# Properties

 Label 368.2.a.h Level $368$ Weight $2$ Character orbit 368.a Self dual yes Analytic conductor $2.938$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 368.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.93849479438$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} + 2 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( -1 - \beta ) q^{5} + ( -1 + \beta ) q^{7} + 2 q^{9} + ( 3 - \beta ) q^{11} + 3 q^{13} + ( 5 + \beta ) q^{15} + ( 3 + \beta ) q^{17} + 2 q^{19} + ( -5 + \beta ) q^{21} - q^{23} + ( 1 + 2 \beta ) q^{25} + \beta q^{27} -3 q^{29} + 3 \beta q^{31} + ( 5 - 3 \beta ) q^{33} -4 q^{35} + ( 1 + \beta ) q^{37} -3 \beta q^{39} + ( 1 + 2 \beta ) q^{41} + ( -2 - 2 \beta ) q^{45} -\beta q^{47} + ( -1 - 2 \beta ) q^{49} + ( -5 - 3 \beta ) q^{51} + ( -4 - 2 \beta ) q^{53} + ( 2 - 2 \beta ) q^{55} -2 \beta q^{57} + ( -2 + 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + ( -2 + 2 \beta ) q^{63} + ( -3 - 3 \beta ) q^{65} + ( 5 + \beta ) q^{67} + \beta q^{69} + ( -10 + \beta ) q^{71} + ( 11 + 2 \beta ) q^{73} + ( -10 - \beta ) q^{75} + ( -8 + 4 \beta ) q^{77} + ( 2 - 4 \beta ) q^{79} -11 q^{81} + ( 11 + \beta ) q^{83} + ( -8 - 4 \beta ) q^{85} + 3 \beta q^{87} + ( -6 + 2 \beta ) q^{89} + ( -3 + 3 \beta ) q^{91} -15 q^{93} + ( -2 - 2 \beta ) q^{95} + ( 11 - 3 \beta ) q^{97} + ( 6 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 2q^{7} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{5} - 2q^{7} + 4q^{9} + 6q^{11} + 6q^{13} + 10q^{15} + 6q^{17} + 4q^{19} - 10q^{21} - 2q^{23} + 2q^{25} - 6q^{29} + 10q^{33} - 8q^{35} + 2q^{37} + 2q^{41} - 4q^{45} - 2q^{49} - 10q^{51} - 8q^{53} + 4q^{55} - 4q^{59} + 4q^{61} - 4q^{63} - 6q^{65} + 10q^{67} - 20q^{71} + 22q^{73} - 20q^{75} - 16q^{77} + 4q^{79} - 22q^{81} + 22q^{83} - 16q^{85} - 12q^{89} - 6q^{91} - 30q^{93} - 4q^{95} + 22q^{97} + 12q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −2.23607 0 −3.23607 0 1.23607 0 2.00000 0
1.2 0 2.23607 0 1.23607 0 −3.23607 0 2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$23$$ $$1$$

## 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 368.2.a.h 2
3.b odd 2 1 3312.2.a.ba 2
4.b odd 2 1 23.2.a.a 2
5.b even 2 1 9200.2.a.bt 2
8.b even 2 1 1472.2.a.s 2
8.d odd 2 1 1472.2.a.t 2
12.b even 2 1 207.2.a.d 2
20.d odd 2 1 575.2.a.f 2
20.e even 4 2 575.2.b.d 4
23.b odd 2 1 8464.2.a.bb 2
28.d even 2 1 1127.2.a.c 2
44.c even 2 1 2783.2.a.c 2
52.b odd 2 1 3887.2.a.i 2
60.h even 2 1 5175.2.a.be 2
68.d odd 2 1 6647.2.a.b 2
76.d even 2 1 8303.2.a.e 2
92.b even 2 1 529.2.a.a 2
92.g odd 22 10 529.2.c.o 20
92.h even 22 10 529.2.c.n 20
276.h odd 2 1 4761.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 4.b odd 2 1
207.2.a.d 2 12.b even 2 1
368.2.a.h 2 1.a even 1 1 trivial
529.2.a.a 2 92.b even 2 1
529.2.c.n 20 92.h even 22 10
529.2.c.o 20 92.g odd 22 10
575.2.a.f 2 20.d odd 2 1
575.2.b.d 4 20.e even 4 2
1127.2.a.c 2 28.d even 2 1
1472.2.a.s 2 8.b even 2 1
1472.2.a.t 2 8.d odd 2 1
2783.2.a.c 2 44.c even 2 1
3312.2.a.ba 2 3.b odd 2 1
3887.2.a.i 2 52.b odd 2 1
4761.2.a.w 2 276.h odd 2 1
5175.2.a.be 2 60.h even 2 1
6647.2.a.b 2 68.d odd 2 1
8303.2.a.e 2 76.d even 2 1
8464.2.a.bb 2 23.b odd 2 1
9200.2.a.bt 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(368))$$:

 $$T_{3}^{2} - 5$$ $$T_{5}^{2} + 2 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-5 + T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$-4 + 2 T + T^{2}$$
$11$ $$4 - 6 T + T^{2}$$
$13$ $$( -3 + T )^{2}$$
$17$ $$4 - 6 T + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$-45 + T^{2}$$
$37$ $$-4 - 2 T + T^{2}$$
$41$ $$-19 - 2 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$-5 + T^{2}$$
$53$ $$-4 + 8 T + T^{2}$$
$59$ $$-16 + 4 T + T^{2}$$
$61$ $$-76 - 4 T + T^{2}$$
$67$ $$20 - 10 T + T^{2}$$
$71$ $$95 + 20 T + T^{2}$$
$73$ $$101 - 22 T + T^{2}$$
$79$ $$-76 - 4 T + T^{2}$$
$83$ $$116 - 22 T + T^{2}$$
$89$ $$16 + 12 T + T^{2}$$
$97$ $$76 - 22 T + T^{2}$$