# Properties

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

# 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: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + ( -2 + \beta ) q^{5} -\beta q^{6} -3 q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} + ( -2 + \beta ) q^{5} -\beta q^{6} -3 q^{7} + ( -1 + 2 \beta ) q^{8} + q^{9} + ( -1 + \beta ) q^{10} + ( -1 + \beta ) q^{12} + ( -3 - 2 \beta ) q^{13} + 3 \beta q^{14} + ( -2 + \beta ) q^{15} -3 \beta q^{16} + ( -1 + \beta ) q^{17} -\beta q^{18} + ( -4 + 3 \beta ) q^{19} + ( 3 - 2 \beta ) q^{20} -3 q^{21} + ( 1 - 4 \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} -3 \beta q^{25} + ( 2 + 5 \beta ) q^{26} + q^{27} + ( 3 - 3 \beta ) q^{28} + ( -2 + 4 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( 1 - 3 \beta ) q^{31} + ( 5 - \beta ) q^{32} - q^{34} + ( 6 - 3 \beta ) q^{35} + ( -1 + \beta ) q^{36} + ( -1 - 2 \beta ) q^{37} + ( -3 + \beta ) q^{38} + ( -3 - 2 \beta ) q^{39} + ( 4 - 3 \beta ) q^{40} + ( 7 - 8 \beta ) q^{41} + 3 \beta q^{42} + ( -5 + 2 \beta ) q^{43} + ( -2 + \beta ) q^{45} + ( 4 + 3 \beta ) q^{46} + ( 1 - \beta ) q^{47} -3 \beta q^{48} + 2 q^{49} + ( 3 + 3 \beta ) q^{50} + ( -1 + \beta ) q^{51} + ( 1 - 3 \beta ) q^{52} + ( -9 + \beta ) q^{53} -\beta q^{54} + ( 3 - 6 \beta ) q^{56} + ( -4 + 3 \beta ) q^{57} + ( -4 - 2 \beta ) q^{58} + ( 6 - 7 \beta ) q^{59} + ( 3 - 2 \beta ) q^{60} + ( -6 + 3 \beta ) q^{61} + ( 3 + 2 \beta ) q^{62} -3 q^{63} + ( 1 + 2 \beta ) q^{64} + ( 4 - \beta ) q^{65} + ( -4 + 9 \beta ) q^{67} + ( 2 - \beta ) q^{68} + ( 1 - 4 \beta ) q^{69} + ( 3 - 3 \beta ) q^{70} + 9 \beta q^{71} + ( -1 + 2 \beta ) q^{72} + ( 2 - 2 \beta ) q^{73} + ( 2 + 3 \beta ) q^{74} -3 \beta q^{75} + ( 7 - 4 \beta ) q^{76} + ( 2 + 5 \beta ) q^{78} + ( -7 + 4 \beta ) q^{79} + ( -3 + 3 \beta ) q^{80} + q^{81} + ( 8 + \beta ) q^{82} + ( 3 + 6 \beta ) q^{83} + ( 3 - 3 \beta ) q^{84} + ( 3 - 2 \beta ) q^{85} + ( -2 + 3 \beta ) q^{86} + ( -2 + 4 \beta ) q^{87} + ( 3 + 4 \beta ) q^{89} + ( -1 + \beta ) q^{90} + ( 9 + 6 \beta ) q^{91} + ( -5 + \beta ) q^{92} + ( 1 - 3 \beta ) q^{93} + q^{94} + ( 11 - 7 \beta ) q^{95} + ( 5 - \beta ) q^{96} + ( -6 + 13 \beta ) q^{97} -2 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 2q^{3} - q^{4} - 3q^{5} - q^{6} - 6q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + 2q^{3} - q^{4} - 3q^{5} - q^{6} - 6q^{7} + 2q^{9} - q^{10} - q^{12} - 8q^{13} + 3q^{14} - 3q^{15} - 3q^{16} - q^{17} - q^{18} - 5q^{19} + 4q^{20} - 6q^{21} - 2q^{23} - 3q^{25} + 9q^{26} + 2q^{27} + 3q^{28} - q^{30} - q^{31} + 9q^{32} - 2q^{34} + 9q^{35} - q^{36} - 4q^{37} - 5q^{38} - 8q^{39} + 5q^{40} + 6q^{41} + 3q^{42} - 8q^{43} - 3q^{45} + 11q^{46} + q^{47} - 3q^{48} + 4q^{49} + 9q^{50} - q^{51} - q^{52} - 17q^{53} - q^{54} - 5q^{57} - 10q^{58} + 5q^{59} + 4q^{60} - 9q^{61} + 8q^{62} - 6q^{63} + 4q^{64} + 7q^{65} + q^{67} + 3q^{68} - 2q^{69} + 3q^{70} + 9q^{71} + 2q^{73} + 7q^{74} - 3q^{75} + 10q^{76} + 9q^{78} - 10q^{79} - 3q^{80} + 2q^{81} + 17q^{82} + 12q^{83} + 3q^{84} + 4q^{85} - q^{86} + 10q^{89} - q^{90} + 24q^{91} - 9q^{92} - q^{93} + 2q^{94} + 15q^{95} + 9q^{96} + q^{97} - 2q^{98} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.61803 1.00000 0.618034 −0.381966 −1.61803 −3.00000 2.23607 1.00000 0.618034
1.2 0.618034 1.00000 −1.61803 −2.61803 0.618034 −3.00000 −2.23607 1.00000 −1.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 363.2.a.e 2
3.b odd 2 1 1089.2.a.s 2
4.b odd 2 1 5808.2.a.bm 2
5.b even 2 1 9075.2.a.bv 2
11.b odd 2 1 363.2.a.h 2
11.c even 5 2 363.2.e.c 4
11.c even 5 2 363.2.e.j 4
11.d odd 10 2 33.2.e.a 4
11.d odd 10 2 363.2.e.h 4
33.d even 2 1 1089.2.a.m 2
33.f even 10 2 99.2.f.b 4
44.c even 2 1 5808.2.a.bl 2
44.g even 10 2 528.2.y.f 4
55.d odd 2 1 9075.2.a.x 2
55.h odd 10 2 825.2.n.f 4
55.l even 20 4 825.2.bx.b 8
99.o odd 30 4 891.2.n.d 8
99.p even 30 4 891.2.n.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 11.d odd 10 2
99.2.f.b 4 33.f even 10 2
363.2.a.e 2 1.a even 1 1 trivial
363.2.a.h 2 11.b odd 2 1
363.2.e.c 4 11.c even 5 2
363.2.e.h 4 11.d odd 10 2
363.2.e.j 4 11.c even 5 2
528.2.y.f 4 44.g even 10 2
825.2.n.f 4 55.h odd 10 2
825.2.bx.b 8 55.l even 20 4
891.2.n.a 8 99.p even 30 4
891.2.n.d 8 99.o odd 30 4
1089.2.a.m 2 33.d even 2 1
1089.2.a.s 2 3.b odd 2 1
5808.2.a.bl 2 44.c even 2 1
5808.2.a.bm 2 4.b odd 2 1
9075.2.a.x 2 55.d odd 2 1
9075.2.a.bv 2 5.b even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(363))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ $$1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 + 3 T + 7 T^{2} )^{2}$$
$11$ 1
$13$ $$1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4}$$
$17$ $$1 + T + 33 T^{2} + 17 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 38 T^{2} + 841 T^{4}$$
$31$ $$1 + T + 51 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T + 73 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 6 T + 11 T^{2} - 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4}$$
$47$ $$1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4}$$
$53$ $$1 + 17 T + 177 T^{2} + 901 T^{3} + 2809 T^{4}$$
$59$ $$1 - 5 T + 63 T^{2} - 295 T^{3} + 3481 T^{4}$$
$61$ $$1 + 9 T + 131 T^{2} + 549 T^{3} + 3721 T^{4}$$
$67$ $$1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4}$$
$71$ $$1 - 9 T + 61 T^{2} - 639 T^{3} + 5041 T^{4}$$
$73$ $$1 - 2 T + 142 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + 10 T + 163 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 157 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4}$$
$97$ $$1 - T - 17 T^{2} - 97 T^{3} + 9409 T^{4}$$