Properties

Label 90.2.l.a
Level $90$
Weight $2$
Character orbit 90.l
Analytic conductor $0.719$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,2,Mod(23,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.l (of order \(12\), degree \(4\), 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: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{2} + 1) q^{3} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{3} + 1) q^{5} + (\zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} - 1) q^{6} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2}) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + 2 \zeta_{24}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{2} + 1) q^{3} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{3} + 1) q^{5} + (\zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} - 1) q^{6} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2}) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + 2 \zeta_{24}) q^{9} + (2 \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{3} + 1) q^{10} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{4} + 2 \zeta_{24} - 4) q^{11} + ( - \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{4} - \zeta_{24}^{2}) q^{12} + (2 \zeta_{24}^{5} - 4 \zeta_{24}) q^{13} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + \zeta_{24}^{2}) q^{14} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{2} - \zeta_{24} + 2) q^{15} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} - 1) q^{17} + (\zeta_{24}^{5} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{2} - 2) q^{18} + ( - 2 \zeta_{24}^{7} - 4 \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}) q^{19} + (\zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{2} - \zeta_{24}) q^{20} + (2 \zeta_{24}^{7} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{3} + \zeta_{24} - 3) q^{21} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} - 2) q^{22} - \zeta_{24}^{5} q^{23} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{24} + ( - 2 \zeta_{24}^{7} - 4 \zeta_{24}^{5} + \zeta_{24}^{4} + 4 \zeta_{24}^{3} + 2 \zeta_{24}) q^{25} + ( - 4 \zeta_{24}^{4} + 2) q^{26} + (\zeta_{24}^{6} + 5 \zeta_{24}^{3} - 1) q^{27} + (2 \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24} + 2) q^{28} + (\zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24}) q^{29} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2 \zeta_{24} - 1) q^{30} + (\zeta_{24}^{7} + \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24}^{3} - 2 \zeta_{24} + 2) q^{31} + (\zeta_{24}^{7} - \zeta_{24}^{3}) q^{32} + (4 \zeta_{24}^{6} - 6 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{2} + 2 \zeta_{24}) q^{33} + (\zeta_{24}^{7} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} - \zeta_{24}) q^{34} + (3 \zeta_{24}^{6} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{2} - 4 \zeta_{24} + 1) q^{35} + ( - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24} - 1) q^{36} + (3 \zeta_{24}^{6} - 3) q^{37} + (\zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} + 4 \zeta_{24} + 2) q^{38} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{39} + ( - \zeta_{24}^{5} - \zeta_{24}^{4} - \zeta_{24}^{2} + 2 \zeta_{24} + 1) q^{40} + ( - 4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} + \zeta_{24}^{4} + 4 \zeta_{24}^{3} + 1) q^{41} + ( - 5 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + \zeta_{24}^{4} + 2 \zeta_{24}^{3} + \zeta_{24}^{2} + 2) q^{42} + (4 \zeta_{24}^{5} - 2 \zeta_{24}) q^{43} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{2} - 2 \zeta_{24}) q^{44} + (4 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 3 \zeta_{24} + 4) q^{45} + q^{46} + 9 \zeta_{24}^{7} q^{47} + (\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24} + 1) q^{48} + (4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{2} + 8 \zeta_{24}) q^{49} + (\zeta_{24}^{7} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 2) q^{50} + (3 \zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{3} - 2 \zeta_{24} + 4) q^{51} + ( - 2 \zeta_{24}^{7} + 4 \zeta_{24}^{3}) q^{52} + ( - 3 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{4} + 6 \zeta_{24}^{2} - 2 \zeta_{24} + 3) q^{53} + ( - \zeta_{24}^{7} + 5 \zeta_{24}^{6} - 5 \zeta_{24}^{2} - \zeta_{24}) q^{54} + ( - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + 4 \zeta_{24} - 6) q^{55} + (2 \zeta_{24}^{7} + \zeta_{24}^{4} - 2 \zeta_{24} - 2) q^{56} + ( - 6 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{2} + 5) q^{57} + ( - \zeta_{24}^{5} + \zeta_{24}^{4} - \zeta_{24}^{2} + 2 \zeta_{24} - 1) q^{58} + ( - \zeta_{24}^{7} + 12 \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - 6 \zeta_{24}^{2}) q^{59} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{60} + (\zeta_{24}^{7} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + \zeta_{24}) q^{61} + (\zeta_{24}^{6} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 1) q^{62} + ( - 4 \zeta_{24}^{6} - 7 \zeta_{24}^{5} - 2 \zeta_{24} - 4) q^{63} - \zeta_{24}^{6} q^{64} + (2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 6 \zeta_{24} - 4) q^{65} + (4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{3} + 4) q^{66} + ( - 3 \zeta_{24}^{7} - \zeta_{24}^{4} - 3 \zeta_{24}^{3} - \zeta_{24}^{2} + 1) q^{67} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{2} - 1) q^{68} + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} - \zeta_{24}) q^{69} + ( - \zeta_{24}^{7} - 3 \zeta_{24}^{6} - 6 \zeta_{24}^{5} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{3} + 3 \zeta_{24}^{2} + 3 \zeta_{24}) q^{70} + ( - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24} + 2) q^{71} + ( - \zeta_{24}^{7} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2}) q^{72} + ( - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{5} - 4 \zeta_{24} - 2) q^{73} + ( - 3 \zeta_{24}^{7} - 3 \zeta_{24}) q^{74} + ( - 8 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{2} - \zeta_{24} - 1) q^{75} + (\zeta_{24}^{7} + \zeta_{24}^{5} + 4 \zeta_{24}^{4} + \zeta_{24}^{3} - 2 \zeta_{24} - 4) q^{76} + (2 \zeta_{24}^{7} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 2) q^{77} + ( - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 4 \zeta_{24} - 2) q^{78} + (\zeta_{24}^{7} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} - \zeta_{24}) q^{79} + ( - \zeta_{24}^{5} + 2 \zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24} - 1) q^{80} + ( - 4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} + 7 \zeta_{24}^{4} + 4 \zeta_{24}^{3} - 7) q^{81} + (2 \zeta_{24}^{7} + 4 \zeta_{24}^{6} - \zeta_{24}^{3} - 4) q^{82} + (\zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} + 3 \zeta_{24} + 2) q^{83} + (3 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + 3 \zeta_{24}^{2} + 2 \zeta_{24}) q^{84} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + \zeta_{24}^{2} + 4 \zeta_{24} - 5) q^{85} + ( - 2 \zeta_{24}^{4} - 2) q^{86} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{3} - 2 \zeta_{24}^{2}) q^{87} + ( - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{88} + (3 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{2} - 2 \zeta_{24}) q^{89} + (4 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{4} - \zeta_{24}^{2} - \zeta_{24} - 4) q^{90} + ( - 8 \zeta_{24}^{7} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{3} + 4 \zeta_{24} + 6) q^{91} + \zeta_{24}^{7} q^{92} + ( - 3 \zeta_{24}^{6} + 4 \zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{2} - 2) q^{93} - 9 \zeta_{24}^{2} q^{94} + (\zeta_{24}^{7} - 8 \zeta_{24}^{6} - 7 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} + 4 \zeta_{24}^{2} - 2) q^{95} + (\zeta_{24}^{7} - \zeta_{24}^{4} - \zeta_{24}) q^{96} + (4 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{4} - 8 \zeta_{24}^{3} + 3 \zeta_{24}^{2}) q^{97} + (4 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{4} - 8 \zeta_{24}^{2} - 4 \zeta_{24} - 4) q^{98} + ( - 6 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{2} - 8 \zeta_{24}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 12 q^{5} - 8 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 12 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{10} - 24 q^{11} - 4 q^{12} + 16 q^{15} + 4 q^{16} - 8 q^{18} - 32 q^{21} - 8 q^{22} + 4 q^{25} - 8 q^{27} + 16 q^{28} - 12 q^{30} + 8 q^{31} + 16 q^{33} - 8 q^{36} - 24 q^{37} + 12 q^{38} + 4 q^{40} + 12 q^{41} + 20 q^{42} + 32 q^{45} + 8 q^{46} + 8 q^{48} + 24 q^{50} + 32 q^{51} - 48 q^{55} - 12 q^{56} + 28 q^{57} - 4 q^{58} - 8 q^{60} + 12 q^{61} - 32 q^{63} - 24 q^{65} + 40 q^{66} + 4 q^{67} - 12 q^{68} - 16 q^{70} + 8 q^{72} - 16 q^{73} + 8 q^{75} - 16 q^{76} + 24 q^{77} - 24 q^{78} - 28 q^{81} - 32 q^{82} + 12 q^{83} - 20 q^{85} - 24 q^{86} - 8 q^{87} - 8 q^{88} - 20 q^{90} + 48 q^{91} - 20 q^{93} - 24 q^{95} - 4 q^{96} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 - \zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.258819 + 0.965926i 1.62484 + 0.599900i −0.866025 0.500000i 2.20711 0.358719i −1.00000 + 1.41421i −4.40508 1.18034i 0.707107 0.707107i 2.28024 + 1.94949i −0.224745 + 2.22474i
23.2 0.258819 0.965926i 1.10721 1.33195i −0.866025 0.500000i 0.792893 + 2.09077i −1.00000 1.41421i −1.05902 0.283763i −0.707107 + 0.707107i −0.548188 2.94949i 2.22474 0.224745i
47.1 −0.258819 0.965926i 1.62484 0.599900i −0.866025 + 0.500000i 2.20711 + 0.358719i −1.00000 1.41421i −4.40508 + 1.18034i 0.707107 + 0.707107i 2.28024 1.94949i −0.224745 2.22474i
47.2 0.258819 + 0.965926i 1.10721 + 1.33195i −0.866025 + 0.500000i 0.792893 2.09077i −1.00000 + 1.41421i −1.05902 + 0.283763i −0.707107 0.707107i −0.548188 + 2.94949i 2.22474 + 0.224745i
77.1 −0.965926 0.258819i 0.599900 1.62484i 0.866025 + 0.500000i 0.792893 + 2.09077i −1.00000 + 1.41421i 1.18034 4.40508i −0.707107 0.707107i −2.28024 1.94949i −0.224745 2.22474i
77.2 0.965926 + 0.258819i −1.33195 1.10721i 0.866025 + 0.500000i 2.20711 0.358719i −1.00000 1.41421i 0.283763 1.05902i 0.707107 + 0.707107i 0.548188 + 2.94949i 2.22474 + 0.224745i
83.1 −0.965926 + 0.258819i 0.599900 + 1.62484i 0.866025 0.500000i 0.792893 2.09077i −1.00000 1.41421i 1.18034 + 4.40508i −0.707107 + 0.707107i −2.28024 + 1.94949i −0.224745 + 2.22474i
83.2 0.965926 0.258819i −1.33195 + 1.10721i 0.866025 0.500000i 2.20711 + 0.358719i −1.00000 + 1.41421i 0.283763 + 1.05902i 0.707107 0.707107i 0.548188 2.94949i 2.22474 0.224745i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.2.l.a 8
3.b odd 2 1 270.2.m.a 8
4.b odd 2 1 720.2.cu.a 8
5.b even 2 1 450.2.p.a 8
5.c odd 4 1 inner 90.2.l.a 8
5.c odd 4 1 450.2.p.a 8
9.c even 3 1 270.2.m.a 8
9.c even 3 1 810.2.f.b 8
9.d odd 6 1 inner 90.2.l.a 8
9.d odd 6 1 810.2.f.b 8
15.d odd 2 1 1350.2.q.g 8
15.e even 4 1 270.2.m.a 8
15.e even 4 1 1350.2.q.g 8
20.e even 4 1 720.2.cu.a 8
36.h even 6 1 720.2.cu.a 8
45.h odd 6 1 450.2.p.a 8
45.j even 6 1 1350.2.q.g 8
45.k odd 12 1 270.2.m.a 8
45.k odd 12 1 810.2.f.b 8
45.k odd 12 1 1350.2.q.g 8
45.l even 12 1 inner 90.2.l.a 8
45.l even 12 1 450.2.p.a 8
45.l even 12 1 810.2.f.b 8
180.v odd 12 1 720.2.cu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.a 8 1.a even 1 1 trivial
90.2.l.a 8 5.c odd 4 1 inner
90.2.l.a 8 9.d odd 6 1 inner
90.2.l.a 8 45.l even 12 1 inner
270.2.m.a 8 3.b odd 2 1
270.2.m.a 8 9.c even 3 1
270.2.m.a 8 15.e even 4 1
270.2.m.a 8 45.k odd 12 1
450.2.p.a 8 5.b even 2 1
450.2.p.a 8 5.c odd 4 1
450.2.p.a 8 45.h odd 6 1
450.2.p.a 8 45.l even 12 1
720.2.cu.a 8 4.b odd 2 1
720.2.cu.a 8 20.e even 4 1
720.2.cu.a 8 36.h even 6 1
720.2.cu.a 8 180.v odd 12 1
810.2.f.b 8 9.c even 3 1
810.2.f.b 8 9.d odd 6 1
810.2.f.b 8 45.k odd 12 1
810.2.f.b 8 45.l even 12 1
1350.2.q.g 8 15.d odd 2 1
1350.2.q.g 8 15.e even 4 1
1350.2.q.g 8 45.j even 6 1
1350.2.q.g 8 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 8T_{7}^{7} + 32T_{7}^{6} + 176T_{7}^{5} + 679T_{7}^{4} + 880T_{7}^{3} + 800T_{7}^{2} + 1000T_{7} + 625 \) acting on \(S_{2}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} + 17 T^{2} - 30 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 32 T^{6} + 176 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{3} + 52 T^{2} + 48 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$17$ \( T^{8} + 392T^{4} + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 44 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{6} + 99 T^{4} + 10 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} - 17 T^{2} + 174 T + 841)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$47$ \( T^{8} - 6561 T^{4} + \cdots + 43046721 \) Copy content Toggle raw display
$53$ \( T^{8} + 8456 T^{4} + \cdots + 6250000 \) Copy content Toggle raw display
$59$ \( T^{8} + 220 T^{6} + \cdots + 126247696 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$71$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 320 T + 1600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + 72 T^{6} - 288 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( (T^{4} - 70 T^{2} + 361)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 810000 \) Copy content Toggle raw display
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