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Properties

Label 3087.1.bj.a
Level $3087$
Weight $1$
Character orbit 3087.bj
Analytic conductor $1.541$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3087 = 3^{2} \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3087.bj (of order \(42\), degree \(12\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.54061369400\)
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: no (minimal twist has level 441)
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

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

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

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times\).

\(n\) \(344\) \(2404\)
\(\chi(n)\) \(1\) \(-\zeta_{42}^{20}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
460.1
−0.988831 − 0.149042i
0.0747301 − 0.997204i
−0.733052 + 0.680173i
0.365341 + 0.930874i
0.0747301 + 0.997204i
−0.733052 − 0.680173i
0.826239 + 0.563320i
0.955573 + 0.294755i
0.955573 − 0.294755i
−0.988831 + 0.149042i
0.365341 − 0.930874i
0.826239 − 0.563320i
0 0 0.733052 − 0.680173i 0 0 0 0 0 0
766.1 0 0 −0.365341 − 0.930874i 0 0 0 0 0 0
901.1 0 0 −0.826239 − 0.563320i 0 0 0 0 0 0
1207.1 0 0 −0.955573 − 0.294755i 0 0 0 0 0 0
1342.1 0 0 −0.365341 + 0.930874i 0 0 0 0 0 0
1648.1 0 0 −0.826239 + 0.563320i 0 0 0 0 0 0
1783.1 0 0 0.988831 + 0.149042i 0 0 0 0 0 0
2089.1 0 0 −0.0747301 + 0.997204i 0 0 0 0 0 0
2224.1 0 0 −0.0747301 − 0.997204i 0 0 0 0 0 0
2530.1 0 0 0.733052 + 0.680173i 0 0 0 0 0 0
2665.1 0 0 −0.955573 + 0.294755i 0 0 0 0 0 0
2971.1 0 0 0.988831 − 0.149042i 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2971.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner
49.f odd 14 1 inner
49.h odd 42 1 inner
147.k even 14 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3087.1.bj.a 12
3.b odd 2 1 CM 3087.1.bj.a 12
7.b odd 2 1 3087.1.bj.b 12
7.c even 3 1 3087.1.v.a 6
7.c even 3 1 inner 3087.1.bj.a 12
7.d odd 6 1 441.1.v.a ✓ 6
7.d odd 6 1 3087.1.bj.b 12
21.c even 2 1 3087.1.bj.b 12
21.g even 6 1 441.1.v.a ✓ 6
21.g even 6 1 3087.1.bj.b 12
21.h odd 6 1 3087.1.v.a 6
21.h odd 6 1 inner 3087.1.bj.a 12
49.e even 7 1 3087.1.bj.b 12
49.f odd 14 1 inner 3087.1.bj.a 12
49.g even 21 1 441.1.v.a ✓ 6
49.g even 21 1 3087.1.bj.b 12
49.h odd 42 1 3087.1.v.a 6
49.h odd 42 1 inner 3087.1.bj.a 12
63.i even 6 1 3969.1.bz.a 12
63.k odd 6 1 3969.1.bz.a 12
63.s even 6 1 3969.1.bz.a 12
63.t odd 6 1 3969.1.bz.a 12
147.k even 14 1 inner 3087.1.bj.a 12
147.l odd 14 1 3087.1.bj.b 12
147.n odd 42 1 441.1.v.a ✓ 6
147.n odd 42 1 3087.1.bj.b 12
147.o even 42 1 3087.1.v.a 6
147.o even 42 1 inner 3087.1.bj.a 12
441.y even 21 1 3969.1.bz.a 12
441.z even 21 1 3969.1.bz.a 12
441.bi odd 42 1 3969.1.bz.a 12
441.bm odd 42 1 3969.1.bz.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a ✓ 6 7.d odd 6 1
441.1.v.a ✓ 6 21.g even 6 1
441.1.v.a ✓ 6 49.g even 21 1
441.1.v.a ✓ 6 147.n odd 42 1
3087.1.v.a 6 7.c even 3 1
3087.1.v.a 6 21.h odd 6 1
3087.1.v.a 6 49.h odd 42 1
3087.1.v.a 6 147.o even 42 1
3087.1.bj.a 12 1.a even 1 1 trivial
3087.1.bj.a 12 3.b odd 2 1 CM
3087.1.bj.a 12 7.c even 3 1 inner
3087.1.bj.a 12 21.h odd 6 1 inner
3087.1.bj.a 12 49.f odd 14 1 inner
3087.1.bj.a 12 49.h odd 42 1 inner
3087.1.bj.a 12 147.k even 14 1 inner
3087.1.bj.a 12 147.o even 42 1 inner
3087.1.bj.b 12 7.b odd 2 1
3087.1.bj.b 12 7.d odd 6 1
3087.1.bj.b 12 21.c even 2 1
3087.1.bj.b 12 21.g even 6 1
3087.1.bj.b 12 49.e even 7 1
3087.1.bj.b 12 49.g even 21 1
3087.1.bj.b 12 147.l odd 14 1
3087.1.bj.b 12 147.n odd 42 1
3969.1.bz.a 12 63.i even 6 1
3969.1.bz.a 12 63.k odd 6 1
3969.1.bz.a 12 63.s even 6 1
3969.1.bz.a 12 63.t odd 6 1
3969.1.bz.a 12 441.y even 21 1
3969.1.bz.a 12 441.z even 21 1
3969.1.bz.a 12 441.bi odd 42 1
3969.1.bz.a 12 441.bm odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} - 7T_{13}^{3} + 7T_{13} + 7 \) acting on \(S_{1}^{\mathrm{new}}(3087, [\chi])\). Copy content Toggle raw display

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
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$37$ \( T^{12} - 5 T^{11} + 14 T^{10} - 29 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( (T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + 9 T^{2} + \cdots + 1)^{2} \) 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
$61$ \( T^{12} + 7 T^{11} + 28 T^{10} + 77 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$67$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 7 T^{8} + 14 T^{7} + 14 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
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