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Properties

Label 2736.2.s.m.577.1
Level $2736$
Weight $2$
Character 2736.577
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [2736,2,Mod(577,2736)] 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("2736.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,8,0,0,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.577
Dual form 2736.2.s.m.1873.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{7} +3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(-3.00000 - 5.19615i) q^{17} +(3.50000 - 2.59808i) q^{19} +(3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} -2.00000 q^{31} -10.0000 q^{37} +(4.50000 + 7.79423i) q^{41} +(-2.00000 - 3.46410i) q^{43} +9.00000 q^{49} +(3.00000 - 5.19615i) q^{53} +(4.50000 + 7.79423i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-3.50000 + 6.06218i) q^{67} +(3.00000 + 5.19615i) q^{71} +(0.500000 + 0.866025i) q^{73} +12.0000 q^{77} +(-2.00000 - 3.46410i) q^{79} +3.00000 q^{83} +(3.00000 - 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(-8.50000 - 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} + 6 q^{11} - 2 q^{13} - 6 q^{17} + 7 q^{19} + 6 q^{23} + 5 q^{25} - 4 q^{31} - 20 q^{37} + 9 q^{41} - 4 q^{43} + 18 q^{49} + 6 q^{53} + 9 q^{59} + 4 q^{61} - 7 q^{67} + 6 q^{71} + q^{73}+ \cdots - 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 − 5.19615i −0.727607 − 1.26025i −0.957892 − 0.287129i \(-0.907299\pi\)
0.230285 − 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) 3.50000 − 2.59808i 0.802955 − 0.596040i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 − 5.19615i 0.625543 − 1.08347i −0.362892 − 0.931831i \(-0.618211\pi\)
0.988436 − 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 + 7.79423i 0.702782 + 1.21725i 0.967486 + 0.252924i \(0.0813924\pi\)
−0.264704 + 0.964330i \(0.585274\pi\)
\(42\) 0 0
\(43\) −2.00000 − 3.46410i −0.304997 − 0.528271i 0.672264 − 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.50000 + 7.79423i 0.585850 + 1.01472i 0.994769 + 0.102151i \(0.0325726\pi\)
−0.408919 + 0.912571i \(0.634094\pi\)
\(60\) 0 0
\(61\) 2.00000 − 3.46410i 0.256074 − 0.443533i −0.709113 − 0.705095i \(-0.750904\pi\)
0.965187 + 0.261562i \(0.0842377\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.50000 + 6.06218i −0.427593 + 0.740613i −0.996659 − 0.0816792i \(-0.973972\pi\)
0.569066 + 0.822292i \(0.307305\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 + 5.19615i 0.356034 + 0.616670i 0.987294 − 0.158901i \(-0.0507952\pi\)
−0.631260 + 0.775571i \(0.717462\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.0585206 + 0.101361i 0.893801 − 0.448463i \(-0.148028\pi\)
−0.835281 + 0.549823i \(0.814695\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −2.00000 − 3.46410i −0.225018 − 0.389742i 0.731307 − 0.682048i \(-0.238911\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 − 5.19615i 0.317999 − 0.550791i −0.662071 − 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) −4.00000 + 6.92820i −0.419314 + 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50000 − 14.7224i −0.863044 − 1.49484i −0.868976 − 0.494854i \(-0.835222\pi\)
0.00593185 − 0.999982i \(-0.498112\pi\)
\(98\) 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 2736.2.s.m.577.1 2
3.2 odd 2 304.2.i.c.273.1 2
4.3 odd 2 342.2.g.b.235.1 2
12.11 even 2 38.2.c.a.7.1 ✓ 2
19.11 even 3 inner 2736.2.s.m.1873.1 2
24.5 odd 2 1216.2.i.d.577.1 2
24.11 even 2 1216.2.i.h.577.1 2
57.11 odd 6 304.2.i.c.49.1 2
57.26 odd 6 5776.2.a.g.1.1 1
57.50 even 6 5776.2.a.n.1.1 1
60.23 odd 4 950.2.j.e.349.2 4
60.47 odd 4 950.2.j.e.349.1 4
60.59 even 2 950.2.e.d.501.1 2
76.7 odd 6 6498.2.a.s.1.1 1
76.11 odd 6 342.2.g.b.163.1 2
76.31 even 6 6498.2.a.e.1.1 1
228.11 even 6 38.2.c.a.11.1 yes 2
228.23 even 18 722.2.e.j.245.1 6
228.35 even 18 722.2.e.j.595.1 6
228.47 even 18 722.2.e.j.415.1 6
228.59 odd 18 722.2.e.i.389.1 6
228.71 odd 18 722.2.e.i.99.1 6
228.83 even 6 722.2.a.c.1.1 1
228.107 odd 6 722.2.a.d.1.1 1
228.119 even 18 722.2.e.j.99.1 6
228.131 even 18 722.2.e.j.389.1 6
228.143 odd 18 722.2.e.i.415.1 6
228.155 odd 18 722.2.e.i.595.1 6
228.167 odd 18 722.2.e.i.245.1 6
228.179 odd 6 722.2.c.b.429.1 2
228.203 odd 18 722.2.e.i.423.1 6
228.215 even 18 722.2.e.j.423.1 6
228.227 odd 2 722.2.c.b.653.1 2
456.11 even 6 1216.2.i.h.961.1 2
456.125 odd 6 1216.2.i.d.961.1 2
1140.239 even 6 950.2.e.d.201.1 2
1140.467 odd 12 950.2.j.e.49.2 4
1140.923 odd 12 950.2.j.e.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.c.a.7.1 ✓ 2 12.11 even 2
38.2.c.a.11.1 yes 2 228.11 even 6
304.2.i.c.49.1 2 57.11 odd 6
304.2.i.c.273.1 2 3.2 odd 2
342.2.g.b.163.1 2 76.11 odd 6
342.2.g.b.235.1 2 4.3 odd 2
722.2.a.c.1.1 1 228.83 even 6
722.2.a.d.1.1 1 228.107 odd 6
722.2.c.b.429.1 2 228.179 odd 6
722.2.c.b.653.1 2 228.227 odd 2
722.2.e.i.99.1 6 228.71 odd 18
722.2.e.i.245.1 6 228.167 odd 18
722.2.e.i.389.1 6 228.59 odd 18
722.2.e.i.415.1 6 228.143 odd 18
722.2.e.i.423.1 6 228.203 odd 18
722.2.e.i.595.1 6 228.155 odd 18
722.2.e.j.99.1 6 228.119 even 18
722.2.e.j.245.1 6 228.23 even 18
722.2.e.j.389.1 6 228.131 even 18
722.2.e.j.415.1 6 228.47 even 18
722.2.e.j.423.1 6 228.215 even 18
722.2.e.j.595.1 6 228.35 even 18
950.2.e.d.201.1 2 1140.239 even 6
950.2.e.d.501.1 2 60.59 even 2
950.2.j.e.49.1 4 1140.923 odd 12
950.2.j.e.49.2 4 1140.467 odd 12
950.2.j.e.349.1 4 60.47 odd 4
950.2.j.e.349.2 4 60.23 odd 4
1216.2.i.d.577.1 2 24.5 odd 2
1216.2.i.d.961.1 2 456.125 odd 6
1216.2.i.h.577.1 2 24.11 even 2
1216.2.i.h.961.1 2 456.11 even 6
2736.2.s.m.577.1 2 1.1 even 1 trivial
2736.2.s.m.1873.1 2 19.11 even 3 inner
5776.2.a.g.1.1 1 57.26 odd 6
5776.2.a.n.1.1 1 57.50 even 6
6498.2.a.e.1.1 1 76.31 even 6
6498.2.a.s.1.1 1 76.7 odd 6