Properties

Label 2268.2.i.l
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
Defining polynomial: \(x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{5} + \beta_{5} q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{5} + \beta_{5} q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{17} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{23} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{25} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{29} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{31} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{43} + ( -3 - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{47} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{49} + ( \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 1 - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{55} + ( -2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{59} + ( 4 + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( 3 - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{65} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{67} + ( -1 + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{71} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{73} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{77} + ( 1 + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{79} + ( -2 \beta_{1} - 10 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{83} + ( 5 + \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{85} + ( -6 - 3 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{89} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{91} + ( 5 + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{95} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} + q^{7} + O(q^{10}) \) \( 8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59} + 24 q^{61} + 24 q^{65} + 14 q^{67} - 22 q^{71} - 10 q^{73} - 11 q^{77} + 35 q^{83} + 13 q^{85} - 18 q^{89} - 9 q^{91} + 20 q^{95} - 19 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 14 \nu^{7} + 23 \nu^{6} - 92 \nu^{5} - 14 \nu^{4} - 391 \nu^{3} + 437 \nu^{2} - 1586 \nu + 92 \)\()/289\)
\(\beta_{2}\)\(=\)\((\)\( 64 \nu^{7} - 60 \nu^{6} + 240 \nu^{5} - 353 \nu^{4} + 1020 \nu^{3} - 1140 \nu^{2} + 305 \nu - 240 \)\()/289\)
\(\beta_{3}\)\(=\)\((\)\( 125 \nu^{7} - 63 \nu^{6} + 541 \nu^{5} - 414 \nu^{4} + 2227 \nu^{3} - 1197 \nu^{2} + 1693 \nu + 326 \)\()/289\)
\(\beta_{4}\)\(=\)\((\)\( -237 \nu^{7} - 121 \nu^{6} - 961 \nu^{5} - 52 \nu^{4} - 3434 \nu^{3} - 854 \nu^{2} - 854 \nu - 773 \)\()/289\)
\(\beta_{5}\)\(=\)\((\)\( 273 \nu^{7} + 15 \nu^{6} + 1096 \nu^{5} - 562 \nu^{4} + 4080 \nu^{3} - 1160 \nu^{2} + 1730 \nu - 229 \)\()/289\)
\(\beta_{6}\)\(=\)\((\)\( -344 \nu^{7} - 111 \nu^{6} - 1290 \nu^{5} + 344 \nu^{4} - 4471 \nu^{3} - 86 \nu^{2} - 86 \nu - 444 \)\()/289\)
\(\beta_{7}\)\(=\)\((\)\( -412 \nu^{7} + 25 \nu^{6} - 1545 \nu^{5} + 990 \nu^{4} - 5916 \nu^{3} + 1920 \nu^{2} - 970 \nu - 189 \)\()/289\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{5} + \beta_{3} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - \beta_{1} - 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 5 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 3\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{7} + \beta_{5} - 8 \beta_{4} + \beta_{3} + 21 \beta_{2} + 5 \beta_{1}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{7} - 20 \beta_{6} - 17 \beta_{5} + 17 \beta_{4} + 6 \beta_{3} - 18 \beta_{2} - 20 \beta_{1} - 18\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-7 \beta_{7} + 24 \beta_{6} + 31 \beta_{5} - 7 \beta_{4} - 38 \beta_{3} + 81\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(73 \beta_{7} + 42 \beta_{5} - 31 \beta_{4} + 42 \beta_{3} + 90 \beta_{2} + 80 \beta_{1}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.346911 0.600868i
−1.03075 + 1.78531i
−0.198169 + 0.343239i
0.882007 1.52768i
0.346911 + 0.600868i
−1.03075 1.78531i
−0.198169 0.343239i
0.882007 + 1.52768i
0 0 0 −2.00677 + 3.47583i 0 −1.89234 1.84906i 0 0 0
865.2 0 0 0 −0.951526 + 1.64809i 0 1.46157 + 2.20541i 0 0 0
865.3 0 0 0 0.705299 1.22161i 0 −0.779537 + 2.52830i 0 0 0
865.4 0 0 0 1.25300 2.17026i 0 1.71031 2.01862i 0 0 0
2053.1 0 0 0 −2.00677 3.47583i 0 −1.89234 + 1.84906i 0 0 0
2053.2 0 0 0 −0.951526 1.64809i 0 1.46157 2.20541i 0 0 0
2053.3 0 0 0 0.705299 + 1.22161i 0 −0.779537 2.52830i 0 0 0
2053.4 0 0 0 1.25300 + 2.17026i 0 1.71031 + 2.01862i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2053.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.l 8
3.b odd 2 1 2268.2.i.m 8
7.c even 3 1 2268.2.l.m 8
9.c even 3 1 2268.2.k.c 8
9.c even 3 1 2268.2.l.m 8
9.d odd 6 1 2268.2.k.d yes 8
9.d odd 6 1 2268.2.l.l 8
21.h odd 6 1 2268.2.l.l 8
63.g even 3 1 2268.2.k.c 8
63.h even 3 1 inner 2268.2.i.l 8
63.j odd 6 1 2268.2.i.m 8
63.n odd 6 1 2268.2.k.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.l 8 1.a even 1 1 trivial
2268.2.i.l 8 63.h even 3 1 inner
2268.2.i.m 8 3.b odd 2 1
2268.2.i.m 8 63.j odd 6 1
2268.2.k.c 8 9.c even 3 1
2268.2.k.c 8 63.g even 3 1
2268.2.k.d yes 8 9.d odd 6 1
2268.2.k.d yes 8 63.n odd 6 1
2268.2.l.l 8 9.d odd 6 1
2268.2.l.l 8 21.h odd 6 1
2268.2.l.m 8 7.c even 3 1
2268.2.l.m 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{8} + 2 T_{5}^{7} + 16 T_{5}^{6} - 6 T_{5}^{5} + 135 T_{5}^{4} + 405 T_{5}^{2} - 243 T_{5} + 729 \)
\(T_{13}^{8} + \cdots\)
\(T_{19}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 729 - 243 T + 405 T^{2} + 135 T^{4} - 6 T^{5} + 16 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( 2401 - 343 T + 490 T^{2} - 35 T^{3} + 101 T^{4} - 5 T^{5} + 10 T^{6} - T^{7} + T^{8} \)
$11$ \( 729 - 972 T + 1134 T^{2} - 486 T^{3} + 243 T^{4} - 42 T^{5} + 31 T^{6} - 5 T^{7} + T^{8} \)
$13$ \( 9 + 48 T + 220 T^{2} + 210 T^{3} + 195 T^{4} - 4 T^{5} + 21 T^{6} + 3 T^{7} + T^{8} \)
$17$ \( 35721 + 3402 T + 6561 T^{2} + 162 T^{3} + 936 T^{4} + 30 T^{5} + 37 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( 97969 - 66043 T + 53911 T^{2} + 1322 T^{3} + 2275 T^{4} + 182 T^{5} + 94 T^{6} + 8 T^{7} + T^{8} \)
$23$ \( 35721 - 39123 T + 32643 T^{2} - 11934 T^{3} + 3519 T^{4} - 306 T^{5} + 58 T^{6} - 2 T^{7} + T^{8} \)
$29$ \( 729 - 486 T + 1944 T^{2} + 972 T^{3} + 3609 T^{4} - 84 T^{5} + 64 T^{6} + 2 T^{7} + T^{8} \)
$31$ \( ( -423 + 340 T - 75 T^{2} + T^{4} )^{2} \)
$37$ \( 361 + 1805 T + 9595 T^{2} - 2698 T^{3} + 1261 T^{4} - 70 T^{5} + 46 T^{6} - 4 T^{7} + T^{8} \)
$41$ \( 59049 + 65610 T + 55404 T^{2} + 20898 T^{3} + 6237 T^{4} + 324 T^{5} + 81 T^{6} + 3 T^{7} + T^{8} \)
$43$ \( 39601 - 18706 T + 14806 T^{2} + 830 T^{3} + 1171 T^{4} + 38 T^{5} + 55 T^{6} + 5 T^{7} + T^{8} \)
$47$ \( ( -81 - 81 T + 45 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$53$ \( 66928761 + 12590559 T + 2957553 T^{2} + 281880 T^{3} + 50301 T^{4} + 4806 T^{5} + 504 T^{6} + 24 T^{7} + T^{8} \)
$59$ \( ( 2403 - 405 T - 90 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$61$ \( ( -4497 + 1594 T - 105 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$67$ \( ( 12527 + 970 T - 252 T^{2} - 7 T^{3} + T^{4} )^{2} \)
$71$ \( ( -7803 - 2295 T - 141 T^{2} + 11 T^{3} + T^{4} )^{2} \)
$73$ \( 2411809 - 1156985 T + 694795 T^{2} + 35990 T^{3} + 13997 T^{4} + 590 T^{5} + 190 T^{6} + 10 T^{7} + T^{8} \)
$79$ \( ( 729 - 227 T - 120 T^{2} + T^{4} )^{2} \)
$83$ \( 729 - 36450 T + 1812132 T^{2} - 516510 T^{3} + 100179 T^{4} - 10740 T^{5} + 841 T^{6} - 35 T^{7} + T^{8} \)
$89$ \( 344065401 + 72619335 T + 12990051 T^{2} + 1161054 T^{3} + 104895 T^{4} + 5562 T^{5} + 450 T^{6} + 18 T^{7} + T^{8} \)
$97$ \( 21520321 + 8257420 T + 2917894 T^{2} + 272402 T^{3} + 41375 T^{4} + 2534 T^{5} + 415 T^{6} + 19 T^{7} + T^{8} \)
show more
show less