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Properties

Label 637.2.k.e
Level $637$
Weight $2$
Character orbit 637.k
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(459,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.459");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.k (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 13x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 1) q^{2} - q^{4} + \beta_1 q^{5} + ( - 2 \beta_{2} + 1) q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 1) q^{2} - q^{4} + \beta_1 q^{5} + ( - 2 \beta_{2} + 1) q^{8} + 3 \beta_{2} q^{9} + ( - 2 \beta_{3} + \beta_1) q^{10} + ( - 2 \beta_{2} - 2) q^{11} + (\beta_{3} - \beta_1) q^{13} - 5 q^{16} + ( - \beta_{3} + 2 \beta_1) q^{17} + ( - 3 \beta_{2} + 6) q^{18} + ( - 2 \beta_{3} + 2 \beta_1) q^{19} - \beta_1 q^{20} + (6 \beta_{2} - 6) q^{22} + 4 q^{23} + 8 \beta_{2} q^{25} + (\beta_{3} + \beta_1) q^{26} - \beta_{2} q^{29} + (6 \beta_{2} - 3) q^{32} - 3 \beta_{3} q^{34} - 3 \beta_{2} q^{36} + ( - 2 \beta_{2} + 1) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} + ( - 2 \beta_{3} + \beta_1) q^{40} + ( - \beta_{3} + \beta_1) q^{41} + (6 \beta_{2} - 6) q^{43} + (2 \beta_{2} + 2) q^{44} + 3 \beta_{3} q^{45} + ( - 8 \beta_{2} + 4) q^{46} - 2 \beta_1 q^{47} + ( - 8 \beta_{2} + 16) q^{50} + ( - \beta_{3} + \beta_1) q^{52} - 5 \beta_{2} q^{53} + ( - 2 \beta_{3} - 2 \beta_1) q^{55} + (\beta_{2} - 2) q^{58} + 2 \beta_{3} q^{59} + (\beta_{3} + \beta_1) q^{61} - q^{64} - 13 q^{65} + ( - 8 \beta_{2} - 8) q^{67} + (\beta_{3} - 2 \beta_1) q^{68} + ( - 6 \beta_{2} - 6) q^{71} + ( - 3 \beta_{2} + 6) q^{72} + (3 \beta_{3} - 3 \beta_1) q^{73} - 3 q^{74} + (2 \beta_{3} - 2 \beta_1) q^{76} + ( - 6 \beta_{2} + 6) q^{79} - 5 \beta_1 q^{80} + (9 \beta_{2} - 9) q^{81} + ( - \beta_{3} - \beta_1) q^{82} + 2 \beta_{3} q^{83} + (13 \beta_{2} + 13) q^{85} + (6 \beta_{2} + 6) q^{86} + (6 \beta_{2} - 6) q^{88} + 2 \beta_{3} q^{89} + ( - 3 \beta_{3} + 6 \beta_1) q^{90} - 4 q^{92} + (4 \beta_{3} - 2 \beta_1) q^{94} + 26 q^{95} - 2 \beta_1 q^{97} + ( - 12 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 6 q^{9} - 12 q^{11} - 20 q^{16} + 18 q^{18} - 12 q^{22} + 16 q^{23} + 16 q^{25} - 2 q^{29} - 6 q^{36} - 12 q^{43} + 12 q^{44} + 48 q^{50} - 10 q^{53} - 6 q^{58} - 4 q^{64} - 52 q^{65} - 48 q^{67} - 36 q^{71} + 18 q^{72} - 12 q^{74} + 12 q^{79} - 18 q^{81} + 78 q^{85} + 36 q^{86} - 12 q^{88} - 16 q^{92} + 104 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
−3.12250 + 1.80278i
3.12250 − 1.80278i
−3.12250 − 1.80278i
3.12250 + 1.80278i
1.73205i 0 −1.00000 −3.12250 + 1.80278i 0 0 1.73205i 1.50000 − 2.59808i −3.12250 − 5.40833i
459.2 1.73205i 0 −1.00000 3.12250 − 1.80278i 0 0 1.73205i 1.50000 − 2.59808i 3.12250 + 5.40833i
569.1 − 1.73205i 0 −1.00000 −3.12250 − 1.80278i 0 0 − 1.73205i 1.50000 + 2.59808i −3.12250 + 5.40833i
569.2 − 1.73205i 0 −1.00000 3.12250 + 1.80278i 0 0 − 1.73205i 1.50000 + 2.59808i 3.12250 − 5.40833i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.k even 6 1 inner
91.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.k.e 4
7.b odd 2 1 inner 637.2.k.e 4
7.c even 3 1 637.2.q.d ✓ 4
7.c even 3 1 637.2.u.f 4
7.d odd 6 1 637.2.q.d ✓ 4
7.d odd 6 1 637.2.u.f 4
13.e even 6 1 637.2.u.f 4
91.k even 6 1 inner 637.2.k.e 4
91.l odd 6 1 inner 637.2.k.e 4
91.p odd 6 1 637.2.q.d ✓ 4
91.t odd 6 1 637.2.u.f 4
91.u even 6 1 637.2.q.d ✓ 4
91.x odd 12 2 8281.2.a.br 4
91.ba even 12 2 8281.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.e 4 1.a even 1 1 trivial
637.2.k.e 4 7.b odd 2 1 inner
637.2.k.e 4 91.k even 6 1 inner
637.2.k.e 4 91.l odd 6 1 inner
637.2.q.d ✓ 4 7.c even 3 1
637.2.q.d ✓ 4 7.d odd 6 1
637.2.q.d ✓ 4 91.p odd 6 1
637.2.q.d ✓ 4 91.u even 6 1
637.2.u.f 4 7.c even 3 1
637.2.u.f 4 7.d odd 6 1
637.2.u.f 4 13.e even 6 1
637.2.u.f 4 91.t odd 6 1
8281.2.a.br 4 91.x odd 12 2
8281.2.a.br 4 91.ba even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 39)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 39T^{2} + 1521 \) Copy content Toggle raw display
$67$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 117 T^{2} + 13689 \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 52T^{2} + 2704 \) Copy content Toggle raw display
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