# Properties

 Label 525.2.a.b Level $525$ Weight $2$ Character orbit 525.a Self dual yes Analytic conductor $4.192$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(1,525)]

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, 0, 0]))

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

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

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## 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: yes Analytic conductor: $$4.19214610612$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - q^6 - q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3 q^{8} + q^{9} - 6 q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{16} + 4 q^{17} - q^{18} - 6 q^{19} - q^{21} + 6 q^{22} + 3 q^{24} + 2 q^{26} + q^{27} + q^{28} - 2 q^{29} - 10 q^{31} - 5 q^{32} - 6 q^{33} - 4 q^{34} - q^{36} - 4 q^{37} + 6 q^{38} - 2 q^{39} + 2 q^{41} + q^{42} - 4 q^{43} + 6 q^{44} - q^{48} + q^{49} + 4 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} - 3 q^{56} - 6 q^{57} + 2 q^{58} - 8 q^{59} - 2 q^{61} + 10 q^{62} - q^{63} + 7 q^{64} + 6 q^{66} - 16 q^{67} - 4 q^{68} + 10 q^{71} + 3 q^{72} - 6 q^{73} + 4 q^{74} + 6 q^{76} + 6 q^{77} + 2 q^{78} + 4 q^{79} + q^{81} - 2 q^{82} + 8 q^{83} + q^{84} + 4 q^{86} - 2 q^{87} - 18 q^{88} + 6 q^{89} + 2 q^{91} - 10 q^{93} - 5 q^{96} - 2 q^{97} - q^{98} - 6 q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 - q^6 - q^7 + 3 * q^8 + q^9 - 6 * q^11 - q^12 - 2 * q^13 + q^14 - q^16 + 4 * q^17 - q^18 - 6 * q^19 - q^21 + 6 * q^22 + 3 * q^24 + 2 * q^26 + q^27 + q^28 - 2 * q^29 - 10 * q^31 - 5 * q^32 - 6 * q^33 - 4 * q^34 - q^36 - 4 * q^37 + 6 * q^38 - 2 * q^39 + 2 * q^41 + q^42 - 4 * q^43 + 6 * q^44 - q^48 + q^49 + 4 * q^51 + 2 * q^52 + 6 * q^53 - q^54 - 3 * q^56 - 6 * q^57 + 2 * q^58 - 8 * q^59 - 2 * q^61 + 10 * q^62 - q^63 + 7 * q^64 + 6 * q^66 - 16 * q^67 - 4 * q^68 + 10 * q^71 + 3 * q^72 - 6 * q^73 + 4 * q^74 + 6 * q^76 + 6 * q^77 + 2 * q^78 + 4 * q^79 + q^81 - 2 * q^82 + 8 * q^83 + q^84 + 4 * q^86 - 2 * q^87 - 18 * q^88 + 6 * q^89 + 2 * q^91 - 10 * q^93 - 5 * q^96 - 2 * q^97 - q^98 - 6 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 −1.00000 0 −1.00000 −1.00000 3.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.a.b 1
3.b odd 2 1 1575.2.a.i 1
4.b odd 2 1 8400.2.a.bj 1
5.b even 2 1 525.2.a.c 1
5.c odd 4 2 105.2.d.a 2
7.b odd 2 1 3675.2.a.d 1
15.d odd 2 1 1575.2.a.e 1
15.e even 4 2 315.2.d.c 2
20.d odd 2 1 8400.2.a.ch 1
20.e even 4 2 1680.2.t.f 2
35.c odd 2 1 3675.2.a.l 1
35.f even 4 2 735.2.d.a 2
35.k even 12 4 735.2.q.b 4
35.l odd 12 4 735.2.q.a 4
60.l odd 4 2 5040.2.t.e 2
105.k odd 4 2 2205.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 5.c odd 4 2
315.2.d.c 2 15.e even 4 2
525.2.a.b 1 1.a even 1 1 trivial
525.2.a.c 1 5.b even 2 1
735.2.d.a 2 35.f even 4 2
735.2.q.a 4 35.l odd 12 4
735.2.q.b 4 35.k even 12 4
1575.2.a.e 1 15.d odd 2 1
1575.2.a.i 1 3.b odd 2 1
1680.2.t.f 2 20.e even 4 2
2205.2.d.f 2 105.k odd 4 2
3675.2.a.d 1 7.b odd 2 1
3675.2.a.l 1 35.c odd 2 1
5040.2.t.e 2 60.l odd 4 2
8400.2.a.bj 1 4.b odd 2 1
8400.2.a.ch 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{11} + 6$$ T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 6$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T + 6$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T + 10$$
$37$ $$T + 4$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 8$$
$61$ $$T + 2$$
$67$ $$T + 16$$
$71$ $$T - 10$$
$73$ $$T + 6$$
$79$ $$T - 4$$
$83$ $$T - 8$$
$89$ $$T - 6$$
$97$ $$T + 2$$