↑

Properties

Label 3060.2.z.g
Level $3060$
Weight $2$
Character orbit 3060.z
Analytic conductor $24.434$
Analytic rank $0$
Dimension $40$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3060,2,Mod(829,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.829"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3060.z (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.4342230185\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 1020)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1 0 0 0 −2.23603 − 0.0136287i 0 2.94725 + 2.94725i 0 0 0
829.2 0 0 0 −2.23291 − 0.118865i 0 −3.33086 − 3.33086i 0 0 0
829.3 0 0 0 −2.11522 − 0.725140i 0 −0.920356 − 0.920356i 0 0 0
829.4 0 0 0 −1.89878 + 1.18095i 0 1.34100 + 1.34100i 0 0 0
829.5 0 0 0 −1.68075 − 1.47482i 0 −0.500854 − 0.500854i 0 0 0
829.6 0 0 0 −1.60010 + 1.56194i 0 −0.816760 − 0.816760i 0 0 0
829.7 0 0 0 −1.47482 − 1.68075i 0 0.500854 + 0.500854i 0 0 0
829.8 0 0 0 −0.725140 − 2.11522i 0 0.920356 + 0.920356i 0 0 0
829.9 0 0 0 −0.637858 + 2.14316i 0 −1.25356 − 1.25356i 0 0 0
829.10 0 0 0 −0.417502 + 2.19675i 0 3.15340 + 3.15340i 0 0 0
829.11 0 0 0 −0.118865 − 2.23291i 0 3.33086 + 3.33086i 0 0 0
829.12 0 0 0 −0.0136287 − 2.23603i 0 −2.94725 − 2.94725i 0 0 0
829.13 0 0 0 0.880193 + 2.05554i 0 −1.19618 − 1.19618i 0 0 0
829.14 0 0 0 1.18095 − 1.89878i 0 −1.34100 − 1.34100i 0 0 0
829.15 0 0 0 1.35208 + 1.78098i 0 −2.38946 − 2.38946i 0 0 0
829.16 0 0 0 1.56194 − 1.60010i 0 0.816760 + 0.816760i 0 0 0
829.17 0 0 0 1.78098 + 1.35208i 0 2.38946 + 2.38946i 0 0 0
829.18 0 0 0 2.05554 + 0.880193i 0 1.19618 + 1.19618i 0 0 0
829.19 0 0 0 2.14316 − 0.637858i 0 1.25356 + 1.25356i 0 0 0
829.20 0 0 0 2.19675 − 0.417502i 0 −3.15340 − 3.15340i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.c even 4 1 inner
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3060.2.z.g 40
3.b odd 2 1 1020.2.y.a ✓ 40
5.b even 2 1 inner 3060.2.z.g 40
15.d odd 2 1 1020.2.y.a ✓ 40
17.c even 4 1 inner 3060.2.z.g 40
51.f odd 4 1 1020.2.y.a ✓ 40
85.j even 4 1 inner 3060.2.z.g 40
255.i odd 4 1 1020.2.y.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.2.y.a ✓ 40 3.b odd 2 1
1020.2.y.a ✓ 40 15.d odd 2 1
1020.2.y.a ✓ 40 51.f odd 4 1
1020.2.y.a ✓ 40 255.i odd 4 1
3060.2.z.g 40 1.a even 1 1 trivial
3060.2.z.g 40 5.b even 2 1 inner
3060.2.z.g 40 17.c even 4 1 inner
3060.2.z.g 40 85.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3060, [\chi])\):

\( T_{7}^{40} + 1356 T_{7}^{36} + 665718 T_{7}^{32} + 141921588 T_{7}^{28} + 12236412993 T_{7}^{24} + \cdots + 10312216477696 \) Copy content Toggle raw display
\( T_{23}^{40} + 18842 T_{23}^{36} + 129473569 T_{23}^{32} + 394353094072 T_{23}^{28} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display