# Properties

 Label 525.2.q.a Level 525 Weight 2 Character orbit 525.q 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.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 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.

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

## Character values

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

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

## 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
−0.866025 + 1.50000i 1.73205 −0.500000 0.866025i 0 −1.50000 + 2.59808i 0.866025 2.50000i −1.73205 3.00000 0
299.2 0.866025 1.50000i −1.73205 −0.500000 0.866025i 0 −1.50000 + 2.59808i −0.866025 + 2.50000i 1.73205 3.00000 0
374.1 −0.866025 1.50000i 1.73205 −0.500000 + 0.866025i 0 −1.50000 2.59808i 0.866025 + 2.50000i −1.73205 3.00000 0
374.2 0.866025 + 1.50000i −1.73205 −0.500000 + 0.866025i 0 −1.50000 2.59808i −0.866025 2.50000i 1.73205 3.00000 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
21.g even 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.a 4
3.b odd 2 1 525.2.q.b 4
5.b even 2 1 inner 525.2.q.a 4
5.c odd 4 1 105.2.s.b yes 2
5.c odd 4 1 525.2.t.a 2
7.d odd 6 1 525.2.q.b 4
15.d odd 2 1 525.2.q.b 4
15.e even 4 1 105.2.s.a 2
15.e even 4 1 525.2.t.e 2
21.g even 6 1 inner 525.2.q.a 4
35.f even 4 1 735.2.s.e 2
35.i odd 6 1 525.2.q.b 4
35.k even 12 1 105.2.s.a 2
35.k even 12 1 525.2.t.e 2
35.k even 12 1 735.2.b.b 2
35.l odd 12 1 735.2.b.a 2
35.l odd 12 1 735.2.s.c 2
105.k odd 4 1 735.2.s.c 2
105.p even 6 1 inner 525.2.q.a 4
105.w odd 12 1 105.2.s.b yes 2
105.w odd 12 1 525.2.t.a 2
105.w odd 12 1 735.2.b.a 2
105.x even 12 1 735.2.b.b 2
105.x even 12 1 735.2.s.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.a 2 15.e even 4 1
105.2.s.a 2 35.k even 12 1
105.2.s.b yes 2 5.c odd 4 1
105.2.s.b yes 2 105.w odd 12 1
525.2.q.a 4 1.a even 1 1 trivial
525.2.q.a 4 5.b even 2 1 inner
525.2.q.a 4 21.g even 6 1 inner
525.2.q.a 4 105.p even 6 1 inner
525.2.q.b 4 3.b odd 2 1
525.2.q.b 4 7.d odd 6 1
525.2.q.b 4 15.d odd 2 1
525.2.q.b 4 35.i odd 6 1
525.2.t.a 2 5.c odd 4 1
525.2.t.a 2 105.w odd 12 1
525.2.t.e 2 15.e even 4 1
525.2.t.e 2 35.k even 12 1
735.2.b.a 2 35.l odd 12 1
735.2.b.a 2 105.w odd 12 1
735.2.b.b 2 35.k even 12 1
735.2.b.b 2 105.x even 12 1
735.2.s.c 2 35.l odd 12 1
735.2.s.c 2 105.k odd 4 1
735.2.s.e 2 35.f even 4 1
735.2.s.e 2 105.x even 12 1

## Hecke kernels

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

 $$T_{2}^{4} + 3 T_{2}^{2} + 9$$ $$T_{11}^{2} - 6 T_{11} + 12$$ $$T_{13}^{2} - 12$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ 
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 55 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 3 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 85 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 94 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 - 134 T^{2} + 12627 T^{4} - 714086 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 85 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 86 T^{2} + 9409 T^{4} )^{2}$$