# Properties

 Label 900.3.p.b Level $900$ Weight $3$ Character orbit 900.p Analytic conductor $24.523$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} - \beta_{2} ) q^{7} + ( -9 + 9 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} - \beta_{2} ) q^{7} + ( -9 + 9 \beta_{1} ) q^{9} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} + ( -10 + 10 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( -19 + \beta_{3} ) q^{19} + ( 12 - 12 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{23} -27 q^{27} + ( -20 + 10 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{31} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{33} -14 q^{37} + ( -30 - 3 \beta_{3} ) q^{39} + ( -15 - 15 \beta_{1} ) q^{41} + ( 11 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -68 + 34 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -27 + 27 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{49} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{51} + ( -14 + 28 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( -57 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -29 - 29 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 22 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 36 + 9 \beta_{3} ) q^{63} + ( -7 + 7 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{67} + ( 12 - 24 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -36 + 72 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 37 + 3 \beta_{3} ) q^{73} + ( -64 - 64 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{77} + ( -80 \beta_{1} - 3 \beta_{2} ) q^{79} -81 \beta_{1} q^{81} + ( 104 - 52 \beta_{1} ) q^{83} + ( -30 - 30 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{87} + ( 24 - 48 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 100 + 14 \beta_{3} ) q^{91} + ( -6 - 21 \beta_{3} ) q^{93} + ( -\beta_{1} + 13 \beta_{2} ) q^{97} + ( -9 + 18 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} - 8q^{7} - 18q^{9} + O(q^{10})$$ $$4q + 6q^{3} - 8q^{7} - 18q^{9} + 6q^{11} - 20q^{13} - 76q^{19} + 24q^{21} - 24q^{23} - 108q^{27} - 60q^{29} - 4q^{31} + 18q^{33} - 56q^{37} - 120q^{39} - 90q^{41} + 22q^{43} - 204q^{47} - 54q^{49} - 18q^{51} - 114q^{57} - 174q^{59} + 44q^{61} + 144q^{63} - 14q^{67} + 148q^{73} - 384q^{77} - 160q^{79} - 162q^{81} + 312q^{83} - 180q^{87} + 400q^{91} - 24q^{93} - 2q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 10 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{3} + 20 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 10 \beta_{2}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1 - \beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.93649 − 1.11803i −1.93649 + 1.11803i 1.93649 + 1.11803i −1.93649 − 1.11803i
0 1.50000 2.59808i 0 0 0 −5.87298 + 10.1723i 0 −4.50000 7.79423i 0
101.2 0 1.50000 2.59808i 0 0 0 1.87298 3.24410i 0 −4.50000 7.79423i 0
401.1 0 1.50000 + 2.59808i 0 0 0 −5.87298 10.1723i 0 −4.50000 + 7.79423i 0
401.2 0 1.50000 + 2.59808i 0 0 0 1.87298 + 3.24410i 0 −4.50000 + 7.79423i 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.b 4
3.b odd 2 1 2700.3.p.a 4
5.b even 2 1 180.3.o.a 4
5.c odd 4 2 900.3.u.b 8
9.c even 3 1 2700.3.p.a 4
9.d odd 6 1 inner 900.3.p.b 4
15.d odd 2 1 540.3.o.a 4
15.e even 4 2 2700.3.u.a 8
20.d odd 2 1 720.3.bs.a 4
45.h odd 6 1 180.3.o.a 4
45.h odd 6 1 1620.3.g.a 4
45.j even 6 1 540.3.o.a 4
45.j even 6 1 1620.3.g.a 4
45.k odd 12 2 2700.3.u.a 8
45.l even 12 2 900.3.u.b 8
60.h even 2 1 2160.3.bs.a 4
180.n even 6 1 720.3.bs.a 4
180.p odd 6 1 2160.3.bs.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 5.b even 2 1
180.3.o.a 4 45.h odd 6 1
540.3.o.a 4 15.d odd 2 1
540.3.o.a 4 45.j even 6 1
720.3.bs.a 4 20.d odd 2 1
720.3.bs.a 4 180.n even 6 1
900.3.p.b 4 1.a even 1 1 trivial
900.3.p.b 4 9.d odd 6 1 inner
900.3.u.b 8 5.c odd 4 2
900.3.u.b 8 45.l even 12 2
1620.3.g.a 4 45.h odd 6 1
1620.3.g.a 4 45.j even 6 1
2160.3.bs.a 4 60.h even 2 1
2160.3.bs.a 4 180.p odd 6 1
2700.3.p.a 4 3.b odd 2 1
2700.3.p.a 4 9.c even 3 1
2700.3.u.a 8 15.e even 4 2
2700.3.u.a 8 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 8 T_{7}^{3} + 108 T_{7}^{2} - 352 T_{7} + 1936$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$1936 - 352 T + 108 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$31329 + 1062 T - 165 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$1600 + 800 T + 360 T^{2} + 20 T^{3} + T^{4}$$
$17$ $$31329 + 366 T^{2} + T^{4}$$
$19$ $$( 301 + 38 T + T^{2} )^{2}$$
$23$ $$451584 - 16128 T - 480 T^{2} + 24 T^{3} + T^{4}$$
$29$ $$176400 - 25200 T + 780 T^{2} + 60 T^{3} + T^{4}$$
$31$ $$8620096 - 11744 T + 2952 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$( 14 + T )^{4}$$
$41$ $$( 675 + 45 T + T^{2} )^{2}$$
$43$ $$703921 + 18458 T + 1323 T^{2} - 22 T^{3} + T^{4}$$
$47$ $$10810944 + 670752 T + 17160 T^{2} + 204 T^{3} + T^{4}$$
$53$ $$166464 + 1536 T^{2} + T^{4}$$
$59$ $$5489649 + 407682 T + 12435 T^{2} + 174 T^{3} + T^{4}$$
$61$ $$3136 + 2464 T + 1992 T^{2} - 44 T^{3} + T^{4}$$
$67$ $$73805281 - 120274 T + 8787 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$5143824 + 11016 T^{2} + T^{4}$$
$73$ $$( 829 - 74 T + T^{2} )^{2}$$
$79$ $$34339600 + 937600 T + 19740 T^{2} + 160 T^{3} + T^{4}$$
$83$ $$( 8112 - 156 T + T^{2} )^{2}$$
$89$ $$1327104 + 9216 T^{2} + T^{4}$$
$97$ $$102799321 - 20278 T + 10143 T^{2} + 2 T^{3} + T^{4}$$