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Properties

Label 2736.2.s.z.1873.1
Level $2736$
Weight $2$
Character 2736.1873
Analytic conductor $21.847$
Analytic rank $0$
Dimension $6$
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 = [6,0,0,0,2,0,2,0,0,0,0,0,1] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.1
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1873
Dual form 2736.2.s.z.577.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.33641 + 2.31473i) q^{5} +3.67282 q^{7} -3.81681 q^{11} +(-0.0719933 - 0.124696i) q^{13} +(-4.24482 - 0.990721i) q^{19} +(-3.76442 - 6.52016i) q^{23} +(-1.07199 - 1.85675i) q^{25} +(-2.67282 - 4.62947i) q^{29} -8.81681 q^{31} +(-4.90841 + 8.50161i) q^{35} -1.00000 q^{37} +(2.67282 - 4.62947i) q^{41} +(1.40841 - 2.43943i) q^{43} +(3.00000 + 5.19615i) q^{47} +6.48963 q^{49} +(4.00924 + 6.94420i) q^{53} +(5.10083 - 8.83490i) q^{55} +(-1.90841 + 3.30545i) q^{59} +(-5.74482 - 9.95031i) q^{61} +0.384851 q^{65} +(2.69243 + 4.66342i) q^{67} +(6.81681 - 11.8071i) q^{71} +(0.172824 - 0.299339i) q^{73} -14.0185 q^{77} +(3.26442 - 5.65414i) q^{79} -2.28797 q^{83} +(-4.33641 - 7.51089i) q^{89} +(-0.264419 - 0.457986i) q^{91} +(7.96608 - 8.50161i) q^{95} +(2.95684 - 5.12140i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(5\) −1.33641 + 2.31473i −0.597662 + 1.03518i 0.395504 + 0.918464i \(0.370570\pi\)
−0.993165 + 0.116716i \(0.962763\pi\)
\(6\) 0 0
\(7\) 3.67282 1.38820 0.694098 − 0.719880i \(-0.255803\pi\)
0.694098 + 0.719880i \(0.255803\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.81681 −1.15081 −0.575406 − 0.817868i \(-0.695156\pi\)
−0.575406 + 0.817868i \(0.695156\pi\)
\(12\) 0 0
\(13\) −0.0719933 − 0.124696i −0.0199673 − 0.0345844i 0.855869 − 0.517193i \(-0.173023\pi\)
−0.875836 + 0.482608i \(0.839690\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −4.24482 − 0.990721i −0.973828 − 0.227287i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.76442 − 6.52016i −0.784936 − 1.35955i −0.929038 − 0.369985i \(-0.879363\pi\)
0.144102 − 0.989563i \(-0.453971\pi\)
\(24\) 0 0
\(25\) −1.07199 − 1.85675i −0.214399 − 0.371349i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.67282 − 4.62947i −0.496331 − 0.859670i 0.503660 − 0.863902i \(-0.331986\pi\)
−0.999991 + 0.00423154i \(0.998653\pi\)
\(30\) 0 0
\(31\) −8.81681 −1.58355 −0.791773 − 0.610816i \(-0.790842\pi\)
−0.791773 + 0.610816i \(0.790842\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.90841 + 8.50161i −0.829672 + 1.43703i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.67282 − 4.62947i 0.417425 − 0.723001i −0.578255 − 0.815856i \(-0.696266\pi\)
0.995680 + 0.0928551i \(0.0295993\pi\)
\(42\) 0 0
\(43\) 1.40841 − 2.43943i 0.214780 − 0.372009i −0.738425 − 0.674336i \(-0.764430\pi\)
0.953204 + 0.302327i \(0.0977633\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) 6.48963 0.927091
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00924 + 6.94420i 0.550711 + 0.953859i 0.998223 + 0.0595815i \(0.0189766\pi\)
−0.447513 + 0.894278i \(0.647690\pi\)
\(54\) 0 0
\(55\) 5.10083 − 8.83490i 0.687796 − 1.19130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.90841 + 3.30545i −0.248453 + 0.430334i −0.963097 − 0.269155i \(-0.913256\pi\)
0.714644 + 0.699489i \(0.246589\pi\)
\(60\) 0 0
\(61\) −5.74482 − 9.95031i −0.735548 − 1.27401i −0.954482 − 0.298268i \(-0.903591\pi\)
0.218934 − 0.975740i \(-0.429742\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.384851 0.0477348
\(66\) 0 0
\(67\) 2.69243 + 4.66342i 0.328932 + 0.569727i 0.982300 − 0.187313i \(-0.0599779\pi\)
−0.653368 + 0.757040i \(0.726645\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.81681 − 11.8071i 0.809007 − 1.40124i −0.104546 − 0.994520i \(-0.533339\pi\)
0.913553 − 0.406720i \(-0.133328\pi\)
\(72\) 0 0
\(73\) 0.172824 − 0.299339i 0.0202275 − 0.0350350i −0.855734 − 0.517415i \(-0.826894\pi\)
0.875962 + 0.482380i \(0.160228\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.0185 −1.59755
\(78\) 0 0
\(79\) 3.26442 − 5.65414i 0.367276 − 0.636140i −0.621863 − 0.783126i \(-0.713624\pi\)
0.989139 + 0.146986i \(0.0469572\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.28797 −0.251138 −0.125569 − 0.992085i \(-0.540076\pi\)
−0.125569 + 0.992085i \(0.540076\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.33641 − 7.51089i −0.459659 − 0.796152i 0.539284 − 0.842124i \(-0.318695\pi\)
−0.998943 + 0.0459717i \(0.985362\pi\)
\(90\) 0 0
\(91\) −0.264419 − 0.457986i −0.0277186 − 0.0480100i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.96608 − 8.50161i 0.817303 − 0.872246i
\(96\) 0 0
\(97\) 2.95684 − 5.12140i 0.300222 − 0.520000i −0.675964 − 0.736935i \(-0.736273\pi\)
0.976186 + 0.216935i \(0.0696059\pi\)
\(98\) 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.z.1873.1 6
3.2 odd 2 912.2.q.l.49.3 6
4.3 odd 2 171.2.f.b.163.2 6
12.11 even 2 57.2.e.b.49.2 yes 6
19.7 even 3 inner 2736.2.s.z.577.1 6
57.26 odd 6 912.2.q.l.577.3 6
76.7 odd 6 171.2.f.b.64.2 6
76.11 odd 6 3249.2.a.y.1.2 3
76.27 even 6 3249.2.a.t.1.2 3
228.11 even 6 1083.2.a.l.1.2 3
228.83 even 6 57.2.e.b.7.2 ✓ 6
228.179 odd 6 1083.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 228.83 even 6
57.2.e.b.49.2 yes 6 12.11 even 2
171.2.f.b.64.2 6 76.7 odd 6
171.2.f.b.163.2 6 4.3 odd 2
912.2.q.l.49.3 6 3.2 odd 2
912.2.q.l.577.3 6 57.26 odd 6
1083.2.a.l.1.2 3 228.11 even 6
1083.2.a.o.1.2 3 228.179 odd 6
2736.2.s.z.577.1 6 19.7 even 3 inner
2736.2.s.z.1873.1 6 1.1 even 1 trivial
3249.2.a.t.1.2 3 76.27 even 6
3249.2.a.y.1.2 3 76.11 odd 6