Properties

 Label 92.2.b.b Level $92$ Weight $2$ Character orbit 92.b Analytic conductor $0.735$ Analytic rank $0$ Dimension $6$ CM discriminant -23 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [92,2,Mod(91,92)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(92, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("92.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$92 = 2^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 92.b (of order $$2$$, degree $$1$$, 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: $$0.734623698596$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8869743.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{3} + 8$$ x^6 - 3*x^3 + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{4} - \beta_{2}) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{3} - 1) q^{8} + (\beta_{4} - \beta_{2} - 2 \beta_1 - 3) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b4 - b2) * q^3 + b2 * q^4 + (-b5 + b4 + b3 + b2 - 1) * q^6 + (-b3 - 1) * q^8 + (b4 - b2 - 2*b1 - 3) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{4} - \beta_{2}) q^{3} + \beta_{2} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{3} - 1) q^{8} + (\beta_{4} - \beta_{2} - 2 \beta_1 - 3) q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 3) q^{12} + (2 \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{5} + \beta_{4} + \beta_1) q^{16} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{18} + ( - 2 \beta_{3} + 1) q^{23} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} - 3 \beta_1) q^{24} + 5 q^{25} + (\beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 5) q^{26} + (3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 2) q^{27} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{29} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1) q^{31} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2}) q^{32} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 5) q^{36} + (2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{39} + ( - 4 \beta_{5} + \beta_{4} + 3 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{5} + 2 \beta_{4} - \beta_1) q^{46} + (2 \beta_{5} + \beta_{4} + 3 \beta_{2} - 4 \beta_1) q^{47} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 7) q^{48} - 7 q^{49} - 5 \beta_1 q^{50} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} + 5 \beta_1 - 1) q^{52} + ( - \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{54} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 1) q^{58} + (4 \beta_{3} - 2) q^{59} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 7) q^{62} + (3 \beta_{3} - 5) q^{64} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{2} - 6 \beta_1) q^{69} + ( - 2 \beta_{5} + 5 \beta_{4} + 3 \beta_{2} + 4 \beta_1) q^{71} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 3) q^{72} + (4 \beta_{5} + \beta_{4} - 5 \beta_{2} - 2 \beta_1) q^{73} + ( - 5 \beta_{4} - 5 \beta_{2}) q^{75} + (3 \beta_{5} + 5 \beta_{4} + \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 9) q^{78} + (4 \beta_{5} - 6 \beta_{4} + 2 \beta_{2} + 12 \beta_1 + 9) q^{81} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + 7) q^{82} + ( - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 4 \beta_1 + 2) q^{87} + (4 \beta_{5} - 4 \beta_{4} - 3 \beta_{2}) q^{92} + (6 \beta_{5} - \beta_{4} - 5 \beta_{2} + 2 \beta_1 - 2) q^{93} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 5) q^{94} + (\beta_{5} + \beta_{4} - \beta_{3} - 7 \beta_1 - 9) q^{96} + 7 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (-b4 - b2) * q^3 + b2 * q^4 + (-b5 + b4 + b3 + b2 - 1) * q^6 + (-b3 - 1) * q^8 + (b4 - b2 - 2*b1 - 3) * q^9 + (-b5 - b4 - b3 + b1 + 3) * q^12 + (2*b5 - b4 - b2 + 2*b1) * q^13 + (b5 + b4 + b1) * q^16 + (b5 - b4 + b3 + b2 + 3*b1 + 3) * q^18 + (-2*b3 + 1) * q^23 + (-b5 + 3*b4 + b2 - 3*b1) * q^24 + 5 * q^25 + (b5 - b4 + b3 - 3*b2 - 5) * q^26 + (3*b4 + 4*b3 + 3*b2 - 2) * q^27 + (-2*b5 - b4 + 3*b2 + 2*b1) * q^29 + (-2*b5 + b4 - b2 + 4*b1) * q^31 + (2*b5 - 2*b4 - 3*b2) * q^32 + (-b5 - b4 - b3 - 3*b2 - 3*b1 - 5) * q^36 + (2*b5 - 3*b4 - 4*b3 - b2 - 4*b1 + 2) * q^39 + (-4*b5 + b4 + 3*b2 - 2*b1) * q^41 + (2*b5 + 2*b4 - b1) * q^46 + (2*b5 + b4 + 3*b2 - 4*b1) * q^47 + (2*b5 - 2*b4 - b3 + b2 + 7) * q^48 - 7 * q^49 - 5*b1 * q^50 + (-b5 - b4 + 3*b3 + 5*b1 - 1) * q^52 + (-b5 - 7*b4 - 3*b3 - 3*b2 + 2*b1 + 3) * q^54 + (-3*b5 + 3*b4 - 3*b3 + b2 - 1) * q^58 + (4*b3 - 2) * q^59 + (-b5 + b4 + b3 - 3*b2 + 7) * q^62 + (3*b3 - 5) * q^64 + (-2*b5 + 3*b4 - b2 - 6*b1) * q^69 + (-2*b5 + 5*b4 + 3*b2 + 4*b1) * q^71 + (-b5 + 3*b4 + 3*b3 + 5*b2 + 5*b1 + 3) * q^72 + (4*b5 + b4 - 5*b2 - 2*b1) * q^73 + (-5*b4 - 5*b2) * q^75 + (3*b5 + 5*b4 + b3 + 5*b2 - 2*b1 - 9) * q^78 + (4*b5 - 6*b4 + 2*b2 + 12*b1 + 9) * q^81 + (-3*b5 + 3*b4 - 3*b3 + 5*b2 + 7) * q^82 + (-2*b5 - 3*b4 - 4*b3 - 5*b2 + 4*b1 + 2) * q^87 + (4*b5 - 4*b4 - 3*b2) * q^92 + (6*b5 - b4 - 5*b2 + 2*b1 - 2) * q^93 + (3*b5 - 3*b4 - 3*b3 + b2 - 5) * q^94 + (b5 + b4 - b3 - 7*b1 - 9) * q^96 + 7*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{6} - 9 q^{8} - 18 q^{9}+O(q^{10})$$ 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9 $$6 q - 3 q^{6} - 9 q^{8} - 18 q^{9} + 15 q^{12} + 21 q^{18} + 30 q^{25} - 27 q^{26} - 33 q^{36} + 39 q^{48} - 42 q^{49} + 3 q^{52} + 9 q^{54} - 15 q^{58} + 45 q^{62} - 21 q^{64} + 27 q^{72} - 51 q^{78} + 54 q^{81} + 33 q^{82} - 12 q^{93} - 39 q^{94} - 57 q^{96}+O(q^{100})$$ 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9 + 15 * q^12 + 21 * q^18 + 30 * q^25 - 27 * q^26 - 33 * q^36 + 39 * q^48 - 42 * q^49 + 3 * q^52 + 9 * q^54 - 15 * q^58 + 45 * q^62 - 21 * q^64 + 27 * q^72 - 51 * q^78 + 54 * q^81 + 33 * q^82 - 12 * q^93 - 39 * q^94 - 57 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{3} + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 1$$ v^3 - 1 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 2\nu^{4} - 3\nu^{2} - 2\nu ) / 4$$ (v^5 + 2*v^4 - 3*v^2 - 2*v) / 4 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} + 3\nu^{2} - 2\nu ) / 4$$ (-v^5 + 2*v^4 + 3*v^2 - 2*v) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 1$$ b3 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_1$$ b5 + b4 + b1 $$\nu^{5}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} + 3\beta_{2}$$ -2*b5 + 2*b4 + 3*b2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/92\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$47$$ $$\chi(n)$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 1.33454 + 0.467979i 1.33454 − 0.467979i −0.261988 + 1.38973i −0.261988 − 1.38973i −1.07255 + 0.921756i −1.07255 − 0.921756i
−1.33454 0.467979i 3.43410i 1.56199 + 1.24907i 0 −1.60709 + 4.58295i 0 −1.50000 2.39792i −8.79306 0
91.2 −1.33454 + 0.467979i 3.43410i 1.56199 1.24907i 0 −1.60709 4.58295i 0 −1.50000 + 2.39792i −8.79306 0
91.3 0.261988 1.38973i 1.32309i −1.86272 0.728188i 0 −1.83875 0.346635i 0 −1.50000 + 2.39792i 1.24943 0
91.4 0.261988 + 1.38973i 1.32309i −1.86272 + 0.728188i 0 −1.83875 + 0.346635i 0 −1.50000 2.39792i 1.24943 0
91.5 1.07255 0.921756i 2.11101i 0.300733 1.97726i 0 1.94584 + 2.26417i 0 −1.50000 2.39792i −1.45636 0
91.6 1.07255 + 0.921756i 2.11101i 0.300733 + 1.97726i 0 1.94584 2.26417i 0 −1.50000 + 2.39792i −1.45636 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
4.b odd 2 1 inner
92.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.2.b.b 6
3.b odd 2 1 828.2.e.b 6
4.b odd 2 1 inner 92.2.b.b 6
8.b even 2 1 1472.2.c.c 6
8.d odd 2 1 1472.2.c.c 6
12.b even 2 1 828.2.e.b 6
23.b odd 2 1 CM 92.2.b.b 6
69.c even 2 1 828.2.e.b 6
92.b even 2 1 inner 92.2.b.b 6
184.e odd 2 1 1472.2.c.c 6
184.h even 2 1 1472.2.c.c 6
276.h odd 2 1 828.2.e.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.b.b 6 1.a even 1 1 trivial
92.2.b.b 6 4.b odd 2 1 inner
92.2.b.b 6 23.b odd 2 1 CM
92.2.b.b 6 92.b even 2 1 inner
828.2.e.b 6 3.b odd 2 1
828.2.e.b 6 12.b even 2 1
828.2.e.b 6 69.c even 2 1
828.2.e.b 6 276.h odd 2 1
1472.2.c.c 6 8.b even 2 1
1472.2.c.c 6 8.d odd 2 1
1472.2.c.c 6 184.e odd 2 1
1472.2.c.c 6 184.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 18T_{3}^{4} + 81T_{3}^{2} + 92$$ acting on $$S_{2}^{\mathrm{new}}(92, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3T^{3} + 8$$
$3$ $$T^{6} + 18 T^{4} + 81 T^{2} + 92$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$(T^{3} - 39 T + 74)^{2}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$(T^{2} + 23)^{3}$$
$29$ $$(T^{3} - 87 T + 282)^{2}$$
$31$ $$T^{6} + 186 T^{4} + 8649 T^{2} + \cdots + 828$$
$37$ $$T^{6}$$
$41$ $$(T^{3} - 123 T - 426)^{2}$$
$43$ $$T^{6}$$
$47$ $$T^{6} + 282 T^{4} + 19881 T^{2} + \cdots + 412988$$
$53$ $$T^{6}$$
$59$ $$(T^{2} + 92)^{3}$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$T^{6} + 426 T^{4} + 45369 T^{2} + \cdots + 48668$$
$73$ $$(T^{3} - 219 T - 1226)^{2}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$