# Properties

 Label 294.4.e.a Level $294$ Weight $4$ Character orbit 294.e Analytic conductor $17.347$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 294.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3465615417$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} + 6 q^{6} + 8 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} + 6 q^{6} + 8 q^{8} -9 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{10} + ( 8 - 8 \zeta_{6} ) q^{11} -12 \zeta_{6} q^{12} -42 q^{13} + 6 q^{15} -16 \zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -18 + 18 \zeta_{6} ) q^{18} + 124 \zeta_{6} q^{19} + 8 q^{20} -16 q^{22} -76 \zeta_{6} q^{23} + ( -24 + 24 \zeta_{6} ) q^{24} + ( 121 - 121 \zeta_{6} ) q^{25} + 84 \zeta_{6} q^{26} + 27 q^{27} + 254 q^{29} -12 \zeta_{6} q^{30} + ( 72 - 72 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 24 \zeta_{6} q^{33} -4 q^{34} + 36 q^{36} -398 \zeta_{6} q^{37} + ( 248 - 248 \zeta_{6} ) q^{38} + ( 126 - 126 \zeta_{6} ) q^{39} -16 \zeta_{6} q^{40} + 462 q^{41} + 212 q^{43} + 32 \zeta_{6} q^{44} + ( -18 + 18 \zeta_{6} ) q^{45} + ( -152 + 152 \zeta_{6} ) q^{46} + 264 \zeta_{6} q^{47} + 48 q^{48} -242 q^{50} + 6 \zeta_{6} q^{51} + ( 168 - 168 \zeta_{6} ) q^{52} + ( 162 - 162 \zeta_{6} ) q^{53} -54 \zeta_{6} q^{54} -16 q^{55} -372 q^{57} -508 \zeta_{6} q^{58} + ( 772 - 772 \zeta_{6} ) q^{59} + ( -24 + 24 \zeta_{6} ) q^{60} -30 \zeta_{6} q^{61} -144 q^{62} + 64 q^{64} + 84 \zeta_{6} q^{65} + ( 48 - 48 \zeta_{6} ) q^{66} + ( 764 - 764 \zeta_{6} ) q^{67} + 8 \zeta_{6} q^{68} + 228 q^{69} -236 q^{71} -72 \zeta_{6} q^{72} + ( -418 + 418 \zeta_{6} ) q^{73} + ( -796 + 796 \zeta_{6} ) q^{74} + 363 \zeta_{6} q^{75} -496 q^{76} -252 q^{78} -552 \zeta_{6} q^{79} + ( -32 + 32 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} -924 \zeta_{6} q^{82} + 1036 q^{83} -4 q^{85} -424 \zeta_{6} q^{86} + ( -762 + 762 \zeta_{6} ) q^{87} + ( 64 - 64 \zeta_{6} ) q^{88} -30 \zeta_{6} q^{89} + 36 q^{90} + 304 q^{92} + 216 \zeta_{6} q^{93} + ( 528 - 528 \zeta_{6} ) q^{94} + ( 248 - 248 \zeta_{6} ) q^{95} -96 \zeta_{6} q^{96} -1190 q^{97} -72 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 3q^{3} - 4q^{4} - 2q^{5} + 12q^{6} + 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 3q^{3} - 4q^{4} - 2q^{5} + 12q^{6} + 16q^{8} - 9q^{9} - 4q^{10} + 8q^{11} - 12q^{12} - 84q^{13} + 12q^{15} - 16q^{16} + 2q^{17} - 18q^{18} + 124q^{19} + 16q^{20} - 32q^{22} - 76q^{23} - 24q^{24} + 121q^{25} + 84q^{26} + 54q^{27} + 508q^{29} - 12q^{30} + 72q^{31} - 32q^{32} + 24q^{33} - 8q^{34} + 72q^{36} - 398q^{37} + 248q^{38} + 126q^{39} - 16q^{40} + 924q^{41} + 424q^{43} + 32q^{44} - 18q^{45} - 152q^{46} + 264q^{47} + 96q^{48} - 484q^{50} + 6q^{51} + 168q^{52} + 162q^{53} - 54q^{54} - 32q^{55} - 744q^{57} - 508q^{58} + 772q^{59} - 24q^{60} - 30q^{61} - 288q^{62} + 128q^{64} + 84q^{65} + 48q^{66} + 764q^{67} + 8q^{68} + 456q^{69} - 472q^{71} - 72q^{72} - 418q^{73} - 796q^{74} + 363q^{75} - 992q^{76} - 504q^{78} - 552q^{79} - 32q^{80} - 81q^{81} - 924q^{82} + 2072q^{83} - 8q^{85} - 424q^{86} - 762q^{87} + 64q^{88} - 30q^{89} + 72q^{90} + 608q^{92} + 216q^{93} + 528q^{94} + 248q^{95} - 96q^{96} - 2380q^{97} - 144q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −1.00000 1.73205i 6.00000 0 8.00000 −4.50000 7.79423i −2.00000 + 3.46410i
79.1 −1.00000 + 1.73205i −1.50000 2.59808i −2.00000 3.46410i −1.00000 + 1.73205i 6.00000 0 8.00000 −4.50000 + 7.79423i −2.00000 3.46410i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.4.e.a 2
3.b odd 2 1 882.4.g.s 2
7.b odd 2 1 294.4.e.d 2
7.c even 3 1 42.4.a.b 1
7.c even 3 1 inner 294.4.e.a 2
7.d odd 6 1 294.4.a.h 1
7.d odd 6 1 294.4.e.d 2
21.c even 2 1 882.4.g.r 2
21.g even 6 1 882.4.a.d 1
21.g even 6 1 882.4.g.r 2
21.h odd 6 1 126.4.a.c 1
21.h odd 6 1 882.4.g.s 2
28.f even 6 1 2352.4.a.ba 1
28.g odd 6 1 336.4.a.d 1
35.j even 6 1 1050.4.a.d 1
35.l odd 12 2 1050.4.g.n 2
56.k odd 6 1 1344.4.a.t 1
56.p even 6 1 1344.4.a.f 1
84.n even 6 1 1008.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 7.c even 3 1
126.4.a.c 1 21.h odd 6 1
294.4.a.h 1 7.d odd 6 1
294.4.e.a 2 1.a even 1 1 trivial
294.4.e.a 2 7.c even 3 1 inner
294.4.e.d 2 7.b odd 2 1
294.4.e.d 2 7.d odd 6 1
336.4.a.d 1 28.g odd 6 1
882.4.a.d 1 21.g even 6 1
882.4.g.r 2 21.c even 2 1
882.4.g.r 2 21.g even 6 1
882.4.g.s 2 3.b odd 2 1
882.4.g.s 2 21.h odd 6 1
1008.4.a.j 1 84.n even 6 1
1050.4.a.d 1 35.j even 6 1
1050.4.g.n 2 35.l odd 12 2
1344.4.a.f 1 56.p even 6 1
1344.4.a.t 1 56.k odd 6 1
2352.4.a.ba 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} - 8 T_{11} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$64 - 8 T + T^{2}$$
$13$ $$( 42 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$15376 - 124 T + T^{2}$$
$23$ $$5776 + 76 T + T^{2}$$
$29$ $$( -254 + T )^{2}$$
$31$ $$5184 - 72 T + T^{2}$$
$37$ $$158404 + 398 T + T^{2}$$
$41$ $$( -462 + T )^{2}$$
$43$ $$( -212 + T )^{2}$$
$47$ $$69696 - 264 T + T^{2}$$
$53$ $$26244 - 162 T + T^{2}$$
$59$ $$595984 - 772 T + T^{2}$$
$61$ $$900 + 30 T + T^{2}$$
$67$ $$583696 - 764 T + T^{2}$$
$71$ $$( 236 + T )^{2}$$
$73$ $$174724 + 418 T + T^{2}$$
$79$ $$304704 + 552 T + T^{2}$$
$83$ $$( -1036 + T )^{2}$$
$89$ $$900 + 30 T + T^{2}$$
$97$ $$( 1190 + T )^{2}$$