# Properties

 Label 294.3.h.g Level $294$ Weight $3$ Character orbit 294.h Analytic conductor $8.011$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## 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: 8.0.12745506816.5 Defining polynomial: $$x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81$$ x^8 - 8*x^6 + 55*x^4 - 72*x^2 + 81 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_{4}) q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{4}) q^{5} + ( - \beta_{7} - \beta_{3} + 2) q^{6} - 2 \beta_{6} q^{8} + ( - 2 \beta_{7} + 5 \beta_{2} - 5) q^{9}+O(q^{10})$$ q + (-b6 + b4) * q^2 + (-b5 + b4) * q^3 + 2*b2 * q^4 + (-b7 + 2*b6 - 2*b4) * q^5 + (-b7 - b3 + 2) * q^6 - 2*b6 * q^8 + (-2*b7 + 5*b2 - 5) * q^9 $$q + ( - \beta_{6} + \beta_{4}) q^{2} + ( - \beta_{5} + \beta_{4}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{4}) q^{5} + ( - \beta_{7} - \beta_{3} + 2) q^{6} - 2 \beta_{6} q^{8} + ( - 2 \beta_{7} + 5 \beta_{2} - 5) q^{9} + ( - 2 \beta_{5} - 4 \beta_{2}) q^{10} + (5 \beta_{4} + 2 \beta_{3}) q^{11} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{12} + (4 \beta_1 - 10) q^{13} + (2 \beta_{7} - 7 \beta_{6} + 2 \beta_{3} + 2 \beta_1 - 4) q^{15} + (4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{4} - 5 \beta_{3}) q^{17} + ( - 4 \beta_{5} - 5 \beta_{4}) q^{18} + (16 \beta_{2} - 16) q^{19} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{3}) q^{20} + (4 \beta_1 + 10) q^{22} + ( - 10 \beta_{7} - \beta_{6} + \beta_{4}) q^{23} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{24} + (8 \beta_{5} - 3 \beta_{2}) q^{25} + ( - 4 \beta_{7} + 10 \beta_{6} - 10 \beta_{4}) q^{26} + ( - 19 \beta_{6} - \beta_1) q^{27} + (2 \beta_{7} + 20 \beta_{6} + 2 \beta_{3}) q^{29} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 14 \beta_{2} + \cdots - 14) q^{30}+ \cdots + ( - 10 \beta_{7} - 25 \beta_{6} - 10 \beta_{3} + 20 \beta_1 + 56) q^{99}+O(q^{100})$$ q + (-b6 + b4) * q^2 + (-b5 + b4) * q^3 + 2*b2 * q^4 + (-b7 + 2*b6 - 2*b4) * q^5 + (-b7 - b3 + 2) * q^6 - 2*b6 * q^8 + (-2*b7 + 5*b2 - 5) * q^9 + (-2*b5 - 4*b2) * q^10 + (5*b4 + 2*b3) * q^11 + (-2*b6 - 2*b5 + 2*b4 - 2*b1) * q^12 + (4*b1 - 10) * q^13 + (2*b7 - 7*b6 + 2*b3 + 2*b1 - 4) * q^15 + (4*b2 - 4) * q^16 + (-2*b4 - 5*b3) * q^17 + (-4*b5 - 5*b4) * q^18 + (16*b2 - 16) * q^19 + (-2*b7 + 4*b6 - 2*b3) * q^20 + (4*b1 + 10) * q^22 + (-10*b7 - b6 + b4) * q^23 + (-2*b3 + 4*b2) * q^24 + (8*b5 - 3*b2) * q^25 + (-4*b7 + 10*b6 - 10*b4) * q^26 + (-19*b6 - b1) * q^27 + (2*b7 + 20*b6 + 2*b3) * q^29 + (-2*b7 + 4*b6 + 4*b5 - 4*b4 + 14*b2 + 4*b1 - 14) * q^30 + (10*b5 - 32*b2) * q^31 - 4*b4 * q^32 + (-5*b7 - 14*b6 + 4*b5 + 14*b4 - 10*b2 + 4*b1 + 10) * q^33 + (-10*b1 - 4) * q^34 + (-4*b7 - 4*b3 - 10) * q^36 + (20*b2 - 20) * q^37 - 16*b4 * q^38 + (10*b5 - 10*b4 + 4*b3 + 28*b2) * q^39 + (-4*b5 - 8*b2 - 4*b1 + 8) * q^40 + (-9*b7 + 30*b6 - 9*b3) * q^41 + (-12*b1 + 20) * q^43 + (-4*b7 - 10*b6 + 10*b4) * q^44 + (8*b5 + 10*b4 - 5*b3 + 28*b2) * q^45 + (-20*b5 + 2*b2) * q^46 + (-6*b6 + 6*b4) * q^47 + (-4*b6 - 4*b1) * q^48 + (8*b7 + 3*b6 + 8*b3) * q^50 + (2*b7 + 35*b6 - 10*b5 - 35*b4 + 4*b2 - 10*b1 - 4) * q^51 + (-8*b5 - 20*b2) * q^52 + 36*b4 * q^53 + (b7 + 38*b2 - 38) * q^54 + (2*b1 + 8) * q^55 + (-16*b6 - 16*b1) * q^57 + (4*b5 - 40*b2 + 4*b1 + 40) * q^58 + (20*b4 + 8*b3) * q^59 + (-4*b5 - 14*b4 + 4*b3 - 8*b2) * q^60 + (20*b5 + 14*b2 + 20*b1 - 14) * q^61 + (10*b7 + 32*b6 + 10*b3) * q^62 - 8 * q^64 + (18*b7 - 48*b6 + 48*b4) * q^65 + (-10*b5 + 10*b4 + 4*b3 + 28*b2) * q^66 + (4*b5 - 60*b2) * q^67 + (10*b7 + 4*b6 - 4*b4) * q^68 + (-b7 - 70*b6 - b3 + 20*b1 + 2) * q^69 + (-14*b7 + 25*b6 - 14*b3) * q^71 + (10*b6 - 8*b5 - 10*b4 - 8*b1) * q^72 + (-16*b5 + 30*b2) * q^73 - 20*b4 * q^74 + (8*b7 + 3*b6 + 3*b5 - 3*b4 - 56*b2 + 3*b1 + 56) * q^75 - 32 * q^76 + (10*b7 - 28*b6 + 10*b3 + 8*b1 - 20) * q^78 + (-20*b5 - 32*b2 - 20*b1 + 32) * q^79 + (8*b4 - 4*b3) * q^80 + (-20*b3 + 31*b2) * q^81 + (-18*b5 - 60*b2 - 18*b1 + 60) * q^82 + (20*b7 + 50*b6 + 20*b3) * q^83 + (16*b1 - 62) * q^85 + (12*b7 - 20*b6 + 20*b4) * q^86 + (4*b5 + 14*b4 + 20*b3 - 40*b2) * q^87 + (-8*b5 + 20*b2) * q^88 + (3*b7 + 30*b6 - 30*b4) * q^89 + (8*b7 - 28*b6 + 8*b3 - 10*b1 + 20) * q^90 + (-20*b7 - 2*b6 - 20*b3) * q^92 + (10*b7 + 32*b6 + 32*b5 - 32*b4 - 70*b2 + 32*b1 + 70) * q^93 + 12*b2 * q^94 + (32*b4 - 16*b3) * q^95 + (4*b7 + 8*b2 - 8) * q^96 + (8*b1 + 90) * q^97 + (-10*b7 - 25*b6 - 10*b3 + 20*b1 + 56) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} + 16 q^{6} - 20 q^{9}+O(q^{10})$$ 8 * q + 8 * q^4 + 16 * q^6 - 20 * q^9 $$8 q + 8 q^{4} + 16 q^{6} - 20 q^{9} - 16 q^{10} - 80 q^{13} - 32 q^{15} - 16 q^{16} - 64 q^{19} + 80 q^{22} + 16 q^{24} - 12 q^{25} - 56 q^{30} - 128 q^{31} + 40 q^{33} - 32 q^{34} - 80 q^{36} - 80 q^{37} + 112 q^{39} + 32 q^{40} + 160 q^{43} + 112 q^{45} + 8 q^{46} - 16 q^{51} - 80 q^{52} - 152 q^{54} + 64 q^{55} + 160 q^{58} - 32 q^{60} - 56 q^{61} - 64 q^{64} + 112 q^{66} - 240 q^{67} + 16 q^{69} + 120 q^{73} + 224 q^{75} - 256 q^{76} - 160 q^{78} + 128 q^{79} + 124 q^{81} + 240 q^{82} - 496 q^{85} - 160 q^{87} + 80 q^{88} + 160 q^{90} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 720 q^{97} + 448 q^{99}+O(q^{100})$$ 8 * q + 8 * q^4 + 16 * q^6 - 20 * q^9 - 16 * q^10 - 80 * q^13 - 32 * q^15 - 16 * q^16 - 64 * q^19 + 80 * q^22 + 16 * q^24 - 12 * q^25 - 56 * q^30 - 128 * q^31 + 40 * q^33 - 32 * q^34 - 80 * q^36 - 80 * q^37 + 112 * q^39 + 32 * q^40 + 160 * q^43 + 112 * q^45 + 8 * q^46 - 16 * q^51 - 80 * q^52 - 152 * q^54 + 64 * q^55 + 160 * q^58 - 32 * q^60 - 56 * q^61 - 64 * q^64 + 112 * q^66 - 240 * q^67 + 16 * q^69 + 120 * q^73 + 224 * q^75 - 256 * q^76 - 160 * q^78 + 128 * q^79 + 124 * q^81 + 240 * q^82 - 496 * q^85 - 160 * q^87 + 80 * q^88 + 160 * q^90 + 280 * q^93 + 48 * q^94 - 32 * q^96 + 720 * q^97 + 448 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} - 148 ) / 55$$ (-v^6 - 148) / 55 $$\beta_{2}$$ $$=$$ $$( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495$$ (-8*v^6 + 55*v^4 - 440*v^2 + 576) / 495 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 533\nu ) / 165$$ (v^7 + 533*v) / 165 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 203\nu ) / 165$$ (-v^7 - 203*v) / 165 $$\beta_{5}$$ $$=$$ $$( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495$$ (-23*v^6 + 220*v^4 - 1265*v^2 + 1656) / 495 $$\beta_{6}$$ $$=$$ $$( -8\nu^{7} + 55\nu^{5} - 341\nu^{3} + 81\nu ) / 297$$ (-8*v^7 + 55*v^5 - 341*v^3 + 81*v) / 297 $$\beta_{7}$$ $$=$$ $$( 79\nu^{7} - 605\nu^{5} + 4345\nu^{3} - 5688\nu ) / 1485$$ (79*v^7 - 605*v^5 + 4345*v^3 - 5688*v) / 1485
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 2$$ (b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 4\beta_{2} + \beta _1 + 4$$ b5 - 4*b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$( 5\beta_{7} + 11\beta_{6} + 5\beta_{3} ) / 2$$ (5*b7 + 11*b6 + 5*b3) / 2 $$\nu^{4}$$ $$=$$ $$8\beta_{5} - 23\beta_{2}$$ 8*b5 - 23*b2 $$\nu^{5}$$ $$=$$ $$( 31\beta_{7} + 79\beta_{6} - 79\beta_{4} ) / 2$$ (31*b7 + 79*b6 - 79*b4) / 2 $$\nu^{6}$$ $$=$$ $$-55\beta _1 - 148$$ -55*b1 - 148 $$\nu^{7}$$ $$=$$ $$( -533\beta_{4} - 203\beta_{3} ) / 2$$ (-533*b4 - 203*b3) / 2

## Character values

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

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

## 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
 1.00781 + 0.581861i −2.23256 − 1.28897i −1.00781 − 0.581861i 2.23256 + 1.28897i 1.00781 − 0.581861i −2.23256 + 1.28897i −1.00781 + 0.581861i 2.23256 − 1.28897i
−1.22474 + 0.707107i −2.54762 + 1.58418i 1.00000 1.73205i 5.68986 3.28504i 2.00000 3.74166i 0 2.82843i 3.98074 8.07178i −4.64575 + 8.04668i
263.2 −1.22474 + 0.707107i 0.0981308 2.99839i 1.00000 1.73205i −0.790881 + 0.456615i 2.00000 + 3.74166i 0 2.82843i −8.98074 0.588470i 0.645751 1.11847i
263.3 1.22474 0.707107i −0.0981308 + 2.99839i 1.00000 1.73205i −5.68986 + 3.28504i 2.00000 + 3.74166i 0 2.82843i −8.98074 0.588470i −4.64575 + 8.04668i
263.4 1.22474 0.707107i 2.54762 1.58418i 1.00000 1.73205i 0.790881 0.456615i 2.00000 3.74166i 0 2.82843i 3.98074 8.07178i 0.645751 1.11847i
275.1 −1.22474 0.707107i −2.54762 1.58418i 1.00000 + 1.73205i 5.68986 + 3.28504i 2.00000 + 3.74166i 0 2.82843i 3.98074 + 8.07178i −4.64575 8.04668i
275.2 −1.22474 0.707107i 0.0981308 + 2.99839i 1.00000 + 1.73205i −0.790881 0.456615i 2.00000 3.74166i 0 2.82843i −8.98074 + 0.588470i 0.645751 + 1.11847i
275.3 1.22474 + 0.707107i −0.0981308 2.99839i 1.00000 + 1.73205i −5.68986 3.28504i 2.00000 3.74166i 0 2.82843i −8.98074 + 0.588470i −4.64575 8.04668i
275.4 1.22474 + 0.707107i 2.54762 + 1.58418i 1.00000 + 1.73205i 0.790881 + 0.456615i 2.00000 + 3.74166i 0 2.82843i 3.98074 + 8.07178i 0.645751 + 1.11847i
 $$n$$: 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.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.g 8
3.b odd 2 1 inner 294.3.h.g 8
7.b odd 2 1 294.3.h.d 8
7.c even 3 1 294.3.b.h 4
7.c even 3 1 inner 294.3.h.g 8
7.d odd 6 1 42.3.b.a 4
7.d odd 6 1 294.3.h.d 8
21.c even 2 1 294.3.h.d 8
21.g even 6 1 42.3.b.a 4
21.g even 6 1 294.3.h.d 8
21.h odd 6 1 294.3.b.h 4
21.h odd 6 1 inner 294.3.h.g 8
28.f even 6 1 336.3.d.b 4
35.i odd 6 1 1050.3.e.a 4
35.k even 12 2 1050.3.c.a 8
56.j odd 6 1 1344.3.d.c 4
56.m even 6 1 1344.3.d.e 4
63.i even 6 1 1134.3.q.a 8
63.k odd 6 1 1134.3.q.a 8
63.s even 6 1 1134.3.q.a 8
63.t odd 6 1 1134.3.q.a 8
84.j odd 6 1 336.3.d.b 4
105.p even 6 1 1050.3.e.a 4
105.w odd 12 2 1050.3.c.a 8
168.ba even 6 1 1344.3.d.c 4
168.be odd 6 1 1344.3.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 7.d odd 6 1
42.3.b.a 4 21.g even 6 1
294.3.b.h 4 7.c even 3 1
294.3.b.h 4 21.h odd 6 1
294.3.h.d 8 7.b odd 2 1
294.3.h.d 8 7.d odd 6 1
294.3.h.d 8 21.c even 2 1
294.3.h.d 8 21.g even 6 1
294.3.h.g 8 1.a even 1 1 trivial
294.3.h.g 8 3.b odd 2 1 inner
294.3.h.g 8 7.c even 3 1 inner
294.3.h.g 8 21.h odd 6 1 inner
336.3.d.b 4 28.f even 6 1
336.3.d.b 4 84.j odd 6 1
1050.3.c.a 8 35.k even 12 2
1050.3.c.a 8 105.w odd 12 2
1050.3.e.a 4 35.i odd 6 1
1050.3.e.a 4 105.p even 6 1
1134.3.q.a 8 63.i even 6 1
1134.3.q.a 8 63.k odd 6 1
1134.3.q.a 8 63.s even 6 1
1134.3.q.a 8 63.t odd 6 1
1344.3.d.c 4 56.j odd 6 1
1344.3.d.c 4 168.ba even 6 1
1344.3.d.e 4 56.m even 6 1
1344.3.d.e 4 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{8} - 44T_{5}^{6} + 1900T_{5}^{4} - 1584T_{5}^{2} + 1296$$ T5^8 - 44*T5^6 + 1900*T5^4 - 1584*T5^2 + 1296 $$T_{13}^{2} + 20T_{13} - 12$$ T13^2 + 20*T13 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$3$ $$T^{8} + 10 T^{6} + 19 T^{4} + \cdots + 6561$$
$5$ $$T^{8} - 44 T^{6} + 1900 T^{4} + \cdots + 1296$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 212 T^{6} + 44908 T^{4} + \cdots + 1296$$
$13$ $$(T^{2} + 20 T - 12)^{4}$$
$17$ $$T^{8} - 716 T^{6} + \cdots + 13680577296$$
$19$ $$(T^{2} + 16 T + 256)^{4}$$
$23$ $$T^{8} - 2804 T^{6} + \cdots + 3819694995216$$
$29$ $$(T^{4} + 1712 T^{2} + 553536)^{2}$$
$31$ $$(T^{4} + 64 T^{3} + 3772 T^{2} + \cdots + 104976)^{2}$$
$37$ $$(T^{2} + 20 T + 400)^{4}$$
$41$ $$(T^{4} + 5868 T^{2} + 443556)^{2}$$
$43$ $$(T^{2} - 40 T - 608)^{4}$$
$47$ $$(T^{4} - 72 T^{2} + 5184)^{2}$$
$53$ $$(T^{4} - 2592 T^{2} + 6718464)^{2}$$
$59$ $$T^{8} - 3392 T^{6} + \cdots + 84934656$$
$61$ $$(T^{4} + 28 T^{3} + 3388 T^{2} + \cdots + 6780816)^{2}$$
$67$ $$(T^{4} + 120 T^{3} + 10912 T^{2} + \cdots + 12166144)^{2}$$
$71$ $$(T^{4} + 7988 T^{2} + 2232036)^{2}$$
$73$ $$(T^{4} - 60 T^{3} + 4492 T^{2} + \cdots + 795664)^{2}$$
$79$ $$(T^{4} - 64 T^{3} + 5872 T^{2} + \cdots + 3154176)^{2}$$
$83$ $$(T^{4} + 21200 T^{2} + 360000)^{2}$$
$89$ $$T^{8} - 3852 T^{6} + \cdots + 7852750780176$$
$97$ $$(T^{2} - 180 T + 7652)^{4}$$
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