Properties

Label 2704.2.f.f
Level $2704$
Weight $2$
Character orbit 2704.f
Analytic conductor $21.592$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 0 0 2.00000i 0 2.00000i 0 −3.00000 0
337.2 0 0 0 2.00000i 0 2.00000i 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2704.2.f.f 2
4.b odd 2 1 676.2.d.c 2
12.b even 2 1 6084.2.b.m 2
13.b even 2 1 inner 2704.2.f.f 2
13.d odd 4 1 208.2.a.c 1
13.d odd 4 1 2704.2.a.g 1
39.f even 4 1 1872.2.a.f 1
52.b odd 2 1 676.2.d.c 2
52.f even 4 1 52.2.a.a 1
52.f even 4 1 676.2.a.c 1
52.i odd 6 2 676.2.h.c 4
52.j odd 6 2 676.2.h.c 4
52.l even 12 2 676.2.e.b 2
52.l even 12 2 676.2.e.c 2
65.g odd 4 1 5200.2.a.q 1
104.j odd 4 1 832.2.a.f 1
104.m even 4 1 832.2.a.e 1
156.h even 2 1 6084.2.b.m 2
156.l odd 4 1 468.2.a.b 1
156.l odd 4 1 6084.2.a.m 1
208.l even 4 1 3328.2.b.q 2
208.m odd 4 1 3328.2.b.e 2
208.r odd 4 1 3328.2.b.e 2
208.s even 4 1 3328.2.b.q 2
260.l odd 4 1 1300.2.c.c 2
260.s odd 4 1 1300.2.c.c 2
260.u even 4 1 1300.2.a.d 1
312.w odd 4 1 7488.2.a.bn 1
312.y even 4 1 7488.2.a.bw 1
364.p odd 4 1 2548.2.a.e 1
364.bw odd 12 2 2548.2.j.f 2
364.ce even 12 2 2548.2.j.e 2
468.bs even 12 2 4212.2.i.d 2
468.ch odd 12 2 4212.2.i.i 2
572.k odd 4 1 6292.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 52.f even 4 1
208.2.a.c 1 13.d odd 4 1
468.2.a.b 1 156.l odd 4 1
676.2.a.c 1 52.f even 4 1
676.2.d.c 2 4.b odd 2 1
676.2.d.c 2 52.b odd 2 1
676.2.e.b 2 52.l even 12 2
676.2.e.c 2 52.l even 12 2
676.2.h.c 4 52.i odd 6 2
676.2.h.c 4 52.j odd 6 2
832.2.a.e 1 104.m even 4 1
832.2.a.f 1 104.j odd 4 1
1300.2.a.d 1 260.u even 4 1
1300.2.c.c 2 260.l odd 4 1
1300.2.c.c 2 260.s odd 4 1
1872.2.a.f 1 39.f even 4 1
2548.2.a.e 1 364.p odd 4 1
2548.2.j.e 2 364.ce even 12 2
2548.2.j.f 2 364.bw odd 12 2
2704.2.a.g 1 13.d odd 4 1
2704.2.f.f 2 1.a even 1 1 trivial
2704.2.f.f 2 13.b even 2 1 inner
3328.2.b.e 2 208.m odd 4 1
3328.2.b.e 2 208.r odd 4 1
3328.2.b.q 2 208.l even 4 1
3328.2.b.q 2 208.s even 4 1
4212.2.i.d 2 468.bs even 12 2
4212.2.i.i 2 468.ch odd 12 2
5200.2.a.q 1 65.g odd 4 1
6084.2.a.m 1 156.l odd 4 1
6084.2.b.m 2 12.b even 2 1
6084.2.b.m 2 156.h even 2 1
6292.2.a.g 1 572.k odd 4 1
7488.2.a.bn 1 312.w odd 4 1
7488.2.a.bw 1 312.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2704, [\chi])\):

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