# Properties

 Label 294.3.c.a Level $294$ Weight $3$ Character orbit 294.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01091977219$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{5} + \beta_{3} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} + ( 4 \beta_{2} - 2 \beta_{3} ) q^{10} + 6 q^{11} -2 \beta_{2} q^{12} + ( -\beta_{2} - 8 \beta_{3} ) q^{13} + ( 6 + 6 \beta_{1} ) q^{15} + 4 q^{16} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{17} -3 \beta_{1} q^{18} + ( -7 \beta_{2} + 2 \beta_{3} ) q^{19} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{20} + 6 \beta_{1} q^{22} + ( -12 + 18 \beta_{1} ) q^{23} + 2 \beta_{3} q^{24} + ( -11 - 24 \beta_{1} ) q^{25} + ( 16 \beta_{2} + \beta_{3} ) q^{26} + 3 \beta_{2} q^{27} + 24 \beta_{1} q^{29} + ( 12 + 6 \beta_{1} ) q^{30} + ( -17 \beta_{2} - 6 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} -6 \beta_{2} q^{33} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{34} -6 q^{36} + ( -11 + 12 \beta_{1} ) q^{37} + ( -4 \beta_{2} + 7 \beta_{3} ) q^{38} + ( -3 + 24 \beta_{1} ) q^{39} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{40} + ( -26 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 7 + 6 \beta_{1} ) q^{43} + 12 q^{44} + ( -6 \beta_{2} + 6 \beta_{3} ) q^{45} + ( 36 - 12 \beta_{1} ) q^{46} + ( -22 \beta_{2} - 2 \beta_{3} ) q^{47} -4 \beta_{2} q^{48} + ( -48 - 11 \beta_{1} ) q^{50} + ( 24 + 6 \beta_{1} ) q^{51} + ( -2 \beta_{2} - 16 \beta_{3} ) q^{52} + ( -60 + 18 \beta_{1} ) q^{53} -3 \beta_{3} q^{54} + ( 12 \beta_{2} - 12 \beta_{3} ) q^{55} + ( -21 - 6 \beta_{1} ) q^{57} + 48 q^{58} + ( 4 \beta_{2} + 14 \beta_{3} ) q^{59} + ( 12 + 12 \beta_{1} ) q^{60} + ( -12 \beta_{2} + 8 \beta_{3} ) q^{61} + ( 12 \beta_{2} + 17 \beta_{3} ) q^{62} + 8 q^{64} + ( -90 - 42 \beta_{1} ) q^{65} + 6 \beta_{3} q^{66} + ( -55 - 42 \beta_{1} ) q^{67} + ( 16 \beta_{2} - 4 \beta_{3} ) q^{68} + ( 12 \beta_{2} + 18 \beta_{3} ) q^{69} + ( 78 - 42 \beta_{1} ) q^{71} -6 \beta_{1} q^{72} + ( 11 \beta_{2} - 40 \beta_{3} ) q^{73} + ( 24 - 11 \beta_{1} ) q^{74} + ( 11 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -14 \beta_{2} + 4 \beta_{3} ) q^{76} + ( 48 - 3 \beta_{1} ) q^{78} + ( 5 - 66 \beta_{1} ) q^{79} + ( 8 \beta_{2} - 8 \beta_{3} ) q^{80} + 9 q^{81} + ( 8 \beta_{2} + 26 \beta_{3} ) q^{82} + ( 10 \beta_{2} + 38 \beta_{3} ) q^{83} + ( -72 - 60 \beta_{1} ) q^{85} + ( 12 + 7 \beta_{1} ) q^{86} + 24 \beta_{3} q^{87} + 12 \beta_{1} q^{88} -12 \beta_{2} q^{89} + ( -12 \beta_{2} + 6 \beta_{3} ) q^{90} + ( -24 + 36 \beta_{1} ) q^{92} + ( -51 + 18 \beta_{1} ) q^{93} + ( 4 \beta_{2} + 22 \beta_{3} ) q^{94} + ( 66 + 54 \beta_{1} ) q^{95} + 4 \beta_{3} q^{96} + ( -12 \beta_{2} + 4 \beta_{3} ) q^{97} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 12 q^{9} + O(q^{10})$$ $$4 q + 8 q^{4} - 12 q^{9} + 24 q^{11} + 24 q^{15} + 16 q^{16} - 48 q^{23} - 44 q^{25} + 48 q^{30} - 24 q^{36} - 44 q^{37} - 12 q^{39} + 28 q^{43} + 48 q^{44} + 144 q^{46} - 192 q^{50} + 96 q^{51} - 240 q^{53} - 84 q^{57} + 192 q^{58} + 48 q^{60} + 32 q^{64} - 360 q^{65} - 220 q^{67} + 312 q^{71} + 96 q^{74} + 192 q^{78} + 20 q^{79} + 36 q^{81} - 288 q^{85} + 48 q^{86} - 96 q^{92} - 204 q^{93} + 264 q^{95} - 72 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

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

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i −0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 1.43488i 2.44949i 0 −2.82843 −3.00000 2.02922i
97.2 −1.41421 1.73205i 2.00000 1.43488i 2.44949i 0 −2.82843 −3.00000 2.02922i
97.3 1.41421 1.73205i 2.00000 8.36308i 2.44949i 0 2.82843 −3.00000 11.8272i
97.4 1.41421 1.73205i 2.00000 8.36308i 2.44949i 0 2.82843 −3.00000 11.8272i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.3.c.a 4
3.b odd 2 1 882.3.c.b 4
4.b odd 2 1 2352.3.f.e 4
7.b odd 2 1 inner 294.3.c.a 4
7.c even 3 1 42.3.g.a 4
7.c even 3 1 294.3.g.a 4
7.d odd 6 1 42.3.g.a 4
7.d odd 6 1 294.3.g.a 4
21.c even 2 1 882.3.c.b 4
21.g even 6 1 126.3.n.a 4
21.g even 6 1 882.3.n.e 4
21.h odd 6 1 126.3.n.a 4
21.h odd 6 1 882.3.n.e 4
28.d even 2 1 2352.3.f.e 4
28.f even 6 1 336.3.bh.e 4
28.g odd 6 1 336.3.bh.e 4
35.i odd 6 1 1050.3.p.a 4
35.j even 6 1 1050.3.p.a 4
35.k even 12 2 1050.3.q.a 8
35.l odd 12 2 1050.3.q.a 8
84.j odd 6 1 1008.3.cg.h 4
84.n even 6 1 1008.3.cg.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 7.c even 3 1
42.3.g.a 4 7.d odd 6 1
126.3.n.a 4 21.g even 6 1
126.3.n.a 4 21.h odd 6 1
294.3.c.a 4 1.a even 1 1 trivial
294.3.c.a 4 7.b odd 2 1 inner
294.3.g.a 4 7.c even 3 1
294.3.g.a 4 7.d odd 6 1
336.3.bh.e 4 28.f even 6 1
336.3.bh.e 4 28.g odd 6 1
882.3.c.b 4 3.b odd 2 1
882.3.c.b 4 21.c even 2 1
882.3.n.e 4 21.g even 6 1
882.3.n.e 4 21.h odd 6 1
1008.3.cg.h 4 84.j odd 6 1
1008.3.cg.h 4 84.n even 6 1
1050.3.p.a 4 35.i odd 6 1
1050.3.p.a 4 35.j even 6 1
1050.3.q.a 8 35.k even 12 2
1050.3.q.a 8 35.l odd 12 2
2352.3.f.e 4 4.b odd 2 1
2352.3.f.e 4 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 72 T_{5}^{2} + 144$$ acting on $$S_{3}^{\mathrm{new}}(294, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$144 + 72 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -6 + T )^{4}$$
$13$ $$145161 + 774 T^{2} + T^{4}$$
$17$ $$28224 + 432 T^{2} + T^{4}$$
$19$ $$15129 + 342 T^{2} + T^{4}$$
$23$ $$( -504 + 24 T + T^{2} )^{2}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$423801 + 2166 T^{2} + T^{4}$$
$37$ $$( -167 + 22 T + T^{2} )^{2}$$
$41$ $$3732624 + 4248 T^{2} + T^{4}$$
$43$ $$( -23 - 14 T + T^{2} )^{2}$$
$47$ $$2039184 + 2952 T^{2} + T^{4}$$
$53$ $$( 2952 + 120 T + T^{2} )^{2}$$
$59$ $$1272384 + 2448 T^{2} + T^{4}$$
$61$ $$2304 + 1632 T^{2} + T^{4}$$
$67$ $$( -503 + 110 T + T^{2} )^{2}$$
$71$ $$( 2556 - 156 T + T^{2} )^{2}$$
$73$ $$85322169 + 19926 T^{2} + T^{4}$$
$79$ $$( -8687 - 10 T + T^{2} )^{2}$$
$83$ $$69956496 + 17928 T^{2} + T^{4}$$
$89$ $$( 432 + T^{2} )^{2}$$
$97$ $$112896 + 1056 T^{2} + T^{4}$$