# Properties

 Label 900.2.d.a Level $900$ Weight $2$ Character orbit 900.d Analytic conductor $7.187$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) 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.

 $$f(q)$$ $$=$$ $$q + i q^{7} +O(q^{10})$$ $$q + i q^{7} -6 q^{11} + 5 i q^{13} -6 i q^{17} -5 q^{19} + 6 i q^{23} -6 q^{29} - q^{31} -2 i q^{37} -i q^{43} + 6 i q^{47} + 6 q^{49} + 12 i q^{53} -6 q^{59} -13 q^{61} -11 i q^{67} + 2 i q^{73} -6 i q^{77} -8 q^{79} + 6 i q^{83} -5 q^{91} + 7 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 12q^{11} - 10q^{19} - 12q^{29} - 2q^{31} + 12q^{49} - 12q^{59} - 26q^{61} - 16q^{79} - 10q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 1.00000i 0 0 0
649.2 0 0 0 0 0 1.00000i 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 900.2.d.a 2
3.b odd 2 1 300.2.d.a 2
4.b odd 2 1 3600.2.f.v 2
5.b even 2 1 inner 900.2.d.a 2
5.c odd 4 1 900.2.a.c 1
5.c odd 4 1 900.2.a.e 1
12.b even 2 1 1200.2.f.a 2
15.d odd 2 1 300.2.d.a 2
15.e even 4 1 300.2.a.b 1
15.e even 4 1 300.2.a.c yes 1
20.d odd 2 1 3600.2.f.v 2
20.e even 4 1 3600.2.a.s 1
20.e even 4 1 3600.2.a.z 1
24.f even 2 1 4800.2.f.bi 2
24.h odd 2 1 4800.2.f.b 2
60.h even 2 1 1200.2.f.a 2
60.l odd 4 1 1200.2.a.f 1
60.l odd 4 1 1200.2.a.n 1
120.i odd 2 1 4800.2.f.b 2
120.m even 2 1 4800.2.f.bi 2
120.q odd 4 1 4800.2.a.p 1
120.q odd 4 1 4800.2.a.cf 1
120.w even 4 1 4800.2.a.o 1
120.w even 4 1 4800.2.a.ce 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 15.e even 4 1
300.2.a.c yes 1 15.e even 4 1
300.2.d.a 2 3.b odd 2 1
300.2.d.a 2 15.d odd 2 1
900.2.a.c 1 5.c odd 4 1
900.2.a.e 1 5.c odd 4 1
900.2.d.a 2 1.a even 1 1 trivial
900.2.d.a 2 5.b even 2 1 inner
1200.2.a.f 1 60.l odd 4 1
1200.2.a.n 1 60.l odd 4 1
1200.2.f.a 2 12.b even 2 1
1200.2.f.a 2 60.h even 2 1
3600.2.a.s 1 20.e even 4 1
3600.2.a.z 1 20.e even 4 1
3600.2.f.v 2 4.b odd 2 1
3600.2.f.v 2 20.d odd 2 1
4800.2.a.o 1 120.w even 4 1
4800.2.a.p 1 120.q odd 4 1
4800.2.a.ce 1 120.w even 4 1
4800.2.a.cf 1 120.q odd 4 1
4800.2.f.b 2 24.h odd 2 1
4800.2.f.b 2 120.i odd 2 1
4800.2.f.bi 2 24.f even 2 1
4800.2.f.bi 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$121 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$49 + T^{2}$$