Properties

Label 3267.1.l.a
Level $3267$
Weight $1$
Character orbit 3267.l
Analytic conductor $1.630$
Analytic rank $0$
Dimension $16$
Projective image $D_{12}$
CM discriminant -3
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{4} - \beta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{4} - \beta_{12} q^{7} + \beta_{3} q^{13} - \beta_{8} q^{16} + (\beta_{7} + \beta_{5}) q^{19} - \beta_{13} q^{25} + (\beta_{12} + \beta_{9} + \beta_{5} - \beta_1) q^{28} + \beta_{14} q^{31} + \beta_{11} q^{43} + ( - \beta_{10} + \beta_{8}) q^{49} + \beta_{7} q^{52} + ( - \beta_{15} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{61} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{64} + \beta_{2} q^{67} + ( - \beta_{15} - \beta_{12}) q^{73} - \beta_{11} q^{76} - \beta_{3} q^{79} - \beta_{13} q^{91} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 362 ) / 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 780\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 780\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 780\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{13} - 2911\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{15} - 2911\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 723\nu^{5} ) / 19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} - 7\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - 11\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{15} + 41\beta_{12} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 209\beta_{2} - 362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 209\beta_{3} - 780\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{6} - 1351\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -780\beta_{7} - 2911\beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2911\beta_{10} + 5042\beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2911\beta_{11} + 7953\beta_{9} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
838.1
−1.83730 0.596975i
−0.492303 0.159959i
0.492303 + 0.159959i
1.83730 + 0.596975i
1.13551 + 1.56290i
0.304260 + 0.418778i
−0.304260 0.418778i
−1.13551 1.56290i
1.13551 1.56290i
0.304260 0.418778i
−0.304260 + 0.418778i
−1.13551 + 1.56290i
−1.83730 + 0.596975i
−0.492303 + 0.159959i
0.492303 0.159959i
1.83730 0.596975i
0 0 −0.809017 0.587785i 0 0 −1.13551 + 1.56290i 0 0 0
838.2 0 0 −0.809017 0.587785i 0 0 −0.304260 + 0.418778i 0 0 0
838.3 0 0 −0.809017 0.587785i 0 0 0.304260 0.418778i 0 0 0
838.4 0 0 −0.809017 0.587785i 0 0 1.13551 1.56290i 0 0 0
2296.1 0 0 0.309017 0.951057i 0 0 −1.83730 0.596975i 0 0 0
2296.2 0 0 0.309017 0.951057i 0 0 −0.492303 0.159959i 0 0 0
2296.3 0 0 0.309017 0.951057i 0 0 0.492303 + 0.159959i 0 0 0
2296.4 0 0 0.309017 0.951057i 0 0 1.83730 + 0.596975i 0 0 0
2944.1 0 0 0.309017 + 0.951057i 0 0 −1.83730 + 0.596975i 0 0 0
2944.2 0 0 0.309017 + 0.951057i 0 0 −0.492303 + 0.159959i 0 0 0
2944.3 0 0 0.309017 + 0.951057i 0 0 0.492303 0.159959i 0 0 0
2944.4 0 0 0.309017 + 0.951057i 0 0 1.83730 0.596975i 0 0 0
2998.1 0 0 −0.809017 + 0.587785i 0 0 −1.13551 1.56290i 0 0 0
2998.2 0 0 −0.809017 + 0.587785i 0 0 −0.304260 0.418778i 0 0 0
2998.3 0 0 −0.809017 + 0.587785i 0 0 0.304260 + 0.418778i 0 0 0
2998.4 0 0 −0.809017 + 0.587785i 0 0 1.13551 + 1.56290i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2998.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.l.a 16
3.b odd 2 1 CM 3267.1.l.a 16
11.b odd 2 1 inner 3267.1.l.a 16
11.c even 5 1 3267.1.c.a 4
11.c even 5 3 inner 3267.1.l.a 16
11.d odd 10 1 3267.1.c.a 4
11.d odd 10 3 inner 3267.1.l.a 16
33.d even 2 1 inner 3267.1.l.a 16
33.f even 10 1 3267.1.c.a 4
33.f even 10 3 inner 3267.1.l.a 16
33.h odd 10 1 3267.1.c.a 4
33.h odd 10 3 inner 3267.1.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.c.a 4 11.c even 5 1
3267.1.c.a 4 11.d odd 10 1
3267.1.c.a 4 33.f even 10 1
3267.1.c.a 4 33.h odd 10 1
3267.1.l.a 16 1.a even 1 1 trivial
3267.1.l.a 16 3.b odd 2 1 CM
3267.1.l.a 16 11.b odd 2 1 inner
3267.1.l.a 16 11.c even 5 3 inner
3267.1.l.a 16 11.d odd 10 3 inner
3267.1.l.a 16 33.d even 2 1 inner
3267.1.l.a 16 33.f even 10 3 inner
3267.1.l.a 16 33.h odd 10 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3267, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
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