# Properties

 Label 115.2.a.b Level $115$ Weight $2$ Character orbit 115.a Self dual yes Analytic conductor $0.918$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.918279623245$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} - q^{5} + ( 1 + \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( -1 - 4 \beta ) q^{8} -2 q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} - q^{5} + ( 1 + \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( -1 - 4 \beta ) q^{8} -2 q^{9} + ( 1 + \beta ) q^{10} -2 \beta q^{11} -3 \beta q^{12} + ( -3 - 2 \beta ) q^{13} -2 \beta q^{14} + q^{15} + ( 5 + 3 \beta ) q^{16} + ( -4 + 4 \beta ) q^{17} + ( 2 + 2 \beta ) q^{18} + ( 4 - 6 \beta ) q^{19} -3 \beta q^{20} + ( 2 - 2 \beta ) q^{21} + ( 2 + 4 \beta ) q^{22} - q^{23} + ( 1 + 4 \beta ) q^{24} + q^{25} + ( 5 + 7 \beta ) q^{26} + 5 q^{27} + 6 q^{28} + ( -7 + 4 \beta ) q^{29} + ( -1 - \beta ) q^{30} + ( 1 + 2 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + 2 \beta q^{33} -4 \beta q^{34} + ( 2 - 2 \beta ) q^{35} -6 \beta q^{36} -6 \beta q^{37} + ( 2 + 8 \beta ) q^{38} + ( 3 + 2 \beta ) q^{39} + ( 1 + 4 \beta ) q^{40} + ( -1 - 4 \beta ) q^{41} + 2 \beta q^{42} + ( -6 + 6 \beta ) q^{43} + ( -6 - 6 \beta ) q^{44} + 2 q^{45} + ( 1 + \beta ) q^{46} + ( 3 + 4 \beta ) q^{47} + ( -5 - 3 \beta ) q^{48} + ( 1 - 4 \beta ) q^{49} + ( -1 - \beta ) q^{50} + ( 4 - 4 \beta ) q^{51} + ( -6 - 15 \beta ) q^{52} -6 q^{53} + ( -5 - 5 \beta ) q^{54} + 2 \beta q^{55} + ( -6 - 2 \beta ) q^{56} + ( -4 + 6 \beta ) q^{57} + ( 3 - \beta ) q^{58} + ( -4 + 8 \beta ) q^{59} + 3 \beta q^{60} + ( -4 + 10 \beta ) q^{61} + ( -3 - 5 \beta ) q^{62} + ( 4 - 4 \beta ) q^{63} + ( -1 + 6 \beta ) q^{64} + ( 3 + 2 \beta ) q^{65} + ( -2 - 4 \beta ) q^{66} + 6 \beta q^{67} + 12 q^{68} + q^{69} + 2 \beta q^{70} + ( -3 - 2 \beta ) q^{71} + ( 2 + 8 \beta ) q^{72} + ( 3 - 6 \beta ) q^{73} + ( 6 + 12 \beta ) q^{74} - q^{75} + ( -18 - 6 \beta ) q^{76} -4 q^{77} + ( -5 - 7 \beta ) q^{78} + ( 12 - 2 \beta ) q^{79} + ( -5 - 3 \beta ) q^{80} + q^{81} + ( 5 + 9 \beta ) q^{82} -4 \beta q^{83} -6 q^{84} + ( 4 - 4 \beta ) q^{85} -6 \beta q^{86} + ( 7 - 4 \beta ) q^{87} + ( 8 + 10 \beta ) q^{88} + ( 6 - 2 \beta ) q^{89} + ( -2 - 2 \beta ) q^{90} + ( 2 - 6 \beta ) q^{91} -3 \beta q^{92} + ( -1 - 2 \beta ) q^{93} + ( -7 - 11 \beta ) q^{94} + ( -4 + 6 \beta ) q^{95} + ( 6 + 3 \beta ) q^{96} + ( 10 - 10 \beta ) q^{97} + ( 3 + 7 \beta ) q^{98} + 4 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} - 2q^{5} + 3q^{6} - 2q^{7} - 6q^{8} - 4q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} - 2q^{5} + 3q^{6} - 2q^{7} - 6q^{8} - 4q^{9} + 3q^{10} - 2q^{11} - 3q^{12} - 8q^{13} - 2q^{14} + 2q^{15} + 13q^{16} - 4q^{17} + 6q^{18} + 2q^{19} - 3q^{20} + 2q^{21} + 8q^{22} - 2q^{23} + 6q^{24} + 2q^{25} + 17q^{26} + 10q^{27} + 12q^{28} - 10q^{29} - 3q^{30} + 4q^{31} - 15q^{32} + 2q^{33} - 4q^{34} + 2q^{35} - 6q^{36} - 6q^{37} + 12q^{38} + 8q^{39} + 6q^{40} - 6q^{41} + 2q^{42} - 6q^{43} - 18q^{44} + 4q^{45} + 3q^{46} + 10q^{47} - 13q^{48} - 2q^{49} - 3q^{50} + 4q^{51} - 27q^{52} - 12q^{53} - 15q^{54} + 2q^{55} - 14q^{56} - 2q^{57} + 5q^{58} + 3q^{60} + 2q^{61} - 11q^{62} + 4q^{63} + 4q^{64} + 8q^{65} - 8q^{66} + 6q^{67} + 24q^{68} + 2q^{69} + 2q^{70} - 8q^{71} + 12q^{72} + 24q^{74} - 2q^{75} - 42q^{76} - 8q^{77} - 17q^{78} + 22q^{79} - 13q^{80} + 2q^{81} + 19q^{82} - 4q^{83} - 12q^{84} + 4q^{85} - 6q^{86} + 10q^{87} + 26q^{88} + 10q^{89} - 6q^{90} - 2q^{91} - 3q^{92} - 4q^{93} - 25q^{94} - 2q^{95} + 15q^{96} + 10q^{97} + 13q^{98} + 4q^{99} + O(q^{100})$$

## 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
−2.61803 −1.00000 4.85410 −1.00000 2.61803 1.23607 −7.47214 −2.00000 2.61803
1.2 −0.381966 −1.00000 −1.85410 −1.00000 0.381966 −3.23607 1.47214 −2.00000 0.381966
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$23$$ $$1$$

## 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 115.2.a.b 2
3.b odd 2 1 1035.2.a.m 2
4.b odd 2 1 1840.2.a.p 2
5.b even 2 1 575.2.a.g 2
5.c odd 4 2 575.2.b.c 4
7.b odd 2 1 5635.2.a.n 2
8.b even 2 1 7360.2.a.bt 2
8.d odd 2 1 7360.2.a.bf 2
15.d odd 2 1 5175.2.a.bb 2
20.d odd 2 1 9200.2.a.bm 2
23.b odd 2 1 2645.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.b 2 1.a even 1 1 trivial
575.2.a.g 2 5.b even 2 1
575.2.b.c 4 5.c odd 4 2
1035.2.a.m 2 3.b odd 2 1
1840.2.a.p 2 4.b odd 2 1
2645.2.a.d 2 23.b odd 2 1
5175.2.a.bb 2 15.d odd 2 1
5635.2.a.n 2 7.b odd 2 1
7360.2.a.bf 2 8.d odd 2 1
7360.2.a.bt 2 8.b even 2 1
9200.2.a.bm 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(115))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-4 + 2 T + T^{2}$$
$11$ $$-4 + 2 T + T^{2}$$
$13$ $$11 + 8 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-44 - 2 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$5 + 10 T + T^{2}$$
$31$ $$-1 - 4 T + T^{2}$$
$37$ $$-36 + 6 T + T^{2}$$
$41$ $$-11 + 6 T + T^{2}$$
$43$ $$-36 + 6 T + T^{2}$$
$47$ $$5 - 10 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$-124 - 2 T + T^{2}$$
$67$ $$-36 - 6 T + T^{2}$$
$71$ $$11 + 8 T + T^{2}$$
$73$ $$-45 + T^{2}$$
$79$ $$116 - 22 T + T^{2}$$
$83$ $$-16 + 4 T + T^{2}$$
$89$ $$20 - 10 T + T^{2}$$
$97$ $$-100 - 10 T + T^{2}$$