# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{5} + 6 q^{6} -8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 + 2 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 4 \beta_{2} q^{4} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{5} + 6 q^{6} -8 q^{8} + ( -9 - 9 \beta_{2} ) q^{9} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{11} + ( 12 + 12 \beta_{2} ) q^{12} + ( 24 + 3 \beta_{3} ) q^{13} + ( -18 + 3 \beta_{3} ) q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( \beta_{1} + 42 \beta_{2} + \beta_{3} ) q^{17} -18 \beta_{2} q^{18} + ( 36 - 2 \beta_{1} + 36 \beta_{2} ) q^{19} + ( 24 - 4 \beta_{3} ) q^{20} + ( 4 + 12 \beta_{3} ) q^{22} + ( 154 + 6 \beta_{1} + 154 \beta_{2} ) q^{23} + 24 \beta_{2} q^{24} + ( 12 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} ) q^{25} + ( 48 - 6 \beta_{1} + 48 \beta_{2} ) q^{26} -27 q^{27} + ( -40 + 18 \beta_{3} ) q^{29} + ( -36 - 6 \beta_{1} - 36 \beta_{2} ) q^{30} + ( 6 \beta_{1} + 192 \beta_{2} + 6 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + ( -6 + 18 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -84 + 2 \beta_{3} ) q^{34} + 36 q^{36} + ( -268 - 12 \beta_{1} - 268 \beta_{2} ) q^{37} + ( -4 \beta_{1} + 72 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 9 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{39} + ( 48 + 8 \beta_{1} + 48 \beta_{2} ) q^{40} + ( 378 + 5 \beta_{3} ) q^{41} + ( 200 + 24 \beta_{3} ) q^{43} + ( 8 - 24 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 9 \beta_{1} + 54 \beta_{2} + 9 \beta_{3} ) q^{45} + ( 12 \beta_{1} + 308 \beta_{2} + 12 \beta_{3} ) q^{46} + ( -156 - 10 \beta_{1} - 156 \beta_{2} ) q^{47} -48 q^{48} + ( -18 + 24 \beta_{3} ) q^{50} + ( 126 + 3 \beta_{1} + 126 \beta_{2} ) q^{51} + ( -12 \beta_{1} + 96 \beta_{2} - 12 \beta_{3} ) q^{52} + ( 24 \beta_{1} - 26 \beta_{2} + 24 \beta_{3} ) q^{53} + ( -54 - 54 \beta_{2} ) q^{54} + ( 576 - 34 \beta_{3} ) q^{55} + ( 108 + 6 \beta_{3} ) q^{57} + ( -80 - 36 \beta_{1} - 80 \beta_{2} ) q^{58} + ( -2 \beta_{1} + 432 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -12 \beta_{1} - 72 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -708 - 13 \beta_{1} - 708 \beta_{2} ) q^{61} + ( -384 + 12 \beta_{3} ) q^{62} + 64 q^{64} + ( 150 - 6 \beta_{1} + 150 \beta_{2} ) q^{65} + ( 36 \beta_{1} - 12 \beta_{2} + 36 \beta_{3} ) q^{66} + ( -24 \beta_{1} + 72 \beta_{2} - 24 \beta_{3} ) q^{67} + ( -168 - 4 \beta_{1} - 168 \beta_{2} ) q^{68} + ( 462 - 18 \beta_{3} ) q^{69} + ( -762 - 30 \beta_{3} ) q^{71} + ( 72 + 72 \beta_{2} ) q^{72} + ( -83 \beta_{1} + 372 \beta_{2} - 83 \beta_{3} ) q^{73} + ( -24 \beta_{1} - 536 \beta_{2} - 24 \beta_{3} ) q^{74} + ( 27 + 36 \beta_{1} + 27 \beta_{2} ) q^{75} + ( -144 - 8 \beta_{3} ) q^{76} + ( 144 + 18 \beta_{3} ) q^{78} + ( -488 + 84 \beta_{1} - 488 \beta_{2} ) q^{79} + ( 16 \beta_{1} + 96 \beta_{2} + 16 \beta_{3} ) q^{80} + 81 \beta_{2} q^{81} + ( 756 - 10 \beta_{1} + 756 \beta_{2} ) q^{82} + ( -156 + 136 \beta_{3} ) q^{83} + ( 350 - 48 \beta_{3} ) q^{85} + ( 400 - 48 \beta_{1} + 400 \beta_{2} ) q^{86} + ( 54 \beta_{1} + 120 \beta_{2} + 54 \beta_{3} ) q^{87} + ( -48 \beta_{1} + 16 \beta_{2} - 48 \beta_{3} ) q^{88} + ( -54 - 29 \beta_{1} - 54 \beta_{2} ) q^{89} + ( -108 + 18 \beta_{3} ) q^{90} + ( -616 + 24 \beta_{3} ) q^{92} + ( 576 + 18 \beta_{1} + 576 \beta_{2} ) q^{93} + ( -20 \beta_{1} - 312 \beta_{2} - 20 \beta_{3} ) q^{94} + ( -24 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} ) q^{95} + ( -96 - 96 \beta_{2} ) q^{96} + ( -372 - 125 \beta_{3} ) q^{97} + ( -18 - 54 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} - 12q^{5} + 24q^{6} - 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} - 12q^{5} + 24q^{6} - 32q^{8} - 18q^{9} + 24q^{10} + 4q^{11} + 24q^{12} + 96q^{13} - 72q^{15} - 32q^{16} - 84q^{17} + 36q^{18} + 72q^{19} + 96q^{20} + 16q^{22} + 308q^{23} - 48q^{24} - 18q^{25} + 96q^{26} - 108q^{27} - 160q^{29} - 72q^{30} - 384q^{31} + 64q^{32} - 12q^{33} - 336q^{34} + 144q^{36} - 536q^{37} - 144q^{38} + 144q^{39} + 96q^{40} + 1512q^{41} + 800q^{43} + 16q^{44} - 108q^{45} - 616q^{46} - 312q^{47} - 192q^{48} - 72q^{50} + 252q^{51} - 192q^{52} + 52q^{53} - 108q^{54} + 2304q^{55} + 432q^{57} - 160q^{58} - 864q^{59} + 144q^{60} - 1416q^{61} - 1536q^{62} + 256q^{64} + 300q^{65} + 24q^{66} - 144q^{67} - 336q^{68} + 1848q^{69} - 3048q^{71} + 144q^{72} - 744q^{73} + 1072q^{74} + 54q^{75} - 576q^{76} + 576q^{78} - 976q^{79} - 192q^{80} - 162q^{81} + 1512q^{82} - 624q^{83} + 1400q^{85} + 800q^{86} - 240q^{87} - 32q^{88} - 108q^{89} - 432q^{90} - 2464q^{92} + 1152q^{93} + 624q^{94} + 40q^{95} - 192q^{96} - 1488q^{97} - 72q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$7 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$7 \nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/7$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$$$/7$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i −7.94975 13.7694i 6.00000 0 −8.00000 −4.50000 7.79423i 15.8995 27.5387i
67.2 1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 1.94975 + 3.37706i 6.00000 0 −8.00000 −4.50000 7.79423i −3.89949 + 6.75412i
79.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i −7.94975 + 13.7694i 6.00000 0 −8.00000 −4.50000 + 7.79423i 15.8995 + 27.5387i
79.2 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 1.94975 3.37706i 6.00000 0 −8.00000 −4.50000 + 7.79423i −3.89949 6.75412i
 $$n$$: 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.o 4
3.b odd 2 1 882.4.g.bd 4
7.b odd 2 1 294.4.e.n 4
7.c even 3 1 294.4.a.j 2
7.c even 3 1 inner 294.4.e.o 4
7.d odd 6 1 294.4.a.k yes 2
7.d odd 6 1 294.4.e.n 4
21.c even 2 1 882.4.g.y 4
21.g even 6 1 882.4.a.bi 2
21.g even 6 1 882.4.g.y 4
21.h odd 6 1 882.4.a.bc 2
21.h odd 6 1 882.4.g.bd 4
28.f even 6 1 2352.4.a.bn 2
28.g odd 6 1 2352.4.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 7.c even 3 1
294.4.a.k yes 2 7.d odd 6 1
294.4.e.n 4 7.b odd 2 1
294.4.e.n 4 7.d odd 6 1
294.4.e.o 4 1.a even 1 1 trivial
294.4.e.o 4 7.c even 3 1 inner
882.4.a.bc 2 21.h odd 6 1
882.4.a.bi 2 21.g even 6 1
882.4.g.y 4 21.c even 2 1
882.4.g.y 4 21.g even 6 1
882.4.g.bd 4 3.b odd 2 1
882.4.g.bd 4 21.h odd 6 1
2352.4.a.bn 2 28.f even 6 1
2352.4.a.cd 2 28.g odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 12 T_{5}^{3} + 206 T_{5}^{2} - 744 T_{5} + 3844$$ $$T_{11}^{4} - 4 T_{11}^{3} + 3540 T_{11}^{2} + 14096 T_{11} + 12418576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$3844 - 744 T + 206 T^{2} + 12 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$12418576 + 14096 T + 3540 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( -306 - 48 T + T^{2} )^{2}$$
$17$ $$2775556 + 139944 T + 5390 T^{2} + 84 T^{3} + T^{4}$$
$19$ $$817216 - 65088 T + 4280 T^{2} - 72 T^{3} + T^{4}$$
$23$ $$407555344 - 6217904 T + 74676 T^{2} - 308 T^{3} + T^{4}$$
$29$ $$( -30152 + 80 T + T^{2} )^{2}$$
$31$ $$1111288896 + 12801024 T + 114120 T^{2} + 384 T^{3} + T^{4}$$
$37$ $$3330674944 + 30933632 T + 229584 T^{2} + 536 T^{3} + T^{4}$$
$41$ $$( 140434 - 756 T + T^{2} )^{2}$$
$43$ $$( -16448 - 400 T + T^{2} )^{2}$$
$47$ $$211295296 + 4535232 T + 82808 T^{2} + 312 T^{3} + T^{4}$$
$53$ $$3110515984 + 2900144 T + 58476 T^{2} - 52 T^{3} + T^{4}$$
$59$ $$34682357824 + 160904448 T + 560264 T^{2} + 864 T^{3} + T^{4}$$
$61$ $$234936028804 + 686338032 T + 1520354 T^{2} + 1416 T^{3} + T^{4}$$
$67$ $$2627997696 - 7382016 T + 72000 T^{2} + 144 T^{3} + T^{4}$$
$71$ $$( 492444 + 1524 T + T^{2} )^{2}$$
$73$ $$288087680644 - 399333072 T + 1090274 T^{2} + 744 T^{3} + T^{4}$$
$79$ $$205520782336 - 442463744 T + 1405920 T^{2} + 976 T^{3} + T^{4}$$
$83$ $$( -1788272 + 312 T + T^{2} )^{2}$$
$89$ $$6320568004 - 8586216 T + 91166 T^{2} + 108 T^{3} + T^{4}$$
$97$ $$( -1392866 + 744 T + T^{2} )^{2}$$