# Properties

 Label 294.2.e.b Level 294 Weight 2 Character orbit 294.e Analytic conductor 2.348 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.34760181943$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{6} + q^{8} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( -5 + 5 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} - q^{15} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 8 \zeta_{6} q^{19} - q^{20} + 5 q^{22} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( 4 - 4 \zeta_{6} ) q^{25} + q^{27} -5 q^{29} + \zeta_{6} q^{30} + ( 3 - 3 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -5 \zeta_{6} q^{33} + 4 q^{34} + q^{36} + 4 \zeta_{6} q^{37} + ( 8 - 8 \zeta_{6} ) q^{38} + \zeta_{6} q^{40} + 2 q^{43} -5 \zeta_{6} q^{44} + ( 1 - \zeta_{6} ) q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + q^{48} -4 q^{50} -4 \zeta_{6} q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} -5 q^{55} -8 q^{57} + 5 \zeta_{6} q^{58} + ( -11 + 11 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} -6 \zeta_{6} q^{61} -3 q^{62} + q^{64} + ( -5 + 5 \zeta_{6} ) q^{66} + ( 2 - 2 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} -4 q^{69} + 2 q^{71} -\zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{75} -8 q^{76} -3 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 7 q^{83} -4 q^{85} -2 \zeta_{6} q^{86} + ( 5 - 5 \zeta_{6} ) q^{87} + ( -5 + 5 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} - q^{90} -4 q^{92} + 3 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} + ( -8 + 8 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} -7 q^{97} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} + 2q^{8} - q^{9} + q^{10} - 5q^{11} - q^{12} - 2q^{15} - q^{16} - 4q^{17} - q^{18} + 8q^{19} - 2q^{20} + 10q^{22} + 4q^{23} - q^{24} + 4q^{25} + 2q^{27} - 10q^{29} + q^{30} + 3q^{31} - q^{32} - 5q^{33} + 8q^{34} + 2q^{36} + 4q^{37} + 8q^{38} + q^{40} + 4q^{43} - 5q^{44} + q^{45} + 4q^{46} - 6q^{47} + 2q^{48} - 8q^{50} - 4q^{51} + 9q^{53} - q^{54} - 10q^{55} - 16q^{57} + 5q^{58} - 11q^{59} + q^{60} - 6q^{61} - 6q^{62} + 2q^{64} - 5q^{66} + 2q^{67} - 4q^{68} - 8q^{69} + 4q^{71} - q^{72} + 10q^{73} + 4q^{74} + 4q^{75} - 16q^{76} - 3q^{79} + q^{80} - q^{81} + 14q^{83} - 8q^{85} - 2q^{86} + 5q^{87} - 5q^{88} - 6q^{89} - 2q^{90} - 8q^{92} + 3q^{93} - 6q^{94} - 8q^{95} - q^{96} - 14q^{97} + 10q^{99} + O(q^{100})$$

## 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
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 1.00000 −0.500000 0.866025i 0.500000 0.866025i
79.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 0 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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 294.2.e.b 2
3.b odd 2 1 882.2.g.i 2
4.b odd 2 1 2352.2.q.u 2
7.b odd 2 1 42.2.e.a 2
7.c even 3 1 294.2.a.f 1
7.c even 3 1 inner 294.2.e.b 2
7.d odd 6 1 42.2.e.a 2
7.d odd 6 1 294.2.a.e 1
21.c even 2 1 126.2.g.c 2
21.g even 6 1 126.2.g.c 2
21.g even 6 1 882.2.a.c 1
21.h odd 6 1 882.2.a.d 1
21.h odd 6 1 882.2.g.i 2
28.d even 2 1 336.2.q.b 2
28.f even 6 1 336.2.q.b 2
28.f even 6 1 2352.2.a.t 1
28.g odd 6 1 2352.2.a.f 1
28.g odd 6 1 2352.2.q.u 2
35.c odd 2 1 1050.2.i.l 2
35.f even 4 2 1050.2.o.a 4
35.i odd 6 1 1050.2.i.l 2
35.i odd 6 1 7350.2.a.bl 1
35.j even 6 1 7350.2.a.q 1
35.k even 12 2 1050.2.o.a 4
56.e even 2 1 1344.2.q.s 2
56.h odd 2 1 1344.2.q.g 2
56.j odd 6 1 1344.2.q.g 2
56.j odd 6 1 9408.2.a.ce 1
56.k odd 6 1 9408.2.a.cr 1
56.m even 6 1 1344.2.q.s 2
56.m even 6 1 9408.2.a.q 1
56.p even 6 1 9408.2.a.z 1
63.i even 6 1 1134.2.h.l 2
63.k odd 6 1 1134.2.e.l 2
63.l odd 6 1 1134.2.e.l 2
63.l odd 6 1 1134.2.h.e 2
63.o even 6 1 1134.2.e.e 2
63.o even 6 1 1134.2.h.l 2
63.s even 6 1 1134.2.e.e 2
63.t odd 6 1 1134.2.h.e 2
84.h odd 2 1 1008.2.s.k 2
84.j odd 6 1 1008.2.s.k 2
84.j odd 6 1 7056.2.a.w 1
84.n even 6 1 7056.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 7.b odd 2 1
42.2.e.a 2 7.d odd 6 1
126.2.g.c 2 21.c even 2 1
126.2.g.c 2 21.g even 6 1
294.2.a.e 1 7.d odd 6 1
294.2.a.f 1 7.c even 3 1
294.2.e.b 2 1.a even 1 1 trivial
294.2.e.b 2 7.c even 3 1 inner
336.2.q.b 2 28.d even 2 1
336.2.q.b 2 28.f even 6 1
882.2.a.c 1 21.g even 6 1
882.2.a.d 1 21.h odd 6 1
882.2.g.i 2 3.b odd 2 1
882.2.g.i 2 21.h odd 6 1
1008.2.s.k 2 84.h odd 2 1
1008.2.s.k 2 84.j odd 6 1
1050.2.i.l 2 35.c odd 2 1
1050.2.i.l 2 35.i odd 6 1
1050.2.o.a 4 35.f even 4 2
1050.2.o.a 4 35.k even 12 2
1134.2.e.e 2 63.o even 6 1
1134.2.e.e 2 63.s even 6 1
1134.2.e.l 2 63.k odd 6 1
1134.2.e.l 2 63.l odd 6 1
1134.2.h.e 2 63.l odd 6 1
1134.2.h.e 2 63.t odd 6 1
1134.2.h.l 2 63.i even 6 1
1134.2.h.l 2 63.o even 6 1
1344.2.q.g 2 56.h odd 2 1
1344.2.q.g 2 56.j odd 6 1
1344.2.q.s 2 56.e even 2 1
1344.2.q.s 2 56.m even 6 1
2352.2.a.f 1 28.g odd 6 1
2352.2.a.t 1 28.f even 6 1
2352.2.q.u 2 4.b odd 2 1
2352.2.q.u 2 28.g odd 6 1
7056.2.a.w 1 84.j odd 6 1
7056.2.a.bl 1 84.n even 6 1
7350.2.a.q 1 35.j even 6 1
7350.2.a.bl 1 35.i odd 6 1
9408.2.a.q 1 56.m even 6 1
9408.2.a.z 1 56.p even 6 1
9408.2.a.ce 1 56.j odd 6 1
9408.2.a.cr 1 56.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 5 T + 29 T^{2} )^{2}$$
$31$ $$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 2 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 7 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 7 T + 97 T^{2} )^{2}$$