Properties

Label 2800.2.g.n
Level 2800
Weight 2
Character orbit 2800.g
Analytic conductor 22.358
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + i q^{7} + 3 q^{9} +O(q^{10})\) \( q + i q^{7} + 3 q^{9} -4 q^{11} + 6 i q^{13} + 2 i q^{17} -6 q^{29} -8 q^{31} -10 i q^{37} + 2 q^{41} + 4 i q^{43} -8 i q^{47} - q^{49} + 2 i q^{53} -8 q^{59} -14 q^{61} + 3 i q^{63} + 12 i q^{67} + 16 q^{71} -2 i q^{73} -4 i q^{77} -8 q^{79} + 9 q^{81} + 8 i q^{83} -10 q^{89} -6 q^{91} + 2 i q^{97} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} - 8q^{11} - 12q^{29} - 16q^{31} + 4q^{41} - 2q^{49} - 16q^{59} - 28q^{61} + 32q^{71} - 16q^{79} + 18q^{81} - 20q^{89} - 12q^{91} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 3.00000 0
449.2 0 0 0 0 0 1.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.g.n 2
4.b odd 2 1 350.2.c.b 2
5.b even 2 1 inner 2800.2.g.n 2
5.c odd 4 1 560.2.a.d 1
5.c odd 4 1 2800.2.a.m 1
12.b even 2 1 3150.2.g.c 2
15.e even 4 1 5040.2.a.bm 1
20.d odd 2 1 350.2.c.b 2
20.e even 4 1 70.2.a.a 1
20.e even 4 1 350.2.a.b 1
28.d even 2 1 2450.2.c.k 2
35.f even 4 1 3920.2.a.t 1
40.i odd 4 1 2240.2.a.q 1
40.k even 4 1 2240.2.a.n 1
60.h even 2 1 3150.2.g.c 2
60.l odd 4 1 630.2.a.d 1
60.l odd 4 1 3150.2.a.bj 1
140.c even 2 1 2450.2.c.k 2
140.j odd 4 1 490.2.a.h 1
140.j odd 4 1 2450.2.a.l 1
140.w even 12 2 490.2.e.d 2
140.x odd 12 2 490.2.e.c 2
220.i odd 4 1 8470.2.a.j 1
420.w even 4 1 4410.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 20.e even 4 1
350.2.a.b 1 20.e even 4 1
350.2.c.b 2 4.b odd 2 1
350.2.c.b 2 20.d odd 2 1
490.2.a.h 1 140.j odd 4 1
490.2.e.c 2 140.x odd 12 2
490.2.e.d 2 140.w even 12 2
560.2.a.d 1 5.c odd 4 1
630.2.a.d 1 60.l odd 4 1
2240.2.a.n 1 40.k even 4 1
2240.2.a.q 1 40.i odd 4 1
2450.2.a.l 1 140.j odd 4 1
2450.2.c.k 2 28.d even 2 1
2450.2.c.k 2 140.c even 2 1
2800.2.a.m 1 5.c odd 4 1
2800.2.g.n 2 1.a even 1 1 trivial
2800.2.g.n 2 5.b even 2 1 inner
3150.2.a.bj 1 60.l odd 4 1
3150.2.g.c 2 12.b even 2 1
3150.2.g.c 2 60.h even 2 1
3920.2.a.t 1 35.f even 4 1
4410.2.a.b 1 420.w even 4 1
5040.2.a.bm 1 15.e even 4 1
8470.2.a.j 1 220.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 \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3} \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 36 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 102 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
show more
show less