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Properties

Label 1638.4.a.m.1.1
Level $1638$
Weight $4$
Character 1638.1
Self dual yes
Analytic conductor $96.645$
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] = [1638,4,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,0,8,-5,0,-14,-16,0,10,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(96.6451285894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{105}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.62348\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -7.62348 q^{5} -7.00000 q^{7} -8.00000 q^{8} +15.2470 q^{10} +36.4939 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +90.2348 q^{17} -71.1052 q^{19} -30.4939 q^{20} -72.9878 q^{22} -75.1052 q^{23} -66.8826 q^{25} +26.0000 q^{26} -28.0000 q^{28} -115.081 q^{29} +159.809 q^{31} -32.0000 q^{32} -180.470 q^{34} +53.3643 q^{35} -189.198 q^{37} +142.210 q^{38} +60.9878 q^{40} +418.210 q^{41} -98.3521 q^{43} +145.976 q^{44} +150.210 q^{46} -155.785 q^{47} +49.0000 q^{49} +133.765 q^{50} -52.0000 q^{52} +702.748 q^{53} -278.210 q^{55} +56.0000 q^{56} +230.162 q^{58} +311.741 q^{59} -407.037 q^{61} -319.619 q^{62} +64.0000 q^{64} +99.1052 q^{65} +416.607 q^{67} +360.939 q^{68} -106.729 q^{70} +1097.42 q^{71} -663.883 q^{73} +378.396 q^{74} -284.421 q^{76} -255.457 q^{77} +973.971 q^{79} -121.976 q^{80} -836.421 q^{82} -1180.58 q^{83} -687.902 q^{85} +196.704 q^{86} -291.951 q^{88} -931.145 q^{89} +91.0000 q^{91} -300.421 q^{92} +311.570 q^{94} +542.069 q^{95} +443.404 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} - 14 q^{7} - 16 q^{8} + 10 q^{10} + 32 q^{11} - 26 q^{13} + 28 q^{14} + 32 q^{16} + 78 q^{17} - 9 q^{19} - 20 q^{20} - 64 q^{22} - 17 q^{23} - 185 q^{25} + 52 q^{26} - 56 q^{28}+ \cdots - 196 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −7.62348 −0.681864 −0.340932 − 0.940088i \(-0.610743\pi\)
−0.340932 + 0.940088i \(0.610743\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 15.2470 0.482151
\(11\) 36.4939 1.00030 0.500151 − 0.865938i \(-0.333278\pi\)
0.500151 + 0.865938i \(0.333278\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 90.2348 1.28736 0.643681 − 0.765294i \(-0.277407\pi\)
0.643681 + 0.765294i \(0.277407\pi\)
\(18\) 0 0
\(19\) −71.1052 −0.858560 −0.429280 − 0.903171i \(-0.641233\pi\)
−0.429280 + 0.903171i \(0.641233\pi\)
\(20\) −30.4939 −0.340932
\(21\) 0 0
\(22\) −72.9878 −0.707321
\(23\) −75.1052 −0.680892 −0.340446 − 0.940264i \(-0.610578\pi\)
−0.340446 + 0.940264i \(0.610578\pi\)
\(24\) 0 0
\(25\) −66.8826 −0.535061
\(26\) 26.0000 0.196116
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) −115.081 −0.736895 −0.368448 − 0.929648i \(-0.620111\pi\)
−0.368448 + 0.929648i \(0.620111\pi\)
\(30\) 0 0
\(31\) 159.809 0.925891 0.462946 − 0.886387i \(-0.346793\pi\)
0.462946 + 0.886387i \(0.346793\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −180.470 −0.910302
\(35\) 53.3643 0.257721
\(36\) 0 0
\(37\) −189.198 −0.840648 −0.420324 − 0.907374i \(-0.638084\pi\)
−0.420324 + 0.907374i \(0.638084\pi\)
\(38\) 142.210 0.607094
\(39\) 0 0
\(40\) 60.9878 0.241075
\(41\) 418.210 1.59301 0.796506 − 0.604631i \(-0.206679\pi\)
0.796506 + 0.604631i \(0.206679\pi\)
\(42\) 0 0
\(43\) −98.3521 −0.348804 −0.174402 − 0.984675i \(-0.555799\pi\)
−0.174402 + 0.984675i \(0.555799\pi\)
\(44\) 145.976 0.500151
\(45\) 0 0
\(46\) 150.210 0.481463
\(47\) −155.785 −0.483481 −0.241740 − 0.970341i \(-0.577718\pi\)
−0.241740 + 0.970341i \(0.577718\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 133.765 0.378345
\(51\) 0 0
\(52\) −52.0000 −0.138675
\(53\) 702.748 1.82132 0.910660 − 0.413157i \(-0.135574\pi\)
0.910660 + 0.413157i \(0.135574\pi\)
\(54\) 0 0
\(55\) −278.210 −0.682070
\(56\) 56.0000 0.133631
\(57\) 0 0
\(58\) 230.162 0.521064
\(59\) 311.741 0.687885 0.343942 − 0.938991i \(-0.388238\pi\)
0.343942 + 0.938991i \(0.388238\pi\)
\(60\) 0 0
\(61\) −407.037 −0.854356 −0.427178 − 0.904168i \(-0.640492\pi\)
−0.427178 + 0.904168i \(0.640492\pi\)
\(62\) −319.619 −0.654704
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 99.1052 0.189115
\(66\) 0 0
\(67\) 416.607 0.759651 0.379825 − 0.925058i \(-0.375984\pi\)
0.379825 + 0.925058i \(0.375984\pi\)
\(68\) 360.939 0.643681
\(69\) 0 0
\(70\) −106.729 −0.182236
\(71\) 1097.42 1.83437 0.917185 − 0.398462i \(-0.130456\pi\)
0.917185 + 0.398462i \(0.130456\pi\)
\(72\) 0 0
\(73\) −663.883 −1.06441 −0.532203 − 0.846617i \(-0.678636\pi\)
−0.532203 + 0.846617i \(0.678636\pi\)
\(74\) 378.396 0.594428
\(75\) 0 0
\(76\) −284.421 −0.429280
\(77\) −255.457 −0.378079
\(78\) 0 0
\(79\) 973.971 1.38709 0.693546 − 0.720412i \(-0.256047\pi\)
0.693546 + 0.720412i \(0.256047\pi\)
\(80\) −121.976 −0.170466
\(81\) 0 0
\(82\) −836.421 −1.12643
\(83\) −1180.58 −1.56127 −0.780634 − 0.624988i \(-0.785104\pi\)
−0.780634 + 0.624988i \(0.785104\pi\)
\(84\) 0 0
\(85\) −687.902 −0.877806
\(86\) 196.704 0.246641
\(87\) 0 0
\(88\) −291.951 −0.353660
\(89\) −931.145 −1.10900 −0.554501 − 0.832183i \(-0.687091\pi\)
−0.554501 + 0.832183i \(0.687091\pi\)
\(90\) 0 0
\(91\) 91.0000 0.104828
\(92\) −300.421 −0.340446
\(93\) 0 0
\(94\) 311.570 0.341872
\(95\) 542.069 0.585422
\(96\) 0 0
\(97\) 443.404 0.464132 0.232066 − 0.972700i \(-0.425451\pi\)
0.232066 + 0.972700i \(0.425451\pi\)
\(98\) −98.0000 −0.101015
\(99\) 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 1638.4.a.m.1.1 2
3.2 odd 2 546.4.a.j.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.a.j.1.2 ✓ 2 3.2 odd 2
1638.4.a.m.1.1 2 1.1 even 1 trivial