# Properties

 Label 294.2.e.c 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 2*z * q^5 - q^6 + q^8 - z * q^9 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + q^{8} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} + \zeta_{6} q^{12} + 6 q^{13} + 2 q^{15} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + (\zeta_{6} - 1) q^{18} + 4 \zeta_{6} q^{19} - 2 q^{20} - 4 q^{22} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + ( - \zeta_{6} + 1) q^{25} - 6 \zeta_{6} q^{26} - q^{27} - 2 q^{29} - 2 \zeta_{6} q^{30} + (\zeta_{6} - 1) q^{32} - 4 \zeta_{6} q^{33} + 2 q^{34} + q^{36} + 10 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - 6 \zeta_{6} + 6) q^{39} + 2 \zeta_{6} q^{40} - 6 q^{41} - 4 q^{43} + 4 \zeta_{6} q^{44} + ( - 2 \zeta_{6} + 2) q^{45} + (8 \zeta_{6} - 8) q^{46} - q^{48} - q^{50} + 2 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{52} + (6 \zeta_{6} - 6) q^{53} + \zeta_{6} q^{54} + 8 q^{55} + 4 q^{57} + 2 \zeta_{6} q^{58} + (4 \zeta_{6} - 4) q^{59} + (2 \zeta_{6} - 2) q^{60} - 6 \zeta_{6} q^{61} + q^{64} + 12 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{66} + (4 \zeta_{6} - 4) q^{67} - 2 \zeta_{6} q^{68} - 8 q^{69} + 8 q^{71} - \zeta_{6} q^{72} + (10 \zeta_{6} - 10) q^{73} + ( - 10 \zeta_{6} + 10) q^{74} - \zeta_{6} q^{75} - 4 q^{76} - 6 q^{78} + ( - 2 \zeta_{6} + 2) q^{80} + (\zeta_{6} - 1) q^{81} + 6 \zeta_{6} q^{82} - 4 q^{83} - 4 q^{85} + 4 \zeta_{6} q^{86} + (2 \zeta_{6} - 2) q^{87} + ( - 4 \zeta_{6} + 4) q^{88} + 6 \zeta_{6} q^{89} - 2 q^{90} + 8 q^{92} + (8 \zeta_{6} - 8) q^{95} + \zeta_{6} q^{96} - 14 q^{97} - 4 q^{99} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 2*z * q^5 - q^6 + q^8 - z * q^9 + (-2*z + 2) * q^10 + (-4*z + 4) * q^11 + z * q^12 + 6 * q^13 + 2 * q^15 - z * q^16 + (2*z - 2) * q^17 + (z - 1) * q^18 + 4*z * q^19 - 2 * q^20 - 4 * q^22 - 8*z * q^23 + (-z + 1) * q^24 + (-z + 1) * q^25 - 6*z * q^26 - q^27 - 2 * q^29 - 2*z * q^30 + (z - 1) * q^32 - 4*z * q^33 + 2 * q^34 + q^36 + 10*z * q^37 + (-4*z + 4) * q^38 + (-6*z + 6) * q^39 + 2*z * q^40 - 6 * q^41 - 4 * q^43 + 4*z * q^44 + (-2*z + 2) * q^45 + (8*z - 8) * q^46 - q^48 - q^50 + 2*z * q^51 + (6*z - 6) * q^52 + (6*z - 6) * q^53 + z * q^54 + 8 * q^55 + 4 * q^57 + 2*z * q^58 + (4*z - 4) * q^59 + (2*z - 2) * q^60 - 6*z * q^61 + q^64 + 12*z * q^65 + (4*z - 4) * q^66 + (4*z - 4) * q^67 - 2*z * q^68 - 8 * q^69 + 8 * q^71 - z * q^72 + (10*z - 10) * q^73 + (-10*z + 10) * q^74 - z * q^75 - 4 * q^76 - 6 * q^78 + (-2*z + 2) * q^80 + (z - 1) * q^81 + 6*z * q^82 - 4 * q^83 - 4 * q^85 + 4*z * q^86 + (2*z - 2) * q^87 + (-4*z + 4) * q^88 + 6*z * q^89 - 2 * q^90 + 8 * q^92 + (8*z - 8) * q^95 + z * q^96 - 14 * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 - q^9 $$2 q - q^{2} + q^{3} - q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} - q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + 12 q^{13} + 4 q^{15} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} - 4 q^{20} - 8 q^{22} - 8 q^{23} + q^{24} + q^{25} - 6 q^{26} - 2 q^{27} - 4 q^{29} - 2 q^{30} - q^{32} - 4 q^{33} + 4 q^{34} + 2 q^{36} + 10 q^{37} + 4 q^{38} + 6 q^{39} + 2 q^{40} - 12 q^{41} - 8 q^{43} + 4 q^{44} + 2 q^{45} - 8 q^{46} - 2 q^{48} - 2 q^{50} + 2 q^{51} - 6 q^{52} - 6 q^{53} + q^{54} + 16 q^{55} + 8 q^{57} + 2 q^{58} - 4 q^{59} - 2 q^{60} - 6 q^{61} + 2 q^{64} + 12 q^{65} - 4 q^{66} - 4 q^{67} - 2 q^{68} - 16 q^{69} + 16 q^{71} - q^{72} - 10 q^{73} + 10 q^{74} - q^{75} - 8 q^{76} - 12 q^{78} + 2 q^{80} - q^{81} + 6 q^{82} - 8 q^{83} - 8 q^{85} + 4 q^{86} - 2 q^{87} + 4 q^{88} + 6 q^{89} - 4 q^{90} + 16 q^{92} - 8 q^{95} + q^{96} - 28 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 + 2 * q^5 - 2 * q^6 + 2 * q^8 - q^9 + 2 * q^10 + 4 * q^11 + q^12 + 12 * q^13 + 4 * q^15 - q^16 - 2 * q^17 - q^18 + 4 * q^19 - 4 * q^20 - 8 * q^22 - 8 * q^23 + q^24 + q^25 - 6 * q^26 - 2 * q^27 - 4 * q^29 - 2 * q^30 - q^32 - 4 * q^33 + 4 * q^34 + 2 * q^36 + 10 * q^37 + 4 * q^38 + 6 * q^39 + 2 * q^40 - 12 * q^41 - 8 * q^43 + 4 * q^44 + 2 * q^45 - 8 * q^46 - 2 * q^48 - 2 * q^50 + 2 * q^51 - 6 * q^52 - 6 * q^53 + q^54 + 16 * q^55 + 8 * q^57 + 2 * q^58 - 4 * q^59 - 2 * q^60 - 6 * q^61 + 2 * q^64 + 12 * q^65 - 4 * q^66 - 4 * q^67 - 2 * q^68 - 16 * q^69 + 16 * q^71 - q^72 - 10 * q^73 + 10 * q^74 - q^75 - 8 * q^76 - 12 * q^78 + 2 * q^80 - q^81 + 6 * q^82 - 8 * q^83 - 8 * q^85 + 4 * q^86 - 2 * q^87 + 4 * q^88 + 6 * q^89 - 4 * q^90 + 16 * q^92 - 8 * q^95 + q^96 - 28 * 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 1.00000 + 1.73205i −1.00000 0 1.00000 −0.500000 0.866025i 1.00000 1.73205i
79.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.73205i −1.00000 0 1.00000 −0.500000 + 0.866025i 1.00000 + 1.73205i
 $$n$$: 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.c 2
3.b odd 2 1 882.2.g.h 2
4.b odd 2 1 2352.2.q.i 2
7.b odd 2 1 294.2.e.a 2
7.c even 3 1 42.2.a.a 1
7.c even 3 1 inner 294.2.e.c 2
7.d odd 6 1 294.2.a.g 1
7.d odd 6 1 294.2.e.a 2
21.c even 2 1 882.2.g.j 2
21.g even 6 1 882.2.a.b 1
21.g even 6 1 882.2.g.j 2
21.h odd 6 1 126.2.a.a 1
21.h odd 6 1 882.2.g.h 2
28.d even 2 1 2352.2.q.n 2
28.f even 6 1 2352.2.a.l 1
28.f even 6 1 2352.2.q.n 2
28.g odd 6 1 336.2.a.d 1
28.g odd 6 1 2352.2.q.i 2
35.i odd 6 1 7350.2.a.f 1
35.j even 6 1 1050.2.a.i 1
35.l odd 12 2 1050.2.g.a 2
56.j odd 6 1 9408.2.a.n 1
56.k odd 6 1 1344.2.a.i 1
56.m even 6 1 9408.2.a.bw 1
56.p even 6 1 1344.2.a.q 1
63.g even 3 1 1134.2.f.g 2
63.h even 3 1 1134.2.f.g 2
63.j odd 6 1 1134.2.f.j 2
63.n odd 6 1 1134.2.f.j 2
77.h odd 6 1 5082.2.a.d 1
84.j odd 6 1 7056.2.a.k 1
84.n even 6 1 1008.2.a.j 1
91.r even 6 1 7098.2.a.f 1
105.o odd 6 1 3150.2.a.bo 1
105.x even 12 2 3150.2.g.r 2
112.u odd 12 2 5376.2.c.e 2
112.w even 12 2 5376.2.c.bc 2
140.p odd 6 1 8400.2.a.k 1
168.s odd 6 1 4032.2.a.e 1
168.v even 6 1 4032.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 7.c even 3 1
126.2.a.a 1 21.h odd 6 1
294.2.a.g 1 7.d odd 6 1
294.2.e.a 2 7.b odd 2 1
294.2.e.a 2 7.d odd 6 1
294.2.e.c 2 1.a even 1 1 trivial
294.2.e.c 2 7.c even 3 1 inner
336.2.a.d 1 28.g odd 6 1
882.2.a.b 1 21.g even 6 1
882.2.g.h 2 3.b odd 2 1
882.2.g.h 2 21.h odd 6 1
882.2.g.j 2 21.c even 2 1
882.2.g.j 2 21.g even 6 1
1008.2.a.j 1 84.n even 6 1
1050.2.a.i 1 35.j even 6 1
1050.2.g.a 2 35.l odd 12 2
1134.2.f.g 2 63.g even 3 1
1134.2.f.g 2 63.h even 3 1
1134.2.f.j 2 63.j odd 6 1
1134.2.f.j 2 63.n odd 6 1
1344.2.a.i 1 56.k odd 6 1
1344.2.a.q 1 56.p even 6 1
2352.2.a.l 1 28.f even 6 1
2352.2.q.i 2 4.b odd 2 1
2352.2.q.i 2 28.g odd 6 1
2352.2.q.n 2 28.d even 2 1
2352.2.q.n 2 28.f even 6 1
3150.2.a.bo 1 105.o odd 6 1
3150.2.g.r 2 105.x even 12 2
4032.2.a.e 1 168.s odd 6 1
4032.2.a.m 1 168.v even 6 1
5082.2.a.d 1 77.h odd 6 1
5376.2.c.e 2 112.u odd 12 2
5376.2.c.bc 2 112.w even 12 2
7056.2.a.k 1 84.j odd 6 1
7098.2.a.f 1 91.r even 6 1
7350.2.a.f 1 35.i odd 6 1
8400.2.a.k 1 140.p odd 6 1
9408.2.a.n 1 56.j odd 6 1
9408.2.a.bw 1 56.m even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 100$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 6T + 36$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2}$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 14)^{2}$$