Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2800,2,Mod(1,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 −3.23607 0 0 0 1.00000 0 7.47214 0
1.2 0 1.23607 0 0 0 1.00000 0 −1.47214 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.bh 2
4.b odd 2 1 175.2.a.d 2
5.b even 2 1 2800.2.a.bp 2
5.c odd 4 2 2800.2.g.s 4
12.b even 2 1 1575.2.a.s 2
20.d odd 2 1 175.2.a.e yes 2
20.e even 4 2 175.2.b.c 4
28.d even 2 1 1225.2.a.n 2
60.h even 2 1 1575.2.a.n 2
60.l odd 4 2 1575.2.d.k 4
140.c even 2 1 1225.2.a.u 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 4.b odd 2 1
175.2.a.e yes 2 20.d odd 2 1
175.2.b.c 4 20.e even 4 2
1225.2.a.n 2 28.d even 2 1
1225.2.a.u 2 140.c even 2 1
1225.2.b.k 4 140.j odd 4 2
1575.2.a.n 2 60.h even 2 1
1575.2.a.s 2 12.b even 2 1
1575.2.d.k 4 60.l odd 4 2
2800.2.a.bh 2 1.a even 1 1 trivial
2800.2.a.bp 2 5.b even 2 1
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\):

\( T_{3}^{2} + 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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