Properties

Label 99.8.g.a
Level $99$
Weight $8$
Character orbit 99.g
Analytic conductor $30.926$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,8,Mod(32,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.32");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\), degree \(2\), 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: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (13 \beta_{3} + 48 \beta_{2}) q^{3} + 128 \beta_{2} q^{4} + ( - 71 \beta_{3} + 218 \beta_{2} - 507) q^{5} + (1797 \beta_{2} + 1079 \beta_1 - 1797) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (13 \beta_{3} + 48 \beta_{2}) q^{3} + 128 \beta_{2} q^{4} + ( - 71 \beta_{3} + 218 \beta_{2} - 507) q^{5} + (1797 \beta_{2} + 1079 \beta_1 - 1797) q^{9} + (1331 \beta_{2} + 2662 \beta_1 - 1331) q^{11} + (6144 \beta_{2} + 1664 \beta_1 - 6144) q^{12} + ( - 6591 \beta_{3} - 11103 \beta_{2} + 349 \beta_1 - 13233) q^{15} + (16384 \beta_{2} - 16384) q^{16} + ( - 36992 \beta_{2} - 9088 \beta_1 - 27904) q^{20} + ( - 43706 \beta_{3} - 21853 \beta_{2}) q^{23} + (71994 \beta_{3} - 110526 \beta_{2} - 35997 \beta_1 + 146523) q^{25} + ( - 61126 \beta_{3} + 61126 \beta_1 - 44175) q^{27} + (99327 \beta_{3} - 30131 \beta_{2} - 198654 \beta_1 + 99327) q^{31} + ( - 110473 \beta_{3} + 110473 \beta_1 + 39930) q^{33} + ( - 138112 \beta_{3} + 138112 \beta_1 - 230016) q^{36} + (75891 \beta_{3} + 75891 \beta_{2} + 75891 \beta_1 + 243290) q^{37} + ( - 340736 \beta_{3} + 340736 \beta_1 - 170368) q^{44} + ( - 184244 \beta_{3} - 911079 \beta_{2} - 362809 \beta_1 + 289506) q^{45} + ( - 739685 \beta_{2} - 164446 \beta_1 - 575239) q^{47} + ( - 212992 \beta_{3} + 212992 \beta_1 - 786432) q^{48} - 823543 \beta_{2} q^{49} + ( - 644332 \beta_{3} - 363378 \beta_{2} + 644332 \beta_1 - 140477) q^{53} + ( - 674817 \beta_{3} - 674817 \beta_{2} - 674817 \beta_1 - 182347) q^{55} + (950221 \beta_{3} + 400133 \beta_{2} + 149955) q^{59} + ( - 44672 \beta_{3} - 3115008 \beta_{2} - 798976 \beta_1 + 1421184) q^{60} - 2097152 q^{64} + ( - 1470129 \beta_{3} + 1077400 \beta_{2} + \cdots - 1470129) q^{67}+ \cdots + ( - 3347465 \beta_{3} + 6225087 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 83 q^{3} + 256 q^{4} - 1521 q^{5} - 2515 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 83 q^{3} + 256 q^{4} - 1521 q^{5} - 2515 q^{9} - 10624 q^{12} - 68198 q^{15} - 32768 q^{16} - 194688 q^{20} + 257049 q^{25} - 54448 q^{27} + 39065 q^{31} + 380666 q^{33} - 643840 q^{36} + 1124942 q^{37} - 842699 q^{45} - 3944772 q^{47} - 2719744 q^{48} - 1647086 q^{49} - 2079022 q^{55} + 449865 q^{59} - 1299584 q^{60} - 8388608 q^{64} + 684671 q^{67} - 3124979 q^{69} + 21335067 q^{75} + 3240713 q^{81} - 22926839 q^{93} - 15182479 q^{97} + 15797639 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 2.08017 46.7191i 64.0000 110.851i −278.284 160.667i 0 0 0 −2178.35 194.367i 0
32.2 0 39.4198 25.1610i 64.0000 110.851i −482.216 278.408i 0 0 0 920.846 1983.69i 0
65.1 0 2.08017 + 46.7191i 64.0000 + 110.851i −278.284 + 160.667i 0 0 0 −2178.35 + 194.367i 0
65.2 0 39.4198 + 25.1610i 64.0000 + 110.851i −482.216 + 278.408i 0 0 0 920.846 + 1983.69i 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 \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.g.a 4
9.d odd 6 1 inner 99.8.g.a 4
11.b odd 2 1 CM 99.8.g.a 4
99.g even 6 1 inner 99.8.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.g.a 4 1.a even 1 1 trivial
99.8.g.a 4 9.d odd 6 1 inner
99.8.g.a 4 11.b odd 2 1 CM
99.8.g.a 4 99.g even 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 83 T^{3} + 4702 T^{2} + \cdots + 4782969 \) Copy content Toggle raw display
$5$ \( T^{4} + 1521 T^{3} + \cdots + 32013797776 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 379749833583241 \) 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
$23$ \( T^{4} - 5253089699 T^{2} + \cdots + 27\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 39065 T^{3} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{2} - 562471 T + 31577994442)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3944772 T^{3} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{4} + 2481465850558 T^{2} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{4} - 449865 T^{3} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 684671 T^{3} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + 51014167749871 T^{2} + \cdots + 56\!\cdots\!04 \) 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
$89$ \( (T^{2} + 157352492959475)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 15182479 T^{3} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
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