Properties

 Label 1815.2.a.j Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ 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.

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

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
 −0.618034 1.61803
−0.618034 −1.00000 −1.61803 −1.00000 0.618034 3.23607 2.23607 1.00000 0.618034
1.2 1.61803 −1.00000 0.618034 −1.00000 −1.61803 −1.23607 −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

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

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 1815.2.a.j yes 2
3.b odd 2 1 5445.2.a.o 2
5.b even 2 1 9075.2.a.bd 2
11.b odd 2 1 1815.2.a.f 2
33.d even 2 1 5445.2.a.x 2
55.d odd 2 1 9075.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.f 2 11.b odd 2 1
1815.2.a.j yes 2 1.a even 1 1 trivial
5445.2.a.o 2 3.b odd 2 1
5445.2.a.x 2 33.d even 2 1
9075.2.a.bd 2 5.b even 2 1
9075.2.a.bx 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{7}^{2} - 2 T_{7} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-4 - 2 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$-19 + 2 T + T^{2}$$
$19$ $$-16 - 4 T + T^{2}$$
$23$ $$-41 - 4 T + T^{2}$$
$29$ $$-4 - 2 T + T^{2}$$
$31$ $$-45 + T^{2}$$
$37$ $$4 - 6 T + T^{2}$$
$41$ $$44 - 16 T + T^{2}$$
$43$ $$4 + 6 T + T^{2}$$
$47$ $$-41 - 4 T + T^{2}$$
$53$ $$-79 - 2 T + T^{2}$$
$59$ $$-20 - 10 T + T^{2}$$
$61$ $$-11 - 6 T + T^{2}$$
$67$ $$76 - 18 T + T^{2}$$
$71$ $$80 + 20 T + T^{2}$$
$73$ $$16 + 12 T + T^{2}$$
$79$ $$-9 - 12 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$44 - 14 T + T^{2}$$
$97$ $$220 - 30 T + T^{2}$$