# Properties

 Label 294.3.g.b Level $294$ Weight $3$ Character orbit 294.g Analytic conductor $8.011$ Analytic rank $0$ Dimension $4$ 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.g (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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - 2) * q^3 + 2*b2 * q^4 + (-2*b3 - 2*b2 - b1 + 2) * q^5 + (-b3 - 2*b1) * q^6 + 2*b3 * q^8 + (3*b2 + 3) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{10} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{12} + ( - 2 \beta_{3} - 16 \beta_{2} - 4 \beta_1 - 8) q^{13} + (3 \beta_{3} - 6) q^{15} + ( - 4 \beta_{2} - 4) q^{16} + ( - 11 \beta_{3} - 2 \beta_{2} + 11 \beta_1 - 4) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{18} + ( - 16 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 2) q^{19} + (2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 4) q^{20} + ( - 6 \beta_{3} - 6) q^{22} + ( - 6 \beta_{2} - 9 \beta_1 - 6) q^{23} + ( - 2 \beta_{3} + 2 \beta_1) q^{24} + ( - 12 \beta_{3} + 7 \beta_{2} - 12 \beta_1) q^{25} + ( - 16 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{26} + ( - 6 \beta_{2} - 3) q^{27} + 30 q^{29} + ( - 6 \beta_{2} - 6 \beta_1 - 6) q^{30} + ( - 12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 - 24) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 6) q^{33} + ( - 2 \beta_{3} + 44 \beta_{2} - 4 \beta_1 + 22) q^{34} - 6 q^{36} + (20 \beta_{2} + 36 \beta_1 + 20) q^{37} + ( - 2 \beta_{3} + 16 \beta_{2} + 2 \beta_1 + 32) q^{38} + (6 \beta_{3} + 24 \beta_{2} + 6 \beta_1) q^{39} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{40} + ( - 7 \beta_{3} - 28 \beta_{2} - 14 \beta_1 - 14) q^{41} + ( - 30 \beta_{3} - 32) q^{43} + (12 \beta_{2} - 6 \beta_1 + 12) q^{44} + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 12) q^{45} + ( - 6 \beta_{3} - 18 \beta_{2} - 6 \beta_1) q^{46} + ( - 8 \beta_{3} + 28 \beta_{2} - 4 \beta_1 - 28) q^{47} + (8 \beta_{2} + 4) q^{48} + (7 \beta_{3} + 24) q^{50} + (6 \beta_{2} - 33 \beta_1 + 6) q^{51} + ( - 4 \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 32) q^{52} + (12 \beta_{3} - 54 \beta_{2} + 12 \beta_1) q^{53} + ( - 6 \beta_{3} - 3 \beta_1) q^{54} + ( - 12 \beta_{2} - 6) q^{55} + (24 \beta_{3} - 6) q^{57} + 30 \beta_1 q^{58} + (20 \beta_{3} - 28 \beta_{2} - 20 \beta_1 - 56) q^{59} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{60} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{61} + ( - 12 \beta_{3} + 48 \beta_{2} - 24 \beta_1 + 24) q^{62} + 8 q^{64} + ( - 60 \beta_{2} - 36 \beta_1 - 60) q^{65} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 12) q^{66} + (12 \beta_{3} + 44 \beta_{2} + 12 \beta_1) q^{67} + (44 \beta_{3} - 4 \beta_{2} + 22 \beta_1 + 4) q^{68} + (9 \beta_{3} + 12 \beta_{2} + 18 \beta_1 + 6) q^{69} + (57 \beta_{3} - 30) q^{71} - 6 \beta_1 q^{72} + (26 \beta_{3} + 4 \beta_{2} - 26 \beta_1 + 8) q^{73} + (20 \beta_{3} + 72 \beta_{2} + 20 \beta_1) q^{74} + (24 \beta_{3} - 7 \beta_{2} + 12 \beta_1 + 7) q^{75} + (16 \beta_{3} + 8 \beta_{2} + 32 \beta_1 + 4) q^{76} + (24 \beta_{3} - 12) q^{78} + ( - 32 \beta_{2} + 72 \beta_1 - 32) q^{79} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 16) q^{80} + 9 \beta_{2} q^{81} + ( - 28 \beta_{3} - 14 \beta_{2} - 14 \beta_1 + 14) q^{82} + (20 \beta_{3} - 64 \beta_{2} + 40 \beta_1 - 32) q^{83} + ( - 60 \beta_{3} + 54) q^{85} + (60 \beta_{2} - 32 \beta_1 + 60) q^{86} + ( - 30 \beta_{2} - 60) q^{87} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1) q^{88} + (42 \beta_{3} - 54 \beta_{2} + 21 \beta_1 + 54) q^{89} + (6 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 6) q^{90} + ( - 18 \beta_{3} + 12) q^{92} + (36 \beta_{2} - 36 \beta_1 + 36) q^{93} + (28 \beta_{3} + 8 \beta_{2} - 28 \beta_1 + 16) q^{94} + ( - 54 \beta_{3} - 60 \beta_{2} - 54 \beta_1) q^{95} + (8 \beta_{3} + 4 \beta_1) q^{96} + (10 \beta_{3} - 88 \beta_{2} + 20 \beta_1 - 44) q^{97} + (9 \beta_{3} + 18) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - 2) * q^3 + 2*b2 * q^4 + (-2*b3 - 2*b2 - b1 + 2) * q^5 + (-b3 - 2*b1) * q^6 + 2*b3 * q^8 + (3*b2 + 3) * q^9 + (-2*b3 + 2*b2 + 2*b1 + 4) * q^10 + (3*b3 - 6*b2 + 3*b1) * q^11 + (-2*b2 + 2) * q^12 + (-2*b3 - 16*b2 - 4*b1 - 8) * q^13 + (3*b3 - 6) * q^15 + (-4*b2 - 4) * q^16 + (-11*b3 - 2*b2 + 11*b1 - 4) * q^17 + (3*b3 + 3*b1) * q^18 + (-16*b3 - 2*b2 - 8*b1 + 2) * q^19 + (2*b3 + 8*b2 + 4*b1 + 4) * q^20 + (-6*b3 - 6) * q^22 + (-6*b2 - 9*b1 - 6) * q^23 + (-2*b3 + 2*b1) * q^24 + (-12*b3 + 7*b2 - 12*b1) * q^25 + (-16*b3 - 4*b2 - 8*b1 + 4) * q^26 + (-6*b2 - 3) * q^27 + 30 * q^29 + (-6*b2 - 6*b1 - 6) * q^30 + (-12*b3 - 12*b2 + 12*b1 - 24) * q^31 + (-4*b3 - 4*b1) * q^32 + (-6*b3 + 6*b2 - 3*b1 - 6) * q^33 + (-2*b3 + 44*b2 - 4*b1 + 22) * q^34 - 6 * q^36 + (20*b2 + 36*b1 + 20) * q^37 + (-2*b3 + 16*b2 + 2*b1 + 32) * q^38 + (6*b3 + 24*b2 + 6*b1) * q^39 + (8*b3 + 4*b2 + 4*b1 - 4) * q^40 + (-7*b3 - 28*b2 - 14*b1 - 14) * q^41 + (-30*b3 - 32) * q^43 + (12*b2 - 6*b1 + 12) * q^44 + (-3*b3 + 6*b2 + 3*b1 + 12) * q^45 + (-6*b3 - 18*b2 - 6*b1) * q^46 + (-8*b3 + 28*b2 - 4*b1 - 28) * q^47 + (8*b2 + 4) * q^48 + (7*b3 + 24) * q^50 + (6*b2 - 33*b1 + 6) * q^51 + (-4*b3 + 16*b2 + 4*b1 + 32) * q^52 + (12*b3 - 54*b2 + 12*b1) * q^53 + (-6*b3 - 3*b1) * q^54 + (-12*b2 - 6) * q^55 + (24*b3 - 6) * q^57 + 30*b1 * q^58 + (20*b3 - 28*b2 - 20*b1 - 56) * q^59 + (-6*b3 - 12*b2 - 6*b1) * q^60 + (8*b3 + 4*b2 + 4*b1 - 4) * q^61 + (-12*b3 + 48*b2 - 24*b1 + 24) * q^62 + 8 * q^64 + (-60*b2 - 36*b1 - 60) * q^65 + (6*b3 + 6*b2 - 6*b1 + 12) * q^66 + (12*b3 + 44*b2 + 12*b1) * q^67 + (44*b3 - 4*b2 + 22*b1 + 4) * q^68 + (9*b3 + 12*b2 + 18*b1 + 6) * q^69 + (57*b3 - 30) * q^71 - 6*b1 * q^72 + (26*b3 + 4*b2 - 26*b1 + 8) * q^73 + (20*b3 + 72*b2 + 20*b1) * q^74 + (24*b3 - 7*b2 + 12*b1 + 7) * q^75 + (16*b3 + 8*b2 + 32*b1 + 4) * q^76 + (24*b3 - 12) * q^78 + (-32*b2 + 72*b1 - 32) * q^79 + (4*b3 - 8*b2 - 4*b1 - 16) * q^80 + 9*b2 * q^81 + (-28*b3 - 14*b2 - 14*b1 + 14) * q^82 + (20*b3 - 64*b2 + 40*b1 - 32) * q^83 + (-60*b3 + 54) * q^85 + (60*b2 - 32*b1 + 60) * q^86 + (-30*b2 - 60) * q^87 + (12*b3 - 12*b2 + 12*b1) * q^88 + (42*b3 - 54*b2 + 21*b1 + 54) * q^89 + (6*b3 + 12*b2 + 12*b1 + 6) * q^90 + (-18*b3 + 12) * q^92 + (36*b2 - 36*b1 + 36) * q^93 + (28*b3 + 8*b2 - 28*b1 + 16) * q^94 + (-54*b3 - 60*b2 - 54*b1) * q^95 + (8*b3 + 4*b1) * q^96 + (10*b3 - 88*b2 + 20*b1 - 44) * q^97 + (9*b3 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} - 4 q^{4} + 12 q^{5} + 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^3 - 4 * q^4 + 12 * q^5 + 6 * q^9 $$4 q - 6 q^{3} - 4 q^{4} + 12 q^{5} + 6 q^{9} + 12 q^{10} + 12 q^{11} + 12 q^{12} - 24 q^{15} - 8 q^{16} - 12 q^{17} + 12 q^{19} - 24 q^{22} - 12 q^{23} - 14 q^{25} + 24 q^{26} + 120 q^{29} - 12 q^{30} - 72 q^{31} - 36 q^{33} - 24 q^{36} + 40 q^{37} + 96 q^{38} - 48 q^{39} - 24 q^{40} - 128 q^{43} + 24 q^{44} + 36 q^{45} + 36 q^{46} - 168 q^{47} + 96 q^{50} + 12 q^{51} + 96 q^{52} + 108 q^{53} - 24 q^{57} - 168 q^{59} + 24 q^{60} - 24 q^{61} + 32 q^{64} - 120 q^{65} + 36 q^{66} - 88 q^{67} + 24 q^{68} - 120 q^{71} + 24 q^{73} - 144 q^{74} + 42 q^{75} - 48 q^{78} - 64 q^{79} - 48 q^{80} - 18 q^{81} + 84 q^{82} + 216 q^{85} + 120 q^{86} - 180 q^{87} + 24 q^{88} + 324 q^{89} + 48 q^{92} + 72 q^{93} + 48 q^{94} + 120 q^{95} + 72 q^{99}+O(q^{100})$$ 4 * q - 6 * q^3 - 4 * q^4 + 12 * q^5 + 6 * q^9 + 12 * q^10 + 12 * q^11 + 12 * q^12 - 24 * q^15 - 8 * q^16 - 12 * q^17 + 12 * q^19 - 24 * q^22 - 12 * q^23 - 14 * q^25 + 24 * q^26 + 120 * q^29 - 12 * q^30 - 72 * q^31 - 36 * q^33 - 24 * q^36 + 40 * q^37 + 96 * q^38 - 48 * q^39 - 24 * q^40 - 128 * q^43 + 24 * q^44 + 36 * q^45 + 36 * q^46 - 168 * q^47 + 96 * q^50 + 12 * q^51 + 96 * q^52 + 108 * q^53 - 24 * q^57 - 168 * q^59 + 24 * q^60 - 24 * q^61 + 32 * q^64 - 120 * q^65 + 36 * q^66 - 88 * q^67 + 24 * q^68 - 120 * q^71 + 24 * q^73 - 144 * q^74 + 42 * q^75 - 48 * q^78 - 64 * q^79 - 48 * q^80 - 18 * q^81 + 84 * q^82 + 216 * q^85 + 120 * q^86 - 180 * q^87 + 24 * q^88 + 324 * q^89 + 48 * q^92 + 72 * q^93 + 48 * q^94 + 120 * q^95 + 72 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0.878680 + 0.507306i 2.44949i 0 2.82843 1.50000 2.59808i −1.24264 + 0.717439i
19.2 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 5.12132 + 2.95680i 2.44949i 0 −2.82843 1.50000 2.59808i 7.24264 4.18154i
31.1 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0.878680 0.507306i 2.44949i 0 2.82843 1.50000 + 2.59808i −1.24264 0.717439i
31.2 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 5.12132 2.95680i 2.44949i 0 −2.82843 1.50000 + 2.59808i 7.24264 + 4.18154i
 $$n$$: 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
7.d 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.g.b 4
3.b odd 2 1 882.3.n.a 4
7.b odd 2 1 294.3.g.c 4
7.c even 3 1 42.3.c.a 4
7.c even 3 1 294.3.g.c 4
7.d odd 6 1 42.3.c.a 4
7.d odd 6 1 inner 294.3.g.b 4
21.c even 2 1 882.3.n.d 4
21.g even 6 1 126.3.c.b 4
21.g even 6 1 882.3.n.a 4
21.h odd 6 1 126.3.c.b 4
21.h odd 6 1 882.3.n.d 4
28.f even 6 1 336.3.f.c 4
28.g odd 6 1 336.3.f.c 4
35.i odd 6 1 1050.3.f.a 4
35.j even 6 1 1050.3.f.a 4
35.k even 12 2 1050.3.h.a 8
35.l odd 12 2 1050.3.h.a 8
56.j odd 6 1 1344.3.f.f 4
56.k odd 6 1 1344.3.f.e 4
56.m even 6 1 1344.3.f.e 4
56.p even 6 1 1344.3.f.f 4
84.j odd 6 1 1008.3.f.g 4
84.n even 6 1 1008.3.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 7.c even 3 1
42.3.c.a 4 7.d odd 6 1
126.3.c.b 4 21.g even 6 1
126.3.c.b 4 21.h odd 6 1
294.3.g.b 4 1.a even 1 1 trivial
294.3.g.b 4 7.d odd 6 1 inner
294.3.g.c 4 7.b odd 2 1
294.3.g.c 4 7.c even 3 1
336.3.f.c 4 28.f even 6 1
336.3.f.c 4 28.g odd 6 1
882.3.n.a 4 3.b odd 2 1
882.3.n.a 4 21.g even 6 1
882.3.n.d 4 21.c even 2 1
882.3.n.d 4 21.h odd 6 1
1008.3.f.g 4 84.j odd 6 1
1008.3.f.g 4 84.n even 6 1
1050.3.f.a 4 35.i odd 6 1
1050.3.f.a 4 35.j even 6 1
1050.3.h.a 8 35.k even 12 2
1050.3.h.a 8 35.l odd 12 2
1344.3.f.e 4 56.k odd 6 1
1344.3.f.e 4 56.m even 6 1
1344.3.f.f 4 56.j odd 6 1
1344.3.f.f 4 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 12T_{5}^{3} + 54T_{5}^{2} - 72T_{5} + 36$$ acting on $$S_{3}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$(T^{2} + 3 T + 3)^{2}$$
$5$ $$T^{4} - 12 T^{3} + 54 T^{2} - 72 T + 36$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324$$
$13$ $$T^{4} + 432 T^{2} + 28224$$
$17$ $$T^{4} + 12 T^{3} - 666 T^{2} + \cdots + 509796$$
$19$ $$T^{4} - 12 T^{3} - 324 T^{2} + \cdots + 138384$$
$23$ $$T^{4} + 12 T^{3} + 270 T^{2} + \cdots + 15876$$
$29$ $$(T - 30)^{4}$$
$31$ $$T^{4} + 72 T^{3} + 1296 T^{2} + \cdots + 186624$$
$37$ $$T^{4} - 40 T^{3} + 3792 T^{2} + \cdots + 4804864$$
$41$ $$T^{4} + 1764 T^{2} + 86436$$
$43$ $$(T^{2} + 64 T - 776)^{2}$$
$47$ $$T^{4} + 168 T^{3} + 11664 T^{2} + \cdots + 5089536$$
$53$ $$T^{4} - 108 T^{3} + 9036 T^{2} + \cdots + 6906384$$
$59$ $$T^{4} + 168 T^{3} + 9360 T^{2} + \cdots + 2304$$
$61$ $$T^{4} + 24 T^{3} + 144 T^{2} + \cdots + 2304$$
$67$ $$T^{4} + 88 T^{3} + 6096 T^{2} + \cdots + 2715904$$
$71$ $$(T^{2} + 60 T - 5598)^{2}$$
$73$ $$T^{4} - 24 T^{3} - 3816 T^{2} + \cdots + 16064064$$
$79$ $$T^{4} + 64 T^{3} + 13440 T^{2} + \cdots + 87310336$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 451584$$
$89$ $$T^{4} - 324 T^{3} + \cdots + 37234404$$
$97$ $$T^{4} + 12816 T^{2} + \cdots + 27123264$$