# Properties

 Label 931.2.f.c Level $931$ Weight $2$ Character orbit 931.f Analytic conductor $7.434$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [931,2,Mod(324,931)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(931, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("931.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 931.f (of order $$3$$, degree $$2$$, not 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: $$7.43407242818$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^3 + (-2*z + 2) * q^4 - 3*z * q^5 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} - 4 \zeta_{6} q^{12} - 4 q^{13} - 6 q^{15} - 4 \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - \zeta_{6} q^{19} - 6 q^{20} + (4 \zeta_{6} - 4) q^{25} + 4 q^{27} + 6 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 6 \zeta_{6} q^{33} - 2 q^{36} - 2 \zeta_{6} q^{37} + (8 \zeta_{6} - 8) q^{39} - 6 q^{41} - q^{43} + 6 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{45} + 3 \zeta_{6} q^{47} - 8 q^{48} - 6 \zeta_{6} q^{51} + (8 \zeta_{6} - 8) q^{52} + (12 \zeta_{6} - 12) q^{53} + 9 q^{55} - 2 q^{57} + ( - 6 \zeta_{6} + 6) q^{59} + (12 \zeta_{6} - 12) q^{60} + \zeta_{6} q^{61} - 8 q^{64} + 12 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 6 \zeta_{6} q^{68} + 6 q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + 8 \zeta_{6} q^{75} - 2 q^{76} - 8 \zeta_{6} q^{79} + (12 \zeta_{6} - 12) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 12 q^{83} - 9 q^{85} + ( - 12 \zeta_{6} + 12) q^{87} - 12 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{95} + 8 q^{97} + 3 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^3 + (-2*z + 2) * q^4 - 3*z * q^5 - z * q^9 + (3*z - 3) * q^11 - 4*z * q^12 - 4 * q^13 - 6 * q^15 - 4*z * q^16 + (-3*z + 3) * q^17 - z * q^19 - 6 * q^20 + (4*z - 4) * q^25 + 4 * q^27 + 6 * q^29 + (-4*z + 4) * q^31 + 6*z * q^33 - 2 * q^36 - 2*z * q^37 + (8*z - 8) * q^39 - 6 * q^41 - q^43 + 6*z * q^44 + (3*z - 3) * q^45 + 3*z * q^47 - 8 * q^48 - 6*z * q^51 + (8*z - 8) * q^52 + (12*z - 12) * q^53 + 9 * q^55 - 2 * q^57 + (-6*z + 6) * q^59 + (12*z - 12) * q^60 + z * q^61 - 8 * q^64 + 12*z * q^65 + (-4*z + 4) * q^67 - 6*z * q^68 + 6 * q^71 + (-7*z + 7) * q^73 + 8*z * q^75 - 2 * q^76 - 8*z * q^79 + (12*z - 12) * q^80 + (-11*z + 11) * q^81 + 12 * q^83 - 9 * q^85 + (-12*z + 12) * q^87 - 12*z * q^89 - 8*z * q^93 + (3*z - 3) * q^95 + 8 * q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} - 3 q^{5} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 - 3 * q^5 - q^9 $$2 q + 2 q^{3} + 2 q^{4} - 3 q^{5} - q^{9} - 3 q^{11} - 4 q^{12} - 8 q^{13} - 12 q^{15} - 4 q^{16} + 3 q^{17} - q^{19} - 12 q^{20} - 4 q^{25} + 8 q^{27} + 12 q^{29} + 4 q^{31} + 6 q^{33} - 4 q^{36} - 2 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} + 6 q^{44} - 3 q^{45} + 3 q^{47} - 16 q^{48} - 6 q^{51} - 8 q^{52} - 12 q^{53} + 18 q^{55} - 4 q^{57} + 6 q^{59} - 12 q^{60} + q^{61} - 16 q^{64} + 12 q^{65} + 4 q^{67} - 6 q^{68} + 12 q^{71} + 7 q^{73} + 8 q^{75} - 4 q^{76} - 8 q^{79} - 12 q^{80} + 11 q^{81} + 24 q^{83} - 18 q^{85} + 12 q^{87} - 12 q^{89} - 8 q^{93} - 3 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 - 3 * q^5 - q^9 - 3 * q^11 - 4 * q^12 - 8 * q^13 - 12 * q^15 - 4 * q^16 + 3 * q^17 - q^19 - 12 * q^20 - 4 * q^25 + 8 * q^27 + 12 * q^29 + 4 * q^31 + 6 * q^33 - 4 * q^36 - 2 * q^37 - 8 * q^39 - 12 * q^41 - 2 * q^43 + 6 * q^44 - 3 * q^45 + 3 * q^47 - 16 * q^48 - 6 * q^51 - 8 * q^52 - 12 * q^53 + 18 * q^55 - 4 * q^57 + 6 * q^59 - 12 * q^60 + q^61 - 16 * q^64 + 12 * q^65 + 4 * q^67 - 6 * q^68 + 12 * q^71 + 7 * q^73 + 8 * q^75 - 4 * q^76 - 8 * q^79 - 12 * q^80 + 11 * q^81 + 24 * q^83 - 18 * q^85 + 12 * q^87 - 12 * q^89 - 8 * q^93 - 3 * q^95 + 16 * q^97 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/931\mathbb{Z}\right)^\times$$.

 $$n$$ $$248$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 + 1.73205i 1.00000 + 1.73205i −1.50000 + 2.59808i 0 0 0 −0.500000 + 0.866025i 0
704.1 0 1.00000 1.73205i 1.00000 1.73205i −1.50000 2.59808i 0 0 0 −0.500000 0.866025i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.f.c 2
7.b odd 2 1 931.2.f.b 2
7.c even 3 1 19.2.a.a 1
7.c even 3 1 inner 931.2.f.c 2
7.d odd 6 1 931.2.a.a 1
7.d odd 6 1 931.2.f.b 2
21.g even 6 1 8379.2.a.j 1
21.h odd 6 1 171.2.a.b 1
28.g odd 6 1 304.2.a.f 1
35.j even 6 1 475.2.a.b 1
35.l odd 12 2 475.2.b.a 2
56.k odd 6 1 1216.2.a.b 1
56.p even 6 1 1216.2.a.o 1
77.h odd 6 1 2299.2.a.b 1
84.n even 6 1 2736.2.a.c 1
91.r even 6 1 3211.2.a.a 1
105.o odd 6 1 4275.2.a.i 1
119.j even 6 1 5491.2.a.b 1
133.g even 3 1 361.2.c.c 2
133.h even 3 1 361.2.c.c 2
133.j odd 6 1 361.2.c.a 2
133.n odd 6 1 361.2.c.a 2
133.r odd 6 1 361.2.a.b 1
133.u even 9 3 361.2.e.d 6
133.w even 9 3 361.2.e.d 6
133.bd odd 18 3 361.2.e.e 6
133.be odd 18 3 361.2.e.e 6
140.p odd 6 1 7600.2.a.c 1
399.w even 6 1 3249.2.a.d 1
532.t even 6 1 5776.2.a.c 1
665.x odd 6 1 9025.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 7.c even 3 1
171.2.a.b 1 21.h odd 6 1
304.2.a.f 1 28.g odd 6 1
361.2.a.b 1 133.r odd 6 1
361.2.c.a 2 133.j odd 6 1
361.2.c.a 2 133.n odd 6 1
361.2.c.c 2 133.g even 3 1
361.2.c.c 2 133.h even 3 1
361.2.e.d 6 133.u even 9 3
361.2.e.d 6 133.w even 9 3
361.2.e.e 6 133.bd odd 18 3
361.2.e.e 6 133.be odd 18 3
475.2.a.b 1 35.j even 6 1
475.2.b.a 2 35.l odd 12 2
931.2.a.a 1 7.d odd 6 1
931.2.f.b 2 7.b odd 2 1
931.2.f.b 2 7.d odd 6 1
931.2.f.c 2 1.a even 1 1 trivial
931.2.f.c 2 7.c even 3 1 inner
1216.2.a.b 1 56.k odd 6 1
1216.2.a.o 1 56.p even 6 1
2299.2.a.b 1 77.h odd 6 1
2736.2.a.c 1 84.n even 6 1
3211.2.a.a 1 91.r even 6 1
3249.2.a.d 1 399.w even 6 1
4275.2.a.i 1 105.o odd 6 1
5491.2.a.b 1 119.j even 6 1
5776.2.a.c 1 532.t even 6 1
7600.2.a.c 1 140.p odd 6 1
8379.2.a.j 1 21.g even 6 1
9025.2.a.d 1 665.x odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(931, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$(T - 8)^{2}$$