# Properties

 Label 294.2.a.g 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} - 6 q^{13} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 8 q^{23} + q^{24} - q^{25} - 6 q^{26} + q^{27} - 2 q^{29} + 2 q^{30} + q^{32} - 4 q^{33} - 2 q^{34} + q^{36} - 10 q^{37} + 4 q^{38} - 6 q^{39} + 2 q^{40} + 6 q^{41} - 4 q^{43} - 4 q^{44} + 2 q^{45} + 8 q^{46} + q^{48} - q^{50} - 2 q^{51} - 6 q^{52} + 6 q^{53} + q^{54} - 8 q^{55} + 4 q^{57} - 2 q^{58} - 4 q^{59} + 2 q^{60} - 6 q^{61} + q^{64} - 12 q^{65} - 4 q^{66} + 4 q^{67} - 2 q^{68} + 8 q^{69} + 8 q^{71} + q^{72} - 10 q^{73} - 10 q^{74} - q^{75} + 4 q^{76} - 6 q^{78} + 2 q^{80} + q^{81} + 6 q^{82} + 4 q^{83} - 4 q^{85} - 4 q^{86} - 2 q^{87} - 4 q^{88} + 6 q^{89} + 2 q^{90} + 8 q^{92} + 8 q^{95} + q^{96} + 14 q^{97} - 4 q^{99} + O(q^{100})$$

## 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 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

## Atkin-Lehner signs

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

## 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 294.2.a.g 1
3.b odd 2 1 882.2.a.b 1
4.b odd 2 1 2352.2.a.l 1
5.b even 2 1 7350.2.a.f 1
7.b odd 2 1 42.2.a.a 1
7.c even 3 2 294.2.e.a 2
7.d odd 6 2 294.2.e.c 2
8.b even 2 1 9408.2.a.n 1
8.d odd 2 1 9408.2.a.bw 1
12.b even 2 1 7056.2.a.k 1
21.c even 2 1 126.2.a.a 1
21.g even 6 2 882.2.g.h 2
21.h odd 6 2 882.2.g.j 2
28.d even 2 1 336.2.a.d 1
28.f even 6 2 2352.2.q.i 2
28.g odd 6 2 2352.2.q.n 2
35.c odd 2 1 1050.2.a.i 1
35.f even 4 2 1050.2.g.a 2
56.e even 2 1 1344.2.a.i 1
56.h odd 2 1 1344.2.a.q 1
63.l odd 6 2 1134.2.f.g 2
63.o even 6 2 1134.2.f.j 2
77.b even 2 1 5082.2.a.d 1
84.h odd 2 1 1008.2.a.j 1
91.b odd 2 1 7098.2.a.f 1
105.g even 2 1 3150.2.a.bo 1
105.k odd 4 2 3150.2.g.r 2
112.j even 4 2 5376.2.c.e 2
112.l odd 4 2 5376.2.c.bc 2
140.c even 2 1 8400.2.a.k 1
168.e odd 2 1 4032.2.a.m 1
168.i even 2 1 4032.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 7.b odd 2 1
126.2.a.a 1 21.c even 2 1
294.2.a.g 1 1.a even 1 1 trivial
294.2.e.a 2 7.c even 3 2
294.2.e.c 2 7.d odd 6 2
336.2.a.d 1 28.d even 2 1
882.2.a.b 1 3.b odd 2 1
882.2.g.h 2 21.g even 6 2
882.2.g.j 2 21.h odd 6 2
1008.2.a.j 1 84.h odd 2 1
1050.2.a.i 1 35.c odd 2 1
1050.2.g.a 2 35.f even 4 2
1134.2.f.g 2 63.l odd 6 2
1134.2.f.j 2 63.o even 6 2
1344.2.a.i 1 56.e even 2 1
1344.2.a.q 1 56.h odd 2 1
2352.2.a.l 1 4.b odd 2 1
2352.2.q.i 2 28.f even 6 2
2352.2.q.n 2 28.g odd 6 2
3150.2.a.bo 1 105.g even 2 1
3150.2.g.r 2 105.k odd 4 2
4032.2.a.e 1 168.i even 2 1
4032.2.a.m 1 168.e odd 2 1
5082.2.a.d 1 77.b even 2 1
5376.2.c.e 2 112.j even 4 2
5376.2.c.bc 2 112.l odd 4 2
7056.2.a.k 1 12.b even 2 1
7098.2.a.f 1 91.b odd 2 1
7350.2.a.f 1 5.b even 2 1
8400.2.a.k 1 140.c even 2 1
9408.2.a.n 1 8.b even 2 1
9408.2.a.bw 1 8.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

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