# Properties

 Label 525.4.d.f Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ 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] = [525,4,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, 4, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$30.9760027530$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q - 3 i q^{3} + 8 q^{4} - 7 i q^{7} - 9 q^{9} +O(q^{10})$$ q - 3*i * q^3 + 8 * q^4 - 7*i * q^7 - 9 * q^9 $$q - 3 i q^{3} + 8 q^{4} - 7 i q^{7} - 9 q^{9} + 42 q^{11} - 24 i q^{12} + 20 i q^{13} + 64 q^{16} - 66 i q^{17} - 38 q^{19} - 21 q^{21} + 12 i q^{23} + 27 i q^{27} - 56 i q^{28} + 258 q^{29} + 146 q^{31} - 126 i q^{33} - 72 q^{36} - 434 i q^{37} + 60 q^{39} - 282 q^{41} + 20 i q^{43} + 336 q^{44} + 72 i q^{47} - 192 i q^{48} - 49 q^{49} - 198 q^{51} + 160 i q^{52} + 336 i q^{53} + 114 i q^{57} + 360 q^{59} - 682 q^{61} + 63 i q^{63} + 512 q^{64} - 812 i q^{67} - 528 i q^{68} + 36 q^{69} + 810 q^{71} - 124 i q^{73} - 304 q^{76} - 294 i q^{77} - 1136 q^{79} + 81 q^{81} + 156 i q^{83} - 168 q^{84} - 774 i q^{87} + 1038 q^{89} + 140 q^{91} + 96 i q^{92} - 438 i q^{93} - 1208 i q^{97} - 378 q^{99} +O(q^{100})$$ q - 3*i * q^3 + 8 * q^4 - 7*i * q^7 - 9 * q^9 + 42 * q^11 - 24*i * q^12 + 20*i * q^13 + 64 * q^16 - 66*i * q^17 - 38 * q^19 - 21 * q^21 + 12*i * q^23 + 27*i * q^27 - 56*i * q^28 + 258 * q^29 + 146 * q^31 - 126*i * q^33 - 72 * q^36 - 434*i * q^37 + 60 * q^39 - 282 * q^41 + 20*i * q^43 + 336 * q^44 + 72*i * q^47 - 192*i * q^48 - 49 * q^49 - 198 * q^51 + 160*i * q^52 + 336*i * q^53 + 114*i * q^57 + 360 * q^59 - 682 * q^61 + 63*i * q^63 + 512 * q^64 - 812*i * q^67 - 528*i * q^68 + 36 * q^69 + 810 * q^71 - 124*i * q^73 - 304 * q^76 - 294*i * q^77 - 1136 * q^79 + 81 * q^81 + 156*i * q^83 - 168 * q^84 - 774*i * q^87 + 1038 * q^89 + 140 * q^91 + 96*i * q^92 - 438*i * q^93 - 1208*i * q^97 - 378 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{4} - 18 q^{9}+O(q^{10})$$ 2 * q + 16 * q^4 - 18 * q^9 $$2 q + 16 q^{4} - 18 q^{9} + 84 q^{11} + 128 q^{16} - 76 q^{19} - 42 q^{21} + 516 q^{29} + 292 q^{31} - 144 q^{36} + 120 q^{39} - 564 q^{41} + 672 q^{44} - 98 q^{49} - 396 q^{51} + 720 q^{59} - 1364 q^{61} + 1024 q^{64} + 72 q^{69} + 1620 q^{71} - 608 q^{76} - 2272 q^{79} + 162 q^{81} - 336 q^{84} + 2076 q^{89} + 280 q^{91} - 756 q^{99}+O(q^{100})$$ 2 * q + 16 * q^4 - 18 * q^9 + 84 * q^11 + 128 * q^16 - 76 * q^19 - 42 * q^21 + 516 * q^29 + 292 * q^31 - 144 * q^36 + 120 * q^39 - 564 * q^41 + 672 * q^44 - 98 * q^49 - 396 * q^51 + 720 * q^59 - 1364 * q^61 + 1024 * q^64 + 72 * q^69 + 1620 * q^71 - 608 * q^76 - 2272 * q^79 + 162 * q^81 - 336 * q^84 + 2076 * q^89 + 280 * q^91 - 756 * 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
0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 0
274.2 0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 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 525.4.d.f 2
5.b even 2 1 inner 525.4.d.f 2
5.c odd 4 1 105.4.a.a 1
5.c odd 4 1 525.4.a.e 1
15.e even 4 1 315.4.a.d 1
15.e even 4 1 1575.4.a.f 1
20.e even 4 1 1680.4.a.s 1
35.f even 4 1 735.4.a.c 1
105.k odd 4 1 2205.4.a.o 1

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 49$$
$11$ $$(T - 42)^{2}$$
$13$ $$T^{2} + 400$$
$17$ $$T^{2} + 4356$$
$19$ $$(T + 38)^{2}$$
$23$ $$T^{2} + 144$$
$29$ $$(T - 258)^{2}$$
$31$ $$(T - 146)^{2}$$
$37$ $$T^{2} + 188356$$
$41$ $$(T + 282)^{2}$$
$43$ $$T^{2} + 400$$
$47$ $$T^{2} + 5184$$
$53$ $$T^{2} + 112896$$
$59$ $$(T - 360)^{2}$$
$61$ $$(T + 682)^{2}$$
$67$ $$T^{2} + 659344$$
$71$ $$(T - 810)^{2}$$
$73$ $$T^{2} + 15376$$
$79$ $$(T + 1136)^{2}$$
$83$ $$T^{2} + 24336$$
$89$ $$(T - 1038)^{2}$$
$97$ $$T^{2} + 1459264$$