# Properties

 Label 1014.2.e.c Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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

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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 2 q^{5} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 - 2 * q^5 + z * q^6 + 4*z * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 2 q^{5} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + (4 \zeta_{6} - 4) q^{11} - q^{12} - 4 q^{14} + (2 \zeta_{6} - 2) q^{15} + (\zeta_{6} - 1) q^{16} - 2 \zeta_{6} q^{17} + q^{18} - 8 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} + 4 q^{21} - 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{24} - q^{25} - q^{27} + ( - 4 \zeta_{6} + 4) q^{28} + (6 \zeta_{6} - 6) q^{29} - 2 \zeta_{6} q^{30} + 4 q^{31} - \zeta_{6} q^{32} + 4 \zeta_{6} q^{33} + 2 q^{34} - 8 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{36} + (2 \zeta_{6} - 2) q^{37} + 8 q^{38} - 2 q^{40} + (10 \zeta_{6} - 10) q^{41} + (4 \zeta_{6} - 4) q^{42} - 4 \zeta_{6} q^{43} + 4 q^{44} + 2 \zeta_{6} q^{45} - 8 q^{47} + \zeta_{6} q^{48} + (9 \zeta_{6} - 9) q^{49} + ( - \zeta_{6} + 1) q^{50} - 2 q^{51} - 10 q^{53} + ( - \zeta_{6} + 1) q^{54} + ( - 8 \zeta_{6} + 8) q^{55} + 4 \zeta_{6} q^{56} - 8 q^{57} - 6 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + 2 q^{60} + 2 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 4 \zeta_{6} + 4) q^{63} + q^{64} - 4 q^{66} + (16 \zeta_{6} - 16) q^{67} + (2 \zeta_{6} - 2) q^{68} + 8 q^{70} - 8 \zeta_{6} q^{71} - \zeta_{6} q^{72} - 2 q^{73} - 2 \zeta_{6} q^{74} + (\zeta_{6} - 1) q^{75} + (8 \zeta_{6} - 8) q^{76} - 16 q^{77} + 8 q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + (\zeta_{6} - 1) q^{81} - 10 \zeta_{6} q^{82} - 12 q^{83} - 4 \zeta_{6} q^{84} + 4 \zeta_{6} q^{85} + 4 q^{86} + 6 \zeta_{6} q^{87} + (4 \zeta_{6} - 4) q^{88} + ( - 14 \zeta_{6} + 14) q^{89} - 2 q^{90} + ( - 4 \zeta_{6} + 4) q^{93} + ( - 8 \zeta_{6} + 8) q^{94} + 16 \zeta_{6} q^{95} - q^{96} + 10 \zeta_{6} q^{97} - 9 \zeta_{6} q^{98} + 4 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 - 2 * q^5 + z * q^6 + 4*z * q^7 + q^8 - z * q^9 + (-2*z + 2) * q^10 + (4*z - 4) * q^11 - q^12 - 4 * q^14 + (2*z - 2) * q^15 + (z - 1) * q^16 - 2*z * q^17 + q^18 - 8*z * q^19 + 2*z * q^20 + 4 * q^21 - 4*z * q^22 + (-z + 1) * q^24 - q^25 - q^27 + (-4*z + 4) * q^28 + (6*z - 6) * q^29 - 2*z * q^30 + 4 * q^31 - z * q^32 + 4*z * q^33 + 2 * q^34 - 8*z * q^35 + (z - 1) * q^36 + (2*z - 2) * q^37 + 8 * q^38 - 2 * q^40 + (10*z - 10) * q^41 + (4*z - 4) * q^42 - 4*z * q^43 + 4 * q^44 + 2*z * q^45 - 8 * q^47 + z * q^48 + (9*z - 9) * q^49 + (-z + 1) * q^50 - 2 * q^51 - 10 * q^53 + (-z + 1) * q^54 + (-8*z + 8) * q^55 + 4*z * q^56 - 8 * q^57 - 6*z * q^58 + 4*z * q^59 + 2 * q^60 + 2*z * q^61 + (4*z - 4) * q^62 + (-4*z + 4) * q^63 + q^64 - 4 * q^66 + (16*z - 16) * q^67 + (2*z - 2) * q^68 + 8 * q^70 - 8*z * q^71 - z * q^72 - 2 * q^73 - 2*z * q^74 + (z - 1) * q^75 + (8*z - 8) * q^76 - 16 * q^77 + 8 * q^79 + (-2*z + 2) * q^80 + (z - 1) * q^81 - 10*z * q^82 - 12 * q^83 - 4*z * q^84 + 4*z * q^85 + 4 * q^86 + 6*z * q^87 + (4*z - 4) * q^88 + (-14*z + 14) * q^89 - 2 * q^90 + (-4*z + 4) * q^93 + (-8*z + 8) * q^94 + 16*z * q^95 - q^96 + 10*z * q^97 - 9*z * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 - 4 * q^5 + q^6 + 4 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} + q^{3} - q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + 2 q^{8} - q^{9} + 2 q^{10} - 4 q^{11} - 2 q^{12} - 8 q^{14} - 2 q^{15} - q^{16} - 2 q^{17} + 2 q^{18} - 8 q^{19} + 2 q^{20} + 8 q^{21} - 4 q^{22} + q^{24} - 2 q^{25} - 2 q^{27} + 4 q^{28} - 6 q^{29} - 2 q^{30} + 8 q^{31} - q^{32} + 4 q^{33} + 4 q^{34} - 8 q^{35} - q^{36} - 2 q^{37} + 16 q^{38} - 4 q^{40} - 10 q^{41} - 4 q^{42} - 4 q^{43} + 8 q^{44} + 2 q^{45} - 16 q^{47} + q^{48} - 9 q^{49} + q^{50} - 4 q^{51} - 20 q^{53} + q^{54} + 8 q^{55} + 4 q^{56} - 16 q^{57} - 6 q^{58} + 4 q^{59} + 4 q^{60} + 2 q^{61} - 4 q^{62} + 4 q^{63} + 2 q^{64} - 8 q^{66} - 16 q^{67} - 2 q^{68} + 16 q^{70} - 8 q^{71} - q^{72} - 4 q^{73} - 2 q^{74} - q^{75} - 8 q^{76} - 32 q^{77} + 16 q^{79} + 2 q^{80} - q^{81} - 10 q^{82} - 24 q^{83} - 4 q^{84} + 4 q^{85} + 8 q^{86} + 6 q^{87} - 4 q^{88} + 14 q^{89} - 4 q^{90} + 4 q^{93} + 8 q^{94} + 16 q^{95} - 2 q^{96} + 10 q^{97} - 9 q^{98} + 8 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 - 4 * q^5 + q^6 + 4 * q^7 + 2 * q^8 - q^9 + 2 * q^10 - 4 * q^11 - 2 * q^12 - 8 * q^14 - 2 * q^15 - q^16 - 2 * q^17 + 2 * q^18 - 8 * q^19 + 2 * q^20 + 8 * q^21 - 4 * q^22 + q^24 - 2 * q^25 - 2 * q^27 + 4 * q^28 - 6 * q^29 - 2 * q^30 + 8 * q^31 - q^32 + 4 * q^33 + 4 * q^34 - 8 * q^35 - q^36 - 2 * q^37 + 16 * q^38 - 4 * q^40 - 10 * q^41 - 4 * q^42 - 4 * q^43 + 8 * q^44 + 2 * q^45 - 16 * q^47 + q^48 - 9 * q^49 + q^50 - 4 * q^51 - 20 * q^53 + q^54 + 8 * q^55 + 4 * q^56 - 16 * q^57 - 6 * q^58 + 4 * q^59 + 4 * q^60 + 2 * q^61 - 4 * q^62 + 4 * q^63 + 2 * q^64 - 8 * q^66 - 16 * q^67 - 2 * q^68 + 16 * q^70 - 8 * q^71 - q^72 - 4 * q^73 - 2 * q^74 - q^75 - 8 * q^76 - 32 * q^77 + 16 * q^79 + 2 * q^80 - q^81 - 10 * q^82 - 24 * q^83 - 4 * q^84 + 4 * q^85 + 8 * q^86 + 6 * q^87 - 4 * q^88 + 14 * q^89 - 4 * q^90 + 4 * q^93 + 8 * q^94 + 16 * q^95 - 2 * q^96 + 10 * q^97 - 9 * q^98 + 8 * 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.

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 −2.00000 0.500000 + 0.866025i 2.00000 + 3.46410i 1.00000 −0.500000 0.866025i 1.00000 1.73205i
991.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 2.00000 3.46410i 1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
 $$n$$: 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.c 2
13.b even 2 1 1014.2.e.f 2
13.c even 3 1 1014.2.a.d 1
13.c even 3 1 inner 1014.2.e.c 2
13.d odd 4 2 1014.2.i.d 4
13.e even 6 1 78.2.a.a 1
13.e even 6 1 1014.2.e.f 2
13.f odd 12 2 1014.2.b.b 2
13.f odd 12 2 1014.2.i.d 4
39.h odd 6 1 234.2.a.c 1
39.i odd 6 1 3042.2.a.f 1
39.k even 12 2 3042.2.b.g 2
52.i odd 6 1 624.2.a.h 1
52.j odd 6 1 8112.2.a.v 1
65.l even 6 1 1950.2.a.w 1
65.r odd 12 2 1950.2.e.i 2
91.t odd 6 1 3822.2.a.j 1
104.p odd 6 1 2496.2.a.b 1
104.s even 6 1 2496.2.a.t 1
117.l even 6 1 2106.2.e.q 2
117.m odd 6 1 2106.2.e.j 2
117.r even 6 1 2106.2.e.q 2
117.v odd 6 1 2106.2.e.j 2
143.i odd 6 1 9438.2.a.t 1
156.r even 6 1 1872.2.a.c 1
195.y odd 6 1 5850.2.a.d 1
195.bf even 12 2 5850.2.e.bb 2
312.ba even 6 1 7488.2.a.bk 1
312.bg odd 6 1 7488.2.a.bz 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.e even 6 1
234.2.a.c 1 39.h odd 6 1
624.2.a.h 1 52.i odd 6 1
1014.2.a.d 1 13.c even 3 1
1014.2.b.b 2 13.f odd 12 2
1014.2.e.c 2 1.a even 1 1 trivial
1014.2.e.c 2 13.c even 3 1 inner
1014.2.e.f 2 13.b even 2 1
1014.2.e.f 2 13.e even 6 1
1014.2.i.d 4 13.d odd 4 2
1014.2.i.d 4 13.f odd 12 2
1872.2.a.c 1 156.r even 6 1
1950.2.a.w 1 65.l even 6 1
1950.2.e.i 2 65.r odd 12 2
2106.2.e.j 2 117.m odd 6 1
2106.2.e.j 2 117.v odd 6 1
2106.2.e.q 2 117.l even 6 1
2106.2.e.q 2 117.r even 6 1
2496.2.a.b 1 104.p odd 6 1
2496.2.a.t 1 104.s even 6 1
3042.2.a.f 1 39.i odd 6 1
3042.2.b.g 2 39.k even 12 2
3822.2.a.j 1 91.t odd 6 1
5850.2.a.d 1 195.y odd 6 1
5850.2.e.bb 2 195.bf even 12 2
7488.2.a.bk 1 312.ba even 6 1
7488.2.a.bz 1 312.bg odd 6 1
8112.2.a.v 1 52.j odd 6 1
9438.2.a.t 1 143.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 $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + 8T + 64$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} + 10T + 100$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 16T + 256$$
$71$ $$T^{2} + 8T + 64$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$T^{2} - 10T + 100$$