Properties

Label 637.2.e.c
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
Analytic rank $0$
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] = [637,2,Mod(79,637)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} - 4 \zeta_{6} q^{12} + q^{13} + 6 q^{15} - 4 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + 7 \zeta_{6} q^{19} + 6 q^{20} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 4 q^{27} - 9 q^{29} + (5 \zeta_{6} - 5) q^{31} - 2 q^{36} - 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{39} - 6 q^{41} - q^{43} + ( - 3 \zeta_{6} + 3) q^{45} - 3 \zeta_{6} q^{47} - 8 q^{48} - 12 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + 14 q^{57} + ( - 12 \zeta_{6} + 12) q^{60} + 10 \zeta_{6} q^{61} - 8 q^{64} + 3 \zeta_{6} q^{65} + (14 \zeta_{6} - 14) q^{67} - 12 \zeta_{6} q^{68} - 6 q^{69} - 6 q^{71} + (11 \zeta_{6} - 11) q^{73} + 8 \zeta_{6} q^{75} + 14 q^{76} + \zeta_{6} q^{79} + ( - 12 \zeta_{6} + 12) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 3 q^{83} + 18 q^{85} + (18 \zeta_{6} - 18) q^{87} - 15 \zeta_{6} q^{89} - 6 q^{92} + 10 \zeta_{6} q^{93} + (21 \zeta_{6} - 21) q^{95} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9} - 4 q^{12} + 2 q^{13} + 12 q^{15} - 4 q^{16} + 6 q^{17} + 7 q^{19} + 12 q^{20} - 3 q^{23} - 4 q^{25} + 8 q^{27} - 18 q^{29} - 5 q^{31} - 4 q^{36} - 2 q^{37} + 2 q^{39} - 12 q^{41} - 2 q^{43} + 3 q^{45} - 3 q^{47} - 16 q^{48} - 12 q^{51} + 2 q^{52} + 9 q^{53} + 28 q^{57} + 12 q^{60} + 10 q^{61} - 16 q^{64} + 3 q^{65} - 14 q^{67} - 12 q^{68} - 12 q^{69} - 12 q^{71} - 11 q^{73} + 8 q^{75} + 28 q^{76} + q^{79} + 12 q^{80} + 11 q^{81} + 6 q^{83} + 36 q^{85} - 18 q^{87} - 15 q^{89} - 12 q^{92} + 10 q^{93} - 21 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 −0.500000 + 0.866025i 0
508.1 0 1.00000 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 −0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.c 2
7.b odd 2 1 637.2.e.b 2
7.c even 3 1 91.2.a.b 1
7.c even 3 1 inner 637.2.e.c 2
7.d odd 6 1 637.2.a.b 1
7.d odd 6 1 637.2.e.b 2
21.g even 6 1 5733.2.a.f 1
21.h odd 6 1 819.2.a.c 1
28.g odd 6 1 1456.2.a.k 1
35.j even 6 1 2275.2.a.d 1
56.k odd 6 1 5824.2.a.f 1
56.p even 6 1 5824.2.a.bd 1
91.r even 6 1 1183.2.a.a 1
91.s odd 6 1 8281.2.a.h 1
91.z odd 12 2 1183.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.b 1 7.c even 3 1
637.2.a.b 1 7.d odd 6 1
637.2.e.b 2 7.b odd 2 1
637.2.e.b 2 7.d odd 6 1
637.2.e.c 2 1.a even 1 1 trivial
637.2.e.c 2 7.c even 3 1 inner
819.2.a.c 1 21.h odd 6 1
1183.2.a.a 1 91.r even 6 1
1183.2.c.a 2 91.z odd 12 2
1456.2.a.k 1 28.g odd 6 1
2275.2.a.d 1 35.j even 6 1
5733.2.a.f 1 21.g even 6 1
5824.2.a.f 1 56.k odd 6 1
5824.2.a.bd 1 56.p even 6 1
8281.2.a.h 1 91.s odd 6 1

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
\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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