# Properties

 Label 294.4.e.i 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: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -6 \zeta_{6} q^{5} + 6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} + ( 12 - 12 \zeta_{6} ) q^{10} + ( 30 - 30 \zeta_{6} ) q^{11} + 12 \zeta_{6} q^{12} -53 q^{13} -18 q^{15} -16 \zeta_{6} q^{16} + ( -84 + 84 \zeta_{6} ) q^{17} + ( 18 - 18 \zeta_{6} ) q^{18} -97 \zeta_{6} q^{19} + 24 q^{20} + 60 q^{22} -84 \zeta_{6} q^{23} + ( -24 + 24 \zeta_{6} ) q^{24} + ( 89 - 89 \zeta_{6} ) q^{25} -106 \zeta_{6} q^{26} -27 q^{27} -180 q^{29} -36 \zeta_{6} q^{30} + ( 179 - 179 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -90 \zeta_{6} q^{33} -168 q^{34} + 36 q^{36} + 145 \zeta_{6} q^{37} + ( 194 - 194 \zeta_{6} ) q^{38} + ( -159 + 159 \zeta_{6} ) q^{39} + 48 \zeta_{6} q^{40} -126 q^{41} -325 q^{43} + 120 \zeta_{6} q^{44} + ( -54 + 54 \zeta_{6} ) q^{45} + ( 168 - 168 \zeta_{6} ) q^{46} -366 \zeta_{6} q^{47} -48 q^{48} + 178 q^{50} + 252 \zeta_{6} q^{51} + ( 212 - 212 \zeta_{6} ) q^{52} + ( 768 - 768 \zeta_{6} ) q^{53} -54 \zeta_{6} q^{54} -180 q^{55} -291 q^{57} -360 \zeta_{6} q^{58} + ( -264 + 264 \zeta_{6} ) q^{59} + ( 72 - 72 \zeta_{6} ) q^{60} + 818 \zeta_{6} q^{61} + 358 q^{62} + 64 q^{64} + 318 \zeta_{6} q^{65} + ( 180 - 180 \zeta_{6} ) q^{66} + ( 523 - 523 \zeta_{6} ) q^{67} -336 \zeta_{6} q^{68} -252 q^{69} -342 q^{71} + 72 \zeta_{6} q^{72} + ( -43 + 43 \zeta_{6} ) q^{73} + ( -290 + 290 \zeta_{6} ) q^{74} -267 \zeta_{6} q^{75} + 388 q^{76} -318 q^{78} + 1171 \zeta_{6} q^{79} + ( -96 + 96 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -252 \zeta_{6} q^{82} + 810 q^{83} + 504 q^{85} -650 \zeta_{6} q^{86} + ( -540 + 540 \zeta_{6} ) q^{87} + ( -240 + 240 \zeta_{6} ) q^{88} -600 \zeta_{6} q^{89} -108 q^{90} + 336 q^{92} -537 \zeta_{6} q^{93} + ( 732 - 732 \zeta_{6} ) q^{94} + ( -582 + 582 \zeta_{6} ) q^{95} -96 \zeta_{6} q^{96} -386 q^{97} -270 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} - 4q^{4} - 6q^{5} + 12q^{6} - 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} - 4q^{4} - 6q^{5} + 12q^{6} - 16q^{8} - 9q^{9} + 12q^{10} + 30q^{11} + 12q^{12} - 106q^{13} - 36q^{15} - 16q^{16} - 84q^{17} + 18q^{18} - 97q^{19} + 48q^{20} + 120q^{22} - 84q^{23} - 24q^{24} + 89q^{25} - 106q^{26} - 54q^{27} - 360q^{29} - 36q^{30} + 179q^{31} + 32q^{32} - 90q^{33} - 336q^{34} + 72q^{36} + 145q^{37} + 194q^{38} - 159q^{39} + 48q^{40} - 252q^{41} - 650q^{43} + 120q^{44} - 54q^{45} + 168q^{46} - 366q^{47} - 96q^{48} + 356q^{50} + 252q^{51} + 212q^{52} + 768q^{53} - 54q^{54} - 360q^{55} - 582q^{57} - 360q^{58} - 264q^{59} + 72q^{60} + 818q^{61} + 716q^{62} + 128q^{64} + 318q^{65} + 180q^{66} + 523q^{67} - 336q^{68} - 504q^{69} - 684q^{71} + 72q^{72} - 43q^{73} - 290q^{74} - 267q^{75} + 776q^{76} - 636q^{78} + 1171q^{79} - 96q^{80} - 81q^{81} - 252q^{82} + 1620q^{83} + 1008q^{85} - 650q^{86} - 540q^{87} - 240q^{88} - 600q^{89} - 216q^{90} + 672q^{92} - 537q^{93} + 732q^{94} - 582q^{95} - 96q^{96} - 772q^{97} - 540q^{99} + O(q^{100})$$

## 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 −3.00000 5.19615i 6.00000 0 −8.00000 −4.50000 7.79423i 6.00000 10.3923i
79.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i −3.00000 + 5.19615i 6.00000 0 −8.00000 −4.50000 + 7.79423i 6.00000 + 10.3923i
 $$n$$: 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.i 2
3.b odd 2 1 882.4.g.g 2
7.b odd 2 1 42.4.e.a 2
7.c even 3 1 294.4.a.c 1
7.c even 3 1 inner 294.4.e.i 2
7.d odd 6 1 42.4.e.a 2
7.d odd 6 1 294.4.a.d 1
21.c even 2 1 126.4.g.b 2
21.g even 6 1 126.4.g.b 2
21.g even 6 1 882.4.a.o 1
21.h odd 6 1 882.4.a.l 1
21.h odd 6 1 882.4.g.g 2
28.d even 2 1 336.4.q.f 2
28.f even 6 1 336.4.q.f 2
28.f even 6 1 2352.4.a.f 1
28.g odd 6 1 2352.4.a.bf 1

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

## Hecke kernels

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

 $$T_{5}^{2} + 6 T_{5} + 36$$ $$T_{11}^{2} - 30 T_{11} + 900$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$36 + 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$900 - 30 T + T^{2}$$
$13$ $$( 53 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$9409 + 97 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( 180 + T )^{2}$$
$31$ $$32041 - 179 T + T^{2}$$
$37$ $$21025 - 145 T + T^{2}$$
$41$ $$( 126 + T )^{2}$$
$43$ $$( 325 + T )^{2}$$
$47$ $$133956 + 366 T + T^{2}$$
$53$ $$589824 - 768 T + T^{2}$$
$59$ $$69696 + 264 T + T^{2}$$
$61$ $$669124 - 818 T + T^{2}$$
$67$ $$273529 - 523 T + T^{2}$$
$71$ $$( 342 + T )^{2}$$
$73$ $$1849 + 43 T + T^{2}$$
$79$ $$1371241 - 1171 T + T^{2}$$
$83$ $$( -810 + T )^{2}$$
$89$ $$360000 + 600 T + T^{2}$$
$97$ $$( 386 + T )^{2}$$