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Properties

Label 1456.2.r.l
Level $1456$
Weight $2$
Character orbit 1456.r
Analytic conductor $11.626$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 182)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{7} + (2 \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{7} + (2 \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{11} + q^{13} + (\beta_1 - 2) q^{15} + ( - 2 \beta_{5} - \beta_{4} + \cdots + 2 \beta_1) q^{17}+ \cdots + (8 \beta_{3} - 6 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 4 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} + 4 q^{7} - 9 q^{9} + 2 q^{11} + 6 q^{13} - 14 q^{15} - 4 q^{17} - 12 q^{19} - 25 q^{21} + 2 q^{23} + 5 q^{25} - 6 q^{27} - 4 q^{29} + 13 q^{31} + 22 q^{33} - 4 q^{35} - 8 q^{37} - 8 q^{41} - 40 q^{43} - 13 q^{45} + 6 q^{47} - 12 q^{49} + 25 q^{51} + 12 q^{53} - 4 q^{55} - 6 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} + 14 q^{67} + 16 q^{69} + 4 q^{71} + 8 q^{73} + 4 q^{75} + 20 q^{77} + 24 q^{79} - 27 q^{81} + 8 q^{83} + 26 q^{85} + 22 q^{87} + 18 q^{89} + 4 q^{91} + 8 q^{93} + 8 q^{95} + 36 q^{97} - 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{4}\)

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
0.500000 − 1.41036i
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 + 1.41036i
0.500000 − 0.224437i
0.500000 − 2.05195i
0 −1.47141 + 2.54856i 0 0.380438 + 0.658939i 0 −0.710533 + 2.54856i 0 −2.83009 − 4.90187i 0
417.2 0 −0.0556321 + 0.0963576i 0 1.34981 + 2.33795i 0 2.64400 + 0.0963576i 0 1.49381 + 2.58736i 0
417.3 0 1.52704 − 2.64491i 0 −0.730252 − 1.26483i 0 0.0665372 − 2.64491i 0 −3.16372 − 5.47972i 0
625.1 0 −1.47141 − 2.54856i 0 0.380438 − 0.658939i 0 −0.710533 − 2.54856i 0 −2.83009 + 4.90187i 0
625.2 0 −0.0556321 − 0.0963576i 0 1.34981 − 2.33795i 0 2.64400 − 0.0963576i 0 1.49381 − 2.58736i 0
625.3 0 1.52704 + 2.64491i 0 −0.730252 + 1.26483i 0 0.0665372 + 2.64491i 0 −3.16372 + 5.47972i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.3
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.l 6
4.b odd 2 1 182.2.f.b ✓ 6
7.c even 3 1 inner 1456.2.r.l 6
12.b even 2 1 1638.2.j.p 6
28.d even 2 1 1274.2.f.x 6
28.f even 6 1 1274.2.a.s 3
28.f even 6 1 1274.2.f.x 6
28.g odd 6 1 182.2.f.b ✓ 6
28.g odd 6 1 1274.2.a.r 3
84.n even 6 1 1638.2.j.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.f.b ✓ 6 4.b odd 2 1
182.2.f.b ✓ 6 28.g odd 6 1
1274.2.a.r 3 28.g odd 6 1
1274.2.a.s 3 28.f even 6 1
1274.2.f.x 6 28.d even 2 1
1274.2.f.x 6 28.f even 6 1
1456.2.r.l 6 1.a even 1 1 trivial
1456.2.r.l 6 7.c even 3 1 inner
1638.2.j.p 6 12.b even 2 1
1638.2.j.p 6 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 \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{6} + 9T_{3}^{4} + 2T_{3}^{3} + 81T_{3}^{2} + 9T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} + 7T_{5}^{4} + 15T_{5}^{2} - 9T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} + 28T_{11}^{4} + 96T_{11}^{3} + 528T_{11}^{2} + 576T_{11} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 24 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 13 T^{5} + \cdots + 56169 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} + \cdots + 337561 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 12 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 20 T^{2} + \cdots + 79)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 9801 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 46656 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$67$ \( T^{6} - 14 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} - 2 T^{2} + \cdots + 147)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 8 T^{5} + \cdots + 1806336 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T + 64)^{3} \) Copy content Toggle raw display
$83$ \( (T^{3} - 4 T^{2} + \cdots - 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$97$ \( (T^{3} - 18 T^{2} + \cdots + 808)^{2} \) Copy content Toggle raw display
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