# Properties

 Label 102.2.a.c Level 102 Weight 2 Character orbit 102.a Self dual yes Analytic conductor 0.814 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.814474100617$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - 2q^{15} + q^{16} + q^{17} + q^{18} + 4q^{19} - 2q^{20} - 4q^{22} + q^{24} - q^{25} - 2q^{26} + q^{27} - 10q^{29} - 2q^{30} + 8q^{31} + q^{32} - 4q^{33} + q^{34} + q^{36} - 2q^{37} + 4q^{38} - 2q^{39} - 2q^{40} + 10q^{41} + 12q^{43} - 4q^{44} - 2q^{45} + q^{48} - 7q^{49} - q^{50} + q^{51} - 2q^{52} + 6q^{53} + q^{54} + 8q^{55} + 4q^{57} - 10q^{58} + 12q^{59} - 2q^{60} - 10q^{61} + 8q^{62} + q^{64} + 4q^{65} - 4q^{66} - 12q^{67} + q^{68} + q^{72} + 10q^{73} - 2q^{74} - q^{75} + 4q^{76} - 2q^{78} - 8q^{79} - 2q^{80} + q^{81} + 10q^{82} + 4q^{83} - 2q^{85} + 12q^{86} - 10q^{87} - 4q^{88} - 6q^{89} - 2q^{90} + 8q^{93} - 8q^{95} + q^{96} - 14q^{97} - 7q^{98} - 4q^{99} + O(q^{100})$$

## 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 0 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

## 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 102.2.a.c 1
3.b odd 2 1 306.2.a.b 1
4.b odd 2 1 816.2.a.b 1
5.b even 2 1 2550.2.a.c 1
5.c odd 4 2 2550.2.d.m 2
7.b odd 2 1 4998.2.a.be 1
8.b even 2 1 3264.2.a.m 1
8.d odd 2 1 3264.2.a.bc 1
12.b even 2 1 2448.2.a.p 1
15.d odd 2 1 7650.2.a.ca 1
17.b even 2 1 1734.2.a.j 1
17.c even 4 2 1734.2.b.b 2
17.d even 8 4 1734.2.f.e 4
24.f even 2 1 9792.2.a.l 1
24.h odd 2 1 9792.2.a.k 1
51.c odd 2 1 5202.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 1.a even 1 1 trivial
306.2.a.b 1 3.b odd 2 1
816.2.a.b 1 4.b odd 2 1
1734.2.a.j 1 17.b even 2 1
1734.2.b.b 2 17.c even 4 2
1734.2.f.e 4 17.d even 8 4
2448.2.a.p 1 12.b even 2 1
2550.2.a.c 1 5.b even 2 1
2550.2.d.m 2 5.c odd 4 2
3264.2.a.m 1 8.b even 2 1
3264.2.a.bc 1 8.d odd 2 1
4998.2.a.be 1 7.b odd 2 1
5202.2.a.c 1 51.c odd 2 1
7650.2.a.ca 1 15.d odd 2 1
9792.2.a.k 1 24.h odd 2 1
9792.2.a.l 1 24.f even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(102))$$.

## Hecke characteristic polynomials

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