# Properties

 Label 333.2.c.a Level $333$ Weight $2$ Character orbit 333.c Analytic conductor $2.659$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [333,2,Mod(73,333)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(333, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("333.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$333 = 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 333.c (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: $$2.65901838731$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 37) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} - \beta q^{5} + 3 q^{7} +O(q^{10})$$ q + b * q^2 - 2 * q^4 - b * q^5 + 3 * q^7 $$q + \beta q^{2} - 2 q^{4} - \beta q^{5} + 3 q^{7} + 4 q^{10} + 3 q^{11} + 3 \beta q^{13} + 3 \beta q^{14} - 4 q^{16} + \beta q^{17} - 3 \beta q^{19} + 2 \beta q^{20} + 3 \beta q^{22} + 2 \beta q^{23} + q^{25} - 12 q^{26} - 6 q^{28} - 2 \beta q^{29} - 4 \beta q^{32} - 4 q^{34} - 3 \beta q^{35} + ( - 3 \beta - 1) q^{37} + 12 q^{38} + 3 q^{41} + 3 \beta q^{43} - 6 q^{44} - 8 q^{46} - 3 q^{47} + 2 q^{49} + \beta q^{50} - 6 \beta q^{52} - 9 q^{53} - 3 \beta q^{55} + 8 q^{58} - 2 \beta q^{59} + 8 q^{64} + 12 q^{65} - 12 q^{67} - 2 \beta q^{68} + 12 q^{70} + 3 q^{71} + 9 q^{73} + ( - \beta + 12) q^{74} + 6 \beta q^{76} + 9 q^{77} - 3 \beta q^{79} + 4 \beta q^{80} + 3 \beta q^{82} - 9 q^{83} + 4 q^{85} - 12 q^{86} - 7 \beta q^{89} + 9 \beta q^{91} - 4 \beta q^{92} - 3 \beta q^{94} - 12 q^{95} - 6 \beta q^{97} + 2 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 2 * q^4 - b * q^5 + 3 * q^7 + 4 * q^10 + 3 * q^11 + 3*b * q^13 + 3*b * q^14 - 4 * q^16 + b * q^17 - 3*b * q^19 + 2*b * q^20 + 3*b * q^22 + 2*b * q^23 + q^25 - 12 * q^26 - 6 * q^28 - 2*b * q^29 - 4*b * q^32 - 4 * q^34 - 3*b * q^35 + (-3*b - 1) * q^37 + 12 * q^38 + 3 * q^41 + 3*b * q^43 - 6 * q^44 - 8 * q^46 - 3 * q^47 + 2 * q^49 + b * q^50 - 6*b * q^52 - 9 * q^53 - 3*b * q^55 + 8 * q^58 - 2*b * q^59 + 8 * q^64 + 12 * q^65 - 12 * q^67 - 2*b * q^68 + 12 * q^70 + 3 * q^71 + 9 * q^73 + (-b + 12) * q^74 + 6*b * q^76 + 9 * q^77 - 3*b * q^79 + 4*b * q^80 + 3*b * q^82 - 9 * q^83 + 4 * q^85 - 12 * q^86 - 7*b * q^89 + 9*b * q^91 - 4*b * q^92 - 3*b * q^94 - 12 * q^95 - 6*b * q^97 + 2*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 6 q^{7}+O(q^{10})$$ 2 * q - 4 * q^4 + 6 * q^7 $$2 q - 4 q^{4} + 6 q^{7} + 8 q^{10} + 6 q^{11} - 8 q^{16} + 2 q^{25} - 24 q^{26} - 12 q^{28} - 8 q^{34} - 2 q^{37} + 24 q^{38} + 6 q^{41} - 12 q^{44} - 16 q^{46} - 6 q^{47} + 4 q^{49} - 18 q^{53} + 16 q^{58} + 16 q^{64} + 24 q^{65} - 24 q^{67} + 24 q^{70} + 6 q^{71} + 18 q^{73} + 24 q^{74} + 18 q^{77} - 18 q^{83} + 8 q^{85} - 24 q^{86} - 24 q^{95}+O(q^{100})$$ 2 * q - 4 * q^4 + 6 * q^7 + 8 * q^10 + 6 * q^11 - 8 * q^16 + 2 * q^25 - 24 * q^26 - 12 * q^28 - 8 * q^34 - 2 * q^37 + 24 * q^38 + 6 * q^41 - 12 * q^44 - 16 * q^46 - 6 * q^47 + 4 * q^49 - 18 * q^53 + 16 * q^58 + 16 * q^64 + 24 * q^65 - 24 * q^67 + 24 * q^70 + 6 * q^71 + 18 * q^73 + 24 * q^74 + 18 * q^77 - 18 * q^83 + 8 * q^85 - 24 * q^86 - 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/333\mathbb{Z}\right)^\times$$.

 $$n$$ $$38$$ $$298$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 2.00000i 0 3.00000 0 0 4.00000
73.2 2.00000i 0 −2.00000 2.00000i 0 3.00000 0 0 4.00000
 $$n$$: 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.2.c.a 2
3.b odd 2 1 37.2.b.a 2
4.b odd 2 1 5328.2.h.c 2
12.b even 2 1 592.2.g.b 2
15.d odd 2 1 925.2.c.b 2
15.e even 4 1 925.2.d.a 2
15.e even 4 1 925.2.d.d 2
24.f even 2 1 2368.2.g.b 2
24.h odd 2 1 2368.2.g.f 2
37.b even 2 1 inner 333.2.c.a 2
111.d odd 2 1 37.2.b.a 2
111.g even 4 1 1369.2.a.a 1
111.g even 4 1 1369.2.a.f 1
148.b odd 2 1 5328.2.h.c 2
444.g even 2 1 592.2.g.b 2
555.b odd 2 1 925.2.c.b 2
555.n even 4 1 925.2.d.a 2
555.n even 4 1 925.2.d.d 2
888.c even 2 1 2368.2.g.b 2
888.i odd 2 1 2368.2.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 3.b odd 2 1
37.2.b.a 2 111.d odd 2 1
333.2.c.a 2 1.a even 1 1 trivial
333.2.c.a 2 37.b even 2 1 inner
592.2.g.b 2 12.b even 2 1
592.2.g.b 2 444.g even 2 1
925.2.c.b 2 15.d odd 2 1
925.2.c.b 2 555.b odd 2 1
925.2.d.a 2 15.e even 4 1
925.2.d.a 2 555.n even 4 1
925.2.d.d 2 15.e even 4 1
925.2.d.d 2 555.n even 4 1
1369.2.a.a 1 111.g even 4 1
1369.2.a.f 1 111.g even 4 1
2368.2.g.b 2 24.f even 2 1
2368.2.g.b 2 888.c even 2 1
2368.2.g.f 2 24.h odd 2 1
2368.2.g.f 2 888.i odd 2 1
5328.2.h.c 2 4.b odd 2 1
5328.2.h.c 2 148.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(333, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T - 3)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T + 37$$
$41$ $$(T - 3)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T + 3)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$(T - 3)^{2}$$
$73$ $$(T - 9)^{2}$$
$79$ $$T^{2} + 36$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 144$$
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