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Properties

Label 4840.2.a.k.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $2$
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 = [2,0,0,0,2,0,-4,0,0,0,0,0,0] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +1.00000 q^{5} -3.73205 q^{7} +3.46410 q^{13} +1.73205 q^{15} -2.00000 q^{19} -6.46410 q^{21} +0.535898 q^{23} +1.00000 q^{25} -5.19615 q^{27} -8.92820 q^{29} +5.46410 q^{31} -3.73205 q^{35} -4.00000 q^{37} +6.00000 q^{39} -11.9282 q^{41} +1.73205 q^{43} -10.6603 q^{47} +6.92820 q^{49} -0.535898 q^{53} -3.46410 q^{57} +5.46410 q^{59} -8.46410 q^{61} +3.46410 q^{65} +2.26795 q^{67} +0.928203 q^{69} +9.46410 q^{71} -14.9282 q^{73} +1.73205 q^{75} +6.39230 q^{79} -9.00000 q^{81} +0.535898 q^{83} -15.4641 q^{87} +2.07180 q^{89} -12.9282 q^{91} +9.46410 q^{93} -2.00000 q^{95} +6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{19} - 6 q^{21} + 8 q^{23} + 2 q^{25} - 4 q^{29} + 4 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{39} - 10 q^{41} - 4 q^{47} - 8 q^{53} + 4 q^{59} - 10 q^{61} + 8 q^{67} - 12 q^{69}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(3\) 1.73205 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.73205 −1.41058 −0.705291 − 0.708918i \(-0.749184\pi\)
−0.705291 + 0.708918i \(0.749184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 − 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 1.73205 0.447214
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −6.46410 −1.41058
\(22\) 0 0
\(23\) 0.535898 0.111743 0.0558713 − 0.998438i \(-0.482206\pi\)
0.0558713 + 0.998438i \(0.482206\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) −8.92820 −1.65793 −0.828963 − 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 − 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −11.9282 −1.86287 −0.931436 − 0.363905i \(-0.881443\pi\)
−0.931436 + 0.363905i \(0.881443\pi\)
\(42\) 0 0
\(43\) 1.73205 0.264135 0.132068 − 0.991241i \(-0.457838\pi\)
0.132068 + 0.991241i \(0.457838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.6603 −1.55496 −0.777479 − 0.628909i \(-0.783502\pi\)
−0.777479 + 0.628909i \(0.783502\pi\)
\(48\) 0 0
\(49\) 6.92820 0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.535898 −0.0736113 −0.0368057 − 0.999322i \(-0.511718\pi\)
−0.0368057 + 0.999322i \(0.511718\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) 5.46410 0.711365 0.355683 − 0.934607i \(-0.384248\pi\)
0.355683 + 0.934607i \(0.384248\pi\)
\(60\) 0 0
\(61\) −8.46410 −1.08372 −0.541859 − 0.840470i \(-0.682279\pi\)
−0.541859 + 0.840470i \(0.682279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) 2.26795 0.277074 0.138537 − 0.990357i \(-0.455760\pi\)
0.138537 + 0.990357i \(0.455760\pi\)
\(68\) 0 0
\(69\) 0.928203 0.111743
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 − 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 0 0
\(73\) −14.9282 −1.74721 −0.873607 − 0.486632i \(-0.838225\pi\)
−0.873607 + 0.486632i \(0.838225\pi\)
\(74\) 0 0
\(75\) 1.73205 0.200000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.39230 0.719190 0.359595 − 0.933108i \(-0.382915\pi\)
0.359595 + 0.933108i \(0.382915\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0.535898 0.0588225 0.0294112 − 0.999567i \(-0.490637\pi\)
0.0294112 + 0.999567i \(0.490637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.4641 −1.65793
\(88\) 0 0
\(89\) 2.07180 0.219610 0.109805 − 0.993953i \(-0.464977\pi\)
0.109805 + 0.993953i \(0.464977\pi\)
\(90\) 0 0
\(91\) −12.9282 −1.35524
\(92\) 0 0
\(93\) 9.46410 0.981382
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 6.39230 0.649040 0.324520 − 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 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.k.1.2 ✓ 2
4.3 odd 2 9680.2.a.br.1.1 2
11.10 odd 2 4840.2.a.l.1.2 yes 2
44.43 even 2 9680.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.k.1.2 ✓ 2 1.1 even 1 trivial
4840.2.a.l.1.2 yes 2 11.10 odd 2
9680.2.a.bq.1.1 2 44.43 even 2
9680.2.a.br.1.1 2 4.3 odd 2