Properties

Label 867.2.h.g
Level $867$
Weight $2$
Character orbit 867.h
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(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.

\(f(q)\) \(=\) \( q + (\zeta_{16}^{3} + \zeta_{16}) q^{2} - \zeta_{16}^{7} q^{3} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{4} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} + 1) q^{5} + (\zeta_{16}^{2} + 1) q^{6} + ( - \zeta_{16}^{6} - 2 \zeta_{16}^{5} - \zeta_{16}^{4}) q^{7} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} - \zeta_{16}) q^{8} - \zeta_{16}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{16}^{3} + \zeta_{16}) q^{2} - \zeta_{16}^{7} q^{3} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{4} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} + 1) q^{5} + (\zeta_{16}^{2} + 1) q^{6} + ( - \zeta_{16}^{6} - 2 \zeta_{16}^{5} - \zeta_{16}^{4}) q^{7} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} - \zeta_{16}) q^{8} - \zeta_{16}^{6} q^{9} + (2 \zeta_{16}^{7} + \zeta_{16}^{6} + 2 \zeta_{16}^{5} + \zeta_{16}^{4} + 2 \zeta_{16}^{3}) q^{10} + ( - \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{4} + \zeta_{16}^{3} - \zeta_{16}^{2} + \zeta_{16} - 1) q^{11} + (\zeta_{16}^{5} + \zeta_{16}) q^{12} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{2} - 2 \zeta_{16}) q^{13} + ( - 2 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16} + 2) q^{14} + ( - \zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}^{2} + \zeta_{16}) q^{15} + ( - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{2} + 2) q^{16} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{18} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 3 \zeta_{16}^{2} + 2 \zeta_{16} - 1) q^{19} + (2 \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16} - 2) q^{20} + ( - \zeta_{16}^{5} - 2 \zeta_{16}^{4} - \zeta_{16}^{3}) q^{21} + (\zeta_{16}^{6} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 2 \zeta_{16} + 1) q^{22} + ( - 3 \zeta_{16}^{6} + 3 \zeta_{16}^{4} - \zeta_{16}^{2} - 3 \zeta_{16} - 1) q^{23} + ( - \zeta_{16}^{6} - \zeta_{16}^{4} + \zeta_{16}^{2} - 1) q^{24} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 2 \zeta_{16} - 2) q^{25} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{4} + \zeta_{16}^{3} - \zeta_{16} + 2) q^{26} - \zeta_{16}^{5} q^{27} + ( - 2 \zeta_{16}^{7} - \zeta_{16}^{6} + \zeta_{16}^{4} + 2 \zeta_{16}^{3} + \zeta_{16}^{2} + 1) q^{28} + (3 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 3 \zeta_{16}) q^{29} + (2 \zeta_{16}^{6} + \zeta_{16}^{5} + 2 \zeta_{16}^{4} + \zeta_{16}^{3} + 2 \zeta_{16}^{2}) q^{30} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} + 4 \zeta_{16}^{5} - \zeta_{16}^{4} + \zeta_{16}^{2} - 4 \zeta_{16} - 1) q^{31} + ( - 4 \zeta_{16}^{7} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{32} + (\zeta_{16}^{7} - \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}^{2} - \zeta_{16} + 1) q^{33} + ( - 3 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{2} + 3 \zeta_{16} + 4) q^{35} + (\zeta_{16}^{4} + 1) q^{36} + (6 \zeta_{16}^{7} - \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 2 \zeta_{16} + 1) q^{37} + ( - \zeta_{16}^{7} - 4 \zeta_{16}^{5} + 4 \zeta_{16}^{4} - 4 \zeta_{16}^{3} - \zeta_{16}) q^{38} + ( - 2 \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} - 2) q^{39} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{4} + 2 \zeta_{16}^{3} + \zeta_{16}^{2} + 1) q^{40} + (2 \zeta_{16}^{7} + \zeta_{16}^{6} - 3 \zeta_{16}^{5} + \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 3 \zeta_{16}^{2} + 3) q^{41} + ( - 2 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{5} - \zeta_{16}^{4} + 1) q^{42} + (2 \zeta_{16}^{7} - \zeta_{16}^{6} + 2 \zeta_{16}^{5} + \zeta_{16}^{4} + 4 \zeta_{16}^{3} - 4 \zeta_{16} - 1) q^{43} + (\zeta_{16}^{7} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{4} + \zeta_{16}^{3}) q^{44} + ( - \zeta_{16}^{6} + \zeta_{16}^{4} + \zeta_{16}^{2} + \zeta_{16} + 1) q^{45} + (2 \zeta_{16}^{5} - 3 \zeta_{16}^{4} - 2 \zeta_{16}^{3} - 3 \zeta_{16}^{2} + 2 \zeta_{16}) q^{46} + ( - 2 \zeta_{16}^{7} - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 4 \zeta_{16}^{2} - 2 \zeta_{16}) q^{47} + ( - 2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{48} + ( - \zeta_{16}^{4} - 4 \zeta_{16}^{3} + \zeta_{16}^{2} - 4 \zeta_{16} - 1) q^{49} + (2 \zeta_{16}^{7} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - 4 \zeta_{16}^{2} - 2 \zeta_{16} - 4) q^{50} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - \zeta_{16}^{2} + 2 \zeta_{16} - 2) q^{52} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{4} + 2 \zeta_{16}^{2} - 4) q^{53} + ( - \zeta_{16}^{6} + 1) q^{54} + ( - \zeta_{16}^{6} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - \zeta_{16}^{2}) q^{55} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{2} + 2) q^{56} + (\zeta_{16}^{7} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 3 \zeta_{16} + 2) q^{57} + (\zeta_{16}^{6} + \zeta_{16}^{4} + 3 \zeta_{16}^{2} - 3) q^{58} + (\zeta_{16}^{7} + 6 \zeta_{16}^{6} + \zeta_{16}^{5} - 2 \zeta_{16}^{4} + 2) q^{59} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + \zeta_{16}^{4} - 1) q^{60} + (4 \zeta_{16}^{7} + \zeta_{16}^{6} + 6 \zeta_{16}^{5} + \zeta_{16}^{4} + 4 \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 2) q^{61} + (4 \zeta_{16}^{6} - 4 \zeta_{16}^{4} - 2 \zeta_{16}^{2} - 2 \zeta_{16} - 2) q^{62} + ( - \zeta_{16}^{4} - 2 \zeta_{16}^{3} - \zeta_{16}^{2}) q^{63} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{2}) q^{64} + ( - 3 \zeta_{16}^{7} + 3 \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16}^{4} + \zeta_{16}^{2} - \zeta_{16} - 3) q^{65} + ( - \zeta_{16}^{7} + \zeta_{16}^{5} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{2} + 2 \zeta_{16} - 2) q^{66} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{2} + 2 \zeta_{16} - 2) q^{67} + (\zeta_{16}^{7} - 3 \zeta_{16}^{5} + 3 \zeta_{16}^{3} - \zeta_{16} - 3) q^{69} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 5 \zeta_{16}^{4} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{2} + 6 \zeta_{16} + 5) q^{70} + ( - 6 \zeta_{16}^{7} - 2 \zeta_{16}^{6} - 3 \zeta_{16}^{5} + 3 \zeta_{16} + 2) q^{71} + (\zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16}) q^{72} + ( - 6 \zeta_{16}^{6} - 4 \zeta_{16}^{5} - 4 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 4 \zeta_{16}^{2} - 4 \zeta_{16} - 6) q^{73} + ( - \zeta_{16}^{7} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + \zeta_{16}^{3} - 8 \zeta_{16}^{2} + 2 \zeta_{16} - 8) q^{74} + (2 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{2} - 2) q^{75} + ( - 2 \zeta_{16}^{6} - 3 \zeta_{16}^{4} + 4 \zeta_{16}^{3} - 4 \zeta_{16} + 3) q^{76} + (\zeta_{16}^{3} - \zeta_{16}) q^{77} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{3} + \zeta_{16}^{2} - 1) q^{78} + ( - 2 \zeta_{16}^{7} - 3 \zeta_{16}^{6} + 3 \zeta_{16}^{4} + 2 \zeta_{16}^{3}) q^{79} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{4} + 2 \zeta_{16}^{3} + 6 \zeta_{16}^{2} + 2 \zeta_{16} + 2) q^{80} - \zeta_{16}^{4} q^{81} + (2 \zeta_{16}^{7} - \zeta_{16}^{6} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2} + 2 \zeta_{16} + 1) q^{82} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} + 4 \zeta_{16}^{4} + 4 \zeta_{16}^{3} - 2 \zeta_{16}^{2} + 4 \zeta_{16} + 4) q^{83} + ( - \zeta_{16}^{7} - 2 \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{3} + 2 \zeta_{16}^{2} + \zeta_{16}) q^{84} + (6 \zeta_{16}^{6} + \zeta_{16}^{5} - \zeta_{16}^{3} - 6 \zeta_{16}^{2} - 4) q^{86} + (3 \zeta_{16}^{4} - 2 \zeta_{16}^{2} + 3) q^{87} + ( - \zeta_{16}^{6} + 2 \zeta_{16}^{5} - \zeta_{16}^{4} + \zeta_{16}^{2} - 2 \zeta_{16} + 1) q^{88} + (3 \zeta_{16}^{7} - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + 3 \zeta_{16}) q^{89} + (2 \zeta_{16}^{5} + \zeta_{16}^{4} + 2 \zeta_{16}^{3} + \zeta_{16}^{2} + 2 \zeta_{16}) q^{90} + (3 \zeta_{16}^{6} - 3 \zeta_{16}^{4} + 2 \zeta_{16}^{2} + 2 \zeta_{16} + 2) q^{91} + ( - 3 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + 4) q^{92} + (\zeta_{16}^{7} - 2 \zeta_{16}^{6} + \zeta_{16}^{5} + 4 \zeta_{16}^{4} - \zeta_{16}^{3} + \zeta_{16} - 4) q^{93} + ( - 6 \zeta_{16}^{7} + 4 \zeta_{16}^{6} - 6 \zeta_{16}^{5} - 4 \zeta_{16}^{3} + 4 \zeta_{16}) q^{94} + (3 \zeta_{16}^{7} - 3 \zeta_{16}^{6} - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{4} + 3 \zeta_{16}^{3} - 5 \zeta_{16}^{2} + \cdots + 5) q^{95} + \cdots + ( - \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{2} + \zeta_{16} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 8 q^{6} - 8 q^{11} + 16 q^{14} + 16 q^{16} - 8 q^{19} - 16 q^{20} + 8 q^{22} - 8 q^{23} - 8 q^{24} - 16 q^{25} + 16 q^{26} + 8 q^{28} - 8 q^{31} + 8 q^{33} + 32 q^{35} + 8 q^{36} + 8 q^{37} - 16 q^{39} + 8 q^{40} + 24 q^{41} + 8 q^{42} - 8 q^{43} + 8 q^{45} - 8 q^{49} - 32 q^{50} - 16 q^{52} - 32 q^{53} + 8 q^{54} + 16 q^{56} + 16 q^{57} - 24 q^{58} + 16 q^{59} - 8 q^{60} - 16 q^{61} - 16 q^{62} - 24 q^{65} - 16 q^{66} - 16 q^{67} - 24 q^{69} + 40 q^{70} + 16 q^{71} - 48 q^{73} - 64 q^{74} - 16 q^{75} + 24 q^{76} - 8 q^{78} + 16 q^{80} + 8 q^{82} + 32 q^{83} - 32 q^{86} + 24 q^{87} + 8 q^{88} + 16 q^{91} + 32 q^{92} - 32 q^{93} + 40 q^{95} + 16 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0.382683 + 0.923880i
−0.382683 0.923880i
0.382683 0.923880i
−0.382683 + 0.923880i
−0.923880 0.382683i
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.923880 0.382683i
−0.541196 + 0.541196i 0.382683 0.923880i 1.41421i 0.0761205 + 0.0315301i 0.292893 + 0.707107i −2.55487 + 1.05826i −1.84776 1.84776i −0.707107 0.707107i −0.0582601 + 0.0241321i
688.2 0.541196 0.541196i −0.382683 + 0.923880i 1.41421i 1.92388 + 0.796897i 0.292893 + 0.707107i 1.14065 0.472474i 1.84776 + 1.84776i −0.707107 0.707107i 1.47247 0.609919i
712.1 −0.541196 0.541196i 0.382683 + 0.923880i 1.41421i 0.0761205 0.0315301i 0.292893 0.707107i −2.55487 1.05826i −1.84776 + 1.84776i −0.707107 + 0.707107i −0.0582601 0.0241321i
712.2 0.541196 + 0.541196i −0.382683 0.923880i 1.41421i 1.92388 0.796897i 0.292893 0.707107i 1.14065 + 0.472474i 1.84776 1.84776i −0.707107 + 0.707107i 1.47247 + 0.609919i
733.1 −1.30656 1.30656i −0.923880 + 0.382683i 1.41421i 0.617317 + 1.49033i 1.70711 + 0.707107i −0.0582601 + 0.140652i −0.765367 + 0.765367i 0.707107 0.707107i 1.14065 2.75378i
733.2 1.30656 + 1.30656i 0.923880 0.382683i 1.41421i 1.38268 + 3.33809i 1.70711 + 0.707107i 1.47247 3.55487i 0.765367 0.765367i 0.707107 0.707107i −2.55487 + 6.16799i
757.1 −1.30656 + 1.30656i −0.923880 0.382683i 1.41421i 0.617317 1.49033i 1.70711 0.707107i −0.0582601 0.140652i −0.765367 0.765367i 0.707107 + 0.707107i 1.14065 + 2.75378i
757.2 1.30656 1.30656i 0.923880 + 0.382683i 1.41421i 1.38268 3.33809i 1.70711 0.707107i 1.47247 + 3.55487i 0.765367 + 0.765367i 0.707107 + 0.707107i −2.55487 6.16799i
\(n\): 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.d even 8 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{8} + 12T_{2}^{4} + 4 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{7} + 40T_{5}^{6} - 120T_{5}^{5} + 224T_{5}^{4} - 264T_{5}^{3} + 184T_{5}^{2} - 24T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{7} + 40 T^{6} - 120 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{6} + 40 T^{5} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + 16 T^{6} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + 646 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 134689 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + 88 T^{6} + \cdots + 73441 \) Copy content Toggle raw display
$29$ \( T^{8} - 48 T^{6} + 1152 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + 104 T^{6} + \cdots + 399424 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} - 68 T^{6} + \cdots + 498436 \) Copy content Toggle raw display
$41$ \( T^{8} - 24 T^{7} + 184 T^{6} - 264 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 1682209 \) Copy content Toggle raw display
$47$ \( T^{8} + 208 T^{6} + 13408 T^{4} + \cdots + 565504 \) Copy content Toggle raw display
$53$ \( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$59$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 264196 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + 324 T^{6} + \cdots + 1110916 \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + 4 T^{2} - 16 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 16 T^{7} - 32 T^{6} + \cdots + 4624 \) Copy content Toggle raw display
$73$ \( T^{8} + 48 T^{7} + 1200 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$79$ \( T^{8} + 36 T^{6} - 408 T^{5} + \cdots + 3844 \) Copy content Toggle raw display
$83$ \( T^{8} - 32 T^{7} + 512 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$89$ \( T^{8} + 296 T^{6} + 23956 T^{4} + \cdots + 6543364 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} - 24 T^{6} + \cdots + 399424 \) Copy content Toggle raw display
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