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Properties

Label 4001.2.a.b.1.19
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 1.19
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)
 
\(f(q)\) \(=\) \(q-2.38954 q^{2} +1.02374 q^{3} +3.70989 q^{4} -0.691971 q^{5} -2.44626 q^{6} +1.59189 q^{7} -4.08584 q^{8} -1.95196 q^{9} +1.65349 q^{10} -3.76337 q^{11} +3.79795 q^{12} -6.60109 q^{13} -3.80387 q^{14} -0.708397 q^{15} +2.34348 q^{16} -3.52471 q^{17} +4.66428 q^{18} +8.47101 q^{19} -2.56713 q^{20} +1.62968 q^{21} +8.99271 q^{22} -5.10188 q^{23} -4.18283 q^{24} -4.52118 q^{25} +15.7736 q^{26} -5.06951 q^{27} +5.90572 q^{28} -2.57168 q^{29} +1.69274 q^{30} +3.37313 q^{31} +2.57184 q^{32} -3.85271 q^{33} +8.42243 q^{34} -1.10154 q^{35} -7.24155 q^{36} -1.47696 q^{37} -20.2418 q^{38} -6.75779 q^{39} +2.82728 q^{40} -0.0916968 q^{41} -3.89417 q^{42} -0.702722 q^{43} -13.9617 q^{44} +1.35070 q^{45} +12.1911 q^{46} -10.1367 q^{47} +2.39911 q^{48} -4.46590 q^{49} +10.8035 q^{50} -3.60838 q^{51} -24.4893 q^{52} +3.48172 q^{53} +12.1138 q^{54} +2.60414 q^{55} -6.50419 q^{56} +8.67210 q^{57} +6.14512 q^{58} +11.3479 q^{59} -2.62807 q^{60} +10.2702 q^{61} -8.06021 q^{62} -3.10730 q^{63} -10.8325 q^{64} +4.56776 q^{65} +9.20618 q^{66} +3.97791 q^{67} -13.0763 q^{68} -5.22299 q^{69} +2.63217 q^{70} +1.20708 q^{71} +7.97539 q^{72} +10.3051 q^{73} +3.52925 q^{74} -4.62850 q^{75} +31.4265 q^{76} -5.99086 q^{77} +16.1480 q^{78} +10.0280 q^{79} -1.62162 q^{80} +0.666026 q^{81} +0.219113 q^{82} +11.1174 q^{83} +6.04591 q^{84} +2.43900 q^{85} +1.67918 q^{86} -2.63273 q^{87} +15.3765 q^{88} +9.14236 q^{89} -3.22755 q^{90} -10.5082 q^{91} -18.9274 q^{92} +3.45320 q^{93} +24.2220 q^{94} -5.86169 q^{95} +2.63289 q^{96} +4.87256 q^{97} +10.6714 q^{98} +7.34595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38954 −1.68966 −0.844829 − 0.535037i \(-0.820298\pi\)
−0.844829 + 0.535037i \(0.820298\pi\)
\(3\) 1.02374 0.591056 0.295528 − 0.955334i \(-0.404505\pi\)
0.295528 + 0.955334i \(0.404505\pi\)
\(4\) 3.70989 1.85494
\(5\) −0.691971 −0.309459 −0.154729 − 0.987957i \(-0.549451\pi\)
−0.154729 + 0.987957i \(0.549451\pi\)
\(6\) −2.44626 −0.998682
\(7\) 1.59189 0.601677 0.300838 − 0.953675i \(-0.402734\pi\)
0.300838 + 0.953675i \(0.402734\pi\)
\(8\) −4.08584 −1.44456
\(9\) −1.95196 −0.650653
\(10\) 1.65349 0.522879
\(11\) −3.76337 −1.13470 −0.567349 − 0.823477i \(-0.692031\pi\)
−0.567349 + 0.823477i \(0.692031\pi\)
\(12\) 3.79795 1.09637
\(13\) −6.60109 −1.83081 −0.915407 − 0.402530i \(-0.868131\pi\)
−0.915407 + 0.402530i \(0.868131\pi\)
\(14\) −3.80387 −1.01663
\(15\) −0.708397 −0.182907
\(16\) 2.34348 0.585870
\(17\) −3.52471 −0.854868 −0.427434 − 0.904046i \(-0.640582\pi\)
−0.427434 + 0.904046i \(0.640582\pi\)
\(18\) 4.66428 1.09938
\(19\) 8.47101 1.94338 0.971692 − 0.236252i \(-0.0759189\pi\)
0.971692 + 0.236252i \(0.0759189\pi\)
\(20\) −2.56713 −0.574028
\(21\) 1.62968 0.355624
\(22\) 8.99271 1.91725
\(23\) −5.10188 −1.06382 −0.531908 − 0.846802i \(-0.678525\pi\)
−0.531908 + 0.846802i \(0.678525\pi\)
\(24\) −4.18283 −0.853816
\(25\) −4.52118 −0.904235
\(26\) 15.7736 3.09345
\(27\) −5.06951 −0.975628
\(28\) 5.90572 1.11608
\(29\) −2.57168 −0.477549 −0.238775 − 0.971075i \(-0.576746\pi\)
−0.238775 + 0.971075i \(0.576746\pi\)
\(30\) 1.69274 0.309051
\(31\) 3.37313 0.605832 0.302916 − 0.953017i \(-0.402040\pi\)
0.302916 + 0.953017i \(0.402040\pi\)
\(32\) 2.57184 0.454641
\(33\) −3.85271 −0.670670
\(34\) 8.42243 1.44443
\(35\) −1.10154 −0.186194
\(36\) −7.24155 −1.20692
\(37\) −1.47696 −0.242811 −0.121405 − 0.992603i \(-0.538740\pi\)
−0.121405 + 0.992603i \(0.538740\pi\)
\(38\) −20.2418 −3.28365
\(39\) −6.75779 −1.08211
\(40\) 2.82728 0.447032
\(41\) −0.0916968 −0.0143206 −0.00716032 − 0.999974i \(-0.502279\pi\)
−0.00716032 + 0.999974i \(0.502279\pi\)
\(42\) −3.89417 −0.600884
\(43\) −0.702722 −0.107164 −0.0535820 − 0.998563i \(-0.517064\pi\)
−0.0535820 + 0.998563i \(0.517064\pi\)
\(44\) −13.9617 −2.10480
\(45\) 1.35070 0.201350
\(46\) 12.1911 1.79748
\(47\) −10.1367 −1.47859 −0.739295 − 0.673382i \(-0.764841\pi\)
−0.739295 + 0.673382i \(0.764841\pi\)
\(48\) 2.39911 0.346282
\(49\) −4.46590 −0.637985
\(50\) 10.8035 1.52785
\(51\) −3.60838 −0.505275
\(52\) −24.4893 −3.39606
\(53\) 3.48172 0.478250 0.239125 − 0.970989i \(-0.423139\pi\)
0.239125 + 0.970989i \(0.423139\pi\)
\(54\) 12.1138 1.64848
\(55\) 2.60414 0.351143
\(56\) −6.50419 −0.869159
\(57\) 8.67210 1.14865
\(58\) 6.14512 0.806894
\(59\) 11.3479 1.47737 0.738684 − 0.674052i \(-0.235448\pi\)
0.738684 + 0.674052i \(0.235448\pi\)
\(60\) −2.62807 −0.339283
\(61\) 10.2702 1.31497 0.657483 − 0.753470i \(-0.271621\pi\)
0.657483 + 0.753470i \(0.271621\pi\)
\(62\) −8.06021 −1.02365
\(63\) −3.10730 −0.391483
\(64\) −10.8325 −1.35406
\(65\) 4.56776 0.566561
\(66\) 9.20618 1.13320
\(67\) 3.97791 0.485979 0.242989 − 0.970029i \(-0.421872\pi\)
0.242989 + 0.970029i \(0.421872\pi\)
\(68\) −13.0763 −1.58573
\(69\) −5.22299 −0.628774
\(70\) 2.63217 0.314604
\(71\) 1.20708 0.143254 0.0716268 − 0.997431i \(-0.477181\pi\)
0.0716268 + 0.997431i \(0.477181\pi\)
\(72\) 7.97539 0.939908
\(73\) 10.3051 1.20612 0.603058 − 0.797697i \(-0.293949\pi\)
0.603058 + 0.797697i \(0.293949\pi\)
\(74\) 3.52925 0.410267
\(75\) −4.62850 −0.534453
\(76\) 31.4265 3.60487
\(77\) −5.99086 −0.682722
\(78\) 16.1480 1.82840
\(79\) 10.0280 1.12824 0.564118 − 0.825694i \(-0.309216\pi\)
0.564118 + 0.825694i \(0.309216\pi\)
\(80\) −1.62162 −0.181303
\(81\) 0.666026 0.0740029
\(82\) 0.219113 0.0241970
\(83\) 11.1174 1.22030 0.610149 − 0.792287i \(-0.291110\pi\)
0.610149 + 0.792287i \(0.291110\pi\)
\(84\) 6.04591 0.659663
\(85\) 2.43900 0.264546
\(86\) 1.67918 0.181071
\(87\) −2.63273 −0.282258
\(88\) 15.3765 1.63914
\(89\) 9.14236 0.969089 0.484544 − 0.874767i \(-0.338985\pi\)
0.484544 + 0.874767i \(0.338985\pi\)
\(90\) −3.22755 −0.340213
\(91\) −10.5082 −1.10156
\(92\) −18.9274 −1.97332
\(93\) 3.45320 0.358080
\(94\) 24.2220 2.49831
\(95\) −5.86169 −0.601397
\(96\) 2.63289 0.268718
\(97\) 4.87256 0.494734 0.247367 − 0.968922i \(-0.420435\pi\)
0.247367 + 0.968922i \(0.420435\pi\)
\(98\) 10.6714 1.07798
\(99\) 7.34595 0.738296
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.1.19 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.19 ✓ 184 1.1 even 1 trivial