# Properties

 Label 525.2.bf.d Level 525 Weight 2 Character orbit 525.bf Analytic conductor 4.192 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{2} + ( \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{3} + ( -2 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( 3 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( -2 - 2 \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{2} + ( \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{3} + ( -2 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( 3 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( -2 - 2 \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{9} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{11} + ( 5 \zeta_{24} - 5 \zeta_{24}^{5} ) q^{13} + ( -\zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{14} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{17} + ( -4 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{18} -7 \zeta_{24}^{2} q^{19} + ( 1 + 3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{21} + 6 \zeta_{24}^{3} q^{22} + ( 3 + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} ) q^{23} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{24} + ( -5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{26} + ( 1 - 5 \zeta_{24}^{3} - \zeta_{24}^{6} ) q^{27} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{29} + \zeta_{24}^{4} q^{31} + ( -3 - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{33} -2 \zeta_{24}^{6} q^{34} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} ) q^{37} + ( -7 \zeta_{24}^{2} + 7 \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{38} + ( -5 \zeta_{24} + 5 \zeta_{24}^{2} - 5 \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{39} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{41} + ( 3 - \zeta_{24}^{2} - 6 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{42} + ( -5 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{43} -6 \zeta_{24}^{4} q^{46} + ( -4 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} ) q^{47} + ( 4 - 4 \zeta_{24} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{48} + ( 5 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{49} + ( 2 + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{51} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{53} + ( -2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{54} + ( 4 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{56} + ( -7 - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{6} ) q^{57} -10 \zeta_{24}^{5} q^{58} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{59} + ( -4 + 4 \zeta_{24}^{4} ) q^{61} + ( 1 - \zeta_{24}^{6} ) q^{62} + ( 4 - \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{63} + 8 \zeta_{24}^{6} q^{64} + ( 6 \zeta_{24} + 6 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{66} + \zeta_{24}^{7} q^{67} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{69} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{71} + ( -2 + 2 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 8 \zeta_{24}^{7} ) q^{72} + 7 \zeta_{24} q^{73} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{74} + ( -3 - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 9 \zeta_{24}^{6} ) q^{77} + ( -5 - 10 \zeta_{24} + 10 \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{78} -7 \zeta_{24}^{2} q^{79} + ( -4 \zeta_{24} - 7 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{81} + ( 10 \zeta_{24}^{3} - 10 \zeta_{24}^{7} ) q^{82} + ( 1 + \zeta_{24}^{6} ) q^{83} + ( 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{86} + ( 5 \zeta_{24}^{2} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{6} - 10 \zeta_{24}^{7} ) q^{87} + 12 \zeta_{24} q^{88} + ( -11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} + 11 \zeta_{24}^{7} ) q^{89} + ( 15 - 5 \zeta_{24}^{4} ) q^{91} + ( 1 + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{93} + ( 8 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{94} + 10 \zeta_{24}^{3} q^{97} + ( 3 + 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{98} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{3} - 16q^{6} - 16q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{3} - 16q^{6} - 16q^{8} - 16q^{16} + 4q^{17} + 4q^{18} + 16q^{21} + 12q^{23} + 8q^{27} + 4q^{31} - 12q^{33} + 28q^{38} + 20q^{42} - 24q^{46} - 16q^{47} + 32q^{48} + 8q^{51} - 8q^{53} - 56q^{57} - 16q^{61} + 8q^{62} + 8q^{63} + 24q^{66} - 8q^{72} - 48q^{77} - 40q^{78} - 28q^{81} + 8q^{83} + 20q^{87} + 100q^{91} + 4q^{93} - 8q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$-1$$ $$-1 + \zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
1.36603 + 0.366025i −1.62484 0.599900i 0 0 −2.00000 1.41421i 1.15539 + 2.38014i −2.00000 2.00000i 2.28024 + 1.94949i 0
32.2 1.36603 + 0.366025i −1.10721 + 1.33195i 0 0 −2.00000 + 1.41421i −1.15539 2.38014i −2.00000 2.00000i −0.548188 2.94949i 0
107.1 −0.366025 1.36603i −0.599900 1.62484i 0 0 −2.00000 + 1.41421i −2.38014 1.15539i −2.00000 2.00000i −2.28024 + 1.94949i 0
107.2 −0.366025 1.36603i 1.33195 1.10721i 0 0 −2.00000 1.41421i 2.38014 + 1.15539i −2.00000 2.00000i 0.548188 2.94949i 0
368.1 −0.366025 + 1.36603i −0.599900 + 1.62484i 0 0 −2.00000 1.41421i −2.38014 + 1.15539i −2.00000 + 2.00000i −2.28024 1.94949i 0
368.2 −0.366025 + 1.36603i 1.33195 + 1.10721i 0 0 −2.00000 + 1.41421i 2.38014 1.15539i −2.00000 + 2.00000i 0.548188 + 2.94949i 0
443.1 1.36603 0.366025i −1.62484 + 0.599900i 0 0 −2.00000 + 1.41421i 1.15539 2.38014i −2.00000 + 2.00000i 2.28024 1.94949i 0
443.2 1.36603 0.366025i −1.10721 1.33195i 0 0 −2.00000 1.41421i −1.15539 + 2.38014i −2.00000 + 2.00000i −0.548188 + 2.94949i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.2 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.c odd 4 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.l odd 12 1 inner
105.o odd 6 1 inner
105.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bf.d yes 8
3.b odd 2 1 525.2.bf.a 8
5.b even 2 1 525.2.bf.a 8
5.c odd 4 1 525.2.bf.a 8
5.c odd 4 1 inner 525.2.bf.d yes 8
7.c even 3 1 inner 525.2.bf.d yes 8
15.d odd 2 1 inner 525.2.bf.d yes 8
15.e even 4 1 525.2.bf.a 8
15.e even 4 1 inner 525.2.bf.d yes 8
21.h odd 6 1 525.2.bf.a 8
35.j even 6 1 525.2.bf.a 8
35.l odd 12 1 525.2.bf.a 8
35.l odd 12 1 inner 525.2.bf.d yes 8
105.o odd 6 1 inner 525.2.bf.d yes 8
105.x even 12 1 525.2.bf.a 8
105.x even 12 1 inner 525.2.bf.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.a 8 3.b odd 2 1
525.2.bf.a 8 5.b even 2 1
525.2.bf.a 8 5.c odd 4 1
525.2.bf.a 8 15.e even 4 1
525.2.bf.a 8 21.h odd 6 1
525.2.bf.a 8 35.j even 6 1
525.2.bf.a 8 35.l odd 12 1
525.2.bf.a 8 105.x even 12 1
525.2.bf.d yes 8 1.a even 1 1 trivial
525.2.bf.d yes 8 5.c odd 4 1 inner
525.2.bf.d yes 8 7.c even 3 1 inner
525.2.bf.d yes 8 15.d odd 2 1 inner
525.2.bf.d yes 8 15.e even 4 1 inner
525.2.bf.d yes 8 35.l odd 12 1 inner
525.2.bf.d yes 8 105.o odd 6 1 inner
525.2.bf.d yes 8 105.x even 12 1 inner

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} + 4$$ $$T_{13}^{4} + 625$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{4}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ $$1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 24 T^{5} + 72 T^{6} + 108 T^{7} + 81 T^{8}$$
$5$ 
$7$ $$1 + 23 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 337 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 2 T + 2 T^{2} + 64 T^{3} - 353 T^{4} + 1088 T^{5} + 578 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 6 T + 18 T^{2} + 168 T^{3} - 1033 T^{4} + 3864 T^{5} + 9522 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 8 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} )$$
$41$ $$( 1 - 32 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 8 T + 32 T^{2} - 496 T^{3} - 4193 T^{4} - 23312 T^{5} + 70688 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 4 T + 8 T^{2} - 392 T^{3} - 3593 T^{4} - 20776 T^{5} + 22472 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 - 8711 T^{4} + 55730400 T^{8} - 175536415031 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 - 92 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$
$79$ $$( 1 + 109 T^{2} + 5640 T^{4} + 680269 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 64 T^{2} - 3825 T^{4} + 506944 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2}$$