# Properties

 Label 525.2.d.e Level 525 Weight 2 Character orbit 525.d Analytic conductor 4.192 Analytic rank 0 Dimension 4 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + \beta_{3} q^{3} + 3 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + \beta_{3} q^{3} + 3 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} - q^{9} + ( 1 + 4 \beta_{2} ) q^{11} + 3 \beta_{1} q^{12} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + ( 5 - 3 \beta_{2} ) q^{16} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{3} ) q^{18} -2 \beta_{2} q^{19} - q^{21} + ( -7 \beta_{1} + 3 \beta_{3} ) q^{22} + 5 \beta_{3} q^{23} + ( -1 + 4 \beta_{2} ) q^{24} + 2 q^{26} -\beta_{3} q^{27} + 3 \beta_{1} q^{28} + ( 5 + 6 \beta_{2} ) q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{32} + ( 4 \beta_{1} + \beta_{3} ) q^{33} + ( 4 - 10 \beta_{2} ) q^{34} -3 \beta_{2} q^{36} + ( 4 \beta_{1} + \beta_{3} ) q^{37} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{38} + ( -4 - 2 \beta_{2} ) q^{39} -8 q^{41} + ( -\beta_{1} + \beta_{3} ) q^{42} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{43} + ( 12 - 9 \beta_{2} ) q^{44} + ( 5 - 5 \beta_{2} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{48} - q^{49} + ( 2 + 6 \beta_{2} ) q^{51} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{52} + ( -4 \beta_{1} - 6 \beta_{3} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( -1 + 4 \beta_{2} ) q^{56} -2 \beta_{1} q^{57} + ( -7 \beta_{1} + \beta_{3} ) q^{58} + ( 4 + 2 \beta_{2} ) q^{59} + ( 4 + 12 \beta_{2} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{62} -\beta_{3} q^{63} + ( 1 + 6 \beta_{2} ) q^{64} + ( -3 + 7 \beta_{2} ) q^{66} + ( -6 \beta_{1} - 7 \beta_{3} ) q^{67} + ( 12 \beta_{1} - 18 \beta_{3} ) q^{68} -5 q^{69} + ( -7 - 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} - \beta_{3} ) q^{72} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( -3 + 7 \beta_{2} ) q^{74} + ( -6 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} + \beta_{3} ) q^{77} + 2 \beta_{3} q^{78} + ( -5 - 2 \beta_{2} ) q^{79} + q^{81} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{82} + ( -4 \beta_{1} + 6 \beta_{3} ) q^{83} -3 \beta_{2} q^{84} + ( 7 - 9 \beta_{2} ) q^{86} + ( 6 \beta_{1} + 5 \beta_{3} ) q^{87} + ( 16 \beta_{1} - 15 \beta_{3} ) q^{88} + ( -4 - 6 \beta_{2} ) q^{89} + ( -4 - 2 \beta_{2} ) q^{91} + 15 \beta_{1} q^{92} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{93} + ( -6 + 8 \beta_{2} ) q^{94} + ( 6 - 3 \beta_{2} ) q^{96} + ( 4 \beta_{1} - 6 \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{98} + ( -1 - 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{4} + 6q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 6q^{4} + 6q^{6} - 4q^{9} - 4q^{11} + 6q^{14} + 26q^{16} + 4q^{19} - 4q^{21} - 12q^{24} + 8q^{26} + 8q^{29} + 36q^{34} + 6q^{36} - 12q^{39} - 32q^{41} + 66q^{44} + 30q^{46} - 4q^{49} - 4q^{51} - 6q^{54} - 12q^{56} + 12q^{59} - 8q^{61} - 8q^{64} - 26q^{66} - 20q^{69} - 20q^{71} - 26q^{74} - 36q^{76} - 16q^{79} + 4q^{81} + 6q^{84} + 46q^{86} - 4q^{89} - 12q^{91} - 40q^{94} + 30q^{96} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
2.61803i 1.00000i −4.85410 0 2.61803 1.00000i 7.47214i −1.00000 0
274.2 0.381966i 1.00000i 1.85410 0 0.381966 1.00000i 1.47214i −1.00000 0
274.3 0.381966i 1.00000i 1.85410 0 0.381966 1.00000i 1.47214i −1.00000 0
274.4 2.61803i 1.00000i −4.85410 0 2.61803 1.00000i 7.47214i −1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.d.e 4
3.b odd 2 1 1575.2.d.f 4
5.b even 2 1 inner 525.2.d.e 4
5.c odd 4 1 525.2.a.e 2
5.c odd 4 1 525.2.a.i yes 2
15.d odd 2 1 1575.2.d.f 4
15.e even 4 1 1575.2.a.l 2
15.e even 4 1 1575.2.a.v 2
20.e even 4 1 8400.2.a.cy 2
20.e even 4 1 8400.2.a.da 2
35.f even 4 1 3675.2.a.r 2
35.f even 4 1 3675.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 5.c odd 4 1
525.2.a.i yes 2 5.c odd 4 1
525.2.d.e 4 1.a even 1 1 trivial
525.2.d.e 4 5.b even 2 1 inner
1575.2.a.l 2 15.e even 4 1
1575.2.a.v 2 15.e even 4 1
1575.2.d.f 4 3.b odd 2 1
1575.2.d.f 4 15.d odd 2 1
3675.2.a.r 2 35.f even 4 1
3675.2.a.bh 2 35.f even 4 1
8400.2.a.cy 2 20.e even 4 1
8400.2.a.da 2 20.e even 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 7 T_{2}^{2} + 1$$ $$T_{11}^{2} + 2 T_{11} - 19$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8}$$
$17$ $$1 + 24 T^{2} + 542 T^{4} + 6936 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 21 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 42 T^{2} + 961 T^{4} )^{2}$$
$37$ $$1 - 106 T^{2} + 5467 T^{4} - 145114 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 8 T + 41 T^{2} )^{4}$$
$43$ $$1 - 90 T^{2} + 5003 T^{4} - 166410 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 146 T^{2} + 11427 T^{4} - 655394 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 280 T^{2} + 30238 T^{4} - 1492120 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 8 T + 169 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 164 T^{2} + 15382 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 220 T^{2} + 25798 T^{4} - 2069980 T^{6} + 88529281 T^{8}$$