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 $2$

Related objects

Downloads

Learn more

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:

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

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