Properties

Label 1323.2.a.t
Level $1323$
Weight $2$
Character orbit 1323.a
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} -\beta q^{5} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + q^{4} -\beta q^{5} -\beta q^{8} -3 q^{10} -\beta q^{11} -2 q^{13} -5 q^{16} + 4 \beta q^{17} -5 q^{19} -\beta q^{20} -3 q^{22} + \beta q^{23} -2 q^{25} -2 \beta q^{26} -6 \beta q^{29} -5 q^{31} -3 \beta q^{32} + 12 q^{34} -7 q^{37} -5 \beta q^{38} + 3 q^{40} -3 \beta q^{41} -4 q^{43} -\beta q^{44} + 3 q^{46} -4 \beta q^{47} -2 \beta q^{50} -2 q^{52} + 8 \beta q^{53} + 3 q^{55} -18 q^{58} + 4 \beta q^{59} -8 q^{61} -5 \beta q^{62} + q^{64} + 2 \beta q^{65} + 14 q^{67} + 4 \beta q^{68} + 3 \beta q^{71} + 4 q^{73} -7 \beta q^{74} -5 q^{76} + 8 q^{79} + 5 \beta q^{80} -9 q^{82} + 6 \beta q^{83} -12 q^{85} -4 \beta q^{86} + 3 q^{88} + 5 \beta q^{89} + \beta q^{92} -12 q^{94} + 5 \beta q^{95} + 4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} - 6q^{10} - 4q^{13} - 10q^{16} - 10q^{19} - 6q^{22} - 4q^{25} - 10q^{31} + 24q^{34} - 14q^{37} + 6q^{40} - 8q^{43} + 6q^{46} - 4q^{52} + 6q^{55} - 36q^{58} - 16q^{61} + 2q^{64} + 28q^{67} + 8q^{73} - 10q^{76} + 16q^{79} - 18q^{82} - 24q^{85} + 6q^{88} - 24q^{94} + 8q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 0 1.00000 1.73205 0 0 1.73205 0 −3.00000
1.2 1.73205 0 1.00000 −1.73205 0 0 −1.73205 0 −3.00000
\(n\): 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.t 2
3.b odd 2 1 inner 1323.2.a.t 2
7.b odd 2 1 189.2.a.e 2
21.c even 2 1 189.2.a.e 2
28.d even 2 1 3024.2.a.bg 2
35.c odd 2 1 4725.2.a.ba 2
63.l odd 6 2 567.2.f.k 4
63.o even 6 2 567.2.f.k 4
84.h odd 2 1 3024.2.a.bg 2
105.g even 2 1 4725.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.e 2 7.b odd 2 1
189.2.a.e 2 21.c even 2 1
567.2.f.k 4 63.l odd 6 2
567.2.f.k 4 63.o even 6 2
1323.2.a.t 2 1.a even 1 1 trivial
1323.2.a.t 2 3.b odd 2 1 inner
3024.2.a.bg 2 28.d even 2 1
3024.2.a.bg 2 84.h odd 2 1
4725.2.a.ba 2 35.c odd 2 1
4725.2.a.ba 2 105.g even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -3 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -48 + T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( -3 + T^{2} \)
$29$ \( -108 + T^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( -27 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -192 + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( -14 + T )^{2} \)
$71$ \( -27 + T^{2} \)
$73$ \( ( -4 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( -108 + T^{2} \)
$89$ \( -75 + T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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