# Properties

 Label 900.3.f.b Level $900$ Weight $3$ Character orbit 900.f Analytic conductor $24.523$ Analytic rank $0$ Dimension $2$ CM discriminant -4 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.f (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -4 q^{4} -8 i q^{8} +O(q^{10})$$ $$q + 2 i q^{2} -4 q^{4} -8 i q^{8} -10 i q^{13} + 16 q^{16} + 16 i q^{17} + 20 q^{26} + 40 q^{29} + 32 i q^{32} -32 q^{34} + 70 i q^{37} + 80 q^{41} -49 q^{49} + 40 i q^{52} + 56 i q^{53} + 80 i q^{58} -22 q^{61} -64 q^{64} -64 i q^{68} + 110 i q^{73} -140 q^{74} + 160 i q^{82} + 160 q^{89} + 130 i q^{97} -98 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + O(q^{10})$$ $$2q - 8q^{4} + 32q^{16} + 40q^{26} + 80q^{29} - 64q^{34} + 160q^{41} - 98q^{49} - 44q^{61} - 128q^{64} - 280q^{74} + 320q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
2.00000i 0 −4.00000 0 0 0 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 0 8.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.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.f.b 2
3.b odd 2 1 900.3.f.a 2
4.b odd 2 1 CM 900.3.f.b 2
5.b even 2 1 inner 900.3.f.b 2
5.c odd 4 1 36.3.d.a 1
5.c odd 4 1 900.3.c.c 1
12.b even 2 1 900.3.f.a 2
15.d odd 2 1 900.3.f.a 2
15.e even 4 1 36.3.d.b yes 1
15.e even 4 1 900.3.c.b 1
20.d odd 2 1 inner 900.3.f.b 2
20.e even 4 1 36.3.d.a 1
20.e even 4 1 900.3.c.c 1
40.i odd 4 1 576.3.g.a 1
40.k even 4 1 576.3.g.a 1
45.k odd 12 2 324.3.f.f 2
45.l even 12 2 324.3.f.e 2
60.h even 2 1 900.3.f.a 2
60.l odd 4 1 36.3.d.b yes 1
60.l odd 4 1 900.3.c.b 1
80.i odd 4 1 2304.3.b.d 2
80.j even 4 1 2304.3.b.d 2
80.s even 4 1 2304.3.b.d 2
80.t odd 4 1 2304.3.b.d 2
120.q odd 4 1 576.3.g.c 1
120.w even 4 1 576.3.g.c 1
180.v odd 12 2 324.3.f.e 2
180.x even 12 2 324.3.f.f 2
240.z odd 4 1 2304.3.b.e 2
240.bb even 4 1 2304.3.b.e 2
240.bd odd 4 1 2304.3.b.e 2
240.bf even 4 1 2304.3.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 5.c odd 4 1
36.3.d.a 1 20.e even 4 1
36.3.d.b yes 1 15.e even 4 1
36.3.d.b yes 1 60.l odd 4 1
324.3.f.e 2 45.l even 12 2
324.3.f.e 2 180.v odd 12 2
324.3.f.f 2 45.k odd 12 2
324.3.f.f 2 180.x even 12 2
576.3.g.a 1 40.i odd 4 1
576.3.g.a 1 40.k even 4 1
576.3.g.c 1 120.q odd 4 1
576.3.g.c 1 120.w even 4 1
900.3.c.b 1 15.e even 4 1
900.3.c.b 1 60.l odd 4 1
900.3.c.c 1 5.c odd 4 1
900.3.c.c 1 20.e even 4 1
900.3.f.a 2 3.b odd 2 1
900.3.f.a 2 12.b even 2 1
900.3.f.a 2 15.d odd 2 1
900.3.f.a 2 60.h even 2 1
900.3.f.b 2 1.a even 1 1 trivial
900.3.f.b 2 4.b odd 2 1 CM
900.3.f.b 2 5.b even 2 1 inner
900.3.f.b 2 20.d odd 2 1 inner
2304.3.b.d 2 80.i odd 4 1
2304.3.b.d 2 80.j even 4 1
2304.3.b.d 2 80.s even 4 1
2304.3.b.d 2 80.t odd 4 1
2304.3.b.e 2 240.z odd 4 1
2304.3.b.e 2 240.bb even 4 1
2304.3.b.e 2 240.bd odd 4 1
2304.3.b.e 2 240.bf even 4 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{29} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$256 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -40 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4900 + T^{2}$$
$41$ $$( -80 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$3136 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 22 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$12100 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -160 + T )^{2}$$
$97$ $$16900 + T^{2}$$