# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (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: $$4.19214610612$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} + i q^{7} + 3 i q^{8} - q^{9}+O(q^{10})$$ q + i * q^2 + i * q^3 + q^4 - q^6 + i * q^7 + 3*i * q^8 - q^9 $$q + i q^{2} + i q^{3} + q^{4} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} + 4 q^{11} + i q^{12} - 2 i q^{13} - q^{14} - q^{16} + 6 i q^{17} - i q^{18} - 4 q^{19} - q^{21} + 4 i q^{22} - 3 q^{24} + 2 q^{26} - i q^{27} + i q^{28} + 2 q^{29} + 5 i q^{32} + 4 i q^{33} - 6 q^{34} - q^{36} - 6 i q^{37} - 4 i q^{38} + 2 q^{39} + 2 q^{41} - i q^{42} - 4 i q^{43} + 4 q^{44} - i q^{48} - q^{49} - 6 q^{51} - 2 i q^{52} + 6 i q^{53} + q^{54} - 3 q^{56} - 4 i q^{57} + 2 i q^{58} - 12 q^{59} - 2 q^{61} - i q^{63} - 7 q^{64} - 4 q^{66} - 4 i q^{67} + 6 i q^{68} - 3 i q^{72} - 6 i q^{73} + 6 q^{74} - 4 q^{76} + 4 i q^{77} + 2 i q^{78} + 16 q^{79} + q^{81} + 2 i q^{82} - 12 i q^{83} - q^{84} + 4 q^{86} + 2 i q^{87} + 12 i q^{88} + 14 q^{89} + 2 q^{91} - 5 q^{96} - 18 i q^{97} - i q^{98} - 4 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 + q^4 - q^6 + i * q^7 + 3*i * q^8 - q^9 + 4 * q^11 + i * q^12 - 2*i * q^13 - q^14 - q^16 + 6*i * q^17 - i * q^18 - 4 * q^19 - q^21 + 4*i * q^22 - 3 * q^24 + 2 * q^26 - i * q^27 + i * q^28 + 2 * q^29 + 5*i * q^32 + 4*i * q^33 - 6 * q^34 - q^36 - 6*i * q^37 - 4*i * q^38 + 2 * q^39 + 2 * q^41 - i * q^42 - 4*i * q^43 + 4 * q^44 - i * q^48 - q^49 - 6 * q^51 - 2*i * q^52 + 6*i * q^53 + q^54 - 3 * q^56 - 4*i * q^57 + 2*i * q^58 - 12 * q^59 - 2 * q^61 - i * q^63 - 7 * q^64 - 4 * q^66 - 4*i * q^67 + 6*i * q^68 - 3*i * q^72 - 6*i * q^73 + 6 * q^74 - 4 * q^76 + 4*i * q^77 + 2*i * q^78 + 16 * q^79 + q^81 + 2*i * q^82 - 12*i * q^83 - q^84 + 4 * q^86 + 2*i * q^87 + 12*i * q^88 + 14 * q^89 + 2 * q^91 - 5 * q^96 - 18*i * q^97 - i * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{19} - 2 q^{21} - 6 q^{24} + 4 q^{26} + 4 q^{29} - 12 q^{34} - 2 q^{36} + 4 q^{39} + 4 q^{41} + 8 q^{44} - 2 q^{49} - 12 q^{51} + 2 q^{54} - 6 q^{56} - 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} + 12 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} - 2 q^{84} + 8 q^{86} + 28 q^{89} + 4 q^{91} - 10 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 2 * q^14 - 2 * q^16 - 8 * q^19 - 2 * q^21 - 6 * q^24 + 4 * q^26 + 4 * q^29 - 12 * q^34 - 2 * q^36 + 4 * q^39 + 4 * q^41 + 8 * q^44 - 2 * q^49 - 12 * q^51 + 2 * q^54 - 6 * q^56 - 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 + 12 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 - 2 * q^84 + 8 * q^86 + 28 * q^89 + 4 * q^91 - 10 * q^96 - 8 * q^99

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 1.00000i 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 −1.00000 1.00000i 3.00000i −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.d.a 2
3.b odd 2 1 1575.2.d.a 2
5.b even 2 1 inner 525.2.d.a 2
5.c odd 4 1 21.2.a.a 1
5.c odd 4 1 525.2.a.d 1
15.d odd 2 1 1575.2.d.a 2
15.e even 4 1 63.2.a.a 1
15.e even 4 1 1575.2.a.c 1
20.e even 4 1 336.2.a.a 1
20.e even 4 1 8400.2.a.bn 1
35.f even 4 1 147.2.a.a 1
35.f even 4 1 3675.2.a.n 1
35.k even 12 2 147.2.e.c 2
35.l odd 12 2 147.2.e.b 2
40.i odd 4 1 1344.2.a.g 1
40.k even 4 1 1344.2.a.s 1
45.k odd 12 2 567.2.f.g 2
45.l even 12 2 567.2.f.b 2
55.e even 4 1 2541.2.a.j 1
60.l odd 4 1 1008.2.a.l 1
65.h odd 4 1 3549.2.a.c 1
80.i odd 4 1 5376.2.c.r 2
80.j even 4 1 5376.2.c.l 2
80.s even 4 1 5376.2.c.l 2
80.t odd 4 1 5376.2.c.r 2
85.g odd 4 1 6069.2.a.b 1
95.g even 4 1 7581.2.a.d 1
105.k odd 4 1 441.2.a.f 1
105.w odd 12 2 441.2.e.b 2
105.x even 12 2 441.2.e.a 2
120.q odd 4 1 4032.2.a.k 1
120.w even 4 1 4032.2.a.h 1
140.j odd 4 1 2352.2.a.v 1
140.w even 12 2 2352.2.q.x 2
140.x odd 12 2 2352.2.q.e 2
165.l odd 4 1 7623.2.a.g 1
280.s even 4 1 9408.2.a.bv 1
280.y odd 4 1 9408.2.a.m 1
420.w even 4 1 7056.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 5.c odd 4 1
63.2.a.a 1 15.e even 4 1
147.2.a.a 1 35.f even 4 1
147.2.e.b 2 35.l odd 12 2
147.2.e.c 2 35.k even 12 2
336.2.a.a 1 20.e even 4 1
441.2.a.f 1 105.k odd 4 1
441.2.e.a 2 105.x even 12 2
441.2.e.b 2 105.w odd 12 2
525.2.a.d 1 5.c odd 4 1
525.2.d.a 2 1.a even 1 1 trivial
525.2.d.a 2 5.b even 2 1 inner
567.2.f.b 2 45.l even 12 2
567.2.f.g 2 45.k odd 12 2
1008.2.a.l 1 60.l odd 4 1
1344.2.a.g 1 40.i odd 4 1
1344.2.a.s 1 40.k even 4 1
1575.2.a.c 1 15.e even 4 1
1575.2.d.a 2 3.b odd 2 1
1575.2.d.a 2 15.d odd 2 1
2352.2.a.v 1 140.j odd 4 1
2352.2.q.e 2 140.x odd 12 2
2352.2.q.x 2 140.w even 12 2
2541.2.a.j 1 55.e even 4 1
3549.2.a.c 1 65.h odd 4 1
3675.2.a.n 1 35.f even 4 1
4032.2.a.h 1 120.w even 4 1
4032.2.a.k 1 120.q odd 4 1
5376.2.c.l 2 80.j even 4 1
5376.2.c.l 2 80.s even 4 1
5376.2.c.r 2 80.i odd 4 1
5376.2.c.r 2 80.t odd 4 1
6069.2.a.b 1 85.g odd 4 1
7056.2.a.p 1 420.w even 4 1
7581.2.a.d 1 95.g even 4 1
7623.2.a.g 1 165.l odd 4 1
8400.2.a.bn 1 20.e even 4 1
9408.2.a.m 1 280.y odd 4 1
9408.2.a.bv 1 280.s even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} + 324$$
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