Properties

Label 288.2.i.c
Level 288
Weight 2
Character orbit 288.i
Analytic conductor 2.300
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 288.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + ( -1 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} + ( -1 - 2 \beta_{3} ) q^{9} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{11} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{15} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{17} -4 q^{19} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{21} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{23} + ( 4 - 4 \beta_{2} ) q^{25} + ( 5 + \beta_{3} ) q^{27} + ( -5 - 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{29} + ( \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{31} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{35} + ( 4 + 4 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -8 - \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{39} + ( -2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 5 + 3 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{45} + ( 1 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{47} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 4 - 4 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -1 - 2 \beta_{1} + \beta_{3} ) q^{55} + ( 4 - 4 \beta_{3} ) q^{57} + ( -3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{59} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{61} + ( 3 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{65} + ( -3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -4 + 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{69} + ( 2 + 8 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{73} + ( -4 + 4 \beta_{1} + 4 \beta_{2} ) q^{75} -5 \beta_{2} q^{77} + ( 11 - \beta_{1} - 11 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -7 + 4 \beta_{3} ) q^{81} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{85} + ( 1 - 3 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -8 + 4 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -13 + 6 \beta_{1} - 3 \beta_{3} ) q^{91} + ( -4 - 6 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{93} -4 \beta_{2} q^{95} + ( -1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{97} + ( 5 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 2q^{5} + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 2q^{5} + 2q^{7} - 4q^{9} - 2q^{11} - 2q^{13} - 2q^{15} - 16q^{19} - 14q^{21} - 6q^{23} + 8q^{25} + 20q^{27} - 10q^{29} + 10q^{31} - 10q^{33} + 4q^{35} + 16q^{37} - 22q^{39} + 14q^{41} + 10q^{43} - 2q^{45} + 2q^{47} + 16q^{53} - 4q^{55} + 16q^{57} + 14q^{59} + 6q^{61} + 22q^{63} + 2q^{65} + 10q^{67} - 6q^{69} + 8q^{71} - 8q^{75} - 10q^{77} + 22q^{79} - 28q^{81} - 6q^{83} - 14q^{87} - 32q^{89} - 52q^{91} - 22q^{93} - 8q^{95} - 2q^{97} + 26q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−1.22474 0.707107i
1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 −1.00000 1.41421i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.00000 + 2.82843i 0
97.2 0 −1.00000 + 1.41421i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 −1.00000 2.82843i 0
193.1 0 −1.00000 1.41421i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 −1.00000 + 2.82843i 0
193.2 0 −1.00000 + 1.41421i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.00000 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.i.c 4
3.b odd 2 1 864.2.i.e 4
4.b odd 2 1 288.2.i.e yes 4
8.b even 2 1 576.2.i.m 4
8.d odd 2 1 576.2.i.i 4
9.c even 3 1 inner 288.2.i.c 4
9.c even 3 1 2592.2.a.j 2
9.d odd 6 1 864.2.i.e 4
9.d odd 6 1 2592.2.a.o 2
12.b even 2 1 864.2.i.c 4
24.f even 2 1 1728.2.i.k 4
24.h odd 2 1 1728.2.i.m 4
36.f odd 6 1 288.2.i.e yes 4
36.f odd 6 1 2592.2.a.n 2
36.h even 6 1 864.2.i.c 4
36.h even 6 1 2592.2.a.s 2
72.j odd 6 1 1728.2.i.m 4
72.j odd 6 1 5184.2.a.bj 2
72.l even 6 1 1728.2.i.k 4
72.l even 6 1 5184.2.a.bn 2
72.n even 6 1 576.2.i.m 4
72.n even 6 1 5184.2.a.bu 2
72.p odd 6 1 576.2.i.i 4
72.p odd 6 1 5184.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 1.a even 1 1 trivial
288.2.i.c 4 9.c even 3 1 inner
288.2.i.e yes 4 4.b odd 2 1
288.2.i.e yes 4 36.f odd 6 1
576.2.i.i 4 8.d odd 2 1
576.2.i.i 4 72.p odd 6 1
576.2.i.m 4 8.b even 2 1
576.2.i.m 4 72.n even 6 1
864.2.i.c 4 12.b even 2 1
864.2.i.c 4 36.h even 6 1
864.2.i.e 4 3.b odd 2 1
864.2.i.e 4 9.d odd 6 1
1728.2.i.k 4 24.f even 2 1
1728.2.i.k 4 72.l even 6 1
1728.2.i.m 4 24.h odd 2 1
1728.2.i.m 4 72.j odd 6 1
2592.2.a.j 2 9.c even 3 1
2592.2.a.n 2 36.f odd 6 1
2592.2.a.o 2 9.d odd 6 1
2592.2.a.s 2 36.h even 6 1
5184.2.a.bj 2 72.j odd 6 1
5184.2.a.bn 2 72.l even 6 1
5184.2.a.bu 2 72.n even 6 1
5184.2.a.by 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{4} - 2 T_{7}^{3} + 9 T_{7}^{2} + 10 T_{7} + 25 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 70 T^{5} - 245 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 598 T^{5} + 169 T^{6} + 4394 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{4} \)
$23$ \( 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 414 T^{5} - 6877 T^{6} + 73002 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 10 T + 41 T^{2} + 10 T^{3} - 260 T^{4} + 290 T^{5} + 34481 T^{6} + 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 5890 T^{5} + 18259 T^{6} - 297910 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 - 8 T + 66 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 14350 T^{5} + 149609 T^{6} - 964894 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} ) \)
$47$ \( 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 4982 T^{5} - 81733 T^{6} - 207646 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - 8 T + 98 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 14 T + 83 T^{2} + 70 T^{3} - 2276 T^{4} + 4130 T^{5} + 288923 T^{6} - 2875306 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 5490 T^{5} - 264191 T^{6} - 1361886 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 19430 T^{5} - 22445 T^{6} - 3007630 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 4 T + 50 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 122 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 199870 T^{5} + 1316851 T^{6} - 10846858 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T + 218 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 4462 T^{5} - 1571303 T^{6} + 1825346 T^{7} + 88529281 T^{8} \)
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