Properties

Label 1323.2.a.w
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: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 5 q^{4} + \beta q^{5} + 3 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 5 q^{4} + \beta q^{5} + 3 \beta q^{8} + 7 q^{10} -\beta q^{11} + 2 q^{13} + 11 q^{16} -7 q^{19} + 5 \beta q^{20} -7 q^{22} -3 \beta q^{23} + 2 q^{25} + 2 \beta q^{26} + 2 \beta q^{29} -3 q^{31} + 5 \beta q^{32} -3 q^{37} -7 \beta q^{38} + 21 q^{40} -\beta q^{41} + 8 q^{43} -5 \beta q^{44} -21 q^{46} + 2 \beta q^{50} + 10 q^{52} -7 q^{55} + 14 q^{58} + 8 q^{61} -3 \beta q^{62} + 13 q^{64} + 2 \beta q^{65} -2 q^{67} + 3 \beta q^{71} -3 \beta q^{74} -35 q^{76} -4 q^{79} + 11 \beta q^{80} -7 q^{82} -6 \beta q^{83} + 8 \beta q^{86} -21 q^{88} + 7 \beta q^{89} -15 \beta q^{92} -7 \beta q^{95} + 12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{4} + O(q^{10}) \) \( 2q + 10q^{4} + 14q^{10} + 4q^{13} + 22q^{16} - 14q^{19} - 14q^{22} + 4q^{25} - 6q^{31} - 6q^{37} + 42q^{40} + 16q^{43} - 42q^{46} + 20q^{52} - 14q^{55} + 28q^{58} + 16q^{61} + 26q^{64} - 4q^{67} - 70q^{76} - 8q^{79} - 14q^{82} - 42q^{88} + 24q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
−2.64575 0 5.00000 −2.64575 0 0 −7.93725 0 7.00000
1.2 2.64575 0 5.00000 2.64575 0 0 7.93725 0 7.00000
\(n\): 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.w 2
3.b odd 2 1 inner 1323.2.a.w 2
7.b odd 2 1 189.2.a.f 2
21.c even 2 1 189.2.a.f 2
28.d even 2 1 3024.2.a.bi 2
35.c odd 2 1 4725.2.a.bb 2
63.l odd 6 2 567.2.f.i 4
63.o even 6 2 567.2.f.i 4
84.h odd 2 1 3024.2.a.bi 2
105.g even 2 1 4725.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 7.b odd 2 1
189.2.a.f 2 21.c even 2 1
567.2.f.i 4 63.l odd 6 2
567.2.f.i 4 63.o even 6 2
1323.2.a.w 2 1.a even 1 1 trivial
1323.2.a.w 2 3.b odd 2 1 inner
3024.2.a.bi 2 28.d even 2 1
3024.2.a.bi 2 84.h odd 2 1
4725.2.a.bb 2 35.c odd 2 1
4725.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{2} - 7 \)
\( T_{5}^{2} - 7 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -7 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -7 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( -63 + T^{2} \)
$29$ \( -28 + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( ( 3 + T )^{2} \)
$41$ \( -7 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( -63 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -252 + T^{2} \)
$89$ \( -343 + T^{2} \)
$97$ \( ( -12 + T )^{2} \)
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