# Properties

 Label 525.3.s.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10})$$ q + z * q^2 + (z^3 + z) * q^3 - 3*z^2 * q^4 + (2*z^2 - 1) * q^6 + (-8*z^3 + 5*z) * q^7 - 7*z^3 * q^8 + (3*z^2 - 3) * q^9 $$q + \zeta_{12} q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 3 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - 1) q^{6} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} - 7 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} - 4 \zeta_{12}^{2} q^{11} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{12} + (10 \zeta_{12}^{3} - 20 \zeta_{12}) q^{13} + ( - 3 \zeta_{12}^{2} + 8) q^{14} + (5 \zeta_{12}^{2} - 5) q^{16} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + ( - 4 \zeta_{12}^{2} - 4) q^{19} + (2 \zeta_{12}^{2} + 11) q^{21} - 4 \zeta_{12}^{3} q^{22} - 31 \zeta_{12} q^{23} + ( - 7 \zeta_{12}^{2} + 14) q^{24} + ( - 10 \zeta_{12}^{2} - 10) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (9 \zeta_{12}^{3} - 24 \zeta_{12}) q^{28} - 10 q^{29} + ( - 17 \zeta_{12}^{2} + 34) q^{31} + (33 \zeta_{12}^{3} - 33 \zeta_{12}) q^{32} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{33} + ( - 10 \zeta_{12}^{2} + 5) q^{34} + 9 q^{36} + 50 \zeta_{12} q^{37} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{38} - 30 \zeta_{12}^{2} q^{39} + ( - 62 \zeta_{12}^{2} + 31) q^{41} + (2 \zeta_{12}^{3} + 11 \zeta_{12}) q^{42} - 34 \zeta_{12}^{3} q^{43} + (12 \zeta_{12}^{2} - 12) q^{44} - 31 \zeta_{12}^{2} q^{46} + ( - 50 \zeta_{12}^{3} + 25 \zeta_{12}) q^{47} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{48} + ( - 55 \zeta_{12}^{2} + 16) q^{49} + ( - 15 \zeta_{12}^{2} + 15) q^{51} + (30 \zeta_{12}^{3} + 30 \zeta_{12}) q^{52} + (50 \zeta_{12}^{3} - 50 \zeta_{12}) q^{53} + ( - 3 \zeta_{12}^{2} - 3) q^{54} + ( - 35 \zeta_{12}^{2} - 21) q^{56} - 12 \zeta_{12}^{3} q^{57} - 10 \zeta_{12} q^{58} + ( - 32 \zeta_{12}^{2} + 64) q^{59} + (14 \zeta_{12}^{2} + 14) q^{61} + ( - 17 \zeta_{12}^{3} + 34 \zeta_{12}) q^{62} + (15 \zeta_{12}^{3} + 9 \zeta_{12}) q^{63} - 13 q^{64} + ( - 4 \zeta_{12}^{2} + 8) q^{66} + (50 \zeta_{12}^{3} - 50 \zeta_{12}) q^{67} + (30 \zeta_{12}^{3} - 15 \zeta_{12}) q^{68} + ( - 62 \zeta_{12}^{2} + 31) q^{69} + 97 q^{71} + 21 \zeta_{12} q^{72} + (28 \zeta_{12}^{3} + 28 \zeta_{12}) q^{73} + 50 \zeta_{12}^{2} q^{74} + (24 \zeta_{12}^{2} - 12) q^{76} + (12 \zeta_{12}^{3} - 32 \zeta_{12}) q^{77} - 30 \zeta_{12}^{3} q^{78} + (7 \zeta_{12}^{2} - 7) q^{79} - 9 \zeta_{12}^{2} q^{81} + ( - 62 \zeta_{12}^{3} + 31 \zeta_{12}) q^{82} + (88 \zeta_{12}^{3} - 176 \zeta_{12}) q^{83} + ( - 39 \zeta_{12}^{2} + 6) q^{84} + ( - 34 \zeta_{12}^{2} + 34) q^{86} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{87} + (28 \zeta_{12}^{3} - 28 \zeta_{12}) q^{88} + (91 \zeta_{12}^{2} + 91) q^{89} + (110 \zeta_{12}^{2} - 130) q^{91} + 93 \zeta_{12}^{3} q^{92} + 51 \zeta_{12} q^{93} + ( - 25 \zeta_{12}^{2} + 50) q^{94} + ( - 33 \zeta_{12}^{2} - 33) q^{96} + ( - 65 \zeta_{12}^{3} + 130 \zeta_{12}) q^{97} + ( - 55 \zeta_{12}^{3} + 16 \zeta_{12}) q^{98} + 12 q^{99} +O(q^{100})$$ q + z * q^2 + (z^3 + z) * q^3 - 3*z^2 * q^4 + (2*z^2 - 1) * q^6 + (-8*z^3 + 5*z) * q^7 - 7*z^3 * q^8 + (3*z^2 - 3) * q^9 - 4*z^2 * q^11 + (-6*z^3 + 3*z) * q^12 + (10*z^3 - 20*z) * q^13 + (-3*z^2 + 8) * q^14 + (5*z^2 - 5) * q^16 + (-5*z^3 - 5*z) * q^17 + (3*z^3 - 3*z) * q^18 + (-4*z^2 - 4) * q^19 + (2*z^2 + 11) * q^21 - 4*z^3 * q^22 - 31*z * q^23 + (-7*z^2 + 14) * q^24 + (-10*z^2 - 10) * q^26 + (3*z^3 - 6*z) * q^27 + (9*z^3 - 24*z) * q^28 - 10 * q^29 + (-17*z^2 + 34) * q^31 + (33*z^3 - 33*z) * q^32 + (-8*z^3 + 4*z) * q^33 + (-10*z^2 + 5) * q^34 + 9 * q^36 + 50*z * q^37 + (-4*z^3 - 4*z) * q^38 - 30*z^2 * q^39 + (-62*z^2 + 31) * q^41 + (2*z^3 + 11*z) * q^42 - 34*z^3 * q^43 + (12*z^2 - 12) * q^44 - 31*z^2 * q^46 + (-50*z^3 + 25*z) * q^47 + (5*z^3 - 10*z) * q^48 + (-55*z^2 + 16) * q^49 + (-15*z^2 + 15) * q^51 + (30*z^3 + 30*z) * q^52 + (50*z^3 - 50*z) * q^53 + (-3*z^2 - 3) * q^54 + (-35*z^2 - 21) * q^56 - 12*z^3 * q^57 - 10*z * q^58 + (-32*z^2 + 64) * q^59 + (14*z^2 + 14) * q^61 + (-17*z^3 + 34*z) * q^62 + (15*z^3 + 9*z) * q^63 - 13 * q^64 + (-4*z^2 + 8) * q^66 + (50*z^3 - 50*z) * q^67 + (30*z^3 - 15*z) * q^68 + (-62*z^2 + 31) * q^69 + 97 * q^71 + 21*z * q^72 + (28*z^3 + 28*z) * q^73 + 50*z^2 * q^74 + (24*z^2 - 12) * q^76 + (12*z^3 - 32*z) * q^77 - 30*z^3 * q^78 + (7*z^2 - 7) * q^79 - 9*z^2 * q^81 + (-62*z^3 + 31*z) * q^82 + (88*z^3 - 176*z) * q^83 + (-39*z^2 + 6) * q^84 + (-34*z^2 + 34) * q^86 + (-10*z^3 - 10*z) * q^87 + (28*z^3 - 28*z) * q^88 + (91*z^2 + 91) * q^89 + (110*z^2 - 130) * q^91 + 93*z^3 * q^92 + 51*z * q^93 + (-25*z^2 + 50) * q^94 + (-33*z^2 - 33) * q^96 + (-65*z^3 + 130*z) * q^97 + (-55*z^3 + 16*z) * q^98 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4} - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^4 - 6 * q^9 $$4 q - 6 q^{4} - 6 q^{9} - 8 q^{11} + 26 q^{14} - 10 q^{16} - 24 q^{19} + 48 q^{21} + 42 q^{24} - 60 q^{26} - 40 q^{29} + 102 q^{31} + 36 q^{36} - 60 q^{39} - 24 q^{44} - 62 q^{46} - 46 q^{49} + 30 q^{51} - 18 q^{54} - 154 q^{56} + 192 q^{59} + 84 q^{61} - 52 q^{64} + 24 q^{66} + 388 q^{71} + 100 q^{74} - 14 q^{79} - 18 q^{81} - 54 q^{84} + 68 q^{86} + 546 q^{89} - 300 q^{91} + 150 q^{94} - 198 q^{96} + 48 q^{99}+O(q^{100})$$ 4 * q - 6 * q^4 - 6 * q^9 - 8 * q^11 + 26 * q^14 - 10 * q^16 - 24 * q^19 + 48 * q^21 + 42 * q^24 - 60 * q^26 - 40 * q^29 + 102 * q^31 + 36 * q^36 - 60 * q^39 - 24 * q^44 - 62 * q^46 - 46 * q^49 + 30 * q^51 - 18 * q^54 - 154 * q^56 + 192 * q^59 + 84 * q^61 - 52 * q^64 + 24 * q^66 + 388 * q^71 + 100 * q^74 - 14 * q^79 - 18 * q^81 - 54 * q^84 + 68 * q^86 + 546 * q^89 - 300 * q^91 + 150 * q^94 - 198 * q^96 + 48 * 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
−0.866025 0.500000i −0.866025 1.50000i −1.50000 2.59808i 0 1.73205i −4.33013 + 5.50000i 7.00000i −1.50000 + 2.59808i 0
124.2 0.866025 + 0.500000i 0.866025 + 1.50000i −1.50000 2.59808i 0 1.73205i 4.33013 5.50000i 7.00000i −1.50000 + 2.59808i 0
199.1 −0.866025 + 0.500000i −0.866025 + 1.50000i −1.50000 + 2.59808i 0 1.73205i −4.33013 5.50000i 7.00000i −1.50000 2.59808i 0
199.2 0.866025 0.500000i 0.866025 1.50000i −1.50000 + 2.59808i 0 1.73205i 4.33013 + 5.50000i 7.00000i −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.b 4
5.b even 2 1 inner 525.3.s.b 4
5.c odd 4 1 525.3.o.d 2
5.c odd 4 1 525.3.o.e yes 2
7.d odd 6 1 inner 525.3.s.b 4
35.i odd 6 1 inner 525.3.s.b 4
35.k even 12 1 525.3.o.d 2
35.k even 12 1 525.3.o.e yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.d 2 5.c odd 4 1
525.3.o.d 2 35.k even 12 1
525.3.o.e yes 2 5.c odd 4 1
525.3.o.e yes 2 35.k even 12 1
525.3.s.b 4 1.a even 1 1 trivial
525.3.s.b 4 5.b even 2 1 inner
525.3.s.b 4 7.d odd 6 1 inner
525.3.s.b 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 $$S_{3}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{2} + 1$$ T2^4 - T2^2 + 1 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13}^{2} - 300$$ T13^2 - 300

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 2401$$
$11$ $$(T^{2} + 4 T + 16)^{2}$$
$13$ $$(T^{2} - 300)^{2}$$
$17$ $$T^{4} + 75T^{2} + 5625$$
$19$ $$(T^{2} + 12 T + 48)^{2}$$
$23$ $$T^{4} - 961 T^{2} + 923521$$
$29$ $$(T + 10)^{4}$$
$31$ $$(T^{2} - 51 T + 867)^{2}$$
$37$ $$T^{4} - 2500 T^{2} + \cdots + 6250000$$
$41$ $$(T^{2} + 2883)^{2}$$
$43$ $$(T^{2} + 1156)^{2}$$
$47$ $$T^{4} + 1875 T^{2} + \cdots + 3515625$$
$53$ $$T^{4} - 2500 T^{2} + \cdots + 6250000$$
$59$ $$(T^{2} - 96 T + 3072)^{2}$$
$61$ $$(T^{2} - 42 T + 588)^{2}$$
$67$ $$T^{4} - 2500 T^{2} + \cdots + 6250000$$
$71$ $$(T - 97)^{4}$$
$73$ $$T^{4} + 2352 T^{2} + \cdots + 5531904$$
$79$ $$(T^{2} + 7 T + 49)^{2}$$
$83$ $$(T^{2} - 23232)^{2}$$
$89$ $$(T^{2} - 273 T + 24843)^{2}$$
$97$ $$(T^{2} - 12675)^{2}$$