Properties

Label 121.3.d.b
Level $121$
Weight $3$
Character orbit 121.d
Analytic conductor $3.297$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,3,Mod(40,121)]
 
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([7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 121.d (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: \(3.29701119876\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

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

\(f(q)\) \(=\) \( q + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{3} - 4 \zeta_{10} q^{4} - \zeta_{10}^{2} q^{5} - 16 \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{3} - 4 \zeta_{10} q^{4} - \zeta_{10}^{2} q^{5} - 16 \zeta_{10}^{3} q^{9} - 20 q^{12} - 5 \zeta_{10} q^{15} + 16 \zeta_{10}^{2} q^{16} + 4 \zeta_{10}^{3} q^{20} + 35 q^{23} + ( - 24 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 24 \zeta_{10} + 24) q^{25} - 35 \zeta_{10}^{2} q^{27} + 37 \zeta_{10}^{3} q^{31} + (64 \zeta_{10}^{3} - 64 \zeta_{10}^{2} + 64 \zeta_{10} - 64) q^{36} + 25 \zeta_{10} q^{37} - 16 q^{45} + (50 \zeta_{10}^{3} - 50 \zeta_{10}^{2} + 50 \zeta_{10} - 50) q^{47} + 80 \zeta_{10} q^{48} + 49 \zeta_{10}^{2} q^{49} + 70 \zeta_{10}^{3} q^{53} - 107 \zeta_{10} q^{59} + 20 \zeta_{10}^{2} q^{60} - 64 \zeta_{10}^{3} q^{64} + 35 q^{67} + ( - 175 \zeta_{10}^{3} + 175 \zeta_{10}^{2} - 175 \zeta_{10} + 175) q^{69} - 133 \zeta_{10}^{2} q^{71} - 120 \zeta_{10}^{3} q^{75} + ( - 16 \zeta_{10}^{3} + 16 \zeta_{10}^{2} - 16 \zeta_{10} + 16) q^{80} - 31 \zeta_{10} q^{81} - 97 q^{89} - 140 \zeta_{10} q^{92} + 185 \zeta_{10}^{2} q^{93} - 95 \zeta_{10}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 4 q^{4} + q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 4 q^{4} + q^{5} - 16 q^{9} - 80 q^{12} - 5 q^{15} - 16 q^{16} + 4 q^{20} + 140 q^{23} + 24 q^{25} + 35 q^{27} + 37 q^{31} - 64 q^{36} + 25 q^{37} - 64 q^{45} - 50 q^{47} + 80 q^{48} - 49 q^{49} + 70 q^{53} - 107 q^{59} - 20 q^{60} - 64 q^{64} + 140 q^{67} + 175 q^{69} + 133 q^{71} - 120 q^{75} + 16 q^{80} - 31 q^{81} - 388 q^{89} - 140 q^{92} - 185 q^{93} - 95 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 −1.54508 + 4.75528i 1.23607 + 3.80423i 0.809017 0.587785i 0 0 0 −12.9443 9.40456i 0
94.1 0 4.04508 + 2.93893i −3.23607 + 2.35114i −0.309017 + 0.951057i 0 0 0 4.94427 + 15.2169i 0
112.1 0 4.04508 2.93893i −3.23607 2.35114i −0.309017 0.951057i 0 0 0 4.94427 15.2169i 0
118.1 0 −1.54508 4.75528i 1.23607 3.80423i 0.809017 + 0.587785i 0 0 0 −12.9443 + 9.40456i 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 121.3.d.b 4
11.b odd 2 1 CM 121.3.d.b 4
11.c even 5 1 11.3.b.a 1
11.c even 5 3 inner 121.3.d.b 4
11.d odd 10 1 11.3.b.a 1
11.d odd 10 3 inner 121.3.d.b 4
33.f even 10 1 99.3.c.a 1
33.h odd 10 1 99.3.c.a 1
44.g even 10 1 176.3.h.a 1
44.h odd 10 1 176.3.h.a 1
55.h odd 10 1 275.3.c.a 1
55.j even 10 1 275.3.c.a 1
55.k odd 20 2 275.3.d.a 2
55.l even 20 2 275.3.d.a 2
77.j odd 10 1 539.3.c.a 1
77.l even 10 1 539.3.c.a 1
88.k even 10 1 704.3.h.a 1
88.l odd 10 1 704.3.h.a 1
88.o even 10 1 704.3.h.b 1
88.p odd 10 1 704.3.h.b 1
132.n odd 10 1 1584.3.j.a 1
132.o even 10 1 1584.3.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.b.a 1 11.c even 5 1
11.3.b.a 1 11.d odd 10 1
99.3.c.a 1 33.f even 10 1
99.3.c.a 1 33.h odd 10 1
121.3.d.b 4 1.a even 1 1 trivial
121.3.d.b 4 11.b odd 2 1 CM
121.3.d.b 4 11.c even 5 3 inner
121.3.d.b 4 11.d odd 10 3 inner
176.3.h.a 1 44.g even 10 1
176.3.h.a 1 44.h odd 10 1
275.3.c.a 1 55.h odd 10 1
275.3.c.a 1 55.j even 10 1
275.3.d.a 2 55.k odd 20 2
275.3.d.a 2 55.l even 20 2
539.3.c.a 1 77.j odd 10 1
539.3.c.a 1 77.l even 10 1
704.3.h.a 1 88.k even 10 1
704.3.h.a 1 88.l odd 10 1
704.3.h.b 1 88.o even 10 1
704.3.h.b 1 88.p odd 10 1
1584.3.j.a 1 132.n odd 10 1
1584.3.j.a 1 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + T^{2} - T + 1 \) 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
$23$ \( (T - 35)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 37 T^{3} + 1369 T^{2} + \cdots + 1874161 \) Copy content Toggle raw display
$37$ \( T^{4} - 25 T^{3} + 625 T^{2} + \cdots + 390625 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 50 T^{3} + 2500 T^{2} + \cdots + 6250000 \) Copy content Toggle raw display
$53$ \( T^{4} - 70 T^{3} + 4900 T^{2} + \cdots + 24010000 \) Copy content Toggle raw display
$59$ \( T^{4} + 107 T^{3} + \cdots + 131079601 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T - 35)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 133 T^{3} + \cdots + 312900721 \) 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
$89$ \( (T + 97)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 95 T^{3} + 9025 T^{2} + \cdots + 81450625 \) Copy content Toggle raw display
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