# Properties

 Label 1014.2.a.i Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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 −3.73205 −1.00000 2.73205 −1.00000 1.00000 3.73205
1.2 −1.00000 1.00000 1.00000 −0.267949 −1.00000 −0.732051 −1.00000 1.00000 0.267949
 $$n$$: 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.i 2
3.b odd 2 1 3042.2.a.y 2
4.b odd 2 1 8112.2.a.bj 2
13.b even 2 1 1014.2.a.k 2
13.c even 3 2 1014.2.e.i 4
13.d odd 4 2 1014.2.b.e 4
13.e even 6 2 1014.2.e.g 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.p 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.bp 2
52.l even 12 2 624.2.bv.e 4
65.o even 12 2 1950.2.y.g 4
65.s odd 12 2 1950.2.bc.d 4
65.t even 12 2 1950.2.y.b 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 1.a even 1 1 trivial
1014.2.a.k 2 13.b even 2 1
1014.2.b.e 4 13.d odd 4 2
1014.2.e.g 4 13.e even 6 2
1014.2.e.i 4 13.c even 3 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.t even 12 2
1950.2.y.g 4 65.o even 12 2
1950.2.bc.d 4 65.s odd 12 2
3042.2.a.p 2 39.d odd 2 1
3042.2.a.y 2 3.b odd 2 1
3042.2.b.i 4 39.f even 4 2
8112.2.a.bj 2 4.b odd 2 1
8112.2.a.bp 2 52.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

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