Properties

Label 2400.2.a.bi
Level $2400$
Weight $2$
Character orbit 2400.a
Self dual yes
Analytic conductor $19.164$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
0 −1.00000 0 0 0 −2.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.a.bi 2
3.b odd 2 1 7200.2.a.cc 2
4.b odd 2 1 2400.2.a.bj 2
5.b even 2 1 2400.2.a.bj 2
5.c odd 4 2 480.2.f.e 4
8.b even 2 1 4800.2.a.cv 2
8.d odd 2 1 4800.2.a.cu 2
12.b even 2 1 7200.2.a.cq 2
15.d odd 2 1 7200.2.a.cq 2
15.e even 4 2 1440.2.f.h 4
20.d odd 2 1 inner 2400.2.a.bi 2
20.e even 4 2 480.2.f.e 4
40.e odd 2 1 4800.2.a.cv 2
40.f even 2 1 4800.2.a.cu 2
40.i odd 4 2 960.2.f.k 4
40.k even 4 2 960.2.f.k 4
60.h even 2 1 7200.2.a.cc 2
60.l odd 4 2 1440.2.f.h 4
80.i odd 4 2 3840.2.d.bh 4
80.j even 4 2 3840.2.d.bh 4
80.s even 4 2 3840.2.d.bg 4
80.t odd 4 2 3840.2.d.bg 4
120.q odd 4 2 2880.2.f.v 4
120.w even 4 2 2880.2.f.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 5.c odd 4 2
480.2.f.e 4 20.e even 4 2
960.2.f.k 4 40.i odd 4 2
960.2.f.k 4 40.k even 4 2
1440.2.f.h 4 15.e even 4 2
1440.2.f.h 4 60.l odd 4 2
2400.2.a.bi 2 1.a even 1 1 trivial
2400.2.a.bi 2 20.d odd 2 1 inner
2400.2.a.bj 2 4.b odd 2 1
2400.2.a.bj 2 5.b even 2 1
2880.2.f.v 4 120.q odd 4 2
2880.2.f.v 4 120.w even 4 2
3840.2.d.bg 4 80.s even 4 2
3840.2.d.bg 4 80.t odd 4 2
3840.2.d.bh 4 80.i odd 4 2
3840.2.d.bh 4 80.j even 4 2
4800.2.a.cu 2 8.d odd 2 1
4800.2.a.cu 2 40.f even 2 1
4800.2.a.cv 2 8.b even 2 1
4800.2.a.cv 2 40.e odd 2 1
7200.2.a.cc 2 3.b odd 2 1
7200.2.a.cc 2 60.h even 2 1
7200.2.a.cq 2 12.b even 2 1
7200.2.a.cq 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2400))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 20 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 80 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( T^{2} - 180 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( T^{2} - 80 \) Copy content Toggle raw display
$79$ \( T^{2} - 80 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 320 \) Copy content Toggle raw display
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