# Properties

 Label 525.3.s.f 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: yes 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 5 \zeta_{12}^{2} q^{4} + ( - 6 \zeta_{12}^{2} + 3) q^{6} + 7 \zeta_{12} q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + 3*z * q^2 + (-z^3 - z) * q^3 + 5*z^2 * q^4 + (-6*z^2 + 3) * q^6 + 7*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} + 7 \zeta_{12} q^{7} + 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{12} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + 21 \zeta_{12}^{2} q^{14} + ( - 11 \zeta_{12}^{2} + 11) q^{16} + (9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{17} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{18} + (16 \zeta_{12}^{2} + 16) q^{19} + ( - 14 \zeta_{12}^{2} + 7) q^{21} + 15 \zeta_{12} q^{23} + ( - 3 \zeta_{12}^{2} + 6) q^{24} + (6 \zeta_{12}^{2} + 6) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + 35 \zeta_{12}^{3} q^{28} - 6 q^{29} + ( - 13 \zeta_{12}^{2} + 26) q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} + (54 \zeta_{12}^{2} - 27) q^{34} - 15 q^{36} - 70 \zeta_{12} q^{37} + (48 \zeta_{12}^{3} + 48 \zeta_{12}) q^{38} - 6 \zeta_{12}^{2} q^{39} + (42 \zeta_{12}^{2} - 21) q^{41} + ( - 42 \zeta_{12}^{3} + 21 \zeta_{12}) q^{42} - 34 \zeta_{12}^{3} q^{43} + 45 \zeta_{12}^{2} q^{46} + ( - 30 \zeta_{12}^{3} + 15 \zeta_{12}) q^{47} + (11 \zeta_{12}^{3} - 22 \zeta_{12}) q^{48} + 49 \zeta_{12}^{2} q^{49} + ( - 27 \zeta_{12}^{2} + 27) q^{51} + (10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{52} + (42 \zeta_{12}^{3} - 42 \zeta_{12}) q^{53} + (9 \zeta_{12}^{2} + 9) q^{54} + (21 \zeta_{12}^{2} - 21) q^{56} - 48 \zeta_{12}^{3} q^{57} - 18 \zeta_{12} q^{58} + (24 \zeta_{12}^{2} - 48) q^{59} + (42 \zeta_{12}^{2} + 42) q^{61} + ( - 39 \zeta_{12}^{3} + 78 \zeta_{12}) q^{62} + (21 \zeta_{12}^{3} - 21 \zeta_{12}) q^{63} + 91 q^{64} + (94 \zeta_{12}^{3} - 94 \zeta_{12}) q^{67} + (90 \zeta_{12}^{3} - 45 \zeta_{12}) q^{68} + ( - 30 \zeta_{12}^{2} + 15) q^{69} + 9 q^{71} - 9 \zeta_{12} q^{72} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{73} - 210 \zeta_{12}^{2} q^{74} + (160 \zeta_{12}^{2} - 80) q^{76} - 18 \zeta_{12}^{3} q^{78} + ( - 77 \zeta_{12}^{2} + 77) q^{79} - 9 \zeta_{12}^{2} q^{81} + (126 \zeta_{12}^{3} - 63 \zeta_{12}) q^{82} + (84 \zeta_{12}^{3} - 168 \zeta_{12}) q^{83} + ( - 35 \zeta_{12}^{2} + 70) q^{84} + ( - 102 \zeta_{12}^{2} + 102) q^{86} + (6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} + ( - 33 \zeta_{12}^{2} - 33) q^{89} + (14 \zeta_{12}^{2} + 14) q^{91} + 75 \zeta_{12}^{3} q^{92} - 39 \zeta_{12} q^{93} + ( - 45 \zeta_{12}^{2} + 90) q^{94} + ( - 45 \zeta_{12}^{2} - 45) q^{96} + (57 \zeta_{12}^{3} - 114 \zeta_{12}) q^{97} + 147 \zeta_{12}^{3} q^{98} +O(q^{100})$$ q + 3*z * q^2 + (-z^3 - z) * q^3 + 5*z^2 * q^4 + (-6*z^2 + 3) * q^6 + 7*z * q^7 + 3*z^3 * q^8 + (3*z^2 - 3) * q^9 + (-10*z^3 + 5*z) * q^12 + (-2*z^3 + 4*z) * q^13 + 21*z^2 * q^14 + (-11*z^2 + 11) * q^16 + (9*z^3 + 9*z) * q^17 + (9*z^3 - 9*z) * q^18 + (16*z^2 + 16) * q^19 + (-14*z^2 + 7) * q^21 + 15*z * q^23 + (-3*z^2 + 6) * q^24 + (6*z^2 + 6) * q^26 + (-3*z^3 + 6*z) * q^27 + 35*z^3 * q^28 - 6 * q^29 + (-13*z^2 + 26) * q^31 + (-45*z^3 + 45*z) * q^32 + (54*z^2 - 27) * q^34 - 15 * q^36 - 70*z * q^37 + (48*z^3 + 48*z) * q^38 - 6*z^2 * q^39 + (42*z^2 - 21) * q^41 + (-42*z^3 + 21*z) * q^42 - 34*z^3 * q^43 + 45*z^2 * q^46 + (-30*z^3 + 15*z) * q^47 + (11*z^3 - 22*z) * q^48 + 49*z^2 * q^49 + (-27*z^2 + 27) * q^51 + (10*z^3 + 10*z) * q^52 + (42*z^3 - 42*z) * q^53 + (9*z^2 + 9) * q^54 + (21*z^2 - 21) * q^56 - 48*z^3 * q^57 - 18*z * q^58 + (24*z^2 - 48) * q^59 + (42*z^2 + 42) * q^61 + (-39*z^3 + 78*z) * q^62 + (21*z^3 - 21*z) * q^63 + 91 * q^64 + (94*z^3 - 94*z) * q^67 + (90*z^3 - 45*z) * q^68 + (-30*z^2 + 15) * q^69 + 9 * q^71 - 9*z * q^72 + (4*z^3 + 4*z) * q^73 - 210*z^2 * q^74 + (160*z^2 - 80) * q^76 - 18*z^3 * q^78 + (-77*z^2 + 77) * q^79 - 9*z^2 * q^81 + (126*z^3 - 63*z) * q^82 + (84*z^3 - 168*z) * q^83 + (-35*z^2 + 70) * q^84 + (-102*z^2 + 102) * q^86 + (6*z^3 + 6*z) * q^87 + (-33*z^2 - 33) * q^89 + (14*z^2 + 14) * q^91 + 75*z^3 * q^92 - 39*z * q^93 + (-45*z^2 + 90) * q^94 + (-45*z^2 - 45) * q^96 + (57*z^3 - 114*z) * q^97 + 147*z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} - 6 q^{9}+O(q^{10})$$ 4 * q + 10 * q^4 - 6 * q^9 $$4 q + 10 q^{4} - 6 q^{9} + 42 q^{14} + 22 q^{16} + 96 q^{19} + 18 q^{24} + 36 q^{26} - 24 q^{29} + 78 q^{31} - 60 q^{36} - 12 q^{39} + 90 q^{46} + 98 q^{49} + 54 q^{51} + 54 q^{54} - 42 q^{56} - 144 q^{59} + 252 q^{61} + 364 q^{64} + 36 q^{71} - 420 q^{74} + 154 q^{79} - 18 q^{81} + 210 q^{84} + 204 q^{86} - 198 q^{89} + 84 q^{91} + 270 q^{94} - 270 q^{96}+O(q^{100})$$ 4 * q + 10 * q^4 - 6 * q^9 + 42 * q^14 + 22 * q^16 + 96 * q^19 + 18 * q^24 + 36 * q^26 - 24 * q^29 + 78 * q^31 - 60 * q^36 - 12 * q^39 + 90 * q^46 + 98 * q^49 + 54 * q^51 + 54 * q^54 - 42 * q^56 - 144 * q^59 + 252 * q^61 + 364 * q^64 + 36 * q^71 - 420 * q^74 + 154 * q^79 - 18 * q^81 + 210 * q^84 + 204 * q^86 - 198 * q^89 + 84 * q^91 + 270 * q^94 - 270 * q^96

## 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 −6.06218 3.50000i 3.00000i −1.50000 + 2.59808i 0
124.2 2.59808 + 1.50000i −0.866025 1.50000i 2.50000 + 4.33013i 0 5.19615i 6.06218 + 3.50000i 3.00000i −1.50000 + 2.59808i 0
199.1 −2.59808 + 1.50000i 0.866025 1.50000i 2.50000 4.33013i 0 5.19615i −6.06218 + 3.50000i 3.00000i −1.50000 2.59808i 0
199.2 2.59808 1.50000i −0.866025 + 1.50000i 2.50000 4.33013i 0 5.19615i 6.06218 3.50000i 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.f 4
5.b even 2 1 inner 525.3.s.f 4
5.c odd 4 1 525.3.o.a 2
5.c odd 4 1 525.3.o.i yes 2
7.d odd 6 1 inner 525.3.s.f 4
35.i odd 6 1 inner 525.3.s.f 4
35.k even 12 1 525.3.o.a 2
35.k even 12 1 525.3.o.i yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.a 2 5.c odd 4 1
525.3.o.a 2 35.k even 12 1
525.3.o.i yes 2 5.c odd 4 1
525.3.o.i yes 2 35.k even 12 1
525.3.s.f 4 1.a even 1 1 trivial
525.3.s.f 4 5.b even 2 1 inner
525.3.s.f 4 7.d odd 6 1 inner
525.3.s.f 4 35.i odd 6 1 inner

## 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 $$T_{11}$$ T11 $$T_{13}^{2} - 12$$ T13^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 81$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 49T^{2} + 2401$$
$11$ $$T^{4}$$
$13$ $$(T^{2} - 12)^{2}$$
$17$ $$T^{4} + 243 T^{2} + 59049$$
$19$ $$(T^{2} - 48 T + 768)^{2}$$
$23$ $$T^{4} - 225 T^{2} + 50625$$
$29$ $$(T + 6)^{4}$$
$31$ $$(T^{2} - 39 T + 507)^{2}$$
$37$ $$T^{4} - 4900 T^{2} + \cdots + 24010000$$
$41$ $$(T^{2} + 1323)^{2}$$
$43$ $$(T^{2} + 1156)^{2}$$
$47$ $$T^{4} + 675 T^{2} + 455625$$
$53$ $$T^{4} - 1764 T^{2} + \cdots + 3111696$$
$59$ $$(T^{2} + 72 T + 1728)^{2}$$
$61$ $$(T^{2} - 126 T + 5292)^{2}$$
$67$ $$T^{4} - 8836 T^{2} + \cdots + 78074896$$
$71$ $$(T - 9)^{4}$$
$73$ $$T^{4} + 48T^{2} + 2304$$
$79$ $$(T^{2} - 77 T + 5929)^{2}$$
$83$ $$(T^{2} - 21168)^{2}$$
$89$ $$(T^{2} + 99 T + 3267)^{2}$$
$97$ $$(T^{2} - 9747)^{2}$$