Properties

Label 3332.1.o.d
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
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.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) 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: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} - \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} - \zeta_{6}^{2} q^{9} + q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{17} - \zeta_{6} q^{18} - \zeta_{6} q^{25} - 2 \zeta_{6}^{2} q^{26} + \zeta_{6} q^{32} + q^{34} - q^{36} - q^{50} - 2 \zeta_{6} q^{52} - \zeta_{6} q^{53} + q^{64} - \zeta_{6}^{2} q^{68} + \zeta_{6}^{2} q^{72} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - 2 q^{8} + q^{9} + 4 q^{13} - q^{16} + q^{17} - q^{18} - q^{25} + 2 q^{26} + q^{32} + 2 q^{34} - 2 q^{36} - 2 q^{50} - 2 q^{52} - 2 q^{53} + 2 q^{64} + q^{68} - q^{72} - q^{81} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 0.500000 + 0.866025i 0
2039.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 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 \(\Q(\sqrt{-1}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
7.c even 3 1 inner
28.g odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.d 2
4.b odd 2 1 CM 3332.1.o.d 2
7.b odd 2 1 3332.1.o.c 2
7.c even 3 1 3332.1.g.a 1
7.c even 3 1 inner 3332.1.o.d 2
7.d odd 6 1 68.1.d.a 1
7.d odd 6 1 3332.1.o.c 2
17.b even 2 1 RM 3332.1.o.d 2
21.g even 6 1 612.1.e.a 1
28.d even 2 1 3332.1.o.c 2
28.f even 6 1 68.1.d.a 1
28.f even 6 1 3332.1.o.c 2
28.g odd 6 1 3332.1.g.a 1
28.g odd 6 1 inner 3332.1.o.d 2
35.i odd 6 1 1700.1.h.d 1
35.k even 12 2 1700.1.d.b 2
56.j odd 6 1 1088.1.g.a 1
56.m even 6 1 1088.1.g.a 1
68.d odd 2 1 CM 3332.1.o.d 2
84.j odd 6 1 612.1.e.a 1
119.d odd 2 1 3332.1.o.c 2
119.h odd 6 1 68.1.d.a 1
119.h odd 6 1 3332.1.o.c 2
119.j even 6 1 3332.1.g.a 1
119.j even 6 1 inner 3332.1.o.d 2
119.m odd 12 2 1156.1.c.a 1
119.r odd 24 4 1156.1.f.a 2
119.s even 48 8 1156.1.g.a 4
140.s even 6 1 1700.1.h.d 1
140.x odd 12 2 1700.1.d.b 2
357.s even 6 1 612.1.e.a 1
476.e even 2 1 3332.1.o.c 2
476.o odd 6 1 3332.1.g.a 1
476.o odd 6 1 inner 3332.1.o.d 2
476.q even 6 1 68.1.d.a 1
476.q even 6 1 3332.1.o.c 2
476.z even 12 2 1156.1.c.a 1
476.bj even 24 4 1156.1.f.a 2
476.bk odd 48 8 1156.1.g.a 4
595.bb odd 6 1 1700.1.h.d 1
595.br even 12 2 1700.1.d.b 2
952.bf even 6 1 1088.1.g.a 1
952.bl odd 6 1 1088.1.g.a 1
1428.bl odd 6 1 612.1.e.a 1
2380.bp even 6 1 1700.1.h.d 1
2380.db odd 12 2 1700.1.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 7.d odd 6 1
68.1.d.a 1 28.f even 6 1
68.1.d.a 1 119.h odd 6 1
68.1.d.a 1 476.q even 6 1
612.1.e.a 1 21.g even 6 1
612.1.e.a 1 84.j odd 6 1
612.1.e.a 1 357.s even 6 1
612.1.e.a 1 1428.bl odd 6 1
1088.1.g.a 1 56.j odd 6 1
1088.1.g.a 1 56.m even 6 1
1088.1.g.a 1 952.bf even 6 1
1088.1.g.a 1 952.bl odd 6 1
1156.1.c.a 1 119.m odd 12 2
1156.1.c.a 1 476.z even 12 2
1156.1.f.a 2 119.r odd 24 4
1156.1.f.a 2 476.bj even 24 4
1156.1.g.a 4 119.s even 48 8
1156.1.g.a 4 476.bk odd 48 8
1700.1.d.b 2 35.k even 12 2
1700.1.d.b 2 140.x odd 12 2
1700.1.d.b 2 595.br even 12 2
1700.1.d.b 2 2380.db odd 12 2
1700.1.h.d 1 35.i odd 6 1
1700.1.h.d 1 140.s even 6 1
1700.1.h.d 1 595.bb odd 6 1
1700.1.h.d 1 2380.bp even 6 1
3332.1.g.a 1 7.c even 3 1
3332.1.g.a 1 28.g odd 6 1
3332.1.g.a 1 119.j even 6 1
3332.1.g.a 1 476.o odd 6 1
3332.1.o.c 2 7.b odd 2 1
3332.1.o.c 2 7.d odd 6 1
3332.1.o.c 2 28.d even 2 1
3332.1.o.c 2 28.f even 6 1
3332.1.o.c 2 119.d odd 2 1
3332.1.o.c 2 119.h odd 6 1
3332.1.o.c 2 476.e even 2 1
3332.1.o.c 2 476.q even 6 1
3332.1.o.d 2 1.a even 1 1 trivial
3332.1.o.d 2 4.b odd 2 1 CM
3332.1.o.d 2 7.c even 3 1 inner
3332.1.o.d 2 17.b even 2 1 RM
3332.1.o.d 2 28.g odd 6 1 inner
3332.1.o.d 2 68.d odd 2 1 CM
3332.1.o.d 2 119.j even 6 1 inner
3332.1.o.d 2 476.o odd 6 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

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