# Properties

 Label 363.2.a.f Level $363$ Weight $2$ Character orbit 363.a Self dual yes Analytic conductor $2.899$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.89856959337$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + q^{4} - 3 q^{5} - \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 - q^3 + q^4 - 3 * q^5 - b * q^6 - 2*b * q^7 - b * q^8 + q^9 $$q + \beta q^{2} - q^{3} + q^{4} - 3 q^{5} - \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} - 3 \beta q^{10} - q^{12} - \beta q^{13} - 6 q^{14} + 3 q^{15} - 5 q^{16} + \beta q^{17} + \beta q^{18} + 4 \beta q^{19} - 3 q^{20} + 2 \beta q^{21} - 6 q^{23} + \beta q^{24} + 4 q^{25} - 3 q^{26} - q^{27} - 2 \beta q^{28} - \beta q^{29} + 3 \beta q^{30} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + 6 \beta q^{35} + q^{36} - 11 q^{37} + 12 q^{38} + \beta q^{39} + 3 \beta q^{40} - \beta q^{41} + 6 q^{42} + 2 \beta q^{43} - 3 q^{45} - 6 \beta q^{46} + 5 q^{48} + 5 q^{49} + 4 \beta q^{50} - \beta q^{51} - \beta q^{52} - 9 q^{53} - \beta q^{54} + 6 q^{56} - 4 \beta q^{57} - 3 q^{58} - 6 q^{59} + 3 q^{60} + 4 \beta q^{62} - 2 \beta q^{63} + q^{64} + 3 \beta q^{65} - 2 q^{67} + \beta q^{68} + 6 q^{69} + 18 q^{70} - 6 q^{71} - \beta q^{72} + 4 \beta q^{73} - 11 \beta q^{74} - 4 q^{75} + 4 \beta q^{76} + 3 q^{78} + 15 q^{80} + q^{81} - 3 q^{82} + 2 \beta q^{84} - 3 \beta q^{85} + 6 q^{86} + \beta q^{87} + 9 q^{89} - 3 \beta q^{90} + 6 q^{91} - 6 q^{92} - 4 q^{93} - 12 \beta q^{95} + 3 \beta q^{96} - 7 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 - q^3 + q^4 - 3 * q^5 - b * q^6 - 2*b * q^7 - b * q^8 + q^9 - 3*b * q^10 - q^12 - b * q^13 - 6 * q^14 + 3 * q^15 - 5 * q^16 + b * q^17 + b * q^18 + 4*b * q^19 - 3 * q^20 + 2*b * q^21 - 6 * q^23 + b * q^24 + 4 * q^25 - 3 * q^26 - q^27 - 2*b * q^28 - b * q^29 + 3*b * q^30 + 4 * q^31 - 3*b * q^32 + 3 * q^34 + 6*b * q^35 + q^36 - 11 * q^37 + 12 * q^38 + b * q^39 + 3*b * q^40 - b * q^41 + 6 * q^42 + 2*b * q^43 - 3 * q^45 - 6*b * q^46 + 5 * q^48 + 5 * q^49 + 4*b * q^50 - b * q^51 - b * q^52 - 9 * q^53 - b * q^54 + 6 * q^56 - 4*b * q^57 - 3 * q^58 - 6 * q^59 + 3 * q^60 + 4*b * q^62 - 2*b * q^63 + q^64 + 3*b * q^65 - 2 * q^67 + b * q^68 + 6 * q^69 + 18 * q^70 - 6 * q^71 - b * q^72 + 4*b * q^73 - 11*b * q^74 - 4 * q^75 + 4*b * q^76 + 3 * q^78 + 15 * q^80 + q^81 - 3 * q^82 + 2*b * q^84 - 3*b * q^85 + 6 * q^86 + b * q^87 + 9 * q^89 - 3*b * q^90 + 6 * q^91 - 6 * q^92 - 4 * q^93 - 12*b * q^95 + 3*b * q^96 - 7 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9} - 2 q^{12} - 12 q^{14} + 6 q^{15} - 10 q^{16} - 6 q^{20} - 12 q^{23} + 8 q^{25} - 6 q^{26} - 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} - 22 q^{37} + 24 q^{38} + 12 q^{42} - 6 q^{45} + 10 q^{48} + 10 q^{49} - 18 q^{53} + 12 q^{56} - 6 q^{58} - 12 q^{59} + 6 q^{60} + 2 q^{64} - 4 q^{67} + 12 q^{69} + 36 q^{70} - 12 q^{71} - 8 q^{75} + 6 q^{78} + 30 q^{80} + 2 q^{81} - 6 q^{82} + 12 q^{86} + 18 q^{89} + 12 q^{91} - 12 q^{92} - 8 q^{93} - 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^5 + 2 * q^9 - 2 * q^12 - 12 * q^14 + 6 * q^15 - 10 * q^16 - 6 * q^20 - 12 * q^23 + 8 * q^25 - 6 * q^26 - 2 * q^27 + 8 * q^31 + 6 * q^34 + 2 * q^36 - 22 * q^37 + 24 * q^38 + 12 * q^42 - 6 * q^45 + 10 * q^48 + 10 * q^49 - 18 * q^53 + 12 * q^56 - 6 * q^58 - 12 * q^59 + 6 * q^60 + 2 * q^64 - 4 * q^67 + 12 * q^69 + 36 * q^70 - 12 * q^71 - 8 * q^75 + 6 * q^78 + 30 * q^80 + 2 * q^81 - 6 * q^82 + 12 * q^86 + 18 * q^89 + 12 * q^91 - 12 * q^92 - 8 * q^93 - 14 * q^97

## 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
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 −3.00000 1.73205 3.46410 1.73205 1.00000 5.19615
1.2 1.73205 −1.00000 1.00000 −3.00000 −1.73205 −3.46410 −1.73205 1.00000 −5.19615
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.a.f 2
3.b odd 2 1 1089.2.a.o 2
4.b odd 2 1 5808.2.a.ca 2
5.b even 2 1 9075.2.a.bo 2
11.b odd 2 1 inner 363.2.a.f 2
11.c even 5 4 363.2.e.m 8
11.d odd 10 4 363.2.e.m 8
33.d even 2 1 1089.2.a.o 2
44.c even 2 1 5808.2.a.ca 2
55.d odd 2 1 9075.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 1.a even 1 1 trivial
363.2.a.f 2 11.b odd 2 1 inner
363.2.e.m 8 11.c even 5 4
363.2.e.m 8 11.d odd 10 4
1089.2.a.o 2 3.b odd 2 1
1089.2.a.o 2 33.d even 2 1
5808.2.a.ca 2 4.b odd 2 1
5808.2.a.ca 2 44.c even 2 1
9075.2.a.bo 2 5.b even 2 1
9075.2.a.bo 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(363))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 3$$
$19$ $$T^{2} - 48$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} - 3$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$(T + 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T + 7)^{2}$$