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Properties

Label 99.2.f.c
Level $99$
Weight $2$
Character orbit 99.f
Analytic conductor $0.791$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_{2} + 1) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + ( - 2 \beta_{3} - 3 \beta_{2} - 2) q^{7} + (\beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_{2} + 1) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + ( - 2 \beta_{3} - 3 \beta_{2} - 2) q^{7} + (\beta_{4} + \beta_1) q^{8} + (\beta_{5} - \beta_{2} - 4) q^{10} + (\beta_{6} - 2 \beta_{4} - \beta_1) q^{11} + ( - 3 \beta_{5} + 3 \beta_{3} + \cdots + 1) q^{13}+ \cdots + (8 \beta_{7} + 6 \beta_{4} - 8 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 6 q^{7} - 28 q^{10} - 6 q^{13} + 2 q^{16} + 12 q^{19} + 28 q^{22} + 8 q^{25} + 36 q^{28} + 12 q^{31} - 16 q^{34} - 18 q^{37} - 16 q^{40} - 32 q^{43} - 42 q^{46} - 44 q^{49} + 6 q^{52} - 8 q^{55} + 28 q^{58} + 42 q^{61} + 46 q^{64} + 4 q^{67} + 26 q^{70} + 24 q^{73} - 32 q^{76} + 30 q^{79} + 10 q^{82} + 46 q^{85} - 14 q^{88} + 2 q^{91} - 26 q^{94} - 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 75\beta_{6} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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
−1.73855 − 1.26313i
1.73855 + 1.26313i
−0.476925 + 1.46782i
0.476925 − 1.46782i
−0.476925 − 1.46782i
0.476925 + 1.46782i
−1.73855 + 1.26313i
1.73855 − 1.26313i
−1.73855 − 1.26313i 0 0.809017 + 2.48990i 1.73855 − 1.26313i 0 −1.30902 − 4.02874i 0.410415 − 1.26313i 0 −4.61803
37.2 1.73855 + 1.26313i 0 0.809017 + 2.48990i −1.73855 + 1.26313i 0 −1.30902 − 4.02874i −0.410415 + 1.26313i 0 −4.61803
64.1 −0.476925 + 1.46782i 0 −0.309017 − 0.224514i 0.476925 + 1.46782i 0 −0.190983 − 0.138757i −2.02029 + 1.46782i 0 −2.38197
64.2 0.476925 − 1.46782i 0 −0.309017 − 0.224514i −0.476925 − 1.46782i 0 −0.190983 − 0.138757i 2.02029 − 1.46782i 0 −2.38197
82.1 −0.476925 − 1.46782i 0 −0.309017 + 0.224514i 0.476925 − 1.46782i 0 −0.190983 + 0.138757i −2.02029 − 1.46782i 0 −2.38197
82.2 0.476925 + 1.46782i 0 −0.309017 + 0.224514i −0.476925 + 1.46782i 0 −0.190983 + 0.138757i 2.02029 + 1.46782i 0 −2.38197
91.1 −1.73855 + 1.26313i 0 0.809017 − 2.48990i 1.73855 + 1.26313i 0 −1.30902 + 4.02874i 0.410415 + 1.26313i 0 −4.61803
91.2 1.73855 − 1.26313i 0 0.809017 − 2.48990i −1.73855 − 1.26313i 0 −1.30902 + 4.02874i −0.410415 − 1.26313i 0 −4.61803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.f.c ✓ 8
3.b odd 2 1 inner 99.2.f.c ✓ 8
9.c even 3 2 891.2.n.e 16
9.d odd 6 2 891.2.n.e 16
11.c even 5 1 inner 99.2.f.c ✓ 8
11.c even 5 1 1089.2.a.v 4
11.d odd 10 1 1089.2.a.w 4
33.f even 10 1 1089.2.a.w 4
33.h odd 10 1 inner 99.2.f.c ✓ 8
33.h odd 10 1 1089.2.a.v 4
99.m even 15 2 891.2.n.e 16
99.n odd 30 2 891.2.n.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.f.c ✓ 8 1.a even 1 1 trivial
99.2.f.c ✓ 8 3.b odd 2 1 inner
99.2.f.c ✓ 8 11.c even 5 1 inner
99.2.f.c ✓ 8 33.h odd 10 1 inner
891.2.n.e 16 9.c even 3 2
891.2.n.e 16 9.d odd 6 2
891.2.n.e 16 99.m even 15 2
891.2.n.e 16 99.n odd 30 2
1089.2.a.v 4 11.c even 5 1
1089.2.a.v 4 33.h odd 10 1
1089.2.a.w 4 11.d odd 10 1
1089.2.a.w 4 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{6} + 16T_{2}^{4} + 66T_{2}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{6} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 19 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 34 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 72 T^{2} + 891)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 76 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 9 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 5 T^{6} + \cdots + 75625 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 29)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 19 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{8} + 130 T^{6} + \cdots + 75625 \) Copy content Toggle raw display
$59$ \( T^{8} + 46 T^{6} + \cdots + 15768841 \) Copy content Toggle raw display
$61$ \( (T^{4} - 21 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - T - 11)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 25 T^{6} + \cdots + 47265625 \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 15 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 11 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$89$ \( (T^{4} - 360 T^{2} + 22275)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 27 T^{3} + \cdots + 22801)^{2} \) Copy content Toggle raw display
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