Properties

 Label 525.2.r.c Level 525 Weight 2 Character orbit 525.r Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + q^{6} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + q^{6} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -\zeta_{12} q^{12} -3 \zeta_{12}^{3} q^{13} + ( 1 + 2 \zeta_{12}^{2} ) q^{14} + ( 1 - \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( 1 - \zeta_{12}^{2} ) q^{19} + ( 3 - \zeta_{12}^{2} ) q^{21} -2 \zeta_{12} q^{23} -3 \zeta_{12}^{2} q^{24} + ( 3 - 3 \zeta_{12}^{2} ) q^{26} -\zeta_{12}^{3} q^{27} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + 8 q^{29} + 8 \zeta_{12}^{2} q^{31} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{32} -2 q^{34} - q^{36} + 7 \zeta_{12} q^{37} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{38} -3 \zeta_{12}^{2} q^{39} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{42} -8 \zeta_{12}^{3} q^{43} -2 \zeta_{12}^{2} q^{46} -10 \zeta_{12} q^{47} -\zeta_{12}^{3} q^{48} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( -2 + 2 \zeta_{12}^{2} ) q^{51} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{52} + ( -14 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{53} + ( 1 - \zeta_{12}^{2} ) q^{54} + ( 6 - 9 \zeta_{12}^{2} ) q^{56} -\zeta_{12}^{3} q^{57} + 8 \zeta_{12} q^{58} + 10 \zeta_{12}^{2} q^{59} + ( -7 + 7 \zeta_{12}^{2} ) q^{61} + 8 \zeta_{12}^{3} q^{62} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} -7 q^{64} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + 2 \zeta_{12} q^{68} -2 q^{69} -12 q^{71} -3 \zeta_{12} q^{72} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{73} + 7 \zeta_{12}^{2} q^{74} - q^{76} -3 \zeta_{12}^{3} q^{78} + ( -7 + 7 \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} + 14 \zeta_{12}^{3} q^{83} + ( -1 - 2 \zeta_{12}^{2} ) q^{84} + ( 8 - 8 \zeta_{12}^{2} ) q^{86} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{87} + ( -6 + 6 \zeta_{12}^{2} ) q^{89} + ( 6 - 9 \zeta_{12}^{2} ) q^{91} + 2 \zeta_{12}^{3} q^{92} + 8 \zeta_{12} q^{93} -10 \zeta_{12}^{2} q^{94} + ( -5 + 5 \zeta_{12}^{2} ) q^{96} -9 \zeta_{12}^{3} q^{97} + ( 5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{4} + 4q^{6} + 2q^{9} + 8q^{14} + 2q^{16} + 2q^{19} + 10q^{21} - 6q^{24} + 6q^{26} + 32q^{29} + 16q^{31} - 8q^{34} - 4q^{36} - 6q^{39} - 4q^{46} + 26q^{49} - 4q^{51} + 2q^{54} + 6q^{56} + 20q^{59} - 14q^{61} - 28q^{64} - 8q^{69} - 48q^{71} + 14q^{74} - 4q^{76} - 14q^{79} - 2q^{81} - 8q^{84} + 16q^{86} - 12q^{89} + 6q^{91} - 20q^{94} - 10q^{96} + O(q^{100})$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i 0 1.00000 −2.59808 + 0.500000i 3.00000i 0.500000 + 0.866025i 0
424.2 0.866025 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 0 1.00000 2.59808 0.500000i 3.00000i 0.500000 + 0.866025i 0
499.1 −0.866025 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i 0 1.00000 −2.59808 0.500000i 3.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i 0.866025 0.500000i −0.500000 0.866025i 0 1.00000 2.59808 + 0.500000i 3.00000i 0.500000 0.866025i 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.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.c 4
5.b even 2 1 inner 525.2.r.c 4
5.c odd 4 1 525.2.i.b 2
5.c odd 4 1 525.2.i.d yes 2
7.c even 3 1 inner 525.2.r.c 4
35.j even 6 1 inner 525.2.r.c 4
35.k even 12 1 3675.2.a.g 1
35.k even 12 1 3675.2.a.k 1
35.l odd 12 1 525.2.i.b 2
35.l odd 12 1 525.2.i.d yes 2
35.l odd 12 1 3675.2.a.e 1
35.l odd 12 1 3675.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.b 2 5.c odd 4 1
525.2.i.b 2 35.l odd 12 1
525.2.i.d yes 2 5.c odd 4 1
525.2.i.d yes 2 35.l odd 12 1
525.2.r.c 4 1.a even 1 1 trivial
525.2.r.c 4 5.b even 2 1 inner
525.2.r.c 4 7.c even 3 1 inner
525.2.r.c 4 35.j even 6 1 inner
3675.2.a.e 1 35.l odd 12 1
3675.2.a.g 1 35.k even 12 1
3675.2.a.k 1 35.k even 12 1
3675.2.a.m 1 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 $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{2} + 1$$ $$T_{11}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ 
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 17 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 6 T^{2} - 2173 T^{4} - 13254 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} )$$
$59$ $$( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{4}$$
$73$ $$1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 30 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 113 T^{2} + 9409 T^{4} )^{2}$$