Properties

Label 1710.2.a.d
Level $1710$
Weight $2$
Character orbit 1710.a
Self dual yes
Analytic conductor $13.654$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - q^{13} + q^{14} + q^{16} + 3 q^{17} + q^{19} - q^{20} - 3 q^{23} + q^{25} + q^{26} - q^{28} + 3 q^{29} + 2 q^{31} - q^{32} - 3 q^{34} + q^{35} - 10 q^{37} - q^{38} + q^{40} - 6 q^{41} + 2 q^{43} + 3 q^{46} - 6 q^{49} - q^{50} - q^{52} - 3 q^{53} + q^{56} - 3 q^{58} - 3 q^{59} + 8 q^{61} - 2 q^{62} + q^{64} + q^{65} - 7 q^{67} + 3 q^{68} - q^{70} - 12 q^{71} - 13 q^{73} + 10 q^{74} + q^{76} + 14 q^{79} - q^{80} + 6 q^{82} - 6 q^{83} - 3 q^{85} - 2 q^{86} - 6 q^{89} + q^{91} - 3 q^{92} - q^{95} - 10 q^{97} + 6 q^{98}+O(q^{100}) \) 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 0 1.00000 −1.00000 0 −1.00000 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 1710.2.a.d 1
3.b odd 2 1 190.2.a.c 1
5.b even 2 1 8550.2.a.bd 1
12.b even 2 1 1520.2.a.d 1
15.d odd 2 1 950.2.a.a 1
15.e even 4 2 950.2.b.e 2
21.c even 2 1 9310.2.a.o 1
24.f even 2 1 6080.2.a.p 1
24.h odd 2 1 6080.2.a.h 1
57.d even 2 1 3610.2.a.b 1
60.h even 2 1 7600.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 3.b odd 2 1
950.2.a.a 1 15.d odd 2 1
950.2.b.e 2 15.e even 4 2
1520.2.a.d 1 12.b even 2 1
1710.2.a.d 1 1.a even 1 1 trivial
3610.2.a.b 1 57.d even 2 1
6080.2.a.h 1 24.h odd 2 1
6080.2.a.p 1 24.f even 2 1
7600.2.a.m 1 60.h even 2 1
8550.2.a.bd 1 5.b even 2 1
9310.2.a.o 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{53} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 13 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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