# Properties

 Label 294.4.a.c Level $294$ Weight $4$ Character orbit 294.a Self dual yes Analytic conductor $17.347$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 294.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3465615417$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 8q^{8} + 9q^{9} - 12q^{10} - 30q^{11} - 12q^{12} - 53q^{13} - 18q^{15} + 16q^{16} + 84q^{17} - 18q^{18} + 97q^{19} + 24q^{20} + 60q^{22} + 84q^{23} + 24q^{24} - 89q^{25} + 106q^{26} - 27q^{27} - 180q^{29} + 36q^{30} - 179q^{31} - 32q^{32} + 90q^{33} - 168q^{34} + 36q^{36} - 145q^{37} - 194q^{38} + 159q^{39} - 48q^{40} - 126q^{41} - 325q^{43} - 120q^{44} + 54q^{45} - 168q^{46} + 366q^{47} - 48q^{48} + 178q^{50} - 252q^{51} - 212q^{52} - 768q^{53} + 54q^{54} - 180q^{55} - 291q^{57} + 360q^{58} + 264q^{59} - 72q^{60} - 818q^{61} + 358q^{62} + 64q^{64} - 318q^{65} - 180q^{66} - 523q^{67} + 336q^{68} - 252q^{69} - 342q^{71} - 72q^{72} + 43q^{73} + 290q^{74} + 267q^{75} + 388q^{76} - 318q^{78} - 1171q^{79} + 96q^{80} + 81q^{81} + 252q^{82} + 810q^{83} + 504q^{85} + 650q^{86} + 540q^{87} + 240q^{88} + 600q^{89} - 108q^{90} + 336q^{92} + 537q^{93} - 732q^{94} + 582q^{95} + 96q^{96} - 386q^{97} - 270q^{99} + O(q^{100})$$

## 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
−2.00000 −3.00000 4.00000 6.00000 6.00000 0 −8.00000 9.00000 −12.0000
 $$n$$: 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 294.4.a.c 1
3.b odd 2 1 882.4.a.l 1
4.b odd 2 1 2352.4.a.bf 1
7.b odd 2 1 294.4.a.d 1
7.c even 3 2 294.4.e.i 2
7.d odd 6 2 42.4.e.a 2
21.c even 2 1 882.4.a.o 1
21.g even 6 2 126.4.g.b 2
21.h odd 6 2 882.4.g.g 2
28.d even 2 1 2352.4.a.f 1
28.f even 6 2 336.4.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 7.d odd 6 2
126.4.g.b 2 21.g even 6 2
294.4.a.c 1 1.a even 1 1 trivial
294.4.a.d 1 7.b odd 2 1
294.4.e.i 2 7.c even 3 2
336.4.q.f 2 28.f even 6 2
882.4.a.l 1 3.b odd 2 1
882.4.a.o 1 21.c even 2 1
882.4.g.g 2 21.h odd 6 2
2352.4.a.f 1 28.d even 2 1
2352.4.a.bf 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(294))$$:

 $$T_{5} - 6$$ $$T_{11} + 30$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$3 + T$$
$5$ $$-6 + T$$
$7$ $$T$$
$11$ $$30 + T$$
$13$ $$53 + T$$
$17$ $$-84 + T$$
$19$ $$-97 + T$$
$23$ $$-84 + T$$
$29$ $$180 + T$$
$31$ $$179 + T$$
$37$ $$145 + T$$
$41$ $$126 + T$$
$43$ $$325 + T$$
$47$ $$-366 + T$$
$53$ $$768 + T$$
$59$ $$-264 + T$$
$61$ $$818 + T$$
$67$ $$523 + T$$
$71$ $$342 + T$$
$73$ $$-43 + T$$
$79$ $$1171 + T$$
$83$ $$-810 + T$$
$89$ $$-600 + T$$
$97$ $$386 + T$$