# Properties

 Label 507.4.b.b Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,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, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$29.9139683729$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 8 q^{4} - 6 \beta q^{5} - \beta q^{7} + 9 q^{9} +O(q^{10})$$ q - 3 * q^3 + 8 * q^4 - 6*b * q^5 - b * q^7 + 9 * q^9 $$q - 3 q^{3} + 8 q^{4} - 6 \beta q^{5} - \beta q^{7} + 9 q^{9} + 18 \beta q^{11} - 24 q^{12} + 18 \beta q^{15} + 64 q^{16} + 78 q^{17} + 37 \beta q^{19} - 48 \beta q^{20} + 3 \beta q^{21} + 96 q^{23} - 19 q^{25} - 27 q^{27} - 8 \beta q^{28} + 18 q^{29} - 107 \beta q^{31} - 54 \beta q^{33} - 24 q^{35} + 72 q^{36} + 143 \beta q^{37} - 192 \beta q^{41} - 524 q^{43} + 144 \beta q^{44} - 54 \beta q^{45} - 150 \beta q^{47} - 192 q^{48} + 339 q^{49} - 234 q^{51} + 558 q^{53} + 432 q^{55} - 111 \beta q^{57} - 288 \beta q^{59} + 144 \beta q^{60} + 74 q^{61} - 9 \beta q^{63} + 512 q^{64} + 19 \beta q^{67} + 624 q^{68} - 288 q^{69} - 228 \beta q^{71} + 341 \beta q^{73} + 57 q^{75} + 296 \beta q^{76} + 72 q^{77} + 704 q^{79} - 384 \beta q^{80} + 81 q^{81} - 444 \beta q^{83} + 24 \beta q^{84} - 468 \beta q^{85} - 54 q^{87} + 510 \beta q^{89} + 768 q^{92} + 321 \beta q^{93} + 888 q^{95} + 55 \beta q^{97} + 162 \beta q^{99} +O(q^{100})$$ q - 3 * q^3 + 8 * q^4 - 6*b * q^5 - b * q^7 + 9 * q^9 + 18*b * q^11 - 24 * q^12 + 18*b * q^15 + 64 * q^16 + 78 * q^17 + 37*b * q^19 - 48*b * q^20 + 3*b * q^21 + 96 * q^23 - 19 * q^25 - 27 * q^27 - 8*b * q^28 + 18 * q^29 - 107*b * q^31 - 54*b * q^33 - 24 * q^35 + 72 * q^36 + 143*b * q^37 - 192*b * q^41 - 524 * q^43 + 144*b * q^44 - 54*b * q^45 - 150*b * q^47 - 192 * q^48 + 339 * q^49 - 234 * q^51 + 558 * q^53 + 432 * q^55 - 111*b * q^57 - 288*b * q^59 + 144*b * q^60 + 74 * q^61 - 9*b * q^63 + 512 * q^64 + 19*b * q^67 + 624 * q^68 - 288 * q^69 - 228*b * q^71 + 341*b * q^73 + 57 * q^75 + 296*b * q^76 + 72 * q^77 + 704 * q^79 - 384*b * q^80 + 81 * q^81 - 444*b * q^83 + 24*b * q^84 - 468*b * q^85 - 54 * q^87 + 510*b * q^89 + 768 * q^92 + 321*b * q^93 + 888 * q^95 + 55*b * q^97 + 162*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 $$2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} + 156 q^{17} + 192 q^{23} - 38 q^{25} - 54 q^{27} + 36 q^{29} - 48 q^{35} + 144 q^{36} - 1048 q^{43} - 384 q^{48} + 678 q^{49} - 468 q^{51} + 1116 q^{53} + 864 q^{55} + 148 q^{61} + 1024 q^{64} + 1248 q^{68} - 576 q^{69} + 114 q^{75} + 144 q^{77} + 1408 q^{79} + 162 q^{81} - 108 q^{87} + 1536 q^{92} + 1776 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 + 156 * q^17 + 192 * q^23 - 38 * q^25 - 54 * q^27 + 36 * q^29 - 48 * q^35 + 144 * q^36 - 1048 * q^43 - 384 * q^48 + 678 * q^49 - 468 * q^51 + 1116 * q^53 + 864 * q^55 + 148 * q^61 + 1024 * q^64 + 1248 * q^68 - 576 * q^69 + 114 * q^75 + 144 * q^77 + 1408 * q^79 + 162 * q^81 - 108 * q^87 + 1536 * q^92 + 1776 * q^95

## 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
0 −3.00000 8.00000 12.0000i 0 2.00000i 0 9.00000 0
337.2 0 −3.00000 8.00000 12.0000i 0 2.00000i 0 9.00000 0
 $$n$$: 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 507.4.b.b 2
13.b even 2 1 inner 507.4.b.b 2
13.d odd 4 1 39.4.a.a 1
13.d odd 4 1 507.4.a.c 1
39.f even 4 1 117.4.a.a 1
39.f even 4 1 1521.4.a.f 1
52.f even 4 1 624.4.a.g 1
65.g odd 4 1 975.4.a.e 1
91.i even 4 1 1911.4.a.f 1
104.j odd 4 1 2496.4.a.o 1
104.m even 4 1 2496.4.a.f 1
156.l odd 4 1 1872.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 13.d odd 4 1
117.4.a.a 1 39.f even 4 1
507.4.a.c 1 13.d odd 4 1
507.4.b.b 2 1.a even 1 1 trivial
507.4.b.b 2 13.b even 2 1 inner
624.4.a.g 1 52.f even 4 1
975.4.a.e 1 65.g odd 4 1
1521.4.a.f 1 39.f even 4 1
1872.4.a.m 1 156.l odd 4 1
1911.4.a.f 1 91.i even 4 1
2496.4.a.f 1 104.m even 4 1
2496.4.a.o 1 104.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{2} + 144$$ T5^2 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 144$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 1296$$
$13$ $$T^{2}$$
$17$ $$(T - 78)^{2}$$
$19$ $$T^{2} + 5476$$
$23$ $$(T - 96)^{2}$$
$29$ $$(T - 18)^{2}$$
$31$ $$T^{2} + 45796$$
$37$ $$T^{2} + 81796$$
$41$ $$T^{2} + 147456$$
$43$ $$(T + 524)^{2}$$
$47$ $$T^{2} + 90000$$
$53$ $$(T - 558)^{2}$$
$59$ $$T^{2} + 331776$$
$61$ $$(T - 74)^{2}$$
$67$ $$T^{2} + 1444$$
$71$ $$T^{2} + 207936$$
$73$ $$T^{2} + 465124$$
$79$ $$(T - 704)^{2}$$
$83$ $$T^{2} + 788544$$
$89$ $$T^{2} + 1040400$$
$97$ $$T^{2} + 12100$$