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Properties

Label 637.2.r.b
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM discriminant -91
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(116,637)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (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: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{4} - \beta_1 q^{5} + ( - 3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{4} - \beta_1 q^{5} + ( - 3 \beta_{2} + 3) q^{9} + \beta_{3} q^{13} + (4 \beta_{2} - 4) q^{16} - \beta_1 q^{19} + 2 \beta_{3} q^{20} + (\beta_{2} - 1) q^{23} + 8 \beta_{2} q^{25} - 5 q^{29} + (3 \beta_{3} - 3 \beta_1) q^{31} - 6 q^{36} + 2 \beta_{3} q^{41} + 9 q^{43} + (3 \beta_{3} - 3 \beta_1) q^{45} - \beta_1 q^{47} + ( - 2 \beta_{3} + 2 \beta_1) q^{52} - 11 \beta_{2} q^{53} + ( - 4 \beta_{3} + 4 \beta_1) q^{59} + 8 q^{64} + ( - 13 \beta_{2} + 13) q^{65} + (3 \beta_{3} - 3 \beta_1) q^{73} + 2 \beta_{3} q^{76} + (15 \beta_{2} - 15) q^{79} + ( - 4 \beta_{3} + 4 \beta_1) q^{80} - 9 \beta_{2} q^{81} - 5 \beta_{3} q^{83} - \beta_1 q^{89} + 2 q^{92} + 13 \beta_{2} q^{95} - 5 \beta_{3} q^{97}+O(q^{100}) \) 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 \( 4 q - 4 q^{4} + 6 q^{9} - 8 q^{16} - 2 q^{23} + 16 q^{25} - 20 q^{29} - 24 q^{36} + 36 q^{43} - 22 q^{53} + 32 q^{64} + 26 q^{65} - 30 q^{79} - 18 q^{81} + 8 q^{92} + 26 q^{95}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
3.12250 − 1.80278i
−3.12250 + 1.80278i
3.12250 + 1.80278i
−3.12250 − 1.80278i
0 0 −1.00000 + 1.73205i −3.12250 + 1.80278i 0 0 0 1.50000 + 2.59808i 0
116.2 0 0 −1.00000 + 1.73205i 3.12250 − 1.80278i 0 0 0 1.50000 + 2.59808i 0
324.1 0 0 −1.00000 − 1.73205i −3.12250 − 1.80278i 0 0 0 1.50000 − 2.59808i 0
324.2 0 0 −1.00000 − 1.73205i 3.12250 + 1.80278i 0 0 0 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.b odd 2 1 CM by \(\Q(\sqrt{-91}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
13.b even 2 1 inner
91.r even 6 1 inner
91.s 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.r.b 4
7.b odd 2 1 inner 637.2.r.b 4
7.c even 3 1 637.2.c.b ✓ 2
7.c even 3 1 inner 637.2.r.b 4
7.d odd 6 1 637.2.c.b ✓ 2
7.d odd 6 1 inner 637.2.r.b 4
13.b even 2 1 inner 637.2.r.b 4
91.b odd 2 1 CM 637.2.r.b 4
91.r even 6 1 637.2.c.b ✓ 2
91.r even 6 1 inner 637.2.r.b 4
91.s odd 6 1 637.2.c.b ✓ 2
91.s odd 6 1 inner 637.2.r.b 4
91.z odd 12 2 8281.2.a.u 2
91.bb even 12 2 8281.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b ✓ 2 7.c even 3 1
637.2.c.b ✓ 2 7.d odd 6 1
637.2.c.b ✓ 2 91.r even 6 1
637.2.c.b ✓ 2 91.s odd 6 1
637.2.r.b 4 1.a even 1 1 trivial
637.2.r.b 4 7.b odd 2 1 inner
637.2.r.b 4 7.c even 3 1 inner
637.2.r.b 4 7.d odd 6 1 inner
637.2.r.b 4 13.b even 2 1 inner
637.2.r.b 4 91.b odd 2 1 CM
637.2.r.b 4 91.r even 6 1 inner
637.2.r.b 4 91.s odd 6 1 inner
8281.2.a.u 2 91.z odd 12 2
8281.2.a.u 2 91.bb even 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) 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
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 5)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 117 T^{2} + 13689 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$43$ \( (T - 9)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$53$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 208 T^{2} + 43264 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 117 T^{2} + 13689 \) Copy content Toggle raw display
$79$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 325)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$97$ \( (T^{2} + 325)^{2} \) Copy content Toggle raw display
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