Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} - 2 q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{4} - 2 q^{7} - \beta q^{8} - 2 \beta q^{11} - 2 \beta q^{14} - 5 q^{16} + 4 \beta q^{17} - 2 q^{19} - 6 q^{22} - 4 \beta q^{23} - 5 q^{25} - 2 q^{28} - 4 \beta q^{29} - 2 q^{31} - 3 \beta q^{32} + 12 q^{34} - 2 q^{37} - 2 \beta q^{38} + 4 \beta q^{41} + 8 q^{43} - 2 \beta q^{44} - 12 q^{46} - 6 \beta q^{47} - 3 q^{49} - 5 \beta q^{50} + 2 \beta q^{56} - 12 q^{58} + 2 \beta q^{59} - 10 q^{61} - 2 \beta q^{62} + q^{64} - 14 q^{67} + 4 \beta q^{68} + 2 \beta q^{71} + 10 q^{73} - 2 \beta q^{74} - 2 q^{76} + 4 \beta q^{77} - 4 q^{79} + 12 q^{82} + 6 \beta q^{83} + 8 \beta q^{86} + 6 q^{88} + 4 \beta q^{89} - 4 \beta q^{92} - 18 q^{94} + 10 q^{97} - 3 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{7} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 10 q^{25} - 4 q^{28} - 4 q^{31} + 24 q^{34} - 4 q^{37} + 16 q^{43} - 24 q^{46} - 6 q^{49} - 24 q^{58} - 20 q^{61} + 2 q^{64} - 28 q^{67} + 20 q^{73} - 4 q^{76} - 8 q^{79} + 24 q^{82} + 12 q^{88} - 36 q^{94} + 20 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 0 1.00000 0 0 −2.00000 1.73205 0 0
1.2 1.73205 0 1.00000 0 0 −2.00000 −1.73205 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.a.j 2
3.b odd 2 1 inner 1521.2.a.j 2
13.b even 2 1 117.2.a.b 2
13.d odd 4 2 1521.2.b.i 4
39.d odd 2 1 117.2.a.b 2
39.f even 4 2 1521.2.b.i 4
52.b odd 2 1 1872.2.a.v 2
65.d even 2 1 2925.2.a.y 2
65.h odd 4 2 2925.2.c.s 4
91.b odd 2 1 5733.2.a.t 2
104.e even 2 1 7488.2.a.cq 2
104.h odd 2 1 7488.2.a.cj 2
117.n odd 6 2 1053.2.e.i 4
117.t even 6 2 1053.2.e.i 4
156.h even 2 1 1872.2.a.v 2
195.e odd 2 1 2925.2.a.y 2
195.s even 4 2 2925.2.c.s 4
273.g even 2 1 5733.2.a.t 2
312.b odd 2 1 7488.2.a.cq 2
312.h even 2 1 7488.2.a.cj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.a.b 2 13.b even 2 1
117.2.a.b 2 39.d odd 2 1
1053.2.e.i 4 117.n odd 6 2
1053.2.e.i 4 117.t even 6 2
1521.2.a.j 2 1.a even 1 1 trivial
1521.2.a.j 2 3.b odd 2 1 inner
1521.2.b.i 4 13.d odd 4 2
1521.2.b.i 4 39.f even 4 2
1872.2.a.v 2 52.b odd 2 1
1872.2.a.v 2 156.h even 2 1
2925.2.a.y 2 65.d even 2 1
2925.2.a.y 2 195.e odd 2 1
2925.2.c.s 4 65.h odd 4 2
2925.2.c.s 4 195.s even 4 2
5733.2.a.t 2 91.b odd 2 1
5733.2.a.t 2 273.g even 2 1
7488.2.a.cj 2 104.h odd 2 1
7488.2.a.cj 2 312.h even 2 1
7488.2.a.cq 2 104.e even 2 1
7488.2.a.cq 2 312.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 48 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 48 \) Copy content Toggle raw display
$29$ \( T^{2} - 48 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 48 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 108 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12 \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( (T + 14)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 12 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 108 \) Copy content Toggle raw display
$89$ \( T^{2} - 48 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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