# Properties

 Label 33.2.d.a Level $33$ Weight $2$ Character orbit 33.d Analytic conductor $0.264$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,2,Mod(32,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, 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("33.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (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: $$0.263506326670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 2 q^{4} + ( - 2 \beta + 1) q^{5} + (\beta - 3) q^{9}+O(q^{10})$$ q + b * q^3 - 2 * q^4 + (-2*b + 1) * q^5 + (b - 3) * q^9 $$q + \beta q^{3} - 2 q^{4} + ( - 2 \beta + 1) q^{5} + (\beta - 3) q^{9} + (2 \beta - 1) q^{11} - 2 \beta q^{12} + ( - \beta + 6) q^{15} + 4 q^{16} + (4 \beta - 2) q^{20} + ( - 2 \beta + 1) q^{23} - 6 q^{25} + ( - 2 \beta - 3) q^{27} + 5 q^{31} + (\beta - 6) q^{33} + ( - 2 \beta + 6) q^{36} - 7 q^{37} + ( - 4 \beta + 2) q^{44} + (5 \beta + 3) q^{45} + (4 \beta - 2) q^{47} + 4 \beta q^{48} + 7 q^{49} + ( - 8 \beta + 4) q^{53} + 11 q^{55} + ( - 2 \beta + 1) q^{59} + (2 \beta - 12) q^{60} - 8 q^{64} - 13 q^{67} + ( - \beta + 6) q^{69} + (10 \beta - 5) q^{71} - 6 \beta q^{75} + ( - 8 \beta + 4) q^{80} + ( - 5 \beta + 6) q^{81} + (10 \beta - 5) q^{89} + (4 \beta - 2) q^{92} + 5 \beta q^{93} + 17 q^{97} + ( - 5 \beta - 3) q^{99} +O(q^{100})$$ q + b * q^3 - 2 * q^4 + (-2*b + 1) * q^5 + (b - 3) * q^9 + (2*b - 1) * q^11 - 2*b * q^12 + (-b + 6) * q^15 + 4 * q^16 + (4*b - 2) * q^20 + (-2*b + 1) * q^23 - 6 * q^25 + (-2*b - 3) * q^27 + 5 * q^31 + (b - 6) * q^33 + (-2*b + 6) * q^36 - 7 * q^37 + (-4*b + 2) * q^44 + (5*b + 3) * q^45 + (4*b - 2) * q^47 + 4*b * q^48 + 7 * q^49 + (-8*b + 4) * q^53 + 11 * q^55 + (-2*b + 1) * q^59 + (2*b - 12) * q^60 - 8 * q^64 - 13 * q^67 + (-b + 6) * q^69 + (10*b - 5) * q^71 - 6*b * q^75 + (-8*b + 4) * q^80 + (-5*b + 6) * q^81 + (10*b - 5) * q^89 + (4*b - 2) * q^92 + 5*b * q^93 + 17 * q^97 + (-5*b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 4 q^{4} - 5 q^{9}+O(q^{10})$$ 2 * q + q^3 - 4 * q^4 - 5 * q^9 $$2 q + q^{3} - 4 q^{4} - 5 q^{9} - 2 q^{12} + 11 q^{15} + 8 q^{16} - 12 q^{25} - 8 q^{27} + 10 q^{31} - 11 q^{33} + 10 q^{36} - 14 q^{37} + 11 q^{45} + 4 q^{48} + 14 q^{49} + 22 q^{55} - 22 q^{60} - 16 q^{64} - 26 q^{67} + 11 q^{69} - 6 q^{75} + 7 q^{81} + 5 q^{93} + 34 q^{97} - 11 q^{99}+O(q^{100})$$ 2 * q + q^3 - 4 * q^4 - 5 * q^9 - 2 * q^12 + 11 * q^15 + 8 * q^16 - 12 * q^25 - 8 * q^27 + 10 * q^31 - 11 * q^33 + 10 * q^36 - 14 * q^37 + 11 * q^45 + 4 * q^48 + 14 * q^49 + 22 * q^55 - 22 * q^60 - 16 * q^64 - 26 * q^67 + 11 * q^69 - 6 * q^75 + 7 * q^81 + 5 * q^93 + 34 * q^97 - 11 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 0.5 − 1.65831i 0.5 + 1.65831i
0 0.500000 1.65831i −2.00000 3.31662i 0 0 0 −2.50000 1.65831i 0
32.2 0 0.500000 + 1.65831i −2.00000 3.31662i 0 0 0 −2.50000 + 1.65831i 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 $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.2.d.a 2
3.b odd 2 1 inner 33.2.d.a 2
4.b odd 2 1 528.2.b.a 2
5.b even 2 1 825.2.f.a 2
5.c odd 4 2 825.2.d.a 4
8.b even 2 1 2112.2.b.e 2
8.d odd 2 1 2112.2.b.f 2
9.c even 3 2 891.2.g.a 4
9.d odd 6 2 891.2.g.a 4
11.b odd 2 1 CM 33.2.d.a 2
11.c even 5 4 363.2.f.c 8
11.d odd 10 4 363.2.f.c 8
12.b even 2 1 528.2.b.a 2
15.d odd 2 1 825.2.f.a 2
15.e even 4 2 825.2.d.a 4
24.f even 2 1 2112.2.b.f 2
24.h odd 2 1 2112.2.b.e 2
33.d even 2 1 inner 33.2.d.a 2
33.f even 10 4 363.2.f.c 8
33.h odd 10 4 363.2.f.c 8
44.c even 2 1 528.2.b.a 2
55.d odd 2 1 825.2.f.a 2
55.e even 4 2 825.2.d.a 4
88.b odd 2 1 2112.2.b.e 2
88.g even 2 1 2112.2.b.f 2
99.g even 6 2 891.2.g.a 4
99.h odd 6 2 891.2.g.a 4
132.d odd 2 1 528.2.b.a 2
165.d even 2 1 825.2.f.a 2
165.l odd 4 2 825.2.d.a 4
264.m even 2 1 2112.2.b.e 2
264.p odd 2 1 2112.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 1.a even 1 1 trivial
33.2.d.a 2 3.b odd 2 1 inner
33.2.d.a 2 11.b odd 2 1 CM
33.2.d.a 2 33.d even 2 1 inner
363.2.f.c 8 11.c even 5 4
363.2.f.c 8 11.d odd 10 4
363.2.f.c 8 33.f even 10 4
363.2.f.c 8 33.h odd 10 4
528.2.b.a 2 4.b odd 2 1
528.2.b.a 2 12.b even 2 1
528.2.b.a 2 44.c even 2 1
528.2.b.a 2 132.d odd 2 1
825.2.d.a 4 5.c odd 4 2
825.2.d.a 4 15.e even 4 2
825.2.d.a 4 55.e even 4 2
825.2.d.a 4 165.l odd 4 2
825.2.f.a 2 5.b even 2 1
825.2.f.a 2 15.d odd 2 1
825.2.f.a 2 55.d odd 2 1
825.2.f.a 2 165.d even 2 1
891.2.g.a 4 9.c even 3 2
891.2.g.a 4 9.d odd 6 2
891.2.g.a 4 99.g even 6 2
891.2.g.a 4 99.h odd 6 2
2112.2.b.e 2 8.b even 2 1
2112.2.b.e 2 24.h odd 2 1
2112.2.b.e 2 88.b odd 2 1
2112.2.b.e 2 264.m even 2 1
2112.2.b.f 2 8.d odd 2 1
2112.2.b.f 2 24.f even 2 1
2112.2.b.f 2 88.g even 2 1
2112.2.b.f 2 264.p odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 3$$
$5$ $$T^{2} + 11$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 11$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 11$$
$29$ $$T^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 44$$
$53$ $$T^{2} + 176$$
$59$ $$T^{2} + 11$$
$61$ $$T^{2}$$
$67$ $$(T + 13)^{2}$$
$71$ $$T^{2} + 275$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 275$$
$97$ $$(T - 17)^{2}$$