↑

Properties

Label 484.2.e.d
Level $484$
Weight $2$
Character orbit 484.e
Analytic conductor $3.865$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [484,2,Mod(9,484)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(484, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("484.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,-2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.86475945783\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{5} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{7} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots + 1) q^{9}+ \cdots + (9 \zeta_{10}^{2} - 9 \zeta_{10}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 2 q^{5} - 8 q^{7} + 7 q^{9} + 6 q^{13} - 7 q^{15} + 6 q^{17} + 8 q^{19} - 18 q^{21} + 8 q^{23} + q^{25} - 5 q^{27} + 10 q^{29} + 10 q^{31} - q^{35} + 6 q^{37} + q^{39} + 10 q^{41} - 16 q^{45}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.309017 − 0.951057i
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 2.11803 − 1.53884i 0 −0.500000 − 1.53884i 0 −3.11803 − 2.26538i 0 1.19098 − 3.66547i 0
81.1 0 −0.118034 + 0.363271i 0 −0.500000 + 0.363271i 0 −0.881966 − 2.71441i 0 2.30902 + 1.67760i 0
245.1 0 −0.118034 − 0.363271i 0 −0.500000 − 0.363271i 0 −0.881966 + 2.71441i 0 2.30902 − 1.67760i 0
269.1 0 2.11803 + 1.53884i 0 −0.500000 + 1.53884i 0 −3.11803 + 2.26538i 0 1.19098 + 3.66547i 0
\(n\): e.g. 2-40 or 990-1000
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 484.2.e.d 4
11.b odd 2 1 484.2.e.e 4
11.c even 5 1 484.2.a.c 2
11.c even 5 2 484.2.e.c 4
11.c even 5 1 inner 484.2.e.d 4
11.d odd 10 2 44.2.e.a ✓ 4
11.d odd 10 1 484.2.a.b 2
11.d odd 10 1 484.2.e.e 4
33.f even 10 2 396.2.j.a 4
33.f even 10 1 4356.2.a.t 2
33.h odd 10 1 4356.2.a.u 2
44.g even 10 2 176.2.m.b 4
44.g even 10 1 1936.2.a.ba 2
44.h odd 10 1 1936.2.a.z 2
55.h odd 10 2 1100.2.n.a 4
55.l even 20 4 1100.2.cb.a 8
88.k even 10 2 704.2.m.d 4
88.k even 10 1 7744.2.a.bp 2
88.l odd 10 1 7744.2.a.bo 2
88.o even 10 1 7744.2.a.db 2
88.p odd 10 2 704.2.m.e 4
88.p odd 10 1 7744.2.a.da 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.e.a ✓ 4 11.d odd 10 2
176.2.m.b 4 44.g even 10 2
396.2.j.a 4 33.f even 10 2
484.2.a.b 2 11.d odd 10 1
484.2.a.c 2 11.c even 5 1
484.2.e.c 4 11.c even 5 2
484.2.e.d 4 1.a even 1 1 trivial
484.2.e.d 4 11.c even 5 1 inner
484.2.e.e 4 11.b odd 2 1
484.2.e.e 4 11.d odd 10 1
704.2.m.d 4 88.k even 10 2
704.2.m.e 4 88.p odd 10 2
1100.2.n.a 4 55.h odd 10 2
1100.2.cb.a 8 55.l even 20 4
1936.2.a.z 2 44.h odd 10 1
1936.2.a.ba 2 44.g even 10 1
4356.2.a.t 2 33.f even 10 1
4356.2.a.u 2 33.h odd 10 1
7744.2.a.bo 2 88.l odd 10 1
7744.2.a.bp 2 88.k even 10 1
7744.2.a.da 2 88.p odd 10 1
7744.2.a.db 2 88.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(484, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{3} + 6T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} + 34T_{7}^{2} + 77T_{7} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 10 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{4} + 18 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( T^{4} + 34 T^{3} + \cdots + 43681 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 18 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
show more
show less