Properties

Label 2816.2.c.j
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 11)
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.

\(f(q)\) \(=\) \( q + i q^{3} + i q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + i q^{5} + 2 q^{7} + 2 q^{9} + i q^{11} - 4 i q^{13} - q^{15} - 2 q^{17} + 2 i q^{21} + q^{23} + 4 q^{25} + 5 i q^{27} + 7 q^{31} - q^{33} + 2 i q^{35} + 3 i q^{37} + 4 q^{39} + 8 q^{41} - 6 i q^{43} + 2 i q^{45} + 8 q^{47} - 3 q^{49} - 2 i q^{51} - 6 i q^{53} - q^{55} + 5 i q^{59} - 12 i q^{61} + 4 q^{63} + 4 q^{65} + 7 i q^{67} + i q^{69} + 3 q^{71} - 4 q^{73} + 4 i q^{75} + 2 i q^{77} - 10 q^{79} + q^{81} + 6 i q^{83} - 2 i q^{85} - 15 q^{89} - 8 i q^{91} + 7 i q^{93} - 7 q^{97} + 2 i q^{99} +O(q^{100}) \) 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 \( 2 q + 4 q^{7} + 4 q^{9} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 8 q^{25} + 14 q^{31} - 2 q^{33} + 8 q^{39} + 16 q^{41} + 16 q^{47} - 6 q^{49} - 2 q^{55} + 8 q^{63} + 8 q^{65} + 6 q^{71} - 8 q^{73} - 20 q^{79} + 2 q^{81} - 30 q^{89} - 14 q^{97}+O(q^{100}) \) 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 1.00000i 0 2.00000 0 2.00000 0
1409.2 0 1.00000i 0 1.00000i 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.j 2
4.b odd 2 1 2816.2.c.f 2
8.b even 2 1 inner 2816.2.c.j 2
8.d odd 2 1 2816.2.c.f 2
16.e even 4 1 11.2.a.a 1
16.e even 4 1 704.2.a.h 1
16.f odd 4 1 176.2.a.b 1
16.f odd 4 1 704.2.a.c 1
48.i odd 4 1 99.2.a.d 1
48.i odd 4 1 6336.2.a.br 1
48.k even 4 1 1584.2.a.g 1
48.k even 4 1 6336.2.a.bu 1
80.i odd 4 1 275.2.b.a 2
80.j even 4 1 4400.2.b.h 2
80.k odd 4 1 4400.2.a.i 1
80.q even 4 1 275.2.a.b 1
80.s even 4 1 4400.2.b.h 2
80.t odd 4 1 275.2.b.a 2
112.j even 4 1 8624.2.a.j 1
112.l odd 4 1 539.2.a.a 1
112.w even 12 2 539.2.e.h 2
112.x odd 12 2 539.2.e.g 2
144.w odd 12 2 891.2.e.b 2
144.x even 12 2 891.2.e.k 2
176.i even 4 1 1936.2.a.i 1
176.i even 4 1 7744.2.a.k 1
176.l odd 4 1 121.2.a.d 1
176.l odd 4 1 7744.2.a.x 1
176.u odd 20 4 121.2.c.a 4
176.w even 20 4 121.2.c.e 4
208.p even 4 1 1859.2.a.b 1
240.bb even 4 1 2475.2.c.a 2
240.bf even 4 1 2475.2.c.a 2
240.bm odd 4 1 2475.2.a.a 1
272.r even 4 1 3179.2.a.a 1
304.j odd 4 1 3971.2.a.b 1
336.y even 4 1 4851.2.a.t 1
368.k odd 4 1 5819.2.a.a 1
464.m even 4 1 9251.2.a.d 1
528.x even 4 1 1089.2.a.b 1
880.x odd 4 1 3025.2.a.a 1
1232.x even 4 1 5929.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 16.e even 4 1
99.2.a.d 1 48.i odd 4 1
121.2.a.d 1 176.l odd 4 1
121.2.c.a 4 176.u odd 20 4
121.2.c.e 4 176.w even 20 4
176.2.a.b 1 16.f odd 4 1
275.2.a.b 1 80.q even 4 1
275.2.b.a 2 80.i odd 4 1
275.2.b.a 2 80.t odd 4 1
539.2.a.a 1 112.l odd 4 1
539.2.e.g 2 112.x odd 12 2
539.2.e.h 2 112.w even 12 2
704.2.a.c 1 16.f odd 4 1
704.2.a.h 1 16.e even 4 1
891.2.e.b 2 144.w odd 12 2
891.2.e.k 2 144.x even 12 2
1089.2.a.b 1 528.x even 4 1
1584.2.a.g 1 48.k even 4 1
1859.2.a.b 1 208.p even 4 1
1936.2.a.i 1 176.i even 4 1
2475.2.a.a 1 240.bm odd 4 1
2475.2.c.a 2 240.bb even 4 1
2475.2.c.a 2 240.bf even 4 1
2816.2.c.f 2 4.b odd 2 1
2816.2.c.f 2 8.d odd 2 1
2816.2.c.j 2 1.a even 1 1 trivial
2816.2.c.j 2 8.b even 2 1 inner
3025.2.a.a 1 880.x odd 4 1
3179.2.a.a 1 272.r even 4 1
3971.2.a.b 1 304.j odd 4 1
4400.2.a.i 1 80.k odd 4 1
4400.2.b.h 2 80.j even 4 1
4400.2.b.h 2 80.s even 4 1
4851.2.a.t 1 336.y even 4 1
5819.2.a.a 1 368.k odd 4 1
5929.2.a.h 1 1232.x even 4 1
6336.2.a.br 1 48.i odd 4 1
6336.2.a.bu 1 48.k even 4 1
7744.2.a.k 1 176.i even 4 1
7744.2.a.x 1 176.l odd 4 1
8624.2.a.j 1 112.j even 4 1
9251.2.a.d 1 464.m even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{23} - 1 \) 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
$5$ \( T^{2} + 1 \) 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
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 25 \) Copy content Toggle raw display
$61$ \( T^{2} + 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 49 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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