# Properties

 Label 294.4.e.e 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -15 \zeta_{6} q^{5} -6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} + ( -4 + 4 \zeta_{6} ) q^{4} -15 \zeta_{6} q^{5} -6 q^{6} -8 q^{8} -9 \zeta_{6} q^{9} + ( 30 - 30 \zeta_{6} ) q^{10} + ( 9 - 9 \zeta_{6} ) q^{11} -12 \zeta_{6} q^{12} + 88 q^{13} + 45 q^{15} -16 \zeta_{6} q^{16} + ( -84 + 84 \zeta_{6} ) q^{17} + ( 18 - 18 \zeta_{6} ) q^{18} + 104 \zeta_{6} q^{19} + 60 q^{20} + 18 q^{22} + 84 \zeta_{6} q^{23} + ( 24 - 24 \zeta_{6} ) q^{24} + ( -100 + 100 \zeta_{6} ) q^{25} + 176 \zeta_{6} q^{26} + 27 q^{27} + 51 q^{29} + 90 \zeta_{6} q^{30} + ( 185 - 185 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} + 27 \zeta_{6} q^{33} -168 q^{34} + 36 q^{36} -44 \zeta_{6} q^{37} + ( -208 + 208 \zeta_{6} ) q^{38} + ( -264 + 264 \zeta_{6} ) q^{39} + 120 \zeta_{6} q^{40} + 168 q^{41} + 326 q^{43} + 36 \zeta_{6} q^{44} + ( -135 + 135 \zeta_{6} ) q^{45} + ( -168 + 168 \zeta_{6} ) q^{46} -138 \zeta_{6} q^{47} + 48 q^{48} -200 q^{50} -252 \zeta_{6} q^{51} + ( -352 + 352 \zeta_{6} ) q^{52} + ( -639 + 639 \zeta_{6} ) q^{53} + 54 \zeta_{6} q^{54} -135 q^{55} -312 q^{57} + 102 \zeta_{6} q^{58} + ( 159 - 159 \zeta_{6} ) q^{59} + ( -180 + 180 \zeta_{6} ) q^{60} + 722 \zeta_{6} q^{61} + 370 q^{62} + 64 q^{64} -1320 \zeta_{6} q^{65} + ( -54 + 54 \zeta_{6} ) q^{66} + ( 166 - 166 \zeta_{6} ) q^{67} -336 \zeta_{6} q^{68} -252 q^{69} + 1086 q^{71} + 72 \zeta_{6} q^{72} + ( 218 - 218 \zeta_{6} ) q^{73} + ( 88 - 88 \zeta_{6} ) q^{74} -300 \zeta_{6} q^{75} -416 q^{76} -528 q^{78} + 583 \zeta_{6} q^{79} + ( -240 + 240 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 336 \zeta_{6} q^{82} + 597 q^{83} + 1260 q^{85} + 652 \zeta_{6} q^{86} + ( -153 + 153 \zeta_{6} ) q^{87} + ( -72 + 72 \zeta_{6} ) q^{88} -1038 \zeta_{6} q^{89} -270 q^{90} -336 q^{92} + 555 \zeta_{6} q^{93} + ( 276 - 276 \zeta_{6} ) q^{94} + ( 1560 - 1560 \zeta_{6} ) q^{95} + 96 \zeta_{6} q^{96} + 169 q^{97} -81 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} - 15q^{5} - 12q^{6} - 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} - 4q^{4} - 15q^{5} - 12q^{6} - 16q^{8} - 9q^{9} + 30q^{10} + 9q^{11} - 12q^{12} + 176q^{13} + 90q^{15} - 16q^{16} - 84q^{17} + 18q^{18} + 104q^{19} + 120q^{20} + 36q^{22} + 84q^{23} + 24q^{24} - 100q^{25} + 176q^{26} + 54q^{27} + 102q^{29} + 90q^{30} + 185q^{31} + 32q^{32} + 27q^{33} - 336q^{34} + 72q^{36} - 44q^{37} - 208q^{38} - 264q^{39} + 120q^{40} + 336q^{41} + 652q^{43} + 36q^{44} - 135q^{45} - 168q^{46} - 138q^{47} + 96q^{48} - 400q^{50} - 252q^{51} - 352q^{52} - 639q^{53} + 54q^{54} - 270q^{55} - 624q^{57} + 102q^{58} + 159q^{59} - 180q^{60} + 722q^{61} + 740q^{62} + 128q^{64} - 1320q^{65} - 54q^{66} + 166q^{67} - 336q^{68} - 504q^{69} + 2172q^{71} + 72q^{72} + 218q^{73} + 88q^{74} - 300q^{75} - 832q^{76} - 1056q^{78} + 583q^{79} - 240q^{80} - 81q^{81} + 336q^{82} + 1194q^{83} + 2520q^{85} + 652q^{86} - 153q^{87} - 72q^{88} - 1038q^{89} - 540q^{90} - 672q^{92} + 555q^{93} + 276q^{94} + 1560q^{95} + 96q^{96} + 338q^{97} - 162q^{99} + O(q^{100})$$

## 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 −7.50000 12.9904i −6.00000 0 −8.00000 −4.50000 7.79423i 15.0000 25.9808i
79.1 1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i −7.50000 + 12.9904i −6.00000 0 −8.00000 −4.50000 + 7.79423i 15.0000 + 25.9808i
 $$n$$: 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.e 2
3.b odd 2 1 882.4.g.l 2
7.b odd 2 1 42.4.e.b 2
7.c even 3 1 294.4.a.g 1
7.c even 3 1 inner 294.4.e.e 2
7.d odd 6 1 42.4.e.b 2
7.d odd 6 1 294.4.a.a 1
21.c even 2 1 126.4.g.a 2
21.g even 6 1 126.4.g.a 2
21.g even 6 1 882.4.a.r 1
21.h odd 6 1 882.4.a.h 1
21.h odd 6 1 882.4.g.l 2
28.d even 2 1 336.4.q.d 2
28.f even 6 1 336.4.q.d 2
28.f even 6 1 2352.4.a.u 1
28.g odd 6 1 2352.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 7.b odd 2 1
42.4.e.b 2 7.d odd 6 1
126.4.g.a 2 21.c even 2 1
126.4.g.a 2 21.g even 6 1
294.4.a.a 1 7.d odd 6 1
294.4.a.g 1 7.c even 3 1
294.4.e.e 2 1.a even 1 1 trivial
294.4.e.e 2 7.c even 3 1 inner
336.4.q.d 2 28.d even 2 1
336.4.q.d 2 28.f even 6 1
882.4.a.h 1 21.h odd 6 1
882.4.a.r 1 21.g even 6 1
882.4.g.l 2 3.b odd 2 1
882.4.g.l 2 21.h odd 6 1
2352.4.a.q 1 28.g odd 6 1
2352.4.a.u 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 $$S_{4}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{2} + 15 T_{5} + 225$$ $$T_{11}^{2} - 9 T_{11} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$225 + 15 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$81 - 9 T + T^{2}$$
$13$ $$( -88 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$10816 - 104 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( -51 + T )^{2}$$
$31$ $$34225 - 185 T + T^{2}$$
$37$ $$1936 + 44 T + T^{2}$$
$41$ $$( -168 + T )^{2}$$
$43$ $$( -326 + T )^{2}$$
$47$ $$19044 + 138 T + T^{2}$$
$53$ $$408321 + 639 T + T^{2}$$
$59$ $$25281 - 159 T + T^{2}$$
$61$ $$521284 - 722 T + T^{2}$$
$67$ $$27556 - 166 T + T^{2}$$
$71$ $$( -1086 + T )^{2}$$
$73$ $$47524 - 218 T + T^{2}$$
$79$ $$339889 - 583 T + T^{2}$$
$83$ $$( -597 + T )^{2}$$
$89$ $$1077444 + 1038 T + T^{2}$$
$97$ $$( -169 + T )^{2}$$