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Properties

Label 867.2.h.d
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
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] = [867,2,Mod(688,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.688");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.h (of order \(8\), degree \(4\), 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: \(2\) over \(\Q(\zeta_{8})\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{16}^{2} q^{2} - \zeta_{16}^{3} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{6} - 4 \zeta_{16} q^{7} - 3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{2} q^{2} - \zeta_{16}^{3} q^{3} - \zeta_{16}^{4} q^{4} - \zeta_{16}^{5} q^{6} - 4 \zeta_{16} q^{7} - 3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} + 4 \zeta_{16}^{5} q^{11} + \zeta_{16}^{7} q^{12} - 2 \zeta_{16}^{4} q^{13} - 4 \zeta_{16}^{3} q^{14} + q^{16} - q^{18} - 4 \zeta_{16}^{2} q^{19} + 4 \zeta_{16}^{4} q^{21} + 4 \zeta_{16}^{7} q^{22} + 4 \zeta_{16}^{5} q^{23} - 3 \zeta_{16} q^{24} + 5 \zeta_{16}^{6} q^{25} - 2 \zeta_{16}^{6} q^{26} + \zeta_{16} q^{27} + 4 \zeta_{16}^{5} q^{28} - 4 \zeta_{16}^{3} q^{31} - 5 \zeta_{16}^{2} q^{32} + 4 q^{33} + \zeta_{16}^{2} q^{36} - 8 \zeta_{16}^{3} q^{37} - 4 \zeta_{16}^{4} q^{38} + 2 \zeta_{16}^{7} q^{39} - 8 \zeta_{16} q^{41} + 4 \zeta_{16}^{6} q^{42} + 4 \zeta_{16}^{6} q^{43} + 4 \zeta_{16} q^{44} + 4 \zeta_{16}^{7} q^{46} + 8 \zeta_{16}^{4} q^{47} - \zeta_{16}^{3} q^{48} + 9 \zeta_{16}^{2} q^{49} - 5 q^{50} - 2 q^{52} + 6 \zeta_{16}^{2} q^{53} + \zeta_{16}^{3} q^{54} + 12 \zeta_{16}^{7} q^{56} + 4 \zeta_{16}^{5} q^{57} - 12 \zeta_{16}^{6} q^{59} + 8 \zeta_{16} q^{61} - 4 \zeta_{16}^{5} q^{62} - 4 \zeta_{16}^{7} q^{63} - 7 \zeta_{16}^{4} q^{64} + 4 \zeta_{16}^{2} q^{66} - 12 q^{67} + 4 q^{69} - 12 \zeta_{16}^{3} q^{71} + 3 \zeta_{16}^{4} q^{72} - 8 \zeta_{16}^{5} q^{74} + 5 \zeta_{16} q^{75} + 4 \zeta_{16}^{6} q^{76} - 16 \zeta_{16}^{6} q^{77} - 2 \zeta_{16} q^{78} - 4 \zeta_{16}^{5} q^{79} - \zeta_{16}^{4} q^{81} - 8 \zeta_{16}^{3} q^{82} - 12 \zeta_{16}^{2} q^{83} + 4 q^{84} - 4 q^{86} + 12 \zeta_{16}^{3} q^{88} - 10 \zeta_{16}^{4} q^{89} + 8 \zeta_{16}^{5} q^{91} + 4 \zeta_{16} q^{92} + 4 \zeta_{16}^{6} q^{93} + 8 \zeta_{16}^{6} q^{94} + 5 \zeta_{16}^{5} q^{96} - 16 \zeta_{16}^{7} q^{97} + 9 \zeta_{16}^{4} q^{98} - 4 \zeta_{16}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{16} - 8 q^{18} + 32 q^{33} - 40 q^{50} - 16 q^{52} - 96 q^{67} + 32 q^{69} + 32 q^{84} - 32 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\zeta_{16}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1
0.923880 − 0.382683i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.382683 − 0.923880i
0.382683 + 0.923880i
0.707107 − 0.707107i −0.382683 + 0.923880i 1.00000i 0 0.382683 + 0.923880i −3.69552 + 1.53073i 2.12132 + 2.12132i −0.707107 − 0.707107i 0
688.2 0.707107 − 0.707107i 0.382683 − 0.923880i 1.00000i 0 −0.382683 − 0.923880i 3.69552 − 1.53073i 2.12132 + 2.12132i −0.707107 − 0.707107i 0
712.1 0.707107 + 0.707107i −0.382683 − 0.923880i − 1.00000i 0 0.382683 − 0.923880i −3.69552 − 1.53073i 2.12132 − 2.12132i −0.707107 + 0.707107i 0
712.2 0.707107 + 0.707107i 0.382683 + 0.923880i − 1.00000i 0 −0.382683 + 0.923880i 3.69552 + 1.53073i 2.12132 − 2.12132i −0.707107 + 0.707107i 0
733.1 −0.707107 − 0.707107i −0.923880 + 0.382683i − 1.00000i 0 0.923880 + 0.382683i 1.53073 − 3.69552i −2.12132 + 2.12132i 0.707107 − 0.707107i 0
733.2 −0.707107 − 0.707107i 0.923880 − 0.382683i − 1.00000i 0 −0.923880 − 0.382683i −1.53073 + 3.69552i −2.12132 + 2.12132i 0.707107 − 0.707107i 0
757.1 −0.707107 + 0.707107i −0.923880 − 0.382683i 1.00000i 0 0.923880 − 0.382683i 1.53073 + 3.69552i −2.12132 − 2.12132i 0.707107 + 0.707107i 0
757.2 −0.707107 + 0.707107i 0.923880 + 0.382683i 1.00000i 0 −0.923880 + 0.382683i −1.53073 − 3.69552i −2.12132 − 2.12132i 0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 688.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.d 8
17.b even 2 1 inner 867.2.h.d 8
17.c even 4 2 inner 867.2.h.d 8
17.d even 8 4 inner 867.2.h.d 8
17.e odd 16 2 51.2.d.b ✓ 2
17.e odd 16 1 867.2.a.a 1
17.e odd 16 1 867.2.a.b 1
17.e odd 16 4 867.2.e.d 4
51.i even 16 2 153.2.d.a 2
51.i even 16 1 2601.2.a.i 1
51.i even 16 1 2601.2.a.j 1
68.i even 16 2 816.2.c.c 2
85.o even 16 1 1275.2.d.b 2
85.o even 16 1 1275.2.d.d 2
85.p odd 16 2 1275.2.g.a 2
85.r even 16 1 1275.2.d.b 2
85.r even 16 1 1275.2.d.d 2
136.q odd 16 2 3264.2.c.e 2
136.s even 16 2 3264.2.c.d 2
204.t odd 16 2 2448.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b ✓ 2 17.e odd 16 2
153.2.d.a 2 51.i even 16 2
816.2.c.c 2 68.i even 16 2
867.2.a.a 1 17.e odd 16 1
867.2.a.b 1 17.e odd 16 1
867.2.e.d 4 17.e odd 16 4
867.2.h.d 8 1.a even 1 1 trivial
867.2.h.d 8 17.b even 2 1 inner
867.2.h.d 8 17.c even 4 2 inner
867.2.h.d 8 17.d even 8 4 inner
1275.2.d.b 2 85.o even 16 1
1275.2.d.b 2 85.r even 16 1
1275.2.d.d 2 85.o even 16 1
1275.2.d.d 2 85.r even 16 1
1275.2.g.a 2 85.p odd 16 2
2448.2.c.j 2 204.t odd 16 2
2601.2.a.i 1 51.i even 16 1
2601.2.a.j 1 51.i even 16 1
3264.2.c.d 2 136.s even 16 2
3264.2.c.e 2 136.q odd 16 2

Hecke kernels

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

\( T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 65536 \) Copy content Toggle raw display
$11$ \( T^{8} + 65536 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 65536 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 65536 \) Copy content Toggle raw display
$37$ \( T^{8} + 16777216 \) Copy content Toggle raw display
$41$ \( T^{8} + 16777216 \) Copy content Toggle raw display
$43$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 16777216 \) Copy content Toggle raw display
$67$ \( (T + 12)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 429981696 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + 65536 \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 4294967296 \) Copy content Toggle raw display
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