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Properties

Label 930.2.e.b
Level $930$
Weight $2$
Character orbit 930.e
Analytic conductor $7.426$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,2,Mod(929,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} + 32 q^{4} - 2 q^{5} + 32 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} + 32 q^{4} - 2 q^{5} + 32 q^{8} + 4 q^{9} - 2 q^{10} + 32 q^{16} + 4 q^{18} + 8 q^{19} - 2 q^{20} + 10 q^{25} - 12 q^{31} + 32 q^{32} + 8 q^{33} - 16 q^{35} + 4 q^{36} + 8 q^{38} - 4 q^{39} - 2 q^{40} - 42 q^{45} + 4 q^{47} - 36 q^{49} + 10 q^{50} - 4 q^{51} - 12 q^{62} + 24 q^{63} + 32 q^{64} + 8 q^{66} - 8 q^{69} - 16 q^{70} + 4 q^{72} + 8 q^{76} - 4 q^{78} - 2 q^{80} + 24 q^{81} + 4 q^{87} - 42 q^{90} - 24 q^{93} + 4 q^{94} + 26 q^{95} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1 1.00000 −1.70787 − 0.288390i 1.00000 −2.10277 − 0.760496i −1.70787 − 0.288390i − 4.63015i 1.00000 2.83366 + 0.985067i −2.10277 − 0.760496i
929.2 1.00000 −1.70787 + 0.288390i 1.00000 −2.10277 + 0.760496i −1.70787 + 0.288390i 4.63015i 1.00000 2.83366 − 0.985067i −2.10277 + 0.760496i
929.3 1.00000 −1.69795 − 0.341995i 1.00000 1.23971 − 1.86095i −1.69795 − 0.341995i − 0.843688i 1.00000 2.76608 + 1.16138i 1.23971 − 1.86095i
929.4 1.00000 −1.69795 + 0.341995i 1.00000 1.23971 + 1.86095i −1.69795 + 0.341995i 0.843688i 1.00000 2.76608 − 1.16138i 1.23971 + 1.86095i
929.5 1.00000 −1.63502 − 0.571593i 1.00000 −1.46321 − 1.69086i −1.63502 − 0.571593i 2.83992i 1.00000 2.34656 + 1.86913i −1.46321 − 1.69086i
929.6 1.00000 −1.63502 + 0.571593i 1.00000 −1.46321 + 1.69086i −1.63502 + 0.571593i − 2.83992i 1.00000 2.34656 − 1.86913i −1.46321 + 1.69086i
929.7 1.00000 −1.36636 − 1.06445i 1.00000 2.16820 + 0.546716i −1.36636 − 1.06445i 1.97405i 1.00000 0.733873 + 2.90885i 2.16820 + 0.546716i
929.8 1.00000 −1.36636 + 1.06445i 1.00000 2.16820 − 0.546716i −1.36636 + 1.06445i − 1.97405i 1.00000 0.733873 − 2.90885i 2.16820 − 0.546716i
929.9 1.00000 −1.02858 − 1.39357i 1.00000 −0.467590 + 2.18663i −1.02858 − 1.39357i − 1.43029i 1.00000 −0.884063 + 2.86678i −0.467590 + 2.18663i
929.10 1.00000 −1.02858 + 1.39357i 1.00000 −0.467590 − 2.18663i −1.02858 + 1.39357i 1.43029i 1.00000 −0.884063 − 2.86678i −0.467590 − 2.18663i
929.11 1.00000 −0.880669 − 1.49145i 1.00000 −2.09732 + 0.775395i −0.880669 − 1.49145i 0.657746i 1.00000 −1.44884 + 2.62695i −2.09732 + 0.775395i
929.12 1.00000 −0.880669 + 1.49145i 1.00000 −2.09732 − 0.775395i −0.880669 + 1.49145i − 0.657746i 1.00000 −1.44884 − 2.62695i −2.09732 − 0.775395i
929.13 1.00000 −0.522710 − 1.65129i 1.00000 1.93647 + 1.11808i −0.522710 − 1.65129i − 3.41776i 1.00000 −2.45355 + 1.72630i 1.93647 + 1.11808i
929.14 1.00000 −0.522710 + 1.65129i 1.00000 1.93647 − 1.11808i −0.522710 + 1.65129i 3.41776i 1.00000 −2.45355 − 1.72630i 1.93647 − 1.11808i
929.15 1.00000 −0.230519 − 1.71664i 1.00000 0.286520 − 2.21764i −0.230519 − 1.71664i − 4.09004i 1.00000 −2.89372 + 0.791438i 0.286520 − 2.21764i
929.16 1.00000 −0.230519 + 1.71664i 1.00000 0.286520 + 2.21764i −0.230519 + 1.71664i 4.09004i 1.00000 −2.89372 − 0.791438i 0.286520 + 2.21764i
929.17 1.00000 0.230519 − 1.71664i 1.00000 0.286520 + 2.21764i 0.230519 − 1.71664i 4.09004i 1.00000 −2.89372 − 0.791438i 0.286520 + 2.21764i
929.18 1.00000 0.230519 + 1.71664i 1.00000 0.286520 − 2.21764i 0.230519 + 1.71664i − 4.09004i 1.00000 −2.89372 + 0.791438i 0.286520 − 2.21764i
929.19 1.00000 0.522710 − 1.65129i 1.00000 1.93647 − 1.11808i 0.522710 − 1.65129i 3.41776i 1.00000 −2.45355 − 1.72630i 1.93647 − 1.11808i
929.20 1.00000 0.522710 + 1.65129i 1.00000 1.93647 + 1.11808i 0.522710 + 1.65129i − 3.41776i 1.00000 −2.45355 + 1.72630i 1.93647 + 1.11808i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.b odd 2 1 inner
465.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.e.b yes 32
3.b odd 2 1 930.2.e.a ✓ 32
5.b even 2 1 930.2.e.a ✓ 32
15.d odd 2 1 inner 930.2.e.b yes 32
31.b odd 2 1 inner 930.2.e.b yes 32
93.c even 2 1 930.2.e.a ✓ 32
155.c odd 2 1 930.2.e.a ✓ 32
465.g even 2 1 inner 930.2.e.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.e.a ✓ 32 3.b odd 2 1
930.2.e.a ✓ 32 5.b even 2 1
930.2.e.a ✓ 32 93.c even 2 1
930.2.e.a ✓ 32 155.c odd 2 1
930.2.e.b yes 32 1.a even 1 1 trivial
930.2.e.b yes 32 15.d odd 2 1 inner
930.2.e.b yes 32 31.b odd 2 1 inner
930.2.e.b yes 32 465.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{8} - T_{47}^{7} - 234T_{47}^{6} + 252T_{47}^{5} + 15056T_{47}^{4} - 12776T_{47}^{3} - 268088T_{47}^{2} + 10752T_{47} + 55296 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display