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Properties

Label 1456.2.s.r
Level $1456$
Weight $2$
Character orbit 1456.s
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(113,1456)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.s (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: 8.0.4277552409.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 47x^{4} - 6x^{3} + 32x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 728)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{7} + \beta_1 - 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{7} + \beta_1 - 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \cdots - 1) q^{9}+ \cdots + ( - \beta_{5} + \beta_{2} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} - 10 q^{5} + 4 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} - 10 q^{5} + 4 q^{7} - 5 q^{9} - q^{11} + 11 q^{15} - 12 q^{17} - 3 q^{19} + 6 q^{21} + 6 q^{23} + 2 q^{25} - 24 q^{27} - 11 q^{29} - 8 q^{31} + 13 q^{33} - 5 q^{35} + 8 q^{37} - 26 q^{39} - 2 q^{41} + 17 q^{43} - 11 q^{45} + 32 q^{47} - 4 q^{49} + 10 q^{51} + 48 q^{53} - 11 q^{55} + 30 q^{57} - 4 q^{59} - 8 q^{61} + 5 q^{63} - 13 q^{65} + 22 q^{67} + 6 q^{69} - 4 q^{71} - 41 q^{75} - 2 q^{77} + 16 q^{79} - 12 q^{81} - 24 q^{83} + 29 q^{85} + 29 q^{87} + 15 q^{89} - 17 q^{93} + 21 q^{95} - q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 47x^{4} - 6x^{3} + 32x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{7} - 93\nu^{6} + 310\nu^{5} - 707\nu^{4} + 1271\nu^{3} - 3100\nu^{2} + 7752\nu - 1736 ) / 2160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} - 3\nu^{6} + 10\nu^{5} + 55\nu^{4} + 41\nu^{3} - 100\nu^{2} - 420\nu - 56 ) / 360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -41\nu^{7} + 111\nu^{6} - 370\nu^{5} + 377\nu^{4} - 1517\nu^{3} + 3700\nu^{2} - 912\nu + 2072 ) / 2160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{7} - 3\nu^{6} - 50\nu^{5} - 41\nu^{4} - 439\nu^{3} - 40\nu^{2} - 24\nu + 1144 ) / 360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -89\nu^{7} + 75\nu^{6} - 790\nu^{5} - 367\nu^{4} - 4265\nu^{3} - 1280\nu^{2} - 2928\nu - 760 ) / 2160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -95\nu^{7} + 273\nu^{6} - 910\nu^{5} + 1295\nu^{4} - 3731\nu^{3} + 9100\nu^{2} - 480\nu + 5096 ) / 2160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 8\nu^{5} - 3\nu^{4} - 47\nu^{3} + 6\nu^{2} - 32\nu - 8 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} + 7\beta_{3} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 4\beta_{5} + \beta_{4} - 4\beta_{2} - 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{6} - 49\beta_{3} - 13\beta_{2} - 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{7} + 26\beta_{6} - 63\beta_{5} - 26\beta_{4} - 101\beta_{3} + 127 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41\beta_{7} - 67\beta_{5} - 63\beta_{4} + 67\beta_{2} + 41\beta _1 + 261 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -268\beta_{6} + 965\beta_{3} + 517\beta_{2} + 265\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−0.336130 + 0.582195i
−1.10420 + 1.91254i
1.48752 − 2.57646i
0.452815 − 0.784299i
−0.336130 − 0.582195i
−1.10420 − 1.91254i
1.48752 + 2.57646i
0.452815 + 0.784299i
0 −0.987518 + 1.71043i 0 −3.97504 0 0.500000 + 0.866025i 0 −0.450384 − 0.780088i 0
113.2 0 0.0471847 − 0.0817264i 0 −1.90563 0 0.500000 + 0.866025i 0 1.49555 + 2.59036i 0
113.3 0 0.836130 − 1.44822i 0 −0.327739 0 0.500000 + 0.866025i 0 0.101772 + 0.176274i 0
113.4 0 1.60420 − 2.77856i 0 1.20841 0 0.500000 + 0.866025i 0 −3.64694 − 6.31668i 0
1121.1 0 −0.987518 − 1.71043i 0 −3.97504 0 0.500000 − 0.866025i 0 −0.450384 + 0.780088i 0
1121.2 0 0.0471847 + 0.0817264i 0 −1.90563 0 0.500000 − 0.866025i 0 1.49555 − 2.59036i 0
1121.3 0 0.836130 + 1.44822i 0 −0.327739 0 0.500000 − 0.866025i 0 0.101772 − 0.176274i 0
1121.4 0 1.60420 + 2.77856i 0 1.20841 0 0.500000 − 0.866025i 0 −3.64694 + 6.31668i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.s.r 8
4.b odd 2 1 728.2.s.e ✓ 8
13.c even 3 1 inner 1456.2.s.r 8
52.i odd 6 1 9464.2.a.bc 4
52.j odd 6 1 728.2.s.e ✓ 8
52.j odd 6 1 9464.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.s.e ✓ 8 4.b odd 2 1
728.2.s.e ✓ 8 52.j odd 6 1
1456.2.s.r 8 1.a even 1 1 trivial
1456.2.s.r 8 13.c even 3 1 inner
9464.2.a.bb 4 52.j odd 6 1
9464.2.a.bc 4 52.i odd 6 1

Hecke kernels

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

\( T_{3}^{8} - 3T_{3}^{7} + 13T_{3}^{6} - 10T_{3}^{5} + 50T_{3}^{4} - 50T_{3}^{3} + 117T_{3}^{2} - 11T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 5T_{5}^{3} + 2T_{5}^{2} - 9T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{3} + 2 T^{2} + \cdots - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{4} + 13 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots + 633616 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 29584 \) Copy content Toggle raw display
$29$ \( T^{8} + 11 T^{7} + \cdots + 7862416 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 37 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{8} - 17 T^{7} + \cdots + 26873856 \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} + \cdots - 1284)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 24 T^{3} + \cdots - 1616)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{7} + \cdots + 512656 \) Copy content Toggle raw display
$71$ \( T^{8} + 4 T^{7} + \cdots + 2808976 \) Copy content Toggle raw display
$73$ \( (T^{2} - 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 12 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 15 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 24245776 \) Copy content Toggle raw display
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