Properties

Label 2800.2.g.r
Level $2800$
Weight $2$
Character orbit 2800.g
Analytic conductor $22.358$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -6 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -6 + \beta_{3} ) q^{9} + ( -3 - \beta_{3} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} ) q^{17} + ( 2 - 2 \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} -2 \beta_{1} q^{23} + ( -3 \beta_{1} + 8 \beta_{2} ) q^{27} + ( 1 + \beta_{3} ) q^{29} + 8 q^{31} + ( -3 \beta_{1} - 8 \beta_{2} ) q^{33} -2 \beta_{2} q^{37} + ( 7 + \beta_{3} ) q^{39} + 2 \beta_{3} q^{41} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{43} + 3 \beta_{1} q^{47} - q^{49} + ( 11 - 3 \beta_{3} ) q^{51} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{53} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{57} + 8 q^{59} + ( 4 - 2 \beta_{3} ) q^{61} + ( \beta_{1} - 5 \beta_{2} ) q^{63} + 4 \beta_{2} q^{67} + ( 18 - 2 \beta_{3} ) q^{69} -8 q^{71} + 6 \beta_{2} q^{73} + ( -\beta_{1} - 4 \beta_{2} ) q^{77} + ( 5 + 3 \beta_{3} ) q^{79} + ( 17 - 8 \beta_{3} ) q^{81} -4 \beta_{1} q^{83} + ( \beta_{1} + 8 \beta_{2} ) q^{87} + ( -8 - 2 \beta_{3} ) q^{89} + ( 1 + \beta_{3} ) q^{91} + 8 \beta_{1} q^{93} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 10 + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 22q^{9} + O(q^{10}) \) \( 4q - 22q^{9} - 14q^{11} + 4q^{19} + 2q^{21} + 6q^{29} + 32q^{31} + 30q^{39} + 4q^{41} - 4q^{49} + 38q^{51} + 32q^{59} + 12q^{61} + 68q^{69} - 32q^{71} + 26q^{79} + 52q^{81} - 36q^{89} + 6q^{91} + 44q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 17 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 9\)
\(\nu^{3}\)\(=\)\(8 \beta_{2} - 9 \beta_{1}\)

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
3.37228i
2.37228i
2.37228i
3.37228i
0 3.37228i 0 0 0 1.00000i 0 −8.37228 0
449.2 0 2.37228i 0 0 0 1.00000i 0 −2.62772 0
449.3 0 2.37228i 0 0 0 1.00000i 0 −2.62772 0
449.4 0 3.37228i 0 0 0 1.00000i 0 −8.37228 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.r 4
4.b odd 2 1 1400.2.g.i 4
5.b even 2 1 inner 2800.2.g.r 4
5.c odd 4 1 560.2.a.h 2
5.c odd 4 1 2800.2.a.bk 2
15.e even 4 1 5040.2.a.by 2
20.d odd 2 1 1400.2.g.i 4
20.e even 4 1 280.2.a.c 2
20.e even 4 1 1400.2.a.r 2
35.f even 4 1 3920.2.a.bt 2
40.i odd 4 1 2240.2.a.bg 2
40.k even 4 1 2240.2.a.bk 2
60.l odd 4 1 2520.2.a.x 2
140.j odd 4 1 1960.2.a.s 2
140.j odd 4 1 9800.2.a.bu 2
140.w even 12 2 1960.2.q.t 4
140.x odd 12 2 1960.2.q.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 20.e even 4 1
560.2.a.h 2 5.c odd 4 1
1400.2.a.r 2 20.e even 4 1
1400.2.g.i 4 4.b odd 2 1
1400.2.g.i 4 20.d odd 2 1
1960.2.a.s 2 140.j odd 4 1
1960.2.q.r 4 140.x odd 12 2
1960.2.q.t 4 140.w even 12 2
2240.2.a.bg 2 40.i odd 4 1
2240.2.a.bk 2 40.k even 4 1
2520.2.a.x 2 60.l odd 4 1
2800.2.a.bk 2 5.c odd 4 1
2800.2.g.r 4 1.a even 1 1 trivial
2800.2.g.r 4 5.b even 2 1 inner
3920.2.a.bt 2 35.f even 4 1
5040.2.a.by 2 15.e even 4 1
9800.2.a.bu 2 140.j odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 17 T_{3}^{2} + 64 \)
\( T_{11}^{2} + 7 T_{11} + 4 \)
\( T_{13}^{4} + 21 T_{13}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 64 + 17 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 4 + 7 T + T^{2} )^{2} \)
$13$ \( 36 + 21 T^{2} + T^{4} \)
$17$ \( 4 + 29 T^{2} + T^{4} \)
$19$ \( ( -32 - 2 T + T^{2} )^{2} \)
$23$ \( 1024 + 68 T^{2} + T^{4} \)
$29$ \( ( -6 - 3 T + T^{2} )^{2} \)
$31$ \( ( -8 + T )^{4} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( -32 - 2 T + T^{2} )^{2} \)
$43$ \( 576 + 84 T^{2} + T^{4} \)
$47$ \( 5184 + 153 T^{2} + T^{4} \)
$53$ \( 64 + 116 T^{2} + T^{4} \)
$59$ \( ( -8 + T )^{4} \)
$61$ \( ( -24 - 6 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( 8 + T )^{4} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( ( -32 - 13 T + T^{2} )^{2} \)
$83$ \( 16384 + 272 T^{2} + T^{4} \)
$89$ \( ( 48 + 18 T + T^{2} )^{2} \)
$97$ \( 34596 + 453 T^{2} + T^{4} \)
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