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Properties

Label 648.3.e.d
Level $648$
Weight $3$
Character orbit 648.e
Analytic conductor $17.657$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,3,Mod(161,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.e (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: \(17.6567211305\)
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: \( x^{8} - 26x^{6} + 367x^{4} - 468x^{3} - 1686x^{2} + 2340x + 5238 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{5} - 2) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} + 5) q^{13} + (\beta_{7} + \beta_{6} + \cdots + 5 \beta_1) q^{17}+ \cdots + ( - 2 \beta_{5} - 3 \beta_{4} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7} + 44 q^{13} + 28 q^{19} - 200 q^{25} + 160 q^{31} + 108 q^{37} - 364 q^{43} + 232 q^{49} + 92 q^{55} - 340 q^{61} + 172 q^{67} + 128 q^{73} - 436 q^{79} + 268 q^{85} + 372 q^{91} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 26x^{6} + 367x^{4} - 468x^{3} - 1686x^{2} + 2340x + 5238 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2056 \nu^{7} + 83 \nu^{6} + 37295 \nu^{5} - 47614 \nu^{4} - 474403 \nu^{3} + 1590227 \nu^{2} + \cdots - 4130760 ) / 2430741 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4466 \nu^{7} - 14820 \nu^{6} + 39544 \nu^{5} + 398307 \nu^{4} + 257980 \nu^{3} + \cdots + 24248601 ) / 5097423 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2896309 \nu^{7} - 22648962 \nu^{6} - 83221238 \nu^{5} + 472673118 \nu^{4} + 1166216569 \nu^{3} + \cdots + 27248741811 ) / 2258158389 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5099290 \nu^{7} + 18711204 \nu^{6} - 123770000 \nu^{5} - 359838441 \nu^{4} + 1362307996 \nu^{3} + \cdots + 11137464774 ) / 2258158389 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5217604 \nu^{7} - 8413611 \nu^{6} + 139103174 \nu^{5} + 171383226 \nu^{4} + \cdots - 12044827290 ) / 2258158389 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7477226 \nu^{7} + 2186425 \nu^{6} - 155271280 \nu^{5} + 8777806 \nu^{4} + 2086083770 \nu^{3} + \cdots + 7650902667 ) / 2258158389 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7776961 \nu^{7} + 36211915 \nu^{6} + 270842435 \nu^{5} - 568869662 \nu^{4} + \cdots - 29859888687 ) / 2258158389 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{2} + 4\beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 3\beta_{4} - 6\beta_{3} + \beta_{2} - 10\beta _1 + 39 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} + 48\beta_{6} + 8\beta_{5} - 2\beta_{4} + 9\beta_{3} + 20\beta_{2} + 117\beta _1 + 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -32\beta_{7} - 80\beta_{6} + 24\beta_{5} - 3\beta_{4} - 96\beta_{3} + 83\beta_{2} - 488\beta _1 - 75 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 195\beta_{7} + 532\beta_{6} - 142\beta_{5} - 302\beta_{4} + 195\beta_{3} - 84\beta_{2} + 1253\beta _1 + 1684 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -92\beta_{7} - 700\beta_{6} + 1206\beta_{5} + 1341\beta_{4} - 744\beta_{3} + 1977\beta_{2} - 5256\beta _1 - 8385 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 651 \beta_{7} + 758 \beta_{6} - 4570 \beta_{5} - 7034 \beta_{4} - 987 \beta_{3} - 7028 \beta_{2} + \cdots + 49312 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.58713 − 0.517638i
2.21327 + 1.93185i
−3.94532 − 1.93185i
3.31918 + 0.517638i
3.31918 − 0.517638i
−3.94532 + 1.93185i
2.21327 − 1.93185i
−1.58713 + 0.517638i
0 0 0 − 8.64793i 0 10.9745 0 0 0
161.2 0 0 0 − 7.73217i 0 −6.27041 0 0 0
161.3 0 0 0 − 7.35323i 0 1.53836 0 0 0
161.4 0 0 0 − 3.37001i 0 −12.2424 0 0 0
161.5 0 0 0 3.37001i 0 −12.2424 0 0 0
161.6 0 0 0 7.35323i 0 1.53836 0 0 0
161.7 0 0 0 7.73217i 0 −6.27041 0 0 0
161.8 0 0 0 8.64793i 0 10.9745 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

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 648.3.e.d ✓ 8
3.b odd 2 1 inner 648.3.e.d ✓ 8
4.b odd 2 1 1296.3.e.j 8
9.c even 3 2 648.3.m.g 16
9.d odd 6 2 648.3.m.g 16
12.b even 2 1 1296.3.e.j 8
36.f odd 6 2 1296.3.q.q 16
36.h even 6 2 1296.3.q.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.3.e.d ✓ 8 1.a even 1 1 trivial
648.3.e.d ✓ 8 3.b odd 2 1 inner
648.3.m.g 16 9.c even 3 2
648.3.m.g 16 9.d odd 6 2
1296.3.e.j 8 4.b odd 2 1
1296.3.e.j 8 12.b even 2 1
1296.3.q.q 16 36.f odd 6 2
1296.3.q.q 16 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 200T_{5}^{6} + 13890T_{5}^{4} + 375176T_{5}^{2} + 2745649 \) acting on \(S_{3}^{\mathrm{new}}(648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 200 T^{6} + \cdots + 2745649 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 620 T^{6} + \cdots + 186267904 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} + \cdots - 42383)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 752 T^{6} + \cdots + 83375161 \) Copy content Toggle raw display
$19$ \( (T^{4} - 14 T^{3} + \cdots + 29392)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2361182464 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10755141849 \) Copy content Toggle raw display
$31$ \( (T^{4} - 80 T^{3} + \cdots - 1188608)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 54 T^{3} + \cdots + 6777)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 9680198544 \) Copy content Toggle raw display
$43$ \( (T^{4} + 182 T^{3} + \cdots + 547408)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1556545683456 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 13082949425296 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 206121381597184 \) Copy content Toggle raw display
$61$ \( (T^{4} + 170 T^{3} + \cdots - 56566571)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 86 T^{3} + \cdots + 15037072)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 47188546449664 \) Copy content Toggle raw display
$73$ \( (T^{4} - 64 T^{3} + \cdots + 7840033)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 218 T^{3} + \cdots - 134195696)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 5587436978176 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22188783681 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 22696192)^{2} \) Copy content Toggle raw display
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