Properties

Label 121.4.c.a
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
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] = [121,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(7.13923111069\)
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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + 8 \zeta_{10}^{2} q^{3} + 8 \zeta_{10}^{3} q^{4} - 18 \zeta_{10} q^{5} + (37 \zeta_{10}^{3} - 37 \zeta_{10}^{2} + 37 \zeta_{10} - 37) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 8 \zeta_{10}^{2} q^{3} + 8 \zeta_{10}^{3} q^{4} - 18 \zeta_{10} q^{5} + (37 \zeta_{10}^{3} - 37 \zeta_{10}^{2} + 37 \zeta_{10} - 37) q^{9} - 64 q^{12} - 144 \zeta_{10}^{3} q^{15} - 64 \zeta_{10} q^{16} + ( - 144 \zeta_{10}^{3} + 144 \zeta_{10}^{2} - 144 \zeta_{10} + 144) q^{20} - 108 q^{23} + 199 \zeta_{10}^{2} q^{25} - 80 \zeta_{10} q^{27} + (340 \zeta_{10}^{3} - 340 \zeta_{10}^{2} + 340 \zeta_{10} - 340) q^{31} - 296 \zeta_{10}^{2} q^{36} + 434 \zeta_{10}^{3} q^{37} + 666 q^{45} - 36 \zeta_{10}^{2} q^{47} - 512 \zeta_{10}^{3} q^{48} + 343 \zeta_{10} q^{49} + ( - 738 \zeta_{10}^{3} + 738 \zeta_{10}^{2} - 738 \zeta_{10} + 738) q^{53} + 720 \zeta_{10}^{3} q^{59} + 1152 \zeta_{10} q^{60} + ( - 512 \zeta_{10}^{3} + 512 \zeta_{10}^{2} - 512 \zeta_{10} + 512) q^{64} - 416 q^{67} - 864 \zeta_{10}^{2} q^{69} - 612 \zeta_{10} q^{71} + (1592 \zeta_{10}^{3} - 1592 \zeta_{10}^{2} + 1592 \zeta_{10} - 1592) q^{75} + 1152 \zeta_{10}^{2} q^{80} + 359 \zeta_{10}^{3} q^{81} + 1674 q^{89} - 864 \zeta_{10}^{3} q^{92} - 2720 \zeta_{10} q^{93} + ( - 34 \zeta_{10}^{3} + 34 \zeta_{10}^{2} - 34 \zeta_{10} + 34) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9} - 256 q^{12} - 144 q^{15} - 64 q^{16} + 144 q^{20} - 432 q^{23} - 199 q^{25} - 80 q^{27} - 340 q^{31} + 296 q^{36} + 434 q^{37} + 2664 q^{45} + 36 q^{47} - 512 q^{48} + 343 q^{49} + 738 q^{53} + 720 q^{59} + 1152 q^{60} + 512 q^{64} - 1664 q^{67} + 864 q^{69} - 612 q^{71} - 1592 q^{75} - 1152 q^{80} + 359 q^{81} + 6696 q^{89} - 864 q^{92} - 2720 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 2.47214 + 7.60845i −2.47214 + 7.60845i −14.5623 10.5801i 0 0 0 −29.9336 + 21.7481i 0
9.1 0 −6.47214 + 4.70228i 6.47214 + 4.70228i 5.56231 + 17.1190i 0 0 0 11.4336 35.1891i 0
27.1 0 −6.47214 4.70228i 6.47214 4.70228i 5.56231 17.1190i 0 0 0 11.4336 + 35.1891i 0
81.1 0 2.47214 7.60845i −2.47214 7.60845i −14.5623 + 10.5801i 0 0 0 −29.9336 21.7481i 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 121.4.c.a 4
11.b odd 2 1 CM 121.4.c.a 4
11.c even 5 1 121.4.a.a 1
11.c even 5 3 inner 121.4.c.a 4
11.d odd 10 1 121.4.a.a 1
11.d odd 10 3 inner 121.4.c.a 4
33.f even 10 1 1089.4.a.f 1
33.h odd 10 1 1089.4.a.f 1
44.g even 10 1 1936.4.a.a 1
44.h odd 10 1 1936.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.a 1 11.c even 5 1
121.4.a.a 1 11.d odd 10 1
121.4.c.a 4 1.a even 1 1 trivial
121.4.c.a 4 11.b odd 2 1 CM
121.4.c.a 4 11.c even 5 3 inner
121.4.c.a 4 11.d odd 10 3 inner
1089.4.a.f 1 33.f even 10 1
1089.4.a.f 1 33.h odd 10 1
1936.4.a.a 1 44.g even 10 1
1936.4.a.a 1 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( T^{4} + 18 T^{3} + 324 T^{2} + \cdots + 104976 \) 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
$23$ \( (T + 108)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 340 T^{3} + \cdots + 13363360000 \) Copy content Toggle raw display
$37$ \( T^{4} - 434 T^{3} + \cdots + 35477982736 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 36 T^{3} + 1296 T^{2} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( T^{4} - 738 T^{3} + \cdots + 296637086736 \) Copy content Toggle raw display
$59$ \( T^{4} - 720 T^{3} + \cdots + 268738560000 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T + 416)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 612 T^{3} + \cdots + 140283207936 \) 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
$89$ \( (T - 1674)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 34 T^{3} + 1156 T^{2} + \cdots + 1336336 \) Copy content Toggle raw display
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