↑

Properties

Label 448.9.d.a
Level $448$
Weight $9$
Character orbit 448.d
Analytic conductor $182.506$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,9,Mod(127,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 448.d (of order \(2\), degree \(1\), not 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: \(182.505617307\)
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: \( x^{8} - 2x^{7} - 664x^{6} - 1416x^{5} + 127446x^{4} + 694020x^{3} - 3244793x^{2} - 12838604x + 56693708 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 7^{9} \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} - 116) q^{5} + (\beta_{6} + 4 \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + \cdots + 1344) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{2} - 116) q^{5} + (\beta_{6} + 4 \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + \cdots + 1344) q^{9}+ \cdots + (25251 \beta_{7} - 1755 \beta_{6} + \cdots + 301501 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 924 q^{5} + 10768 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 924 q^{5} + 10768 q^{9} - 95956 q^{13} - 10920 q^{17} + 163268 q^{21} + 1336752 q^{25} + 232680 q^{29} + 3687488 q^{33} - 1427608 q^{37} - 6545784 q^{41} + 13336372 q^{45} - 6588344 q^{49} + 5056080 q^{53} + 4343416 q^{57} - 8276268 q^{61} - 9466440 q^{65} - 76490288 q^{69} + 31954496 q^{73} - 4898040 q^{77} - 143078848 q^{81} - 121576600 q^{85} - 114663696 q^{89} - 275408784 q^{93} - 33074888 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 664x^{6} - 1416x^{5} + 127446x^{4} + 694020x^{3} - 3244793x^{2} - 12838604x + 56693708 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 271230459 \nu^{7} + 3028409627 \nu^{6} + 152401177854 \nu^{5} - 886529379095 \nu^{4} + \cdots - 16\!\cdots\!02 ) / 22406688624185 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 947201981 \nu^{7} + 28843596213 \nu^{6} + 221414224866 \nu^{5} - 11781925496015 \nu^{4} + \cdots + 16\!\cdots\!32 ) / 67220065872555 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2270823673 \nu^{7} + 9878206239 \nu^{6} + 1682263720992 \nu^{5} - 242206349005 \nu^{4} + \cdots + 12\!\cdots\!34 ) / 67220065872555 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6279249662 \nu^{7} - 30784209456 \nu^{6} - 4054343074278 \nu^{5} + 1646251084940 \nu^{4} + \cdots - 67\!\cdots\!41 ) / 67220065872555 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3715597696 \nu^{7} + 34654200278 \nu^{6} - 2876403214230 \nu^{5} - 29173717728650 \nu^{4} + \cdots - 90\!\cdots\!40 ) / 22406688624185 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3996179371 \nu^{7} + 21699632193 \nu^{6} + 2397033957156 \nu^{5} - 1865971167895 \nu^{4} + \cdots + 15\!\cdots\!57 ) / 22406688624185 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14226570440 \nu^{7} + 156484311014 \nu^{6} + 8642546491694 \nu^{5} + \cdots - 19\!\cdots\!47 ) / 67220065872555 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 3\beta_{4} - 2\beta_{3} + \beta_{2} - 4\beta _1 + 172 ) / 686 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -18\beta_{6} - 5\beta_{4} - 85\beta_{3} - 129\beta_{2} + 614\beta _1 + 228545 ) / 1372 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 441 \beta_{7} - 2189 \beta_{6} - 588 \beta_{5} - 2815 \beta_{4} - 3755 \beta_{3} + 2049 \beta_{2} + \cdots + 2827173 ) / 2744 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3087 \beta_{7} - 17067 \beta_{6} - 2744 \beta_{5} - 9299 \beta_{4} - 33721 \beta_{3} + \cdots + 67603183 ) / 1372 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 118335 \beta_{7} - 608513 \beta_{6} - 166355 \beta_{5} - 432084 \beta_{4} - 700391 \beta_{3} + \cdots + 758383372 ) / 1372 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1854405 \beta_{7} - 22269831 \beta_{6} - 4342870 \beta_{5} - 9629461 \beta_{4} + \cdots + 44761180819 ) / 2744 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6368481 \beta_{7} - 44085211 \beta_{6} - 11411953 \beta_{5} - 20276325 \beta_{4} - 36642470 \beta_{3} + \cdots + 45973109323 ) / 196 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−14.8402 − 1.32288i
19.7707 + 1.32288i
−7.16924 − 1.32288i
3.23868 − 1.32288i
3.23868 + 1.32288i
−7.16924 + 1.32288i
19.7707 − 1.32288i
−14.8402 + 1.32288i
0 − 102.339i 0 −1064.06 0 907.493i 0 −3912.19 0
127.2 0 − 80.8052i 0 −354.302 0 − 907.493i 0 31.5265 0
127.3 0 − 61.7478i 0 986.021 0 907.493i 0 2748.21 0
127.4 0 − 6.67430i 0 −29.6556 0 907.493i 0 6516.45 0
127.5 0 6.67430i 0 −29.6556 0 − 907.493i 0 6516.45 0
127.6 0 61.7478i 0 986.021 0 − 907.493i 0 2748.21 0
127.7 0 80.8052i 0 −354.302 0 907.493i 0 31.5265 0
127.8 0 102.339i 0 −1064.06 0 − 907.493i 0 −3912.19 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.9.d.a 8
4.b odd 2 1 inner 448.9.d.a 8
8.b even 2 1 112.9.d.a ✓ 8
8.d odd 2 1 112.9.d.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.9.d.a ✓ 8 8.b even 2 1
112.9.d.a ✓ 8 8.d odd 2 1
448.9.d.a 8 1.a even 1 1 trivial
448.9.d.a 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20860T_{3}^{6} + 134139264T_{3}^{4} + 266669581248T_{3}^{2} + 11614791170304 \) acting on \(S_{9}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 11614791170304 \) Copy content Toggle raw display
$5$ \( (T^{4} + 462 T^{3} + \cdots - 11023865600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 823543)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 51\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 91\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 60\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 84\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 21\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 51\!\cdots\!64)^{2} \) Copy content Toggle raw display
show more
show less