Properties

Label 99.4.a.e
Level $99$
Weight $4$
Character orbit 99.a
Self dual yes
Analytic conductor $5.841$
Analytic rank $1$
Dimension $2$
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] = [99,4,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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + ( - 2 \beta + 2) q^{7} + (7 \beta - 8) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + ( - 2 \beta + 2) q^{7} + (7 \beta - 8) q^{8} + (6 \beta - 32) q^{10} - 11 q^{11} + (8 \beta - 42) q^{13} + 16 q^{14} + ( - 7 \beta - 56) q^{16} + ( - 30 \beta + 28) q^{17} + ( - 18 \beta - 18) q^{19} + ( - 6 \beta + 32) q^{20} + 11 \beta q^{22} - 112 q^{23} + ( - 64 \beta + 103) q^{25} + (34 \beta - 64) q^{26} - 16 q^{28} + ( - 46 \beta - 88) q^{29} + (104 \beta - 72) q^{31} + (7 \beta + 120) q^{32} + (2 \beta + 240) q^{34} + (20 \beta - 84) q^{35} + (44 \beta - 46) q^{37} + (36 \beta + 144) q^{38} + ( - 74 \beta + 304) q^{40} + ( - 2 \beta + 248) q^{41} + ( - 86 \beta + 10) q^{43} - 11 \beta q^{44} + 112 \beta q^{46} + (48 \beta + 8) q^{47} + ( - 4 \beta - 307) q^{49} + ( - 39 \beta + 512) q^{50} + ( - 34 \beta + 64) q^{52} + (128 \beta - 22) q^{53} + ( - 44 \beta + 110) q^{55} + (16 \beta - 128) q^{56} + (134 \beta + 368) q^{58} - 196 q^{59} + ( - 52 \beta - 526) q^{61} + ( - 32 \beta - 832) q^{62} + ( - 71 \beta + 392) q^{64} + ( - 216 \beta + 676) q^{65} + (152 \beta + 388) q^{67} + ( - 2 \beta - 240) q^{68} + (64 \beta - 160) q^{70} + ( - 184 \beta - 136) q^{71} + ( - 252 \beta - 170) q^{73} + (2 \beta - 352) q^{74} + ( - 36 \beta - 144) q^{76} + (22 \beta - 22) q^{77} + ( - 74 \beta - 78) q^{79} + ( - 182 \beta + 336) q^{80} + ( - 246 \beta + 16) q^{82} + (324 \beta - 336) q^{83} + (292 \beta - 1240) q^{85} + (76 \beta + 688) q^{86} + ( - 77 \beta + 88) q^{88} + ( - 8 \beta - 482) q^{89} + (84 \beta - 212) q^{91} - 112 \beta q^{92} + ( - 56 \beta - 384) q^{94} + (36 \beta - 396) q^{95} + (420 \beta - 802) q^{97} + (311 \beta + 32) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8} - 58 q^{10} - 22 q^{11} - 76 q^{13} + 32 q^{14} - 119 q^{16} + 26 q^{17} - 54 q^{19} + 58 q^{20} + 11 q^{22} - 224 q^{23} + 142 q^{25} - 94 q^{26} - 32 q^{28} - 222 q^{29} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} - 48 q^{37} + 324 q^{38} + 534 q^{40} + 494 q^{41} - 66 q^{43} - 11 q^{44} + 112 q^{46} + 64 q^{47} - 618 q^{49} + 985 q^{50} + 94 q^{52} + 84 q^{53} + 176 q^{55} - 240 q^{56} + 870 q^{58} - 392 q^{59} - 1104 q^{61} - 1696 q^{62} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} - 256 q^{70} - 456 q^{71} - 592 q^{73} - 702 q^{74} - 324 q^{76} - 22 q^{77} - 230 q^{79} + 490 q^{80} - 214 q^{82} - 348 q^{83} - 2188 q^{85} + 1452 q^{86} + 99 q^{88} - 972 q^{89} - 340 q^{91} - 112 q^{92} - 824 q^{94} - 756 q^{95} - 1184 q^{97} + 375 q^{98}+O(q^{100}) \) 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
3.37228
−2.37228
−3.37228 0 3.37228 3.48913 0 −4.74456 15.6060 0 −11.7663
1.2 2.37228 0 −2.37228 −19.4891 0 6.74456 −24.6060 0 −46.2337
\(n\): 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.e 2
3.b odd 2 1 33.4.a.d 2
4.b odd 2 1 1584.4.a.x 2
5.b even 2 1 2475.4.a.o 2
11.b odd 2 1 1089.4.a.t 2
12.b even 2 1 528.4.a.o 2
15.d odd 2 1 825.4.a.k 2
15.e even 4 2 825.4.c.i 4
21.c even 2 1 1617.4.a.j 2
24.f even 2 1 2112.4.a.bh 2
24.h odd 2 1 2112.4.a.ba 2
33.d even 2 1 363.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 3.b odd 2 1
99.4.a.e 2 1.a even 1 1 trivial
363.4.a.j 2 33.d even 2 1
528.4.a.o 2 12.b even 2 1
825.4.a.k 2 15.d odd 2 1
825.4.c.i 4 15.e even 4 2
1089.4.a.t 2 11.b odd 2 1
1584.4.a.x 2 4.b odd 2 1
1617.4.a.j 2 21.c even 2 1
2112.4.a.ba 2 24.h odd 2 1
2112.4.a.bh 2 24.f even 2 1
2475.4.a.o 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 16T - 68 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 76T + 916 \) Copy content Toggle raw display
$17$ \( T^{2} - 26T - 7256 \) Copy content Toggle raw display
$19$ \( T^{2} + 54T - 1944 \) Copy content Toggle raw display
$23$ \( (T + 112)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 222T - 5136 \) Copy content Toggle raw display
$31$ \( T^{2} + 40T - 88832 \) Copy content Toggle raw display
$37$ \( T^{2} + 48T - 15396 \) Copy content Toggle raw display
$41$ \( T^{2} - 494T + 60976 \) Copy content Toggle raw display
$43$ \( T^{2} + 66T - 59928 \) Copy content Toggle raw display
$47$ \( T^{2} - 64T - 17984 \) Copy content Toggle raw display
$53$ \( T^{2} - 84T - 133404 \) Copy content Toggle raw display
$59$ \( (T + 196)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 1104 T + 282396 \) Copy content Toggle raw display
$67$ \( T^{2} - 928T + 24688 \) Copy content Toggle raw display
$71$ \( T^{2} + 456T - 227328 \) Copy content Toggle raw display
$73$ \( T^{2} + 592T - 436292 \) Copy content Toggle raw display
$79$ \( T^{2} + 230T - 31952 \) Copy content Toggle raw display
$83$ \( T^{2} + 348T - 835776 \) Copy content Toggle raw display
$89$ \( T^{2} + 972T + 235668 \) Copy content Toggle raw display
$97$ \( T^{2} + 1184 T - 1104836 \) Copy content Toggle raw display
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