Properties

 Label 294.2.e.d 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: $$x^{2} - x + 1$$ 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.

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

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.d 2
3.b odd 2 1 882.2.g.a 2
4.b odd 2 1 2352.2.q.y 2
7.b odd 2 1 294.2.e.e 2
7.c even 3 1 294.2.a.c yes 1
7.c even 3 1 inner 294.2.e.d 2
7.d odd 6 1 294.2.a.b 1
7.d odd 6 1 294.2.e.e 2
21.c even 2 1 882.2.g.f 2
21.g even 6 1 882.2.a.f 1
21.g even 6 1 882.2.g.f 2
21.h odd 6 1 882.2.a.l 1
21.h odd 6 1 882.2.g.a 2
28.d even 2 1 2352.2.q.a 2
28.f even 6 1 2352.2.a.y 1
28.f even 6 1 2352.2.q.a 2
28.g odd 6 1 2352.2.a.b 1
28.g odd 6 1 2352.2.q.y 2
35.i odd 6 1 7350.2.a.cj 1
35.j even 6 1 7350.2.a.br 1
56.j odd 6 1 9408.2.a.br 1
56.k odd 6 1 9408.2.a.de 1
56.m even 6 1 9408.2.a.b 1
56.p even 6 1 9408.2.a.bo 1
84.j odd 6 1 7056.2.a.a 1
84.n even 6 1 7056.2.a.ca 1

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$T^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 16T + 256$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 8T + 64$$
$97$ $$(T + 8)^{2}$$