Properties

 Label 1150.2.a.j Level $1150$ Weight $2$ Character orbit 1150.a Self dual yes Analytic conductor $9.183$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$9.18279623245$$ Analytic rank: $$1$$ 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: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

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

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
−1.00000 −1.61803 1.00000 0 1.61803 0.618034 −1.00000 −0.381966 0
1.2 −1.00000 0.618034 1.00000 0 −0.618034 −1.61803 −1.00000 −2.61803 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
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

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 1150.2.a.j 2
4.b odd 2 1 9200.2.a.bu 2
5.b even 2 1 230.2.a.c 2
5.c odd 4 2 1150.2.b.i 4
15.d odd 2 1 2070.2.a.u 2
20.d odd 2 1 1840.2.a.l 2
40.e odd 2 1 7360.2.a.bn 2
40.f even 2 1 7360.2.a.bh 2
115.c odd 2 1 5290.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 5.b even 2 1
1150.2.a.j 2 1.a even 1 1 trivial
1150.2.b.i 4 5.c odd 4 2
1840.2.a.l 2 20.d odd 2 1
2070.2.a.u 2 15.d odd 2 1
5290.2.a.o 2 115.c odd 2 1
7360.2.a.bh 2 40.f even 2 1
7360.2.a.bn 2 40.e odd 2 1
9200.2.a.bu 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1150))$$:

 $$T_{3}^{2} + T_{3} - 1$$ $$T_{7}^{2} + T_{7} - 1$$ $$T_{11}^{2} - T_{11} - 11$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + T + T^{2}$$
$11$ $$-11 - T + T^{2}$$
$13$ $$-29 - 3 T + T^{2}$$
$17$ $$-31 + T + T^{2}$$
$19$ $$-9 + 3 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$44 + 14 T + T^{2}$$
$31$ $$-19 - 7 T + T^{2}$$
$37$ $$-16 + 4 T + T^{2}$$
$41$ $$-41 + 9 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$-36 + 6 T + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-20 + 10 T + T^{2}$$
$61$ $$-59 + 3 T + T^{2}$$
$67$ $$80 + 20 T + T^{2}$$
$71$ $$-29 - 3 T + T^{2}$$
$73$ $$-4 + 2 T + T^{2}$$
$79$ $$16 - 12 T + T^{2}$$
$83$ $$-76 + 4 T + T^{2}$$
$89$ $$16 + 12 T + T^{2}$$
$97$ $$181 + 27 T + T^{2}$$