# Properties

 Label 294.2.a.c Level 294 Weight 2 Character orbit 294.a Self dual yes Analytic conductor 2.348 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 294.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.34760181943$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} - q^{8} + q^{9} + 4q^{10} - 4q^{11} + q^{12} - 4q^{13} - 4q^{15} + q^{16} - q^{18} - 4q^{19} - 4q^{20} + 4q^{22} - q^{24} + 11q^{25} + 4q^{26} + q^{27} + 2q^{29} + 4q^{30} - 8q^{31} - q^{32} - 4q^{33} + q^{36} - 6q^{37} + 4q^{38} - 4q^{39} + 4q^{40} + 4q^{43} - 4q^{44} - 4q^{45} + 8q^{47} + q^{48} - 11q^{50} - 4q^{52} - 10q^{53} - q^{54} + 16q^{55} - 4q^{57} - 2q^{58} - 4q^{59} - 4q^{60} + 4q^{61} + 8q^{62} + q^{64} + 16q^{65} + 4q^{66} + 4q^{67} + 8q^{71} - q^{72} + 16q^{73} + 6q^{74} + 11q^{75} - 4q^{76} + 4q^{78} - 8q^{79} - 4q^{80} + q^{81} + 12q^{83} - 4q^{86} + 2q^{87} + 4q^{88} - 8q^{89} + 4q^{90} - 8q^{93} - 8q^{94} + 16q^{95} - q^{96} - 8q^{97} - 4q^{99} + O(q^{100})$$

## 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 −4.00000 −1.00000 0 −1.00000 1.00000 4.00000
 $$n$$: 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 294.2.a.c yes 1
3.b odd 2 1 882.2.a.l 1
4.b odd 2 1 2352.2.a.b 1
5.b even 2 1 7350.2.a.br 1
7.b odd 2 1 294.2.a.b 1
7.c even 3 2 294.2.e.d 2
7.d odd 6 2 294.2.e.e 2
8.b even 2 1 9408.2.a.bo 1
8.d odd 2 1 9408.2.a.de 1
12.b even 2 1 7056.2.a.ca 1
21.c even 2 1 882.2.a.f 1
21.g even 6 2 882.2.g.f 2
21.h odd 6 2 882.2.g.a 2
28.d even 2 1 2352.2.a.y 1
28.f even 6 2 2352.2.q.a 2
28.g odd 6 2 2352.2.q.y 2
35.c odd 2 1 7350.2.a.cj 1
56.e even 2 1 9408.2.a.b 1
56.h odd 2 1 9408.2.a.br 1
84.h odd 2 1 7056.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.a.b 1 7.b odd 2 1
294.2.a.c yes 1 1.a even 1 1 trivial
294.2.e.d 2 7.c even 3 2
294.2.e.e 2 7.d odd 6 2
882.2.a.f 1 21.c even 2 1
882.2.a.l 1 3.b odd 2 1
882.2.g.a 2 21.h odd 6 2
882.2.g.f 2 21.g even 6 2
2352.2.a.b 1 4.b odd 2 1
2352.2.a.y 1 28.d even 2 1
2352.2.q.a 2 28.f even 6 2
2352.2.q.y 2 28.g odd 6 2
7056.2.a.a 1 84.h odd 2 1
7056.2.a.ca 1 12.b even 2 1
7350.2.a.br 1 5.b even 2 1
7350.2.a.cj 1 35.c odd 2 1
9408.2.a.b 1 56.e even 2 1
9408.2.a.bo 1 8.b even 2 1
9408.2.a.br 1 56.h odd 2 1
9408.2.a.de 1 8.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(294))$$.

## Hecke characteristic polynomials

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