# Properties

 Label 1014.2.a.o Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.80194 0.445042 −1.24698
1.00000 1.00000 1.00000 −4.04892 1.00000 −0.692021 1.00000 1.00000 −4.04892
1.2 1.00000 1.00000 1.00000 0.356896 1.00000 4.04892 1.00000 1.00000 0.356896
1.3 1.00000 1.00000 1.00000 0.692021 1.00000 −0.356896 1.00000 1.00000 0.692021
 $$n$$: 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.o yes 3
3.b odd 2 1 3042.2.a.bd 3
4.b odd 2 1 8112.2.a.bz 3
13.b even 2 1 1014.2.a.m 3
13.c even 3 2 1014.2.e.k 6
13.d odd 4 2 1014.2.b.g 6
13.e even 6 2 1014.2.e.m 6
13.f odd 12 4 1014.2.i.g 12
39.d odd 2 1 3042.2.a.be 3
39.f even 4 2 3042.2.b.r 6
52.b odd 2 1 8112.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 13.b even 2 1
1014.2.a.o yes 3 1.a even 1 1 trivial
1014.2.b.g 6 13.d odd 4 2
1014.2.e.k 6 13.c even 3 2
1014.2.e.m 6 13.e even 6 2
1014.2.i.g 12 13.f odd 12 4
3042.2.a.bd 3 3.b odd 2 1
3042.2.a.be 3 39.d odd 2 1
3042.2.b.r 6 39.f even 4 2
8112.2.a.bz 3 4.b odd 2 1
8112.2.a.ce 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1014))$$:

 $$T_{5}^{3} + 3 T_{5}^{2} - 4 T_{5} + 1$$ $$T_{7}^{3} - 3 T_{7}^{2} - 4 T_{7} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$1 - 4 T + 3 T^{2} + T^{3}$$
$7$ $$-1 - 4 T - 3 T^{2} + T^{3}$$
$11$ $$-13 - 16 T - T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$104 + 20 T - 12 T^{2} + T^{3}$$
$19$ $$64 - 32 T - 4 T^{2} + T^{3}$$
$23$ $$-104 + 76 T - 16 T^{2} + T^{3}$$
$29$ $$223 + 12 T - 13 T^{2} + T^{3}$$
$31$ $$-29 - 22 T + 9 T^{2} + T^{3}$$
$37$ $$8 + 20 T + 12 T^{2} + T^{3}$$
$41$ $$56 + 56 T + 14 T^{2} + T^{3}$$
$43$ $$-344 - 44 T + 8 T^{2} + T^{3}$$
$47$ $$64 - 32 T - 4 T^{2} + T^{3}$$
$53$ $$1247 - 72 T - 15 T^{2} + T^{3}$$
$59$ $$13 + 20 T + 9 T^{2} + T^{3}$$
$61$ $$8 + 24 T + 10 T^{2} + T^{3}$$
$67$ $$-1112 - 184 T + 6 T^{2} + T^{3}$$
$71$ $$104 - 72 T - 6 T^{2} + T^{3}$$
$73$ $$13 - 22 T - 5 T^{2} + T^{3}$$
$79$ $$-1469 - 204 T + 5 T^{2} + T^{3}$$
$83$ $$-1477 - 224 T + 7 T^{2} + T^{3}$$
$89$ $$8 + 24 T + 10 T^{2} + T^{3}$$
$97$ $$7 - 14 T - 7 T^{2} + T^{3}$$