# Properties

 Label 2400.2.a.bi Level $2400$ Weight $2$ Character orbit 2400.a Self dual yes Analytic conductor $19.164$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1640964851$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^7 + q^9 $$q - q^{3} - 2 q^{7} + q^{9} - \beta q^{11} + \beta q^{13} - \beta q^{17} + 2 q^{21} - 4 q^{23} - q^{27} + 4 q^{29} + 2 \beta q^{31} + \beta q^{33} - \beta q^{37} - \beta q^{39} + 10 q^{41} + 4 q^{43} - 8 q^{47} - 3 q^{49} + \beta q^{51} - \beta q^{53} + 3 \beta q^{59} + 10 q^{61} - 2 q^{63} + 8 q^{67} + 4 q^{69} - 2 \beta q^{71} - 2 \beta q^{73} + 2 \beta q^{77} - 2 \beta q^{79} + q^{81} + 4 q^{83} - 4 q^{87} + 6 q^{89} - 2 \beta q^{91} - 2 \beta q^{93} + 4 \beta q^{97} - \beta q^{99} +O(q^{100})$$ q - q^3 - 2 * q^7 + q^9 - b * q^11 + b * q^13 - b * q^17 + 2 * q^21 - 4 * q^23 - q^27 + 4 * q^29 + 2*b * q^31 + b * q^33 - b * q^37 - b * q^39 + 10 * q^41 + 4 * q^43 - 8 * q^47 - 3 * q^49 + b * q^51 - b * q^53 + 3*b * q^59 + 10 * q^61 - 2 * q^63 + 8 * q^67 + 4 * q^69 - 2*b * q^71 - 2*b * q^73 + 2*b * q^77 - 2*b * q^79 + q^81 + 4 * q^83 - 4 * q^87 + 6 * q^89 - 2*b * q^91 - 2*b * q^93 + 4*b * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 4 q^{7} + 2 q^{9} + 4 q^{21} - 8 q^{23} - 2 q^{27} + 8 q^{29} + 20 q^{41} + 8 q^{43} - 16 q^{47} - 6 q^{49} + 20 q^{61} - 4 q^{63} + 16 q^{67} + 8 q^{69} + 2 q^{81} + 8 q^{83} - 8 q^{87} + 12 q^{89}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^7 + 2 * q^9 + 4 * q^21 - 8 * q^23 - 2 * q^27 + 8 * q^29 + 20 * q^41 + 8 * q^43 - 16 * q^47 - 6 * q^49 + 20 * q^61 - 4 * q^63 + 16 * q^67 + 8 * q^69 + 2 * q^81 + 8 * q^83 - 8 * q^87 + 12 * q^89

## 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 −1.00000 0 0 0 −2.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 −2.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.a.bi 2
3.b odd 2 1 7200.2.a.cc 2
4.b odd 2 1 2400.2.a.bj 2
5.b even 2 1 2400.2.a.bj 2
5.c odd 4 2 480.2.f.e 4
8.b even 2 1 4800.2.a.cv 2
8.d odd 2 1 4800.2.a.cu 2
12.b even 2 1 7200.2.a.cq 2
15.d odd 2 1 7200.2.a.cq 2
15.e even 4 2 1440.2.f.h 4
20.d odd 2 1 inner 2400.2.a.bi 2
20.e even 4 2 480.2.f.e 4
40.e odd 2 1 4800.2.a.cv 2
40.f even 2 1 4800.2.a.cu 2
40.i odd 4 2 960.2.f.k 4
40.k even 4 2 960.2.f.k 4
60.h even 2 1 7200.2.a.cc 2
60.l odd 4 2 1440.2.f.h 4
80.i odd 4 2 3840.2.d.bh 4
80.j even 4 2 3840.2.d.bh 4
80.s even 4 2 3840.2.d.bg 4
80.t odd 4 2 3840.2.d.bg 4
120.q odd 4 2 2880.2.f.v 4
120.w even 4 2 2880.2.f.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 5.c odd 4 2
480.2.f.e 4 20.e even 4 2
960.2.f.k 4 40.i odd 4 2
960.2.f.k 4 40.k even 4 2
1440.2.f.h 4 15.e even 4 2
1440.2.f.h 4 60.l odd 4 2
2400.2.a.bi 2 1.a even 1 1 trivial
2400.2.a.bi 2 20.d odd 2 1 inner
2400.2.a.bj 2 4.b odd 2 1
2400.2.a.bj 2 5.b even 2 1
2880.2.f.v 4 120.q odd 4 2
2880.2.f.v 4 120.w even 4 2
3840.2.d.bg 4 80.s even 4 2
3840.2.d.bg 4 80.t odd 4 2
3840.2.d.bh 4 80.i odd 4 2
3840.2.d.bh 4 80.j even 4 2
4800.2.a.cu 2 8.d odd 2 1
4800.2.a.cu 2 40.f even 2 1
4800.2.a.cv 2 8.b even 2 1
4800.2.a.cv 2 40.e odd 2 1
7200.2.a.cc 2 3.b odd 2 1
7200.2.a.cc 2 60.h even 2 1
7200.2.a.cq 2 12.b even 2 1
7200.2.a.cq 2 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2400))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} - 20$$ T11^2 - 20 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 20$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} - 80$$
$37$ $$T^{2} - 20$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 20$$
$59$ $$T^{2} - 180$$
$61$ $$(T - 10)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 80$$
$73$ $$T^{2} - 80$$
$79$ $$T^{2} - 80$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 320$$