# Properties

 Label 525.2.q.d Level 525 Weight 2 Character orbit 525.q Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM discriminant -3 Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{12} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( -4 + 4 \zeta_{12}^{2} ) q^{16} + ( 3 + 3 \zeta_{12}^{2} ) q^{19} + ( -1 - 4 \zeta_{12}^{2} ) q^{21} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + ( 10 - 5 \zeta_{12}^{2} ) q^{31} -6 q^{36} + \zeta_{12} q^{37} -3 \zeta_{12}^{2} q^{39} -5 \zeta_{12}^{3} q^{43} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( 8 - 3 \zeta_{12}^{2} ) q^{49} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{52} -9 \zeta_{12}^{3} q^{57} + ( 4 + 4 \zeta_{12}^{2} ) q^{61} + ( -3 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} -8 q^{64} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{67} + ( -9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{73} + ( -6 + 12 \zeta_{12}^{2} ) q^{76} + ( -13 + 13 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 8 - 10 \zeta_{12}^{2} ) q^{84} + ( 5 - \zeta_{12}^{2} ) q^{91} -15 \zeta_{12} q^{93} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 6q^{9} + O(q^{10})$$ $$4q + 4q^{4} - 6q^{9} - 8q^{16} + 18q^{19} - 12q^{21} + 30q^{31} - 24q^{36} - 6q^{39} + 26q^{49} + 24q^{61} - 32q^{64} - 26q^{79} - 18q^{81} + 12q^{84} + 18q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 1.00000 + 1.73205i 0 0 2.59808 0.500000i 0 −1.50000 + 2.59808i 0
299.2 0 0.866025 + 1.50000i 1.00000 + 1.73205i 0 0 −2.59808 + 0.500000i 0 −1.50000 + 2.59808i 0
374.1 0 −0.866025 + 1.50000i 1.00000 1.73205i 0 0 2.59808 + 0.500000i 0 −1.50000 2.59808i 0
374.2 0 0.866025 1.50000i 1.00000 1.73205i 0 0 −2.59808 0.500000i 0 −1.50000 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.q.d 4
3.b odd 2 1 CM 525.2.q.d 4
5.b even 2 1 inner 525.2.q.d 4
5.c odd 4 1 21.2.g.a 2
5.c odd 4 1 525.2.t.c 2
7.d odd 6 1 inner 525.2.q.d 4
15.d odd 2 1 inner 525.2.q.d 4
15.e even 4 1 21.2.g.a 2
15.e even 4 1 525.2.t.c 2
20.e even 4 1 336.2.bc.c 2
21.g even 6 1 inner 525.2.q.d 4
35.f even 4 1 147.2.g.a 2
35.i odd 6 1 inner 525.2.q.d 4
35.k even 12 1 21.2.g.a 2
35.k even 12 1 147.2.c.a 2
35.k even 12 1 525.2.t.c 2
35.l odd 12 1 147.2.c.a 2
35.l odd 12 1 147.2.g.a 2
45.k odd 12 1 567.2.i.b 2
45.k odd 12 1 567.2.s.a 2
45.l even 12 1 567.2.i.b 2
45.l even 12 1 567.2.s.a 2
60.l odd 4 1 336.2.bc.c 2
105.k odd 4 1 147.2.g.a 2
105.p even 6 1 inner 525.2.q.d 4
105.w odd 12 1 21.2.g.a 2
105.w odd 12 1 147.2.c.a 2
105.w odd 12 1 525.2.t.c 2
105.x even 12 1 147.2.c.a 2
105.x even 12 1 147.2.g.a 2
140.w even 12 1 2352.2.k.c 2
140.x odd 12 1 336.2.bc.c 2
140.x odd 12 1 2352.2.k.c 2
315.bs even 12 1 567.2.s.a 2
315.bu odd 12 1 567.2.s.a 2
315.bw odd 12 1 567.2.i.b 2
315.cg even 12 1 567.2.i.b 2
420.bp odd 12 1 2352.2.k.c 2
420.br even 12 1 336.2.bc.c 2
420.br even 12 1 2352.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a 2 5.c odd 4 1
21.2.g.a 2 15.e even 4 1
21.2.g.a 2 35.k even 12 1
21.2.g.a 2 105.w odd 12 1
147.2.c.a 2 35.k even 12 1
147.2.c.a 2 35.l odd 12 1
147.2.c.a 2 105.w odd 12 1
147.2.c.a 2 105.x even 12 1
147.2.g.a 2 35.f even 4 1
147.2.g.a 2 35.l odd 12 1
147.2.g.a 2 105.k odd 4 1
147.2.g.a 2 105.x even 12 1
336.2.bc.c 2 20.e even 4 1
336.2.bc.c 2 60.l odd 4 1
336.2.bc.c 2 140.x odd 12 1
336.2.bc.c 2 420.br even 12 1
525.2.q.d 4 1.a even 1 1 trivial
525.2.q.d 4 3.b odd 2 1 CM
525.2.q.d 4 5.b even 2 1 inner
525.2.q.d 4 7.d odd 6 1 inner
525.2.q.d 4 15.d odd 2 1 inner
525.2.q.d 4 21.g even 6 1 inner
525.2.q.d 4 35.i odd 6 1 inner
525.2.q.d 4 105.p even 6 1 inner
525.2.t.c 2 5.c odd 4 1
525.2.t.c 2 15.e even 4 1
525.2.t.c 2 35.k even 12 1
525.2.t.c 2 105.w odd 12 1
567.2.i.b 2 45.k odd 12 1
567.2.i.b 2 45.l even 12 1
567.2.i.b 2 315.bw odd 12 1
567.2.i.b 2 315.cg even 12 1
567.2.s.a 2 45.k odd 12 1
567.2.s.a 2 45.l even 12 1
567.2.s.a 2 315.bs even 12 1
567.2.s.a 2 315.bu odd 12 1
2352.2.k.c 2 140.w even 12 1
2352.2.k.c 2 140.x odd 12 1
2352.2.k.c 2 420.bp odd 12 1
2352.2.k.c 2 420.br even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}$$ $$T_{11}$$ $$T_{13}^{2} - 3$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ 
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 23 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} )$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 61 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 46 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} )$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 2 T^{2} + 9409 T^{4} )^{2}$$