# Properties

 Label 525.2.i.g Level 525 Weight 2 Character orbit 525.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

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

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

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 1.22474i −0.500000 + 0.866025i 0 0 1.41421 2.62132 0.358719i −2.82843 −0.500000 0.866025i 0
151.2 0.707107 + 1.22474i −0.500000 + 0.866025i 0 0 −1.41421 −1.62132 + 2.09077i 2.82843 −0.500000 0.866025i 0
226.1 −0.707107 + 1.22474i −0.500000 0.866025i 0 0 1.41421 2.62132 + 0.358719i −2.82843 −0.500000 + 0.866025i 0
226.2 0.707107 1.22474i −0.500000 0.866025i 0 0 −1.41421 −1.62132 2.09077i 2.82843 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.g 4
5.b even 2 1 105.2.i.c 4
5.c odd 4 2 525.2.r.g 8
7.c even 3 1 inner 525.2.i.g 4
7.c even 3 1 3675.2.a.z 2
7.d odd 6 1 3675.2.a.x 2
15.d odd 2 1 315.2.j.d 4
20.d odd 2 1 1680.2.bg.p 4
35.c odd 2 1 735.2.i.j 4
35.i odd 6 1 735.2.a.j 2
35.i odd 6 1 735.2.i.j 4
35.j even 6 1 105.2.i.c 4
35.j even 6 1 735.2.a.i 2
35.l odd 12 2 525.2.r.g 8
105.o odd 6 1 315.2.j.d 4
105.o odd 6 1 2205.2.a.u 2
105.p even 6 1 2205.2.a.s 2
140.p odd 6 1 1680.2.bg.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.c 4 5.b even 2 1
105.2.i.c 4 35.j even 6 1
315.2.j.d 4 15.d odd 2 1
315.2.j.d 4 105.o odd 6 1
525.2.i.g 4 1.a even 1 1 trivial
525.2.i.g 4 7.c even 3 1 inner
525.2.r.g 8 5.c odd 4 2
525.2.r.g 8 35.l odd 12 2
735.2.a.i 2 35.j even 6 1
735.2.a.j 2 35.i odd 6 1
735.2.i.j 4 35.c odd 2 1
735.2.i.j 4 35.i odd 6 1
1680.2.bg.p 4 20.d odd 2 1
1680.2.bg.p 4 140.p odd 6 1
2205.2.a.s 2 105.p even 6 1
2205.2.a.u 2 105.o odd 6 1
3675.2.a.x 2 7.d odd 6 1
3675.2.a.z 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2 T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}( 1 + 2 T^{2} + 4 T^{4} )$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ 
$7$ $$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 88 T^{5} - 968 T^{6} + 5324 T^{7} + 14641 T^{8}$$
$13$ $$( 1 - 6 T + 33 T^{2} - 78 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 1178 T^{5} - 1083 T^{6} - 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 1288 T^{5} - 8464 T^{6} + 48668 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8}$$
$37$ $$1 + 14 T + 75 T^{2} + 658 T^{3} + 6020 T^{4} + 24346 T^{5} + 102675 T^{6} + 709142 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 18 T + 165 T^{2} - 774 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 23312 T^{5} + 101614 T^{6} + 415292 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8}$$
$61$ $$1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 21574 T^{5} + 139159 T^{6} + 601526 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 + 10 T - 21 T^{2} - 250 T^{3} + 2012 T^{4} - 18250 T^{5} - 111909 T^{6} + 3890170 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 4898 T^{5} - 767643 T^{6} + 986078 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 8 T + 180 T^{2} + 664 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 65504 T^{5} + 253472 T^{6} - 11279504 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 - 16 T + 250 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2}$$