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Properties

Label 1456.2.r.m
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 9x^{6} + 56x^{4} - 8x^{3} + 112x^{2} + 48x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{6} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{6} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{9} + (\beta_{7} - \beta_{6} + 2 \beta_{3} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - q^{5} - q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - q^{5} - q^{7} - 5 q^{9} - 8 q^{11} + 8 q^{13} + 18 q^{15} + 4 q^{17} + q^{21} - 9 q^{23} - 5 q^{25} + 14 q^{27} + 22 q^{29} + 7 q^{31} + 7 q^{33} + 15 q^{35} + q^{37} - q^{39} + 20 q^{43} - 26 q^{45} - q^{49} - 7 q^{51} + 2 q^{53} - 2 q^{55} - 34 q^{57} - 12 q^{59} + 26 q^{61} + 17 q^{63} - q^{65} - 38 q^{67} + 6 q^{69} + 50 q^{71} - 3 q^{73} - 28 q^{75} - 33 q^{77} - 25 q^{79} + 20 q^{81} - 68 q^{83} - 34 q^{85} + 14 q^{87} - 3 q^{89} - q^{91} - 15 q^{93} - 13 q^{95} + 20 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 9x^{6} + 56x^{4} - 8x^{3} + 112x^{2} + 48x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13\nu^{6} + 7\nu^{5} + 6\nu^{4} + 232\nu^{3} + 20\nu^{2} + 24\nu - 1456 ) / 736 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{7} + 31\nu^{6} - 93\nu^{5} + 147\nu^{4} - 434\nu^{3} + 1364\nu^{2} - 700\nu + 2232 ) / 2208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{7} - 28\nu^{6} + 84\nu^{5} - 411\nu^{4} + 392\nu^{3} - 1232\nu^{2} + 5164\nu - 2016 ) / 2208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 41\nu^{6} + 123\nu^{5} - 348\nu^{4} + 574\nu^{3} - 1804\nu^{2} + 80\nu - 2952 ) / 1104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - \nu^{6} + 49\nu^{5} + 42\nu^{4} + 428\nu^{3} + 140\nu^{2} + 168\nu + 480 ) / 736 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{7} - 58\nu^{6} + 588\nu^{5} + 21\nu^{4} + 3434\nu^{3} + 208\nu^{2} + 6892\nu + 3000 ) / 2208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 4\beta_{3} - \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 7\beta_{6} + 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{5} + 2\beta_{4} - 22\beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{7} - 51\beta_{6} - 13\beta_{5} - 22\beta_{3} + 13\beta_{2} - 51\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26\beta_{7} - 99\beta_{6} - 26\beta_{4} + 69\beta_{2} + 142 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{5} - 138\beta_{4} + 206\beta_{3} + 379\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
1.40014 + 2.42511i
0.814732 + 1.41116i
−0.578647 − 1.00225i
−1.13622 − 1.96799i
1.40014 − 2.42511i
0.814732 − 1.41116i
−0.578647 + 1.00225i
−1.13622 + 1.96799i
0 −1.40014 + 2.42511i 0 −1.92076 − 3.32686i 0 −2.62908 + 0.296507i 0 −2.42076 − 4.19288i 0
417.2 0 −0.814732 + 1.41116i 0 0.672424 + 1.16467i 0 −0.103098 + 2.64374i 0 0.172424 + 0.298648i 0
417.3 0 0.578647 − 1.00225i 0 1.33034 + 2.30421i 0 −0.356316 − 2.62165i 0 0.830336 + 1.43818i 0
417.4 0 1.13622 − 1.96799i 0 −0.581998 − 1.00805i 0 2.58850 + 0.547425i 0 −1.08200 − 1.87408i 0
625.1 0 −1.40014 − 2.42511i 0 −1.92076 + 3.32686i 0 −2.62908 − 0.296507i 0 −2.42076 + 4.19288i 0
625.2 0 −0.814732 − 1.41116i 0 0.672424 − 1.16467i 0 −0.103098 − 2.64374i 0 0.172424 − 0.298648i 0
625.3 0 0.578647 + 1.00225i 0 1.33034 − 2.30421i 0 −0.356316 + 2.62165i 0 0.830336 − 1.43818i 0
625.4 0 1.13622 + 1.96799i 0 −0.581998 + 1.00805i 0 2.58850 − 0.547425i 0 −1.08200 + 1.87408i 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.m 8
4.b odd 2 1 728.2.r.e ✓ 8
7.c even 3 1 inner 1456.2.r.m 8
28.f even 6 1 5096.2.a.v 4
28.g odd 6 1 728.2.r.e ✓ 8
28.g odd 6 1 5096.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.r.e ✓ 8 4.b odd 2 1
728.2.r.e ✓ 8 28.g odd 6 1
1456.2.r.m 8 1.a even 1 1 trivial
1456.2.r.m 8 7.c even 3 1 inner
5096.2.a.u 4 28.g odd 6 1
5096.2.a.v 4 28.f even 6 1

Hecke kernels

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

\( T_{3}^{8} + T_{3}^{7} + 9T_{3}^{6} + 56T_{3}^{4} + 8T_{3}^{3} + 112T_{3}^{2} - 48T_{3} + 144 \) Copy content Toggle raw display
\( T_{5}^{8} + T_{5}^{7} + 13T_{5}^{6} - 12T_{5}^{5} + 128T_{5}^{4} - 32T_{5}^{3} + 192T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} + 8T_{11}^{7} + 50T_{11}^{6} + 124T_{11}^{5} + 245T_{11}^{4} - 68T_{11}^{3} + 50T_{11}^{2} + 6T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 144 \) 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
$11$ \( T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 28 T^{6} + \cdots + 2209 \) Copy content Toggle raw display
$23$ \( T^{8} + 9 T^{7} + \cdots + 51984 \) Copy content Toggle raw display
$29$ \( (T^{4} - 11 T^{3} + 37 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 7 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + \cdots + 38416 \) Copy content Toggle raw display
$41$ \( (T^{4} - 40 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 8590761 \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + \cdots + 97239321 \) Copy content Toggle raw display
$67$ \( T^{8} + 38 T^{7} + \cdots + 12524521 \) Copy content Toggle raw display
$71$ \( (T^{4} - 25 T^{3} + \cdots - 20936)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 3 T^{7} + \cdots + 23104 \) Copy content Toggle raw display
$79$ \( T^{8} + 25 T^{7} + \cdots + 1201216 \) Copy content Toggle raw display
$83$ \( (T^{4} + 34 T^{3} + \cdots - 12592)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 3 T^{7} + \cdots + 898704 \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + \cdots + 552)^{2} \) Copy content Toggle raw display
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