Properties

Label 637.2.u.a
Level $637$
Weight $2$
Character orbit 637.u
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(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 91)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{2} + q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} + 1) q^{5} + (\zeta_{6} - 2) q^{6} + ( - 2 \zeta_{6} + 1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{2} + q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} + 1) q^{5} + (\zeta_{6} - 2) q^{6} + ( - 2 \zeta_{6} + 1) q^{8} - 2 q^{9} - 3 q^{10} + ( - 6 \zeta_{6} + 3) q^{11} + ( - \zeta_{6} + 1) q^{12} + ( - 4 \zeta_{6} + 3) q^{13} + (\zeta_{6} + 1) q^{15} + 5 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 2 \zeta_{6} + 4) q^{18} + ( - 2 \zeta_{6} + 1) q^{19} + ( - \zeta_{6} + 2) q^{20} + 9 \zeta_{6} q^{22} + ( - 2 \zeta_{6} + 1) q^{24} - 2 \zeta_{6} q^{25} + (7 \zeta_{6} - 2) q^{26} - 5 q^{27} + (3 \zeta_{6} - 3) q^{29} - 3 q^{30} + (\zeta_{6} - 2) q^{31} + ( - 3 \zeta_{6} - 3) q^{32} + ( - 6 \zeta_{6} + 3) q^{33} + (12 \zeta_{6} - 6) q^{34} + (2 \zeta_{6} - 2) q^{36} + 3 \zeta_{6} q^{38} + ( - 4 \zeta_{6} + 3) q^{39} + ( - 3 \zeta_{6} + 3) q^{40} + (3 \zeta_{6} + 3) q^{41} - 11 \zeta_{6} q^{43} + ( - 3 \zeta_{6} - 3) q^{44} + ( - 2 \zeta_{6} - 2) q^{45} + (5 \zeta_{6} + 5) q^{47} + 5 \zeta_{6} q^{48} + (2 \zeta_{6} + 2) q^{50} + ( - 6 \zeta_{6} + 6) q^{51} + ( - 3 \zeta_{6} - 1) q^{52} + 9 \zeta_{6} q^{53} + ( - 5 \zeta_{6} + 10) q^{54} + ( - 9 \zeta_{6} + 9) q^{55} + ( - 2 \zeta_{6} + 1) q^{57} + ( - 6 \zeta_{6} + 3) q^{58} + (2 \zeta_{6} + 2) q^{59} + ( - \zeta_{6} + 2) q^{60} - 7 q^{61} + ( - 3 \zeta_{6} + 3) q^{62} - q^{64} + ( - 5 \zeta_{6} + 7) q^{65} + 9 \zeta_{6} q^{66} + (10 \zeta_{6} - 5) q^{67} - 6 \zeta_{6} q^{68} + ( - \zeta_{6} + 2) q^{71} + (4 \zeta_{6} - 2) q^{72} + ( - 5 \zeta_{6} + 10) q^{73} - 2 \zeta_{6} q^{75} + ( - \zeta_{6} - 1) q^{76} + (7 \zeta_{6} - 2) q^{78} + ( - 5 \zeta_{6} + 5) q^{79} + (10 \zeta_{6} - 5) q^{80} + q^{81} - 9 q^{82} + (4 \zeta_{6} - 2) q^{83} + ( - 6 \zeta_{6} + 12) q^{85} + (11 \zeta_{6} + 11) q^{86} + (3 \zeta_{6} - 3) q^{87} - 9 q^{88} + ( - 4 \zeta_{6} + 8) q^{89} + 6 q^{90} + (\zeta_{6} - 2) q^{93} - 15 q^{94} + ( - 3 \zeta_{6} + 3) q^{95} + ( - 3 \zeta_{6} - 3) q^{96} + ( - 3 \zeta_{6} + 6) q^{97} + (12 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 3 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 3 q^{6} - 4 q^{9} - 6 q^{10} + q^{12} + 2 q^{13} + 3 q^{15} + 5 q^{16} + 6 q^{17} + 6 q^{18} + 3 q^{20} + 9 q^{22} - 2 q^{25} + 3 q^{26} - 10 q^{27} - 3 q^{29} - 6 q^{30} - 3 q^{31} - 9 q^{32} - 2 q^{36} + 3 q^{38} + 2 q^{39} + 3 q^{40} + 9 q^{41} - 11 q^{43} - 9 q^{44} - 6 q^{45} + 15 q^{47} + 5 q^{48} + 6 q^{50} + 6 q^{51} - 5 q^{52} + 9 q^{53} + 15 q^{54} + 9 q^{55} + 6 q^{59} + 3 q^{60} - 14 q^{61} + 3 q^{62} - 2 q^{64} + 9 q^{65} + 9 q^{66} - 6 q^{68} + 3 q^{71} + 15 q^{73} - 2 q^{75} - 3 q^{76} + 3 q^{78} + 5 q^{79} + 2 q^{81} - 18 q^{82} + 18 q^{85} + 33 q^{86} - 3 q^{87} - 18 q^{88} + 12 q^{89} + 12 q^{90} - 3 q^{93} - 30 q^{94} + 3 q^{95} - 9 q^{96} + 9 q^{97}+O(q^{100}) \) 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 1.00000 0.500000 0.866025i 1.50000 + 0.866025i −1.50000 + 0.866025i 0 1.73205i −2.00000 −3.00000
361.1 −1.50000 0.866025i 1.00000 0.500000 + 0.866025i 1.50000 0.866025i −1.50000 0.866025i 0 1.73205i −2.00000 −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.a 2
7.b odd 2 1 91.2.u.a yes 2
7.c even 3 1 637.2.k.b 2
7.c even 3 1 637.2.q.b 2
7.d odd 6 1 91.2.k.a 2
7.d odd 6 1 637.2.q.c 2
13.e even 6 1 637.2.k.b 2
21.c even 2 1 819.2.do.c 2
21.g even 6 1 819.2.bm.a 2
91.k even 6 1 637.2.q.b 2
91.l odd 6 1 637.2.q.c 2
91.p odd 6 1 91.2.u.a yes 2
91.t odd 6 1 91.2.k.a 2
91.u even 6 1 inner 637.2.u.a 2
91.w even 12 2 8281.2.a.s 2
91.ba even 12 2 1183.2.e.e 4
91.bc even 12 2 1183.2.e.e 4
91.bd odd 12 2 8281.2.a.w 2
273.u even 6 1 819.2.bm.a 2
273.y even 6 1 819.2.do.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 7.d odd 6 1
91.2.k.a 2 91.t odd 6 1
91.2.u.a yes 2 7.b odd 2 1
91.2.u.a yes 2 91.p odd 6 1
637.2.k.b 2 7.c even 3 1
637.2.k.b 2 13.e even 6 1
637.2.q.b 2 7.c even 3 1
637.2.q.b 2 91.k even 6 1
637.2.q.c 2 7.d odd 6 1
637.2.q.c 2 91.l odd 6 1
637.2.u.a 2 1.a even 1 1 trivial
637.2.u.a 2 91.u even 6 1 inner
819.2.bm.a 2 21.g even 6 1
819.2.bm.a 2 273.u even 6 1
819.2.do.c 2 21.c even 2 1
819.2.do.c 2 273.y even 6 1
1183.2.e.e 4 91.ba even 12 2
1183.2.e.e 4 91.bc even 12 2
8281.2.a.s 2 91.w even 12 2
8281.2.a.w 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
\( T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 27 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 3 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$73$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
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