Properties

Label 2816.2.c.k
Level $2816$
Weight $2$
Character orbit 2816.c
Analytic conductor $22.486$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.4858732092\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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.

\(f(q)\) \(=\) \( q + i q^{3} - 3 i q^{5} + 2 q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 3 i q^{5} + 2 q^{7} + 2 q^{9} + i q^{11} + 4 i q^{13} + 3 q^{15} + 6 q^{17} + 8 i q^{19} + 2 i q^{21} - 3 q^{23} - 4 q^{25} + 5 i q^{27} - 5 q^{31} - q^{33} - 6 i q^{35} - i q^{37} - 4 q^{39} + 10 i q^{43} - 6 i q^{45} - 3 q^{49} + 6 i q^{51} - 6 i q^{53} + 3 q^{55} - 8 q^{57} - 3 i q^{59} + 4 i q^{61} + 4 q^{63} + 12 q^{65} - i q^{67} - 3 i q^{69} + 15 q^{71} + 4 q^{73} - 4 i q^{75} + 2 i q^{77} - 2 q^{79} + q^{81} + 6 i q^{83} - 18 i q^{85} + 9 q^{89} + 8 i q^{91} - 5 i q^{93} + 24 q^{95} - 7 q^{97} + 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} + 4 q^{9} + 6 q^{15} + 12 q^{17} - 6 q^{23} - 8 q^{25} - 10 q^{31} - 2 q^{33} - 8 q^{39} - 6 q^{49} + 6 q^{55} - 16 q^{57} + 8 q^{63} + 24 q^{65} + 30 q^{71} + 8 q^{73} - 4 q^{79} + 2 q^{81} + 18 q^{89} + 48 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1409.1
1.00000i
1.00000i
0 1.00000i 0 3.00000i 0 2.00000 0 2.00000 0
1409.2 0 1.00000i 0 3.00000i 0 2.00000 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
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 2816.2.c.k 2
4.b odd 2 1 2816.2.c.e 2
8.b even 2 1 inner 2816.2.c.k 2
8.d odd 2 1 2816.2.c.e 2
16.e even 4 1 176.2.a.a 1
16.e even 4 1 704.2.a.i 1
16.f odd 4 1 44.2.a.a 1
16.f odd 4 1 704.2.a.f 1
48.i odd 4 1 1584.2.a.p 1
48.i odd 4 1 6336.2.a.i 1
48.k even 4 1 396.2.a.c 1
48.k even 4 1 6336.2.a.j 1
80.i odd 4 1 4400.2.b.k 2
80.j even 4 1 1100.2.b.c 2
80.k odd 4 1 1100.2.a.b 1
80.q even 4 1 4400.2.a.v 1
80.s even 4 1 1100.2.b.c 2
80.t odd 4 1 4400.2.b.k 2
112.j even 4 1 2156.2.a.a 1
112.l odd 4 1 8624.2.a.w 1
112.u odd 12 2 2156.2.i.b 2
112.v even 12 2 2156.2.i.c 2
144.u even 12 2 3564.2.i.a 2
144.v odd 12 2 3564.2.i.j 2
176.i even 4 1 484.2.a.a 1
176.i even 4 1 7744.2.a.m 1
176.l odd 4 1 1936.2.a.c 1
176.l odd 4 1 7744.2.a.bc 1
176.v odd 20 4 484.2.e.a 4
176.x even 20 4 484.2.e.b 4
208.o odd 4 1 7436.2.a.d 1
240.t even 4 1 9900.2.a.h 1
240.z odd 4 1 9900.2.c.g 2
240.bd odd 4 1 9900.2.c.g 2
528.s odd 4 1 4356.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 16.f odd 4 1
176.2.a.a 1 16.e even 4 1
396.2.a.c 1 48.k even 4 1
484.2.a.a 1 176.i even 4 1
484.2.e.a 4 176.v odd 20 4
484.2.e.b 4 176.x even 20 4
704.2.a.f 1 16.f odd 4 1
704.2.a.i 1 16.e even 4 1
1100.2.a.b 1 80.k odd 4 1
1100.2.b.c 2 80.j even 4 1
1100.2.b.c 2 80.s even 4 1
1584.2.a.p 1 48.i odd 4 1
1936.2.a.c 1 176.l odd 4 1
2156.2.a.a 1 112.j even 4 1
2156.2.i.b 2 112.u odd 12 2
2156.2.i.c 2 112.v even 12 2
2816.2.c.e 2 4.b odd 2 1
2816.2.c.e 2 8.d odd 2 1
2816.2.c.k 2 1.a even 1 1 trivial
2816.2.c.k 2 8.b even 2 1 inner
3564.2.i.a 2 144.u even 12 2
3564.2.i.j 2 144.v odd 12 2
4356.2.a.j 1 528.s odd 4 1
4400.2.a.v 1 80.q even 4 1
4400.2.b.k 2 80.i odd 4 1
4400.2.b.k 2 80.t odd 4 1
6336.2.a.i 1 48.i odd 4 1
6336.2.a.j 1 48.k even 4 1
7436.2.a.d 1 208.o odd 4 1
7744.2.a.m 1 176.i even 4 1
7744.2.a.bc 1 176.l odd 4 1
8624.2.a.w 1 112.l odd 4 1
9900.2.a.h 1 240.t even 4 1
9900.2.c.g 2 240.z odd 4 1
9900.2.c.g 2 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2816, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{23} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T - 15)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( (T + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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