# Properties

 Label 900.3.p.a Level $900$ Weight $3$ Character orbit 900.p Analytic conductor $24.523$ Analytic rank $0$ Dimension $4$ 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.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 3 \beta_{3} ) q^{7} + ( -1 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 3 \beta_{3} ) q^{7} + ( -1 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} + ( -10 + 7 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{13} + ( -7 + 16 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - 3 \beta_{2} ) q^{19} + ( 26 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -16 - 17 \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -15 + 24 \beta_{1} - 6 \beta_{2} ) q^{27} + ( -28 + 7 \beta_{1} - 14 \beta_{2} + 7 \beta_{3} ) q^{29} + ( -8 - \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{31} + ( 21 - 39 \beta_{1} - 9 \beta_{2} + 12 \beta_{3} ) q^{33} + ( 10 - 12 \beta_{2} ) q^{37} + ( 4 + 23 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{39} + ( -7 + \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{41} + 23 \beta_{1} q^{43} + ( 24 - 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -27 + 24 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{49} + ( 6 - 24 \beta_{1} + 6 \beta_{2} + 9 \beta_{3} ) q^{51} + ( 16 - 16 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{53} + ( -24 + 24 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{57} + ( 25 + 17 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{59} + ( 25 \beta_{1} - 9 \beta_{3} ) q^{61} + ( -10 - 17 \beta_{1} + 22 \beta_{2} - 23 \beta_{3} ) q^{63} + ( -67 + 49 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{67} + ( 9 \beta_{1} - 33 \beta_{2} + 15 \beta_{3} ) q^{69} + ( -14 + 32 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -17 + 9 \beta_{2} ) q^{73} + ( 38 + 55 \beta_{1} - 17 \beta_{2} - 17 \beta_{3} ) q^{77} + ( 49 \beta_{1} - 15 \beta_{3} ) q^{79} + ( -24 + 24 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{81} + ( 24 - 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -42 + 105 \beta_{1} + 21 \beta_{3} ) q^{87} + ( -56 + 128 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{89} + ( -80 - 9 \beta_{2} ) q^{91} + ( 8 + 73 \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{93} + ( 95 \beta_{1} + 6 \beta_{3} ) q^{97} + ( -3 + 111 \beta_{1} + 3 \beta_{2} - 33 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{3} + q^{7} - 15q^{9} + O(q^{10})$$ $$4q - 3q^{3} + q^{7} - 15q^{9} - 36q^{11} - 5q^{13} + 2q^{19} + 99q^{21} - 99q^{23} - 63q^{29} - 7q^{31} + 36q^{33} + 64q^{37} + 57q^{39} - 18q^{41} + 46q^{43} + 81q^{47} - 51q^{49} - 27q^{51} - 51q^{57} + 126q^{59} + 41q^{61} - 141q^{63} - 116q^{67} + 99q^{69} - 86q^{73} + 279q^{77} + 83q^{79} - 63q^{81} + 81q^{83} + 63q^{87} - 302q^{91} + 159q^{93} + 196q^{97} + 171q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1 - \beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 −2.18614 2.05446i 0 0 0 −4.05842 + 7.02939i 0 0.558422 + 8.98266i 0
101.2 0 0.686141 + 2.92048i 0 0 0 4.55842 7.89542i 0 −8.05842 + 4.00772i 0
401.1 0 −2.18614 + 2.05446i 0 0 0 −4.05842 7.02939i 0 0.558422 8.98266i 0
401.2 0 0.686141 2.92048i 0 0 0 4.55842 + 7.89542i 0 −8.05842 4.00772i 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.a 4
3.b odd 2 1 2700.3.p.b 4
5.b even 2 1 36.3.g.a 4
5.c odd 4 2 900.3.u.a 8
9.c even 3 1 2700.3.p.b 4
9.d odd 6 1 inner 900.3.p.a 4
15.d odd 2 1 108.3.g.a 4
15.e even 4 2 2700.3.u.b 8
20.d odd 2 1 144.3.q.b 4
40.e odd 2 1 576.3.q.g 4
40.f even 2 1 576.3.q.d 4
45.h odd 6 1 36.3.g.a 4
45.h odd 6 1 324.3.c.b 4
45.j even 6 1 108.3.g.a 4
45.j even 6 1 324.3.c.b 4
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
60.h even 2 1 432.3.q.b 4
120.i odd 2 1 1728.3.q.g 4
120.m even 2 1 1728.3.q.h 4
180.n even 6 1 144.3.q.b 4
180.n even 6 1 1296.3.e.e 4
180.p odd 6 1 432.3.q.b 4
180.p odd 6 1 1296.3.e.e 4
360.z odd 6 1 1728.3.q.h 4
360.bd even 6 1 576.3.q.g 4
360.bh odd 6 1 576.3.q.d 4
360.bk even 6 1 1728.3.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 5.b even 2 1
36.3.g.a 4 45.h odd 6 1
108.3.g.a 4 15.d odd 2 1
108.3.g.a 4 45.j even 6 1
144.3.q.b 4 20.d odd 2 1
144.3.q.b 4 180.n even 6 1
324.3.c.b 4 45.h odd 6 1
324.3.c.b 4 45.j even 6 1
432.3.q.b 4 60.h even 2 1
432.3.q.b 4 180.p odd 6 1
576.3.q.d 4 40.f even 2 1
576.3.q.d 4 360.bh odd 6 1
576.3.q.g 4 40.e odd 2 1
576.3.q.g 4 360.bd even 6 1
900.3.p.a 4 1.a even 1 1 trivial
900.3.p.a 4 9.d odd 6 1 inner
900.3.u.a 8 5.c odd 4 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 180.n even 6 1
1296.3.e.e 4 180.p odd 6 1
1728.3.q.g 4 120.i odd 2 1
1728.3.q.g 4 360.bk even 6 1
1728.3.q.h 4 120.m even 2 1
1728.3.q.h 4 360.z odd 6 1
2700.3.p.b 4 3.b odd 2 1
2700.3.p.b 4 9.c even 3 1
2700.3.u.b 8 15.e even 4 2
2700.3.u.b 8 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - T_{7}^{3} + 75 T_{7}^{2} + 74 T_{7} + 5476$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 + 27 T + 12 T^{2} + 3 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$5476 + 74 T + 75 T^{2} - T^{3} + T^{4}$$
$11$ $$81 + 324 T + 441 T^{2} + 36 T^{3} + T^{4}$$
$13$ $$4624 - 340 T + 93 T^{2} + 5 T^{3} + T^{4}$$
$17$ $$20736 + 387 T^{2} + T^{4}$$
$19$ $$( -74 - T + T^{2} )^{2}$$
$23$ $$627264 + 78408 T + 4059 T^{2} + 99 T^{3} + T^{4}$$
$29$ $$777924 - 55566 T + 441 T^{2} + 63 T^{3} + T^{4}$$
$31$ $$430336 - 4592 T + 705 T^{2} + 7 T^{3} + T^{4}$$
$37$ $$( -932 - 32 T + T^{2} )^{2}$$
$41$ $$2424249 - 28026 T - 1449 T^{2} + 18 T^{3} + T^{4}$$
$43$ $$( 529 - 23 T + T^{2} )^{2}$$
$47$ $$104976 - 26244 T + 2511 T^{2} - 81 T^{3} + T^{4}$$
$53$ $$1327104 + 4032 T^{2} + T^{4}$$
$59$ $$68121 + 32886 T + 5031 T^{2} - 126 T^{3} + T^{4}$$
$61$ $$61504 + 10168 T + 1929 T^{2} - 41 T^{3} + T^{4}$$
$67$ $$477481 + 80156 T + 12765 T^{2} + 116 T^{3} + T^{4}$$
$71$ $$331776 + 1548 T^{2} + T^{4}$$
$73$ $$( -206 + 43 T + T^{2} )^{2}$$
$79$ $$17956 + 11122 T + 7023 T^{2} - 83 T^{3} + T^{4}$$
$83$ $$104976 - 26244 T + 2511 T^{2} - 81 T^{3} + T^{4}$$
$89$ $$84934656 + 24768 T^{2} + T^{4}$$
$97$ $$86620249 - 1824172 T + 29109 T^{2} - 196 T^{3} + T^{4}$$