# Properties

 Label 294.2.f.a Level $294$ Weight $2$ Character orbit 294.f Analytic conductor $2.348$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 294.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.34760181943$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + (2 \zeta_{12}^{2} - 1) q^{6} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + z * q^2 + (z^3 + z) * q^3 + z^2 * q^4 + (-2*z^3 + z) * q^5 + (2*z^2 - 1) * q^6 + z^3 * q^8 + (3*z^2 - 3) * q^9 $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + (2 \zeta_{12}^{2} - 1) q^{6} + \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - \zeta_{12}^{2} + 2) q^{10} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{12} + (4 \zeta_{12}^{2} - 2) q^{13} + 3 q^{15} + (\zeta_{12}^{2} - 1) q^{16} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + ( - 2 \zeta_{12}^{2} - 2) q^{19} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{20} + 3 q^{22} - 6 \zeta_{12} q^{23} + (\zeta_{12}^{2} - 2) q^{24} + 2 \zeta_{12}^{2} q^{25} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 3 \zeta_{12}^{3} q^{29} + 3 \zeta_{12} q^{30} + ( - \zeta_{12}^{2} + 2) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (3 \zeta_{12}^{2} + 3) q^{33} + ( - 4 \zeta_{12}^{2} + 2) q^{34} - 3 q^{36} + ( - 2 \zeta_{12}^{2} + 2) q^{37} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{38} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{39} + (\zeta_{12}^{2} + 1) q^{40} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{41} - 8 q^{43} + 3 \zeta_{12} q^{44} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{45} - 6 \zeta_{12}^{2} q^{46} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{47} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{48} + 2 \zeta_{12}^{3} q^{50} + ( - 6 \zeta_{12}^{2} + 6) q^{51} + (2 \zeta_{12}^{2} - 4) q^{52} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{53} + ( - 3 \zeta_{12}^{2} - 3) q^{54} + ( - 6 \zeta_{12}^{2} + 3) q^{55} - 6 \zeta_{12}^{3} q^{57} + ( - 3 \zeta_{12}^{2} + 3) q^{58} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{59} + 3 \zeta_{12}^{2} q^{60} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{62} - q^{64} + 6 \zeta_{12} q^{65} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{66} - 2 \zeta_{12}^{2} q^{67} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + ( - 12 \zeta_{12}^{2} + 6) q^{69} + 12 \zeta_{12}^{3} q^{71} - 3 \zeta_{12} q^{72} + (4 \zeta_{12}^{2} - 8) q^{73} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{74} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{75} + ( - 4 \zeta_{12}^{2} + 2) q^{76} - 6 q^{78} + ( - \zeta_{12}^{2} + 1) q^{79} + (\zeta_{12}^{3} + \zeta_{12}) q^{80} - 9 \zeta_{12}^{2} q^{81} + (4 \zeta_{12}^{2} + 4) q^{82} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{83} - 6 q^{85} - 8 \zeta_{12} q^{86} + ( - 3 \zeta_{12}^{2} + 6) q^{87} + 3 \zeta_{12}^{2} q^{88} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} + (6 \zeta_{12}^{2} - 3) q^{90} - 6 \zeta_{12}^{3} q^{92} + 3 \zeta_{12} q^{93} + ( - 4 \zeta_{12}^{2} + 8) q^{94} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{95} + ( - \zeta_{12}^{2} - 1) q^{96} + (6 \zeta_{12}^{2} - 3) q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + z * q^2 + (z^3 + z) * q^3 + z^2 * q^4 + (-2*z^3 + z) * q^5 + (2*z^2 - 1) * q^6 + z^3 * q^8 + (3*z^2 - 3) * q^9 + (-z^2 + 2) * q^10 + (-3*z^3 + 3*z) * q^11 + (2*z^3 - z) * q^12 + (4*z^2 - 2) * q^13 + 3 * q^15 + (z^2 - 1) * q^16 + (-2*z^3 - 2*z) * q^17 + (3*z^3 - 3*z) * q^18 + (-2*z^2 - 2) * q^19 + (-z^3 + 2*z) * q^20 + 3 * q^22 - 6*z * q^23 + (z^2 - 2) * q^24 + 2*z^2 * q^25 + (4*z^3 - 2*z) * q^26 + (3*z^3 - 6*z) * q^27 - 3*z^3 * q^29 + 3*z * q^30 + (-z^2 + 2) * q^31 + (z^3 - z) * q^32 + (3*z^2 + 3) * q^33 + (-4*z^2 + 2) * q^34 - 3 * q^36 + (-2*z^2 + 2) * q^37 + (-2*z^3 - 2*z) * q^38 + (6*z^3 - 6*z) * q^39 + (z^2 + 1) * q^40 + (-4*z^3 + 8*z) * q^41 - 8 * q^43 + 3*z * q^44 + (3*z^3 + 3*z) * q^45 - 6*z^2 * q^46 + (-8*z^3 + 4*z) * q^47 + (z^3 - 2*z) * q^48 + 2*z^3 * q^50 + (-6*z^2 + 6) * q^51 + (2*z^2 - 4) * q^52 + (-9*z^3 + 9*z) * q^53 + (-3*z^2 - 3) * q^54 + (-6*z^2 + 3) * q^55 - 6*z^3 * q^57 + (-3*z^2 + 3) * q^58 + (-z^3 - z) * q^59 + 3*z^2 * q^60 + (-z^3 + 2*z) * q^62 - q^64 + 6*z * q^65 + (3*z^3 + 3*z) * q^66 - 2*z^2 * q^67 + (-4*z^3 + 2*z) * q^68 + (-12*z^2 + 6) * q^69 + 12*z^3 * q^71 - 3*z * q^72 + (4*z^2 - 8) * q^73 + (-2*z^3 + 2*z) * q^74 + (4*z^3 - 2*z) * q^75 + (-4*z^2 + 2) * q^76 - 6 * q^78 + (-z^2 + 1) * q^79 + (z^3 + z) * q^80 - 9*z^2 * q^81 + (4*z^2 + 4) * q^82 + (5*z^3 - 10*z) * q^83 - 6 * q^85 - 8*z * q^86 + (-3*z^2 + 6) * q^87 + 3*z^2 * q^88 + (12*z^3 - 6*z) * q^89 + (6*z^2 - 3) * q^90 - 6*z^3 * q^92 + 3*z * q^93 + (-4*z^2 + 8) * q^94 + (6*z^3 - 6*z) * q^95 + (-z^2 - 1) * q^96 + (6*z^2 - 3) * q^97 + 9*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 6 * q^9 $$4 q + 2 q^{4} - 6 q^{9} + 6 q^{10} + 12 q^{15} - 2 q^{16} - 12 q^{19} + 12 q^{22} - 6 q^{24} + 4 q^{25} + 6 q^{31} + 18 q^{33} - 12 q^{36} + 4 q^{37} + 6 q^{40} - 32 q^{43} - 12 q^{46} + 12 q^{51} - 12 q^{52} - 18 q^{54} + 6 q^{58} + 6 q^{60} - 4 q^{64} - 4 q^{67} - 24 q^{73} - 24 q^{78} + 2 q^{79} - 18 q^{81} + 24 q^{82} - 24 q^{85} + 18 q^{87} + 6 q^{88} + 24 q^{94} - 6 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 - 6 * q^9 + 6 * q^10 + 12 * q^15 - 2 * q^16 - 12 * q^19 + 12 * q^22 - 6 * q^24 + 4 * q^25 + 6 * q^31 + 18 * q^33 - 12 * q^36 + 4 * q^37 + 6 * q^40 - 32 * q^43 - 12 * q^46 + 12 * q^51 - 12 * q^52 - 18 * q^54 + 6 * q^58 + 6 * q^60 - 4 * q^64 - 4 * q^67 - 24 * q^73 - 24 * q^78 + 2 * q^79 - 18 * q^81 + 24 * q^82 - 24 * q^85 + 18 * q^87 + 6 * q^88 + 24 * q^94 - 6 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i 0 1.00000i −1.50000 + 2.59808i 1.50000 0.866025i
215.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i 0 1.00000i −1.50000 + 2.59808i 1.50000 0.866025i
227.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i 0 1.00000i −1.50000 2.59808i 1.50000 + 0.866025i
227.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i 0 1.00000i −1.50000 2.59808i 1.50000 + 0.866025i
 $$n$$: 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.f.a 4
3.b odd 2 1 inner 294.2.f.a 4
7.b odd 2 1 42.2.f.a 4
7.c even 3 1 42.2.f.a 4
7.c even 3 1 294.2.d.a 4
7.d odd 6 1 294.2.d.a 4
7.d odd 6 1 inner 294.2.f.a 4
21.c even 2 1 42.2.f.a 4
21.g even 6 1 294.2.d.a 4
21.g even 6 1 inner 294.2.f.a 4
21.h odd 6 1 42.2.f.a 4
21.h odd 6 1 294.2.d.a 4
28.d even 2 1 336.2.bc.e 4
28.f even 6 1 2352.2.k.e 4
28.g odd 6 1 336.2.bc.e 4
28.g odd 6 1 2352.2.k.e 4
35.c odd 2 1 1050.2.s.b 4
35.f even 4 1 1050.2.u.a 4
35.f even 4 1 1050.2.u.d 4
35.j even 6 1 1050.2.s.b 4
35.l odd 12 1 1050.2.u.a 4
35.l odd 12 1 1050.2.u.d 4
63.g even 3 1 1134.2.l.c 4
63.h even 3 1 1134.2.t.d 4
63.j odd 6 1 1134.2.t.d 4
63.l odd 6 1 1134.2.l.c 4
63.l odd 6 1 1134.2.t.d 4
63.n odd 6 1 1134.2.l.c 4
63.o even 6 1 1134.2.l.c 4
63.o even 6 1 1134.2.t.d 4
84.h odd 2 1 336.2.bc.e 4
84.j odd 6 1 2352.2.k.e 4
84.n even 6 1 336.2.bc.e 4
84.n even 6 1 2352.2.k.e 4
105.g even 2 1 1050.2.s.b 4
105.k odd 4 1 1050.2.u.a 4
105.k odd 4 1 1050.2.u.d 4
105.o odd 6 1 1050.2.s.b 4
105.x even 12 1 1050.2.u.a 4
105.x even 12 1 1050.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 7.b odd 2 1
42.2.f.a 4 7.c even 3 1
42.2.f.a 4 21.c even 2 1
42.2.f.a 4 21.h odd 6 1
294.2.d.a 4 7.c even 3 1
294.2.d.a 4 7.d odd 6 1
294.2.d.a 4 21.g even 6 1
294.2.d.a 4 21.h odd 6 1
294.2.f.a 4 1.a even 1 1 trivial
294.2.f.a 4 3.b odd 2 1 inner
294.2.f.a 4 7.d odd 6 1 inner
294.2.f.a 4 21.g even 6 1 inner
336.2.bc.e 4 28.d even 2 1
336.2.bc.e 4 28.g odd 6 1
336.2.bc.e 4 84.h odd 2 1
336.2.bc.e 4 84.n even 6 1
1050.2.s.b 4 35.c odd 2 1
1050.2.s.b 4 35.j even 6 1
1050.2.s.b 4 105.g even 2 1
1050.2.s.b 4 105.o odd 6 1
1050.2.u.a 4 35.f even 4 1
1050.2.u.a 4 35.l odd 12 1
1050.2.u.a 4 105.k odd 4 1
1050.2.u.a 4 105.x even 12 1
1050.2.u.d 4 35.f even 4 1
1050.2.u.d 4 35.l odd 12 1
1050.2.u.d 4 105.k odd 4 1
1050.2.u.d 4 105.x even 12 1
1134.2.l.c 4 63.g even 3 1
1134.2.l.c 4 63.l odd 6 1
1134.2.l.c 4 63.n odd 6 1
1134.2.l.c 4 63.o even 6 1
1134.2.t.d 4 63.h even 3 1
1134.2.t.d 4 63.j odd 6 1
1134.2.t.d 4 63.l odd 6 1
1134.2.t.d 4 63.o even 6 1
2352.2.k.e 4 28.f even 6 1
2352.2.k.e 4 28.g odd 6 1
2352.2.k.e 4 84.j odd 6 1
2352.2.k.e 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$T^{4} + 12T^{2} + 144$$
$19$ $$(T^{2} + 6 T + 12)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 9)^{2}$$
$31$ $$(T^{2} - 3 T + 3)^{2}$$
$37$ $$(T^{2} - 2 T + 4)^{2}$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$T^{4} + 48T^{2} + 2304$$
$53$ $$T^{4} - 81T^{2} + 6561$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 2 T + 4)^{2}$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 12 T + 48)^{2}$$
$79$ $$(T^{2} - T + 1)^{2}$$
$83$ $$(T^{2} - 75)^{2}$$
$89$ $$T^{4} + 108 T^{2} + 11664$$
$97$ $$(T^{2} + 27)^{2}$$