Properties

Label 637.2.u.b
Level $637$
Weight $2$
Character orbit 637.u
Analytic conductor $5.086$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(30,637)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.30");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.u (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: \(2\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 2) q^{2} - 2 q^{3} + ( - \zeta_{6} + 1) q^{4} + ( - \zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (2 \zeta_{6} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 2) q^{2} - 2 q^{3} + ( - \zeta_{6} + 1) q^{4} + ( - \zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (2 \zeta_{6} - 1) q^{8} + q^{9} - 3 q^{10} + (2 \zeta_{6} - 2) q^{12} + (3 \zeta_{6} + 1) q^{13} + (2 \zeta_{6} + 2) q^{15} + 5 \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + ( - \zeta_{6} + 2) q^{18} + (4 \zeta_{6} - 2) q^{19} + (\zeta_{6} - 2) q^{20} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 2) q^{24} - 2 \zeta_{6} q^{25} + (2 \zeta_{6} + 5) q^{26} + 4 q^{27} + (3 \zeta_{6} - 3) q^{29} + 6 q^{30} + (2 \zeta_{6} - 4) q^{31} + (3 \zeta_{6} + 3) q^{32} + (6 \zeta_{6} - 3) q^{34} + ( - \zeta_{6} + 1) q^{36} + (5 \zeta_{6} - 10) q^{37} + 6 \zeta_{6} q^{38} + ( - 6 \zeta_{6} - 2) q^{39} + ( - 3 \zeta_{6} + 3) q^{40} + (3 \zeta_{6} + 3) q^{41} - 8 \zeta_{6} q^{43} + ( - \zeta_{6} - 1) q^{45} + (6 \zeta_{6} + 6) q^{46} + ( - 2 \zeta_{6} - 2) q^{47} - 10 \zeta_{6} q^{48} + ( - 2 \zeta_{6} - 2) q^{50} + ( - 6 \zeta_{6} + 6) q^{51} + ( - \zeta_{6} + 4) q^{52} + 3 \zeta_{6} q^{53} + ( - 4 \zeta_{6} + 8) q^{54} + ( - 8 \zeta_{6} + 4) q^{57} + (6 \zeta_{6} - 3) q^{58} + (4 \zeta_{6} + 4) q^{59} + ( - 2 \zeta_{6} + 4) q^{60} - q^{61} + (6 \zeta_{6} - 6) q^{62} - q^{64} + ( - 7 \zeta_{6} + 2) q^{65} + ( - 4 \zeta_{6} + 2) q^{67} + 3 \zeta_{6} q^{68} - 12 \zeta_{6} q^{69} + ( - 2 \zeta_{6} + 4) q^{71} + (2 \zeta_{6} - 1) q^{72} + (\zeta_{6} - 2) q^{73} + (15 \zeta_{6} - 15) q^{74} + 4 \zeta_{6} q^{75} + (2 \zeta_{6} + 2) q^{76} + ( - 4 \zeta_{6} - 10) q^{78} + (4 \zeta_{6} - 4) q^{79} + ( - 10 \zeta_{6} + 5) q^{80} - 11 q^{81} + 9 q^{82} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + ( - 8 \zeta_{6} - 8) q^{86} + ( - 6 \zeta_{6} + 6) q^{87} + (4 \zeta_{6} - 8) q^{89} - 3 q^{90} + 6 q^{92} + ( - 4 \zeta_{6} + 8) q^{93} - 6 q^{94} + ( - 6 \zeta_{6} + 6) q^{95} + ( - 6 \zeta_{6} - 6) q^{96} + (4 \zeta_{6} - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 4 q^{3} + q^{4} - 3 q^{5} - 6 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - 4 q^{3} + q^{4} - 3 q^{5} - 6 q^{6} + 2 q^{9} - 6 q^{10} - 2 q^{12} + 5 q^{13} + 6 q^{15} + 5 q^{16} - 3 q^{17} + 3 q^{18} - 3 q^{20} + 6 q^{23} - 2 q^{25} + 12 q^{26} + 8 q^{27} - 3 q^{29} + 12 q^{30} - 6 q^{31} + 9 q^{32} + q^{36} - 15 q^{37} + 6 q^{38} - 10 q^{39} + 3 q^{40} + 9 q^{41} - 8 q^{43} - 3 q^{45} + 18 q^{46} - 6 q^{47} - 10 q^{48} - 6 q^{50} + 6 q^{51} + 7 q^{52} + 3 q^{53} + 12 q^{54} + 12 q^{59} + 6 q^{60} - 2 q^{61} - 6 q^{62} - 2 q^{64} - 3 q^{65} + 3 q^{68} - 12 q^{69} + 6 q^{71} - 3 q^{73} - 15 q^{74} + 4 q^{75} + 6 q^{76} - 24 q^{78} - 4 q^{79} - 22 q^{81} + 18 q^{82} + 9 q^{85} - 24 q^{86} + 6 q^{87} - 12 q^{89} - 6 q^{90} + 12 q^{92} + 12 q^{93} - 12 q^{94} + 6 q^{95} - 18 q^{96} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 0.866025i −2.00000 0.500000 0.866025i −1.50000 0.866025i −3.00000 + 1.73205i 0 1.73205i 1.00000 −3.00000
361.1 1.50000 + 0.866025i −2.00000 0.500000 + 0.866025i −1.50000 + 0.866025i −3.00000 1.73205i 0 1.73205i 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.u.b 2
7.b odd 2 1 637.2.u.c 2
7.c even 3 1 637.2.k.c 2
7.c even 3 1 637.2.q.a 2
7.d odd 6 1 13.2.e.a 2
7.d odd 6 1 637.2.k.a 2
13.e even 6 1 637.2.k.c 2
21.g even 6 1 117.2.q.c 2
28.f even 6 1 208.2.w.b 2
35.i odd 6 1 325.2.n.a 2
35.k even 12 2 325.2.m.a 4
56.j odd 6 1 832.2.w.d 2
56.m even 6 1 832.2.w.a 2
84.j odd 6 1 1872.2.by.d 2
91.k even 6 1 637.2.q.a 2
91.l odd 6 1 13.2.e.a 2
91.m odd 6 1 169.2.b.a 2
91.p odd 6 1 169.2.b.a 2
91.p odd 6 1 637.2.u.c 2
91.s odd 6 1 169.2.e.a 2
91.t odd 6 1 637.2.k.a 2
91.u even 6 1 inner 637.2.u.b 2
91.v odd 6 1 169.2.e.a 2
91.w even 12 2 169.2.a.a 2
91.ba even 12 2 169.2.c.a 4
91.bb even 12 2 169.2.c.a 4
91.bd odd 12 2 8281.2.a.q 2
273.y even 6 1 1521.2.b.a 2
273.bf even 6 1 1521.2.b.a 2
273.br even 6 1 117.2.q.c 2
273.ch odd 12 2 1521.2.a.k 2
364.w even 6 1 208.2.w.b 2
364.bp even 6 1 2704.2.f.b 2
364.br even 6 1 2704.2.f.b 2
364.cg odd 12 2 2704.2.a.o 2
455.bx odd 6 1 325.2.n.a 2
455.cr even 12 2 325.2.m.a 4
455.dn even 12 2 4225.2.a.v 2
728.bw odd 6 1 832.2.w.d 2
728.cv even 6 1 832.2.w.a 2
1092.cy odd 6 1 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 7.d odd 6 1
13.2.e.a 2 91.l odd 6 1
117.2.q.c 2 21.g even 6 1
117.2.q.c 2 273.br even 6 1
169.2.a.a 2 91.w even 12 2
169.2.b.a 2 91.m odd 6 1
169.2.b.a 2 91.p odd 6 1
169.2.c.a 4 91.ba even 12 2
169.2.c.a 4 91.bb even 12 2
169.2.e.a 2 91.s odd 6 1
169.2.e.a 2 91.v odd 6 1
208.2.w.b 2 28.f even 6 1
208.2.w.b 2 364.w even 6 1
325.2.m.a 4 35.k even 12 2
325.2.m.a 4 455.cr even 12 2
325.2.n.a 2 35.i odd 6 1
325.2.n.a 2 455.bx odd 6 1
637.2.k.a 2 7.d odd 6 1
637.2.k.a 2 91.t odd 6 1
637.2.k.c 2 7.c even 3 1
637.2.k.c 2 13.e even 6 1
637.2.q.a 2 7.c even 3 1
637.2.q.a 2 91.k even 6 1
637.2.u.b 2 1.a even 1 1 trivial
637.2.u.b 2 91.u even 6 1 inner
637.2.u.c 2 7.b odd 2 1
637.2.u.c 2 91.p odd 6 1
832.2.w.a 2 56.m even 6 1
832.2.w.a 2 728.cv even 6 1
832.2.w.d 2 56.j odd 6 1
832.2.w.d 2 728.bw odd 6 1
1521.2.a.k 2 273.ch odd 12 2
1521.2.b.a 2 273.y even 6 1
1521.2.b.a 2 273.bf even 6 1
1872.2.by.d 2 84.j odd 6 1
1872.2.by.d 2 1092.cy odd 6 1
2704.2.a.o 2 364.cg odd 12 2
2704.2.f.b 2 364.bp even 6 1
2704.2.f.b 2 364.br even 6 1
4225.2.a.v 2 455.dn even 12 2
8281.2.a.q 2 91.bd odd 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
show more
show less