
$978$	0	0
$979$	0.431277	0.0137837
$980$	31.9006	1.01903
$981$	0	0
$982$	71.6622	2.28683
$983$	−21.8391	−0.696559	−0.348280	−	0.937391i	$-0.613234\pi$
$983$	−21.8391	−0.696559	−0.348280	+	0.937391i	$0.613234\pi$
$984$	0	0
$985$	27.1793	0.866006
$986$	53.3253	1.69822
$987$	0	0
$988$	−16.1294	−0.513145
$989$	−2.78394	−0.0885242
$990$	0	0
$991$	−21.7822	−0.691935	−0.345967	−	0.938247i	$-0.612449\pi$
$991$	−21.7822	−0.691935	−0.345967	+	0.938247i	$0.612449\pi$
$992$	22.2676	0.706998
$993$	0	0
$994$	19.8196	0.628641
$995$	18.6358	0.590793
$996$	0	0
$997$	−55.0499	−1.74345	−0.871724	−	0.489998i	$-0.836998\pi$
$997$	−55.0499	−1.74345	−0.871724	+	0.489998i	$0.836998\pi$
$998$	−30.8855	−0.977664
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.3		13
3.2	odd	2		2013.2.a.g.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.11	✓	13	3.2	odd	2
6039.2.a.h.1.3		13	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.25716 q^{2} +3.09478 q^{4} -1.69786 q^{5} +0.963803 q^{7} -2.47110 q^{8} +O(q^{10})\)	\(q-2.25716 q^{2} +3.09478 q^{4} -1.69786 q^{5} +0.963803 q^{7} -2.47110 q^{8} +3.83236 q^{10} -1.00000 q^{11} -4.15967 q^{13} -2.17546 q^{14} -0.611895 q^{16} -6.69466 q^{17} +1.25294 q^{19} -5.25452 q^{20} +2.25716 q^{22} -0.411188 q^{23} -2.11726 q^{25} +9.38905 q^{26} +2.98276 q^{28} +3.52892 q^{29} +3.52150 q^{31} +6.32334 q^{32} +15.1109 q^{34} -1.63641 q^{35} +11.0031 q^{37} -2.82809 q^{38} +4.19559 q^{40} +8.91329 q^{41} +6.77048 q^{43} -3.09478 q^{44} +0.928119 q^{46} +5.52308 q^{47} -6.07108 q^{49} +4.77899 q^{50} -12.8733 q^{52} +0.321548 q^{53} +1.69786 q^{55} -2.38165 q^{56} -7.96534 q^{58} +10.3468 q^{59} +1.00000 q^{61} -7.94859 q^{62} -13.0490 q^{64} +7.06255 q^{65} -12.8483 q^{67} -20.7185 q^{68} +3.69364 q^{70} -9.11055 q^{71} +15.1546 q^{73} -24.8358 q^{74} +3.87757 q^{76} -0.963803 q^{77} -6.47197 q^{79} +1.03891 q^{80} -20.1187 q^{82} +4.87774 q^{83} +11.3666 q^{85} -15.2821 q^{86} +2.47110 q^{88} -0.431277 q^{89} -4.00910 q^{91} -1.27254 q^{92} -12.4665 q^{94} -2.12732 q^{95} -4.74378 q^{97} +13.7034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Introduction

L-functions

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