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Properties

Label 690.2.h.a
Level $690$
Weight $2$
Character orbit 690.h
Analytic conductor $5.510$
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] = [690,2,Mod(689,690)]
 
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([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("690.689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.h (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: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 2 q^{3} + 24 q^{4} - 2 q^{6} - 24 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 2 q^{3} + 24 q^{4} - 2 q^{6} - 24 q^{8} + 6 q^{9} + 2 q^{12} + 24 q^{16} - 6 q^{18} - 4 q^{23} - 2 q^{24} + 12 q^{25} + 2 q^{27} - 28 q^{31} - 24 q^{32} + 8 q^{35} + 6 q^{36} + 4 q^{46} + 16 q^{47} + 2 q^{48} - 4 q^{49} - 12 q^{50} - 2 q^{54} + 4 q^{55} + 28 q^{62} + 24 q^{64} - 8 q^{69} - 8 q^{70} - 6 q^{72} + 14 q^{75} + 8 q^{77} + 14 q^{81} - 44 q^{85} + 28 q^{87} - 4 q^{92} - 4 q^{93} - 16 q^{94} + 4 q^{95} - 2 q^{96} + 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1 −1.00000 −1.65921 − 0.497008i 1.00000 −1.15459 + 1.91492i 1.65921 + 0.497008i −3.15825 −1.00000 2.50597 + 1.64928i 1.15459 − 1.91492i
689.2 −1.00000 −1.65921 − 0.497008i 1.00000 1.15459 − 1.91492i 1.65921 + 0.497008i 3.15825 −1.00000 2.50597 + 1.64928i −1.15459 + 1.91492i
689.3 −1.00000 −1.65921 + 0.497008i 1.00000 −1.15459 − 1.91492i 1.65921 − 0.497008i −3.15825 −1.00000 2.50597 − 1.64928i 1.15459 + 1.91492i
689.4 −1.00000 −1.65921 + 0.497008i 1.00000 1.15459 + 1.91492i 1.65921 − 0.497008i 3.15825 −1.00000 2.50597 − 1.64928i −1.15459 − 1.91492i
689.5 −1.00000 −1.29400 − 1.15133i 1.00000 −2.22484 − 0.223804i 1.29400 + 1.15133i 0.666856 −1.00000 0.348885 + 2.97964i 2.22484 + 0.223804i
689.6 −1.00000 −1.29400 − 1.15133i 1.00000 2.22484 + 0.223804i 1.29400 + 1.15133i −0.666856 −1.00000 0.348885 + 2.97964i −2.22484 − 0.223804i
689.7 −1.00000 −1.29400 + 1.15133i 1.00000 −2.22484 + 0.223804i 1.29400 − 1.15133i 0.666856 −1.00000 0.348885 − 2.97964i 2.22484 − 0.223804i
689.8 −1.00000 −1.29400 + 1.15133i 1.00000 2.22484 − 0.223804i 1.29400 − 1.15133i −0.666856 −1.00000 0.348885 − 2.97964i −2.22484 + 0.223804i
689.9 −1.00000 −0.250553 − 1.71383i 1.00000 −0.545034 − 2.16863i 0.250553 + 1.71383i 1.86244 −1.00000 −2.87445 + 0.858813i 0.545034 + 2.16863i
689.10 −1.00000 −0.250553 − 1.71383i 1.00000 0.545034 + 2.16863i 0.250553 + 1.71383i −1.86244 −1.00000 −2.87445 + 0.858813i −0.545034 − 2.16863i
689.11 −1.00000 −0.250553 + 1.71383i 1.00000 −0.545034 + 2.16863i 0.250553 − 1.71383i 1.86244 −1.00000 −2.87445 − 0.858813i 0.545034 − 2.16863i
689.12 −1.00000 −0.250553 + 1.71383i 1.00000 0.545034 − 2.16863i 0.250553 − 1.71383i −1.86244 −1.00000 −2.87445 − 0.858813i −0.545034 + 2.16863i
689.13 −1.00000 0.571841 − 1.63493i 1.00000 −2.12720 + 0.689226i −0.571841 + 1.63493i 1.28874 −1.00000 −2.34600 − 1.86984i 2.12720 − 0.689226i
689.14 −1.00000 0.571841 − 1.63493i 1.00000 2.12720 − 0.689226i −0.571841 + 1.63493i −1.28874 −1.00000 −2.34600 − 1.86984i −2.12720 + 0.689226i
689.15 −1.00000 0.571841 + 1.63493i 1.00000 −2.12720 − 0.689226i −0.571841 − 1.63493i 1.28874 −1.00000 −2.34600 + 1.86984i 2.12720 + 0.689226i
689.16 −1.00000 0.571841 + 1.63493i 1.00000 2.12720 + 0.689226i −0.571841 − 1.63493i −1.28874 −1.00000 −2.34600 + 1.86984i −2.12720 − 0.689226i
689.17 −1.00000 1.44698 − 0.951967i 1.00000 −1.42959 − 1.71938i −1.44698 + 0.951967i −4.73682 −1.00000 1.18752 − 2.75496i 1.42959 + 1.71938i
689.18 −1.00000 1.44698 − 0.951967i 1.00000 1.42959 + 1.71938i −1.44698 + 0.951967i 4.73682 −1.00000 1.18752 − 2.75496i −1.42959 − 1.71938i
689.19 −1.00000 1.44698 + 0.951967i 1.00000 −1.42959 + 1.71938i −1.44698 − 0.951967i −4.73682 −1.00000 1.18752 + 2.75496i 1.42959 − 1.71938i
689.20 −1.00000 1.44698 + 0.951967i 1.00000 1.42959 − 1.71938i −1.44698 − 0.951967i 4.73682 −1.00000 1.18752 + 2.75496i −1.42959 + 1.71938i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 689.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
23.b odd 2 1 inner
345.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.h.a ✓ 24
3.b odd 2 1 690.2.h.b yes 24
5.b even 2 1 690.2.h.b yes 24
15.d odd 2 1 inner 690.2.h.a ✓ 24
23.b odd 2 1 inner 690.2.h.a ✓ 24
69.c even 2 1 690.2.h.b yes 24
115.c odd 2 1 690.2.h.b yes 24
345.h even 2 1 inner 690.2.h.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.h.a ✓ 24 1.a even 1 1 trivial
690.2.h.a ✓ 24 15.d odd 2 1 inner
690.2.h.a ✓ 24 23.b odd 2 1 inner
690.2.h.a ✓ 24 345.h even 2 1 inner
690.2.h.b yes 24 3.b odd 2 1
690.2.h.b yes 24 5.b even 2 1
690.2.h.b yes 24 69.c even 2 1
690.2.h.b yes 24 115.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{6} - 4T_{47}^{5} - 132T_{47}^{4} + 472T_{47}^{3} + 2368T_{47}^{2} - 12288T_{47} + 13824 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display