↑

Properties

Label 264.1.r.a
Level $264$
Weight $1$
Character orbit 264.r
Analytic conductor $0.132$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -8
Inner twists $4$

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] = [264,1,Mod(35,264)] 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("264.35"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(264, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 5, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 264.r (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1] 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: \(0.131753163335\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.2346931359387648.3

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10}^{2} q^{2} + \zeta_{10}^{3} q^{3} + \zeta_{10}^{4} q^{4} - q^{6} - \zeta_{10} q^{8} - \zeta_{10} q^{9} + \zeta_{10} q^{11} - \zeta_{10}^{2} q^{12} - \zeta_{10}^{3} q^{16} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{17} + \cdots - \zeta_{10}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} - q^{4} - 4 q^{6} - q^{8} - q^{9} + q^{11} + q^{12} - q^{16} + 2 q^{17} - q^{18} - 5 q^{19} + q^{22} + q^{24} + q^{25} + q^{27} + 4 q^{32} - q^{33} + 2 q^{34} + 4 q^{36}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/264\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(133\) \(145\) \(199\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{10}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.309017 − 0.951057i −0.309017 − 0.951057i −0.809017 − 0.587785i 0 −1.00000 0 −0.809017 + 0.587785i −0.809017 + 0.587785i 0
83.1 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i 0 −1.00000 0 −0.809017 − 0.587785i −0.809017 − 0.587785i 0
107.1 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 − 0.951057i 0 −1.00000 0 0.309017 + 0.951057i 0.309017 + 0.951057i 0
227.1 −0.809017 − 0.587785i 0.809017 − 0.587785i 0.309017 + 0.951057i 0 −1.00000 0 0.309017 − 0.951057i 0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
33.f even 10 1 inner
264.r odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 264.1.r.a ✓ 4
3.b odd 2 1 264.1.r.b yes 4
4.b odd 2 1 1056.1.bh.a 4
8.b even 2 1 1056.1.bh.a 4
8.d odd 2 1 CM 264.1.r.a ✓ 4
11.b odd 2 1 2904.1.r.h 4
11.c even 5 1 2904.1.p.c 4
11.c even 5 1 2904.1.r.a 4
11.c even 5 1 2904.1.r.c 4
11.c even 5 1 2904.1.r.d 4
11.d odd 10 1 264.1.r.b yes 4
11.d odd 10 1 2904.1.p.a 4
11.d odd 10 1 2904.1.r.e 4
11.d odd 10 1 2904.1.r.g 4
12.b even 2 1 1056.1.bh.b 4
24.f even 2 1 264.1.r.b yes 4
24.h odd 2 1 1056.1.bh.b 4
33.d even 2 1 2904.1.r.d 4
33.f even 10 1 inner 264.1.r.a ✓ 4
33.f even 10 1 2904.1.p.c 4
33.f even 10 1 2904.1.r.a 4
33.f even 10 1 2904.1.r.c 4
33.h odd 10 1 2904.1.p.a 4
33.h odd 10 1 2904.1.r.e 4
33.h odd 10 1 2904.1.r.g 4
33.h odd 10 1 2904.1.r.h 4
44.g even 10 1 1056.1.bh.b 4
88.g even 2 1 2904.1.r.h 4
88.k even 10 1 264.1.r.b yes 4
88.k even 10 1 2904.1.p.a 4
88.k even 10 1 2904.1.r.e 4
88.k even 10 1 2904.1.r.g 4
88.l odd 10 1 2904.1.p.c 4
88.l odd 10 1 2904.1.r.a 4
88.l odd 10 1 2904.1.r.c 4
88.l odd 10 1 2904.1.r.d 4
88.p odd 10 1 1056.1.bh.b 4
132.n odd 10 1 1056.1.bh.a 4
264.p odd 2 1 2904.1.r.d 4
264.r odd 10 1 inner 264.1.r.a ✓ 4
264.r odd 10 1 2904.1.p.c 4
264.r odd 10 1 2904.1.r.a 4
264.r odd 10 1 2904.1.r.c 4
264.u even 10 1 1056.1.bh.a 4
264.w even 10 1 2904.1.p.a 4
264.w even 10 1 2904.1.r.e 4
264.w even 10 1 2904.1.r.g 4
264.w even 10 1 2904.1.r.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.1.r.a ✓ 4 1.a even 1 1 trivial
264.1.r.a ✓ 4 8.d odd 2 1 CM
264.1.r.a ✓ 4 33.f even 10 1 inner
264.1.r.a ✓ 4 264.r odd 10 1 inner
264.1.r.b yes 4 3.b odd 2 1
264.1.r.b yes 4 11.d odd 10 1
264.1.r.b yes 4 24.f even 2 1
264.1.r.b yes 4 88.k even 10 1
1056.1.bh.a 4 4.b odd 2 1
1056.1.bh.a 4 8.b even 2 1
1056.1.bh.a 4 132.n odd 10 1
1056.1.bh.a 4 264.u even 10 1
1056.1.bh.b 4 12.b even 2 1
1056.1.bh.b 4 24.h odd 2 1
1056.1.bh.b 4 44.g even 10 1
1056.1.bh.b 4 88.p odd 10 1
2904.1.p.a 4 11.d odd 10 1
2904.1.p.a 4 33.h odd 10 1
2904.1.p.a 4 88.k even 10 1
2904.1.p.a 4 264.w even 10 1
2904.1.p.c 4 11.c even 5 1
2904.1.p.c 4 33.f even 10 1
2904.1.p.c 4 88.l odd 10 1
2904.1.p.c 4 264.r odd 10 1
2904.1.r.a 4 11.c even 5 1
2904.1.r.a 4 33.f even 10 1
2904.1.r.a 4 88.l odd 10 1
2904.1.r.a 4 264.r odd 10 1
2904.1.r.c 4 11.c even 5 1
2904.1.r.c 4 33.f even 10 1
2904.1.r.c 4 88.l odd 10 1
2904.1.r.c 4 264.r odd 10 1
2904.1.r.d 4 11.c even 5 1
2904.1.r.d 4 33.d even 2 1
2904.1.r.d 4 88.l odd 10 1
2904.1.r.d 4 264.p odd 2 1
2904.1.r.e 4 11.d odd 10 1
2904.1.r.e 4 33.h odd 10 1
2904.1.r.e 4 88.k even 10 1
2904.1.r.e 4 264.w even 10 1
2904.1.r.g 4 11.d odd 10 1
2904.1.r.g 4 33.h odd 10 1
2904.1.r.g 4 88.k even 10 1
2904.1.r.g 4 264.w even 10 1
2904.1.r.h 4 11.b odd 2 1
2904.1.r.h 4 33.h odd 10 1
2904.1.r.h 4 88.g even 2 1
2904.1.r.h 4 264.w even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{4} - 2T_{17}^{3} + 4T_{17}^{2} - 3T_{17} + 1 \) acting on \(S_{1}^{\mathrm{new}}(264, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{4} + 5T^{2} + 5 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 5T + 5 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{4} + 5T^{2} + 5 \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
show more
show less