# Properties

 Label 900.3.c.g Level $900$ Weight $3$ Character orbit 900.c Analytic conductor $24.523$ Analytic rank $0$ Dimension $2$ CM discriminant -15 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-15})$$ Defining polynomial: $$x^{2} - x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-15})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 + 3 \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 + 3 \beta ) q^{8} + ( 12 - 7 \beta ) q^{16} -14 q^{17} + ( 8 - 16 \beta ) q^{19} + ( -8 + 16 \beta ) q^{23} + ( -16 + 32 \beta ) q^{31} + ( -28 - 5 \beta ) q^{32} + 14 \beta q^{34} + ( -64 + 8 \beta ) q^{38} + ( 64 - 8 \beta ) q^{46} + ( -24 + 48 \beta ) q^{47} + 49 q^{49} -86 q^{53} + 118 q^{61} + ( 128 - 16 \beta ) q^{62} + ( -20 + 33 \beta ) q^{64} + ( 56 - 14 \beta ) q^{68} + ( 32 + 56 \beta ) q^{76} + ( -32 + 64 \beta ) q^{79} + ( -16 + 32 \beta ) q^{83} + ( -32 - 56 \beta ) q^{92} + ( 192 - 24 \beta ) q^{94} -49 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 7q^{4} + 11q^{8} + O(q^{10})$$ $$2q - q^{2} - 7q^{4} + 11q^{8} + 17q^{16} - 28q^{17} - 61q^{32} + 14q^{34} - 120q^{38} + 120q^{46} + 98q^{49} - 172q^{53} + 236q^{61} + 240q^{62} - 7q^{64} + 98q^{68} + 120q^{76} - 120q^{92} + 360q^{94} - 49q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 1.93649i 0.5 − 1.93649i
−0.500000 1.93649i 0 −3.50000 + 1.93649i 0 0 0 5.50000 + 5.80948i 0 0
451.2 −0.500000 + 1.93649i 0 −3.50000 1.93649i 0 0 0 5.50000 5.80948i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
4.b odd 2 1 inner
60.h even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.f 4 5.c odd 4 2
180.3.f.f 4 15.e even 4 2
180.3.f.f 4 20.e even 4 2
180.3.f.f 4 60.l odd 4 2
900.3.c.g 2 1.a even 1 1 trivial
900.3.c.g 2 4.b odd 2 1 inner
900.3.c.g 2 15.d odd 2 1 CM
900.3.c.g 2 60.h even 2 1 inner
900.3.c.i 2 3.b odd 2 1
900.3.c.i 2 5.b even 2 1
900.3.c.i 2 12.b even 2 1
900.3.c.i 2 20.d odd 2 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{13}$$ $$T_{17} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 14 + T )^{2}$$
$19$ $$960 + T^{2}$$
$23$ $$960 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$3840 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$8640 + T^{2}$$
$53$ $$( 86 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$( -118 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$15360 + T^{2}$$
$83$ $$3840 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$