Properties

Label 3332.1.bc.c
Level $3332$
Weight $1$
Character orbit 3332.bc
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $C_3\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{5} - \zeta_{12}^{3} q^{8} - \zeta_{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{5} - \zeta_{12}^{3} q^{8} - \zeta_{12} q^{9} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{10} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{2} q^{17} + \zeta_{12}^{2} q^{18} + (\zeta_{12}^{3} + 1) q^{20} - \zeta_{12}^{5} q^{25} + (\zeta_{12}^{3} + 1) q^{29} - \zeta_{12}^{5} q^{32} - \zeta_{12}^{3} q^{34} - \zeta_{12}^{3} q^{36} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{37} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{40} + ( - \zeta_{12}^{3} + 1) q^{41} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{45} - q^{50} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{58} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{61} - q^{64} + \zeta_{12}^{4} q^{68} + \zeta_{12}^{4} q^{72} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{73} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{74} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{80} + \zeta_{12}^{2} q^{81} + (\zeta_{12}^{4} - \zeta_{12}) q^{82} + (\zeta_{12}^{3} + 1) q^{85} + (\zeta_{12}^{3} + 1) q^{90} + ( - \zeta_{12}^{3} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{5} - 2 q^{10} - 2 q^{16} + 2 q^{17} + 2 q^{18} + 4 q^{20} + 4 q^{29} + 2 q^{37} + 2 q^{40} + 4 q^{41} - 2 q^{45} - 4 q^{50} + 2 q^{58} + 2 q^{61} - 4 q^{64} - 2 q^{68} - 2 q^{72} - 2 q^{73} - 2 q^{74} + 2 q^{80} + 2 q^{81} - 2 q^{82} + 4 q^{85} + 4 q^{90} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
667.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.366025 1.36603i 0 0 1.00000i 0.866025 + 0.500000i 0.366025 1.36603i
863.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.36603 0.366025i 0 0 1.00000i −0.866025 0.500000i −1.36603 0.366025i
2027.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.36603 + 0.366025i 0 0 1.00000i −0.866025 + 0.500000i −1.36603 + 0.366025i
2223.1 0.866025 0.500000i 0 0.500000 0.866025i −0.366025 + 1.36603i 0 0 1.00000i 0.866025 0.500000i 0.366025 + 1.36603i
\(n\): 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 \(\Q(\sqrt{-1}) \)
7.c even 3 1 inner
17.c even 4 1 inner
28.g odd 6 1 inner
68.f odd 4 1 inner
119.n even 12 1 inner
476.bb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bc.c 4
4.b odd 2 1 CM 3332.1.bc.c 4
7.b odd 2 1 3332.1.bc.b 4
7.c even 3 1 68.1.f.a 2
7.c even 3 1 inner 3332.1.bc.c 4
7.d odd 6 1 3332.1.m.b 2
7.d odd 6 1 3332.1.bc.b 4
17.c even 4 1 inner 3332.1.bc.c 4
21.h odd 6 1 612.1.l.a 2
28.d even 2 1 3332.1.bc.b 4
28.f even 6 1 3332.1.m.b 2
28.f even 6 1 3332.1.bc.b 4
28.g odd 6 1 68.1.f.a 2
28.g odd 6 1 inner 3332.1.bc.c 4
35.j even 6 1 1700.1.p.a 2
35.l odd 12 1 1700.1.n.a 2
35.l odd 12 1 1700.1.n.b 2
56.k odd 6 1 1088.1.p.a 2
56.p even 6 1 1088.1.p.a 2
68.f odd 4 1 inner 3332.1.bc.c 4
84.n even 6 1 612.1.l.a 2
119.f odd 4 1 3332.1.bc.b 4
119.j even 6 1 1156.1.f.b 2
119.m odd 12 1 3332.1.m.b 2
119.m odd 12 1 3332.1.bc.b 4
119.n even 12 1 68.1.f.a 2
119.n even 12 1 1156.1.f.b 2
119.n even 12 1 inner 3332.1.bc.c 4
119.q even 24 2 1156.1.c.b 2
119.q even 24 2 1156.1.d.a 2
119.t odd 48 8 1156.1.g.b 8
140.p odd 6 1 1700.1.p.a 2
140.w even 12 1 1700.1.n.a 2
140.w even 12 1 1700.1.n.b 2
357.z odd 12 1 612.1.l.a 2
476.k even 4 1 3332.1.bc.b 4
476.o odd 6 1 1156.1.f.b 2
476.z even 12 1 3332.1.m.b 2
476.z even 12 1 3332.1.bc.b 4
476.bb odd 12 1 68.1.f.a 2
476.bb odd 12 1 1156.1.f.b 2
476.bb odd 12 1 inner 3332.1.bc.c 4
476.bg odd 24 2 1156.1.c.b 2
476.bg odd 24 2 1156.1.d.a 2
476.bm even 48 8 1156.1.g.b 8
595.bk even 12 1 1700.1.p.a 2
595.bm odd 12 1 1700.1.n.a 2
595.bs odd 12 1 1700.1.n.b 2
952.bw even 12 1 1088.1.p.a 2
952.by odd 12 1 1088.1.p.a 2
1428.ca even 12 1 612.1.l.a 2
2380.cx even 12 1 1700.1.n.b 2
2380.dc odd 12 1 1700.1.p.a 2
2380.dq even 12 1 1700.1.n.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 7.c even 3 1
68.1.f.a 2 28.g odd 6 1
68.1.f.a 2 119.n even 12 1
68.1.f.a 2 476.bb odd 12 1
612.1.l.a 2 21.h odd 6 1
612.1.l.a 2 84.n even 6 1
612.1.l.a 2 357.z odd 12 1
612.1.l.a 2 1428.ca even 12 1
1088.1.p.a 2 56.k odd 6 1
1088.1.p.a 2 56.p even 6 1
1088.1.p.a 2 952.bw even 12 1
1088.1.p.a 2 952.by odd 12 1
1156.1.c.b 2 119.q even 24 2
1156.1.c.b 2 476.bg odd 24 2
1156.1.d.a 2 119.q even 24 2
1156.1.d.a 2 476.bg odd 24 2
1156.1.f.b 2 119.j even 6 1
1156.1.f.b 2 119.n even 12 1
1156.1.f.b 2 476.o odd 6 1
1156.1.f.b 2 476.bb odd 12 1
1156.1.g.b 8 119.t odd 48 8
1156.1.g.b 8 476.bm even 48 8
1700.1.n.a 2 35.l odd 12 1
1700.1.n.a 2 140.w even 12 1
1700.1.n.a 2 595.bm odd 12 1
1700.1.n.a 2 2380.dq even 12 1
1700.1.n.b 2 35.l odd 12 1
1700.1.n.b 2 140.w even 12 1
1700.1.n.b 2 595.bs odd 12 1
1700.1.n.b 2 2380.cx even 12 1
1700.1.p.a 2 35.j even 6 1
1700.1.p.a 2 140.p odd 6 1
1700.1.p.a 2 595.bk even 12 1
1700.1.p.a 2 2380.dc odd 12 1
3332.1.m.b 2 7.d odd 6 1
3332.1.m.b 2 28.f even 6 1
3332.1.m.b 2 119.m odd 12 1
3332.1.m.b 2 476.z even 12 1
3332.1.bc.b 4 7.b odd 2 1
3332.1.bc.b 4 7.d odd 6 1
3332.1.bc.b 4 28.d even 2 1
3332.1.bc.b 4 28.f even 6 1
3332.1.bc.b 4 119.f odd 4 1
3332.1.bc.b 4 119.m odd 12 1
3332.1.bc.b 4 476.k even 4 1
3332.1.bc.b 4 476.z even 12 1
3332.1.bc.c 4 1.a even 1 1 trivial
3332.1.bc.c 4 4.b odd 2 1 CM
3332.1.bc.c 4 7.c even 3 1 inner
3332.1.bc.c 4 17.c even 4 1 inner
3332.1.bc.c 4 28.g odd 6 1 inner
3332.1.bc.c 4 68.f odd 4 1 inner
3332.1.bc.c 4 119.n even 12 1 inner
3332.1.bc.c 4 476.bb odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} - 4T_{5} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

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