Properties

 Label 6400.2.a.cg Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6400 = 2^{8} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6400.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$51.1042572936$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta q^{7} - 2 q^{9} +O(q^{10})$$ q + q^3 - b * q^7 - 2 * q^9 $$q + q^{3} - \beta q^{7} - 2 q^{9} + 3 q^{11} - \beta q^{13} + 3 q^{17} - q^{19} - \beta q^{21} - 5 q^{27} - 3 \beta q^{29} - 2 \beta q^{31} + 3 q^{33} + 3 \beta q^{37} - \beta q^{39} + 9 q^{41} + 4 q^{43} + 3 \beta q^{47} + 5 q^{49} + 3 q^{51} - q^{57} - 12 q^{59} + \beta q^{61} + 2 \beta q^{63} + 11 q^{67} + 3 \beta q^{71} - 7 q^{73} - 3 \beta q^{77} + 3 \beta q^{79} + q^{81} + 15 q^{83} - 3 \beta q^{87} + 3 q^{89} + 12 q^{91} - 2 \beta q^{93} + 14 q^{97} - 6 q^{99} +O(q^{100})$$ q + q^3 - b * q^7 - 2 * q^9 + 3 * q^11 - b * q^13 + 3 * q^17 - q^19 - b * q^21 - 5 * q^27 - 3*b * q^29 - 2*b * q^31 + 3 * q^33 + 3*b * q^37 - b * q^39 + 9 * q^41 + 4 * q^43 + 3*b * q^47 + 5 * q^49 + 3 * q^51 - q^57 - 12 * q^59 + b * q^61 + 2*b * q^63 + 11 * q^67 + 3*b * q^71 - 7 * q^73 - 3*b * q^77 + 3*b * q^79 + q^81 + 15 * q^83 - 3*b * q^87 + 3 * q^89 + 12 * q^91 - 2*b * q^93 + 14 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^9 $$2 q + 2 q^{3} - 4 q^{9} + 6 q^{11} + 6 q^{17} - 2 q^{19} - 10 q^{27} + 6 q^{33} + 18 q^{41} + 8 q^{43} + 10 q^{49} + 6 q^{51} - 2 q^{57} - 24 q^{59} + 22 q^{67} - 14 q^{73} + 2 q^{81} + 30 q^{83} + 6 q^{89} + 24 q^{91} + 28 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^11 + 6 * q^17 - 2 * q^19 - 10 * q^27 + 6 * q^33 + 18 * q^41 + 8 * q^43 + 10 * q^49 + 6 * q^51 - 2 * q^57 - 24 * q^59 + 22 * q^67 - 14 * q^73 + 2 * q^81 + 30 * q^83 + 6 * q^89 + 24 * q^91 + 28 * q^97 - 12 * q^99

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
 1.73205 −1.73205
0 1.00000 0 0 0 −3.46410 0 −2.00000 0
1.2 0 1.00000 0 0 0 3.46410 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
$$2$$ $$-1$$
$$5$$ $$1$$

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.cg 2
4.b odd 2 1 6400.2.a.ba 2
5.b even 2 1 6400.2.a.bb 2
8.b even 2 1 6400.2.a.ba 2
8.d odd 2 1 inner 6400.2.a.cg 2
16.e even 4 2 1600.2.d.f yes 4
16.f odd 4 2 1600.2.d.f yes 4
20.d odd 2 1 6400.2.a.cf 2
40.e odd 2 1 6400.2.a.bb 2
40.f even 2 1 6400.2.a.cf 2
80.i odd 4 2 1600.2.f.c 4
80.j even 4 2 1600.2.f.c 4
80.k odd 4 2 1600.2.d.e 4
80.q even 4 2 1600.2.d.e 4
80.s even 4 2 1600.2.f.g 4
80.t odd 4 2 1600.2.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.e 4 80.k odd 4 2
1600.2.d.e 4 80.q even 4 2
1600.2.d.f yes 4 16.e even 4 2
1600.2.d.f yes 4 16.f odd 4 2
1600.2.f.c 4 80.i odd 4 2
1600.2.f.c 4 80.j even 4 2
1600.2.f.g 4 80.s even 4 2
1600.2.f.g 4 80.t odd 4 2
6400.2.a.ba 2 4.b odd 2 1
6400.2.a.ba 2 8.b even 2 1
6400.2.a.bb 2 5.b even 2 1
6400.2.a.bb 2 40.e odd 2 1
6400.2.a.cf 2 20.d odd 2 1
6400.2.a.cf 2 40.f even 2 1
6400.2.a.cg 2 1.a even 1 1 trivial
6400.2.a.cg 2 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} - 12$$ T13^2 - 12 $$T_{17} - 3$$ T17 - 3 $$T_{29}^{2} - 108$$ T29^2 - 108 $$T_{31}^{2} - 48$$ T31^2 - 48

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} - 12$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 108$$
$31$ $$T^{2} - 48$$
$37$ $$T^{2} - 108$$
$41$ $$(T - 9)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 108$$
$53$ $$T^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$T^{2} - 12$$
$67$ $$(T - 11)^{2}$$
$71$ $$T^{2} - 108$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} - 108$$
$83$ $$(T - 15)^{2}$$
$89$ $$(T - 3)^{2}$$
$97$ $$(T - 14)^{2}$$