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Properties

Label 867.2.e.h
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
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] = [867,2,Mod(616,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.616");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.e (of order \(4\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 51)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4}) q^{2} - \beta_1 q^{3} + (\beta_{5} - \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4}) q^{2} - \beta_1 q^{3} + (\beta_{5} - \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{10} + 8 q^{11} + 8 q^{12} + 8 q^{13} + 8 q^{14} - 16 q^{16} + 8 q^{20} + 16 q^{21} + 16 q^{22} + 16 q^{28} + 24 q^{29} + 16 q^{30} - 32 q^{31} - 8 q^{33} + 32 q^{35} + 16 q^{37} - 32 q^{38} - 8 q^{39} - 16 q^{40} + 16 q^{41} - 8 q^{44} + 16 q^{46} - 16 q^{47} + 16 q^{48} + 32 q^{50} - 16 q^{52} + 24 q^{55} + 8 q^{57} - 32 q^{61} - 8 q^{65} + 16 q^{67} - 24 q^{69} - 24 q^{71} + 32 q^{73} - 8 q^{74} + 16 q^{75} + 16 q^{78} + 16 q^{79} + 16 q^{80} - 8 q^{81} + 16 q^{82} - 32 q^{86} + 16 q^{88} - 32 q^{89} - 16 q^{90} - 24 q^{92} - 24 q^{95} - 16 q^{97} - 64 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
616.1
0.923880 − 0.382683i
0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.923880 + 0.382683i
−0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 − 0.923880i
0.923880 + 0.382683i
− 1.84776i −0.707107 + 0.707107i −1.41421 1.14065 − 1.14065i 1.30656 + 1.30656i −0.107651 − 0.107651i − 1.08239i − 1.00000i −2.10765 − 2.10765i
616.2 − 0.765367i 0.707107 − 0.707107i 1.41421 1.47247 − 1.47247i −0.541196 − 0.541196i 0.873017 + 0.873017i − 2.61313i − 1.00000i −1.12698 − 1.12698i
616.3 0.765367i 0.707107 − 0.707107i 1.41421 −0.0582601 + 0.0582601i 0.541196 + 0.541196i 1.95541 + 1.95541i 2.61313i − 1.00000i −0.0445903 − 0.0445903i
616.4 1.84776i −0.707107 + 0.707107i −1.41421 −2.55487 + 2.55487i −1.30656 − 1.30656i −2.72078 − 2.72078i 1.08239i − 1.00000i −4.72078 − 4.72078i
829.1 − 1.84776i −0.707107 − 0.707107i −1.41421 −2.55487 − 2.55487i −1.30656 + 1.30656i −2.72078 + 2.72078i − 1.08239i 1.00000i −4.72078 + 4.72078i
829.2 − 0.765367i 0.707107 + 0.707107i 1.41421 −0.0582601 − 0.0582601i 0.541196 − 0.541196i 1.95541 − 1.95541i − 2.61313i 1.00000i −0.0445903 + 0.0445903i
829.3 0.765367i 0.707107 + 0.707107i 1.41421 1.47247 + 1.47247i −0.541196 + 0.541196i 0.873017 − 0.873017i 2.61313i 1.00000i −1.12698 + 1.12698i
829.4 1.84776i −0.707107 − 0.707107i −1.41421 1.14065 + 1.14065i 1.30656 − 1.30656i −0.107651 + 0.107651i 1.08239i 1.00000i −2.10765 + 2.10765i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 616.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.e.h 8
17.b even 2 1 867.2.e.i 8
17.c even 4 1 inner 867.2.e.h 8
17.c even 4 1 867.2.e.i 8
17.d even 8 1 867.2.a.m 4
17.d even 8 1 867.2.a.n 4
17.d even 8 2 867.2.d.e 8
17.e odd 16 2 51.2.h.a ✓ 8
17.e odd 16 2 867.2.h.b 8
17.e odd 16 2 867.2.h.f 8
17.e odd 16 2 867.2.h.g 8
51.g odd 8 1 2601.2.a.bc 4
51.g odd 8 1 2601.2.a.bd 4
51.i even 16 2 153.2.l.e 8
68.i even 16 2 816.2.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.h.a ✓ 8 17.e odd 16 2
153.2.l.e 8 51.i even 16 2
816.2.bq.a 8 68.i even 16 2
867.2.a.m 4 17.d even 8 1
867.2.a.n 4 17.d even 8 1
867.2.d.e 8 17.d even 8 2
867.2.e.h 8 1.a even 1 1 trivial
867.2.e.h 8 17.c even 4 1 inner
867.2.e.i 8 17.b even 2 1
867.2.e.i 8 17.c even 4 1
867.2.h.b 8 17.e odd 16 2
867.2.h.f 8 17.e odd 16 2
867.2.h.g 8 17.e odd 16 2
2601.2.a.bc 4 51.g odd 8 1
2601.2.a.bd 4 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{4} + 4T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{8} - 16T_{5}^{5} + 98T_{5}^{4} - 160T_{5}^{3} + 128T_{5}^{2} + 16T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - 16 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} - 14 T^{2} + \cdots - 47)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 108 T^{6} + \cdots + 134689 \) Copy content Toggle raw display
$23$ \( T^{8} - 336 T^{5} + \cdots + 73441 \) Copy content Toggle raw display
$29$ \( (T^{4} - 12 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 32 T^{7} + \cdots + 399424 \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + \cdots + 498436 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} + 172 T^{6} + \cdots + 1682209 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 176 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$59$ \( T^{8} + 184 T^{6} + \cdots + 264196 \) Copy content Toggle raw display
$61$ \( T^{8} + 32 T^{7} + \cdots + 1110916 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$73$ \( T^{8} - 32 T^{7} + \cdots + 565504 \) Copy content Toggle raw display
$79$ \( T^{8} - 16 T^{7} + \cdots + 3844 \) Copy content Toggle raw display
$83$ \( T^{8} + 304 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots - 2558)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots + 399424 \) Copy content Toggle raw display
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