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Properties

Label 63.9.d.d
Level $63$
Weight $9$
Character orbit 63.d
Analytic conductor $25.665$
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] = [63,9,Mod(55,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.55");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 63.d (of order \(2\), degree \(1\), 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: \(25.6648524339\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 1521 x^{6} + 98406 x^{5} + 5244543 x^{4} + 169934616 x^{3} + 9910518704 x^{2} + \cdots + 4400369870814 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 153) q^{4} - \beta_{3} q^{5} + (\beta_{5} - 4 \beta_{2} + 648) q^{7} + ( - 2 \beta_{4} - 86 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 153) q^{4} - \beta_{3} q^{5} + (\beta_{5} - 4 \beta_{2} + 648) q^{7} + ( - 2 \beta_{4} - 86 \beta_1) q^{8} + (\beta_{7} + 2 \beta_{5} - \beta_{2} + 1) q^{10} + ( - 3 \beta_{4} + 161 \beta_1) q^{11} + (\beta_{7} + 4 \beta_{5} - 2 \beta_{2} + 2) q^{13} + (\beta_{6} + 7 \beta_{4} + \cdots + 14 \beta_1) q^{14}+ \cdots + (42 \beta_{6} + 18130 \beta_{4} + \cdots + 4750109 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1224 q^{4} + 5180 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1224 q^{4} + 5180 q^{7} - 33392 q^{16} - 528952 q^{22} - 777376 q^{25} - 1366848 q^{28} - 576392 q^{37} - 20759672 q^{43} - 6802312 q^{46} - 24294592 q^{49} + 73039232 q^{58} - 45928288 q^{64} + 47907832 q^{67} - 69284040 q^{70} + 291994504 q^{79} - 41008680 q^{85} + 158749648 q^{88} - 139697040 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 1521 x^{6} + 98406 x^{5} + 5244543 x^{4} + 169934616 x^{3} + 9910518704 x^{2} + \cdots + 4400369870814 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 78\!\cdots\!93 \nu^{7} + \cdots + 15\!\cdots\!06 ) / 19\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14375092284 \nu^{7} - 543702823894 \nu^{6} - 15259249006356 \nu^{5} + \cdots - 63\!\cdots\!17 ) / 20\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35\!\cdots\!83 \nu^{7} + \cdots - 32\!\cdots\!28 ) / 64\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!04 \nu^{7} + \cdots - 11\!\cdots\!24 ) / 64\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1416896411522 \nu^{7} - 26565744846669 \nu^{6} + \cdots + 19\!\cdots\!73 ) / 61\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!75 \nu^{7} + \cdots - 22\!\cdots\!76 ) / 64\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3354124793468 \nu^{7} + 24222560977296 \nu^{6} + \cdots - 57\!\cdots\!84 ) / 20\!\cdots\!23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{6} + 41\beta_{3} + 588\beta_{2} + 3528\beta _1 + 1764 ) / 7056 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 28 \beta_{7} - 17 \beta_{6} + 336 \beta_{5} + 168 \beta_{4} + 1853 \beta_{3} - 2100 \beta_{2} + \cdots - 382620 ) / 1008 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12495 \beta_{7} - 1774 \beta_{6} + 59976 \beta_{5} - 22050 \beta_{4} - 33434 \beta_{3} + \cdots - 134182188 ) / 3528 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13328 \beta_{7} + 32113 \beta_{6} + 217056 \beta_{5} - 817824 \beta_{4} - 878509 \beta_{3} + \cdots - 2162411580 ) / 1008 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 56252490 \beta_{7} + 17535403 \beta_{6} - 443822400 \beta_{5} - 122991372 \beta_{4} + \cdots - 117206332236 ) / 7056 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 272908426 \beta_{7} + 62410451 \beta_{6} - 2086973616 \beta_{5} + 145114284 \beta_{4} + \cdots + 749781996600 ) / 504 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 92339739324 \beta_{7} + 27644748385 \beta_{6} - 563967042912 \beta_{5} + 766608349248 \beta_{4} + \cdots + 19\!\cdots\!88 ) / 7056 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
36.4948 + 40.3843i
36.4948 − 40.3843i
−17.1631 − 20.1086i
−17.1631 + 20.1086i
−28.6214 − 20.1086i
−28.6214 + 20.1086i
10.2897 + 40.3843i
10.2897 − 40.3843i
−26.2051 0 430.707 − 546.007i 0 −324.473 + 2378.97i −4578.20 0 14308.2i
55.2 −26.2051 0 430.707 546.007i 0 −324.473 − 2378.97i −4578.20 0 − 14308.2i
55.3 −11.4583 0 −124.707 − 823.086i 0 1619.47 − 1772.60i 4362.26 0 9431.19i
55.4 −11.4583 0 −124.707 823.086i 0 1619.47 + 1772.60i 4362.26 0 − 9431.19i
55.5 11.4583 0 −124.707 − 823.086i 0 1619.47 + 1772.60i −4362.26 0 − 9431.19i
55.6 11.4583 0 −124.707 823.086i 0 1619.47 − 1772.60i −4362.26 0 9431.19i
55.7 26.2051 0 430.707 − 546.007i 0 −324.473 − 2378.97i 4578.20 0 − 14308.2i
55.8 26.2051 0 430.707 546.007i 0 −324.473 + 2378.97i 4578.20 0 14308.2i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.9.d.d ✓ 8
3.b odd 2 1 inner 63.9.d.d ✓ 8
7.b odd 2 1 inner 63.9.d.d ✓ 8
21.c even 2 1 inner 63.9.d.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.9.d.d ✓ 8 1.a even 1 1 trivial
63.9.d.d ✓ 8 3.b odd 2 1 inner
63.9.d.d ✓ 8 7.b odd 2 1 inner
63.9.d.d ✓ 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 818T_{2}^{2} + 90160 \) acting on \(S_{9}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 818 T^{2} + 90160)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 975594 T^{2} + 201969815040)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 33232930569601)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 52183468738240)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 11\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 6433237811648)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 6654673032232)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 17350291961512)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 34\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 61551717656368)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
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