Properties

Label 99.6.a.g
Level $99$
Weight $6$
Character orbit 99.a
Self dual yes
Analytic conductor $15.878$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,6,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

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: yes
Analytic conductor: \(15.8779981615\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + ( - 10 \beta_{2} - 10 \beta_1 + 28) q^{7} + ( - 26 \beta_{2} + 188) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + ( - 10 \beta_{2} - 10 \beta_1 + 28) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 9 \beta_{2} + 14 \beta_1 - 138) q^{10} - 121 q^{11} + (70 \beta_{2} + 6 \beta_1 + 162) q^{13} + ( - 138 \beta_{2} + 20 \beta_1 + 340) q^{14} + ( - 164 \beta_{2} + 12 \beta_1 + 664) q^{16} + ( - 132 \beta_{2} + 20 \beta_1 - 362) q^{17} + ( - 60 \beta_{2} - 20 \beta_1 + 460) q^{19} + (168 \beta_{2} + 22 \beta_1 + 1160) q^{20} + 121 \beta_{2} q^{22} + (21 \beta_{2} - 167 \beta_1 + 1022) q^{23} + (265 \beta_{2} + 85 \beta_1 - 19) q^{25} + (160 \beta_{2} + 116 \beta_1 - 4044) q^{26} + ( - 432 \beta_{2} - 36 \beta_1 + 7904) q^{28} + (182 \beta_{2} + 46 \beta_1 + 1142) q^{29} + ( - 101 \beta_{2} + 23 \beta_1 - 1366) q^{31} + ( - 404 \beta_{2} - 376 \beta_1 + 4136) q^{32} + ( - 26 \beta_{2} - 344 \beta_1 + 8440) q^{34} + (818 \beta_{2} + 306 \beta_1 + 8076) q^{35} + (937 \beta_{2} + 197 \beta_1 + 5908) q^{37} + ( - 840 \beta_{2} - 40 \beta_1 + 3080) q^{38} + ( - 46 \beta_{2} - 200 \beta_1 - 5092) q^{40} + (1378 \beta_{2} - 94 \beta_1 - 1998) q^{41} + (1190 \beta_{2} - 370 \beta_1 - 8736) q^{43} + (484 \beta_{2} + 242 \beta_1 - 3388) q^{44} + ( - 2107 \beta_{2} + 710 \beta_1 - 5602) q^{46} + (600 \beta_{2} + 424 \beta_1 + 5744) q^{47} + (340 \beta_{2} + 740 \beta_1 + 16177) q^{49} + (1674 \beta_{2} + 190 \beta_1 - 13690) q^{50} + (3256 \beta_{2} - 336 \beta_1 - 11768) q^{52} + (476 \beta_{2} + 540 \beta_1 - 16862) q^{53} + ( - 121 \beta_{2} + 363 \beta_1 + 968) q^{55} + ( - 5468 \beta_{2} - 1360 \beta_1 + 14104) q^{56} + ( - 92 \beta_{2} + 180 \beta_1 - 9724) q^{58} + ( - 3141 \beta_{2} - 249 \beta_1 + 1246) q^{59} + ( - 1466 \beta_{2} - 1346 \beta_1 + 6162) q^{61} + (1123 \beta_{2} - 294 \beta_1 + 6658) q^{62} + ( - 3136 \beta_{2} + 312 \beta_1 - 6784) q^{64} + (264 \beta_{2} - 1616 \beta_1 + 2556) q^{65} + ( - 6575 \beta_{2} - 907 \beta_1 - 15918) q^{67} + ( - 6728 \beta_{2} + 684 \beta_1 + 4200) q^{68} + ( - 2662 \beta_{2} + 412 \beta_1 - 41124) q^{70} + (3935 \beta_{2} + 891 \beta_1 - 13094) q^{71} + (5370 \beta_{2} - 1174 \beta_1 + 5142) q^{73} + ( - 781 \beta_{2} + 1086 \beta_1 - 51098) q^{74} + ( - 4800 \beta_{2} - 880 \beta_1 + 34640) q^{76} + (1210 \beta_{2} + 1210 \beta_1 - 3388) q^{77} + (3902 \beta_{2} + 454 \beta_1 + 41716) q^{79} + ( - 1868 \beta_{2} + 4 \beta_1 - 39560) q^{80} + (6852 \beta_{2} + 3132 \beta_1 - 85124) q^{82} + ( - 3150 \beta_{2} - 3750 \beta_1 + 47976) q^{83} + ( - 3310 \beta_{2} + 2434 \beta_1 - 34680) q^{85} + (10906 \beta_{2} + 3860 \beta_1 - 81020) q^{86} + (3146 \beta_{2} - 22748) q^{88} + (5167 \beta_{2} + 1523 \beta_1 + 35608) q^{89} + (6840 \beta_{2} - 3512 \beta_1 - 36544) q^{91} + (1472 \beta_{2} - 1710 \beta_1 + 112176) q^{92} + ( - 376 \beta_{2} - 496 \beta_1 - 24976) q^{94} + (1680 \beta_{2} - 40 \beta_1 + 7400) q^{95} + (123 \beta_{2} + 2143 \beta_1 + 3228) q^{97} + ( - 9637 \beta_{2} - 2280 \beta_1 - 1160) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8} - 414 q^{10} - 363 q^{11} + 486 q^{13} + 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} + 23712 q^{28} + 3426 q^{29} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 17724 q^{37} + 9240 q^{38} - 15276 q^{40} - 5994 q^{41} - 26208 q^{43} - 10164 q^{44} - 16806 q^{46} + 17232 q^{47} + 48531 q^{49} - 41070 q^{50} - 35304 q^{52} - 50586 q^{53} + 2904 q^{55} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} + 18486 q^{61} + 19974 q^{62} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} - 123372 q^{70} - 39282 q^{71} + 15426 q^{73} - 153294 q^{74} + 103920 q^{76} - 10164 q^{77} + 125148 q^{79} - 118680 q^{80} - 255372 q^{82} + 143928 q^{83} - 104040 q^{85} - 243060 q^{86} - 68244 q^{88} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} - 74928 q^{94} + 22200 q^{95} + 9684 q^{97} - 3480 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x - 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 34 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + \beta _1 + 35 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−6.29828
8.04796
−0.749680
−8.18772 0 35.0388 59.8722 0 145.071 −24.8808 0 −490.217
1.2 −2.20859 0 −27.1221 −75.2230 0 −225.525 130.577 0 166.137
1.3 10.3963 0 76.0833 −8.64919 0 164.454 458.304 0 −89.9197
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.6.a.g 3
3.b odd 2 1 11.6.a.b 3
11.b odd 2 1 1089.6.a.r 3
12.b even 2 1 176.6.a.i 3
15.d odd 2 1 275.6.a.b 3
15.e even 4 2 275.6.b.b 6
21.c even 2 1 539.6.a.e 3
24.f even 2 1 704.6.a.t 3
24.h odd 2 1 704.6.a.q 3
33.d even 2 1 121.6.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 3.b odd 2 1
99.6.a.g 3 1.a even 1 1 trivial
121.6.a.d 3 33.d even 2 1
176.6.a.i 3 12.b even 2 1
275.6.a.b 3 15.d odd 2 1
275.6.b.b 6 15.e even 4 2
539.6.a.e 3 21.c even 2 1
704.6.a.q 3 24.h odd 2 1
704.6.a.t 3 24.f even 2 1
1089.6.a.r 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 90T_{2} - 188 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 90T - 188 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 24 T^{2} + \cdots - 38954 \) Copy content Toggle raw display
$7$ \( T^{3} - 84 T^{2} + \cdots + 5380448 \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 486 T^{2} + \cdots + 164136608 \) Copy content Toggle raw display
$17$ \( T^{3} + 1086 T^{2} + \cdots - 331752056 \) Copy content Toggle raw display
$19$ \( T^{3} - 1380 T^{2} + \cdots + 57024000 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 17004325928 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 4029189120 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 1094344400 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 541788167034 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 201929821568 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 2443875098544 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 70174939136 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 1850911309656 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 7759637437060 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 15233874751008 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 147288561330212 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 1290398551704 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 34539701265952 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 1279883216320 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 411597824719824 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 90320980174650 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 10221902527106 \) Copy content Toggle raw display
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