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Properties

Label 1584.4.a.bm.1.3
Level $1584$
Weight $4$
Character 1584.1
Self dual yes
Analytic conductor $93.459$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [1584,4,Mod(1,1584)] 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("1584.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-8,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(93.4590254491\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.4364.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 19x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 792)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.52203\) of defining polynomial
Character \(\chi\) \(=\) 1584.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.04406 q^{5} +1.85343 q^{7} -11.0000 q^{11} -21.3372 q^{13} -133.719 q^{17} -88.1943 q^{19} +72.4290 q^{23} -75.3813 q^{25} +98.1062 q^{29} +251.825 q^{31} +13.0557 q^{35} +234.671 q^{37} +421.545 q^{41} +468.102 q^{43} +314.237 q^{47} -339.565 q^{49} -247.632 q^{53} -77.4846 q^{55} -463.160 q^{59} +641.754 q^{61} -150.300 q^{65} +20.1049 q^{67} +342.821 q^{71} -1164.40 q^{73} -20.3877 q^{77} +1349.69 q^{79} -12.8829 q^{83} -941.927 q^{85} +1189.45 q^{89} -39.5470 q^{91} -621.246 q^{95} -624.485 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 8 q^{5} - 10 q^{7} - 33 q^{11} - 66 q^{13} - 38 q^{17} - 10 q^{19} - 80 q^{23} - 199 q^{25} + 98 q^{29} + 196 q^{31} + 16 q^{35} - 162 q^{37} + 342 q^{41} + 338 q^{43} + 268 q^{47} - 393 q^{49} - 100 q^{53}+ \cdots - 2282 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 7.04406 0.630040 0.315020 − 0.949085i \(-0.397989\pi\)
0.315020 + 0.949085i \(0.397989\pi\)
\(6\) 0 0
\(7\) 1.85343 0.100076 0.0500379 − 0.998747i \(-0.484066\pi\)
0.0500379 + 0.998747i \(0.484066\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −21.3372 −0.455221 −0.227611 − 0.973752i \(-0.573091\pi\)
−0.227611 + 0.973752i \(0.573091\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −133.719 −1.90775 −0.953873 − 0.300209i \(-0.902944\pi\)
−0.953873 + 0.300209i \(0.902944\pi\)
\(18\) 0 0
\(19\) −88.1943 −1.06490 −0.532451 − 0.846461i \(-0.678729\pi\)
−0.532451 + 0.846461i \(0.678729\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 72.4290 0.656630 0.328315 − 0.944568i \(-0.393519\pi\)
0.328315 + 0.944568i \(0.393519\pi\)
\(24\) 0 0
\(25\) −75.3813 −0.603050
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 98.1062 0.628202 0.314101 − 0.949390i \(-0.398297\pi\)
0.314101 + 0.949390i \(0.398297\pi\)
\(30\) 0 0
\(31\) 251.825 1.45901 0.729503 − 0.683977i \(-0.239751\pi\)
0.729503 + 0.683977i \(0.239751\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.0557 0.0630517
\(36\) 0 0
\(37\) 234.671 1.04269 0.521346 − 0.853345i \(-0.325430\pi\)
0.521346 + 0.853345i \(0.325430\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 421.545 1.60571 0.802857 − 0.596172i \(-0.203312\pi\)
0.802857 + 0.596172i \(0.203312\pi\)
\(42\) 0 0
\(43\) 468.102 1.66011 0.830057 − 0.557678i \(-0.188308\pi\)
0.830057 + 0.557678i \(0.188308\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 314.237 0.975237 0.487619 − 0.873057i \(-0.337866\pi\)
0.487619 + 0.873057i \(0.337866\pi\)
\(48\) 0 0
\(49\) −339.565 −0.989985
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −247.632 −0.641790 −0.320895 − 0.947115i \(-0.603984\pi\)
−0.320895 + 0.947115i \(0.603984\pi\)
\(54\) 0 0
\(55\) −77.4846 −0.189964
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −463.160 −1.02201 −0.511003 − 0.859579i \(-0.670726\pi\)
−0.511003 + 0.859579i \(0.670726\pi\)
\(60\) 0 0
\(61\) 641.754 1.34702 0.673510 − 0.739178i \(-0.264786\pi\)
0.673510 + 0.739178i \(0.264786\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −150.300 −0.286807
\(66\) 0 0
\(67\) 20.1049 0.0366598 0.0183299 − 0.999832i \(-0.494165\pi\)
0.0183299 + 0.999832i \(0.494165\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 342.821 0.573033 0.286516 − 0.958075i \(-0.407503\pi\)
0.286516 + 0.958075i \(0.407503\pi\)
\(72\) 0 0
\(73\) −1164.40 −1.86688 −0.933442 − 0.358728i \(-0.883210\pi\)
−0.933442 + 0.358728i \(0.883210\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −20.3877 −0.0301740
\(78\) 0 0
\(79\) 1349.69 1.92218 0.961090 − 0.276234i \(-0.0890865\pi\)
0.961090 + 0.276234i \(0.0890865\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.8829 −0.0170371 −0.00851854 − 0.999964i \(-0.502712\pi\)
−0.00851854 + 0.999964i \(0.502712\pi\)
\(84\) 0 0
\(85\) −941.927 −1.20196
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1189.45 1.41664 0.708320 − 0.705892i \(-0.249453\pi\)
0.708320 + 0.705892i \(0.249453\pi\)
\(90\) 0 0
\(91\) −39.5470 −0.0455566
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −621.246 −0.670931
\(96\) 0 0
\(97\) −624.485 −0.653679 −0.326840 − 0.945080i \(-0.605984\pi\)
−0.326840 + 0.945080i \(0.605984\pi\)
\(98\) 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 1584.4.a.bm.1.3 3
3.2 odd 2 1584.4.a.bp.1.1 3
4.3 odd 2 792.4.a.k.1.3 ✓ 3
12.11 even 2 792.4.a.o.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
792.4.a.k.1.3 ✓ 3 4.3 odd 2
792.4.a.o.1.1 yes 3 12.11 even 2
1584.4.a.bm.1.3 3 1.1 even 1 trivial
1584.4.a.bp.1.1 3 3.2 odd 2