# Properties

 Label 513.2.g.b Level $513$ Weight $2$ Character orbit 513.g Analytic conductor $4.096$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [513,2,Mod(64,513)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(513, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 2]))

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

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

chi := DirichletCharacter("513.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$513 = 3^{3} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 513.g (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$4.09632562369$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 171) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{8} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^4 + q^5 + (3*z - 3) * q^7 - 3 * q^8 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{8} - \zeta_{6} q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 6 \zeta_{6} + 6) q^{13} + 3 q^{14} + \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + ( - 2 \zeta_{6} - 3) q^{19} + ( - \zeta_{6} + 1) q^{20} - 3 q^{22} + ( - 8 \zeta_{6} + 8) q^{23} - 4 q^{25} - 6 q^{26} + 3 \zeta_{6} q^{28} + 5 q^{29} + 7 \zeta_{6} q^{31} + (5 \zeta_{6} - 5) q^{32} - 3 q^{34} + (3 \zeta_{6} - 3) q^{35} + 2 q^{37} + (5 \zeta_{6} - 2) q^{38} - 3 q^{40} + q^{41} - 8 \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} - 8 q^{46} - 9 q^{47} - 2 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} - 6 \zeta_{6} q^{52} + 3 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + ( - 9 \zeta_{6} + 9) q^{56} - 5 \zeta_{6} q^{58} - 3 q^{59} + 7 q^{61} + ( - 7 \zeta_{6} + 7) q^{62} + 7 q^{64} + ( - 6 \zeta_{6} + 6) q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 3 \zeta_{6} q^{68} + 3 q^{70} + (15 \zeta_{6} - 15) q^{71} + ( - 5 \zeta_{6} + 5) q^{73} - 2 \zeta_{6} q^{74} + (3 \zeta_{6} - 5) q^{76} + 9 \zeta_{6} q^{77} + 12 \zeta_{6} q^{79} + \zeta_{6} q^{80} - \zeta_{6} q^{82} + (\zeta_{6} - 1) q^{83} + ( - 3 \zeta_{6} + 3) q^{85} + (8 \zeta_{6} - 8) q^{86} + (9 \zeta_{6} - 9) q^{88} - \zeta_{6} q^{89} + 18 \zeta_{6} q^{91} - 8 \zeta_{6} q^{92} + 9 \zeta_{6} q^{94} + ( - 2 \zeta_{6} - 3) q^{95} + 2 \zeta_{6} q^{97} + (2 \zeta_{6} - 2) q^{98} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^4 + q^5 + (3*z - 3) * q^7 - 3 * q^8 - z * q^10 + (-3*z + 3) * q^11 + (-6*z + 6) * q^13 + 3 * q^14 + z * q^16 + (-3*z + 3) * q^17 + (-2*z - 3) * q^19 + (-z + 1) * q^20 - 3 * q^22 + (-8*z + 8) * q^23 - 4 * q^25 - 6 * q^26 + 3*z * q^28 + 5 * q^29 + 7*z * q^31 + (5*z - 5) * q^32 - 3 * q^34 + (3*z - 3) * q^35 + 2 * q^37 + (5*z - 2) * q^38 - 3 * q^40 + q^41 - 8*z * q^43 - 3*z * q^44 - 8 * q^46 - 9 * q^47 - 2*z * q^49 + 4*z * q^50 - 6*z * q^52 + 3*z * q^53 + (-3*z + 3) * q^55 + (-9*z + 9) * q^56 - 5*z * q^58 - 3 * q^59 + 7 * q^61 + (-7*z + 7) * q^62 + 7 * q^64 + (-6*z + 6) * q^65 + (-4*z + 4) * q^67 - 3*z * q^68 + 3 * q^70 + (15*z - 15) * q^71 + (-5*z + 5) * q^73 - 2*z * q^74 + (3*z - 5) * q^76 + 9*z * q^77 + 12*z * q^79 + z * q^80 - z * q^82 + (z - 1) * q^83 + (-3*z + 3) * q^85 + (8*z - 8) * q^86 + (9*z - 9) * q^88 - z * q^89 + 18*z * q^91 - 8*z * q^92 + 9*z * q^94 + (-2*z - 3) * q^95 + 2*z * q^97 + (2*z - 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} + 2 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 + 2 * q^5 - 3 * q^7 - 6 * q^8 $$2 q - q^{2} + q^{4} + 2 q^{5} - 3 q^{7} - 6 q^{8} - q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + q^{16} + 3 q^{17} - 8 q^{19} + q^{20} - 6 q^{22} + 8 q^{23} - 8 q^{25} - 12 q^{26} + 3 q^{28} + 10 q^{29} + 7 q^{31} - 5 q^{32} - 6 q^{34} - 3 q^{35} + 4 q^{37} + q^{38} - 6 q^{40} + 2 q^{41} - 8 q^{43} - 3 q^{44} - 16 q^{46} - 18 q^{47} - 2 q^{49} + 4 q^{50} - 6 q^{52} + 3 q^{53} + 3 q^{55} + 9 q^{56} - 5 q^{58} - 6 q^{59} + 14 q^{61} + 7 q^{62} + 14 q^{64} + 6 q^{65} + 4 q^{67} - 3 q^{68} + 6 q^{70} - 15 q^{71} + 5 q^{73} - 2 q^{74} - 7 q^{76} + 9 q^{77} + 12 q^{79} + q^{80} - q^{82} - q^{83} + 3 q^{85} - 8 q^{86} - 9 q^{88} - q^{89} + 18 q^{91} - 8 q^{92} + 9 q^{94} - 8 q^{95} + 2 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 + 2 * q^5 - 3 * q^7 - 6 * q^8 - q^10 + 3 * q^11 + 6 * q^13 + 6 * q^14 + q^16 + 3 * q^17 - 8 * q^19 + q^20 - 6 * q^22 + 8 * q^23 - 8 * q^25 - 12 * q^26 + 3 * q^28 + 10 * q^29 + 7 * q^31 - 5 * q^32 - 6 * q^34 - 3 * q^35 + 4 * q^37 + q^38 - 6 * q^40 + 2 * q^41 - 8 * q^43 - 3 * q^44 - 16 * q^46 - 18 * q^47 - 2 * q^49 + 4 * q^50 - 6 * q^52 + 3 * q^53 + 3 * q^55 + 9 * q^56 - 5 * q^58 - 6 * q^59 + 14 * q^61 + 7 * q^62 + 14 * q^64 + 6 * q^65 + 4 * q^67 - 3 * q^68 + 6 * q^70 - 15 * q^71 + 5 * q^73 - 2 * q^74 - 7 * q^76 + 9 * q^77 + 12 * q^79 + q^80 - q^82 - q^83 + 3 * q^85 - 8 * q^86 - 9 * q^88 - q^89 + 18 * q^91 - 8 * q^92 + 9 * q^94 - 8 * q^95 + 2 * q^97 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 −1.50000 2.59808i −3.00000 0 −0.500000 + 0.866025i
505.1 −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 −1.50000 + 2.59808i −3.00000 0 −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.g.b 2
3.b odd 2 1 171.2.g.b 2
9.c even 3 1 513.2.h.a 2
9.d odd 6 1 171.2.h.b yes 2
19.c even 3 1 513.2.h.a 2
57.h odd 6 1 171.2.h.b yes 2
171.g even 3 1 inner 513.2.g.b 2
171.n odd 6 1 171.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.b 2 3.b odd 2 1
171.2.g.b 2 171.n odd 6 1
171.2.h.b yes 2 9.d odd 6 1
171.2.h.b yes 2 57.h odd 6 1
513.2.g.b 2 1.a even 1 1 trivial
513.2.g.b 2 171.g even 3 1 inner
513.2.h.a 2 9.c even 3 1
513.2.h.a 2 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(513, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2} - 7T + 49$$
$37$ $$(T - 2)^{2}$$
$41$ $$(T - 1)^{2}$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$(T + 9)^{2}$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2} + 15T + 225$$
$73$ $$T^{2} - 5T + 25$$
$79$ $$T^{2} - 12T + 144$$
$83$ $$T^{2} + T + 1$$
$89$ $$T^{2} + T + 1$$
$97$ $$T^{2} - 2T + 4$$