# Properties

 Label 294.4.a.o 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 6 + \beta ) q^{5} + 6 q^{6} + 8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( 6 + \beta ) q^{5} + 6 q^{6} + 8 q^{8} + 9 q^{9} + ( 12 + 2 \beta ) q^{10} + ( 2 + 6 \beta ) q^{11} + 12 q^{12} + ( 24 - 15 \beta ) q^{13} + ( 18 + 3 \beta ) q^{15} + 16 q^{16} + ( 66 - 11 \beta ) q^{17} + 18 q^{18} + ( 60 + 46 \beta ) q^{19} + ( 24 + 4 \beta ) q^{20} + ( 4 + 12 \beta ) q^{22} + ( -38 - 102 \beta ) q^{23} + 24 q^{24} + ( -87 + 12 \beta ) q^{25} + ( 48 - 30 \beta ) q^{26} + 27 q^{27} + ( -56 + 150 \beta ) q^{29} + ( 36 + 6 \beta ) q^{30} + ( 216 - 54 \beta ) q^{31} + 32 q^{32} + ( 6 + 18 \beta ) q^{33} + ( 132 - 22 \beta ) q^{34} + 36 q^{36} + ( -140 - 180 \beta ) q^{37} + ( 120 + 92 \beta ) q^{38} + ( 72 - 45 \beta ) q^{39} + ( 48 + 8 \beta ) q^{40} + ( 18 + 127 \beta ) q^{41} + ( -64 + 288 \beta ) q^{43} + ( 8 + 24 \beta ) q^{44} + ( 54 + 9 \beta ) q^{45} + ( -76 - 204 \beta ) q^{46} + ( -132 - 338 \beta ) q^{47} + 48 q^{48} + ( -174 + 24 \beta ) q^{50} + ( 198 - 33 \beta ) q^{51} + ( 96 - 60 \beta ) q^{52} + ( 134 - 192 \beta ) q^{53} + 54 q^{54} + ( 24 + 38 \beta ) q^{55} + ( 180 + 138 \beta ) q^{57} + ( -112 + 300 \beta ) q^{58} + ( 168 + 298 \beta ) q^{59} + ( 72 + 12 \beta ) q^{60} + ( -252 + 353 \beta ) q^{61} + ( 432 - 108 \beta ) q^{62} + 64 q^{64} + ( 114 - 66 \beta ) q^{65} + ( 12 + 36 \beta ) q^{66} + ( -192 - 144 \beta ) q^{67} + ( 264 - 44 \beta ) q^{68} + ( -114 - 306 \beta ) q^{69} + ( -198 + 342 \beta ) q^{71} + 72 q^{72} + ( -156 - 595 \beta ) q^{73} + ( -280 - 360 \beta ) q^{74} + ( -261 + 36 \beta ) q^{75} + ( 240 + 184 \beta ) q^{76} + ( 144 - 90 \beta ) q^{78} + ( -424 - 300 \beta ) q^{79} + ( 96 + 16 \beta ) q^{80} + 81 q^{81} + ( 36 + 254 \beta ) q^{82} + ( -324 + 80 \beta ) q^{83} + 374 q^{85} + ( -128 + 576 \beta ) q^{86} + ( -168 + 450 \beta ) q^{87} + ( 16 + 48 \beta ) q^{88} + ( -306 - 175 \beta ) q^{89} + ( 108 + 18 \beta ) q^{90} + ( -152 - 408 \beta ) q^{92} + ( 648 - 162 \beta ) q^{93} + ( -264 - 676 \beta ) q^{94} + ( 452 + 336 \beta ) q^{95} + 96 q^{96} + ( -1092 + 133 \beta ) q^{97} + ( 18 + 54 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 4q^{2} + 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} + 16q^{8} + 18q^{9} + 24q^{10} + 4q^{11} + 24q^{12} + 48q^{13} + 36q^{15} + 32q^{16} + 132q^{17} + 36q^{18} + 120q^{19} + 48q^{20} + 8q^{22} - 76q^{23} + 48q^{24} - 174q^{25} + 96q^{26} + 54q^{27} - 112q^{29} + 72q^{30} + 432q^{31} + 64q^{32} + 12q^{33} + 264q^{34} + 72q^{36} - 280q^{37} + 240q^{38} + 144q^{39} + 96q^{40} + 36q^{41} - 128q^{43} + 16q^{44} + 108q^{45} - 152q^{46} - 264q^{47} + 96q^{48} - 348q^{50} + 396q^{51} + 192q^{52} + 268q^{53} + 108q^{54} + 48q^{55} + 360q^{57} - 224q^{58} + 336q^{59} + 144q^{60} - 504q^{61} + 864q^{62} + 128q^{64} + 228q^{65} + 24q^{66} - 384q^{67} + 528q^{68} - 228q^{69} - 396q^{71} + 144q^{72} - 312q^{73} - 560q^{74} - 522q^{75} + 480q^{76} + 288q^{78} - 848q^{79} + 192q^{80} + 162q^{81} + 72q^{82} - 648q^{83} + 748q^{85} - 256q^{86} - 336q^{87} + 32q^{88} - 612q^{89} + 216q^{90} - 304q^{92} + 1296q^{93} - 528q^{94} + 904q^{95} + 192q^{96} - 2184q^{97} + 36q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
2.00000 3.00000 4.00000 4.58579 6.00000 0 8.00000 9.00000 9.17157
1.2 2.00000 3.00000 4.00000 7.41421 6.00000 0 8.00000 9.00000 14.8284
 $$n$$: 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.o yes 2
3.b odd 2 1 882.4.a.t 2
4.b odd 2 1 2352.4.a.bu 2
7.b odd 2 1 294.4.a.l 2
7.c even 3 2 294.4.e.k 4
7.d odd 6 2 294.4.e.m 4
21.c even 2 1 882.4.a.bb 2
21.g even 6 2 882.4.g.be 4
21.h odd 6 2 882.4.g.bk 4
28.d even 2 1 2352.4.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 7.b odd 2 1
294.4.a.o yes 2 1.a even 1 1 trivial
294.4.e.k 4 7.c even 3 2
294.4.e.m 4 7.d odd 6 2
882.4.a.t 2 3.b odd 2 1
882.4.a.bb 2 21.c even 2 1
882.4.g.be 4 21.g even 6 2
882.4.g.bk 4 21.h odd 6 2
2352.4.a.bu 2 4.b odd 2 1
2352.4.a.bw 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(294))$$:

 $$T_{5}^{2} - 12 T_{5} + 34$$ $$T_{11}^{2} - 4 T_{11} - 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$34 - 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-68 - 4 T + T^{2}$$
$13$ $$126 - 48 T + T^{2}$$
$17$ $$4114 - 132 T + T^{2}$$
$19$ $$-632 - 120 T + T^{2}$$
$23$ $$-19364 + 76 T + T^{2}$$
$29$ $$-41864 + 112 T + T^{2}$$
$31$ $$40824 - 432 T + T^{2}$$
$37$ $$-45200 + 280 T + T^{2}$$
$41$ $$-31934 - 36 T + T^{2}$$
$43$ $$-161792 + 128 T + T^{2}$$
$47$ $$-211064 + 264 T + T^{2}$$
$53$ $$-55772 - 268 T + T^{2}$$
$59$ $$-149384 - 336 T + T^{2}$$
$61$ $$-185714 + 504 T + T^{2}$$
$67$ $$-4608 + 384 T + T^{2}$$
$71$ $$-194724 + 396 T + T^{2}$$
$73$ $$-683714 + 312 T + T^{2}$$
$79$ $$-224 + 848 T + T^{2}$$
$83$ $$92176 + 648 T + T^{2}$$
$89$ $$32386 + 612 T + T^{2}$$
$97$ $$1157086 + 2184 T + T^{2}$$