Properties

Label 2800.1.bf.a
Level $2800$
Weight $1$
Character orbit 2800.bf
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
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.bf (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.14336.1
Artin image: $C_4^3.C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + i q^{9} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{7} - q^{8} + i q^{9} + ( -1 - i ) q^{11} + q^{14} + q^{16} -i q^{18} + ( 1 + i ) q^{22} - q^{28} + ( 1 - i ) q^{29} - q^{32} + i q^{36} + ( 1 - i ) q^{37} + ( 1 - i ) q^{43} + ( -1 - i ) q^{44} + q^{49} + ( 1 - i ) q^{53} + q^{56} + ( -1 + i ) q^{58} -i q^{63} + q^{64} + ( 1 + i ) q^{67} -2 i q^{71} -i q^{72} + ( -1 + i ) q^{74} + ( 1 + i ) q^{77} - q^{81} + ( -1 + i ) q^{86} + ( 1 + i ) q^{88} - q^{98} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} - 2q^{11} + 2q^{14} + 2q^{16} + 2q^{22} - 2q^{28} + 2q^{29} - 2q^{32} + 2q^{37} + 2q^{43} - 2q^{44} + 2q^{49} + 2q^{53} + 2q^{56} - 2q^{58} + 2q^{64} + 2q^{67} - 2q^{74} + 2q^{77} - 2q^{81} - 2q^{86} + 2q^{88} - 2q^{98} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(-1\) \(i\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.00000i
1.00000i
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 1.00000i 0
1749.1 −1.00000 0 1.00000 0 0 −1.00000 −1.00000 1.00000i 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 \(\Q(\sqrt{-7}) \)
80.q even 4 1 inner
560.bf odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.bf.a 2
5.b even 2 1 2800.1.bf.b 2
5.c odd 4 1 112.1.l.a 2
5.c odd 4 1 2800.1.z.a 2
7.b odd 2 1 CM 2800.1.bf.a 2
15.e even 4 1 1008.1.u.b 2
16.e even 4 1 2800.1.bf.b 2
20.e even 4 1 448.1.l.a 2
35.c odd 2 1 2800.1.bf.b 2
35.f even 4 1 112.1.l.a 2
35.f even 4 1 2800.1.z.a 2
35.k even 12 2 784.1.y.a 4
35.l odd 12 2 784.1.y.a 4
40.i odd 4 1 896.1.l.b 2
40.k even 4 1 896.1.l.a 2
80.i odd 4 1 112.1.l.a 2
80.j even 4 1 896.1.l.a 2
80.q even 4 1 inner 2800.1.bf.a 2
80.s even 4 1 448.1.l.a 2
80.t odd 4 1 896.1.l.b 2
80.t odd 4 1 2800.1.z.a 2
105.k odd 4 1 1008.1.u.b 2
112.l odd 4 1 2800.1.bf.b 2
140.j odd 4 1 448.1.l.a 2
140.w even 12 2 3136.1.bc.a 4
140.x odd 12 2 3136.1.bc.a 4
240.bb even 4 1 1008.1.u.b 2
280.s even 4 1 896.1.l.b 2
280.y odd 4 1 896.1.l.a 2
560.r even 4 1 896.1.l.b 2
560.r even 4 1 2800.1.z.a 2
560.u odd 4 1 448.1.l.a 2
560.bf odd 4 1 inner 2800.1.bf.a 2
560.bm odd 4 1 896.1.l.a 2
560.bn even 4 1 112.1.l.a 2
560.cf even 12 2 3136.1.bc.a 4
560.ch even 12 2 784.1.y.a 4
560.cy odd 12 2 784.1.y.a 4
560.da odd 12 2 3136.1.bc.a 4
1680.cn odd 4 1 1008.1.u.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 5.c odd 4 1
112.1.l.a 2 35.f even 4 1
112.1.l.a 2 80.i odd 4 1
112.1.l.a 2 560.bn even 4 1
448.1.l.a 2 20.e even 4 1
448.1.l.a 2 80.s even 4 1
448.1.l.a 2 140.j odd 4 1
448.1.l.a 2 560.u odd 4 1
784.1.y.a 4 35.k even 12 2
784.1.y.a 4 35.l odd 12 2
784.1.y.a 4 560.ch even 12 2
784.1.y.a 4 560.cy odd 12 2
896.1.l.a 2 40.k even 4 1
896.1.l.a 2 80.j even 4 1
896.1.l.a 2 280.y odd 4 1
896.1.l.a 2 560.bm odd 4 1
896.1.l.b 2 40.i odd 4 1
896.1.l.b 2 80.t odd 4 1
896.1.l.b 2 280.s even 4 1
896.1.l.b 2 560.r even 4 1
1008.1.u.b 2 15.e even 4 1
1008.1.u.b 2 105.k odd 4 1
1008.1.u.b 2 240.bb even 4 1
1008.1.u.b 2 1680.cn odd 4 1
2800.1.z.a 2 5.c odd 4 1
2800.1.z.a 2 35.f even 4 1
2800.1.z.a 2 80.t odd 4 1
2800.1.z.a 2 560.r even 4 1
2800.1.bf.a 2 1.a even 1 1 trivial
2800.1.bf.a 2 7.b odd 2 1 CM
2800.1.bf.a 2 80.q even 4 1 inner
2800.1.bf.a 2 560.bf odd 4 1 inner
2800.1.bf.b 2 5.b even 2 1
2800.1.bf.b 2 16.e even 4 1
2800.1.bf.b 2 35.c odd 2 1
2800.1.bf.b 2 112.l odd 4 1
3136.1.bc.a 4 140.w even 12 2
3136.1.bc.a 4 140.x odd 12 2
3136.1.bc.a 4 560.cf even 12 2
3136.1.bc.a 4 560.da odd 12 2

Hecke kernels

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

\( T_{11}^{2} + 2 T_{11} + 2 \)
\( T_{37}^{2} - 2 T_{37} + 2 \)

Hecke characteristic polynomials

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