Properties

Label 3332.1.g.a
Level $3332$
Weight $1$
Character orbit 3332.g
Self dual yes
Analytic conductor $1.663$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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: $D_4$
Artin field: Galois closure of 4.0.13328.1

$q$-expansion

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

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0
−1.00000 0 1.00000 0 0 0 −1.00000 −1.00000 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 \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.a 1
4.b odd 2 1 CM 3332.1.g.a 1
7.b odd 2 1 68.1.d.a 1
7.c even 3 2 3332.1.o.d 2
7.d odd 6 2 3332.1.o.c 2
17.b even 2 1 RM 3332.1.g.a 1
21.c even 2 1 612.1.e.a 1
28.d even 2 1 68.1.d.a 1
28.f even 6 2 3332.1.o.c 2
28.g odd 6 2 3332.1.o.d 2
35.c odd 2 1 1700.1.h.d 1
35.f even 4 2 1700.1.d.b 2
56.e even 2 1 1088.1.g.a 1
56.h odd 2 1 1088.1.g.a 1
68.d odd 2 1 CM 3332.1.g.a 1
84.h odd 2 1 612.1.e.a 1
119.d odd 2 1 68.1.d.a 1
119.f odd 4 2 1156.1.c.a 1
119.h odd 6 2 3332.1.o.c 2
119.j even 6 2 3332.1.o.d 2
119.l odd 8 4 1156.1.f.a 2
119.p even 16 8 1156.1.g.a 4
140.c even 2 1 1700.1.h.d 1
140.j odd 4 2 1700.1.d.b 2
357.c even 2 1 612.1.e.a 1
476.e even 2 1 68.1.d.a 1
476.k even 4 2 1156.1.c.a 1
476.o odd 6 2 3332.1.o.d 2
476.q even 6 2 3332.1.o.c 2
476.w even 8 4 1156.1.f.a 2
476.bf odd 16 8 1156.1.g.a 4
595.b odd 2 1 1700.1.h.d 1
595.p even 4 2 1700.1.d.b 2
952.e odd 2 1 1088.1.g.a 1
952.k even 2 1 1088.1.g.a 1
1428.b odd 2 1 612.1.e.a 1
2380.p even 2 1 1700.1.h.d 1
2380.bi odd 4 2 1700.1.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 7.b odd 2 1
68.1.d.a 1 28.d even 2 1
68.1.d.a 1 119.d odd 2 1
68.1.d.a 1 476.e even 2 1
612.1.e.a 1 21.c even 2 1
612.1.e.a 1 84.h odd 2 1
612.1.e.a 1 357.c even 2 1
612.1.e.a 1 1428.b odd 2 1
1088.1.g.a 1 56.e even 2 1
1088.1.g.a 1 56.h odd 2 1
1088.1.g.a 1 952.e odd 2 1
1088.1.g.a 1 952.k even 2 1
1156.1.c.a 1 119.f odd 4 2
1156.1.c.a 1 476.k even 4 2
1156.1.f.a 2 119.l odd 8 4
1156.1.f.a 2 476.w even 8 4
1156.1.g.a 4 119.p even 16 8
1156.1.g.a 4 476.bf odd 16 8
1700.1.d.b 2 35.f even 4 2
1700.1.d.b 2 140.j odd 4 2
1700.1.d.b 2 595.p even 4 2
1700.1.d.b 2 2380.bi odd 4 2
1700.1.h.d 1 35.c odd 2 1
1700.1.h.d 1 140.c even 2 1
1700.1.h.d 1 595.b odd 2 1
1700.1.h.d 1 2380.p even 2 1
3332.1.g.a 1 1.a even 1 1 trivial
3332.1.g.a 1 4.b odd 2 1 CM
3332.1.g.a 1 17.b even 2 1 RM
3332.1.g.a 1 68.d odd 2 1 CM
3332.1.o.c 2 7.d odd 6 2
3332.1.o.c 2 28.f even 6 2
3332.1.o.c 2 119.h odd 6 2
3332.1.o.c 2 476.q even 6 2
3332.1.o.d 2 7.c even 3 2
3332.1.o.d 2 28.g odd 6 2
3332.1.o.d 2 119.j even 6 2
3332.1.o.d 2 476.o odd 6 2

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
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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