# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 6 - 12 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 6 - 12 \zeta_{6} ) q^{7} + 8 q^{8} + ( -6 + 12 \zeta_{6} ) q^{11} + 18 q^{13} + ( -24 + 12 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} -10 q^{17} + ( 8 - 16 \zeta_{6} ) q^{19} + ( 24 - 12 \zeta_{6} ) q^{22} + ( -4 + 8 \zeta_{6} ) q^{23} -36 \zeta_{6} q^{26} + ( 24 + 24 \zeta_{6} ) q^{28} + 36 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} + 20 \zeta_{6} q^{34} + 54 q^{37} + ( -32 + 16 \zeta_{6} ) q^{38} -18 q^{41} + ( 12 - 24 \zeta_{6} ) q^{43} + ( -24 - 24 \zeta_{6} ) q^{44} + ( 16 - 8 \zeta_{6} ) q^{46} -59 q^{49} + ( -72 + 72 \zeta_{6} ) q^{52} + 26 q^{53} + ( 48 - 96 \zeta_{6} ) q^{56} -72 \zeta_{6} q^{58} + ( 18 - 36 \zeta_{6} ) q^{59} -74 q^{61} + ( 16 - 8 \zeta_{6} ) q^{62} + 64 q^{64} + ( 24 - 48 \zeta_{6} ) q^{67} + ( 40 - 40 \zeta_{6} ) q^{68} + ( 60 - 120 \zeta_{6} ) q^{71} + 36 q^{73} -108 \zeta_{6} q^{74} + ( 32 + 32 \zeta_{6} ) q^{76} + 108 q^{77} + ( 52 - 104 \zeta_{6} ) q^{79} + 36 \zeta_{6} q^{82} + ( 52 - 104 \zeta_{6} ) q^{83} + ( -48 + 24 \zeta_{6} ) q^{86} + ( -48 + 96 \zeta_{6} ) q^{88} + 18 q^{89} + ( 108 - 216 \zeta_{6} ) q^{91} + ( -16 - 16 \zeta_{6} ) q^{92} -72 q^{97} + 118 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} + 16q^{8} + 36q^{13} - 36q^{14} - 16q^{16} - 20q^{17} + 36q^{22} - 36q^{26} + 72q^{28} + 72q^{29} - 32q^{32} + 20q^{34} + 108q^{37} - 48q^{38} - 36q^{41} - 72q^{44} + 24q^{46} - 118q^{49} - 72q^{52} + 52q^{53} - 72q^{58} - 148q^{61} + 24q^{62} + 128q^{64} + 40q^{68} + 72q^{73} - 108q^{74} + 96q^{76} + 216q^{77} + 36q^{82} - 72q^{86} + 36q^{89} - 48q^{92} - 144q^{97} + 118q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 0 0 10.3923i 8.00000 0 0
451.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0 0 10.3923i 8.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.f 2
3.b odd 2 1 300.3.c.c 2
4.b odd 2 1 inner 900.3.c.f 2
5.b even 2 1 900.3.c.j 2
5.c odd 4 2 180.3.f.e 4
12.b even 2 1 300.3.c.c 2
15.d odd 2 1 300.3.c.a 2
15.e even 4 2 60.3.f.a 4
20.d odd 2 1 900.3.c.j 2
20.e even 4 2 180.3.f.e 4
60.h even 2 1 300.3.c.a 2
60.l odd 4 2 60.3.f.a 4
120.q odd 4 2 960.3.j.b 4
120.w even 4 2 960.3.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 15.e even 4 2
60.3.f.a 4 60.l odd 4 2
180.3.f.e 4 5.c odd 4 2
180.3.f.e 4 20.e even 4 2
300.3.c.a 2 15.d odd 2 1
300.3.c.a 2 60.h even 2 1
300.3.c.c 2 3.b odd 2 1
300.3.c.c 2 12.b even 2 1
900.3.c.f 2 1.a even 1 1 trivial
900.3.c.f 2 4.b odd 2 1 inner
900.3.c.j 2 5.b even 2 1
900.3.c.j 2 20.d odd 2 1
960.3.j.b 4 120.q odd 4 2
960.3.j.b 4 120.w even 4 2

## Hecke kernels

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

 $$T_{7}^{2} + 108$$ $$T_{13} - 18$$ $$T_{17} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$108 + T^{2}$$
$11$ $$108 + T^{2}$$
$13$ $$( -18 + T )^{2}$$
$17$ $$( 10 + T )^{2}$$
$19$ $$192 + T^{2}$$
$23$ $$48 + T^{2}$$
$29$ $$( -36 + T )^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$( -54 + T )^{2}$$
$41$ $$( 18 + T )^{2}$$
$43$ $$432 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -26 + T )^{2}$$
$59$ $$972 + T^{2}$$
$61$ $$( 74 + T )^{2}$$
$67$ $$1728 + T^{2}$$
$71$ $$10800 + T^{2}$$
$73$ $$( -36 + T )^{2}$$
$79$ $$8112 + T^{2}$$
$83$ $$8112 + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$( 72 + T )^{2}$$