# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{7} -\beta_{3} q^{11} + ( 6 + 6 \beta_{1} ) q^{13} + ( -\beta_{2} + \beta_{3} ) q^{17} + 14 \beta_{1} q^{19} + ( \beta_{2} + \beta_{3} ) q^{23} + \beta_{2} q^{29} + 16 q^{31} + ( 30 - 30 \beta_{1} ) q^{37} + 2 \beta_{3} q^{41} + ( 54 + 54 \beta_{1} ) q^{43} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{47} + 47 \beta_{1} q^{49} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{53} -5 \beta_{2} q^{59} + 54 q^{61} + ( 34 - 34 \beta_{1} ) q^{67} + 4 \beta_{3} q^{71} + ( 65 + 65 \beta_{1} ) q^{73} + ( \beta_{2} - \beta_{3} ) q^{77} + 108 \beta_{1} q^{79} + ( 3 \beta_{2} + 3 \beta_{3} ) q^{83} + 8 \beta_{2} q^{89} + 12 q^{91} + ( 69 - 69 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 24q^{13} + 64q^{31} + 120q^{37} + 216q^{43} + 216q^{61} + 136q^{67} + 260q^{73} + 48q^{91} + 276q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 1.58114 − 1.58114i −1.58114 + 1.58114i 1.58114 + 1.58114i −1.58114 − 1.58114i
0 0 0 0 0 1.00000 + 1.00000i 0 0 0
757.2 0 0 0 0 0 1.00000 + 1.00000i 0 0 0
793.1 0 0 0 0 0 1.00000 1.00000i 0 0 0
793.2 0 0 0 0 0 1.00000 1.00000i 0 0 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.h 4
3.b odd 2 1 inner 900.3.l.h 4
5.b even 2 1 180.3.l.c 4
5.c odd 4 1 180.3.l.c 4
5.c odd 4 1 inner 900.3.l.h 4
15.d odd 2 1 180.3.l.c 4
15.e even 4 1 180.3.l.c 4
15.e even 4 1 inner 900.3.l.h 4
20.d odd 2 1 720.3.bh.h 4
20.e even 4 1 720.3.bh.h 4
60.h even 2 1 720.3.bh.h 4
60.l odd 4 1 720.3.bh.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.l.c 4 5.b even 2 1
180.3.l.c 4 5.c odd 4 1
180.3.l.c 4 15.d odd 2 1
180.3.l.c 4 15.e even 4 1
720.3.bh.h 4 20.d odd 2 1
720.3.bh.h 4 20.e even 4 1
720.3.bh.h 4 60.h even 2 1
720.3.bh.h 4 60.l odd 4 1
900.3.l.h 4 1.a even 1 1 trivial
900.3.l.h 4 3.b odd 2 1 inner
900.3.l.h 4 5.c odd 4 1 inner
900.3.l.h 4 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} - 2 T_{7} + 2$$ $$T_{11}^{2} - 250$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 - 2 T + T^{2} )^{2}$$
$11$ $$( -250 + T^{2} )^{2}$$
$13$ $$( 72 - 12 T + T^{2} )^{2}$$
$17$ $$250000 + T^{4}$$
$19$ $$( 196 + T^{2} )^{2}$$
$23$ $$250000 + T^{4}$$
$29$ $$( 250 + T^{2} )^{2}$$
$31$ $$( -16 + T )^{4}$$
$37$ $$( 1800 - 60 T + T^{2} )^{2}$$
$41$ $$( -1000 + T^{2} )^{2}$$
$43$ $$( 5832 - 108 T + T^{2} )^{2}$$
$47$ $$20250000 + T^{4}$$
$53$ $$64000000 + T^{4}$$
$59$ $$( 6250 + T^{2} )^{2}$$
$61$ $$( -54 + T )^{4}$$
$67$ $$( 2312 - 68 T + T^{2} )^{2}$$
$71$ $$( -4000 + T^{2} )^{2}$$
$73$ $$( 8450 - 130 T + T^{2} )^{2}$$
$79$ $$( 11664 + T^{2} )^{2}$$
$83$ $$20250000 + T^{4}$$
$89$ $$( 16000 + T^{2} )^{2}$$
$97$ $$( 9522 - 138 T + T^{2} )^{2}$$