# Properties

 Label 525.2.r.a Level $525$ Weight $2$ Character orbit 525.r Analytic conductor $4.192$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.r (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( 3 - \zeta_{12}^{2} ) q^{7} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} + ( 3 - \zeta_{12}^{2} ) q^{7} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + ( \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{11} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{12} + ( 1 - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{13} + ( -4 + 3 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14} + ( -8 + 4 \zeta_{12} + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{16} + ( 2 + 5 \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{17} + ( -2 + \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{18} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{21} -2 \zeta_{12}^{3} q^{22} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{23} + ( -2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + ( 1 - 3 \zeta_{12} - \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{26} -\zeta_{12}^{3} q^{27} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( 16 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{32} + ( 1 + \zeta_{12} + \zeta_{12}^{2} ) q^{33} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{34} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{36} + ( -3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{37} + ( -6 + 7 \zeta_{12} + 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{38} + ( -\zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{39} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{41} + ( 3 - 5 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{42} + ( -3 + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + ( 4 - 4 \zeta_{12}^{2} ) q^{44} + ( 4 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{46} -2 \zeta_{12} q^{47} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{48} + ( 8 - 5 \zeta_{12}^{2} ) q^{49} + ( 5 + \zeta_{12} - 5 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} + ( -12 + 2 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( 12 - 2 \zeta_{12} - 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{54} + ( 2 + 6 \zeta_{12} - 10 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{56} + ( -2 + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{57} + ( -2 + 8 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{58} + ( 3 \zeta_{12} + 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{59} + ( -4 + 4 \zeta_{12}^{2} ) q^{61} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{62} + ( 2 - 3 \zeta_{12}^{2} ) q^{63} + ( -16 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{64} -2 \zeta_{12}^{2} q^{66} + ( 10 - 6 \zeta_{12} - 5 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{67} + ( -8 + 4 \zeta_{12} - 8 \zeta_{12}^{2} ) q^{68} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{69} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -2 + 6 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{72} + ( 10 - 4 \zeta_{12} - 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( -\zeta_{12} + 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{74} + ( 14 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{76} + ( 1 + 4 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( -3 + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{78} + ( 3 + 6 \zeta_{12} - 3 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} -2 \zeta_{12} q^{82} + ( -7 + 14 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{83} + ( -8 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{84} + ( 7 - \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{86} + ( -6 - \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{87} + ( -8 + 4 \zeta_{12}^{2} ) q^{88} + ( 3 - 7 \zeta_{12} - 3 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{89} + ( 1 - 4 \zeta_{12} - 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{91} + ( -8 + 16 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{92} + ( -2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{93} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} + ( -8 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{96} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} + ( -13 + 8 \zeta_{12} + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{2} + 4q^{4} + 4q^{6} + 10q^{7} + 2q^{9} + O(q^{10})$$ $$4q - 6q^{2} + 4q^{4} + 4q^{6} + 10q^{7} + 2q^{9} + 2q^{11} - 12q^{12} - 18q^{14} - 16q^{16} + 6q^{17} - 6q^{18} + 2q^{19} + 6q^{23} + 12q^{24} + 2q^{26} + 16q^{28} - 4q^{29} - 6q^{31} + 48q^{32} + 6q^{33} + 8q^{34} + 8q^{36} - 18q^{37} - 18q^{38} - 8q^{39} + 4q^{41} + 10q^{42} + 8q^{44} - 12q^{46} + 22q^{49} + 10q^{51} - 36q^{52} + 36q^{53} + 2q^{54} - 12q^{56} - 12q^{58} + 10q^{59} - 8q^{61} + 2q^{63} - 64q^{64} - 4q^{66} + 30q^{67} - 48q^{68} - 12q^{69} + 4q^{71} - 12q^{72} + 30q^{73} + 14q^{74} + 56q^{76} + 8q^{77} + 6q^{79} - 2q^{81} - 36q^{84} + 14q^{86} - 18q^{87} - 24q^{88} + 6q^{89} - 6q^{91} - 12q^{93} - 4q^{94} - 16q^{96} - 48q^{98} + 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−2.36603 + 1.36603i −0.866025 0.500000i 2.73205 4.73205i 0 2.73205 2.50000 + 0.866025i 9.46410i 0.500000 + 0.866025i 0
424.2 −0.633975 + 0.366025i 0.866025 + 0.500000i −0.732051 + 1.26795i 0 −0.732051 2.50000 + 0.866025i 2.53590i 0.500000 + 0.866025i 0
499.1 −2.36603 1.36603i −0.866025 + 0.500000i 2.73205 + 4.73205i 0 2.73205 2.50000 0.866025i 9.46410i 0.500000 0.866025i 0
499.2 −0.633975 0.366025i 0.866025 0.500000i −0.732051 1.26795i 0 −0.732051 2.50000 0.866025i 2.53590i 0.500000 0.866025i 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
35.j 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.r.a 4
5.b even 2 1 525.2.r.f 4
5.c odd 4 1 105.2.i.d 4
5.c odd 4 1 525.2.i.f 4
7.c even 3 1 525.2.r.f 4
15.e even 4 1 315.2.j.c 4
20.e even 4 1 1680.2.bg.o 4
35.f even 4 1 735.2.i.l 4
35.j even 6 1 inner 525.2.r.a 4
35.k even 12 1 735.2.a.h 2
35.k even 12 1 735.2.i.l 4
35.k even 12 1 3675.2.a.be 2
35.l odd 12 1 105.2.i.d 4
35.l odd 12 1 525.2.i.f 4
35.l odd 12 1 735.2.a.g 2
35.l odd 12 1 3675.2.a.bg 2
105.w odd 12 1 2205.2.a.ba 2
105.x even 12 1 315.2.j.c 4
105.x even 12 1 2205.2.a.z 2
140.w even 12 1 1680.2.bg.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 5.c odd 4 1
105.2.i.d 4 35.l odd 12 1
315.2.j.c 4 15.e even 4 1
315.2.j.c 4 105.x even 12 1
525.2.i.f 4 5.c odd 4 1
525.2.i.f 4 35.l odd 12 1
525.2.r.a 4 1.a even 1 1 trivial
525.2.r.a 4 35.j even 6 1 inner
525.2.r.f 4 5.b even 2 1
525.2.r.f 4 7.c even 3 1
735.2.a.g 2 35.l odd 12 1
735.2.a.h 2 35.k even 12 1
735.2.i.l 4 35.f even 4 1
735.2.i.l 4 35.k even 12 1
1680.2.bg.o 4 20.e even 4 1
1680.2.bg.o 4 140.w even 12 1
2205.2.a.z 2 105.x even 12 1
2205.2.a.ba 2 105.w odd 12 1
3675.2.a.be 2 35.k even 12 1
3675.2.a.bg 2 35.l odd 12 1

## Hecke kernels

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

 $$T_{2}^{4} + 6 T_{2}^{3} + 14 T_{2}^{2} + 12 T_{2} + 4$$ $$T_{11}^{4} - 2 T_{11}^{3} + 6 T_{11}^{2} + 4 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - 5 T + T^{2} )^{2}$$
$11$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$169 + 38 T^{2} + T^{4}$$
$17$ $$484 + 132 T - 10 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$( -26 + 2 T + T^{2} )^{2}$$
$31$ $$9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$529 + 414 T + 131 T^{2} + 18 T^{3} + T^{4}$$
$41$ $$( -2 - 2 T + T^{2} )^{2}$$
$43$ $$529 + 62 T^{2} + T^{4}$$
$47$ $$16 - 4 T^{2} + T^{4}$$
$53$ $$10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4}$$
$59$ $$4 + 20 T + 102 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$( 16 + 4 T + T^{2} )^{2}$$
$67$ $$1521 - 1170 T + 339 T^{2} - 30 T^{3} + T^{4}$$
$71$ $$( -26 - 2 T + T^{2} )^{2}$$
$73$ $$3481 - 1770 T + 359 T^{2} - 30 T^{3} + T^{4}$$
$79$ $$9801 + 594 T + 135 T^{2} - 6 T^{3} + T^{4}$$
$83$ $$19044 + 312 T^{2} + T^{4}$$
$89$ $$19044 + 828 T + 174 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$256 + 224 T^{2} + T^{4}$$