# Properties

 Label 525.3.s.e Level $525$ Weight $3$ Character orbit 525.s Analytic conductor $14.305$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 525.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3052138789$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + 3*z * q^2 + (-z^3 - z) * q^3 + 5*z^2 * q^4 + (-6*z^2 + 3) * q^6 + (-5*z^3 - 3*z) * q^7 + 3*z^3 * q^8 + (3*z^2 - 3) * q^9 $$q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} - 15 \zeta_{12}^{2} q^{11} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{12} + (8 \zeta_{12}^{3} - 16 \zeta_{12}) q^{13} + ( - 24 \zeta_{12}^{2} + 15) q^{14} + ( - 11 \zeta_{12}^{2} + 11) q^{16} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{17} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{18} + (6 \zeta_{12}^{2} + 6) q^{19} + (11 \zeta_{12}^{2} - 13) q^{21} - 45 \zeta_{12}^{3} q^{22} + ( - 3 \zeta_{12}^{2} + 6) q^{24} + ( - 24 \zeta_{12}^{2} - 24) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 40 \zeta_{12}^{3} + 25 \zeta_{12}) q^{28} + 9 q^{29} + (7 \zeta_{12}^{2} - 14) q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} + (30 \zeta_{12}^{3} - 15 \zeta_{12}) q^{33} + ( - 36 \zeta_{12}^{2} + 18) q^{34} - 15 q^{36} + 10 \zeta_{12} q^{37} + (18 \zeta_{12}^{3} + 18 \zeta_{12}) q^{38} + 24 \zeta_{12}^{2} q^{39} + (12 \zeta_{12}^{2} - 6) q^{41} + (33 \zeta_{12}^{3} - 39 \zeta_{12}) q^{42} - 74 \zeta_{12}^{3} q^{43} + ( - 75 \zeta_{12}^{2} + 75) q^{44} + (11 \zeta_{12}^{3} - 22 \zeta_{12}) q^{48} + (39 \zeta_{12}^{2} - 55) q^{49} + (18 \zeta_{12}^{2} - 18) q^{51} + ( - 40 \zeta_{12}^{3} - 40 \zeta_{12}) q^{52} + ( - 33 \zeta_{12}^{3} + 33 \zeta_{12}) q^{53} + (9 \zeta_{12}^{2} + 9) q^{54} + ( - 9 \zeta_{12}^{2} + 24) q^{56} - 18 \zeta_{12}^{3} q^{57} + 27 \zeta_{12} q^{58} + (9 \zeta_{12}^{2} - 18) q^{59} + (52 \zeta_{12}^{2} + 52) q^{61} + (21 \zeta_{12}^{3} - 42 \zeta_{12}) q^{62} + ( - 9 \zeta_{12}^{3} + 24 \zeta_{12}) q^{63} + 91 q^{64} + (45 \zeta_{12}^{2} - 90) q^{66} + ( - 76 \zeta_{12}^{3} + 76 \zeta_{12}) q^{67} + ( - 60 \zeta_{12}^{3} + 30 \zeta_{12}) q^{68} + 84 q^{71} - 9 \zeta_{12} q^{72} + ( - 36 \zeta_{12}^{3} - 36 \zeta_{12}) q^{73} + 30 \zeta_{12}^{2} q^{74} + (60 \zeta_{12}^{2} - 30) q^{76} + (120 \zeta_{12}^{3} - 75 \zeta_{12}) q^{77} + 72 \zeta_{12}^{3} q^{78} + (43 \zeta_{12}^{2} - 43) q^{79} - 9 \zeta_{12}^{2} q^{81} + (36 \zeta_{12}^{3} - 18 \zeta_{12}) q^{82} + (69 \zeta_{12}^{3} - 138 \zeta_{12}) q^{83} + ( - 10 \zeta_{12}^{2} - 55) q^{84} + ( - 222 \zeta_{12}^{2} + 222) q^{86} + ( - 9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{87} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{88} + (42 \zeta_{12}^{2} + 42) q^{89} + (104 \zeta_{12}^{2} - 16) q^{91} + 21 \zeta_{12} q^{93} + ( - 45 \zeta_{12}^{2} - 45) q^{96} + (107 \zeta_{12}^{3} - 214 \zeta_{12}) q^{97} + (117 \zeta_{12}^{3} - 165 \zeta_{12}) q^{98} + 45 q^{99} +O(q^{100})$$ q + 3*z * q^2 + (-z^3 - z) * q^3 + 5*z^2 * q^4 + (-6*z^2 + 3) * q^6 + (-5*z^3 - 3*z) * q^7 + 3*z^3 * q^8 + (3*z^2 - 3) * q^9 - 15*z^2 * q^11 + (-10*z^3 + 5*z) * q^12 + (8*z^3 - 16*z) * q^13 + (-24*z^2 + 15) * q^14 + (-11*z^2 + 11) * q^16 + (-6*z^3 - 6*z) * q^17 + (9*z^3 - 9*z) * q^18 + (6*z^2 + 6) * q^19 + (11*z^2 - 13) * q^21 - 45*z^3 * q^22 + (-3*z^2 + 6) * q^24 + (-24*z^2 - 24) * q^26 + (-3*z^3 + 6*z) * q^27 + (-40*z^3 + 25*z) * q^28 + 9 * q^29 + (7*z^2 - 14) * q^31 + (-45*z^3 + 45*z) * q^32 + (30*z^3 - 15*z) * q^33 + (-36*z^2 + 18) * q^34 - 15 * q^36 + 10*z * q^37 + (18*z^3 + 18*z) * q^38 + 24*z^2 * q^39 + (12*z^2 - 6) * q^41 + (33*z^3 - 39*z) * q^42 - 74*z^3 * q^43 + (-75*z^2 + 75) * q^44 + (11*z^3 - 22*z) * q^48 + (39*z^2 - 55) * q^49 + (18*z^2 - 18) * q^51 + (-40*z^3 - 40*z) * q^52 + (-33*z^3 + 33*z) * q^53 + (9*z^2 + 9) * q^54 + (-9*z^2 + 24) * q^56 - 18*z^3 * q^57 + 27*z * q^58 + (9*z^2 - 18) * q^59 + (52*z^2 + 52) * q^61 + (21*z^3 - 42*z) * q^62 + (-9*z^3 + 24*z) * q^63 + 91 * q^64 + (45*z^2 - 90) * q^66 + (-76*z^3 + 76*z) * q^67 + (-60*z^3 + 30*z) * q^68 + 84 * q^71 - 9*z * q^72 + (-36*z^3 - 36*z) * q^73 + 30*z^2 * q^74 + (60*z^2 - 30) * q^76 + (120*z^3 - 75*z) * q^77 + 72*z^3 * q^78 + (43*z^2 - 43) * q^79 - 9*z^2 * q^81 + (36*z^3 - 18*z) * q^82 + (69*z^3 - 138*z) * q^83 + (-10*z^2 - 55) * q^84 + (-222*z^2 + 222) * q^86 + (-9*z^3 - 9*z) * q^87 + (-45*z^3 + 45*z) * q^88 + (42*z^2 + 42) * q^89 + (104*z^2 - 16) * q^91 + 21*z * q^93 + (-45*z^2 - 45) * q^96 + (107*z^3 - 214*z) * q^97 + (117*z^3 - 165*z) * q^98 + 45 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} - 6 q^{9}+O(q^{10})$$ 4 * q + 10 * q^4 - 6 * q^9 $$4 q + 10 q^{4} - 6 q^{9} - 30 q^{11} + 12 q^{14} + 22 q^{16} + 36 q^{19} - 30 q^{21} + 18 q^{24} - 144 q^{26} + 36 q^{29} - 42 q^{31} - 60 q^{36} + 48 q^{39} + 150 q^{44} - 142 q^{49} - 36 q^{51} + 54 q^{54} + 78 q^{56} - 54 q^{59} + 312 q^{61} + 364 q^{64} - 270 q^{66} + 336 q^{71} + 60 q^{74} - 86 q^{79} - 18 q^{81} - 240 q^{84} + 444 q^{86} + 252 q^{89} + 144 q^{91} - 270 q^{96} + 180 q^{99}+O(q^{100})$$ 4 * q + 10 * q^4 - 6 * q^9 - 30 * q^11 + 12 * q^14 + 22 * q^16 + 36 * q^19 - 30 * q^21 + 18 * q^24 - 144 * q^26 + 36 * q^29 - 42 * q^31 - 60 * q^36 + 48 * q^39 + 150 * q^44 - 142 * q^49 - 36 * q^51 + 54 * q^54 + 78 * q^56 - 54 * q^59 + 312 * q^61 + 364 * q^64 - 270 * q^66 + 336 * q^71 + 60 * q^74 - 86 * q^79 - 18 * q^81 - 240 * q^84 + 444 * q^86 + 252 * q^89 + 144 * q^91 - 270 * q^96 + 180 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−2.59808 1.50000i 0.866025 + 1.50000i 2.50000 + 4.33013i 0 5.19615i 2.59808 + 6.50000i 3.00000i −1.50000 + 2.59808i 0
124.2 2.59808 + 1.50000i −0.866025 1.50000i 2.50000 + 4.33013i 0 5.19615i −2.59808 6.50000i 3.00000i −1.50000 + 2.59808i 0
199.1 −2.59808 + 1.50000i 0.866025 1.50000i 2.50000 4.33013i 0 5.19615i 2.59808 6.50000i 3.00000i −1.50000 2.59808i 0
199.2 2.59808 1.50000i −0.866025 + 1.50000i 2.50000 4.33013i 0 5.19615i −2.59808 + 6.50000i 3.00000i −1.50000 2.59808i 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
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.s.e 4
5.b even 2 1 inner 525.3.s.e 4
5.c odd 4 1 21.3.f.a 2
5.c odd 4 1 525.3.o.h 2
7.d odd 6 1 inner 525.3.s.e 4
15.e even 4 1 63.3.m.d 2
20.e even 4 1 336.3.bh.d 2
35.f even 4 1 147.3.f.a 2
35.i odd 6 1 inner 525.3.s.e 4
35.k even 12 1 21.3.f.a 2
35.k even 12 1 147.3.d.c 2
35.k even 12 1 525.3.o.h 2
35.l odd 12 1 147.3.d.c 2
35.l odd 12 1 147.3.f.a 2
60.l odd 4 1 1008.3.cg.a 2
105.k odd 4 1 441.3.m.g 2
105.w odd 12 1 63.3.m.d 2
105.w odd 12 1 441.3.d.a 2
105.x even 12 1 441.3.d.a 2
105.x even 12 1 441.3.m.g 2
140.w even 12 1 2352.3.f.a 2
140.x odd 12 1 336.3.bh.d 2
140.x odd 12 1 2352.3.f.a 2
420.br even 12 1 1008.3.cg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 5.c odd 4 1
21.3.f.a 2 35.k even 12 1
63.3.m.d 2 15.e even 4 1
63.3.m.d 2 105.w odd 12 1
147.3.d.c 2 35.k even 12 1
147.3.d.c 2 35.l odd 12 1
147.3.f.a 2 35.f even 4 1
147.3.f.a 2 35.l odd 12 1
336.3.bh.d 2 20.e even 4 1
336.3.bh.d 2 140.x odd 12 1
441.3.d.a 2 105.w odd 12 1
441.3.d.a 2 105.x even 12 1
441.3.m.g 2 105.k odd 4 1
441.3.m.g 2 105.x even 12 1
525.3.o.h 2 5.c odd 4 1
525.3.o.h 2 35.k even 12 1
525.3.s.e 4 1.a even 1 1 trivial
525.3.s.e 4 5.b even 2 1 inner
525.3.s.e 4 7.d odd 6 1 inner
525.3.s.e 4 35.i odd 6 1 inner
1008.3.cg.a 2 60.l odd 4 1
1008.3.cg.a 2 420.br even 12 1
2352.3.f.a 2 140.w even 12 1
2352.3.f.a 2 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} - 9T_{2}^{2} + 81$$ T2^4 - 9*T2^2 + 81 $$T_{11}^{2} + 15T_{11} + 225$$ T11^2 + 15*T11 + 225 $$T_{13}^{2} - 192$$ T13^2 - 192

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 81$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 71T^{2} + 2401$$
$11$ $$(T^{2} + 15 T + 225)^{2}$$
$13$ $$(T^{2} - 192)^{2}$$
$17$ $$T^{4} + 108 T^{2} + 11664$$
$19$ $$(T^{2} - 18 T + 108)^{2}$$
$23$ $$T^{4}$$
$29$ $$(T - 9)^{4}$$
$31$ $$(T^{2} + 21 T + 147)^{2}$$
$37$ $$T^{4} - 100 T^{2} + 10000$$
$41$ $$(T^{2} + 108)^{2}$$
$43$ $$(T^{2} + 5476)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 1089 T^{2} + \cdots + 1185921$$
$59$ $$(T^{2} + 27 T + 243)^{2}$$
$61$ $$(T^{2} - 156 T + 8112)^{2}$$
$67$ $$T^{4} - 5776 T^{2} + \cdots + 33362176$$
$71$ $$(T - 84)^{4}$$
$73$ $$T^{4} + 3888 T^{2} + \cdots + 15116544$$
$79$ $$(T^{2} + 43 T + 1849)^{2}$$
$83$ $$(T^{2} - 14283)^{2}$$
$89$ $$(T^{2} - 126 T + 5292)^{2}$$
$97$ $$(T^{2} - 34347)^{2}$$