# Properties

 Label 294.6.e.a Level 294 Weight 6 Character orbit 294.e Analytic conductor 47.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 294.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.1528430250$$ 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 6) 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.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{2} + ( -9 + 9 \zeta_{6} ) q^{3} + ( -16 + 16 \zeta_{6} ) q^{4} -66 \zeta_{6} q^{5} + 36 q^{6} + 64 q^{8} -81 \zeta_{6} q^{9} +O(q^{10})$$ $$q -4 \zeta_{6} q^{2} + ( -9 + 9 \zeta_{6} ) q^{3} + ( -16 + 16 \zeta_{6} ) q^{4} -66 \zeta_{6} q^{5} + 36 q^{6} + 64 q^{8} -81 \zeta_{6} q^{9} + ( -264 + 264 \zeta_{6} ) q^{10} + ( 60 - 60 \zeta_{6} ) q^{11} -144 \zeta_{6} q^{12} + 658 q^{13} + 594 q^{15} -256 \zeta_{6} q^{16} + ( -414 + 414 \zeta_{6} ) q^{17} + ( -324 + 324 \zeta_{6} ) q^{18} + 956 \zeta_{6} q^{19} + 1056 q^{20} -240 q^{22} -600 \zeta_{6} q^{23} + ( -576 + 576 \zeta_{6} ) q^{24} + ( -1231 + 1231 \zeta_{6} ) q^{25} -2632 \zeta_{6} q^{26} + 729 q^{27} + 5574 q^{29} -2376 \zeta_{6} q^{30} + ( -3592 + 3592 \zeta_{6} ) q^{31} + ( -1024 + 1024 \zeta_{6} ) q^{32} + 540 \zeta_{6} q^{33} + 1656 q^{34} + 1296 q^{36} + 8458 \zeta_{6} q^{37} + ( 3824 - 3824 \zeta_{6} ) q^{38} + ( -5922 + 5922 \zeta_{6} ) q^{39} -4224 \zeta_{6} q^{40} -19194 q^{41} + 13316 q^{43} + 960 \zeta_{6} q^{44} + ( -5346 + 5346 \zeta_{6} ) q^{45} + ( -2400 + 2400 \zeta_{6} ) q^{46} -19680 \zeta_{6} q^{47} + 2304 q^{48} + 4924 q^{50} -3726 \zeta_{6} q^{51} + ( -10528 + 10528 \zeta_{6} ) q^{52} + ( 31266 - 31266 \zeta_{6} ) q^{53} -2916 \zeta_{6} q^{54} -3960 q^{55} -8604 q^{57} -22296 \zeta_{6} q^{58} + ( 26340 - 26340 \zeta_{6} ) q^{59} + ( -9504 + 9504 \zeta_{6} ) q^{60} -31090 \zeta_{6} q^{61} + 14368 q^{62} + 4096 q^{64} -43428 \zeta_{6} q^{65} + ( 2160 - 2160 \zeta_{6} ) q^{66} + ( 16804 - 16804 \zeta_{6} ) q^{67} -6624 \zeta_{6} q^{68} + 5400 q^{69} + 6120 q^{71} -5184 \zeta_{6} q^{72} + ( -25558 + 25558 \zeta_{6} ) q^{73} + ( 33832 - 33832 \zeta_{6} ) q^{74} -11079 \zeta_{6} q^{75} -15296 q^{76} + 23688 q^{78} -74408 \zeta_{6} q^{79} + ( -16896 + 16896 \zeta_{6} ) q^{80} + ( -6561 + 6561 \zeta_{6} ) q^{81} + 76776 \zeta_{6} q^{82} + 6468 q^{83} + 27324 q^{85} -53264 \zeta_{6} q^{86} + ( -50166 + 50166 \zeta_{6} ) q^{87} + ( 3840 - 3840 \zeta_{6} ) q^{88} -32742 \zeta_{6} q^{89} + 21384 q^{90} + 9600 q^{92} -32328 \zeta_{6} q^{93} + ( -78720 + 78720 \zeta_{6} ) q^{94} + ( 63096 - 63096 \zeta_{6} ) q^{95} -9216 \zeta_{6} q^{96} -166082 q^{97} -4860 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 9q^{3} - 16q^{4} - 66q^{5} + 72q^{6} + 128q^{8} - 81q^{9} + O(q^{10})$$ $$2q - 4q^{2} - 9q^{3} - 16q^{4} - 66q^{5} + 72q^{6} + 128q^{8} - 81q^{9} - 264q^{10} + 60q^{11} - 144q^{12} + 1316q^{13} + 1188q^{15} - 256q^{16} - 414q^{17} - 324q^{18} + 956q^{19} + 2112q^{20} - 480q^{22} - 600q^{23} - 576q^{24} - 1231q^{25} - 2632q^{26} + 1458q^{27} + 11148q^{29} - 2376q^{30} - 3592q^{31} - 1024q^{32} + 540q^{33} + 3312q^{34} + 2592q^{36} + 8458q^{37} + 3824q^{38} - 5922q^{39} - 4224q^{40} - 38388q^{41} + 26632q^{43} + 960q^{44} - 5346q^{45} - 2400q^{46} - 19680q^{47} + 4608q^{48} + 9848q^{50} - 3726q^{51} - 10528q^{52} + 31266q^{53} - 2916q^{54} - 7920q^{55} - 17208q^{57} - 22296q^{58} + 26340q^{59} - 9504q^{60} - 31090q^{61} + 28736q^{62} + 8192q^{64} - 43428q^{65} + 2160q^{66} + 16804q^{67} - 6624q^{68} + 10800q^{69} + 12240q^{71} - 5184q^{72} - 25558q^{73} + 33832q^{74} - 11079q^{75} - 30592q^{76} + 47376q^{78} - 74408q^{79} - 16896q^{80} - 6561q^{81} + 76776q^{82} + 12936q^{83} + 54648q^{85} - 53264q^{86} - 50166q^{87} + 3840q^{88} - 32742q^{89} + 42768q^{90} + 19200q^{92} - 32328q^{93} - 78720q^{94} + 63096q^{95} - 9216q^{96} - 332164q^{97} - 9720q^{99} + O(q^{100})$$

## 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
−2.00000 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i −33.0000 57.1577i 36.0000 0 64.0000 −40.5000 70.1481i −132.000 + 228.631i
79.1 −2.00000 + 3.46410i −4.50000 7.79423i −8.00000 13.8564i −33.0000 + 57.1577i 36.0000 0 64.0000 −40.5000 + 70.1481i −132.000 228.631i
 $$n$$: 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 294.6.e.a 2
7.b odd 2 1 294.6.e.g 2
7.c even 3 1 294.6.a.m 1
7.c even 3 1 inner 294.6.e.a 2
7.d odd 6 1 6.6.a.a 1
7.d odd 6 1 294.6.e.g 2
21.g even 6 1 18.6.a.b 1
21.h odd 6 1 882.6.a.a 1
28.f even 6 1 48.6.a.c 1
35.i odd 6 1 150.6.a.d 1
35.k even 12 2 150.6.c.b 2
56.j odd 6 1 192.6.a.o 1
56.m even 6 1 192.6.a.g 1
63.i even 6 1 162.6.c.h 2
63.k odd 6 1 162.6.c.e 2
63.s even 6 1 162.6.c.h 2
63.t odd 6 1 162.6.c.e 2
77.i even 6 1 726.6.a.a 1
84.j odd 6 1 144.6.a.j 1
91.s odd 6 1 1014.6.a.c 1
105.p even 6 1 450.6.a.m 1
105.w odd 12 2 450.6.c.j 2
112.v even 12 2 768.6.d.p 2
112.x odd 12 2 768.6.d.c 2
168.ba even 6 1 576.6.a.j 1
168.be odd 6 1 576.6.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 7.d odd 6 1
18.6.a.b 1 21.g even 6 1
48.6.a.c 1 28.f even 6 1
144.6.a.j 1 84.j odd 6 1
150.6.a.d 1 35.i odd 6 1
150.6.c.b 2 35.k even 12 2
162.6.c.e 2 63.k odd 6 1
162.6.c.e 2 63.t odd 6 1
162.6.c.h 2 63.i even 6 1
162.6.c.h 2 63.s even 6 1
192.6.a.g 1 56.m even 6 1
192.6.a.o 1 56.j odd 6 1
294.6.a.m 1 7.c even 3 1
294.6.e.a 2 1.a even 1 1 trivial
294.6.e.a 2 7.c even 3 1 inner
294.6.e.g 2 7.b odd 2 1
294.6.e.g 2 7.d odd 6 1
450.6.a.m 1 105.p even 6 1
450.6.c.j 2 105.w odd 12 2
576.6.a.i 1 168.be odd 6 1
576.6.a.j 1 168.ba even 6 1
726.6.a.a 1 77.i even 6 1
768.6.d.c 2 112.x odd 12 2
768.6.d.p 2 112.v even 12 2
882.6.a.a 1 21.h odd 6 1
1014.6.a.c 1 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 66 T_{5} + 4356$$ $$T_{11}^{2} - 60 T_{11} + 3600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 16 T^{2}$$
$3$ $$1 + 9 T + 81 T^{2}$$
$5$ $$1 + 66 T + 1231 T^{2} + 206250 T^{3} + 9765625 T^{4}$$
$7$ 1
$11$ $$1 - 60 T - 157451 T^{2} - 9663060 T^{3} + 25937424601 T^{4}$$
$13$ $$( 1 - 658 T + 371293 T^{2} )^{2}$$
$17$ $$1 + 414 T - 1248461 T^{2} + 587820798 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 - 956 T - 1562163 T^{2} - 2367150644 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 600 T - 6076343 T^{2} + 3861805800 T^{3} + 41426511213649 T^{4}$$
$29$ $$( 1 - 5574 T + 20511149 T^{2} )^{2}$$
$31$ $$1 + 3592 T - 15726687 T^{2} + 102835910392 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 8458 T + 2193807 T^{2} - 586511188306 T^{3} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 19194 T + 115856201 T^{2} )^{2}$$
$43$ $$( 1 - 13316 T + 147008443 T^{2} )^{2}$$
$47$ $$1 + 19680 T + 157957393 T^{2} + 4513509737760 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 31266 T + 559367263 T^{2} - 13075300284138 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 26340 T - 21128699 T^{2} - 18831106035660 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 31090 T + 121991799 T^{2} + 26258498998090 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 16804 T - 1067750691 T^{2} - 22687502298028 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 6120 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 + 25558 T - 1419860229 T^{2} + 52983563773894 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 + 74408 T + 2459494065 T^{2} + 228957612536792 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$( 1 - 6468 T + 3939040643 T^{2} )^{2}$$
$89$ $$1 + 32742 T - 4512020885 T^{2} + 182833274479158 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$( 1 + 166082 T + 8587340257 T^{2} )^{2}$$