# Properties

 Label 900.3.u.b 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: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 180) 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + ( 4 \beta_{1} + \beta_{6} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + ( 4 \beta_{1} + \beta_{6} ) q^{7} -9 \beta_{2} q^{9} + ( 1 - \beta_{2} - \beta_{5} - \beta_{7} ) q^{11} + ( 10 \beta_{1} + 10 \beta_{3} - \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{17} + ( 19 + \beta_{5} ) q^{19} + ( -12 \beta_{2} - 3 \beta_{7} ) q^{21} + ( -4 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} ) q^{23} -27 \beta_{3} q^{27} + ( 10 - 10 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{29} + ( 2 \beta_{2} + 7 \beta_{7} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{6} ) q^{33} -14 \beta_{3} q^{37} + ( 30 - 3 \beta_{5} ) q^{39} + ( -30 - 15 \beta_{2} ) q^{41} + ( 11 \beta_{1} - 4 \beta_{6} ) q^{43} + ( -34 \beta_{1} - 68 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{47} + ( -27 \beta_{2} - 8 \beta_{7} ) q^{49} + ( -3 + 3 \beta_{2} + 3 \beta_{5} + 3 \beta_{7} ) q^{51} + ( 28 \beta_{1} + 14 \beta_{3} - \beta_{4} + \beta_{6} ) q^{53} + ( 57 \beta_{1} - 3 \beta_{6} ) q^{57} + ( 58 + 29 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{59} + ( 22 + 22 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{61} + ( -36 \beta_{3} + 9 \beta_{4} + 9 \beta_{6} ) q^{63} + ( -7 \beta_{1} - 7 \beta_{3} - 12 \beta_{4} ) q^{67} + ( 12 + 24 \beta_{2} - 6 \beta_{5} + 12 \beta_{7} ) q^{69} + ( 36 + 72 \beta_{2} + 3 \beta_{5} - 6 \beta_{7} ) q^{71} + ( -37 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{73} + ( 64 \beta_{1} - 64 \beta_{3} + 5 \beta_{4} + 10 \beta_{6} ) q^{77} + ( 80 + 80 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{79} + ( -81 - 81 \beta_{2} ) q^{81} + ( -52 \beta_{1} - 104 \beta_{3} ) q^{83} + ( 30 \beta_{1} - 30 \beta_{3} - 6 \beta_{4} - 12 \beta_{6} ) q^{87} + ( 24 + 48 \beta_{2} + 4 \beta_{5} - 8 \beta_{7} ) q^{89} + ( 100 - 14 \beta_{5} ) q^{91} + ( 6 \beta_{3} - 21 \beta_{4} - 21 \beta_{6} ) q^{93} + ( \beta_{1} - 13 \beta_{6} ) q^{97} + ( -9 - 18 \beta_{2} + 9 \beta_{5} - 18 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 36q^{9} + O(q^{10})$$ $$8q + 36q^{9} + 12q^{11} + 152q^{19} + 48q^{21} + 120q^{29} - 8q^{31} + 240q^{39} - 180q^{41} + 108q^{49} - 36q^{51} + 348q^{59} + 88q^{61} + 320q^{79} - 324q^{81} + 800q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 1$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 39$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 44 \nu^{3} + 31 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-11 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} - 15$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{7} - 32 \nu^{5} + 88 \nu^{3} + 25 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} - 6 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} + \beta_{4} - 18 \beta_{2}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - \beta_{5} + 12 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{4} - 14 \beta_{2} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 10 \beta_{5} + 66 \beta_{3} + 66 \beta_{1}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{6} + 2 \beta_{4} - 27$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} - 13 \beta_{5} + 174 \beta_{1}$$$$)/12$$

## Character values

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

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.535233 + 0.309017i 1.40126 − 0.809017i 0.535233 − 0.309017i −1.40126 + 0.809017i −0.535233 − 0.309017i 1.40126 + 0.809017i 0.535233 + 0.309017i −1.40126 − 0.809017i
0 −2.59808 + 1.50000i 0 0 0 −10.1723 + 5.87298i 0 4.50000 7.79423i 0
149.2 0 −2.59808 + 1.50000i 0 0 0 3.24410 1.87298i 0 4.50000 7.79423i 0
149.3 0 2.59808 1.50000i 0 0 0 −3.24410 + 1.87298i 0 4.50000 7.79423i 0
149.4 0 2.59808 1.50000i 0 0 0 10.1723 5.87298i 0 4.50000 7.79423i 0
749.1 0 −2.59808 1.50000i 0 0 0 −10.1723 5.87298i 0 4.50000 + 7.79423i 0
749.2 0 −2.59808 1.50000i 0 0 0 3.24410 + 1.87298i 0 4.50000 + 7.79423i 0
749.3 0 2.59808 + 1.50000i 0 0 0 −3.24410 1.87298i 0 4.50000 + 7.79423i 0
749.4 0 2.59808 + 1.50000i 0 0 0 10.1723 + 5.87298i 0 4.50000 + 7.79423i 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.b 8
3.b odd 2 1 2700.3.u.a 8
5.b even 2 1 inner 900.3.u.b 8
5.c odd 4 1 180.3.o.a 4
5.c odd 4 1 900.3.p.b 4
9.c even 3 1 2700.3.u.a 8
9.d odd 6 1 inner 900.3.u.b 8
15.d odd 2 1 2700.3.u.a 8
15.e even 4 1 540.3.o.a 4
15.e even 4 1 2700.3.p.a 4
20.e even 4 1 720.3.bs.a 4
45.h odd 6 1 inner 900.3.u.b 8
45.j even 6 1 2700.3.u.a 8
45.k odd 12 1 540.3.o.a 4
45.k odd 12 1 1620.3.g.a 4
45.k odd 12 1 2700.3.p.a 4
45.l even 12 1 180.3.o.a 4
45.l even 12 1 900.3.p.b 4
45.l even 12 1 1620.3.g.a 4
60.l odd 4 1 2160.3.bs.a 4
180.v odd 12 1 720.3.bs.a 4
180.x even 12 1 2160.3.bs.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 5.c odd 4 1
180.3.o.a 4 45.l even 12 1
540.3.o.a 4 15.e even 4 1
540.3.o.a 4 45.k odd 12 1
720.3.bs.a 4 20.e even 4 1
720.3.bs.a 4 180.v odd 12 1
900.3.p.b 4 5.c odd 4 1
900.3.p.b 4 45.l even 12 1
900.3.u.b 8 1.a even 1 1 trivial
900.3.u.b 8 5.b even 2 1 inner
900.3.u.b 8 9.d odd 6 1 inner
900.3.u.b 8 45.h odd 6 1 inner
1620.3.g.a 4 45.k odd 12 1
1620.3.g.a 4 45.l even 12 1
2160.3.bs.a 4 60.l odd 4 1
2160.3.bs.a 4 180.x even 12 1
2700.3.p.a 4 15.e even 4 1
2700.3.p.a 4 45.k odd 12 1
2700.3.u.a 8 3.b odd 2 1
2700.3.u.a 8 9.c even 3 1
2700.3.u.a 8 15.d odd 2 1
2700.3.u.a 8 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 152 T_{7}^{6} + 21168 T_{7}^{4} - 294272 T_{7}^{2} + 3748096$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 81 - 9 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$3748096 - 294272 T^{2} + 21168 T^{4} - 152 T^{6} + T^{8}$$
$11$ $$( 31329 + 1062 T - 165 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$13$ $$2560000 - 512000 T^{2} + 100800 T^{4} - 320 T^{6} + T^{8}$$
$17$ $$( 31329 - 366 T^{2} + T^{4} )^{2}$$
$19$ $$( 301 - 38 T + T^{2} )^{4}$$
$23$ $$203928109056 + 693633024 T^{2} + 1907712 T^{4} + 1536 T^{6} + T^{8}$$
$29$ $$( 176400 + 25200 T + 780 T^{2} - 60 T^{3} + T^{4} )^{2}$$
$31$ $$( 8620096 - 11744 T + 2952 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 196 + T^{2} )^{4}$$
$41$ $$( 675 + 45 T + T^{2} )^{4}$$
$43$ $$495504774241 - 1521877202 T^{2} + 3970323 T^{4} - 2162 T^{6} + T^{8}$$
$47$ $$116876510171136 + 78876647424 T^{2} + 42420672 T^{4} + 7296 T^{6} + T^{8}$$
$53$ $$( 166464 - 1536 T^{2} + T^{4} )^{2}$$
$59$ $$( 5489649 - 407682 T + 12435 T^{2} - 174 T^{3} + T^{4} )^{2}$$
$61$ $$( 3136 + 2464 T + 1992 T^{2} - 44 T^{3} + T^{4} )^{2}$$
$67$ $$5447219503488961 - 1282588173218 T^{2} + 228189603 T^{4} - 17378 T^{6} + T^{8}$$
$71$ $$( 5143824 + 11016 T^{2} + T^{4} )^{2}$$
$73$ $$( 687241 + 3818 T^{2} + T^{4} )^{2}$$
$79$ $$( 34339600 - 937600 T + 19740 T^{2} - 160 T^{3} + T^{4} )^{2}$$
$83$ $$( 65804544 + 8112 T^{2} + T^{4} )^{2}$$
$89$ $$( 1327104 + 9216 T^{2} + T^{4} )^{2}$$
$97$ $$10567700398061041 - 2084975828522 T^{2} + 308560203 T^{4} - 20282 T^{6} + T^{8}$$