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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Newspace parameters

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

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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{2} + 2q^{3} + 24q^{4} - 2q^{6} - 24q^{8} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 24q^{2} + 2q^{3} + 24q^{4} - 2q^{6} - 24q^{8} + 6q^{9} + 2q^{12} + 24q^{16} - 6q^{18} - 4q^{23} - 2q^{24} + 12q^{25} + 2q^{27} - 28q^{31} - 24q^{32} + 8q^{35} + 6q^{36} + 4q^{46} + 16q^{47} + 2q^{48} - 4q^{49} - 12q^{50} - 2q^{54} + 4q^{55} + 28q^{62} + 24q^{64} - 8q^{69} - 8q^{70} - 6q^{72} + 14q^{75} + 8q^{77} + 14q^{81} - 44q^{85} + 28q^{87} - 4q^{92} - 4q^{93} - 16q^{94} + 4q^{95} - 2q^{96} + 4q^{98} + O(q^{100}) \)

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.

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 \( T_{47}^{6} - 4 T_{47}^{5} - 132 T_{47}^{4} + 472 T_{47}^{3} + 2368 T_{47}^{2} - 12288 T_{47} + 13824 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).