# Properties

 Label 294.3.h.e Level $294$ Weight $3$ Character orbit 294.h Analytic conductor $8.011$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ 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: $$8$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + (2 \beta_{7} - 2 \beta_{4} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 4 \beta_{3} + 4 \beta_1) q^{5} + (\beta_{7} - 4 \beta_{3}) q^{6} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8} + (4 \beta_{5} - 7 \beta_{2}) q^{9}+O(q^{10})$$ q + b5 * q^2 + (2*b7 - 2*b4 + b1) * q^3 + (-2*b2 + 2) * q^4 + (-4*b3 + 4*b1) * q^5 + (b7 - 4*b3) * q^6 + (-2*b6 + 2*b5) * q^8 + (4*b5 - 7*b2) * q^9 $$q + \beta_{5} q^{2} + (2 \beta_{7} - 2 \beta_{4} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 4 \beta_{3} + 4 \beta_1) q^{5} + (\beta_{7} - 4 \beta_{3}) q^{6} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8} + (4 \beta_{5} - 7 \beta_{2}) q^{9} + (4 \beta_{7} - 4 \beta_{4}) q^{10} - 7 \beta_{6} q^{11} + ( - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{12} - 9 \beta_{7} q^{13} + ( - 8 \beta_{6} + 8 \beta_{5} + 4) q^{15} - 4 \beta_{2} q^{16} + 32 \beta_1 q^{17} + ( - 7 \beta_{6} - 8 \beta_{2} + 8) q^{18} + 20 \beta_{4} q^{19} - 8 \beta_{3} q^{20} - 14 q^{22} - \beta_{5} q^{23} + (2 \beta_{7} - 2 \beta_{4} - 8 \beta_1) q^{24} + (9 \beta_{2} - 9) q^{25} + (18 \beta_{3} - 18 \beta_1) q^{26} + ( - 10 \beta_{7} - 23 \beta_{3}) q^{27} + ( - 40 \beta_{6} + 40 \beta_{5}) q^{29} + (4 \beta_{5} - 16 \beta_{2}) q^{30} - 4 \beta_{6} q^{32} + ( - 7 \beta_{4} + 28 \beta_{3} - 28 \beta_1) q^{33} + 32 \beta_{7} q^{34} + ( - 8 \beta_{6} + 8 \beta_{5} - 14) q^{36} + 50 \beta_{2} q^{37} + 40 \beta_1 q^{38} + ( - 9 \beta_{6} + 36 \beta_{2} - 36) q^{39} - 8 \beta_{4} q^{40} + 16 \beta_{3} q^{41} - 52 q^{43} - 14 \beta_{5} q^{44} + (16 \beta_{7} - 16 \beta_{4} - 28 \beta_1) q^{45} + (2 \beta_{2} - 2) q^{46} + ( - 34 \beta_{3} + 34 \beta_1) q^{47} + ( - 8 \beta_{7} - 4 \beta_{3}) q^{48} + (9 \beta_{6} - 9 \beta_{5}) q^{50} + (64 \beta_{5} + 32 \beta_{2}) q^{51} + ( - 18 \beta_{7} + 18 \beta_{4}) q^{52} + 32 \beta_{6} q^{53} + ( - 23 \beta_{4} + 20 \beta_{3} - 20 \beta_1) q^{54} - 28 \beta_{7} q^{55} + (20 \beta_{6} - 20 \beta_{5} + 80) q^{57} - 80 \beta_{2} q^{58} + 32 \beta_1 q^{59} + ( - 16 \beta_{6} - 8 \beta_{2} + 8) q^{60} - 23 \beta_{4} q^{61} - 8 q^{64} - 36 \beta_{5} q^{65} + ( - 28 \beta_{7} + 28 \beta_{4} - 14 \beta_1) q^{66} + ( - 12 \beta_{2} + 12) q^{67} + ( - 64 \beta_{3} + 64 \beta_1) q^{68} + ( - \beta_{7} + 4 \beta_{3}) q^{69} + (63 \beta_{6} - 63 \beta_{5}) q^{71} + ( - 14 \beta_{5} - 16 \beta_{2}) q^{72} + ( - 23 \beta_{7} + 23 \beta_{4}) q^{73} + 50 \beta_{6} q^{74} + (18 \beta_{4} + 9 \beta_{3} - 9 \beta_1) q^{75} + 40 \beta_{7} q^{76} + (36 \beta_{6} - 36 \beta_{5} - 18) q^{78} + 40 \beta_{2} q^{79} - 16 \beta_1 q^{80} + ( - 56 \beta_{6} + 17 \beta_{2} - 17) q^{81} + 16 \beta_{4} q^{82} + 62 \beta_{3} q^{83} + 128 q^{85} - 52 \beta_{5} q^{86} + (40 \beta_{7} - 40 \beta_{4} - 160 \beta_1) q^{87} + (28 \beta_{2} - 28) q^{88} + ( - 56 \beta_{3} + 56 \beta_1) q^{89} + ( - 28 \beta_{7} - 32 \beta_{3}) q^{90} + (2 \beta_{6} - 2 \beta_{5}) q^{92} + (34 \beta_{7} - 34 \beta_{4}) q^{94} + 80 \beta_{6} q^{95} + ( - 4 \beta_{4} + 16 \beta_{3} - 16 \beta_1) q^{96} - 17 \beta_{7} q^{97} + (49 \beta_{6} - 49 \beta_{5} - 56) q^{99}+O(q^{100})$$ q + b5 * q^2 + (2*b7 - 2*b4 + b1) * q^3 + (-2*b2 + 2) * q^4 + (-4*b3 + 4*b1) * q^5 + (b7 - 4*b3) * q^6 + (-2*b6 + 2*b5) * q^8 + (4*b5 - 7*b2) * q^9 + (4*b7 - 4*b4) * q^10 - 7*b6 * q^11 + (-4*b4 - 2*b3 + 2*b1) * q^12 - 9*b7 * q^13 + (-8*b6 + 8*b5 + 4) * q^15 - 4*b2 * q^16 + 32*b1 * q^17 + (-7*b6 - 8*b2 + 8) * q^18 + 20*b4 * q^19 - 8*b3 * q^20 - 14 * q^22 - b5 * q^23 + (2*b7 - 2*b4 - 8*b1) * q^24 + (9*b2 - 9) * q^25 + (18*b3 - 18*b1) * q^26 + (-10*b7 - 23*b3) * q^27 + (-40*b6 + 40*b5) * q^29 + (4*b5 - 16*b2) * q^30 - 4*b6 * q^32 + (-7*b4 + 28*b3 - 28*b1) * q^33 + 32*b7 * q^34 + (-8*b6 + 8*b5 - 14) * q^36 + 50*b2 * q^37 + 40*b1 * q^38 + (-9*b6 + 36*b2 - 36) * q^39 - 8*b4 * q^40 + 16*b3 * q^41 - 52 * q^43 - 14*b5 * q^44 + (16*b7 - 16*b4 - 28*b1) * q^45 + (2*b2 - 2) * q^46 + (-34*b3 + 34*b1) * q^47 + (-8*b7 - 4*b3) * q^48 + (9*b6 - 9*b5) * q^50 + (64*b5 + 32*b2) * q^51 + (-18*b7 + 18*b4) * q^52 + 32*b6 * q^53 + (-23*b4 + 20*b3 - 20*b1) * q^54 - 28*b7 * q^55 + (20*b6 - 20*b5 + 80) * q^57 - 80*b2 * q^58 + 32*b1 * q^59 + (-16*b6 - 8*b2 + 8) * q^60 - 23*b4 * q^61 - 8 * q^64 - 36*b5 * q^65 + (-28*b7 + 28*b4 - 14*b1) * q^66 + (-12*b2 + 12) * q^67 + (-64*b3 + 64*b1) * q^68 + (-b7 + 4*b3) * q^69 + (63*b6 - 63*b5) * q^71 + (-14*b5 - 16*b2) * q^72 + (-23*b7 + 23*b4) * q^73 + 50*b6 * q^74 + (18*b4 + 9*b3 - 9*b1) * q^75 + 40*b7 * q^76 + (36*b6 - 36*b5 - 18) * q^78 + 40*b2 * q^79 - 16*b1 * q^80 + (-56*b6 + 17*b2 - 17) * q^81 + 16*b4 * q^82 + 62*b3 * q^83 + 128 * q^85 - 52*b5 * q^86 + (40*b7 - 40*b4 - 160*b1) * q^87 + (28*b2 - 28) * q^88 + (-56*b3 + 56*b1) * q^89 + (-28*b7 - 32*b3) * q^90 + (2*b6 - 2*b5) * q^92 + (34*b7 - 34*b4) * q^94 + 80*b6 * q^95 + (-4*b4 + 16*b3 - 16*b1) * q^96 - 17*b7 * q^97 + (49*b6 - 49*b5 - 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} - 28 q^{9}+O(q^{10})$$ 8 * q + 8 * q^4 - 28 * q^9 $$8 q + 8 q^{4} - 28 q^{9} + 32 q^{15} - 16 q^{16} + 32 q^{18} - 112 q^{22} - 36 q^{25} - 64 q^{30} - 112 q^{36} + 200 q^{37} - 144 q^{39} - 416 q^{43} - 8 q^{46} + 128 q^{51} + 640 q^{57} - 320 q^{58} + 32 q^{60} - 64 q^{64} + 48 q^{67} - 64 q^{72} - 144 q^{78} + 160 q^{79} - 68 q^{81} + 1024 q^{85} - 112 q^{88} - 448 q^{99}+O(q^{100})$$ 8 * q + 8 * q^4 - 28 * q^9 + 32 * q^15 - 16 * q^16 + 32 * q^18 - 112 * q^22 - 36 * q^25 - 64 * q^30 - 112 * q^36 + 200 * q^37 - 144 * q^39 - 416 * q^43 - 8 * q^46 + 128 * q^51 + 640 * q^57 - 320 * q^58 + 32 * q^60 - 64 * q^64 + 48 * q^67 - 64 * q^72 - 144 * q^78 + 160 * q^79 - 68 * q^81 + 1024 * q^85 - 112 * q^88 - 448 * q^99

Basis of coefficient ring

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 −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 0.965926 − 0.258819i
−1.22474 + 0.707107i −0.548188 + 2.94949i 1.00000 1.73205i 3.46410 2.00000i −1.41421 4.00000i 0 2.82843i −8.39898 3.23375i −2.82843 + 4.89898i
263.2 −1.22474 + 0.707107i 0.548188 2.94949i 1.00000 1.73205i −3.46410 + 2.00000i 1.41421 + 4.00000i 0 2.82843i −8.39898 3.23375i 2.82843 4.89898i
263.3 1.22474 0.707107i −2.28024 + 1.94949i 1.00000 1.73205i −3.46410 + 2.00000i −1.41421 + 4.00000i 0 2.82843i 1.39898 8.89060i −2.82843 + 4.89898i
263.4 1.22474 0.707107i 2.28024 1.94949i 1.00000 1.73205i 3.46410 2.00000i 1.41421 4.00000i 0 2.82843i 1.39898 8.89060i 2.82843 4.89898i
275.1 −1.22474 0.707107i −0.548188 2.94949i 1.00000 + 1.73205i 3.46410 + 2.00000i −1.41421 + 4.00000i 0 2.82843i −8.39898 + 3.23375i −2.82843 4.89898i
275.2 −1.22474 0.707107i 0.548188 + 2.94949i 1.00000 + 1.73205i −3.46410 2.00000i 1.41421 4.00000i 0 2.82843i −8.39898 + 3.23375i 2.82843 + 4.89898i
275.3 1.22474 + 0.707107i −2.28024 1.94949i 1.00000 + 1.73205i −3.46410 2.00000i −1.41421 4.00000i 0 2.82843i 1.39898 + 8.89060i −2.82843 4.89898i
275.4 1.22474 + 0.707107i 2.28024 + 1.94949i 1.00000 + 1.73205i 3.46410 + 2.00000i 1.41421 + 4.00000i 0 2.82843i 1.39898 + 8.89060i 2.82843 + 4.89898i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.4 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 294.3.h.e 8
3.b odd 2 1 inner 294.3.h.e 8
7.b odd 2 1 inner 294.3.h.e 8
7.c even 3 1 294.3.b.g 4
7.c even 3 1 inner 294.3.h.e 8
7.d odd 6 1 294.3.b.g 4
7.d odd 6 1 inner 294.3.h.e 8
21.c even 2 1 inner 294.3.h.e 8
21.g even 6 1 294.3.b.g 4
21.g even 6 1 inner 294.3.h.e 8
21.h odd 6 1 294.3.b.g 4
21.h odd 6 1 inner 294.3.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.3.b.g 4 7.c even 3 1
294.3.b.g 4 7.d odd 6 1
294.3.b.g 4 21.g even 6 1
294.3.b.g 4 21.h odd 6 1
294.3.h.e 8 1.a even 1 1 trivial
294.3.h.e 8 3.b odd 2 1 inner
294.3.h.e 8 7.b odd 2 1 inner
294.3.h.e 8 7.c even 3 1 inner
294.3.h.e 8 7.d odd 6 1 inner
294.3.h.e 8 21.c even 2 1 inner
294.3.h.e 8 21.g even 6 1 inner
294.3.h.e 8 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 $$S_{3}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{4} - 16T_{5}^{2} + 256$$ T5^4 - 16*T5^2 + 256 $$T_{13}^{2} - 162$$ T13^2 - 162

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$3$ $$T^{8} + 14 T^{6} + 115 T^{4} + \cdots + 6561$$
$5$ $$(T^{4} - 16 T^{2} + 256)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 98 T^{2} + 9604)^{2}$$
$13$ $$(T^{2} - 162)^{4}$$
$17$ $$(T^{4} - 1024 T^{2} + 1048576)^{2}$$
$19$ $$(T^{4} + 800 T^{2} + 640000)^{2}$$
$23$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$29$ $$(T^{2} + 3200)^{4}$$
$31$ $$T^{8}$$
$37$ $$(T^{2} - 50 T + 2500)^{4}$$
$41$ $$(T^{2} + 256)^{4}$$
$43$ $$(T + 52)^{8}$$
$47$ $$(T^{4} - 1156 T^{2} + 1336336)^{2}$$
$53$ $$(T^{4} - 2048 T^{2} + 4194304)^{2}$$
$59$ $$(T^{4} - 1024 T^{2} + 1048576)^{2}$$
$61$ $$(T^{4} + 1058 T^{2} + 1119364)^{2}$$
$67$ $$(T^{2} - 12 T + 144)^{4}$$
$71$ $$(T^{2} + 7938)^{4}$$
$73$ $$(T^{4} + 1058 T^{2} + 1119364)^{2}$$
$79$ $$(T^{2} - 40 T + 1600)^{4}$$
$83$ $$(T^{2} + 3844)^{4}$$
$89$ $$(T^{4} - 3136 T^{2} + 9834496)^{2}$$
$97$ $$(T^{2} - 578)^{4}$$