# Properties

 Label 294.4.e.f Level $294$ Weight $4$ Character orbit 294.e Analytic conductor $17.347$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 294.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3465615417$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -8 \zeta_{6} q^{5} -6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -8 \zeta_{6} q^{5} -6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} + ( 16 - 16 \zeta_{6} ) q^{10} + ( -40 + 40 \zeta_{6} ) q^{11} -12 \zeta_{6} q^{12} + 4 q^{13} + 24 q^{15} -16 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} + ( 18 - 18 \zeta_{6} ) q^{18} -148 \zeta_{6} q^{19} + 32 q^{20} -80 q^{22} -84 \zeta_{6} q^{23} + ( 24 - 24 \zeta_{6} ) q^{24} + ( 61 - 61 \zeta_{6} ) q^{25} + 8 \zeta_{6} q^{26} + 27 q^{27} + 58 q^{29} + 48 \zeta_{6} q^{30} + ( 136 - 136 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -120 \zeta_{6} q^{33} + 168 q^{34} + 36 q^{36} + 222 \zeta_{6} q^{37} + ( 296 - 296 \zeta_{6} ) q^{38} + ( -12 + 12 \zeta_{6} ) q^{39} + 64 \zeta_{6} q^{40} + 420 q^{41} -164 q^{43} -160 \zeta_{6} q^{44} + ( -72 + 72 \zeta_{6} ) q^{45} + ( 168 - 168 \zeta_{6} ) q^{46} -488 \zeta_{6} q^{47} + 48 q^{48} + 122 q^{50} + 252 \zeta_{6} q^{51} + ( -16 + 16 \zeta_{6} ) q^{52} + ( -478 + 478 \zeta_{6} ) q^{53} + 54 \zeta_{6} q^{54} + 320 q^{55} + 444 q^{57} + 116 \zeta_{6} q^{58} + ( -548 + 548 \zeta_{6} ) q^{59} + ( -96 + 96 \zeta_{6} ) q^{60} -692 \zeta_{6} q^{61} + 272 q^{62} + 64 q^{64} -32 \zeta_{6} q^{65} + ( 240 - 240 \zeta_{6} ) q^{66} + ( 908 - 908 \zeta_{6} ) q^{67} + 336 \zeta_{6} q^{68} + 252 q^{69} -524 q^{71} + 72 \zeta_{6} q^{72} + ( -440 + 440 \zeta_{6} ) q^{73} + ( -444 + 444 \zeta_{6} ) q^{74} + 183 \zeta_{6} q^{75} + 592 q^{76} -24 q^{78} -1216 \zeta_{6} q^{79} + ( -128 + 128 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 840 \zeta_{6} q^{82} -684 q^{83} -672 q^{85} -328 \zeta_{6} q^{86} + ( -174 + 174 \zeta_{6} ) q^{87} + ( 320 - 320 \zeta_{6} ) q^{88} -604 \zeta_{6} q^{89} -144 q^{90} + 336 q^{92} + 408 \zeta_{6} q^{93} + ( 976 - 976 \zeta_{6} ) q^{94} + ( -1184 + 1184 \zeta_{6} ) q^{95} + 96 \zeta_{6} q^{96} -832 q^{97} + 360 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} - 8q^{5} - 12q^{6} - 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} - 8q^{5} - 12q^{6} - 16q^{8} - 9q^{9} + 16q^{10} - 40q^{11} - 12q^{12} + 8q^{13} + 48q^{15} - 16q^{16} + 84q^{17} + 18q^{18} - 148q^{19} + 64q^{20} - 160q^{22} - 84q^{23} + 24q^{24} + 61q^{25} + 8q^{26} + 54q^{27} + 116q^{29} + 48q^{30} + 136q^{31} + 32q^{32} - 120q^{33} + 336q^{34} + 72q^{36} + 222q^{37} + 296q^{38} - 12q^{39} + 64q^{40} + 840q^{41} - 328q^{43} - 160q^{44} - 72q^{45} + 168q^{46} - 488q^{47} + 96q^{48} + 244q^{50} + 252q^{51} - 16q^{52} - 478q^{53} + 54q^{54} + 640q^{55} + 888q^{57} + 116q^{58} - 548q^{59} - 96q^{60} - 692q^{61} + 544q^{62} + 128q^{64} - 32q^{65} + 240q^{66} + 908q^{67} + 336q^{68} + 504q^{69} - 1048q^{71} + 72q^{72} - 440q^{73} - 444q^{74} + 183q^{75} + 1184q^{76} - 48q^{78} - 1216q^{79} - 128q^{80} - 81q^{81} + 840q^{82} - 1368q^{83} - 1344q^{85} - 328q^{86} - 174q^{87} + 320q^{88} - 604q^{89} - 288q^{90} + 672q^{92} + 408q^{93} + 976q^{94} - 1184q^{95} + 96q^{96} - 1664q^{97} + 720q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −4.00000 6.92820i −6.00000 0 −8.00000 −4.50000 7.79423i 8.00000 13.8564i
79.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i −4.00000 + 6.92820i −6.00000 0 −8.00000 −4.50000 + 7.79423i 8.00000 + 13.8564i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.4.e.f 2
3.b odd 2 1 882.4.g.j 2
7.b odd 2 1 294.4.e.j 2
7.c even 3 1 294.4.a.f yes 1
7.c even 3 1 inner 294.4.e.f 2
7.d odd 6 1 294.4.a.b 1
7.d odd 6 1 294.4.e.j 2
21.c even 2 1 882.4.g.c 2
21.g even 6 1 882.4.a.q 1
21.g even 6 1 882.4.g.c 2
21.h odd 6 1 882.4.a.j 1
21.h odd 6 1 882.4.g.j 2
28.f even 6 1 2352.4.a.z 1
28.g odd 6 1 2352.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.b 1 7.d odd 6 1
294.4.a.f yes 1 7.c even 3 1
294.4.e.f 2 1.a even 1 1 trivial
294.4.e.f 2 7.c even 3 1 inner
294.4.e.j 2 7.b odd 2 1
294.4.e.j 2 7.d odd 6 1
882.4.a.j 1 21.h odd 6 1
882.4.a.q 1 21.g even 6 1
882.4.g.c 2 21.c even 2 1
882.4.g.c 2 21.g even 6 1
882.4.g.j 2 3.b odd 2 1
882.4.g.j 2 21.h odd 6 1
2352.4.a.m 1 28.g odd 6 1
2352.4.a.z 1 28.f even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 8 T_{5} + 64$$ $$T_{11}^{2} + 40 T_{11} + 1600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$64 + 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1600 + 40 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$21904 + 148 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( -58 + T )^{2}$$
$31$ $$18496 - 136 T + T^{2}$$
$37$ $$49284 - 222 T + T^{2}$$
$41$ $$( -420 + T )^{2}$$
$43$ $$( 164 + T )^{2}$$
$47$ $$238144 + 488 T + T^{2}$$
$53$ $$228484 + 478 T + T^{2}$$
$59$ $$300304 + 548 T + T^{2}$$
$61$ $$478864 + 692 T + T^{2}$$
$67$ $$824464 - 908 T + T^{2}$$
$71$ $$( 524 + T )^{2}$$
$73$ $$193600 + 440 T + T^{2}$$
$79$ $$1478656 + 1216 T + T^{2}$$
$83$ $$( 684 + T )^{2}$$
$89$ $$364816 + 604 T + T^{2}$$
$97$ $$( 832 + T )^{2}$$