# Properties

 Label 525.2.bc.a Level 525 Weight 2 Character orbit 525.bc 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.bc (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.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{7} q^{3} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{4} + ( \zeta_{24} - 3 \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{24}^{7} q^{3} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{4} + ( \zeta_{24} - 3 \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{2} q^{9} -6 \zeta_{24}^{4} q^{11} -2 \zeta_{24}^{5} q^{12} + 2 \zeta_{24}^{3} q^{13} + ( 4 - 4 \zeta_{24}^{4} ) q^{16} + 6 \zeta_{24} q^{17} + ( 2 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{19} + ( 2 + \zeta_{24}^{4} ) q^{21} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{27} + ( 4 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{28} + ( -6 + 3 \zeta_{24}^{4} ) q^{31} + ( 6 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{33} + 2 q^{36} + ( 10 \zeta_{24} - 5 \zeta_{24}^{5} ) q^{37} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{39} + ( 6 - 12 \zeta_{24}^{4} ) q^{41} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{43} + 12 \zeta_{24}^{2} q^{44} -6 \zeta_{24}^{5} q^{47} + 4 \zeta_{24}^{3} q^{48} + ( -8 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{49} + ( -6 + 6 \zeta_{24}^{4} ) q^{51} -4 \zeta_{24} q^{52} + ( -12 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{57} + ( -6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{59} + ( 5 + 5 \zeta_{24}^{4} ) q^{61} + ( -\zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{63} + 8 \zeta_{24}^{6} q^{64} + ( -2 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{67} + ( -12 \zeta_{24}^{3} + 12 \zeta_{24}^{7} ) q^{68} -6 q^{71} -\zeta_{24}^{7} q^{73} + ( -4 + 8 \zeta_{24}^{4} ) q^{76} + ( -18 \zeta_{24} + 12 \zeta_{24}^{5} ) q^{77} + 5 \zeta_{24}^{2} q^{79} + \zeta_{24}^{4} q^{81} -6 \zeta_{24}^{3} q^{83} + ( -6 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{84} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{89} + ( 6 - 4 \zeta_{24}^{4} ) q^{91} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{93} + ( -13 \zeta_{24} + 13 \zeta_{24}^{5} ) q^{97} + 6 \zeta_{24}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{11} + 16q^{16} + 20q^{21} - 36q^{31} + 16q^{36} - 24q^{51} + 60q^{61} - 48q^{71} + 4q^{81} + 32q^{91} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
0 −0.258819 + 0.965926i −1.73205 + 1.00000i 0 0 0.189469 2.63896i 0 −0.866025 0.500000i 0
82.2 0 0.258819 0.965926i −1.73205 + 1.00000i 0 0 −0.189469 + 2.63896i 0 −0.866025 0.500000i 0
157.1 0 −0.965926 + 0.258819i 1.73205 + 1.00000i 0 0 −2.63896 + 0.189469i 0 0.866025 0.500000i 0
157.2 0 0.965926 0.258819i 1.73205 + 1.00000i 0 0 2.63896 0.189469i 0 0.866025 0.500000i 0
418.1 0 −0.965926 0.258819i 1.73205 1.00000i 0 0 −2.63896 0.189469i 0 0.866025 + 0.500000i 0
418.2 0 0.965926 + 0.258819i 1.73205 1.00000i 0 0 2.63896 + 0.189469i 0 0.866025 + 0.500000i 0
493.1 0 −0.258819 0.965926i −1.73205 1.00000i 0 0 0.189469 + 2.63896i 0 −0.866025 + 0.500000i 0
493.2 0 0.258819 + 0.965926i −1.73205 1.00000i 0 0 −0.189469 2.63896i 0 −0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.2 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.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bc.a 8
5.b even 2 1 inner 525.2.bc.a 8
5.c odd 4 2 inner 525.2.bc.a 8
7.d odd 6 1 inner 525.2.bc.a 8
35.i odd 6 1 inner 525.2.bc.a 8
35.k even 12 2 inner 525.2.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bc.a 8 1.a even 1 1 trivial
525.2.bc.a 8 5.b even 2 1 inner
525.2.bc.a 8 5.c odd 4 2 inner
525.2.bc.a 8 7.d odd 6 1 inner
525.2.bc.a 8 35.i odd 6 1 inner
525.2.bc.a 8 35.k even 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ 1
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 146 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 8 T + 32 T^{2} + 16 T^{3} - 353 T^{4} + 272 T^{5} + 9248 T^{6} - 39304 T^{7} + 83521 T^{8} )( 1 + 8 T + 32 T^{2} - 16 T^{3} - 353 T^{4} - 272 T^{5} + 9248 T^{6} + 39304 T^{7} + 83521 T^{8} )$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 11 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 529 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{8}$$
$31$ $$( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{4}$$
$37$ $$1 + 2737 T^{4} + 5617008 T^{8} + 5129578657 T^{12} + 3512479453921 T^{16}$$
$41$ $$( 1 + 26 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 217 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$
$53$ $$1 + 5614 T^{4} + 23626515 T^{8} + 44297160334 T^{12} + 62259690411361 T^{16}$$
$59$ $$( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{4}( 1 - T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 8809 T^{4} + 20151121 T^{8} )( 1 + 2903 T^{4} + 20151121 T^{8} )$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{8}$$
$73$ $$1 - 10367 T^{4} + 79076448 T^{8} - 294404564447 T^{12} + 806460091894081 T^{16}$$
$79$ $$( 1 + 133 T^{2} + 11448 T^{4} + 830053 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 3122 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 18193 T^{4} + 88529281 T^{8} )^{2}$$