Properties

Label 1100.2.cb.a
Level $1100$
Weight $2$
Character orbit 1100.cb
Analytic conductor $8.784$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,2,Mod(49,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.cb (of order \(10\), 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: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{3}+ \cdots + (3 \zeta_{20}^{4} - \zeta_{20}^{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{3}+ \cdots + (10 \zeta_{20}^{6} - 5 \zeta_{20}^{4} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} - 8 q^{11} - 14 q^{19} + 36 q^{21} + 10 q^{29} - 10 q^{31} - 18 q^{39} - 10 q^{41} + 24 q^{49} - 2 q^{51} - 34 q^{59} - 2 q^{61} - 16 q^{69} + 10 q^{71} - 6 q^{79} - 32 q^{81} + 32 q^{89} - 54 q^{91} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(\zeta_{20}^{4}\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
0 −0.224514 + 0.309017i 0 0 0 1.67760 + 2.30902i 0 0.881966 + 2.71441i 0
49.2 0 0.224514 0.309017i 0 0 0 −1.67760 2.30902i 0 0.881966 + 2.71441i 0
449.1 0 −0.224514 0.309017i 0 0 0 1.67760 2.30902i 0 0.881966 2.71441i 0
449.2 0 0.224514 + 0.309017i 0 0 0 −1.67760 + 2.30902i 0 0.881966 2.71441i 0
949.1 0 −2.48990 + 0.809017i 0 0 0 −3.66547 1.19098i 0 3.11803 2.26538i 0
949.2 0 2.48990 0.809017i 0 0 0 3.66547 + 1.19098i 0 3.11803 2.26538i 0
1049.1 0 −2.48990 0.809017i 0 0 0 −3.66547 + 1.19098i 0 3.11803 + 2.26538i 0
1049.2 0 2.48990 + 0.809017i 0 0 0 3.66547 1.19098i 0 3.11803 + 2.26538i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.cb.a 8
5.b even 2 1 inner 1100.2.cb.a 8
5.c odd 4 1 44.2.e.a 4
5.c odd 4 1 1100.2.n.a 4
11.c even 5 1 inner 1100.2.cb.a 8
15.e even 4 1 396.2.j.a 4
20.e even 4 1 176.2.m.b 4
40.i odd 4 1 704.2.m.e 4
40.k even 4 1 704.2.m.d 4
55.e even 4 1 484.2.e.c 4
55.j even 10 1 inner 1100.2.cb.a 8
55.k odd 20 1 44.2.e.a 4
55.k odd 20 1 484.2.a.b 2
55.k odd 20 2 484.2.e.e 4
55.k odd 20 1 1100.2.n.a 4
55.l even 20 1 484.2.a.c 2
55.l even 20 1 484.2.e.c 4
55.l even 20 2 484.2.e.d 4
165.u odd 20 1 4356.2.a.u 2
165.v even 20 1 396.2.j.a 4
165.v even 20 1 4356.2.a.t 2
220.v even 20 1 176.2.m.b 4
220.v even 20 1 1936.2.a.ba 2
220.w odd 20 1 1936.2.a.z 2
440.bp odd 20 1 704.2.m.e 4
440.bp odd 20 1 7744.2.a.da 2
440.br odd 20 1 7744.2.a.bo 2
440.bs even 20 1 704.2.m.d 4
440.bs even 20 1 7744.2.a.bp 2
440.bu even 20 1 7744.2.a.db 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.e.a 4 5.c odd 4 1
44.2.e.a 4 55.k odd 20 1
176.2.m.b 4 20.e even 4 1
176.2.m.b 4 220.v even 20 1
396.2.j.a 4 15.e even 4 1
396.2.j.a 4 165.v even 20 1
484.2.a.b 2 55.k odd 20 1
484.2.a.c 2 55.l even 20 1
484.2.e.c 4 55.e even 4 1
484.2.e.c 4 55.l even 20 1
484.2.e.d 4 55.l even 20 2
484.2.e.e 4 55.k odd 20 2
704.2.m.d 4 40.k even 4 1
704.2.m.d 4 440.bs even 20 1
704.2.m.e 4 40.i odd 4 1
704.2.m.e 4 440.bp odd 20 1
1100.2.n.a 4 5.c odd 4 1
1100.2.n.a 4 55.k odd 20 1
1100.2.cb.a 8 1.a even 1 1 trivial
1100.2.cb.a 8 5.b even 2 1 inner
1100.2.cb.a 8 11.c even 5 1 inner
1100.2.cb.a 8 55.j even 10 1 inner
1936.2.a.z 2 220.w odd 20 1
1936.2.a.ba 2 220.v even 20 1
4356.2.a.t 2 165.v even 20 1
4356.2.a.u 2 165.u odd 20 1
7744.2.a.bo 2 440.br odd 20 1
7744.2.a.bp 2 440.bs even 20 1
7744.2.a.da 2 440.bp odd 20 1
7744.2.a.db 2 440.bu even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 11T_{3}^{6} + 46T_{3}^{4} + 4T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 19 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 31 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{8} - 31 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$19$ \( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 48 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 9 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( (T^{4} + 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 5 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$53$ \( T^{8} - 71 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$59$ \( (T^{4} + 17 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} + 16 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 192 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 89 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$79$ \( (T^{4} + 3 T^{3} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1908029761 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 4)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 81 T^{6} + \cdots + 43046721 \) Copy content Toggle raw display
show more
show less