# Properties

 Label 1014.2.e.a Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ 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] = [1014,2,Mod(529,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("1014.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.e (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: $$8.09683076496$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - 3 q^{5} - \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 - 3 * q^5 - z * q^6 + 2*z * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - 3 q^{5} - \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + ( - 6 \zeta_{6} + 6) q^{11} + q^{12} - 2 q^{14} + ( - 3 \zeta_{6} + 3) q^{15} + (\zeta_{6} - 1) q^{16} + 3 \zeta_{6} q^{17} + q^{18} + 2 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} - 2 q^{21} + 6 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + (\zeta_{6} - 1) q^{24} + 4 q^{25} + q^{27} + ( - 2 \zeta_{6} + 2) q^{28} + (3 \zeta_{6} - 3) q^{29} + 3 \zeta_{6} q^{30} + 4 q^{31} - \zeta_{6} q^{32} + 6 \zeta_{6} q^{33} - 3 q^{34} - 6 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{36} + (7 \zeta_{6} - 7) q^{37} - 2 q^{38} - 3 q^{40} + (3 \zeta_{6} - 3) q^{41} + ( - 2 \zeta_{6} + 2) q^{42} + 10 \zeta_{6} q^{43} - 6 q^{44} + 3 \zeta_{6} q^{45} + 6 \zeta_{6} q^{46} - 6 q^{47} - \zeta_{6} q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} - 3 q^{51} + 3 q^{53} + (\zeta_{6} - 1) q^{54} + (18 \zeta_{6} - 18) q^{55} + 2 \zeta_{6} q^{56} - 2 q^{57} - 3 \zeta_{6} q^{58} - 3 q^{60} + 7 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 2 \zeta_{6} + 2) q^{63} + q^{64} - 6 q^{66} + (10 \zeta_{6} - 10) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + 6 \zeta_{6} q^{69} + 6 q^{70} + 6 \zeta_{6} q^{71} - \zeta_{6} q^{72} + 13 q^{73} - 7 \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} + ( - 2 \zeta_{6} + 2) q^{76} + 12 q^{77} - 4 q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + (\zeta_{6} - 1) q^{81} - 3 \zeta_{6} q^{82} + 6 q^{83} + 2 \zeta_{6} q^{84} - 9 \zeta_{6} q^{85} - 10 q^{86} - 3 \zeta_{6} q^{87} + ( - 6 \zeta_{6} + 6) q^{88} + ( - 18 \zeta_{6} + 18) q^{89} - 3 q^{90} - 6 q^{92} + (4 \zeta_{6} - 4) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} - 6 \zeta_{6} q^{95} + q^{96} + 14 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} - 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 - 3 * q^5 - z * q^6 + 2*z * q^7 + q^8 - z * q^9 + (-3*z + 3) * q^10 + (-6*z + 6) * q^11 + q^12 - 2 * q^14 + (-3*z + 3) * q^15 + (z - 1) * q^16 + 3*z * q^17 + q^18 + 2*z * q^19 + 3*z * q^20 - 2 * q^21 + 6*z * q^22 + (-6*z + 6) * q^23 + (z - 1) * q^24 + 4 * q^25 + q^27 + (-2*z + 2) * q^28 + (3*z - 3) * q^29 + 3*z * q^30 + 4 * q^31 - z * q^32 + 6*z * q^33 - 3 * q^34 - 6*z * q^35 + (z - 1) * q^36 + (7*z - 7) * q^37 - 2 * q^38 - 3 * q^40 + (3*z - 3) * q^41 + (-2*z + 2) * q^42 + 10*z * q^43 - 6 * q^44 + 3*z * q^45 + 6*z * q^46 - 6 * q^47 - z * q^48 + (-3*z + 3) * q^49 + (4*z - 4) * q^50 - 3 * q^51 + 3 * q^53 + (z - 1) * q^54 + (18*z - 18) * q^55 + 2*z * q^56 - 2 * q^57 - 3*z * q^58 - 3 * q^60 + 7*z * q^61 + (4*z - 4) * q^62 + (-2*z + 2) * q^63 + q^64 - 6 * q^66 + (10*z - 10) * q^67 + (-3*z + 3) * q^68 + 6*z * q^69 + 6 * q^70 + 6*z * q^71 - z * q^72 + 13 * q^73 - 7*z * q^74 + (4*z - 4) * q^75 + (-2*z + 2) * q^76 + 12 * q^77 - 4 * q^79 + (-3*z + 3) * q^80 + (z - 1) * q^81 - 3*z * q^82 + 6 * q^83 + 2*z * q^84 - 9*z * q^85 - 10 * q^86 - 3*z * q^87 + (-6*z + 6) * q^88 + (-18*z + 18) * q^89 - 3 * q^90 - 6 * q^92 + (4*z - 4) * q^93 + (-6*z + 6) * q^94 - 6*z * q^95 + q^96 + 14*z * q^97 + 3*z * q^98 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} - 6 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 - 6 * q^5 - q^6 + 2 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} - 6 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - q^{9} + 3 q^{10} + 6 q^{11} + 2 q^{12} - 4 q^{14} + 3 q^{15} - q^{16} + 3 q^{17} + 2 q^{18} + 2 q^{19} + 3 q^{20} - 4 q^{21} + 6 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 2 q^{27} + 2 q^{28} - 3 q^{29} + 3 q^{30} + 8 q^{31} - q^{32} + 6 q^{33} - 6 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 4 q^{38} - 6 q^{40} - 3 q^{41} + 2 q^{42} + 10 q^{43} - 12 q^{44} + 3 q^{45} + 6 q^{46} - 12 q^{47} - q^{48} + 3 q^{49} - 4 q^{50} - 6 q^{51} + 6 q^{53} - q^{54} - 18 q^{55} + 2 q^{56} - 4 q^{57} - 3 q^{58} - 6 q^{60} + 7 q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} - 12 q^{66} - 10 q^{67} + 3 q^{68} + 6 q^{69} + 12 q^{70} + 6 q^{71} - q^{72} + 26 q^{73} - 7 q^{74} - 4 q^{75} + 2 q^{76} + 24 q^{77} - 8 q^{79} + 3 q^{80} - q^{81} - 3 q^{82} + 12 q^{83} + 2 q^{84} - 9 q^{85} - 20 q^{86} - 3 q^{87} + 6 q^{88} + 18 q^{89} - 6 q^{90} - 12 q^{92} - 4 q^{93} + 6 q^{94} - 6 q^{95} + 2 q^{96} + 14 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 - 6 * q^5 - q^6 + 2 * q^7 + 2 * q^8 - q^9 + 3 * q^10 + 6 * q^11 + 2 * q^12 - 4 * q^14 + 3 * q^15 - q^16 + 3 * q^17 + 2 * q^18 + 2 * q^19 + 3 * q^20 - 4 * q^21 + 6 * q^22 + 6 * q^23 - q^24 + 8 * q^25 + 2 * q^27 + 2 * q^28 - 3 * q^29 + 3 * q^30 + 8 * q^31 - q^32 + 6 * q^33 - 6 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 4 * q^38 - 6 * q^40 - 3 * q^41 + 2 * q^42 + 10 * q^43 - 12 * q^44 + 3 * q^45 + 6 * q^46 - 12 * q^47 - q^48 + 3 * q^49 - 4 * q^50 - 6 * q^51 + 6 * q^53 - q^54 - 18 * q^55 + 2 * q^56 - 4 * q^57 - 3 * q^58 - 6 * q^60 + 7 * q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 - 12 * q^66 - 10 * q^67 + 3 * q^68 + 6 * q^69 + 12 * q^70 + 6 * q^71 - q^72 + 26 * q^73 - 7 * q^74 - 4 * q^75 + 2 * q^76 + 24 * q^77 - 8 * q^79 + 3 * q^80 - q^81 - 3 * q^82 + 12 * q^83 + 2 * q^84 - 9 * q^85 - 20 * q^86 - 3 * q^87 + 6 * q^88 + 18 * q^89 - 6 * q^90 - 12 * q^92 - 4 * q^93 + 6 * q^94 - 6 * q^95 + 2 * q^96 + 14 * q^97 + 3 * q^98 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.00000 −0.500000 0.866025i 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i 1.50000 2.59808i
991.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.00000 −0.500000 + 0.866025i 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.e.a 2
13.b even 2 1 78.2.e.a 2
13.c even 3 1 1014.2.a.f 1
13.c even 3 1 inner 1014.2.e.a 2
13.d odd 4 2 1014.2.i.b 4
13.e even 6 1 78.2.e.a 2
13.e even 6 1 1014.2.a.c 1
13.f odd 12 2 1014.2.b.c 2
13.f odd 12 2 1014.2.i.b 4
39.d odd 2 1 234.2.h.a 2
39.h odd 6 1 234.2.h.a 2
39.h odd 6 1 3042.2.a.i 1
39.i odd 6 1 3042.2.a.h 1
39.k even 12 2 3042.2.b.h 2
52.b odd 2 1 624.2.q.g 2
52.i odd 6 1 624.2.q.g 2
52.i odd 6 1 8112.2.a.m 1
52.j odd 6 1 8112.2.a.c 1
65.d even 2 1 1950.2.i.m 2
65.h odd 4 2 1950.2.z.g 4
65.l even 6 1 1950.2.i.m 2
65.r odd 12 2 1950.2.z.g 4
156.h even 2 1 1872.2.t.c 2
156.r even 6 1 1872.2.t.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 13.b even 2 1
78.2.e.a 2 13.e even 6 1
234.2.h.a 2 39.d odd 2 1
234.2.h.a 2 39.h odd 6 1
624.2.q.g 2 52.b odd 2 1
624.2.q.g 2 52.i odd 6 1
1014.2.a.c 1 13.e even 6 1
1014.2.a.f 1 13.c even 3 1
1014.2.b.c 2 13.f odd 12 2
1014.2.e.a 2 1.a even 1 1 trivial
1014.2.e.a 2 13.c even 3 1 inner
1014.2.i.b 4 13.d odd 4 2
1014.2.i.b 4 13.f odd 12 2
1872.2.t.c 2 156.h even 2 1
1872.2.t.c 2 156.r even 6 1
1950.2.i.m 2 65.d even 2 1
1950.2.i.m 2 65.l even 6 1
1950.2.z.g 4 65.h odd 4 2
1950.2.z.g 4 65.r odd 12 2
3042.2.a.h 1 39.i odd 6 1
3042.2.a.i 1 39.h odd 6 1
3042.2.b.h 2 39.k even 12 2
8112.2.a.c 1 52.j odd 6 1
8112.2.a.m 1 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 10T + 100$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T - 13)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$T^{2} - 14T + 196$$