Properties

Label 1323.2.a.v
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 4 q^{4} + \beta q^{5} + 2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 4 q^{4} + \beta q^{5} + 2 \beta q^{8} + 6 q^{10} + 2 \beta q^{11} -4 q^{13} + 4 q^{16} -\beta q^{17} - q^{19} + 4 \beta q^{20} + 12 q^{22} + \beta q^{23} + q^{25} -4 \beta q^{26} -3 \beta q^{29} -7 q^{31} -6 q^{34} + 8 q^{37} -\beta q^{38} + 12 q^{40} + 3 \beta q^{41} - q^{43} + 8 \beta q^{44} + 6 q^{46} + \beta q^{47} + \beta q^{50} -16 q^{52} -\beta q^{53} + 12 q^{55} -18 q^{58} -4 \beta q^{59} + 5 q^{61} -7 \beta q^{62} -8 q^{64} -4 \beta q^{65} + 2 q^{67} -4 \beta q^{68} - q^{73} + 8 \beta q^{74} -4 q^{76} -4 q^{79} + 4 \beta q^{80} + 18 q^{82} + 6 \beta q^{83} -6 q^{85} -\beta q^{86} + 24 q^{88} + \beta q^{89} + 4 \beta q^{92} + 6 q^{94} -\beta q^{95} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + O(q^{10}) \) \( 2q + 8q^{4} + 12q^{10} - 8q^{13} + 8q^{16} - 2q^{19} + 24q^{22} + 2q^{25} - 14q^{31} - 12q^{34} + 16q^{37} + 24q^{40} - 2q^{43} + 12q^{46} - 32q^{52} + 24q^{55} - 36q^{58} + 10q^{61} - 16q^{64} + 4q^{67} - 2q^{73} - 8q^{76} - 8q^{79} + 36q^{82} - 12q^{85} + 48q^{88} + 12q^{94} - 2q^{97} + O(q^{100}) \)

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
−2.44949
2.44949
−2.44949 0 4.00000 −2.44949 0 0 −4.89898 0 6.00000
1.2 2.44949 0 4.00000 2.44949 0 0 4.89898 0 6.00000
\(n\): 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
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 1323.2.a.v 2
3.b odd 2 1 inner 1323.2.a.v 2
7.b odd 2 1 1323.2.a.u 2
7.c even 3 2 189.2.e.d 4
21.c even 2 1 1323.2.a.u 2
21.h odd 6 2 189.2.e.d 4
63.g even 3 2 567.2.g.g 4
63.h even 3 2 567.2.h.g 4
63.j odd 6 2 567.2.h.g 4
63.n odd 6 2 567.2.g.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.d 4 7.c even 3 2
189.2.e.d 4 21.h odd 6 2
567.2.g.g 4 63.g even 3 2
567.2.g.g 4 63.n odd 6 2
567.2.h.g 4 63.h even 3 2
567.2.h.g 4 63.j odd 6 2
1323.2.a.u 2 7.b odd 2 1
1323.2.a.u 2 21.c even 2 1
1323.2.a.v 2 1.a even 1 1 trivial
1323.2.a.v 2 3.b odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -24 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( -6 + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -6 + T^{2} \)
$29$ \( -54 + T^{2} \)
$31$ \( ( 7 + T )^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( -54 + T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -6 + T^{2} \)
$53$ \( -6 + T^{2} \)
$59$ \( -96 + T^{2} \)
$61$ \( ( -5 + T )^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -216 + T^{2} \)
$89$ \( -6 + T^{2} \)
$97$ \( ( 1 + T )^{2} \)
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