Properties

Label 2800.2.g.d
Level $2800$
Weight $2$
Character orbit 2800.g
Analytic conductor $22.358$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -i q^{7} - q^{9} -5 q^{11} + 8 i q^{17} -2 q^{19} + 2 q^{21} -7 i q^{23} + 4 i q^{27} + 3 q^{29} -4 q^{31} -10 i q^{33} -i q^{37} -2 q^{41} + 3 i q^{43} -6 i q^{47} - q^{49} -16 q^{51} -10 i q^{53} -4 i q^{57} -4 q^{59} -6 q^{61} + i q^{63} -13 i q^{67} + 14 q^{69} -5 q^{71} -6 i q^{73} + 5 i q^{77} -13 q^{79} -11 q^{81} + 16 i q^{83} + 6 i q^{87} -8 i q^{93} -12 i q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 10q^{11} - 4q^{19} + 4q^{21} + 6q^{29} - 8q^{31} - 4q^{41} - 2q^{49} - 32q^{51} - 8q^{59} - 12q^{61} + 28q^{69} - 10q^{71} - 26q^{79} - 22q^{81} + 10q^{99} + O(q^{100}) \)

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 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 1.00000i 0 −1.00000 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.d 2
4.b odd 2 1 1400.2.g.d 2
5.b even 2 1 inner 2800.2.g.d 2
5.c odd 4 1 2800.2.a.d 1
5.c odd 4 1 2800.2.a.bc 1
20.d odd 2 1 1400.2.g.d 2
20.e even 4 1 1400.2.a.b 1
20.e even 4 1 1400.2.a.m yes 1
140.j odd 4 1 9800.2.a.l 1
140.j odd 4 1 9800.2.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.b 1 20.e even 4 1
1400.2.a.m yes 1 20.e even 4 1
1400.2.g.d 2 4.b odd 2 1
1400.2.g.d 2 20.d odd 2 1
2800.2.a.d 1 5.c odd 4 1
2800.2.a.bc 1 5.c odd 4 1
2800.2.g.d 2 1.a even 1 1 trivial
2800.2.g.d 2 5.b even 2 1 inner
9800.2.a.l 1 140.j odd 4 1
9800.2.a.bo 1 140.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 \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{11} + 5 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 49 + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 9 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( 5 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 13 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 144 + T^{2} \)
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