# Properties

 Label 1014.2.a.d Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 −1.00000 1.00000 −2.00000 −1.00000 −4.00000 1.00000 1.00000 −2.00000
 $$n$$: 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.d 1
3.b odd 2 1 3042.2.a.f 1
4.b odd 2 1 8112.2.a.v 1
13.b even 2 1 78.2.a.a 1
13.c even 3 2 1014.2.e.c 2
13.d odd 4 2 1014.2.b.b 2
13.e even 6 2 1014.2.e.f 2
13.f odd 12 4 1014.2.i.d 4
39.d odd 2 1 234.2.a.c 1
39.f even 4 2 3042.2.b.g 2
52.b odd 2 1 624.2.a.h 1
65.d even 2 1 1950.2.a.w 1
65.h odd 4 2 1950.2.e.i 2
91.b odd 2 1 3822.2.a.j 1
104.e even 2 1 2496.2.a.t 1
104.h odd 2 1 2496.2.a.b 1
117.n odd 6 2 2106.2.e.j 2
117.t even 6 2 2106.2.e.q 2
143.d odd 2 1 9438.2.a.t 1
156.h even 2 1 1872.2.a.c 1
195.e odd 2 1 5850.2.a.d 1
195.s even 4 2 5850.2.e.bb 2
312.b odd 2 1 7488.2.a.bz 1
312.h even 2 1 7488.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.b even 2 1
234.2.a.c 1 39.d odd 2 1
624.2.a.h 1 52.b odd 2 1
1014.2.a.d 1 1.a even 1 1 trivial
1014.2.b.b 2 13.d odd 4 2
1014.2.e.c 2 13.c even 3 2
1014.2.e.f 2 13.e even 6 2
1014.2.i.d 4 13.f odd 12 4
1872.2.a.c 1 156.h even 2 1
1950.2.a.w 1 65.d even 2 1
1950.2.e.i 2 65.h odd 4 2
2106.2.e.j 2 117.n odd 6 2
2106.2.e.q 2 117.t even 6 2
2496.2.a.b 1 104.h odd 2 1
2496.2.a.t 1 104.e even 2 1
3042.2.a.f 1 3.b odd 2 1
3042.2.b.g 2 39.f even 4 2
3822.2.a.j 1 91.b odd 2 1
5850.2.a.d 1 195.e odd 2 1
5850.2.e.bb 2 195.s even 4 2
7488.2.a.bk 1 312.h even 2 1
7488.2.a.bz 1 312.b odd 2 1
8112.2.a.v 1 4.b odd 2 1
9438.2.a.t 1 143.d odd 2 1

## Hecke kernels

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

 $$T_{5} + 2$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

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