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Properties

Label 1638.4.a.v.1.1
Level $1638$
Weight $4$
Character 1638.1
Self dual yes
Analytic conductor $96.645$
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] = [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 = [3,-6,0,12,-7,0,-21,-24,0,14,47] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.118088.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 50x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.645376\) 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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -17.5010 q^{5} -7.00000 q^{7} -8.00000 q^{8} +35.0020 q^{10} +3.08051 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -124.540 q^{17} -78.9607 q^{19} -70.0040 q^{20} -6.16102 q^{22} -126.218 q^{23} +181.285 q^{25} -26.0000 q^{26} -28.0000 q^{28} +25.3812 q^{29} -215.249 q^{31} -32.0000 q^{32} +249.080 q^{34} +122.507 q^{35} +370.949 q^{37} +157.921 q^{38} +140.008 q^{40} +215.291 q^{41} -461.369 q^{43} +12.3220 q^{44} +252.436 q^{46} -478.874 q^{47} +49.0000 q^{49} -362.569 q^{50} +52.0000 q^{52} -503.038 q^{53} -53.9120 q^{55} +56.0000 q^{56} -50.7624 q^{58} -681.315 q^{59} -96.3991 q^{61} +430.499 q^{62} +64.0000 q^{64} -227.513 q^{65} +489.363 q^{67} -498.161 q^{68} -245.014 q^{70} -271.106 q^{71} -529.647 q^{73} -741.897 q^{74} -315.843 q^{76} -21.5636 q^{77} -910.161 q^{79} -280.016 q^{80} -430.581 q^{82} -1051.22 q^{83} +2179.58 q^{85} +922.738 q^{86} -24.6441 q^{88} +305.279 q^{89} -91.0000 q^{91} -504.873 q^{92} +957.748 q^{94} +1381.89 q^{95} +1128.48 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 12 q^{4} - 7 q^{5} - 21 q^{7} - 24 q^{8} + 14 q^{10} + 47 q^{11} + 39 q^{13} + 42 q^{14} + 48 q^{16} - 119 q^{17} + 101 q^{19} - 28 q^{20} - 94 q^{22} + 27 q^{23} + 266 q^{25} - 78 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 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
\(5\) −17.5010 −1.56534 −0.782668 − 0.622439i \(-0.786142\pi\)
−0.782668 + 0.622439i \(0.786142\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 35.0020 1.10686
\(11\) 3.08051 0.0844372 0.0422186 − 0.999108i \(-0.486557\pi\)
0.0422186 + 0.999108i \(0.486557\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −124.540 −1.77679 −0.888395 − 0.459079i \(-0.848179\pi\)
−0.888395 + 0.459079i \(0.848179\pi\)
\(18\) 0 0
\(19\) −78.9607 −0.953412 −0.476706 − 0.879063i \(-0.658169\pi\)
−0.476706 + 0.879063i \(0.658169\pi\)
\(20\) −70.0040 −0.782668
\(21\) 0 0
\(22\) −6.16102 −0.0597061
\(23\) −126.218 −1.14427 −0.572137 − 0.820158i \(-0.693886\pi\)
−0.572137 + 0.820158i \(0.693886\pi\)
\(24\) 0 0
\(25\) 181.285 1.45028
\(26\) −26.0000 −0.196116
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) 25.3812 0.162523 0.0812615 − 0.996693i \(-0.474105\pi\)
0.0812615 + 0.996693i \(0.474105\pi\)
\(30\) 0 0
\(31\) −215.249 −1.24709 −0.623547 − 0.781786i \(-0.714309\pi\)
−0.623547 + 0.781786i \(0.714309\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 249.080 1.25638
\(35\) 122.507 0.591642
\(36\) 0 0
\(37\) 370.949 1.64821 0.824103 − 0.566441i \(-0.191680\pi\)
0.824103 + 0.566441i \(0.191680\pi\)
\(38\) 157.921 0.674164
\(39\) 0 0
\(40\) 140.008 0.553430
\(41\) 215.291 0.820067 0.410034 − 0.912070i \(-0.365517\pi\)
0.410034 + 0.912070i \(0.365517\pi\)
\(42\) 0 0
\(43\) −461.369 −1.63624 −0.818118 − 0.575051i \(-0.804982\pi\)
−0.818118 + 0.575051i \(0.804982\pi\)
\(44\) 12.3220 0.0422186
\(45\) 0 0
\(46\) 252.436 0.809124
\(47\) −478.874 −1.48619 −0.743095 − 0.669185i \(-0.766643\pi\)
−0.743095 + 0.669185i \(0.766643\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −362.569 −1.02550
\(51\) 0 0
\(52\) 52.0000 0.138675
\(53\) −503.038 −1.30373 −0.651864 − 0.758336i \(-0.726013\pi\)
−0.651864 + 0.758336i \(0.726013\pi\)
\(54\) 0 0
\(55\) −53.9120 −0.132173
\(56\) 56.0000 0.133631
\(57\) 0 0
\(58\) −50.7624 −0.114921
\(59\) −681.315 −1.50338 −0.751692 − 0.659515i \(-0.770762\pi\)
−0.751692 + 0.659515i \(0.770762\pi\)
\(60\) 0 0
\(61\) −96.3991 −0.202338 −0.101169 − 0.994869i \(-0.532258\pi\)
−0.101169 + 0.994869i \(0.532258\pi\)
\(62\) 430.499 0.881829
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −227.513 −0.434146
\(66\) 0 0
\(67\) 489.363 0.892316 0.446158 − 0.894954i \(-0.352792\pi\)
0.446158 + 0.894954i \(0.352792\pi\)
\(68\) −498.161 −0.888395
\(69\) 0 0
\(70\) −245.014 −0.418354
\(71\) −271.106 −0.453160 −0.226580 − 0.973993i \(-0.572754\pi\)
−0.226580 + 0.973993i \(0.572754\pi\)
\(72\) 0 0
\(73\) −529.647 −0.849185 −0.424593 − 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(74\) −741.897 −1.16546
\(75\) 0 0
\(76\) −315.843 −0.476706
\(77\) −21.5636 −0.0319143
\(78\) 0 0
\(79\) −910.161 −1.29622 −0.648108 − 0.761548i \(-0.724440\pi\)
−0.648108 + 0.761548i \(0.724440\pi\)
\(80\) −280.016 −0.391334
\(81\) 0 0
\(82\) −430.581 −0.579875
\(83\) −1051.22 −1.39020 −0.695100 − 0.718913i \(-0.744640\pi\)
−0.695100 + 0.718913i \(0.744640\pi\)
\(84\) 0 0
\(85\) 2179.58 2.78128
\(86\) 922.738 1.15699
\(87\) 0 0
\(88\) −24.6441 −0.0298531
\(89\) 305.279 0.363590 0.181795 − 0.983336i \(-0.441809\pi\)
0.181795 + 0.983336i \(0.441809\pi\)
\(90\) 0 0
\(91\) −91.0000 −0.104828
\(92\) −504.873 −0.572137
\(93\) 0 0
\(94\) 957.748 1.05090
\(95\) 1381.89 1.49241
\(96\) 0 0
\(97\) 1128.48 1.18124 0.590619 − 0.806951i \(-0.298884\pi\)
0.590619 + 0.806951i \(0.298884\pi\)
\(98\) −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.v.1.1 3
3.2 odd 2 546.4.a.p.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.a.p.1.3 ✓ 3 3.2 odd 2
1638.4.a.v.1.1 3 1.1 even 1 trivial