# Properties

 Label 900.3.l.a Level $900$ Weight $3$ Character orbit 900.l Analytic conductor $24.523$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 7 + 7 i ) q^{7} +O(q^{10})$$ $$q + ( 7 + 7 i ) q^{7} -10 q^{11} + ( -9 + 9 i ) q^{13} + ( 1 + i ) q^{17} -8 i q^{19} + ( -23 + 23 i ) q^{23} + 8 i q^{29} -14 q^{31} + ( -33 - 33 i ) q^{37} + 14 q^{41} + ( 15 - 15 i ) q^{43} + ( -39 - 39 i ) q^{47} + 49 i q^{49} + ( -7 + 7 i ) q^{53} + 56 i q^{59} + 42 q^{61} + ( 7 + 7 i ) q^{67} -98 q^{71} + ( -49 + 49 i ) q^{73} + ( -70 - 70 i ) q^{77} + 96 i q^{79} + ( -63 + 63 i ) q^{83} + 112 i q^{89} -126 q^{91} + ( -33 - 33 i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{7} + O(q^{10})$$ $$2q + 14q^{7} - 20q^{11} - 18q^{13} + 2q^{17} - 46q^{23} - 28q^{31} - 66q^{37} + 28q^{41} + 30q^{43} - 78q^{47} - 14q^{53} + 84q^{61} + 14q^{67} - 196q^{71} - 98q^{73} - 140q^{77} - 126q^{83} - 252q^{91} - 66q^{97} + O(q^{100})$$

## Character values

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

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

## 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
 1.00000i − 1.00000i
0 0 0 0 0 7.00000 + 7.00000i 0 0 0
793.1 0 0 0 0 0 7.00000 7.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.a 2
3.b odd 2 1 100.3.f.a 2
5.b even 2 1 180.3.l.a 2
5.c odd 4 1 180.3.l.a 2
5.c odd 4 1 inner 900.3.l.a 2
12.b even 2 1 400.3.p.d 2
15.d odd 2 1 20.3.f.a 2
15.e even 4 1 20.3.f.a 2
15.e even 4 1 100.3.f.a 2
20.d odd 2 1 720.3.bh.e 2
20.e even 4 1 720.3.bh.e 2
60.h even 2 1 80.3.p.a 2
60.l odd 4 1 80.3.p.a 2
60.l odd 4 1 400.3.p.d 2
105.g even 2 1 980.3.l.a 2
105.k odd 4 1 980.3.l.a 2
120.i odd 2 1 320.3.p.c 2
120.m even 2 1 320.3.p.g 2
120.q odd 4 1 320.3.p.g 2
120.w even 4 1 320.3.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 15.d odd 2 1
20.3.f.a 2 15.e even 4 1
80.3.p.a 2 60.h even 2 1
80.3.p.a 2 60.l odd 4 1
100.3.f.a 2 3.b odd 2 1
100.3.f.a 2 15.e even 4 1
180.3.l.a 2 5.b even 2 1
180.3.l.a 2 5.c odd 4 1
320.3.p.c 2 120.i odd 2 1
320.3.p.c 2 120.w even 4 1
320.3.p.g 2 120.m even 2 1
320.3.p.g 2 120.q odd 4 1
400.3.p.d 2 12.b even 2 1
400.3.p.d 2 60.l odd 4 1
720.3.bh.e 2 20.d odd 2 1
720.3.bh.e 2 20.e even 4 1
900.3.l.a 2 1.a even 1 1 trivial
900.3.l.a 2 5.c odd 4 1 inner
980.3.l.a 2 105.g even 2 1
980.3.l.a 2 105.k odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} - 14 T_{7} + 98$$ $$T_{11} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$98 - 14 T + T^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$162 + 18 T + T^{2}$$
$17$ $$2 - 2 T + T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$1058 + 46 T + T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( 14 + T )^{2}$$
$37$ $$2178 + 66 T + T^{2}$$
$41$ $$( -14 + T )^{2}$$
$43$ $$450 - 30 T + T^{2}$$
$47$ $$3042 + 78 T + T^{2}$$
$53$ $$98 + 14 T + T^{2}$$
$59$ $$3136 + T^{2}$$
$61$ $$( -42 + T )^{2}$$
$67$ $$98 - 14 T + T^{2}$$
$71$ $$( 98 + T )^{2}$$
$73$ $$4802 + 98 T + T^{2}$$
$79$ $$9216 + T^{2}$$
$83$ $$7938 + 126 T + T^{2}$$
$89$ $$12544 + T^{2}$$
$97$ $$2178 + 66 T + T^{2}$$