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Properties

Label 1170.2.v.d
Level $1170$
Weight $2$
Character orbit 1170.v
Analytic conductor $9.342$
Analytic rank $0$
Dimension $24$
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] = [1170,2,Mod(233,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.233");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.v (of order \(4\), degree \(2\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{10} + 8 q^{13} - 24 q^{16} + 16 q^{19} - 8 q^{22} + 16 q^{25} - 32 q^{37} - 32 q^{43} - 24 q^{52} - 24 q^{55} + 80 q^{61} + 8 q^{67} + 8 q^{70} + 8 q^{73} - 64 q^{82} + 120 q^{85} + 8 q^{88} + 16 q^{91} + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1 −0.707107 + 0.707107i 0 − 1.00000i −1.61477 + 1.54678i 0 −1.84213 − 1.84213i 0.707107 + 0.707107i 0 0.0480793 − 2.23555i
233.2 −0.707107 + 0.707107i 0 − 1.00000i 0.119982 − 2.23285i 0 −0.759484 − 0.759484i 0.707107 + 0.707107i 0 1.49402 + 1.66370i
233.3 −0.707107 + 0.707107i 0 − 1.00000i 2.02676 − 0.944579i 0 −0.571888 − 0.571888i 0.707107 + 0.707107i 0 −0.765220 + 2.10106i
233.4 −0.707107 + 0.707107i 0 − 1.00000i 0.870010 + 2.05987i 0 −1.52271 − 1.52271i 0.707107 + 0.707107i 0 −2.07174 − 0.841361i
233.5 −0.707107 + 0.707107i 0 − 1.00000i 2.18710 + 0.465418i 0 1.92569 + 1.92569i 0.707107 + 0.707107i 0 −1.87561 + 1.21741i
233.6 −0.707107 + 0.707107i 0 − 1.00000i −2.17487 + 0.519571i 0 2.77052 + 2.77052i 0.707107 + 0.707107i 0 1.17047 − 1.90526i
233.7 0.707107 − 0.707107i 0 − 1.00000i −2.02676 + 0.944579i 0 −0.571888 − 0.571888i −0.707107 − 0.707107i 0 −0.765220 + 2.10106i
233.8 0.707107 − 0.707107i 0 − 1.00000i 1.61477 − 1.54678i 0 −1.84213 − 1.84213i −0.707107 − 0.707107i 0 0.0480793 − 2.23555i
233.9 0.707107 − 0.707107i 0 − 1.00000i 2.17487 − 0.519571i 0 2.77052 + 2.77052i −0.707107 − 0.707107i 0 1.17047 − 1.90526i
233.10 0.707107 − 0.707107i 0 − 1.00000i −2.18710 − 0.465418i 0 1.92569 + 1.92569i −0.707107 − 0.707107i 0 −1.87561 + 1.21741i
233.11 0.707107 − 0.707107i 0 − 1.00000i −0.870010 − 2.05987i 0 −1.52271 − 1.52271i −0.707107 − 0.707107i 0 −2.07174 − 0.841361i
233.12 0.707107 − 0.707107i 0 − 1.00000i −0.119982 + 2.23285i 0 −0.759484 − 0.759484i −0.707107 − 0.707107i 0 1.49402 + 1.66370i
467.1 −0.707107 − 0.707107i 0 1.00000i 0.119982 + 2.23285i 0 −0.759484 + 0.759484i 0.707107 − 0.707107i 0 1.49402 − 1.66370i
467.2 −0.707107 − 0.707107i 0 1.00000i −1.61477 − 1.54678i 0 −1.84213 + 1.84213i 0.707107 − 0.707107i 0 0.0480793 + 2.23555i
467.3 −0.707107 − 0.707107i 0 1.00000i 2.02676 + 0.944579i 0 −0.571888 + 0.571888i 0.707107 − 0.707107i 0 −0.765220 − 2.10106i
467.4 −0.707107 − 0.707107i 0 1.00000i 0.870010 − 2.05987i 0 −1.52271 + 1.52271i 0.707107 − 0.707107i 0 −2.07174 + 0.841361i
467.5 −0.707107 − 0.707107i 0 1.00000i 2.18710 − 0.465418i 0 1.92569 − 1.92569i 0.707107 − 0.707107i 0 −1.87561 − 1.21741i
467.6 −0.707107 − 0.707107i 0 1.00000i −2.17487 − 0.519571i 0 2.77052 − 2.77052i 0.707107 − 0.707107i 0 1.17047 + 1.90526i
467.7 0.707107 + 0.707107i 0 1.00000i −2.02676 − 0.944579i 0 −0.571888 + 0.571888i −0.707107 + 0.707107i 0 −0.765220 − 2.10106i
467.8 0.707107 + 0.707107i 0 1.00000i 1.61477 + 1.54678i 0 −1.84213 + 1.84213i −0.707107 + 0.707107i 0 0.0480793 + 2.23555i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
65.h odd 4 1 inner
195.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.v.d yes 24
3.b odd 2 1 inner 1170.2.v.d yes 24
5.c odd 4 1 1170.2.v.c ✓ 24
13.b even 2 1 1170.2.v.c ✓ 24
15.e even 4 1 1170.2.v.c ✓ 24
39.d odd 2 1 1170.2.v.c ✓ 24
65.h odd 4 1 inner 1170.2.v.d yes 24
195.s even 4 1 inner 1170.2.v.d yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.v.c ✓ 24 5.c odd 4 1
1170.2.v.c ✓ 24 13.b even 2 1
1170.2.v.c ✓ 24 15.e even 4 1
1170.2.v.c ✓ 24 39.d odd 2 1
1170.2.v.d yes 24 1.a even 1 1 trivial
1170.2.v.d yes 24 3.b odd 2 1 inner
1170.2.v.d yes 24 65.h odd 4 1 inner
1170.2.v.d yes 24 195.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 24 T_{7}^{9} + 180 T_{7}^{8} + 256 T_{7}^{7} + 288 T_{7}^{6} + 1440 T_{7}^{5} + 7056 T_{7}^{4} + \cdots + 2704 \) acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\). Copy content Toggle raw display