# Properties

 Label 294.2.a.f Level 294 Weight 2 Character orbit 294.a Self dual yes Analytic conductor 2.348 Analytic rank 0 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
 $$n$$: 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.f 1
3.b odd 2 1 882.2.a.d 1
4.b odd 2 1 2352.2.a.f 1
5.b even 2 1 7350.2.a.q 1
7.b odd 2 1 294.2.a.e 1
7.c even 3 2 294.2.e.b 2
7.d odd 6 2 42.2.e.a 2
8.b even 2 1 9408.2.a.z 1
8.d odd 2 1 9408.2.a.cr 1
12.b even 2 1 7056.2.a.bl 1
21.c even 2 1 882.2.a.c 1
21.g even 6 2 126.2.g.c 2
21.h odd 6 2 882.2.g.i 2
28.d even 2 1 2352.2.a.t 1
28.f even 6 2 336.2.q.b 2
28.g odd 6 2 2352.2.q.u 2
35.c odd 2 1 7350.2.a.bl 1
35.i odd 6 2 1050.2.i.l 2
35.k even 12 4 1050.2.o.a 4
56.e even 2 1 9408.2.a.q 1
56.h odd 2 1 9408.2.a.ce 1
56.j odd 6 2 1344.2.q.g 2
56.m even 6 2 1344.2.q.s 2
63.i even 6 2 1134.2.e.e 2
63.k odd 6 2 1134.2.h.e 2
63.s even 6 2 1134.2.h.l 2
63.t odd 6 2 1134.2.e.l 2
84.h odd 2 1 7056.2.a.w 1
84.j odd 6 2 1008.2.s.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 7.d odd 6 2
126.2.g.c 2 21.g even 6 2
294.2.a.e 1 7.b odd 2 1
294.2.a.f 1 1.a even 1 1 trivial
294.2.e.b 2 7.c even 3 2
336.2.q.b 2 28.f even 6 2
882.2.a.c 1 21.c even 2 1
882.2.a.d 1 3.b odd 2 1
882.2.g.i 2 21.h odd 6 2
1008.2.s.k 2 84.j odd 6 2
1050.2.i.l 2 35.i odd 6 2
1050.2.o.a 4 35.k even 12 4
1134.2.e.e 2 63.i even 6 2
1134.2.e.l 2 63.t odd 6 2
1134.2.h.e 2 63.k odd 6 2
1134.2.h.l 2 63.s even 6 2
1344.2.q.g 2 56.j odd 6 2
1344.2.q.s 2 56.m even 6 2
2352.2.a.f 1 4.b odd 2 1
2352.2.a.t 1 28.d even 2 1
2352.2.q.u 2 28.g odd 6 2
7056.2.a.w 1 84.h odd 2 1
7056.2.a.bl 1 12.b even 2 1
7350.2.a.q 1 5.b even 2 1
7350.2.a.bl 1 35.c odd 2 1
9408.2.a.q 1 56.e even 2 1
9408.2.a.z 1 8.b even 2 1
9408.2.a.ce 1 56.h odd 2 1
9408.2.a.cr 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 $$T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(294))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - T$$
$5$ $$1 + T + 5 T^{2}$$
$7$ 1
$11$ $$1 - 5 T + 11 T^{2}$$
$13$ $$1 + 13 T^{2}$$
$17$ $$1 - 4 T + 17 T^{2}$$
$19$ $$1 + 8 T + 19 T^{2}$$
$23$ $$1 + 4 T + 23 T^{2}$$
$29$ $$1 + 5 T + 29 T^{2}$$
$31$ $$1 + 3 T + 31 T^{2}$$
$37$ $$1 + 4 T + 37 T^{2}$$
$41$ $$1 + 41 T^{2}$$
$43$ $$1 - 2 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 + 9 T + 53 T^{2}$$
$59$ $$1 - 11 T + 59 T^{2}$$
$61$ $$1 - 6 T + 61 T^{2}$$
$67$ $$1 + 2 T + 67 T^{2}$$
$71$ $$1 - 2 T + 71 T^{2}$$
$73$ $$1 + 10 T + 73 T^{2}$$
$79$ $$1 - 3 T + 79 T^{2}$$
$83$ $$1 - 7 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 + 7 T + 97 T^{2}$$