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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4056,2,Mod(337,4056)] 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("4056.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,-6,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4056.337
Dual form 4056.2.c.b.337.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} -3.00000i q^{15} -3.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} +8.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -5.00000 q^{29} -8.00000i q^{31} +4.00000i q^{33} -12.0000 q^{35} -7.00000i q^{37} -9.00000i q^{41} -8.00000 q^{43} +3.00000i q^{45} +4.00000i q^{47} -9.00000 q^{49} +3.00000 q^{51} -5.00000 q^{53} +12.0000 q^{55} +4.00000i q^{57} -4.00000i q^{59} -5.00000 q^{61} +4.00000i q^{63} +8.00000i q^{67} -8.00000 q^{69} -4.00000i q^{71} -11.0000i q^{73} +4.00000 q^{75} +16.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -9.00000i q^{85} +5.00000 q^{87} +6.00000i q^{89} +8.00000i q^{93} +12.0000 q^{95} -14.0000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{17} + 16 q^{23} - 8 q^{25} - 2 q^{27} - 10 q^{29} - 24 q^{35} - 16 q^{43} - 18 q^{49} + 6 q^{51} - 10 q^{53} + 24 q^{55} - 10 q^{61} - 16 q^{69} + 8 q^{75} + 32 q^{77} - 8 q^{79}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 3.00000i − 0.774597i
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) − 8.00000i − 1.43684i −0.695608 − 0.718421i \(-0.744865\pi\)
0.695608 − 0.718421i \(-0.255135\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) − 7.00000i − 1.15079i −0.817875 − 0.575396i \(-0.804848\pi\)
0.817875 − 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.00000i − 1.40556i −0.711405 − 0.702782i \(-0.751941\pi\)
0.711405 − 0.702782i \(-0.248059\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.00000i 0.447214i
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) − 4.00000i − 0.474713i −0.971423 − 0.237356i \(-0.923719\pi\)
0.971423 − 0.237356i \(-0.0762809\pi\)
\(72\) 0 0
\(73\) − 11.0000i − 1.28745i −0.765256 − 0.643726i \(-0.777388\pi\)
0.765256 − 0.643726i \(-0.222612\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) − 9.00000i − 0.976187i
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 − 0.710742i \(-0.748359\pi\)
0.703452 − 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
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 4056.2.c.b.337.2 2
13.2 odd 12 312.2.q.c.217.1 ✓ 2
13.5 odd 4 4056.2.a.j.1.1 1
13.6 odd 12 312.2.q.c.289.1 yes 2
13.8 odd 4 4056.2.a.b.1.1 1
13.12 even 2 inner 4056.2.c.b.337.1 2
39.2 even 12 936.2.t.b.217.1 2
39.32 even 12 936.2.t.b.289.1 2
52.15 even 12 624.2.q.e.529.1 2
52.19 even 12 624.2.q.e.289.1 2
52.31 even 4 8112.2.a.bh.1.1 1
52.47 even 4 8112.2.a.r.1.1 1
156.71 odd 12 1872.2.t.a.289.1 2
156.119 odd 12 1872.2.t.a.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.c.217.1 ✓ 2 13.2 odd 12
312.2.q.c.289.1 yes 2 13.6 odd 12
624.2.q.e.289.1 2 52.19 even 12
624.2.q.e.529.1 2 52.15 even 12
936.2.t.b.217.1 2 39.2 even 12
936.2.t.b.289.1 2 39.32 even 12
1872.2.t.a.289.1 2 156.71 odd 12
1872.2.t.a.1153.1 2 156.119 odd 12
4056.2.a.b.1.1 1 13.8 odd 4
4056.2.a.j.1.1 1 13.5 odd 4
4056.2.c.b.337.1 2 13.12 even 2 inner
4056.2.c.b.337.2 2 1.1 even 1 trivial
8112.2.a.r.1.1 1 52.47 even 4
8112.2.a.bh.1.1 1 52.31 even 4