# Properties

 Label 1014.2.a.k Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
1.00000 1.00000 1.00000 0.267949 1.00000 0.732051 1.00000 1.00000 0.267949
1.2 1.00000 1.00000 1.00000 3.73205 1.00000 −2.73205 1.00000 1.00000 3.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.a.k 2
3.b odd 2 1 3042.2.a.p 2
4.b odd 2 1 8112.2.a.bp 2
13.b even 2 1 1014.2.a.i 2
13.c even 3 2 1014.2.e.g 4
13.d odd 4 2 1014.2.b.e 4
13.e even 6 2 1014.2.e.i 4
13.f odd 12 2 78.2.i.a 4
13.f odd 12 2 1014.2.i.a 4
39.d odd 2 1 3042.2.a.y 2
39.f even 4 2 3042.2.b.i 4
39.k even 12 2 234.2.l.c 4
52.b odd 2 1 8112.2.a.bj 2
52.l even 12 2 624.2.bv.e 4
65.o even 12 2 1950.2.y.b 4
65.s odd 12 2 1950.2.bc.d 4
65.t even 12 2 1950.2.y.g 4
156.v odd 12 2 1872.2.by.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 13.f odd 12 2
234.2.l.c 4 39.k even 12 2
624.2.bv.e 4 52.l even 12 2
1014.2.a.i 2 13.b even 2 1
1014.2.a.k 2 1.a even 1 1 trivial
1014.2.b.e 4 13.d odd 4 2
1014.2.e.g 4 13.c even 3 2
1014.2.e.i 4 13.e even 6 2
1014.2.i.a 4 13.f odd 12 2
1872.2.by.h 4 156.v odd 12 2
1950.2.y.b 4 65.o even 12 2
1950.2.y.g 4 65.t even 12 2
1950.2.bc.d 4 65.s odd 12 2
3042.2.a.p 2 3.b odd 2 1
3042.2.a.y 2 39.d odd 2 1
3042.2.b.i 4 39.f even 4 2
8112.2.a.bj 2 52.b odd 2 1
8112.2.a.bp 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1014))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 - 4 T + T^{2}$$
$7$ $$-2 + 2 T + T^{2}$$
$11$ $$6 - 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$13 + 8 T + T^{2}$$
$19$ $$6 - 6 T + T^{2}$$
$23$ $$-26 + 2 T + T^{2}$$
$29$ $$-11 + 2 T + T^{2}$$
$31$ $$-8 + 4 T + T^{2}$$
$37$ $$37 - 14 T + T^{2}$$
$41$ $$-107 - 2 T + T^{2}$$
$43$ $$-74 + 2 T + T^{2}$$
$47$ $$-18 + 6 T + T^{2}$$
$53$ $$-3 + 6 T + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$-11 + 8 T + T^{2}$$
$67$ $$-146 - 2 T + T^{2}$$
$71$ $$6 - 6 T + T^{2}$$
$73$ $$61 + 16 T + T^{2}$$
$79$ $$24 + 12 T + T^{2}$$
$83$ $$-2 - 10 T + T^{2}$$
$89$ $$24 + 12 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$