# Properties

 Label 507.2.b.a Level $507$ Weight $2$ Character orbit 507.b Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, 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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} + q^{4} + 2 i q^{5} - i q^{6} + 4 i q^{7} + 3 i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 - q^3 + q^4 + 2*i * q^5 - i * q^6 + 4*i * q^7 + 3*i * q^8 + q^9 $$q + i q^{2} - q^{3} + q^{4} + 2 i q^{5} - i q^{6} + 4 i q^{7} + 3 i q^{8} + q^{9} - 2 q^{10} - 4 i q^{11} - q^{12} - 4 q^{14} - 2 i q^{15} - q^{16} - 2 q^{17} + i q^{18} + 2 i q^{20} - 4 i q^{21} + 4 q^{22} - 3 i q^{24} + q^{25} - q^{27} + 4 i q^{28} - 10 q^{29} + 2 q^{30} + 4 i q^{31} + 5 i q^{32} + 4 i q^{33} - 2 i q^{34} - 8 q^{35} + q^{36} + 2 i q^{37} - 6 q^{40} + 6 i q^{41} + 4 q^{42} + 12 q^{43} - 4 i q^{44} + 2 i q^{45} + q^{48} - 9 q^{49} + i q^{50} + 2 q^{51} + 6 q^{53} - i q^{54} + 8 q^{55} - 12 q^{56} - 10 i q^{58} - 12 i q^{59} - 2 i q^{60} - 2 q^{61} - 4 q^{62} + 4 i q^{63} - 7 q^{64} - 4 q^{66} - 8 i q^{67} - 2 q^{68} - 8 i q^{70} + 3 i q^{72} - 2 i q^{73} - 2 q^{74} - q^{75} + 16 q^{77} + 8 q^{79} - 2 i q^{80} + q^{81} - 6 q^{82} + 4 i q^{83} - 4 i q^{84} - 4 i q^{85} + 12 i q^{86} + 10 q^{87} + 12 q^{88} + 2 i q^{89} - 2 q^{90} - 4 i q^{93} - 5 i q^{96} + 10 i q^{97} - 9 i q^{98} - 4 i q^{99} +O(q^{100})$$ q + i * q^2 - q^3 + q^4 + 2*i * q^5 - i * q^6 + 4*i * q^7 + 3*i * q^8 + q^9 - 2 * q^10 - 4*i * q^11 - q^12 - 4 * q^14 - 2*i * q^15 - q^16 - 2 * q^17 + i * q^18 + 2*i * q^20 - 4*i * q^21 + 4 * q^22 - 3*i * q^24 + q^25 - q^27 + 4*i * q^28 - 10 * q^29 + 2 * q^30 + 4*i * q^31 + 5*i * q^32 + 4*i * q^33 - 2*i * q^34 - 8 * q^35 + q^36 + 2*i * q^37 - 6 * q^40 + 6*i * q^41 + 4 * q^42 + 12 * q^43 - 4*i * q^44 + 2*i * q^45 + q^48 - 9 * q^49 + i * q^50 + 2 * q^51 + 6 * q^53 - i * q^54 + 8 * q^55 - 12 * q^56 - 10*i * q^58 - 12*i * q^59 - 2*i * q^60 - 2 * q^61 - 4 * q^62 + 4*i * q^63 - 7 * q^64 - 4 * q^66 - 8*i * q^67 - 2 * q^68 - 8*i * q^70 + 3*i * q^72 - 2*i * q^73 - 2 * q^74 - q^75 + 16 * q^77 + 8 * q^79 - 2*i * q^80 + q^81 - 6 * q^82 + 4*i * q^83 - 4*i * q^84 - 4*i * q^85 + 12*i * q^86 + 10 * q^87 + 12 * q^88 + 2*i * q^89 - 2 * q^90 - 4*i * q^93 - 5*i * q^96 + 10*i * q^97 - 9*i * q^98 - 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 8 q^{14} - 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} - 20 q^{29} + 4 q^{30} - 16 q^{35} + 2 q^{36} - 12 q^{40} + 8 q^{42} + 24 q^{43} + 2 q^{48} - 18 q^{49} + 4 q^{51} + 12 q^{53} + 16 q^{55} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} - 8 q^{66} - 4 q^{68} - 4 q^{74} - 2 q^{75} + 32 q^{77} + 16 q^{79} + 2 q^{81} - 12 q^{82} + 20 q^{87} + 24 q^{88} - 4 q^{90}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 - 4 * q^10 - 2 * q^12 - 8 * q^14 - 2 * q^16 - 4 * q^17 + 8 * q^22 + 2 * q^25 - 2 * q^27 - 20 * q^29 + 4 * q^30 - 16 * q^35 + 2 * q^36 - 12 * q^40 + 8 * q^42 + 24 * q^43 + 2 * q^48 - 18 * q^49 + 4 * q^51 + 12 * q^53 + 16 * q^55 - 24 * q^56 - 4 * q^61 - 8 * q^62 - 14 * q^64 - 8 * q^66 - 4 * q^68 - 4 * q^74 - 2 * q^75 + 32 * q^77 + 16 * q^79 + 2 * q^81 - 12 * q^82 + 20 * q^87 + 24 * q^88 - 4 * q^90

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i −1.00000 1.00000 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
337.2 1.00000i −1.00000 1.00000 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.b.a 2
3.b odd 2 1 1521.2.b.b 2
13.b even 2 1 inner 507.2.b.a 2
13.c even 3 2 507.2.j.e 4
13.d odd 4 1 39.2.a.a 1
13.d odd 4 1 507.2.a.a 1
13.e even 6 2 507.2.j.e 4
13.f odd 12 2 507.2.e.a 2
13.f odd 12 2 507.2.e.b 2
39.d odd 2 1 1521.2.b.b 2
39.f even 4 1 117.2.a.a 1
39.f even 4 1 1521.2.a.e 1
52.f even 4 1 624.2.a.i 1
52.f even 4 1 8112.2.a.s 1
65.f even 4 1 975.2.c.f 2
65.g odd 4 1 975.2.a.f 1
65.k even 4 1 975.2.c.f 2
91.i even 4 1 1911.2.a.f 1
104.j odd 4 1 2496.2.a.q 1
104.m even 4 1 2496.2.a.e 1
117.y odd 12 2 1053.2.e.b 2
117.z even 12 2 1053.2.e.d 2
143.g even 4 1 4719.2.a.c 1
156.l odd 4 1 1872.2.a.h 1
195.j odd 4 1 2925.2.c.e 2
195.n even 4 1 2925.2.a.p 1
195.u odd 4 1 2925.2.c.e 2
273.o odd 4 1 5733.2.a.e 1
312.w odd 4 1 7488.2.a.by 1
312.y even 4 1 7488.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.d odd 4 1
117.2.a.a 1 39.f even 4 1
507.2.a.a 1 13.d odd 4 1
507.2.b.a 2 1.a even 1 1 trivial
507.2.b.a 2 13.b even 2 1 inner
507.2.e.a 2 13.f odd 12 2
507.2.e.b 2 13.f odd 12 2
507.2.j.e 4 13.c even 3 2
507.2.j.e 4 13.e even 6 2
624.2.a.i 1 52.f even 4 1
975.2.a.f 1 65.g odd 4 1
975.2.c.f 2 65.f even 4 1
975.2.c.f 2 65.k even 4 1
1053.2.e.b 2 117.y odd 12 2
1053.2.e.d 2 117.z even 12 2
1521.2.a.e 1 39.f even 4 1
1521.2.b.b 2 3.b odd 2 1
1521.2.b.b 2 39.d odd 2 1
1872.2.a.h 1 156.l odd 4 1
1911.2.a.f 1 91.i even 4 1
2496.2.a.e 1 104.m even 4 1
2496.2.a.q 1 104.j odd 4 1
2925.2.a.p 1 195.n even 4 1
2925.2.c.e 2 195.j odd 4 1
2925.2.c.e 2 195.u odd 4 1
4719.2.a.c 1 143.g even 4 1
5733.2.a.e 1 273.o odd 4 1
7488.2.a.bl 1 312.y even 4 1
7488.2.a.by 1 312.w odd 4 1
8112.2.a.s 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{5}^{2} + 4$$ T5^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 36$$
$43$ $$(T - 12)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 100$$
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