# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{2} + \zeta_{24}^{7} q^{3} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( -1 + 2 \zeta_{24}^{4} ) q^{6} + ( -3 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{2} + \zeta_{24}^{7} q^{3} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( -1 + 2 \zeta_{24}^{4} ) q^{6} + ( -3 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{8} -\zeta_{24}^{2} q^{9} + 6 \zeta_{24}^{4} q^{11} + \zeta_{24}^{5} q^{12} + 5 \zeta_{24}^{3} q^{13} + ( -4 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{14} + ( -5 + 5 \zeta_{24}^{4} ) q^{16} + ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{18} + ( 3 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{19} + ( 1 - 3 \zeta_{24}^{4} ) q^{21} + ( 6 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{22} + ( -2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{23} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{24} + ( 5 + 5 \zeta_{24}^{4} ) q^{26} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{27} + ( -\zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{28} + ( 4 - 2 \zeta_{24}^{4} ) q^{31} + ( -3 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{32} + ( -6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{33} - q^{36} + ( -10 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{37} -9 \zeta_{24}^{7} q^{38} + ( -5 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{39} + ( 6 - 12 \zeta_{24}^{4} ) q^{41} + ( -\zeta_{24} - 4 \zeta_{24}^{5} ) q^{42} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43} + 6 \zeta_{24}^{2} q^{44} -6 \zeta_{24}^{4} q^{46} -5 \zeta_{24}^{3} q^{48} + ( 5 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{49} + 5 \zeta_{24} q^{52} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{54} + ( 4 - 5 \zeta_{24}^{4} ) q^{56} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{57} + ( 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{59} + ( -1 - \zeta_{24}^{4} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{62} + ( 3 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{63} -\zeta_{24}^{6} q^{64} + ( -12 + 6 \zeta_{24}^{4} ) q^{66} + ( -5 \zeta_{24} + 10 \zeta_{24}^{5} ) q^{67} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{69} + 12 q^{71} + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{72} -7 \zeta_{24}^{7} q^{73} + ( -15 \zeta_{24}^{2} + 15 \zeta_{24}^{6} ) q^{74} + ( 3 - 6 \zeta_{24}^{4} ) q^{76} + ( -12 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{77} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{78} + 11 \zeta_{24}^{2} q^{79} + \zeta_{24}^{4} q^{81} -18 \zeta_{24}^{5} q^{82} + 12 \zeta_{24}^{3} q^{83} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{84} + ( -6 + 6 \zeta_{24}^{4} ) q^{86} + ( -12 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{88} + ( 8 \zeta_{24}^{2} - 16 \zeta_{24}^{6} ) q^{89} + ( -10 - 5 \zeta_{24}^{4} ) q^{91} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{92} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{93} + ( -3 - 3 \zeta_{24}^{4} ) q^{96} + ( 5 \zeta_{24} - 5 \zeta_{24}^{5} ) q^{97} + ( 2 \zeta_{24}^{3} - 13 \zeta_{24}^{7} ) q^{98} -6 \zeta_{24}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 24q^{11} - 20q^{16} - 4q^{21} + 60q^{26} + 24q^{31} - 8q^{36} - 24q^{46} + 12q^{56} - 12q^{61} - 72q^{66} + 96q^{71} + 4q^{81} - 24q^{86} - 100q^{91} - 36q^{96} + O(q^{100})$$

## 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
−1.67303 + 0.448288i 0.258819 0.965926i 0.866025 0.500000i 0 1.73205i 2.38014 1.15539i 1.22474 1.22474i −0.866025 0.500000i 0
82.2 1.67303 0.448288i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.73205i −2.38014 + 1.15539i −1.22474 + 1.22474i −0.866025 0.500000i 0
157.1 −0.448288 + 1.67303i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.73205i 1.15539 2.38014i −1.22474 + 1.22474i 0.866025 0.500000i 0
157.2 0.448288 1.67303i 0.965926 0.258819i −0.866025 0.500000i 0 1.73205i −1.15539 + 2.38014i 1.22474 1.22474i 0.866025 0.500000i 0
418.1 −0.448288 1.67303i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.73205i 1.15539 + 2.38014i −1.22474 1.22474i 0.866025 + 0.500000i 0
418.2 0.448288 + 1.67303i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.73205i −1.15539 2.38014i 1.22474 + 1.22474i 0.866025 + 0.500000i 0
493.1 −1.67303 0.448288i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.73205i 2.38014 + 1.15539i 1.22474 + 1.22474i −0.866025 + 0.500000i 0
493.2 1.67303 + 0.448288i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.73205i −2.38014 1.15539i −1.22474 1.22474i −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.b 8
5.b even 2 1 inner 525.2.bc.b 8
5.c odd 4 2 inner 525.2.bc.b 8
7.d odd 6 1 inner 525.2.bc.b 8
35.i odd 6 1 inner 525.2.bc.b 8
35.k even 12 2 inner 525.2.bc.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 9 T_{2}^{4} + 81$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T^{4} + 33 T^{8} + 112 T^{12} + 256 T^{16}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ 
$7$ $$1 + 23 T^{4} + 2401 T^{8}$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 337 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 289 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 - 98 T^{4} - 270237 T^{8} - 27424418 T^{12} + 78310985281 T^{16}$$
$29$ $$( 1 - 29 T^{2} )^{8}$$
$31$ $$( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{4}$$
$37$ $$1 + 2737 T^{4} + 5617008 T^{8} + 5129578657 T^{12} + 3512479453921 T^{16}$$
$41$ $$( 1 + 26 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 1778 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 2209 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 2809 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 + 5497 T^{4} + 10065888 T^{8} + 110770712137 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{8}$$
$73$ $$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$
$79$ $$( 1 + 37 T^{2} - 4872 T^{4} + 230917 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 13294 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 14 T^{2} - 7725 T^{4} + 110894 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 9743 T^{4} + 88529281 T^{8} )^{2}$$
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