# Properties

 Label 294.3.h.c 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{3} + 2 \beta_{2} q^{4} + \beta_1 q^{5} + (\beta_{3} + 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-2*b3 + b2 + 2*b1) * q^3 + 2*b2 * q^4 + 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} + \beta_1 q^{5} + (\beta_{3} + 4) q^{6} + 2 \beta_{3} q^{8} + ( - 7 \beta_{2} + 4 \beta_1 + 7) q^{9} + 2 \beta_{2} q^{10} + (7 \beta_{3} - 7 \beta_1) q^{11} + (2 \beta_{2} + 4 \beta_1 - 2) q^{12} + 20 q^{13} + (\beta_{3} + 4) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 15 \beta_{3} + 15 \beta_1) q^{17} + ( - 7 \beta_{3} + 8 \beta_{2} + 7 \beta_1) q^{18} + (32 \beta_{2} - 32) q^{19} + 2 \beta_{3} q^{20} - 14 q^{22} - 15 \beta_1 q^{23} + (2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{24} - 23 \beta_{2} q^{25} + 20 \beta_1 q^{26} + ( - 10 \beta_{3} + 23) q^{27} - 16 \beta_{3} q^{29} + (2 \beta_{2} + 4 \beta_1 - 2) q^{30} + 14 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (28 \beta_{2} - 7 \beta_1 - 28) q^{33} + 30 q^{34} + (8 \beta_{3} + 14) q^{36} + (48 \beta_{2} - 48) q^{37} + (32 \beta_{3} - 32 \beta_1) q^{38} + ( - 40 \beta_{3} + 20 \beta_{2} + 40 \beta_1) q^{39} + (4 \beta_{2} - 4) q^{40} - 39 \beta_{3} q^{41} + 4 q^{43} - 14 \beta_1 q^{44} + ( - 7 \beta_{3} + 8 \beta_{2} + 7 \beta_1) q^{45} - 30 \beta_{2} q^{46} - 44 \beta_1 q^{47} + (8 \beta_{3} - 4) q^{48} - 23 \beta_{3} q^{50} + ( - 60 \beta_{2} + 15 \beta_1 + 60) q^{51} + 40 \beta_{2} q^{52} + (52 \beta_{3} - 52 \beta_1) q^{53} + ( - 20 \beta_{2} + 23 \beta_1 + 20) q^{54} - 14 q^{55} + (64 \beta_{3} - 32) q^{57} + ( - 32 \beta_{2} + 32) q^{58} + ( - 8 \beta_{3} + 8 \beta_1) q^{59} + (2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{60} + ( - 20 \beta_{2} + 20) q^{61} + 14 \beta_{3} q^{62} - 8 q^{64} + 20 \beta_1 q^{65} + (28 \beta_{3} - 14 \beta_{2} - 28 \beta_1) q^{66} + 68 \beta_{2} q^{67} + 30 \beta_1 q^{68} + ( - 15 \beta_{3} - 60) q^{69} - 49 \beta_{3} q^{71} + (16 \beta_{2} + 14 \beta_1 - 16) q^{72} - 64 \beta_{2} q^{73} + (48 \beta_{3} - 48 \beta_1) q^{74} + ( - 23 \beta_{2} - 46 \beta_1 + 23) q^{75} - 64 q^{76} + (20 \beta_{3} + 80) q^{78} + (16 \beta_{2} - 16) q^{79} + (4 \beta_{3} - 4 \beta_1) q^{80} + ( - 56 \beta_{3} - 17 \beta_{2} + 56 \beta_1) q^{81} + ( - 78 \beta_{2} + 78) q^{82} + 72 \beta_{3} q^{83} + 30 q^{85} + 4 \beta_1 q^{86} + ( - 16 \beta_{3} - 64 \beta_{2} + 16 \beta_1) q^{87} - 28 \beta_{2} q^{88} - 7 \beta_1 q^{89} + (8 \beta_{3} + 14) q^{90} - 30 \beta_{3} q^{92} + (14 \beta_{2} + 28 \beta_1 - 14) q^{93} - 88 \beta_{2} q^{94} + (32 \beta_{3} - 32 \beta_1) q^{95} + (16 \beta_{2} - 4 \beta_1 - 16) q^{96} - 152 q^{97} + (49 \beta_{3} - 56) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-2*b3 + b2 + 2*b1) * q^3 + 2*b2 * q^4 + b1 * q^5 + (b3 + 4) * q^6 + 2*b3 * q^8 + (-7*b2 + 4*b1 + 7) * q^9 + 2*b2 * q^10 + (7*b3 - 7*b1) * q^11 + (2*b2 + 4*b1 - 2) * q^12 + 20 * q^13 + (b3 + 4) * q^15 + (4*b2 - 4) * q^16 + (-15*b3 + 15*b1) * q^17 + (-7*b3 + 8*b2 + 7*b1) * q^18 + (32*b2 - 32) * q^19 + 2*b3 * q^20 - 14 * q^22 - 15*b1 * q^23 + (2*b3 + 8*b2 - 2*b1) * q^24 - 23*b2 * q^25 + 20*b1 * q^26 + (-10*b3 + 23) * q^27 - 16*b3 * q^29 + (2*b2 + 4*b1 - 2) * q^30 + 14*b2 * q^31 + (4*b3 - 4*b1) * q^32 + (28*b2 - 7*b1 - 28) * q^33 + 30 * q^34 + (8*b3 + 14) * q^36 + (48*b2 - 48) * q^37 + (32*b3 - 32*b1) * q^38 + (-40*b3 + 20*b2 + 40*b1) * q^39 + (4*b2 - 4) * q^40 - 39*b3 * q^41 + 4 * q^43 - 14*b1 * q^44 + (-7*b3 + 8*b2 + 7*b1) * q^45 - 30*b2 * q^46 - 44*b1 * q^47 + (8*b3 - 4) * q^48 - 23*b3 * q^50 + (-60*b2 + 15*b1 + 60) * q^51 + 40*b2 * q^52 + (52*b3 - 52*b1) * q^53 + (-20*b2 + 23*b1 + 20) * q^54 - 14 * q^55 + (64*b3 - 32) * q^57 + (-32*b2 + 32) * q^58 + (-8*b3 + 8*b1) * q^59 + (2*b3 + 8*b2 - 2*b1) * q^60 + (-20*b2 + 20) * q^61 + 14*b3 * q^62 - 8 * q^64 + 20*b1 * q^65 + (28*b3 - 14*b2 - 28*b1) * q^66 + 68*b2 * q^67 + 30*b1 * q^68 + (-15*b3 - 60) * q^69 - 49*b3 * q^71 + (16*b2 + 14*b1 - 16) * q^72 - 64*b2 * q^73 + (48*b3 - 48*b1) * q^74 + (-23*b2 - 46*b1 + 23) * q^75 - 64 * q^76 + (20*b3 + 80) * q^78 + (16*b2 - 16) * q^79 + (4*b3 - 4*b1) * q^80 + (-56*b3 - 17*b2 + 56*b1) * q^81 + (-78*b2 + 78) * q^82 + 72*b3 * q^83 + 30 * q^85 + 4*b1 * q^86 + (-16*b3 - 64*b2 + 16*b1) * q^87 - 28*b2 * q^88 - 7*b1 * q^89 + (8*b3 + 14) * q^90 - 30*b3 * q^92 + (14*b2 + 28*b1 - 14) * q^93 - 88*b2 * q^94 + (32*b3 - 32*b1) * q^95 + (16*b2 - 4*b1 - 16) * q^96 - 152 * q^97 + (49*b3 - 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{4} + 16 q^{6} + 14 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^4 + 16 * q^6 + 14 * q^9 $$4 q + 2 q^{3} + 4 q^{4} + 16 q^{6} + 14 q^{9} + 4 q^{10} - 4 q^{12} + 80 q^{13} + 16 q^{15} - 8 q^{16} + 16 q^{18} - 64 q^{19} - 56 q^{22} + 16 q^{24} - 46 q^{25} + 92 q^{27} - 4 q^{30} + 28 q^{31} - 56 q^{33} + 120 q^{34} + 56 q^{36} - 96 q^{37} + 40 q^{39} - 8 q^{40} + 16 q^{43} + 16 q^{45} - 60 q^{46} - 16 q^{48} + 120 q^{51} + 80 q^{52} + 40 q^{54} - 56 q^{55} - 128 q^{57} + 64 q^{58} + 16 q^{60} + 40 q^{61} - 32 q^{64} - 28 q^{66} + 136 q^{67} - 240 q^{69} - 32 q^{72} - 128 q^{73} + 46 q^{75} - 256 q^{76} + 320 q^{78} - 32 q^{79} - 34 q^{81} + 156 q^{82} + 120 q^{85} - 128 q^{87} - 56 q^{88} + 56 q^{90} - 28 q^{93} - 176 q^{94} - 32 q^{96} - 608 q^{97} - 224 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^4 + 16 * q^6 + 14 * q^9 + 4 * q^10 - 4 * q^12 + 80 * q^13 + 16 * q^15 - 8 * q^16 + 16 * q^18 - 64 * q^19 - 56 * q^22 + 16 * q^24 - 46 * q^25 + 92 * q^27 - 4 * q^30 + 28 * q^31 - 56 * q^33 + 120 * q^34 + 56 * q^36 - 96 * q^37 + 40 * q^39 - 8 * q^40 + 16 * q^43 + 16 * q^45 - 60 * q^46 - 16 * q^48 + 120 * q^51 + 80 * q^52 + 40 * q^54 - 56 * q^55 - 128 * q^57 + 64 * q^58 + 16 * q^60 + 40 * q^61 - 32 * q^64 - 28 * q^66 + 136 * q^67 - 240 * q^69 - 32 * q^72 - 128 * q^73 + 46 * q^75 - 256 * q^76 + 320 * q^78 - 32 * q^79 - 34 * q^81 + 156 * q^82 + 120 * q^85 - 128 * q^87 - 56 * q^88 + 56 * q^90 - 28 * q^93 - 176 * q^94 - 32 * q^96 - 608 * q^97 - 224 * q^99

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 −1.22474 + 0.707107i 4.00000 + 1.41421i 0 2.82843i −1.39898 + 8.89060i 1.00000 1.73205i
263.2 1.22474 0.707107i 2.94949 + 0.548188i 1.00000 1.73205i 1.22474 0.707107i 4.00000 1.41421i 0 2.82843i 8.39898 + 3.23375i 1.00000 1.73205i
275.1 −1.22474 0.707107i −1.94949 + 2.28024i 1.00000 + 1.73205i −1.22474 0.707107i 4.00000 1.41421i 0 2.82843i −1.39898 8.89060i 1.00000 + 1.73205i
275.2 1.22474 + 0.707107i 2.94949 0.548188i 1.00000 + 1.73205i 1.22474 + 0.707107i 4.00000 + 1.41421i 0 2.82843i 8.39898 3.23375i 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
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.c 4
3.b odd 2 1 inner 294.3.h.c 4
7.b odd 2 1 294.3.h.a 4
7.c even 3 1 294.3.b.a 2
7.c even 3 1 inner 294.3.h.c 4
7.d odd 6 1 294.3.b.d yes 2
7.d odd 6 1 294.3.h.a 4
21.c even 2 1 294.3.h.a 4
21.g even 6 1 294.3.b.d yes 2
21.g even 6 1 294.3.h.a 4
21.h odd 6 1 294.3.b.a 2
21.h odd 6 1 inner 294.3.h.c 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{13} - 20$$ T13 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4} - 2 T^{3} - 5 T^{2} - 18 T + 81$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 98T^{2} + 9604$$
$13$ $$(T - 20)^{4}$$
$17$ $$T^{4} - 450 T^{2} + 202500$$
$19$ $$(T^{2} + 32 T + 1024)^{2}$$
$23$ $$T^{4} - 450 T^{2} + 202500$$
$29$ $$(T^{2} + 512)^{2}$$
$31$ $$(T^{2} - 14 T + 196)^{2}$$
$37$ $$(T^{2} + 48 T + 2304)^{2}$$
$41$ $$(T^{2} + 3042)^{2}$$
$43$ $$(T - 4)^{4}$$
$47$ $$T^{4} - 3872 T^{2} + \cdots + 14992384$$
$53$ $$T^{4} - 5408 T^{2} + \cdots + 29246464$$
$59$ $$T^{4} - 128 T^{2} + 16384$$
$61$ $$(T^{2} - 20 T + 400)^{2}$$
$67$ $$(T^{2} - 68 T + 4624)^{2}$$
$71$ $$(T^{2} + 4802)^{2}$$
$73$ $$(T^{2} + 64 T + 4096)^{2}$$
$79$ $$(T^{2} + 16 T + 256)^{2}$$
$83$ $$(T^{2} + 10368)^{2}$$
$89$ $$T^{4} - 98T^{2} + 9604$$
$97$ $$(T + 152)^{4}$$