Properties

Label 2450.2.a.br
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2450,2,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 4 q^{9} + 5 q^{11} + \beta q^{12} - 2 \beta q^{13} + q^{16} + \beta q^{17} + 4 q^{18} + 3 \beta q^{19} + 5 q^{22} - 4 q^{23} + \beta q^{24} - 2 \beta q^{26} + \beta q^{27} + 6 q^{29} - 4 \beta q^{31} + q^{32} + 5 \beta q^{33} + \beta q^{34} + 4 q^{36} - 4 q^{37} + 3 \beta q^{38} - 14 q^{39} - \beta q^{41} - 8 q^{43} + 5 q^{44} - 4 q^{46} + 2 \beta q^{47} + \beta q^{48} + 7 q^{51} - 2 \beta q^{52} + 4 q^{53} + \beta q^{54} + 21 q^{57} + 6 q^{58} + 2 \beta q^{59} - 2 \beta q^{61} - 4 \beta q^{62} + q^{64} + 5 \beta q^{66} + 5 q^{67} + \beta q^{68} - 4 \beta q^{69} + 6 q^{71} + 4 q^{72} - \beta q^{73} - 4 q^{74} + 3 \beta q^{76} - 14 q^{78} + 10 q^{79} - 5 q^{81} - \beta q^{82} - \beta q^{83} - 8 q^{86} + 6 \beta q^{87} + 5 q^{88} - 5 \beta q^{89} - 4 q^{92} - 28 q^{93} + 2 \beta q^{94} + \beta q^{96} + 20 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 8 q^{9} + 10 q^{11} + 2 q^{16} + 8 q^{18} + 10 q^{22} - 8 q^{23} + 12 q^{29} + 2 q^{32} + 8 q^{36} - 8 q^{37} - 28 q^{39} - 16 q^{43} + 10 q^{44} - 8 q^{46} + 14 q^{51} + 8 q^{53} + 42 q^{57} + 12 q^{58} + 2 q^{64} + 10 q^{67} + 12 q^{71} + 8 q^{72} - 8 q^{74} - 28 q^{78} + 20 q^{79} - 10 q^{81} - 16 q^{86} + 10 q^{88} - 8 q^{92} - 56 q^{93} + 40 q^{99}+O(q^{100}) \) 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
−2.64575
2.64575
1.00000 −2.64575 1.00000 0 −2.64575 0 1.00000 4.00000 0
1.2 1.00000 2.64575 1.00000 0 2.64575 0 1.00000 4.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.br yes 2
5.b even 2 1 2450.2.a.bm 2
5.c odd 4 2 2450.2.c.u 4
7.b odd 2 1 inner 2450.2.a.br yes 2
35.c odd 2 1 2450.2.a.bm 2
35.f even 4 2 2450.2.c.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2450.2.a.bm 2 5.b even 2 1
2450.2.a.bm 2 35.c odd 2 1
2450.2.a.br yes 2 1.a even 1 1 trivial
2450.2.a.br yes 2 7.b odd 2 1 inner
2450.2.c.u 4 5.c odd 4 2
2450.2.c.u 4 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 7 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} - 28 \) Copy content Toggle raw display
\( T_{17}^{2} - 7 \) Copy content Toggle raw display
\( T_{19}^{2} - 63 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display
\( T_{37} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 7 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 28 \) Copy content Toggle raw display
$17$ \( T^{2} - 7 \) Copy content Toggle raw display
$19$ \( T^{2} - 63 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 112 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 28 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 28 \) Copy content Toggle raw display
$67$ \( (T - 5)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 7 \) Copy content Toggle raw display
$89$ \( T^{2} - 175 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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