# Properties

 Label 525.6.d.d Level $525$ Weight $6$ Character orbit 525.d Analytic conductor $84.202$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$84.2015054018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 9 i q^{3} + 31 q^{4} - 9 q^{6} - 49 i q^{7} + 63 i q^{8} - 81 q^{9} +O(q^{10})$$ q + i * q^2 + 9*i * q^3 + 31 * q^4 - 9 * q^6 - 49*i * q^7 + 63*i * q^8 - 81 * q^9 $$q + i q^{2} + 9 i q^{3} + 31 q^{4} - 9 q^{6} - 49 i q^{7} + 63 i q^{8} - 81 q^{9} - 340 q^{11} + 279 i q^{12} - 454 i q^{13} + 49 q^{14} + 929 q^{16} - 798 i q^{17} - 81 i q^{18} - 892 q^{19} + 441 q^{21} - 340 i q^{22} + 3192 i q^{23} - 567 q^{24} + 454 q^{26} - 729 i q^{27} - 1519 i q^{28} + 8242 q^{29} - 2496 q^{31} + 2945 i q^{32} - 3060 i q^{33} + 798 q^{34} - 2511 q^{36} + 9798 i q^{37} - 892 i q^{38} + 4086 q^{39} + 19834 q^{41} + 441 i q^{42} + 17236 i q^{43} - 10540 q^{44} - 3192 q^{46} + 8928 i q^{47} + 8361 i q^{48} - 2401 q^{49} + 7182 q^{51} - 14074 i q^{52} - 150 i q^{53} + 729 q^{54} + 3087 q^{56} - 8028 i q^{57} + 8242 i q^{58} + 42396 q^{59} + 14758 q^{61} - 2496 i q^{62} + 3969 i q^{63} + 26783 q^{64} + 3060 q^{66} - 1676 i q^{67} - 24738 i q^{68} - 28728 q^{69} + 14568 q^{71} - 5103 i q^{72} - 78378 i q^{73} - 9798 q^{74} - 27652 q^{76} + 16660 i q^{77} + 4086 i q^{78} + 2272 q^{79} + 6561 q^{81} + 19834 i q^{82} + 37764 i q^{83} + 13671 q^{84} - 17236 q^{86} + 74178 i q^{87} - 21420 i q^{88} + 117286 q^{89} - 22246 q^{91} + 98952 i q^{92} - 22464 i q^{93} - 8928 q^{94} - 26505 q^{96} + 10002 i q^{97} - 2401 i q^{98} + 27540 q^{99} +O(q^{100})$$ q + i * q^2 + 9*i * q^3 + 31 * q^4 - 9 * q^6 - 49*i * q^7 + 63*i * q^8 - 81 * q^9 - 340 * q^11 + 279*i * q^12 - 454*i * q^13 + 49 * q^14 + 929 * q^16 - 798*i * q^17 - 81*i * q^18 - 892 * q^19 + 441 * q^21 - 340*i * q^22 + 3192*i * q^23 - 567 * q^24 + 454 * q^26 - 729*i * q^27 - 1519*i * q^28 + 8242 * q^29 - 2496 * q^31 + 2945*i * q^32 - 3060*i * q^33 + 798 * q^34 - 2511 * q^36 + 9798*i * q^37 - 892*i * q^38 + 4086 * q^39 + 19834 * q^41 + 441*i * q^42 + 17236*i * q^43 - 10540 * q^44 - 3192 * q^46 + 8928*i * q^47 + 8361*i * q^48 - 2401 * q^49 + 7182 * q^51 - 14074*i * q^52 - 150*i * q^53 + 729 * q^54 + 3087 * q^56 - 8028*i * q^57 + 8242*i * q^58 + 42396 * q^59 + 14758 * q^61 - 2496*i * q^62 + 3969*i * q^63 + 26783 * q^64 + 3060 * q^66 - 1676*i * q^67 - 24738*i * q^68 - 28728 * q^69 + 14568 * q^71 - 5103*i * q^72 - 78378*i * q^73 - 9798 * q^74 - 27652 * q^76 + 16660*i * q^77 + 4086*i * q^78 + 2272 * q^79 + 6561 * q^81 + 19834*i * q^82 + 37764*i * q^83 + 13671 * q^84 - 17236 * q^86 + 74178*i * q^87 - 21420*i * q^88 + 117286 * q^89 - 22246 * q^91 + 98952*i * q^92 - 22464*i * q^93 - 8928 * q^94 - 26505 * q^96 + 10002*i * q^97 - 2401*i * q^98 + 27540 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 62 q^{4} - 18 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 $$2 q + 62 q^{4} - 18 q^{6} - 162 q^{9} - 680 q^{11} + 98 q^{14} + 1858 q^{16} - 1784 q^{19} + 882 q^{21} - 1134 q^{24} + 908 q^{26} + 16484 q^{29} - 4992 q^{31} + 1596 q^{34} - 5022 q^{36} + 8172 q^{39} + 39668 q^{41} - 21080 q^{44} - 6384 q^{46} - 4802 q^{49} + 14364 q^{51} + 1458 q^{54} + 6174 q^{56} + 84792 q^{59} + 29516 q^{61} + 53566 q^{64} + 6120 q^{66} - 57456 q^{69} + 29136 q^{71} - 19596 q^{74} - 55304 q^{76} + 4544 q^{79} + 13122 q^{81} + 27342 q^{84} - 34472 q^{86} + 234572 q^{89} - 44492 q^{91} - 17856 q^{94} - 53010 q^{96} + 55080 q^{99}+O(q^{100})$$ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 - 680 * q^11 + 98 * q^14 + 1858 * q^16 - 1784 * q^19 + 882 * q^21 - 1134 * q^24 + 908 * q^26 + 16484 * q^29 - 4992 * q^31 + 1596 * q^34 - 5022 * q^36 + 8172 * q^39 + 39668 * q^41 - 21080 * q^44 - 6384 * q^46 - 4802 * q^49 + 14364 * q^51 + 1458 * q^54 + 6174 * q^56 + 84792 * q^59 + 29516 * q^61 + 53566 * q^64 + 6120 * q^66 - 57456 * q^69 + 29136 * q^71 - 19596 * q^74 - 55304 * q^76 + 4544 * q^79 + 13122 * q^81 + 27342 * q^84 - 34472 * q^86 + 234572 * q^89 - 44492 * q^91 - 17856 * q^94 - 53010 * q^96 + 55080 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
1.00000i 9.00000i 31.0000 0 −9.00000 49.0000i 63.0000i −81.0000 0
274.2 1.00000i 9.00000i 31.0000 0 −9.00000 49.0000i 63.0000i −81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.6.d.d 2
5.b even 2 1 inner 525.6.d.d 2
5.c odd 4 1 21.6.a.b 1
5.c odd 4 1 525.6.a.c 1
15.e even 4 1 63.6.a.c 1
20.e even 4 1 336.6.a.l 1
35.f even 4 1 147.6.a.e 1
35.k even 12 2 147.6.e.e 2
35.l odd 12 2 147.6.e.f 2
60.l odd 4 1 1008.6.a.t 1
105.k odd 4 1 441.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 5.c odd 4 1
63.6.a.c 1 15.e even 4 1
147.6.a.e 1 35.f even 4 1
147.6.e.e 2 35.k even 12 2
147.6.e.f 2 35.l odd 12 2
336.6.a.l 1 20.e even 4 1
441.6.a.d 1 105.k odd 4 1
525.6.a.c 1 5.c odd 4 1
525.6.d.d 2 1.a even 1 1 trivial
525.6.d.d 2 5.b even 2 1 inner
1008.6.a.t 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{6}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 340)^{2}$$
$13$ $$T^{2} + 206116$$
$17$ $$T^{2} + 636804$$
$19$ $$(T + 892)^{2}$$
$23$ $$T^{2} + 10188864$$
$29$ $$(T - 8242)^{2}$$
$31$ $$(T + 2496)^{2}$$
$37$ $$T^{2} + 96000804$$
$41$ $$(T - 19834)^{2}$$
$43$ $$T^{2} + 297079696$$
$47$ $$T^{2} + 79709184$$
$53$ $$T^{2} + 22500$$
$59$ $$(T - 42396)^{2}$$
$61$ $$(T - 14758)^{2}$$
$67$ $$T^{2} + 2808976$$
$71$ $$(T - 14568)^{2}$$
$73$ $$T^{2} + 6143110884$$
$79$ $$(T - 2272)^{2}$$
$83$ $$T^{2} + 1426119696$$
$89$ $$(T - 117286)^{2}$$
$97$ $$T^{2} + 100040004$$