Properties

Label 2800.1.f.b
Level $2800$
Weight $1$
Character orbit 2800.f
Self dual yes
Analytic conductor $1.397$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -7
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.175.1
Artin image $D_6$
Artin field Galois closure of 6.0.9800000.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} + q^{9} + O(q^{10}) \) \( q + q^{7} + q^{9} + q^{11} - q^{23} - q^{29} + q^{37} - q^{43} + q^{49} - 2q^{53} + q^{63} - q^{67} + q^{71} + q^{77} + q^{79} + q^{81} + q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2001.1
0
0 0 0 0 0 1.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.f.b 1
4.b odd 2 1 175.1.d.b yes 1
5.b even 2 1 2800.1.f.a 1
5.c odd 4 2 2800.1.p.a 2
7.b odd 2 1 CM 2800.1.f.b 1
12.b even 2 1 1575.1.h.a 1
20.d odd 2 1 175.1.d.a 1
20.e even 4 2 175.1.c.a 2
28.d even 2 1 175.1.d.b yes 1
28.f even 6 2 1225.1.i.a 2
28.g odd 6 2 1225.1.i.a 2
35.c odd 2 1 2800.1.f.a 1
35.f even 4 2 2800.1.p.a 2
60.h even 2 1 1575.1.h.c 1
60.l odd 4 2 1575.1.e.a 2
84.h odd 2 1 1575.1.h.a 1
140.c even 2 1 175.1.d.a 1
140.j odd 4 2 175.1.c.a 2
140.p odd 6 2 1225.1.i.b 2
140.s even 6 2 1225.1.i.b 2
140.w even 12 4 1225.1.j.a 4
140.x odd 12 4 1225.1.j.a 4
420.o odd 2 1 1575.1.h.c 1
420.w even 4 2 1575.1.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 20.e even 4 2
175.1.c.a 2 140.j odd 4 2
175.1.d.a 1 20.d odd 2 1
175.1.d.a 1 140.c even 2 1
175.1.d.b yes 1 4.b odd 2 1
175.1.d.b yes 1 28.d even 2 1
1225.1.i.a 2 28.f even 6 2
1225.1.i.a 2 28.g odd 6 2
1225.1.i.b 2 140.p odd 6 2
1225.1.i.b 2 140.s even 6 2
1225.1.j.a 4 140.w even 12 4
1225.1.j.a 4 140.x odd 12 4
1575.1.e.a 2 60.l odd 4 2
1575.1.e.a 2 420.w even 4 2
1575.1.h.a 1 12.b even 2 1
1575.1.h.a 1 84.h odd 2 1
1575.1.h.c 1 60.h even 2 1
1575.1.h.c 1 420.o odd 2 1
2800.1.f.a 1 5.b even 2 1
2800.1.f.a 1 35.c odd 2 1
2800.1.f.b 1 1.a even 1 1 trivial
2800.1.f.b 1 7.b odd 2 1 CM
2800.1.p.a 2 5.c odd 4 2
2800.1.p.a 2 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3} \)
\( T_{23} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )( 1 + T ) \)
$5$ 1
$7$ \( 1 - T \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 1 - T )( 1 + T ) \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( 1 + T + T^{2} \)
$29$ \( 1 + T + T^{2} \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( 1 - T + T^{2} \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( 1 + T + T^{2} \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( 1 + T + T^{2} \)
$71$ \( 1 - T + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( ( 1 - T )( 1 + T ) \)
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