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Properties

Label 4840.2.a.bd.1.4
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
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] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,6,0,4,0,4,0,0,0,0] 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: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.90131\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.614975 q^{3} +1.00000 q^{5} +2.24598 q^{7} -2.62181 q^{9} -5.19447 q^{13} +0.614975 q^{15} +3.22995 q^{17} +7.19447 q^{19} +1.38122 q^{21} +5.11377 q^{23} +1.00000 q^{25} -3.45727 q^{27} -9.61890 q^{29} +0.491468 q^{31} +2.24598 q^{35} +0.350828 q^{37} -3.19447 q^{39} +4.42090 q^{41} +10.9537 q^{43} -2.62181 q^{45} -1.03420 q^{47} -1.95557 q^{49} +1.98634 q^{51} +3.75348 q^{53} +4.42442 q^{57} +12.7082 q^{59} +6.38983 q^{61} -5.88852 q^{63} -5.19447 q^{65} -8.44020 q^{67} +3.14484 q^{69} -12.4475 q^{71} +12.0273 q^{73} +0.614975 q^{75} -2.13034 q^{79} +5.73928 q^{81} +10.7270 q^{83} +3.22995 q^{85} -5.91538 q^{87} +9.90118 q^{89} -11.6667 q^{91} +0.302241 q^{93} +7.19447 q^{95} -15.9253 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) \) 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
\(3\) 0.614975 0.355056 0.177528 − 0.984116i \(-0.443190\pi\)
0.177528 + 0.984116i \(0.443190\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.24598 0.848901 0.424450 − 0.905451i \(-0.360467\pi\)
0.424450 + 0.905451i \(0.360467\pi\)
\(8\) 0 0
\(9\) −2.62181 −0.873935
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.19447 −1.44069 −0.720344 − 0.693617i \(-0.756016\pi\)
−0.720344 + 0.693617i \(0.756016\pi\)
\(14\) 0 0
\(15\) 0.614975 0.158786
\(16\) 0 0
\(17\) 3.22995 0.783378 0.391689 − 0.920098i \(-0.371891\pi\)
0.391689 + 0.920098i \(0.371891\pi\)
\(18\) 0 0
\(19\) 7.19447 1.65053 0.825263 − 0.564749i \(-0.191027\pi\)
0.825263 + 0.564749i \(0.191027\pi\)
\(20\) 0 0
\(21\) 1.38122 0.301408
\(22\) 0 0
\(23\) 5.11377 1.06629 0.533147 − 0.846023i \(-0.321009\pi\)
0.533147 + 0.846023i \(0.321009\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.45727 −0.665352
\(28\) 0 0
\(29\) −9.61890 −1.78618 −0.893092 − 0.449874i \(-0.851469\pi\)
−0.893092 + 0.449874i \(0.851469\pi\)
\(30\) 0 0
\(31\) 0.491468 0.0882703 0.0441352 − 0.999026i \(-0.485947\pi\)
0.0441352 + 0.999026i \(0.485947\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.24598 0.379640
\(36\) 0 0
\(37\) 0.350828 0.0576758 0.0288379 − 0.999584i \(-0.490819\pi\)
0.0288379 + 0.999584i \(0.490819\pi\)
\(38\) 0 0
\(39\) −3.19447 −0.511525
\(40\) 0 0
\(41\) 4.42090 0.690429 0.345214 − 0.938524i \(-0.387806\pi\)
0.345214 + 0.938524i \(0.387806\pi\)
\(42\) 0 0
\(43\) 10.9537 1.67042 0.835211 − 0.549929i \(-0.185345\pi\)
0.835211 + 0.549929i \(0.185345\pi\)
\(44\) 0 0
\(45\) −2.62181 −0.390836
\(46\) 0 0
\(47\) −1.03420 −0.150853 −0.0754265 − 0.997151i \(-0.524032\pi\)
−0.0754265 + 0.997151i \(0.524032\pi\)
\(48\) 0 0
\(49\) −1.95557 −0.279367
\(50\) 0 0
\(51\) 1.98634 0.278143
\(52\) 0 0
\(53\) 3.75348 0.515580 0.257790 − 0.966201i \(-0.417006\pi\)
0.257790 + 0.966201i \(0.417006\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.42442 0.586029
\(58\) 0 0
\(59\) 12.7082 1.65447 0.827234 − 0.561858i \(-0.189913\pi\)
0.827234 + 0.561858i \(0.189913\pi\)
\(60\) 0 0
\(61\) 6.38983 0.818134 0.409067 − 0.912504i \(-0.365854\pi\)
0.409067 + 0.912504i \(0.365854\pi\)
\(62\) 0 0
\(63\) −5.88852 −0.741884
\(64\) 0 0
\(65\) −5.19447 −0.644295
\(66\) 0 0
\(67\) −8.44020 −1.03114 −0.515568 − 0.856849i \(-0.672419\pi\)
−0.515568 + 0.856849i \(0.672419\pi\)
\(68\) 0 0
\(69\) 3.14484 0.378594
\(70\) 0 0
\(71\) −12.4475 −1.47725 −0.738625 − 0.674116i \(-0.764525\pi\)
−0.738625 + 0.674116i \(0.764525\pi\)
\(72\) 0 0
\(73\) 12.0273 1.40769 0.703846 − 0.710353i \(-0.251465\pi\)
0.703846 + 0.710353i \(0.251465\pi\)
\(74\) 0 0
\(75\) 0.614975 0.0710112
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.13034 −0.239682 −0.119841 − 0.992793i \(-0.538238\pi\)
−0.119841 + 0.992793i \(0.538238\pi\)
\(80\) 0 0
\(81\) 5.73928 0.637698
\(82\) 0 0
\(83\) 10.7270 1.17744 0.588719 − 0.808338i \(-0.299633\pi\)
0.588719 + 0.808338i \(0.299633\pi\)
\(84\) 0 0
\(85\) 3.22995 0.350337
\(86\) 0 0
\(87\) −5.91538 −0.634196
\(88\) 0 0
\(89\) 9.90118 1.04952 0.524762 − 0.851249i \(-0.324154\pi\)
0.524762 + 0.851249i \(0.324154\pi\)
\(90\) 0 0
\(91\) −11.6667 −1.22300
\(92\) 0 0
\(93\) 0.302241 0.0313409
\(94\) 0 0
\(95\) 7.19447 0.738137
\(96\) 0 0
\(97\) −15.9253 −1.61697 −0.808484 − 0.588518i \(-0.799712\pi\)
−0.808484 + 0.588518i \(0.799712\pi\)
\(98\) 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 4840.2.a.bd.1.4 yes 6
4.3 odd 2 9680.2.a.da.1.3 6
11.10 odd 2 4840.2.a.bc.1.4 ✓ 6
44.43 even 2 9680.2.a.db.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.4 ✓ 6 11.10 odd 2
4840.2.a.bd.1.4 yes 6 1.1 even 1 trivial
9680.2.a.da.1.3 6 4.3 odd 2
9680.2.a.db.1.3 6 44.43 even 2