# Properties

 Label 294.4.a.n 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 3 - \beta ) q^{5} + 6 q^{6} + 8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 3 - \beta ) q^{5} + 6 q^{6} + 8 q^{8} + 9 q^{9} + ( 6 - 2 \beta ) q^{10} + ( 33 + \beta ) q^{11} + 12 q^{12} + ( 20 + \beta ) q^{13} + ( 9 - 3 \beta ) q^{15} + 16 q^{16} + ( -48 + 4 \beta ) q^{17} + 18 q^{18} + ( 20 + 3 \beta ) q^{19} + ( 12 - 4 \beta ) q^{20} + ( 66 + 2 \beta ) q^{22} + ( 72 + 4 \beta ) q^{23} + 24 q^{24} + ( 220 - 5 \beta ) q^{25} + ( 40 + 2 \beta ) q^{26} + 27 q^{27} + ( 33 + 11 \beta ) q^{29} + ( 18 - 6 \beta ) q^{30} + ( -259 - 2 \beta ) q^{31} + 32 q^{32} + ( 99 + 3 \beta ) q^{33} + ( -96 + 8 \beta ) q^{34} + 36 q^{36} + ( 8 - 9 \beta ) q^{37} + ( 40 + 6 \beta ) q^{38} + ( 60 + 3 \beta ) q^{39} + ( 24 - 8 \beta ) q^{40} + ( -216 + 6 \beta ) q^{41} + ( -46 - 15 \beta ) q^{43} + ( 132 + 4 \beta ) q^{44} + ( 27 - 9 \beta ) q^{45} + ( 144 + 8 \beta ) q^{46} + ( 294 - 12 \beta ) q^{47} + 48 q^{48} + ( 440 - 10 \beta ) q^{50} + ( -144 + 12 \beta ) q^{51} + ( 80 + 4 \beta ) q^{52} + ( -123 + 3 \beta ) q^{53} + 54 q^{54} + ( -237 - 31 \beta ) q^{55} + ( 60 + 9 \beta ) q^{57} + ( 66 + 22 \beta ) q^{58} + ( 9 - 25 \beta ) q^{59} + ( 36 - 12 \beta ) q^{60} + ( 110 + 4 \beta ) q^{61} + ( -518 - 4 \beta ) q^{62} + 64 q^{64} + ( -276 - 18 \beta ) q^{65} + ( 198 + 6 \beta ) q^{66} + ( 338 + 11 \beta ) q^{67} + ( -192 + 16 \beta ) q^{68} + ( 216 + 12 \beta ) q^{69} + ( 246 - 20 \beta ) q^{71} + 72 q^{72} + ( -478 + 35 \beta ) q^{73} + ( 16 - 18 \beta ) q^{74} + ( 660 - 15 \beta ) q^{75} + ( 80 + 12 \beta ) q^{76} + ( 120 + 6 \beta ) q^{78} + ( -259 - 8 \beta ) q^{79} + ( 48 - 16 \beta ) q^{80} + 81 q^{81} + ( -432 + 12 \beta ) q^{82} + ( -123 + 25 \beta ) q^{83} + ( -1488 + 56 \beta ) q^{85} + ( -92 - 30 \beta ) q^{86} + ( 99 + 33 \beta ) q^{87} + ( 264 + 8 \beta ) q^{88} + ( 366 + 42 \beta ) q^{89} + ( 54 - 18 \beta ) q^{90} + ( 288 + 16 \beta ) q^{92} + ( -777 - 6 \beta ) q^{93} + ( 588 - 24 \beta ) q^{94} + ( -948 - 14 \beta ) q^{95} + 96 q^{96} + ( 959 + 35 \beta ) q^{97} + ( 297 + 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 5q^{5} + 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 5q^{5} + 12q^{6} + 16q^{8} + 18q^{9} + 10q^{10} + 67q^{11} + 24q^{12} + 41q^{13} + 15q^{15} + 32q^{16} - 92q^{17} + 36q^{18} + 43q^{19} + 20q^{20} + 134q^{22} + 148q^{23} + 48q^{24} + 435q^{25} + 82q^{26} + 54q^{27} + 77q^{29} + 30q^{30} - 520q^{31} + 64q^{32} + 201q^{33} - 184q^{34} + 72q^{36} + 7q^{37} + 86q^{38} + 123q^{39} + 40q^{40} - 426q^{41} - 107q^{43} + 268q^{44} + 45q^{45} + 296q^{46} + 576q^{47} + 96q^{48} + 870q^{50} - 276q^{51} + 164q^{52} - 243q^{53} + 108q^{54} - 505q^{55} + 129q^{57} + 154q^{58} - 7q^{59} + 60q^{60} + 224q^{61} - 1040q^{62} + 128q^{64} - 570q^{65} + 402q^{66} + 687q^{67} - 368q^{68} + 444q^{69} + 472q^{71} + 144q^{72} - 921q^{73} + 14q^{74} + 1305q^{75} + 172q^{76} + 246q^{78} - 526q^{79} + 80q^{80} + 162q^{81} - 852q^{82} - 221q^{83} - 2920q^{85} - 214q^{86} + 231q^{87} + 536q^{88} + 774q^{89} + 90q^{90} + 592q^{92} - 1560q^{93} + 1152q^{94} - 1910q^{95} + 192q^{96} + 1953q^{97} + 603q^{99} + O(q^{100})$$

## 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 −15.8371 6.00000 0 8.00000 9.00000 −31.6742
1.2 2.00000 3.00000 4.00000 20.8371 6.00000 0 8.00000 9.00000 41.6742
 $$n$$: 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.n 2
3.b odd 2 1 882.4.a.v 2
4.b odd 2 1 2352.4.a.bq 2
7.b odd 2 1 294.4.a.m 2
7.c even 3 2 42.4.e.c 4
7.d odd 6 2 294.4.e.l 4
21.c even 2 1 882.4.a.z 2
21.g even 6 2 882.4.g.bf 4
21.h odd 6 2 126.4.g.g 4
28.d even 2 1 2352.4.a.ca 2
28.g odd 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.c even 3 2
126.4.g.g 4 21.h odd 6 2
294.4.a.m 2 7.b odd 2 1
294.4.a.n 2 1.a even 1 1 trivial
294.4.e.l 4 7.d odd 6 2
336.4.q.j 4 28.g odd 6 2
882.4.a.v 2 3.b odd 2 1
882.4.a.z 2 21.c even 2 1
882.4.g.bf 4 21.g even 6 2
2352.4.a.bq 2 4.b odd 2 1
2352.4.a.ca 2 28.d even 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

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