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Properties

Label 3720.2.k.f.1489.15
Level $3720$
Weight $2$
Character 3720.1489
Analytic conductor $29.704$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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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] = [3720,2,Mod(1489,3720)] 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("3720.1489"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,-4,0,0,0,-18,0,-14,0,0,0,-4,0,0,0,22,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} + 14 x^{16} - 28 x^{15} + 43 x^{14} + 8 x^{13} - 155 x^{12} + 316 x^{11} + \cdots + 1953125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1489.15
Root \(2.20257 + 0.385590i\) of defining polynomial
Character \(\chi\) \(=\) 3720.1489
Dual form 3720.2.k.f.1489.6

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(0.385590 + 2.20257i) q^{5} -3.73454i q^{7} -1.00000 q^{9} -0.318171 q^{11} +3.89868i q^{13} +(-2.20257 + 0.385590i) q^{15} +3.60457i q^{17} +6.58912 q^{19} +3.73454 q^{21} -3.53464i q^{23} +(-4.70264 + 1.69858i) q^{25} -1.00000i q^{27} +2.97904 q^{29} +1.00000 q^{31} -0.318171i q^{33} +(8.22560 - 1.44000i) q^{35} +0.0225089i q^{37} -3.89868 q^{39} -2.85483 q^{41} -9.11596i q^{43} +(-0.385590 - 2.20257i) q^{45} +11.4069i q^{47} -6.94682 q^{49} -3.60457 q^{51} +6.55008i q^{53} +(-0.122684 - 0.700794i) q^{55} +6.58912i q^{57} +4.20786 q^{59} +1.52555 q^{61} +3.73454i q^{63} +(-8.58713 + 1.50329i) q^{65} +11.6958i q^{67} +3.53464 q^{69} +3.37824 q^{71} +0.978663i q^{73} +(-1.69858 - 4.70264i) q^{75} +1.18822i q^{77} -4.63012 q^{79} +1.00000 q^{81} +4.18367i q^{83} +(-7.93933 + 1.38989i) q^{85} +2.97904i q^{87} +7.61936 q^{89} +14.5598 q^{91} +1.00000i q^{93} +(2.54070 + 14.5130i) q^{95} +1.25089i q^{97} +0.318171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{5} - 18 q^{9} - 14 q^{11} - 4 q^{15} + 22 q^{19} - 14 q^{21} + 12 q^{25} + 8 q^{29} + 18 q^{31} + 6 q^{35} - 12 q^{39} - 12 q^{41} + 4 q^{45} - 16 q^{49} - 4 q^{51} - 10 q^{55} + 52 q^{59} - 60 q^{61}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3720\mathbb{Z}\right)^\times\).

\(n\) \(1241\) \(1801\) \(1861\) \(2791\) \(2977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0.385590 + 2.20257i 0.172441 + 0.985020i
\(6\) 0 0
\(7\) − 3.73454i − 1.41152i −0.708449 − 0.705762i \(-0.750605\pi\)
0.708449 − 0.705762i \(-0.249395\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −0.318171 −0.0959322 −0.0479661 − 0.998849i \(-0.515274\pi\)
−0.0479661 + 0.998849i \(0.515274\pi\)
\(12\) 0 0
\(13\) 3.89868i 1.08130i 0.841248 + 0.540650i \(0.181822\pi\)
−0.841248 + 0.540650i \(0.818178\pi\)
\(14\) 0 0
\(15\) −2.20257 + 0.385590i −0.568701 + 0.0995589i
\(16\) 0 0
\(17\) 3.60457i 0.874238i 0.899404 + 0.437119i \(0.144001\pi\)
−0.899404 + 0.437119i \(0.855999\pi\)
\(18\) 0 0
\(19\) 6.58912 1.51165 0.755824 − 0.654775i \(-0.227236\pi\)
0.755824 + 0.654775i \(0.227236\pi\)
\(20\) 0 0
\(21\) 3.73454 0.814944
\(22\) 0 0
\(23\) − 3.53464i − 0.737024i −0.929623 − 0.368512i \(-0.879867\pi\)
0.929623 − 0.368512i \(-0.120133\pi\)
\(24\) 0 0
\(25\) −4.70264 + 1.69858i −0.940528 + 0.339716i
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.97904 0.553194 0.276597 − 0.960986i \(-0.410793\pi\)
0.276597 + 0.960986i \(0.410793\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) − 0.318171i − 0.0553865i
\(34\) 0 0
\(35\) 8.22560 − 1.44000i 1.39038 − 0.243405i
\(36\) 0 0
\(37\) 0.0225089i 0.00370045i 0.999998 + 0.00185022i \(0.000588945\pi\)
−0.999998 + 0.00185022i \(0.999411\pi\)
\(38\) 0 0
\(39\) −3.89868 −0.624289
\(40\) 0 0
\(41\) −2.85483 −0.445849 −0.222925 − 0.974836i \(-0.571560\pi\)
−0.222925 + 0.974836i \(0.571560\pi\)
\(42\) 0 0
\(43\) − 9.11596i − 1.39017i −0.718927 − 0.695085i \(-0.755367\pi\)
0.718927 − 0.695085i \(-0.244633\pi\)
\(44\) 0 0
\(45\) −0.385590 − 2.20257i −0.0574803 − 0.328340i
\(46\) 0 0
\(47\) 11.4069i 1.66386i 0.554878 + 0.831932i \(0.312765\pi\)
−0.554878 + 0.831932i \(0.687235\pi\)
\(48\) 0 0
\(49\) −6.94682 −0.992403
\(50\) 0 0
\(51\) −3.60457 −0.504741
\(52\) 0 0
\(53\) 6.55008i 0.899723i 0.893098 + 0.449861i \(0.148527\pi\)
−0.893098 + 0.449861i \(0.851473\pi\)
\(54\) 0 0
\(55\) −0.122684 − 0.700794i −0.0165426 − 0.0944951i
\(56\) 0 0
\(57\) 6.58912i 0.872751i
\(58\) 0 0
\(59\) 4.20786 0.547817 0.273908 − 0.961756i \(-0.411684\pi\)
0.273908 + 0.961756i \(0.411684\pi\)
\(60\) 0 0
\(61\) 1.52555 0.195326 0.0976630 − 0.995220i \(-0.468863\pi\)
0.0976630 + 0.995220i \(0.468863\pi\)
\(62\) 0 0
\(63\) 3.73454i 0.470508i
\(64\) 0 0
\(65\) −8.58713 + 1.50329i −1.06510 + 0.186460i
\(66\) 0 0
\(67\) 11.6958i 1.42886i 0.699705 + 0.714432i \(0.253315\pi\)
−0.699705 + 0.714432i \(0.746685\pi\)
\(68\) 0 0
\(69\) 3.53464 0.425521
\(70\) 0 0
\(71\) 3.37824 0.400923 0.200462 − 0.979702i \(-0.435756\pi\)
0.200462 + 0.979702i \(0.435756\pi\)
\(72\) 0 0
\(73\) 0.978663i 0.114544i 0.998359 + 0.0572719i \(0.0182402\pi\)
−0.998359 + 0.0572719i \(0.981760\pi\)
\(74\) 0 0
\(75\) −1.69858 − 4.70264i −0.196135 − 0.543014i
\(76\) 0 0
\(77\) 1.18822i 0.135411i
\(78\) 0 0
\(79\) −4.63012 −0.520929 −0.260465 − 0.965483i \(-0.583876\pi\)
−0.260465 + 0.965483i \(0.583876\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.18367i 0.459218i 0.973283 + 0.229609i \(0.0737447\pi\)
−0.973283 + 0.229609i \(0.926255\pi\)
\(84\) 0 0
\(85\) −7.93933 + 1.38989i −0.861141 + 0.150754i
\(86\) 0 0
\(87\) 2.97904i 0.319387i
\(88\) 0 0
\(89\) 7.61936 0.807651 0.403825 − 0.914836i \(-0.367680\pi\)
0.403825 + 0.914836i \(0.367680\pi\)
\(90\) 0 0
\(91\) 14.5598 1.52628
\(92\) 0 0
\(93\) 1.00000i 0.103695i
\(94\) 0 0
\(95\) 2.54070 + 14.5130i 0.260670 + 1.48900i
\(96\) 0 0
\(97\) 1.25089i 0.127008i 0.997982 + 0.0635042i \(0.0202276\pi\)
−0.997982 + 0.0635042i \(0.979772\pi\)
\(98\) 0 0
\(99\) 0.318171 0.0319774
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 3720.2.k.f.1489.15 yes 18
5.4 even 2 inner 3720.2.k.f.1489.6 ✓ 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.k.f.1489.6 ✓ 18 5.4 even 2 inner
3720.2.k.f.1489.15 yes 18 1.1 even 1 trivial