Properties

Label 2800.1.p.a
Level $2800$
Weight $1$
Character orbit 2800.p
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{7} - q^{9} + q^{11} - i q^{23} + q^{29} - i q^{37} - i q^{43} - q^{49} - i q^{53} + i q^{63} + i q^{67} + q^{71} - i q^{77} - q^{79} + q^{81} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{11} + 2 q^{29} - 2 q^{49} + 2 q^{71} - 2 q^{79} + 2 q^{81} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
2449.2 0 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
35.c odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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