Properties

Label 99.2.f.a
Level $99$
Weight $2$
Character orbit 99.f
Analytic conductor $0.791$
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] = [99,2,Mod(37,99)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), 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: \(0.790518980011\)
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 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{2} + (3 \zeta_{10} - 3) q^{4} + (\zeta_{10}^{2} + 1) q^{5} + \zeta_{10}^{3} q^{7} + ( - 4 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 4 \zeta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{2} + (3 \zeta_{10} - 3) q^{4} + (\zeta_{10}^{2} + 1) q^{5} + \zeta_{10}^{3} q^{7} + ( - 4 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 4 \zeta_{10}) q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{10} + (2 \zeta_{10}^{3} + \zeta_{10} + 2) q^{11} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 3) q^{13} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{14} + (3 \zeta_{10}^{2} - 8 \zeta_{10} + 3) q^{16} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{17} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3 \zeta_{10}) q^{19} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{20} + (3 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{22} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{23} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{25} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{26} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{28} - 6 \zeta_{10}^{3} q^{29} + (2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} - 2) q^{31} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 6) q^{32} + 3 q^{34} + (\zeta_{10}^{3} - 1) q^{35} + (3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{37} + (7 \zeta_{10}^{2} - 11 \zeta_{10} + 7) q^{38} + ( - 3 \zeta_{10}^{3} + \zeta_{10} - 1) q^{40} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} + 2 \zeta_{10}) q^{41} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 3) q^{43} + ( - 3 \zeta_{10}^{2} + 9 \zeta_{10} - 12) q^{44} + (\zeta_{10}^{3} - 1) q^{46} + ( - 5 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 5 \zeta_{10}) q^{47} + 6 \zeta_{10} q^{49} + ( - 5 \zeta_{10}^{2} + 9 \zeta_{10} - 5) q^{50} + (3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{52} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{53} + (3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \zeta_{10}) q^{55} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 1) q^{56} + ( - 6 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 6 \zeta_{10}) q^{58} + (9 \zeta_{10}^{3} + \zeta_{10} - 1) q^{59} + ( - 9 \zeta_{10}^{2} + 6 \zeta_{10} - 9) q^{61} + (3 \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{62} + ( - \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{64} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 5) q^{65} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 3) q^{67} + ( - 9 \zeta_{10}^{3} + 9) q^{68} + ( - \zeta_{10} + 1) q^{70} + ( - 7 \zeta_{10}^{2} + 6 \zeta_{10} - 7) q^{71} + ( - 4 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{73} + (7 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 7 \zeta_{10}) q^{74} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 9) q^{76} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{77} + (11 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 11 \zeta_{10} - 11) q^{79} + ( - 5 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 5 \zeta_{10}) q^{80} + ( - \zeta_{10}^{2} - 1) q^{82} + ( - 4 \zeta_{10}^{2} - \zeta_{10} - 4) q^{83} + ( - 6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 6 \zeta_{10}) q^{85} + ( - 3 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 3) q^{86} + ( - 15 \zeta_{10}^{3} + 18 \zeta_{10}^{2} - 12 \zeta_{10} + 2) q^{88} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 5) q^{89} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} + 2 \zeta_{10}) q^{91} + (6 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{92} + (12 \zeta_{10}^{2} - 19 \zeta_{10} + 12) q^{94} + ( - 2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{95} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{97} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 9 q^{4} + 3 q^{5} + q^{7} - 13 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 9 q^{4} + 3 q^{5} + q^{7} - 13 q^{8} + 2 q^{10} + 11 q^{11} + 7 q^{13} + 4 q^{14} + q^{16} - 12 q^{17} - 10 q^{19} - 3 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 6 q^{28} - 6 q^{29} - 12 q^{31} + 30 q^{32} + 12 q^{34} - 3 q^{35} + 9 q^{37} + 10 q^{38} - 6 q^{40} + 3 q^{41} - 36 q^{44} - 3 q^{46} - 17 q^{47} + 6 q^{49} - 6 q^{50} + 3 q^{52} - 4 q^{53} + 2 q^{55} - 12 q^{56} - 24 q^{58} + 6 q^{59} - 21 q^{61} + 27 q^{62} + 13 q^{64} + 14 q^{65} - 6 q^{67} + 27 q^{68} + 3 q^{70} - 15 q^{71} + 14 q^{73} + 26 q^{74} + 60 q^{76} - q^{77} - 11 q^{79} - 13 q^{80} - 3 q^{82} - 13 q^{83} - 9 q^{85} - 15 q^{86} - 37 q^{88} + 24 q^{89} + 3 q^{91} - 3 q^{92} + 17 q^{94} - 5 q^{95} + 3 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.224514i 0 −0.572949 1.76336i 1.30902 0.951057i 0 −0.309017 0.951057i −0.454915 + 1.40008i 0 −0.618034
64.1 0.809017 2.48990i 0 −3.92705 2.85317i 0.190983 + 0.587785i 0 0.809017 + 0.587785i −6.04508 + 4.39201i 0 1.61803
82.1 0.809017 + 2.48990i 0 −3.92705 + 2.85317i 0.190983 0.587785i 0 0.809017 0.587785i −6.04508 4.39201i 0 1.61803
91.1 −0.309017 + 0.224514i 0 −0.572949 + 1.76336i 1.30902 + 0.951057i 0 −0.309017 + 0.951057i −0.454915 1.40008i 0 −0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.f.a 4
3.b odd 2 1 33.2.e.b 4
9.c even 3 2 891.2.n.b 8
9.d odd 6 2 891.2.n.c 8
11.c even 5 1 inner 99.2.f.a 4
11.c even 5 1 1089.2.a.t 2
11.d odd 10 1 1089.2.a.l 2
12.b even 2 1 528.2.y.b 4
15.d odd 2 1 825.2.n.c 4
15.e even 4 2 825.2.bx.d 8
33.d even 2 1 363.2.e.f 4
33.f even 10 1 363.2.a.i 2
33.f even 10 2 363.2.e.b 4
33.f even 10 1 363.2.e.f 4
33.h odd 10 1 33.2.e.b 4
33.h odd 10 1 363.2.a.d 2
33.h odd 10 2 363.2.e.k 4
99.m even 15 2 891.2.n.b 8
99.n odd 30 2 891.2.n.c 8
132.n odd 10 1 5808.2.a.ci 2
132.o even 10 1 528.2.y.b 4
132.o even 10 1 5808.2.a.cj 2
165.o odd 10 1 825.2.n.c 4
165.o odd 10 1 9075.2.a.cb 2
165.r even 10 1 9075.2.a.u 2
165.v even 20 2 825.2.bx.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 3.b odd 2 1
33.2.e.b 4 33.h odd 10 1
99.2.f.a 4 1.a even 1 1 trivial
99.2.f.a 4 11.c even 5 1 inner
363.2.a.d 2 33.h odd 10 1
363.2.a.i 2 33.f even 10 1
363.2.e.b 4 33.f even 10 2
363.2.e.f 4 33.d even 2 1
363.2.e.f 4 33.f even 10 1
363.2.e.k 4 33.h odd 10 2
528.2.y.b 4 12.b even 2 1
528.2.y.b 4 132.o even 10 1
825.2.n.c 4 15.d odd 2 1
825.2.n.c 4 165.o odd 10 1
825.2.bx.d 8 15.e even 4 2
825.2.bx.d 8 165.v even 20 2
891.2.n.b 8 9.c even 3 2
891.2.n.b 8 99.m even 15 2
891.2.n.c 8 9.d odd 6 2
891.2.n.c 8 99.n odd 30 2
1089.2.a.l 2 11.d odd 10 1
1089.2.a.t 2 11.c even 5 1
5808.2.a.ci 2 132.n odd 10 1
5808.2.a.cj 2 132.o even 10 1
9075.2.a.u 2 165.r even 10 1
9075.2.a.cb 2 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$41$ \( T^{4} - 3 T^{3} + 19 T^{2} - 7 T + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} + 21 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$67$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 15 T^{3} + 190 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$83$ \( T^{4} + 13 T^{3} + 69 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 31)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \) Copy content Toggle raw display
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