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Properties

Label 4001.2.a.b.1.17
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 1.17
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)
 
\(f(q)\) \(=\) \(q-2.42514 q^{2} -1.95794 q^{3} +3.88128 q^{4} +1.35530 q^{5} +4.74827 q^{6} +4.91337 q^{7} -4.56237 q^{8} +0.833523 q^{9} -3.28679 q^{10} -3.37448 q^{11} -7.59931 q^{12} +5.27920 q^{13} -11.9156 q^{14} -2.65360 q^{15} +3.30180 q^{16} +0.0667348 q^{17} -2.02141 q^{18} +1.69482 q^{19} +5.26031 q^{20} -9.62007 q^{21} +8.18357 q^{22} -2.77558 q^{23} +8.93284 q^{24} -3.16316 q^{25} -12.8028 q^{26} +4.24183 q^{27} +19.0702 q^{28} -7.59144 q^{29} +6.43533 q^{30} -7.71655 q^{31} +1.11743 q^{32} +6.60702 q^{33} -0.161841 q^{34} +6.65909 q^{35} +3.23514 q^{36} -1.61453 q^{37} -4.11016 q^{38} -10.3363 q^{39} -6.18339 q^{40} -5.37816 q^{41} +23.3300 q^{42} +4.89344 q^{43} -13.0973 q^{44} +1.12967 q^{45} +6.73117 q^{46} +1.94532 q^{47} -6.46472 q^{48} +17.1412 q^{49} +7.67109 q^{50} -0.130663 q^{51} +20.4901 q^{52} +3.19462 q^{53} -10.2870 q^{54} -4.57344 q^{55} -22.4166 q^{56} -3.31835 q^{57} +18.4103 q^{58} +5.14216 q^{59} -10.2994 q^{60} +2.84575 q^{61} +18.7137 q^{62} +4.09540 q^{63} -9.31352 q^{64} +7.15490 q^{65} -16.0229 q^{66} +7.90903 q^{67} +0.259017 q^{68} +5.43442 q^{69} -16.1492 q^{70} +11.1443 q^{71} -3.80284 q^{72} +8.94215 q^{73} +3.91545 q^{74} +6.19327 q^{75} +6.57806 q^{76} -16.5801 q^{77} +25.0670 q^{78} +0.733789 q^{79} +4.47493 q^{80} -10.8058 q^{81} +13.0428 q^{82} -5.25827 q^{83} -37.3382 q^{84} +0.0904458 q^{85} -11.8672 q^{86} +14.8636 q^{87} +15.3956 q^{88} +11.9131 q^{89} -2.73962 q^{90} +25.9386 q^{91} -10.7728 q^{92} +15.1085 q^{93} -4.71766 q^{94} +2.29699 q^{95} -2.18786 q^{96} +8.48979 q^{97} -41.5696 q^{98} -2.81271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.42514 −1.71483 −0.857415 − 0.514626i \(-0.827931\pi\)
−0.857415 + 0.514626i \(0.827931\pi\)
\(3\) −1.95794 −1.13042 −0.565208 − 0.824948i \(-0.691204\pi\)
−0.565208 + 0.824948i \(0.691204\pi\)
\(4\) 3.88128 1.94064
\(5\) 1.35530 0.606109 0.303055 − 0.952973i \(-0.401994\pi\)
0.303055 + 0.952973i \(0.401994\pi\)
\(6\) 4.74827 1.93847
\(7\) 4.91337 1.85708 0.928539 − 0.371235i \(-0.121066\pi\)
0.928539 + 0.371235i \(0.121066\pi\)
\(8\) −4.56237 −1.61304
\(9\) 0.833523 0.277841
\(10\) −3.28679 −1.03937
\(11\) −3.37448 −1.01744 −0.508722 − 0.860931i \(-0.669882\pi\)
−0.508722 + 0.860931i \(0.669882\pi\)
\(12\) −7.59931 −2.19373
\(13\) 5.27920 1.46419 0.732093 − 0.681205i \(-0.238544\pi\)
0.732093 + 0.681205i \(0.238544\pi\)
\(14\) −11.9156 −3.18457
\(15\) −2.65360 −0.685156
\(16\) 3.30180 0.825449
\(17\) 0.0667348 0.0161856 0.00809278 − 0.999967i \(-0.497424\pi\)
0.00809278 + 0.999967i \(0.497424\pi\)
\(18\) −2.02141 −0.476450
\(19\) 1.69482 0.388818 0.194409 − 0.980921i \(-0.437721\pi\)
0.194409 + 0.980921i \(0.437721\pi\)
\(20\) 5.26031 1.17624
\(21\) −9.62007 −2.09927
\(22\) 8.18357 1.74474
\(23\) −2.77558 −0.578749 −0.289375 − 0.957216i \(-0.593447\pi\)
−0.289375 + 0.957216i \(0.593447\pi\)
\(24\) 8.93284 1.82341
\(25\) −3.16316 −0.632631
\(26\) −12.8028 −2.51083
\(27\) 4.24183 0.816340
\(28\) 19.0702 3.60392
\(29\) −7.59144 −1.40969 −0.704847 − 0.709359i \(-0.748984\pi\)
−0.704847 + 0.709359i \(0.748984\pi\)
\(30\) 6.43533 1.17493
\(31\) −7.71655 −1.38593 −0.692966 − 0.720970i \(-0.743697\pi\)
−0.692966 + 0.720970i \(0.743697\pi\)
\(32\) 1.11743 0.197536
\(33\) 6.60702 1.15014
\(34\) −0.161841 −0.0277555
\(35\) 6.65909 1.12559
\(36\) 3.23514 0.539190
\(37\) −1.61453 −0.265427 −0.132714 − 0.991154i \(-0.542369\pi\)
−0.132714 + 0.991154i \(0.542369\pi\)
\(38\) −4.11016 −0.666756
\(39\) −10.3363 −1.65514
\(40\) −6.18339 −0.977679
\(41\) −5.37816 −0.839928 −0.419964 − 0.907541i \(-0.637957\pi\)
−0.419964 + 0.907541i \(0.637957\pi\)
\(42\) 23.3300 3.59989
\(43\) 4.89344 0.746242 0.373121 − 0.927783i \(-0.378288\pi\)
0.373121 + 0.927783i \(0.378288\pi\)
\(44\) −13.0973 −1.97449
\(45\) 1.12967 0.168402
\(46\) 6.73117 0.992457
\(47\) 1.94532 0.283754 0.141877 − 0.989884i \(-0.454686\pi\)
0.141877 + 0.989884i \(0.454686\pi\)
\(48\) −6.46472 −0.933101
\(49\) 17.1412 2.44874
\(50\) 7.67109 1.08486
\(51\) −0.130663 −0.0182964
\(52\) 20.4901 2.84146
\(53\) 3.19462 0.438814 0.219407 − 0.975633i \(-0.429588\pi\)
0.219407 + 0.975633i \(0.429588\pi\)
\(54\) −10.2870 −1.39988
\(55\) −4.57344 −0.616682
\(56\) −22.4166 −2.99554
\(57\) −3.31835 −0.439526
\(58\) 18.4103 2.41739
\(59\) 5.14216 0.669453 0.334726 − 0.942315i \(-0.391356\pi\)
0.334726 + 0.942315i \(0.391356\pi\)
\(60\) −10.2994 −1.32964
\(61\) 2.84575 0.364360 0.182180 − 0.983265i \(-0.441685\pi\)
0.182180 + 0.983265i \(0.441685\pi\)
\(62\) 18.7137 2.37664
\(63\) 4.09540 0.515972
\(64\) −9.31352 −1.16419
\(65\) 7.15490 0.887457
\(66\) −16.0229 −1.97229
\(67\) 7.90903 0.966242 0.483121 − 0.875554i \(-0.339503\pi\)
0.483121 + 0.875554i \(0.339503\pi\)
\(68\) 0.259017 0.0314104
\(69\) 5.43442 0.654228
\(70\) −16.1492 −1.93020
\(71\) 11.1443 1.32258 0.661292 − 0.750128i \(-0.270008\pi\)
0.661292 + 0.750128i \(0.270008\pi\)
\(72\) −3.80284 −0.448169
\(73\) 8.94215 1.04660 0.523300 − 0.852149i \(-0.324701\pi\)
0.523300 + 0.852149i \(0.324701\pi\)
\(74\) 3.91545 0.455162
\(75\) 6.19327 0.715137
\(76\) 6.57806 0.754556
\(77\) −16.5801 −1.88947
\(78\) 25.0670 2.83828
\(79\) 0.733789 0.0825577 0.0412789 − 0.999148i \(-0.486857\pi\)
0.0412789 + 0.999148i \(0.486857\pi\)
\(80\) 4.47493 0.500313
\(81\) −10.8058 −1.20065
\(82\) 13.0428 1.44033
\(83\) −5.25827 −0.577170 −0.288585 − 0.957454i \(-0.593185\pi\)
−0.288585 + 0.957454i \(0.593185\pi\)
\(84\) −37.3382 −4.07393
\(85\) 0.0904458 0.00981023
\(86\) −11.8672 −1.27968
\(87\) 14.8636 1.59354
\(88\) 15.3956 1.64118
\(89\) 11.9131 1.26279 0.631395 − 0.775461i \(-0.282483\pi\)
0.631395 + 0.775461i \(0.282483\pi\)
\(90\) −2.73962 −0.288781
\(91\) 25.9386 2.71911
\(92\) −10.7728 −1.12315
\(93\) 15.1085 1.56668
\(94\) −4.71766 −0.486589
\(95\) 2.29699 0.235666
\(96\) −2.18786 −0.223298
\(97\) 8.48979 0.862007 0.431004 − 0.902350i \(-0.358160\pi\)
0.431004 + 0.902350i \(0.358160\pi\)
\(98\) −41.5696 −4.19917
\(99\) −2.81271 −0.282688
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.1.17 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.17 ✓ 184 1.1 even 1 trivial