# Properties

 Label 507.2.b.e 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: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 1.00000 −3.82843 2.82843i 2.41421i 2.82843i 4.41421i 1.00000 6.82843
337.2 0.414214i 1.00000 1.82843 2.82843i 0.414214i 2.82843i 1.58579i 1.00000 1.17157
337.3 0.414214i 1.00000 1.82843 2.82843i 0.414214i 2.82843i 1.58579i 1.00000 1.17157
337.4 2.41421i 1.00000 −3.82843 2.82843i 2.41421i 2.82843i 4.41421i 1.00000 6.82843
 $$n$$: 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.e 4
3.b odd 2 1 1521.2.b.j 4
13.b even 2 1 inner 507.2.b.e 4
13.c even 3 2 507.2.j.f 8
13.d odd 4 1 39.2.a.b 2
13.d odd 4 1 507.2.a.h 2
13.e even 6 2 507.2.j.f 8
13.f odd 12 2 507.2.e.d 4
13.f odd 12 2 507.2.e.h 4
39.d odd 2 1 1521.2.b.j 4
39.f even 4 1 117.2.a.c 2
39.f even 4 1 1521.2.a.f 2
52.f even 4 1 624.2.a.k 2
52.f even 4 1 8112.2.a.bm 2
65.f even 4 1 975.2.c.h 4
65.g odd 4 1 975.2.a.l 2
65.k even 4 1 975.2.c.h 4
91.i even 4 1 1911.2.a.h 2
104.j odd 4 1 2496.2.a.bf 2
104.m even 4 1 2496.2.a.bi 2
117.y odd 12 2 1053.2.e.m 4
117.z even 12 2 1053.2.e.e 4
143.g even 4 1 4719.2.a.p 2
156.l odd 4 1 1872.2.a.w 2
195.j odd 4 1 2925.2.c.u 4
195.n even 4 1 2925.2.a.v 2
195.u odd 4 1 2925.2.c.u 4
273.o odd 4 1 5733.2.a.u 2
312.w odd 4 1 7488.2.a.co 2
312.y even 4 1 7488.2.a.cl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.d odd 4 1
117.2.a.c 2 39.f even 4 1
507.2.a.h 2 13.d odd 4 1
507.2.b.e 4 1.a even 1 1 trivial
507.2.b.e 4 13.b even 2 1 inner
507.2.e.d 4 13.f odd 12 2
507.2.e.h 4 13.f odd 12 2
507.2.j.f 8 13.c even 3 2
507.2.j.f 8 13.e even 6 2
624.2.a.k 2 52.f even 4 1
975.2.a.l 2 65.g odd 4 1
975.2.c.h 4 65.f even 4 1
975.2.c.h 4 65.k even 4 1
1053.2.e.e 4 117.z even 12 2
1053.2.e.m 4 117.y odd 12 2
1521.2.a.f 2 39.f even 4 1
1521.2.b.j 4 3.b odd 2 1
1521.2.b.j 4 39.d odd 2 1
1872.2.a.w 2 156.l odd 4 1
1911.2.a.h 2 91.i even 4 1
2496.2.a.bf 2 104.j odd 4 1
2496.2.a.bi 2 104.m even 4 1
2925.2.a.v 2 195.n even 4 1
2925.2.c.u 4 195.j odd 4 1
2925.2.c.u 4 195.u odd 4 1
4719.2.a.p 2 143.g even 4 1
5733.2.a.u 2 273.o odd 4 1
7488.2.a.cl 2 312.y even 4 1
7488.2.a.co 2 312.w odd 4 1
8112.2.a.bm 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{5}^{2} + 8$$ T5^2 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 4 T - 28)^{2}$$
$19$ $$(T^{2} + 8)^{2}$$
$23$ $$(T - 4)^{4}$$
$29$ $$(T - 2)^{4}$$
$31$ $$T^{4} + 48T^{2} + 64$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$T^{4} + 144T^{2} + 3136$$
$43$ $$(T^{2} + 8 T - 16)^{2}$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$(T + 2)^{4}$$
$59$ $$T^{4} + 72T^{2} + 784$$
$61$ $$(T^{2} - 4 T - 124)^{2}$$
$67$ $$T^{4} + 48T^{2} + 64$$
$71$ $$(T^{2} + 4)^{2}$$
$73$ $$T^{4} + 136T^{2} + 16$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 72T^{2} + 784$$
$89$ $$T^{4} + 304 T^{2} + 18496$$
$97$ $$T^{4} + 72T^{2} + 784$$