# Properties

 Label 510.2.a.h Level $510$ Weight $2$ Character orbit 510.a Self dual yes Analytic conductor $4.072$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(1,510)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$510 = 2 \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 510.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.07237050309$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.44949 2.44949
−1.00000 −1.00000 1.00000 1.00000 1.00000 −4.89898 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 4.89898 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$17$$ $$1$$

## 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 510.2.a.h 2
3.b odd 2 1 1530.2.a.s 2
4.b odd 2 1 4080.2.a.bq 2
5.b even 2 1 2550.2.a.bl 2
5.c odd 4 2 2550.2.d.u 4
15.d odd 2 1 7650.2.a.cu 2
17.b even 2 1 8670.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.h 2 1.a even 1 1 trivial
1530.2.a.s 2 3.b odd 2 1
2550.2.a.bl 2 5.b even 2 1
2550.2.d.u 4 5.c odd 4 2
4080.2.a.bq 2 4.b odd 2 1
7650.2.a.cu 2 15.d odd 2 1
8670.2.a.be 2 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(510))$$:

 $$T_{7}^{2} - 24$$ T7^2 - 24 $$T_{11}$$ T11 $$T_{13}^{2} - 4T_{13} - 20$$ T13^2 - 4*T13 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 24$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T - 20$$
$17$ $$(T + 1)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} - 4T - 20$$
$43$ $$T^{2} - 8T - 8$$
$47$ $$T^{2} - 96$$
$53$ $$T^{2} - 4T - 92$$
$59$ $$T^{2} - 24$$
$61$ $$T^{2} - 4T - 92$$
$67$ $$T^{2} + 8T - 8$$
$71$ $$T^{2} + 8T - 8$$
$73$ $$T^{2} + 12T + 12$$
$79$ $$T^{2} - 8T - 80$$
$83$ $$T^{2} - 8T - 80$$
$89$ $$T^{2} - 4T - 92$$
$97$ $$T^{2} - 4T - 212$$