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Properties

Label 4840.2.a.bf.1.6
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,-6,0,6,0,10,0,0,0,-6] 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: 6.6.45753625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.6
Root \(-2.04842\) 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)
 
\(f(q)\) \(=\) \(q+3.31441 q^{3} -1.00000 q^{5} +0.185958 q^{7} +7.98529 q^{9} +3.01352 q^{13} -3.31441 q^{15} +5.11840 q^{17} -4.86246 q^{19} +0.616340 q^{21} +6.20391 q^{23} +1.00000 q^{25} +16.5233 q^{27} -0.190393 q^{29} +8.06089 q^{31} -0.185958 q^{35} -2.55988 q^{37} +9.98803 q^{39} -8.95655 q^{41} -7.83103 q^{43} -7.98529 q^{45} -5.30075 q^{47} -6.96542 q^{49} +16.9645 q^{51} +1.30089 q^{53} -16.1162 q^{57} -0.593594 q^{59} -1.13498 q^{61} +1.48493 q^{63} -3.01352 q^{65} -1.73474 q^{67} +20.5623 q^{69} +13.0710 q^{71} +0.894075 q^{73} +3.31441 q^{75} +13.3047 q^{79} +30.8090 q^{81} -16.9499 q^{83} -5.11840 q^{85} -0.631040 q^{87} -7.22907 q^{89} +0.560388 q^{91} +26.7171 q^{93} +4.86246 q^{95} +3.39538 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(3\) 3.31441 1.91357 0.956787 − 0.290790i \(-0.0939182\pi\)
0.956787 + 0.290790i \(0.0939182\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.185958 0.0702854 0.0351427 − 0.999382i \(-0.488811\pi\)
0.0351427 + 0.999382i \(0.488811\pi\)
\(8\) 0 0
\(9\) 7.98529 2.66176
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.01352 0.835800 0.417900 − 0.908493i \(-0.362766\pi\)
0.417900 + 0.908493i \(0.362766\pi\)
\(14\) 0 0
\(15\) −3.31441 −0.855776
\(16\) 0 0
\(17\) 5.11840 1.24139 0.620697 − 0.784051i \(-0.286850\pi\)
0.620697 + 0.784051i \(0.286850\pi\)
\(18\) 0 0
\(19\) −4.86246 −1.11552 −0.557762 − 0.830001i \(-0.688340\pi\)
−0.557762 + 0.830001i \(0.688340\pi\)
\(20\) 0 0
\(21\) 0.616340 0.134496
\(22\) 0 0
\(23\) 6.20391 1.29361 0.646803 − 0.762657i \(-0.276106\pi\)
0.646803 + 0.762657i \(0.276106\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 16.5233 3.17991
\(28\) 0 0
\(29\) −0.190393 −0.0353551 −0.0176776 − 0.999844i \(-0.505627\pi\)
−0.0176776 + 0.999844i \(0.505627\pi\)
\(30\) 0 0
\(31\) 8.06089 1.44778 0.723889 − 0.689916i \(-0.242353\pi\)
0.723889 + 0.689916i \(0.242353\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.185958 −0.0314326
\(36\) 0 0
\(37\) −2.55988 −0.420841 −0.210421 − 0.977611i \(-0.567483\pi\)
−0.210421 + 0.977611i \(0.567483\pi\)
\(38\) 0 0
\(39\) 9.98803 1.59937
\(40\) 0 0
\(41\) −8.95655 −1.39878 −0.699389 − 0.714741i \(-0.746544\pi\)
−0.699389 + 0.714741i \(0.746544\pi\)
\(42\) 0 0
\(43\) −7.83103 −1.19422 −0.597111 − 0.802159i \(-0.703685\pi\)
−0.597111 + 0.802159i \(0.703685\pi\)
\(44\) 0 0
\(45\) −7.98529 −1.19038
\(46\) 0 0
\(47\) −5.30075 −0.773193 −0.386597 − 0.922249i \(-0.626349\pi\)
−0.386597 + 0.922249i \(0.626349\pi\)
\(48\) 0 0
\(49\) −6.96542 −0.995060
\(50\) 0 0
\(51\) 16.9645 2.37550
\(52\) 0 0
\(53\) 1.30089 0.178690 0.0893452 − 0.996001i \(-0.471523\pi\)
0.0893452 + 0.996001i \(0.471523\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −16.1162 −2.13464
\(58\) 0 0
\(59\) −0.593594 −0.0772793 −0.0386397 − 0.999253i \(-0.512302\pi\)
−0.0386397 + 0.999253i \(0.512302\pi\)
\(60\) 0 0
\(61\) −1.13498 −0.145319 −0.0726595 − 0.997357i \(-0.523149\pi\)
−0.0726595 + 0.997357i \(0.523149\pi\)
\(62\) 0 0
\(63\) 1.48493 0.187083
\(64\) 0 0
\(65\) −3.01352 −0.373781
\(66\) 0 0
\(67\) −1.73474 −0.211932 −0.105966 − 0.994370i \(-0.533793\pi\)
−0.105966 + 0.994370i \(0.533793\pi\)
\(68\) 0 0
\(69\) 20.5623 2.47541
\(70\) 0 0
\(71\) 13.0710 1.55125 0.775623 − 0.631197i \(-0.217436\pi\)
0.775623 + 0.631197i \(0.217436\pi\)
\(72\) 0 0
\(73\) 0.894075 0.104644 0.0523218 − 0.998630i \(-0.483338\pi\)
0.0523218 + 0.998630i \(0.483338\pi\)
\(74\) 0 0
\(75\) 3.31441 0.382715
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.3047 1.49689 0.748446 − 0.663196i \(-0.230800\pi\)
0.748446 + 0.663196i \(0.230800\pi\)
\(80\) 0 0
\(81\) 30.8090 3.42322
\(82\) 0 0
\(83\) −16.9499 −1.86049 −0.930246 − 0.366937i \(-0.880407\pi\)
−0.930246 + 0.366937i \(0.880407\pi\)
\(84\) 0 0
\(85\) −5.11840 −0.555168
\(86\) 0 0
\(87\) −0.631040 −0.0676546
\(88\) 0 0
\(89\) −7.22907 −0.766280 −0.383140 − 0.923690i \(-0.625157\pi\)
−0.383140 + 0.923690i \(0.625157\pi\)
\(90\) 0 0
\(91\) 0.560388 0.0587446
\(92\) 0 0
\(93\) 26.7171 2.77043
\(94\) 0 0
\(95\) 4.86246 0.498878
\(96\) 0 0
\(97\) 3.39538 0.344748 0.172374 − 0.985032i \(-0.444856\pi\)
0.172374 + 0.985032i \(0.444856\pi\)
\(98\) 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.bf.1.6 6
4.3 odd 2 9680.2.a.cx.1.1 6
11.3 even 5 440.2.y.b.361.1 ✓ 12
11.4 even 5 440.2.y.b.401.1 yes 12
11.10 odd 2 4840.2.a.be.1.6 6
44.3 odd 10 880.2.bo.j.801.3 12
44.15 odd 10 880.2.bo.j.401.3 12
44.43 even 2 9680.2.a.cy.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.1 ✓ 12 11.3 even 5
440.2.y.b.401.1 yes 12 11.4 even 5
880.2.bo.j.401.3 12 44.15 odd 10
880.2.bo.j.801.3 12 44.3 odd 10
4840.2.a.be.1.6 6 11.10 odd 2
4840.2.a.bf.1.6 6 1.1 even 1 trivial
9680.2.a.cx.1.1 6 4.3 odd 2
9680.2.a.cy.1.1 6 44.43 even 2