# 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 $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## 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: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 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.

 $$f(q)$$ $$=$$ $$q + (13 \beta_{3} + 48 \beta_{2}) q^{3} + 128 \beta_{2} q^{4} + ( - 71 \beta_{3} + 218 \beta_{2} - 507) q^{5} + (1797 \beta_{2} + 1079 \beta_1 - 1797) q^{9}+O(q^{10})$$ q + (13*b3 + 48*b2) * q^3 + 128*b2 * q^4 + (-71*b3 + 218*b2 - 507) * q^5 + (1797*b2 + 1079*b1 - 1797) * q^9 $$q + (13 \beta_{3} + 48 \beta_{2}) q^{3} + 128 \beta_{2} q^{4} + ( - 71 \beta_{3} + 218 \beta_{2} - 507) q^{5} + (1797 \beta_{2} + 1079 \beta_1 - 1797) q^{9} + (1331 \beta_{2} + 2662 \beta_1 - 1331) q^{11} + (6144 \beta_{2} + 1664 \beta_1 - 6144) q^{12} + ( - 6591 \beta_{3} - 11103 \beta_{2} + 349 \beta_1 - 13233) q^{15} + (16384 \beta_{2} - 16384) q^{16} + ( - 36992 \beta_{2} - 9088 \beta_1 - 27904) q^{20} + ( - 43706 \beta_{3} - 21853 \beta_{2}) q^{23} + (71994 \beta_{3} - 110526 \beta_{2} - 35997 \beta_1 + 146523) q^{25} + ( - 61126 \beta_{3} + 61126 \beta_1 - 44175) q^{27} + (99327 \beta_{3} - 30131 \beta_{2} - 198654 \beta_1 + 99327) q^{31} + ( - 110473 \beta_{3} + 110473 \beta_1 + 39930) q^{33} + ( - 138112 \beta_{3} + 138112 \beta_1 - 230016) q^{36} + (75891 \beta_{3} + 75891 \beta_{2} + 75891 \beta_1 + 243290) q^{37} + ( - 340736 \beta_{3} + 340736 \beta_1 - 170368) q^{44} + ( - 184244 \beta_{3} - 911079 \beta_{2} - 362809 \beta_1 + 289506) q^{45} + ( - 739685 \beta_{2} - 164446 \beta_1 - 575239) q^{47} + ( - 212992 \beta_{3} + 212992 \beta_1 - 786432) q^{48} - 823543 \beta_{2} q^{49} + ( - 644332 \beta_{3} - 363378 \beta_{2} + 644332 \beta_1 - 140477) q^{53} + ( - 674817 \beta_{3} - 674817 \beta_{2} - 674817 \beta_1 - 182347) q^{55} + (950221 \beta_{3} + 400133 \beta_{2} + 149955) q^{59} + ( - 44672 \beta_{3} - 3115008 \beta_{2} - 798976 \beta_1 + 1421184) q^{60} - 2097152 q^{64} + ( - 1470129 \beta_{3} + 1077400 \beta_{2} + \cdots - 1470129) q^{67}+ \cdots + ( - 3347465 \beta_{3} + 6225087 \beta_{2}) q^{99}+O(q^{100})$$ q + (13*b3 + 48*b2) * q^3 + 128*b2 * q^4 + (-71*b3 + 218*b2 - 507) * q^5 + (1797*b2 + 1079*b1 - 1797) * q^9 + (1331*b2 + 2662*b1 - 1331) * q^11 + (6144*b2 + 1664*b1 - 6144) * q^12 + (-6591*b3 - 11103*b2 + 349*b1 - 13233) * q^15 + (16384*b2 - 16384) * q^16 + (-36992*b2 - 9088*b1 - 27904) * q^20 + (-43706*b3 - 21853*b2) * q^23 + (71994*b3 - 110526*b2 - 35997*b1 + 146523) * q^25 + (-61126*b3 + 61126*b1 - 44175) * q^27 + (99327*b3 - 30131*b2 - 198654*b1 + 99327) * q^31 + (-110473*b3 + 110473*b1 + 39930) * q^33 + (-138112*b3 + 138112*b1 - 230016) * q^36 + (75891*b3 + 75891*b2 + 75891*b1 + 243290) * q^37 + (-340736*b3 + 340736*b1 - 170368) * q^44 + (-184244*b3 - 911079*b2 - 362809*b1 + 289506) * q^45 + (-739685*b2 - 164446*b1 - 575239) * q^47 + (-212992*b3 + 212992*b1 - 786432) * q^48 - 823543*b2 * q^49 + (-644332*b3 - 363378*b2 + 644332*b1 - 140477) * q^53 + (-674817*b3 - 674817*b2 - 674817*b1 - 182347) * q^55 + (950221*b3 + 400133*b2 + 149955) * q^59 + (-44672*b3 - 3115008*b2 - 798976*b1 + 1421184) * q^60 - 2097152 * q^64 + (-1470129*b3 + 1077400*b2 + 2940258*b1 - 1470129) * q^67 + (655590*b2 - 1813799*b1 - 655590) * q^69 + (-568615*b3 + 5729217*b2 + 568615*b1 - 3148916) * q^71 + (3164694*b3 - 1079910*b2 - 176943*b1 + 6709131) * q^75 + (1163264*b3 - 8306688*b2 - 1163264*b1 + 4734976) * q^80 + (-2713685*b3 + 263514*b2) * q^81 + (-7564330*b3 + 7564330*b1 - 3782165) * q^89 + (-2797184*b2 - 5594368*b1 + 2797184) * q^92 + (8244141*b3 - 552345*b2 - 3868148*b1 - 2427465) * q^93 + (5805498*b3 + 9042614*b2 - 2902749*b1 - 6139865) * q^97 + (-3347465*b3 + 6225087*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 83 q^{3} + 256 q^{4} - 1521 q^{5} - 2515 q^{9}+O(q^{10})$$ 4 * q + 83 * q^3 + 256 * q^4 - 1521 * q^5 - 2515 * q^9 $$4 q + 83 q^{3} + 256 q^{4} - 1521 q^{5} - 2515 q^{9} - 10624 q^{12} - 68198 q^{15} - 32768 q^{16} - 194688 q^{20} + 257049 q^{25} - 54448 q^{27} + 39065 q^{31} + 380666 q^{33} - 643840 q^{36} + 1124942 q^{37} - 842699 q^{45} - 3944772 q^{47} - 2719744 q^{48} - 1647086 q^{49} - 2079022 q^{55} + 449865 q^{59} - 1299584 q^{60} - 8388608 q^{64} + 684671 q^{67} - 3124979 q^{69} + 21335067 q^{75} + 3240713 q^{81} - 22926839 q^{93} - 15182479 q^{97} + 15797639 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2}$$ b3 + 3*b2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 2\beta _1 + 3$$ -2*b3 + 2*b1 + 3

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{8}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 83 T^{3} + 4702 T^{2} + \cdots + 4782969$$
$5$ $$T^{4} + 1521 T^{3} + \cdots + 32013797776$$
$7$ $$T^{4}$$
$11$ $$T^{4} + \cdots + 379749833583241$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 5253089699 T^{2} + \cdots + 27\!\cdots\!01$$
$29$ $$T^{4}$$
$31$ $$T^{4} - 39065 T^{3} + \cdots + 65\!\cdots\!64$$
$37$ $$(T^{2} - 562471 T + 31577994442)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 3944772 T^{3} + \cdots + 14\!\cdots\!69$$
$53$ $$T^{4} + 2481465850558 T^{2} + \cdots + 10\!\cdots\!09$$
$59$ $$T^{4} - 449865 T^{3} + \cdots + 60\!\cdots\!36$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 684671 T^{3} + \cdots + 31\!\cdots\!84$$
$71$ $$T^{4} + 51014167749871 T^{2} + \cdots + 56\!\cdots\!04$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 157352492959475)^{2}$$
$97$ $$T^{4} + 15182479 T^{3} + \cdots + 14\!\cdots\!04$$