Properties

Label 1617.2.a.j
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: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 3 q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 2 q^{13} + 2 q^{15} - q^{16} + 2 q^{17} + q^{18} - 2 q^{20} + q^{22} + 8 q^{23} - 3 q^{24} - q^{25} + 2 q^{26} + q^{27} - 6 q^{29} + 2 q^{30} + 8 q^{31} + 5 q^{32} + q^{33} + 2 q^{34} - q^{36} + 6 q^{37} + 2 q^{39} - 6 q^{40} + 2 q^{41} - q^{44} + 2 q^{45} + 8 q^{46} - 8 q^{47} - q^{48} - q^{50} + 2 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + 2 q^{55} - 6 q^{58} + 4 q^{59} - 2 q^{60} - 6 q^{61} + 8 q^{62} + 7 q^{64} + 4 q^{65} + q^{66} - 4 q^{67} - 2 q^{68} + 8 q^{69} - 3 q^{72} + 14 q^{73} + 6 q^{74} - q^{75} + 2 q^{78} - 4 q^{79} - 2 q^{80} + q^{81} + 2 q^{82} - 12 q^{83} + 4 q^{85} - 6 q^{87} - 3 q^{88} + 6 q^{89} + 2 q^{90} - 8 q^{92} + 8 q^{93} - 8 q^{94} + 5 q^{96} - 2 q^{97} + q^{99}+O(q^{100}) \) 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 2.00000 1.00000 0 −3.00000 1.00000 2.00000
\(n\): 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.j 1
3.b odd 2 1 4851.2.a.b 1
7.b odd 2 1 33.2.a.a 1
21.c even 2 1 99.2.a.b 1
28.d even 2 1 528.2.a.g 1
35.c odd 2 1 825.2.a.a 1
35.f even 4 2 825.2.c.a 2
56.e even 2 1 2112.2.a.j 1
56.h odd 2 1 2112.2.a.bb 1
63.l odd 6 2 891.2.e.e 2
63.o even 6 2 891.2.e.g 2
77.b even 2 1 363.2.a.b 1
77.j odd 10 4 363.2.e.e 4
77.l even 10 4 363.2.e.g 4
84.h odd 2 1 1584.2.a.o 1
91.b odd 2 1 5577.2.a.a 1
105.g even 2 1 2475.2.a.g 1
105.k odd 4 2 2475.2.c.d 2
119.d odd 2 1 9537.2.a.m 1
168.e odd 2 1 6336.2.a.n 1
168.i even 2 1 6336.2.a.x 1
231.h odd 2 1 1089.2.a.j 1
308.g odd 2 1 5808.2.a.t 1
385.h even 2 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 7.b odd 2 1
99.2.a.b 1 21.c even 2 1
363.2.a.b 1 77.b even 2 1
363.2.e.e 4 77.j odd 10 4
363.2.e.g 4 77.l even 10 4
528.2.a.g 1 28.d even 2 1
825.2.a.a 1 35.c odd 2 1
825.2.c.a 2 35.f even 4 2
891.2.e.e 2 63.l odd 6 2
891.2.e.g 2 63.o even 6 2
1089.2.a.j 1 231.h odd 2 1
1584.2.a.o 1 84.h odd 2 1
1617.2.a.j 1 1.a even 1 1 trivial
2112.2.a.j 1 56.e even 2 1
2112.2.a.bb 1 56.h odd 2 1
2475.2.a.g 1 105.g even 2 1
2475.2.c.d 2 105.k odd 4 2
4851.2.a.b 1 3.b odd 2 1
5577.2.a.a 1 91.b odd 2 1
5808.2.a.t 1 308.g odd 2 1
6336.2.a.n 1 168.e odd 2 1
6336.2.a.x 1 168.i even 2 1
9075.2.a.q 1 385.h even 2 1
9537.2.a.m 1 119.d 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
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 2 \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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