# Properties

 Label 1014.2.e.b 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + 2 q^{5} - \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 + 2 * q^5 - z * q^6 - 2*z * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + 2 q^{5} - \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + q^{12} + 2 q^{14} + (2 \zeta_{6} - 2) q^{15} + (\zeta_{6} - 1) q^{16} - 2 \zeta_{6} q^{17} + q^{18} + 6 \zeta_{6} q^{19} - 2 \zeta_{6} q^{20} + 2 q^{21} + ( - 4 \zeta_{6} + 4) q^{23} + (\zeta_{6} - 1) q^{24} - q^{25} + q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - 10 \zeta_{6} + 10) q^{29} - 2 \zeta_{6} q^{30} + 10 q^{31} - \zeta_{6} q^{32} + 2 q^{34} - 4 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{36} + ( - 8 \zeta_{6} + 8) q^{37} - 6 q^{38} + 2 q^{40} + (10 \zeta_{6} - 10) q^{41} + (2 \zeta_{6} - 2) q^{42} + 4 \zeta_{6} q^{43} - 2 \zeta_{6} q^{45} + 4 \zeta_{6} q^{46} + 12 q^{47} - \zeta_{6} q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + ( - \zeta_{6} + 1) q^{50} + 2 q^{51} - 6 q^{53} + (\zeta_{6} - 1) q^{54} - 2 \zeta_{6} q^{56} - 6 q^{57} + 10 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + 2 q^{60} - 2 \zeta_{6} q^{61} + (10 \zeta_{6} - 10) q^{62} + (2 \zeta_{6} - 2) q^{63} + q^{64} + ( - 2 \zeta_{6} + 2) q^{67} + (2 \zeta_{6} - 2) q^{68} + 4 \zeta_{6} q^{69} + 4 q^{70} - \zeta_{6} q^{72} + 4 q^{73} + 8 \zeta_{6} q^{74} + ( - \zeta_{6} + 1) q^{75} + ( - 6 \zeta_{6} + 6) q^{76} + (2 \zeta_{6} - 2) q^{80} + (\zeta_{6} - 1) q^{81} - 10 \zeta_{6} q^{82} - 4 q^{83} - 2 \zeta_{6} q^{84} - 4 \zeta_{6} q^{85} - 4 q^{86} + 10 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{89} + 2 q^{90} - 4 q^{92} + (10 \zeta_{6} - 10) q^{93} + (12 \zeta_{6} - 12) q^{94} + 12 \zeta_{6} q^{95} + q^{96} + 12 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 + 2 * q^5 - z * q^6 - 2*z * q^7 + q^8 - z * q^9 + (2*z - 2) * q^10 + q^12 + 2 * q^14 + (2*z - 2) * q^15 + (z - 1) * q^16 - 2*z * q^17 + q^18 + 6*z * q^19 - 2*z * q^20 + 2 * q^21 + (-4*z + 4) * q^23 + (z - 1) * q^24 - q^25 + q^27 + (2*z - 2) * q^28 + (-10*z + 10) * q^29 - 2*z * q^30 + 10 * q^31 - z * q^32 + 2 * q^34 - 4*z * q^35 + (z - 1) * q^36 + (-8*z + 8) * q^37 - 6 * q^38 + 2 * q^40 + (10*z - 10) * q^41 + (2*z - 2) * q^42 + 4*z * q^43 - 2*z * q^45 + 4*z * q^46 + 12 * q^47 - z * q^48 + (-3*z + 3) * q^49 + (-z + 1) * q^50 + 2 * q^51 - 6 * q^53 + (z - 1) * q^54 - 2*z * q^56 - 6 * q^57 + 10*z * q^58 + 4*z * q^59 + 2 * q^60 - 2*z * q^61 + (10*z - 10) * q^62 + (2*z - 2) * q^63 + q^64 + (-2*z + 2) * q^67 + (2*z - 2) * q^68 + 4*z * q^69 + 4 * q^70 - z * q^72 + 4 * q^73 + 8*z * q^74 + (-z + 1) * q^75 + (-6*z + 6) * q^76 + (2*z - 2) * q^80 + (z - 1) * q^81 - 10*z * q^82 - 4 * q^83 - 2*z * q^84 - 4*z * q^85 - 4 * q^86 + 10*z * q^87 + (6*z - 6) * q^89 + 2 * q^90 - 4 * q^92 + (10*z - 10) * q^93 + (12*z - 12) * q^94 + 12*z * q^95 + q^96 + 12*z * q^97 + 3*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + 4 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 + 4 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} + 4 q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{20} + 4 q^{21} + 4 q^{23} - q^{24} - 2 q^{25} + 2 q^{27} - 2 q^{28} + 10 q^{29} - 2 q^{30} + 20 q^{31} - q^{32} + 4 q^{34} - 4 q^{35} - q^{36} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 10 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{45} + 4 q^{46} + 24 q^{47} - q^{48} + 3 q^{49} + q^{50} + 4 q^{51} - 12 q^{53} - q^{54} - 2 q^{56} - 12 q^{57} + 10 q^{58} + 4 q^{59} + 4 q^{60} - 2 q^{61} - 10 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{67} - 2 q^{68} + 4 q^{69} + 8 q^{70} - q^{72} + 8 q^{73} + 8 q^{74} + q^{75} + 6 q^{76} - 2 q^{80} - q^{81} - 10 q^{82} - 8 q^{83} - 2 q^{84} - 4 q^{85} - 8 q^{86} + 10 q^{87} - 6 q^{89} + 4 q^{90} - 8 q^{92} - 10 q^{93} - 12 q^{94} + 12 q^{95} + 2 q^{96} + 12 q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + 4 * q^5 - q^6 - 2 * q^7 + 2 * q^8 - q^9 - 2 * q^10 + 2 * q^12 + 4 * q^14 - 2 * q^15 - q^16 - 2 * q^17 + 2 * q^18 + 6 * q^19 - 2 * q^20 + 4 * q^21 + 4 * q^23 - q^24 - 2 * q^25 + 2 * q^27 - 2 * q^28 + 10 * q^29 - 2 * q^30 + 20 * q^31 - q^32 + 4 * q^34 - 4 * q^35 - q^36 + 8 * q^37 - 12 * q^38 + 4 * q^40 - 10 * q^41 - 2 * q^42 + 4 * q^43 - 2 * q^45 + 4 * q^46 + 24 * q^47 - q^48 + 3 * q^49 + q^50 + 4 * q^51 - 12 * q^53 - q^54 - 2 * q^56 - 12 * q^57 + 10 * q^58 + 4 * q^59 + 4 * q^60 - 2 * q^61 - 10 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^67 - 2 * q^68 + 4 * q^69 + 8 * q^70 - q^72 + 8 * q^73 + 8 * q^74 + q^75 + 6 * q^76 - 2 * q^80 - q^81 - 10 * q^82 - 8 * q^83 - 2 * q^84 - 4 * q^85 - 8 * q^86 + 10 * q^87 - 6 * q^89 + 4 * q^90 - 8 * q^92 - 10 * q^93 - 12 * q^94 + 12 * q^95 + 2 * q^96 + 12 * q^97 + 3 * q^98

## 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 −1.00000 1.73205i 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 −1.00000 + 1.73205i 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.b 2
13.b even 2 1 1014.2.e.e 2
13.c even 3 1 1014.2.a.g 1
13.c even 3 1 inner 1014.2.e.b 2
13.d odd 4 2 1014.2.i.c 4
13.e even 6 1 1014.2.a.b 1
13.e even 6 1 1014.2.e.e 2
13.f odd 12 2 78.2.b.a 2
13.f odd 12 2 1014.2.i.c 4
39.h odd 6 1 3042.2.a.n 1
39.i odd 6 1 3042.2.a.c 1
39.k even 12 2 234.2.b.a 2
52.i odd 6 1 8112.2.a.g 1
52.j odd 6 1 8112.2.a.j 1
52.l even 12 2 624.2.c.a 2
65.o even 12 2 1950.2.f.g 2
65.s odd 12 2 1950.2.b.c 2
65.t even 12 2 1950.2.f.d 2
91.bc even 12 2 3822.2.c.d 2
104.u even 12 2 2496.2.c.m 2
104.x odd 12 2 2496.2.c.f 2
156.v odd 12 2 1872.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 13.f odd 12 2
234.2.b.a 2 39.k even 12 2
624.2.c.a 2 52.l even 12 2
1014.2.a.b 1 13.e even 6 1
1014.2.a.g 1 13.c even 3 1
1014.2.e.b 2 1.a even 1 1 trivial
1014.2.e.b 2 13.c even 3 1 inner
1014.2.e.e 2 13.b even 2 1
1014.2.e.e 2 13.e even 6 1
1014.2.i.c 4 13.d odd 4 2
1014.2.i.c 4 13.f odd 12 2
1872.2.c.b 2 156.v odd 12 2
1950.2.b.c 2 65.s odd 12 2
1950.2.f.d 2 65.t even 12 2
1950.2.f.g 2 65.o even 12 2
2496.2.c.f 2 104.x odd 12 2
2496.2.c.m 2 104.u even 12 2
3042.2.a.c 1 39.i odd 6 1
3042.2.a.n 1 39.h odd 6 1
3822.2.c.d 2 91.bc even 12 2
8112.2.a.g 1 52.i odd 6 1
8112.2.a.j 1 52.j 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 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} - 8T + 64$$
$41$ $$T^{2} + 10T + 100$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$(T - 12)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$T^{2}$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2}$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} - 12T + 144$$