Properties

Label 3328.2.b.e
Level $3328$
Weight $2$
Character orbit 3328.b
Analytic conductor $26.574$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 i q^{5} - 2 q^{7} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{5} - 2 q^{7} + 3 q^{9} - 2 i q^{11} - i q^{13} + 6 q^{17} + 6 i q^{19} + 8 q^{23} + q^{25} + 2 i q^{29} - 10 q^{31} + 4 i q^{35} + 6 i q^{37} + 6 q^{41} + 4 i q^{43} - 6 i q^{45} + 2 q^{47} - 3 q^{49} - 6 i q^{53} - 4 q^{55} - 10 i q^{59} - 2 i q^{61} - 6 q^{63} - 2 q^{65} - 10 i q^{67} + 10 q^{71} - 2 q^{73} + 4 i q^{77} + 4 q^{79} + 9 q^{81} + 6 i q^{83} - 12 i q^{85} + 6 q^{89} + 2 i q^{91} + 12 q^{95} + 2 q^{97} - 6 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} + 6 q^{9} + 12 q^{17} + 16 q^{23} + 2 q^{25} - 20 q^{31} + 12 q^{41} + 4 q^{47} - 6 q^{49} - 8 q^{55} - 12 q^{63} - 4 q^{65} + 20 q^{71} - 4 q^{73} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 24 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1665.1
1.00000i
1.00000i
0 0 0 2.00000i 0 −2.00000 0 3.00000 0
1665.2 0 0 0 2.00000i 0 −2.00000 0 3.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.e 2
4.b odd 2 1 3328.2.b.q 2
8.b even 2 1 inner 3328.2.b.e 2
8.d odd 2 1 3328.2.b.q 2
16.e even 4 1 208.2.a.c 1
16.e even 4 1 832.2.a.f 1
16.f odd 4 1 52.2.a.a 1
16.f odd 4 1 832.2.a.e 1
48.i odd 4 1 1872.2.a.f 1
48.i odd 4 1 7488.2.a.bw 1
48.k even 4 1 468.2.a.b 1
48.k even 4 1 7488.2.a.bn 1
80.j even 4 1 1300.2.c.c 2
80.k odd 4 1 1300.2.a.d 1
80.q even 4 1 5200.2.a.q 1
80.s even 4 1 1300.2.c.c 2
112.j even 4 1 2548.2.a.e 1
112.u odd 12 2 2548.2.j.e 2
112.v even 12 2 2548.2.j.f 2
144.u even 12 2 4212.2.i.i 2
144.v odd 12 2 4212.2.i.d 2
176.i even 4 1 6292.2.a.g 1
208.l even 4 1 676.2.d.c 2
208.m odd 4 1 2704.2.f.f 2
208.o odd 4 1 676.2.a.c 1
208.p even 4 1 2704.2.a.g 1
208.r odd 4 1 2704.2.f.f 2
208.s even 4 1 676.2.d.c 2
208.bf even 12 2 676.2.h.c 4
208.bg odd 12 2 676.2.e.c 2
208.bi odd 12 2 676.2.e.b 2
208.bk even 12 2 676.2.h.c 4
624.s odd 4 1 6084.2.b.m 2
624.v even 4 1 6084.2.a.m 1
624.bo odd 4 1 6084.2.b.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 16.f odd 4 1
208.2.a.c 1 16.e even 4 1
468.2.a.b 1 48.k even 4 1
676.2.a.c 1 208.o odd 4 1
676.2.d.c 2 208.l even 4 1
676.2.d.c 2 208.s even 4 1
676.2.e.b 2 208.bi odd 12 2
676.2.e.c 2 208.bg odd 12 2
676.2.h.c 4 208.bf even 12 2
676.2.h.c 4 208.bk even 12 2
832.2.a.e 1 16.f odd 4 1
832.2.a.f 1 16.e even 4 1
1300.2.a.d 1 80.k odd 4 1
1300.2.c.c 2 80.j even 4 1
1300.2.c.c 2 80.s even 4 1
1872.2.a.f 1 48.i odd 4 1
2548.2.a.e 1 112.j even 4 1
2548.2.j.e 2 112.u odd 12 2
2548.2.j.f 2 112.v even 12 2
2704.2.a.g 1 208.p even 4 1
2704.2.f.f 2 208.m odd 4 1
2704.2.f.f 2 208.r odd 4 1
3328.2.b.e 2 1.a even 1 1 trivial
3328.2.b.e 2 8.b even 2 1 inner
3328.2.b.q 2 4.b odd 2 1
3328.2.b.q 2 8.d odd 2 1
4212.2.i.d 2 144.v odd 12 2
4212.2.i.i 2 144.u even 12 2
5200.2.a.q 1 80.q even 4 1
6084.2.a.m 1 624.v even 4 1
6084.2.b.m 2 624.s odd 4 1
6084.2.b.m 2 624.bo odd 4 1
6292.2.a.g 1 176.i even 4 1
7488.2.a.bn 1 48.k even 4 1
7488.2.a.bw 1 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3328, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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