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Properties

Label 4840.2.a.ba.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
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,-3,0,-6,0,-7,0,5,0,0,0,1] 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: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.29288\) 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)
 
\(f(q)\) \(=\) \(q-3.29288 q^{3} -1.00000 q^{5} -3.51508 q^{7} +7.84307 q^{9} +4.27118 q^{13} +3.29288 q^{15} -1.58998 q^{17} -5.55094 q^{19} +11.5747 q^{21} -6.39042 q^{23} +1.00000 q^{25} -15.9477 q^{27} +0.381220 q^{29} -0.760465 q^{31} +3.51508 q^{35} -1.45191 q^{37} -14.0645 q^{39} +6.12208 q^{41} +7.19067 q^{43} -7.84307 q^{45} +4.32164 q^{47} +5.35577 q^{49} +5.23561 q^{51} -6.94904 q^{53} +18.2786 q^{57} +12.7110 q^{59} +8.89599 q^{61} -27.5690 q^{63} -4.27118 q^{65} +12.5135 q^{67} +21.0429 q^{69} +7.35890 q^{71} -7.03210 q^{73} -3.29288 q^{75} +6.99381 q^{79} +28.9846 q^{81} +10.7509 q^{83} +1.58998 q^{85} -1.25531 q^{87} +0.451594 q^{89} -15.0135 q^{91} +2.50412 q^{93} +5.55094 q^{95} -2.77956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 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
\(3\) −3.29288 −1.90115 −0.950573 − 0.310501i \(-0.899503\pi\)
−0.950573 + 0.310501i \(0.899503\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.51508 −1.32857 −0.664287 − 0.747477i \(-0.731265\pi\)
−0.664287 + 0.747477i \(0.731265\pi\)
\(8\) 0 0
\(9\) 7.84307 2.61436
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 4.27118 1.18461 0.592306 − 0.805713i \(-0.298218\pi\)
0.592306 + 0.805713i \(0.298218\pi\)
\(14\) 0 0
\(15\) 3.29288 0.850218
\(16\) 0 0
\(17\) −1.58998 −0.385626 −0.192813 − 0.981236i \(-0.561761\pi\)
−0.192813 + 0.981236i \(0.561761\pi\)
\(18\) 0 0
\(19\) −5.55094 −1.27347 −0.636736 − 0.771082i \(-0.719716\pi\)
−0.636736 + 0.771082i \(0.719716\pi\)
\(20\) 0 0
\(21\) 11.5747 2.52581
\(22\) 0 0
\(23\) −6.39042 −1.33249 −0.666247 − 0.745731i \(-0.732100\pi\)
−0.666247 + 0.745731i \(0.732100\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −15.9477 −3.06913
\(28\) 0 0
\(29\) 0.381220 0.0707908 0.0353954 − 0.999373i \(-0.488731\pi\)
0.0353954 + 0.999373i \(0.488731\pi\)
\(30\) 0 0
\(31\) −0.760465 −0.136584 −0.0682918 − 0.997665i \(-0.521755\pi\)
−0.0682918 + 0.997665i \(0.521755\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.51508 0.594156
\(36\) 0 0
\(37\) −1.45191 −0.238692 −0.119346 − 0.992853i \(-0.538080\pi\)
−0.119346 + 0.992853i \(0.538080\pi\)
\(38\) 0 0
\(39\) −14.0645 −2.25212
\(40\) 0 0
\(41\) 6.12208 0.956108 0.478054 − 0.878330i \(-0.341342\pi\)
0.478054 + 0.878330i \(0.341342\pi\)
\(42\) 0 0
\(43\) 7.19067 1.09657 0.548284 − 0.836293i \(-0.315281\pi\)
0.548284 + 0.836293i \(0.315281\pi\)
\(44\) 0 0
\(45\) −7.84307 −1.16918
\(46\) 0 0
\(47\) 4.32164 0.630376 0.315188 − 0.949029i \(-0.397932\pi\)
0.315188 + 0.949029i \(0.397932\pi\)
\(48\) 0 0
\(49\) 5.35577 0.765110
\(50\) 0 0
\(51\) 5.23561 0.733132
\(52\) 0 0
\(53\) −6.94904 −0.954523 −0.477262 − 0.878761i \(-0.658371\pi\)
−0.477262 + 0.878761i \(0.658371\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.2786 2.42106
\(58\) 0 0
\(59\) 12.7110 1.65484 0.827419 − 0.561586i \(-0.189808\pi\)
0.827419 + 0.561586i \(0.189808\pi\)
\(60\) 0 0
\(61\) 8.89599 1.13901 0.569507 − 0.821986i \(-0.307134\pi\)
0.569507 + 0.821986i \(0.307134\pi\)
\(62\) 0 0
\(63\) −27.5690 −3.47337
\(64\) 0 0
\(65\) −4.27118 −0.529775
\(66\) 0 0
\(67\) 12.5135 1.52877 0.764383 − 0.644763i \(-0.223044\pi\)
0.764383 + 0.644763i \(0.223044\pi\)
\(68\) 0 0
\(69\) 21.0429 2.53327
\(70\) 0 0
\(71\) 7.35890 0.873340 0.436670 − 0.899622i \(-0.356158\pi\)
0.436670 + 0.899622i \(0.356158\pi\)
\(72\) 0 0
\(73\) −7.03210 −0.823045 −0.411523 − 0.911399i \(-0.635003\pi\)
−0.411523 + 0.911399i \(0.635003\pi\)
\(74\) 0 0
\(75\) −3.29288 −0.380229
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.99381 0.786865 0.393432 − 0.919354i \(-0.371288\pi\)
0.393432 + 0.919354i \(0.371288\pi\)
\(80\) 0 0
\(81\) 28.9846 3.22051
\(82\) 0 0
\(83\) 10.7509 1.18007 0.590034 − 0.807379i \(-0.299114\pi\)
0.590034 + 0.807379i \(0.299114\pi\)
\(84\) 0 0
\(85\) 1.58998 0.172457
\(86\) 0 0
\(87\) −1.25531 −0.134584
\(88\) 0 0
\(89\) 0.451594 0.0478689 0.0239344 − 0.999714i \(-0.492381\pi\)
0.0239344 + 0.999714i \(0.492381\pi\)
\(90\) 0 0
\(91\) −15.0135 −1.57385
\(92\) 0 0
\(93\) 2.50412 0.259665
\(94\) 0 0
\(95\) 5.55094 0.569514
\(96\) 0 0
\(97\) −2.77956 −0.282222 −0.141111 − 0.989994i \(-0.545067\pi\)
−0.141111 + 0.989994i \(0.545067\pi\)
\(98\) 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.ba.1.1 6
4.3 odd 2 9680.2.a.dd.1.6 6
11.2 odd 10 440.2.y.c.81.1 ✓ 12
11.6 odd 10 440.2.y.c.201.1 yes 12
11.10 odd 2 4840.2.a.bb.1.1 6
44.35 even 10 880.2.bo.i.81.3 12
44.39 even 10 880.2.bo.i.641.3 12
44.43 even 2 9680.2.a.dc.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.1 ✓ 12 11.2 odd 10
440.2.y.c.201.1 yes 12 11.6 odd 10
880.2.bo.i.81.3 12 44.35 even 10
880.2.bo.i.641.3 12 44.39 even 10
4840.2.a.ba.1.1 6 1.1 even 1 trivial
4840.2.a.bb.1.1 6 11.10 odd 2
9680.2.a.dc.1.6 6 44.43 even 2
9680.2.a.dd.1.6 6 4.3 odd 2