# Properties

 Label 294.2.e.e 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: 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} + q^{6} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} + q^{6} - q^{8} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 4 q^{13} -4 q^{15} -\zeta_{6} q^{16} + ( 1 - \zeta_{6} ) q^{18} -4 \zeta_{6} q^{19} + 4 q^{20} + 4 q^{22} + ( -1 + \zeta_{6} ) q^{24} + ( -11 + 11 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} - q^{27} + 2 q^{29} -4 \zeta_{6} q^{30} + ( -8 + 8 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -4 \zeta_{6} q^{33} + q^{36} + 6 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} + 4 \zeta_{6} q^{40} + 4 q^{43} + 4 \zeta_{6} q^{44} + ( -4 + 4 \zeta_{6} ) q^{45} + 8 \zeta_{6} q^{47} - q^{48} -11 q^{50} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 10 - 10 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} -16 q^{55} -4 q^{57} + 2 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( 4 - 4 \zeta_{6} ) q^{60} + 4 \zeta_{6} q^{61} -8 q^{62} + q^{64} -16 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{66} + ( -4 + 4 \zeta_{6} ) q^{67} + 8 q^{71} + \zeta_{6} q^{72} + ( 16 - 16 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} + 11 \zeta_{6} q^{75} + 4 q^{76} + 4 q^{78} + 8 \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} + 4 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{87} + ( -4 + 4 \zeta_{6} ) q^{88} -8 \zeta_{6} q^{89} -4 q^{90} + 8 \zeta_{6} q^{93} + ( -8 + 8 \zeta_{6} ) q^{94} + ( -16 + 16 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} + 8 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{3} - q^{4} - 4q^{5} + 2q^{6} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} + q^{3} - q^{4} - 4q^{5} + 2q^{6} - 2q^{8} - q^{9} + 4q^{10} + 4q^{11} + q^{12} + 8q^{13} - 8q^{15} - q^{16} + q^{18} - 4q^{19} + 8q^{20} + 8q^{22} - q^{24} - 11q^{25} + 4q^{26} - 2q^{27} + 4q^{29} - 4q^{30} - 8q^{31} + q^{32} - 4q^{33} + 2q^{36} + 6q^{37} + 4q^{38} + 4q^{39} + 4q^{40} + 8q^{43} + 4q^{44} - 4q^{45} + 8q^{47} - 2q^{48} - 22q^{50} - 4q^{52} + 10q^{53} - q^{54} - 32q^{55} - 8q^{57} + 2q^{58} - 4q^{59} + 4q^{60} + 4q^{61} - 16q^{62} + 2q^{64} - 16q^{65} + 4q^{66} - 4q^{67} + 16q^{71} + q^{72} + 16q^{73} - 6q^{74} + 11q^{75} + 8q^{76} + 8q^{78} + 8q^{79} - 4q^{80} - q^{81} - 24q^{83} + 4q^{86} + 2q^{87} - 4q^{88} - 8q^{89} - 8q^{90} + 8q^{93} - 8q^{94} - 16q^{95} - q^{96} + 16q^{97} - 8q^{99} + O(q^{100})$$

## 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 −2.00000 3.46410i 1.00000 0 −1.00000 −0.500000 0.866025i 2.00000 3.46410i
79.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −2.00000 + 3.46410i 1.00000 0 −1.00000 −0.500000 + 0.866025i 2.00000 + 3.46410i
 $$n$$: 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.e 2
3.b odd 2 1 882.2.g.f 2
4.b odd 2 1 2352.2.q.a 2
7.b odd 2 1 294.2.e.d 2
7.c even 3 1 294.2.a.b 1
7.c even 3 1 inner 294.2.e.e 2
7.d odd 6 1 294.2.a.c yes 1
7.d odd 6 1 294.2.e.d 2
21.c even 2 1 882.2.g.a 2
21.g even 6 1 882.2.a.l 1
21.g even 6 1 882.2.g.a 2
21.h odd 6 1 882.2.a.f 1
21.h odd 6 1 882.2.g.f 2
28.d even 2 1 2352.2.q.y 2
28.f even 6 1 2352.2.a.b 1
28.f even 6 1 2352.2.q.y 2
28.g odd 6 1 2352.2.a.y 1
28.g odd 6 1 2352.2.q.a 2
35.i odd 6 1 7350.2.a.br 1
35.j even 6 1 7350.2.a.cj 1
56.j odd 6 1 9408.2.a.bo 1
56.k odd 6 1 9408.2.a.b 1
56.m even 6 1 9408.2.a.de 1
56.p even 6 1 9408.2.a.br 1
84.j odd 6 1 7056.2.a.ca 1
84.n even 6 1 7056.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.a.b 1 7.c even 3 1
294.2.a.c yes 1 7.d odd 6 1
294.2.e.d 2 7.b odd 2 1
294.2.e.d 2 7.d odd 6 1
294.2.e.e 2 1.a even 1 1 trivial
294.2.e.e 2 7.c even 3 1 inner
882.2.a.f 1 21.h odd 6 1
882.2.a.l 1 21.g even 6 1
882.2.g.a 2 21.c even 2 1
882.2.g.a 2 21.g even 6 1
882.2.g.f 2 3.b odd 2 1
882.2.g.f 2 21.h odd 6 1
2352.2.a.b 1 28.f even 6 1
2352.2.a.y 1 28.g odd 6 1
2352.2.q.a 2 4.b odd 2 1
2352.2.q.a 2 28.g odd 6 1
2352.2.q.y 2 28.d even 2 1
2352.2.q.y 2 28.f even 6 1
7056.2.a.a 1 84.n even 6 1
7056.2.a.ca 1 84.j odd 6 1
7350.2.a.br 1 35.i odd 6 1
7350.2.a.cj 1 35.j even 6 1
9408.2.a.b 1 56.k odd 6 1
9408.2.a.bo 1 56.j odd 6 1
9408.2.a.br 1 56.p even 6 1
9408.2.a.de 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 + 8 T - 25 T^{2} + 712 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{2}$$