Properties

Label 2800.2.a.bp
Level $2800$
Weight $2$
Character orbit 2800.a
Self dual yes
Analytic conductor $22.358$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} - q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} - q^{7} + ( 3 + 2 \beta ) q^{9} + ( -2 + \beta ) q^{11} + ( -1 + \beta ) q^{13} + ( -2 + 2 \beta ) q^{17} + 2 \beta q^{19} + ( -1 - \beta ) q^{21} + ( -4 - \beta ) q^{23} + ( 10 + 2 \beta ) q^{27} + 5 q^{29} + ( 3 - 3 \beta ) q^{31} + ( 3 - \beta ) q^{33} + 3 q^{37} + 4 q^{39} + ( 7 - \beta ) q^{41} + ( -4 + \beta ) q^{43} + 2 q^{47} + q^{49} + 8 q^{51} + ( 4 + 2 \beta ) q^{53} + ( 10 + 2 \beta ) q^{57} + ( -5 - 3 \beta ) q^{59} + ( -3 - 3 \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + ( 2 + \beta ) q^{67} + ( -9 - 5 \beta ) q^{69} + ( -2 - 3 \beta ) q^{71} + ( -11 + \beta ) q^{73} + ( 2 - \beta ) q^{77} + 5 \beta q^{79} + ( 11 + 6 \beta ) q^{81} + ( 1 + 3 \beta ) q^{83} + ( 5 + 5 \beta ) q^{87} + ( 15 + \beta ) q^{89} + ( 1 - \beta ) q^{91} -12 q^{93} + ( 3 + \beta ) q^{97} + ( 4 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{7} + 6q^{9} - 4q^{11} - 2q^{13} - 4q^{17} - 2q^{21} - 8q^{23} + 20q^{27} + 10q^{29} + 6q^{31} + 6q^{33} + 6q^{37} + 8q^{39} + 14q^{41} - 8q^{43} + 4q^{47} + 2q^{49} + 16q^{51} + 8q^{53} + 20q^{57} - 10q^{59} - 6q^{61} - 6q^{63} + 4q^{67} - 18q^{69} - 4q^{71} - 22q^{73} + 4q^{77} + 22q^{81} + 2q^{83} + 10q^{87} + 30q^{89} + 2q^{91} - 24q^{93} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −1.23607 0 0 0 −1.00000 0 −1.47214 0
1.2 0 3.23607 0 0 0 −1.00000 0 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.a.bp 2
4.b odd 2 1 175.2.a.e yes 2
5.b even 2 1 2800.2.a.bh 2
5.c odd 4 2 2800.2.g.s 4
12.b even 2 1 1575.2.a.n 2
20.d odd 2 1 175.2.a.d 2
20.e even 4 2 175.2.b.c 4
28.d even 2 1 1225.2.a.u 2
60.h even 2 1 1575.2.a.s 2
60.l odd 4 2 1575.2.d.k 4
140.c even 2 1 1225.2.a.n 2
140.j odd 4 2 1225.2.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 20.d odd 2 1
175.2.a.e yes 2 4.b odd 2 1
175.2.b.c 4 20.e even 4 2
1225.2.a.n 2 140.c even 2 1
1225.2.a.u 2 28.d even 2 1
1225.2.b.k 4 140.j odd 4 2
1575.2.a.n 2 12.b even 2 1
1575.2.a.s 2 60.h even 2 1
1575.2.d.k 4 60.l odd 4 2
2800.2.a.bh 2 5.b even 2 1
2800.2.a.bp 2 1.a even 1 1 trivial
2800.2.g.s 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\):

\( T_{3}^{2} - 2 T_{3} - 4 \)
\( T_{11}^{2} + 4 T_{11} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -1 + 4 T + T^{2} \)
$13$ \( -4 + 2 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( 11 + 8 T + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( -36 - 6 T + T^{2} \)
$37$ \( ( -3 + T )^{2} \)
$41$ \( 44 - 14 T + T^{2} \)
$43$ \( 11 + 8 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( -20 + 10 T + T^{2} \)
$61$ \( -36 + 6 T + T^{2} \)
$67$ \( -1 - 4 T + T^{2} \)
$71$ \( -41 + 4 T + T^{2} \)
$73$ \( 116 + 22 T + T^{2} \)
$79$ \( -125 + T^{2} \)
$83$ \( -44 - 2 T + T^{2} \)
$89$ \( 220 - 30 T + T^{2} \)
$97$ \( 4 - 6 T + T^{2} \)
show more
show less