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Properties

Label 63.5.m.f
Level $63$
Weight $5$
Character orbit 63.m
Analytic conductor $6.512$
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] = [63,5,Mod(10,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), 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: \(6.51230767428\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 59x^{6} + 2739x^{4} + 43778x^{2} + 550564 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{6} - 13 \beta_{3} + 2 \beta_{2} - 1) q^{4} + (\beta_{7} - 2 \beta_{4} + 4 \beta_1) q^{5} + ( - 7 \beta_{6} - 7 \beta_{2} + 21) q^{7} + (2 \beta_{7} + \beta_{5} + \cdots + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{6} - 13 \beta_{3} + 2 \beta_{2} - 1) q^{4} + (\beta_{7} - 2 \beta_{4} + 4 \beta_1) q^{5} + ( - 7 \beta_{6} - 7 \beta_{2} + 21) q^{7} + (2 \beta_{7} + \beta_{5} + \cdots + 2 \beta_1) q^{8}+ \cdots + ( - 245 \beta_{7} - 1421 \beta_{4} - 735 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 54 q^{4} + 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 54 q^{4} + 140 q^{7} - 690 q^{10} + 646 q^{16} - 1122 q^{19} - 1852 q^{22} + 3146 q^{25} + 2646 q^{28} - 5640 q^{31} + 2686 q^{37} + 14178 q^{40} + 1180 q^{43} - 2452 q^{46} - 14308 q^{49} - 11880 q^{52} + 19586 q^{58} + 19812 q^{61} - 13132 q^{64} - 8666 q^{67} - 49182 q^{70} + 4494 q^{73} + 10384 q^{79} + 72900 q^{82} - 48432 q^{85} + 14614 q^{88} - 2394 q^{91} - 102948 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 59x^{6} + 2739x^{4} + 43778x^{2} + 550564 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{6} + 1826\nu^{4} + 59345\nu^{2} + 1675171 ) / 145167 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\nu^{6} + 2739\nu^{4} + 161601\nu^{2} + 2582902 ) / 2032338 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59\nu^{7} + 2739\nu^{5} + 161601\nu^{3} + 2582902\nu ) / 2032338 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 87\nu^{7} + 15521\nu^{5} + 577016\nu^{3} + 15325268\nu ) / 1016169 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 857\nu^{6} + 28303\nu^{4} + 992431\nu^{2} - 2451568 ) / 2032338 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 458\nu^{7} + 15521\nu^{5} + 577016\nu^{3} - 2982840\nu ) / 1016169 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{6} + 30\beta_{3} - \beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - \beta_{5} + 34\beta_{4} - 34\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 59\beta_{6} - 969\beta_{3} + 118\beta_{2} - 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 59\beta_{7} + 118\beta_{5} - 1264\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2739\beta_{6} - 2739\beta_{3} - 2739\beta_{2} + 38392 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2739\beta_{7} - 2739\beta_{5} + 49348\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1 - \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
3.19471 − 5.53340i
2.13162 − 3.69208i
−2.13162 + 3.69208i
−3.19471 + 5.53340i
3.19471 + 5.53340i
2.13162 + 3.69208i
−2.13162 − 3.69208i
−3.19471 − 5.53340i
−3.19471 − 5.53340i 0 −12.4124 + 21.4989i 40.9714 − 23.6548i 0 17.5000 − 45.7684i 56.3851 0 −261.784 − 151.141i
10.2 −2.13162 − 3.69208i 0 −1.08762 + 1.88382i −20.9427 + 12.0912i 0 17.5000 + 45.7684i −58.9383 0 89.2837 + 51.5479i
10.3 2.13162 + 3.69208i 0 −1.08762 + 1.88382i 20.9427 − 12.0912i 0 17.5000 + 45.7684i 58.9383 0 89.2837 + 51.5479i
10.4 3.19471 + 5.53340i 0 −12.4124 + 21.4989i −40.9714 + 23.6548i 0 17.5000 − 45.7684i −56.3851 0 −261.784 − 151.141i
19.1 −3.19471 + 5.53340i 0 −12.4124 − 21.4989i 40.9714 + 23.6548i 0 17.5000 + 45.7684i 56.3851 0 −261.784 + 151.141i
19.2 −2.13162 + 3.69208i 0 −1.08762 − 1.88382i −20.9427 − 12.0912i 0 17.5000 − 45.7684i −58.9383 0 89.2837 − 51.5479i
19.3 2.13162 − 3.69208i 0 −1.08762 − 1.88382i 20.9427 + 12.0912i 0 17.5000 − 45.7684i 58.9383 0 89.2837 − 51.5479i
19.4 3.19471 − 5.53340i 0 −12.4124 − 21.4989i −40.9714 − 23.6548i 0 17.5000 + 45.7684i −56.3851 0 −261.784 + 151.141i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.5.m.f ✓ 8
3.b odd 2 1 inner 63.5.m.f ✓ 8
7.c even 3 1 441.5.d.f 8
7.d odd 6 1 inner 63.5.m.f ✓ 8
7.d odd 6 1 441.5.d.f 8
21.g even 6 1 inner 63.5.m.f ✓ 8
21.g even 6 1 441.5.d.f 8
21.h odd 6 1 441.5.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.5.m.f ✓ 8 1.a even 1 1 trivial
63.5.m.f ✓ 8 3.b odd 2 1 inner
63.5.m.f ✓ 8 7.d odd 6 1 inner
63.5.m.f ✓ 8 21.g even 6 1 inner
441.5.d.f 8 7.c even 3 1
441.5.d.f 8 7.d odd 6 1
441.5.d.f 8 21.g even 6 1
441.5.d.f 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 59T_{2}^{6} + 2739T_{2}^{4} + 43778T_{2}^{2} + 550564 \) acting on \(S_{5}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 59 T^{6} + \cdots + 550564 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 1713187796544 \) Copy content Toggle raw display
$7$ \( (T^{2} - 35 T + 2401)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 99687284859904 \) Copy content Toggle raw display
$13$ \( (T^{4} + 35199 T^{2} + 306740196)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{4} + 561 T^{3} + \cdots + 413227584)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2715797 T^{2} + 574768206208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2820 T^{3} + \cdots + 140811812001)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 1275219113536)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 15095164468608)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 295 T - 5473628)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 48\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 66867123181824)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 6214171494976)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 94679506404448)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 514104290081424)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 23\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
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