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Properties

Label 4001.2.a.b.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.80114 q^{2} -3.24231 q^{3} +5.84639 q^{4} -0.362239 q^{5} +9.08218 q^{6} +0.683027 q^{7} -10.7743 q^{8} +7.51260 q^{9} +1.01468 q^{10} -2.13162 q^{11} -18.9558 q^{12} -1.68516 q^{13} -1.91326 q^{14} +1.17449 q^{15} +18.4875 q^{16} -5.09163 q^{17} -21.0439 q^{18} -0.801920 q^{19} -2.11779 q^{20} -2.21459 q^{21} +5.97097 q^{22} -8.57971 q^{23} +34.9336 q^{24} -4.86878 q^{25} +4.72037 q^{26} -14.6313 q^{27} +3.99325 q^{28} -3.50963 q^{29} -3.28992 q^{30} +1.80809 q^{31} -30.2376 q^{32} +6.91138 q^{33} +14.2624 q^{34} -0.247419 q^{35} +43.9216 q^{36} -5.00800 q^{37} +2.24629 q^{38} +5.46382 q^{39} +3.90287 q^{40} -9.31222 q^{41} +6.20338 q^{42} -7.60034 q^{43} -12.4623 q^{44} -2.72136 q^{45} +24.0330 q^{46} +1.24331 q^{47} -59.9424 q^{48} -6.53347 q^{49} +13.6381 q^{50} +16.5087 q^{51} -9.85210 q^{52} +7.78403 q^{53} +40.9843 q^{54} +0.772157 q^{55} -7.35914 q^{56} +2.60008 q^{57} +9.83096 q^{58} -1.98291 q^{59} +6.86655 q^{60} -3.99389 q^{61} -5.06473 q^{62} +5.13131 q^{63} +47.7248 q^{64} +0.610431 q^{65} -19.3598 q^{66} +5.71728 q^{67} -29.7677 q^{68} +27.8181 q^{69} +0.693057 q^{70} +8.29768 q^{71} -80.9430 q^{72} -1.66674 q^{73} +14.0281 q^{74} +15.7861 q^{75} -4.68834 q^{76} -1.45595 q^{77} -15.3049 q^{78} +1.42417 q^{79} -6.69691 q^{80} +24.9014 q^{81} +26.0849 q^{82} -1.86129 q^{83} -12.9474 q^{84} +1.84439 q^{85} +21.2896 q^{86} +11.3793 q^{87} +22.9667 q^{88} -6.35369 q^{89} +7.62291 q^{90} -1.15101 q^{91} -50.1604 q^{92} -5.86241 q^{93} -3.48270 q^{94} +0.290487 q^{95} +98.0399 q^{96} +10.6003 q^{97} +18.3012 q^{98} -16.0140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.80114 −1.98071 −0.990353 − 0.138567i \(-0.955750\pi\)
−0.990353 + 0.138567i \(0.955750\pi\)
\(3\) −3.24231 −1.87195 −0.935975 − 0.352065i \(-0.885480\pi\)
−0.935975 + 0.352065i \(0.885480\pi\)
\(4\) 5.84639 2.92320
\(5\) −0.362239 −0.161998 −0.0809992 − 0.996714i \(-0.525811\pi\)
−0.0809992 + 0.996714i \(0.525811\pi\)
\(6\) 9.08218 3.70778
\(7\) 0.683027 0.258160 0.129080 − 0.991634i \(-0.458798\pi\)
0.129080 + 0.991634i \(0.458798\pi\)
\(8\) −10.7743 −3.80929
\(9\) 7.51260 2.50420
\(10\) 1.01468 0.320871
\(11\) −2.13162 −0.642708 −0.321354 − 0.946959i \(-0.604138\pi\)
−0.321354 + 0.946959i \(0.604138\pi\)
\(12\) −18.9558 −5.47208
\(13\) −1.68516 −0.467379 −0.233690 − 0.972311i \(-0.575080\pi\)
−0.233690 + 0.972311i \(0.575080\pi\)
\(14\) −1.91326 −0.511339
\(15\) 1.17449 0.303253
\(16\) 18.4875 4.62188
\(17\) −5.09163 −1.23490 −0.617450 − 0.786610i \(-0.711834\pi\)
−0.617450 + 0.786610i \(0.711834\pi\)
\(18\) −21.0439 −4.96009
\(19\) −0.801920 −0.183973 −0.0919865 − 0.995760i \(-0.529322\pi\)
−0.0919865 + 0.995760i \(0.529322\pi\)
\(20\) −2.11779 −0.473553
\(21\) −2.21459 −0.483263
\(22\) 5.97097 1.27302
\(23\) −8.57971 −1.78899 −0.894497 − 0.447075i \(-0.852466\pi\)
−0.894497 + 0.447075i \(0.852466\pi\)
\(24\) 34.9336 7.13080
\(25\) −4.86878 −0.973757
\(26\) 4.72037 0.925741
\(27\) −14.6313 −2.81579
\(28\) 3.99325 0.754653
\(29\) −3.50963 −0.651721 −0.325861 − 0.945418i \(-0.605654\pi\)
−0.325861 + 0.945418i \(0.605654\pi\)
\(30\) −3.28992 −0.600655
\(31\) 1.80809 0.324743 0.162372 − 0.986730i \(-0.448086\pi\)
0.162372 + 0.986730i \(0.448086\pi\)
\(32\) −30.2376 −5.34531
\(33\) 6.91138 1.20312
\(34\) 14.2624 2.44598
\(35\) −0.247419 −0.0418215
\(36\) 43.9216 7.32027
\(37\) −5.00800 −0.823310 −0.411655 − 0.911340i \(-0.635049\pi\)
−0.411655 + 0.911340i \(0.635049\pi\)
\(38\) 2.24629 0.364396
\(39\) 5.46382 0.874911
\(40\) 3.90287 0.617098
\(41\) −9.31222 −1.45432 −0.727162 − 0.686465i \(-0.759161\pi\)
−0.727162 + 0.686465i \(0.759161\pi\)
\(42\) 6.20338 0.957202
\(43\) −7.60034 −1.15904 −0.579520 − 0.814958i \(-0.696760\pi\)
−0.579520 + 0.814958i \(0.696760\pi\)
\(44\) −12.4623 −1.87876
\(45\) −2.72136 −0.405676
\(46\) 24.0330 3.54347
\(47\) 1.24331 0.181356 0.0906780 − 0.995880i \(-0.471097\pi\)
0.0906780 + 0.995880i \(0.471097\pi\)
\(48\) −59.9424 −8.65194
\(49\) −6.53347 −0.933353
\(50\) 13.6381 1.92873
\(51\) 16.5087 2.31167
\(52\) −9.85210 −1.36624
\(53\) 7.78403 1.06922 0.534609 − 0.845099i \(-0.320459\pi\)
0.534609 + 0.845099i \(0.320459\pi\)
\(54\) 40.9843 5.57725
\(55\) 0.772157 0.104118
\(56\) −7.35914 −0.983406
\(57\) 2.60008 0.344388
\(58\) 9.83096 1.29087
\(59\) −1.98291 −0.258153 −0.129076 − 0.991635i \(-0.541201\pi\)
−0.129076 + 0.991635i \(0.541201\pi\)
\(60\) 6.86655 0.886468
\(61\) −3.99389 −0.511366 −0.255683 − 0.966761i \(-0.582300\pi\)
−0.255683 + 0.966761i \(0.582300\pi\)
\(62\) −5.06473 −0.643221
\(63\) 5.13131 0.646484
\(64\) 47.7248 5.96560
\(65\) 0.610431 0.0757146
\(66\) −19.3598 −2.38302
\(67\) 5.71728 0.698476 0.349238 − 0.937034i \(-0.386440\pi\)
0.349238 + 0.937034i \(0.386440\pi\)
\(68\) −29.7677 −3.60986
\(69\) 27.8181 3.34891
\(70\) 0.693057 0.0828361
\(71\) 8.29768 0.984753 0.492377 − 0.870382i \(-0.336128\pi\)
0.492377 + 0.870382i \(0.336128\pi\)
\(72\) −80.9430 −9.53922
\(73\) −1.66674 −0.195077 −0.0975385 − 0.995232i \(-0.531097\pi\)
−0.0975385 + 0.995232i \(0.531097\pi\)
\(74\) 14.0281 1.63074
\(75\) 15.7861 1.82282
\(76\) −4.68834 −0.537789
\(77\) −1.45595 −0.165921
\(78\) −15.3049 −1.73294
\(79\) 1.42417 0.160232 0.0801160 − 0.996786i \(-0.474471\pi\)
0.0801160 + 0.996786i \(0.474471\pi\)
\(80\) −6.69691 −0.748738
\(81\) 24.9014 2.76682
\(82\) 26.0849 2.88059
\(83\) −1.86129 −0.204303 −0.102152 − 0.994769i \(-0.532573\pi\)
−0.102152 + 0.994769i \(0.532573\pi\)
\(84\) −12.9474 −1.41267
\(85\) 1.84439 0.200052
\(86\) 21.2896 2.29572
\(87\) 11.3793 1.21999
\(88\) 22.9667 2.44826
\(89\) −6.35369 −0.673490 −0.336745 − 0.941596i \(-0.609326\pi\)
−0.336745 + 0.941596i \(0.609326\pi\)
\(90\) 7.62291 0.803526
\(91\) −1.15101 −0.120659
\(92\) −50.1604 −5.22958
\(93\) −5.86241 −0.607904
\(94\) −3.48270 −0.359213
\(95\) 0.290487 0.0298033
\(96\) 98.0399 10.0062
\(97\) 10.6003 1.07630 0.538150 − 0.842849i \(-0.319123\pi\)
0.538150 + 0.842849i \(0.319123\pi\)
\(98\) 18.3012 1.84870
\(99\) −16.0140 −1.60947
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.1.1 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.1 ✓ 184 1.1 even 1 trivial