# Properties

 Label 294.2.d.a Level $294$ Weight $2$ Character orbit 294.d Analytic conductor $2.348$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 294.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.34760181943$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
1.00000i −1.73205 −1.00000 −1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.2 1.00000i 1.73205 −1.00000 1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.3 1.00000i −1.73205 −1.00000 −1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
293.4 1.00000i 1.73205 −1.00000 1.73205 1.73205i 0 1.00000i 3.00000 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.d.a 4
3.b odd 2 1 inner 294.2.d.a 4
4.b odd 2 1 2352.2.k.e 4
7.b odd 2 1 inner 294.2.d.a 4
7.c even 3 1 42.2.f.a 4
7.c even 3 1 294.2.f.a 4
7.d odd 6 1 42.2.f.a 4
7.d odd 6 1 294.2.f.a 4
12.b even 2 1 2352.2.k.e 4
21.c even 2 1 inner 294.2.d.a 4
21.g even 6 1 42.2.f.a 4
21.g even 6 1 294.2.f.a 4
21.h odd 6 1 42.2.f.a 4
21.h odd 6 1 294.2.f.a 4
28.d even 2 1 2352.2.k.e 4
28.f even 6 1 336.2.bc.e 4
28.g odd 6 1 336.2.bc.e 4
35.i odd 6 1 1050.2.s.b 4
35.j even 6 1 1050.2.s.b 4
35.k even 12 1 1050.2.u.a 4
35.k even 12 1 1050.2.u.d 4
35.l odd 12 1 1050.2.u.a 4
35.l odd 12 1 1050.2.u.d 4
63.g even 3 1 1134.2.t.d 4
63.h even 3 1 1134.2.l.c 4
63.i even 6 1 1134.2.l.c 4
63.j odd 6 1 1134.2.l.c 4
63.k odd 6 1 1134.2.t.d 4
63.n odd 6 1 1134.2.t.d 4
63.s even 6 1 1134.2.t.d 4
63.t odd 6 1 1134.2.l.c 4
84.h odd 2 1 2352.2.k.e 4
84.j odd 6 1 336.2.bc.e 4
84.n even 6 1 336.2.bc.e 4
105.o odd 6 1 1050.2.s.b 4
105.p even 6 1 1050.2.s.b 4
105.w odd 12 1 1050.2.u.a 4
105.w odd 12 1 1050.2.u.d 4
105.x even 12 1 1050.2.u.a 4
105.x even 12 1 1050.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 7.c even 3 1
42.2.f.a 4 7.d odd 6 1
42.2.f.a 4 21.g even 6 1
42.2.f.a 4 21.h odd 6 1
294.2.d.a 4 1.a even 1 1 trivial
294.2.d.a 4 3.b odd 2 1 inner
294.2.d.a 4 7.b odd 2 1 inner
294.2.d.a 4 21.c even 2 1 inner
294.2.f.a 4 7.c even 3 1
294.2.f.a 4 7.d odd 6 1
294.2.f.a 4 21.g even 6 1
294.2.f.a 4 21.h odd 6 1
336.2.bc.e 4 28.f even 6 1
336.2.bc.e 4 28.g odd 6 1
336.2.bc.e 4 84.j odd 6 1
336.2.bc.e 4 84.n even 6 1
1050.2.s.b 4 35.i odd 6 1
1050.2.s.b 4 35.j even 6 1
1050.2.s.b 4 105.o odd 6 1
1050.2.s.b 4 105.p even 6 1
1050.2.u.a 4 35.k even 12 1
1050.2.u.a 4 35.l odd 12 1
1050.2.u.a 4 105.w odd 12 1
1050.2.u.a 4 105.x even 12 1
1050.2.u.d 4 35.k even 12 1
1050.2.u.d 4 35.l odd 12 1
1050.2.u.d 4 105.w odd 12 1
1050.2.u.d 4 105.x even 12 1
1134.2.l.c 4 63.h even 3 1
1134.2.l.c 4 63.i even 6 1
1134.2.l.c 4 63.j odd 6 1
1134.2.l.c 4 63.t odd 6 1
1134.2.t.d 4 63.g even 3 1
1134.2.t.d 4 63.k odd 6 1
1134.2.t.d 4 63.n odd 6 1
1134.2.t.d 4 63.s even 6 1
2352.2.k.e 4 4.b odd 2 1
2352.2.k.e 4 12.b even 2 1
2352.2.k.e 4 28.d even 2 1
2352.2.k.e 4 84.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 9)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 9)^{2}$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$(T + 2)^{4}$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$(T^{2} - 48)^{2}$$
$53$ $$(T^{2} + 81)^{2}$$
$59$ $$(T^{2} - 3)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T - 2)^{4}$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T + 1)^{4}$$
$83$ $$(T^{2} - 75)^{2}$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$(T^{2} + 27)^{2}$$