Properties

Label 4050.2.a.br
Level $4050$
Weight $2$
Character orbit 4050.a
Self dual yes
Analytic conductor $32.339$
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] = [4050,2,Mod(1,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, 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("4050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta + 2) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta + 2) q^{7} - q^{8} + 2 \beta q^{11} + (\beta + 2) q^{13} + ( - \beta - 2) q^{14} + q^{16} - 2 \beta q^{17} + ( - \beta + 5) q^{19} - 2 \beta q^{22} + \beta q^{23} + ( - \beta - 2) q^{26} + (\beta + 2) q^{28} + \beta q^{29} + ( - \beta + 2) q^{31} - q^{32} + 2 \beta q^{34} + (3 \beta - 4) q^{37} + (\beta - 5) q^{38} + 9 q^{41} + (\beta + 5) q^{43} + 2 \beta q^{44} - \beta q^{46} + (2 \beta - 6) q^{47} + (4 \beta + 3) q^{49} + (\beta + 2) q^{52} + ( - \beta - 6) q^{53} + ( - \beta - 2) q^{56} - \beta q^{58} + (\beta - 3) q^{59} + 8 q^{61} + (\beta - 2) q^{62} + q^{64} + ( - 3 \beta - 7) q^{67} - 2 \beta q^{68} + ( - 3 \beta + 6) q^{71} - q^{73} + ( - 3 \beta + 4) q^{74} + ( - \beta + 5) q^{76} + (4 \beta + 12) q^{77} + ( - 6 \beta + 2) q^{79} - 9 q^{82} + (\beta - 3) q^{83} + ( - \beta - 5) q^{86} - 2 \beta q^{88} - 9 q^{89} + (4 \beta + 10) q^{91} + \beta q^{92} + ( - 2 \beta + 6) q^{94} + ( - 4 \beta - 1) q^{97} + ( - 4 \beta - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 10 q^{19} - 4 q^{26} + 4 q^{28} + 4 q^{31} - 2 q^{32} - 8 q^{37} - 10 q^{38} + 18 q^{41} + 10 q^{43} - 12 q^{47} + 6 q^{49} + 4 q^{52} - 12 q^{53} - 4 q^{56} - 6 q^{59} + 16 q^{61} - 4 q^{62} + 2 q^{64} - 14 q^{67} + 12 q^{71} - 2 q^{73} + 8 q^{74} + 10 q^{76} + 24 q^{77} + 4 q^{79} - 18 q^{82} - 6 q^{83} - 10 q^{86} - 18 q^{89} + 20 q^{91} + 12 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
−1.00000 0 1.00000 0 0 −0.449490 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 4.44949 −1.00000 0 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

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 4050.2.a.br 2
3.b odd 2 1 4050.2.a.by 2
5.b even 2 1 4050.2.a.bu 2
5.c odd 4 2 4050.2.c.y 4
9.c even 3 2 450.2.e.m yes 4
9.d odd 6 2 1350.2.e.k 4
15.d odd 2 1 4050.2.a.bl 2
15.e even 4 2 4050.2.c.w 4
45.h odd 6 2 1350.2.e.n 4
45.j even 6 2 450.2.e.l 4
45.k odd 12 4 450.2.j.f 8
45.l even 12 4 1350.2.j.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 45.j even 6 2
450.2.e.m yes 4 9.c even 3 2
450.2.j.f 8 45.k odd 12 4
1350.2.e.k 4 9.d odd 6 2
1350.2.e.n 4 45.h odd 6 2
1350.2.j.g 8 45.l even 12 4
4050.2.a.bl 2 15.d odd 2 1
4050.2.a.br 2 1.a even 1 1 trivial
4050.2.a.bu 2 5.b even 2 1
4050.2.a.by 2 3.b odd 2 1
4050.2.c.w 4 15.e even 4 2
4050.2.c.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7}^{2} - 4T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 24 \) Copy content Toggle raw display
\( T_{23}^{2} - 6 \) Copy content Toggle raw display
\( T_{41} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 24 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 24 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 6 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 38 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 19 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 30 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T - 5 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T - 18 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 212 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 95 \) Copy content Toggle raw display
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