Properties

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

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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 350)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} -i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -i q^{7} -6 q^{9} + 5 q^{11} -6 i q^{13} + i q^{17} -3 q^{19} + 3 q^{21} -9 i q^{27} + 6 q^{29} + 4 q^{31} + 15 i q^{33} -8 i q^{37} + 18 q^{39} + 11 q^{41} + 8 i q^{43} + 2 i q^{47} - q^{49} -3 q^{51} + 4 i q^{53} -9 i q^{57} + 4 q^{59} -2 q^{61} + 6 i q^{63} + 9 i q^{67} + 10 q^{71} -7 i q^{73} -5 i q^{77} -2 q^{79} + 9 q^{81} -11 i q^{83} + 18 i q^{87} + 11 q^{89} -6 q^{91} + 12 i q^{93} + 10 i q^{97} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{9} + O(q^{10}) \) \( 2q - 12q^{9} + 10q^{11} - 6q^{19} + 6q^{21} + 12q^{29} + 8q^{31} + 36q^{39} + 22q^{41} - 2q^{49} - 6q^{51} + 8q^{59} - 4q^{61} + 20q^{71} - 4q^{79} + 18q^{81} + 22q^{89} - 12q^{91} - 60q^{99} + O(q^{100}) \)

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 3.00000i 0 0 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 1.00000i 0 −6.00000 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.a 2
4.b odd 2 1 350.2.c.a 2
5.b even 2 1 inner 2800.2.g.a 2
5.c odd 4 1 2800.2.a.b 1
5.c odd 4 1 2800.2.a.bg 1
12.b even 2 1 3150.2.g.v 2
20.d odd 2 1 350.2.c.a 2
20.e even 4 1 350.2.a.c 1
20.e even 4 1 350.2.a.d yes 1
28.d even 2 1 2450.2.c.r 2
60.h even 2 1 3150.2.g.v 2
60.l odd 4 1 3150.2.a.j 1
60.l odd 4 1 3150.2.a.bq 1
140.c even 2 1 2450.2.c.r 2
140.j odd 4 1 2450.2.a.a 1
140.j odd 4 1 2450.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 20.e even 4 1
350.2.a.d yes 1 20.e even 4 1
350.2.c.a 2 4.b odd 2 1
350.2.c.a 2 20.d odd 2 1
2450.2.a.a 1 140.j odd 4 1
2450.2.a.bg 1 140.j odd 4 1
2450.2.c.r 2 28.d even 2 1
2450.2.c.r 2 140.c even 2 1
2800.2.a.b 1 5.c odd 4 1
2800.2.a.bg 1 5.c odd 4 1
2800.2.g.a 2 1.a even 1 1 trivial
2800.2.g.a 2 5.b even 2 1 inner
3150.2.a.j 1 60.l odd 4 1
3150.2.a.bq 1 60.l odd 4 1
3150.2.g.v 2 12.b even 2 1
3150.2.g.v 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{11} - 5 \)
\( T_{13}^{2} + 36 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 5 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( 1 - 33 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 3 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 11 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 90 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 53 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 97 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 2 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 45 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 11 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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