Properties

Label 1323.2.c.b
Level $1323$
Weight $2$
Character orbit 1323.c
Analytic conductor $10.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} - 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 10 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{2} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(5 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1322.1
−1.93649 1.11803i
1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
2.23607i 0 −3.00000 0 0 0 2.23607i 0 0
1322.2 2.23607i 0 −3.00000 0 0 0 2.23607i 0 0
1322.3 2.23607i 0 −3.00000 0 0 0 2.23607i 0 0
1322.4 2.23607i 0 −3.00000 0 0 0 2.23607i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
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.c.b 4
3.b odd 2 1 inner 1323.2.c.b 4
7.b odd 2 1 inner 1323.2.c.b 4
7.c even 3 1 189.2.p.c 4
7.d odd 6 1 189.2.p.c 4
21.c even 2 1 inner 1323.2.c.b 4
21.g even 6 1 189.2.p.c 4
21.h odd 6 1 189.2.p.c 4
63.g even 3 1 567.2.s.e 4
63.h even 3 1 567.2.i.c 4
63.i even 6 1 567.2.i.c 4
63.j odd 6 1 567.2.i.c 4
63.k odd 6 1 567.2.s.e 4
63.n odd 6 1 567.2.s.e 4
63.s even 6 1 567.2.s.e 4
63.t odd 6 1 567.2.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c 4 7.c even 3 1
189.2.p.c 4 7.d odd 6 1
189.2.p.c 4 21.g even 6 1
189.2.p.c 4 21.h odd 6 1
567.2.i.c 4 63.h even 3 1
567.2.i.c 4 63.i even 6 1
567.2.i.c 4 63.j odd 6 1
567.2.i.c 4 63.t odd 6 1
567.2.s.e 4 63.g even 3 1
567.2.s.e 4 63.k odd 6 1
567.2.s.e 4 63.n odd 6 1
567.2.s.e 4 63.s even 6 1
1323.2.c.b 4 1.a even 1 1 trivial
1323.2.c.b 4 3.b odd 2 1 inner
1323.2.c.b 4 7.b odd 2 1 inner
1323.2.c.b 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 5 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( ( -60 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( 20 + T^{2} )^{2} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( ( -5 + T )^{4} \)
$41$ \( ( -60 + T^{2} )^{2} \)
$43$ \( ( 7 + T )^{4} \)
$47$ \( ( -60 + T^{2} )^{2} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( ( -60 + T^{2} )^{2} \)
$61$ \( ( 75 + T^{2} )^{2} \)
$67$ \( ( 1 + T )^{4} \)
$71$ \( ( 80 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( -11 + T )^{4} \)
$83$ \( ( -60 + T^{2} )^{2} \)
$89$ \( ( -240 + T^{2} )^{2} \)
$97$ \( ( 3 + T^{2} )^{2} \)
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