Properties

Label 2800.2.a.bl
Level 2800
Weight 2
Character orbit 2800.a
Self dual yes
Analytic conductor 22.358
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{7} + 3 q^{9} -2 \beta q^{11} + ( -2 + \beta ) q^{13} -2 q^{17} + ( -4 - \beta ) q^{19} -\beta q^{21} + ( -2 - 2 \beta ) q^{23} + ( 2 - 2 \beta ) q^{29} + ( -4 + 2 \beta ) q^{31} -12 q^{33} -2 q^{37} + ( 6 - 2 \beta ) q^{39} + ( -6 - 2 \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} + ( 4 - 2 \beta ) q^{47} + q^{49} -2 \beta q^{51} + ( 6 - 2 \beta ) q^{53} + ( -6 - 4 \beta ) q^{57} + ( 4 + \beta ) q^{59} + ( 6 + \beta ) q^{61} -3 q^{63} -8 q^{67} + ( -12 - 2 \beta ) q^{69} + ( 6 + 2 \beta ) q^{71} + ( 2 + 2 \beta ) q^{73} + 2 \beta q^{77} + ( -2 + 2 \beta ) q^{79} -9 q^{81} + \beta q^{83} + ( -12 + 2 \beta ) q^{87} -10 q^{89} + ( 2 - \beta ) q^{91} + ( 12 - 4 \beta ) q^{93} + ( -6 + 4 \beta ) q^{97} -6 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{7} + 6q^{9} - 4q^{13} - 4q^{17} - 8q^{19} - 4q^{23} + 4q^{29} - 8q^{31} - 24q^{33} - 4q^{37} + 12q^{39} - 12q^{41} + 8q^{43} + 8q^{47} + 2q^{49} + 12q^{53} - 12q^{57} + 8q^{59} + 12q^{61} - 6q^{63} - 16q^{67} - 24q^{69} + 12q^{71} + 4q^{73} - 4q^{79} - 18q^{81} - 24q^{87} - 20q^{89} + 4q^{91} + 24q^{93} - 12q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
0 −2.44949 0 0 0 −1.00000 0 3.00000 0
1.2 0 2.44949 0 0 0 −1.00000 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.a.bl 2
4.b odd 2 1 350.2.a.h 2
5.b even 2 1 2800.2.a.bm 2
5.c odd 4 2 560.2.g.e 4
12.b even 2 1 3150.2.a.bs 2
15.e even 4 2 5040.2.t.t 4
20.d odd 2 1 350.2.a.g 2
20.e even 4 2 70.2.c.a 4
28.d even 2 1 2450.2.a.bq 2
40.i odd 4 2 2240.2.g.i 4
40.k even 4 2 2240.2.g.j 4
60.h even 2 1 3150.2.a.bt 2
60.l odd 4 2 630.2.g.g 4
140.c even 2 1 2450.2.a.bl 2
140.j odd 4 2 490.2.c.e 4
140.w even 12 4 490.2.i.c 8
140.x odd 12 4 490.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 20.e even 4 2
350.2.a.g 2 20.d odd 2 1
350.2.a.h 2 4.b odd 2 1
490.2.c.e 4 140.j odd 4 2
490.2.i.c 8 140.w even 12 4
490.2.i.f 8 140.x odd 12 4
560.2.g.e 4 5.c odd 4 2
630.2.g.g 4 60.l odd 4 2
2240.2.g.i 4 40.i odd 4 2
2240.2.g.j 4 40.k even 4 2
2450.2.a.bl 2 140.c even 2 1
2450.2.a.bq 2 28.d even 2 1
2800.2.a.bl 2 1.a even 1 1 trivial
2800.2.a.bm 2 5.b even 2 1
3150.2.a.bs 2 12.b even 2 1
3150.2.a.bt 2 60.h even 2 1
5040.2.t.t 4 15.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\):

\( T_{3}^{2} - 6 \)
\( T_{11}^{2} - 24 \)
\( T_{13}^{2} + 4 T_{13} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T^{4} \)
$5$ 1
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 2 T^{2} + 121 T^{4} \)
$13$ \( 1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 4 T + 26 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 + 8 T + 54 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 78 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 86 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 118 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 8 T + 128 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 12 T + 154 T^{2} - 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T + 126 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 160 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4} \)
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