# Properties

 Label 294.2.e.f 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: 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 3*z * q^5 + q^6 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{10} + (3 \zeta_{6} - 3) q^{11} + \zeta_{6} q^{12} + 4 q^{13} + 3 q^{15} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} - 3 q^{20} - 3 q^{22} + (\zeta_{6} - 1) q^{24} + (4 \zeta_{6} - 4) q^{25} + 4 \zeta_{6} q^{26} - q^{27} + 9 q^{29} + 3 \zeta_{6} q^{30} + (\zeta_{6} - 1) q^{31} + ( - \zeta_{6} + 1) q^{32} + 3 \zeta_{6} q^{33} + q^{36} - 8 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - 4 \zeta_{6} + 4) q^{39} - 3 \zeta_{6} q^{40} - 10 q^{43} - 3 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{45} - 6 \zeta_{6} q^{47} - q^{48} - 4 q^{50} + (4 \zeta_{6} - 4) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} - \zeta_{6} q^{54} - 9 q^{55} - 4 q^{57} + 9 \zeta_{6} q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + (3 \zeta_{6} - 3) q^{60} - 10 \zeta_{6} q^{61} - q^{62} + q^{64} + 12 \zeta_{6} q^{65} + (3 \zeta_{6} - 3) q^{66} + ( - 10 \zeta_{6} + 10) q^{67} - 6 q^{71} + \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + 4 \zeta_{6} q^{75} + 4 q^{76} + 4 q^{78} + \zeta_{6} q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + (\zeta_{6} - 1) q^{81} + 9 q^{83} - 10 \zeta_{6} q^{86} + ( - 9 \zeta_{6} + 9) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} + 6 \zeta_{6} q^{89} + 3 q^{90} + \zeta_{6} q^{93} + ( - 6 \zeta_{6} + 6) q^{94} + ( - 12 \zeta_{6} + 12) q^{95} - \zeta_{6} q^{96} + q^{97} + 3 q^{99} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 3*z * q^5 + q^6 - q^8 - z * q^9 + (3*z - 3) * q^10 + (3*z - 3) * q^11 + z * q^12 + 4 * q^13 + 3 * q^15 - z * q^16 + (-z + 1) * q^18 - 4*z * q^19 - 3 * q^20 - 3 * q^22 + (z - 1) * q^24 + (4*z - 4) * q^25 + 4*z * q^26 - q^27 + 9 * q^29 + 3*z * q^30 + (z - 1) * q^31 + (-z + 1) * q^32 + 3*z * q^33 + q^36 - 8*z * q^37 + (-4*z + 4) * q^38 + (-4*z + 4) * q^39 - 3*z * q^40 - 10 * q^43 - 3*z * q^44 + (-3*z + 3) * q^45 - 6*z * q^47 - q^48 - 4 * q^50 + (4*z - 4) * q^52 + (-3*z + 3) * q^53 - z * q^54 - 9 * q^55 - 4 * q^57 + 9*z * q^58 + (-3*z + 3) * q^59 + (3*z - 3) * q^60 - 10*z * q^61 - q^62 + q^64 + 12*z * q^65 + (3*z - 3) * q^66 + (-10*z + 10) * q^67 - 6 * q^71 + z * q^72 + (-2*z + 2) * q^73 + (-8*z + 8) * q^74 + 4*z * q^75 + 4 * q^76 + 4 * q^78 + z * q^79 + (-3*z + 3) * q^80 + (z - 1) * q^81 + 9 * q^83 - 10*z * q^86 + (-9*z + 9) * q^87 + (-3*z + 3) * q^88 + 6*z * q^89 + 3 * q^90 + z * q^93 + (-6*z + 6) * q^94 + (-12*z + 12) * q^95 - z * q^96 + q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 + 3 * q^5 + 2 * q^6 - 2 * q^8 - q^9 $$2 q + q^{2} + q^{3} - q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - q^{9} - 3 q^{10} - 3 q^{11} + q^{12} + 8 q^{13} + 6 q^{15} - q^{16} + q^{18} - 4 q^{19} - 6 q^{20} - 6 q^{22} - q^{24} - 4 q^{25} + 4 q^{26} - 2 q^{27} + 18 q^{29} + 3 q^{30} - q^{31} + q^{32} + 3 q^{33} + 2 q^{36} - 8 q^{37} + 4 q^{38} + 4 q^{39} - 3 q^{40} - 20 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{47} - 2 q^{48} - 8 q^{50} - 4 q^{52} + 3 q^{53} - q^{54} - 18 q^{55} - 8 q^{57} + 9 q^{58} + 3 q^{59} - 3 q^{60} - 10 q^{61} - 2 q^{62} + 2 q^{64} + 12 q^{65} - 3 q^{66} + 10 q^{67} - 12 q^{71} + q^{72} + 2 q^{73} + 8 q^{74} + 4 q^{75} + 8 q^{76} + 8 q^{78} + q^{79} + 3 q^{80} - q^{81} + 18 q^{83} - 10 q^{86} + 9 q^{87} + 3 q^{88} + 6 q^{89} + 6 q^{90} + q^{93} + 6 q^{94} + 12 q^{95} - q^{96} + 2 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 3 * q^5 + 2 * q^6 - 2 * q^8 - q^9 - 3 * q^10 - 3 * q^11 + q^12 + 8 * q^13 + 6 * q^15 - q^16 + q^18 - 4 * q^19 - 6 * q^20 - 6 * q^22 - q^24 - 4 * q^25 + 4 * q^26 - 2 * q^27 + 18 * q^29 + 3 * q^30 - q^31 + q^32 + 3 * q^33 + 2 * q^36 - 8 * q^37 + 4 * q^38 + 4 * q^39 - 3 * q^40 - 20 * q^43 - 3 * q^44 + 3 * q^45 - 6 * q^47 - 2 * q^48 - 8 * q^50 - 4 * q^52 + 3 * q^53 - q^54 - 18 * q^55 - 8 * q^57 + 9 * q^58 + 3 * q^59 - 3 * q^60 - 10 * q^61 - 2 * q^62 + 2 * q^64 + 12 * q^65 - 3 * q^66 + 10 * q^67 - 12 * q^71 + q^72 + 2 * q^73 + 8 * q^74 + 4 * q^75 + 8 * q^76 + 8 * q^78 + q^79 + 3 * q^80 - q^81 + 18 * q^83 - 10 * q^86 + 9 * q^87 + 3 * q^88 + 6 * q^89 + 6 * q^90 + q^93 + 6 * q^94 + 12 * q^95 - q^96 + 2 * q^97 + 6 * 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 1.50000 + 2.59808i 1.00000 0 −1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
79.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.50000 2.59808i 1.00000 0 −1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
 $$n$$: 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.f 2
3.b odd 2 1 882.2.g.b 2
4.b odd 2 1 2352.2.q.m 2
7.b odd 2 1 42.2.e.b 2
7.c even 3 1 294.2.a.a 1
7.c even 3 1 inner 294.2.e.f 2
7.d odd 6 1 42.2.e.b 2
7.d odd 6 1 294.2.a.d 1
21.c even 2 1 126.2.g.b 2
21.g even 6 1 126.2.g.b 2
21.g even 6 1 882.2.a.g 1
21.h odd 6 1 882.2.a.k 1
21.h odd 6 1 882.2.g.b 2
28.d even 2 1 336.2.q.d 2
28.f even 6 1 336.2.q.d 2
28.f even 6 1 2352.2.a.m 1
28.g odd 6 1 2352.2.a.n 1
28.g odd 6 1 2352.2.q.m 2
35.c odd 2 1 1050.2.i.e 2
35.f even 4 2 1050.2.o.b 4
35.i odd 6 1 1050.2.i.e 2
35.i odd 6 1 7350.2.a.ce 1
35.j even 6 1 7350.2.a.cw 1
35.k even 12 2 1050.2.o.b 4
56.e even 2 1 1344.2.q.j 2
56.h odd 2 1 1344.2.q.v 2
56.j odd 6 1 1344.2.q.v 2
56.j odd 6 1 9408.2.a.d 1
56.k odd 6 1 9408.2.a.bm 1
56.m even 6 1 1344.2.q.j 2
56.m even 6 1 9408.2.a.bu 1
56.p even 6 1 9408.2.a.db 1
63.i even 6 1 1134.2.h.a 2
63.k odd 6 1 1134.2.e.a 2
63.l odd 6 1 1134.2.e.a 2
63.l odd 6 1 1134.2.h.p 2
63.o even 6 1 1134.2.e.p 2
63.o even 6 1 1134.2.h.a 2
63.s even 6 1 1134.2.e.p 2
63.t odd 6 1 1134.2.h.p 2
84.h odd 2 1 1008.2.s.n 2
84.j odd 6 1 1008.2.s.n 2
84.j odd 6 1 7056.2.a.g 1
84.n even 6 1 7056.2.a.bz 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + T + 1$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$T^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 1)^{2}$$