# Properties

 Label 525.2.b.g Level 525 Weight 2 Character orbit 525.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - \beta_{2} ) q^{7} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{6} + ( 2 - \beta_{2} ) q^{7} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + ( \beta_{1} - \beta_{3} ) q^{11} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{13} + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{14} + ( 3 - \beta_{1} - \beta_{3} ) q^{16} + ( 2 + \beta_{1} + \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{18} -2 \beta_{2} q^{19} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} + 2 q^{22} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 1 + 5 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{24} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{26} + ( 4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{27} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{28} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + 2 \beta_{2} q^{31} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{32} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + 2 \beta_{2} q^{34} + ( -5 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{36} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{38} + ( 4 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{39} + 6 q^{41} + ( -2 - \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{43} -2 \beta_{2} q^{44} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{46} + ( -4 + \beta_{1} + \beta_{3} ) q^{47} + ( 4 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{48} + ( 1 - 4 \beta_{2} ) q^{49} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{51} + ( 6 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{52} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 5 - 5 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{54} + ( 7 + \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{56} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 4 - 2 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 4 + 2 \beta_{1} + 2 \beta_{3} ) q^{59} + 4 \beta_{2} q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{62} + ( 6 - 3 \beta_{1} + 2 \beta_{3} ) q^{63} + ( -1 - \beta_{1} - \beta_{3} ) q^{64} -2 \beta_{3} q^{66} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{67} + 6 q^{68} + ( -6 - 6 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -8 + 5 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{72} -4 \beta_{2} q^{73} + ( 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{76} + ( -2 + \beta_{1} - 3 \beta_{3} ) q^{77} + ( -8 + 2 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{78} + ( -\beta_{1} - \beta_{3} ) q^{79} + ( -3 + 3 \beta_{2} - 5 \beta_{3} ) q^{81} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{82} + ( -8 - 4 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -11 + 5 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{84} + ( -4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{86} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{88} + ( -10 - 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -3 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} ) q^{91} + ( 6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{92} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 6 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{94} + ( 5 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{96} + ( -3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -4 + 3 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{98} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} - 6q^{4} + 4q^{6} + 8q^{7} + 5q^{9} + O(q^{10})$$ $$4q + q^{3} - 6q^{4} + 4q^{6} + 8q^{7} + 5q^{9} - 18q^{12} - 6q^{14} + 14q^{16} + 6q^{17} + 2q^{18} - q^{21} + 8q^{22} + 2q^{24} - 36q^{26} + 16q^{27} - 12q^{28} - 7q^{33} - 24q^{36} + 4q^{37} - 12q^{38} + 15q^{39} + 24q^{41} - 10q^{42} + 4q^{43} + 4q^{46} - 18q^{47} + 20q^{48} + 4q^{49} - 15q^{51} + 22q^{54} + 30q^{56} - 6q^{57} + 20q^{58} + 12q^{59} + 12q^{62} + 25q^{63} - 2q^{64} + 2q^{66} + 4q^{67} + 24q^{68} - 20q^{69} - 32q^{72} - 6q^{77} - 42q^{78} + 2q^{79} - 7q^{81} - 24q^{83} - 48q^{84} - q^{87} + 4q^{88} - 36q^{89} - 6q^{91} + 6q^{93} + 18q^{96} - 24q^{98} + 13q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 4 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 2 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{2} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i 1.68614 + 0.396143i −4.37228 0 1.00000 4.25639i 2.00000 1.73205i 5.98844i 2.68614 + 1.33591i 0
251.2 0.792287i −1.18614 + 1.26217i 1.37228 0 1.00000 + 0.939764i 2.00000 + 1.73205i 2.67181i −0.186141 2.99422i 0
251.3 0.792287i −1.18614 1.26217i 1.37228 0 1.00000 0.939764i 2.00000 1.73205i 2.67181i −0.186141 + 2.99422i 0
251.4 2.52434i 1.68614 0.396143i −4.37228 0 1.00000 + 4.25639i 2.00000 + 1.73205i 5.98844i 2.68614 1.33591i 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.c 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.b.g 4
3.b odd 2 1 525.2.b.e 4
5.b even 2 1 105.2.b.c 4
5.c odd 4 2 525.2.g.e 8
7.b odd 2 1 525.2.b.e 4
15.d odd 2 1 105.2.b.d yes 4
15.e even 4 2 525.2.g.d 8
20.d odd 2 1 1680.2.f.h 4
21.c even 2 1 inner 525.2.b.g 4
35.c odd 2 1 105.2.b.d yes 4
35.f even 4 2 525.2.g.d 8
35.i odd 6 1 735.2.s.g 4
35.i odd 6 1 735.2.s.j 4
35.j even 6 1 735.2.s.h 4
35.j even 6 1 735.2.s.i 4
60.h even 2 1 1680.2.f.g 4
105.g even 2 1 105.2.b.c 4
105.k odd 4 2 525.2.g.e 8
105.o odd 6 1 735.2.s.g 4
105.o odd 6 1 735.2.s.j 4
105.p even 6 1 735.2.s.h 4
105.p even 6 1 735.2.s.i 4
140.c even 2 1 1680.2.f.g 4
420.o odd 2 1 1680.2.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 5.b even 2 1
105.2.b.c 4 105.g even 2 1
105.2.b.d yes 4 15.d odd 2 1
105.2.b.d yes 4 35.c odd 2 1
525.2.b.e 4 3.b odd 2 1
525.2.b.e 4 7.b odd 2 1
525.2.b.g 4 1.a even 1 1 trivial
525.2.b.g 4 21.c even 2 1 inner
525.2.g.d 8 15.e even 4 2
525.2.g.d 8 35.f even 4 2
525.2.g.e 8 5.c odd 4 2
525.2.g.e 8 105.k odd 4 2
735.2.s.g 4 35.i odd 6 1
735.2.s.g 4 105.o odd 6 1
735.2.s.h 4 35.j even 6 1
735.2.s.h 4 105.p even 6 1
735.2.s.i 4 35.j even 6 1
735.2.s.i 4 105.p even 6 1
735.2.s.j 4 35.i odd 6 1
735.2.s.j 4 105.o odd 6 1
1680.2.f.g 4 60.h even 2 1
1680.2.f.g 4 140.c even 2 1
1680.2.f.h 4 20.d odd 2 1
1680.2.f.h 4 420.o odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 7 T_{2}^{2} + 4$$ $$T_{11}^{4} + 7 T_{11}^{2} + 4$$ $$T_{17}^{2} - 3 T_{17} - 6$$ $$T_{37}^{2} - 2 T_{37} - 32$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 4 T^{6} + 16 T^{8}$$
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ 
$7$ $$( 1 - 4 T + 7 T^{2} )^{2}$$
$11$ $$1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8}$$
$13$ $$1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$1 - 16 T^{2} - 66 T^{4} - 8464 T^{6} + 279841 T^{8}$$
$29$ $$1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8}$$
$31$ $$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 2 T + 42 T^{2} - 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 2 T + 54 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 9 T + 106 T^{2} + 423 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 6 T + 94 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 2 T + 102 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - T + 150 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 12 T + 70 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8}$$