# Properties

 Label 1014.2.e.d 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{6} -2 \zeta_{6} q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{6} -2 \zeta_{6} q^{7} + q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 2 - 2 \zeta_{6} ) q^{11} - q^{12} + 2 q^{14} + ( 1 - \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -5 \zeta_{6} q^{17} + q^{18} -2 \zeta_{6} q^{19} -\zeta_{6} q^{20} -2 q^{21} + 2 \zeta_{6} q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} -4 q^{25} - q^{27} + ( -2 + 2 \zeta_{6} ) q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} + \zeta_{6} q^{30} + 4 q^{31} -\zeta_{6} q^{32} -2 \zeta_{6} q^{33} + 5 q^{34} -2 \zeta_{6} q^{35} + ( -1 + \zeta_{6} ) q^{36} + ( -11 + 11 \zeta_{6} ) q^{37} + 2 q^{38} + q^{40} + ( 5 - 5 \zeta_{6} ) q^{41} + ( 2 - 2 \zeta_{6} ) q^{42} -10 \zeta_{6} q^{43} -2 q^{44} -\zeta_{6} q^{45} -6 \zeta_{6} q^{46} -2 q^{47} + \zeta_{6} q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} -5 q^{51} - q^{53} + ( 1 - \zeta_{6} ) q^{54} + ( 2 - 2 \zeta_{6} ) q^{55} -2 \zeta_{6} q^{56} -2 q^{57} + 9 \zeta_{6} q^{58} -8 \zeta_{6} q^{59} - q^{60} + 11 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} + ( -2 + 2 \zeta_{6} ) q^{63} + q^{64} + 2 q^{66} + ( 2 - 2 \zeta_{6} ) q^{67} + ( -5 + 5 \zeta_{6} ) q^{68} + 6 \zeta_{6} q^{69} + 2 q^{70} -14 \zeta_{6} q^{71} -\zeta_{6} q^{72} + 13 q^{73} -11 \zeta_{6} q^{74} + ( -4 + 4 \zeta_{6} ) q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} -4 q^{77} -4 q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 5 \zeta_{6} q^{82} -6 q^{83} + 2 \zeta_{6} q^{84} -5 \zeta_{6} q^{85} + 10 q^{86} -9 \zeta_{6} q^{87} + ( 2 - 2 \zeta_{6} ) q^{88} + ( 2 - 2 \zeta_{6} ) q^{89} + q^{90} + 6 q^{92} + ( 4 - 4 \zeta_{6} ) q^{93} + ( 2 - 2 \zeta_{6} ) q^{94} -2 \zeta_{6} q^{95} - q^{96} -2 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 2 q^{7} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q - q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 2 q^{7} + 2 q^{8} - q^{9} - q^{10} + 2 q^{11} - 2 q^{12} + 4 q^{14} + q^{15} - q^{16} - 5 q^{17} + 2 q^{18} - 2 q^{19} - q^{20} - 4 q^{21} + 2 q^{22} - 6 q^{23} + q^{24} - 8 q^{25} - 2 q^{27} - 2 q^{28} + 9 q^{29} + q^{30} + 8 q^{31} - q^{32} - 2 q^{33} + 10 q^{34} - 2 q^{35} - q^{36} - 11 q^{37} + 4 q^{38} + 2 q^{40} + 5 q^{41} + 2 q^{42} - 10 q^{43} - 4 q^{44} - q^{45} - 6 q^{46} - 4 q^{47} + q^{48} + 3 q^{49} + 4 q^{50} - 10 q^{51} - 2 q^{53} + q^{54} + 2 q^{55} - 2 q^{56} - 4 q^{57} + 9 q^{58} - 8 q^{59} - 2 q^{60} + 11 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} + 2 q^{67} - 5 q^{68} + 6 q^{69} + 4 q^{70} - 14 q^{71} - q^{72} + 26 q^{73} - 11 q^{74} - 4 q^{75} - 2 q^{76} - 8 q^{77} - 8 q^{79} - q^{80} - q^{81} + 5 q^{82} - 12 q^{83} + 2 q^{84} - 5 q^{85} + 20 q^{86} - 9 q^{87} + 2 q^{88} + 2 q^{89} + 2 q^{90} + 12 q^{92} + 4 q^{93} + 2 q^{94} - 2 q^{95} - 2 q^{96} - 2 q^{97} + 3 q^{98} - 4 q^{99} + O(q^{100})$$

## 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 1.00000 0.500000 + 0.866025i −1.00000 1.73205i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
991.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.500000 0.866025i −1.00000 + 1.73205i 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
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.d 2
13.b even 2 1 78.2.e.b 2
13.c even 3 1 1014.2.a.e 1
13.c even 3 1 inner 1014.2.e.d 2
13.d odd 4 2 1014.2.i.e 4
13.e even 6 1 78.2.e.b 2
13.e even 6 1 1014.2.a.a 1
13.f odd 12 2 1014.2.b.a 2
13.f odd 12 2 1014.2.i.e 4
39.d odd 2 1 234.2.h.b 2
39.h odd 6 1 234.2.h.b 2
39.h odd 6 1 3042.2.a.m 1
39.i odd 6 1 3042.2.a.d 1
39.k even 12 2 3042.2.b.d 2
52.b odd 2 1 624.2.q.b 2
52.i odd 6 1 624.2.q.b 2
52.i odd 6 1 8112.2.a.x 1
52.j odd 6 1 8112.2.a.bb 1
65.d even 2 1 1950.2.i.b 2
65.h odd 4 2 1950.2.z.b 4
65.l even 6 1 1950.2.i.b 2
65.r odd 12 2 1950.2.z.b 4
156.h even 2 1 1872.2.t.i 2
156.r even 6 1 1872.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 13.b even 2 1
78.2.e.b 2 13.e even 6 1
234.2.h.b 2 39.d odd 2 1
234.2.h.b 2 39.h odd 6 1
624.2.q.b 2 52.b odd 2 1
624.2.q.b 2 52.i odd 6 1
1014.2.a.a 1 13.e even 6 1
1014.2.a.e 1 13.c even 3 1
1014.2.b.a 2 13.f odd 12 2
1014.2.e.d 2 1.a even 1 1 trivial
1014.2.e.d 2 13.c even 3 1 inner
1014.2.i.e 4 13.d odd 4 2
1014.2.i.e 4 13.f odd 12 2
1872.2.t.i 2 156.h even 2 1
1872.2.t.i 2 156.r even 6 1
1950.2.i.b 2 65.d even 2 1
1950.2.i.b 2 65.l even 6 1
1950.2.z.b 4 65.h odd 4 2
1950.2.z.b 4 65.r odd 12 2
3042.2.a.d 1 39.i odd 6 1
3042.2.a.m 1 39.h odd 6 1
3042.2.b.d 2 39.k even 12 2
8112.2.a.x 1 52.i odd 6 1
8112.2.a.bb 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 $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5} - 1$$ $$T_{7}^{2} + 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$25 + 5 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$121 + 11 T + T^{2}$$
$41$ $$25 - 5 T + T^{2}$$
$43$ $$100 + 10 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$121 - 11 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$196 + 14 T + T^{2}$$
$73$ $$( -13 + T )^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$4 - 2 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$