Properties

 Label 555.2.c.a Level $555$ Weight $2$ Character orbit 555.c Analytic conductor $4.432$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [555,2,Mod(334,555)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(555, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("555.334");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$555 = 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 555.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: $$4.43169731218$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + i q^{3} - 2 q^{4} + (2 i - 1) q^{5} - 2 q^{6} - q^{9}+O(q^{10})$$ q + 2*i * q^2 + i * q^3 - 2 * q^4 + (2*i - 1) * q^5 - 2 * q^6 - q^9 $$q + 2 i q^{2} + i q^{3} - 2 q^{4} + (2 i - 1) q^{5} - 2 q^{6} - q^{9} + ( - 2 i - 4) q^{10} - 2 q^{11} - 2 i q^{12} - i q^{13} + ( - i - 2) q^{15} - 4 q^{16} - 2 i q^{18} + 2 q^{19} + ( - 4 i + 2) q^{20} - 4 i q^{22} + 6 i q^{23} + ( - 4 i - 3) q^{25} + 2 q^{26} - i q^{27} + 3 q^{29} + ( - 4 i + 2) q^{30} - 2 q^{31} - 8 i q^{32} - 2 i q^{33} + 2 q^{36} + i q^{37} + 4 i q^{38} + q^{39} - 2 q^{41} - 5 i q^{43} + 4 q^{44} + ( - 2 i + 1) q^{45} - 12 q^{46} + 3 i q^{47} - 4 i q^{48} + 7 q^{49} + ( - 6 i + 8) q^{50} + 2 i q^{52} + 7 i q^{53} + 2 q^{54} + ( - 4 i + 2) q^{55} + 2 i q^{57} + 6 i q^{58} + 3 q^{59} + (2 i + 4) q^{60} - 8 q^{61} - 4 i q^{62} + 8 q^{64} + (i + 2) q^{65} + 4 q^{66} + 10 i q^{67} - 6 q^{69} + 4 q^{71} + 10 i q^{73} - 2 q^{74} + ( - 3 i + 4) q^{75} - 4 q^{76} + 2 i q^{78} + 2 q^{79} + ( - 8 i + 4) q^{80} + q^{81} - 4 i q^{82} + 7 i q^{83} + 10 q^{86} + 3 i q^{87} + 5 q^{89} + (2 i + 4) q^{90} - 12 i q^{92} - 2 i q^{93} - 6 q^{94} + (4 i - 2) q^{95} + 8 q^{96} + 2 i q^{97} + 14 i q^{98} + 2 q^{99} +O(q^{100})$$ q + 2*i * q^2 + i * q^3 - 2 * q^4 + (2*i - 1) * q^5 - 2 * q^6 - q^9 + (-2*i - 4) * q^10 - 2 * q^11 - 2*i * q^12 - i * q^13 + (-i - 2) * q^15 - 4 * q^16 - 2*i * q^18 + 2 * q^19 + (-4*i + 2) * q^20 - 4*i * q^22 + 6*i * q^23 + (-4*i - 3) * q^25 + 2 * q^26 - i * q^27 + 3 * q^29 + (-4*i + 2) * q^30 - 2 * q^31 - 8*i * q^32 - 2*i * q^33 + 2 * q^36 + i * q^37 + 4*i * q^38 + q^39 - 2 * q^41 - 5*i * q^43 + 4 * q^44 + (-2*i + 1) * q^45 - 12 * q^46 + 3*i * q^47 - 4*i * q^48 + 7 * q^49 + (-6*i + 8) * q^50 + 2*i * q^52 + 7*i * q^53 + 2 * q^54 + (-4*i + 2) * q^55 + 2*i * q^57 + 6*i * q^58 + 3 * q^59 + (2*i + 4) * q^60 - 8 * q^61 - 4*i * q^62 + 8 * q^64 + (i + 2) * q^65 + 4 * q^66 + 10*i * q^67 - 6 * q^69 + 4 * q^71 + 10*i * q^73 - 2 * q^74 + (-3*i + 4) * q^75 - 4 * q^76 + 2*i * q^78 + 2 * q^79 + (-8*i + 4) * q^80 + q^81 - 4*i * q^82 + 7*i * q^83 + 10 * q^86 + 3*i * q^87 + 5 * q^89 + (2*i + 4) * q^90 - 12*i * q^92 - 2*i * q^93 - 6 * q^94 + (4*i - 2) * q^95 + 8 * q^96 + 2*i * q^97 + 14*i * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9} - 8 q^{10} - 4 q^{11} - 4 q^{15} - 8 q^{16} + 4 q^{19} + 4 q^{20} - 6 q^{25} + 4 q^{26} + 6 q^{29} + 4 q^{30} - 4 q^{31} + 4 q^{36} + 2 q^{39} - 4 q^{41} + 8 q^{44} + 2 q^{45} - 24 q^{46} + 14 q^{49} + 16 q^{50} + 4 q^{54} + 4 q^{55} + 6 q^{59} + 8 q^{60} - 16 q^{61} + 16 q^{64} + 4 q^{65} + 8 q^{66} - 12 q^{69} + 8 q^{71} - 4 q^{74} + 8 q^{75} - 8 q^{76} + 4 q^{79} + 8 q^{80} + 2 q^{81} + 20 q^{86} + 10 q^{89} + 8 q^{90} - 12 q^{94} - 4 q^{95} + 16 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9 - 8 * q^10 - 4 * q^11 - 4 * q^15 - 8 * q^16 + 4 * q^19 + 4 * q^20 - 6 * q^25 + 4 * q^26 + 6 * q^29 + 4 * q^30 - 4 * q^31 + 4 * q^36 + 2 * q^39 - 4 * q^41 + 8 * q^44 + 2 * q^45 - 24 * q^46 + 14 * q^49 + 16 * q^50 + 4 * q^54 + 4 * q^55 + 6 * q^59 + 8 * q^60 - 16 * q^61 + 16 * q^64 + 4 * q^65 + 8 * q^66 - 12 * q^69 + 8 * q^71 - 4 * q^74 + 8 * q^75 - 8 * q^76 + 4 * q^79 + 8 * q^80 + 2 * q^81 + 20 * q^86 + 10 * q^89 + 8 * q^90 - 12 * q^94 - 4 * q^95 + 16 * q^96 + 4 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/555\mathbb{Z}\right)^\times$$.

 $$n$$ $$76$$ $$112$$ $$371$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
334.1
 − 1.00000i 1.00000i
2.00000i 1.00000i −2.00000 −1.00000 2.00000i −2.00000 0 0 −1.00000 −4.00000 + 2.00000i
334.2 2.00000i 1.00000i −2.00000 −1.00000 + 2.00000i −2.00000 0 0 −1.00000 −4.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.c.a 2
3.b odd 2 1 1665.2.c.a 2
5.b even 2 1 inner 555.2.c.a 2
5.c odd 4 1 2775.2.a.a 1
5.c odd 4 1 2775.2.a.j 1
15.d odd 2 1 1665.2.c.a 2
15.e even 4 1 8325.2.a.d 1
15.e even 4 1 8325.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.a 2 1.a even 1 1 trivial
555.2.c.a 2 5.b even 2 1 inner
1665.2.c.a 2 3.b odd 2 1
1665.2.c.a 2 15.d odd 2 1
2775.2.a.a 1 5.c odd 4 1
2775.2.a.j 1 5.c odd 4 1
8325.2.a.d 1 15.e even 4 1
8325.2.a.bc 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(555, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 49$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 49$$
$89$ $$(T - 5)^{2}$$
$97$ $$T^{2} + 4$$
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