# Properties

 Label 33.2.a.a Level $33$ Weight $2$ Character orbit 33.a Self dual yes Analytic conductor $0.264$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,2,Mod(1,33)]

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([0, 0]))

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

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.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: $$0.263506326670$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^4 - 2 * q^5 - q^6 + 4 * q^7 - 3 * q^8 + q^9 $$q + q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - 2 q^{10} + q^{11} + q^{12} - 2 q^{13} + 4 q^{14} + 2 q^{15} - q^{16} - 2 q^{17} + q^{18} + 2 q^{20} - 4 q^{21} + q^{22} + 8 q^{23} + 3 q^{24} - q^{25} - 2 q^{26} - q^{27} - 4 q^{28} - 6 q^{29} + 2 q^{30} - 8 q^{31} + 5 q^{32} - q^{33} - 2 q^{34} - 8 q^{35} - q^{36} + 6 q^{37} + 2 q^{39} + 6 q^{40} - 2 q^{41} - 4 q^{42} - q^{44} - 2 q^{45} + 8 q^{46} + 8 q^{47} + q^{48} + 9 q^{49} - q^{50} + 2 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} - 2 q^{55} - 12 q^{56} - 6 q^{58} - 4 q^{59} - 2 q^{60} + 6 q^{61} - 8 q^{62} + 4 q^{63} + 7 q^{64} + 4 q^{65} - q^{66} - 4 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{70} - 3 q^{72} - 14 q^{73} + 6 q^{74} + q^{75} + 4 q^{77} + 2 q^{78} - 4 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} + 12 q^{83} + 4 q^{84} + 4 q^{85} + 6 q^{87} - 3 q^{88} - 6 q^{89} - 2 q^{90} - 8 q^{91} - 8 q^{92} + 8 q^{93} + 8 q^{94} - 5 q^{96} + 2 q^{97} + 9 q^{98} + q^{99}+O(q^{100})$$ q + q^2 - q^3 - q^4 - 2 * q^5 - q^6 + 4 * q^7 - 3 * q^8 + q^9 - 2 * q^10 + q^11 + q^12 - 2 * q^13 + 4 * q^14 + 2 * q^15 - q^16 - 2 * q^17 + q^18 + 2 * q^20 - 4 * q^21 + q^22 + 8 * q^23 + 3 * q^24 - q^25 - 2 * q^26 - q^27 - 4 * q^28 - 6 * q^29 + 2 * q^30 - 8 * q^31 + 5 * q^32 - q^33 - 2 * q^34 - 8 * q^35 - q^36 + 6 * q^37 + 2 * q^39 + 6 * q^40 - 2 * q^41 - 4 * q^42 - q^44 - 2 * q^45 + 8 * q^46 + 8 * q^47 + q^48 + 9 * q^49 - q^50 + 2 * q^51 + 2 * q^52 + 6 * q^53 - q^54 - 2 * q^55 - 12 * q^56 - 6 * q^58 - 4 * q^59 - 2 * q^60 + 6 * q^61 - 8 * q^62 + 4 * q^63 + 7 * q^64 + 4 * q^65 - q^66 - 4 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^70 - 3 * q^72 - 14 * q^73 + 6 * q^74 + q^75 + 4 * q^77 + 2 * q^78 - 4 * q^79 + 2 * q^80 + q^81 - 2 * q^82 + 12 * q^83 + 4 * q^84 + 4 * q^85 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 2 * q^90 - 8 * q^91 - 8 * q^92 + 8 * q^93 + 8 * q^94 - 5 * q^96 + 2 * q^97 + 9 * q^98 + q^99

## 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
 0
1.00000 −1.00000 −1.00000 −2.00000 −1.00000 4.00000 −3.00000 1.00000 −2.00000
 $$n$$: 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 33.2.a.a 1
3.b odd 2 1 99.2.a.b 1
4.b odd 2 1 528.2.a.g 1
5.b even 2 1 825.2.a.a 1
5.c odd 4 2 825.2.c.a 2
7.b odd 2 1 1617.2.a.j 1
8.b even 2 1 2112.2.a.bb 1
8.d odd 2 1 2112.2.a.j 1
9.c even 3 2 891.2.e.e 2
9.d odd 6 2 891.2.e.g 2
11.b odd 2 1 363.2.a.b 1
11.c even 5 4 363.2.e.e 4
11.d odd 10 4 363.2.e.g 4
12.b even 2 1 1584.2.a.o 1
13.b even 2 1 5577.2.a.a 1
15.d odd 2 1 2475.2.a.g 1
15.e even 4 2 2475.2.c.d 2
17.b even 2 1 9537.2.a.m 1
21.c even 2 1 4851.2.a.b 1
24.f even 2 1 6336.2.a.n 1
24.h odd 2 1 6336.2.a.x 1
33.d even 2 1 1089.2.a.j 1
44.c even 2 1 5808.2.a.t 1
55.d odd 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 1.a even 1 1 trivial
99.2.a.b 1 3.b odd 2 1
363.2.a.b 1 11.b odd 2 1
363.2.e.e 4 11.c even 5 4
363.2.e.g 4 11.d odd 10 4
528.2.a.g 1 4.b odd 2 1
825.2.a.a 1 5.b even 2 1
825.2.c.a 2 5.c odd 4 2
891.2.e.e 2 9.c even 3 2
891.2.e.g 2 9.d odd 6 2
1089.2.a.j 1 33.d even 2 1
1584.2.a.o 1 12.b even 2 1
1617.2.a.j 1 7.b odd 2 1
2112.2.a.j 1 8.d odd 2 1
2112.2.a.bb 1 8.b even 2 1
2475.2.a.g 1 15.d odd 2 1
2475.2.c.d 2 15.e even 4 2
4851.2.a.b 1 21.c even 2 1
5577.2.a.a 1 13.b even 2 1
5808.2.a.t 1 44.c even 2 1
6336.2.a.n 1 24.f even 2 1
6336.2.a.x 1 24.h odd 2 1
9075.2.a.q 1 55.d odd 2 1
9537.2.a.m 1 17.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T - 4$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T + 2$$
$43$ $$T$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T - 6$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 4$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T - 2$$