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Properties

Label 6840.2.a.bm.1.3
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
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] = [6840,2,Mod(1,6840)] 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("6840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-1,0,0,0,0,0,-11] 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: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +2.71982 q^{7} +5.55691 q^{11} +2.02760 q^{13} +3.77846 q^{17} +1.00000 q^{19} +5.77846 q^{23} +1.00000 q^{25} +5.66119 q^{29} +7.55691 q^{31} +2.71982 q^{35} -3.75086 q^{37} +12.6155 q^{41} -9.43965 q^{43} -11.1138 q^{47} +0.397442 q^{49} +8.85170 q^{53} +5.55691 q^{55} -11.4526 q^{59} -10.6155 q^{61} +2.02760 q^{65} -11.5845 q^{67} -9.45264 q^{73} +15.1138 q^{77} +8.94137 q^{79} +4.94137 q^{83} +3.77846 q^{85} -15.4948 q^{89} +5.51471 q^{91} +1.00000 q^{95} +10.8647 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.71982 1.02800 0.513998 − 0.857791i \(-0.328164\pi\)
0.513998 + 0.857791i \(0.328164\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.55691 1.67547 0.837736 − 0.546075i \(-0.183879\pi\)
0.837736 + 0.546075i \(0.183879\pi\)
\(12\) 0 0
\(13\) 2.02760 0.562354 0.281177 − 0.959656i \(-0.409275\pi\)
0.281177 + 0.959656i \(0.409275\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.77846 0.916410 0.458205 − 0.888846i \(-0.348492\pi\)
0.458205 + 0.888846i \(0.348492\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.77846 1.20489 0.602446 − 0.798160i \(-0.294193\pi\)
0.602446 + 0.798160i \(0.294193\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.66119 1.05126 0.525628 − 0.850714i \(-0.323830\pi\)
0.525628 + 0.850714i \(0.323830\pi\)
\(30\) 0 0
\(31\) 7.55691 1.35726 0.678631 − 0.734479i \(-0.262574\pi\)
0.678631 + 0.734479i \(0.262574\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.71982 0.459734
\(36\) 0 0
\(37\) −3.75086 −0.616637 −0.308319 − 0.951283i \(-0.599766\pi\)
−0.308319 + 0.951283i \(0.599766\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.6155 1.97022 0.985109 − 0.171932i \(-0.0550011\pi\)
0.985109 + 0.171932i \(0.0550011\pi\)
\(42\) 0 0
\(43\) −9.43965 −1.43953 −0.719766 − 0.694216i \(-0.755751\pi\)
−0.719766 + 0.694216i \(0.755751\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1138 −1.62112 −0.810559 − 0.585657i \(-0.800837\pi\)
−0.810559 + 0.585657i \(0.800837\pi\)
\(48\) 0 0
\(49\) 0.397442 0.0567775
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.85170 1.21587 0.607937 − 0.793985i \(-0.291997\pi\)
0.607937 + 0.793985i \(0.291997\pi\)
\(54\) 0 0
\(55\) 5.55691 0.749294
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.4526 −1.49101 −0.745503 − 0.666502i \(-0.767791\pi\)
−0.745503 + 0.666502i \(0.767791\pi\)
\(60\) 0 0
\(61\) −10.6155 −1.35918 −0.679591 − 0.733591i \(-0.737843\pi\)
−0.679591 + 0.733591i \(0.737843\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.02760 0.251493
\(66\) 0 0
\(67\) −11.5845 −1.41527 −0.707637 − 0.706576i \(-0.750239\pi\)
−0.707637 + 0.706576i \(0.750239\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −9.45264 −1.10635 −0.553174 − 0.833066i \(-0.686583\pi\)
−0.553174 + 0.833066i \(0.686583\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.1138 1.72238
\(78\) 0 0
\(79\) 8.94137 1.00598 0.502991 − 0.864292i \(-0.332233\pi\)
0.502991 + 0.864292i \(0.332233\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.94137 0.542385 0.271193 − 0.962525i \(-0.412582\pi\)
0.271193 + 0.962525i \(0.412582\pi\)
\(84\) 0 0
\(85\) 3.77846 0.409831
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.4948 −1.64245 −0.821225 − 0.570604i \(-0.806709\pi\)
−0.821225 + 0.570604i \(0.806709\pi\)
\(90\) 0 0
\(91\) 5.51471 0.578099
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 10.8647 1.10314 0.551571 − 0.834128i \(-0.314029\pi\)
0.551571 + 0.834128i \(0.314029\pi\)
\(98\) 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 6840.2.a.bm.1.3 3
3.2 odd 2 760.2.a.i.1.2 ✓ 3
12.11 even 2 1520.2.a.q.1.2 3
15.2 even 4 3800.2.d.n.3649.4 6
15.8 even 4 3800.2.d.n.3649.3 6
15.14 odd 2 3800.2.a.w.1.2 3
24.5 odd 2 6080.2.a.bx.1.2 3
24.11 even 2 6080.2.a.br.1.2 3
60.59 even 2 7600.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 ✓ 3 3.2 odd 2
1520.2.a.q.1.2 3 12.11 even 2
3800.2.a.w.1.2 3 15.14 odd 2
3800.2.d.n.3649.3 6 15.8 even 4
3800.2.d.n.3649.4 6 15.2 even 4
6080.2.a.br.1.2 3 24.11 even 2
6080.2.a.bx.1.2 3 24.5 odd 2
6840.2.a.bm.1.3 3 1.1 even 1 trivial
7600.2.a.bp.1.2 3 60.59 even 2