# Properties

 Label 294.4.a.m Level $294$ Weight $4$ Character orbit 294.a Self dual yes Analytic conductor $17.347$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 294.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3465615417$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1345})$$ Defining polynomial: $$x^{2} - x - 336$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1345})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 - \beta ) q^{5} -6 q^{6} + 8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 - \beta ) q^{5} -6 q^{6} + 8 q^{8} + 9 q^{9} + ( -4 - 2 \beta ) q^{10} + ( 34 - \beta ) q^{11} -12 q^{12} + ( -21 + \beta ) q^{13} + ( 6 + 3 \beta ) q^{15} + 16 q^{16} + ( 44 + 4 \beta ) q^{17} + 18 q^{18} + ( -23 + 3 \beta ) q^{19} + ( -8 - 4 \beta ) q^{20} + ( 68 - 2 \beta ) q^{22} + ( 76 - 4 \beta ) q^{23} -24 q^{24} + ( 215 + 5 \beta ) q^{25} + ( -42 + 2 \beta ) q^{26} -27 q^{27} + ( 44 - 11 \beta ) q^{29} + ( 12 + 6 \beta ) q^{30} + ( 261 - 2 \beta ) q^{31} + 32 q^{32} + ( -102 + 3 \beta ) q^{33} + ( 88 + 8 \beta ) q^{34} + 36 q^{36} + ( -1 + 9 \beta ) q^{37} + ( -46 + 6 \beta ) q^{38} + ( 63 - 3 \beta ) q^{39} + ( -16 - 8 \beta ) q^{40} + ( 210 + 6 \beta ) q^{41} + ( -61 + 15 \beta ) q^{43} + ( 136 - 4 \beta ) q^{44} + ( -18 - 9 \beta ) q^{45} + ( 152 - 8 \beta ) q^{46} + ( -282 - 12 \beta ) q^{47} -48 q^{48} + ( 430 + 10 \beta ) q^{50} + ( -132 - 12 \beta ) q^{51} + ( -84 + 4 \beta ) q^{52} + ( -120 - 3 \beta ) q^{53} -54 q^{54} + ( 268 - 31 \beta ) q^{55} + ( 69 - 9 \beta ) q^{57} + ( 88 - 22 \beta ) q^{58} + ( 16 - 25 \beta ) q^{59} + ( 24 + 12 \beta ) q^{60} + ( -114 + 4 \beta ) q^{61} + ( 522 - 4 \beta ) q^{62} + 64 q^{64} + ( -294 + 18 \beta ) q^{65} + ( -204 + 6 \beta ) q^{66} + ( 349 - 11 \beta ) q^{67} + ( 176 + 16 \beta ) q^{68} + ( -228 + 12 \beta ) q^{69} + ( 226 + 20 \beta ) q^{71} + 72 q^{72} + ( 443 + 35 \beta ) q^{73} + ( -2 + 18 \beta ) q^{74} + ( -645 - 15 \beta ) q^{75} + ( -92 + 12 \beta ) q^{76} + ( 126 - 6 \beta ) q^{78} + ( -267 + 8 \beta ) q^{79} + ( -32 - 16 \beta ) q^{80} + 81 q^{81} + ( 420 + 12 \beta ) q^{82} + ( 98 + 25 \beta ) q^{83} + ( -1432 - 56 \beta ) q^{85} + ( -122 + 30 \beta ) q^{86} + ( -132 + 33 \beta ) q^{87} + ( 272 - 8 \beta ) q^{88} + ( -408 + 42 \beta ) q^{89} + ( -36 - 18 \beta ) q^{90} + ( 304 - 16 \beta ) q^{92} + ( -783 + 6 \beta ) q^{93} + ( -564 - 24 \beta ) q^{94} + ( -962 + 14 \beta ) q^{95} -96 q^{96} + ( -994 + 35 \beta ) q^{97} + ( 306 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 16q^{8} + 18q^{9} - 10q^{10} + 67q^{11} - 24q^{12} - 41q^{13} + 15q^{15} + 32q^{16} + 92q^{17} + 36q^{18} - 43q^{19} - 20q^{20} + 134q^{22} + 148q^{23} - 48q^{24} + 435q^{25} - 82q^{26} - 54q^{27} + 77q^{29} + 30q^{30} + 520q^{31} + 64q^{32} - 201q^{33} + 184q^{34} + 72q^{36} + 7q^{37} - 86q^{38} + 123q^{39} - 40q^{40} + 426q^{41} - 107q^{43} + 268q^{44} - 45q^{45} + 296q^{46} - 576q^{47} - 96q^{48} + 870q^{50} - 276q^{51} - 164q^{52} - 243q^{53} - 108q^{54} + 505q^{55} + 129q^{57} + 154q^{58} + 7q^{59} + 60q^{60} - 224q^{61} + 1040q^{62} + 128q^{64} - 570q^{65} - 402q^{66} + 687q^{67} + 368q^{68} - 444q^{69} + 472q^{71} + 144q^{72} + 921q^{73} + 14q^{74} - 1305q^{75} - 172q^{76} + 246q^{78} - 526q^{79} - 80q^{80} + 162q^{81} + 852q^{82} + 221q^{83} - 2920q^{85} - 214q^{86} - 231q^{87} + 536q^{88} - 774q^{89} - 90q^{90} + 592q^{92} - 1560q^{93} - 1152q^{94} - 1910q^{95} - 192q^{96} - 1953q^{97} + 603q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 18.8371 −17.8371
2.00000 −3.00000 4.00000 −20.8371 −6.00000 0 8.00000 9.00000 −41.6742
1.2 2.00000 −3.00000 4.00000 15.8371 −6.00000 0 8.00000 9.00000 31.6742
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.4.a.m 2
3.b odd 2 1 882.4.a.z 2
4.b odd 2 1 2352.4.a.ca 2
7.b odd 2 1 294.4.a.n 2
7.c even 3 2 294.4.e.l 4
7.d odd 6 2 42.4.e.c 4
21.c even 2 1 882.4.a.v 2
21.g even 6 2 126.4.g.g 4
21.h odd 6 2 882.4.g.bf 4
28.d even 2 1 2352.4.a.bq 2
28.f even 6 2 336.4.q.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 7.d odd 6 2
126.4.g.g 4 21.g even 6 2
294.4.a.m 2 1.a even 1 1 trivial
294.4.a.n 2 7.b odd 2 1
294.4.e.l 4 7.c even 3 2
336.4.q.j 4 28.f even 6 2
882.4.a.v 2 21.c even 2 1
882.4.a.z 2 3.b odd 2 1
882.4.g.bf 4 21.h odd 6 2
2352.4.a.bq 2 28.d even 2 1
2352.4.a.ca 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(294))$$:

 $$T_{5}^{2} + 5 T_{5} - 330$$ $$T_{11}^{2} - 67 T_{11} + 786$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-330 + 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$786 - 67 T + T^{2}$$
$13$ $$84 + 41 T + T^{2}$$
$17$ $$-3264 - 92 T + T^{2}$$
$19$ $$-2564 + 43 T + T^{2}$$
$23$ $$96 - 148 T + T^{2}$$
$29$ $$-39204 - 77 T + T^{2}$$
$31$ $$66255 - 520 T + T^{2}$$
$37$ $$-27224 - 7 T + T^{2}$$
$41$ $$33264 - 426 T + T^{2}$$
$43$ $$-72794 + 107 T + T^{2}$$
$47$ $$34524 + 576 T + T^{2}$$
$53$ $$11736 + 243 T + T^{2}$$
$59$ $$-210144 - 7 T + T^{2}$$
$61$ $$7164 + 224 T + T^{2}$$
$67$ $$77306 - 687 T + T^{2}$$
$71$ $$-78804 - 472 T + T^{2}$$
$73$ $$-199846 - 921 T + T^{2}$$
$79$ $$47649 + 526 T + T^{2}$$
$83$ $$-197946 - 221 T + T^{2}$$
$89$ $$-443376 + 774 T + T^{2}$$
$97$ $$541646 + 1953 T + T^{2}$$