# Properties

 Label 294.2.f.b Level $294$ Weight $2$ Character orbit 294.f Analytic conductor $2.348$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} - \beta_{5}) q^{6} + \beta_{4} q^{8} + ( - 3 \beta_{4} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (b4 - b1) * q^2 - b6 * q^3 + (-b2 + 1) * q^4 + (b7 - b6 + b3) * q^5 + (b7 - b5) * q^6 + b4 * q^8 + (-3*b4 + 3*b1) * q^9 $$q + (\beta_{4} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} - \beta_{5}) q^{6} + \beta_{4} q^{8} + ( - 3 \beta_{4} + 3 \beta_1) q^{9} + ( - \beta_{6} - \beta_{5}) q^{10} + ( - \beta_{6} + \beta_{3}) q^{12} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{13} + ( - 3 \beta_{4} - 3) q^{15} - \beta_{2} q^{16} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{17} + (3 \beta_{2} - 3) q^{18} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{19} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{20} + (6 \beta_{4} - 6 \beta_1) q^{23} - \beta_{5} q^{24} + (\beta_{2} - 1) q^{25} + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{26} + ( - 3 \beta_{7} + 3 \beta_{5}) q^{27} + 6 \beta_{4} q^{29} + ( - 3 \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{30} + \beta_1 q^{32} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{34} - 3 \beta_{4} q^{36} + 2 \beta_{2} q^{37} + (\beta_{6} - \beta_{5}) q^{38} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{39} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{40} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{41} + 4 q^{43} + (3 \beta_{6} + 3 \beta_{5}) q^{45} + ( - 6 \beta_{2} + 6) q^{46} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{47} + \beta_{3} q^{48} - \beta_{4} q^{50} + ( - 6 \beta_{4} - 6 \beta_{2} + 6 \beta_1) q^{51} + (\beta_{6} + \beta_{5}) q^{52} + 6 \beta_1 q^{53} + (3 \beta_{6} - 3 \beta_{3}) q^{54} + ( - 3 \beta_{4} + 3) q^{57} - 6 \beta_{2} q^{58} + (5 \beta_{6} - 5 \beta_{5}) q^{59} + (3 \beta_{2} - 3 \beta_1 - 3) q^{60} + (5 \beta_{7} + 5 \beta_{6} - 5 \beta_{3}) q^{61} - q^{64} + (6 \beta_{4} - 6 \beta_1) q^{65} + (8 \beta_{2} - 8) q^{67} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{68} + (6 \beta_{7} - 6 \beta_{5}) q^{69} + 3 \beta_{2} q^{72} + (4 \beta_{6} + 4 \beta_{5}) q^{73} - 2 \beta_1 q^{74} + (\beta_{6} - \beta_{3}) q^{75} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{76} + (3 \beta_{4} + 3) q^{78} + 10 \beta_{2} q^{79} + (\beta_{6} - \beta_{5}) q^{80} + ( - 9 \beta_{2} + 9) q^{81} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3}) q^{82} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{83} - 12 q^{85} + (4 \beta_{4} - 4 \beta_1) q^{86} - 6 \beta_{5} q^{87} + ( - 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3}) q^{90} + 6 \beta_{4} q^{92} + (2 \beta_{6} + 2 \beta_{5}) q^{94} - 6 \beta_1 q^{95} - \beta_{7} q^{96} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{97}+O(q^{100})$$ q + (b4 - b1) * q^2 - b6 * q^3 + (-b2 + 1) * q^4 + (b7 - b6 + b3) * q^5 + (b7 - b5) * q^6 + b4 * q^8 + (-3*b4 + 3*b1) * q^9 + (-b6 - b5) * q^10 + (-b6 + b3) * q^12 + (-b7 + b5 + b3) * q^13 + (-3*b4 - 3) * q^15 - b2 * q^16 + (-2*b6 + 2*b5) * q^17 + (3*b2 - 3) * q^18 + (-b7 - b6 + b3) * q^19 + (b7 - b5 + b3) * q^20 + (6*b4 - 6*b1) * q^23 - b5 * q^24 + (b2 - 1) * q^25 + (-b7 + b6 - b3) * q^26 + (-3*b7 + 3*b5) * q^27 + 6*b4 * q^29 + (-3*b4 + 3*b2 + 3*b1) * q^30 + b1 * q^32 + (2*b7 - 2*b5 - 2*b3) * q^34 - 3*b4 * q^36 + 2*b2 * q^37 + (b6 - b5) * q^38 + (-3*b2 - 3*b1 + 3) * q^39 + (-b7 - b6 + b3) * q^40 + (-2*b7 + 2*b5 - 2*b3) * q^41 + 4 * q^43 + (3*b6 + 3*b5) * q^45 + (-6*b2 + 6) * q^46 + (-2*b7 + 2*b6 - 2*b3) * q^47 + b3 * q^48 - b4 * q^50 + (-6*b4 - 6*b2 + 6*b1) * q^51 + (b6 + b5) * q^52 + 6*b1 * q^53 + (3*b6 - 3*b3) * q^54 + (-3*b4 + 3) * q^57 - 6*b2 * q^58 + (5*b6 - 5*b5) * q^59 + (3*b2 - 3*b1 - 3) * q^60 + (5*b7 + 5*b6 - 5*b3) * q^61 - q^64 + (6*b4 - 6*b1) * q^65 + (8*b2 - 8) * q^67 + (2*b7 - 2*b6 + 2*b3) * q^68 + (6*b7 - 6*b5) * q^69 + 3*b2 * q^72 + (4*b6 + 4*b5) * q^73 - 2*b1 * q^74 + (b6 - b3) * q^75 + (-b7 + b5 + b3) * q^76 + (3*b4 + 3) * q^78 + 10*b2 * q^79 + (b6 - b5) * q^80 + (-9*b2 + 9) * q^81 + (2*b7 + 2*b6 - 2*b3) * q^82 + (b7 - b5 + b3) * q^83 - 12 * q^85 + (4*b4 - 4*b1) * q^86 - 6*b5 * q^87 + (-3*b7 + 3*b5 - 3*b3) * q^90 + 6*b4 * q^92 + (2*b6 + 2*b5) * q^94 - 6*b1 * q^95 - b7 * q^96 + (2*b7 - 2*b5 - 2*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4}+O(q^{10})$$ 8 * q + 4 * q^4 $$8 q + 4 q^{4} - 24 q^{15} - 4 q^{16} - 12 q^{18} - 4 q^{25} + 12 q^{30} + 8 q^{37} + 12 q^{39} + 32 q^{43} + 24 q^{46} - 24 q^{51} + 24 q^{57} - 24 q^{58} - 12 q^{60} - 8 q^{64} - 32 q^{67} + 12 q^{72} + 24 q^{78} + 40 q^{79} + 36 q^{81} - 96 q^{85}+O(q^{100})$$ 8 * q + 4 * q^4 - 24 * q^15 - 4 * q^16 - 12 * q^18 - 4 * q^25 + 12 * q^30 + 8 * q^37 + 12 * q^39 + 32 * q^43 + 24 * q^46 - 24 * q^51 + 24 * q^57 - 24 * q^58 - 12 * q^60 - 8 * q^64 - 32 * q^67 + 12 * q^72 + 24 * q^78 + 40 * q^79 + 36 * q^81 - 96 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{5} + \zeta_{24}$$ v^5 + v $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{5}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}^{3}$$ v^7 + v^3 $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{5} + 2\zeta_{24}$$ -v^5 + 2*v $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{3}$$ -v^7 + 2*v^3
 $$\zeta_{24}$$ $$=$$ $$( \beta_{6} + \beta_{3} ) / 3$$ (b6 + b3) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{5} ) / 3$$ (b7 + b5) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{6} + 2\beta_{3} ) / 3$$ (-b6 + 2*b3) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{4}$$ b4 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} ) / 3$$ (-b7 + 2*b5) / 3

## Character values

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

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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
 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
−0.866025 0.500000i −1.67303 0.448288i 0.500000 + 0.866025i 1.22474 2.12132i 1.22474 + 1.22474i 0 1.00000i 2.59808 + 1.50000i −2.12132 + 1.22474i
215.2 −0.866025 0.500000i 1.67303 + 0.448288i 0.500000 + 0.866025i −1.22474 + 2.12132i −1.22474 1.22474i 0 1.00000i 2.59808 + 1.50000i 2.12132 1.22474i
215.3 0.866025 + 0.500000i −0.448288 + 1.67303i 0.500000 + 0.866025i −1.22474 + 2.12132i −1.22474 + 1.22474i 0 1.00000i −2.59808 1.50000i −2.12132 + 1.22474i
215.4 0.866025 + 0.500000i 0.448288 1.67303i 0.500000 + 0.866025i 1.22474 2.12132i 1.22474 1.22474i 0 1.00000i −2.59808 1.50000i 2.12132 1.22474i
227.1 −0.866025 + 0.500000i −1.67303 + 0.448288i 0.500000 0.866025i 1.22474 + 2.12132i 1.22474 1.22474i 0 1.00000i 2.59808 1.50000i −2.12132 1.22474i
227.2 −0.866025 + 0.500000i 1.67303 0.448288i 0.500000 0.866025i −1.22474 2.12132i −1.22474 + 1.22474i 0 1.00000i 2.59808 1.50000i 2.12132 + 1.22474i
227.3 0.866025 0.500000i −0.448288 1.67303i 0.500000 0.866025i −1.22474 2.12132i −1.22474 1.22474i 0 1.00000i −2.59808 + 1.50000i −2.12132 1.22474i
227.4 0.866025 0.500000i 0.448288 + 1.67303i 0.500000 0.866025i 1.22474 + 2.12132i 1.22474 + 1.22474i 0 1.00000i −2.59808 + 1.50000i 2.12132 + 1.22474i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 227.4 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 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.b 8
3.b odd 2 1 inner 294.2.f.b 8
7.b odd 2 1 inner 294.2.f.b 8
7.c even 3 1 42.2.d.a 4
7.c even 3 1 inner 294.2.f.b 8
7.d odd 6 1 42.2.d.a 4
7.d odd 6 1 inner 294.2.f.b 8
21.c even 2 1 inner 294.2.f.b 8
21.g even 6 1 42.2.d.a 4
21.g even 6 1 inner 294.2.f.b 8
21.h odd 6 1 42.2.d.a 4
21.h odd 6 1 inner 294.2.f.b 8
28.f even 6 1 336.2.k.b 4
28.g odd 6 1 336.2.k.b 4
35.i odd 6 1 1050.2.b.b 4
35.j even 6 1 1050.2.b.b 4
35.k even 12 1 1050.2.d.b 4
35.k even 12 1 1050.2.d.e 4
35.l odd 12 1 1050.2.d.b 4
35.l odd 12 1 1050.2.d.e 4
56.j odd 6 1 1344.2.k.c 4
56.k odd 6 1 1344.2.k.d 4
56.m even 6 1 1344.2.k.d 4
56.p even 6 1 1344.2.k.c 4
63.g even 3 1 1134.2.m.g 8
63.h even 3 1 1134.2.m.g 8
63.i even 6 1 1134.2.m.g 8
63.j odd 6 1 1134.2.m.g 8
63.k odd 6 1 1134.2.m.g 8
63.n odd 6 1 1134.2.m.g 8
63.s even 6 1 1134.2.m.g 8
63.t odd 6 1 1134.2.m.g 8
84.j odd 6 1 336.2.k.b 4
84.n even 6 1 336.2.k.b 4
105.o odd 6 1 1050.2.b.b 4
105.p even 6 1 1050.2.b.b 4
105.w odd 12 1 1050.2.d.b 4
105.w odd 12 1 1050.2.d.e 4
105.x even 12 1 1050.2.d.b 4
105.x even 12 1 1050.2.d.e 4
168.s odd 6 1 1344.2.k.c 4
168.v even 6 1 1344.2.k.d 4
168.ba even 6 1 1344.2.k.c 4
168.be odd 6 1 1344.2.k.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 7.c even 3 1
42.2.d.a 4 7.d odd 6 1
42.2.d.a 4 21.g even 6 1
42.2.d.a 4 21.h odd 6 1
294.2.f.b 8 1.a even 1 1 trivial
294.2.f.b 8 3.b odd 2 1 inner
294.2.f.b 8 7.b odd 2 1 inner
294.2.f.b 8 7.c even 3 1 inner
294.2.f.b 8 7.d odd 6 1 inner
294.2.f.b 8 21.c even 2 1 inner
294.2.f.b 8 21.g even 6 1 inner
294.2.f.b 8 21.h odd 6 1 inner
336.2.k.b 4 28.f even 6 1
336.2.k.b 4 28.g odd 6 1
336.2.k.b 4 84.j odd 6 1
336.2.k.b 4 84.n even 6 1
1050.2.b.b 4 35.i odd 6 1
1050.2.b.b 4 35.j even 6 1
1050.2.b.b 4 105.o odd 6 1
1050.2.b.b 4 105.p even 6 1
1050.2.d.b 4 35.k even 12 1
1050.2.d.b 4 35.l odd 12 1
1050.2.d.b 4 105.w odd 12 1
1050.2.d.b 4 105.x even 12 1
1050.2.d.e 4 35.k even 12 1
1050.2.d.e 4 35.l odd 12 1
1050.2.d.e 4 105.w odd 12 1
1050.2.d.e 4 105.x even 12 1
1134.2.m.g 8 63.g even 3 1
1134.2.m.g 8 63.h even 3 1
1134.2.m.g 8 63.i even 6 1
1134.2.m.g 8 63.j odd 6 1
1134.2.m.g 8 63.k odd 6 1
1134.2.m.g 8 63.n odd 6 1
1134.2.m.g 8 63.s even 6 1
1134.2.m.g 8 63.t odd 6 1
1344.2.k.c 4 56.j odd 6 1
1344.2.k.c 4 56.p even 6 1
1344.2.k.c 4 168.s odd 6 1
1344.2.k.c 4 168.ba even 6 1
1344.2.k.d 4 56.k odd 6 1
1344.2.k.d 4 56.m even 6 1
1344.2.k.d 4 168.v even 6 1
1344.2.k.d 4 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 6T_{5}^{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8} - 9T^{4} + 81$$
$5$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$(T^{2} + 6)^{4}$$
$17$ $$(T^{4} + 24 T^{2} + 576)^{2}$$
$19$ $$(T^{4} - 6 T^{2} + 36)^{2}$$
$23$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$29$ $$(T^{2} + 36)^{4}$$
$31$ $$T^{8}$$
$37$ $$(T^{2} - 2 T + 4)^{4}$$
$41$ $$(T^{2} - 24)^{4}$$
$43$ $$(T - 4)^{8}$$
$47$ $$(T^{4} + 24 T^{2} + 576)^{2}$$
$53$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$59$ $$(T^{4} + 150 T^{2} + 22500)^{2}$$
$61$ $$(T^{4} - 150 T^{2} + 22500)^{2}$$
$67$ $$(T^{2} + 8 T + 64)^{4}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} - 96 T^{2} + 9216)^{2}$$
$79$ $$(T^{2} - 10 T + 100)^{4}$$
$83$ $$(T^{2} - 6)^{4}$$
$89$ $$T^{8}$$
$97$ $$(T^{2} + 24)^{4}$$