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Properties

Label 4001.2.a.b.1.2
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 1.2
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)
 
\(f(q)\) \(=\) \(q-2.79357 q^{2} +0.695080 q^{3} +5.80404 q^{4} -1.43429 q^{5} -1.94176 q^{6} +3.16464 q^{7} -10.6269 q^{8} -2.51686 q^{9} +4.00680 q^{10} -2.99882 q^{11} +4.03427 q^{12} +2.89469 q^{13} -8.84066 q^{14} -0.996950 q^{15} +18.0788 q^{16} -7.99885 q^{17} +7.03104 q^{18} +2.23544 q^{19} -8.32470 q^{20} +2.19968 q^{21} +8.37743 q^{22} +5.98316 q^{23} -7.38652 q^{24} -2.94280 q^{25} -8.08651 q^{26} -3.83466 q^{27} +18.3677 q^{28} +10.3919 q^{29} +2.78505 q^{30} -6.69270 q^{31} -29.2507 q^{32} -2.08442 q^{33} +22.3454 q^{34} -4.53903 q^{35} -14.6080 q^{36} +3.17307 q^{37} -6.24487 q^{38} +2.01204 q^{39} +15.2421 q^{40} -2.39388 q^{41} -6.14497 q^{42} -3.70255 q^{43} -17.4053 q^{44} +3.60992 q^{45} -16.7144 q^{46} +1.51431 q^{47} +12.5662 q^{48} +3.01497 q^{49} +8.22092 q^{50} -5.55984 q^{51} +16.8009 q^{52} -2.05987 q^{53} +10.7124 q^{54} +4.30120 q^{55} -33.6302 q^{56} +1.55381 q^{57} -29.0305 q^{58} +4.95271 q^{59} -5.78634 q^{60} -7.27148 q^{61} +18.6965 q^{62} -7.96498 q^{63} +45.5564 q^{64} -4.15183 q^{65} +5.82298 q^{66} +5.32889 q^{67} -46.4257 q^{68} +4.15877 q^{69} +12.6801 q^{70} +2.28191 q^{71} +26.7464 q^{72} -0.565151 q^{73} -8.86419 q^{74} -2.04548 q^{75} +12.9746 q^{76} -9.49021 q^{77} -5.62077 q^{78} +3.79783 q^{79} -25.9303 q^{80} +4.88519 q^{81} +6.68748 q^{82} +8.06607 q^{83} +12.7670 q^{84} +11.4727 q^{85} +10.3433 q^{86} +7.22321 q^{87} +31.8681 q^{88} -3.99387 q^{89} -10.0846 q^{90} +9.16065 q^{91} +34.7265 q^{92} -4.65197 q^{93} -4.23033 q^{94} -3.20628 q^{95} -20.3316 q^{96} -3.82157 q^{97} -8.42254 q^{98} +7.54763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.79357 −1.97535 −0.987677 − 0.156508i \(-0.949976\pi\)
−0.987677 + 0.156508i \(0.949976\pi\)
\(3\) 0.695080 0.401305 0.200652 − 0.979663i \(-0.435694\pi\)
0.200652 + 0.979663i \(0.435694\pi\)
\(4\) 5.80404 2.90202
\(5\) −1.43429 −0.641436 −0.320718 − 0.947175i \(-0.603924\pi\)
−0.320718 + 0.947175i \(0.603924\pi\)
\(6\) −1.94176 −0.792719
\(7\) 3.16464 1.19612 0.598062 − 0.801450i \(-0.295938\pi\)
0.598062 + 0.801450i \(0.295938\pi\)
\(8\) −10.6269 −3.75716
\(9\) −2.51686 −0.838955
\(10\) 4.00680 1.26706
\(11\) −2.99882 −0.904179 −0.452090 − 0.891972i \(-0.649321\pi\)
−0.452090 + 0.891972i \(0.649321\pi\)
\(12\) 4.03427 1.16459
\(13\) 2.89469 0.802841 0.401421 − 0.915894i \(-0.368517\pi\)
0.401421 + 0.915894i \(0.368517\pi\)
\(14\) −8.84066 −2.36277
\(15\) −0.996950 −0.257411
\(16\) 18.0788 4.51970
\(17\) −7.99885 −1.94001 −0.970003 − 0.243091i \(-0.921839\pi\)
−0.970003 + 0.243091i \(0.921839\pi\)
\(18\) 7.03104 1.65723
\(19\) 2.23544 0.512846 0.256423 − 0.966565i \(-0.417456\pi\)
0.256423 + 0.966565i \(0.417456\pi\)
\(20\) −8.32470 −1.86146
\(21\) 2.19968 0.480010
\(22\) 8.37743 1.78607
\(23\) 5.98316 1.24757 0.623787 − 0.781594i \(-0.285593\pi\)
0.623787 + 0.781594i \(0.285593\pi\)
\(24\) −7.38652 −1.50777
\(25\) −2.94280 −0.588560
\(26\) −8.08651 −1.58590
\(27\) −3.83466 −0.737981
\(28\) 18.3677 3.47117
\(29\) 10.3919 1.92973 0.964865 − 0.262747i \(-0.0846285\pi\)
0.964865 + 0.262747i \(0.0846285\pi\)
\(30\) 2.78505 0.508478
\(31\) −6.69270 −1.20205 −0.601023 − 0.799232i \(-0.705240\pi\)
−0.601023 + 0.799232i \(0.705240\pi\)
\(32\) −29.2507 −5.17085
\(33\) −2.08442 −0.362851
\(34\) 22.3454 3.83220
\(35\) −4.53903 −0.767236
\(36\) −14.6080 −2.43466
\(37\) 3.17307 0.521649 0.260824 − 0.965386i \(-0.416006\pi\)
0.260824 + 0.965386i \(0.416006\pi\)
\(38\) −6.24487 −1.01305
\(39\) 2.01204 0.322184
\(40\) 15.2421 2.40998
\(41\) −2.39388 −0.373861 −0.186931 − 0.982373i \(-0.559854\pi\)
−0.186931 + 0.982373i \(0.559854\pi\)
\(42\) −6.14497 −0.948189
\(43\) −3.70255 −0.564633 −0.282317 − 0.959321i \(-0.591103\pi\)
−0.282317 + 0.959321i \(0.591103\pi\)
\(44\) −17.4053 −2.62395
\(45\) 3.60992 0.538136
\(46\) −16.7144 −2.46440
\(47\) 1.51431 0.220885 0.110442 − 0.993883i \(-0.464773\pi\)
0.110442 + 0.993883i \(0.464773\pi\)
\(48\) 12.5662 1.81378
\(49\) 3.01497 0.430710
\(50\) 8.22092 1.16261
\(51\) −5.55984 −0.778534
\(52\) 16.8009 2.32986
\(53\) −2.05987 −0.282945 −0.141472 − 0.989942i \(-0.545184\pi\)
−0.141472 + 0.989942i \(0.545184\pi\)
\(54\) 10.7124 1.45777
\(55\) 4.30120 0.579973
\(56\) −33.6302 −4.49403
\(57\) 1.55381 0.205807
\(58\) −29.0305 −3.81190
\(59\) 4.95271 0.644788 0.322394 − 0.946606i \(-0.395512\pi\)
0.322394 + 0.946606i \(0.395512\pi\)
\(60\) −5.78634 −0.747013
\(61\) −7.27148 −0.931018 −0.465509 − 0.885043i \(-0.654129\pi\)
−0.465509 + 0.885043i \(0.654129\pi\)
\(62\) 18.6965 2.37446
\(63\) −7.96498 −1.00349
\(64\) 45.5564 5.69455
\(65\) −4.15183 −0.514971
\(66\) 5.82298 0.716760
\(67\) 5.32889 0.651027 0.325514 − 0.945537i \(-0.394463\pi\)
0.325514 + 0.945537i \(0.394463\pi\)
\(68\) −46.4257 −5.62994
\(69\) 4.15877 0.500657
\(70\) 12.6801 1.51556
\(71\) 2.28191 0.270813 0.135406 − 0.990790i \(-0.456766\pi\)
0.135406 + 0.990790i \(0.456766\pi\)
\(72\) 26.7464 3.15209
\(73\) −0.565151 −0.0661459 −0.0330729 − 0.999453i \(-0.510529\pi\)
−0.0330729 + 0.999453i \(0.510529\pi\)
\(74\) −8.86419 −1.03044
\(75\) −2.04548 −0.236192
\(76\) 12.9746 1.48829
\(77\) −9.49021 −1.08151
\(78\) −5.62077 −0.636427
\(79\) 3.79783 0.427290 0.213645 − 0.976911i \(-0.431466\pi\)
0.213645 + 0.976911i \(0.431466\pi\)
\(80\) −25.9303 −2.89910
\(81\) 4.88519 0.542799
\(82\) 6.68748 0.738508
\(83\) 8.06607 0.885367 0.442683 − 0.896678i \(-0.354027\pi\)
0.442683 + 0.896678i \(0.354027\pi\)
\(84\) 12.7670 1.39300
\(85\) 11.4727 1.24439
\(86\) 10.3433 1.11535
\(87\) 7.22321 0.774410
\(88\) 31.8681 3.39715
\(89\) −3.99387 −0.423349 −0.211675 − 0.977340i \(-0.567892\pi\)
−0.211675 + 0.977340i \(0.567892\pi\)
\(90\) −10.0846 −1.06301
\(91\) 9.16065 0.960297
\(92\) 34.7265 3.62049
\(93\) −4.65197 −0.482386
\(94\) −4.23033 −0.436325
\(95\) −3.20628 −0.328958
\(96\) −20.3316 −2.07509
\(97\) −3.82157 −0.388022 −0.194011 − 0.980999i \(-0.562150\pi\)
−0.194011 + 0.980999i \(0.562150\pi\)
\(98\) −8.42254 −0.850805
\(99\) 7.54763 0.758565
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.1.2 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.2 ✓ 184 1.1 even 1 trivial