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Properties

Label 4001.2.a.b.1.10
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 1.10
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)
 
\(f(q)\) \(=\) \(q-2.65266 q^{2} +3.05290 q^{3} +5.03661 q^{4} -1.94746 q^{5} -8.09830 q^{6} +2.48037 q^{7} -8.05510 q^{8} +6.32018 q^{9} +5.16596 q^{10} -2.94497 q^{11} +15.3763 q^{12} +0.192789 q^{13} -6.57957 q^{14} -5.94540 q^{15} +11.2942 q^{16} +4.76794 q^{17} -16.7653 q^{18} -2.90135 q^{19} -9.80861 q^{20} +7.57230 q^{21} +7.81201 q^{22} -3.08821 q^{23} -24.5914 q^{24} -1.20739 q^{25} -0.511405 q^{26} +10.1362 q^{27} +12.4926 q^{28} +2.23235 q^{29} +15.7711 q^{30} -3.07974 q^{31} -13.8496 q^{32} -8.99069 q^{33} -12.6477 q^{34} -4.83042 q^{35} +31.8323 q^{36} +8.80848 q^{37} +7.69631 q^{38} +0.588566 q^{39} +15.6870 q^{40} -2.72877 q^{41} -20.0867 q^{42} +11.9799 q^{43} -14.8327 q^{44} -12.3083 q^{45} +8.19199 q^{46} -4.38663 q^{47} +34.4801 q^{48} -0.847785 q^{49} +3.20280 q^{50} +14.5560 q^{51} +0.971006 q^{52} +14.3594 q^{53} -26.8878 q^{54} +5.73522 q^{55} -19.9796 q^{56} -8.85753 q^{57} -5.92167 q^{58} -3.28792 q^{59} -29.9447 q^{60} +12.6102 q^{61} +8.16952 q^{62} +15.6764 q^{63} +14.1497 q^{64} -0.375450 q^{65} +23.8493 q^{66} +4.80775 q^{67} +24.0143 q^{68} -9.42800 q^{69} +12.8135 q^{70} +0.546449 q^{71} -50.9097 q^{72} +4.67236 q^{73} -23.3659 q^{74} -3.68604 q^{75} -14.6130 q^{76} -7.30461 q^{77} -1.56127 q^{78} -4.07327 q^{79} -21.9951 q^{80} +11.9841 q^{81} +7.23849 q^{82} -0.610918 q^{83} +38.1387 q^{84} -9.28538 q^{85} -31.7787 q^{86} +6.81513 q^{87} +23.7220 q^{88} +4.34903 q^{89} +32.6498 q^{90} +0.478188 q^{91} -15.5541 q^{92} -9.40214 q^{93} +11.6363 q^{94} +5.65028 q^{95} -42.2813 q^{96} +6.32560 q^{97} +2.24889 q^{98} -18.6127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.65266 −1.87571 −0.937857 − 0.347021i \(-0.887193\pi\)
−0.937857 + 0.347021i \(0.887193\pi\)
\(3\) 3.05290 1.76259 0.881295 − 0.472566i \(-0.156672\pi\)
0.881295 + 0.472566i \(0.156672\pi\)
\(4\) 5.03661 2.51831
\(5\) −1.94746 −0.870932 −0.435466 − 0.900205i \(-0.643416\pi\)
−0.435466 + 0.900205i \(0.643416\pi\)
\(6\) −8.09830 −3.30612
\(7\) 2.48037 0.937490 0.468745 − 0.883333i \(-0.344706\pi\)
0.468745 + 0.883333i \(0.344706\pi\)
\(8\) −8.05510 −2.84791
\(9\) 6.32018 2.10673
\(10\) 5.16596 1.63362
\(11\) −2.94497 −0.887942 −0.443971 − 0.896041i \(-0.646431\pi\)
−0.443971 + 0.896041i \(0.646431\pi\)
\(12\) 15.3763 4.43874
\(13\) 0.192789 0.0534702 0.0267351 − 0.999643i \(-0.491489\pi\)
0.0267351 + 0.999643i \(0.491489\pi\)
\(14\) −6.57957 −1.75846
\(15\) −5.94540 −1.53510
\(16\) 11.2942 2.82356
\(17\) 4.76794 1.15640 0.578198 − 0.815897i \(-0.303756\pi\)
0.578198 + 0.815897i \(0.303756\pi\)
\(18\) −16.7653 −3.95162
\(19\) −2.90135 −0.665616 −0.332808 − 0.942995i \(-0.607996\pi\)
−0.332808 + 0.942995i \(0.607996\pi\)
\(20\) −9.80861 −2.19327
\(21\) 7.57230 1.65241
\(22\) 7.81201 1.66553
\(23\) −3.08821 −0.643937 −0.321969 − 0.946750i \(-0.604345\pi\)
−0.321969 + 0.946750i \(0.604345\pi\)
\(24\) −24.5914 −5.01970
\(25\) −1.20739 −0.241478
\(26\) −0.511405 −0.100295
\(27\) 10.1362 1.95070
\(28\) 12.4926 2.36089
\(29\) 2.23235 0.414537 0.207268 − 0.978284i \(-0.433543\pi\)
0.207268 + 0.978284i \(0.433543\pi\)
\(30\) 15.7711 2.87940
\(31\) −3.07974 −0.553138 −0.276569 − 0.960994i \(-0.589197\pi\)
−0.276569 + 0.960994i \(0.589197\pi\)
\(32\) −13.8496 −2.44828
\(33\) −8.99069 −1.56508
\(34\) −12.6477 −2.16907
\(35\) −4.83042 −0.816490
\(36\) 31.8323 5.30538
\(37\) 8.80848 1.44810 0.724052 − 0.689745i \(-0.242277\pi\)
0.724052 + 0.689745i \(0.242277\pi\)
\(38\) 7.69631 1.24851
\(39\) 0.588566 0.0942460
\(40\) 15.6870 2.48033
\(41\) −2.72877 −0.426162 −0.213081 − 0.977035i \(-0.568350\pi\)
−0.213081 + 0.977035i \(0.568350\pi\)
\(42\) −20.0867 −3.09945
\(43\) 11.9799 1.82692 0.913460 − 0.406929i \(-0.133400\pi\)
0.913460 + 0.406929i \(0.133400\pi\)
\(44\) −14.8327 −2.23611
\(45\) −12.3083 −1.83481
\(46\) 8.19199 1.20784
\(47\) −4.38663 −0.639856 −0.319928 − 0.947442i \(-0.603659\pi\)
−0.319928 + 0.947442i \(0.603659\pi\)
\(48\) 34.4801 4.97678
\(49\) −0.847785 −0.121112
\(50\) 3.20280 0.452944
\(51\) 14.5560 2.03825
\(52\) 0.971006 0.134654
\(53\) 14.3594 1.97242 0.986209 − 0.165504i \(-0.0529250\pi\)
0.986209 + 0.165504i \(0.0529250\pi\)
\(54\) −26.8878 −3.65896
\(55\) 5.73522 0.773337
\(56\) −19.9796 −2.66989
\(57\) −8.85753 −1.17321
\(58\) −5.92167 −0.777553
\(59\) −3.28792 −0.428050 −0.214025 − 0.976828i \(-0.568657\pi\)
−0.214025 + 0.976828i \(0.568657\pi\)
\(60\) −29.9447 −3.86584
\(61\) 12.6102 1.61458 0.807288 − 0.590158i \(-0.200935\pi\)
0.807288 + 0.590158i \(0.200935\pi\)
\(62\) 8.16952 1.03753
\(63\) 15.6764 1.97503
\(64\) 14.1497 1.76872
\(65\) −0.375450 −0.0465689
\(66\) 23.8493 2.93564
\(67\) 4.80775 0.587360 0.293680 − 0.955904i \(-0.405120\pi\)
0.293680 + 0.955904i \(0.405120\pi\)
\(68\) 24.0143 2.91216
\(69\) −9.42800 −1.13500
\(70\) 12.8135 1.53150
\(71\) 0.546449 0.0648516 0.0324258 − 0.999474i \(-0.489677\pi\)
0.0324258 + 0.999474i \(0.489677\pi\)
\(72\) −50.9097 −5.99976
\(73\) 4.67236 0.546858 0.273429 − 0.961892i \(-0.411842\pi\)
0.273429 + 0.961892i \(0.411842\pi\)
\(74\) −23.3659 −2.71623
\(75\) −3.68604 −0.425627
\(76\) −14.6130 −1.67622
\(77\) −7.30461 −0.832437
\(78\) −1.56127 −0.176779
\(79\) −4.07327 −0.458279 −0.229140 − 0.973394i \(-0.573591\pi\)
−0.229140 + 0.973394i \(0.573591\pi\)
\(80\) −21.9951 −2.45913
\(81\) 11.9841 1.33157
\(82\) 7.23849 0.799357
\(83\) −0.610918 −0.0670570 −0.0335285 − 0.999438i \(-0.510674\pi\)
−0.0335285 + 0.999438i \(0.510674\pi\)
\(84\) 38.1387 4.16128
\(85\) −9.28538 −1.00714
\(86\) −31.7787 −3.42678
\(87\) 6.81513 0.730659
\(88\) 23.7220 2.52878
\(89\) 4.34903 0.460996 0.230498 − 0.973073i \(-0.425965\pi\)
0.230498 + 0.973073i \(0.425965\pi\)
\(90\) 32.6498 3.44159
\(91\) 0.478188 0.0501278
\(92\) −15.5541 −1.62163
\(93\) −9.40214 −0.974956
\(94\) 11.6363 1.20019
\(95\) 5.65028 0.579706
\(96\) −42.2813 −4.31532
\(97\) 6.32560 0.642267 0.321134 − 0.947034i \(-0.395936\pi\)
0.321134 + 0.947034i \(0.395936\pi\)
\(98\) 2.24889 0.227172
\(99\) −18.6127 −1.87065
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.1.10 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.10 ✓ 184 1.1 even 1 trivial