Properties

Label 1170.2.a.d
Level $1170$
Weight $2$
Character orbit 1170.a
Self dual yes
Analytic conductor $9.342$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.34249703649\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - 4 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} - 4 q^{7} - q^{8} - q^{10} + 2 q^{11} - q^{13} + 4 q^{14} + q^{16} - 2 q^{17} + 6 q^{19} + q^{20} - 2 q^{22} - 6 q^{23} + q^{25} + q^{26} - 4 q^{28} - 2 q^{29} - 6 q^{31} - q^{32} + 2 q^{34} - 4 q^{35} - 2 q^{37} - 6 q^{38} - q^{40} - 10 q^{41} - 10 q^{43} + 2 q^{44} + 6 q^{46} + 12 q^{47} + 9 q^{49} - q^{50} - q^{52} - 2 q^{53} + 2 q^{55} + 4 q^{56} + 2 q^{58} - 10 q^{59} + 2 q^{61} + 6 q^{62} + q^{64} - q^{65} - 12 q^{67} - 2 q^{68} + 4 q^{70} - 10 q^{71} + 10 q^{73} + 2 q^{74} + 6 q^{76} - 8 q^{77} - 4 q^{79} + q^{80} + 10 q^{82} - 2 q^{85} + 10 q^{86} - 2 q^{88} + 14 q^{89} + 4 q^{91} - 6 q^{92} - 12 q^{94} + 6 q^{95} + 14 q^{97} - 9 q^{98}+O(q^{100}) \) 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 −4.00000 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 1170.2.a.d 1
3.b odd 2 1 130.2.a.c 1
4.b odd 2 1 9360.2.a.by 1
5.b even 2 1 5850.2.a.cb 1
5.c odd 4 2 5850.2.e.u 2
12.b even 2 1 1040.2.a.b 1
15.d odd 2 1 650.2.a.c 1
15.e even 4 2 650.2.b.g 2
21.c even 2 1 6370.2.a.l 1
24.f even 2 1 4160.2.a.t 1
24.h odd 2 1 4160.2.a.c 1
39.d odd 2 1 1690.2.a.e 1
39.f even 4 2 1690.2.d.e 2
39.h odd 6 2 1690.2.e.g 2
39.i odd 6 2 1690.2.e.a 2
39.k even 12 4 1690.2.l.a 4
60.h even 2 1 5200.2.a.bd 1
195.e odd 2 1 8450.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.c 1 3.b odd 2 1
650.2.a.c 1 15.d odd 2 1
650.2.b.g 2 15.e even 4 2
1040.2.a.b 1 12.b even 2 1
1170.2.a.d 1 1.a even 1 1 trivial
1690.2.a.e 1 39.d odd 2 1
1690.2.d.e 2 39.f even 4 2
1690.2.e.a 2 39.i odd 6 2
1690.2.e.g 2 39.h odd 6 2
1690.2.l.a 4 39.k even 12 4
4160.2.a.c 1 24.h odd 2 1
4160.2.a.t 1 24.f even 2 1
5200.2.a.bd 1 60.h even 2 1
5850.2.a.cb 1 5.b even 2 1
5850.2.e.u 2 5.c odd 4 2
6370.2.a.l 1 21.c even 2 1
8450.2.a.n 1 195.e odd 2 1
9360.2.a.by 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{31} + 6 \) 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
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T + 6 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 10 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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