# 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: $$x^{4} + 2 x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( -2 \beta_{1} - \beta_{3} ) q^{6} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( -2 \beta_{1} - \beta_{3} ) q^{6} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{11} + ( 2 - 2 \beta_{2} ) q^{12} + ( -8 - 4 \beta_{1} - 16 \beta_{2} - 2 \beta_{3} ) q^{13} + ( -6 + 3 \beta_{3} ) q^{15} + ( -4 - 4 \beta_{2} ) q^{16} + ( -4 + 11 \beta_{1} - 2 \beta_{2} - 11 \beta_{3} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{18} + ( 2 - 8 \beta_{1} - 2 \beta_{2} - 16 \beta_{3} ) q^{19} + ( 4 + 4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{20} + ( -6 - 6 \beta_{3} ) q^{22} + ( -6 - 9 \beta_{1} - 6 \beta_{2} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{24} + ( -12 \beta_{1} + 7 \beta_{2} - 12 \beta_{3} ) q^{25} + ( 4 - 8 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} ) q^{26} + ( -3 - 6 \beta_{2} ) q^{27} + 30 q^{29} + ( -6 - 6 \beta_{1} - 6 \beta_{2} ) q^{30} + ( -24 + 12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{33} + ( 22 - 4 \beta_{1} + 44 \beta_{2} - 2 \beta_{3} ) q^{34} -6 q^{36} + ( 20 + 36 \beta_{1} + 20 \beta_{2} ) q^{37} + ( 32 + 2 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 6 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} ) q^{39} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{40} + ( -14 - 14 \beta_{1} - 28 \beta_{2} - 7 \beta_{3} ) q^{41} + ( -32 - 30 \beta_{3} ) q^{43} + ( 12 - 6 \beta_{1} + 12 \beta_{2} ) q^{44} + ( 12 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} ) q^{46} + ( -28 - 4 \beta_{1} + 28 \beta_{2} - 8 \beta_{3} ) q^{47} + ( 4 + 8 \beta_{2} ) q^{48} + ( 24 + 7 \beta_{3} ) q^{50} + ( 6 - 33 \beta_{1} + 6 \beta_{2} ) q^{51} + ( 32 + 4 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} ) q^{52} + ( 12 \beta_{1} - 54 \beta_{2} + 12 \beta_{3} ) q^{53} + ( -3 \beta_{1} - 6 \beta_{3} ) q^{54} + ( -6 - 12 \beta_{2} ) q^{55} + ( -6 + 24 \beta_{3} ) q^{57} + 30 \beta_{1} q^{58} + ( -56 - 20 \beta_{1} - 28 \beta_{2} + 20 \beta_{3} ) q^{59} + ( -6 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} ) q^{60} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{61} + ( 24 - 24 \beta_{1} + 48 \beta_{2} - 12 \beta_{3} ) q^{62} + 8 q^{64} + ( -60 - 36 \beta_{1} - 60 \beta_{2} ) q^{65} + ( 12 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{66} + ( 12 \beta_{1} + 44 \beta_{2} + 12 \beta_{3} ) q^{67} + ( 4 + 22 \beta_{1} - 4 \beta_{2} + 44 \beta_{3} ) q^{68} + ( 6 + 18 \beta_{1} + 12 \beta_{2} + 9 \beta_{3} ) q^{69} + ( -30 + 57 \beta_{3} ) q^{71} -6 \beta_{1} q^{72} + ( 8 - 26 \beta_{1} + 4 \beta_{2} + 26 \beta_{3} ) q^{73} + ( 20 \beta_{1} + 72 \beta_{2} + 20 \beta_{3} ) q^{74} + ( 7 + 12 \beta_{1} - 7 \beta_{2} + 24 \beta_{3} ) q^{75} + ( 4 + 32 \beta_{1} + 8 \beta_{2} + 16 \beta_{3} ) q^{76} + ( -12 + 24 \beta_{3} ) q^{78} + ( -32 + 72 \beta_{1} - 32 \beta_{2} ) q^{79} + ( -16 - 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{80} + 9 \beta_{2} q^{81} + ( 14 - 14 \beta_{1} - 14 \beta_{2} - 28 \beta_{3} ) q^{82} + ( -32 + 40 \beta_{1} - 64 \beta_{2} + 20 \beta_{3} ) q^{83} + ( 54 - 60 \beta_{3} ) q^{85} + ( 60 - 32 \beta_{1} + 60 \beta_{2} ) q^{86} + ( -60 - 30 \beta_{2} ) q^{87} + ( 12 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{88} + ( 54 + 21 \beta_{1} - 54 \beta_{2} + 42 \beta_{3} ) q^{89} + ( 6 + 12 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{90} + ( 12 - 18 \beta_{3} ) q^{92} + ( 36 - 36 \beta_{1} + 36 \beta_{2} ) q^{93} + ( 16 - 28 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} ) q^{94} + ( -54 \beta_{1} - 60 \beta_{2} - 54 \beta_{3} ) q^{95} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{96} + ( -44 + 20 \beta_{1} - 88 \beta_{2} + 10 \beta_{3} ) q^{97} + ( 18 + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 4q^{4} + 12q^{5} + 6q^{9} + O(q^{10})$$ $$4q - 6q^{3} - 4q^{4} + 12q^{5} + 6q^{9} + 12q^{10} + 12q^{11} + 12q^{12} - 24q^{15} - 8q^{16} - 12q^{17} + 12q^{19} - 24q^{22} - 12q^{23} - 14q^{25} + 24q^{26} + 120q^{29} - 12q^{30} - 72q^{31} - 36q^{33} - 24q^{36} + 40q^{37} + 96q^{38} - 48q^{39} - 24q^{40} - 128q^{43} + 24q^{44} + 36q^{45} + 36q^{46} - 168q^{47} + 96q^{50} + 12q^{51} + 96q^{52} + 108q^{53} - 24q^{57} - 168q^{59} + 24q^{60} - 24q^{61} + 32q^{64} - 120q^{65} + 36q^{66} - 88q^{67} + 24q^{68} - 120q^{71} + 24q^{73} - 144q^{74} + 42q^{75} - 48q^{78} - 64q^{79} - 48q^{80} - 18q^{81} + 84q^{82} + 216q^{85} + 120q^{86} - 180q^{87} + 24q^{88} + 324q^{89} + 48q^{92} + 72q^{93} + 48q^{94} + 120q^{95} + 72q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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 $$T_{5}^{4} - 12 T_{5}^{3} + 54 T_{5}^{2} - 72 T_{5} + 36$$ acting on $$S_{3}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$36 - 72 T + 54 T^{2} - 12 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4}$$
$13$ $$28224 + 432 T^{2} + T^{4}$$
$17$ $$509796 - 8568 T - 666 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$138384 + 4464 T - 324 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$15876 - 1512 T + 270 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( -30 + T )^{4}$$
$31$ $$186624 - 31104 T + 1296 T^{2} + 72 T^{3} + T^{4}$$
$37$ $$4804864 + 87680 T + 3792 T^{2} - 40 T^{3} + T^{4}$$
$41$ $$86436 + 1764 T^{2} + T^{4}$$
$43$ $$( -776 + 64 T + T^{2} )^{2}$$
$47$ $$5089536 + 379008 T + 11664 T^{2} + 168 T^{3} + T^{4}$$
$53$ $$6906384 - 283824 T + 9036 T^{2} - 108 T^{3} + T^{4}$$
$59$ $$2304 - 8064 T + 9360 T^{2} + 168 T^{3} + T^{4}$$
$61$ $$2304 - 1152 T + 144 T^{2} + 24 T^{3} + T^{4}$$
$67$ $$2715904 + 145024 T + 6096 T^{2} + 88 T^{3} + T^{4}$$
$71$ $$( -5598 + 60 T + T^{2} )^{2}$$
$73$ $$16064064 + 96192 T - 3816 T^{2} - 24 T^{3} + T^{4}$$
$79$ $$87310336 - 598016 T + 13440 T^{2} + 64 T^{3} + T^{4}$$
$83$ $$451584 + 10944 T^{2} + T^{4}$$
$89$ $$37234404 - 1977048 T + 41094 T^{2} - 324 T^{3} + T^{4}$$
$97$ $$27123264 + 12816 T^{2} + T^{4}$$