# Properties

 Label 900.3.l.f Level $900$ Weight $3$ Character orbit 900.l Analytic conductor $24.523$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{49}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{7} -\beta_{3} q^{13} + 26 \beta_{2} q^{19} + 46 q^{31} -5 \beta_{1} q^{37} + 6 \beta_{3} q^{43} + 143 \beta_{2} q^{49} + 74 q^{61} + 4 \beta_{1} q^{67} -10 \beta_{3} q^{73} + 142 \beta_{2} q^{79} + 192 q^{91} + 14 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 184q^{31} + 296q^{61} + 768q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$8 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/8$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$$$/8$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
0 0 0 0 0 −9.79796 9.79796i 0 0 0
757.2 0 0 0 0 0 9.79796 + 9.79796i 0 0 0
793.1 0 0 0 0 0 −9.79796 + 9.79796i 0 0 0
793.2 0 0 0 0 0 9.79796 9.79796i 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 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.f 4
3.b odd 2 1 CM 900.3.l.f 4
5.b even 2 1 inner 900.3.l.f 4
5.c odd 4 2 inner 900.3.l.f 4
15.d odd 2 1 inner 900.3.l.f 4
15.e even 4 2 inner 900.3.l.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.l.f 4 1.a even 1 1 trivial
900.3.l.f 4 3.b odd 2 1 CM
900.3.l.f 4 5.b even 2 1 inner
900.3.l.f 4 5.c odd 4 2 inner
900.3.l.f 4 15.d odd 2 1 inner
900.3.l.f 4 15.e even 4 2 inner

## Hecke kernels

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

 $$T_{7}^{4} + 36864$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$36864 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$36864 + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( 676 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -46 + T )^{4}$$
$37$ $$23040000 + T^{4}$$
$41$ $$T^{4}$$
$43$ $$47775744 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -74 + T )^{4}$$
$67$ $$9437184 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$368640000 + T^{4}$$
$79$ $$( 20164 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$1416167424 + T^{4}$$