# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 4) q^{7}+O(q^{10})$$ q + (-b2 + 4) * q^7 $$q + ( - \beta_{2} + 4) q^{7} + (\beta_{3} - \beta_1) q^{11} + ( - \beta_{2} - 2) q^{13} - 4 \beta_1 q^{17} + (2 \beta_{2} + 8) q^{19} + ( - \beta_{3} + 2 \beta_1) q^{23} + (\beta_{3} - 8 \beta_1) q^{29} + (2 \beta_{2} + 2) q^{31} + (3 \beta_{2} + 34) q^{37} + ( - 4 \beta_{3} + 3 \beta_1) q^{41} + (2 \beta_{2} - 20) q^{43} + ( - 2 \beta_{3} - 14 \beta_1) q^{47} + ( - 8 \beta_{2} + 57) q^{49} + (4 \beta_{3} + 10 \beta_1) q^{53} + ( - 3 \beta_{3} - 17 \beta_1) q^{59} + ( - 6 \beta_{2} - 10) q^{61} + 76 q^{67} + (2 \beta_{3} - 12 \beta_1) q^{71} + (6 \beta_{2} - 38) q^{73} + (7 \beta_{3} - 34 \beta_1) q^{77} + (6 \beta_{2} + 50) q^{79} + ( - 4 \beta_{3} + 2 \beta_1) q^{83} + ( - 2 \beta_{3} - 21 \beta_1) q^{89} + ( - 2 \beta_{2} + 82) q^{91} + ( - 2 \beta_{2} + 106) q^{97}+O(q^{100})$$ q + (-b2 + 4) * q^7 + (b3 - b1) * q^11 + (-b2 - 2) * q^13 - 4*b1 * q^17 + (2*b2 + 8) * q^19 + (-b3 + 2*b1) * q^23 + (b3 - 8*b1) * q^29 + (2*b2 + 2) * q^31 + (3*b2 + 34) * q^37 + (-4*b3 + 3*b1) * q^41 + (2*b2 - 20) * q^43 + (-2*b3 - 14*b1) * q^47 + (-8*b2 + 57) * q^49 + (4*b3 + 10*b1) * q^53 + (-3*b3 - 17*b1) * q^59 + (-6*b2 - 10) * q^61 + 76 * q^67 + (2*b3 - 12*b1) * q^71 + (6*b2 - 38) * q^73 + (7*b3 - 34*b1) * q^77 + (6*b2 + 50) * q^79 + (-4*b3 + 2*b1) * q^83 + (-2*b3 - 21*b1) * q^89 + (-2*b2 + 82) * q^91 + (-2*b2 + 106) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{7}+O(q^{10})$$ 4 * q + 16 * q^7 $$4 q + 16 q^{7} - 8 q^{13} + 32 q^{19} + 8 q^{31} + 136 q^{37} - 80 q^{43} + 228 q^{49} - 40 q^{61} + 304 q^{67} - 152 q^{73} + 200 q^{79} + 328 q^{91} + 424 q^{97}+O(q^{100})$$ 4 * q + 16 * q^7 - 8 * q^13 + 32 * q^19 + 8 * q^31 + 136 * q^37 - 80 * q^43 + 228 * q^49 - 40 * q^61 + 304 * q^67 - 152 * q^73 + 200 * q^79 + 328 * q^91 + 424 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ v^3 - v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 7\nu$$ -v^3 + 7*v $$\beta_{3}$$ $$=$$ $$6\nu^{2} - 12$$ 6*v^2 - 12
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 6$$ (b2 + b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 12 ) / 6$$ (b3 + 12) / 6 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 7\beta_1 ) / 6$$ (b2 + 7*b1) / 6

## Character values

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

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

## 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
 1.58114 − 0.707107i 1.58114 + 0.707107i −1.58114 + 0.707107i −1.58114 − 0.707107i
0 0 0 0 0 −5.48683 0 0 0
701.2 0 0 0 0 0 −5.48683 0 0 0
701.3 0 0 0 0 0 13.4868 0 0 0
701.4 0 0 0 0 0 13.4868 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.g.d 4
3.b odd 2 1 inner 900.3.g.d 4
4.b odd 2 1 3600.3.l.n 4
5.b even 2 1 180.3.g.a 4
5.c odd 4 2 900.3.b.b 8
12.b even 2 1 3600.3.l.n 4
15.d odd 2 1 180.3.g.a 4
15.e even 4 2 900.3.b.b 8
20.d odd 2 1 720.3.l.c 4
20.e even 4 2 3600.3.c.k 8
40.e odd 2 1 2880.3.l.f 4
40.f even 2 1 2880.3.l.b 4
45.h odd 6 2 1620.3.o.f 8
45.j even 6 2 1620.3.o.f 8
60.h even 2 1 720.3.l.c 4
60.l odd 4 2 3600.3.c.k 8
120.i odd 2 1 2880.3.l.b 4
120.m even 2 1 2880.3.l.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 5.b even 2 1
180.3.g.a 4 15.d odd 2 1
720.3.l.c 4 20.d odd 2 1
720.3.l.c 4 60.h even 2 1
900.3.b.b 8 5.c odd 4 2
900.3.b.b 8 15.e even 4 2
900.3.g.d 4 1.a even 1 1 trivial
900.3.g.d 4 3.b odd 2 1 inner
1620.3.o.f 8 45.h odd 6 2
1620.3.o.f 8 45.j even 6 2
2880.3.l.b 4 40.f even 2 1
2880.3.l.b 4 120.i odd 2 1
2880.3.l.f 4 40.e odd 2 1
2880.3.l.f 4 120.m even 2 1
3600.3.c.k 8 20.e even 4 2
3600.3.c.k 8 60.l odd 4 2
3600.3.l.n 4 4.b odd 2 1
3600.3.l.n 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 8T_{7} - 74$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 8 T - 74)^{2}$$
$11$ $$T^{4} + 396 T^{2} + 26244$$
$13$ $$(T^{2} + 4 T - 86)^{2}$$
$17$ $$(T^{2} + 288)^{2}$$
$19$ $$(T^{2} - 16 T - 296)^{2}$$
$23$ $$T^{4} + 504 T^{2} + 11664$$
$29$ $$T^{4} + 2664 T^{2} + 944784$$
$31$ $$(T^{2} - 4 T - 356)^{2}$$
$37$ $$(T^{2} - 68 T + 346)^{2}$$
$41$ $$T^{4} + 6084 T^{2} + \cdots + 7387524$$
$43$ $$(T^{2} + 40 T + 40)^{2}$$
$47$ $$T^{4} + 8496 T^{2} + \cdots + 7884864$$
$53$ $$T^{4} + 9360 T^{2} + \cdots + 1166400$$
$59$ $$T^{4} + 13644 T^{2} + \cdots + 12830724$$
$61$ $$(T^{2} + 20 T - 3140)^{2}$$
$67$ $$(T - 76)^{4}$$
$71$ $$T^{4} + 6624 T^{2} + \cdots + 3504384$$
$73$ $$(T^{2} + 76 T - 1796)^{2}$$
$79$ $$(T^{2} - 100 T - 740)^{2}$$
$83$ $$T^{4} + 5904 T^{2} + \cdots + 7884864$$
$89$ $$T^{4} + 17316 T^{2} + \cdots + 52099524$$
$97$ $$(T^{2} - 212 T + 10876)^{2}$$