# 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 $1$

# Related objects

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: $$x^{2} - x - 8$$ x^2 - x - 8 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + ( - 2 \beta + 2) q^{7} + (7 \beta - 8) q^{8} +O(q^{10})$$ q - b * q^2 + b * q^4 + (4*b - 10) * q^5 + (-2*b + 2) * q^7 + (7*b - 8) * q^8 $$q - \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + ( - 2 \beta + 2) q^{7} + (7 \beta - 8) q^{8} + (6 \beta - 32) q^{10} - 11 q^{11} + (8 \beta - 42) q^{13} + 16 q^{14} + ( - 7 \beta - 56) q^{16} + ( - 30 \beta + 28) q^{17} + ( - 18 \beta - 18) q^{19} + ( - 6 \beta + 32) q^{20} + 11 \beta q^{22} - 112 q^{23} + ( - 64 \beta + 103) q^{25} + (34 \beta - 64) q^{26} - 16 q^{28} + ( - 46 \beta - 88) q^{29} + (104 \beta - 72) q^{31} + (7 \beta + 120) q^{32} + (2 \beta + 240) q^{34} + (20 \beta - 84) q^{35} + (44 \beta - 46) q^{37} + (36 \beta + 144) q^{38} + ( - 74 \beta + 304) q^{40} + ( - 2 \beta + 248) q^{41} + ( - 86 \beta + 10) q^{43} - 11 \beta q^{44} + 112 \beta q^{46} + (48 \beta + 8) q^{47} + ( - 4 \beta - 307) q^{49} + ( - 39 \beta + 512) q^{50} + ( - 34 \beta + 64) q^{52} + (128 \beta - 22) q^{53} + ( - 44 \beta + 110) q^{55} + (16 \beta - 128) q^{56} + (134 \beta + 368) q^{58} - 196 q^{59} + ( - 52 \beta - 526) q^{61} + ( - 32 \beta - 832) q^{62} + ( - 71 \beta + 392) q^{64} + ( - 216 \beta + 676) q^{65} + (152 \beta + 388) q^{67} + ( - 2 \beta - 240) q^{68} + (64 \beta - 160) q^{70} + ( - 184 \beta - 136) q^{71} + ( - 252 \beta - 170) q^{73} + (2 \beta - 352) q^{74} + ( - 36 \beta - 144) q^{76} + (22 \beta - 22) q^{77} + ( - 74 \beta - 78) q^{79} + ( - 182 \beta + 336) q^{80} + ( - 246 \beta + 16) q^{82} + (324 \beta - 336) q^{83} + (292 \beta - 1240) q^{85} + (76 \beta + 688) q^{86} + ( - 77 \beta + 88) q^{88} + ( - 8 \beta - 482) q^{89} + (84 \beta - 212) q^{91} - 112 \beta q^{92} + ( - 56 \beta - 384) q^{94} + (36 \beta - 396) q^{95} + (420 \beta - 802) q^{97} + (311 \beta + 32) q^{98} +O(q^{100})$$ q - b * q^2 + b * q^4 + (4*b - 10) * q^5 + (-2*b + 2) * q^7 + (7*b - 8) * q^8 + (6*b - 32) * q^10 - 11 * q^11 + (8*b - 42) * q^13 + 16 * q^14 + (-7*b - 56) * q^16 + (-30*b + 28) * q^17 + (-18*b - 18) * q^19 + (-6*b + 32) * q^20 + 11*b * q^22 - 112 * q^23 + (-64*b + 103) * q^25 + (34*b - 64) * q^26 - 16 * q^28 + (-46*b - 88) * q^29 + (104*b - 72) * q^31 + (7*b + 120) * q^32 + (2*b + 240) * q^34 + (20*b - 84) * q^35 + (44*b - 46) * q^37 + (36*b + 144) * q^38 + (-74*b + 304) * q^40 + (-2*b + 248) * q^41 + (-86*b + 10) * q^43 - 11*b * q^44 + 112*b * q^46 + (48*b + 8) * q^47 + (-4*b - 307) * q^49 + (-39*b + 512) * q^50 + (-34*b + 64) * q^52 + (128*b - 22) * q^53 + (-44*b + 110) * q^55 + (16*b - 128) * q^56 + (134*b + 368) * q^58 - 196 * q^59 + (-52*b - 526) * q^61 + (-32*b - 832) * q^62 + (-71*b + 392) * q^64 + (-216*b + 676) * q^65 + (152*b + 388) * q^67 + (-2*b - 240) * q^68 + (64*b - 160) * q^70 + (-184*b - 136) * q^71 + (-252*b - 170) * q^73 + (2*b - 352) * q^74 + (-36*b - 144) * q^76 + (22*b - 22) * q^77 + (-74*b - 78) * q^79 + (-182*b + 336) * q^80 + (-246*b + 16) * q^82 + (324*b - 336) * q^83 + (292*b - 1240) * q^85 + (76*b + 688) * q^86 + (-77*b + 88) * q^88 + (-8*b - 482) * q^89 + (84*b - 212) * q^91 - 112*b * q^92 + (-56*b - 384) * q^94 + (36*b - 396) * q^95 + (420*b - 802) * q^97 + (311*b + 32) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 - 16 * q^5 + 2 * q^7 - 9 * q^8 $$2 q - q^{2} + q^{4} - 16 q^{5} + 2 q^{7} - 9 q^{8} - 58 q^{10} - 22 q^{11} - 76 q^{13} + 32 q^{14} - 119 q^{16} + 26 q^{17} - 54 q^{19} + 58 q^{20} + 11 q^{22} - 224 q^{23} + 142 q^{25} - 94 q^{26} - 32 q^{28} - 222 q^{29} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} - 48 q^{37} + 324 q^{38} + 534 q^{40} + 494 q^{41} - 66 q^{43} - 11 q^{44} + 112 q^{46} + 64 q^{47} - 618 q^{49} + 985 q^{50} + 94 q^{52} + 84 q^{53} + 176 q^{55} - 240 q^{56} + 870 q^{58} - 392 q^{59} - 1104 q^{61} - 1696 q^{62} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} - 256 q^{70} - 456 q^{71} - 592 q^{73} - 702 q^{74} - 324 q^{76} - 22 q^{77} - 230 q^{79} + 490 q^{80} - 214 q^{82} - 348 q^{83} - 2188 q^{85} + 1452 q^{86} + 99 q^{88} - 972 q^{89} - 340 q^{91} - 112 q^{92} - 824 q^{94} - 756 q^{95} - 1184 q^{97} + 375 q^{98}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

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