Properties

Label 675.2.a.l
Level $675$
Weight $2$
Character orbit 675.a
Self dual yes
Analytic conductor $5.390$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(1,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, 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("675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.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: \(5.38990213644\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 3 q^{7} - 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 3 q^{7} - 2 \beta q^{8} - 3 \beta q^{11} - 3 q^{13} - 3 \beta q^{14} - 4 q^{16} - 2 \beta q^{17} + q^{19} - 6 q^{22} + 5 \beta q^{23} - 3 \beta q^{26} + 3 \beta q^{29} + 2 q^{31} - 4 q^{34} - 9 q^{37} + \beta q^{38} + 3 \beta q^{41} - 6 q^{43} + 10 q^{46} - 2 \beta q^{47} + 2 q^{49} - 7 \beta q^{53} + 6 \beta q^{56} + 6 q^{58} + 6 \beta q^{59} - 13 q^{61} + 2 \beta q^{62} + 8 q^{64} + 3 q^{67} + 9 \beta q^{71} - 9 q^{73} - 9 \beta q^{74} + 9 \beta q^{77} - 5 q^{79} + 6 q^{82} - \beta q^{83} - 6 \beta q^{86} + 12 q^{88} + 9 q^{91} - 4 q^{94} - 3 q^{97} + 2 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{7} - 6 q^{13} - 8 q^{16} + 2 q^{19} - 12 q^{22} + 4 q^{31} - 8 q^{34} - 18 q^{37} - 12 q^{43} + 20 q^{46} + 4 q^{49} + 12 q^{58} - 26 q^{61} + 16 q^{64} + 6 q^{67} - 18 q^{73} - 10 q^{79} + 12 q^{82} + 24 q^{88} + 18 q^{91} - 8 q^{94} - 6 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−1.41421 0 0 0 0 −3.00000 2.82843 0 0
1.2 1.41421 0 0 0 0 −3.00000 −2.82843 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 675.2.a.l 2
3.b odd 2 1 inner 675.2.a.l 2
5.b even 2 1 675.2.a.m 2
5.c odd 4 2 135.2.b.b 4
15.d odd 2 1 675.2.a.m 2
15.e even 4 2 135.2.b.b 4
20.e even 4 2 2160.2.f.k 4
45.k odd 12 4 405.2.j.f 8
45.l even 12 4 405.2.j.f 8
60.l odd 4 2 2160.2.f.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b 4 5.c odd 4 2
135.2.b.b 4 15.e even 4 2
405.2.j.f 8 45.k odd 12 4
405.2.j.f 8 45.l even 12 4
675.2.a.l 2 1.a even 1 1 trivial
675.2.a.l 2 3.b odd 2 1 inner
675.2.a.m 2 5.b even 2 1
675.2.a.m 2 15.d odd 2 1
2160.2.f.k 4 20.e even 4 2
2160.2.f.k 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 50 \) Copy content Toggle raw display
$29$ \( T^{2} - 18 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 18 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 98 \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( (T - 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 162 \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 3)^{2} \) Copy content Toggle raw display
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