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Properties

Label 690.3.c.a
Level $690$
Weight $3$
Character orbit 690.c
Analytic conductor $18.801$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [690,3,Mod(91,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 690.c (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: \(18.8011382409\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.41421 −1.73205 2.00000 − 2.23607i 2.44949 − 12.6313i −2.82843 3.00000 3.16228i
91.2 −1.41421 −1.73205 2.00000 − 2.23607i 2.44949 − 8.87387i −2.82843 3.00000 3.16228i
91.3 −1.41421 −1.73205 2.00000 − 2.23607i 2.44949 1.52229i −2.82843 3.00000 3.16228i
91.4 −1.41421 −1.73205 2.00000 − 2.23607i 2.44949 5.52876i −2.82843 3.00000 3.16228i
91.5 −1.41421 −1.73205 2.00000 2.23607i 2.44949 − 5.52876i −2.82843 3.00000 − 3.16228i
91.6 −1.41421 −1.73205 2.00000 2.23607i 2.44949 − 1.52229i −2.82843 3.00000 − 3.16228i
91.7 −1.41421 −1.73205 2.00000 2.23607i 2.44949 8.87387i −2.82843 3.00000 − 3.16228i
91.8 −1.41421 −1.73205 2.00000 2.23607i 2.44949 12.6313i −2.82843 3.00000 − 3.16228i
91.9 −1.41421 1.73205 2.00000 − 2.23607i −2.44949 − 7.36146i −2.82843 3.00000 3.16228i
91.10 −1.41421 1.73205 2.00000 − 2.23607i −2.44949 − 3.49916i −2.82843 3.00000 3.16228i
91.11 −1.41421 1.73205 2.00000 − 2.23607i −2.44949 − 0.411309i −2.82843 3.00000 3.16228i
91.12 −1.41421 1.73205 2.00000 − 2.23607i −2.44949 12.3097i −2.82843 3.00000 3.16228i
91.13 −1.41421 1.73205 2.00000 2.23607i −2.44949 − 12.3097i −2.82843 3.00000 − 3.16228i
91.14 −1.41421 1.73205 2.00000 2.23607i −2.44949 0.411309i −2.82843 3.00000 − 3.16228i
91.15 −1.41421 1.73205 2.00000 2.23607i −2.44949 3.49916i −2.82843 3.00000 − 3.16228i
91.16 −1.41421 1.73205 2.00000 2.23607i −2.44949 7.36146i −2.82843 3.00000 − 3.16228i
91.17 1.41421 −1.73205 2.00000 − 2.23607i −2.44949 − 11.8194i 2.82843 3.00000 − 3.16228i
91.18 1.41421 −1.73205 2.00000 − 2.23607i −2.44949 − 7.96402i 2.82843 3.00000 − 3.16228i
91.19 1.41421 −1.73205 2.00000 − 2.23607i −2.44949 1.37049i 2.82843 3.00000 − 3.16228i
91.20 1.41421 −1.73205 2.00000 − 2.23607i −2.44949 3.95881i 2.82843 3.00000 − 3.16228i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.c.a ✓ 32
3.b odd 2 1 2070.3.c.b 32
23.b odd 2 1 inner 690.3.c.a ✓ 32
69.c even 2 1 2070.3.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.c.a ✓ 32 1.a even 1 1 trivial
690.3.c.a ✓ 32 23.b odd 2 1 inner
2070.3.c.b 32 3.b odd 2 1
2070.3.c.b 32 69.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(690, [\chi])\).