↑

Properties

Label 4001.2.a.b.1.5
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 1.5
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)
 
\(f(q)\) \(=\) \(q-2.73134 q^{2} +2.60660 q^{3} +5.46019 q^{4} -2.78013 q^{5} -7.11951 q^{6} -4.77010 q^{7} -9.45095 q^{8} +3.79438 q^{9} +7.59346 q^{10} -5.83664 q^{11} +14.2326 q^{12} -1.87301 q^{13} +13.0288 q^{14} -7.24669 q^{15} +14.8933 q^{16} -7.42699 q^{17} -10.3637 q^{18} -2.84199 q^{19} -15.1800 q^{20} -12.4338 q^{21} +15.9418 q^{22} +0.931598 q^{23} -24.6349 q^{24} +2.72912 q^{25} +5.11582 q^{26} +2.07064 q^{27} -26.0457 q^{28} -7.87193 q^{29} +19.7932 q^{30} +9.52117 q^{31} -21.7768 q^{32} -15.2138 q^{33} +20.2856 q^{34} +13.2615 q^{35} +20.7181 q^{36} -9.70110 q^{37} +7.76244 q^{38} -4.88220 q^{39} +26.2749 q^{40} +3.79143 q^{41} +33.9608 q^{42} +4.05023 q^{43} -31.8692 q^{44} -10.5489 q^{45} -2.54451 q^{46} +1.91568 q^{47} +38.8210 q^{48} +15.7539 q^{49} -7.45413 q^{50} -19.3592 q^{51} -10.2270 q^{52} +0.163432 q^{53} -5.65561 q^{54} +16.2266 q^{55} +45.0820 q^{56} -7.40795 q^{57} +21.5009 q^{58} -1.71736 q^{59} -39.5684 q^{60} -4.72755 q^{61} -26.0055 q^{62} -18.0996 q^{63} +29.6930 q^{64} +5.20721 q^{65} +41.5540 q^{66} -15.7470 q^{67} -40.5528 q^{68} +2.42831 q^{69} -36.2216 q^{70} -4.17639 q^{71} -35.8605 q^{72} -7.50657 q^{73} +26.4970 q^{74} +7.11372 q^{75} -15.5178 q^{76} +27.8414 q^{77} +13.3349 q^{78} +4.79640 q^{79} -41.4054 q^{80} -5.98581 q^{81} -10.3557 q^{82} +1.13275 q^{83} -67.8908 q^{84} +20.6480 q^{85} -11.0625 q^{86} -20.5190 q^{87} +55.1618 q^{88} -6.41276 q^{89} +28.8125 q^{90} +8.93446 q^{91} +5.08671 q^{92} +24.8179 q^{93} -5.23237 q^{94} +7.90111 q^{95} -56.7634 q^{96} -3.59427 q^{97} -43.0292 q^{98} -22.1464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73134 −1.93135 −0.965673 − 0.259761i \(-0.916356\pi\)
−0.965673 + 0.259761i \(0.916356\pi\)
\(3\) 2.60660 1.50492 0.752462 − 0.658636i \(-0.228866\pi\)
0.752462 + 0.658636i \(0.228866\pi\)
\(4\) 5.46019 2.73010
\(5\) −2.78013 −1.24331 −0.621656 − 0.783291i \(-0.713540\pi\)
−0.621656 + 0.783291i \(0.713540\pi\)
\(6\) −7.11951 −2.90653
\(7\) −4.77010 −1.80293 −0.901465 − 0.432852i \(-0.857507\pi\)
−0.901465 + 0.432852i \(0.857507\pi\)
\(8\) −9.45095 −3.34142
\(9\) 3.79438 1.26479
\(10\) 7.59346 2.40126
\(11\) −5.83664 −1.75981 −0.879907 − 0.475147i \(-0.842395\pi\)
−0.879907 + 0.475147i \(0.842395\pi\)
\(12\) 14.2326 4.10859
\(13\) −1.87301 −0.519480 −0.259740 − 0.965679i \(-0.583637\pi\)
−0.259740 + 0.965679i \(0.583637\pi\)
\(14\) 13.0288 3.48208
\(15\) −7.24669 −1.87109
\(16\) 14.8933 3.72333
\(17\) −7.42699 −1.80131 −0.900655 − 0.434534i \(-0.856913\pi\)
−0.900655 + 0.434534i \(0.856913\pi\)
\(18\) −10.3637 −2.44275
\(19\) −2.84199 −0.651998 −0.325999 − 0.945370i \(-0.605701\pi\)
−0.325999 + 0.945370i \(0.605701\pi\)
\(20\) −15.1800 −3.39436
\(21\) −12.4338 −2.71327
\(22\) 15.9418 3.39881
\(23\) 0.931598 0.194252 0.0971258 − 0.995272i \(-0.469035\pi\)
0.0971258 + 0.995272i \(0.469035\pi\)
\(24\) −24.6349 −5.02857
\(25\) 2.72912 0.545823
\(26\) 5.11582 1.00330
\(27\) 2.07064 0.398495
\(28\) −26.0457 −4.92217
\(29\) −7.87193 −1.46178 −0.730890 − 0.682495i \(-0.760895\pi\)
−0.730890 + 0.682495i \(0.760895\pi\)
\(30\) 19.7932 3.61372
\(31\) 9.52117 1.71005 0.855026 − 0.518585i \(-0.173541\pi\)
0.855026 + 0.518585i \(0.173541\pi\)
\(32\) −21.7768 −3.84963
\(33\) −15.2138 −2.64838
\(34\) 20.2856 3.47895
\(35\) 13.2615 2.24160
\(36\) 20.7181 3.45301
\(37\) −9.70110 −1.59485 −0.797426 − 0.603417i \(-0.793806\pi\)
−0.797426 + 0.603417i \(0.793806\pi\)
\(38\) 7.76244 1.25923
\(39\) −4.88220 −0.781777
\(40\) 26.2749 4.15442
\(41\) 3.79143 0.592122 0.296061 − 0.955169i \(-0.404327\pi\)
0.296061 + 0.955169i \(0.404327\pi\)
\(42\) 33.9608 5.24027
\(43\) 4.05023 0.617654 0.308827 − 0.951118i \(-0.400064\pi\)
0.308827 + 0.951118i \(0.400064\pi\)
\(44\) −31.8692 −4.80446
\(45\) −10.5489 −1.57253
\(46\) −2.54451 −0.375167
\(47\) 1.91568 0.279431 0.139715 − 0.990192i \(-0.455381\pi\)
0.139715 + 0.990192i \(0.455381\pi\)
\(48\) 38.8210 5.60333
\(49\) 15.7539 2.25056
\(50\) −7.45413 −1.05417
\(51\) −19.3592 −2.71083
\(52\) −10.2270 −1.41823
\(53\) 0.163432 0.0224492 0.0112246 − 0.999937i \(-0.496427\pi\)
0.0112246 + 0.999937i \(0.496427\pi\)
\(54\) −5.65561 −0.769631
\(55\) 16.2266 2.18800
\(56\) 45.0820 6.02434
\(57\) −7.40795 −0.981207
\(58\) 21.5009 2.82320
\(59\) −1.71736 −0.223581 −0.111791 − 0.993732i \(-0.535659\pi\)
−0.111791 + 0.993732i \(0.535659\pi\)
\(60\) −39.5684 −5.10825
\(61\) −4.72755 −0.605300 −0.302650 − 0.953102i \(-0.597871\pi\)
−0.302650 + 0.953102i \(0.597871\pi\)
\(62\) −26.0055 −3.30270
\(63\) −18.0996 −2.28034
\(64\) 29.6930 3.71163
\(65\) 5.20721 0.645875
\(66\) 41.5540 5.11495
\(67\) −15.7470 −1.92380 −0.961900 − 0.273402i \(-0.911851\pi\)
−0.961900 + 0.273402i \(0.911851\pi\)
\(68\) −40.5528 −4.91775
\(69\) 2.42831 0.292334
\(70\) −36.2216 −4.32931
\(71\) −4.17639 −0.495646 −0.247823 − 0.968805i \(-0.579715\pi\)
−0.247823 + 0.968805i \(0.579715\pi\)
\(72\) −35.8605 −4.22620
\(73\) −7.50657 −0.878578 −0.439289 − 0.898346i \(-0.644770\pi\)
−0.439289 + 0.898346i \(0.644770\pi\)
\(74\) 26.4970 3.08021
\(75\) 7.11372 0.821422
\(76\) −15.5178 −1.78002
\(77\) 27.8414 3.17282
\(78\) 13.3349 1.50988
\(79\) 4.79640 0.539637 0.269818 − 0.962911i \(-0.413036\pi\)
0.269818 + 0.962911i \(0.413036\pi\)
\(80\) −41.4054 −4.62926
\(81\) −5.98581 −0.665090
\(82\) −10.3557 −1.14359
\(83\) 1.13275 0.124335 0.0621677 − 0.998066i \(-0.480199\pi\)
0.0621677 + 0.998066i \(0.480199\pi\)
\(84\) −67.8908 −7.40749
\(85\) 20.6480 2.23959
\(86\) −11.0625 −1.19290
\(87\) −20.5190 −2.19987
\(88\) 55.1618 5.88027
\(89\) −6.41276 −0.679752 −0.339876 − 0.940470i \(-0.610385\pi\)
−0.339876 + 0.940470i \(0.610385\pi\)
\(90\) 28.8125 3.03710
\(91\) 8.93446 0.936586
\(92\) 5.08671 0.530326
\(93\) 24.8179 2.57350
\(94\) −5.23237 −0.539677
\(95\) 7.90111 0.810636
\(96\) −56.7634 −5.79339
\(97\) −3.59427 −0.364942 −0.182471 − 0.983211i \(-0.558410\pi\)
−0.182471 + 0.983211i \(0.558410\pi\)
\(98\) −43.0292 −4.34660
\(99\) −22.1464 −2.22580
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.1.5 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.5 ✓ 184 1.1 even 1 trivial