Properties

Label 1521.2.a.a
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + q^{5} + 2 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 2 q^{14} - q^{16} + 7 q^{17} - 6 q^{19} - q^{20} - 2 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{28} + q^{29} + 4 q^{31} - 5 q^{32} - 7 q^{34} + 2 q^{35} + q^{37} + 6 q^{38} + 3 q^{40} - 9 q^{41} + 6 q^{43} - 2 q^{44} - 6 q^{46} - 6 q^{47} - 3 q^{49} + 4 q^{50} + 9 q^{53} + 2 q^{55} + 6 q^{56} - q^{58} + q^{61} - 4 q^{62} + 7 q^{64} - 2 q^{67} - 7 q^{68} - 2 q^{70} - 6 q^{71} + 11 q^{73} - q^{74} + 6 q^{76} + 4 q^{77} - 4 q^{79} - q^{80} + 9 q^{82} + 14 q^{83} + 7 q^{85} - 6 q^{86} + 6 q^{88} + 14 q^{89} - 6 q^{92} + 6 q^{94} - 6 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) 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 2.00000 3.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1521.2.a.a 1
3.b odd 2 1 507.2.a.c 1
12.b even 2 1 8112.2.a.w 1
13.b even 2 1 1521.2.a.d 1
13.c even 3 2 117.2.g.b 2
13.d odd 4 2 1521.2.b.c 2
39.d odd 2 1 507.2.a.b 1
39.f even 4 2 507.2.b.b 2
39.h odd 6 2 507.2.e.c 2
39.i odd 6 2 39.2.e.a 2
39.k even 12 4 507.2.j.d 4
52.j odd 6 2 1872.2.t.j 2
156.h even 2 1 8112.2.a.bc 1
156.p even 6 2 624.2.q.c 2
195.x odd 6 2 975.2.i.f 2
195.bl even 12 4 975.2.bb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 39.i odd 6 2
117.2.g.b 2 13.c even 3 2
507.2.a.b 1 39.d odd 2 1
507.2.a.c 1 3.b odd 2 1
507.2.b.b 2 39.f even 4 2
507.2.e.c 2 39.h odd 6 2
507.2.j.d 4 39.k even 12 4
624.2.q.c 2 156.p even 6 2
975.2.i.f 2 195.x odd 6 2
975.2.bb.d 4 195.bl even 12 4
1521.2.a.a 1 1.a even 1 1 trivial
1521.2.a.d 1 13.b even 2 1
1521.2.b.c 2 13.d odd 4 2
1872.2.t.j 2 52.j odd 6 2
8112.2.a.w 1 12.b even 2 1
8112.2.a.bc 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) 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
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 7 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 14 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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