# Properties

 Label 900.4.d.d Level $900$ Weight $4$ Character orbit 900.d Analytic conductor $53.102$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$53.1017190052$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{7}+O(q^{10})$$ q + b * q^7 $$q + \beta q^{7} - 30 q^{11} + 2 \beta q^{13} - 45 \beta q^{17} + 28 q^{19} + 60 \beta q^{23} + 210 q^{29} - 4 q^{31} + 100 \beta q^{37} - 240 q^{41} + 68 \beta q^{43} + 60 \beta q^{47} + 339 q^{49} - 15 \beta q^{53} - 450 q^{59} - 166 q^{61} + 454 \beta q^{67} + 1020 q^{71} + 125 \beta q^{73} - 30 \beta q^{77} + 916 q^{79} - 570 \beta q^{83} - 420 q^{89} - 8 q^{91} + 769 \beta q^{97} +O(q^{100})$$ q + b * q^7 - 30 * q^11 + 2*b * q^13 - 45*b * q^17 + 28 * q^19 + 60*b * q^23 + 210 * q^29 - 4 * q^31 + 100*b * q^37 - 240 * q^41 + 68*b * q^43 + 60*b * q^47 + 339 * q^49 - 15*b * q^53 - 450 * q^59 - 166 * q^61 + 454*b * q^67 + 1020 * q^71 + 125*b * q^73 - 30*b * q^77 + 916 * q^79 - 570*b * q^83 - 420 * q^89 - 8 * q^91 + 769*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 60 q^{11} + 56 q^{19} + 420 q^{29} - 8 q^{31} - 480 q^{41} + 678 q^{49} - 900 q^{59} - 332 q^{61} + 2040 q^{71} + 1832 q^{79} - 840 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q - 60 * q^11 + 56 * q^19 + 420 * q^29 - 8 * q^31 - 480 * q^41 + 678 * q^49 - 900 * q^59 - 332 * q^61 + 2040 * q^71 + 1832 * q^79 - 840 * q^89 - 16 * q^91

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.4.d.d 2
3.b odd 2 1 900.4.d.i 2
5.b even 2 1 inner 900.4.d.d 2
5.c odd 4 1 180.4.a.b 1
5.c odd 4 1 900.4.a.i 1
15.d odd 2 1 900.4.d.i 2
15.e even 4 1 180.4.a.e yes 1
15.e even 4 1 900.4.a.j 1
20.e even 4 1 720.4.a.h 1
45.k odd 12 2 1620.4.i.i 2
45.l even 12 2 1620.4.i.c 2
60.l odd 4 1 720.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 5.c odd 4 1
180.4.a.e yes 1 15.e even 4 1
720.4.a.h 1 20.e even 4 1
720.4.a.w 1 60.l odd 4 1
900.4.a.i 1 5.c odd 4 1
900.4.a.j 1 15.e even 4 1
900.4.d.d 2 1.a even 1 1 trivial
900.4.d.d 2 5.b even 2 1 inner
900.4.d.i 2 3.b odd 2 1
900.4.d.i 2 15.d odd 2 1
1620.4.i.c 2 45.l even 12 2
1620.4.i.i 2 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 30$$ T11 + 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 30)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 8100$$
$19$ $$(T - 28)^{2}$$
$23$ $$T^{2} + 14400$$
$29$ $$(T - 210)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 40000$$
$41$ $$(T + 240)^{2}$$
$43$ $$T^{2} + 18496$$
$47$ $$T^{2} + 14400$$
$53$ $$T^{2} + 900$$
$59$ $$(T + 450)^{2}$$
$61$ $$(T + 166)^{2}$$
$67$ $$T^{2} + 824464$$
$71$ $$(T - 1020)^{2}$$
$73$ $$T^{2} + 62500$$
$79$ $$(T - 916)^{2}$$
$83$ $$T^{2} + 1299600$$
$89$ $$(T + 420)^{2}$$
$97$ $$T^{2} + 2365444$$