# Properties

 Label 228.2.q.a Level $228$ Weight $2$ Character orbit 228.q Analytic conductor $1.821$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$228 = 2^{2} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 228.q (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.82058916609$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{4} q^{3} + ( 1 + \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 1 + 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{4} q^{3} + ( 1 + \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 1 + 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{9} + ( \zeta_{18} - 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{13} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{15} + ( 1 + \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{17} + ( 2 + 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( 2 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{21} + ( -3 + \zeta_{18} - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{23} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} -\zeta_{18}^{3} q^{27} + ( -4 - 4 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{29} + ( -2 - 3 \zeta_{18} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{31} + ( 2 - \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{33} + ( 1 - \zeta_{18}^{2} - 4 \zeta_{18}^{3} - \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{35} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( -3 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} ) q^{39} + ( -3 + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{41} + ( -2 - 7 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 7 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{43} + ( -1 - \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{45} + ( 5 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{47} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{49} + ( -1 - 2 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{51} + ( -3 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{53} + ( -5 - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{55} + ( 1 + 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{57} + ( 5 - 3 \zeta_{18} + \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{59} + ( -1 - \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{61} + ( -2 + \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{63} + ( 3 \zeta_{18} + 2 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{65} + ( 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{67} + ( 4 - \zeta_{18} + 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{69} + ( -6 + 3 \zeta_{18} - 6 \zeta_{18}^{2} + 6 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{71} + ( 3 - 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{73} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + ( -8 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{77} + ( 2 - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 9 \zeta_{18}^{4} ) q^{79} + ( \zeta_{18} - \zeta_{18}^{4} ) q^{81} + ( 5 + 3 \zeta_{18} - 7 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{83} + ( 5 + 5 \zeta_{18} + 6 \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{85} + ( -4 \zeta_{18} - \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{87} + ( -1 - 6 \zeta_{18} + 7 \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{89} + ( 1 + 8 \zeta_{18} + 3 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{91} + ( -3 - 2 \zeta_{18} - 3 \zeta_{18}^{2} ) q^{93} + ( 7 + \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{3} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{95} + ( 5 - 2 \zeta_{18} - 2 \zeta_{18}^{2} - 7 \zeta_{18}^{3} + 7 \zeta_{18}^{5} ) q^{97} + ( -1 + 3 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{5} + 3 q^{7} + O(q^{10})$$ $$6 q + 6 q^{5} + 3 q^{7} + 3 q^{11} + 6 q^{13} + 6 q^{15} + 12 q^{17} + 6 q^{19} + 9 q^{21} - 15 q^{23} - 3 q^{27} - 12 q^{29} - 6 q^{31} + 3 q^{33} - 6 q^{35} - 12 q^{37} - 18 q^{39} - 18 q^{41} - 18 q^{43} - 3 q^{45} - 3 q^{47} + 3 q^{51} - 24 q^{53} - 27 q^{55} + 9 q^{57} + 18 q^{59} - 9 q^{61} - 9 q^{63} + 21 q^{65} - 6 q^{67} + 12 q^{69} - 18 q^{71} + 21 q^{73} - 12 q^{75} - 48 q^{77} + 6 q^{79} + 15 q^{83} + 27 q^{85} - 3 q^{87} + 15 q^{89} + 30 q^{91} - 18 q^{93} + 24 q^{95} + 9 q^{97} + 3 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$115$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.766044 + 0.642788i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i 0.939693 − 0.342020i −0.173648 + 0.984808i
0 −0.939693 0.342020i 0 0.233956 + 1.32683i 0 −1.20574 + 2.08840i 0 0.766044 + 0.642788i 0
61.1 0 0.173648 + 0.984808i 0 1.93969 1.62760i 0 1.61334 2.79439i 0 −0.939693 + 0.342020i 0
73.1 0 −0.939693 + 0.342020i 0 0.233956 1.32683i 0 −1.20574 2.08840i 0 0.766044 0.642788i 0
85.1 0 0.766044 0.642788i 0 0.826352 + 0.300767i 0 1.09240 1.89209i 0 0.173648 0.984808i 0
157.1 0 0.173648 0.984808i 0 1.93969 + 1.62760i 0 1.61334 + 2.79439i 0 −0.939693 0.342020i 0
169.1 0 0.766044 + 0.642788i 0 0.826352 0.300767i 0 1.09240 + 1.89209i 0 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.2.q.a 6
3.b odd 2 1 684.2.bo.a 6
4.b odd 2 1 912.2.bo.e 6
19.e even 9 1 inner 228.2.q.a 6
19.e even 9 1 4332.2.a.o 3
19.f odd 18 1 4332.2.a.n 3
57.l odd 18 1 684.2.bo.a 6
76.l odd 18 1 912.2.bo.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.q.a 6 1.a even 1 1 trivial
228.2.q.a 6 19.e even 9 1 inner
684.2.bo.a 6 3.b odd 2 1
684.2.bo.a 6 57.l odd 18 1
912.2.bo.e 6 4.b odd 2 1
912.2.bo.e 6 76.l odd 18 1
4332.2.a.n 3 19.f odd 18 1
4332.2.a.o 3 19.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 6 T_{5}^{5} + 18 T_{5}^{4} - 30 T_{5}^{3} + 36 T_{5}^{2} - 27 T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(228, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + T^{3} + T^{6}$$
$5$ $$9 - 27 T + 36 T^{2} - 30 T^{3} + 18 T^{4} - 6 T^{5} + T^{6}$$
$7$ $$289 - 102 T + 87 T^{2} - 16 T^{3} + 15 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$9 - 54 T + 333 T^{2} + 48 T^{3} + 27 T^{4} - 3 T^{5} + T^{6}$$
$13$ $$289 - 714 T + 786 T^{2} - 271 T^{3} + 42 T^{4} - 6 T^{5} + T^{6}$$
$17$ $$9 + 81 T + 306 T^{2} - 132 T^{3} + 54 T^{4} - 12 T^{5} + T^{6}$$
$19$ $$6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$3249 + 5643 T + 3465 T^{2} + 840 T^{3} + 144 T^{4} + 15 T^{5} + T^{6}$$
$29$ $$12321 - 3996 T + 360 T^{2} + 321 T^{3} + 72 T^{4} + 12 T^{5} + T^{6}$$
$31$ $$361 + 285 T + 339 T^{2} - 52 T^{3} + 51 T^{4} + 6 T^{5} + T^{6}$$
$37$ $$( -17 - 9 T + 6 T^{2} + T^{3} )^{2}$$
$41$ $$729 - 3645 T + 4860 T^{2} + 945 T^{3} + 189 T^{4} + 18 T^{5} + T^{6}$$
$43$ $$361 - 3762 T + 10782 T^{2} + 2141 T^{3} + 270 T^{4} + 18 T^{5} + T^{6}$$
$47$ $$12321 - 999 T + 522 T^{2} - 192 T^{3} - 9 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$45369 + 17253 T + 5220 T^{2} + 1353 T^{3} + 243 T^{4} + 24 T^{5} + T^{6}$$
$59$ $$81 - 567 T + 1134 T^{2} - 72 T^{3} + 144 T^{4} - 18 T^{5} + T^{6}$$
$61$ $$5041 - 3834 T + 1161 T^{2} + 287 T^{3} + 36 T^{4} + 9 T^{5} + T^{6}$$
$67$ $$64 - 96 T + 96 T^{2} - 64 T^{3} + 12 T^{4} + 6 T^{5} + T^{6}$$
$71$ $$263169 + 138510 T + 29646 T^{2} + 3375 T^{3} + 270 T^{4} + 18 T^{5} + T^{6}$$
$73$ $$361 - 456 T + 672 T^{2} - 442 T^{3} + 156 T^{4} - 21 T^{5} + T^{6}$$
$79$ $$375769 + 47814 T + 30 T^{2} - 1333 T^{3} + 78 T^{4} - 6 T^{5} + T^{6}$$
$83$ $$9 + 108 T + 1251 T^{2} + 546 T^{3} + 261 T^{4} - 15 T^{5} + T^{6}$$
$89$ $$145161 + 72009 T + 11133 T^{2} + 672 T^{3} + 108 T^{4} - 15 T^{5} + T^{6}$$
$97$ $$143641 + 68220 T + 7614 T^{2} - 244 T^{3} + 90 T^{4} - 9 T^{5} + T^{6}$$
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