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Properties

Label 63.18.a.g
Level $63$
Weight $18$
Character orbit 63.a
Self dual yes
Analytic conductor $115.430$
Analytic rank $1$
Dimension $8$
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] = [63,18,Mod(1,63)]
 
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, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

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: yes
Analytic conductor: \(115.429915027\)
Analytic rank: \(1\)
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: \( x^{8} - 210535x^{6} + 13496218908x^{4} - 265411061483008x^{2} + 172877539344498688 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{16}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 79463) q^{4} + ( - \beta_{2} + 212 \beta_1) q^{5} - 5764801 q^{7} + (\beta_{4} + 19 \beta_{2} + 67170 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 79463) q^{4} + ( - \beta_{2} + 212 \beta_1) q^{5} - 5764801 q^{7} + (\beta_{4} + 19 \beta_{2} + 67170 \beta_1) q^{8} + (\beta_{6} - 482 \beta_{3} + 44729993) q^{10} + ( - \beta_{5} - 7 \beta_{4} + \cdots + 151356 \beta_1) q^{11}+ \cdots + 33232930569601 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 635704 q^{4} - 46118408 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 635704 q^{4} - 46118408 q^{7} + 357839944 q^{10} - 2382276064 q^{13} + 29794658080 q^{16} + 5512205552 q^{19} + 255043291240 q^{22} + 1227756615304 q^{25} - 3664707054904 q^{28} - 19056047398416 q^{31} - 63038639885928 q^{34} - 46923380151072 q^{37} - 114863457057888 q^{40} - 23387392458272 q^{43} - 137344901588712 q^{46} + 265863444556808 q^{49} + 28810933077808 q^{52} + 11\!\cdots\!48 q^{55}+ \cdots - 45\!\cdots\!44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 210535x^{6} + 13496218908x^{4} - 265411061483008x^{2} + 172877539344498688 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -221\nu^{7} + 57753387\nu^{5} - 4555341849324\nu^{3} + 101884158493478144\nu ) / 2195100923904 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 210535 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4199\nu^{7} - 1097314353\nu^{5} + 104112302528388\nu^{3} - 3381553942685128448\nu ) / 2195100923904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 71035\nu^{7} - 13597110333\nu^{5} + 742068725093556\nu^{3} - 11132884329041558272\nu ) / 2195100923904 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 140103\nu^{4} - 3579596412\nu^{2} - 17699220800912 ) / 97776 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{6} + 896067\nu^{4} - 39534050892\nu^{2} + 202937707262528 ) / 195552 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 210535 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} + 19\beta_{2} + 329314\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 10\beta_{6} + 110641\beta_{3} + 17332548409 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3536\beta_{5} + 145407\beta_{4} + 3899293\beta_{2} + 30474978542\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 560412\beta_{7} - 1792134\beta_{6} + 11921539611\beta_{3} + 1603914815942059 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 924054192\beta_{5} + 17386447485\beta_{4} + 547896267543\beta_{2} + 3020054378997514\beta_1 ) / 8 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−329.712
−254.343
−190.946
−25.9660
25.9660
190.946
254.343
329.712
−659.425 0 303769. 652008. 0 −5.76480e6 −1.13881e8 0 −4.29950e8
1.2 −508.685 0 127688. −1.37575e6 0 −5.76480e6 1.72114e6 0 6.99821e8
1.3 −381.892 0 14769.5 80664.4 0 −5.76480e6 4.44150e7 0 −3.08051e7
1.4 −51.9319 0 −128375. 1.15816e6 0 −5.76480e6 1.34736e7 0 −6.01456e7
1.5 51.9319 0 −128375. −1.15816e6 0 −5.76480e6 −1.34736e7 0 −6.01456e7
1.6 381.892 0 14769.5 −80664.4 0 −5.76480e6 −4.44150e7 0 −3.08051e7
1.7 508.685 0 127688. 1.37575e6 0 −5.76480e6 −1.72114e6 0 6.99821e8
1.8 659.425 0 303769. −652008. 0 −5.76480e6 1.13881e8 0 −4.29950e8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.18.a.g ✓ 8
3.b odd 2 1 inner 63.18.a.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.18.a.g ✓ 8 1.a even 1 1 trivial
63.18.a.g ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 842140T_{2}^{6} + 215939502528T_{2}^{4} - 16986307934912512T_{2}^{2} + 44256650072191664128 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 5764801)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 79\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 38\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 77\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 57\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 88\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 71\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 44\!\cdots\!92)^{2} \) Copy content Toggle raw display
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