Properties

Label 1323.2.c.a
Level $1323$
Weight $2$
Character orbit 1323.c
Analytic conductor $10.564$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} +O(q^{10})\) \( q + 2 q^{4} + ( 4 - 8 \zeta_{6} ) q^{13} + 4 q^{16} + ( 3 - 6 \zeta_{6} ) q^{19} -5 q^{25} + ( 1 - 2 \zeta_{6} ) q^{31} + 10 q^{37} + 13 q^{43} + ( 8 - 16 \zeta_{6} ) q^{52} + ( -5 + 10 \zeta_{6} ) q^{61} + 8 q^{64} -16 q^{67} + ( -9 + 18 \zeta_{6} ) q^{73} + ( 6 - 12 \zeta_{6} ) q^{76} -4 q^{79} + ( 11 - 22 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + O(q^{10}) \) \( 2q + 4q^{4} + 8q^{16} - 10q^{25} + 20q^{37} + 26q^{43} + 16q^{64} - 32q^{67} - 8q^{79} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 2.00000 0 0 0 0 0 0
1322.2 0 0 2.00000 0 0 0 0 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 CM by \(\Q(\sqrt{-3}) \)
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.a 2
3.b odd 2 1 CM 1323.2.c.a 2
7.b odd 2 1 inner 1323.2.c.a 2
7.c even 3 1 189.2.p.a 2
7.d odd 6 1 189.2.p.a 2
21.c even 2 1 inner 1323.2.c.a 2
21.g even 6 1 189.2.p.a 2
21.h odd 6 1 189.2.p.a 2
63.g even 3 1 567.2.s.b 2
63.h even 3 1 567.2.i.a 2
63.i even 6 1 567.2.i.a 2
63.j odd 6 1 567.2.i.a 2
63.k odd 6 1 567.2.s.b 2
63.n odd 6 1 567.2.s.b 2
63.s even 6 1 567.2.s.b 2
63.t odd 6 1 567.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.a 2 7.c even 3 1
189.2.p.a 2 7.d odd 6 1
189.2.p.a 2 21.g even 6 1
189.2.p.a 2 21.h odd 6 1
567.2.i.a 2 63.h even 3 1
567.2.i.a 2 63.i even 6 1
567.2.i.a 2 63.j odd 6 1
567.2.i.a 2 63.t odd 6 1
567.2.s.b 2 63.g even 3 1
567.2.s.b 2 63.k odd 6 1
567.2.s.b 2 63.n odd 6 1
567.2.s.b 2 63.s even 6 1
1323.2.c.a 2 1.a even 1 1 trivial
1323.2.c.a 2 3.b odd 2 1 CM
1323.2.c.a 2 7.b odd 2 1 inner
1323.2.c.a 2 21.c even 2 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -13 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 75 + T^{2} \)
$67$ \( ( 16 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 243 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 363 + T^{2} \)
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