Properties

Label 1323.2.a.bb
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.5642081874\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{10} + ( -\beta_{1} - \beta_{2} ) q^{11} + \beta_{1} q^{13} -5 q^{17} + ( -3 \beta_{1} + \beta_{2} ) q^{19} + ( -7 - 3 \beta_{3} ) q^{20} + ( -4 - 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 4 + 4 \beta_{3} ) q^{25} + ( 3 + \beta_{3} ) q^{26} + ( -5 \beta_{1} + 3 \beta_{2} ) q^{29} + ( -\beta_{1} + 4 \beta_{2} ) q^{31} -4 \beta_{2} q^{32} -5 \beta_{1} q^{34} + ( 2 - 3 \beta_{3} ) q^{37} + ( -8 - 2 \beta_{3} ) q^{38} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{40} + ( -2 + 3 \beta_{3} ) q^{41} + \beta_{3} q^{43} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 3 - \beta_{3} ) q^{46} + ( -3 + 2 \beta_{3} ) q^{47} + ( 8 \beta_{1} + 8 \beta_{2} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{52} + ( 2 \beta_{1} - \beta_{2} ) q^{53} + ( 5 \beta_{1} + 3 \beta_{2} ) q^{55} + ( -12 - 2 \beta_{3} ) q^{58} + ( -5 + 4 \beta_{3} ) q^{59} + ( -\beta_{1} - \beta_{2} ) q^{61} + ( 1 + 3 \beta_{3} ) q^{62} + ( -4 - 4 \beta_{3} ) q^{64} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -9 - \beta_{3} ) q^{67} + ( -5 - 5 \beta_{3} ) q^{68} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -\beta_{1} - 6 \beta_{2} ) q^{74} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -7 - 2 \beta_{3} ) q^{79} + ( \beta_{1} + 6 \beta_{2} ) q^{82} + ( -2 + \beta_{3} ) q^{83} + ( 10 + 5 \beta_{3} ) q^{85} + ( \beta_{1} + 2 \beta_{2} ) q^{86} + ( -6 - 2 \beta_{3} ) q^{88} + ( -5 + 3 \beta_{3} ) q^{89} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -\beta_{1} + 4 \beta_{2} ) q^{94} + ( 7 \beta_{1} + 5 \beta_{2} ) q^{95} + ( \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 8q^{5} + O(q^{10}) \) \( 4q + 4q^{4} - 8q^{5} - 20q^{17} - 28q^{20} - 16q^{22} + 16q^{25} + 12q^{26} + 8q^{37} - 32q^{38} - 8q^{41} + 12q^{46} - 12q^{47} - 48q^{58} - 20q^{59} + 4q^{62} - 16q^{64} - 36q^{67} - 20q^{68} - 28q^{79} - 8q^{83} + 40q^{85} - 24q^{88} - 20q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 4 \beta_{1}\)

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
−2.28825
−0.874032
0.874032
2.28825
−2.28825 0 3.23607 −4.23607 0 0 −2.82843 0 9.69316
1.2 −0.874032 0 −1.23607 0.236068 0 0 2.82843 0 −0.206331
1.3 0.874032 0 −1.23607 0.236068 0 0 −2.82843 0 0.206331
1.4 2.28825 0 3.23607 −4.23607 0 0 2.82843 0 −9.69316
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.bb 4
3.b odd 2 1 1323.2.a.be yes 4
7.b odd 2 1 1323.2.a.be yes 4
21.c even 2 1 inner 1323.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.a.bb 4 1.a even 1 1 trivial
1323.2.a.bb 4 21.c even 2 1 inner
1323.2.a.be yes 4 3.b odd 2 1
1323.2.a.be yes 4 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{4} - 6 T_{2}^{2} + 4 \)
\( T_{5}^{2} + 4 T_{5} - 1 \)
\( T_{13}^{4} - 6 T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 6 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -1 + 4 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 4 - 14 T^{2} + T^{4} \)
$13$ \( 4 - 6 T^{2} + T^{4} \)
$17$ \( ( 5 + T )^{4} \)
$19$ \( 484 - 46 T^{2} + T^{4} \)
$23$ \( 4 - 36 T^{2} + T^{4} \)
$29$ \( 3844 - 126 T^{2} + T^{4} \)
$31$ \( 484 - 54 T^{2} + T^{4} \)
$37$ \( ( -41 - 4 T + T^{2} )^{2} \)
$41$ \( ( -41 + 4 T + T^{2} )^{2} \)
$43$ \( ( -5 + T^{2} )^{2} \)
$47$ \( ( -11 + 6 T + T^{2} )^{2} \)
$53$ \( ( -10 + T^{2} )^{2} \)
$59$ \( ( -55 + 10 T + T^{2} )^{2} \)
$61$ \( 4 - 14 T^{2} + T^{4} \)
$67$ \( ( 76 + 18 T + T^{2} )^{2} \)
$71$ \( 7744 - 216 T^{2} + T^{4} \)
$73$ \( 100 - 180 T^{2} + T^{4} \)
$79$ \( ( 29 + 14 T + T^{2} )^{2} \)
$83$ \( ( -1 + 4 T + T^{2} )^{2} \)
$89$ \( ( -20 + 10 T + T^{2} )^{2} \)
$97$ \( 100 - 30 T^{2} + T^{4} \)
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