# Properties

 Label 525.2.r.e Level $525$ Weight $2$ Character orbit 525.r Analytic conductor $4.192$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(424,525)]

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, 3, 4]))

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

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

chi := DirichletCharacter("525.424");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.r (of order $$6$$, degree $$2$$, 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: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12}^{2} q^{4} + 2 q^{6} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10})$$ q + 2*z * q^2 + (-z^3 + z) * q^3 + 2*z^2 * q^4 + 2 * q^6 + (2*z^3 + z) * q^7 + (-z^2 + 1) * q^9 $$q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12}^{2} q^{4} + 2 q^{6} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9} + 2 \zeta_{12}^{2} q^{11} + 2 \zeta_{12} q^{12} + \zeta_{12}^{3} q^{13} + (6 \zeta_{12}^{2} - 4) q^{14} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{18} + ( - \zeta_{12}^{2} + 1) q^{19} + (2 \zeta_{12}^{2} + 1) q^{21} + 4 \zeta_{12}^{3} q^{22} + (2 \zeta_{12}^{2} - 2) q^{26} - \zeta_{12}^{3} q^{27} + (6 \zeta_{12}^{3} - 4 \zeta_{12}) q^{28} - 4 q^{29} - 9 \zeta_{12}^{2} q^{31} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{32} + 2 \zeta_{12} q^{33} + 2 q^{36} + 3 \zeta_{12} q^{37} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{38} + \zeta_{12}^{2} q^{39} - 10 q^{41} + (4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{42} + 5 \zeta_{12}^{3} q^{43} + (4 \zeta_{12}^{2} - 4) q^{44} - 6 \zeta_{12} q^{47} - 4 \zeta_{12}^{3} q^{48} + (5 \zeta_{12}^{2} - 8) q^{49} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{52} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{53} + ( - 2 \zeta_{12}^{2} + 2) q^{54} - \zeta_{12}^{3} q^{57} - 8 \zeta_{12} q^{58} - 12 \zeta_{12}^{2} q^{59} + (10 \zeta_{12}^{2} - 10) q^{61} - 18 \zeta_{12}^{3} q^{62} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} + 8 q^{64} + 4 \zeta_{12}^{2} q^{66} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{67} - 6 q^{71} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{73} + 6 \zeta_{12}^{2} q^{74} + 2 q^{76} + (6 \zeta_{12}^{3} - 4 \zeta_{12}) q^{77} + 2 \zeta_{12}^{3} q^{78} + (\zeta_{12}^{2} - 1) q^{79} - \zeta_{12}^{2} q^{81} - 20 \zeta_{12} q^{82} + 6 \zeta_{12}^{3} q^{83} + (6 \zeta_{12}^{2} - 4) q^{84} + (10 \zeta_{12}^{2} - 10) q^{86} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{87} + ( - 16 \zeta_{12}^{2} + 16) q^{89} + (\zeta_{12}^{2} - 3) q^{91} - 9 \zeta_{12} q^{93} - 12 \zeta_{12}^{2} q^{94} + ( - 8 \zeta_{12}^{2} + 8) q^{96} + 6 \zeta_{12}^{3} q^{97} + (10 \zeta_{12}^{3} - 16 \zeta_{12}) q^{98} + 2 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (-z^3 + z) * q^3 + 2*z^2 * q^4 + 2 * q^6 + (2*z^3 + z) * q^7 + (-z^2 + 1) * q^9 + 2*z^2 * q^11 + 2*z * q^12 + z^3 * q^13 + (6*z^2 - 4) * q^14 + (-4*z^2 + 4) * q^16 + (-2*z^3 + 2*z) * q^18 + (-z^2 + 1) * q^19 + (2*z^2 + 1) * q^21 + 4*z^3 * q^22 + (2*z^2 - 2) * q^26 - z^3 * q^27 + (6*z^3 - 4*z) * q^28 - 4 * q^29 - 9*z^2 * q^31 + (-8*z^3 + 8*z) * q^32 + 2*z * q^33 + 2 * q^36 + 3*z * q^37 + (-2*z^3 + 2*z) * q^38 + z^2 * q^39 - 10 * q^41 + (4*z^3 + 2*z) * q^42 + 5*z^3 * q^43 + (4*z^2 - 4) * q^44 - 6*z * q^47 - 4*z^3 * q^48 + (5*z^2 - 8) * q^49 + (2*z^3 - 2*z) * q^52 + (-12*z^3 + 12*z) * q^53 + (-2*z^2 + 2) * q^54 - z^3 * q^57 - 8*z * q^58 - 12*z^2 * q^59 + (10*z^2 - 10) * q^61 - 18*z^3 * q^62 + (-z^3 + 3*z) * q^63 + 8 * q^64 + 4*z^2 * q^66 + (-5*z^3 + 5*z) * q^67 - 6 * q^71 + (3*z^3 - 3*z) * q^73 + 6*z^2 * q^74 + 2 * q^76 + (6*z^3 - 4*z) * q^77 + 2*z^3 * q^78 + (z^2 - 1) * q^79 - z^2 * q^81 - 20*z * q^82 + 6*z^3 * q^83 + (6*z^2 - 4) * q^84 + (10*z^2 - 10) * q^86 + (4*z^3 - 4*z) * q^87 + (-16*z^2 + 16) * q^89 + (z^2 - 3) * q^91 - 9*z * q^93 - 12*z^2 * q^94 + (-8*z^2 + 8) * q^96 + 6*z^3 * q^97 + (10*z^3 - 16*z) * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 8 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 + 8 * q^6 + 2 * q^9 $$4 q + 4 q^{4} + 8 q^{6} + 2 q^{9} + 4 q^{11} - 4 q^{14} + 8 q^{16} + 2 q^{19} + 8 q^{21} - 4 q^{26} - 16 q^{29} - 18 q^{31} + 8 q^{36} + 2 q^{39} - 40 q^{41} - 8 q^{44} - 22 q^{49} + 4 q^{54} - 24 q^{59} - 20 q^{61} + 32 q^{64} + 8 q^{66} - 24 q^{71} + 12 q^{74} + 8 q^{76} - 2 q^{79} - 2 q^{81} - 4 q^{84} - 20 q^{86} + 32 q^{89} - 10 q^{91} - 24 q^{94} + 16 q^{96} + 8 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 + 8 * q^6 + 2 * q^9 + 4 * q^11 - 4 * q^14 + 8 * q^16 + 2 * q^19 + 8 * q^21 - 4 * q^26 - 16 * q^29 - 18 * q^31 + 8 * q^36 + 2 * q^39 - 40 * q^41 - 8 * q^44 - 22 * q^49 + 4 * q^54 - 24 * q^59 - 20 * q^61 + 32 * q^64 + 8 * q^66 - 24 * q^71 + 12 * q^74 + 8 * q^76 - 2 * q^79 - 2 * q^81 - 4 * q^84 - 20 * q^86 + 32 * q^89 - 10 * q^91 - 24 * q^94 + 16 * q^96 + 8 * q^99

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i −0.866025 0.500000i 1.00000 1.73205i 0 2.00000 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
424.2 1.73205 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i 0 2.00000 0.866025 2.50000i 0 0.500000 + 0.866025i 0
499.1 −1.73205 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i 0 2.00000 −0.866025 2.50000i 0 0.500000 0.866025i 0
499.2 1.73205 + 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i 0 2.00000 0.866025 + 2.50000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} - 4T_{2}^{2} + 16$$ T2^4 - 4*T2^2 + 16 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - T + 1)^{2}$$
$23$ $$T^{4}$$
$29$ $$(T + 4)^{4}$$
$31$ $$(T^{2} + 9 T + 81)^{2}$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$(T + 10)^{4}$$
$43$ $$(T^{2} + 25)^{2}$$
$47$ $$T^{4} - 36T^{2} + 1296$$
$53$ $$T^{4} - 144 T^{2} + 20736$$
$59$ $$(T^{2} + 12 T + 144)^{2}$$
$61$ $$(T^{2} + 10 T + 100)^{2}$$
$67$ $$T^{4} - 25T^{2} + 625$$
$71$ $$(T + 6)^{4}$$
$73$ $$T^{4} - 9T^{2} + 81$$
$79$ $$(T^{2} + T + 1)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 16 T + 256)^{2}$$
$97$ $$(T^{2} + 36)^{2}$$