Properties

Label 1521.1.bd.b
Level $1521$
Weight $1$
Character orbit 1521.bd
Analytic conductor $0.759$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1521.bd (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.759077884215\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.6591.1
Artin image: $C_3\times C_4{\rm wrC}_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12} q^{4} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{7} +O(q^{10})\) \( q + \zeta_{12} q^{4} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{7} + \zeta_{12}^{2} q^{16} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{19} -\zeta_{12}^{3} q^{25} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{28} + ( -1 - \zeta_{12}^{3} ) q^{31} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{37} + \zeta_{12}^{5} q^{49} + \zeta_{12}^{3} q^{64} + ( -\zeta_{12}^{2} - \zeta_{12}^{5} ) q^{67} + ( 1 - \zeta_{12}^{3} ) q^{73} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{76} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{7} + O(q^{10}) \) \( 4q - 2q^{7} + 2q^{16} - 2q^{19} + 2q^{28} - 4q^{31} + 2q^{37} - 2q^{67} + 4q^{73} - 2q^{76} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0.866025 0.500000i 0 0 0.366025 1.36603i 0 0 0
1333.1 0 0 −0.866025 + 0.500000i 0 0 −1.36603 0.366025i 0 0 0
1432.1 0 0 −0.866025 0.500000i 0 0 −1.36603 + 0.366025i 0 0 0
1441.1 0 0 0.866025 + 0.500000i 0 0 0.366025 + 1.36603i 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 \(\Q(\sqrt{-3}) \)
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
39.f even 4 1 inner
39.i odd 6 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.1.bd.b 4
3.b odd 2 1 CM 1521.1.bd.b 4
13.b even 2 1 1521.1.bd.c 4
13.c even 3 1 1521.1.j.b 2
13.c even 3 1 inner 1521.1.bd.b 4
13.d odd 4 1 inner 1521.1.bd.b 4
13.d odd 4 1 1521.1.bd.c 4
13.e even 6 1 117.1.j.a 2
13.e even 6 1 1521.1.bd.c 4
13.f odd 12 1 117.1.j.a 2
13.f odd 12 1 1521.1.j.b 2
13.f odd 12 1 inner 1521.1.bd.b 4
13.f odd 12 1 1521.1.bd.c 4
39.d odd 2 1 1521.1.bd.c 4
39.f even 4 1 inner 1521.1.bd.b 4
39.f even 4 1 1521.1.bd.c 4
39.h odd 6 1 117.1.j.a 2
39.h odd 6 1 1521.1.bd.c 4
39.i odd 6 1 1521.1.j.b 2
39.i odd 6 1 inner 1521.1.bd.b 4
39.k even 12 1 117.1.j.a 2
39.k even 12 1 1521.1.j.b 2
39.k even 12 1 inner 1521.1.bd.b 4
39.k even 12 1 1521.1.bd.c 4
52.i odd 6 1 1872.1.bd.a 2
52.l even 12 1 1872.1.bd.a 2
65.l even 6 1 2925.1.s.a 2
65.o even 12 1 2925.1.t.a 2
65.r odd 12 1 2925.1.t.a 2
65.r odd 12 1 2925.1.t.b 2
65.s odd 12 1 2925.1.s.a 2
65.t even 12 1 2925.1.t.b 2
117.l even 6 1 1053.1.bb.a 4
117.m odd 6 1 1053.1.bb.a 4
117.r even 6 1 1053.1.bb.a 4
117.v odd 6 1 1053.1.bb.a 4
117.w odd 12 1 1053.1.bb.a 4
117.x even 12 1 1053.1.bb.a 4
117.bb odd 12 1 1053.1.bb.a 4
117.bc even 12 1 1053.1.bb.a 4
156.r even 6 1 1872.1.bd.a 2
156.v odd 12 1 1872.1.bd.a 2
195.y odd 6 1 2925.1.s.a 2
195.bc odd 12 1 2925.1.t.b 2
195.bf even 12 1 2925.1.t.a 2
195.bf even 12 1 2925.1.t.b 2
195.bh even 12 1 2925.1.s.a 2
195.bn odd 12 1 2925.1.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 13.e even 6 1
117.1.j.a 2 13.f odd 12 1
117.1.j.a 2 39.h odd 6 1
117.1.j.a 2 39.k even 12 1
1053.1.bb.a 4 117.l even 6 1
1053.1.bb.a 4 117.m odd 6 1
1053.1.bb.a 4 117.r even 6 1
1053.1.bb.a 4 117.v odd 6 1
1053.1.bb.a 4 117.w odd 12 1
1053.1.bb.a 4 117.x even 12 1
1053.1.bb.a 4 117.bb odd 12 1
1053.1.bb.a 4 117.bc even 12 1
1521.1.j.b 2 13.c even 3 1
1521.1.j.b 2 13.f odd 12 1
1521.1.j.b 2 39.i odd 6 1
1521.1.j.b 2 39.k even 12 1
1521.1.bd.b 4 1.a even 1 1 trivial
1521.1.bd.b 4 3.b odd 2 1 CM
1521.1.bd.b 4 13.c even 3 1 inner
1521.1.bd.b 4 13.d odd 4 1 inner
1521.1.bd.b 4 13.f odd 12 1 inner
1521.1.bd.b 4 39.f even 4 1 inner
1521.1.bd.b 4 39.i odd 6 1 inner
1521.1.bd.b 4 39.k even 12 1 inner
1521.1.bd.c 4 13.b even 2 1
1521.1.bd.c 4 13.d odd 4 1
1521.1.bd.c 4 13.e even 6 1
1521.1.bd.c 4 13.f odd 12 1
1521.1.bd.c 4 39.d odd 2 1
1521.1.bd.c 4 39.f even 4 1
1521.1.bd.c 4 39.h odd 6 1
1521.1.bd.c 4 39.k even 12 1
1872.1.bd.a 2 52.i odd 6 1
1872.1.bd.a 2 52.l even 12 1
1872.1.bd.a 2 156.r even 6 1
1872.1.bd.a 2 156.v odd 12 1
2925.1.s.a 2 65.l even 6 1
2925.1.s.a 2 65.s odd 12 1
2925.1.s.a 2 195.y odd 6 1
2925.1.s.a 2 195.bh even 12 1
2925.1.t.a 2 65.o even 12 1
2925.1.t.a 2 65.r odd 12 1
2925.1.t.a 2 195.bf even 12 1
2925.1.t.a 2 195.bn odd 12 1
2925.1.t.b 2 65.r odd 12 1
2925.1.t.b 2 65.t even 12 1
2925.1.t.b 2 195.bc odd 12 1
2925.1.t.b 2 195.bf even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2} \)
\( T_{7}^{4} + 2 T_{7}^{3} + 2 T_{7}^{2} + 4 T_{7} + 4 \)

Hecke characteristic polynomials

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