# Properties

 Label 294.2.e.a 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} + q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} + q^{6} + q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} -6 q^{13} + 2 q^{15} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -4 \zeta_{6} q^{19} + 2 q^{20} -4 q^{22} -8 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} + q^{27} -2 q^{29} -2 \zeta_{6} q^{30} + ( -1 + \zeta_{6} ) q^{32} + 4 \zeta_{6} q^{33} -2 q^{34} + q^{36} + 10 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + ( 6 - 6 \zeta_{6} ) q^{39} -2 \zeta_{6} q^{40} + 6 q^{41} -4 q^{43} + 4 \zeta_{6} q^{44} + ( -2 + 2 \zeta_{6} ) q^{45} + ( -8 + 8 \zeta_{6} ) q^{46} + q^{48} - q^{50} + 2 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} -8 q^{55} + 4 q^{57} + 2 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} + ( -2 + 2 \zeta_{6} ) q^{60} + 6 \zeta_{6} q^{61} + q^{64} + 12 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{66} + ( -4 + 4 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 8 q^{69} + 8 q^{71} -\zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} + 4 q^{76} -6 q^{78} + ( -2 + 2 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 \zeta_{6} q^{82} + 4 q^{83} -4 q^{85} + 4 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{87} + ( 4 - 4 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + 2 q^{90} + 8 q^{92} + ( -8 + 8 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} + 14 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q - q^{2} - q^{3} - q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} - q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 12 q^{13} + 4 q^{15} - q^{16} + 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 8 q^{23} - q^{24} + q^{25} + 6 q^{26} + 2 q^{27} - 4 q^{29} - 2 q^{30} - q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} + 10 q^{37} - 4 q^{38} + 6 q^{39} - 2 q^{40} + 12 q^{41} - 8 q^{43} + 4 q^{44} - 2 q^{45} - 8 q^{46} + 2 q^{48} - 2 q^{50} + 2 q^{51} + 6 q^{52} - 6 q^{53} - q^{54} - 16 q^{55} + 8 q^{57} + 2 q^{58} + 4 q^{59} - 2 q^{60} + 6 q^{61} + 2 q^{64} + 12 q^{65} + 4 q^{66} - 4 q^{67} + 2 q^{68} + 16 q^{69} + 16 q^{71} - q^{72} + 10 q^{73} + 10 q^{74} + q^{75} + 8 q^{76} - 12 q^{78} - 2 q^{80} - q^{81} - 6 q^{82} + 8 q^{83} - 8 q^{85} + 4 q^{86} + 2 q^{87} + 4 q^{88} - 6 q^{89} + 4 q^{90} + 16 q^{92} - 8 q^{95} - q^{96} + 28 q^{97} - 8 q^{99} + O(q^{100})$$

## 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.a 2
3.b odd 2 1 882.2.g.j 2
4.b odd 2 1 2352.2.q.n 2
7.b odd 2 1 294.2.e.c 2
7.c even 3 1 294.2.a.g 1
7.c even 3 1 inner 294.2.e.a 2
7.d odd 6 1 42.2.a.a 1
7.d odd 6 1 294.2.e.c 2
21.c even 2 1 882.2.g.h 2
21.g even 6 1 126.2.a.a 1
21.g even 6 1 882.2.g.h 2
21.h odd 6 1 882.2.a.b 1
21.h odd 6 1 882.2.g.j 2
28.d even 2 1 2352.2.q.i 2
28.f even 6 1 336.2.a.d 1
28.f even 6 1 2352.2.q.i 2
28.g odd 6 1 2352.2.a.l 1
28.g odd 6 1 2352.2.q.n 2
35.i odd 6 1 1050.2.a.i 1
35.j even 6 1 7350.2.a.f 1
35.k even 12 2 1050.2.g.a 2
56.j odd 6 1 1344.2.a.q 1
56.k odd 6 1 9408.2.a.bw 1
56.m even 6 1 1344.2.a.i 1
56.p even 6 1 9408.2.a.n 1
63.i even 6 1 1134.2.f.j 2
63.k odd 6 1 1134.2.f.g 2
63.s even 6 1 1134.2.f.j 2
63.t odd 6 1 1134.2.f.g 2
77.i even 6 1 5082.2.a.d 1
84.j odd 6 1 1008.2.a.j 1
84.n even 6 1 7056.2.a.k 1
91.s odd 6 1 7098.2.a.f 1
105.p even 6 1 3150.2.a.bo 1
105.w odd 12 2 3150.2.g.r 2
112.v even 12 2 5376.2.c.e 2
112.x odd 12 2 5376.2.c.bc 2
140.s even 6 1 8400.2.a.k 1
168.ba even 6 1 4032.2.a.e 1
168.be odd 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.d odd 6 1
126.2.a.a 1 21.g even 6 1
294.2.a.g 1 7.c even 3 1
294.2.e.a 2 1.a even 1 1 trivial
294.2.e.a 2 7.c even 3 1 inner
294.2.e.c 2 7.b odd 2 1
294.2.e.c 2 7.d odd 6 1
336.2.a.d 1 28.f even 6 1
882.2.a.b 1 21.h odd 6 1
882.2.g.h 2 21.c even 2 1
882.2.g.h 2 21.g even 6 1
882.2.g.j 2 3.b odd 2 1
882.2.g.j 2 21.h odd 6 1
1008.2.a.j 1 84.j odd 6 1
1050.2.a.i 1 35.i odd 6 1
1050.2.g.a 2 35.k even 12 2
1134.2.f.g 2 63.k odd 6 1
1134.2.f.g 2 63.t odd 6 1
1134.2.f.j 2 63.i even 6 1
1134.2.f.j 2 63.s even 6 1
1344.2.a.i 1 56.m even 6 1
1344.2.a.q 1 56.j odd 6 1
2352.2.a.l 1 28.g odd 6 1
2352.2.q.i 2 28.d even 2 1
2352.2.q.i 2 28.f even 6 1
2352.2.q.n 2 4.b odd 2 1
2352.2.q.n 2 28.g odd 6 1
3150.2.a.bo 1 105.p even 6 1
3150.2.g.r 2 105.w odd 12 2
4032.2.a.e 1 168.ba even 6 1
4032.2.a.m 1 168.be odd 6 1
5082.2.a.d 1 77.i even 6 1
5376.2.c.e 2 112.v even 12 2
5376.2.c.bc 2 112.x odd 12 2
7056.2.a.k 1 84.n even 6 1
7098.2.a.f 1 91.s odd 6 1
7350.2.a.f 1 35.j even 6 1
8400.2.a.k 1 140.s even 6 1
9408.2.a.n 1 56.p even 6 1
9408.2.a.bw 1 56.k odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

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