Properties

Label 1323.1.br.a
Level $1323$
Weight $1$
Character orbit 1323.br
Analytic conductor $0.660$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1323.br (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.660263011713\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{42}^{13} q^{4} - \zeta_{42}^{3} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{42}^{13} q^{4} - \zeta_{42}^{3} q^{7} + ( - \zeta_{42}^{5} - \zeta_{42}) q^{13} - \zeta_{42}^{5} q^{16} + ( - \zeta_{42}^{17} - \zeta_{42}^{11}) q^{19} + \zeta_{42}^{4} q^{25} + \zeta_{42}^{16} q^{28} + (\zeta_{42}^{20} - \zeta_{42}^{15}) q^{31} + (\zeta_{42}^{16} + 1) q^{37} + (\zeta_{42}^{2} - \zeta_{42}) q^{43} + \zeta_{42}^{6} q^{49} + (\zeta_{42}^{18} + \zeta_{42}^{14}) q^{52} + ( - \zeta_{42}^{9} - \zeta_{42}^{7}) q^{61} + \zeta_{42}^{18} q^{64} + (\zeta_{42}^{12} + \zeta_{42}^{2}) q^{67} + ( - \zeta_{42}^{19} + \zeta_{42}^{10}) q^{73} + ( - \zeta_{42}^{9} - \zeta_{42}^{3}) q^{76} + ( - \zeta_{42}^{15} - \zeta_{42}^{13}) q^{79} + (\zeta_{42}^{8} + \zeta_{42}^{4}) q^{91} + ( - \zeta_{42}^{11} + \zeta_{42}^{10}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{4} - 2 q^{7} + 2 q^{13} + q^{16} + 2 q^{19} + q^{25} + q^{28} - q^{31} + 13 q^{37} + 2 q^{43} - 2 q^{49} - 8 q^{52} - 8 q^{61} - 2 q^{64} - q^{67} + 2 q^{73} - 4 q^{76} - q^{79} + 2 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-\zeta_{42}^{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.988831 + 0.149042i
0.826239 + 0.563320i
0.955573 + 0.294755i
0.0747301 + 0.997204i
−0.733052 0.680173i
−0.733052 + 0.680173i
0.365341 + 0.930874i
−0.988831 0.149042i
0.0747301 0.997204i
0.365341 0.930874i
0.826239 0.563320i
0.955573 0.294755i
0 0 0.365341 + 0.930874i 0 0 −0.900969 + 0.433884i 0 0 0
107.1 0 0 0.0747301 + 0.997204i 0 0 −0.222521 + 0.974928i 0 0 0
242.1 0 0 −0.733052 0.680173i 0 0 0.623490 + 0.781831i 0 0 0
296.1 0 0 0.826239 + 0.563320i 0 0 −0.222521 0.974928i 0 0 0
431.1 0 0 0.955573 + 0.294755i 0 0 0.623490 0.781831i 0 0 0
485.1 0 0 0.955573 0.294755i 0 0 0.623490 + 0.781831i 0 0 0
620.1 0 0 −0.988831 + 0.149042i 0 0 −0.900969 0.433884i 0 0 0
674.1 0 0 0.365341 0.930874i 0 0 −0.900969 0.433884i 0 0 0
809.1 0 0 0.826239 0.563320i 0 0 −0.222521 + 0.974928i 0 0 0
1052.1 0 0 −0.988831 0.149042i 0 0 −0.900969 + 0.433884i 0 0 0
1187.1 0 0 0.0747301 0.997204i 0 0 −0.222521 0.974928i 0 0 0
1241.1 0 0 −0.733052 + 0.680173i 0 0 0.623490 0.781831i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.1.br.a 12
3.b odd 2 1 CM 1323.1.br.a 12
9.c even 3 1 3969.1.bx.a 12
9.c even 3 1 3969.1.cb.a 12
9.d odd 6 1 3969.1.bx.a 12
9.d odd 6 1 3969.1.cb.a 12
49.g even 21 1 inner 1323.1.br.a 12
147.n odd 42 1 inner 1323.1.br.a 12
441.y even 21 1 3969.1.bx.a 12
441.z even 21 1 3969.1.cb.a 12
441.bi odd 42 1 3969.1.cb.a 12
441.bm odd 42 1 3969.1.bx.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.1.br.a 12 1.a even 1 1 trivial
1323.1.br.a 12 3.b odd 2 1 CM
1323.1.br.a 12 49.g even 21 1 inner
1323.1.br.a 12 147.n odd 42 1 inner
3969.1.bx.a 12 9.c even 3 1
3969.1.bx.a 12 9.d odd 6 1
3969.1.bx.a 12 441.y even 21 1
3969.1.bx.a 12 441.bm odd 42 1
3969.1.cb.a 12 9.c even 3 1
3969.1.cb.a 12 9.d odd 6 1
3969.1.cb.a 12 441.z even 21 1
3969.1.cb.a 12 441.bi odd 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1323, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + 3 T^{10} + 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} - 13 T^{11} + 77 T^{10} - 274 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + 3 T^{10} - 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + 35 T^{10} + 104 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 2 T^{11} + 8 T^{9} - 9 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} - T^{5} - 6 T^{4} + 6 T^{3} + 8 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
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