# Properties

 Label 900.3.g.c Level $900$ Weight $3$ Character orbit 900.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-23})$$ Defining polynomial: $$x^{4} - 2 x^{3} + 17 x^{2} - 16 x + 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{7} -7 \beta_{1} q^{11} + 3 \beta_{2} q^{13} -2 \beta_{3} q^{17} -12 q^{19} + \beta_{3} q^{23} -6 \beta_{1} q^{29} -38 q^{31} -\beta_{2} q^{37} -49 \beta_{1} q^{41} -10 \beta_{2} q^{43} + 8 \beta_{3} q^{47} -3 q^{49} -59 \beta_{1} q^{59} -70 q^{61} + 16 \beta_{2} q^{67} -84 \beta_{1} q^{71} + 2 \beta_{2} q^{73} -7 \beta_{3} q^{77} -30 q^{79} -14 \beta_{3} q^{83} + 23 \beta_{1} q^{89} -138 q^{91} -14 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 48q^{19} - 152q^{31} - 12q^{49} - 280q^{61} - 120q^{79} - 552q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 17 x^{2} - 16 x + 18$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} - 25 \nu + 12$$$$)/15$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu + 8$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{3} - 12 \nu^{2} + 160 \nu - 78$$$$)/15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} + 4 \beta_{1} - 30$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-11 \beta_{3} + 6 \beta_{2} - 74 \beta_{1} - 46$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 0.5 − 0.983702i 0.5 + 0.983702i 0.5 + 3.81213i 0.5 − 3.81213i
0 0 0 0 0 −6.78233 0 0 0
701.2 0 0 0 0 0 −6.78233 0 0 0
701.3 0 0 0 0 0 6.78233 0 0 0
701.4 0 0 0 0 0 6.78233 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.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.g.c 4
3.b odd 2 1 inner 900.3.g.c 4
4.b odd 2 1 3600.3.l.r 4
5.b even 2 1 inner 900.3.g.c 4
5.c odd 4 2 180.3.b.a 4
12.b even 2 1 3600.3.l.r 4
15.d odd 2 1 inner 900.3.g.c 4
15.e even 4 2 180.3.b.a 4
20.d odd 2 1 3600.3.l.r 4
20.e even 4 2 720.3.c.b 4
40.i odd 4 2 2880.3.c.c 4
40.k even 4 2 2880.3.c.f 4
45.k odd 12 4 1620.3.t.c 8
45.l even 12 4 1620.3.t.c 8
60.h even 2 1 3600.3.l.r 4
60.l odd 4 2 720.3.c.b 4
120.q odd 4 2 2880.3.c.f 4
120.w even 4 2 2880.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 5.c odd 4 2
180.3.b.a 4 15.e even 4 2
720.3.c.b 4 20.e even 4 2
720.3.c.b 4 60.l odd 4 2
900.3.g.c 4 1.a even 1 1 trivial
900.3.g.c 4 3.b odd 2 1 inner
900.3.g.c 4 5.b even 2 1 inner
900.3.g.c 4 15.d odd 2 1 inner
1620.3.t.c 8 45.k odd 12 4
1620.3.t.c 8 45.l even 12 4
2880.3.c.c 4 40.i odd 4 2
2880.3.c.c 4 120.w even 4 2
2880.3.c.f 4 40.k even 4 2
2880.3.c.f 4 120.q odd 4 2
3600.3.l.r 4 4.b odd 2 1
3600.3.l.r 4 12.b even 2 1
3600.3.l.r 4 20.d odd 2 1
3600.3.l.r 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 46$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -46 + T^{2} )^{2}$$
$11$ $$( 98 + T^{2} )^{2}$$
$13$ $$( -414 + T^{2} )^{2}$$
$17$ $$( 368 + T^{2} )^{2}$$
$19$ $$( 12 + T )^{4}$$
$23$ $$( 92 + T^{2} )^{2}$$
$29$ $$( 72 + T^{2} )^{2}$$
$31$ $$( 38 + T )^{4}$$
$37$ $$( -46 + T^{2} )^{2}$$
$41$ $$( 4802 + T^{2} )^{2}$$
$43$ $$( -4600 + T^{2} )^{2}$$
$47$ $$( 5888 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 6962 + T^{2} )^{2}$$
$61$ $$( 70 + T )^{4}$$
$67$ $$( -11776 + T^{2} )^{2}$$
$71$ $$( 14112 + T^{2} )^{2}$$
$73$ $$( -184 + T^{2} )^{2}$$
$79$ $$( 30 + T )^{4}$$
$83$ $$( 18032 + T^{2} )^{2}$$
$89$ $$( 1058 + T^{2} )^{2}$$
$97$ $$( -9016 + T^{2} )^{2}$$