Properties

Label 3332.1.m.a
Level $3332$
Weight $1$
Character orbit 3332.m
Analytic conductor $1.663$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.962948.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + ( - i - 1) q^{5} + i q^{8} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + ( - i - 1) q^{5} + i q^{8} + i q^{9} + (i - 1) q^{10} + q^{13} + q^{16} - i q^{17} + q^{18} + (i + 1) q^{20} + i q^{25} - 2 i q^{26} + ( - i - 1) q^{29} - i q^{32} - q^{34} - i q^{36} + (i + 1) q^{37} + ( - i + 1) q^{40} + ( - i + 1) q^{41} + ( - i + 1) q^{45} + q^{50} - 2 q^{52} + (i - 1) q^{58} + ( - i + 1) q^{61} - q^{64} + ( - 2 i - 2) q^{65} + i q^{68} - q^{72} + ( - i - 1) q^{73} + ( - i + 1) q^{74} + ( - i - 1) q^{80} - q^{81} + ( - i - 1) q^{82} + (i - 1) q^{85} - q^{89} + ( - i - 1) q^{90} + ( - i - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} - 2 q^{10} + 4 q^{13} + 2 q^{16} + 2 q^{18} + 2 q^{20} - 2 q^{29} - 2 q^{34} + 2 q^{37} + 2 q^{40} + 2 q^{41} + 2 q^{45} + 2 q^{50} - 4 q^{52} - 2 q^{58} + 2 q^{61} - 2 q^{64} - 4 q^{65} - 2 q^{72} - 2 q^{73} + 2 q^{74} - 2 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{85} - 4 q^{89} - 2 q^{90} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2843.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 1.00000i −1.00000 1.00000i
3039.1 1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 1.00000i −1.00000 + 1.00000i
\(n\): 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.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.m.a 2
4.b odd 2 1 CM 3332.1.m.a 2
7.b odd 2 1 3332.1.m.c yes 2
7.c even 3 2 3332.1.bc.d 4
7.d odd 6 2 3332.1.bc.a 4
17.c even 4 1 inner 3332.1.m.a 2
28.d even 2 1 3332.1.m.c yes 2
28.f even 6 2 3332.1.bc.a 4
28.g odd 6 2 3332.1.bc.d 4
68.f odd 4 1 inner 3332.1.m.a 2
119.f odd 4 1 3332.1.m.c yes 2
119.m odd 12 2 3332.1.bc.a 4
119.n even 12 2 3332.1.bc.d 4
476.k even 4 1 3332.1.m.c yes 2
476.z even 12 2 3332.1.bc.a 4
476.bb odd 12 2 3332.1.bc.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.m.a 2 1.a even 1 1 trivial
3332.1.m.a 2 4.b odd 2 1 CM
3332.1.m.a 2 17.c even 4 1 inner
3332.1.m.a 2 68.f odd 4 1 inner
3332.1.m.c yes 2 7.b odd 2 1
3332.1.m.c yes 2 28.d even 2 1
3332.1.m.c yes 2 119.f odd 4 1
3332.1.m.c yes 2 476.k even 4 1
3332.1.bc.a 4 7.d odd 6 2
3332.1.bc.a 4 28.f even 6 2
3332.1.bc.a 4 119.m odd 12 2
3332.1.bc.a 4 476.z even 12 2
3332.1.bc.d 4 7.c even 3 2
3332.1.bc.d 4 28.g odd 6 2
3332.1.bc.d 4 119.n even 12 2
3332.1.bc.d 4 476.bb odd 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

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