Properties

Label 1521.2.a.v
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.1997632.1
Defining polynomial: \( x^{6} - 8x^{4} + 19x^{2} - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{4} - 2 \beta_{2}) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{4} - 2 \beta_{2}) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{5} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{3} + \beta_1) q^{8} + (4 \beta_{2} - 1) q^{10} + (\beta_{5} - 2 \beta_1) q^{11} + ( - 2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{14} + (2 \beta_{4} - 3 \beta_{2} - 2) q^{16} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{17} + (\beta_{4} + 2 \beta_{2} - 4) q^{19} + ( - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{20} + ( - 3 \beta_{4} + 2) q^{22} + (\beta_{5} + \beta_{3} - 3 \beta_1) q^{23} + (3 \beta_{4} - \beta_{2} + 3) q^{25} + ( - 3 \beta_{4} + 7 \beta_{2} - 1) q^{28} + ( - \beta_{5} - 3 \beta_{3} + 4 \beta_1) q^{29} + (\beta_{4} - 3 \beta_{2} - 8) q^{31} + (\beta_{5} + \beta_{3}) q^{32} + ( - 3 \beta_{4} + 2 \beta_{2} - 5) q^{34} + (\beta_{5} - 5 \beta_{3}) q^{35} + (5 \beta_{4} + 2 \beta_{2} - 3) q^{37} + ( - 6 \beta_{5} + 2 \beta_{3} + \beta_1) q^{38} + ( - 9 \beta_{4} + 6 \beta_{2} - 2) q^{40} + ( - \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{41} + ( - 2 \beta_{4} - 3) q^{43} + \beta_1 q^{44} + ( - 6 \beta_{4} + 4 \beta_{2} + 3) q^{46} + (\beta_{3} + \beta_1) q^{47} + (5 \beta_{4} - 3 \beta_{2} - 1) q^{49} + (4 \beta_{5} - \beta_{3} + 3 \beta_1) q^{50} + (\beta_{5} + 4 \beta_1) q^{53} + (2 \beta_{4} - 4 \beta_{2} - 9) q^{55} + ( - 4 \beta_{5} + 5 \beta_{3} + \beta_1) q^{56} + (10 \beta_{4} - 11 \beta_{2} - 5) q^{58} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{59} + ( - 3 \beta_{4} + 2 \beta_{2} + 8) q^{61} + ( - 5 \beta_{5} - 3 \beta_{3} + \beta_1) q^{62} + ( - 4 \beta_{4} + 7 \beta_{2} + 7) q^{64} + ( - 5 \beta_{2} - 4) q^{67} + ( - 3 \beta_{5} + 4 \beta_{3} - \beta_1) q^{68} + (6 \beta_{4} - 17 \beta_{2} - 3) q^{70} + (3 \beta_{5} + \beta_{3} + 2 \beta_1) q^{71} + ( - 3 \beta_{2} - 5) q^{73} + ( - 5 \beta_{5} + 2 \beta_{3} + 5 \beta_1) q^{74} + ( - 8 \beta_{4} + 13 \beta_{2} - 2) q^{76} + (2 \beta_{5} + 3 \beta_{3}) q^{77} + ( - 4 \beta_{4} + 7 \beta_{2} + 2) q^{79} + ( - 2 \beta_{5} + 2 \beta_{3} - 5 \beta_1) q^{80} + (7 \beta_{4} - 12 \beta_{2} - 6) q^{82} + ( - 4 \beta_{5} - 2 \beta_{3} - \beta_1) q^{83} + ( - 3 \beta_{4} - 11 \beta_{2} - 5) q^{85} + ( - 3 \beta_{5} - 2 \beta_1) q^{86} + (8 \beta_{4} - \beta_{2} - 4) q^{88} + ( - 5 \beta_{5} - \beta_1) q^{89} + ( - 3 \beta_{5} + 2 \beta_{3}) q^{92} + (\beta_{4} + 2 \beta_{2} + 1) q^{94} + (9 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{95} + (8 \beta_{4} - 8 \beta_{2} - 7) q^{97} + (2 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{4} - 12 q^{7} - 14 q^{10} - 2 q^{16} - 26 q^{19} + 6 q^{22} + 26 q^{25} - 26 q^{28} - 40 q^{31} - 40 q^{34} - 12 q^{37} - 42 q^{40} - 22 q^{43} - 2 q^{46} + 10 q^{49} - 42 q^{55} + 12 q^{58} + 38 q^{61} + 20 q^{64} - 14 q^{67} + 28 q^{70} - 24 q^{73} - 54 q^{76} - 10 q^{79} + 2 q^{82} - 14 q^{85} - 6 q^{88} + 4 q^{94} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} + 19x^{2} - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 6\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 6\beta_{3} + 10\beta_1 \) 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
−1.09456
1.59842
−2.06082
2.06082
−1.59842
1.09456
−2.45945 0 4.04892 3.33722 0 −3.69202 −5.03922 0 −8.20775
1.2 −1.28184 0 −0.356896 2.16889 0 1.04892 3.02115 0 −2.78017
1.3 −1.14367 0 −0.692021 −3.48695 0 −3.35690 3.07878 0 3.98792
1.4 1.14367 0 −0.692021 3.48695 0 −3.35690 −3.07878 0 3.98792
1.5 1.28184 0 −0.356896 −2.16889 0 1.04892 −3.02115 0 −2.78017
1.6 2.45945 0 4.04892 −3.33722 0 −3.69202 5.03922 0 −8.20775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.v 6
3.b odd 2 1 inner 1521.2.a.v 6
13.b even 2 1 1521.2.a.w yes 6
13.d odd 4 2 1521.2.b.n 12
39.d odd 2 1 1521.2.a.w yes 6
39.f even 4 2 1521.2.b.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1521.2.a.v 6 1.a even 1 1 trivial
1521.2.a.v 6 3.b odd 2 1 inner
1521.2.a.w yes 6 13.b even 2 1
1521.2.a.w yes 6 39.d odd 2 1
1521.2.b.n 12 13.d odd 4 2
1521.2.b.n 12 39.f even 4 2

Hecke kernels

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

\( T_{2}^{6} - 9T_{2}^{4} + 20T_{2}^{2} - 13 \) Copy content Toggle raw display
\( T_{5}^{6} - 28T_{5}^{4} + 245T_{5}^{2} - 637 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} + 5T_{7} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 9 T^{4} + 20 T^{2} - 13 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 28 T^{4} + 245 T^{2} + \cdots - 637 \) Copy content Toggle raw display
$7$ \( (T^{3} + 6 T^{2} + 5 T - 13)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 29 T^{4} + 180 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 67 T^{4} + 962 T^{2} + \cdots - 2197 \) Copy content Toggle raw display
$19$ \( (T^{3} + 13 T^{2} + 40 T - 13)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 60 T^{4} + 689 T^{2} + \cdots - 2197 \) Copy content Toggle raw display
$29$ \( T^{6} - 158 T^{4} + 5967 T^{2} + \cdots - 2197 \) Copy content Toggle raw display
$31$ \( (T^{3} + 20 T^{2} + 117 T + 169)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} - 79 T - 377)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 165 T^{4} + 9068 T^{2} + \cdots - 165997 \) Copy content Toggle raw display
$43$ \( (T^{3} + 11 T^{2} + 31 T + 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 23 T^{4} + 34 T^{2} - 13 \) Copy content Toggle raw display
$53$ \( T^{6} - 161 T^{4} + 7644 T^{2} + \cdots - 107653 \) Copy content Toggle raw display
$59$ \( T^{6} - 120 T^{4} + 1328 T^{2} + \cdots - 832 \) Copy content Toggle raw display
$61$ \( (T^{3} - 19 T^{2} + 104 T - 169)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 7 T^{2} - 42 T - 91)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 162 T^{4} + 6207 T^{2} + \cdots - 10933 \) Copy content Toggle raw display
$73$ \( (T^{3} + 12 T^{2} + 27 T - 13)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 78 T + 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 212 T^{4} + 11367 T^{2} + \cdots - 165997 \) Copy content Toggle raw display
$89$ \( T^{6} - 263 T^{4} + 16404 T^{2} + \cdots - 251173 \) Copy content Toggle raw display
$97$ \( (T^{3} + 5 T^{2} - 141 T - 377)^{2} \) Copy content Toggle raw display
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