Properties

Label 1323.2.a.ba
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( 2 + \beta_{1} ) q^{11} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{13} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{16} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} + ( -5 + 2 \beta_{1} ) q^{20} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{22} + ( 2 + \beta_{1} + \beta_{2} ) q^{23} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{25} + ( 6 + \beta_{2} ) q^{26} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{31} + ( 10 - 4 \beta_{1} + \beta_{2} ) q^{32} + ( 6 - 3 \beta_{1} + 3 \beta_{2} ) q^{34} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - 2 \beta_{2} ) q^{38} + ( -9 + 3 \beta_{1} ) q^{40} + ( 2 - 5 \beta_{1} - \beta_{2} ) q^{41} + ( -4 + 3 \beta_{1} ) q^{43} + \beta_{2} q^{44} -3 \beta_{1} q^{46} + ( 2 + 4 \beta_{1} + \beta_{2} ) q^{47} + ( 3 \beta_{1} - \beta_{2} ) q^{50} + ( 5 - 3 \beta_{1} + 3 \beta_{2} ) q^{52} + ( 7 + 2 \beta_{1} - \beta_{2} ) q^{53} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{55} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( -5 - \beta_{1} - \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -8 + 2 \beta_{1} - \beta_{2} ) q^{62} + ( 13 - 3 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} ) q^{67} + ( 16 - 7 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{71} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -2 + 5 \beta_{1} - \beta_{2} ) q^{74} + ( -7 + 3 \beta_{1} ) q^{76} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -8 + 5 \beta_{1} - 3 \beta_{2} ) q^{80} + ( 16 - \beta_{1} + 4 \beta_{2} ) q^{82} + ( 6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -9 + 3 \beta_{1} + 3 \beta_{2} ) q^{85} + ( -13 + 4 \beta_{1} - 3 \beta_{2} ) q^{86} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{88} + ( -3 - 4 \beta_{2} ) q^{89} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{92} + ( -9 - 3 \beta_{1} - 3 \beta_{2} ) q^{94} + ( 5 - 2 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 4q^{4} + q^{5} + 9q^{8} + O(q^{10}) \) \( 3q + 2q^{2} + 4q^{4} + q^{5} + 9q^{8} - q^{10} + 7q^{11} + 2q^{13} + 10q^{16} + 5q^{19} - 13q^{20} - 4q^{22} + 6q^{23} - 2q^{25} + 17q^{26} + 13q^{29} - 8q^{31} + 25q^{32} + 12q^{34} - 8q^{37} - 7q^{38} - 24q^{40} + 2q^{41} - 9q^{43} - q^{44} - 3q^{46} + 9q^{47} + 4q^{50} + 9q^{52} + 24q^{53} + 4q^{55} + 14q^{58} - 15q^{59} - q^{61} - 21q^{62} + 33q^{64} + 10q^{65} + 14q^{67} + 39q^{68} - 3q^{71} + 7q^{73} - 18q^{76} + 6q^{79} - 16q^{80} + 43q^{82} + 3q^{83} - 27q^{85} - 32q^{86} + 9q^{88} - 5q^{89} + 12q^{92} - 27q^{94} + 16q^{95} - 14q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

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

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
2.46050
0.239123
−1.69963
−1.46050 0 0.133074 −0.593579 0 0 2.72665 0 0.866926
1.2 0.760877 0 −1.42107 3.18194 0 0 −2.60301 0 2.42107
1.3 2.69963 0 5.28799 −1.58836 0 0 8.87636 0 −4.28799
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1323.2.a.ba 3
3.b odd 2 1 1323.2.a.x 3
7.b odd 2 1 1323.2.a.z 3
7.c even 3 2 189.2.e.e 6
21.c even 2 1 1323.2.a.y 3
21.h odd 6 2 189.2.e.f yes 6
63.g even 3 2 567.2.g.h 6
63.h even 3 2 567.2.h.i 6
63.j odd 6 2 567.2.h.h 6
63.n odd 6 2 567.2.g.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e 6 7.c even 3 2
189.2.e.f yes 6 21.h odd 6 2
567.2.g.h 6 63.g even 3 2
567.2.g.i 6 63.n odd 6 2
567.2.h.h 6 63.j odd 6 2
567.2.h.i 6 63.h even 3 2
1323.2.a.x 3 3.b odd 2 1
1323.2.a.y 3 21.c even 2 1
1323.2.a.z 3 7.b odd 2 1
1323.2.a.ba 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 3 T_{2} + 3 \)
\( T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 3 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 19 T_{13} + 47 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 3 T - 2 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -3 - 6 T - T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -3 + 12 T - 7 T^{2} + T^{3} \)
$13$ \( 47 - 19 T - 2 T^{2} + T^{3} \)
$17$ \( -9 - 33 T + T^{3} \)
$19$ \( 29 - 4 T - 5 T^{2} + T^{3} \)
$23$ \( 9 + 3 T - 6 T^{2} + T^{3} \)
$29$ \( -9 + 30 T - 13 T^{2} + T^{3} \)
$31$ \( -69 + T + 8 T^{2} + T^{3} \)
$37$ \( -93 - 5 T + 8 T^{2} + T^{3} \)
$41$ \( 387 - 105 T - 2 T^{2} + T^{3} \)
$43$ \( -101 - 12 T + 9 T^{2} + T^{3} \)
$47$ \( -9 - 42 T - 9 T^{2} + T^{3} \)
$53$ \( -243 + 165 T - 24 T^{2} + T^{3} \)
$59$ \( 81 + 66 T + 15 T^{2} + T^{3} \)
$61$ \( -121 - 49 T + T^{2} + T^{3} \)
$67$ \( -31 + 53 T - 14 T^{2} + T^{3} \)
$71$ \( 243 - 108 T + 3 T^{2} + T^{3} \)
$73$ \( 981 - 134 T - 7 T^{2} + T^{3} \)
$79$ \( 127 - 69 T - 6 T^{2} + T^{3} \)
$83$ \( -729 - 180 T - 3 T^{2} + T^{3} \)
$89$ \( -489 - 93 T + 5 T^{2} + T^{3} \)
$97$ \( -24 + 16 T + 14 T^{2} + T^{3} \)
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