Properties

Label 300.6.a.f
Level $300$
Weight $6$
Character orbit 300.a
Self dual yes
Analytic conductor $48.115$
Analytic rank $1$
Dimension $1$
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] = [300,6,Mod(1,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 300.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: \(48.1151459439\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} + 91 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + 91 q^{7} + 81 q^{9} - 174 q^{11} - 785 q^{13} - 1794 q^{17} - 925 q^{19} + 819 q^{21} + 2346 q^{23} + 729 q^{27} - 726 q^{29} - 811 q^{31} - 1566 q^{33} - 7922 q^{37} - 7065 q^{39} - 360 q^{41} + 4951 q^{43} - 9906 q^{47} - 8526 q^{49} - 16146 q^{51} + 8292 q^{53} - 8325 q^{57} + 7014 q^{59} - 51433 q^{61} + 7371 q^{63} - 581 q^{67} + 21114 q^{69} - 56520 q^{71} + 42478 q^{73} - 15834 q^{77} - 28912 q^{79} + 6561 q^{81} + 104586 q^{83} - 6534 q^{87} - 118080 q^{89} - 71435 q^{91} - 7299 q^{93} - 110273 q^{97} - 14094 q^{99}+O(q^{100}) \) 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
0
0 9.00000 0 0 0 91.0000 0 81.0000 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 300.6.a.f yes 1
3.b odd 2 1 900.6.a.i 1
5.b even 2 1 300.6.a.a 1
5.c odd 4 2 300.6.d.b 2
15.d odd 2 1 900.6.a.d 1
15.e even 4 2 900.6.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.6.a.a 1 5.b even 2 1
300.6.a.f yes 1 1.a even 1 1 trivial
300.6.d.b 2 5.c odd 4 2
900.6.a.d 1 15.d odd 2 1
900.6.a.i 1 3.b odd 2 1
900.6.d.g 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 91 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(300))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 91 \) Copy content Toggle raw display
$11$ \( T + 174 \) Copy content Toggle raw display
$13$ \( T + 785 \) Copy content Toggle raw display
$17$ \( T + 1794 \) Copy content Toggle raw display
$19$ \( T + 925 \) Copy content Toggle raw display
$23$ \( T - 2346 \) Copy content Toggle raw display
$29$ \( T + 726 \) Copy content Toggle raw display
$31$ \( T + 811 \) Copy content Toggle raw display
$37$ \( T + 7922 \) Copy content Toggle raw display
$41$ \( T + 360 \) Copy content Toggle raw display
$43$ \( T - 4951 \) Copy content Toggle raw display
$47$ \( T + 9906 \) Copy content Toggle raw display
$53$ \( T - 8292 \) Copy content Toggle raw display
$59$ \( T - 7014 \) Copy content Toggle raw display
$61$ \( T + 51433 \) Copy content Toggle raw display
$67$ \( T + 581 \) Copy content Toggle raw display
$71$ \( T + 56520 \) Copy content Toggle raw display
$73$ \( T - 42478 \) Copy content Toggle raw display
$79$ \( T + 28912 \) Copy content Toggle raw display
$83$ \( T - 104586 \) Copy content Toggle raw display
$89$ \( T + 118080 \) Copy content Toggle raw display
$97$ \( T + 110273 \) Copy content Toggle raw display
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