LMFDB - Embedded newform 6050.2.a.de.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.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:	$48.3094932229$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.35136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 6x + 9 $ x^4 - 2x^3 - 8x^2 + 6*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.839882$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.57193 q^{3} +1.00000 q^{4} +2.57193 q^{6} +3.57193 q^{7} -1.00000 q^{8} +3.61484 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.57193 q^{3} +1.00000 q^{4} +2.57193 q^{6} +3.57193 q^{7} -1.00000 q^{8} +3.61484 q^{9} -2.57193 q^{12} +5.30398 q^{13} -3.57193 q^{14} +1.00000 q^{16} +2.61484 q^{17} -3.61484 q^{18} -7.34689 q^{19} -9.18677 q^{21} +0.957096 q^{23} +2.57193 q^{24} -5.30398 q^{26} -1.58132 q^{27} +3.57193 q^{28} +10.3306 q^{29} -7.13448 q^{31} -1.00000 q^{32} -2.61484 q^{34} +3.61484 q^{36} -5.14386 q^{37} +7.34689 q^{38} -13.6415 q^{39} -4.56255 q^{41} +9.18677 q^{42} -0.356271 q^{43} -0.957096 q^{46} -10.3642 q^{47} -2.57193 q^{48} +5.75870 q^{49} -6.72518 q^{51} +5.30398 q^{52} +1.56255 q^{53} +1.58132 q^{54} -3.57193 q^{56} +18.8957 q^{57} -10.3306 q^{58} -3.66712 q^{59} -13.4212 q^{61} +7.13448 q^{62} +12.9119 q^{63} +1.00000 q^{64} -5.56255 q^{67} +2.61484 q^{68} -2.46159 q^{69} -14.5697 q^{71} -3.61484 q^{72} -7.73205 q^{73} +5.14386 q^{74} -7.34689 q^{76} +13.6415 q^{78} -2.32024 q^{79} -6.77747 q^{81} +4.56255 q^{82} -1.14386 q^{83} -9.18677 q^{84} +0.356271 q^{86} -26.5697 q^{87} -10.1439 q^{89} +18.9455 q^{91} +0.957096 q^{92} +18.3494 q^{93} +10.3642 q^{94} +2.57193 q^{96} +12.6032 q^{97} -5.75870 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 16 q^{19} - 18 q^{21} - 2 q^{23} - 2 q^{24} - 2 q^{26} + 8 q^{27} + 2 q^{28} - 2 q^{29} - 6 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{36} + 4 q^{37} + 16 q^{38} - 24 q^{39} - 8 q^{41} + 18 q^{42} + 14 q^{43} + 2 q^{46} - 6 q^{47} + 2 q^{48} - 8 q^{49} + 12 q^{51} + 2 q^{52} - 4 q^{53} - 8 q^{54} - 2 q^{56} - 12 q^{57} + 2 q^{58} - 12 q^{59} - 34 q^{61} + 6 q^{62} - 6 q^{63} + 4 q^{64} - 12 q^{67} + 4 q^{68} - 30 q^{69} - 8 q^{72} - 24 q^{73} - 4 q^{74} - 16 q^{76} + 24 q^{78} - 20 q^{79} + 8 q^{81} + 8 q^{82} + 20 q^{83} - 18 q^{84} - 14 q^{86} - 48 q^{87} - 16 q^{89} + 26 q^{91} - 2 q^{92} + 26 q^{93} + 6 q^{94} - 2 q^{96} + 8 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 16 * q^19 - 18 * q^21 - 2 * q^23 - 2 * q^24 - 2 * q^26 + 8 * q^27 + 2 * q^28 - 2 * q^29 - 6 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^36 + 4 * q^37 + 16 * q^38 - 24 * q^39 - 8 * q^41 + 18 * q^42 + 14 * q^43 + 2 * q^46 - 6 * q^47 + 2 * q^48 - 8 * q^49 + 12 * q^51 + 2 * q^52 - 4 * q^53 - 8 * q^54 - 2 * q^56 - 12 * q^57 + 2 * q^58 - 12 * q^59 - 34 * q^61 + 6 * q^62 - 6 * q^63 + 4 * q^64 - 12 * q^67 + 4 * q^68 - 30 * q^69 - 8 * q^72 - 24 * q^73 - 4 * q^74 - 16 * q^76 + 24 * q^78 - 20 * q^79 + 8 * q^81 + 8 * q^82 + 20 * q^83 - 18 * q^84 - 14 * q^86 - 48 * q^87 - 16 * q^89 + 26 * q^91 - 2 * q^92 + 26 * q^93 + 6 * q^94 - 2 * q^96 + 8 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.57193	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$3$	−2.57193	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.57193	1.04999
$7$	3.57193	1.35006	0.675032	−	0.737789i	$-0.264130\pi$
$7$	3.57193	1.35006	0.675032	+	0.737789i	$0.264130\pi$
$8$	−1.00000	−0.353553
$9$	3.61484	1.20495
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−2.57193	−0.742453
$13$	5.30398	1.47106	0.735530	−	0.677492i	$-0.236933\pi$
$13$	5.30398	1.47106	0.735530	+	0.677492i	$0.236933\pi$
$14$	−3.57193	−0.954639
$15$	0	0
$16$	1.00000	0.250000
$17$	2.61484	0.634191	0.317095	−	0.948394i	$-0.397292\pi$
$17$	2.61484	0.634191	0.317095	+	0.948394i	$0.397292\pi$
$18$	−3.61484	−0.852025
$19$	−7.34689	−1.68549	−0.842746	−	0.538312i	$-0.819062\pi$
$19$	−7.34689	−1.68549	−0.842746	+	0.538312i	$0.819062\pi$
$20$	0	0
$21$	−9.18677	−2.00472
$22$	0	0
$23$	0.957096	0.199568	0.0997842	−	0.995009i	$-0.468185\pi$
$23$	0.957096	0.199568	0.0997842	+	0.995009i	$0.468185\pi$
$24$	2.57193	0.524993
$25$	0	0
$26$	−5.30398	−1.04020
$27$	−1.58132	−0.304324
$28$	3.57193	0.675032
$29$	10.3306	1.91835	0.959175	−	0.282813i	$-0.0912674\pi$
$29$	10.3306	1.91835	0.959175	+	0.282813i	$0.0912674\pi$
$30$	0	0
$31$	−7.13448	−1.28139	−0.640695	−	0.767795i	$-0.721354\pi$
$31$	−7.13448	−1.28139	−0.640695	+	0.767795i	$0.721354\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.61484	−0.448441
$35$	0	0
$36$	3.61484	0.602473
$37$	−5.14386	−0.845646	−0.422823	−	0.906212i	$-0.638961\pi$
$37$	−5.14386	−0.845646	−0.422823	+	0.906212i	$0.638961\pi$
$38$	7.34689	1.19182
$39$	−13.6415	−2.18439
$40$	0	0
$41$	−4.56255	−0.712550	−0.356275	−	0.934381i	$-0.615953\pi$
$41$	−4.56255	−0.712550	−0.356275	+	0.934381i	$0.615953\pi$
$42$	9.18677	1.41755
$43$	−0.356271	−0.0543308	−0.0271654	−	0.999631i	$-0.508648\pi$
$43$	−0.356271	−0.0543308	−0.0271654	+	0.999631i	$0.508648\pi$
$44$	0	0
$45$	0	0
$46$	−0.957096	−0.141116
$47$	−10.3642	−1.51177	−0.755884	−	0.654706i	$-0.772792\pi$
$47$	−10.3642	−1.51177	−0.755884	+	0.654706i	$0.772792\pi$
$48$	−2.57193	−0.371226
$49$	5.75870	0.822672
$50$	0	0
$51$	−6.72518	−0.941714
$52$	5.30398	0.735530
$53$	1.56255	0.214633	0.107316	−	0.994225i	$-0.465774\pi$
$53$	1.56255	0.214633	0.107316	+	0.994225i	$0.465774\pi$
$54$	1.58132	0.215190
$55$	0	0
$56$	−3.57193	−0.477320
$57$	18.8957	2.50280
$58$	−10.3306	−1.35648
$59$	−3.66712	−0.477419	−0.238709	−	0.971091i	$-0.576724\pi$
$59$	−3.66712	−0.477419	−0.238709	+	0.971091i	$0.576724\pi$
$60$	0	0
$61$	−13.4212	−1.71841	−0.859204	−	0.511633i	$-0.829041\pi$
$61$	−13.4212	−1.71841	−0.859204	+	0.511633i	$0.829041\pi$
$62$	7.13448	0.906080
$63$	12.9119	1.62675
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.56255	−0.679573	−0.339787	−	0.940503i	$-0.610355\pi$
$67$	−5.56255	−0.679573	−0.339787	+	0.940503i	$0.610355\pi$
$68$	2.61484	0.317095
$69$	−2.46159	−0.296340
$70$	0	0
$71$	−14.5697	−1.72910	−0.864552	−	0.502543i	$-0.832398\pi$
$71$	−14.5697	−1.72910	−0.864552	+	0.502543i	$0.832398\pi$
$72$	−3.61484	−0.426013
$73$	−7.73205	−0.904968	−0.452484	−	0.891772i	$-0.649462\pi$
$73$	−7.73205	−0.904968	−0.452484	+	0.891772i	$0.649462\pi$
$74$	5.14386	0.597962
$75$	0	0
$76$	−7.34689	−0.842746
$77$	0	0
$78$	13.6415	1.54459
$79$	−2.32024	−0.261047	−0.130524	−	0.991445i	$-0.541666\pi$
$79$	−2.32024	−0.261047	−0.130524	+	0.991445i	$0.541666\pi$
$80$	0	0
$81$	−6.77747	−0.753052
$82$	4.56255	0.503849
$83$	−1.14386	−0.125555	−0.0627777	−	0.998028i	$-0.519996\pi$
$83$	−1.14386	−0.125555	−0.0627777	+	0.998028i	$0.519996\pi$
$84$	−9.18677	−1.00236
$85$	0	0
$86$	0.356271	0.0384177
$87$	−26.5697	−2.84857
$88$	0	0
$89$	−10.1439	−1.07525	−0.537624	−	0.843185i	$-0.680678\pi$
$89$	−10.1439	−1.07525	−0.537624	+	0.843185i	$0.680678\pi$
$90$	0	0
$91$	18.9455	1.98602
$92$	0.957096	0.0997842
$93$	18.3494	1.90274
$94$	10.3642	1.06898
$95$	0	0
$96$	2.57193	0.262497
$97$	12.6032	1.27966	0.639831	−	0.768516i	$-0.279004\pi$
$97$	12.6032	1.27966	0.639831	+	0.768516i	$0.279004\pi$
$98$	−5.75870	−0.581717
$99$	0	0
$100$	0	0
$101$	−14.7064	−1.46334	−0.731671	−	0.681658i	$-0.761259\pi$
$101$	−14.7064	−1.46334	−0.731671	+	0.681658i	$0.761259\pi$
$102$	6.72518	0.665892
$103$	1.15325	0.113633	0.0568165	−	0.998385i	$-0.481905\pi$
$103$	1.15325	0.113633	0.0568165	+	0.998385i	$0.481905\pi$
$104$	−5.30398	−0.520098
$105$	0	0
$106$	−1.56255	−0.151768
$107$	14.2877	1.38125	0.690623	−	0.723215i	$-0.257336\pi$
$107$	14.2877	1.38125	0.690623	+	0.723215i	$0.257336\pi$
$108$	−1.58132	−0.152162
$109$	−0.770328	−0.0737840	−0.0368920	−	0.999319i	$-0.511746\pi$
$109$	−0.770328	−0.0737840	−0.0368920	+	0.999319i	$0.511746\pi$
$110$	0	0
$111$	13.2297	1.25570
$112$	3.57193	0.337516
$113$	20.6729	1.94474	0.972371	−	0.233440i	$-0.0749984\pi$
$113$	20.6729	1.94474	0.972371	+	0.233440i	$0.0749984\pi$
$114$	−18.8957	−1.76974
$115$	0	0
$116$	10.3306	0.959175
$117$	19.1730	1.77255
$118$	3.66712	0.337586
$119$	9.34002	0.856198
$120$	0	0
$121$	0	0
$122$	13.4212	1.21510
$123$	11.7346	1.05807
$124$	−7.13448	−0.640695
$125$	0	0
$126$	−12.9119	−1.15029
$127$	−21.4478	−1.90319	−0.951594	−	0.307357i	$-0.900556\pi$
$127$	−21.4478	−1.90319	−0.951594	+	0.307357i	$0.900556\pi$
$128$	−1.00000	−0.0883883
$129$	0.916305	0.0806761
$130$	0	0
$131$	−5.77971	−0.504976	−0.252488	−	0.967600i	$-0.581249\pi$
$131$	−5.77971	−0.504976	−0.252488	+	0.967600i	$0.581249\pi$
$132$	0	0
$133$	−26.2426	−2.27552
$134$	5.56255	0.480531
$135$	0	0
$136$	−2.61484	−0.224220
$137$	12.5103	1.06882	0.534412	−	0.845224i	$-0.320533\pi$
$137$	12.5103	1.06882	0.534412	+	0.845224i	$0.320533\pi$
$138$	2.46159	0.209544
$139$	19.0987	1.61993	0.809966	−	0.586477i	$-0.199485\pi$
$139$	19.0987	1.61993	0.809966	+	0.586477i	$0.199485\pi$
$140$	0	0
$141$	26.6559	2.24483
$142$	14.5697	1.22266
$143$	0	0
$144$	3.61484	0.301236
$145$	0	0
$146$	7.73205	0.639909
$147$	−14.8110	−1.22159
$148$	−5.14386	−0.422823
$149$	6.43521	0.527193	0.263596	−	0.964633i	$-0.415091\pi$
$149$	6.43521	0.527193	0.263596	+	0.964633i	$0.415091\pi$
$150$	0	0
$151$	−2.02202	−0.164550	−0.0822750	−	0.996610i	$-0.526219\pi$
$151$	−2.02202	−0.164550	−0.0822750	+	0.996610i	$0.526219\pi$
$152$	7.34689	0.595911
$153$	9.45220	0.764165
$154$	0	0
$155$	0	0
$156$	−13.6415	−1.09219
$157$	−11.8074	−0.942331	−0.471166	−	0.882045i	$-0.656167\pi$
$157$	−11.8074	−0.942331	−0.471166	+	0.882045i	$0.656167\pi$
$158$	2.32024	0.184588
$159$	−4.01877	−0.318709
$160$	0	0
$161$	3.41868	0.269430
$162$	6.77747	0.532488
$163$	−11.8204	−0.925843	−0.462922	−	0.886399i	$-0.653199\pi$
$163$	−11.8204	−0.925843	−0.462922	+	0.886399i	$0.653199\pi$
$164$	−4.56255	−0.356275
$165$	0	0
$166$	1.14386	0.0887811
$167$	−12.5384	−0.970252	−0.485126	−	0.874444i	$-0.661226\pi$
$167$	−12.5384	−0.970252	−0.485126	+	0.874444i	$0.661226\pi$
$168$	9.18677	0.708775
$169$	15.1322	1.16402
$170$	0	0
$171$	−26.5578	−2.03092
$172$	−0.356271	−0.0271654
$173$	−5.51964	−0.419651	−0.209825	−	0.977739i	$-0.567290\pi$
$173$	−5.51964	−0.419651	−0.209825	+	0.977739i	$0.567290\pi$
$174$	26.5697	2.01424
$175$	0	0
$176$	0	0
$177$	9.43159	0.708922
$178$	10.1439	0.760315
$179$	7.29711	0.545412	0.272706	−	0.962097i	$-0.412081\pi$
$179$	7.29711	0.545412	0.272706	+	0.962097i	$0.412081\pi$
$180$	0	0
$181$	13.4580	1.00032	0.500162	−	0.865932i	$-0.333274\pi$
$181$	13.4580	1.00032	0.500162	+	0.865932i	$0.333274\pi$
$182$	−18.9455	−1.40433
$183$	34.5184	2.55167
$184$	−0.957096	−0.0705581
$185$	0	0
$186$	−18.3494	−1.34544
$187$	0	0
$188$	−10.3642	−0.755884
$189$	−5.64836	−0.410857
$190$	0	0
$191$	11.0393	0.798775	0.399387	−	0.916782i	$-0.369223\pi$
$191$	11.0393	0.798775	0.399387	+	0.916782i	$0.369223\pi$
$192$	−2.57193	−0.185613
$193$	−3.87116	−0.278652	−0.139326	−	0.990247i	$-0.544494\pi$
$193$	−3.87116	−0.278652	−0.139326	+	0.990247i	$0.544494\pi$
$194$	−12.6032	−0.904858
$195$	0	0
$196$	5.75870	0.411336
$197$	−13.2055	−0.940856	−0.470428	−	0.882439i	$-0.655900\pi$
$197$	−13.2055	−0.940856	−0.470428	+	0.882439i	$0.655900\pi$
$198$	0	0
$199$	−8.95710	−0.634952	−0.317476	−	0.948266i	$-0.602835\pi$
$199$	−8.95710	−0.634952	−0.317476	+	0.948266i	$0.602835\pi$
$200$	0	0
$201$	14.3065	1.00910
$202$	14.7064	1.03474
$203$	36.9003	2.58989
$204$	−6.72518	−0.470857
$205$	0	0
$206$	−1.15325	−0.0803507
$207$	3.45975	0.240469
$208$	5.30398	0.367765
$209$	0	0
$210$	0	0
$211$	6.94547	0.478146	0.239073	−	0.971002i	$-0.423157\pi$
$211$	6.94547	0.478146	0.239073	+	0.971002i	$0.423157\pi$
$212$	1.56255	0.107316
$213$	37.4723	2.56756
$214$	−14.2877	−0.976689
$215$	0	0
$216$	1.58132	0.107595
$217$	−25.4839	−1.72996
$218$	0.770328	0.0521732
$219$	19.8863	1.34379
$220$	0	0
$221$	13.8690	0.932933
$222$	−13.2297	−0.887917
$223$	5.04867	0.338084	0.169042	−	0.985609i	$-0.445933\pi$
$223$	5.04867	0.338084	0.169042	+	0.985609i	$0.445933\pi$
$224$	−3.57193	−0.238660
$225$	0	0
$226$	−20.6729	−1.37514
$227$	−10.6876	−0.709364	−0.354682	−	0.934987i	$-0.615411\pi$
$227$	−10.6876	−0.709364	−0.354682	+	0.934987i	$0.615411\pi$
$228$	18.8957	1.25140
$229$	11.0987	0.733424	0.366712	−	0.930335i	$-0.380483\pi$
$229$	11.0987	0.733424	0.366712	+	0.930335i	$0.380483\pi$
$230$	0	0
$231$	0	0
$232$	−10.3306	−0.678239
$233$	−3.72518	−0.244045	−0.122022	−	0.992527i	$-0.538938\pi$
$233$	−3.72518	−0.244045	−0.122022	+	0.992527i	$0.538938\pi$
$234$	−19.1730	−1.25338
$235$	0	0
$236$	−3.66712	−0.238709
$237$	5.96749	0.387630
$238$	−9.34002	−0.605423
$239$	3.28997	0.212811	0.106405	−	0.994323i	$-0.466066\pi$
$239$	3.28997	0.212811	0.106405	+	0.994323i	$0.466066\pi$
$240$	0	0
$241$	−2.75055	−0.177178	−0.0885892	−	0.996068i	$-0.528236\pi$
$241$	−2.75055	−0.177178	−0.0885892	+	0.996068i	$0.528236\pi$
$242$	0	0
$243$	22.1751	1.42254
$244$	−13.4212	−0.859204
$245$	0	0
$246$	−11.7346	−0.748169
$247$	−38.9678	−2.47946
$248$	7.13448	0.453040
$249$	2.94194	0.186438
$250$	0	0
$251$	27.0049	1.70453	0.852267	−	0.523107i	$-0.175227\pi$
$251$	27.0049	1.70453	0.852267	+	0.523107i	$0.175227\pi$
$252$	12.9119	0.813376
$253$	0	0
$254$	21.4478	1.34576
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.9213	−0.930767	−0.465384	−	0.885109i	$-0.654084\pi$
$257$	−14.9213	−0.930767	−0.465384	+	0.885109i	$0.654084\pi$
$258$	−0.916305	−0.0570466
$259$	−18.3735	−1.14168
$260$	0	0
$261$	37.3435	2.31151
$262$	5.77971	0.357072
$263$	9.28997	0.572844	0.286422	−	0.958104i	$-0.407534\pi$
$263$	9.28997	0.572844	0.286422	+	0.958104i	$0.407534\pi$
$264$	0	0
$265$	0	0
$266$	26.2426	1.60904
$267$	26.0893	1.59664
$268$	−5.56255	−0.339787
$269$	−17.6649	−1.07705	−0.538523	−	0.842611i	$-0.681018\pi$
$269$	−17.6649	−1.07705	−0.538523	+	0.842611i	$0.681018\pi$
$270$	0	0
$271$	−16.2274	−0.985746	−0.492873	−	0.870101i	$-0.664053\pi$
$271$	−16.2274	−0.985746	−0.492873	+	0.870101i	$0.664053\pi$
$272$	2.61484	0.158548
$273$	−48.7265	−2.94906
$274$	−12.5103	−0.755772
$275$	0	0
$276$	−2.46159	−0.148170
$277$	−10.1946	−0.612537	−0.306269	−	0.951945i	$-0.599081\pi$
$277$	−10.1946	−0.612537	−0.306269	+	0.951945i	$0.599081\pi$
$278$	−19.0987	−1.14546
$279$	−25.7900	−1.54401
$280$	0	0
$281$	−15.2149	−0.907646	−0.453823	−	0.891092i	$-0.649940\pi$
$281$	−15.2149	−0.907646	−0.453823	+	0.891092i	$0.649940\pi$
$282$	−26.6559	−1.58734
$283$	−16.4268	−0.976474	−0.488237	−	0.872711i	$-0.662360\pi$
$283$	−16.4268	−0.976474	−0.488237	+	0.872711i	$0.662360\pi$
$284$	−14.5697	−0.864552
$285$	0	0
$286$	0	0
$287$	−16.2971	−0.961988
$288$	−3.61484	−0.213006
$289$	−10.1626	−0.597802
$290$	0	0
$291$	−32.4146	−1.90018
$292$	−7.73205	−0.452484
$293$	5.77170	0.337186	0.168593	−	0.985686i	$-0.446078\pi$
$293$	5.77170	0.337186	0.168593	+	0.985686i	$0.446078\pi$
$294$	14.8110	0.863794
$295$	0	0
$296$	5.14386	0.298981
$297$	0	0
$298$	−6.43521	−0.372782
$299$	5.07642	0.293577
$300$	0	0
$301$	−1.27258	−0.0733500
$302$	2.02202	0.116354
$303$	37.8239	2.17293
$304$	−7.34689	−0.421373
$305$	0	0
$306$	−9.45220	−0.540346
$307$	4.31873	0.246483	0.123242	−	0.992377i	$-0.460671\pi$
$307$	4.31873	0.246483	0.123242	+	0.992377i	$0.460671\pi$
$308$	0	0
$309$	−2.96608	−0.168734
$310$	0	0
$311$	−3.70866	−0.210299	−0.105149	−	0.994456i	$-0.533532\pi$
$311$	−3.70866	−0.210299	−0.105149	+	0.994456i	$0.533532\pi$
$312$	13.6415	0.772297
$313$	−4.44322	−0.251146	−0.125573	−	0.992084i	$-0.540077\pi$
$313$	−4.44322	−0.251146	−0.125573	+	0.992084i	$0.540077\pi$
$314$	11.8074	0.666329
$315$	0	0
$316$	−2.32024	−0.130524
$317$	4.45398	0.250160	0.125080	−	0.992147i	$-0.460081\pi$
$317$	4.45398	0.250160	0.125080	+	0.992147i	$0.460081\pi$
$318$	4.01877	0.225361
$319$	0	0
$320$	0	0
$321$	−36.7471	−2.05102
$322$	−3.41868	−0.190516
$323$	−19.2109	−1.06892
$324$	−6.77747	−0.376526
$325$	0	0
$326$	11.8204	0.654670
$327$	1.98123	0.109562
$328$	4.56255	0.251925
$329$	−37.0201	−2.04098
$330$	0	0
$331$	−2.84089	−0.156150	−0.0780748	−	0.996948i	$-0.524877\pi$
$331$	−2.84089	−0.156150	−0.0780748	+	0.996948i	$0.524877\pi$
$332$	−1.14386	−0.0627777
$333$	−18.5942	−1.01896
$334$	12.5384	0.686071
$335$	0	0
$336$	−9.18677	−0.501179
$337$	−0.953482	−0.0519395	−0.0259697	−	0.999663i	$-0.508267\pi$
$337$	−0.953482	−0.0519395	−0.0259697	+	0.999663i	$0.508267\pi$
$338$	−15.1322	−0.823085
$339$	−53.1693	−2.88776
$340$	0	0
$341$	0	0
$342$	26.5578	1.43608
$343$	−4.43384	−0.239405
$344$	0.356271	0.0192088
$345$	0	0
$346$	5.51964	0.296738
$347$	27.1581	1.45793	0.728963	−	0.684553i	$-0.240003\pi$
$347$	27.1581	1.45793	0.728963	+	0.684553i	$0.240003\pi$
$348$	−26.5697	−1.42428
$349$	−27.8595	−1.49129	−0.745643	−	0.666345i	$-0.767858\pi$
$349$	−27.8595	−1.49129	−0.745643	+	0.666345i	$0.767858\pi$
$350$	0	0
$351$	−8.38728	−0.447680
$352$	0	0
$353$	13.0071	0.692300	0.346150	−	0.938179i	$-0.387489\pi$
$353$	13.0071	0.692300	0.346150	+	0.938179i	$0.387489\pi$
$354$	−9.43159	−0.501284
$355$	0	0
$356$	−10.1439	−0.537624
$357$	−24.0219	−1.27137
$358$	−7.29711	−0.385664
$359$	11.6765	0.616263	0.308131	−	0.951344i	$-0.400296\pi$
$359$	11.6765	0.616263	0.308131	+	0.951344i	$0.400296\pi$
$360$	0	0
$361$	34.9767	1.84088
$362$	−13.4580	−0.707335
$363$	0	0
$364$	18.9455	0.993012
$365$	0	0
$366$	−34.5184	−1.80431
$367$	−26.7752	−1.39766	−0.698828	−	0.715290i	$-0.746295\pi$
$367$	−26.7752	−1.39766	−0.698828	+	0.715290i	$0.746295\pi$
$368$	0.957096	0.0498921
$369$	−16.4929	−0.858584
$370$	0	0
$371$	5.58132	0.289768
$372$	18.3494	0.951372
$373$	−10.7493	−0.556579	−0.278289	−	0.960497i	$-0.589767\pi$
$373$	−10.7493	−0.556579	−0.278289	+	0.960497i	$0.589767\pi$
$374$	0	0
$375$	0	0
$376$	10.3642	0.534490
$377$	54.7935	2.82201
$378$	5.64836	0.290520
$379$	−30.1112	−1.54671	−0.773355	−	0.633973i	$-0.781423\pi$
$379$	−30.1112	−1.54671	−0.773355	+	0.633973i	$0.781423\pi$
$380$	0	0
$381$	55.1624	2.82606
$382$	−11.0393	−0.564819
$383$	−2.10458	−0.107539	−0.0537694	−	0.998553i	$-0.517124\pi$
$383$	−2.10458	−0.107539	−0.0537694	+	0.998553i	$0.517124\pi$
$384$	2.57193	0.131248
$385$	0	0
$386$	3.87116	0.197037
$387$	−1.28786	−0.0654656
$388$	12.6032	0.639831
$389$	25.8293	1.30960	0.654798	−	0.755804i	$-0.272754\pi$
$389$	25.8293	1.30960	0.654798	+	0.755804i	$0.272754\pi$
$390$	0	0
$391$	2.50265	0.126564
$392$	−5.75870	−0.290858
$393$	14.8650	0.749842
$394$	13.2055	0.665285
$395$	0	0
$396$	0	0
$397$	4.80875	0.241344	0.120672	−	0.992692i	$-0.461495\pi$
$397$	4.80875	0.241344	0.120672	+	0.992692i	$0.461495\pi$
$398$	8.95710	0.448979
$399$	67.4941	3.37893
$400$	0	0
$401$	2.10633	0.105185	0.0525925	−	0.998616i	$-0.483252\pi$
$401$	2.10633	0.105185	0.0525925	+	0.998616i	$0.483252\pi$
$402$	−14.3065	−0.713543
$403$	−37.8412	−1.88500
$404$	−14.7064	−0.731671
$405$	0	0
$406$	−36.9003	−1.83133
$407$	0	0
$408$	6.72518	0.332946
$409$	−33.9754	−1.67997	−0.839987	−	0.542606i	$-0.817438\pi$
$409$	−33.9754	−1.67997	−0.839987	+	0.542606i	$0.817438\pi$
$410$	0	0
$411$	−32.1755	−1.58710
$412$	1.15325	0.0568165
$413$	−13.0987	−0.644546
$414$	−3.45975	−0.170037
$415$	0	0
$416$	−5.30398	−0.260049
$417$	−49.1206	−2.40545
$418$	0	0
$419$	29.7565	1.45370	0.726849	−	0.686797i	$-0.240984\pi$
$419$	29.7565	1.45370	0.726849	+	0.686797i	$0.240984\pi$
$420$	0	0
$421$	−23.1416	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$421$	−23.1416	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$422$	−6.94547	−0.338100
$423$	−37.4647	−1.82160
$424$	−1.56255	−0.0758840
$425$	0	0
$426$	−37.4723	−1.81554
$427$	−47.9396	−2.31996
$428$	14.2877	0.690623
$429$	0	0
$430$	0	0
$431$	18.0183	0.867910	0.433955	−	0.900935i	$-0.357118\pi$
$431$	18.0183	0.867910	0.433955	+	0.900935i	$0.357118\pi$
$432$	−1.58132	−0.0760811
$433$	26.2556	1.26176	0.630881	−	0.775879i	$-0.282693\pi$
$433$	26.2556	1.26176	0.630881	+	0.775879i	$0.282693\pi$
$434$	25.4839	1.22327
$435$	0	0
$436$	−0.770328	−0.0368920
$437$	−7.03168	−0.336371
$438$	−19.8863	−0.950205
$439$	−8.33649	−0.397879	−0.198939	−	0.980012i	$-0.563750\pi$
$439$	−8.33649	−0.397879	−0.198939	+	0.980012i	$0.563750\pi$
$440$	0	0
$441$	20.8168	0.991274
$442$	−13.8690	−0.659683
$443$	−6.55004	−0.311202	−0.155601	−	0.987820i	$-0.549731\pi$
$443$	−6.55004	−0.311202	−0.155601	+	0.987820i	$0.549731\pi$
$444$	13.2297	0.627852
$445$	0	0
$446$	−5.04867	−0.239062
$447$	−16.5509	−0.782832
$448$	3.57193	0.168758
$449$	11.8633	0.559863	0.279931	−	0.960020i	$-0.409688\pi$
$449$	11.8633	0.559863	0.279931	+	0.960020i	$0.409688\pi$
$450$	0	0
$451$	0	0
$452$	20.6729	0.972371
$453$	5.20051	0.244341
$454$	10.6876	0.501596
$455$	0	0
$456$	−18.8957	−0.884872
$457$	14.6948	0.687393	0.343697	−	0.939081i	$-0.388321\pi$
$457$	14.6948	0.687393	0.343697	+	0.939081i	$0.388321\pi$
$458$	−11.0987	−0.518609
$459$	−4.13488	−0.193000
$460$	0	0
$461$	−2.20953	−0.102908	−0.0514541	−	0.998675i	$-0.516386\pi$
$461$	−2.20953	−0.102908	−0.0514541	+	0.998675i	$0.516386\pi$
$462$	0	0
$463$	−0.859750	−0.0399560	−0.0199780	−	0.999800i	$-0.506360\pi$
$463$	−0.859750	−0.0399560	−0.0199780	+	0.999800i	$0.506360\pi$
$464$	10.3306	0.479588
$465$	0	0
$466$	3.72518	0.172566
$467$	21.8204	1.00973	0.504863	−	0.863199i	$-0.331543\pi$
$467$	21.8204	1.00973	0.504863	+	0.863199i	$0.331543\pi$
$468$	19.1730	0.886274
$469$	−19.8690	−0.917467
$470$	0	0
$471$	30.3678	1.39927
$472$	3.66712	0.168793
$473$	0	0
$474$	−5.96749	−0.274096
$475$	0	0
$476$	9.34002	0.428099
$477$	5.64836	0.258620
$478$	−3.28997	−0.150480
$479$	1.65774	0.0757441	0.0378720	−	0.999283i	$-0.487942\pi$
$479$	1.65774	0.0757441	0.0378720	+	0.999283i	$0.487942\pi$
$480$	0	0
$481$	−27.2830	−1.24400
$482$	2.75055	0.125284
$483$	−8.79262	−0.400078
$484$	0	0
$485$	0	0
$486$	−22.1751	−1.00588
$487$	−10.8901	−0.493476	−0.246738	−	0.969082i	$-0.579359\pi$
$487$	−10.8901	−0.493476	−0.246738	+	0.969082i	$0.579359\pi$
$488$	13.4212	0.607549
$489$	30.4012	1.37479
$490$	0	0
$491$	−19.8610	−0.896316	−0.448158	−	0.893954i	$-0.647920\pi$
$491$	−19.8610	−0.896316	−0.448158	+	0.893954i	$0.647920\pi$
$492$	11.7346	0.529035
$493$	27.0129	1.21660
$494$	38.9678	1.75324
$495$	0	0
$496$	−7.13448	−0.320348
$497$	−52.0419	−2.33440
$498$	−2.94194	−0.131832
$499$	32.3472	1.44806	0.724029	−	0.689770i	$-0.242288\pi$
$499$	32.3472	1.44806	0.724029	+	0.689770i	$0.242288\pi$
$500$	0	0
$501$	32.2479	1.44073
$502$	−27.0049	−1.20529
$503$	−33.7381	−1.50431	−0.752154	−	0.658988i	$-0.770985\pi$
$503$	−33.7381	−1.50431	−0.752154	+	0.658988i	$0.770985\pi$
$504$	−12.9119	−0.575144
$505$	0	0
$506$	0	0
$507$	−38.9191	−1.72846
$508$	−21.4478	−0.951594
$509$	34.2855	1.51968	0.759839	−	0.650112i	$-0.225278\pi$
$509$	34.2855	1.51968	0.759839	+	0.650112i	$0.225278\pi$
$510$	0	0
$511$	−27.6184	−1.22176
$512$	−1.00000	−0.0441942
$513$	11.6178	0.512936
$514$	14.9213	0.658152
$515$	0	0
$516$	0.916305	0.0403381
$517$	0	0
$518$	18.3735	0.807287
$519$	14.1962	0.623142
$520$	0	0
$521$	−30.4083	−1.33221	−0.666107	−	0.745856i	$-0.732040\pi$
$521$	−30.4083	−1.33221	−0.666107	+	0.745856i	$0.732040\pi$
$522$	−37.3435	−1.63448
$523$	29.3922	1.28523	0.642615	−	0.766189i	$-0.277849\pi$
$523$	29.3922	1.28523	0.642615	+	0.766189i	$0.277849\pi$
$524$	−5.77971	−0.252488
$525$	0	0
$526$	−9.28997	−0.405062
$527$	−18.6555	−0.812646
$528$	0	0
$529$	−22.0840	−0.960172
$530$	0	0
$531$	−13.2561	−0.575264
$532$	−26.2426	−1.13776
$533$	−24.1997	−1.04820
$534$	−26.0893	−1.12900
$535$	0	0
$536$	5.56255	0.240265
$537$	−18.7677	−0.809885
$538$	17.6649	0.761587
$539$	0	0
$540$	0	0
$541$	−2.24593	−0.0965599	−0.0482799	−	0.998834i	$-0.515374\pi$
$541$	−2.24593	−0.0965599	−0.0482799	+	0.998834i	$0.515374\pi$
$542$	16.2274	0.697028
$543$	−34.6130	−1.48539
$544$	−2.61484	−0.112110
$545$	0	0
$546$	48.7265	2.08530
$547$	−40.7877	−1.74396	−0.871979	−	0.489544i	$-0.837163\pi$
$547$	−40.7877	−1.74396	−0.871979	+	0.489544i	$0.837163\pi$
$548$	12.5103	0.534412
$549$	−48.5154	−2.07059
$550$	0	0
$551$	−75.8980	−3.23336
$552$	2.46159	0.104772
$553$	−8.28773	−0.352430
$554$	10.1946	0.433129
$555$	0	0
$556$	19.0987	0.809966
$557$	29.7484	1.26048	0.630241	−	0.776400i	$-0.282956\pi$
$557$	29.7484	1.26048	0.630241	+	0.776400i	$0.282956\pi$
$558$	25.7900	1.09178
$559$	−1.88965	−0.0799239
$560$	0	0
$561$	0	0
$562$	15.2149	0.641803
$563$	−12.9375	−0.545249	−0.272624	−	0.962121i	$-0.587892\pi$
$563$	−12.9375	−0.545249	−0.272624	+	0.962121i	$0.587892\pi$
$564$	26.6559	1.12242
$565$	0	0
$566$	16.4268	0.690471
$567$	−24.2087	−1.01667
$568$	14.5697	0.611331
$569$	14.1845	0.594646	0.297323	−	0.954777i	$-0.403906\pi$
$569$	14.1845	0.594646	0.297323	+	0.954777i	$0.403906\pi$
$570$	0	0
$571$	−15.4767	−0.647681	−0.323841	−	0.946112i	$-0.604974\pi$
$571$	−15.4767	−0.647681	−0.323841	+	0.946112i	$0.604974\pi$
$572$	0	0
$573$	−28.3923	−1.18611
$574$	16.2971	0.680228
$575$	0	0
$576$	3.61484	0.150618
$577$	−24.8633	−1.03507	−0.517536	−	0.855662i	$-0.673151\pi$
$577$	−24.8633	−1.03507	−0.517536	+	0.855662i	$0.673151\pi$
$578$	10.1626	0.422710
$579$	9.95636	0.413772
$580$	0	0
$581$	−4.08581	−0.169508
$582$	32.4146	1.34363
$583$	0	0
$584$	7.73205	0.319955
$585$	0	0
$586$	−5.77170	−0.238427
$587$	22.3020	0.920503	0.460251	−	0.887789i	$-0.347759\pi$
$587$	22.3020	0.920503	0.460251	+	0.887789i	$0.347759\pi$
$588$	−14.8110	−0.610795
$589$	52.4162	2.15977
$590$	0	0
$591$	33.9637	1.39708
$592$	−5.14386	−0.211412
$593$	12.4634	0.511809	0.255904	−	0.966702i	$-0.417627\pi$
$593$	12.4634	0.511809	0.255904	+	0.966702i	$0.417627\pi$
$594$	0	0
$595$	0	0
$596$	6.43521	0.263596
$597$	23.0370	0.942844
$598$	−5.07642	−0.207590
$599$	−16.8538	−0.688628	−0.344314	−	0.938855i	$-0.611888\pi$
$599$	−16.8538	−0.688628	−0.344314	+	0.938855i	$0.611888\pi$
$600$	0	0
$601$	−6.11510	−0.249440	−0.124720	−	0.992192i	$-0.539803\pi$
$601$	−6.11510	−0.249440	−0.124720	+	0.992192i	$0.539803\pi$
$602$	1.27258	0.0518663
$603$	−20.1077	−0.818849
$604$	−2.02202	−0.0822750
$605$	0	0
$606$	−37.8239	−1.53649
$607$	19.8727	0.806606	0.403303	−	0.915066i	$-0.367862\pi$
$607$	19.8727	0.806606	0.403303	+	0.915066i	$0.367862\pi$
$608$	7.34689	0.297956
$609$	−94.9051	−3.84575
$610$	0	0
$611$	−54.9713	−2.22390
$612$	9.45220	0.382083
$613$	−20.9551	−0.846369	−0.423185	−	0.906043i	$-0.639088\pi$
$613$	−20.9551	−0.846369	−0.423185	+	0.906043i	$0.639088\pi$
$614$	−4.31873	−0.174290
$615$	0	0
$616$	0	0
$617$	−8.86328	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$617$	−8.86328	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$618$	2.96608	0.119313
$619$	−44.8958	−1.80452	−0.902258	−	0.431197i	$-0.858092\pi$
$619$	−44.8958	−1.80452	−0.902258	+	0.431197i	$0.858092\pi$
$620$	0	0
$621$	−1.51347	−0.0607336
$622$	3.70866	0.148704
$623$	−36.2332	−1.45165
$624$	−13.6415	−0.546096
$625$	0	0
$626$	4.44322	0.177587
$627$	0	0
$628$	−11.8074	−0.471166
$629$	−13.4504	−0.536301
$630$	0	0
$631$	−16.9383	−0.674304	−0.337152	−	0.941450i	$-0.609464\pi$
$631$	−16.9383	−0.674304	−0.337152	+	0.941450i	$0.609464\pi$
$632$	2.32024	0.0922941
$633$	−17.8633	−0.710001
$634$	−4.45398	−0.176890
$635$	0	0
$636$	−4.01877	−0.159355
$637$	30.5441	1.21020
$638$	0	0
$639$	−52.6670	−2.08348
$640$	0	0
$641$	5.10458	0.201619	0.100809	−	0.994906i	$-0.467857\pi$
$641$	5.10458	0.201619	0.100809	+	0.994906i	$0.467857\pi$
$642$	36.7471	1.45029
$643$	45.8493	1.80812	0.904060	−	0.427405i	$-0.140572\pi$
$643$	45.8493	1.80812	0.904060	+	0.427405i	$0.140572\pi$
$644$	3.41868	0.134715
$645$	0	0
$646$	19.2109	0.755843
$647$	−32.4990	−1.27767	−0.638834	−	0.769344i	$-0.720583\pi$
$647$	−32.4990	−1.27767	−0.638834	+	0.769344i	$0.720583\pi$
$648$	6.77747	0.266244
$649$	0	0
$650$	0	0
$651$	65.5428	2.56883
$652$	−11.8204	−0.462922
$653$	−26.4330	−1.03440	−0.517201	−	0.855864i	$-0.673026\pi$
$653$	−26.4330	−1.03440	−0.517201	+	0.855864i	$0.673026\pi$
$654$	−1.98123	−0.0774723
$655$	0	0
$656$	−4.56255	−0.178138
$657$	−27.9501	−1.09044
$658$	37.0201	1.44319
$659$	3.55902	0.138640	0.0693199	−	0.997594i	$-0.477917\pi$
$659$	3.55902	0.138640	0.0693199	+	0.997594i	$0.477917\pi$
$660$	0	0
$661$	29.7622	1.15762	0.578808	−	0.815464i	$-0.303518\pi$
$661$	29.7622	1.15762	0.578808	+	0.815464i	$0.303518\pi$
$662$	2.84089	0.110414
$663$	−35.6702	−1.38532
$664$	1.14386	0.0443906
$665$	0	0
$666$	18.5942	0.720512
$667$	9.88741	0.382842
$668$	−12.5384	−0.485126
$669$	−12.9848	−0.502023
$670$	0	0
$671$	0	0
$672$	9.18677	0.354387
$673$	−17.8958	−0.689832	−0.344916	−	0.938634i	$-0.612093\pi$
$673$	−17.8958	−0.689832	−0.344916	+	0.938634i	$0.612093\pi$
$674$	0.953482	0.0367268
$675$	0	0
$676$	15.1322	0.582009
$677$	−0.792220	−0.0304475	−0.0152237	−	0.999884i	$-0.504846\pi$
$677$	−0.792220	−0.0304475	−0.0152237	+	0.999884i	$0.504846\pi$
$678$	53.1693	2.04195
$679$	45.0178	1.72762
$680$	0	0
$681$	27.4879	1.05334
$682$	0	0
$683$	−2.90481	−0.111149	−0.0555747	−	0.998455i	$-0.517699\pi$
$683$	−2.90481	−0.111149	−0.0555747	+	0.998455i	$0.517699\pi$
$684$	−26.5578	−1.01546
$685$	0	0
$686$	4.43384	0.169285
$687$	−28.5452	−1.08907
$688$	−0.356271	−0.0135827
$689$	8.28773	0.315737
$690$	0	0
$691$	−7.49062	−0.284957	−0.142478	−	0.989798i	$-0.545507\pi$
$691$	−7.49062	−0.284957	−0.142478	+	0.989798i	$0.545507\pi$
$692$	−5.51964	−0.209825
$693$	0	0
$694$	−27.1581	−1.03091
$695$	0	0
$696$	26.5697	1.00712
$697$	−11.9303	−0.451893
$698$	27.8595	1.05450
$699$	9.58091	0.362383
$700$	0	0
$701$	−32.9731	−1.24538	−0.622689	−	0.782469i	$-0.713960\pi$
$701$	−32.9731	−1.24538	−0.622689	+	0.782469i	$0.713960\pi$
$702$	8.38728	0.316557
$703$	37.7914	1.42533
$704$	0	0
$705$	0	0
$706$	−13.0071	−0.489530
$707$	−52.5303	−1.97561
$708$	9.43159	0.354461
$709$	−9.79085	−0.367703	−0.183852	−	0.982954i	$-0.558857\pi$
$709$	−9.79085	−0.367703	−0.183852	+	0.982954i	$0.558857\pi$
$710$	0	0
$711$	−8.38728	−0.314547
$712$	10.1439	0.380157
$713$	−6.82839	−0.255725
$714$	24.0219	0.898997
$715$	0	0
$716$	7.29711	0.272706
$717$	−8.46159	−0.316004
$718$	−11.6765	−0.435763
$719$	−22.8856	−0.853488	−0.426744	−	0.904373i	$-0.640339\pi$
$719$	−22.8856	−0.853488	−0.426744	+	0.904373i	$0.640339\pi$
$720$	0	0
$721$	4.11933	0.153412
$722$	−34.9767	−1.30170
$723$	7.07422	0.263093
$724$	13.4580	0.500162
$725$	0	0
$726$	0	0
$727$	−5.29758	−0.196476	−0.0982382	−	0.995163i	$-0.531321\pi$
$727$	−5.29758	−0.196476	−0.0982382	+	0.995163i	$0.531321\pi$
$728$	−18.9455	−0.702166
$729$	−36.7006	−1.35928
$730$	0	0
$731$	−0.931590	−0.0344561
$732$	34.5184	1.27584
$733$	10.2927	0.380170	0.190085	−	0.981768i	$-0.439124\pi$
$733$	10.2927	0.380170	0.190085	+	0.981768i	$0.439124\pi$
$734$	26.7752	0.988292
$735$	0	0
$736$	−0.957096	−0.0352790
$737$	0	0
$738$	16.4929	0.607111
$739$	20.4801	0.753374	0.376687	−	0.926341i	$-0.377063\pi$
$739$	20.4801	0.753374	0.376687	+	0.926341i	$0.377063\pi$
$740$	0	0
$741$	100.222	3.68176
$742$	−5.58132	−0.204897
$743$	8.85165	0.324735	0.162368	−	0.986730i	$-0.448087\pi$
$743$	8.85165	0.324735	0.162368	+	0.986730i	$0.448087\pi$
$744$	−18.3494	−0.672722
$745$	0	0
$746$	10.7493	0.393560
$747$	−4.13488	−0.151287
$748$	0	0
$749$	51.0348	1.86477
$750$	0	0
$751$	−46.2010	−1.68590	−0.842950	−	0.537992i	$-0.819183\pi$
$751$	−46.2010	−1.68590	−0.842950	+	0.537992i	$0.819183\pi$
$752$	−10.3642	−0.377942
$753$	−69.4548	−2.53107
$754$	−54.7935	−1.99546
$755$	0	0
$756$	−5.64836	−0.205429
$757$	0.558934	0.0203148	0.0101574	−	0.999948i	$-0.496767\pi$
$757$	0.558934	0.0203148	0.0101574	+	0.999948i	$0.496767\pi$
$758$	30.1112	1.09369
$759$	0	0
$760$	0	0
$761$	−54.8735	−1.98916	−0.994581	−	0.103969i	$-0.966846\pi$
$761$	−54.8735	−1.98916	−0.994581	+	0.103969i	$0.966846\pi$
$762$	−55.1624	−1.99832
$763$	−2.75156	−0.0996132
$764$	11.0393	0.399387
$765$	0	0
$766$	2.10458	0.0760414
$767$	−19.4504	−0.702312
$768$	−2.57193	−0.0928066
$769$	1.91520	0.0690640	0.0345320	−	0.999404i	$-0.489006\pi$
$769$	1.91520	0.0690640	0.0345320	+	0.999404i	$0.489006\pi$
$770$	0	0
$771$	38.3767	1.38210
$772$	−3.87116	−0.139326
$773$	35.3006	1.26968	0.634838	−	0.772645i	$-0.281067\pi$
$773$	35.3006	1.26968	0.634838	+	0.772645i	$0.281067\pi$
$774$	1.28786	0.0462912
$775$	0	0
$776$	−12.6032	−0.452429
$777$	47.2555	1.69528
$778$	−25.8293	−0.926024
$779$	33.5205	1.20100
$780$	0	0
$781$	0	0
$782$	−2.50265	−0.0894946
$783$	−16.3360	−0.583801
$784$	5.75870	0.205668
$785$	0	0
$786$	−14.8650	−0.530218
$787$	−17.0784	−0.608781	−0.304390	−	0.952547i	$-0.598453\pi$
$787$	−17.0784	−0.608781	−0.304390	+	0.952547i	$0.598453\pi$
$788$	−13.2055	−0.470428
$789$	−23.8932	−0.850620
$790$	0	0
$791$	73.8422	2.62553
$792$	0	0
$793$	−71.1858	−2.52788
$794$	−4.80875	−0.170656
$795$	0	0
$796$	−8.95710	−0.317476
$797$	35.5415	1.25895	0.629473	−	0.777022i	$-0.283271\pi$
$797$	35.5415	1.25895	0.629473	+	0.777022i	$0.283271\pi$
$798$	−67.4941	−2.38927
$799$	−27.1006	−0.958749
$800$	0	0
$801$	−36.6684	−1.29561
$802$	−2.10633	−0.0743770
$803$	0	0
$804$	14.3065	0.504551
$805$	0	0
$806$	37.8412	1.33290
$807$	45.4329	1.59931
$808$	14.7064	0.517370
$809$	27.4517	0.965152	0.482576	−	0.875854i	$-0.339701\pi$
$809$	27.4517	0.965152	0.482576	+	0.875854i	$0.339701\pi$
$810$	0	0
$811$	−6.57193	−0.230772	−0.115386	−	0.993321i	$-0.536810\pi$
$811$	−6.57193	−0.230772	−0.115386	+	0.993321i	$0.536810\pi$
$812$	36.9003	1.29495
$813$	41.7358	1.46374
$814$	0	0
$815$	0	0
$816$	−6.72518	−0.235428
$817$	2.61748	0.0915741
$818$	33.9754	1.18792
$819$	68.4848	2.39305
$820$	0	0
$821$	−24.4897	−0.854695	−0.427347	−	0.904088i	$-0.640552\pi$
$821$	−24.4897	−0.854695	−0.427347	+	0.904088i	$0.640552\pi$
$822$	32.1755	1.12225
$823$	21.1430	0.736998	0.368499	−	0.929628i	$-0.379872\pi$
$823$	21.1430	0.736998	0.368499	+	0.929628i	$0.379872\pi$
$824$	−1.15325	−0.0401753
$825$	0	0
$826$	13.0987	0.455763
$827$	33.6898	1.17151	0.585755	−	0.810488i	$-0.300798\pi$
$827$	33.6898	1.17151	0.585755	+	0.810488i	$0.300798\pi$
$828$	3.45975	0.120234
$829$	−1.63407	−0.0567537	−0.0283769	−	0.999597i	$-0.509034\pi$
$829$	−1.63407	−0.0567537	−0.0283769	+	0.999597i	$0.509034\pi$
$830$	0	0
$831$	26.2199	0.909560
$832$	5.30398	0.183883
$833$	15.0581	0.521731
$834$	49.1206	1.70091
$835$	0	0
$836$	0	0
$837$	11.2819	0.389959
$838$	−29.7565	−1.02792
$839$	−7.41860	−0.256118	−0.128059	−	0.991767i	$-0.540875\pi$
$839$	−7.41860	−0.256118	−0.128059	+	0.991767i	$0.540875\pi$
$840$	0	0
$841$	77.7220	2.68007
$842$	23.1416	0.797513
$843$	39.1317	1.34777
$844$	6.94547	0.239073
$845$	0	0
$846$	37.4647	1.28806
$847$	0	0
$848$	1.56255	0.0536581
$849$	42.2487	1.44997
$850$	0	0
$851$	−4.92317	−0.168764
$852$	37.4723	1.28378
$853$	−32.0886	−1.09869	−0.549346	−	0.835595i	$-0.685123\pi$
$853$	−32.0886	−1.09869	−0.549346	+	0.835595i	$0.685123\pi$
$854$	47.9396	1.64046
$855$	0	0
$856$	−14.2877	−0.488345
$857$	51.5786	1.76189	0.880945	−	0.473218i	$-0.156908\pi$
$857$	51.5786	1.76189	0.880945	+	0.473218i	$0.156908\pi$
$858$	0	0
$859$	−9.50626	−0.324350	−0.162175	−	0.986762i	$-0.551851\pi$
$859$	−9.50626	−0.324350	−0.162175	+	0.986762i	$0.551851\pi$
$860$	0	0
$861$	41.9151	1.42846
$862$	−18.0183	−0.613705
$863$	−21.5586	−0.733862	−0.366931	−	0.930248i	$-0.619591\pi$
$863$	−21.5586	−0.733862	−0.366931	+	0.930248i	$0.619591\pi$
$864$	1.58132	0.0537975
$865$	0	0
$866$	−26.2556	−0.892201
$867$	26.1376	0.887680
$868$	−25.4839	−0.864979
$869$	0	0
$870$	0	0
$871$	−29.5037	−0.999693
$872$	0.770328	0.0260866
$873$	45.5585	1.54192
$874$	7.03168	0.237850
$875$	0	0
$876$	19.8863	0.671896
$877$	44.3514	1.49764	0.748821	−	0.662773i	$-0.230620\pi$
$877$	44.3514	1.49764	0.748821	+	0.662773i	$0.230620\pi$
$878$	8.33649	0.281343
$879$	−14.8444	−0.500690
$880$	0	0
$881$	−0.269049	−0.00906450	−0.00453225	−	0.999990i	$-0.501443\pi$
$881$	−0.269049	−0.00906450	−0.00453225	+	0.999990i	$0.501443\pi$
$882$	−20.8168	−0.700937
$883$	−7.60810	−0.256033	−0.128016	−	0.991772i	$-0.540861\pi$
$883$	−7.60810	−0.256033	−0.128016	+	0.991772i	$0.540861\pi$
$884$	13.8690	0.466466
$885$	0	0
$886$	6.55004	0.220053
$887$	37.5920	1.26222	0.631108	−	0.775695i	$-0.282601\pi$
$887$	37.5920	1.26222	0.631108	+	0.775695i	$0.282601\pi$
$888$	−13.2297	−0.443959
$889$	−76.6103	−2.56943
$890$	0	0
$891$	0	0
$892$	5.04867	0.169042
$893$	76.1443	2.54807
$894$	16.5509	0.553546
$895$	0	0
$896$	−3.57193	−0.119330
$897$	−13.0562	−0.435934
$898$	−11.8633	−0.395883
$899$	−73.7037	−2.45816
$900$	0	0
$901$	4.08581	0.136118
$902$	0	0
$903$	3.27298	0.108918
$904$	−20.6729	−0.687570
$905$	0	0
$906$	−5.20051	−0.172775
$907$	24.5000	0.813509	0.406755	−	0.913537i	$-0.366660\pi$
$907$	24.5000	0.813509	0.406755	+	0.913537i	$0.366660\pi$
$908$	−10.6876	−0.354682
$909$	−53.1613	−1.76325
$910$	0	0
$911$	−12.3101	−0.407852	−0.203926	−	0.978986i	$-0.565370\pi$
$911$	−12.3101	−0.407852	−0.203926	+	0.978986i	$0.565370\pi$
$912$	18.8957	0.625699
$913$	0	0
$914$	−14.6948	−0.486060
$915$	0	0
$916$	11.0987	0.366712
$917$	−20.6447	−0.681749
$918$	4.13488	0.136471
$919$	3.19117	0.105267	0.0526334	−	0.998614i	$-0.483239\pi$
$919$	3.19117	0.105267	0.0526334	+	0.998614i	$0.483239\pi$
$920$	0	0
$921$	−11.1075	−0.366004
$922$	2.20953	0.0727671
$923$	−77.2774	−2.54362
$924$	0	0
$925$	0	0
$926$	0.859750	0.0282531
$927$	4.16881	0.136922
$928$	−10.3306	−0.339120
$929$	0.193029	0.00633307	0.00316653	−	0.999995i	$-0.498992\pi$
$929$	0.193029	0.00633307	0.00316653	+	0.999995i	$0.498992\pi$
$930$	0	0
$931$	−42.3085	−1.38661
$932$	−3.72518	−0.122022
$933$	9.53841	0.312274
$934$	−21.8204	−0.713984
$935$	0	0
$936$	−19.1730	−0.626690
$937$	33.3284	1.08879	0.544396	−	0.838829i	$-0.316759\pi$
$937$	33.3284	1.08879	0.544396	+	0.838829i	$0.316759\pi$
$938$	19.8690	0.648747
$939$	11.4277	0.372928
$940$	0	0
$941$	36.0598	1.17552	0.587758	−	0.809037i	$-0.300011\pi$
$941$	36.0598	1.17552	0.587758	+	0.809037i	$0.300011\pi$
$942$	−30.3678	−0.989435
$943$	−4.36680	−0.142203
$944$	−3.66712	−0.119355
$945$	0	0
$946$	0	0
$947$	−5.36768	−0.174426	−0.0872131	−	0.996190i	$-0.527796\pi$
$947$	−5.36768	−0.174426	−0.0872131	+	0.996190i	$0.527796\pi$
$948$	5.96749	0.193815
$949$	−41.0107	−1.33126
$950$	0	0
$951$	−11.4553	−0.371464
$952$	−9.34002	−0.302712
$953$	51.4646	1.66710	0.833552	−	0.552442i	$-0.186304\pi$
$953$	51.4646	1.66710	0.833552	+	0.552442i	$0.186304\pi$
$954$	−5.64836	−0.182872
$955$	0	0
$956$	3.28997	0.106405
$957$	0	0
$958$	−1.65774	−0.0535591
$959$	44.6858	1.44298
$960$	0	0
$961$	19.9008	0.641962
$962$	27.2830	0.879638
$963$	51.6478	1.66433
$964$	−2.75055	−0.0885892
$965$	0	0
$966$	8.79262	0.282898
$967$	22.0033	0.707577	0.353788	−	0.935325i	$-0.384893\pi$
$967$	22.0033	0.707577	0.353788	+	0.935325i	$0.384893\pi$
$968$	0	0
$969$	49.4091	1.58725
$970$	0	0
$971$	−48.4146	−1.55370	−0.776849	−	0.629687i	$-0.783183\pi$
$971$	−48.4146	−1.55370	−0.776849	+	0.629687i	$0.783183\pi$
$972$	22.1751	0.711268
$973$	68.2193	2.18701
$974$	10.8901	0.348940
$975$	0	0
$976$	−13.4212	−0.429602
$977$	21.8185	0.698036	0.349018	−	0.937116i	$-0.386515\pi$
$977$	21.8185	0.698036	0.349018	+	0.937116i	$0.386515\pi$
$978$	−30.4012	−0.972123
$979$	0	0
$980$	0	0
$981$	−2.78461	−0.0889057
$982$	19.8610	0.633791
$983$	−51.5902	−1.64547	−0.822736	−	0.568424i	$-0.807554\pi$
$983$	−51.5902	−1.64547	−0.822736	+	0.568424i	$0.807554\pi$
$984$	−11.7346	−0.374084
$985$	0	0
$986$	−27.0129	−0.860266
$987$	95.2131	3.03067
$988$	−38.9678	−1.23973
$989$	−0.340986	−0.0108427
$990$	0	0
$991$	16.6751	0.529703	0.264852	−	0.964289i	$-0.414677\pi$
$991$	16.6751	0.529703	0.264852	+	0.964289i	$0.414677\pi$
$992$	7.13448	0.226520
$993$	7.30659	0.231867
$994$	52.0419	1.65067
$995$	0	0
$996$	2.94194	0.0932190
$997$	57.3666	1.81682	0.908409	−	0.418083i	$-0.137298\pi$
$997$	57.3666	1.81682	0.908409	+	0.418083i	$0.137298\pi$
$998$	−32.3472	−1.02393
$999$	8.13408	0.257351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.de.1.1	yes	4
5.4	even	2		6050.2.a.dj.1.4	yes	4
11.10	odd	2		6050.2.a.dm.1.1	yes	4
55.54	odd	2		6050.2.a.db.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6050.2.a.db.1.4	✓	4	55.54	odd	2
6050.2.a.de.1.1	yes	4	1.1	even	1	trivial
6050.2.a.dj.1.4	yes	4	5.4	even	2
6050.2.a.dm.1.1	yes	4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.57193 q^{3} +1.00000 q^{4} +2.57193 q^{6} +3.57193 q^{7} -1.00000 q^{8} +3.61484 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.57193 q^{3} +1.00000 q^{4} +2.57193 q^{6} +3.57193 q^{7} -1.00000 q^{8} +3.61484 q^{9} -2.57193 q^{12} +5.30398 q^{13} -3.57193 q^{14} +1.00000 q^{16} +2.61484 q^{17} -3.61484 q^{18} -7.34689 q^{19} -9.18677 q^{21} +0.957096 q^{23} +2.57193 q^{24} -5.30398 q^{26} -1.58132 q^{27} +3.57193 q^{28} +10.3306 q^{29} -7.13448 q^{31} -1.00000 q^{32} -2.61484 q^{34} +3.61484 q^{36} -5.14386 q^{37} +7.34689 q^{38} -13.6415 q^{39} -4.56255 q^{41} +9.18677 q^{42} -0.356271 q^{43} -0.957096 q^{46} -10.3642 q^{47} -2.57193 q^{48} +5.75870 q^{49} -6.72518 q^{51} +5.30398 q^{52} +1.56255 q^{53} +1.58132 q^{54} -3.57193 q^{56} +18.8957 q^{57} -10.3306 q^{58} -3.66712 q^{59} -13.4212 q^{61} +7.13448 q^{62} +12.9119 q^{63} +1.00000 q^{64} -5.56255 q^{67} +2.61484 q^{68} -2.46159 q^{69} -14.5697 q^{71} -3.61484 q^{72} -7.73205 q^{73} +5.14386 q^{74} -7.34689 q^{76} +13.6415 q^{78} -2.32024 q^{79} -6.77747 q^{81} +4.56255 q^{82} -1.14386 q^{83} -9.18677 q^{84} +0.356271 q^{86} -26.5697 q^{87} -10.1439 q^{89} +18.9455 q^{91} +0.957096 q^{92} +18.3494 q^{93} +10.3642 q^{94} +2.57193 q^{96} +12.6032 q^{97} -5.75870 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} + 8 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 16 q^{19} - 18 q^{21} - 2 q^{23} - 2 q^{24} - 2 q^{26} + 8 q^{27} + 2 q^{28} - 2 q^{29} - 6 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{36} + 4 q^{37} + 16 q^{38} - 24 q^{39} - 8 q^{41} + 18 q^{42} + 14 q^{43} + 2 q^{46} - 6 q^{47} + 2 q^{48} - 8 q^{49} + 12 q^{51} + 2 q^{52} - 4 q^{53} - 8 q^{54} - 2 q^{56} - 12 q^{57} + 2 q^{58} - 12 q^{59} - 34 q^{61} + 6 q^{62} - 6 q^{63} + 4 q^{64} - 12 q^{67} + 4 q^{68} - 30 q^{69} - 8 q^{72} - 24 q^{73} - 4 q^{74} - 16 q^{76} + 24 q^{78} - 20 q^{79} + 8 q^{81} + 8 q^{82} + 20 q^{83} - 18 q^{84} - 14 q^{86} - 48 q^{87} - 16 q^{89} + 26 q^{91} - 2 q^{92} + 26 q^{93} + 6 q^{94} - 2 q^{96} + 8 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 + 2 * q^7 - 4 * q^8 + 8 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 16 * q^19 - 18 * q^21 - 2 * q^23 - 2 * q^24 - 2 * q^26 + 8 * q^27 + 2 * q^28 - 2 * q^29 - 6 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^36 + 4 * q^37 + 16 * q^38 - 24 * q^39 - 8 * q^41 + 18 * q^42 + 14 * q^43 + 2 * q^46 - 6 * q^47 + 2 * q^48 - 8 * q^49 + 12 * q^51 + 2 * q^52 - 4 * q^53 - 8 * q^54 - 2 * q^56 - 12 * q^57 + 2 * q^58 - 12 * q^59 - 34 * q^61 + 6 * q^62 - 6 * q^63 + 4 * q^64 - 12 * q^67 + 4 * q^68 - 30 * q^69 - 8 * q^72 - 24 * q^73 - 4 * q^74 - 16 * q^76 + 24 * q^78 - 20 * q^79 + 8 * q^81 + 8 * q^82 + 20 * q^83 - 18 * q^84 - 14 * q^86 - 48 * q^87 - 16 * q^89 + 26 * q^91 - 2 * q^92 + 26 * q^93 + 6 * q^94 - 2 * q^96 + 8 * q^98

Introduction

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