# Properties

 Label 525.4.a.e Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ 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] = [525,4,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, 4, names="a")

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

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$30.9760027530$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 8 q^{4} - 7 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 8 * q^4 - 7 * q^7 + 9 * q^9 $$q + 3 q^{3} - 8 q^{4} - 7 q^{7} + 9 q^{9} + 42 q^{11} - 24 q^{12} - 20 q^{13} + 64 q^{16} - 66 q^{17} + 38 q^{19} - 21 q^{21} - 12 q^{23} + 27 q^{27} + 56 q^{28} - 258 q^{29} + 146 q^{31} + 126 q^{33} - 72 q^{36} - 434 q^{37} - 60 q^{39} - 282 q^{41} - 20 q^{43} - 336 q^{44} + 72 q^{47} + 192 q^{48} + 49 q^{49} - 198 q^{51} + 160 q^{52} - 336 q^{53} + 114 q^{57} - 360 q^{59} - 682 q^{61} - 63 q^{63} - 512 q^{64} - 812 q^{67} + 528 q^{68} - 36 q^{69} + 810 q^{71} + 124 q^{73} - 304 q^{76} - 294 q^{77} + 1136 q^{79} + 81 q^{81} - 156 q^{83} + 168 q^{84} - 774 q^{87} - 1038 q^{89} + 140 q^{91} + 96 q^{92} + 438 q^{93} - 1208 q^{97} + 378 q^{99}+O(q^{100})$$ q + 3 * q^3 - 8 * q^4 - 7 * q^7 + 9 * q^9 + 42 * q^11 - 24 * q^12 - 20 * q^13 + 64 * q^16 - 66 * q^17 + 38 * q^19 - 21 * q^21 - 12 * q^23 + 27 * q^27 + 56 * q^28 - 258 * q^29 + 146 * q^31 + 126 * q^33 - 72 * q^36 - 434 * q^37 - 60 * q^39 - 282 * q^41 - 20 * q^43 - 336 * q^44 + 72 * q^47 + 192 * q^48 + 49 * q^49 - 198 * q^51 + 160 * q^52 - 336 * q^53 + 114 * q^57 - 360 * q^59 - 682 * q^61 - 63 * q^63 - 512 * q^64 - 812 * q^67 + 528 * q^68 - 36 * q^69 + 810 * q^71 + 124 * q^73 - 304 * q^76 - 294 * q^77 + 1136 * q^79 + 81 * q^81 - 156 * q^83 + 168 * q^84 - 774 * q^87 - 1038 * q^89 + 140 * q^91 + 96 * q^92 + 438 * q^93 - 1208 * q^97 + 378 * 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
0 3.00000 −8.00000 0 0 −7.00000 0 9.00000 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 525.4.a.e 1
3.b odd 2 1 1575.4.a.f 1
5.b even 2 1 105.4.a.a 1
5.c odd 4 2 525.4.d.f 2
15.d odd 2 1 315.4.a.d 1
20.d odd 2 1 1680.4.a.s 1
35.c odd 2 1 735.4.a.c 1
105.g even 2 1 2205.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 5.b even 2 1
315.4.a.d 1 15.d odd 2 1
525.4.a.e 1 1.a even 1 1 trivial
525.4.d.f 2 5.c odd 4 2
735.4.a.c 1 35.c odd 2 1
1575.4.a.f 1 3.b odd 2 1
1680.4.a.s 1 20.d odd 2 1
2205.4.a.o 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}$$ T2 $$T_{11} - 42$$ T11 - 42

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 7$$
$11$ $$T - 42$$
$13$ $$T + 20$$
$17$ $$T + 66$$
$19$ $$T - 38$$
$23$ $$T + 12$$
$29$ $$T + 258$$
$31$ $$T - 146$$
$37$ $$T + 434$$
$41$ $$T + 282$$
$43$ $$T + 20$$
$47$ $$T - 72$$
$53$ $$T + 336$$
$59$ $$T + 360$$
$61$ $$T + 682$$
$67$ $$T + 812$$
$71$ $$T - 810$$
$73$ $$T - 124$$
$79$ $$T - 1136$$
$83$ $$T + 156$$
$89$ $$T + 1038$$
$97$ $$T + 1208$$