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Properties

Label 637.2.g.k
Level $637$
Weight $2$
Character orbit 637.g
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
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(263,637)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.g (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.59066497296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \cdots - 1) q^{4} + (2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{5} + ( - \beta_{4} + \beta_1) q^{6} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8}+ \cdots + ( - 6 \beta_{6} - \beta_{3} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} - 5 q^{4} + 7 q^{5} + 5 q^{6} - 12 q^{8} + 14 q^{9} - 22 q^{10} - 2 q^{11} - 12 q^{12} + 4 q^{13} - 3 q^{15} - 19 q^{16} + 4 q^{17} + 3 q^{18} + 2 q^{19} + 2 q^{20} - 5 q^{22} + 2 q^{23}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 7\beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(\beta_{4}\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
−1.11000 + 1.92258i
−0.115680 + 0.200364i
0.355143 − 0.615126i
1.37054 − 2.37385i
−1.11000 − 1.92258i
−0.115680 − 0.200364i
0.355143 + 0.615126i
1.37054 + 2.37385i
−1.11000 + 1.92258i 0.549551 −1.46422 − 2.53610i 2.11000 + 3.65463i −0.610004 + 1.05656i 0 2.06113 −2.69799 −9.36845
263.2 −0.115680 + 0.200364i −3.32225 0.973236 + 1.68569i 1.11568 + 1.93242i 0.384320 − 0.665661i 0 −0.913059 8.03736 −0.516249
263.3 0.355143 − 0.615126i 2.40788 0.747746 + 1.29513i 0.644857 + 1.11692i 0.855143 − 1.48115i 0 2.48280 2.79790 0.916066
263.4 1.37054 − 2.37385i 1.36482 −2.75677 − 4.77486i −0.370541 − 0.641796i 1.87054 − 3.23987i 0 −9.63087 −1.13727 −2.03137
373.1 −1.11000 − 1.92258i 0.549551 −1.46422 + 2.53610i 2.11000 − 3.65463i −0.610004 − 1.05656i 0 2.06113 −2.69799 −9.36845
373.2 −0.115680 − 0.200364i −3.32225 0.973236 − 1.68569i 1.11568 − 1.93242i 0.384320 + 0.665661i 0 −0.913059 8.03736 −0.516249
373.3 0.355143 + 0.615126i 2.40788 0.747746 − 1.29513i 0.644857 − 1.11692i 0.855143 + 1.48115i 0 2.48280 2.79790 0.916066
373.4 1.37054 + 2.37385i 1.36482 −2.75677 + 4.77486i −0.370541 + 0.641796i 1.87054 + 3.23987i 0 −9.63087 −1.13727 −2.03137
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.g.k 8
7.b odd 2 1 637.2.g.j 8
7.c even 3 1 91.2.f.c ✓ 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.f.i 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 637.2.h.h 8
21.h odd 6 1 819.2.o.h 8
28.g odd 6 1 1456.2.s.q 8
91.g even 3 1 inner 637.2.g.k 8
91.g even 3 1 1183.2.a.k 4
91.h even 3 1 91.2.f.c ✓ 8
91.m odd 6 1 637.2.g.j 8
91.m odd 6 1 8281.2.a.bp 4
91.n odd 6 1 637.2.h.i 8
91.p odd 6 1 8281.2.a.bt 4
91.u even 6 1 1183.2.a.l 4
91.v odd 6 1 637.2.f.i 8
91.bd odd 12 2 1183.2.c.g 8
273.s odd 6 1 819.2.o.h 8
364.bi odd 6 1 1456.2.s.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c ✓ 8 7.c even 3 1
91.2.f.c ✓ 8 91.h even 3 1
637.2.f.i 8 7.d odd 6 1
637.2.f.i 8 91.v odd 6 1
637.2.g.j 8 7.b odd 2 1
637.2.g.j 8 91.m odd 6 1
637.2.g.k 8 1.a even 1 1 trivial
637.2.g.k 8 91.g even 3 1 inner
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 13.c even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.n odd 6 1
819.2.o.h 8 21.h odd 6 1
819.2.o.h 8 273.s odd 6 1
1183.2.a.k 4 91.g even 3 1
1183.2.a.l 4 91.u even 6 1
1183.2.c.g 8 91.bd odd 12 2
1456.2.s.q 8 28.g odd 6 1
1456.2.s.q 8 364.bi odd 6 1
8281.2.a.bp 4 91.m odd 6 1
8281.2.a.bt 4 91.p odd 6 1

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 38T_{2}^{4} - 16T_{2}^{3} + 15T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{3} - 9T_{3}^{2} + 16T_{3} - 6 \) Copy content Toggle raw display
\( T_{5}^{8} - 7T_{5}^{7} + 37T_{5}^{6} - 86T_{5}^{5} + 160T_{5}^{4} - 114T_{5}^{3} + 109T_{5}^{2} - 9T_{5} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} - 9 T^{2} + \cdots - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 9 T^{2} + \cdots - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 2809 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 55 T^{2} + 500)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + \cdots + 318096 \) Copy content Toggle raw display
$43$ \( T^{8} + 3 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{8} + 2 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + \cdots + 1929321 \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + \cdots + 498436 \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots - 100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} + \cdots + 11010)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 4129024 \) Copy content Toggle raw display
$73$ \( T^{8} - 8 T^{7} + \cdots + 3139984 \) Copy content Toggle raw display
$79$ \( T^{8} - 26 T^{7} + \cdots + 58982400 \) Copy content Toggle raw display
$83$ \( (T^{4} - 97 T^{2} + \cdots - 426)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + \cdots + 246238864 \) Copy content Toggle raw display
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