Properties

 Label 900.3.u.a Level $900$ Weight $3$ Character orbit 900.u Analytic conductor $24.523$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -\beta_{2} - 7 \beta_{3} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{3} + ( -3 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -\beta_{2} - 7 \beta_{3} - \beta_{6} ) q^{9} + ( -12 - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{6} ) q^{11} + ( -\beta_{1} + 3 \beta_{7} ) q^{13} + ( 9 \beta_{1} + \beta_{4} - 7 \beta_{5} + 2 \beta_{7} ) q^{17} + ( -2 - 3 \beta_{6} ) q^{19} + ( 27 - 3 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{21} + ( -15 \beta_{1} + 2 \beta_{4} - 32 \beta_{5} + \beta_{7} ) q^{23} + ( 15 \beta_{1} + 6 \beta_{4} - 9 \beta_{5} ) q^{27} + ( 21 - 7 \beta_{2} + 14 \beta_{3} - 7 \beta_{6} ) q^{29} + ( -8 + 9 \beta_{2} - 8 \beta_{3} ) q^{31} + ( 33 \beta_{1} + 3 \beta_{4} - 6 \beta_{5} + 12 \beta_{7} ) q^{33} + ( -10 \beta_{1} + 12 \beta_{4} - 10 \beta_{5} ) q^{37} + ( -\beta_{2} + 24 \beta_{3} + 4 \beta_{6} ) q^{39} + ( 9 - 8 \beta_{2} + 7 \beta_{3} + 16 \beta_{6} ) q^{41} + 23 \beta_{5} q^{43} + ( -27 \beta_{1} + 3 \beta_{4} - 12 \beta_{5} - 3 \beta_{7} ) q^{47} + ( 27 - 3 \beta_{2} + 27 \beta_{3} ) q^{49} + ( -9 + 9 \beta_{2} + 15 \beta_{3} - 15 \beta_{6} ) q^{51} + ( -8 \beta_{4} - 16 \beta_{5} - 16 \beta_{7} ) q^{53} + ( 23 \beta_{1} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{57} + ( -9 - 8 \beta_{2} + 25 \beta_{3} + 16 \beta_{6} ) q^{59} + ( 9 - 9 \beta_{2} - 16 \beta_{3} + 9 \beta_{6} ) q^{61} + ( -33 \beta_{1} - \beta_{4} - 50 \beta_{5} - 23 \beta_{7} ) q^{63} + ( 49 \beta_{1} - 18 \beta_{7} ) q^{67} + ( -18 - 15 \beta_{2} + 24 \beta_{3} - 18 \beta_{6} ) q^{69} + ( -12 - 4 \beta_{2} - 28 \beta_{3} + 2 \beta_{6} ) q^{71} + ( -17 \beta_{1} + 9 \beta_{4} - 17 \beta_{5} ) q^{73} + ( -21 \beta_{1} + 34 \beta_{4} - 76 \beta_{5} + 17 \beta_{7} ) q^{77} + ( -15 + 15 \beta_{2} + 34 \beta_{3} - 15 \beta_{6} ) q^{79} + ( -54 + 15 \beta_{2} - 39 \beta_{3} - 30 \beta_{6} ) q^{81} + ( 27 \beta_{1} - 3 \beta_{4} + 12 \beta_{5} + 3 \beta_{7} ) q^{83} + ( 21 \beta_{1} - 21 \beta_{4} - 84 \beta_{5} - 21 \beta_{7} ) q^{87} + ( 48 + 16 \beta_{2} + 112 \beta_{3} - 8 \beta_{6} ) q^{89} + ( -71 + 9 \beta_{6} ) q^{91} + ( 7 \beta_{1} + 8 \beta_{4} + 80 \beta_{5} - \beta_{7} ) q^{93} + ( -6 \beta_{1} - 6 \beta_{4} - 101 \beta_{5} - 6 \beta_{7} ) q^{97} + ( -27 + 33 \beta_{2} + 78 \beta_{3} - 30 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 30q^{9} + O(q^{10})$$ $$8q + 30q^{9} - 72q^{11} - 4q^{19} + 198q^{21} + 126q^{29} - 14q^{31} - 114q^{39} - 36q^{41} + 102q^{49} - 54q^{51} - 252q^{59} + 82q^{61} - 198q^{69} - 166q^{79} - 126q^{81} - 604q^{91} - 342q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 16 \nu^{4} - 32 \nu^{2} - 51$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 5 \nu^{4} - 7 \nu^{2} - 27$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 10 \nu^{3} - 3 \nu$$$$)/18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{4}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 7 \beta_{3} - \beta_{2} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} - 11 \beta_{5} + 2 \beta_{4} - 9 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{6} + 13 \beta_{3} - 5 \beta_{2}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} - 2 \beta_{4} + 45 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{6} - 16 \beta_{3} + 32 \beta_{2} - 39$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} + 118 \beta_{5} - 13 \beta_{4}$$$$)/3$$

Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1 + \beta_{3}$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i
0 −2.92048 0.686141i 0 0 0 −7.89542 + 4.55842i 0 8.05842 + 4.00772i 0
149.2 0 −2.05446 2.18614i 0 0 0 −7.02939 + 4.05842i 0 −0.558422 + 8.98266i 0
149.3 0 2.05446 + 2.18614i 0 0 0 7.02939 4.05842i 0 −0.558422 + 8.98266i 0
149.4 0 2.92048 + 0.686141i 0 0 0 7.89542 4.55842i 0 8.05842 + 4.00772i 0
749.1 0 −2.92048 + 0.686141i 0 0 0 −7.89542 4.55842i 0 8.05842 4.00772i 0
749.2 0 −2.05446 + 2.18614i 0 0 0 −7.02939 4.05842i 0 −0.558422 8.98266i 0
749.3 0 2.05446 2.18614i 0 0 0 7.02939 + 4.05842i 0 −0.558422 8.98266i 0
749.4 0 2.92048 0.686141i 0 0 0 7.89542 + 4.55842i 0 8.05842 4.00772i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 749.4 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.d odd 6 1 inner
45.h 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.u.a 8
3.b odd 2 1 2700.3.u.b 8
5.b even 2 1 inner 900.3.u.a 8
5.c odd 4 1 36.3.g.a 4
5.c odd 4 1 900.3.p.a 4
9.c even 3 1 2700.3.u.b 8
9.d odd 6 1 inner 900.3.u.a 8
15.d odd 2 1 2700.3.u.b 8
15.e even 4 1 108.3.g.a 4
15.e even 4 1 2700.3.p.b 4
20.e even 4 1 144.3.q.b 4
40.i odd 4 1 576.3.q.d 4
40.k even 4 1 576.3.q.g 4
45.h odd 6 1 inner 900.3.u.a 8
45.j even 6 1 2700.3.u.b 8
45.k odd 12 1 108.3.g.a 4
45.k odd 12 1 324.3.c.b 4
45.k odd 12 1 2700.3.p.b 4
45.l even 12 1 36.3.g.a 4
45.l even 12 1 324.3.c.b 4
45.l even 12 1 900.3.p.a 4
60.l odd 4 1 432.3.q.b 4
120.q odd 4 1 1728.3.q.h 4
120.w even 4 1 1728.3.q.g 4
180.v odd 12 1 144.3.q.b 4
180.v odd 12 1 1296.3.e.e 4
180.x even 12 1 432.3.q.b 4
180.x even 12 1 1296.3.e.e 4
360.bo even 12 1 1728.3.q.h 4
360.br even 12 1 576.3.q.d 4
360.bt odd 12 1 576.3.q.g 4
360.bu odd 12 1 1728.3.q.g 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 149 T_{7}^{6} + 16725 T_{7}^{4} - 815924 T_{7}^{2} + 29986576$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 - 1215 T^{2} + 144 T^{4} - 15 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$29986576 - 815924 T^{2} + 16725 T^{4} - 149 T^{6} + T^{8}$$
$11$ $$( 81 + 324 T + 441 T^{2} + 36 T^{3} + T^{4} )^{2}$$
$13$ $$21381376 - 744464 T^{2} + 21297 T^{4} - 161 T^{6} + T^{8}$$
$17$ $$( 20736 - 387 T^{2} + T^{4} )^{2}$$
$19$ $$( -74 + T + T^{2} )^{4}$$
$23$ $$393460125696 + 1055685312 T^{2} + 2205225 T^{4} + 1683 T^{6} + T^{8}$$
$29$ $$( 777924 + 55566 T + 441 T^{2} - 63 T^{3} + T^{4} )^{2}$$
$31$ $$( 430336 - 4592 T + 705 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$37$ $$( 868624 + 2888 T^{2} + T^{4} )^{2}$$
$41$ $$( 2424249 - 28026 T - 1449 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$43$ $$( 279841 - 529 T^{2} + T^{4} )^{2}$$
$47$ $$11019960576 + 161558064 T^{2} + 2263545 T^{4} + 1539 T^{6} + T^{8}$$
$53$ $$( 1327104 - 4032 T^{2} + T^{4} )^{2}$$
$59$ $$( 68121 - 32886 T + 5031 T^{2} + 126 T^{3} + T^{4} )^{2}$$
$61$ $$( 61504 + 10168 T + 1929 T^{2} - 41 T^{3} + T^{4} )^{2}$$
$67$ $$227988105361 - 5765105594 T^{2} + 145303995 T^{4} - 12074 T^{6} + T^{8}$$
$71$ $$( 331776 + 1548 T^{2} + T^{4} )^{2}$$
$73$ $$( 42436 + 2261 T^{2} + T^{4} )^{2}$$
$79$ $$( 17956 - 11122 T + 7023 T^{2} + 83 T^{3} + T^{4} )^{2}$$
$83$ $$11019960576 + 161558064 T^{2} + 2263545 T^{4} + 1539 T^{6} + T^{8}$$
$89$ $$( 84934656 + 24768 T^{2} + T^{4} )^{2}$$
$97$ $$7503067536822001 - 1715254170698 T^{2} + 305498955 T^{4} - 19802 T^{6} + T^{8}$$