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Properties

Label 8064.2.j.i
Level $8064$
Weight $2$
Character orbit 8064.j
Analytic conductor $64.391$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8064,2,Mod(575,8064)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8064, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8064.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8064 = 2^{7} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8064.j (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: \(64.3913641900\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{26} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{5} - \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{5} - \beta_{5} q^{7} + ( - \beta_{12} - \beta_{7}) q^{11} - \beta_{11} q^{13} - \beta_{14} q^{17} + ( - \beta_{8} - \beta_1) q^{19} + (\beta_{15} - \beta_{3}) q^{23} + q^{25} - \beta_{6} q^{29} + (\beta_{9} - 2 \beta_{5}) q^{31} - \beta_{12} q^{35} + (\beta_{14} + 2 \beta_{2}) q^{41} - 4 \beta_{3} q^{47} - q^{49} + (2 \beta_{13} - \beta_{6}) q^{53} + ( - \beta_{9} - 6 \beta_{5}) q^{55} - 4 \beta_{12} q^{59} + ( - \beta_{11} - \beta_{4}) q^{61} + 2 \beta_{14} q^{65} + ( - 3 \beta_{8} - \beta_1) q^{67} + ( - \beta_{15} + \beta_{3}) q^{71} + ( - \beta_{10} + 2) q^{73} + ( - \beta_{13} + \beta_{6}) q^{77} + ( - \beta_{9} - 2 \beta_{5}) q^{79} + ( - 2 \beta_{12} + 2 \beta_{7}) q^{83} + 3 \beta_{11} q^{85} + ( - \beta_{14} + 2 \beta_{2}) q^{89} + \beta_1 q^{91} + ( - 2 \beta_{15} - 6 \beta_{3}) q^{95} + (\beta_{10} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} - 16 q^{49} + 32 q^{73} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{12} + 161 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} - 305\nu^{3} + 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} + 305\nu^{3} + 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{12} - 48\nu^{8} + 336\nu^{4} - 25 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} + 341\nu^{3} + 133\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} - 341\nu^{3} + 133\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -9\nu^{14} + 64\nu^{10} - 440\nu^{6} + 127\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{12} + 28\nu^{8} - 188\nu^{4} + 14 ) / 3 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5\nu^{14} + 36\nu^{10} - 246\nu^{6} + 71\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{14} + 48\nu^{10} - 328\nu^{6} + \nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\nu^{15} + 3\nu^{13} - 104\nu^{11} - 20\nu^{9} + 712\nu^{7} + 136\nu^{5} - 53\nu^{3} + 19\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -15\nu^{15} + 3\nu^{13} + 104\nu^{11} - 20\nu^{9} - 712\nu^{7} + 136\nu^{5} + 53\nu^{3} + 19\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 67\nu^{15} + 13\nu^{13} - 464\nu^{11} - 88\nu^{9} + 3184\nu^{7} + 608\nu^{5} - 237\nu^{3} + 85\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -67\nu^{15} + 13\nu^{13} + 464\nu^{11} - 88\nu^{9} - 3184\nu^{7} + 608\nu^{5} + 237\nu^{3} + 85\nu ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + 3\beta_{8} + 6\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 2\beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{9} + 7\beta_{4} - 3\beta _1 + 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{15} + 5\beta_{14} - 11\beta_{13} - 11\beta_{12} - 5\beta_{7} - 5\beta_{6} + 11\beta_{3} + 11\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} + 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13 \beta_{15} + 13 \beta_{14} + 29 \beta_{13} - 29 \beta_{12} - 13 \beta_{7} + 13 \beta_{6} + \cdots + 29 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{9} + 47\beta_{4} + 21\beta _1 - 94 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -17\beta_{7} - 17\beta_{6} + 38\beta_{3} + 38\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 55\beta_{11} + 55\beta_{10} - 123\beta_{8} + 246\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89 \beta_{15} + 89 \beta_{14} + 199 \beta_{13} - 199 \beta_{12} + 89 \beta_{7} - 89 \beta_{6} + \cdots - 199 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 36\beta _1 - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 233 \beta_{15} - 233 \beta_{14} + 521 \beta_{13} + 521 \beta_{12} - 233 \beta_{7} + \cdots + 521 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -377\beta_{11} + 377\beta_{10} - 843\beta_{8} - 1686\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 305\beta_{7} - 305\beta_{6} + 682\beta_{3} - 682\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8064\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(1793\) \(4609\) \(7813\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
0.418778 + 1.56290i
−0.596975 − 0.159959i
1.56290 + 0.418778i
−0.159959 − 0.596975i
−0.159959 + 0.596975i
1.56290 − 0.418778i
−0.596975 + 0.159959i
0.418778 − 1.56290i
−1.56290 − 0.418778i
0.159959 + 0.596975i
−0.418778 − 1.56290i
0.596975 + 0.159959i
0.596975 − 0.159959i
−0.418778 + 1.56290i
0.159959 − 0.596975i
−1.56290 + 0.418778i
0 0 0 −2.44949 0 − 1.00000i 0 0 0
575.2 0 0 0 −2.44949 0 − 1.00000i 0 0 0
575.3 0 0 0 −2.44949 0 − 1.00000i 0 0 0
575.4 0 0 0 −2.44949 0 − 1.00000i 0 0 0
575.5 0 0 0 −2.44949 0 1.00000i 0 0 0
575.6 0 0 0 −2.44949 0 1.00000i 0 0 0
575.7 0 0 0 −2.44949 0 1.00000i 0 0 0
575.8 0 0 0 −2.44949 0 1.00000i 0 0 0
575.9 0 0 0 2.44949 0 − 1.00000i 0 0 0
575.10 0 0 0 2.44949 0 − 1.00000i 0 0 0
575.11 0 0 0 2.44949 0 − 1.00000i 0 0 0
575.12 0 0 0 2.44949 0 − 1.00000i 0 0 0
575.13 0 0 0 2.44949 0 1.00000i 0 0 0
575.14 0 0 0 2.44949 0 1.00000i 0 0 0
575.15 0 0 0 2.44949 0 1.00000i 0 0 0
575.16 0 0 0 2.44949 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8064.2.j.i ✓ 16
3.b odd 2 1 inner 8064.2.j.i ✓ 16
4.b odd 2 1 inner 8064.2.j.i ✓ 16
8.b even 2 1 inner 8064.2.j.i ✓ 16
8.d odd 2 1 inner 8064.2.j.i ✓ 16
12.b even 2 1 inner 8064.2.j.i ✓ 16
24.f even 2 1 inner 8064.2.j.i ✓ 16
24.h odd 2 1 inner 8064.2.j.i ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8064.2.j.i ✓ 16 1.a even 1 1 trivial
8064.2.j.i ✓ 16 3.b odd 2 1 inner
8064.2.j.i ✓ 16 4.b odd 2 1 inner
8064.2.j.i ✓ 16 8.b even 2 1 inner
8064.2.j.i ✓ 16 8.d odd 2 1 inner
8064.2.j.i ✓ 16 12.b even 2 1 inner
8064.2.j.i ✓ 16 24.f even 2 1 inner
8064.2.j.i ✓ 16 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8064, [\chi])\):

\( T_{5}^{2} - 6 \) Copy content Toggle raw display
\( T_{19}^{4} - 64T_{19}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 30)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{2} + 64)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 64 T^{2} + 784)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 10)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 128 T^{2} + 3136)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} + 76 T^{2} + 484)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{2} - 32)^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} - 68 T^{2} + 196)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 96)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 64 T^{2} + 64)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 256 T^{2} + 7744)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 64 T^{2} + 784)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 56)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 128 T^{2} + 3136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 128 T^{2} + 256)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 76 T^{2} + 484)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T - 24)^{8} \) Copy content Toggle raw display
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