# Properties

 Label 507.2.b.d Level $507$ Weight $2$ Character orbit 507.b Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b3 - 3) * q^4 + (2*b2 + b1) * q^5 + b1 * q^6 + (-b2 + b1) * q^7 + (4*b2 - b1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + q^{9} + ( - \beta_{3} - 3) q^{10} - 2 \beta_{2} q^{11} + (\beta_{3} - 3) q^{12} + (2 \beta_{3} - 6) q^{14} + (2 \beta_{2} + \beta_1) q^{15} + ( - 3 \beta_{3} + 3) q^{16} + (\beta_{3} - 1) q^{17} + \beta_1 q^{18} + ( - 4 \beta_{2} - 2 \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1) q^{21} + (2 \beta_{3} - 2) q^{22} - 2 q^{23} + (4 \beta_{2} - \beta_1) q^{24} - 3 \beta_{3} q^{25} + q^{27} + (6 \beta_{2} - 4 \beta_1) q^{28} + (3 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 3) q^{30} + ( - \beta_{2} - \beta_1) q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} - 2 \beta_{2} q^{33} + (4 \beta_{2} - \beta_1) q^{34} - 2 q^{35} + (\beta_{3} - 3) q^{36} + (6 \beta_{2} + \beta_1) q^{37} + (2 \beta_{3} + 6) q^{38} + ( - 3 \beta_{3} - 1) q^{40} + \beta_1 q^{41} + (2 \beta_{3} - 6) q^{42} + ( - \beta_{3} - 2) q^{43} + (4 \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{2} + \beta_1) q^{45} - 2 \beta_1 q^{46} + (2 \beta_{2} + 4 \beta_1) q^{47} + ( - 3 \beta_{3} + 3) q^{48} + (3 \beta_{3} - 1) q^{49} - 12 \beta_{2} q^{50} + (\beta_{3} - 1) q^{51} + ( - 3 \beta_{3} + 7) q^{53} + \beta_1 q^{54} + (2 \beta_{3} + 2) q^{55} + ( - 6 \beta_{3} + 14) q^{56} + ( - 4 \beta_{2} - 2 \beta_1) q^{57} + (12 \beta_{2} - \beta_1) q^{58} + ( - 6 \beta_{2} + 2 \beta_1) q^{59} - \beta_1 q^{60} + ( - 2 \beta_{3} + 9) q^{61} + 4 q^{62} + ( - \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} - 3) q^{64} + (2 \beta_{3} - 2) q^{66} + ( - 3 \beta_{2} - \beta_1) q^{67} + ( - 3 \beta_{3} + 7) q^{68} - 2 q^{69} - 2 \beta_1 q^{70} - 14 \beta_{2} q^{71} + (4 \beta_{2} - \beta_1) q^{72} + ( - 7 \beta_{2} - 2 \beta_1) q^{73} + ( - 5 \beta_{3} + 1) q^{74} - 3 \beta_{3} q^{75} + 2 \beta_1 q^{76} + (2 \beta_{3} - 4) q^{77} + ( - \beta_{3} + 8) q^{79} + ( - 12 \beta_{2} - 3 \beta_1) q^{80} + q^{81} + (\beta_{3} - 5) q^{82} + (4 \beta_{2} - 2 \beta_1) q^{83} + (6 \beta_{2} - 4 \beta_1) q^{84} + (4 \beta_{2} + \beta_1) q^{85} + ( - 4 \beta_{2} - 2 \beta_1) q^{86} + (3 \beta_{3} - 1) q^{87} + ( - 2 \beta_{3} + 10) q^{88} + (8 \beta_{2} - 2 \beta_1) q^{89} + ( - \beta_{3} - 3) q^{90} + ( - 2 \beta_{3} + 6) q^{92} + ( - \beta_{2} - \beta_1) q^{93} + (2 \beta_{3} - 18) q^{94} + (6 \beta_{3} + 10) q^{95} + ( - 4 \beta_{2} + \beta_1) q^{96} + (7 \beta_{2} + \beta_1) q^{97} + (12 \beta_{2} - \beta_1) q^{98} - 2 \beta_{2} q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b3 - 3) * q^4 + (2*b2 + b1) * q^5 + b1 * q^6 + (-b2 + b1) * q^7 + (4*b2 - b1) * q^8 + q^9 + (-b3 - 3) * q^10 - 2*b2 * q^11 + (b3 - 3) * q^12 + (2*b3 - 6) * q^14 + (2*b2 + b1) * q^15 + (-3*b3 + 3) * q^16 + (b3 - 1) * q^17 + b1 * q^18 + (-4*b2 - 2*b1) * q^19 - b1 * q^20 + (-b2 + b1) * q^21 + (2*b3 - 2) * q^22 - 2 * q^23 + (4*b2 - b1) * q^24 - 3*b3 * q^25 + q^27 + (6*b2 - 4*b1) * q^28 + (3*b3 - 1) * q^29 + (-b3 - 3) * q^30 + (-b2 - b1) * q^31 + (-4*b2 + b1) * q^32 - 2*b2 * q^33 + (4*b2 - b1) * q^34 - 2 * q^35 + (b3 - 3) * q^36 + (6*b2 + b1) * q^37 + (2*b3 + 6) * q^38 + (-3*b3 - 1) * q^40 + b1 * q^41 + (2*b3 - 6) * q^42 + (-b3 - 2) * q^43 + (4*b2 - 2*b1) * q^44 + (2*b2 + b1) * q^45 - 2*b1 * q^46 + (2*b2 + 4*b1) * q^47 + (-3*b3 + 3) * q^48 + (3*b3 - 1) * q^49 - 12*b2 * q^50 + (b3 - 1) * q^51 + (-3*b3 + 7) * q^53 + b1 * q^54 + (2*b3 + 2) * q^55 + (-6*b3 + 14) * q^56 + (-4*b2 - 2*b1) * q^57 + (12*b2 - b1) * q^58 + (-6*b2 + 2*b1) * q^59 - b1 * q^60 + (-2*b3 + 9) * q^61 + 4 * q^62 + (-b2 + b1) * q^63 + (-b3 - 3) * q^64 + (2*b3 - 2) * q^66 + (-3*b2 - b1) * q^67 + (-3*b3 + 7) * q^68 - 2 * q^69 - 2*b1 * q^70 - 14*b2 * q^71 + (4*b2 - b1) * q^72 + (-7*b2 - 2*b1) * q^73 + (-5*b3 + 1) * q^74 - 3*b3 * q^75 + 2*b1 * q^76 + (2*b3 - 4) * q^77 + (-b3 + 8) * q^79 + (-12*b2 - 3*b1) * q^80 + q^81 + (b3 - 5) * q^82 + (4*b2 - 2*b1) * q^83 + (6*b2 - 4*b1) * q^84 + (4*b2 + b1) * q^85 + (-4*b2 - 2*b1) * q^86 + (3*b3 - 1) * q^87 + (-2*b3 + 10) * q^88 + (8*b2 - 2*b1) * q^89 + (-b3 - 3) * q^90 + (-2*b3 + 6) * q^92 + (-b2 - b1) * q^93 + (2*b3 - 18) * q^94 + (6*b3 + 10) * q^95 + (-4*b2 + b1) * q^96 + (7*b2 + b1) * q^97 + (12*b2 - b1) * q^98 - 2*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 10 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 $$4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40} - 20 q^{42} - 10 q^{43} + 6 q^{48} + 2 q^{49} - 2 q^{51} + 22 q^{53} + 12 q^{55} + 44 q^{56} + 32 q^{61} + 16 q^{62} - 14 q^{64} - 4 q^{66} + 22 q^{68} - 8 q^{69} - 6 q^{74} - 6 q^{75} - 12 q^{77} + 30 q^{79} + 4 q^{81} - 18 q^{82} + 2 q^{87} + 36 q^{88} - 14 q^{90} + 20 q^{92} - 68 q^{94} + 52 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 - 14 * q^10 - 10 * q^12 - 20 * q^14 + 6 * q^16 - 2 * q^17 - 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^27 + 2 * q^29 - 14 * q^30 - 8 * q^35 - 10 * q^36 + 28 * q^38 - 10 * q^40 - 20 * q^42 - 10 * q^43 + 6 * q^48 + 2 * q^49 - 2 * q^51 + 22 * q^53 + 12 * q^55 + 44 * q^56 + 32 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^66 + 22 * q^68 - 8 * q^69 - 6 * q^74 - 6 * q^75 - 12 * q^77 + 30 * q^79 + 4 * q^81 - 18 * q^82 + 2 * q^87 + 36 * q^88 - 14 * q^90 + 20 * q^92 - 68 * q^94 + 52 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
337.2 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.3 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.4 2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
 $$n$$: 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.d 4
3.b odd 2 1 1521.2.b.h 4
13.b even 2 1 inner 507.2.b.d 4
13.c even 3 2 507.2.j.g 8
13.d odd 4 1 507.2.a.d 2
13.d odd 4 1 507.2.a.g 2
13.e even 6 2 507.2.j.g 8
13.f odd 12 2 39.2.e.b 4
13.f odd 12 2 507.2.e.g 4
39.d odd 2 1 1521.2.b.h 4
39.f even 4 1 1521.2.a.g 2
39.f even 4 1 1521.2.a.m 2
39.k even 12 2 117.2.g.c 4
52.f even 4 1 8112.2.a.bk 2
52.f even 4 1 8112.2.a.bo 2
52.l even 12 2 624.2.q.h 4
65.o even 12 2 975.2.bb.i 8
65.s odd 12 2 975.2.i.k 4
65.t even 12 2 975.2.bb.i 8
156.v odd 12 2 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.f odd 12 2
117.2.g.c 4 39.k even 12 2
507.2.a.d 2 13.d odd 4 1
507.2.a.g 2 13.d odd 4 1
507.2.b.d 4 1.a even 1 1 trivial
507.2.b.d 4 13.b even 2 1 inner
507.2.e.g 4 13.f odd 12 2
507.2.j.g 8 13.c even 3 2
507.2.j.g 8 13.e even 6 2
624.2.q.h 4 52.l even 12 2
975.2.i.k 4 65.s odd 12 2
975.2.bb.i 8 65.o even 12 2
975.2.bb.i 8 65.t even 12 2
1521.2.a.g 2 39.f even 4 1
1521.2.a.m 2 39.f even 4 1
1521.2.b.h 4 3.b odd 2 1
1521.2.b.h 4 39.d odd 2 1
1872.2.t.r 4 156.v odd 12 2
8112.2.a.bk 2 52.f even 4 1
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{2} + 16$$ T2^4 + 9*T2^2 + 16 $$T_{5}^{4} + 13T_{5}^{2} + 4$$ T5^4 + 13*T5^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 13T^{2} + 4$$
$7$ $$T^{4} + 13T^{2} + 4$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + T - 4)^{2}$$
$19$ $$T^{4} + 52T^{2} + 64$$
$23$ $$(T + 2)^{4}$$
$29$ $$(T^{2} - T - 38)^{2}$$
$31$ $$T^{4} + 9T^{2} + 16$$
$37$ $$T^{4} + 69T^{2} + 676$$
$41$ $$T^{4} + 9T^{2} + 16$$
$43$ $$(T^{2} + 5 T + 2)^{2}$$
$47$ $$(T^{2} + 68)^{2}$$
$53$ $$(T^{2} - 11 T - 8)^{2}$$
$59$ $$T^{4} + 132T^{2} + 1024$$
$61$ $$(T^{2} - 16 T + 47)^{2}$$
$67$ $$T^{4} + 21T^{2} + 4$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$T^{4} + 106T^{2} + 361$$
$79$ $$(T^{2} - 15 T + 52)^{2}$$
$83$ $$T^{4} + 84T^{2} + 64$$
$89$ $$T^{4} + 196T^{2} + 4096$$
$97$ $$T^{4} + 93T^{2} + 1444$$