# Properties

 Label 525.6.a.d Level $525$ Weight $6$ Character orbit 525.a Self dual yes Analytic conductor $84.202$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,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, 6, names="a")

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

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$84.2015054018$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 6 q^{2} + 9 q^{3} + 4 q^{4} + 54 q^{6} - 49 q^{7} - 168 q^{8} + 81 q^{9}+O(q^{10})$$ q + 6 * q^2 + 9 * q^3 + 4 * q^4 + 54 * q^6 - 49 * q^7 - 168 * q^8 + 81 * q^9 $$q + 6 q^{2} + 9 q^{3} + 4 q^{4} + 54 q^{6} - 49 q^{7} - 168 q^{8} + 81 q^{9} + 444 q^{11} + 36 q^{12} + 442 q^{13} - 294 q^{14} - 1136 q^{16} + 126 q^{17} + 486 q^{18} + 2684 q^{19} - 441 q^{21} + 2664 q^{22} - 4200 q^{23} - 1512 q^{24} + 2652 q^{26} + 729 q^{27} - 196 q^{28} - 5442 q^{29} + 80 q^{31} - 1440 q^{32} + 3996 q^{33} + 756 q^{34} + 324 q^{36} + 5434 q^{37} + 16104 q^{38} + 3978 q^{39} + 7962 q^{41} - 2646 q^{42} + 11524 q^{43} + 1776 q^{44} - 25200 q^{46} + 13920 q^{47} - 10224 q^{48} + 2401 q^{49} + 1134 q^{51} + 1768 q^{52} + 9594 q^{53} + 4374 q^{54} + 8232 q^{56} + 24156 q^{57} - 32652 q^{58} + 27492 q^{59} + 49478 q^{61} + 480 q^{62} - 3969 q^{63} + 27712 q^{64} + 23976 q^{66} + 59356 q^{67} + 504 q^{68} - 37800 q^{69} + 32040 q^{71} - 13608 q^{72} + 61846 q^{73} + 32604 q^{74} + 10736 q^{76} - 21756 q^{77} + 23868 q^{78} - 65776 q^{79} + 6561 q^{81} + 47772 q^{82} - 40188 q^{83} - 1764 q^{84} + 69144 q^{86} - 48978 q^{87} - 74592 q^{88} - 7974 q^{89} - 21658 q^{91} - 16800 q^{92} + 720 q^{93} + 83520 q^{94} - 12960 q^{96} + 143662 q^{97} + 14406 q^{98} + 35964 q^{99}+O(q^{100})$$ q + 6 * q^2 + 9 * q^3 + 4 * q^4 + 54 * q^6 - 49 * q^7 - 168 * q^8 + 81 * q^9 + 444 * q^11 + 36 * q^12 + 442 * q^13 - 294 * q^14 - 1136 * q^16 + 126 * q^17 + 486 * q^18 + 2684 * q^19 - 441 * q^21 + 2664 * q^22 - 4200 * q^23 - 1512 * q^24 + 2652 * q^26 + 729 * q^27 - 196 * q^28 - 5442 * q^29 + 80 * q^31 - 1440 * q^32 + 3996 * q^33 + 756 * q^34 + 324 * q^36 + 5434 * q^37 + 16104 * q^38 + 3978 * q^39 + 7962 * q^41 - 2646 * q^42 + 11524 * q^43 + 1776 * q^44 - 25200 * q^46 + 13920 * q^47 - 10224 * q^48 + 2401 * q^49 + 1134 * q^51 + 1768 * q^52 + 9594 * q^53 + 4374 * q^54 + 8232 * q^56 + 24156 * q^57 - 32652 * q^58 + 27492 * q^59 + 49478 * q^61 + 480 * q^62 - 3969 * q^63 + 27712 * q^64 + 23976 * q^66 + 59356 * q^67 + 504 * q^68 - 37800 * q^69 + 32040 * q^71 - 13608 * q^72 + 61846 * q^73 + 32604 * q^74 + 10736 * q^76 - 21756 * q^77 + 23868 * q^78 - 65776 * q^79 + 6561 * q^81 + 47772 * q^82 - 40188 * q^83 - 1764 * q^84 + 69144 * q^86 - 48978 * q^87 - 74592 * q^88 - 7974 * q^89 - 21658 * q^91 - 16800 * q^92 + 720 * q^93 + 83520 * q^94 - 12960 * q^96 + 143662 * q^97 + 14406 * q^98 + 35964 * 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
6.00000 9.00000 4.00000 0 54.0000 −49.0000 −168.000 81.0000 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 525.6.a.d 1
5.b even 2 1 21.6.a.a 1
5.c odd 4 2 525.6.d.b 2
15.d odd 2 1 63.6.a.d 1
20.d odd 2 1 336.6.a.r 1
35.c odd 2 1 147.6.a.b 1
35.i odd 6 2 147.6.e.i 2
35.j even 6 2 147.6.e.j 2
60.h even 2 1 1008.6.a.c 1
105.g even 2 1 441.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 5.b even 2 1
63.6.a.d 1 15.d odd 2 1
147.6.a.b 1 35.c odd 2 1
147.6.e.i 2 35.i odd 6 2
147.6.e.j 2 35.j even 6 2
336.6.a.r 1 20.d odd 2 1
441.6.a.j 1 105.g even 2 1
525.6.a.d 1 1.a even 1 1 trivial
525.6.d.b 2 5.c odd 4 2
1008.6.a.c 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(525))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 6$$
$3$ $$T - 9$$
$5$ $$T$$
$7$ $$T + 49$$
$11$ $$T - 444$$
$13$ $$T - 442$$
$17$ $$T - 126$$
$19$ $$T - 2684$$
$23$ $$T + 4200$$
$29$ $$T + 5442$$
$31$ $$T - 80$$
$37$ $$T - 5434$$
$41$ $$T - 7962$$
$43$ $$T - 11524$$
$47$ $$T - 13920$$
$53$ $$T - 9594$$
$59$ $$T - 27492$$
$61$ $$T - 49478$$
$67$ $$T - 59356$$
$71$ $$T - 32040$$
$73$ $$T - 61846$$
$79$ $$T + 65776$$
$83$ $$T + 40188$$
$89$ $$T + 7974$$
$97$ $$T - 143662$$
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