# Properties

 Label 294.4.e.b 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} + 6 q^{6} + 8 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} + 18 \zeta_{6} q^{5} + 6 q^{6} + 8 q^{8} -9 \zeta_{6} q^{9} + ( 36 - 36 \zeta_{6} ) q^{10} + ( 72 - 72 \zeta_{6} ) q^{11} -12 \zeta_{6} q^{12} + 34 q^{13} -54 q^{15} -16 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -18 + 18 \zeta_{6} ) q^{18} + 92 \zeta_{6} q^{19} -72 q^{20} -144 q^{22} + 180 \zeta_{6} q^{23} + ( -24 + 24 \zeta_{6} ) q^{24} + ( -199 + 199 \zeta_{6} ) q^{25} -68 \zeta_{6} q^{26} + 27 q^{27} -114 q^{29} + 108 \zeta_{6} q^{30} + ( 56 - 56 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 216 \zeta_{6} q^{33} -12 q^{34} + 36 q^{36} + 34 \zeta_{6} q^{37} + ( 184 - 184 \zeta_{6} ) q^{38} + ( -102 + 102 \zeta_{6} ) q^{39} + 144 \zeta_{6} q^{40} -6 q^{41} + 164 q^{43} + 288 \zeta_{6} q^{44} + ( 162 - 162 \zeta_{6} ) q^{45} + ( 360 - 360 \zeta_{6} ) q^{46} + 168 \zeta_{6} q^{47} + 48 q^{48} + 398 q^{50} + 18 \zeta_{6} q^{51} + ( -136 + 136 \zeta_{6} ) q^{52} + ( -654 + 654 \zeta_{6} ) q^{53} -54 \zeta_{6} q^{54} + 1296 q^{55} -276 q^{57} + 228 \zeta_{6} q^{58} + ( -492 + 492 \zeta_{6} ) q^{59} + ( 216 - 216 \zeta_{6} ) q^{60} -250 \zeta_{6} q^{61} -112 q^{62} + 64 q^{64} + 612 \zeta_{6} q^{65} + ( 432 - 432 \zeta_{6} ) q^{66} + ( 124 - 124 \zeta_{6} ) q^{67} + 24 \zeta_{6} q^{68} -540 q^{69} + 36 q^{71} -72 \zeta_{6} q^{72} + ( 1010 - 1010 \zeta_{6} ) q^{73} + ( 68 - 68 \zeta_{6} ) q^{74} -597 \zeta_{6} q^{75} -368 q^{76} + 204 q^{78} -56 \zeta_{6} q^{79} + ( 288 - 288 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 12 \zeta_{6} q^{82} -228 q^{83} + 108 q^{85} -328 \zeta_{6} q^{86} + ( 342 - 342 \zeta_{6} ) q^{87} + ( 576 - 576 \zeta_{6} ) q^{88} + 390 \zeta_{6} q^{89} -324 q^{90} -720 q^{92} + 168 \zeta_{6} q^{93} + ( 336 - 336 \zeta_{6} ) q^{94} + ( -1656 + 1656 \zeta_{6} ) q^{95} -96 \zeta_{6} q^{96} + 70 q^{97} -648 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 3q^{3} - 4q^{4} + 18q^{5} + 12q^{6} + 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 3q^{3} - 4q^{4} + 18q^{5} + 12q^{6} + 16q^{8} - 9q^{9} + 36q^{10} + 72q^{11} - 12q^{12} + 68q^{13} - 108q^{15} - 16q^{16} + 6q^{17} - 18q^{18} + 92q^{19} - 144q^{20} - 288q^{22} + 180q^{23} - 24q^{24} - 199q^{25} - 68q^{26} + 54q^{27} - 228q^{29} + 108q^{30} + 56q^{31} - 32q^{32} + 216q^{33} - 24q^{34} + 72q^{36} + 34q^{37} + 184q^{38} - 102q^{39} + 144q^{40} - 12q^{41} + 328q^{43} + 288q^{44} + 162q^{45} + 360q^{46} + 168q^{47} + 96q^{48} + 796q^{50} + 18q^{51} - 136q^{52} - 654q^{53} - 54q^{54} + 2592q^{55} - 552q^{57} + 228q^{58} - 492q^{59} + 216q^{60} - 250q^{61} - 224q^{62} + 128q^{64} + 612q^{65} + 432q^{66} + 124q^{67} + 24q^{68} - 1080q^{69} + 72q^{71} - 72q^{72} + 1010q^{73} + 68q^{74} - 597q^{75} - 736q^{76} + 408q^{78} - 56q^{79} + 288q^{80} - 81q^{81} + 12q^{82} - 456q^{83} + 216q^{85} - 328q^{86} + 342q^{87} + 576q^{88} + 390q^{89} - 648q^{90} - 1440q^{92} + 168q^{93} + 336q^{94} - 1656q^{95} - 96q^{96} + 140q^{97} - 1296q^{99} + O(q^{100})$$

## 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 9.00000 + 15.5885i 6.00000 0 8.00000 −4.50000 7.79423i 18.0000 31.1769i
79.1 −1.00000 + 1.73205i −1.50000 2.59808i −2.00000 3.46410i 9.00000 15.5885i 6.00000 0 8.00000 −4.50000 + 7.79423i 18.0000 + 31.1769i
 $$n$$: 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.b 2
3.b odd 2 1 882.4.g.o 2
7.b odd 2 1 294.4.e.c 2
7.c even 3 1 294.4.a.i 1
7.c even 3 1 inner 294.4.e.b 2
7.d odd 6 1 42.4.a.a 1
7.d odd 6 1 294.4.e.c 2
21.c even 2 1 882.4.g.w 2
21.g even 6 1 126.4.a.a 1
21.g even 6 1 882.4.g.w 2
21.h odd 6 1 882.4.a.g 1
21.h odd 6 1 882.4.g.o 2
28.f even 6 1 336.4.a.l 1
28.g odd 6 1 2352.4.a.a 1
35.i odd 6 1 1050.4.a.g 1
35.k even 12 2 1050.4.g.a 2
56.j odd 6 1 1344.4.a.o 1
56.m even 6 1 1344.4.a.a 1
84.j odd 6 1 1008.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 7.d odd 6 1
126.4.a.a 1 21.g even 6 1
294.4.a.i 1 7.c even 3 1
294.4.e.b 2 1.a even 1 1 trivial
294.4.e.b 2 7.c even 3 1 inner
294.4.e.c 2 7.b odd 2 1
294.4.e.c 2 7.d odd 6 1
336.4.a.l 1 28.f even 6 1
882.4.a.g 1 21.h odd 6 1
882.4.g.o 2 3.b odd 2 1
882.4.g.o 2 21.h odd 6 1
882.4.g.w 2 21.c even 2 1
882.4.g.w 2 21.g even 6 1
1008.4.a.b 1 84.j odd 6 1
1050.4.a.g 1 35.i odd 6 1
1050.4.g.a 2 35.k even 12 2
1344.4.a.a 1 56.m even 6 1
1344.4.a.o 1 56.j odd 6 1
2352.4.a.a 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 $$S_{4}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} - 18 T_{5} + 324$$ $$T_{11}^{2} - 72 T_{11} + 5184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$324 - 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$5184 - 72 T + T^{2}$$
$13$ $$( -34 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$8464 - 92 T + T^{2}$$
$23$ $$32400 - 180 T + T^{2}$$
$29$ $$( 114 + T )^{2}$$
$31$ $$3136 - 56 T + T^{2}$$
$37$ $$1156 - 34 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -164 + T )^{2}$$
$47$ $$28224 - 168 T + T^{2}$$
$53$ $$427716 + 654 T + T^{2}$$
$59$ $$242064 + 492 T + T^{2}$$
$61$ $$62500 + 250 T + T^{2}$$
$67$ $$15376 - 124 T + T^{2}$$
$71$ $$( -36 + T )^{2}$$
$73$ $$1020100 - 1010 T + T^{2}$$
$79$ $$3136 + 56 T + T^{2}$$
$83$ $$( 228 + T )^{2}$$
$89$ $$152100 - 390 T + T^{2}$$
$97$ $$( -70 + T )^{2}$$