# Properties

 Label 525.2.t.e Level 525 Weight 2 Character orbit 525.t Analytic conductor 4.192 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + ( 3 - 3 \zeta_{6} ) q^{6} + ( 3 - \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + ( 3 - 3 \zeta_{6} ) q^{6} + ( 3 - \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( -2 - 2 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + ( -2 + 4 \zeta_{6} ) q^{13} + ( 5 - 4 \zeta_{6} ) q^{14} + 5 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( 3 - 6 \zeta_{6} ) q^{18} + ( -8 + 4 \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} -6 q^{22} + ( -2 + \zeta_{6} ) q^{23} + 3 \zeta_{6} q^{24} + 6 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 - 3 \zeta_{6} ) q^{28} + ( -1 + 2 \zeta_{6} ) q^{29} + ( -2 - 2 \zeta_{6} ) q^{31} + ( 3 + 3 \zeta_{6} ) q^{32} -6 q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} -3 \zeta_{6} q^{36} + 4 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{38} + 6 \zeta_{6} q^{39} -3 q^{41} + ( 6 - 9 \zeta_{6} ) q^{42} - q^{43} + ( -4 + 2 \zeta_{6} ) q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + ( 5 + 5 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 6 - 12 \zeta_{6} ) q^{51} + ( 2 + 2 \zeta_{6} ) q^{52} -9 \zeta_{6} q^{54} + ( -1 + 5 \zeta_{6} ) q^{56} + ( -12 + 12 \zeta_{6} ) q^{57} + 3 \zeta_{6} q^{58} + ( -6 + 3 \zeta_{6} ) q^{61} -6 q^{62} + ( 6 - 9 \zeta_{6} ) q^{63} - q^{64} + ( -12 + 6 \zeta_{6} ) q^{66} + ( -13 + 13 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( -3 + 3 \zeta_{6} ) q^{69} + ( -4 + 8 \zeta_{6} ) q^{71} + ( 3 + 3 \zeta_{6} ) q^{72} + ( -2 - 2 \zeta_{6} ) q^{73} + ( 4 + 4 \zeta_{6} ) q^{74} + ( -4 + 8 \zeta_{6} ) q^{76} + ( -8 - 2 \zeta_{6} ) q^{77} + ( 6 + 6 \zeta_{6} ) q^{78} + 16 \zeta_{6} q^{79} -9 \zeta_{6} q^{81} + ( -6 + 3 \zeta_{6} ) q^{82} + 9 q^{83} + ( 1 - 5 \zeta_{6} ) q^{84} + ( -2 + \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{87} + ( 6 - 6 \zeta_{6} ) q^{88} -3 \zeta_{6} q^{89} + ( -2 + 10 \zeta_{6} ) q^{91} + ( -1 + 2 \zeta_{6} ) q^{92} -6 q^{93} + 9 q^{96} + ( 6 - 12 \zeta_{6} ) q^{97} + ( 11 - 13 \zeta_{6} ) q^{98} + ( -12 + 6 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} + 3q^{3} + q^{4} + 3q^{6} + 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{2} + 3q^{3} + q^{4} + 3q^{6} + 5q^{7} + 3q^{9} - 6q^{11} + 6q^{14} + 5q^{16} + 6q^{17} - 12q^{19} + 6q^{21} - 12q^{22} - 3q^{23} + 3q^{24} + 6q^{26} + q^{28} - 6q^{31} + 9q^{32} - 12q^{33} - 3q^{36} + 4q^{37} - 12q^{38} + 6q^{39} - 6q^{41} + 3q^{42} - 2q^{43} - 6q^{44} - 3q^{46} + 15q^{48} + 11q^{49} + 6q^{52} - 9q^{54} + 3q^{56} - 12q^{57} + 3q^{58} - 9q^{61} - 12q^{62} + 3q^{63} - 2q^{64} - 18q^{66} - 13q^{67} - 6q^{68} - 3q^{69} + 9q^{72} - 6q^{73} + 12q^{74} - 18q^{77} + 18q^{78} + 16q^{79} - 9q^{81} - 9q^{82} + 18q^{83} - 3q^{84} - 3q^{86} + 3q^{87} + 6q^{88} - 3q^{89} + 6q^{91} - 12q^{93} + 18q^{96} + 9q^{98} - 18q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 + 0.866025i 1.50000 + 0.866025i 0.500000 + 0.866025i 0 1.50000 + 2.59808i 2.50000 + 0.866025i 1.73205i 1.50000 + 2.59808i 0
101.1 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i 0 1.50000 2.59808i 2.50000 0.866025i 1.73205i 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
21.g 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.t.e 2
3.b odd 2 1 525.2.t.a 2
5.b even 2 1 105.2.s.a 2
5.c odd 4 2 525.2.q.b 4
7.d odd 6 1 525.2.t.a 2
15.d odd 2 1 105.2.s.b yes 2
15.e even 4 2 525.2.q.a 4
21.g even 6 1 inner 525.2.t.e 2
35.c odd 2 1 735.2.s.c 2
35.i odd 6 1 105.2.s.b yes 2
35.i odd 6 1 735.2.b.a 2
35.j even 6 1 735.2.b.b 2
35.j even 6 1 735.2.s.e 2
35.k even 12 2 525.2.q.a 4
105.g even 2 1 735.2.s.e 2
105.o odd 6 1 735.2.b.a 2
105.o odd 6 1 735.2.s.c 2
105.p even 6 1 105.2.s.a 2
105.p even 6 1 735.2.b.b 2
105.w odd 12 2 525.2.q.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.a 2 5.b even 2 1
105.2.s.a 2 105.p even 6 1
105.2.s.b yes 2 15.d odd 2 1
105.2.s.b yes 2 35.i odd 6 1
525.2.q.a 4 15.e even 4 2
525.2.q.a 4 35.k even 12 2
525.2.q.b 4 5.c odd 4 2
525.2.q.b 4 105.w odd 12 2
525.2.t.a 2 3.b odd 2 1
525.2.t.a 2 7.d odd 6 1
525.2.t.e 2 1.a even 1 1 trivial
525.2.t.e 2 21.g even 6 1 inner
735.2.b.a 2 35.i odd 6 1
735.2.b.a 2 105.o odd 6 1
735.2.b.b 2 35.j even 6 1
735.2.b.b 2 105.p even 6 1
735.2.s.c 2 35.c odd 2 1
735.2.s.c 2 105.o odd 6 1
735.2.s.e 2 35.j even 6 1
735.2.s.e 2 105.g even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3 T_{2} + 3$$ $$T_{13}^{2} + 12$$ $$T_{37}^{2} - 4 T_{37} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}$$
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ 1
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}$$
$13$ $$1 - 14 T^{2} + 169 T^{4}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4}$$
$23$ $$1 + 3 T + 26 T^{2} + 69 T^{3} + 529 T^{4}$$
$29$ $$1 - 55 T^{2} + 841 T^{4}$$
$31$ $$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4}$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$1 - 94 T^{2} + 5041 T^{4}$$
$73$ $$1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4}$$
$79$ $$1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 9 T + 83 T^{2} )^{2}$$
$89$ $$1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4}$$
$97$ $$1 - 86 T^{2} + 9409 T^{4}$$