# Properties

 Label 510.2.a.c Level $510$ Weight $2$ Character orbit 510.a Self dual yes Analytic conductor $4.072$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + q^{15} + q^{16} + q^{17} + q^{18} - 4 q^{19} - q^{20} + 4 q^{21} - 4 q^{22} - 4 q^{23} - q^{24} + q^{25} - 2 q^{26} - q^{27} - 4 q^{28} + 2 q^{29} + q^{30} + 4 q^{31} + q^{32} + 4 q^{33} + q^{34} + 4 q^{35} + q^{36} - 6 q^{37} - 4 q^{38} + 2 q^{39} - q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 4 q^{44} - q^{45} - 4 q^{46} + 8 q^{47} - q^{48} + 9 q^{49} + q^{50} - q^{51} - 2 q^{52} - 2 q^{53} - q^{54} + 4 q^{55} - 4 q^{56} + 4 q^{57} + 2 q^{58} + 12 q^{59} + q^{60} + 2 q^{61} + 4 q^{62} - 4 q^{63} + q^{64} + 2 q^{65} + 4 q^{66} + 4 q^{67} + q^{68} + 4 q^{69} + 4 q^{70} - 4 q^{71} + q^{72} - 14 q^{73} - 6 q^{74} - q^{75} - 4 q^{76} + 16 q^{77} + 2 q^{78} - 12 q^{79} - q^{80} + q^{81} + 2 q^{82} - 4 q^{83} + 4 q^{84} - q^{85} - 12 q^{86} - 2 q^{87} - 4 q^{88} + 10 q^{89} - q^{90} + 8 q^{91} - 4 q^{92} - 4 q^{93} + 8 q^{94} + 4 q^{95} - q^{96} + 18 q^{97} + 9 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 - 4 * q^7 + q^8 + q^9 - q^10 - 4 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + q^15 + q^16 + q^17 + q^18 - 4 * q^19 - q^20 + 4 * q^21 - 4 * q^22 - 4 * q^23 - q^24 + q^25 - 2 * q^26 - q^27 - 4 * q^28 + 2 * q^29 + q^30 + 4 * q^31 + q^32 + 4 * q^33 + q^34 + 4 * q^35 + q^36 - 6 * q^37 - 4 * q^38 + 2 * q^39 - q^40 + 2 * q^41 + 4 * q^42 - 12 * q^43 - 4 * q^44 - q^45 - 4 * q^46 + 8 * q^47 - q^48 + 9 * q^49 + q^50 - q^51 - 2 * q^52 - 2 * q^53 - q^54 + 4 * q^55 - 4 * q^56 + 4 * q^57 + 2 * q^58 + 12 * q^59 + q^60 + 2 * q^61 + 4 * q^62 - 4 * q^63 + q^64 + 2 * q^65 + 4 * q^66 + 4 * q^67 + q^68 + 4 * q^69 + 4 * q^70 - 4 * q^71 + q^72 - 14 * q^73 - 6 * q^74 - q^75 - 4 * q^76 + 16 * q^77 + 2 * q^78 - 12 * q^79 - q^80 + q^81 + 2 * q^82 - 4 * q^83 + 4 * q^84 - q^85 - 12 * q^86 - 2 * q^87 - 4 * q^88 + 10 * q^89 - q^90 + 8 * q^91 - 4 * q^92 - 4 * q^93 + 8 * q^94 + 4 * q^95 - q^96 + 18 * q^97 + 9 * q^98 - 4 * 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.

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
 0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −4.00000 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.c 1
3.b odd 2 1 1530.2.a.d 1
4.b odd 2 1 4080.2.a.x 1
5.b even 2 1 2550.2.a.n 1
5.c odd 4 2 2550.2.d.b 2
15.d odd 2 1 7650.2.a.cn 1
17.b even 2 1 8670.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.c 1 1.a even 1 1 trivial
1530.2.a.d 1 3.b odd 2 1
2550.2.a.n 1 5.b even 2 1
2550.2.d.b 2 5.c odd 4 2
4080.2.a.x 1 4.b odd 2 1
7650.2.a.cn 1 15.d odd 2 1
8670.2.a.bb 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(510))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 1$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T + 12$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T - 12$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T + 4$$
$73$ $$T + 14$$
$79$ $$T + 12$$
$83$ $$T + 4$$
$89$ $$T - 10$$
$97$ $$T - 18$$