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Properties

Label 6003.2.a.t.1.8
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6003,2,Mod(1,6003)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 6003.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.24737 q^{2} -0.444068 q^{4} -1.99313 q^{5} -4.26394 q^{7} +3.04866 q^{8} +2.48617 q^{10} -2.17945 q^{11} +0.563290 q^{13} +5.31871 q^{14} -2.91467 q^{16} -2.08203 q^{17} +7.02640 q^{19} +0.885084 q^{20} +2.71858 q^{22} +1.00000 q^{23} -1.02744 q^{25} -0.702630 q^{26} +1.89348 q^{28} +1.00000 q^{29} -5.22820 q^{31} -2.46165 q^{32} +2.59706 q^{34} +8.49858 q^{35} -3.21412 q^{37} -8.76453 q^{38} -6.07636 q^{40} +4.83996 q^{41} -12.2884 q^{43} +0.967826 q^{44} -1.24737 q^{46} -6.11511 q^{47} +11.1812 q^{49} +1.28160 q^{50} -0.250139 q^{52} +8.92910 q^{53} +4.34393 q^{55} -12.9993 q^{56} -1.24737 q^{58} +6.86887 q^{59} +1.96328 q^{61} +6.52150 q^{62} +8.89992 q^{64} -1.12271 q^{65} +13.0683 q^{67} +0.924563 q^{68} -10.6009 q^{70} +9.69199 q^{71} +7.14377 q^{73} +4.00920 q^{74} -3.12020 q^{76} +9.29306 q^{77} +11.3234 q^{79} +5.80930 q^{80} -6.03722 q^{82} +6.81709 q^{83} +4.14975 q^{85} +15.3282 q^{86} -6.64441 q^{88} +5.39987 q^{89} -2.40183 q^{91} -0.444068 q^{92} +7.62781 q^{94} -14.0045 q^{95} -3.65219 q^{97} -13.9471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.24737 −0.882024 −0.441012 − 0.897501i \(-0.645380\pi\)
−0.441012 + 0.897501i \(0.645380\pi\)
\(3\) 0 0
\(4\) −0.444068 −0.222034
\(5\) −1.99313 −0.891354 −0.445677 − 0.895194i \(-0.647037\pi\)
−0.445677 + 0.895194i \(0.647037\pi\)
\(6\) 0 0
\(7\) −4.26394 −1.61162 −0.805809 − 0.592175i \(-0.798269\pi\)
−0.805809 + 0.592175i \(0.798269\pi\)
\(8\) 3.04866 1.07786
\(9\) 0 0
\(10\) 2.48617 0.786195
\(11\) −2.17945 −0.657130 −0.328565 − 0.944481i \(-0.606565\pi\)
−0.328565 + 0.944481i \(0.606565\pi\)
\(12\) 0 0
\(13\) 0.563290 0.156228 0.0781142 − 0.996944i \(-0.475110\pi\)
0.0781142 + 0.996944i \(0.475110\pi\)
\(14\) 5.31871 1.42149
\(15\) 0 0
\(16\) −2.91467 −0.728667
\(17\) −2.08203 −0.504966 −0.252483 − 0.967601i \(-0.581247\pi\)
−0.252483 + 0.967601i \(0.581247\pi\)
\(18\) 0 0
\(19\) 7.02640 1.61197 0.805984 − 0.591938i \(-0.201637\pi\)
0.805984 + 0.591938i \(0.201637\pi\)
\(20\) 0.885084 0.197911
\(21\) 0 0
\(22\) 2.71858 0.579604
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.02744 −0.205489
\(26\) −0.702630 −0.137797
\(27\) 0 0
\(28\) 1.89348 0.357834
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.22820 −0.939012 −0.469506 − 0.882929i \(-0.655568\pi\)
−0.469506 + 0.882929i \(0.655568\pi\)
\(32\) −2.46165 −0.435162
\(33\) 0 0
\(34\) 2.59706 0.445392
\(35\) 8.49858 1.43652
\(36\) 0 0
\(37\) −3.21412 −0.528398 −0.264199 − 0.964468i \(-0.585108\pi\)
−0.264199 + 0.964468i \(0.585108\pi\)
\(38\) −8.76453 −1.42179
\(39\) 0 0
\(40\) −6.07636 −0.960757
\(41\) 4.83996 0.755875 0.377937 − 0.925831i \(-0.376633\pi\)
0.377937 + 0.925831i \(0.376633\pi\)
\(42\) 0 0
\(43\) −12.2884 −1.87397 −0.936985 − 0.349368i \(-0.886396\pi\)
−0.936985 + 0.349368i \(0.886396\pi\)
\(44\) 0.967826 0.145905
\(45\) 0 0
\(46\) −1.24737 −0.183915
\(47\) −6.11511 −0.891981 −0.445990 − 0.895038i \(-0.647148\pi\)
−0.445990 + 0.895038i \(0.647148\pi\)
\(48\) 0 0
\(49\) 11.1812 1.59731
\(50\) 1.28160 0.181246
\(51\) 0 0
\(52\) −0.250139 −0.0346880
\(53\) 8.92910 1.22651 0.613253 − 0.789887i \(-0.289861\pi\)
0.613253 + 0.789887i \(0.289861\pi\)
\(54\) 0 0
\(55\) 4.34393 0.585735
\(56\) −12.9993 −1.73710
\(57\) 0 0
\(58\) −1.24737 −0.163788
\(59\) 6.86887 0.894251 0.447126 − 0.894471i \(-0.352448\pi\)
0.447126 + 0.894471i \(0.352448\pi\)
\(60\) 0 0
\(61\) 1.96328 0.251372 0.125686 − 0.992070i \(-0.459887\pi\)
0.125686 + 0.992070i \(0.459887\pi\)
\(62\) 6.52150 0.828231
\(63\) 0 0
\(64\) 8.89992 1.11249
\(65\) −1.12271 −0.139255
\(66\) 0 0
\(67\) 13.0683 1.59655 0.798276 − 0.602292i \(-0.205746\pi\)
0.798276 + 0.602292i \(0.205746\pi\)
\(68\) 0.924563 0.112120
\(69\) 0 0
\(70\) −10.6009 −1.26705
\(71\) 9.69199 1.15023 0.575114 − 0.818073i \(-0.304958\pi\)
0.575114 + 0.818073i \(0.304958\pi\)
\(72\) 0 0
\(73\) 7.14377 0.836115 0.418057 − 0.908421i \(-0.362711\pi\)
0.418057 + 0.908421i \(0.362711\pi\)
\(74\) 4.00920 0.466060
\(75\) 0 0
\(76\) −3.12020 −0.357912
\(77\) 9.29306 1.05904
\(78\) 0 0
\(79\) 11.3234 1.27399 0.636994 − 0.770869i \(-0.280178\pi\)
0.636994 + 0.770869i \(0.280178\pi\)
\(80\) 5.80930 0.649500
\(81\) 0 0
\(82\) −6.03722 −0.666699
\(83\) 6.81709 0.748273 0.374136 − 0.927374i \(-0.377939\pi\)
0.374136 + 0.927374i \(0.377939\pi\)
\(84\) 0 0
\(85\) 4.14975 0.450103
\(86\) 15.3282 1.65289
\(87\) 0 0
\(88\) −6.64441 −0.708296
\(89\) 5.39987 0.572385 0.286192 − 0.958172i \(-0.407610\pi\)
0.286192 + 0.958172i \(0.407610\pi\)
\(90\) 0 0
\(91\) −2.40183 −0.251781
\(92\) −0.444068 −0.0462973
\(93\) 0 0
\(94\) 7.62781 0.786748
\(95\) −14.0045 −1.43683
\(96\) 0 0
\(97\) −3.65219 −0.370824 −0.185412 − 0.982661i \(-0.559362\pi\)
−0.185412 + 0.982661i \(0.559362\pi\)
\(98\) −13.9471 −1.40887
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6003.2.a.t.1.8 ✓ 22
3.2 odd 2 6003.2.a.u.1.15 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.8 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.15 yes 22 3.2 odd 2