Properties

Label 1521.2.a.u
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 3 q^{4} + \beta_{3} q^{5} + 2 \beta_1 q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 3 q^{4} + \beta_{3} q^{5} + 2 \beta_1 q^{7} + \beta_{3} q^{8} + 5 q^{10} + 2 \beta_{3} q^{11} - 2 \beta_{2} q^{14} - q^{16} + \beta_{2} q^{17} - 2 \beta_1 q^{19} + 3 \beta_{3} q^{20} + 10 q^{22} + 2 \beta_{2} q^{23} + 6 \beta_1 q^{28} + \beta_{2} q^{29} - 3 \beta_{3} q^{32} - 5 \beta_1 q^{34} - 2 \beta_{2} q^{35} + \beta_1 q^{37} + 2 \beta_{2} q^{38} + 5 q^{40} + \beta_{3} q^{41} - 2 q^{43} + 6 \beta_{3} q^{44} - 10 \beta_1 q^{46} + 2 \beta_{3} q^{47} + 5 q^{49} - 3 \beta_{2} q^{53} + 10 q^{55} - 2 \beta_{2} q^{56} - 5 \beta_1 q^{58} - 4 \beta_{3} q^{59} - 7 q^{61} - 13 q^{64} + 2 \beta_1 q^{67} + 3 \beta_{2} q^{68} + 10 \beta_1 q^{70} - 2 \beta_{3} q^{71} + 9 \beta_1 q^{73} - \beta_{2} q^{74} - 6 \beta_1 q^{76} - 4 \beta_{2} q^{77} + 8 q^{79} - \beta_{3} q^{80} + 5 q^{82} - 2 \beta_{3} q^{83} - 5 \beta_1 q^{85} - 2 \beta_{3} q^{86} + 10 q^{88} + 2 \beta_{3} q^{89} + 6 \beta_{2} q^{92} + 10 q^{94} + 2 \beta_{2} q^{95} + 4 \beta_1 q^{97} + 5 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} - 8 q^{43} + 20 q^{49} + 40 q^{55} - 28 q^{61} - 52 q^{64} + 32 q^{79} + 20 q^{82} + 40 q^{88} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 19\nu + 10 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} - 24\nu + 13 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - 2\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 3\beta_{2} - 15\beta _1 + 13 ) / 2 \) 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
1.11402
−2.35008
3.35008
−0.114017
−2.23607 0 3.00000 −2.23607 0 −3.46410 −2.23607 0 5.00000
1.2 −2.23607 0 3.00000 −2.23607 0 3.46410 −2.23607 0 5.00000
1.3 2.23607 0 3.00000 2.23607 0 −3.46410 2.23607 0 5.00000
1.4 2.23607 0 3.00000 2.23607 0 3.46410 2.23607 0 5.00000
\(n\): 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
13.b even 2 1 inner
39.d 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.u 4
3.b odd 2 1 inner 1521.2.a.u 4
13.b even 2 1 inner 1521.2.a.u 4
13.d odd 4 2 1521.2.b.g 4
13.f odd 12 2 117.2.q.d 4
39.d odd 2 1 inner 1521.2.a.u 4
39.f even 4 2 1521.2.b.g 4
39.k even 12 2 117.2.q.d 4
52.l even 12 2 1872.2.by.l 4
156.v odd 12 2 1872.2.by.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.d 4 13.f odd 12 2
117.2.q.d 4 39.k even 12 2
1521.2.a.u 4 1.a even 1 1 trivial
1521.2.a.u 4 3.b odd 2 1 inner
1521.2.a.u 4 13.b even 2 1 inner
1521.2.a.u 4 39.d odd 2 1 inner
1521.2.b.g 4 13.d odd 4 2
1521.2.b.g 4 39.f even 4 2
1872.2.by.l 4 52.l even 12 2
1872.2.by.l 4 156.v odd 12 2

Hecke kernels

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

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{5}^{2} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 135)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 243)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
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