Properties

Label 6075.2.a.bn
Level $6075$
Weight $2$
Character orbit 6075.a
Self dual yes
Analytic conductor $48.509$
Analytic rank $1$
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] = [6075,2,Mod(1,6075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6075, 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("6075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6075 = 3^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6075.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: \(48.5091192279\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + 4 q^{4} - 2 q^{7} + 2 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 4 q^{4} - 2 q^{7} + 2 \beta q^{8} - \beta q^{11} + q^{13} - 2 \beta q^{14} + 4 q^{16} - 3 \beta q^{17} - q^{19} - 6 q^{22} - \beta q^{23} + \beta q^{26} - 8 q^{28} + 2 \beta q^{29} - q^{31} - 18 q^{34} - 8 q^{37} - \beta q^{38} - 2 \beta q^{41} - 11 q^{43} - 4 \beta q^{44} - 6 q^{46} + 4 \beta q^{47} - 3 q^{49} + 4 q^{52} + 3 \beta q^{53} - 4 \beta q^{56} + 12 q^{58} + \beta q^{59} + 5 q^{61} - \beta q^{62} - 8 q^{64} + 7 q^{67} - 12 \beta q^{68} - 3 \beta q^{71} - 11 q^{73} - 8 \beta q^{74} - 4 q^{76} + 2 \beta q^{77} - 7 q^{79} - 12 q^{82} - 5 \beta q^{83} - 11 \beta q^{86} - 12 q^{88} - 2 q^{91} - 4 \beta q^{92} + 24 q^{94} + 7 q^{97} - 3 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{19} - 12 q^{22} - 16 q^{28} - 2 q^{31} - 36 q^{34} - 16 q^{37} - 22 q^{43} - 12 q^{46} - 6 q^{49} + 8 q^{52} + 24 q^{58} + 10 q^{61} - 16 q^{64} + 14 q^{67} - 22 q^{73} - 8 q^{76} - 14 q^{79} - 24 q^{82} - 24 q^{88} - 4 q^{91} + 48 q^{94} + 14 q^{97}+O(q^{100}) \) 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
−2.44949 0 4.00000 0 0 −2.00000 −4.89898 0 0
1.2 2.44949 0 4.00000 0 0 −2.00000 4.89898 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6075.2.a.bn 2
3.b odd 2 1 inner 6075.2.a.bn 2
5.b even 2 1 243.2.a.d 2
15.d odd 2 1 243.2.a.d 2
20.d odd 2 1 3888.2.a.z 2
45.h odd 6 2 243.2.c.c 4
45.j even 6 2 243.2.c.c 4
60.h even 2 1 3888.2.a.z 2
135.n odd 18 6 729.2.e.p 12
135.p even 18 6 729.2.e.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.d 2 5.b even 2 1
243.2.a.d 2 15.d odd 2 1
243.2.c.c 4 45.h odd 6 2
243.2.c.c 4 45.j even 6 2
729.2.e.p 12 135.n odd 18 6
729.2.e.p 12 135.p even 18 6
3888.2.a.z 2 20.d odd 2 1
3888.2.a.z 2 60.h even 2 1
6075.2.a.bn 2 1.a even 1 1 trivial
6075.2.a.bn 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6075))\):

\( T_{2}^{2} - 6 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 54 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6 \) Copy content Toggle raw display
$29$ \( T^{2} - 24 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 24 \) Copy content Toggle raw display
$43$ \( (T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 96 \) Copy content Toggle raw display
$53$ \( T^{2} - 54 \) Copy content Toggle raw display
$59$ \( T^{2} - 6 \) Copy content Toggle raw display
$61$ \( (T - 5)^{2} \) Copy content Toggle raw display
$67$ \( (T - 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 54 \) Copy content Toggle raw display
$73$ \( (T + 11)^{2} \) Copy content Toggle raw display
$79$ \( (T + 7)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 150 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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