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Properties

Label 3024.2.cx.j.2575.2
Level $3024$
Weight $2$
Character 3024.2575
Analytic conductor $24.147$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3024,2,Mod(559,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3024.559"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cx (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,6,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2575.2
Character \(\chi\) \(=\) 3024.2575
Dual form 3024.2.cx.j.559.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.37919 - 1.95098i) q^{5} +(-2.55088 - 0.702155i) q^{7} +(-2.79983 + 1.61648i) q^{11} +(4.41092 + 2.54665i) q^{13} +5.22255i q^{17} +6.24838 q^{19} +(0.0232032 + 0.0133964i) q^{23} +(5.11261 + 8.85531i) q^{25} +(-2.19378 - 3.79973i) q^{29} +(1.42877 - 2.47471i) q^{31} +(7.25001 + 7.34942i) q^{35} -9.43733 q^{37} +(3.54513 + 2.04678i) q^{41} +(-2.33333 + 1.34715i) q^{43} +(-3.46834 - 6.00735i) q^{47} +(6.01396 + 3.58223i) q^{49} +2.98191 q^{53} +12.6149 q^{55} +(2.27240 - 3.93592i) q^{59} +(-0.402478 + 0.232371i) q^{61} +(-9.93690 - 17.2112i) q^{65} +(-5.69642 - 3.28883i) q^{67} -1.27973i q^{71} -6.76224i q^{73} +(8.27704 - 2.15753i) q^{77} +(2.02181 - 1.16729i) q^{79} +(-7.18933 - 12.4523i) q^{83} +(10.1891 - 17.6480i) q^{85} -4.82670i q^{89} +(-9.46359 - 9.59334i) q^{91} +(-21.1145 - 12.1904i) q^{95} +(2.00464 - 1.15738i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{7} + 18 q^{23} + 24 q^{25} + 6 q^{29} - 12 q^{37} - 42 q^{43} + 12 q^{49} - 96 q^{53} - 42 q^{65} - 36 q^{67} + 18 q^{77} + 60 q^{79} - 6 q^{85} - 126 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −3.37919 − 1.95098i −1.51122 − 0.872503i −0.999914 − 0.0131039i \(-0.995829\pi\)
−0.511305 − 0.859399i \(-0.670838\pi\)
\(6\) 0 0
\(7\) −2.55088 − 0.702155i −0.964141 − 0.265390i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.79983 + 1.61648i −0.844180 + 0.487387i −0.858683 − 0.512507i \(-0.828717\pi\)
0.0145030 + 0.999895i \(0.495383\pi\)
\(12\) 0 0
\(13\) 4.41092 + 2.54665i 1.22337 + 0.706313i 0.965635 − 0.259902i \(-0.0836904\pi\)
0.257735 + 0.966216i \(0.417024\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.22255i 1.26665i 0.773884 + 0.633327i \(0.218311\pi\)
−0.773884 + 0.633327i \(0.781689\pi\)
\(18\) 0 0
\(19\) 6.24838 1.43348 0.716739 − 0.697342i \(-0.245634\pi\)
0.716739 + 0.697342i \(0.245634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.0232032 + 0.0133964i 0.00483820 + 0.00279334i 0.502417 − 0.864625i \(-0.332444\pi\)
−0.497579 + 0.867419i \(0.665778\pi\)
\(24\) 0 0
\(25\) 5.11261 + 8.85531i 1.02252 + 1.77106i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.19378 − 3.79973i −0.407374 − 0.705592i 0.587221 − 0.809427i \(-0.300222\pi\)
−0.994595 + 0.103835i \(0.966889\pi\)
\(30\) 0 0
\(31\) 1.42877 − 2.47471i 0.256615 − 0.444471i −0.708718 − 0.705492i \(-0.750726\pi\)
0.965333 + 0.261021i \(0.0840593\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.25001 + 7.34942i 1.22548 + 1.24228i
\(36\) 0 0
\(37\) −9.43733 −1.55149 −0.775744 − 0.631048i \(-0.782625\pi\)
−0.775744 + 0.631048i \(0.782625\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.54513 + 2.04678i 0.553656 + 0.319653i 0.750595 − 0.660762i \(-0.229767\pi\)
−0.196939 + 0.980416i \(0.563100\pi\)
\(42\) 0 0
\(43\) −2.33333 + 1.34715i −0.355829 + 0.205438i −0.667250 − 0.744834i \(-0.732529\pi\)
0.311420 + 0.950272i \(0.399195\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.46834 − 6.00735i −0.505910 − 0.876262i −0.999977 − 0.00683782i \(-0.997823\pi\)
0.494067 − 0.869424i \(-0.335510\pi\)
\(48\) 0 0
\(49\) 6.01396 + 3.58223i 0.859137 + 0.511746i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.98191 0.409597 0.204799 − 0.978804i \(-0.434346\pi\)
0.204799 + 0.978804i \(0.434346\pi\)
\(54\) 0 0
\(55\) 12.6149 1.70099
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.27240 − 3.93592i 0.295842 − 0.512413i −0.679339 − 0.733825i \(-0.737733\pi\)
0.975180 + 0.221412i \(0.0710666\pi\)
\(60\) 0 0
\(61\) −0.402478 + 0.232371i −0.0515320 + 0.0297520i −0.525545 − 0.850766i \(-0.676138\pi\)
0.474013 + 0.880518i \(0.342805\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −9.93690 − 17.2112i −1.23252 − 2.13479i
\(66\) 0 0
\(67\) −5.69642 − 3.28883i −0.695929 − 0.401795i 0.109900 − 0.993943i \(-0.464947\pi\)
−0.805829 + 0.592148i \(0.798280\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.27973i − 0.151876i −0.997113 − 0.0759381i \(-0.975805\pi\)
0.997113 − 0.0759381i \(-0.0241951\pi\)
\(72\) 0 0
\(73\) − 6.76224i − 0.791460i −0.918367 − 0.395730i \(-0.870492\pi\)
0.918367 − 0.395730i \(-0.129508\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.27704 − 2.15753i 0.943256 − 0.245874i
\(78\) 0 0
\(79\) 2.02181 − 1.16729i 0.227471 − 0.131331i −0.381934 − 0.924190i \(-0.624742\pi\)
0.609405 + 0.792859i \(0.291408\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.18933 − 12.4523i −0.789131 − 1.36682i −0.926500 − 0.376295i \(-0.877198\pi\)
0.137369 − 0.990520i \(-0.456135\pi\)
\(84\) 0 0
\(85\) 10.1891 − 17.6480i 1.10516 − 1.91419i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.82670i − 0.511629i −0.966726 − 0.255815i \(-0.917656\pi\)
0.966726 − 0.255815i \(-0.0823437\pi\)
\(90\) 0 0
\(91\) −9.46359 − 9.59334i −0.992053 − 1.00566i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.1145 − 12.1904i −2.16630 − 1.25071i
\(96\) 0 0
\(97\) 2.00464 − 1.15738i 0.203540 − 0.117514i −0.394765 − 0.918782i \(-0.629174\pi\)
0.598306 + 0.801268i \(0.295841\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3024.2.cx.j.2575.2 24
3.2 odd 2 1008.2.cx.j.895.4 yes 24
4.3 odd 2 3024.2.cx.i.2575.2 24
7.6 odd 2 inner 3024.2.cx.j.2575.11 24
9.2 odd 6 1008.2.cx.i.223.4 ✓ 24
9.7 even 3 3024.2.cx.i.559.11 24
12.11 even 2 1008.2.cx.i.895.9 yes 24
21.20 even 2 1008.2.cx.j.895.9 yes 24
28.27 even 2 3024.2.cx.i.2575.11 24
36.7 odd 6 inner 3024.2.cx.j.559.11 24
36.11 even 6 1008.2.cx.j.223.9 yes 24
63.20 even 6 1008.2.cx.i.223.9 yes 24
63.34 odd 6 3024.2.cx.i.559.2 24
84.83 odd 2 1008.2.cx.i.895.4 yes 24
252.83 odd 6 1008.2.cx.j.223.4 yes 24
252.223 even 6 inner 3024.2.cx.j.559.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cx.i.223.4 ✓ 24 9.2 odd 6
1008.2.cx.i.223.9 yes 24 63.20 even 6
1008.2.cx.i.895.4 yes 24 84.83 odd 2
1008.2.cx.i.895.9 yes 24 12.11 even 2
1008.2.cx.j.223.4 yes 24 252.83 odd 6
1008.2.cx.j.223.9 yes 24 36.11 even 6
1008.2.cx.j.895.4 yes 24 3.2 odd 2
1008.2.cx.j.895.9 yes 24 21.20 even 2
3024.2.cx.i.559.2 24 63.34 odd 6
3024.2.cx.i.559.11 24 9.7 even 3
3024.2.cx.i.2575.2 24 4.3 odd 2
3024.2.cx.i.2575.11 24 28.27 even 2
3024.2.cx.j.559.2 24 252.223 even 6 inner
3024.2.cx.j.559.11 24 36.7 odd 6 inner
3024.2.cx.j.2575.2 24 1.1 even 1 trivial
3024.2.cx.j.2575.11 24 7.6 odd 2 inner