# Properties

 Label 900.2.s.b Level $900$ Weight $2$ Character orbit 900.s Analytic conductor $7.187$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12} q^{7} + 3 q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{13} + 6 \zeta_{12}^{3} q^{17} + 4 q^{19} + ( 1 + \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 3 - 3 \zeta_{12}^{2} ) q^{29} -5 \zeta_{12}^{2} q^{31} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{33} + 2 \zeta_{12}^{3} q^{37} + ( 2 - \zeta_{12}^{2} ) q^{39} -3 \zeta_{12}^{2} q^{41} -\zeta_{12} q^{43} + 9 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} + ( -6 + 12 \zeta_{12}^{2} ) q^{51} + 6 \zeta_{12}^{3} q^{53} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{57} -3 \zeta_{12}^{2} q^{59} + ( 13 - 13 \zeta_{12}^{2} ) q^{61} + 3 \zeta_{12} q^{63} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} + ( 6 - 3 \zeta_{12}^{2} ) q^{69} -12 q^{71} + 10 \zeta_{12}^{3} q^{73} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + ( 11 - 11 \zeta_{12}^{2} ) q^{79} + 9 q^{81} -9 \zeta_{12} q^{83} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{87} -6 q^{89} + q^{91} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{93} -11 \zeta_{12} q^{97} + ( -9 + 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 6q^{11} + 16q^{19} + 6q^{21} + 6q^{29} - 10q^{31} + 6q^{39} - 6q^{41} - 12q^{49} - 6q^{59} + 26q^{61} + 18q^{69} - 48q^{71} + 22q^{79} + 36q^{81} - 24q^{89} + 4q^{91} - 18q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −1.73205 0 0 0 −0.866025 + 0.500000i 0 3.00000 0
49.2 0 1.73205 0 0 0 0.866025 0.500000i 0 3.00000 0
349.1 0 −1.73205 0 0 0 −0.866025 0.500000i 0 3.00000 0
349.2 0 1.73205 0 0 0 0.866025 + 0.500000i 0 3.00000 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.s.b 4
3.b odd 2 1 2700.2.s.b 4
5.b even 2 1 inner 900.2.s.b 4
5.c odd 4 1 36.2.e.a 2
5.c odd 4 1 900.2.i.b 2
9.c even 3 1 inner 900.2.s.b 4
9.c even 3 1 8100.2.d.h 2
9.d odd 6 1 2700.2.s.b 4
9.d odd 6 1 8100.2.d.c 2
15.d odd 2 1 2700.2.s.b 4
15.e even 4 1 108.2.e.a 2
15.e even 4 1 2700.2.i.b 2
20.e even 4 1 144.2.i.a 2
35.f even 4 1 1764.2.j.b 2
35.k even 12 1 1764.2.i.c 2
35.k even 12 1 1764.2.l.a 2
35.l odd 12 1 1764.2.i.a 2
35.l odd 12 1 1764.2.l.c 2
40.i odd 4 1 576.2.i.f 2
40.k even 4 1 576.2.i.e 2
45.h odd 6 1 2700.2.s.b 4
45.h odd 6 1 8100.2.d.c 2
45.j even 6 1 inner 900.2.s.b 4
45.j even 6 1 8100.2.d.h 2
45.k odd 12 1 36.2.e.a 2
45.k odd 12 1 324.2.a.c 1
45.k odd 12 1 900.2.i.b 2
45.k odd 12 1 8100.2.a.j 1
45.l even 12 1 108.2.e.a 2
45.l even 12 1 324.2.a.a 1
45.l even 12 1 2700.2.i.b 2
45.l even 12 1 8100.2.a.g 1
60.l odd 4 1 432.2.i.c 2
105.k odd 4 1 5292.2.j.a 2
105.w odd 12 1 5292.2.i.a 2
105.w odd 12 1 5292.2.l.c 2
105.x even 12 1 5292.2.i.c 2
105.x even 12 1 5292.2.l.a 2
120.q odd 4 1 1728.2.i.c 2
120.w even 4 1 1728.2.i.d 2
180.v odd 12 1 432.2.i.c 2
180.v odd 12 1 1296.2.a.b 1
180.x even 12 1 144.2.i.a 2
180.x even 12 1 1296.2.a.k 1
315.bs even 12 1 1764.2.l.a 2
315.bt odd 12 1 1764.2.l.c 2
315.bu odd 12 1 5292.2.l.c 2
315.bv even 12 1 5292.2.l.a 2
315.bw odd 12 1 5292.2.i.a 2
315.bx even 12 1 5292.2.i.c 2
315.cb even 12 1 1764.2.j.b 2
315.cf odd 12 1 5292.2.j.a 2
315.cg even 12 1 1764.2.i.c 2
315.ch odd 12 1 1764.2.i.a 2
360.bo even 12 1 576.2.i.e 2
360.bo even 12 1 5184.2.a.f 1
360.br even 12 1 1728.2.i.d 2
360.br even 12 1 5184.2.a.ba 1
360.bt odd 12 1 1728.2.i.c 2
360.bt odd 12 1 5184.2.a.bb 1
360.bu odd 12 1 576.2.i.f 2
360.bu odd 12 1 5184.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 5.c odd 4 1
36.2.e.a 2 45.k odd 12 1
108.2.e.a 2 15.e even 4 1
108.2.e.a 2 45.l even 12 1
144.2.i.a 2 20.e even 4 1
144.2.i.a 2 180.x even 12 1
324.2.a.a 1 45.l even 12 1
324.2.a.c 1 45.k odd 12 1
432.2.i.c 2 60.l odd 4 1
432.2.i.c 2 180.v odd 12 1
576.2.i.e 2 40.k even 4 1
576.2.i.e 2 360.bo even 12 1
576.2.i.f 2 40.i odd 4 1
576.2.i.f 2 360.bu odd 12 1
900.2.i.b 2 5.c odd 4 1
900.2.i.b 2 45.k odd 12 1
900.2.s.b 4 1.a even 1 1 trivial
900.2.s.b 4 5.b even 2 1 inner
900.2.s.b 4 9.c even 3 1 inner
900.2.s.b 4 45.j even 6 1 inner
1296.2.a.b 1 180.v odd 12 1
1296.2.a.k 1 180.x even 12 1
1728.2.i.c 2 120.q odd 4 1
1728.2.i.c 2 360.bt odd 12 1
1728.2.i.d 2 120.w even 4 1
1728.2.i.d 2 360.br even 12 1
1764.2.i.a 2 35.l odd 12 1
1764.2.i.a 2 315.ch odd 12 1
1764.2.i.c 2 35.k even 12 1
1764.2.i.c 2 315.cg even 12 1
1764.2.j.b 2 35.f even 4 1
1764.2.j.b 2 315.cb even 12 1
1764.2.l.a 2 35.k even 12 1
1764.2.l.a 2 315.bs even 12 1
1764.2.l.c 2 35.l odd 12 1
1764.2.l.c 2 315.bt odd 12 1
2700.2.i.b 2 15.e even 4 1
2700.2.i.b 2 45.l even 12 1
2700.2.s.b 4 3.b odd 2 1
2700.2.s.b 4 9.d odd 6 1
2700.2.s.b 4 15.d odd 2 1
2700.2.s.b 4 45.h odd 6 1
5184.2.a.e 1 360.bu odd 12 1
5184.2.a.f 1 360.bo even 12 1
5184.2.a.ba 1 360.br even 12 1
5184.2.a.bb 1 360.bt odd 12 1
5292.2.i.a 2 105.w odd 12 1
5292.2.i.a 2 315.bw odd 12 1
5292.2.i.c 2 105.x even 12 1
5292.2.i.c 2 315.bx even 12 1
5292.2.j.a 2 105.k odd 4 1
5292.2.j.a 2 315.cf odd 12 1
5292.2.l.a 2 105.x even 12 1
5292.2.l.a 2 315.bv even 12 1
5292.2.l.c 2 105.w odd 12 1
5292.2.l.c 2 315.bu odd 12 1
8100.2.a.g 1 45.l even 12 1
8100.2.a.j 1 45.k odd 12 1
8100.2.d.c 2 9.d odd 6 1
8100.2.d.c 2 45.h odd 6 1
8100.2.d.h 2 9.c even 3 1
8100.2.d.h 2 45.j even 6 1

## Hecke kernels

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

 $$T_{7}^{4} - T_{7}^{2} + 1$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( -4 + T )^{4}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( 9 - 3 T + T^{2} )^{2}$$
$31$ $$( 25 + 5 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 9 + 3 T + T^{2} )^{2}$$
$43$ $$1 - T^{2} + T^{4}$$
$47$ $$6561 - 81 T^{2} + T^{4}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 9 + 3 T + T^{2} )^{2}$$
$61$ $$( 169 - 13 T + T^{2} )^{2}$$
$67$ $$2401 - 49 T^{2} + T^{4}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$( 100 + T^{2} )^{2}$$
$79$ $$( 121 - 11 T + T^{2} )^{2}$$
$83$ $$6561 - 81 T^{2} + T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$14641 - 121 T^{2} + T^{4}$$
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