# Properties

 Label 525.4.a.g 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 21) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 3 q^{3} + q^{4} + 9 q^{6} - 7 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10})$$ q + 3 * q^2 + 3 * q^3 + q^4 + 9 * q^6 - 7 * q^7 - 21 * q^8 + 9 * q^9 $$q + 3 q^{2} + 3 q^{3} + q^{4} + 9 q^{6} - 7 q^{7} - 21 q^{8} + 9 q^{9} - 36 q^{11} + 3 q^{12} + 34 q^{13} - 21 q^{14} - 71 q^{16} - 42 q^{17} + 27 q^{18} - 124 q^{19} - 21 q^{21} - 108 q^{22} - 63 q^{24} + 102 q^{26} + 27 q^{27} - 7 q^{28} + 102 q^{29} - 160 q^{31} - 45 q^{32} - 108 q^{33} - 126 q^{34} + 9 q^{36} - 398 q^{37} - 372 q^{38} + 102 q^{39} - 318 q^{41} - 63 q^{42} + 268 q^{43} - 36 q^{44} - 240 q^{47} - 213 q^{48} + 49 q^{49} - 126 q^{51} + 34 q^{52} + 498 q^{53} + 81 q^{54} + 147 q^{56} - 372 q^{57} + 306 q^{58} - 132 q^{59} + 398 q^{61} - 480 q^{62} - 63 q^{63} + 433 q^{64} - 324 q^{66} - 92 q^{67} - 42 q^{68} - 720 q^{71} - 189 q^{72} + 502 q^{73} - 1194 q^{74} - 124 q^{76} + 252 q^{77} + 306 q^{78} - 1024 q^{79} + 81 q^{81} - 954 q^{82} + 204 q^{83} - 21 q^{84} + 804 q^{86} + 306 q^{87} + 756 q^{88} + 354 q^{89} - 238 q^{91} - 480 q^{93} - 720 q^{94} - 135 q^{96} + 286 q^{97} + 147 q^{98} - 324 q^{99}+O(q^{100})$$ q + 3 * q^2 + 3 * q^3 + q^4 + 9 * q^6 - 7 * q^7 - 21 * q^8 + 9 * q^9 - 36 * q^11 + 3 * q^12 + 34 * q^13 - 21 * q^14 - 71 * q^16 - 42 * q^17 + 27 * q^18 - 124 * q^19 - 21 * q^21 - 108 * q^22 - 63 * q^24 + 102 * q^26 + 27 * q^27 - 7 * q^28 + 102 * q^29 - 160 * q^31 - 45 * q^32 - 108 * q^33 - 126 * q^34 + 9 * q^36 - 398 * q^37 - 372 * q^38 + 102 * q^39 - 318 * q^41 - 63 * q^42 + 268 * q^43 - 36 * q^44 - 240 * q^47 - 213 * q^48 + 49 * q^49 - 126 * q^51 + 34 * q^52 + 498 * q^53 + 81 * q^54 + 147 * q^56 - 372 * q^57 + 306 * q^58 - 132 * q^59 + 398 * q^61 - 480 * q^62 - 63 * q^63 + 433 * q^64 - 324 * q^66 - 92 * q^67 - 42 * q^68 - 720 * q^71 - 189 * q^72 + 502 * q^73 - 1194 * q^74 - 124 * q^76 + 252 * q^77 + 306 * q^78 - 1024 * q^79 + 81 * q^81 - 954 * q^82 + 204 * q^83 - 21 * q^84 + 804 * q^86 + 306 * q^87 + 756 * q^88 + 354 * q^89 - 238 * q^91 - 480 * q^93 - 720 * q^94 - 135 * q^96 + 286 * q^97 + 147 * q^98 - 324 * 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
3.00000 3.00000 1.00000 0 9.00000 −7.00000 −21.0000 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.g 1
3.b odd 2 1 1575.4.a.b 1
5.b even 2 1 21.4.a.a 1
5.c odd 4 2 525.4.d.c 2
15.d odd 2 1 63.4.a.c 1
20.d odd 2 1 336.4.a.f 1
35.c odd 2 1 147.4.a.c 1
35.i odd 6 2 147.4.e.g 2
35.j even 6 2 147.4.e.i 2
40.e odd 2 1 1344.4.a.n 1
40.f even 2 1 1344.4.a.ba 1
60.h even 2 1 1008.4.a.v 1
105.g even 2 1 441.4.a.j 1
105.o odd 6 2 441.4.e.b 2
105.p even 6 2 441.4.e.d 2
140.c even 2 1 2352.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 5.b even 2 1
63.4.a.c 1 15.d odd 2 1
147.4.a.c 1 35.c odd 2 1
147.4.e.g 2 35.i odd 6 2
147.4.e.i 2 35.j even 6 2
336.4.a.f 1 20.d odd 2 1
441.4.a.j 1 105.g even 2 1
441.4.e.b 2 105.o odd 6 2
441.4.e.d 2 105.p even 6 2
525.4.a.g 1 1.a even 1 1 trivial
525.4.d.c 2 5.c odd 4 2
1008.4.a.v 1 60.h even 2 1
1344.4.a.n 1 40.e odd 2 1
1344.4.a.ba 1 40.f even 2 1
1575.4.a.b 1 3.b odd 2 1
2352.4.a.r 1 140.c even 2 1

## Hecke kernels

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

 $$T_{2} - 3$$ T2 - 3 $$T_{11} + 36$$ T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 7$$
$11$ $$T + 36$$
$13$ $$T - 34$$
$17$ $$T + 42$$
$19$ $$T + 124$$
$23$ $$T$$
$29$ $$T - 102$$
$31$ $$T + 160$$
$37$ $$T + 398$$
$41$ $$T + 318$$
$43$ $$T - 268$$
$47$ $$T + 240$$
$53$ $$T - 498$$
$59$ $$T + 132$$
$61$ $$T - 398$$
$67$ $$T + 92$$
$71$ $$T + 720$$
$73$ $$T - 502$$
$79$ $$T + 1024$$
$83$ $$T - 204$$
$89$ $$T - 354$$
$97$ $$T - 286$$