# 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: $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - \beta_{5} - \beta_1) q^{3} + ( - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{6} - 7 \beta_{3} - \beta_{2}) q^{9}+O(q^{10})$$ q + (-b7 - b5 - b1) * q^3 + (-3*b7 - 2*b5 - 3*b4 - 3*b1) * q^7 + (-b6 - 7*b3 - b2) * q^9 $$q + ( - \beta_{7} - \beta_{5} - \beta_1) q^{3} + ( - 3 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{6} - 7 \beta_{3} - \beta_{2}) q^{9} + ( - 2 \beta_{6} - 5 \beta_{3} - 2 \beta_{2} - 12) q^{11} + (3 \beta_{7} - \beta_1) q^{13} + (2 \beta_{7} - 7 \beta_{5} + \beta_{4} + 9 \beta_1) q^{17} + ( - 3 \beta_{6} - 2) q^{19} + (\beta_{6} + 2 \beta_{3} - 3 \beta_{2} + 27) q^{21} + (\beta_{7} - 32 \beta_{5} + 2 \beta_{4} - 15 \beta_1) q^{23} + ( - 9 \beta_{5} + 6 \beta_{4} + 15 \beta_1) q^{27} + ( - 7 \beta_{6} + 14 \beta_{3} - 7 \beta_{2} + 21) q^{29} + ( - 8 \beta_{3} + 9 \beta_{2} - 8) q^{31} + (12 \beta_{7} - 6 \beta_{5} + 3 \beta_{4} + 33 \beta_1) q^{33} + ( - 10 \beta_{5} + 12 \beta_{4} - 10 \beta_1) q^{37} + (4 \beta_{6} + 24 \beta_{3} - \beta_{2}) q^{39} + (16 \beta_{6} + 7 \beta_{3} - 8 \beta_{2} + 9) q^{41} + 23 \beta_{5} q^{43} + ( - 3 \beta_{7} - 12 \beta_{5} + 3 \beta_{4} - 27 \beta_1) q^{47} + (27 \beta_{3} - 3 \beta_{2} + 27) q^{49} + ( - 15 \beta_{6} + 15 \beta_{3} + 9 \beta_{2} - 9) q^{51} + ( - 16 \beta_{7} - 16 \beta_{5} - 8 \beta_{4}) q^{53} + ( - \beta_{7} - \beta_{5} - 3 \beta_{4} + 23 \beta_1) q^{57} + (16 \beta_{6} + 25 \beta_{3} - 8 \beta_{2} - 9) q^{59} + (9 \beta_{6} - 16 \beta_{3} - 9 \beta_{2} + 9) q^{61} + ( - 23 \beta_{7} - 50 \beta_{5} - \beta_{4} - 33 \beta_1) q^{63} + ( - 18 \beta_{7} + 49 \beta_1) q^{67} + ( - 18 \beta_{6} + 24 \beta_{3} - 15 \beta_{2} - 18) q^{69} + (2 \beta_{6} - 28 \beta_{3} - 4 \beta_{2} - 12) q^{71} + ( - 17 \beta_{5} + 9 \beta_{4} - 17 \beta_1) q^{73} + (17 \beta_{7} - 76 \beta_{5} + 34 \beta_{4} - 21 \beta_1) q^{77} + ( - 15 \beta_{6} + 34 \beta_{3} + 15 \beta_{2} - 15) q^{79} + ( - 30 \beta_{6} - 39 \beta_{3} + 15 \beta_{2} - 54) q^{81} + (3 \beta_{7} + 12 \beta_{5} - 3 \beta_{4} + 27 \beta_1) q^{83} + ( - 21 \beta_{7} - 84 \beta_{5} - 21 \beta_{4} + 21 \beta_1) q^{87} + ( - 8 \beta_{6} + 112 \beta_{3} + 16 \beta_{2} + 48) q^{89} + (9 \beta_{6} - 71) q^{91} + ( - \beta_{7} + 80 \beta_{5} + 8 \beta_{4} + 7 \beta_1) q^{93} + ( - 6 \beta_{7} - 101 \beta_{5} - 6 \beta_{4} - 6 \beta_1) q^{97} + ( - 30 \beta_{6} + 78 \beta_{3} + 33 \beta_{2} - 27) q^{99}+O(q^{100})$$ q + (-b7 - b5 - b1) * q^3 + (-3*b7 - 2*b5 - 3*b4 - 3*b1) * q^7 + (-b6 - 7*b3 - b2) * q^9 + (-2*b6 - 5*b3 - 2*b2 - 12) * q^11 + (3*b7 - b1) * q^13 + (2*b7 - 7*b5 + b4 + 9*b1) * q^17 + (-3*b6 - 2) * q^19 + (b6 + 2*b3 - 3*b2 + 27) * q^21 + (b7 - 32*b5 + 2*b4 - 15*b1) * q^23 + (-9*b5 + 6*b4 + 15*b1) * q^27 + (-7*b6 + 14*b3 - 7*b2 + 21) * q^29 + (-8*b3 + 9*b2 - 8) * q^31 + (12*b7 - 6*b5 + 3*b4 + 33*b1) * q^33 + (-10*b5 + 12*b4 - 10*b1) * q^37 + (4*b6 + 24*b3 - b2) * q^39 + (16*b6 + 7*b3 - 8*b2 + 9) * q^41 + 23*b5 * q^43 + (-3*b7 - 12*b5 + 3*b4 - 27*b1) * q^47 + (27*b3 - 3*b2 + 27) * q^49 + (-15*b6 + 15*b3 + 9*b2 - 9) * q^51 + (-16*b7 - 16*b5 - 8*b4) * q^53 + (-b7 - b5 - 3*b4 + 23*b1) * q^57 + (16*b6 + 25*b3 - 8*b2 - 9) * q^59 + (9*b6 - 16*b3 - 9*b2 + 9) * q^61 + (-23*b7 - 50*b5 - b4 - 33*b1) * q^63 + (-18*b7 + 49*b1) * q^67 + (-18*b6 + 24*b3 - 15*b2 - 18) * q^69 + (2*b6 - 28*b3 - 4*b2 - 12) * q^71 + (-17*b5 + 9*b4 - 17*b1) * q^73 + (17*b7 - 76*b5 + 34*b4 - 21*b1) * q^77 + (-15*b6 + 34*b3 + 15*b2 - 15) * q^79 + (-30*b6 - 39*b3 + 15*b2 - 54) * q^81 + (3*b7 + 12*b5 - 3*b4 + 27*b1) * q^83 + (-21*b7 - 84*b5 - 21*b4 + 21*b1) * q^87 + (-8*b6 + 112*b3 + 16*b2 + 48) * q^89 + (9*b6 - 71) * q^91 + (-b7 + 80*b5 + 8*b4 + 7*b1) * q^93 + (-6*b7 - 101*b5 - 6*b4 - 6*b1) * q^97 + (-30*b6 + 78*b3 + 33*b2 - 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 30 q^{9}+O(q^{10})$$ 8 * q + 30 * q^9 $$8 q + 30 q^{9} - 72 q^{11} - 4 q^{19} + 198 q^{21} + 126 q^{29} - 14 q^{31} - 114 q^{39} - 36 q^{41} + 102 q^{49} - 54 q^{51} - 252 q^{59} + 82 q^{61} - 198 q^{69} - 166 q^{79} - 126 q^{81} - 604 q^{91} - 342 q^{99}+O(q^{100})$$ 8 * q + 30 * q^9 - 72 * q^11 - 4 * q^19 + 198 * q^21 + 126 * q^29 - 14 * q^31 - 114 * q^39 - 36 * q^41 + 102 * q^49 - 54 * q^51 - 252 * q^59 + 82 * q^61 - 198 * q^69 - 166 * q^79 - 126 * q^81 - 604 * q^91 - 342 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432$$ (v^7 + 32*v^5 + 16*v^3 + 45*v) / 432 $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48$$ (v^6 - 16*v^4 - 32*v^2 - 51) / 48 $$\beta_{3}$$ $$=$$ $$( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144$$ (-5*v^6 - 16*v^4 - 80*v^2 - 225) / 144 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72$$ (v^7 + 8*v^5 + 40*v^3 + 165*v) / 72 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 48$$ (v^7 + 13*v) / 48 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9$$ (-v^6 - 5*v^4 - 7*v^2 - 27) / 9 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18$$ (v^7 + 2*v^5 + 10*v^3 - 3*v) / 18
 $$\nu$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3$$ (-b7 + 2*b5 + b4) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3$$ (2*b6 - 7*b3 - b2 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3$$ (4*b7 - 11*b5 + 2*b4 - 9*b1) / 3 $$\nu^{4}$$ $$=$$ $$( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3$$ (-5*b6 + 13*b3 - 5*b2) / 3 $$\nu^{5}$$ $$=$$ $$( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3$$ (-b7 - b5 - 2*b4 + 45*b1) / 3 $$\nu^{6}$$ $$=$$ $$( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3$$ (-16*b6 - 16*b3 + 32*b2 - 39) / 3 $$\nu^{7}$$ $$=$$ $$( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3$$ (13*b7 + 118*b5 - 13*b4) / 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 $$T_{7}^{8} - 149T_{7}^{6} + 16725T_{7}^{4} - 815924T_{7}^{2} + 29986576$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 15 T^{6} + 144 T^{4} + \cdots + 6561$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 149 T^{6} + \cdots + 29986576$$
$11$ $$(T^{4} + 36 T^{3} + 441 T^{2} + 324 T + 81)^{2}$$
$13$ $$T^{8} - 161 T^{6} + \cdots + 21381376$$
$17$ $$(T^{4} - 387 T^{2} + 20736)^{2}$$
$19$ $$(T^{2} + T - 74)^{4}$$
$23$ $$T^{8} + 1683 T^{6} + \cdots + 393460125696$$
$29$ $$(T^{4} - 63 T^{3} + 441 T^{2} + \cdots + 777924)^{2}$$
$31$ $$(T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336)^{2}$$
$37$ $$(T^{4} + 2888 T^{2} + 868624)^{2}$$
$41$ $$(T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249)^{2}$$
$43$ $$(T^{4} - 529 T^{2} + 279841)^{2}$$
$47$ $$T^{8} + 1539 T^{6} + \cdots + 11019960576$$
$53$ $$(T^{4} - 4032 T^{2} + 1327104)^{2}$$
$59$ $$(T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121)^{2}$$
$61$ $$(T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504)^{2}$$
$67$ $$T^{8} - 12074 T^{6} + \cdots + 227988105361$$
$71$ $$(T^{4} + 1548 T^{2} + 331776)^{2}$$
$73$ $$(T^{4} + 2261 T^{2} + 42436)^{2}$$
$79$ $$(T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956)^{2}$$
$83$ $$T^{8} + 1539 T^{6} + \cdots + 11019960576$$
$89$ $$(T^{4} + 24768 T^{2} + 84934656)^{2}$$
$97$ $$T^{8} - 19802 T^{6} + \cdots + 75\!\cdots\!01$$