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Properties

Label 648.3.e.c
Level $648$
Weight $3$
Character orbit 648.e
Analytic conductor $17.657$
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] = [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: 8.0.19269881856.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{5} + (\beta_{5} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{5} + (\beta_{5} - 2) q^{7} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{11}+ \cdots + ( - 4 \beta_{5} + 6 \beta_{2} + 61) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{7} - 28 q^{13} + 4 q^{19} - 20 q^{25} + 100 q^{31} + 120 q^{37} + 56 q^{43} - 188 q^{49} - 244 q^{55} - 28 q^{61} + 40 q^{67} - 76 q^{73} - 52 q^{79} + 448 q^{85} + 108 q^{91} + 472 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 203\nu^{7} - 428\nu^{6} + 2917\nu^{5} - 664\nu^{4} + 26635\nu^{3} - 26042\nu^{2} + 72336\nu + 10668 ) / 17700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -429\nu^{7} + 454\nu^{6} - 5031\nu^{5} - 7098\nu^{4} - 39605\nu^{3} - 27144\nu^{2} - 16848\nu - 145224 ) / 35400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{7} + 38\nu^{6} - 207\nu^{5} + 194\nu^{4} - 1585\nu^{3} + 2532\nu^{2} - 2856\nu + 72 ) / 900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 561\nu^{7} + 314\nu^{6} + 6579\nu^{5} + 9282\nu^{4} + 93545\nu^{3} + 35496\nu^{2} + 22032\nu - 245784 ) / 35400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2233 \nu^{7} + 1758 \nu^{6} - 26187 \nu^{5} - 36946 \nu^{4} - 269385 \nu^{3} - 141288 \nu^{2} + \cdots - 736848 ) / 106200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -677\nu^{7} + 1827\nu^{6} - 10353\nu^{5} + 6901\nu^{4} - 82215\nu^{3} + 132153\nu^{2} - 156924\nu + 738 ) / 26550 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -773\nu^{7} + 1778\nu^{6} - 12927\nu^{5} + 5554\nu^{4} - 112165\nu^{3} + 85932\nu^{2} - 316776\nu - 53928 ) / 21240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{2} + 3\beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{6} - 2\beta_{5} - \beta_{4} - 9\beta_{3} + \beta_{2} + 3\beta _1 - 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{5} - 2\beta_{4} + 13\beta_{2} - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{7} - 79\beta_{6} - 28\beta_{5} - 15\beta_{4} + 123\beta_{3} + 25\beta_{2} - 63\beta _1 - 196 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -90\beta_{7} - 205\beta_{6} + 119\beta_{5} + 40\beta_{4} + 270\beta_{3} - 169\beta_{2} - 417\beta _1 + 417 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 414\beta_{5} + 209\beta_{4} - 445\beta_{2} + 2498 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1254 \beta_{7} + 3229 \beta_{6} + 1693 \beta_{5} + 654 \beta_{4} - 4632 \beta_{3} - 2293 \beta_{2} + \cdots + 6955 ) / 6 \) 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
−1.41950 − 2.45865i
0.831167 − 1.43962i
1.91950 − 3.32468i
−0.331167 − 0.573598i
−0.331167 + 0.573598i
1.91950 + 3.32468i
0.831167 + 1.43962i
−1.41950 + 2.45865i
0 0 0 − 9.47779i 0 −2.11341 0 0 0
161.2 0 0 0 − 3.97562i 0 3.60938 0 0 0
161.3 0 0 0 − 2.08888i 0 1.56290 0 0 0
161.4 0 0 0 − 0.0508200i 0 −9.05887 0 0 0
161.5 0 0 0 0.0508200i 0 −9.05887 0 0 0
161.6 0 0 0 2.08888i 0 1.56290 0 0 0
161.7 0 0 0 3.97562i 0 3.60938 0 0 0
161.8 0 0 0 9.47779i 0 −2.11341 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.c 8
3.b odd 2 1 inner 648.3.e.c 8
4.b odd 2 1 1296.3.e.i 8
9.c even 3 1 72.3.m.b ✓ 8
9.c even 3 1 216.3.m.b 8
9.d odd 6 1 72.3.m.b ✓ 8
9.d odd 6 1 216.3.m.b 8
12.b even 2 1 1296.3.e.i 8
36.f odd 6 1 144.3.q.e 8
36.f odd 6 1 432.3.q.e 8
36.h even 6 1 144.3.q.e 8
36.h even 6 1 432.3.q.e 8
72.j odd 6 1 576.3.q.i 8
72.j odd 6 1 1728.3.q.j 8
72.l even 6 1 576.3.q.j 8
72.l even 6 1 1728.3.q.i 8
72.n even 6 1 576.3.q.i 8
72.n even 6 1 1728.3.q.j 8
72.p odd 6 1 576.3.q.j 8
72.p odd 6 1 1728.3.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b ✓ 8 9.c even 3 1
72.3.m.b ✓ 8 9.d odd 6 1
144.3.q.e 8 36.f odd 6 1
144.3.q.e 8 36.h even 6 1
216.3.m.b 8 9.c even 3 1
216.3.m.b 8 9.d odd 6 1
432.3.q.e 8 36.f odd 6 1
432.3.q.e 8 36.h even 6 1
576.3.q.i 8 72.j odd 6 1
576.3.q.i 8 72.n even 6 1
576.3.q.j 8 72.l even 6 1
576.3.q.j 8 72.p odd 6 1
648.3.e.c 8 1.a even 1 1 trivial
648.3.e.c 8 3.b odd 2 1 inner
1296.3.e.i 8 4.b odd 2 1
1296.3.e.i 8 12.b even 2 1
1728.3.q.i 8 72.l even 6 1
1728.3.q.i 8 72.p odd 6 1
1728.3.q.j 8 72.j odd 6 1
1728.3.q.j 8 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 110T_{5}^{6} + 1881T_{5}^{4} + 6200T_{5}^{2} + 16 \) 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
$5$ \( T^{8} + 110 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + \cdots + 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 608 T^{6} + \cdots + 105616729 \) Copy content Toggle raw display
$13$ \( (T^{4} + 14 T^{3} + \cdots - 1616)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 7020428944 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} + \cdots + 226348)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11198718976 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 106450807824 \) Copy content Toggle raw display
$31$ \( (T^{4} - 50 T^{3} + \cdots + 390784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 60 T^{3} + \cdots + 206496)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1919025613521 \) Copy content Toggle raw display
$43$ \( (T^{4} - 28 T^{3} + \cdots - 1163069)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 4615347568896 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 78435844096 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 127589696809 \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots - 365504)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + \cdots + 131875)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 114698616545536 \) Copy content Toggle raw display
$73$ \( (T^{4} + 38 T^{3} + \cdots + 2961976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 26 T^{3} + \cdots + 10174924)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1085363908864 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 309931236458496 \) Copy content Toggle raw display
$97$ \( (T^{4} - 236 T^{3} + \cdots + 1853377)^{2} \) Copy content Toggle raw display
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