# Properties

 Label 294.3.b.b Level $294$ Weight $3$ Character orbit 294.b Analytic conductor $8.011$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 294.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01091977219$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} - 6 \beta q^{5} + ( - \beta + 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} +O(q^{10})$$ q + b * q^2 + (-2*b - 1) * q^3 - 2 * q^4 - 6*b * q^5 + (-b + 4) * q^6 - 2*b * q^8 + (4*b - 7) * q^9 $$q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} - 6 \beta q^{5} + ( - \beta + 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} + 12 q^{10} + (4 \beta + 2) q^{12} - q^{13} + (6 \beta - 24) q^{15} + 4 q^{16} + 6 \beta q^{17} + ( - 7 \beta - 8) q^{18} - 31 q^{19} + 12 \beta q^{20} + 6 \beta q^{23} + (2 \beta - 8) q^{24} - 47 q^{25} - \beta q^{26} + (10 \beta + 23) q^{27} - 12 \beta q^{29} + ( - 24 \beta - 12) q^{30} - 7 q^{31} + 4 \beta q^{32} - 12 q^{34} + ( - 8 \beta + 14) q^{36} - q^{37} - 31 \beta q^{38} + (2 \beta + 1) q^{39} - 24 q^{40} - 24 \beta q^{41} - 31 q^{43} + (42 \beta + 48) q^{45} - 12 q^{46} - 30 \beta q^{47} + ( - 8 \beta - 4) q^{48} - 47 \beta q^{50} + ( - 6 \beta + 24) q^{51} + 2 q^{52} - 18 \beta q^{53} + (23 \beta - 20) q^{54} + (62 \beta + 31) q^{57} + 24 q^{58} + 6 \beta q^{59} + ( - 12 \beta + 48) q^{60} + 50 q^{61} - 7 \beta q^{62} - 8 q^{64} + 6 \beta q^{65} + 65 q^{67} - 12 \beta q^{68} + ( - 6 \beta + 24) q^{69} + 42 \beta q^{71} + (14 \beta + 16) q^{72} - 97 q^{73} - \beta q^{74} + (94 \beta + 47) q^{75} + 62 q^{76} + (\beta - 4) q^{78} - 103 q^{79} - 24 \beta q^{80} + ( - 56 \beta + 17) q^{81} + 48 q^{82} - 30 \beta q^{83} + 72 q^{85} - 31 \beta q^{86} + (12 \beta - 48) q^{87} - 84 \beta q^{89} + (48 \beta - 84) q^{90} - 12 \beta q^{92} + (14 \beta + 7) q^{93} + 60 q^{94} + 186 \beta q^{95} + ( - 4 \beta + 16) q^{96} - 166 q^{97} +O(q^{100})$$ q + b * q^2 + (-2*b - 1) * q^3 - 2 * q^4 - 6*b * q^5 + (-b + 4) * q^6 - 2*b * q^8 + (4*b - 7) * q^9 + 12 * q^10 + (4*b + 2) * q^12 - q^13 + (6*b - 24) * q^15 + 4 * q^16 + 6*b * q^17 + (-7*b - 8) * q^18 - 31 * q^19 + 12*b * q^20 + 6*b * q^23 + (2*b - 8) * q^24 - 47 * q^25 - b * q^26 + (10*b + 23) * q^27 - 12*b * q^29 + (-24*b - 12) * q^30 - 7 * q^31 + 4*b * q^32 - 12 * q^34 + (-8*b + 14) * q^36 - q^37 - 31*b * q^38 + (2*b + 1) * q^39 - 24 * q^40 - 24*b * q^41 - 31 * q^43 + (42*b + 48) * q^45 - 12 * q^46 - 30*b * q^47 + (-8*b - 4) * q^48 - 47*b * q^50 + (-6*b + 24) * q^51 + 2 * q^52 - 18*b * q^53 + (23*b - 20) * q^54 + (62*b + 31) * q^57 + 24 * q^58 + 6*b * q^59 + (-12*b + 48) * q^60 + 50 * q^61 - 7*b * q^62 - 8 * q^64 + 6*b * q^65 + 65 * q^67 - 12*b * q^68 + (-6*b + 24) * q^69 + 42*b * q^71 + (14*b + 16) * q^72 - 97 * q^73 - b * q^74 + (94*b + 47) * q^75 + 62 * q^76 + (b - 4) * q^78 - 103 * q^79 - 24*b * q^80 + (-56*b + 17) * q^81 + 48 * q^82 - 30*b * q^83 + 72 * q^85 - 31*b * q^86 + (12*b - 48) * q^87 - 84*b * q^89 + (48*b - 84) * q^90 - 12*b * q^92 + (14*b + 7) * q^93 + 60 * q^94 + 186*b * q^95 + (-4*b + 16) * q^96 - 166 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^4 + 8 * q^6 - 14 * q^9 $$2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{9} + 24 q^{10} + 4 q^{12} - 2 q^{13} - 48 q^{15} + 8 q^{16} - 16 q^{18} - 62 q^{19} - 16 q^{24} - 94 q^{25} + 46 q^{27} - 24 q^{30} - 14 q^{31} - 24 q^{34} + 28 q^{36} - 2 q^{37} + 2 q^{39} - 48 q^{40} - 62 q^{43} + 96 q^{45} - 24 q^{46} - 8 q^{48} + 48 q^{51} + 4 q^{52} - 40 q^{54} + 62 q^{57} + 48 q^{58} + 96 q^{60} + 100 q^{61} - 16 q^{64} + 130 q^{67} + 48 q^{69} + 32 q^{72} - 194 q^{73} + 94 q^{75} + 124 q^{76} - 8 q^{78} - 206 q^{79} + 34 q^{81} + 96 q^{82} + 144 q^{85} - 96 q^{87} - 168 q^{90} + 14 q^{93} + 120 q^{94} + 32 q^{96} - 332 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^4 + 8 * q^6 - 14 * q^9 + 24 * q^10 + 4 * q^12 - 2 * q^13 - 48 * q^15 + 8 * q^16 - 16 * q^18 - 62 * q^19 - 16 * q^24 - 94 * q^25 + 46 * q^27 - 24 * q^30 - 14 * q^31 - 24 * q^34 + 28 * q^36 - 2 * q^37 + 2 * q^39 - 48 * q^40 - 62 * q^43 + 96 * q^45 - 24 * q^46 - 8 * q^48 + 48 * q^51 + 4 * q^52 - 40 * q^54 + 62 * q^57 + 48 * q^58 + 96 * q^60 + 100 * q^61 - 16 * q^64 + 130 * q^67 + 48 * q^69 + 32 * q^72 - 194 * q^73 + 94 * q^75 + 124 * q^76 - 8 * q^78 - 206 * q^79 + 34 * q^81 + 96 * q^82 + 144 * q^85 - 96 * q^87 - 168 * q^90 + 14 * q^93 + 120 * q^94 + 32 * q^96 - 332 * q^97

## Character values

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

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 − 1.41421i 1.41421i
1.41421i −1.00000 + 2.82843i −2.00000 8.48528i 4.00000 + 1.41421i 0 2.82843i −7.00000 5.65685i 12.0000
197.2 1.41421i −1.00000 2.82843i −2.00000 8.48528i 4.00000 1.41421i 0 2.82843i −7.00000 + 5.65685i 12.0000
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.3.b.b 2
3.b odd 2 1 inner 294.3.b.b 2
7.b odd 2 1 294.3.b.c 2
7.c even 3 2 42.3.h.a 4
7.d odd 6 2 294.3.h.b 4
21.c even 2 1 294.3.b.c 2
21.g even 6 2 294.3.h.b 4
21.h odd 6 2 42.3.h.a 4
28.g odd 6 2 336.3.bn.c 4
84.n even 6 2 336.3.bn.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.a 4 7.c even 3 2
42.3.h.a 4 21.h odd 6 2
294.3.b.b 2 1.a even 1 1 trivial
294.3.b.b 2 3.b odd 2 1 inner
294.3.b.c 2 7.b odd 2 1
294.3.b.c 2 21.c even 2 1
294.3.h.b 4 7.d odd 6 2
294.3.h.b 4 21.g even 6 2
336.3.bn.c 4 28.g odd 6 2
336.3.bn.c 4 84.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 72$$ T5^2 + 72 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2} + 2T + 9$$
$5$ $$T^{2} + 72$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 72$$
$19$ $$(T + 31)^{2}$$
$23$ $$T^{2} + 72$$
$29$ $$T^{2} + 288$$
$31$ $$(T + 7)^{2}$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 1152$$
$43$ $$(T + 31)^{2}$$
$47$ $$T^{2} + 1800$$
$53$ $$T^{2} + 648$$
$59$ $$T^{2} + 72$$
$61$ $$(T - 50)^{2}$$
$67$ $$(T - 65)^{2}$$
$71$ $$T^{2} + 3528$$
$73$ $$(T + 97)^{2}$$
$79$ $$(T + 103)^{2}$$
$83$ $$T^{2} + 1800$$
$89$ $$T^{2} + 14112$$
$97$ $$(T + 166)^{2}$$