# Properties

 Label 294.4.a.a Level $294$ Weight $4$ Character orbit 294.a Self dual yes Analytic conductor $17.347$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} - 15q^{5} + 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} - 15q^{5} + 6q^{6} - 8q^{8} + 9q^{9} + 30q^{10} - 9q^{11} - 12q^{12} - 88q^{13} + 45q^{15} + 16q^{16} - 84q^{17} - 18q^{18} + 104q^{19} - 60q^{20} + 18q^{22} - 84q^{23} + 24q^{24} + 100q^{25} + 176q^{26} - 27q^{27} + 51q^{29} - 90q^{30} + 185q^{31} - 32q^{32} + 27q^{33} + 168q^{34} + 36q^{36} + 44q^{37} - 208q^{38} + 264q^{39} + 120q^{40} - 168q^{41} + 326q^{43} - 36q^{44} - 135q^{45} + 168q^{46} - 138q^{47} - 48q^{48} - 200q^{50} + 252q^{51} - 352q^{52} + 639q^{53} + 54q^{54} + 135q^{55} - 312q^{57} - 102q^{58} + 159q^{59} + 180q^{60} + 722q^{61} - 370q^{62} + 64q^{64} + 1320q^{65} - 54q^{66} - 166q^{67} - 336q^{68} + 252q^{69} + 1086q^{71} - 72q^{72} + 218q^{73} - 88q^{74} - 300q^{75} + 416q^{76} - 528q^{78} - 583q^{79} - 240q^{80} + 81q^{81} + 336q^{82} - 597q^{83} + 1260q^{85} - 652q^{86} - 153q^{87} + 72q^{88} - 1038q^{89} + 270q^{90} - 336q^{92} - 555q^{93} + 276q^{94} - 1560q^{95} + 96q^{96} - 169q^{97} - 81q^{99} + O(q^{100})$$

## 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 −15.0000 6.00000 0 −8.00000 9.00000 30.0000
 $$n$$: 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.a 1
3.b odd 2 1 882.4.a.r 1
4.b odd 2 1 2352.4.a.u 1
7.b odd 2 1 294.4.a.g 1
7.c even 3 2 42.4.e.b 2
7.d odd 6 2 294.4.e.e 2
21.c even 2 1 882.4.a.h 1
21.g even 6 2 882.4.g.l 2
21.h odd 6 2 126.4.g.a 2
28.d even 2 1 2352.4.a.q 1
28.g odd 6 2 336.4.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 7.c even 3 2
126.4.g.a 2 21.h odd 6 2
294.4.a.a 1 1.a even 1 1 trivial
294.4.a.g 1 7.b odd 2 1
294.4.e.e 2 7.d odd 6 2
336.4.q.d 2 28.g odd 6 2
882.4.a.h 1 21.c even 2 1
882.4.a.r 1 3.b odd 2 1
882.4.g.l 2 21.g even 6 2
2352.4.a.q 1 28.d even 2 1
2352.4.a.u 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(294))$$:

 $$T_{5} + 15$$ $$T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$3 + T$$
$5$ $$15 + T$$
$7$ $$T$$
$11$ $$9 + T$$
$13$ $$88 + T$$
$17$ $$84 + T$$
$19$ $$-104 + T$$
$23$ $$84 + T$$
$29$ $$-51 + T$$
$31$ $$-185 + T$$
$37$ $$-44 + T$$
$41$ $$168 + T$$
$43$ $$-326 + T$$
$47$ $$138 + T$$
$53$ $$-639 + T$$
$59$ $$-159 + T$$
$61$ $$-722 + T$$
$67$ $$166 + T$$
$71$ $$-1086 + T$$
$73$ $$-218 + T$$
$79$ $$583 + T$$
$83$ $$597 + T$$
$89$ $$1038 + T$$
$97$ $$169 + T$$