Properties

 Label 525.2.i.e Level $525$ Weight $2$ Character orbit 525.i Analytic conductor $4.192$ 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] = [525,2,Mod(151,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("525.151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.i (of order $$3$$, degree $$2$$, 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: $$4.19214610612$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 + 2 * q^6 + (z + 2) * q^7 - z * q^9 $$q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + 2 q^{6} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + 2 \zeta_{6} q^{12} - q^{13} + (6 \zeta_{6} - 2) q^{14} + 4 \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{18} - \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} + 4 q^{22} - 2 \zeta_{6} q^{26} - q^{27} + (4 \zeta_{6} - 6) q^{28} + 4 q^{29} + (9 \zeta_{6} - 9) q^{31} + (8 \zeta_{6} - 8) q^{32} - 2 \zeta_{6} q^{33} + 2 q^{36} + 3 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + (\zeta_{6} - 1) q^{39} - 10 q^{41} + (2 \zeta_{6} + 4) q^{42} - 5 q^{43} + 4 \zeta_{6} q^{44} - 6 \zeta_{6} q^{47} + 4 q^{48} + (5 \zeta_{6} + 3) q^{49} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 12 \zeta_{6} + 12) q^{53} - 2 \zeta_{6} q^{54} - q^{57} + 8 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} - 10 \zeta_{6} q^{61} - 18 q^{62} + ( - 3 \zeta_{6} + 1) q^{63} - 8 q^{64} + ( - 4 \zeta_{6} + 4) q^{66} + (5 \zeta_{6} - 5) q^{67} - 6 q^{71} + (3 \zeta_{6} - 3) q^{73} + (6 \zeta_{6} - 6) q^{74} + 2 q^{76} + ( - 4 \zeta_{6} + 6) q^{77} - 2 q^{78} + \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 20 \zeta_{6} q^{82} - 6 q^{83} + (6 \zeta_{6} - 2) q^{84} - 10 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{87} - 16 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} + 9 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{94} + 8 \zeta_{6} q^{96} + 6 q^{97} + (16 \zeta_{6} - 10) q^{98} - 2 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 + 2 * q^6 + (z + 2) * q^7 - z * q^9 + (-2*z + 2) * q^11 + 2*z * q^12 - q^13 + (6*z - 2) * q^14 + 4*z * q^16 + (-2*z + 2) * q^18 - z * q^19 + (-2*z + 3) * q^21 + 4 * q^22 - 2*z * q^26 - q^27 + (4*z - 6) * q^28 + 4 * q^29 + (9*z - 9) * q^31 + (8*z - 8) * q^32 - 2*z * q^33 + 2 * q^36 + 3*z * q^37 + (-2*z + 2) * q^38 + (z - 1) * q^39 - 10 * q^41 + (2*z + 4) * q^42 - 5 * q^43 + 4*z * q^44 - 6*z * q^47 + 4 * q^48 + (5*z + 3) * q^49 + (-2*z + 2) * q^52 + (-12*z + 12) * q^53 - 2*z * q^54 - q^57 + 8*z * q^58 + (-12*z + 12) * q^59 - 10*z * q^61 - 18 * q^62 + (-3*z + 1) * q^63 - 8 * q^64 + (-4*z + 4) * q^66 + (5*z - 5) * q^67 - 6 * q^71 + (3*z - 3) * q^73 + (6*z - 6) * q^74 + 2 * q^76 + (-4*z + 6) * q^77 - 2 * q^78 + z * q^79 + (z - 1) * q^81 - 20*z * q^82 - 6 * q^83 + (6*z - 2) * q^84 - 10*z * q^86 + (-4*z + 4) * q^87 - 16*z * q^89 + (-z - 2) * q^91 + 9*z * q^93 + (-12*z + 12) * q^94 + 8*z * q^96 + 6 * q^97 + (16*z - 10) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 + 4 * q^6 + 5 * q^7 - q^9 $$2 q + 2 q^{2} + q^{3} - 2 q^{4} + 4 q^{6} + 5 q^{7} - q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + 4 q^{16} + 2 q^{18} - q^{19} + 4 q^{21} + 8 q^{22} - 2 q^{26} - 2 q^{27} - 8 q^{28} + 8 q^{29} - 9 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{36} + 3 q^{37} + 2 q^{38} - q^{39} - 20 q^{41} + 10 q^{42} - 10 q^{43} + 4 q^{44} - 6 q^{47} + 8 q^{48} + 11 q^{49} + 2 q^{52} + 12 q^{53} - 2 q^{54} - 2 q^{57} + 8 q^{58} + 12 q^{59} - 10 q^{61} - 36 q^{62} - q^{63} - 16 q^{64} + 4 q^{66} - 5 q^{67} - 12 q^{71} - 3 q^{73} - 6 q^{74} + 4 q^{76} + 8 q^{77} - 4 q^{78} + q^{79} - q^{81} - 20 q^{82} - 12 q^{83} + 2 q^{84} - 10 q^{86} + 4 q^{87} - 16 q^{89} - 5 q^{91} + 9 q^{93} + 12 q^{94} + 8 q^{96} + 12 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 + 4 * q^6 + 5 * q^7 - q^9 + 2 * q^11 + 2 * q^12 - 2 * q^13 + 2 * q^14 + 4 * q^16 + 2 * q^18 - q^19 + 4 * q^21 + 8 * q^22 - 2 * q^26 - 2 * q^27 - 8 * q^28 + 8 * q^29 - 9 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^36 + 3 * q^37 + 2 * q^38 - q^39 - 20 * q^41 + 10 * q^42 - 10 * q^43 + 4 * q^44 - 6 * q^47 + 8 * q^48 + 11 * q^49 + 2 * q^52 + 12 * q^53 - 2 * q^54 - 2 * q^57 + 8 * q^58 + 12 * q^59 - 10 * q^61 - 36 * q^62 - q^63 - 16 * q^64 + 4 * q^66 - 5 * q^67 - 12 * q^71 - 3 * q^73 - 6 * q^74 + 4 * q^76 + 8 * q^77 - 4 * q^78 + q^79 - q^81 - 20 * q^82 - 12 * q^83 + 2 * q^84 - 10 * q^86 + 4 * q^87 - 16 * q^89 - 5 * q^91 + 9 * q^93 + 12 * q^94 + 8 * q^96 + 12 * q^97 - 4 * q^98 - 4 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
226.1 1.00000 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 2.50000 0.866025i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.e 2
5.b even 2 1 21.2.e.a 2
5.c odd 4 2 525.2.r.e 4
7.c even 3 1 inner 525.2.i.e 2
7.c even 3 1 3675.2.a.a 1
7.d odd 6 1 3675.2.a.c 1
15.d odd 2 1 63.2.e.b 2
20.d odd 2 1 336.2.q.f 2
35.c odd 2 1 147.2.e.a 2
35.i odd 6 1 147.2.a.b 1
35.i odd 6 1 147.2.e.a 2
35.j even 6 1 21.2.e.a 2
35.j even 6 1 147.2.a.c 1
35.l odd 12 2 525.2.r.e 4
40.e odd 2 1 1344.2.q.c 2
40.f even 2 1 1344.2.q.m 2
45.h odd 6 1 567.2.g.f 2
45.h odd 6 1 567.2.h.a 2
45.j even 6 1 567.2.g.a 2
45.j even 6 1 567.2.h.f 2
60.h even 2 1 1008.2.s.d 2
105.g even 2 1 441.2.e.e 2
105.o odd 6 1 63.2.e.b 2
105.o odd 6 1 441.2.a.b 1
105.p even 6 1 441.2.a.a 1
105.p even 6 1 441.2.e.e 2
140.c even 2 1 2352.2.q.c 2
140.p odd 6 1 336.2.q.f 2
140.p odd 6 1 2352.2.a.d 1
140.s even 6 1 2352.2.a.w 1
140.s even 6 1 2352.2.q.c 2
280.ba even 6 1 9408.2.a.k 1
280.bf even 6 1 1344.2.q.m 2
280.bf even 6 1 9408.2.a.bg 1
280.bi odd 6 1 1344.2.q.c 2
280.bi odd 6 1 9408.2.a.cv 1
280.bk odd 6 1 9408.2.a.bz 1
315.r even 6 1 567.2.g.a 2
315.v odd 6 1 567.2.h.a 2
315.bo even 6 1 567.2.h.f 2
315.br odd 6 1 567.2.g.f 2
420.ba even 6 1 1008.2.s.d 2
420.ba even 6 1 7056.2.a.bp 1
420.be odd 6 1 7056.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 5.b even 2 1
21.2.e.a 2 35.j even 6 1
63.2.e.b 2 15.d odd 2 1
63.2.e.b 2 105.o odd 6 1
147.2.a.b 1 35.i odd 6 1
147.2.a.c 1 35.j even 6 1
147.2.e.a 2 35.c odd 2 1
147.2.e.a 2 35.i odd 6 1
336.2.q.f 2 20.d odd 2 1
336.2.q.f 2 140.p odd 6 1
441.2.a.a 1 105.p even 6 1
441.2.a.b 1 105.o odd 6 1
441.2.e.e 2 105.g even 2 1
441.2.e.e 2 105.p even 6 1
525.2.i.e 2 1.a even 1 1 trivial
525.2.i.e 2 7.c even 3 1 inner
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 45.j even 6 1
567.2.g.a 2 315.r even 6 1
567.2.g.f 2 45.h odd 6 1
567.2.g.f 2 315.br odd 6 1
567.2.h.a 2 45.h odd 6 1
567.2.h.a 2 315.v odd 6 1
567.2.h.f 2 45.j even 6 1
567.2.h.f 2 315.bo even 6 1
1008.2.s.d 2 60.h even 2 1
1008.2.s.d 2 420.ba even 6 1
1344.2.q.c 2 40.e odd 2 1
1344.2.q.c 2 280.bi odd 6 1
1344.2.q.m 2 40.f even 2 1
1344.2.q.m 2 280.bf even 6 1
2352.2.a.d 1 140.p odd 6 1
2352.2.a.w 1 140.s even 6 1
2352.2.q.c 2 140.c even 2 1
2352.2.q.c 2 140.s even 6 1
3675.2.a.a 1 7.c even 3 1
3675.2.a.c 1 7.d odd 6 1
7056.2.a.m 1 420.be odd 6 1
7056.2.a.bp 1 420.ba even 6 1
9408.2.a.k 1 280.ba even 6 1
9408.2.a.bg 1 280.bf even 6 1
9408.2.a.bz 1 280.bk odd 6 1
9408.2.a.cv 1 280.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} + 9T + 81$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$(T + 10)^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 3T + 9$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 16T + 256$$
$97$ $$(T - 6)^{2}$$