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Properties

Label 63.14.a.h
Level $63$
Weight $14$
Character orbit 63.a
Self dual yes
Analytic conductor $67.555$
Analytic rank $0$
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] = [63,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(67.5554852397\)
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: \( x^{8} - 60255x^{6} + 1064194308x^{4} - 5389786688832x^{2} + 4953486795522048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{16}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 6872) q^{4} + ( - \beta_{2} - 78 \beta_1) q^{5} + 117649 q^{7} + (\beta_{4} + 15 \beta_{2} + 8550 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 6872) q^{4} + ( - \beta_{2} - 78 \beta_1) q^{5} + 117649 q^{7} + (\beta_{4} + 15 \beta_{2} + 8550 \beta_1) q^{8} + ( - \beta_{6} - 250 \beta_{3} - 1172902) q^{10} + (\beta_{5} + 5 \beta_{4} - 46 \beta_{2} + 11815 \beta_1) q^{11} + (\beta_{7} + \beta_{6} + 566 \beta_{3} + 7250573) q^{13} + 117649 \beta_1 q^{14} + ( - 11 \beta_{7} + 16 \beta_{6} + 8949 \beta_{3} + \cdots + 72473489) q^{16}+ \cdots + 13841287201 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 54974 q^{4} + 941192 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 54974 q^{4} + 941192 q^{7} - 9382716 q^{10} + 58003456 q^{13} + 579769970 q^{16} - 3111248 q^{19} + 1424676060 q^{22} + 4617676904 q^{25} + 6467636126 q^{28} + 11832391984 q^{31} - 10802211588 q^{34} + 37394398528 q^{37} - 285048532188 q^{40} - 45810897632 q^{43} + 113905125828 q^{46} + 110730297608 q^{49} + 1071682647868 q^{52} + 512232649488 q^{55} + 556812835488 q^{58} + 1902487553344 q^{61} + 6191199391958 q^{64} + 254203233088 q^{67} - 1103867154684 q^{70} + 1384002998224 q^{73} + 4025422746184 q^{76} + 4201902434080 q^{79} + 22286304331956 q^{82} + 18233112235248 q^{85} + 40686221311740 q^{88} + 6824048594944 q^{91} + 51480064359864 q^{94} + 24376061490256 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 60255x^{6} + 1064194308x^{4} - 5389786688832x^{2} + 4953486795522048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1319\nu^{7} - 68833977\nu^{5} + 963893849436\nu^{3} - 3229282193102784\nu ) / 2370291420672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 15064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6595\nu^{7} + 344169885\nu^{5} - 4029372106956\nu^{3} - 3553871128831296\nu ) / 790097140224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14843\nu^{7} - 924357093\nu^{5} + 16711990167852\nu^{3} - 76773841412829120\nu ) / 1185145710336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{6} - 454557\nu^{4} + 3588836580\nu^{2} - 400264541712 ) / 2449944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{6} - 110487\nu^{4} + 1585860885\nu^{2} - 3676506884127 ) / 306243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 15064 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 15\beta_{2} + 24934\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{7} + 16\beta_{6} + 33525\beta_{3} + 375577489 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7914\beta_{5} + 39165\beta_{4} + 765591\beta_{2} + 706535532\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -454557\beta_{7} + 883896\beta_{6} + 1059107895\beta_{3} + 10641764269815 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -413003862\beta_{5} + 1313107551\beta_{4} + 30788898381\beta_{2} + 21098713806396\beta_1 \) 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
−179.982
−142.307
−80.2305
−34.2499
34.2499
80.2305
142.307
179.982
−179.982 0 24201.7 59573.6 0 117649. −2.88146e6 0 −1.07222e7
1.2 −142.307 0 12059.4 −47993.5 0 117649. −550353. 0 6.82983e6
1.3 −80.2305 0 −1755.06 22334.4 0 117649. 798058. 0 −1.79190e6
1.4 −34.2499 0 −7018.95 −28990.2 0 117649. 520973. 0 992910.
1.5 34.2499 0 −7018.95 28990.2 0 117649. −520973. 0 992910.
1.6 80.2305 0 −1755.06 −22334.4 0 117649. −798058. 0 −1.79190e6
1.7 142.307 0 12059.4 47993.5 0 117649. 550353. 0 6.82983e6
1.8 179.982 0 24201.7 −59573.6 0 117649. 2.88146e6 0 −1.07222e7
\(n\): 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 63.14.a.h ✓ 8
3.b odd 2 1 inner 63.14.a.h ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.14.a.h ✓ 8 1.a even 1 1 trivial
63.14.a.h ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 60255T_{2}^{6} + 1064194308T_{2}^{4} - 5389786688832T_{2}^{2} + 4953486795522048 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 60255 T^{6} + \cdots + 49\!\cdots\!48 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 7191650952 T^{6} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 117649)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 125985204265896 T^{6} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} - 29001728 T^{3} + \cdots - 48\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( (T^{4} + 1555624 T^{3} + \cdots + 14\!\cdots\!60)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5916195992 T^{3} + \cdots + 91\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18697199264 T^{3} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{4} + 22905448816 T^{3} + \cdots + 36\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 80\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 70\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 47\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( (T^{4} - 951243776672 T^{3} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 127101616544 T^{3} + \cdots + 39\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 71\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( (T^{4} - 692001499112 T^{3} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2100951217040 T^{3} + \cdots - 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 63\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( (T^{4} - 12188030745128 T^{3} + \cdots - 30\!\cdots\!88)^{2} \) Copy content Toggle raw display
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