Properties

Label 3234.2.e.d
Level $3234$
Weight $2$
Character orbit 3234.e
Analytic conductor $25.824$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8236200137\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9} + 24 q^{16} - 16 q^{17} + 32 q^{19} - 8 q^{22} - 24 q^{24} - 8 q^{25} + 8 q^{33} + 24 q^{36} + 16 q^{37} + 16 q^{41} - 24 q^{54} - 16 q^{55} + 16 q^{62} - 24 q^{64} - 64 q^{67} + 16 q^{68} + 64 q^{71} - 32 q^{76} + 24 q^{81} + 16 q^{83} + 8 q^{88} - 16 q^{93} - 64 q^{94} + 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2155.1 1.00000i 1.00000i −1.00000 4.05345i 1.00000 0 1.00000i −1.00000 −4.05345
2155.2 1.00000i 1.00000i −1.00000 3.45711i 1.00000 0 1.00000i −1.00000 −3.45711
2155.3 1.00000i 1.00000i −1.00000 2.66749i 1.00000 0 1.00000i −1.00000 −2.66749
2155.4 1.00000i 1.00000i −1.00000 0.516505i 1.00000 0 1.00000i −1.00000 −0.516505
2155.5 1.00000i 1.00000i −1.00000 0.357203i 1.00000 0 1.00000i −1.00000 −0.357203
2155.6 1.00000i 1.00000i −1.00000 0.313884i 1.00000 0 1.00000i −1.00000 −0.313884
2155.7 1.00000i 1.00000i −1.00000 0.500309i 1.00000 0 1.00000i −1.00000 0.500309
2155.8 1.00000i 1.00000i −1.00000 0.942563i 1.00000 0 1.00000i −1.00000 0.942563
2155.9 1.00000i 1.00000i −1.00000 1.58222i 1.00000 0 1.00000i −1.00000 1.58222
2155.10 1.00000i 1.00000i −1.00000 1.98215i 1.00000 0 1.00000i −1.00000 1.98215
2155.11 1.00000i 1.00000i −1.00000 2.84462i 1.00000 0 1.00000i −1.00000 2.84462
2155.12 1.00000i 1.00000i −1.00000 3.51377i 1.00000 0 1.00000i −1.00000 3.51377
2155.13 1.00000i 1.00000i −1.00000 3.51377i 1.00000 0 1.00000i −1.00000 3.51377
2155.14 1.00000i 1.00000i −1.00000 2.84462i 1.00000 0 1.00000i −1.00000 2.84462
2155.15 1.00000i 1.00000i −1.00000 1.98215i 1.00000 0 1.00000i −1.00000 1.98215
2155.16 1.00000i 1.00000i −1.00000 1.58222i 1.00000 0 1.00000i −1.00000 1.58222
2155.17 1.00000i 1.00000i −1.00000 0.942563i 1.00000 0 1.00000i −1.00000 0.942563
2155.18 1.00000i 1.00000i −1.00000 0.500309i 1.00000 0 1.00000i −1.00000 0.500309
2155.19 1.00000i 1.00000i −1.00000 0.313884i 1.00000 0 1.00000i −1.00000 −0.313884
2155.20 1.00000i 1.00000i −1.00000 0.357203i 1.00000 0 1.00000i −1.00000 −0.357203
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2155.24
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.e.d yes 24
7.b odd 2 1 3234.2.e.c 24
11.b odd 2 1 3234.2.e.c 24
77.b even 2 1 inner 3234.2.e.d yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.e.c 24 7.b odd 2 1
3234.2.e.c 24 11.b odd 2 1
3234.2.e.d yes 24 1.a even 1 1 trivial
3234.2.e.d yes 24 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3234, [\chi])\):

\( T_{5}^{24} + 64 T_{5}^{22} + 1696 T_{5}^{20} + 24160 T_{5}^{18} + 201024 T_{5}^{16} + 995200 T_{5}^{14} + 2870592 T_{5}^{12} + 4574208 T_{5}^{10} + 3727360 T_{5}^{8} + 1498112 T_{5}^{6} + 301056 T_{5}^{4} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{13}^{12} - 76 T_{13}^{10} - 48 T_{13}^{9} + 2134 T_{13}^{8} + 2432 T_{13}^{7} - 26512 T_{13}^{6} - 40000 T_{13}^{5} + 138220 T_{13}^{4} + 247808 T_{13}^{3} - 211312 T_{13}^{2} - 510912 T_{13} - 207224 \) Copy content Toggle raw display