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Properties

Label 4001.2.a.b.1.20
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $0$
Dimension $184$
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] = [4001,2,Mod(1,4001)] 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("4001.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4001, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [184] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.9481458487\)
Analytic rank: \(0\)
Dimension: \(184\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 4001.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.38517 q^{2} +0.919938 q^{3} +3.68905 q^{4} +3.44498 q^{5} -2.19421 q^{6} -3.87561 q^{7} -4.02868 q^{8} -2.15371 q^{9} -8.21687 q^{10} -2.95884 q^{11} +3.39370 q^{12} -7.06133 q^{13} +9.24401 q^{14} +3.16917 q^{15} +2.23100 q^{16} +5.48855 q^{17} +5.13698 q^{18} -1.68327 q^{19} +12.7087 q^{20} -3.56532 q^{21} +7.05734 q^{22} +1.14696 q^{23} -3.70614 q^{24} +6.86788 q^{25} +16.8425 q^{26} -4.74110 q^{27} -14.2973 q^{28} +7.01474 q^{29} -7.55901 q^{30} +10.3580 q^{31} +2.73604 q^{32} -2.72195 q^{33} -13.0911 q^{34} -13.3514 q^{35} -7.94517 q^{36} -10.6986 q^{37} +4.01490 q^{38} -6.49598 q^{39} -13.8787 q^{40} -6.98313 q^{41} +8.50391 q^{42} -4.35611 q^{43} -10.9153 q^{44} -7.41950 q^{45} -2.73570 q^{46} +0.995509 q^{47} +2.05238 q^{48} +8.02038 q^{49} -16.3811 q^{50} +5.04912 q^{51} -26.0496 q^{52} +14.3843 q^{53} +11.3083 q^{54} -10.1931 q^{55} +15.6136 q^{56} -1.54851 q^{57} -16.7314 q^{58} +1.51056 q^{59} +11.6912 q^{60} +3.59272 q^{61} -24.7057 q^{62} +8.34696 q^{63} -10.9879 q^{64} -24.3261 q^{65} +6.49232 q^{66} +1.34754 q^{67} +20.2475 q^{68} +1.05513 q^{69} +31.8454 q^{70} +2.71774 q^{71} +8.67663 q^{72} -13.8312 q^{73} +25.5181 q^{74} +6.31803 q^{75} -6.20968 q^{76} +11.4673 q^{77} +15.4940 q^{78} +2.81997 q^{79} +7.68576 q^{80} +2.09963 q^{81} +16.6560 q^{82} +12.1277 q^{83} -13.1527 q^{84} +18.9079 q^{85} +10.3901 q^{86} +6.45312 q^{87} +11.9202 q^{88} +14.7548 q^{89} +17.6968 q^{90} +27.3670 q^{91} +4.23120 q^{92} +9.52873 q^{93} -2.37446 q^{94} -5.79884 q^{95} +2.51698 q^{96} -16.0606 q^{97} -19.1300 q^{98} +6.37249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38517 −1.68657 −0.843286 − 0.537465i \(-0.819382\pi\)
−0.843286 + 0.537465i \(0.819382\pi\)
\(3\) 0.919938 0.531126 0.265563 − 0.964093i \(-0.414442\pi\)
0.265563 + 0.964093i \(0.414442\pi\)
\(4\) 3.68905 1.84453
\(5\) 3.44498 1.54064 0.770321 − 0.637657i \(-0.220096\pi\)
0.770321 + 0.637657i \(0.220096\pi\)
\(6\) −2.19421 −0.895783
\(7\) −3.87561 −1.46484 −0.732422 − 0.680851i \(-0.761610\pi\)
−0.732422 + 0.680851i \(0.761610\pi\)
\(8\) −4.02868 −1.42435
\(9\) −2.15371 −0.717905
\(10\) −8.21687 −2.59840
\(11\) −2.95884 −0.892123 −0.446062 − 0.895002i \(-0.647174\pi\)
−0.446062 + 0.895002i \(0.647174\pi\)
\(12\) 3.39370 0.979676
\(13\) −7.06133 −1.95846 −0.979230 − 0.202751i \(-0.935012\pi\)
−0.979230 + 0.202751i \(0.935012\pi\)
\(14\) 9.24401 2.47057
\(15\) 3.16917 0.818275
\(16\) 2.23100 0.557751
\(17\) 5.48855 1.33117 0.665584 − 0.746323i \(-0.268183\pi\)
0.665584 + 0.746323i \(0.268183\pi\)
\(18\) 5.13698 1.21080
\(19\) −1.68327 −0.386169 −0.193085 − 0.981182i \(-0.561849\pi\)
−0.193085 + 0.981182i \(0.561849\pi\)
\(20\) 12.7087 2.84175
\(21\) −3.56532 −0.778017
\(22\) 7.05734 1.50463
\(23\) 1.14696 0.239158 0.119579 − 0.992825i \(-0.461846\pi\)
0.119579 + 0.992825i \(0.461846\pi\)
\(24\) −3.70614 −0.756512
\(25\) 6.86788 1.37358
\(26\) 16.8425 3.30309
\(27\) −4.74110 −0.912424
\(28\) −14.2973 −2.70194
\(29\) 7.01474 1.30260 0.651302 − 0.758819i \(-0.274223\pi\)
0.651302 + 0.758819i \(0.274223\pi\)
\(30\) −7.55901 −1.38008
\(31\) 10.3580 1.86036 0.930178 − 0.367110i \(-0.119653\pi\)
0.930178 + 0.367110i \(0.119653\pi\)
\(32\) 2.73604 0.483668
\(33\) −2.72195 −0.473830
\(34\) −13.0911 −2.24511
\(35\) −13.3514 −2.25680
\(36\) −7.94517 −1.32419
\(37\) −10.6986 −1.75884 −0.879421 − 0.476045i \(-0.842070\pi\)
−0.879421 + 0.476045i \(0.842070\pi\)
\(38\) 4.01490 0.651302
\(39\) −6.49598 −1.04019
\(40\) −13.8787 −2.19442
\(41\) −6.98313 −1.09058 −0.545291 − 0.838247i \(-0.683581\pi\)
−0.545291 + 0.838247i \(0.683581\pi\)
\(42\) 8.50391 1.31218
\(43\) −4.35611 −0.664300 −0.332150 − 0.943227i \(-0.607774\pi\)
−0.332150 + 0.943227i \(0.607774\pi\)
\(44\) −10.9153 −1.64554
\(45\) −7.41950 −1.10603
\(46\) −2.73570 −0.403357
\(47\) 0.995509 0.145210 0.0726049 − 0.997361i \(-0.476869\pi\)
0.0726049 + 0.997361i \(0.476869\pi\)
\(48\) 2.05238 0.296236
\(49\) 8.02038 1.14577
\(50\) −16.3811 −2.31664
\(51\) 5.04912 0.707018
\(52\) −26.0496 −3.61243
\(53\) 14.3843 1.97583 0.987916 − 0.154991i \(-0.0495349\pi\)
0.987916 + 0.154991i \(0.0495349\pi\)
\(54\) 11.3083 1.53887
\(55\) −10.1931 −1.37444
\(56\) 15.6136 2.08646
\(57\) −1.54851 −0.205105
\(58\) −16.7314 −2.19694
\(59\) 1.51056 0.196658 0.0983291 − 0.995154i \(-0.468650\pi\)
0.0983291 + 0.995154i \(0.468650\pi\)
\(60\) 11.6912 1.50933
\(61\) 3.59272 0.460001 0.230000 − 0.973191i \(-0.426127\pi\)
0.230000 + 0.973191i \(0.426127\pi\)
\(62\) −24.7057 −3.13762
\(63\) 8.34696 1.05162
\(64\) −10.9879 −1.37349
\(65\) −24.3261 −3.01729
\(66\) 6.49232 0.799149
\(67\) 1.34754 0.164628 0.0823139 − 0.996606i \(-0.473769\pi\)
0.0823139 + 0.996606i \(0.473769\pi\)
\(68\) 20.2475 2.45537
\(69\) 1.05513 0.127023
\(70\) 31.8454 3.80626
\(71\) 2.71774 0.322537 0.161268 − 0.986911i \(-0.448442\pi\)
0.161268 + 0.986911i \(0.448442\pi\)
\(72\) 8.67663 1.02255
\(73\) −13.8312 −1.61882 −0.809409 − 0.587245i \(-0.800213\pi\)
−0.809409 + 0.587245i \(0.800213\pi\)
\(74\) 25.5181 2.96641
\(75\) 6.31803 0.729543
\(76\) −6.20968 −0.712299
\(77\) 11.4673 1.30682
\(78\) 15.4940 1.75436
\(79\) 2.81997 0.317271 0.158636 − 0.987337i \(-0.449291\pi\)
0.158636 + 0.987337i \(0.449291\pi\)
\(80\) 7.68576 0.859294
\(81\) 2.09963 0.233292
\(82\) 16.6560 1.83935
\(83\) 12.1277 1.33118 0.665592 − 0.746316i \(-0.268179\pi\)
0.665592 + 0.746316i \(0.268179\pi\)
\(84\) −13.1527 −1.43507
\(85\) 18.9079 2.05085
\(86\) 10.3901 1.12039
\(87\) 6.45312 0.691847
\(88\) 11.9202 1.27070
\(89\) 14.7548 1.56400 0.782000 − 0.623278i \(-0.214199\pi\)
0.782000 + 0.623278i \(0.214199\pi\)
\(90\) 17.6968 1.86541
\(91\) 27.3670 2.86884
\(92\) 4.23120 0.441133
\(93\) 9.52873 0.988084
\(94\) −2.37446 −0.244907
\(95\) −5.79884 −0.594948
\(96\) 2.51698 0.256889
\(97\) −16.0606 −1.63071 −0.815353 − 0.578964i \(-0.803457\pi\)
−0.815353 + 0.578964i \(0.803457\pi\)
\(98\) −19.1300 −1.93242
\(99\) 6.37249 0.640460
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 4001.2.a.b.1.20 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.20 ✓ 184 1.1 even 1 trivial