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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [891,2,Mod(82,891)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("891.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 2) q^{2} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{4} + ( - 2 \zeta_{10}^{2} - 2) q^{5} + \zeta_{10}^{3} q^{7} + ( - 4 \zeta_{10}^{3} + \cdots - 4 \zeta_{10}) q^{8}+ \cdots + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{4} - 6 q^{5} + q^{7} - 7 q^{8} - 16 q^{10} - q^{11} - 8 q^{13} + q^{14} - 14 q^{16} + 6 q^{17} - q^{19} - 24 q^{20} - 11 q^{22} + 14 q^{23} + 9 q^{25} - 8 q^{26} + 9 q^{28} + 9 q^{29}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.309017 + 0.951057i
−0.309017 − 0.951057i
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.118034 − 0.363271i 0 1.50000 − 1.08981i −0.381966 + 1.17557i 0 0.809017 − 0.587785i −1.19098 − 0.865300i 0 0.472136
163.1 −0.118034 + 0.363271i 0 1.50000 + 1.08981i −0.381966 − 1.17557i 0 0.809017 + 0.587785i −1.19098 + 0.865300i 0 0.472136
487.1 2.11803 − 1.53884i 0 1.50000 − 4.61653i −2.61803 − 1.90211i 0 −0.309017 + 0.951057i −2.30902 − 7.10642i 0 −8.47214
730.1 2.11803 + 1.53884i 0 1.50000 + 4.61653i −2.61803 + 1.90211i 0 −0.309017 − 0.951057i −2.30902 + 7.10642i 0 −8.47214
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.f.b 4
3.b odd 2 1 891.2.f.a 4
9.c even 3 2 99.2.m.a ✓ 8
9.d odd 6 2 297.2.n.a 8
11.c even 5 1 inner 891.2.f.b 4
11.c even 5 1 9801.2.a.n 2
11.d odd 10 1 9801.2.a.bc 2
33.f even 10 1 9801.2.a.m 2
33.h odd 10 1 891.2.f.a 4
33.h odd 10 1 9801.2.a.bb 2
99.m even 15 2 99.2.m.a ✓ 8
99.m even 15 2 1089.2.e.g 4
99.n odd 30 2 297.2.n.a 8
99.o odd 30 2 1089.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.a ✓ 8 9.c even 3 2
99.2.m.a ✓ 8 99.m even 15 2
297.2.n.a 8 9.d odd 6 2
297.2.n.a 8 99.n odd 30 2
891.2.f.a 4 3.b odd 2 1
891.2.f.a 4 33.h odd 10 1
891.2.f.b 4 1.a even 1 1 trivial
891.2.f.b 4 11.c even 5 1 inner
1089.2.e.d 4 99.o odd 30 2
1089.2.e.g 4 99.m even 15 2
9801.2.a.m 2 33.f even 10 1
9801.2.a.n 2 11.c even 5 1
9801.2.a.bb 2 33.h odd 10 1
9801.2.a.bc 2 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{3} + 6T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{2} - 7 T + 11)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( T^{4} + 3 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 23 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$53$ \( T^{4} - 14 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$59$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{4} + 90 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$67$ \( (T - 6)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 21 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{4} - 22 T^{3} + \cdots + 10201 \) Copy content Toggle raw display
$83$ \( T^{4} - 17 T^{3} + \cdots + 19321 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
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