# Properties

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

# 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} -2 q^{6} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} -2 q^{6} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} + 6 \zeta_{12}^{2} q^{11} -2 \zeta_{12} q^{12} + 3 \zeta_{12}^{3} q^{13} + ( 4 + 2 \zeta_{12}^{2} ) q^{14} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{17} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{18} + ( 1 - \zeta_{12}^{2} ) q^{19} + ( -3 + 2 \zeta_{12}^{2} ) q^{21} + 12 \zeta_{12}^{3} q^{22} -4 \zeta_{12} q^{23} + ( -6 + 6 \zeta_{12}^{2} ) q^{26} + \zeta_{12}^{3} q^{27} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + 8 q^{29} -\zeta_{12}^{2} q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -6 \zeta_{12} q^{33} -8 q^{34} + 2 q^{36} -7 \zeta_{12} q^{37} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} -3 \zeta_{12}^{2} q^{39} -6 q^{41} + ( -6 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{42} -\zeta_{12}^{3} q^{43} + ( -12 + 12 \zeta_{12}^{2} ) q^{44} -8 \zeta_{12}^{2} q^{46} -2 \zeta_{12} q^{47} + 4 \zeta_{12}^{3} q^{48} + ( 8 - 3 \zeta_{12}^{2} ) q^{49} + ( 4 - 4 \zeta_{12}^{2} ) q^{51} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{52} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{53} + ( -2 + 2 \zeta_{12}^{2} ) q^{54} + \zeta_{12}^{3} q^{57} + 16 \zeta_{12} q^{58} -8 \zeta_{12}^{2} q^{59} + ( 14 - 14 \zeta_{12}^{2} ) q^{61} -2 \zeta_{12}^{3} q^{62} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + 8 q^{64} -12 \zeta_{12}^{2} q^{66} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{67} -8 \zeta_{12} q^{68} + 4 q^{69} + 6 q^{71} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{73} -14 \zeta_{12}^{2} q^{74} + 2 q^{76} + ( 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -6 \zeta_{12}^{3} q^{78} + ( -1 + \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} -12 \zeta_{12} q^{82} -2 \zeta_{12}^{3} q^{83} + ( -4 - 2 \zeta_{12}^{2} ) q^{84} + ( 2 - 2 \zeta_{12}^{2} ) q^{86} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{87} + ( -12 + 12 \zeta_{12}^{2} ) q^{89} + ( -3 + 9 \zeta_{12}^{2} ) q^{91} -8 \zeta_{12}^{3} q^{92} + \zeta_{12} q^{93} -4 \zeta_{12}^{2} q^{94} + ( -8 + 8 \zeta_{12}^{2} ) q^{96} -6 \zeta_{12}^{3} q^{97} + ( 16 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{98} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 8q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 4q^{4} - 8q^{6} + 2q^{9} + 12q^{11} + 20q^{14} + 8q^{16} + 2q^{19} - 8q^{21} - 12q^{26} + 32q^{29} - 2q^{31} - 32q^{34} + 8q^{36} - 6q^{39} - 24q^{41} - 24q^{44} - 16q^{46} + 26q^{49} + 8q^{51} - 4q^{54} - 16q^{59} + 28q^{61} + 32q^{64} - 24q^{66} + 16q^{69} + 24q^{71} - 28q^{74} + 8q^{76} - 2q^{79} - 2q^{81} - 20q^{84} + 4q^{86} - 24q^{89} + 6q^{91} - 8q^{94} - 16q^{96} + 24q^{99} + O(q^{100})$$

## 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
−1.73205 + 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i 0 −2.00000 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
424.2 1.73205 1.00000i −0.866025 0.500000i 1.00000 1.73205i 0 −2.00000 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
499.1 −1.73205 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i 0 −2.00000 −2.59808 + 0.500000i 0 0.500000 0.866025i 0
499.2 1.73205 + 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i 0 −2.00000 2.59808 0.500000i 0 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
5.b even 2 1 inner
7.c even 3 1 inner
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.d 4
5.b even 2 1 inner 525.2.r.d 4
5.c odd 4 1 105.2.i.b 2
5.c odd 4 1 525.2.i.a 2
7.c even 3 1 inner 525.2.r.d 4
15.e even 4 1 315.2.j.a 2
20.e even 4 1 1680.2.bg.l 2
35.f even 4 1 735.2.i.f 2
35.j even 6 1 inner 525.2.r.d 4
35.k even 12 1 735.2.a.a 1
35.k even 12 1 735.2.i.f 2
35.k even 12 1 3675.2.a.p 1
35.l odd 12 1 105.2.i.b 2
35.l odd 12 1 525.2.i.a 2
35.l odd 12 1 735.2.a.b 1
35.l odd 12 1 3675.2.a.o 1
105.w odd 12 1 2205.2.a.m 1
105.x even 12 1 315.2.j.a 2
105.x even 12 1 2205.2.a.k 1
140.w even 12 1 1680.2.bg.l 2

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

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ $$T_{11}^{2} - 6 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ 1
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 17 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 85 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} )$$
$59$ $$( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2}$$
$67$ $$1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$1 + 145 T^{2} + 15696 T^{4} + 772705 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 162 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 158 T^{2} + 9409 T^{4} )^{2}$$