Properties

Label 2496.4.a.c.1.1
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
Analytic rank $1$
Dimension $1$
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] = [2496,4,Mod(1,2496)] 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("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,-4,0,4,0,9,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2496.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} -2.00000 q^{11} +13.0000 q^{13} +12.0000 q^{15} -6.00000 q^{17} +36.0000 q^{19} -12.0000 q^{21} -20.0000 q^{23} -109.000 q^{25} -27.0000 q^{27} +14.0000 q^{29} -152.000 q^{31} +6.00000 q^{33} -16.0000 q^{35} +258.000 q^{37} -39.0000 q^{39} +84.0000 q^{41} +188.000 q^{43} -36.0000 q^{45} +254.000 q^{47} -327.000 q^{49} +18.0000 q^{51} -366.000 q^{53} +8.00000 q^{55} -108.000 q^{57} -550.000 q^{59} +14.0000 q^{61} +36.0000 q^{63} -52.0000 q^{65} -448.000 q^{67} +60.0000 q^{69} +926.000 q^{71} +254.000 q^{73} +327.000 q^{75} -8.00000 q^{77} +1328.00 q^{79} +81.0000 q^{81} -186.000 q^{83} +24.0000 q^{85} -42.0000 q^{87} -336.000 q^{89} +52.0000 q^{91} +456.000 q^{93} -144.000 q^{95} +614.000 q^{97} -18.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −4.00000 −0.357771 −0.178885 − 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 − 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.0548202 −0.0274101 − 0.999624i \(-0.508726\pi\)
−0.0274101 + 0.999624i \(0.508726\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 12.0000 0.206559
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 − 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) 36.0000 0.434682 0.217341 − 0.976096i \(-0.430262\pi\)
0.217341 + 0.976096i \(0.430262\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) −20.0000 −0.181317 −0.0906584 − 0.995882i \(-0.528897\pi\)
−0.0906584 + 0.995882i \(0.528897\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 14.0000 0.0896460 0.0448230 − 0.998995i \(-0.485728\pi\)
0.0448230 + 0.998995i \(0.485728\pi\)
\(30\) 0 0
\(31\) −152.000 −0.880645 −0.440323 − 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) 0 0
\(33\) 6.00000 0.0316505
\(34\) 0 0
\(35\) −16.0000 −0.0772712
\(36\) 0 0
\(37\) 258.000 1.14635 0.573175 − 0.819433i \(-0.305712\pi\)
0.573175 + 0.819433i \(0.305712\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 84.0000 0.319966 0.159983 − 0.987120i \(-0.448856\pi\)
0.159983 + 0.987120i \(0.448856\pi\)
\(42\) 0 0
\(43\) 188.000 0.666738 0.333369 − 0.942796i \(-0.391815\pi\)
0.333369 + 0.942796i \(0.391815\pi\)
\(44\) 0 0
\(45\) −36.0000 −0.119257
\(46\) 0 0
\(47\) 254.000 0.788292 0.394146 − 0.919048i \(-0.371040\pi\)
0.394146 + 0.919048i \(0.371040\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) 18.0000 0.0494217
\(52\) 0 0
\(53\) −366.000 −0.948565 −0.474283 − 0.880373i \(-0.657293\pi\)
−0.474283 + 0.880373i \(0.657293\pi\)
\(54\) 0 0
\(55\) 8.00000 0.0196131
\(56\) 0 0
\(57\) −108.000 −0.250964
\(58\) 0 0
\(59\) −550.000 −1.21363 −0.606813 − 0.794845i \(-0.707552\pi\)
−0.606813 + 0.794845i \(0.707552\pi\)
\(60\) 0 0
\(61\) 14.0000 0.0293855 0.0146928 − 0.999892i \(-0.495323\pi\)
0.0146928 + 0.999892i \(0.495323\pi\)
\(62\) 0 0
\(63\) 36.0000 0.0719932
\(64\) 0 0
\(65\) −52.0000 −0.0992278
\(66\) 0 0
\(67\) −448.000 −0.816894 −0.408447 − 0.912782i \(-0.633930\pi\)
−0.408447 + 0.912782i \(0.633930\pi\)
\(68\) 0 0
\(69\) 60.0000 0.104683
\(70\) 0 0
\(71\) 926.000 1.54783 0.773915 − 0.633289i \(-0.218296\pi\)
0.773915 + 0.633289i \(0.218296\pi\)
\(72\) 0 0
\(73\) 254.000 0.407239 0.203620 − 0.979050i \(-0.434729\pi\)
0.203620 + 0.979050i \(0.434729\pi\)
\(74\) 0 0
\(75\) 327.000 0.503449
\(76\) 0 0
\(77\) −8.00000 −0.0118401
\(78\) 0 0
\(79\) 1328.00 1.89129 0.945644 − 0.325205i \(-0.105433\pi\)
0.945644 + 0.325205i \(0.105433\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −186.000 −0.245978 −0.122989 − 0.992408i \(-0.539248\pi\)
−0.122989 + 0.992408i \(0.539248\pi\)
\(84\) 0 0
\(85\) 24.0000 0.0306255
\(86\) 0 0
\(87\) −42.0000 −0.0517572
\(88\) 0 0
\(89\) −336.000 −0.400179 −0.200089 − 0.979778i \(-0.564123\pi\)
−0.200089 + 0.979778i \(0.564123\pi\)
\(90\) 0 0
\(91\) 52.0000 0.0599020
\(92\) 0 0
\(93\) 456.000 0.508441
\(94\) 0 0
\(95\) −144.000 −0.155517
\(96\) 0 0
\(97\) 614.000 0.642704 0.321352 − 0.946960i \(-0.395863\pi\)
0.321352 + 0.946960i \(0.395863\pi\)
\(98\) 0 0
\(99\) −18.0000 −0.0182734
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 2496.4.a.c.1.1 1
4.3 odd 2 2496.4.a.l.1.1 1
8.3 odd 2 624.4.a.c.1.1 1
8.5 even 2 78.4.a.f.1.1 ✓ 1
24.5 odd 2 234.4.a.c.1.1 1
24.11 even 2 1872.4.a.f.1.1 1
40.29 even 2 1950.4.a.a.1.1 1
104.5 odd 4 1014.4.b.g.337.1 2
104.21 odd 4 1014.4.b.g.337.2 2
104.77 even 2 1014.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 ✓ 1 8.5 even 2
234.4.a.c.1.1 1 24.5 odd 2
624.4.a.c.1.1 1 8.3 odd 2
1014.4.a.e.1.1 1 104.77 even 2
1014.4.b.g.337.1 2 104.5 odd 4
1014.4.b.g.337.2 2 104.21 odd 4
1872.4.a.f.1.1 1 24.11 even 2
1950.4.a.a.1.1 1 40.29 even 2
2496.4.a.c.1.1 1 1.1 even 1 trivial
2496.4.a.l.1.1 1 4.3 odd 2