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Properties

Label 726.2.b.f
Level $726$
Weight $2$
Character orbit 726.b
Analytic conductor $5.797$
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] = [726,2,Mod(725,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.725");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.b (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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3588489216.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{6} + \beta_1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_1) q^{6} + (\beta_{4} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{7} - \beta_{4} + \beta_{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{6} + \beta_1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_1) q^{6} + (\beta_{4} + \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{7} - \beta_{4} + \beta_{2} + \cdots - 1) q^{9}+ \cdots + ( - \beta_{7} - \beta_{5} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{8} - 4 q^{9} + 4 q^{12} - 8 q^{15} + 8 q^{16} - 16 q^{17} - 4 q^{18} - 12 q^{21} + 4 q^{24} - 40 q^{25} + 16 q^{27} + 8 q^{29} - 8 q^{30} + 24 q^{31} + 8 q^{32} - 16 q^{34} - 4 q^{36} + 24 q^{37} + 28 q^{39} - 12 q^{42} - 16 q^{45} + 4 q^{48} - 40 q^{50} - 28 q^{51} + 16 q^{54} + 16 q^{57} + 8 q^{58} - 8 q^{60} + 24 q^{62} - 12 q^{63} + 8 q^{64} - 32 q^{65} + 8 q^{67} - 16 q^{68} + 40 q^{69} - 4 q^{72} + 24 q^{74} - 12 q^{75} + 28 q^{78} + 8 q^{81} - 16 q^{83} - 12 q^{84} + 32 q^{87} - 16 q^{90} + 8 q^{91} + 64 q^{95} + 4 q^{96} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 10x^{6} - 8x^{5} + 8x^{4} + 4x^{3} + 16x^{2} + 32x + 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + 7\nu^{3} - \nu^{2} - 2\nu - 1 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 2\nu^{3} - 11\nu^{2} + 2\nu - 2 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 7\nu^{5} - \nu^{4} + 7\nu^{3} + 11\nu^{2} + 27\nu + 59 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 10\nu^{5} - 13\nu^{4} + 28\nu^{3} - 19\nu^{2} + 48\nu + 29 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{6} - 19\nu^{5} + 31\nu^{4} - 37\nu^{3} + 19\nu^{2} - 30\nu - 23 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 10\nu^{5} + 10\nu^{4} - 10\nu^{3} + \nu^{2} - 12\nu - 18 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 3\beta_{6} + \beta_{5} + 5\beta_{4} - 4\beta_{2} - 7\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 10\beta_{5} - 7\beta_{3} - 13\beta_{2} - 18\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{6} + 30\beta_{5} - 21\beta_{4} - 18\beta_{3} - 12\beta_{2} - 21\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -39\beta_{7} + 66\beta_{6} + 39\beta_{5} - 63\beta_{4} - 12\beta_{3} + 33\beta_{2} + 25\beta _1 + 75 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(607\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
725.1
−0.697085 + 0.346269i
−0.697085 − 0.346269i
−0.252986 − 1.19584i
−0.252986 + 1.19584i
1.25299 − 1.71348i
1.25299 + 1.71348i
1.69709 + 1.58558i
1.69709 − 1.58558i
1.00000 −0.697085 − 1.58558i 1.00000 2.86823i −0.697085 − 1.58558i 0.692538i 1.00000 −2.02814 + 2.21057i 2.86823i
725.2 1.00000 −0.697085 + 1.58558i 1.00000 − 2.86823i −0.697085 + 1.58558i − 0.692538i 1.00000 −2.02814 − 2.21057i − 2.86823i
725.3 1.00000 −0.252986 − 1.71348i 1.00000 − 4.06162i −0.252986 − 1.71348i − 2.39167i 1.00000 −2.87200 + 0.866969i − 4.06162i
725.4 1.00000 −0.252986 + 1.71348i 1.00000 4.06162i −0.252986 + 1.71348i 2.39167i 1.00000 −2.87200 − 0.866969i 4.06162i
725.5 1.00000 1.25299 − 1.19584i 1.00000 − 0.197913i 1.25299 − 1.19584i − 3.42695i 1.00000 0.139946 − 2.99673i − 0.197913i
725.6 1.00000 1.25299 + 1.19584i 1.00000 0.197913i 1.25299 + 1.19584i 3.42695i 1.00000 0.139946 + 2.99673i 0.197913i
725.7 1.00000 1.69709 − 0.346269i 1.00000 − 3.90351i 1.69709 − 0.346269i 3.17117i 1.00000 2.76020 − 1.17530i − 3.90351i
725.8 1.00000 1.69709 + 0.346269i 1.00000 3.90351i 1.69709 + 0.346269i − 3.17117i 1.00000 2.76020 + 1.17530i 3.90351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 725.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.b.f yes 8
3.b odd 2 1 726.2.b.d ✓ 8
11.b odd 2 1 726.2.b.d ✓ 8
11.c even 5 4 726.2.h.k 32
11.d odd 10 4 726.2.h.l 32
33.d even 2 1 inner 726.2.b.f yes 8
33.f even 10 4 726.2.h.k 32
33.h odd 10 4 726.2.h.l 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.b.d ✓ 8 3.b odd 2 1
726.2.b.d ✓ 8 11.b odd 2 1
726.2.b.f yes 8 1.a even 1 1 trivial
726.2.b.f yes 8 33.d even 2 1 inner
726.2.h.k 32 11.c even 5 4
726.2.h.k 32 33.f even 10 4
726.2.h.l 32 11.d odd 10 4
726.2.h.l 32 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\):

\( T_{5}^{8} + 40T_{5}^{6} + 514T_{5}^{4} + 2088T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{17}^{4} + 8T_{17}^{3} - 8T_{17}^{2} - 96T_{17} - 99 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 40 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 324 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 68 T^{6} + \cdots + 6889 \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} - 8 T^{2} + \cdots - 99)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 68 T^{6} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{8} + 124 T^{6} + \cdots + 39204 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 38 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + \cdots - 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots - 243)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 120 T^{2} + \cdots + 1269)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 112 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$47$ \( T^{8} + 148 T^{6} + \cdots + 1205604 \) Copy content Toggle raw display
$53$ \( T^{8} + 184 T^{6} + \cdots + 762129 \) Copy content Toggle raw display
$59$ \( T^{8} + 136 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{8} + 256 T^{6} + \cdots + 1354896 \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 1878)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 176 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$79$ \( T^{8} + 404 T^{6} + \cdots + 111556 \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} + 10 T^{2} + \cdots - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 76 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 1401)^{2} \) Copy content Toggle raw display
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