# Properties

 Label 525.2.b.h Level 525 Weight 2 Character orbit 525.b Analytic conductor 4.192 Analytic rank 0 Dimension 8 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.624529833984.7 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{7} q^{3} + ( -2 + \beta_{4} ) q^{4} + \beta_{3} q^{6} + ( -1 - \beta_{1} ) q^{7} + ( -\beta_{2} + \beta_{6} ) q^{8} + ( -\beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{7} q^{3} + ( -2 + \beta_{4} ) q^{4} + \beta_{3} q^{6} + ( -1 - \beta_{1} ) q^{7} + ( -\beta_{2} + \beta_{6} ) q^{8} + ( -\beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{2} + \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{12} + \beta_{1} q^{13} + ( -\beta_{2} + \beta_{3} - \beta_{5} ) q^{14} + ( 1 - \beta_{4} ) q^{16} + ( \beta_{3} - \beta_{5} ) q^{17} + ( 4 - \beta_{2} - \beta_{4} + \beta_{6} ) q^{18} + ( -\beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{19} + ( -2 + \beta_{4} + \beta_{6} + \beta_{7} ) q^{21} + ( 5 - 3 \beta_{4} ) q^{22} + ( \beta_{2} + \beta_{6} ) q^{23} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{24} + ( -\beta_{3} + \beta_{5} ) q^{26} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{27} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{28} + ( -\beta_{2} - \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{31} + \beta_{6} q^{32} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{33} + ( 3 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{34} + ( 5 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{36} + ( 3 + 2 \beta_{4} ) q^{37} + ( \beta_{1} + 3 \beta_{3} - \beta_{5} - 4 \beta_{7} ) q^{38} + ( 2 - \beta_{4} - \beta_{6} ) q^{39} + ( \beta_{1} + \beta_{3} + \beta_{5} - 4 \beta_{7} ) q^{41} + ( 1 - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{42} + ( -1 - 2 \beta_{4} ) q^{43} + ( 6 \beta_{2} - \beta_{6} ) q^{44} + ( -3 - \beta_{4} ) q^{46} + ( \beta_{3} - \beta_{5} ) q^{47} + ( -\beta_{1} - \beta_{5} ) q^{48} + ( -5 + 2 \beta_{1} ) q^{49} + ( 1 - 3 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{51} + ( -\beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{52} + ( 4 \beta_{2} - 2 \beta_{6} ) q^{53} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{7} ) q^{54} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} + 4 \beta_{7} ) q^{56} + ( -7 - 3 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{57} + ( 3 + \beta_{4} ) q^{58} + ( -\beta_{1} - 2 \beta_{3} + 4 \beta_{7} ) q^{59} + ( -\beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{61} + ( \beta_{1} + 2 \beta_{3} - 4 \beta_{7} ) q^{62} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{63} + ( 3 - 4 \beta_{4} ) q^{64} + ( -3 \beta_{1} - 3 \beta_{5} - 2 \beta_{7} ) q^{66} + ( -7 + 4 \beta_{4} ) q^{67} + ( -\beta_{1} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} ) q^{68} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{69} + ( -\beta_{2} + \beta_{6} ) q^{71} + ( -5 + 4 \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{72} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} ) q^{73} + ( \beta_{2} + 2 \beta_{6} ) q^{74} + ( 5 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} ) q^{76} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} + 4 \beta_{7} ) q^{77} + ( -1 + 3 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{78} + ( -5 + 2 \beta_{4} ) q^{79} + ( 1 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{81} + ( -\beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} ) q^{82} + ( -\beta_{1} + \beta_{3} - 3 \beta_{5} + 4 \beta_{7} ) q^{83} + ( 9 - \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{84} + ( \beta_{2} - 2 \beta_{6} ) q^{86} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{87} + ( -15 + 2 \beta_{4} ) q^{88} + ( \beta_{1} + 2 \beta_{3} - 4 \beta_{7} ) q^{89} + ( 6 - \beta_{1} ) q^{91} + \beta_{6} q^{92} + ( -5 - 3 \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{93} + ( 3 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{94} + ( -2 \beta_{1} + \beta_{5} + \beta_{7} ) q^{96} + ( -3 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{97} + ( -5 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{98} + ( -5 + 4 \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{4} - 8q^{7} + 4q^{9} + O(q^{10})$$ $$8q - 12q^{4} - 8q^{7} + 4q^{9} + 4q^{16} + 28q^{18} - 12q^{21} + 28q^{22} + 12q^{28} + 36q^{36} + 32q^{37} + 12q^{39} - 16q^{43} - 28q^{46} - 40q^{49} - 48q^{57} + 28q^{58} - 4q^{63} + 8q^{64} - 40q^{67} - 28q^{72} - 32q^{79} + 16q^{81} + 60q^{84} - 112q^{88} + 48q^{91} - 36q^{93} - 28q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} - 2 x^{4} - 18 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu^{5} - 7 \nu^{3} + 63 \nu$$$$)/54$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{4} + 7 \nu^{2} - 18$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} + 5 \nu^{5} - 22 \nu^{3} + 27 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} + 11 \nu^{2} + 18$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{5} + 25 \nu^{3} + 36 \nu$$$$)/54$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{6} + 5 \nu^{4} - 4 \nu^{2} - 45$$$$)/18$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{5} - 2 \nu^{3} - 18 \nu$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} + \beta_{3} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 4 \beta_{5} - 2 \beta_{3} - 2 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 1$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} + \beta_{5} + 7 \beta_{3} - 11 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{6} - 3 \beta_{4} + 7 \beta_{2} + 20$$ $$\nu^{7}$$ $$=$$ $$($$$$35 \beta_{7} + 28 \beta_{5} + 28 \beta_{3} - 8 \beta_{1}$$$$)/3$$

## 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
 −0.777403 + 1.54779i 0.777403 − 1.54779i −1.70166 + 0.323042i 1.70166 − 0.323042i −1.70166 − 0.323042i 1.70166 + 0.323042i −0.777403 − 1.54779i 0.777403 + 1.54779i
2.40651i −0.777403 1.54779i −3.79129 0 −3.72476 + 1.87083i −1.00000 2.44949i 4.31075i −1.79129 + 2.40651i 0
251.2 2.40651i 0.777403 + 1.54779i −3.79129 0 3.72476 1.87083i −1.00000 + 2.44949i 4.31075i −1.79129 + 2.40651i 0
251.3 1.09941i −1.70166 0.323042i 0.791288 0 −0.355157 + 1.87083i −1.00000 + 2.44949i 3.06878i 2.79129 + 1.09941i 0
251.4 1.09941i 1.70166 + 0.323042i 0.791288 0 0.355157 1.87083i −1.00000 2.44949i 3.06878i 2.79129 + 1.09941i 0
251.5 1.09941i −1.70166 + 0.323042i 0.791288 0 −0.355157 1.87083i −1.00000 2.44949i 3.06878i 2.79129 1.09941i 0
251.6 1.09941i 1.70166 0.323042i 0.791288 0 0.355157 + 1.87083i −1.00000 + 2.44949i 3.06878i 2.79129 1.09941i 0
251.7 2.40651i −0.777403 + 1.54779i −3.79129 0 −3.72476 1.87083i −1.00000 + 2.44949i 4.31075i −1.79129 2.40651i 0
251.8 2.40651i 0.777403 1.54779i −3.79129 0 3.72476 + 1.87083i −1.00000 2.44949i 4.31075i −1.79129 2.40651i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
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.h 8
3.b odd 2 1 inner 525.2.b.h 8
5.b even 2 1 525.2.b.i yes 8
5.c odd 4 2 525.2.g.f 16
7.b odd 2 1 inner 525.2.b.h 8
15.d odd 2 1 525.2.b.i yes 8
15.e even 4 2 525.2.g.f 16
21.c even 2 1 inner 525.2.b.h 8
35.c odd 2 1 525.2.b.i yes 8
35.f even 4 2 525.2.g.f 16
105.g even 2 1 525.2.b.i yes 8
105.k odd 4 2 525.2.g.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.h 8 1.a even 1 1 trivial
525.2.b.h 8 3.b odd 2 1 inner
525.2.b.h 8 7.b odd 2 1 inner
525.2.b.h 8 21.c even 2 1 inner
525.2.b.i yes 8 5.b even 2 1
525.2.b.i yes 8 15.d odd 2 1
525.2.b.i yes 8 35.c odd 2 1
525.2.b.i yes 8 105.g even 2 1
525.2.g.f 16 5.c odd 4 2
525.2.g.f 16 15.e even 4 2
525.2.g.f 16 35.f even 4 2
525.2.g.f 16 105.k odd 4 2

## Hecke kernels

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

 $$T_{2}^{4} + 7 T_{2}^{2} + 7$$ $$T_{11}^{4} + 28 T_{11}^{2} + 175$$ $$T_{17}^{4} - 42 T_{17}^{2} + 252$$ $$T_{37}^{2} - 8 T_{37} - 5$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + 3 T^{4} - 4 T^{6} + 16 T^{8} )^{2}$$
$3$ $$1 - 2 T^{2} - 2 T^{4} - 18 T^{6} + 81 T^{8}$$
$5$ 
$7$ $$( 1 + 2 T + 7 T^{2} )^{4}$$
$11$ $$( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 26 T^{2} + 558 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 64 T^{2} + 1893 T^{4} - 33856 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 69 T^{2} - 296 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 4 T + 69 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 146 T^{2} + 9558 T^{4} + 322514 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 16 T^{2} - 1122 T^{4} - 44944 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 10 T + 75 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 8 T^{2} - 1422 T^{4} + 42632 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 8 T + 153 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 38 T^{2} + 4878 T^{4} + 261782 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 250 T^{2} + 29718 T^{4} - 2352250 T^{6} + 88529281 T^{8} )^{2}$$