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Properties

Label 637.2.f.h
Level $637$
Weight $2$
Character orbit 637.f
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(295,637)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.f (of order \(3\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.100088711424.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 13x^{6} + 130x^{4} - 507x^{2} + 1521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{3}) q^{3} + (\beta_{5} + 2 \beta_{2}) q^{4} + \beta_{3} q^{5} + (2 \beta_{4} + \beta_1) q^{6} + 3 q^{8} + ( - \beta_{5} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{3}) q^{3} + (\beta_{5} + 2 \beta_{2}) q^{4} + \beta_{3} q^{5} + (2 \beta_{4} + \beta_1) q^{6} + 3 q^{8} + ( - \beta_{5} + 3 \beta_{2}) q^{9} + ( - \beta_{4} + \beta_1) q^{10} + ( - 3 \beta_{6} - \beta_{2} - 1) q^{11} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{12} + (\beta_{7} + \beta_1) q^{13} + (\beta_{6} - 6 \beta_{2} - 6) q^{15} + ( - \beta_{6} + \beta_{2} + 1) q^{16} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{17} + (3 \beta_{6} - 3 \beta_{5}) q^{18} - \beta_{7} q^{19} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{20} + (\beta_{5} + 10 \beta_{2}) q^{22} + (2 \beta_{6} - 2 \beta_{2} - 2) q^{23} + (3 \beta_{7} - 3 \beta_{3}) q^{24} + ( - \beta_{6} + \beta_{5} + 1) q^{25} + ( - 2 \beta_{7} + 2 \beta_{4} + \cdots + 2 \beta_1) q^{26}+ \cdots + (11 \beta_{6} - 11 \beta_{5} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 6 q^{4} + 24 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 6 q^{4} + 24 q^{8} - 14 q^{9} + 2 q^{11} - 26 q^{15} + 6 q^{16} - 12 q^{18} - 38 q^{22} - 12 q^{23} + 12 q^{25} + 2 q^{29} + 14 q^{32} - 8 q^{36} - 20 q^{37} - 52 q^{39} + 14 q^{43} + 72 q^{44} + 20 q^{46} + 10 q^{50} + 52 q^{51} + 48 q^{53} + 52 q^{57} + 14 q^{58} + 52 q^{60} - 16 q^{64} + 26 q^{65} - 4 q^{67} - 16 q^{71} - 42 q^{72} - 36 q^{74} + 78 q^{78} - 24 q^{79} + 16 q^{81} - 26 q^{85} - 144 q^{86} + 6 q^{88} - 16 q^{92} + 26 q^{93} - 26 q^{95} - 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 13x^{6} + 130x^{4} - 507x^{2} + 1521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 130\nu^{2} - 507 ) / 390 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 70\nu^{5} - 520\nu^{3} + 3237\nu ) / 1170 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 10\nu^{5} - 130\nu^{3} + 117\nu ) / 390 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 35\nu^{4} - 260\nu^{2} + 1014 ) / 195 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} + 70\nu^{4} - 520\nu^{2} + 819 ) / 390 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} + 140\nu^{5} - 1040\nu^{3} + 2847\nu ) / 1170 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 7\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 7\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{5} + 52\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} - 52\beta_{4} + 13\beta_{3} - 52\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -130\beta_{6} + 130\beta_{5} - 403 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -130\beta_{7} + 260\beta_{3} - 403\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
−2.49541 + 1.44073i
2.49541 − 1.44073i
1.87694 − 1.08365i
−1.87694 + 1.08365i
−2.49541 − 1.44073i
2.49541 + 1.44073i
1.87694 + 1.08365i
−1.87694 − 1.08365i
−1.15139 + 1.99426i −1.08365 + 1.87694i −1.65139 − 2.86029i 2.16731 −2.49541 − 4.32218i 0 3.00000 −0.848612 − 1.46984i −2.49541 + 4.32218i
295.2 −1.15139 + 1.99426i 1.08365 − 1.87694i −1.65139 − 2.86029i −2.16731 2.49541 + 4.32218i 0 3.00000 −0.848612 − 1.46984i 2.49541 − 4.32218i
295.3 0.651388 − 1.12824i −1.44073 + 2.49541i 0.151388 + 0.262211i 2.88145 1.87694 + 3.25096i 0 3.00000 −2.65139 − 4.59234i 1.87694 − 3.25096i
295.4 0.651388 − 1.12824i 1.44073 − 2.49541i 0.151388 + 0.262211i −2.88145 −1.87694 − 3.25096i 0 3.00000 −2.65139 − 4.59234i −1.87694 + 3.25096i
393.1 −1.15139 − 1.99426i −1.08365 − 1.87694i −1.65139 + 2.86029i 2.16731 −2.49541 + 4.32218i 0 3.00000 −0.848612 + 1.46984i −2.49541 − 4.32218i
393.2 −1.15139 − 1.99426i 1.08365 + 1.87694i −1.65139 + 2.86029i −2.16731 2.49541 − 4.32218i 0 3.00000 −0.848612 + 1.46984i 2.49541 + 4.32218i
393.3 0.651388 + 1.12824i −1.44073 − 2.49541i 0.151388 − 0.262211i 2.88145 1.87694 − 3.25096i 0 3.00000 −2.65139 + 4.59234i 1.87694 + 3.25096i
393.4 0.651388 + 1.12824i 1.44073 + 2.49541i 0.151388 − 0.262211i −2.88145 −1.87694 + 3.25096i 0 3.00000 −2.65139 + 4.59234i −1.87694 − 3.25096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n 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.f.h ✓ 8
7.b odd 2 1 inner 637.2.f.h ✓ 8
7.c even 3 1 637.2.g.i 8
7.c even 3 1 637.2.h.j 8
7.d odd 6 1 637.2.g.i 8
7.d odd 6 1 637.2.h.j 8
13.c even 3 1 inner 637.2.f.h ✓ 8
13.c even 3 1 8281.2.a.bu 4
13.e even 6 1 8281.2.a.bo 4
91.g even 3 1 637.2.h.j 8
91.h even 3 1 637.2.g.i 8
91.m odd 6 1 637.2.h.j 8
91.n odd 6 1 inner 637.2.f.h ✓ 8
91.n odd 6 1 8281.2.a.bu 4
91.t odd 6 1 8281.2.a.bo 4
91.v odd 6 1 637.2.g.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h ✓ 8 1.a even 1 1 trivial
637.2.f.h ✓ 8 7.b odd 2 1 inner
637.2.f.h ✓ 8 13.c even 3 1 inner
637.2.f.h ✓ 8 91.n odd 6 1 inner
637.2.g.i 8 7.c even 3 1
637.2.g.i 8 7.d odd 6 1
637.2.g.i 8 91.h even 3 1
637.2.g.i 8 91.v odd 6 1
637.2.h.j 8 7.c even 3 1
637.2.h.j 8 7.d odd 6 1
637.2.h.j 8 91.g even 3 1
637.2.h.j 8 91.m odd 6 1
8281.2.a.bo 4 13.e even 6 1
8281.2.a.bo 4 91.t odd 6 1
8281.2.a.bu 4 13.c even 3 1
8281.2.a.bu 4 91.n odd 6 1

Hecke kernels

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

\( T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{8} + 13T_{3}^{6} + 130T_{3}^{4} + 507T_{3}^{2} + 1521 \) Copy content Toggle raw display
\( T_{5}^{4} - 13T_{5}^{2} + 39 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + 4 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 13 T^{6} + \cdots + 1521 \) Copy content Toggle raw display
$5$ \( (T^{4} - 13 T^{2} + 39)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + 30 T^{2} + \cdots + 841)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 13T^{4} + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 52 T^{6} + \cdots + 1521 \) Copy content Toggle raw display
$19$ \( T^{8} + 13 T^{6} + \cdots + 1521 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + 40 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - T^{3} + 4 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 52 T^{2} + 39)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 156 T^{6} + \cdots + 31539456 \) Copy content Toggle raw display
$43$ \( (T^{4} - 7 T^{3} + \cdots + 4761)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 156 T^{2} + 351)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 23)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 52 T^{6} + \cdots + 123201 \) Copy content Toggle raw display
$61$ \( T^{8} + 52 T^{6} + \cdots + 389376 \) Copy content Toggle raw display
$67$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 52 T^{2} + 624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 52 T^{2} + 351)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 117 T^{6} + \cdots + 9979281 \) Copy content Toggle raw display
$97$ \( T^{8} + 247 T^{6} + \cdots + 127035441 \) Copy content Toggle raw display
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