↑

Properties

Label 121.2.c.c
Level $121$
Weight $2$
Character orbit 121.c
Analytic conductor $0.966$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,2,Mod(3,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), 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: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 2) q^{9} + \cdots + (17 \zeta_{10}^{3} - 17 \zeta_{10}^{2} + \cdots - 17) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9} + 8 q^{12} - 3 q^{15} - 4 q^{16} - 6 q^{20} - 36 q^{23} - 4 q^{25} - 5 q^{27} + 5 q^{31} - 4 q^{36} - 7 q^{37} + 24 q^{45} + 12 q^{47} + 4 q^{48} + 7 q^{49} - 6 q^{53} + 15 q^{59} + 6 q^{60} + 8 q^{64} + 52 q^{67} - 9 q^{69} + 3 q^{71} + 4 q^{75} + 12 q^{80} - q^{81} - 36 q^{89} - 18 q^{92} - 5 q^{93} - 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 + 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.809017 − 0.587785i
0 −0.309017 − 0.951057i −0.618034 + 1.90211i 2.42705 + 1.76336i 0 0 0 1.61803 − 1.17557i 0
9.1 0 0.809017 − 0.587785i 1.61803 + 1.17557i −0.927051 − 2.85317i 0 0 0 −0.618034 + 1.90211i 0
27.1 0 0.809017 + 0.587785i 1.61803 − 1.17557i −0.927051 + 2.85317i 0 0 0 −0.618034 − 1.90211i 0
81.1 0 −0.309017 + 0.951057i −0.618034 − 1.90211i 2.42705 − 1.76336i 0 0 0 1.61803 + 1.17557i 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.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.c.c 4
11.b odd 2 1 CM 121.2.c.c 4
11.c even 5 1 121.2.a.b ✓ 1
11.c even 5 3 inner 121.2.c.c 4
11.d odd 10 1 121.2.a.b ✓ 1
11.d odd 10 3 inner 121.2.c.c 4
33.f even 10 1 1089.2.a.g 1
33.h odd 10 1 1089.2.a.g 1
44.g even 10 1 1936.2.a.h 1
44.h odd 10 1 1936.2.a.h 1
55.h odd 10 1 3025.2.a.d 1
55.j even 10 1 3025.2.a.d 1
77.j odd 10 1 5929.2.a.e 1
77.l even 10 1 5929.2.a.e 1
88.k even 10 1 7744.2.a.n 1
88.l odd 10 1 7744.2.a.n 1
88.o even 10 1 7744.2.a.bb 1
88.p odd 10 1 7744.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.b ✓ 1 11.c even 5 1
121.2.a.b ✓ 1 11.d odd 10 1
121.2.c.c 4 1.a even 1 1 trivial
121.2.c.c 4 11.b odd 2 1 CM
121.2.c.c 4 11.c even 5 3 inner
121.2.c.c 4 11.d odd 10 3 inner
1089.2.a.g 1 33.f even 10 1
1089.2.a.g 1 33.h odd 10 1
1936.2.a.h 1 44.g even 10 1
1936.2.a.h 1 44.h odd 10 1
3025.2.a.d 1 55.h odd 10 1
3025.2.a.d 1 55.j even 10 1
5929.2.a.e 1 77.j odd 10 1
5929.2.a.e 1 77.l even 10 1
7744.2.a.n 1 88.k even 10 1
7744.2.a.n 1 88.l odd 10 1
7744.2.a.bb 1 88.o even 10 1
7744.2.a.bb 1 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T + 9)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 5 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 15 T^{3} + \cdots + 50625 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T - 13)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T + 9)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 17 T^{3} + \cdots + 83521 \) Copy content Toggle raw display
show more
show less