Properties

Label 99.4.a.c
Level $99$
Weight $4$
Character orbit 99.a
Self dual yes
Analytic conductor $5.841$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 9 \beta + 25) q^{10} + 11 q^{11} + (20 \beta + 40) q^{13} + (6 \beta + 2) q^{14} + (32 \beta - 4) q^{16} + (12 \beta + 62) q^{17} + ( - 60 \beta + 36) q^{19} + ( - 30 \beta - 44) q^{20} + (11 \beta - 11) q^{22} + ( - 36 \beta + 49) q^{23} + ( - 16 \beta + 68) q^{25} + (20 \beta + 20) q^{26} + ( - 36 \beta - 64) q^{28} + ( - 56 \beta - 72) q^{29} + ( - 28 \beta - 17) q^{31} + (44 \beta + 52) q^{32} + (50 \beta - 26) q^{34} + (76 \beta + 86) q^{35} + (8 \beta + 27) q^{37} + (96 \beta - 216) q^{38} + (58 \beta - 246) q^{40} + ( - 4 \beta - 268) q^{41} + (16 \beta - 30) q^{43} + ( - 22 \beta - 44) q^{44} + (85 \beta - 157) q^{46} + ( - 120 \beta + 136) q^{47} + (80 \beta - 195) q^{49} + (84 \beta - 116) q^{50} + ( - 160 \beta - 280) q^{52} + ( - 56 \beta + 246) q^{53} + (88 \beta - 11) q^{55} + ( - 76 \beta - 60) q^{56} + ( - 16 \beta - 96) q^{58} + ( - 132 \beta - 317) q^{59} + ( - 184 \beta + 420) q^{61} + (11 \beta - 67) q^{62} + ( - 248 \beta + 112) q^{64} + (300 \beta + 440) q^{65} + (20 \beta + 377) q^{67} + ( - 172 \beta - 320) q^{68} + (10 \beta + 142) q^{70} + (76 \beta + 339) q^{71} + (468 \beta - 200) q^{73} + (19 \beta - 3) q^{74} + (168 \beta + 216) q^{76} + (44 \beta + 110) q^{77} + ( - 656 \beta + 158) q^{79} + ( - 64 \beta + 772) q^{80} + ( - 264 \beta + 256) q^{82} + (120 \beta - 234) q^{83} + (484 \beta + 226) q^{85} + ( - 46 \beta + 78) q^{86} + ( - 110 \beta + 66) q^{88} + ( - 328 \beta + 921) q^{89} + (360 \beta + 640) q^{91} + (46 \beta + 20) q^{92} + (256 \beta - 496) q^{94} + (348 \beta - 1476) q^{95} + ( - 144 \beta + 1097) q^{97} + ( - 275 \beta + 435) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−2.73205 0 −0.535898 −14.8564 0 3.07180 23.3205 0 40.5885
1.2 0.732051 0 −7.46410 12.8564 0 16.9282 −11.3205 0 9.41154
\(n\): 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 99.4.a.c 2
3.b odd 2 1 11.4.a.a 2
4.b odd 2 1 1584.4.a.bc 2
5.b even 2 1 2475.4.a.q 2
11.b odd 2 1 1089.4.a.v 2
12.b even 2 1 176.4.a.i 2
15.d odd 2 1 275.4.a.b 2
15.e even 4 2 275.4.b.c 4
21.c even 2 1 539.4.a.e 2
24.f even 2 1 704.4.a.n 2
24.h odd 2 1 704.4.a.p 2
33.d even 2 1 121.4.a.c 2
33.f even 10 4 121.4.c.f 8
33.h odd 10 4 121.4.c.c 8
39.d odd 2 1 1859.4.a.a 2
132.d odd 2 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 3.b odd 2 1
99.4.a.c 2 1.a even 1 1 trivial
121.4.a.c 2 33.d even 2 1
121.4.c.c 8 33.h odd 10 4
121.4.c.f 8 33.f even 10 4
176.4.a.i 2 12.b even 2 1
275.4.a.b 2 15.d odd 2 1
275.4.b.c 4 15.e even 4 2
539.4.a.e 2 21.c even 2 1
704.4.a.n 2 24.f even 2 1
704.4.a.p 2 24.h odd 2 1
1089.4.a.v 2 11.b odd 2 1
1584.4.a.bc 2 4.b odd 2 1
1859.4.a.a 2 39.d odd 2 1
1936.4.a.w 2 132.d odd 2 1
2475.4.a.q 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 52 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 400 \) Copy content Toggle raw display
$17$ \( T^{2} - 124T + 3412 \) Copy content Toggle raw display
$19$ \( T^{2} - 72T - 9504 \) Copy content Toggle raw display
$23$ \( T^{2} - 98T - 1487 \) Copy content Toggle raw display
$29$ \( T^{2} + 144T - 4224 \) Copy content Toggle raw display
$31$ \( T^{2} + 34T - 2063 \) Copy content Toggle raw display
$37$ \( T^{2} - 54T + 537 \) Copy content Toggle raw display
$41$ \( T^{2} + 536T + 71776 \) Copy content Toggle raw display
$43$ \( T^{2} + 60T + 132 \) Copy content Toggle raw display
$47$ \( T^{2} - 272T - 24704 \) Copy content Toggle raw display
$53$ \( T^{2} - 492T + 51108 \) Copy content Toggle raw display
$59$ \( T^{2} + 634T + 48217 \) Copy content Toggle raw display
$61$ \( T^{2} - 840T + 74832 \) Copy content Toggle raw display
$67$ \( T^{2} - 754T + 140929 \) Copy content Toggle raw display
$71$ \( T^{2} - 678T + 97593 \) Copy content Toggle raw display
$73$ \( T^{2} + 400T - 617072 \) Copy content Toggle raw display
$79$ \( T^{2} - 316 T - 1266044 \) Copy content Toggle raw display
$83$ \( T^{2} + 468T + 11556 \) Copy content Toggle raw display
$89$ \( T^{2} - 1842 T + 525489 \) Copy content Toggle raw display
$97$ \( T^{2} - 2194 T + 1141201 \) Copy content Toggle raw display
show more
show less