↑

Properties

Label 4001.2.a.b.1.16
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $0$
Dimension $184$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.16
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)
 
\(f(q)\) \(=\) \(q-2.43211 q^{2} +0.645473 q^{3} +3.91518 q^{4} -1.15456 q^{5} -1.56986 q^{6} +1.52754 q^{7} -4.65795 q^{8} -2.58336 q^{9} +2.80801 q^{10} -3.11685 q^{11} +2.52715 q^{12} +4.55740 q^{13} -3.71516 q^{14} -0.745234 q^{15} +3.49829 q^{16} -2.58969 q^{17} +6.28304 q^{18} -7.28248 q^{19} -4.52030 q^{20} +0.985988 q^{21} +7.58053 q^{22} -6.08960 q^{23} -3.00658 q^{24} -3.66700 q^{25} -11.0841 q^{26} -3.60391 q^{27} +5.98061 q^{28} -10.5062 q^{29} +1.81250 q^{30} +7.22937 q^{31} +0.807642 q^{32} -2.01184 q^{33} +6.29842 q^{34} -1.76363 q^{35} -10.1143 q^{36} +10.7652 q^{37} +17.7118 q^{38} +2.94168 q^{39} +5.37786 q^{40} +5.88882 q^{41} -2.39804 q^{42} -4.57214 q^{43} -12.2030 q^{44} +2.98264 q^{45} +14.8106 q^{46} +10.7979 q^{47} +2.25805 q^{48} -4.66661 q^{49} +8.91857 q^{50} -1.67157 q^{51} +17.8431 q^{52} -2.00845 q^{53} +8.76513 q^{54} +3.59857 q^{55} -7.11521 q^{56} -4.70065 q^{57} +25.5523 q^{58} -5.67677 q^{59} -2.91773 q^{60} -3.17694 q^{61} -17.5827 q^{62} -3.94620 q^{63} -8.96086 q^{64} -5.26177 q^{65} +4.89303 q^{66} -1.68173 q^{67} -10.1391 q^{68} -3.93067 q^{69} +4.28936 q^{70} -13.0240 q^{71} +12.0332 q^{72} +14.1461 q^{73} -26.1821 q^{74} -2.36695 q^{75} -28.5123 q^{76} -4.76112 q^{77} -7.15450 q^{78} +15.4580 q^{79} -4.03897 q^{80} +5.42387 q^{81} -14.3223 q^{82} +11.0513 q^{83} +3.86032 q^{84} +2.98994 q^{85} +11.1200 q^{86} -6.78148 q^{87} +14.5181 q^{88} +15.1818 q^{89} -7.25412 q^{90} +6.96162 q^{91} -23.8419 q^{92} +4.66637 q^{93} -26.2617 q^{94} +8.40803 q^{95} +0.521311 q^{96} +1.72861 q^{97} +11.3497 q^{98} +8.05195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.43211 −1.71976 −0.859882 − 0.510492i \(-0.829463\pi\)
−0.859882 + 0.510492i \(0.829463\pi\)
\(3\) 0.645473 0.372664 0.186332 − 0.982487i \(-0.440340\pi\)
0.186332 + 0.982487i \(0.440340\pi\)
\(4\) 3.91518 1.95759
\(5\) −1.15456 −0.516333 −0.258166 − 0.966100i \(-0.583118\pi\)
−0.258166 + 0.966100i \(0.583118\pi\)
\(6\) −1.56986 −0.640895
\(7\) 1.52754 0.577357 0.288679 − 0.957426i \(-0.406784\pi\)
0.288679 + 0.957426i \(0.406784\pi\)
\(8\) −4.65795 −1.64683
\(9\) −2.58336 −0.861121
\(10\) 2.80801 0.887971
\(11\) −3.11685 −0.939764 −0.469882 − 0.882729i \(-0.655704\pi\)
−0.469882 + 0.882729i \(0.655704\pi\)
\(12\) 2.52715 0.729524
\(13\) 4.55740 1.26400 0.631998 − 0.774970i \(-0.282235\pi\)
0.631998 + 0.774970i \(0.282235\pi\)
\(14\) −3.71516 −0.992918
\(15\) −0.745234 −0.192419
\(16\) 3.49829 0.874573
\(17\) −2.58969 −0.628092 −0.314046 − 0.949408i \(-0.601685\pi\)
−0.314046 + 0.949408i \(0.601685\pi\)
\(18\) 6.28304 1.48093
\(19\) −7.28248 −1.67072 −0.835358 − 0.549706i \(-0.814740\pi\)
−0.835358 + 0.549706i \(0.814740\pi\)
\(20\) −4.52030 −1.01077
\(21\) 0.985988 0.215160
\(22\) 7.58053 1.61617
\(23\) −6.08960 −1.26977 −0.634885 − 0.772607i \(-0.718952\pi\)
−0.634885 + 0.772607i \(0.718952\pi\)
\(24\) −3.00658 −0.613715
\(25\) −3.66700 −0.733400
\(26\) −11.0841 −2.17377
\(27\) −3.60391 −0.693573
\(28\) 5.98061 1.13023
\(29\) −10.5062 −1.95096 −0.975478 − 0.220097i \(-0.929362\pi\)
−0.975478 + 0.220097i \(0.929362\pi\)
\(30\) 1.81250 0.330915
\(31\) 7.22937 1.29843 0.649217 − 0.760603i \(-0.275097\pi\)
0.649217 + 0.760603i \(0.275097\pi\)
\(32\) 0.807642 0.142772
\(33\) −2.01184 −0.350216
\(34\) 6.29842 1.08017
\(35\) −1.76363 −0.298108
\(36\) −10.1143 −1.68572
\(37\) 10.7652 1.76978 0.884890 − 0.465800i \(-0.154233\pi\)
0.884890 + 0.465800i \(0.154233\pi\)
\(38\) 17.7118 2.87324
\(39\) 2.94168 0.471046
\(40\) 5.37786 0.850314
\(41\) 5.88882 0.919679 0.459839 − 0.888002i \(-0.347907\pi\)
0.459839 + 0.888002i \(0.347907\pi\)
\(42\) −2.39804 −0.370025
\(43\) −4.57214 −0.697245 −0.348622 − 0.937263i \(-0.613350\pi\)
−0.348622 + 0.937263i \(0.613350\pi\)
\(44\) −12.2030 −1.83967
\(45\) 2.98264 0.444625
\(46\) 14.8106 2.18371
\(47\) 10.7979 1.57503 0.787516 − 0.616294i \(-0.211367\pi\)
0.787516 + 0.616294i \(0.211367\pi\)
\(48\) 2.25805 0.325922
\(49\) −4.66661 −0.666659
\(50\) 8.91857 1.26128
\(51\) −1.67157 −0.234067
\(52\) 17.8431 2.47439
\(53\) −2.00845 −0.275882 −0.137941 − 0.990440i \(-0.544048\pi\)
−0.137941 + 0.990440i \(0.544048\pi\)
\(54\) 8.76513 1.19278
\(55\) 3.59857 0.485231
\(56\) −7.11521 −0.950810
\(57\) −4.70065 −0.622616
\(58\) 25.5523 3.35519
\(59\) −5.67677 −0.739053 −0.369526 − 0.929220i \(-0.620480\pi\)
−0.369526 + 0.929220i \(0.620480\pi\)
\(60\) −2.91773 −0.376677
\(61\) −3.17694 −0.406766 −0.203383 − 0.979099i \(-0.565194\pi\)
−0.203383 + 0.979099i \(0.565194\pi\)
\(62\) −17.5827 −2.23300
\(63\) −3.94620 −0.497175
\(64\) −8.96086 −1.12011
\(65\) −5.26177 −0.652642
\(66\) 4.89303 0.602290
\(67\) −1.68173 −0.205456 −0.102728 − 0.994709i \(-0.532757\pi\)
−0.102728 + 0.994709i \(0.532757\pi\)
\(68\) −10.1391 −1.22955
\(69\) −3.93067 −0.473198
\(70\) 4.28936 0.512676
\(71\) −13.0240 −1.54566 −0.772830 − 0.634613i \(-0.781159\pi\)
−0.772830 + 0.634613i \(0.781159\pi\)
\(72\) 12.0332 1.41812
\(73\) 14.1461 1.65567 0.827836 − 0.560969i \(-0.189572\pi\)
0.827836 + 0.560969i \(0.189572\pi\)
\(74\) −26.1821 −3.04361
\(75\) −2.36695 −0.273312
\(76\) −28.5123 −3.27058
\(77\) −4.76112 −0.542580
\(78\) −7.15450 −0.810088
\(79\) 15.4580 1.73916 0.869579 − 0.493793i \(-0.164390\pi\)
0.869579 + 0.493793i \(0.164390\pi\)
\(80\) −4.03897 −0.451571
\(81\) 5.42387 0.602652
\(82\) −14.3223 −1.58163
\(83\) 11.0513 1.21304 0.606520 − 0.795068i \(-0.292565\pi\)
0.606520 + 0.795068i \(0.292565\pi\)
\(84\) 3.86032 0.421196
\(85\) 2.98994 0.324305
\(86\) 11.1200 1.19910
\(87\) −6.78148 −0.727051
\(88\) 14.5181 1.54763
\(89\) 15.1818 1.60927 0.804634 − 0.593771i \(-0.202362\pi\)
0.804634 + 0.593771i \(0.202362\pi\)
\(90\) −7.25412 −0.764651
\(91\) 6.96162 0.729776
\(92\) −23.8419 −2.48569
\(93\) 4.66637 0.483880
\(94\) −26.2617 −2.70869
\(95\) 8.40803 0.862646
\(96\) 0.521311 0.0532061
\(97\) 1.72861 0.175513 0.0877567 − 0.996142i \(-0.472030\pi\)
0.0877567 + 0.996142i \(0.472030\pi\)
\(98\) 11.3497 1.14650
\(99\) 8.05195 0.809251
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.1.16 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.16 ✓ 184 1.1 even 1 trivial