Properties

Label 2704.2.f.d
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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.

\(f(q)\) \(=\) \( q - q^{3} - 3 i q^{5} - i q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 3 i q^{5} - i q^{7} - 2 q^{9} + 6 i q^{11} + 3 i q^{15} + 3 q^{17} - 2 i q^{19} + i q^{21} - 4 q^{25} + 5 q^{27} + 6 q^{29} + 4 i q^{31} - 6 i q^{33} - 3 q^{35} + 7 i q^{37} - q^{43} + 6 i q^{45} + 3 i q^{47} + 6 q^{49} - 3 q^{51} + 18 q^{55} + 2 i q^{57} - 6 i q^{59} + 8 q^{61} + 2 i q^{63} - 14 i q^{67} + 3 i q^{71} - 2 i q^{73} + 4 q^{75} + 6 q^{77} - 8 q^{79} + q^{81} - 12 i q^{83} - 9 i q^{85} - 6 q^{87} + 6 i q^{89} - 4 i q^{93} - 6 q^{95} - 10 i q^{97} - 12 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{9} + 6 q^{17} - 8 q^{25} + 10 q^{27} + 12 q^{29} - 6 q^{35} - 2 q^{43} + 12 q^{49} - 6 q^{51} + 36 q^{55} + 16 q^{61} + 8 q^{75} + 12 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) 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 −1.00000 0 3.00000i 0 1.00000i 0 −2.00000 0
337.2 0 −1.00000 0 3.00000i 0 1.00000i 0 −2.00000 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.d 2
4.b odd 2 1 338.2.b.c 2
12.b even 2 1 3042.2.b.a 2
13.b even 2 1 inner 2704.2.f.d 2
13.d odd 4 1 208.2.a.a 1
13.d odd 4 1 2704.2.a.f 1
39.f even 4 1 1872.2.a.q 1
52.b odd 2 1 338.2.b.c 2
52.f even 4 1 26.2.a.a 1
52.f even 4 1 338.2.a.f 1
52.i odd 6 2 338.2.e.a 4
52.j odd 6 2 338.2.e.a 4
52.l even 12 2 338.2.c.a 2
52.l even 12 2 338.2.c.d 2
65.g odd 4 1 5200.2.a.x 1
104.j odd 4 1 832.2.a.i 1
104.m even 4 1 832.2.a.d 1
156.h even 2 1 3042.2.b.a 2
156.l odd 4 1 234.2.a.e 1
156.l odd 4 1 3042.2.a.a 1
208.l even 4 1 3328.2.b.m 2
208.m odd 4 1 3328.2.b.j 2
208.r odd 4 1 3328.2.b.j 2
208.s even 4 1 3328.2.b.m 2
260.l odd 4 1 650.2.b.d 2
260.s odd 4 1 650.2.b.d 2
260.u even 4 1 650.2.a.j 1
260.u even 4 1 8450.2.a.c 1
312.w odd 4 1 7488.2.a.g 1
312.y even 4 1 7488.2.a.h 1
364.p odd 4 1 1274.2.a.d 1
364.bw odd 12 2 1274.2.f.r 2
364.ce even 12 2 1274.2.f.p 2
468.bs even 12 2 2106.2.e.ba 2
468.ch odd 12 2 2106.2.e.b 2
572.k odd 4 1 3146.2.a.n 1
780.u even 4 1 5850.2.e.a 2
780.bb odd 4 1 5850.2.a.p 1
780.bn even 4 1 5850.2.e.a 2
884.t even 4 1 7514.2.a.c 1
988.p odd 4 1 9386.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 52.f even 4 1
208.2.a.a 1 13.d odd 4 1
234.2.a.e 1 156.l odd 4 1
338.2.a.f 1 52.f even 4 1
338.2.b.c 2 4.b odd 2 1
338.2.b.c 2 52.b odd 2 1
338.2.c.a 2 52.l even 12 2
338.2.c.d 2 52.l even 12 2
338.2.e.a 4 52.i odd 6 2
338.2.e.a 4 52.j odd 6 2
650.2.a.j 1 260.u even 4 1
650.2.b.d 2 260.l odd 4 1
650.2.b.d 2 260.s odd 4 1
832.2.a.d 1 104.m even 4 1
832.2.a.i 1 104.j odd 4 1
1274.2.a.d 1 364.p odd 4 1
1274.2.f.p 2 364.ce even 12 2
1274.2.f.r 2 364.bw odd 12 2
1872.2.a.q 1 39.f even 4 1
2106.2.e.b 2 468.ch odd 12 2
2106.2.e.ba 2 468.bs even 12 2
2704.2.a.f 1 13.d odd 4 1
2704.2.f.d 2 1.a even 1 1 trivial
2704.2.f.d 2 13.b even 2 1 inner
3042.2.a.a 1 156.l odd 4 1
3042.2.b.a 2 12.b even 2 1
3042.2.b.a 2 156.h even 2 1
3146.2.a.n 1 572.k odd 4 1
3328.2.b.j 2 208.m odd 4 1
3328.2.b.j 2 208.r odd 4 1
3328.2.b.m 2 208.l even 4 1
3328.2.b.m 2 208.s even 4 1
5200.2.a.x 1 65.g odd 4 1
5850.2.a.p 1 780.bb odd 4 1
5850.2.e.a 2 780.u even 4 1
5850.2.e.a 2 780.bn even 4 1
7488.2.a.g 1 312.w odd 4 1
7488.2.a.h 1 312.y even 4 1
7514.2.a.c 1 884.t even 4 1
8450.2.a.c 1 260.u even 4 1
9386.2.a.j 1 988.p odd 4 1

Hecke kernels

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

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( T^{2} + 9 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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