# Properties

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

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## 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + ( -\beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{5} -6 q^{6} + 8 q^{8} + 9 \beta_{2} q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( 3 + 3 \beta_{2} ) q^{3} + ( -4 - 4 \beta_{2} ) q^{4} + ( -\beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{5} -6 q^{6} + 8 q^{8} + 9 \beta_{2} q^{9} + ( 12 + 2 \beta_{1} + 12 \beta_{2} ) q^{10} + ( -2 - 6 \beta_{1} - 2 \beta_{2} ) q^{11} -12 \beta_{2} q^{12} + ( -24 - 15 \beta_{3} ) q^{13} + ( 18 - 3 \beta_{3} ) q^{15} + 16 \beta_{2} q^{16} + ( 66 - 11 \beta_{1} + 66 \beta_{2} ) q^{17} + ( -18 - 18 \beta_{2} ) q^{18} + ( -46 \beta_{1} - 60 \beta_{2} - 46 \beta_{3} ) q^{19} + ( -24 + 4 \beta_{3} ) q^{20} + ( 4 - 12 \beta_{3} ) q^{22} + ( -102 \beta_{1} - 38 \beta_{2} - 102 \beta_{3} ) q^{23} + ( 24 + 24 \beta_{2} ) q^{24} + ( 87 - 12 \beta_{1} + 87 \beta_{2} ) q^{25} + ( 30 \beta_{1} - 48 \beta_{2} + 30 \beta_{3} ) q^{26} -27 q^{27} + ( -56 - 150 \beta_{3} ) q^{29} + ( 6 \beta_{1} + 36 \beta_{2} + 6 \beta_{3} ) q^{30} + ( 216 - 54 \beta_{1} + 216 \beta_{2} ) q^{31} + ( -32 - 32 \beta_{2} ) q^{32} + ( -18 \beta_{1} - 6 \beta_{2} - 18 \beta_{3} ) q^{33} + ( -132 - 22 \beta_{3} ) q^{34} + 36 q^{36} + ( -180 \beta_{1} - 140 \beta_{2} - 180 \beta_{3} ) q^{37} + ( 120 + 92 \beta_{1} + 120 \beta_{2} ) q^{38} + ( -72 + 45 \beta_{1} - 72 \beta_{2} ) q^{39} + ( -8 \beta_{1} - 48 \beta_{2} - 8 \beta_{3} ) q^{40} + ( -18 + 127 \beta_{3} ) q^{41} + ( -64 - 288 \beta_{3} ) q^{43} + ( 24 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{44} + ( 54 + 9 \beta_{1} + 54 \beta_{2} ) q^{45} + ( 76 + 204 \beta_{1} + 76 \beta_{2} ) q^{46} + ( 338 \beta_{1} + 132 \beta_{2} + 338 \beta_{3} ) q^{47} -48 q^{48} + ( -174 - 24 \beta_{3} ) q^{50} + ( -33 \beta_{1} + 198 \beta_{2} - 33 \beta_{3} ) q^{51} + ( 96 - 60 \beta_{1} + 96 \beta_{2} ) q^{52} + ( -134 + 192 \beta_{1} - 134 \beta_{2} ) q^{53} -54 \beta_{2} q^{54} + ( -24 + 38 \beta_{3} ) q^{55} + ( 180 - 138 \beta_{3} ) q^{57} + ( 300 \beta_{1} - 112 \beta_{2} + 300 \beta_{3} ) q^{58} + ( 168 + 298 \beta_{1} + 168 \beta_{2} ) q^{59} + ( -72 - 12 \beta_{1} - 72 \beta_{2} ) q^{60} + ( -353 \beta_{1} + 252 \beta_{2} - 353 \beta_{3} ) q^{61} + ( -432 - 108 \beta_{3} ) q^{62} + 64 q^{64} + ( -66 \beta_{1} + 114 \beta_{2} - 66 \beta_{3} ) q^{65} + ( 12 + 36 \beta_{1} + 12 \beta_{2} ) q^{66} + ( 192 + 144 \beta_{1} + 192 \beta_{2} ) q^{67} + ( 44 \beta_{1} - 264 \beta_{2} + 44 \beta_{3} ) q^{68} + ( 114 - 306 \beta_{3} ) q^{69} + ( -198 - 342 \beta_{3} ) q^{71} + 72 \beta_{2} q^{72} + ( -156 - 595 \beta_{1} - 156 \beta_{2} ) q^{73} + ( 280 + 360 \beta_{1} + 280 \beta_{2} ) q^{74} + ( -36 \beta_{1} + 261 \beta_{2} - 36 \beta_{3} ) q^{75} + ( -240 + 184 \beta_{3} ) q^{76} + ( 144 + 90 \beta_{3} ) q^{78} + ( -300 \beta_{1} - 424 \beta_{2} - 300 \beta_{3} ) q^{79} + ( 96 + 16 \beta_{1} + 96 \beta_{2} ) q^{80} + ( -81 - 81 \beta_{2} ) q^{81} + ( -254 \beta_{1} - 36 \beta_{2} - 254 \beta_{3} ) q^{82} + ( 324 + 80 \beta_{3} ) q^{83} + 374 q^{85} + ( 576 \beta_{1} - 128 \beta_{2} + 576 \beta_{3} ) q^{86} + ( -168 + 450 \beta_{1} - 168 \beta_{2} ) q^{87} + ( -16 - 48 \beta_{1} - 16 \beta_{2} ) q^{88} + ( 175 \beta_{1} + 306 \beta_{2} + 175 \beta_{3} ) q^{89} + ( -108 + 18 \beta_{3} ) q^{90} + ( -152 + 408 \beta_{3} ) q^{92} + ( -162 \beta_{1} + 648 \beta_{2} - 162 \beta_{3} ) q^{93} + ( -264 - 676 \beta_{1} - 264 \beta_{2} ) q^{94} + ( -452 - 336 \beta_{1} - 452 \beta_{2} ) q^{95} -96 \beta_{2} q^{96} + ( 1092 + 133 \beta_{3} ) q^{97} + ( 18 - 54 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} + 12q^{5} - 24q^{6} + 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 6q^{3} - 8q^{4} + 12q^{5} - 24q^{6} + 32q^{8} - 18q^{9} + 24q^{10} - 4q^{11} + 24q^{12} - 96q^{13} + 72q^{15} - 32q^{16} + 132q^{17} - 36q^{18} + 120q^{19} - 96q^{20} + 16q^{22} + 76q^{23} + 48q^{24} + 174q^{25} + 96q^{26} - 108q^{27} - 224q^{29} - 72q^{30} + 432q^{31} - 64q^{32} + 12q^{33} - 528q^{34} + 144q^{36} + 280q^{37} + 240q^{38} - 144q^{39} + 96q^{40} - 72q^{41} - 256q^{43} - 16q^{44} + 108q^{45} + 152q^{46} - 264q^{47} - 192q^{48} - 696q^{50} - 396q^{51} + 192q^{52} - 268q^{53} + 108q^{54} - 96q^{55} + 720q^{57} + 224q^{58} + 336q^{59} - 144q^{60} - 504q^{61} - 1728q^{62} + 256q^{64} - 228q^{65} + 24q^{66} + 384q^{67} + 528q^{68} + 456q^{69} - 792q^{71} - 144q^{72} - 312q^{73} + 560q^{74} - 522q^{75} - 960q^{76} + 576q^{78} + 848q^{79} + 192q^{80} - 162q^{81} + 72q^{82} + 1296q^{83} + 1496q^{85} + 256q^{86} - 336q^{87} - 32q^{88} - 612q^{89} - 432q^{90} - 608q^{92} - 1296q^{93} - 528q^{94} - 904q^{95} + 192q^{96} + 4368q^{97} + 72q^{99} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$197$$ $$199$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 2.29289 + 3.97141i −6.00000 0 8.00000 −4.50000 7.79423i 4.58579 7.94282i
67.2 −1.00000 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 3.70711 + 6.42090i −6.00000 0 8.00000 −4.50000 7.79423i 7.41421 12.8418i
79.1 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.29289 3.97141i −6.00000 0 8.00000 −4.50000 + 7.79423i 4.58579 + 7.94282i
79.2 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 3.70711 6.42090i −6.00000 0 8.00000 −4.50000 + 7.79423i 7.41421 + 12.8418i
 $$n$$: 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.m 4
3.b odd 2 1 882.4.g.be 4
7.b odd 2 1 294.4.e.k 4
7.c even 3 1 294.4.a.l 2
7.c even 3 1 inner 294.4.e.m 4
7.d odd 6 1 294.4.a.o yes 2
7.d odd 6 1 294.4.e.k 4
21.c even 2 1 882.4.g.bk 4
21.g even 6 1 882.4.a.t 2
21.g even 6 1 882.4.g.bk 4
21.h odd 6 1 882.4.a.bb 2
21.h odd 6 1 882.4.g.be 4
28.f even 6 1 2352.4.a.bu 2
28.g odd 6 1 2352.4.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 7.c even 3 1
294.4.a.o yes 2 7.d odd 6 1
294.4.e.k 4 7.b odd 2 1
294.4.e.k 4 7.d odd 6 1
294.4.e.m 4 1.a even 1 1 trivial
294.4.e.m 4 7.c even 3 1 inner
882.4.a.t 2 21.g even 6 1
882.4.a.bb 2 21.h odd 6 1
882.4.g.be 4 3.b odd 2 1
882.4.g.be 4 21.h odd 6 1
882.4.g.bk 4 21.c even 2 1
882.4.g.bk 4 21.g even 6 1
2352.4.a.bu 2 28.f even 6 1
2352.4.a.bw 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 $$S_{4}^{\mathrm{new}}(294, [\chi])$$:

 $$T_{5}^{4} - 12 T_{5}^{3} + 110 T_{5}^{2} - 408 T_{5} + 1156$$ $$T_{11}^{4} + 4 T_{11}^{3} + 84 T_{11}^{2} - 272 T_{11} + 4624$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$1156 - 408 T + 110 T^{2} - 12 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( 126 + 48 T + T^{2} )^{2}$$
$17$ $$16924996 - 543048 T + 13310 T^{2} - 132 T^{3} + T^{4}$$
$19$ $$399424 + 75840 T + 15032 T^{2} - 120 T^{3} + T^{4}$$
$23$ $$374964496 + 1471664 T + 25140 T^{2} - 76 T^{3} + T^{4}$$
$29$ $$( -41864 + 112 T + T^{2} )^{2}$$
$31$ $$1666598976 - 17635968 T + 145800 T^{2} - 432 T^{3} + T^{4}$$
$37$ $$2043040000 + 12656000 T + 123600 T^{2} - 280 T^{3} + T^{4}$$
$41$ $$( -31934 + 36 T + T^{2} )^{2}$$
$43$ $$( -161792 + 128 T + T^{2} )^{2}$$
$47$ $$44548012096 - 55720896 T + 280760 T^{2} + 264 T^{3} + T^{4}$$
$53$ $$3110515984 - 14946896 T + 127596 T^{2} + 268 T^{3} + T^{4}$$
$59$ $$22315579456 + 50193024 T + 262280 T^{2} - 336 T^{3} + T^{4}$$
$61$ $$34489689796 - 93599856 T + 439730 T^{2} + 504 T^{3} + T^{4}$$
$67$ $$21233664 + 1769472 T + 152064 T^{2} - 384 T^{3} + T^{4}$$
$71$ $$( -194724 + 396 T + T^{2} )^{2}$$
$73$ $$467464833796 - 213318768 T + 781058 T^{2} + 312 T^{3} + T^{4}$$
$79$ $$50176 + 189952 T + 719328 T^{2} - 848 T^{3} + T^{4}$$
$83$ $$( 92176 - 648 T + T^{2} )^{2}$$
$89$ $$1048852996 + 19820232 T + 342158 T^{2} + 612 T^{3} + T^{4}$$
$97$ $$( 1157086 - 2184 T + T^{2} )^{2}$$
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