# Properties

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

# Related objects

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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} - 9 q^{3} - 7 q^{4} + 45 q^{6} + 49 q^{7} + 195 q^{8} + 81 q^{9}+O(q^{10})$$ q - 5 * q^2 - 9 * q^3 - 7 * q^4 + 45 * q^6 + 49 * q^7 + 195 * q^8 + 81 * q^9 $$q - 5 q^{2} - 9 q^{3} - 7 q^{4} + 45 q^{6} + 49 q^{7} + 195 q^{8} + 81 q^{9} + 52 q^{11} + 63 q^{12} + 770 q^{13} - 245 q^{14} - 751 q^{16} + 2022 q^{17} - 405 q^{18} + 1732 q^{19} - 441 q^{21} - 260 q^{22} + 576 q^{23} - 1755 q^{24} - 3850 q^{26} - 729 q^{27} - 343 q^{28} + 5518 q^{29} + 6336 q^{31} - 2485 q^{32} - 468 q^{33} - 10110 q^{34} - 567 q^{36} + 7338 q^{37} - 8660 q^{38} - 6930 q^{39} - 3262 q^{41} + 2205 q^{42} - 5420 q^{43} - 364 q^{44} - 2880 q^{46} - 864 q^{47} + 6759 q^{48} + 2401 q^{49} - 18198 q^{51} - 5390 q^{52} - 4182 q^{53} + 3645 q^{54} + 9555 q^{56} - 15588 q^{57} - 27590 q^{58} - 11220 q^{59} - 45602 q^{61} - 31680 q^{62} + 3969 q^{63} + 36457 q^{64} + 2340 q^{66} - 1396 q^{67} - 14154 q^{68} - 5184 q^{69} + 18720 q^{71} + 15795 q^{72} - 46362 q^{73} - 36690 q^{74} - 12124 q^{76} + 2548 q^{77} + 34650 q^{78} + 97424 q^{79} + 6561 q^{81} + 16310 q^{82} + 81228 q^{83} + 3087 q^{84} + 27100 q^{86} - 49662 q^{87} + 10140 q^{88} - 3182 q^{89} + 37730 q^{91} - 4032 q^{92} - 57024 q^{93} + 4320 q^{94} + 22365 q^{96} - 4914 q^{97} - 12005 q^{98} + 4212 q^{99}+O(q^{100})$$ q - 5 * q^2 - 9 * q^3 - 7 * q^4 + 45 * q^6 + 49 * q^7 + 195 * q^8 + 81 * q^9 + 52 * q^11 + 63 * q^12 + 770 * q^13 - 245 * q^14 - 751 * q^16 + 2022 * q^17 - 405 * q^18 + 1732 * q^19 - 441 * q^21 - 260 * q^22 + 576 * q^23 - 1755 * q^24 - 3850 * q^26 - 729 * q^27 - 343 * q^28 + 5518 * q^29 + 6336 * q^31 - 2485 * q^32 - 468 * q^33 - 10110 * q^34 - 567 * q^36 + 7338 * q^37 - 8660 * q^38 - 6930 * q^39 - 3262 * q^41 + 2205 * q^42 - 5420 * q^43 - 364 * q^44 - 2880 * q^46 - 864 * q^47 + 6759 * q^48 + 2401 * q^49 - 18198 * q^51 - 5390 * q^52 - 4182 * q^53 + 3645 * q^54 + 9555 * q^56 - 15588 * q^57 - 27590 * q^58 - 11220 * q^59 - 45602 * q^61 - 31680 * q^62 + 3969 * q^63 + 36457 * q^64 + 2340 * q^66 - 1396 * q^67 - 14154 * q^68 - 5184 * q^69 + 18720 * q^71 + 15795 * q^72 - 46362 * q^73 - 36690 * q^74 - 12124 * q^76 + 2548 * q^77 + 34650 * q^78 + 97424 * q^79 + 6561 * q^81 + 16310 * q^82 + 81228 * q^83 + 3087 * q^84 + 27100 * q^86 - 49662 * q^87 + 10140 * q^88 - 3182 * q^89 + 37730 * q^91 - 4032 * q^92 - 57024 * q^93 + 4320 * q^94 + 22365 * q^96 - 4914 * q^97 - 12005 * q^98 + 4212 * 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
−5.00000 −9.00000 −7.00000 0 45.0000 49.0000 195.000 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.b 1
5.b even 2 1 21.6.a.c 1
5.c odd 4 2 525.6.d.c 2
15.d odd 2 1 63.6.a.b 1
20.d odd 2 1 336.6.a.i 1
35.c odd 2 1 147.6.a.f 1
35.i odd 6 2 147.6.e.d 2
35.j even 6 2 147.6.e.c 2
60.h even 2 1 1008.6.a.a 1
105.g even 2 1 441.6.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 5$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(525))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T + 9$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T - 52$$
$13$ $$T - 770$$
$17$ $$T - 2022$$
$19$ $$T - 1732$$
$23$ $$T - 576$$
$29$ $$T - 5518$$
$31$ $$T - 6336$$
$37$ $$T - 7338$$
$41$ $$T + 3262$$
$43$ $$T + 5420$$
$47$ $$T + 864$$
$53$ $$T + 4182$$
$59$ $$T + 11220$$
$61$ $$T + 45602$$
$67$ $$T + 1396$$
$71$ $$T - 18720$$
$73$ $$T + 46362$$
$79$ $$T - 97424$$
$83$ $$T - 81228$$
$89$ $$T + 3182$$
$97$ $$T + 4914$$