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Properties

Label 637.2.h.e
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
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(165,637)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{6} + \beta_{3} q^{8} + (2 \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{6} + \beta_{3} q^{8} + (2 \beta_{2} - \beta_1) q^{9} + ( - 3 \beta_1 + 3) q^{10} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{13} + (\beta_{2} - 3 \beta_1) q^{15} - 5 q^{16} + ( - \beta_{3} + 6) q^{17} + (\beta_{2} - 6 \beta_1) q^{18} + 2 \beta_1 q^{19} + ( - \beta_{3} + \beta_{2}) q^{20} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - \beta_{3} + 3) q^{23} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{24} + 2 \beta_1 q^{25} + (2 \beta_{2} + 3 \beta_1 - 6) q^{26} - 4 q^{27} + 3 \beta_1 q^{29} + (3 \beta_{2} - 3 \beta_1) q^{30} + ( - 3 \beta_{2} - \beta_1) q^{31} + 3 \beta_{3} q^{32} - 2 \beta_{2} q^{33} + ( - 6 \beta_{3} + 3) q^{34} + (2 \beta_{2} - \beta_1) q^{36} - 7 q^{37} - 2 \beta_{2} q^{38} + (3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 5) q^{39} + (3 \beta_1 - 3) q^{40} + 3 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 5) q^{43}+ \cdots + ( - 5 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{4} + 6 q^{6} - 2 q^{9} + 6 q^{10} - 6 q^{11} + 2 q^{12} - 4 q^{13} - 6 q^{15} - 20 q^{16} + 24 q^{17} - 12 q^{18} + 4 q^{19} + 6 q^{22} + 12 q^{23} - 6 q^{24} + 4 q^{25} - 18 q^{26} - 16 q^{27} + 6 q^{29} - 6 q^{30} - 2 q^{31} + 12 q^{34} - 2 q^{36} - 28 q^{37} - 8 q^{39} - 6 q^{40} - 10 q^{43} - 6 q^{44} - 24 q^{45} + 12 q^{46} + 12 q^{47} - 10 q^{48} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 6 q^{55} + 8 q^{57} - 36 q^{59} - 6 q^{60} - 20 q^{61} + 18 q^{62} + 4 q^{64} + 12 q^{66} + 2 q^{67} + 24 q^{68} + 12 q^{69} - 12 q^{71} + 12 q^{72} + 4 q^{73} + 8 q^{75} + 4 q^{76} - 24 q^{78} - 22 q^{79} - 2 q^{81} - 18 q^{82} - 12 q^{83} + 6 q^{85} + 18 q^{86} + 12 q^{87} - 6 q^{88} - 24 q^{89} - 12 q^{90} + 12 q^{92} + 32 q^{93} + 24 q^{94} - 18 q^{96} - 8 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−1.73205 −0.366025 − 0.633975i 1.00000 −0.866025 − 1.50000i 0.633975 + 1.09808i 0 1.73205 1.23205 − 2.13397i 1.50000 + 2.59808i
165.2 1.73205 1.36603 + 2.36603i 1.00000 0.866025 + 1.50000i 2.36603 + 4.09808i 0 −1.73205 −2.23205 + 3.86603i 1.50000 + 2.59808i
471.1 −1.73205 −0.366025 + 0.633975i 1.00000 −0.866025 + 1.50000i 0.633975 − 1.09808i 0 1.73205 1.23205 + 2.13397i 1.50000 − 2.59808i
471.2 1.73205 1.36603 − 2.36603i 1.00000 0.866025 − 1.50000i 2.36603 − 4.09808i 0 −1.73205 −2.23205 − 3.86603i 1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.h.e 4
7.b odd 2 1 637.2.h.d 4
7.c even 3 1 637.2.f.d 4
7.c even 3 1 637.2.g.d 4
7.d odd 6 1 91.2.f.b ✓ 4
7.d odd 6 1 637.2.g.e 4
13.c even 3 1 637.2.g.d 4
21.g even 6 1 819.2.o.b 4
28.f even 6 1 1456.2.s.o 4
91.g even 3 1 637.2.f.d 4
91.h even 3 1 inner 637.2.h.e 4
91.h even 3 1 8281.2.a.r 2
91.k even 6 1 8281.2.a.t 2
91.l odd 6 1 1183.2.a.e 2
91.m odd 6 1 91.2.f.b ✓ 4
91.n odd 6 1 637.2.g.e 4
91.v odd 6 1 637.2.h.d 4
91.v odd 6 1 1183.2.a.f 2
91.ba even 12 2 1183.2.c.e 4
273.bf even 6 1 819.2.o.b 4
364.br even 6 1 1456.2.s.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b ✓ 4 7.d odd 6 1
91.2.f.b ✓ 4 91.m odd 6 1
637.2.f.d 4 7.c even 3 1
637.2.f.d 4 91.g even 3 1
637.2.g.d 4 7.c even 3 1
637.2.g.d 4 13.c even 3 1
637.2.g.e 4 7.d odd 6 1
637.2.g.e 4 91.n odd 6 1
637.2.h.d 4 7.b odd 2 1
637.2.h.d 4 91.v odd 6 1
637.2.h.e 4 1.a even 1 1 trivial
637.2.h.e 4 91.h even 3 1 inner
819.2.o.b 4 21.g even 6 1
819.2.o.b 4 273.bf even 6 1
1183.2.a.e 2 91.l odd 6 1
1183.2.a.f 2 91.v odd 6 1
1183.2.c.e 4 91.ba even 12 2
1456.2.s.o 4 28.f even 6 1
1456.2.s.o 4 364.br even 6 1
8281.2.a.r 2 91.h even 3 1
8281.2.a.t 2 91.k even 6 1

Hecke kernels

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

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 3T_{5}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T + 33)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$37$ \( (T + 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$59$ \( (T^{2} + 18 T + 78)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + \cdots + 8836 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
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