# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.8405.1 Defining polynomial: $$x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 100) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{11} + ( -6 \beta_{2} - 2 \beta_{3} ) q^{13} + ( 8 + 2 \beta_{2} + 4 \beta_{3} ) q^{14} + ( -9 - 3 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{16} + ( -5 - 6 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} ) q^{19} + ( -10 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{22} + ( 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{23} + ( -20 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{26} + ( 30 + 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{28} + ( -12 - 6 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{31} + ( -5 + 9 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{32} + ( -20 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{34} + ( -30 + 6 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 30 + 14 \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{38} -17 q^{41} + ( 12 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -25 + \beta_{1} - 11 \beta_{2} - 3 \beta_{3} ) q^{44} + ( 12 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{46} + ( -8 \beta_{1} - 14 \beta_{2} + 6 \beta_{3} ) q^{47} + ( -27 - 24 \beta_{2} - 8 \beta_{3} ) q^{49} + ( 20 - 4 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 30 - 24 \beta_{2} - 8 \beta_{3} ) q^{53} + ( 18 - 10 \beta_{1} + 22 \beta_{2} - 6 \beta_{3} ) q^{56} + ( -20 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 24 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} ) q^{59} + ( -38 - 30 \beta_{2} - 10 \beta_{3} ) q^{61} + ( -20 - 4 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} ) q^{62} + ( -9 - 3 \beta_{1} + \beta_{2} + 15 \beta_{3} ) q^{64} + ( 11 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{67} + ( 15 + \beta_{1} - 15 \beta_{2} + 9 \beta_{3} ) q^{68} + ( -32 \beta_{1} - 18 \beta_{2} - 14 \beta_{3} ) q^{71} + ( 55 - 6 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 20 - 4 \beta_{1} - 34 \beta_{2} - 4 \beta_{3} ) q^{74} + ( 25 - 17 \beta_{1} + 27 \beta_{2} + 11 \beta_{3} ) q^{76} + ( 70 + 30 \beta_{2} + 10 \beta_{3} ) q^{77} + ( -2 \beta_{1} + 12 \beta_{2} - 14 \beta_{3} ) q^{79} -17 \beta_{2} q^{82} + ( 17 \beta_{1} + 11 \beta_{2} + 6 \beta_{3} ) q^{83} + ( -12 - 20 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{86} + ( -35 + 7 \beta_{1} - 17 \beta_{2} + 9 \beta_{3} ) q^{88} + ( 53 - 6 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 20 \beta_{2} - 20 \beta_{3} ) q^{91} + ( 70 + 10 \beta_{1} + 18 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 52 + 28 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 90 + 36 \beta_{2} + 12 \beta_{3} ) q^{97} + ( -80 + 16 \beta_{1} - 11 \beta_{2} + 16 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + 5q^{4} - 19q^{8} + O(q^{10})$$ $$4q - q^{2} + 5q^{4} - 19q^{8} + 8q^{13} + 26q^{14} - 31q^{16} - 12q^{17} - 35q^{22} - 84q^{26} + 110q^{28} - 40q^{29} - 11q^{32} - 79q^{34} - 128q^{37} + 115q^{38} - 68q^{41} - 85q^{44} + 34q^{46} - 76q^{49} + 92q^{52} + 152q^{53} + 46q^{56} - 72q^{58} - 112q^{61} - 70q^{62} - 55q^{64} + 67q^{68} + 228q^{73} + 114q^{74} + 45q^{76} + 240q^{77} + 17q^{82} - 64q^{86} - 125q^{88} + 220q^{89} + 270q^{92} + 204q^{94} + 312q^{97} - 309q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + \nu^{2} - 5 \nu + 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} + 7 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3} + 5 \nu^{2} - 23 \nu + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} + \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} - 6 \beta_{1} - 8$$$$)/2$$

## 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 − 2.53999i 0.5 + 2.53999i 0.5 − 0.220086i 0.5 + 0.220086i
−1.85078 0.758030i 0 2.85078 + 2.80590i 0 0 4.09573i −3.14922 7.35408i 0 0
451.2 −1.85078 + 0.758030i 0 2.85078 2.80590i 0 0 4.09573i −3.14922 + 7.35408i 0 0
451.3 1.35078 1.47492i 0 −0.350781 3.98459i 0 0 10.9190i −6.35078 4.86493i 0 0
451.4 1.35078 + 1.47492i 0 −0.350781 + 3.98459i 0 0 10.9190i −6.35078 + 4.86493i 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.l 4
3.b odd 2 1 100.3.b.e yes 4
4.b odd 2 1 inner 900.3.c.l 4
5.b even 2 1 900.3.c.m 4
5.c odd 4 2 900.3.f.d 8
12.b even 2 1 100.3.b.e yes 4
15.d odd 2 1 100.3.b.d 4
15.e even 4 2 100.3.d.a 8
20.d odd 2 1 900.3.c.m 4
20.e even 4 2 900.3.f.d 8
24.f even 2 1 1600.3.b.o 4
24.h odd 2 1 1600.3.b.o 4
60.h even 2 1 100.3.b.d 4
60.l odd 4 2 100.3.d.a 8
120.i odd 2 1 1600.3.b.p 4
120.m even 2 1 1600.3.b.p 4
120.q odd 4 2 1600.3.h.o 8
120.w even 4 2 1600.3.h.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.b.d 4 15.d odd 2 1
100.3.b.d 4 60.h even 2 1
100.3.b.e yes 4 3.b odd 2 1
100.3.b.e yes 4 12.b even 2 1
100.3.d.a 8 15.e even 4 2
100.3.d.a 8 60.l odd 4 2
900.3.c.l 4 1.a even 1 1 trivial
900.3.c.l 4 4.b odd 2 1 inner
900.3.c.m 4 5.b even 2 1
900.3.c.m 4 20.d odd 2 1
900.3.f.d 8 5.c odd 4 2
900.3.f.d 8 20.e even 4 2
1600.3.b.o 4 24.f even 2 1
1600.3.b.o 4 24.h odd 2 1
1600.3.b.p 4 120.i odd 2 1
1600.3.b.p 4 120.m even 2 1
1600.3.h.o 8 120.q odd 4 2
1600.3.h.o 8 120.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 $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{4} + 136 T_{7}^{2} + 2000$$ $$T_{13}^{2} - 4 T_{13} - 160$$ $$T_{17}^{2} + 6 T_{17} - 155$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T - 2 T^{2} + T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$2000 + 136 T^{2} + T^{4}$$
$11$ $$125 + 130 T^{2} + T^{4}$$
$13$ $$( -160 - 4 T + T^{2} )^{2}$$
$17$ $$( -155 + 6 T + T^{2} )^{2}$$
$19$ $$465125 + 1370 T^{2} + T^{4}$$
$23$ $$147920 + 776 T^{2} + T^{4}$$
$29$ $$( -64 + 20 T + T^{2} )^{2}$$
$31$ $$2000 + 520 T^{2} + T^{4}$$
$37$ $$( 860 + 64 T + T^{2} )^{2}$$
$41$ $$( 17 + T )^{4}$$
$43$ $$128000 + 3056 T^{2} + T^{4}$$
$47$ $$5242880 + 4976 T^{2} + T^{4}$$
$53$ $$( -1180 - 76 T + T^{2} )^{2}$$
$59$ $$41472000 + 13680 T^{2} + T^{4}$$
$61$ $$( -3316 + 56 T + T^{2} )^{2}$$
$67$ $$1915805 + 3586 T^{2} + T^{4}$$
$71$ $$67712000 + 19120 T^{2} + T^{4}$$
$73$ $$( 3085 - 114 T + T^{2} )^{2}$$
$79$ $$6962000 + 7720 T^{2} + T^{4}$$
$83$ $$3621005 + 5426 T^{2} + T^{4}$$
$89$ $$( 2861 - 110 T + T^{2} )^{2}$$
$97$ $$( 180 - 156 T + T^{2} )^{2}$$