# Properties

 Label 294.4.a.k Level $294$ Weight $4$ Character orbit 294.a Self dual yes Analytic conductor $17.347$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 7\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -6 + \beta ) q^{5} -6 q^{6} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -6 + \beta ) q^{5} -6 q^{6} -8 q^{8} + 9 q^{9} + ( 12 - 2 \beta ) q^{10} + ( -2 - 6 \beta ) q^{11} + 12 q^{12} + ( -24 - 3 \beta ) q^{13} + ( -18 + 3 \beta ) q^{15} + 16 q^{16} + ( -42 + \beta ) q^{17} -18 q^{18} + ( 36 + 2 \beta ) q^{19} + ( -24 + 4 \beta ) q^{20} + ( 4 + 12 \beta ) q^{22} + ( -154 + 6 \beta ) q^{23} -24 q^{24} + ( 9 - 12 \beta ) q^{25} + ( 48 + 6 \beta ) q^{26} + 27 q^{27} + ( -40 + 18 \beta ) q^{29} + ( 36 - 6 \beta ) q^{30} + ( -192 + 6 \beta ) q^{31} -32 q^{32} + ( -6 - 18 \beta ) q^{33} + ( 84 - 2 \beta ) q^{34} + 36 q^{36} + ( 268 - 12 \beta ) q^{37} + ( -72 - 4 \beta ) q^{38} + ( -72 - 9 \beta ) q^{39} + ( 48 - 8 \beta ) q^{40} + ( -378 - 5 \beta ) q^{41} + ( 200 + 24 \beta ) q^{43} + ( -8 - 24 \beta ) q^{44} + ( -54 + 9 \beta ) q^{45} + ( 308 - 12 \beta ) q^{46} + ( -156 + 10 \beta ) q^{47} + 48 q^{48} + ( -18 + 24 \beta ) q^{50} + ( -126 + 3 \beta ) q^{51} + ( -96 - 12 \beta ) q^{52} + ( -26 - 24 \beta ) q^{53} -54 q^{54} + ( -576 + 34 \beta ) q^{55} + ( 108 + 6 \beta ) q^{57} + ( 80 - 36 \beta ) q^{58} + ( -432 - 2 \beta ) q^{59} + ( -72 + 12 \beta ) q^{60} + ( -708 + 13 \beta ) q^{61} + ( 384 - 12 \beta ) q^{62} + 64 q^{64} + ( -150 - 6 \beta ) q^{65} + ( 12 + 36 \beta ) q^{66} + ( 72 + 24 \beta ) q^{67} + ( -168 + 4 \beta ) q^{68} + ( -462 + 18 \beta ) q^{69} + ( -762 - 30 \beta ) q^{71} -72 q^{72} + ( -372 - 83 \beta ) q^{73} + ( -536 + 24 \beta ) q^{74} + ( 27 - 36 \beta ) q^{75} + ( 144 + 8 \beta ) q^{76} + ( 144 + 18 \beta ) q^{78} + ( 488 + 84 \beta ) q^{79} + ( -96 + 16 \beta ) q^{80} + 81 q^{81} + ( 756 + 10 \beta ) q^{82} + ( 156 - 136 \beta ) q^{83} + ( 350 - 48 \beta ) q^{85} + ( -400 - 48 \beta ) q^{86} + ( -120 + 54 \beta ) q^{87} + ( 16 + 48 \beta ) q^{88} + ( -54 + 29 \beta ) q^{89} + ( 108 - 18 \beta ) q^{90} + ( -616 + 24 \beta ) q^{92} + ( -576 + 18 \beta ) q^{93} + ( 312 - 20 \beta ) q^{94} + ( -20 + 24 \beta ) q^{95} -96 q^{96} + ( 372 + 125 \beta ) q^{97} + ( -18 - 54 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} - 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 4q^{2} + 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} - 16q^{8} + 18q^{9} + 24q^{10} - 4q^{11} + 24q^{12} - 48q^{13} - 36q^{15} + 32q^{16} - 84q^{17} - 36q^{18} + 72q^{19} - 48q^{20} + 8q^{22} - 308q^{23} - 48q^{24} + 18q^{25} + 96q^{26} + 54q^{27} - 80q^{29} + 72q^{30} - 384q^{31} - 64q^{32} - 12q^{33} + 168q^{34} + 72q^{36} + 536q^{37} - 144q^{38} - 144q^{39} + 96q^{40} - 756q^{41} + 400q^{43} - 16q^{44} - 108q^{45} + 616q^{46} - 312q^{47} + 96q^{48} - 36q^{50} - 252q^{51} - 192q^{52} - 52q^{53} - 108q^{54} - 1152q^{55} + 216q^{57} + 160q^{58} - 864q^{59} - 144q^{60} - 1416q^{61} + 768q^{62} + 128q^{64} - 300q^{65} + 24q^{66} + 144q^{67} - 336q^{68} - 924q^{69} - 1524q^{71} - 144q^{72} - 744q^{73} - 1072q^{74} + 54q^{75} + 288q^{76} + 288q^{78} + 976q^{79} - 192q^{80} + 162q^{81} + 1512q^{82} + 312q^{83} + 700q^{85} - 800q^{86} - 240q^{87} + 32q^{88} - 108q^{89} + 216q^{90} - 1232q^{92} - 1152q^{93} + 624q^{94} - 40q^{95} - 192q^{96} + 744q^{97} - 36q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.00000 3.00000 4.00000 −15.8995 −6.00000 0 −8.00000 9.00000 31.7990
1.2 −2.00000 3.00000 4.00000 3.89949 −6.00000 0 −8.00000 9.00000 −7.79899
 $$n$$: 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.k yes 2
3.b odd 2 1 882.4.a.bi 2
4.b odd 2 1 2352.4.a.bn 2
7.b odd 2 1 294.4.a.j 2
7.c even 3 2 294.4.e.n 4
7.d odd 6 2 294.4.e.o 4
21.c even 2 1 882.4.a.bc 2
21.g even 6 2 882.4.g.bd 4
21.h odd 6 2 882.4.g.y 4
28.d even 2 1 2352.4.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 7.b odd 2 1
294.4.a.k yes 2 1.a even 1 1 trivial
294.4.e.n 4 7.c even 3 2
294.4.e.o 4 7.d odd 6 2
882.4.a.bc 2 21.c even 2 1
882.4.a.bi 2 3.b odd 2 1
882.4.g.y 4 21.h odd 6 2
882.4.g.bd 4 21.g even 6 2
2352.4.a.bn 2 4.b odd 2 1
2352.4.a.cd 2 28.d even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12 T_{5} - 62$$ $$T_{11}^{2} + 4 T_{11} - 3524$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-62 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3524 + 4 T + T^{2}$$
$13$ $$-306 + 48 T + T^{2}$$
$17$ $$1666 + 84 T + T^{2}$$
$19$ $$904 - 72 T + T^{2}$$
$23$ $$20188 + 308 T + T^{2}$$
$29$ $$-30152 + 80 T + T^{2}$$
$31$ $$33336 + 384 T + T^{2}$$
$37$ $$57712 - 536 T + T^{2}$$
$41$ $$140434 + 756 T + T^{2}$$
$43$ $$-16448 - 400 T + T^{2}$$
$47$ $$14536 + 312 T + T^{2}$$
$53$ $$-55772 + 52 T + T^{2}$$
$59$ $$186232 + 864 T + T^{2}$$
$61$ $$484702 + 1416 T + T^{2}$$
$67$ $$-51264 - 144 T + T^{2}$$
$71$ $$492444 + 1524 T + T^{2}$$
$73$ $$-536738 + 744 T + T^{2}$$
$79$ $$-453344 - 976 T + T^{2}$$
$83$ $$-1788272 - 312 T + T^{2}$$
$89$ $$-79502 + 108 T + T^{2}$$
$97$ $$-1392866 - 744 T + T^{2}$$