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Properties

Label 4840.2.a.k.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +1.00000 q^{5} -0.267949 q^{7} -3.46410 q^{13} -1.73205 q^{15} -2.00000 q^{19} +0.464102 q^{21} +7.46410 q^{23} +1.00000 q^{25} +5.19615 q^{27} +4.92820 q^{29} -1.46410 q^{31} -0.267949 q^{35} -4.00000 q^{37} +6.00000 q^{39} +1.92820 q^{41} -1.73205 q^{43} +6.66025 q^{47} -6.92820 q^{49} -7.46410 q^{53} +3.46410 q^{57} -1.46410 q^{59} -1.53590 q^{61} -3.46410 q^{65} +5.73205 q^{67} -12.9282 q^{69} +2.53590 q^{71} -1.07180 q^{73} -1.73205 q^{75} -14.3923 q^{79} -9.00000 q^{81} +7.46410 q^{83} -8.53590 q^{87} +15.9282 q^{89} +0.928203 q^{91} +2.53590 q^{93} -2.00000 q^{95} -14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{19} - 6 q^{21} + 8 q^{23} + 2 q^{25} - 4 q^{29} + 4 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{39} - 10 q^{41} - 4 q^{47} - 8 q^{53} + 4 q^{59} - 10 q^{61} + 8 q^{67} - 12 q^{69}+ \cdots - 8 q^{97}+O(q^{100}) \) 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 \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.267949 −0.101275 −0.0506376 − 0.998717i \(-0.516125\pi\)
−0.0506376 + 0.998717i \(0.516125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 − 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 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
\(21\) 0.464102 0.101275
\(22\) 0 0
\(23\) 7.46410 1.55637 0.778186 − 0.628033i \(-0.216140\pi\)
0.778186 + 0.628033i \(0.216140\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 4.92820 0.915144 0.457572 − 0.889172i \(-0.348719\pi\)
0.457572 + 0.889172i \(0.348719\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 − 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.267949 −0.0452917
\(36\) 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
\(41\) 1.92820 0.301135 0.150567 − 0.988600i \(-0.451890\pi\)
0.150567 + 0.988600i \(0.451890\pi\)
\(42\) 0 0
\(43\) −1.73205 −0.264135 −0.132068 − 0.991241i \(-0.542162\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.66025 0.971498 0.485749 − 0.874098i \(-0.338547\pi\)
0.485749 + 0.874098i \(0.338547\pi\)
\(48\) 0 0
\(49\) −6.92820 −0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.46410 −1.02527 −0.512637 − 0.858606i \(-0.671331\pi\)
−0.512637 + 0.858606i \(0.671331\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −1.46410 −0.190610 −0.0953049 − 0.995448i \(-0.530383\pi\)
−0.0953049 + 0.995448i \(0.530383\pi\)
\(60\) 0 0
\(61\) −1.53590 −0.196652 −0.0983258 − 0.995154i \(-0.531349\pi\)
−0.0983258 + 0.995154i \(0.531349\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 5.73205 0.700281 0.350141 − 0.936697i \(-0.386134\pi\)
0.350141 + 0.936697i \(0.386134\pi\)
\(68\) 0 0
\(69\) −12.9282 −1.55637
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −1.07180 −0.125444 −0.0627222 − 0.998031i \(-0.519978\pi\)
−0.0627222 + 0.998031i \(0.519978\pi\)
\(74\) 0 0
\(75\) −1.73205 −0.200000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.3923 −1.61926 −0.809630 − 0.586940i \(-0.800332\pi\)
−0.809630 + 0.586940i \(0.800332\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 7.46410 0.819292 0.409646 − 0.912245i \(-0.365652\pi\)
0.409646 + 0.912245i \(0.365652\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.53590 −0.915144
\(88\) 0 0
\(89\) 15.9282 1.68839 0.844193 − 0.536039i \(-0.180080\pi\)
0.844193 + 0.536039i \(0.180080\pi\)
\(90\) 0 0
\(91\) 0.928203 0.0973021
\(92\) 0 0
\(93\) 2.53590 0.262960
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −14.3923 −1.46132 −0.730659 − 0.682743i \(-0.760787\pi\)
−0.730659 + 0.682743i \(0.760787\pi\)
\(98\) 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.1.1 ✓ 2
4.3 odd 2 9680.2.a.br.1.2 2
11.10 odd 2 4840.2.a.l.1.1 yes 2
44.43 even 2 9680.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.k.1.1 ✓ 2 1.1 even 1 trivial
4840.2.a.l.1.1 yes 2 11.10 odd 2
9680.2.a.bq.1.2 2 44.43 even 2
9680.2.a.br.1.2 2 4.3 odd 2