Properties

Label 1617.2.a.f
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} + q^{9} - q^{11} + q^{12} + 4 q^{13} - q^{16} - 4 q^{17} + q^{18} + 8 q^{19} - q^{22} - 8 q^{23} + 3 q^{24} - 5 q^{25} + 4 q^{26} - q^{27} + 2 q^{29} + 4 q^{31} + 5 q^{32} + q^{33} - 4 q^{34} - q^{36} + 10 q^{37} + 8 q^{38} - 4 q^{39} - 4 q^{41} + 4 q^{43} + q^{44} - 8 q^{46} + 12 q^{47} + q^{48} - 5 q^{50} + 4 q^{51} - 4 q^{52} + 6 q^{53} - q^{54} - 8 q^{57} + 2 q^{58} + 12 q^{59} + 12 q^{61} + 4 q^{62} + 7 q^{64} + q^{66} + 4 q^{67} + 4 q^{68} + 8 q^{69} - 3 q^{72} + 4 q^{73} + 10 q^{74} + 5 q^{75} - 8 q^{76} - 4 q^{78} - 8 q^{79} + q^{81} - 4 q^{82} + 8 q^{83} + 4 q^{86} - 2 q^{87} + 3 q^{88} + 8 q^{92} - 4 q^{93} + 12 q^{94} - 5 q^{96} - 8 q^{97} - q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −1.00000 −1.00000 0 −1.00000 0 −3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.2.a.f 1
3.b odd 2 1 4851.2.a.f 1
7.b odd 2 1 1617.2.a.i yes 1
21.c even 2 1 4851.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.f 1 1.a even 1 1 trivial
1617.2.a.i yes 1 7.b odd 2 1
4851.2.a.c 1 21.c even 2 1
4851.2.a.f 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 4 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 12 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 8 \) Copy content Toggle raw display
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