# Properties

 Label 930.2.i.b 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \zeta_{6} + 1) q^{3} + q^{4} + \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + (3 \zeta_{6} - 3) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q - q^2 + (-z + 1) * q^3 + q^4 + z * q^5 + (z - 1) * q^6 + (3*z - 3) * q^7 - q^8 - z * q^9 $$q - q^{2} + ( - \zeta_{6} + 1) q^{3} + q^{4} + \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{6} + (3 \zeta_{6} - 3) q^{7} - q^{8} - \zeta_{6} q^{9} - \zeta_{6} q^{10} + \zeta_{6} q^{11} + ( - \zeta_{6} + 1) q^{12} + ( - 3 \zeta_{6} + 3) q^{14} + q^{15} + q^{16} + (2 \zeta_{6} - 2) q^{17} + \zeta_{6} q^{18} + (2 \zeta_{6} - 2) q^{19} + \zeta_{6} q^{20} + 3 \zeta_{6} q^{21} - \zeta_{6} q^{22} - 4 q^{23} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - q^{27} + (3 \zeta_{6} - 3) q^{28} - 7 q^{29} - q^{30} + (5 \zeta_{6} + 1) q^{31} - q^{32} + q^{33} + ( - 2 \zeta_{6} + 2) q^{34} - 3 q^{35} - \zeta_{6} q^{36} + (4 \zeta_{6} - 4) q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - \zeta_{6} q^{40} - 3 \zeta_{6} q^{42} + (8 \zeta_{6} - 8) q^{43} + \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} + 4 q^{46} - 6 q^{47} + ( - \zeta_{6} + 1) q^{48} - 2 \zeta_{6} q^{49} + ( - \zeta_{6} + 1) q^{50} + 2 \zeta_{6} q^{51} - 5 \zeta_{6} q^{53} + q^{54} + (\zeta_{6} - 1) q^{55} + ( - 3 \zeta_{6} + 3) q^{56} + 2 \zeta_{6} q^{57} + 7 q^{58} + ( - 7 \zeta_{6} + 7) q^{59} + q^{60} + 12 q^{61} + ( - 5 \zeta_{6} - 1) q^{62} + 3 q^{63} + q^{64} - q^{66} + 8 \zeta_{6} q^{67} + (2 \zeta_{6} - 2) q^{68} + (4 \zeta_{6} - 4) q^{69} + 3 q^{70} + \zeta_{6} q^{72} + 6 \zeta_{6} q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + \zeta_{6} q^{75} + (2 \zeta_{6} - 2) q^{76} - 3 q^{77} + (8 \zeta_{6} - 8) q^{79} + \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} + 9 \zeta_{6} q^{83} + 3 \zeta_{6} q^{84} - 2 q^{85} + ( - 8 \zeta_{6} + 8) q^{86} + (7 \zeta_{6} - 7) q^{87} - \zeta_{6} q^{88} + 6 q^{89} + (\zeta_{6} - 1) q^{90} - 4 q^{92} + ( - \zeta_{6} + 6) q^{93} + 6 q^{94} - 2 q^{95} + (\zeta_{6} - 1) q^{96} - 5 q^{97} + 2 \zeta_{6} q^{98} + ( - \zeta_{6} + 1) q^{99} +O(q^{100})$$ q - q^2 + (-z + 1) * q^3 + q^4 + z * q^5 + (z - 1) * q^6 + (3*z - 3) * q^7 - q^8 - z * q^9 - z * q^10 + z * q^11 + (-z + 1) * q^12 + (-3*z + 3) * q^14 + q^15 + q^16 + (2*z - 2) * q^17 + z * q^18 + (2*z - 2) * q^19 + z * q^20 + 3*z * q^21 - z * q^22 - 4 * q^23 + (z - 1) * q^24 + (z - 1) * q^25 - q^27 + (3*z - 3) * q^28 - 7 * q^29 - q^30 + (5*z + 1) * q^31 - q^32 + q^33 + (-2*z + 2) * q^34 - 3 * q^35 - z * q^36 + (4*z - 4) * q^37 + (-2*z + 2) * q^38 - z * q^40 - 3*z * q^42 + (8*z - 8) * q^43 + z * q^44 + (-z + 1) * q^45 + 4 * q^46 - 6 * q^47 + (-z + 1) * q^48 - 2*z * q^49 + (-z + 1) * q^50 + 2*z * q^51 - 5*z * q^53 + q^54 + (z - 1) * q^55 + (-3*z + 3) * q^56 + 2*z * q^57 + 7 * q^58 + (-7*z + 7) * q^59 + q^60 + 12 * q^61 + (-5*z - 1) * q^62 + 3 * q^63 + q^64 - q^66 + 8*z * q^67 + (2*z - 2) * q^68 + (4*z - 4) * q^69 + 3 * q^70 + z * q^72 + 6*z * q^73 + (-4*z + 4) * q^74 + z * q^75 + (2*z - 2) * q^76 - 3 * q^77 + (8*z - 8) * q^79 + z * q^80 + (z - 1) * q^81 + 9*z * q^83 + 3*z * q^84 - 2 * q^85 + (-8*z + 8) * q^86 + (7*z - 7) * q^87 - z * q^88 + 6 * q^89 + (z - 1) * q^90 - 4 * q^92 + (-z + 6) * q^93 + 6 * q^94 - 2 * q^95 + (z - 1) * q^96 - 5 * q^97 + 2*z * q^98 + (-z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 3 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 3 * q^7 - 2 * q^8 - q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 3 q^{7} - 2 q^{8} - q^{9} - q^{10} + q^{11} + q^{12} + 3 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + q^{18} - 2 q^{19} + q^{20} + 3 q^{21} - q^{22} - 8 q^{23} - q^{24} - q^{25} - 2 q^{27} - 3 q^{28} - 14 q^{29} - 2 q^{30} + 7 q^{31} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 6 q^{35} - q^{36} - 4 q^{37} + 2 q^{38} - q^{40} - 3 q^{42} - 8 q^{43} + q^{44} + q^{45} + 8 q^{46} - 12 q^{47} + q^{48} - 2 q^{49} + q^{50} + 2 q^{51} - 5 q^{53} + 2 q^{54} - q^{55} + 3 q^{56} + 2 q^{57} + 14 q^{58} + 7 q^{59} + 2 q^{60} + 24 q^{61} - 7 q^{62} + 6 q^{63} + 2 q^{64} - 2 q^{66} + 8 q^{67} - 2 q^{68} - 4 q^{69} + 6 q^{70} + q^{72} + 6 q^{73} + 4 q^{74} + q^{75} - 2 q^{76} - 6 q^{77} - 8 q^{79} + q^{80} - q^{81} + 9 q^{83} + 3 q^{84} - 4 q^{85} + 8 q^{86} - 7 q^{87} - q^{88} + 12 q^{89} - q^{90} - 8 q^{92} + 11 q^{93} + 12 q^{94} - 4 q^{95} - q^{96} - 10 q^{97} + 2 q^{98} + q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 3 * q^7 - 2 * q^8 - q^9 - q^10 + q^11 + q^12 + 3 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + q^18 - 2 * q^19 + q^20 + 3 * q^21 - q^22 - 8 * q^23 - q^24 - q^25 - 2 * q^27 - 3 * q^28 - 14 * q^29 - 2 * q^30 + 7 * q^31 - 2 * q^32 + 2 * q^33 + 2 * q^34 - 6 * q^35 - q^36 - 4 * q^37 + 2 * q^38 - q^40 - 3 * q^42 - 8 * q^43 + q^44 + q^45 + 8 * q^46 - 12 * q^47 + q^48 - 2 * q^49 + q^50 + 2 * q^51 - 5 * q^53 + 2 * q^54 - q^55 + 3 * q^56 + 2 * q^57 + 14 * q^58 + 7 * q^59 + 2 * q^60 + 24 * q^61 - 7 * q^62 + 6 * q^63 + 2 * q^64 - 2 * q^66 + 8 * q^67 - 2 * q^68 - 4 * q^69 + 6 * q^70 + q^72 + 6 * q^73 + 4 * q^74 + q^75 - 2 * q^76 - 6 * q^77 - 8 * q^79 + q^80 - q^81 + 9 * q^83 + 3 * q^84 - 4 * q^85 + 8 * q^86 - 7 * q^87 - q^88 + 12 * q^89 - q^90 - 8 * q^92 + 11 * q^93 + 12 * q^94 - 4 * q^95 - q^96 - 10 * q^97 + 2 * 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 −1.50000 2.59808i −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 −1.50000 + 2.59808i −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.b 2
31.c even 3 1 inner 930.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.b 2 1.a even 1 1 trivial
930.2.i.b 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{2} + 3T_{7} + 9$$ T7^2 + 3*T7 + 9 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 7)^{2}$$
$31$ $$T^{2} - 7T + 31$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 5T + 25$$
$59$ $$T^{2} - 7T + 49$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 5)^{2}$$