# Properties

 Label 525.2.d.d 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: $$\Q(i, \sqrt{13})$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{2} q^{7} + 3 \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{2} q^{7} + 3 \beta_{2} q^{8} - q^{9} -3 q^{11} + ( \beta_{1} - \beta_{2} ) q^{12} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( -1 - \beta_{3} ) q^{16} + 2 \beta_{1} q^{17} -\beta_{1} q^{18} + ( -4 + 2 \beta_{3} ) q^{19} + q^{21} -3 \beta_{1} q^{22} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{23} -3 q^{24} -6 q^{26} -\beta_{2} q^{27} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 5 - 2 \beta_{3} ) q^{29} + ( -2 + 4 \beta_{3} ) q^{31} + ( -\beta_{1} + 3 \beta_{2} ) q^{32} -3 \beta_{2} q^{33} + ( -8 + 2 \beta_{3} ) q^{34} + ( 2 - \beta_{3} ) q^{36} + ( -4 \beta_{1} - 5 \beta_{2} ) q^{37} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{38} -2 \beta_{3} q^{39} + \beta_{1} q^{42} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{43} + ( 6 - 3 \beta_{3} ) q^{44} + ( -13 + \beta_{3} ) q^{46} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{47} + ( -\beta_{1} - 2 \beta_{2} ) q^{48} - q^{49} + ( 2 - 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{52} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{53} + ( -1 + \beta_{3} ) q^{54} + 3 q^{56} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{57} + ( 5 \beta_{1} - 6 \beta_{2} ) q^{58} + ( 8 - 2 \beta_{3} ) q^{59} + ( 4 + 4 \beta_{3} ) q^{61} + ( -2 \beta_{1} + 12 \beta_{2} ) q^{62} + \beta_{2} q^{63} + ( 5 - 6 \beta_{3} ) q^{64} + ( -3 + 3 \beta_{3} ) q^{66} + ( 2 \beta_{1} - 11 \beta_{2} ) q^{67} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{68} + ( 1 - 4 \beta_{3} ) q^{69} -3 q^{71} -3 \beta_{2} q^{72} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 11 + \beta_{3} ) q^{74} + ( 14 - 6 \beta_{3} ) q^{76} + 3 \beta_{2} q^{77} -6 \beta_{2} q^{78} + ( 7 - 6 \beta_{3} ) q^{79} + q^{81} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -2 + \beta_{3} ) q^{84} + ( 13 - 7 \beta_{3} ) q^{86} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{87} -9 \beta_{2} q^{88} + 6 \beta_{3} q^{89} + 2 \beta_{3} q^{91} + ( -5 \beta_{1} + 9 \beta_{2} ) q^{92} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -14 + 8 \beta_{3} ) q^{94} + ( -4 + \beta_{3} ) q^{96} + ( 4 \beta_{1} + 10 \beta_{2} ) q^{97} -\beta_{1} q^{98} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{4} + 2q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 6q^{4} + 2q^{6} - 4q^{9} - 12q^{11} - 2q^{14} - 6q^{16} - 12q^{19} + 4q^{21} - 12q^{24} - 24q^{26} + 16q^{29} - 28q^{34} + 6q^{36} - 4q^{39} + 18q^{44} - 50q^{46} - 4q^{49} + 4q^{51} - 2q^{54} + 12q^{56} + 28q^{59} + 24q^{61} + 8q^{64} - 6q^{66} - 4q^{69} - 12q^{71} + 46q^{74} + 44q^{76} + 16q^{79} + 4q^{81} - 6q^{84} + 38q^{86} + 12q^{89} + 4q^{91} - 40q^{94} - 14q^{96} + 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} - 4 \beta_{1}$$

## 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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
2.30278i 1.00000i −3.30278 0 2.30278 1.00000i 3.00000i −1.00000 0
274.2 1.30278i 1.00000i 0.302776 0 −1.30278 1.00000i 3.00000i −1.00000 0
274.3 1.30278i 1.00000i 0.302776 0 −1.30278 1.00000i 3.00000i −1.00000 0
274.4 2.30278i 1.00000i −3.30278 0 2.30278 1.00000i 3.00000i −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.d 4
3.b odd 2 1 1575.2.d.g 4
5.b even 2 1 inner 525.2.d.d 4
5.c odd 4 1 525.2.a.f 2
5.c odd 4 1 525.2.a.h yes 2
15.d odd 2 1 1575.2.d.g 4
15.e even 4 1 1575.2.a.o 2
15.e even 4 1 1575.2.a.t 2
20.e even 4 1 8400.2.a.cw 2
20.e even 4 1 8400.2.a.df 2
35.f even 4 1 3675.2.a.w 2
35.f even 4 1 3675.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.f 2 5.c odd 4 1
525.2.a.h yes 2 5.c odd 4 1
525.2.d.d 4 1.a even 1 1 trivial
525.2.d.d 4 5.b even 2 1 inner
1575.2.a.o 2 15.e even 4 1
1575.2.a.t 2 15.e even 4 1
1575.2.d.g 4 3.b odd 2 1
1575.2.d.g 4 15.d odd 2 1
3675.2.a.w 2 35.f even 4 1
3675.2.a.bb 2 35.f even 4 1
8400.2.a.cw 2 20.e even 4 1
8400.2.a.df 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 $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} + 7 T_{2}^{2} + 9$$ $$T_{11} + 3$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 5 T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{4}$$
$13$ $$1 - 24 T^{2} + 430 T^{4} - 4056 T^{6} + 28561 T^{8}$$
$17$ $$1 - 40 T^{2} + 926 T^{4} - 11560 T^{6} + 83521 T^{8}$$
$19$ $$( 1 + 6 T + 34 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 14 T^{2} + 899 T^{4} + 7406 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 10 T^{2} + 961 T^{4} )^{2}$$
$37$ $$1 - 26 T^{2} + 1035 T^{4} - 35594 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$1 - 74 T^{2} + 3195 T^{4} - 136826 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 64 T^{2} + 2894 T^{4} - 141376 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 76 T^{2} + 3734 T^{4} - 213484 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 46 T^{2} + 2019 T^{4} + 206494 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 3 T + 71 T^{2} )^{4}$$
$73$ $$1 - 248 T^{2} + 25566 T^{4} - 1321592 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 8 T + 57 T^{2} - 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 196 T^{2} + 20054 T^{4} - 1350244 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 156 T^{2} + 11590 T^{4} - 1467804 T^{6} + 88529281 T^{8}$$