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Properties

Label 1456.2.r.o
Level $1456$
Weight $2$
Character orbit 1456.r
Analytic conductor $11.626$
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] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), degree \(2\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + (4 \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} + q^{5} - q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} + q^{5} - q^{7} - 9 q^{9} + 4 q^{11} - 8 q^{13} + 18 q^{15} + 3 q^{21} + 11 q^{23} - 5 q^{25} - 54 q^{27} + 22 q^{29} - 5 q^{31} + 15 q^{33} + 25 q^{35} - 27 q^{37} - 3 q^{39} + 12 q^{41} - 8 q^{43} - 6 q^{45} + 4 q^{47} - 25 q^{49} + 9 q^{51} - 18 q^{53} - 46 q^{55} - 18 q^{57} - 16 q^{59} - 10 q^{61} + 33 q^{63} - q^{65} - 2 q^{67} + 54 q^{69} + 10 q^{71} + 7 q^{73} - 6 q^{75} + 31 q^{77} + 17 q^{79} - 24 q^{81} + 12 q^{83} - 30 q^{85} - 12 q^{87} + 21 q^{89} + q^{91} - 33 q^{93} + 9 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 2\nu^{3} + 22\nu^{2} + 10\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} - 34\nu^{3} - 2\nu^{2} - 20\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 10\nu^{5} + 2\nu^{4} - 29\nu^{3} + 10\nu^{2} - 26\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} - 34\nu^{3} + 2\nu^{2} - 20\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 11\nu^{5} - 4\nu^{4} + 34\nu^{3} - 22\nu^{2} + 20\nu - 12 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - 2\beta_{4} - \beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 8\beta_{6} + 10\beta_{4} + 3\beta_{3} + 8\beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 6\beta_{5} + 5\beta_{3} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} - 46\beta_{6} - 50\beta_{4} - 21\beta_{3} - 46\beta_{2} + 16\beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} - 2\beta_{6} + 32\beta_{5} - 25\beta_{3} + 2\beta_{2} - 2\beta _1 - 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 125\beta_{7} + 254\beta_{6} - 6\beta_{5} + 250\beta_{4} + 123\beta_{3} + 254\beta_{2} - 128\beta _1 + 62 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
− 0.385731i
− 1.07834i
2.06288i
− 2.33086i
0.385731i
1.07834i
− 2.06288i
2.33086i
0 −0.925606 + 1.60320i 0 −1.56470 − 2.71015i 0 −0.165947 + 2.64054i 0 −0.213493 − 0.369780i 0
417.2 0 −0.418594 + 0.725026i 0 0.812371 + 1.40707i 0 0.433868 − 2.60993i 0 1.14956 + 1.99109i 0
417.3 0 1.12774 − 1.95330i 0 1.71189 + 2.96508i 0 −2.28651 − 1.33112i 0 −1.04359 − 1.80755i 0
417.4 0 1.71646 − 2.97300i 0 −0.459555 − 0.795973i 0 1.51859 + 2.16654i 0 −4.39248 − 7.60799i 0
625.1 0 −0.925606 − 1.60320i 0 −1.56470 + 2.71015i 0 −0.165947 − 2.64054i 0 −0.213493 + 0.369780i 0
625.2 0 −0.418594 − 0.725026i 0 0.812371 − 1.40707i 0 0.433868 + 2.60993i 0 1.14956 − 1.99109i 0
625.3 0 1.12774 + 1.95330i 0 1.71189 − 2.96508i 0 −2.28651 + 1.33112i 0 −1.04359 + 1.80755i 0
625.4 0 1.71646 + 2.97300i 0 −0.459555 + 0.795973i 0 1.51859 − 2.16654i 0 −4.39248 + 7.60799i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.r.o 8
4.b odd 2 1 364.2.j.e ✓ 8
7.c even 3 1 inner 1456.2.r.o 8
12.b even 2 1 3276.2.r.j 8
28.d even 2 1 2548.2.j.q 8
28.f even 6 1 2548.2.a.p 4
28.f even 6 1 2548.2.j.q 8
28.g odd 6 1 364.2.j.e ✓ 8
28.g odd 6 1 2548.2.a.q 4
84.n even 6 1 3276.2.r.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.j.e ✓ 8 4.b odd 2 1
364.2.j.e ✓ 8 28.g odd 6 1
1456.2.r.o 8 1.a even 1 1 trivial
1456.2.r.o 8 7.c even 3 1 inner
2548.2.a.p 4 28.f even 6 1
2548.2.a.q 4 28.g odd 6 1
2548.2.j.q 8 28.d even 2 1
2548.2.j.q 8 28.f even 6 1
3276.2.r.j 8 12.b even 2 1
3276.2.r.j 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{8} - 3T_{3}^{7} + 15T_{3}^{6} - 6T_{3}^{5} + 60T_{3}^{4} + 216T_{3}^{2} + 144T_{3} + 144 \) Copy content Toggle raw display
\( T_{5}^{8} - T_{5}^{7} + 13T_{5}^{6} - 4T_{5}^{5} + 136T_{5}^{4} - 64T_{5}^{3} + 256T_{5}^{2} + 128T_{5} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} - 4T_{11}^{7} + 34T_{11}^{6} - 124T_{11}^{5} + 817T_{11}^{4} - 2572T_{11}^{3} + 7786T_{11}^{2} - 9898T_{11} + 10201 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 42 T^{6} + \cdots + 8649 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$23$ \( T^{8} - 11 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{4} - 11 T^{3} + \cdots - 746)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + \cdots + 71824 \) Copy content Toggle raw display
$37$ \( T^{8} + 27 T^{7} + \cdots + 2782224 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 36 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} - 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{7} + \cdots + 337561 \) Copy content Toggle raw display
$53$ \( T^{8} + 18 T^{7} + \cdots + 1580049 \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + \cdots + 6436369 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots + 528529 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + \cdots + 2809 \) Copy content Toggle raw display
$71$ \( (T^{4} - 5 T^{3} + \cdots + 12208)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 7 T^{7} + \cdots + 15178816 \) Copy content Toggle raw display
$79$ \( T^{8} - 17 T^{7} + \cdots + 817216 \) Copy content Toggle raw display
$83$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 21 T^{7} + \cdots + 747913104 \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + \cdots - 248)^{2} \) Copy content Toggle raw display
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