Properties

Label 3328.1.v.b
Level $3328$
Weight $1$
Character orbit 3328.v
Analytic conductor $1.661$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3328.v (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{5} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{5} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{5} q^{13} -\zeta_{12}^{4} q^{17} -\zeta_{12}^{5} q^{29} + \zeta_{12}^{5} q^{37} -\zeta_{12}^{2} q^{41} -\zeta_{12} q^{45} -\zeta_{12}^{2} q^{49} -\zeta_{12}^{3} q^{53} -\zeta_{12} q^{61} + \zeta_{12}^{2} q^{65} + q^{73} -\zeta_{12}^{2} q^{81} -\zeta_{12} q^{85} + 2 \zeta_{12}^{2} q^{89} + 2 \zeta_{12}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} + 2q^{17} - 2q^{41} - 2q^{49} + 2q^{65} + 4q^{73} - 2q^{81} + 4q^{89} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1407.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 1.00000i 0 0 0 0.500000 0.866025i 0
1407.2 0 0 0 1.00000i 0 0 0 0.500000 0.866025i 0
2687.1 0 0 0 1.00000i 0 0 0 0.500000 + 0.866025i 0
2687.2 0 0 0 1.00000i 0 0 0 0.500000 + 0.866025i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 2 1 inner
13.c even 3 1 inner
52.j odd 6 1 inner
104.n odd 6 1 inner
104.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.1.v.b 4
4.b odd 2 1 CM 3328.1.v.b 4
8.b even 2 1 inner 3328.1.v.b 4
8.d odd 2 1 inner 3328.1.v.b 4
13.c even 3 1 inner 3328.1.v.b 4
16.e even 4 1 52.1.j.a 2
16.e even 4 1 832.1.bb.a 2
16.f odd 4 1 52.1.j.a 2
16.f odd 4 1 832.1.bb.a 2
48.i odd 4 1 468.1.br.a 2
48.k even 4 1 468.1.br.a 2
52.j odd 6 1 inner 3328.1.v.b 4
80.i odd 4 1 1300.1.w.a 4
80.j even 4 1 1300.1.w.a 4
80.k odd 4 1 1300.1.bc.a 2
80.q even 4 1 1300.1.bc.a 2
80.s even 4 1 1300.1.w.a 4
80.t odd 4 1 1300.1.w.a 4
104.n odd 6 1 inner 3328.1.v.b 4
104.r even 6 1 inner 3328.1.v.b 4
112.j even 4 1 2548.1.bn.a 2
112.l odd 4 1 2548.1.bn.a 2
112.u odd 12 1 2548.1.q.b 2
112.u odd 12 1 2548.1.bi.b 2
112.v even 12 1 2548.1.q.a 2
112.v even 12 1 2548.1.bi.a 2
112.w even 12 1 2548.1.q.b 2
112.w even 12 1 2548.1.bi.b 2
112.x odd 12 1 2548.1.q.a 2
112.x odd 12 1 2548.1.bi.a 2
208.l even 4 1 676.1.i.a 4
208.m odd 4 1 676.1.i.a 4
208.o odd 4 1 676.1.j.a 2
208.p even 4 1 676.1.j.a 2
208.r odd 4 1 676.1.i.a 4
208.s even 4 1 676.1.i.a 4
208.be odd 12 1 676.1.b.a 2
208.be odd 12 1 676.1.i.a 4
208.bf even 12 1 676.1.b.a 2
208.bf even 12 1 676.1.i.a 4
208.bg odd 12 1 52.1.j.a 2
208.bg odd 12 1 676.1.c.b 1
208.bg odd 12 1 832.1.bb.a 2
208.bh even 12 1 676.1.c.a 1
208.bh even 12 1 676.1.j.a 2
208.bi odd 12 1 676.1.c.a 1
208.bi odd 12 1 676.1.j.a 2
208.bj even 12 1 52.1.j.a 2
208.bj even 12 1 676.1.c.b 1
208.bj even 12 1 832.1.bb.a 2
208.bk even 12 1 676.1.b.a 2
208.bk even 12 1 676.1.i.a 4
208.bl odd 12 1 676.1.b.a 2
208.bl odd 12 1 676.1.i.a 4
624.cl even 12 1 468.1.br.a 2
624.cw odd 12 1 468.1.br.a 2
1040.dp even 12 1 1300.1.w.a 4
1040.dr odd 12 1 1300.1.w.a 4
1040.ec even 12 1 1300.1.bc.a 2
1040.fb odd 12 1 1300.1.bc.a 2
1040.fh even 12 1 1300.1.w.a 4
1040.fj odd 12 1 1300.1.w.a 4
1456.fh even 12 1 2548.1.bi.b 2
1456.fj odd 12 1 2548.1.bi.a 2
1456.fk odd 12 1 2548.1.bi.b 2
1456.fm even 12 1 2548.1.bi.a 2
1456.fs odd 12 1 2548.1.q.a 2
1456.fx odd 12 1 2548.1.bn.a 2
1456.fy even 12 1 2548.1.q.b 2
1456.gb even 12 1 2548.1.q.a 2
1456.ge even 12 1 2548.1.bn.a 2
1456.gh odd 12 1 2548.1.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 16.e even 4 1
52.1.j.a 2 16.f odd 4 1
52.1.j.a 2 208.bg odd 12 1
52.1.j.a 2 208.bj even 12 1
468.1.br.a 2 48.i odd 4 1
468.1.br.a 2 48.k even 4 1
468.1.br.a 2 624.cl even 12 1
468.1.br.a 2 624.cw odd 12 1
676.1.b.a 2 208.be odd 12 1
676.1.b.a 2 208.bf even 12 1
676.1.b.a 2 208.bk even 12 1
676.1.b.a 2 208.bl odd 12 1
676.1.c.a 1 208.bh even 12 1
676.1.c.a 1 208.bi odd 12 1
676.1.c.b 1 208.bg odd 12 1
676.1.c.b 1 208.bj even 12 1
676.1.i.a 4 208.l even 4 1
676.1.i.a 4 208.m odd 4 1
676.1.i.a 4 208.r odd 4 1
676.1.i.a 4 208.s even 4 1
676.1.i.a 4 208.be odd 12 1
676.1.i.a 4 208.bf even 12 1
676.1.i.a 4 208.bk even 12 1
676.1.i.a 4 208.bl odd 12 1
676.1.j.a 2 208.o odd 4 1
676.1.j.a 2 208.p even 4 1
676.1.j.a 2 208.bh even 12 1
676.1.j.a 2 208.bi odd 12 1
832.1.bb.a 2 16.e even 4 1
832.1.bb.a 2 16.f odd 4 1
832.1.bb.a 2 208.bg odd 12 1
832.1.bb.a 2 208.bj even 12 1
1300.1.w.a 4 80.i odd 4 1
1300.1.w.a 4 80.j even 4 1
1300.1.w.a 4 80.s even 4 1
1300.1.w.a 4 80.t odd 4 1
1300.1.w.a 4 1040.dp even 12 1
1300.1.w.a 4 1040.dr odd 12 1
1300.1.w.a 4 1040.fh even 12 1
1300.1.w.a 4 1040.fj odd 12 1
1300.1.bc.a 2 80.k odd 4 1
1300.1.bc.a 2 80.q even 4 1
1300.1.bc.a 2 1040.ec even 12 1
1300.1.bc.a 2 1040.fb odd 12 1
2548.1.q.a 2 112.v even 12 1
2548.1.q.a 2 112.x odd 12 1
2548.1.q.a 2 1456.fs odd 12 1
2548.1.q.a 2 1456.gb even 12 1
2548.1.q.b 2 112.u odd 12 1
2548.1.q.b 2 112.w even 12 1
2548.1.q.b 2 1456.fy even 12 1
2548.1.q.b 2 1456.gh odd 12 1
2548.1.bi.a 2 112.v even 12 1
2548.1.bi.a 2 112.x odd 12 1
2548.1.bi.a 2 1456.fj odd 12 1
2548.1.bi.a 2 1456.fm even 12 1
2548.1.bi.b 2 112.u odd 12 1
2548.1.bi.b 2 112.w even 12 1
2548.1.bi.b 2 1456.fh even 12 1
2548.1.bi.b 2 1456.fk odd 12 1
2548.1.bn.a 2 112.j even 4 1
2548.1.bn.a 2 112.l odd 4 1
2548.1.bn.a 2 1456.fx odd 12 1
2548.1.bn.a 2 1456.ge even 12 1
3328.1.v.b 4 1.a even 1 1 trivial
3328.1.v.b 4 4.b odd 2 1 CM
3328.1.v.b 4 8.b even 2 1 inner
3328.1.v.b 4 8.d odd 2 1 inner
3328.1.v.b 4 13.c even 3 1 inner
3328.1.v.b 4 52.j odd 6 1 inner
3328.1.v.b 4 104.n odd 6 1 inner
3328.1.v.b 4 104.r even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(3328, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 1 - T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( 1 - T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -1 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 4 - 2 T + T^{2} )^{2} \)
$97$ \( ( 4 + 2 T + T^{2} )^{2} \)
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