# Properties

 Label 525.2.r.g Level 525 Weight 2 Character orbit 525.r Analytic conductor 4.192 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{2} q^{3} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{2} q^{3} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + \zeta_{24}^{4} q^{9} + ( -2 + \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{13} + ( 2 - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{14} + 4 \zeta_{24}^{4} q^{16} + ( 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{18} + ( 4 \zeta_{24} - \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{19} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{22} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{23} + ( -2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{24} + ( -3 \zeta_{24} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{6} q^{27} + ( -4 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{29} + ( 3 + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{31} + ( -\zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{33} + ( 6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{34} + ( -\zeta_{24} - 7 \zeta_{24}^{2} + 7 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{37} + ( -8 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{38} + ( 3 + \zeta_{24}^{3} - 3 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{39} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{41} + ( \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{42} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 9 \zeta_{24}^{6} ) q^{43} + ( 6 - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{46} + ( -8 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{47} -4 \zeta_{24}^{6} q^{48} + ( -5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( 3 \zeta_{24} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{51} + ( 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{54} + ( -8 - 2 \zeta_{24} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{56} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{57} + ( 4 \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{58} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{59} + ( 6 \zeta_{24} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{62} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{66} + ( \zeta_{24}^{2} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{67} + ( -2 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{69} + ( -2 + \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{71} + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{72} + ( -5 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{73} + ( 2 + 7 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 7 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{74} + ( 3 \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{78} + ( 4 \zeta_{24} + \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{79} + ( -1 + \zeta_{24}^{4} ) q^{81} + ( 2 \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{82} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{83} + ( -9 \zeta_{24} + 2 \zeta_{24}^{4} - 9 \zeta_{24}^{7} ) q^{86} + ( 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{87} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{88} + ( -3 \zeta_{24} - 8 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{89} + ( 4 + 4 \zeta_{24} + 6 \zeta_{24}^{3} + \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -2 \zeta_{24} - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{93} + ( 16 + 2 \zeta_{24}^{3} - 16 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{94} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{97} + ( 5 \zeta_{24} + 8 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{98} + ( -2 + \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} - 8q^{11} + 24q^{14} + 16q^{16} - 4q^{19} - 8q^{21} - 8q^{26} - 32q^{29} + 12q^{31} + 48q^{34} + 12q^{39} - 16q^{41} + 24q^{46} - 20q^{49} - 8q^{51} - 48q^{56} - 16q^{61} - 64q^{64} + 8q^{66} - 16q^{69} - 16q^{71} + 8q^{74} + 4q^{79} - 4q^{81} + 8q^{86} - 32q^{89} + 36q^{91} + 64q^{94} - 16q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{24}^{2}$$

## 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
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i
−1.22474 + 0.707107i −0.866025 0.500000i 0 0 1.41421 −0.358719 2.62132i 2.82843i 0.500000 + 0.866025i 0
424.2 −1.22474 + 0.707107i 0.866025 + 0.500000i 0 0 −1.41421 −2.09077 1.62132i 2.82843i 0.500000 + 0.866025i 0
424.3 1.22474 0.707107i −0.866025 0.500000i 0 0 −1.41421 2.09077 + 1.62132i 2.82843i 0.500000 + 0.866025i 0
424.4 1.22474 0.707107i 0.866025 + 0.500000i 0 0 1.41421 0.358719 + 2.62132i 2.82843i 0.500000 + 0.866025i 0
499.1 −1.22474 0.707107i −0.866025 + 0.500000i 0 0 1.41421 −0.358719 + 2.62132i 2.82843i 0.500000 0.866025i 0
499.2 −1.22474 0.707107i 0.866025 0.500000i 0 0 −1.41421 −2.09077 + 1.62132i 2.82843i 0.500000 0.866025i 0
499.3 1.22474 + 0.707107i −0.866025 + 0.500000i 0 0 −1.41421 2.09077 1.62132i 2.82843i 0.500000 0.866025i 0
499.4 1.22474 + 0.707107i 0.866025 0.500000i 0 0 1.41421 0.358719 2.62132i 2.82843i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 499.4 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.g 8
5.b even 2 1 inner 525.2.r.g 8
5.c odd 4 1 105.2.i.c 4
5.c odd 4 1 525.2.i.g 4
7.c even 3 1 inner 525.2.r.g 8
15.e even 4 1 315.2.j.d 4
20.e even 4 1 1680.2.bg.p 4
35.f even 4 1 735.2.i.j 4
35.j even 6 1 inner 525.2.r.g 8
35.k even 12 1 735.2.a.j 2
35.k even 12 1 735.2.i.j 4
35.k even 12 1 3675.2.a.x 2
35.l odd 12 1 105.2.i.c 4
35.l odd 12 1 525.2.i.g 4
35.l odd 12 1 735.2.a.i 2
35.l odd 12 1 3675.2.a.z 2
105.w odd 12 1 2205.2.a.s 2
105.x even 12 1 315.2.j.d 4
105.x even 12 1 2205.2.a.u 2
140.w even 12 1 1680.2.bg.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.c 4 5.c odd 4 1
105.2.i.c 4 35.l odd 12 1
315.2.j.d 4 15.e even 4 1
315.2.j.d 4 105.x even 12 1
525.2.i.g 4 5.c odd 4 1
525.2.i.g 4 35.l odd 12 1
525.2.r.g 8 1.a even 1 1 trivial
525.2.r.g 8 5.b even 2 1 inner
525.2.r.g 8 7.c even 3 1 inner
525.2.r.g 8 35.j even 6 1 inner
735.2.a.i 2 35.l odd 12 1
735.2.a.j 2 35.k even 12 1
735.2.i.j 4 35.f even 4 1
735.2.i.j 4 35.k even 12 1
1680.2.bg.p 4 20.e even 4 1
1680.2.bg.p 4 140.w even 12 1
2205.2.a.s 2 105.w odd 12 1
2205.2.a.u 2 105.x even 12 1
3675.2.a.x 2 35.k even 12 1
3675.2.a.z 2 35.l odd 12 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{2} + 4$$ $$T_{11}^{4} + 4 T_{11}^{3} + 14 T_{11}^{2} + 8 T_{11} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{4}( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ 
$7$ $$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$
$11$ $$( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 88 T^{5} - 968 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 30 T^{2} + 491 T^{4} - 5070 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$1 + 24 T^{2} + 142 T^{4} - 3456 T^{6} - 61629 T^{8} - 998784 T^{10} + 11859982 T^{12} + 579301656 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 + 2 T - 3 T^{2} - 62 T^{3} - 388 T^{4} - 1178 T^{5} - 1083 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 + 48 T^{2} + 958 T^{4} + 13824 T^{6} + 313059 T^{8} + 7312896 T^{10} + 268087678 T^{12} + 7105722672 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 + 8 T + 56 T^{2} + 232 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 + 46 T^{2} - 759 T^{4} + 6302 T^{6} + 3494660 T^{8} + 8627438 T^{10} - 1422488199 T^{12} + 118023414814 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 6 T^{2} + 3059 T^{4} - 11094 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 76 T^{2} + 1962 T^{4} + 45904 T^{6} - 3092269 T^{8} + 101401936 T^{10} + 9573934122 T^{12} - 819220365004 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 164 T^{2} + 15066 T^{4} + 1018768 T^{6} + 56507555 T^{8} + 2861719312 T^{10} + 118877986746 T^{12} + 3634955225156 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 58 T^{2} - 5807 T^{4} - 11194 T^{6} + 48855124 T^{8} - 50249866 T^{10} - 117017559647 T^{12} - 5246586165802 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$1 + 142 T^{2} + 9465 T^{4} + 5822 T^{6} - 21383596 T^{8} + 31025438 T^{10} + 268789351065 T^{12} + 21489460133038 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 - 2 T - 123 T^{2} + 62 T^{3} + 9572 T^{4} + 4898 T^{5} - 767643 T^{6} - 986078 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 296 T^{2} + 35554 T^{4} - 2039144 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 16 T + 32 T^{2} + 736 T^{3} + 19471 T^{4} + 65504 T^{5} + 253472 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 244 T^{2} + 31654 T^{4} - 2295796 T^{6} + 88529281 T^{8} )^{2}$$