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Properties

Label 384.8.d.d
Level $384$
Weight $8$
Character orbit 384.d
Analytic conductor $119.956$
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] = [384,8,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(119.955849786\)
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: \( x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + (\beta_{3} + 360) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + (\beta_{3} + 360) q^{7} - 729 q^{9} + (\beta_{7} - 3 \beta_{6} + \cdots - 36 \beta_1) q^{11}+ \cdots + ( - 729 \beta_{7} + 2187 \beta_{6} + \cdots + 26244 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2880 q^{7} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 131\nu^{7} + 58684\nu^{5} + 6584732\nu^{3} + 5063679\nu ) / 1626480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\nu^{6} + 27156\nu^{4} + 2829796\nu^{2} + 2486853 ) / 9036 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{6} + 24308\nu^{4} + 2496836\nu^{2} - 8497827 ) / 9036 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 553\nu^{6} + 249076\nu^{4} + 28104132\nu^{2} + 20876013 ) / 4518 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -713\nu^{7} - 320092\nu^{5} - 36039416\nu^{3} - 40729437\nu ) / 101655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7231\nu^{7} + 3248924\nu^{5} + 366814612\nu^{3} + 540861579\nu ) / 101655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9022\nu^{7} + 4049168\nu^{5} + 455154004\nu^{3} + 338539158\nu ) / 101655 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - 25\beta_{5} - 192\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 73\beta_{3} + 47\beta_{2} - 86208 ) / 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 71\beta_{7} + 143\beta_{6} + 2783\beta_{5} + 37824\beta_1 ) / 384 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{4} + 16303\beta_{3} - 13769\beta_{2} + 19264704 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -22159\beta_{7} - 63991\beta_{6} - 1252807\beta_{5} - 28166592\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3823\beta_{4} - 454133\beta_{3} + 476635\beta_{2} - 540598368 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2827561\beta_{7} + 14328841\beta_{6} + 282409921\beta_{5} + 8832273600\beta_1 ) / 768 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
15.2503i
1.01122i
− 0.596953i
− 14.6646i
14.6646i
0.596953i
− 1.01122i
− 15.2503i
0 − 27.0000i 0 − 522.419i 0 1309.01 0 −729.000 0
193.2 0 − 27.0000i 0 − 69.8856i 0 −860.192 0 −729.000 0
193.3 0 − 27.0000i 0 135.999i 0 −678.568 0 −729.000 0
193.4 0 − 27.0000i 0 344.306i 0 1669.75 0 −729.000 0
193.5 0 27.0000i 0 − 344.306i 0 1669.75 0 −729.000 0
193.6 0 27.0000i 0 − 135.999i 0 −678.568 0 −729.000 0
193.7 0 27.0000i 0 69.8856i 0 −860.192 0 −729.000 0
193.8 0 27.0000i 0 522.419i 0 1309.01 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.d.d yes 8
4.b odd 2 1 384.8.d.c ✓ 8
8.b even 2 1 inner 384.8.d.d yes 8
8.d odd 2 1 384.8.d.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.c ✓ 8 4.b odd 2 1
384.8.d.c ✓ 8 8.d odd 2 1
384.8.d.d yes 8 1.a even 1 1 trivial
384.8.d.d yes 8 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{8} + 414848T_{5}^{6} + 41596678144T_{5}^{4} + 791786461593600T_{5}^{2} + 2922622746624000000 \) Copy content Toggle raw display
\( T_{7}^{4} - 1440T_{7}^{3} - 1814176T_{7}^{2} + 1624601088T_{7} + 1275801663744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 1275801663744)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 58\!\cdots\!48)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 45\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 32\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots - 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 15\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 79\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 69\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
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