Properties

Label 1617.2.a.d
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - 4 q^{5} - q^{6} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - 4 q^{5} - q^{6} + 3 q^{8} + q^{9} + 4 q^{10} - q^{11} - q^{12} - 4 q^{15} - q^{16} + 7 q^{17} - q^{18} + 5 q^{19} + 4 q^{20} + q^{22} - 9 q^{23} + 3 q^{24} + 11 q^{25} + q^{27} + q^{29} + 4 q^{30} - 2 q^{31} - 5 q^{32} - q^{33} - 7 q^{34} - q^{36} - 3 q^{37} - 5 q^{38} - 12 q^{40} + 2 q^{41} - q^{43} + q^{44} - 4 q^{45} + 9 q^{46} - 7 q^{47} - q^{48} - 11 q^{50} + 7 q^{51} - q^{54} + 4 q^{55} + 5 q^{57} - q^{58} - 7 q^{59} + 4 q^{60} + 10 q^{61} + 2 q^{62} + 7 q^{64} + q^{66} - 12 q^{67} - 7 q^{68} - 9 q^{69} - 15 q^{71} + 3 q^{72} + 4 q^{73} + 3 q^{74} + 11 q^{75} - 5 q^{76} - 8 q^{79} + 4 q^{80} + q^{81} - 2 q^{82} - 4 q^{83} - 28 q^{85} + q^{86} + q^{87} - 3 q^{88} - 12 q^{89} + 4 q^{90} + 9 q^{92} - 2 q^{93} + 7 q^{94} - 20 q^{95} - 5 q^{96} + q^{97} - q^{99}+O(q^{100}) \) 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 −4.00000 −1.00000 0 3.00000 1.00000 4.00000
\(n\): 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.d 1
3.b odd 2 1 4851.2.a.q 1
7.b odd 2 1 1617.2.a.c 1
7.d odd 6 2 231.2.i.b 2
21.c even 2 1 4851.2.a.n 1
21.g even 6 2 693.2.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.b 2 7.d odd 6 2
693.2.i.b 2 21.g even 6 2
1617.2.a.c 1 7.b odd 2 1
1617.2.a.d 1 1.a even 1 1 trivial
4851.2.a.n 1 21.c even 2 1
4851.2.a.q 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
\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} - 7 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 7 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T + 9 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T + 7 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 7 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 15 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T - 1 \) Copy content Toggle raw display
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