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Properties

Label 4056.2.c.j.337.2
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [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 = [2,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,12,0,0,0,0,0,-8] 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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.j.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} -2.00000i q^{11} +4.00000i q^{15} +6.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -8.00000i q^{31} -2.00000i q^{33} +16.0000 q^{35} -10.0000i q^{37} +4.00000i q^{41} +4.00000 q^{43} +4.00000i q^{45} -6.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} +8.00000 q^{55} -4.00000i q^{57} -6.00000i q^{59} -6.00000 q^{61} -4.00000i q^{63} -4.00000 q^{69} -10.0000i q^{71} -2.00000i q^{73} -11.0000 q^{75} -8.00000 q^{77} +1.00000 q^{81} +10.0000i q^{83} +24.0000i q^{85} -6.00000 q^{87} +8.00000i q^{89} -8.00000i q^{93} +16.0000 q^{95} +10.0000i q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{23} - 22 q^{25} + 2 q^{27} - 12 q^{29} + 32 q^{35} + 8 q^{43} - 18 q^{49} + 12 q^{51} + 12 q^{53} + 16 q^{55} - 12 q^{61} - 8 q^{69} - 22 q^{75} - 16 q^{77} + 2 q^{81}+ \cdots + 32 q^{95}+O(q^{100}) \) 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
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 4.00000i 1.03280i
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 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
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) − 8.00000i − 1.43684i −0.695608 − 0.718421i \(-0.744865\pi\)
0.695608 − 0.718421i \(-0.255135\pi\)
\(32\) 0 0
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 16.0000 2.70449
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 4.00000i 0.596285i
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 − 0.437595i \(-0.855830\pi\)
0.899172 − 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 − 0.390567i \(-0.872279\pi\)
0.920575 − 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) − 10.0000i − 1.18678i −0.804914 − 0.593391i \(-0.797789\pi\)
0.804914 − 0.593391i \(-0.202211\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 − 0.117041i \(-0.962659\pi\)
0.993127 − 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0000i 1.09764i 0.835940 + 0.548821i \(0.184923\pi\)
−0.835940 + 0.548821i \(0.815077\pi\)
\(84\) 0 0
\(85\) 24.0000i 2.60317i
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 8.00000i 0.847998i 0.905663 + 0.423999i \(0.139374\pi\)
−0.905663 + 0.423999i \(0.860626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) − 2.00000i − 0.201008i
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.j.337.2 2
13.5 odd 4 4056.2.a.s.1.1 1
13.8 odd 4 312.2.a.d.1.1 ✓ 1
13.12 even 2 inner 4056.2.c.j.337.1 2
39.8 even 4 936.2.a.i.1.1 1
52.31 even 4 8112.2.a.o.1.1 1
52.47 even 4 624.2.a.a.1.1 1
65.34 odd 4 7800.2.a.j.1.1 1
104.21 odd 4 2496.2.a.n.1.1 1
104.99 even 4 2496.2.a.bd.1.1 1
156.47 odd 4 1872.2.a.t.1.1 1
312.125 even 4 7488.2.a.b.1.1 1
312.203 odd 4 7488.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.a.d.1.1 ✓ 1 13.8 odd 4
624.2.a.a.1.1 1 52.47 even 4
936.2.a.i.1.1 1 39.8 even 4
1872.2.a.t.1.1 1 156.47 odd 4
2496.2.a.n.1.1 1 104.21 odd 4
2496.2.a.bd.1.1 1 104.99 even 4
4056.2.a.s.1.1 1 13.5 odd 4
4056.2.c.j.337.1 2 13.12 even 2 inner
4056.2.c.j.337.2 2 1.1 even 1 trivial
7488.2.a.b.1.1 1 312.125 even 4
7488.2.a.e.1.1 1 312.203 odd 4
7800.2.a.j.1.1 1 65.34 odd 4
8112.2.a.o.1.1 1 52.31 even 4