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Properties

Label 99.3.k.a
Level $99$
Weight $3$
Character orbit 99.k
Analytic conductor $2.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,3,Mod(19,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 99.k (of order \(10\), degree \(4\), 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: \(2.69755461717\)
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: $\mathrm{SU}(2)[C_{10}]$

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

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{2} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} - 4 \zeta_{10}^{2} q^{5} + ( - 8 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 6) q^{7} + \cdots + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 3) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{2} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} - 4 \zeta_{10}^{2} q^{5} + ( - 8 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 6) q^{7} + \cdots + ( - 44 \zeta_{10}^{3} + 86 \zeta_{10}^{2} + \cdots + 65) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} - 9 q^{4} + 4 q^{5} + 10 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} - 9 q^{4} + 4 q^{5} + 10 q^{7} - 15 q^{8} - q^{11} - 20 q^{13} + 10 q^{14} + 19 q^{16} + 25 q^{19} - 44 q^{20} - 35 q^{22} + 20 q^{23} + 9 q^{25} + 10 q^{26} - 60 q^{28} + 40 q^{29} - 58 q^{31} + 130 q^{34} - 80 q^{35} + 90 q^{37} + 60 q^{38} - 60 q^{40} + 80 q^{41} - 24 q^{44} + 30 q^{46} + 30 q^{47} - 109 q^{49} + 45 q^{50} + 110 q^{52} - 120 q^{53} - 76 q^{55} - 100 q^{56} + 40 q^{58} - 23 q^{59} + 10 q^{61} - 200 q^{62} - 149 q^{64} - 230 q^{67} + 260 q^{68} - 40 q^{70} - 148 q^{71} + 300 q^{73} + 270 q^{74} + 200 q^{77} + 70 q^{79} + 84 q^{80} + 25 q^{82} - 225 q^{83} + 260 q^{85} - 175 q^{86} + 55 q^{88} - 122 q^{89} - 80 q^{91} - 40 q^{92} + 120 q^{94} + 100 q^{95} - 165 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(\zeta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.309017 + 0.951057i
0.809017 − 0.587785i
0.809017 + 0.587785i
−0.309017 − 0.951057i
1.80902 + 2.48990i 0 −1.69098 + 5.20431i 3.23607 + 2.35114i 0 −0.854102 − 0.277515i −4.30902 + 1.40008i 0 12.3107i
28.1 0.690983 − 0.224514i 0 −2.80902 + 2.04087i −1.23607 + 3.80423i 0 5.85410 + 8.05748i −3.19098 + 4.39201i 0 2.90617i
46.1 0.690983 + 0.224514i 0 −2.80902 − 2.04087i −1.23607 − 3.80423i 0 5.85410 − 8.05748i −3.19098 − 4.39201i 0 − 2.90617i
73.1 1.80902 − 2.48990i 0 −1.69098 − 5.20431i 3.23607 − 2.35114i 0 −0.854102 + 0.277515i −4.30902 − 1.40008i 0 − 12.3107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.k.a 4
3.b odd 2 1 11.3.d.a ✓ 4
11.c even 5 1 1089.3.c.e 4
11.d odd 10 1 inner 99.3.k.a 4
11.d odd 10 1 1089.3.c.e 4
12.b even 2 1 176.3.n.a 4
15.d odd 2 1 275.3.x.e 4
15.e even 4 2 275.3.q.d 8
33.d even 2 1 121.3.d.d 4
33.f even 10 1 11.3.d.a ✓ 4
33.f even 10 1 121.3.b.b 4
33.f even 10 1 121.3.d.a 4
33.f even 10 1 121.3.d.c 4
33.h odd 10 1 121.3.b.b 4
33.h odd 10 1 121.3.d.a 4
33.h odd 10 1 121.3.d.c 4
33.h odd 10 1 121.3.d.d 4
132.n odd 10 1 176.3.n.a 4
165.r even 10 1 275.3.x.e 4
165.u odd 20 2 275.3.q.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a ✓ 4 3.b odd 2 1
11.3.d.a ✓ 4 33.f even 10 1
99.3.k.a 4 1.a even 1 1 trivial
99.3.k.a 4 11.d odd 10 1 inner
121.3.b.b 4 33.f even 10 1
121.3.b.b 4 33.h odd 10 1
121.3.d.a 4 33.f even 10 1
121.3.d.a 4 33.h odd 10 1
121.3.d.c 4 33.f even 10 1
121.3.d.c 4 33.h odd 10 1
121.3.d.d 4 33.d even 2 1
121.3.d.d 4 33.h odd 10 1
176.3.n.a 4 12.b even 2 1
176.3.n.a 4 132.n odd 10 1
275.3.q.d 8 15.e even 4 2
275.3.q.d 8 165.u odd 20 2
275.3.x.e 4 15.d odd 2 1
275.3.x.e 4 165.r even 10 1
1089.3.c.e 4 11.c even 5 1
1089.3.c.e 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{3} + 15T_{2}^{2} - 15T_{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{4} - 10 T^{3} + \cdots + 80 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 20 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
$17$ \( T^{4} + 10985 T + 142805 \) Copy content Toggle raw display
$19$ \( T^{4} - 25 T^{3} + \cdots + 605 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 40 T^{3} + \cdots + 9680 \) Copy content Toggle raw display
$31$ \( T^{4} + 58 T^{3} + \cdots + 55696 \) Copy content Toggle raw display
$37$ \( T^{4} - 90 T^{3} + \cdots + 2624400 \) Copy content Toggle raw display
$41$ \( T^{4} - 80 T^{3} + \cdots + 8405 \) Copy content Toggle raw display
$43$ \( T^{4} + 1625 T^{2} + 581405 \) Copy content Toggle raw display
$47$ \( T^{4} - 30 T^{3} + \cdots + 384400 \) Copy content Toggle raw display
$53$ \( T^{4} + 120 T^{3} + \cdots + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 23 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + \cdots + 403280 \) Copy content Toggle raw display
$67$ \( (T^{2} + 115 T + 2945)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 148 T^{3} + \cdots + 22619536 \) Copy content Toggle raw display
$73$ \( T^{4} - 300 T^{3} + \cdots + 93787805 \) Copy content Toggle raw display
$79$ \( T^{4} - 70 T^{3} + \cdots + 67280 \) Copy content Toggle raw display
$83$ \( T^{4} + 225 T^{3} + \cdots + 22281605 \) Copy content Toggle raw display
$89$ \( (T^{2} + 61 T - 7681)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 165 T^{3} + \cdots + 31416025 \) Copy content Toggle raw display
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