# Properties

 Label 294.3.h.b Level $294$ Weight $3$ Character orbit 294.h Analytic conductor $8.011$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.01091977219$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{3} + 2 \beta_{2} q^{4} + 6 \beta_1 q^{5} + ( - \beta_{3} - 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9}+O(q^{10})$$ q + b1 * q^2 + (2*b3 - b2 - 2*b1) * q^3 + 2*b2 * q^4 + 6*b1 * q^5 + (-b3 - 4) * q^6 + 2*b3 * q^8 + (-7*b2 + 4*b1 + 7) * q^9 $$q + \beta_1 q^{2} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{3} + 2 \beta_{2} q^{4} + 6 \beta_1 q^{5} + ( - \beta_{3} - 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9} + 12 \beta_{2} q^{10} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{12} + q^{13} + ( - 6 \beta_{3} - 24) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{3} + 6 \beta_1) q^{17} + ( - 7 \beta_{3} + 8 \beta_{2} + 7 \beta_1) q^{18} + (31 \beta_{2} - 31) q^{19} + 12 \beta_{3} q^{20} + 6 \beta_1 q^{23} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{24} + 47 \beta_{2} q^{25} + \beta_1 q^{26} + (10 \beta_{3} - 23) q^{27} + 12 \beta_{3} q^{29} + ( - 12 \beta_{2} - 24 \beta_1 + 12) q^{30} - 7 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + 12 q^{34} + (8 \beta_{3} + 14) q^{36} + ( - \beta_{2} + 1) q^{37} + (31 \beta_{3} - 31 \beta_1) q^{38} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{39} + (24 \beta_{2} - 24) q^{40} - 24 \beta_{3} q^{41} - 31 q^{43} + ( - 42 \beta_{3} + 48 \beta_{2} + 42 \beta_1) q^{45} + 12 \beta_{2} q^{46} + 30 \beta_1 q^{47} + ( - 8 \beta_{3} + 4) q^{48} + 47 \beta_{3} q^{50} + (24 \beta_{2} - 6 \beta_1 - 24) q^{51} + 2 \beta_{2} q^{52} + ( - 18 \beta_{3} + 18 \beta_1) q^{53} + (20 \beta_{2} - 23 \beta_1 - 20) q^{54} + ( - 62 \beta_{3} + 31) q^{57} + (24 \beta_{2} - 24) q^{58} + ( - 6 \beta_{3} + 6 \beta_1) q^{59} + ( - 12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{60} + ( - 50 \beta_{2} + 50) q^{61} - 7 \beta_{3} q^{62} - 8 q^{64} + 6 \beta_1 q^{65} - 65 \beta_{2} q^{67} + 12 \beta_1 q^{68} + ( - 6 \beta_{3} - 24) q^{69} - 42 \beta_{3} q^{71} + (16 \beta_{2} + 14 \beta_1 - 16) q^{72} - 97 \beta_{2} q^{73} + ( - \beta_{3} + \beta_1) q^{74} + ( - 47 \beta_{2} - 94 \beta_1 + 47) q^{75} - 62 q^{76} + ( - \beta_{3} - 4) q^{78} + ( - 103 \beta_{2} + 103) q^{79} + (24 \beta_{3} - 24 \beta_1) q^{80} + ( - 56 \beta_{3} - 17 \beta_{2} + 56 \beta_1) q^{81} + ( - 48 \beta_{2} + 48) q^{82} - 30 \beta_{3} q^{83} + 72 q^{85} - 31 \beta_1 q^{86} + ( - 12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{87} + 84 \beta_1 q^{89} + (48 \beta_{3} + 84) q^{90} + 12 \beta_{3} q^{92} + (7 \beta_{2} + 14 \beta_1 - 7) q^{93} + 60 \beta_{2} q^{94} + (186 \beta_{3} - 186 \beta_1) q^{95} + ( - 16 \beta_{2} + 4 \beta_1 + 16) q^{96} + 166 q^{97}+O(q^{100})$$ q + b1 * q^2 + (2*b3 - b2 - 2*b1) * q^3 + 2*b2 * q^4 + 6*b1 * q^5 + (-b3 - 4) * q^6 + 2*b3 * q^8 + (-7*b2 + 4*b1 + 7) * q^9 + 12*b2 * q^10 + (-2*b2 - 4*b1 + 2) * q^12 + q^13 + (-6*b3 - 24) * q^15 + (4*b2 - 4) * q^16 + (-6*b3 + 6*b1) * q^17 + (-7*b3 + 8*b2 + 7*b1) * q^18 + (31*b2 - 31) * q^19 + 12*b3 * q^20 + 6*b1 * q^23 + (-2*b3 - 8*b2 + 2*b1) * q^24 + 47*b2 * q^25 + b1 * q^26 + (10*b3 - 23) * q^27 + 12*b3 * q^29 + (-12*b2 - 24*b1 + 12) * q^30 - 7*b2 * q^31 + (4*b3 - 4*b1) * q^32 + 12 * q^34 + (8*b3 + 14) * q^36 + (-b2 + 1) * q^37 + (31*b3 - 31*b1) * q^38 + (2*b3 - b2 - 2*b1) * q^39 + (24*b2 - 24) * q^40 - 24*b3 * q^41 - 31 * q^43 + (-42*b3 + 48*b2 + 42*b1) * q^45 + 12*b2 * q^46 + 30*b1 * q^47 + (-8*b3 + 4) * q^48 + 47*b3 * q^50 + (24*b2 - 6*b1 - 24) * q^51 + 2*b2 * q^52 + (-18*b3 + 18*b1) * q^53 + (20*b2 - 23*b1 - 20) * q^54 + (-62*b3 + 31) * q^57 + (24*b2 - 24) * q^58 + (-6*b3 + 6*b1) * q^59 + (-12*b3 - 48*b2 + 12*b1) * q^60 + (-50*b2 + 50) * q^61 - 7*b3 * q^62 - 8 * q^64 + 6*b1 * q^65 - 65*b2 * q^67 + 12*b1 * q^68 + (-6*b3 - 24) * q^69 - 42*b3 * q^71 + (16*b2 + 14*b1 - 16) * q^72 - 97*b2 * q^73 + (-b3 + b1) * q^74 + (-47*b2 - 94*b1 + 47) * q^75 - 62 * q^76 + (-b3 - 4) * q^78 + (-103*b2 + 103) * q^79 + (24*b3 - 24*b1) * q^80 + (-56*b3 - 17*b2 + 56*b1) * q^81 + (-48*b2 + 48) * q^82 - 30*b3 * q^83 + 72 * q^85 - 31*b1 * q^86 + (-12*b3 - 48*b2 + 12*b1) * q^87 + 84*b1 * q^89 + (48*b3 + 84) * q^90 + 12*b3 * q^92 + (7*b2 + 14*b1 - 7) * q^93 + 60*b2 * q^94 + (186*b3 - 186*b1) * q^95 + (-16*b2 + 4*b1 + 16) * q^96 + 166 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{4} - 16 q^{6} + 14 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 4 * q^4 - 16 * q^6 + 14 * q^9 $$4 q - 2 q^{3} + 4 q^{4} - 16 q^{6} + 14 q^{9} + 24 q^{10} + 4 q^{12} + 4 q^{13} - 96 q^{15} - 8 q^{16} + 16 q^{18} - 62 q^{19} - 16 q^{24} + 94 q^{25} - 92 q^{27} + 24 q^{30} - 14 q^{31} + 48 q^{34} + 56 q^{36} + 2 q^{37} - 2 q^{39} - 48 q^{40} - 124 q^{43} + 96 q^{45} + 24 q^{46} + 16 q^{48} - 48 q^{51} + 4 q^{52} - 40 q^{54} + 124 q^{57} - 48 q^{58} - 96 q^{60} + 100 q^{61} - 32 q^{64} - 130 q^{67} - 96 q^{69} - 32 q^{72} - 194 q^{73} + 94 q^{75} - 248 q^{76} - 16 q^{78} + 206 q^{79} - 34 q^{81} + 96 q^{82} + 288 q^{85} - 96 q^{87} + 336 q^{90} - 14 q^{93} + 120 q^{94} + 32 q^{96} + 664 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 4 * q^4 - 16 * q^6 + 14 * q^9 + 24 * q^10 + 4 * q^12 + 4 * q^13 - 96 * q^15 - 8 * q^16 + 16 * q^18 - 62 * q^19 - 16 * q^24 + 94 * q^25 - 92 * q^27 + 24 * q^30 - 14 * q^31 + 48 * q^34 + 56 * q^36 + 2 * q^37 - 2 * q^39 - 48 * q^40 - 124 * q^43 + 96 * q^45 + 24 * q^46 + 16 * q^48 - 48 * q^51 + 4 * q^52 - 40 * q^54 + 124 * q^57 - 48 * q^58 - 96 * q^60 + 100 * q^61 - 32 * q^64 - 130 * q^67 - 96 * q^69 - 32 * q^72 - 194 * q^73 + 94 * q^75 - 248 * q^76 - 16 * q^78 + 206 * q^79 - 34 * q^81 + 96 * q^82 + 288 * q^85 - 96 * q^87 + 336 * q^90 - 14 * q^93 + 120 * q^94 + 32 * q^96 + 664 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 1.94949 + 2.28024i 1.00000 1.73205i −7.34847 + 4.24264i −4.00000 1.41421i 0 2.82843i −1.39898 + 8.89060i 6.00000 10.3923i
263.2 1.22474 0.707107i −2.94949 0.548188i 1.00000 1.73205i 7.34847 4.24264i −4.00000 + 1.41421i 0 2.82843i 8.39898 + 3.23375i 6.00000 10.3923i
275.1 −1.22474 0.707107i 1.94949 2.28024i 1.00000 + 1.73205i −7.34847 4.24264i −4.00000 + 1.41421i 0 2.82843i −1.39898 8.89060i 6.00000 + 10.3923i
275.2 1.22474 + 0.707107i −2.94949 + 0.548188i 1.00000 + 1.73205i 7.34847 + 4.24264i −4.00000 1.41421i 0 2.82843i 8.39898 3.23375i 6.00000 + 10.3923i
 $$n$$: 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.c even 3 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 294.3.h.b 4
3.b odd 2 1 inner 294.3.h.b 4
7.b odd 2 1 42.3.h.a 4
7.c even 3 1 294.3.b.c 2
7.c even 3 1 inner 294.3.h.b 4
7.d odd 6 1 42.3.h.a 4
7.d odd 6 1 294.3.b.b 2
21.c even 2 1 42.3.h.a 4
21.g even 6 1 42.3.h.a 4
21.g even 6 1 294.3.b.b 2
21.h odd 6 1 294.3.b.c 2
21.h odd 6 1 inner 294.3.h.b 4
28.d even 2 1 336.3.bn.c 4
28.f even 6 1 336.3.bn.c 4
84.h odd 2 1 336.3.bn.c 4
84.j odd 6 1 336.3.bn.c 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 72T_{5}^{2} + 5184$$ T5^4 - 72*T5^2 + 5184 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4} + 2 T^{3} - 5 T^{2} + 18 T + 81$$
$5$ $$T^{4} - 72T^{2} + 5184$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} - 72T^{2} + 5184$$
$19$ $$(T^{2} + 31 T + 961)^{2}$$
$23$ $$T^{4} - 72T^{2} + 5184$$
$29$ $$(T^{2} + 288)^{2}$$
$31$ $$(T^{2} + 7 T + 49)^{2}$$
$37$ $$(T^{2} - T + 1)^{2}$$
$41$ $$(T^{2} + 1152)^{2}$$
$43$ $$(T + 31)^{4}$$
$47$ $$T^{4} - 1800 T^{2} + \cdots + 3240000$$
$53$ $$T^{4} - 648 T^{2} + 419904$$
$59$ $$T^{4} - 72T^{2} + 5184$$
$61$ $$(T^{2} - 50 T + 2500)^{2}$$
$67$ $$(T^{2} + 65 T + 4225)^{2}$$
$71$ $$(T^{2} + 3528)^{2}$$
$73$ $$(T^{2} + 97 T + 9409)^{2}$$
$79$ $$(T^{2} - 103 T + 10609)^{2}$$
$83$ $$(T^{2} + 1800)^{2}$$
$89$ $$T^{4} - 14112 T^{2} + \cdots + 199148544$$
$97$ $$(T - 166)^{4}$$