# Properties

 Label 525.2.g.d Level 525 Weight 2 Character orbit 525.g 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{2} + \beta_{7} ) q^{4} + ( -1 - \beta_{2} - \beta_{7} ) q^{6} + ( \beta_{4} + \beta_{6} ) q^{7} + ( \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{8} + ( -1 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{2} + \beta_{7} ) q^{4} + ( -1 - \beta_{2} - \beta_{7} ) q^{6} + ( \beta_{4} + \beta_{6} ) q^{7} + ( \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{8} + ( -1 + \beta_{2} + \beta_{3} ) q^{9} + \beta_{3} q^{11} + ( -\beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{12} + ( 3 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} ) q^{13} + ( -1 + \beta_{3} - 2 \beta_{7} ) q^{14} + ( 3 - \beta_{2} + \beta_{7} ) q^{16} + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{17} + ( -\beta_{1} + 3 \beta_{4} + 3 \beta_{5} ) q^{18} + ( \beta_{2} + \beta_{3} + \beta_{7} ) q^{19} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{21} + \beta_{6} q^{22} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{23} + ( 1 - \beta_{2} + 2 \beta_{3} - 3 \beta_{7} ) q^{24} + ( 8 - 2 \beta_{2} + 2 \beta_{7} ) q^{26} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{27} + ( -5 \beta_{1} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{28} + ( \beta_{2} + \beta_{7} ) q^{29} + ( \beta_{2} + \beta_{3} + \beta_{7} ) q^{31} + ( 3 \beta_{1} - \beta_{4} - 3 \beta_{5} ) q^{32} + ( \beta_{4} - \beta_{5} - \beta_{6} ) q^{33} + ( -\beta_{2} - \beta_{3} - \beta_{7} ) q^{34} + ( -5 + 3 \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{36} + ( 2 \beta_{1} + 2 \beta_{5} + \beta_{6} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( -4 - 3 \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{39} -6 q^{41} + ( -\beta_{1} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{42} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{43} + ( -\beta_{2} - \beta_{3} - \beta_{7} ) q^{44} + ( 2 + 2 \beta_{2} - 2 \beta_{7} ) q^{46} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{47} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{48} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{49} + ( -4 + 2 \beta_{3} + \beta_{7} ) q^{51} + ( 6 \beta_{1} - 12 \beta_{4} - 6 \beta_{5} ) q^{52} + ( 2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 5 + \beta_{2} - 2 \beta_{3} + 3 \beta_{7} ) q^{54} + ( -7 - 2 \beta_{2} - \beta_{3} - 4 \beta_{7} ) q^{56} + ( 2 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{58} + ( 4 + 2 \beta_{2} - 2 \beta_{7} ) q^{59} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{5} - \beta_{6} ) q^{62} + ( 2 \beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{6} ) q^{63} + ( 1 + \beta_{2} - \beta_{7} ) q^{64} + ( \beta_{2} - \beta_{3} - \beta_{7} ) q^{66} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{67} + 3 \beta_{6} q^{68} + ( -6 - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{7} ) q^{69} + ( -3 \beta_{2} + \beta_{3} - 3 \beta_{7} ) q^{71} + ( -5 \beta_{1} + 7 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} ) q^{72} -4 \beta_{4} q^{73} + ( 3 \beta_{2} + \beta_{3} + 3 \beta_{7} ) q^{74} + ( 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{7} ) q^{76} + ( -\beta_{1} + 3 \beta_{5} + \beta_{6} ) q^{77} + ( -8 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{78} + ( \beta_{2} - \beta_{7} ) q^{79} + ( -3 - 4 \beta_{2} + \beta_{3} + \beta_{7} ) q^{81} + ( -6 \beta_{1} + 6 \beta_{4} + 6 \beta_{5} ) q^{82} + ( -4 \beta_{1} - 4 \beta_{5} - 4 \beta_{6} ) q^{83} + ( -5 + 6 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{84} + ( 4 \beta_{2} + 4 \beta_{7} ) q^{86} + ( 2 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{5} - \beta_{6} ) q^{88} + ( -10 - 2 \beta_{2} + 2 \beta_{7} ) q^{89} + ( \beta_{2} + 4 \beta_{3} - 5 \beta_{7} ) q^{91} + ( -6 \beta_{1} + 2 \beta_{4} + 6 \beta_{5} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{93} + ( -4 \beta_{2} + 2 \beta_{3} - 4 \beta_{7} ) q^{94} + ( -5 - 3 \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{96} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{97} + ( 3 \beta_{1} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{98} + ( -4 - \beta_{2} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{4} - 8q^{6} - 10q^{9} + O(q^{10})$$ $$8q + 12q^{4} - 8q^{6} - 10q^{9} - 12q^{14} + 28q^{16} - 10q^{21} + 4q^{24} + 72q^{26} - 48q^{36} - 30q^{39} - 48q^{41} + 8q^{46} - 8q^{49} - 30q^{51} + 44q^{54} - 60q^{56} + 24q^{59} + 4q^{64} - 4q^{66} - 40q^{69} - 4q^{79} - 14q^{81} - 48q^{84} - 72q^{89} - 12q^{91} - 36q^{96} - 26q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 4 \nu^{4} + 16 \nu^{2} + 9$$$$)/36$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27$$$$)/36$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} - 2 \nu^{3} + 9 \nu$$$$)/54$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/27$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} - 16 \nu^{5} - 8 \nu^{3} - 81 \nu$$$$)/108$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 44 \nu^{2} + 117$$$$)/36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{3} - 4 \beta_{2} - 3$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{6} - \beta_{5} - 8 \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{7} - 12 \beta_{3} + 4 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$-12 \beta_{6} + 24 \beta_{4} - 13 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
524.1
 −0.396143 + 1.68614i −0.396143 − 1.68614i −1.26217 + 1.18614i −1.26217 − 1.18614i 1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 + 1.68614i 0.396143 − 1.68614i
−2.52434 0.396143 1.68614i 4.37228 0 −1.00000 + 4.25639i 1.73205 2.00000i −5.98844 −2.68614 1.33591i 0
524.2 −2.52434 0.396143 + 1.68614i 4.37228 0 −1.00000 4.25639i 1.73205 + 2.00000i −5.98844 −2.68614 + 1.33591i 0
524.3 −0.792287 1.26217 1.18614i −1.37228 0 −1.00000 + 0.939764i −1.73205 + 2.00000i 2.67181 0.186141 2.99422i 0
524.4 −0.792287 1.26217 + 1.18614i −1.37228 0 −1.00000 0.939764i −1.73205 2.00000i 2.67181 0.186141 + 2.99422i 0
524.5 0.792287 −1.26217 1.18614i −1.37228 0 −1.00000 0.939764i 1.73205 + 2.00000i −2.67181 0.186141 + 2.99422i 0
524.6 0.792287 −1.26217 + 1.18614i −1.37228 0 −1.00000 + 0.939764i 1.73205 2.00000i −2.67181 0.186141 2.99422i 0
524.7 2.52434 −0.396143 1.68614i 4.37228 0 −1.00000 4.25639i −1.73205 2.00000i 5.98844 −2.68614 + 1.33591i 0
524.8 2.52434 −0.396143 + 1.68614i 4.37228 0 −1.00000 + 4.25639i −1.73205 + 2.00000i 5.98844 −2.68614 1.33591i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 524.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.c even 2 1 inner
105.g 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.g.d 8
3.b odd 2 1 525.2.g.e 8
5.b even 2 1 inner 525.2.g.d 8
5.c odd 4 1 105.2.b.d yes 4
5.c odd 4 1 525.2.b.e 4
7.b odd 2 1 525.2.g.e 8
15.d odd 2 1 525.2.g.e 8
15.e even 4 1 105.2.b.c 4
15.e even 4 1 525.2.b.g 4
20.e even 4 1 1680.2.f.g 4
21.c even 2 1 inner 525.2.g.d 8
35.c odd 2 1 525.2.g.e 8
35.f even 4 1 105.2.b.c 4
35.f even 4 1 525.2.b.g 4
35.k even 12 1 735.2.s.h 4
35.k even 12 1 735.2.s.i 4
35.l odd 12 1 735.2.s.g 4
35.l odd 12 1 735.2.s.j 4
60.l odd 4 1 1680.2.f.h 4
105.g even 2 1 inner 525.2.g.d 8
105.k odd 4 1 105.2.b.d yes 4
105.k odd 4 1 525.2.b.e 4
105.w odd 12 1 735.2.s.g 4
105.w odd 12 1 735.2.s.j 4
105.x even 12 1 735.2.s.h 4
105.x even 12 1 735.2.s.i 4
140.j odd 4 1 1680.2.f.h 4
420.w even 4 1 1680.2.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 15.e even 4 1
105.2.b.c 4 35.f even 4 1
105.2.b.d yes 4 5.c odd 4 1
105.2.b.d yes 4 105.k odd 4 1
525.2.b.e 4 5.c odd 4 1
525.2.b.e 4 105.k odd 4 1
525.2.b.g 4 15.e even 4 1
525.2.b.g 4 35.f even 4 1
525.2.g.d 8 1.a even 1 1 trivial
525.2.g.d 8 5.b even 2 1 inner
525.2.g.d 8 21.c even 2 1 inner
525.2.g.d 8 105.g even 2 1 inner
525.2.g.e 8 3.b odd 2 1
525.2.g.e 8 7.b odd 2 1
525.2.g.e 8 15.d odd 2 1
525.2.g.e 8 35.c odd 2 1
735.2.s.g 4 35.l odd 12 1
735.2.s.g 4 105.w odd 12 1
735.2.s.h 4 35.k even 12 1
735.2.s.h 4 105.x even 12 1
735.2.s.i 4 35.k even 12 1
735.2.s.i 4 105.x even 12 1
735.2.s.j 4 35.l odd 12 1
735.2.s.j 4 105.w odd 12 1
1680.2.f.g 4 20.e even 4 1
1680.2.f.g 4 420.w even 4 1
1680.2.f.h 4 60.l odd 4 1
1680.2.f.h 4 140.j odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} - 7 T_{2}^{2} + 4$$ $$T_{41} + 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{6} + 16 T^{8} )^{2}$$
$3$ $$1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8}$$
$5$ 
$7$ $$( 1 + 2 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + T^{2} + 264 T^{4} + 169 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 47 T^{2} + 1056 T^{4} - 13583 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4}$$
$23$ $$( 1 + 16 T^{2} - 66 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 50 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 80 T^{2} + 4206 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{8}$$
$43$ $$( 1 - 104 T^{2} + 6270 T^{4} - 192296 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 131 T^{2} + 8040 T^{4} - 289379 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 136 T^{2} + 9054 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 6 T + 94 T^{2} - 354 T^{3} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 200 T^{2} + 18846 T^{4} - 897800 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 98 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + T + 150 T^{2} + 79 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 4 T^{2} - 5226 T^{4} + 27556 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 265 T^{2} + 32736 T^{4} + 2493385 T^{6} + 88529281 T^{8} )^{2}$$