Properties

 Label 930.2.i.e Level $930$ Weight $2$ Character orbit 930.i Analytic conductor $7.426$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(211,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("930.211");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.i (of order $$3$$, degree $$2$$, 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: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\zeta_{6} - 1) q^{3} + q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + q^2 + (z - 1) * q^3 + q^4 - z * q^5 + (z - 1) * q^6 + q^8 - z * q^9 $$q + q^{2} + (\zeta_{6} - 1) q^{3} + q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + q^{8} - \zeta_{6} q^{9} - \zeta_{6} q^{10} + \zeta_{6} q^{11} + (\zeta_{6} - 1) q^{12} + 6 \zeta_{6} q^{13} + q^{15} + q^{16} + ( - \zeta_{6} + 1) q^{17} - \zeta_{6} q^{18} + (4 \zeta_{6} - 4) q^{19} - \zeta_{6} q^{20} + \zeta_{6} q^{22} + 7 q^{23} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} + 6 \zeta_{6} q^{26} + q^{27} + 6 q^{29} + q^{30} + ( - 6 \zeta_{6} + 5) q^{31} + q^{32} - q^{33} + ( - \zeta_{6} + 1) q^{34} - \zeta_{6} q^{36} + (7 \zeta_{6} - 7) q^{37} + (4 \zeta_{6} - 4) q^{38} - 6 q^{39} - \zeta_{6} q^{40} + 2 \zeta_{6} q^{41} + ( - 7 \zeta_{6} + 7) q^{43} + \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + 7 q^{46} + 3 q^{47} + (\zeta_{6} - 1) q^{48} + 7 \zeta_{6} q^{49} + (\zeta_{6} - 1) q^{50} + \zeta_{6} q^{51} + 6 \zeta_{6} q^{52} + 4 \zeta_{6} q^{53} + q^{54} + ( - \zeta_{6} + 1) q^{55} - 4 \zeta_{6} q^{57} + 6 q^{58} + (12 \zeta_{6} - 12) q^{59} + q^{60} - 14 q^{61} + ( - 6 \zeta_{6} + 5) q^{62} + q^{64} + ( - 6 \zeta_{6} + 6) q^{65} - q^{66} - 5 \zeta_{6} q^{67} + ( - \zeta_{6} + 1) q^{68} + (7 \zeta_{6} - 7) q^{69} - 6 \zeta_{6} q^{71} - \zeta_{6} q^{72} - 2 \zeta_{6} q^{73} + (7 \zeta_{6} - 7) q^{74} - \zeta_{6} q^{75} + (4 \zeta_{6} - 4) q^{76} - 6 q^{78} + (3 \zeta_{6} - 3) q^{79} - \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} + 2 \zeta_{6} q^{82} - 16 \zeta_{6} q^{83} - q^{85} + ( - 7 \zeta_{6} + 7) q^{86} + (6 \zeta_{6} - 6) q^{87} + \zeta_{6} q^{88} + 4 q^{89} + (\zeta_{6} - 1) q^{90} + 7 q^{92} + (5 \zeta_{6} + 1) q^{93} + 3 q^{94} + 4 q^{95} + (\zeta_{6} - 1) q^{96} - 6 q^{97} + 7 \zeta_{6} q^{98} + ( - \zeta_{6} + 1) q^{99} +O(q^{100})$$ q + q^2 + (z - 1) * q^3 + q^4 - z * q^5 + (z - 1) * q^6 + q^8 - z * q^9 - z * q^10 + z * q^11 + (z - 1) * q^12 + 6*z * q^13 + q^15 + q^16 + (-z + 1) * q^17 - z * q^18 + (4*z - 4) * q^19 - z * q^20 + z * q^22 + 7 * q^23 + (z - 1) * q^24 + (z - 1) * q^25 + 6*z * q^26 + q^27 + 6 * q^29 + q^30 + (-6*z + 5) * q^31 + q^32 - q^33 + (-z + 1) * q^34 - z * q^36 + (7*z - 7) * q^37 + (4*z - 4) * q^38 - 6 * q^39 - z * q^40 + 2*z * q^41 + (-7*z + 7) * q^43 + z * q^44 + (z - 1) * q^45 + 7 * q^46 + 3 * q^47 + (z - 1) * q^48 + 7*z * q^49 + (z - 1) * q^50 + z * q^51 + 6*z * q^52 + 4*z * q^53 + q^54 + (-z + 1) * q^55 - 4*z * q^57 + 6 * q^58 + (12*z - 12) * q^59 + q^60 - 14 * q^61 + (-6*z + 5) * q^62 + q^64 + (-6*z + 6) * q^65 - q^66 - 5*z * q^67 + (-z + 1) * q^68 + (7*z - 7) * q^69 - 6*z * q^71 - z * q^72 - 2*z * q^73 + (7*z - 7) * q^74 - z * q^75 + (4*z - 4) * q^76 - 6 * q^78 + (3*z - 3) * q^79 - z * q^80 + (z - 1) * q^81 + 2*z * q^82 - 16*z * q^83 - q^85 + (-7*z + 7) * q^86 + (6*z - 6) * q^87 + z * q^88 + 4 * q^89 + (z - 1) * q^90 + 7 * q^92 + (5*z + 1) * q^93 + 3 * q^94 + 4 * q^95 + (z - 1) * q^96 - 6 * q^97 + 7*z * q^98 + (-z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - q^6 + 2 * q^8 - q^9 $$2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + 2 q^{8} - q^{9} - q^{10} + q^{11} - q^{12} + 6 q^{13} + 2 q^{15} + 2 q^{16} + q^{17} - q^{18} - 4 q^{19} - q^{20} + q^{22} + 14 q^{23} - q^{24} - q^{25} + 6 q^{26} + 2 q^{27} + 12 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} - 2 q^{33} + q^{34} - q^{36} - 7 q^{37} - 4 q^{38} - 12 q^{39} - q^{40} + 2 q^{41} + 7 q^{43} + q^{44} - q^{45} + 14 q^{46} + 6 q^{47} - q^{48} + 7 q^{49} - q^{50} + q^{51} + 6 q^{52} + 4 q^{53} + 2 q^{54} + q^{55} - 4 q^{57} + 12 q^{58} - 12 q^{59} + 2 q^{60} - 28 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{66} - 5 q^{67} + q^{68} - 7 q^{69} - 6 q^{71} - q^{72} - 2 q^{73} - 7 q^{74} - q^{75} - 4 q^{76} - 12 q^{78} - 3 q^{79} - q^{80} - q^{81} + 2 q^{82} - 16 q^{83} - 2 q^{85} + 7 q^{86} - 6 q^{87} + q^{88} + 8 q^{89} - q^{90} + 14 q^{92} + 7 q^{93} + 6 q^{94} + 8 q^{95} - q^{96} - 12 q^{97} + 7 q^{98} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - q^6 + 2 * q^8 - q^9 - q^10 + q^11 - q^12 + 6 * q^13 + 2 * q^15 + 2 * q^16 + q^17 - q^18 - 4 * q^19 - q^20 + q^22 + 14 * q^23 - q^24 - q^25 + 6 * q^26 + 2 * q^27 + 12 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 - 2 * q^33 + q^34 - q^36 - 7 * q^37 - 4 * q^38 - 12 * q^39 - q^40 + 2 * q^41 + 7 * q^43 + q^44 - q^45 + 14 * q^46 + 6 * q^47 - q^48 + 7 * q^49 - q^50 + q^51 + 6 * q^52 + 4 * q^53 + 2 * q^54 + q^55 - 4 * q^57 + 12 * q^58 - 12 * q^59 + 2 * q^60 - 28 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^66 - 5 * q^67 + q^68 - 7 * q^69 - 6 * q^71 - q^72 - 2 * q^73 - 7 * q^74 - q^75 - 4 * q^76 - 12 * q^78 - 3 * q^79 - q^80 - q^81 + 2 * q^82 - 16 * q^83 - 2 * q^85 + 7 * q^86 - 6 * q^87 + q^88 + 8 * q^89 - q^90 + 14 * q^92 + 7 * q^93 + 6 * q^94 + 8 * q^95 - q^96 - 12 * q^97 + 7 * q^98 + q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
811.1 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000 −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: 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
31.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.e 2
31.c even 3 1 inner 930.2.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.e 2 1.a even 1 1 trivial
930.2.i.e 2 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13}^{2} - 6T_{13} + 36$$ T13^2 - 6*T13 + 36

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$(T - 7)^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 4T + 31$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} - 2T + 4$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$(T - 3)^{2}$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 3T + 9$$
$83$ $$T^{2} + 16T + 256$$
$89$ $$(T - 4)^{2}$$
$97$ $$(T + 6)^{2}$$