Properties

Label 816.2.bf.b
Level $816$
Weight $2$
Character orbit 816.bf
Analytic conductor $6.516$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(47,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bf (of order \(4\), degree \(2\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_1 q^{5} + (\beta_{6} - \beta_{4}) q^{7} - 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + \beta_1 q^{5} + (\beta_{6} - \beta_{4}) q^{7} - 3 \beta_{3} q^{9} - 3 \beta_{2} q^{11} + 3 q^{13} - \beta_{6} q^{15} + ( - \beta_{5} - 4 \beta_1) q^{17} - 3 \beta_{6} q^{19} + ( - 3 \beta_{5} + 3 \beta_1) q^{21} - \beta_{2} q^{23} - 4 \beta_{3} q^{25} + 3 \beta_{2} q^{27} + 4 \beta_1 q^{29} + ( - 4 \beta_{6} - 4 \beta_{4}) q^{31} + 9 q^{33} + ( - \beta_{7} + \beta_{2}) q^{35} + ( - 5 \beta_{3} + 5) q^{37} + 3 \beta_{7} q^{39} + 5 \beta_{5} q^{41} + 3 \beta_{6} q^{43} + 3 \beta_{5} q^{45} + ( - 4 \beta_{7} + 4 \beta_{2}) q^{47} + \beta_{3} q^{49} + (4 \beta_{6} - \beta_{4}) q^{51} + ( - 8 \beta_{5} - 8 \beta_1) q^{53} - 3 \beta_{4} q^{55} + 9 \beta_{5} q^{57} + ( - \beta_{7} - \beta_{2}) q^{59} + ( - \beta_{3} - 1) q^{61} + ( - 3 \beta_{6} - 3 \beta_{4}) q^{63} + 3 \beta_1 q^{65} + 8 \beta_{4} q^{67} + 3 q^{69} - 4 \beta_{7} q^{71} + ( - 8 \beta_{3} + 8) q^{73} + 4 \beta_{2} q^{75} + ( - 9 \beta_{5} - 9 \beta_1) q^{77} + ( - \beta_{6} + \beta_{4}) q^{79} - 9 q^{81} + ( - 4 \beta_{7} - 4 \beta_{2}) q^{83} + ( - 4 \beta_{3} - 1) q^{85} - 4 \beta_{6} q^{87} + (7 \beta_{5} - 7 \beta_1) q^{89} + (3 \beta_{6} - 3 \beta_{4}) q^{91} + (12 \beta_{5} + 12 \beta_1) q^{93} - 3 \beta_{2} q^{95} + (8 \beta_{3} - 8) q^{97} + 9 \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{13} + 72 q^{33} + 40 q^{37} - 8 q^{61} + 24 q^{69} + 64 q^{73} - 72 q^{81} - 8 q^{85} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0 −1.22474 1.22474i 0 −0.707107 + 0.707107i 0 −1.73205 1.73205i 0 3.00000i 0
47.2 0 −1.22474 1.22474i 0 0.707107 0.707107i 0 1.73205 + 1.73205i 0 3.00000i 0
47.3 0 1.22474 + 1.22474i 0 −0.707107 + 0.707107i 0 1.73205 + 1.73205i 0 3.00000i 0
47.4 0 1.22474 + 1.22474i 0 0.707107 0.707107i 0 −1.73205 1.73205i 0 3.00000i 0
191.1 0 −1.22474 + 1.22474i 0 −0.707107 0.707107i 0 −1.73205 + 1.73205i 0 3.00000i 0
191.2 0 −1.22474 + 1.22474i 0 0.707107 + 0.707107i 0 1.73205 1.73205i 0 3.00000i 0
191.3 0 1.22474 1.22474i 0 −0.707107 0.707107i 0 1.73205 1.73205i 0 3.00000i 0
191.4 0 1.22474 1.22474i 0 0.707107 + 0.707107i 0 −1.73205 + 1.73205i 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
17.c even 4 1 inner
51.f odd 4 1 inner
68.f odd 4 1 inner
204.l even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bf.b 8
3.b odd 2 1 inner 816.2.bf.b 8
4.b odd 2 1 inner 816.2.bf.b 8
12.b even 2 1 inner 816.2.bf.b 8
17.c even 4 1 inner 816.2.bf.b 8
51.f odd 4 1 inner 816.2.bf.b 8
68.f odd 4 1 inner 816.2.bf.b 8
204.l even 4 1 inner 816.2.bf.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.bf.b 8 1.a even 1 1 trivial
816.2.bf.b 8 3.b odd 2 1 inner
816.2.bf.b 8 4.b odd 2 1 inner
816.2.bf.b 8 12.b even 2 1 inner
816.2.bf.b 8 17.c even 4 1 inner
816.2.bf.b 8 51.f odd 4 1 inner
816.2.bf.b 8 68.f odd 4 1 inner
816.2.bf.b 8 204.l even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5}^{4} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$13$ \( (T - 3)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 9216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 192)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T + 128)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T + 128)^{4} \) Copy content Toggle raw display
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