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Properties

Label 4001.2.a.b.1.15
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 1.15
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)
 
\(f(q)\) \(=\) \(q-2.55462 q^{2} +2.40673 q^{3} +4.52611 q^{4} +3.51168 q^{5} -6.14829 q^{6} +3.52515 q^{7} -6.45325 q^{8} +2.79234 q^{9} -8.97103 q^{10} -5.09895 q^{11} +10.8931 q^{12} -0.353446 q^{13} -9.00544 q^{14} +8.45166 q^{15} +7.43343 q^{16} +1.22514 q^{17} -7.13338 q^{18} +0.445154 q^{19} +15.8942 q^{20} +8.48408 q^{21} +13.0259 q^{22} +9.35822 q^{23} -15.5312 q^{24} +7.33191 q^{25} +0.902921 q^{26} -0.499785 q^{27} +15.9552 q^{28} -6.41904 q^{29} -21.5908 q^{30} +4.59805 q^{31} -6.08311 q^{32} -12.2718 q^{33} -3.12979 q^{34} +12.3792 q^{35} +12.6384 q^{36} -0.0828449 q^{37} -1.13720 q^{38} -0.850647 q^{39} -22.6618 q^{40} +4.27416 q^{41} -21.6736 q^{42} -9.28723 q^{43} -23.0784 q^{44} +9.80581 q^{45} -23.9067 q^{46} +2.53171 q^{47} +17.8902 q^{48} +5.42670 q^{49} -18.7303 q^{50} +2.94859 q^{51} -1.59973 q^{52} -1.29833 q^{53} +1.27676 q^{54} -17.9059 q^{55} -22.7487 q^{56} +1.07136 q^{57} +16.3982 q^{58} -7.77128 q^{59} +38.2531 q^{60} +7.07624 q^{61} -11.7463 q^{62} +9.84342 q^{63} +0.673206 q^{64} -1.24119 q^{65} +31.3498 q^{66} +15.5801 q^{67} +5.54514 q^{68} +22.5227 q^{69} -31.6243 q^{70} +6.33364 q^{71} -18.0197 q^{72} +12.7305 q^{73} +0.211638 q^{74} +17.6459 q^{75} +2.01482 q^{76} -17.9746 q^{77} +2.17308 q^{78} -0.799467 q^{79} +26.1038 q^{80} -9.57986 q^{81} -10.9189 q^{82} +2.66341 q^{83} +38.3999 q^{84} +4.30232 q^{85} +23.7254 q^{86} -15.4489 q^{87} +32.9048 q^{88} -12.6340 q^{89} -25.0502 q^{90} -1.24595 q^{91} +42.3563 q^{92} +11.0663 q^{93} -6.46758 q^{94} +1.56324 q^{95} -14.6404 q^{96} +7.86963 q^{97} -13.8632 q^{98} -14.2380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.55462 −1.80639 −0.903196 − 0.429228i \(-0.858786\pi\)
−0.903196 + 0.429228i \(0.858786\pi\)
\(3\) 2.40673 1.38952 0.694762 − 0.719239i \(-0.255510\pi\)
0.694762 + 0.719239i \(0.255510\pi\)
\(4\) 4.52611 2.26305
\(5\) 3.51168 1.57047 0.785236 − 0.619196i \(-0.212542\pi\)
0.785236 + 0.619196i \(0.212542\pi\)
\(6\) −6.14829 −2.51003
\(7\) 3.52515 1.33238 0.666191 − 0.745781i \(-0.267923\pi\)
0.666191 + 0.745781i \(0.267923\pi\)
\(8\) −6.45325 −2.28157
\(9\) 2.79234 0.930779
\(10\) −8.97103 −2.83689
\(11\) −5.09895 −1.53739 −0.768696 − 0.639615i \(-0.779094\pi\)
−0.768696 + 0.639615i \(0.779094\pi\)
\(12\) 10.8931 3.14457
\(13\) −0.353446 −0.0980281 −0.0490141 − 0.998798i \(-0.515608\pi\)
−0.0490141 + 0.998798i \(0.515608\pi\)
\(14\) −9.00544 −2.40681
\(15\) 8.45166 2.18221
\(16\) 7.43343 1.85836
\(17\) 1.22514 0.297141 0.148571 − 0.988902i \(-0.452533\pi\)
0.148571 + 0.988902i \(0.452533\pi\)
\(18\) −7.13338 −1.68135
\(19\) 0.445154 0.102125 0.0510627 − 0.998695i \(-0.483739\pi\)
0.0510627 + 0.998695i \(0.483739\pi\)
\(20\) 15.8942 3.55406
\(21\) 8.48408 1.85138
\(22\) 13.0259 2.77713
\(23\) 9.35822 1.95132 0.975662 − 0.219280i \(-0.0703710\pi\)
0.975662 + 0.219280i \(0.0703710\pi\)
\(24\) −15.5312 −3.17030
\(25\) 7.33191 1.46638
\(26\) 0.902921 0.177077
\(27\) −0.499785 −0.0961838
\(28\) 15.9552 3.01525
\(29\) −6.41904 −1.19199 −0.595993 − 0.802990i \(-0.703241\pi\)
−0.595993 + 0.802990i \(0.703241\pi\)
\(30\) −21.5908 −3.94193
\(31\) 4.59805 0.825834 0.412917 − 0.910769i \(-0.364510\pi\)
0.412917 + 0.910769i \(0.364510\pi\)
\(32\) −6.08311 −1.07535
\(33\) −12.2718 −2.13624
\(34\) −3.12979 −0.536754
\(35\) 12.3792 2.09247
\(36\) 12.6384 2.10640
\(37\) −0.0828449 −0.0136196 −0.00680981 − 0.999977i \(-0.502168\pi\)
−0.00680981 + 0.999977i \(0.502168\pi\)
\(38\) −1.13720 −0.184478
\(39\) −0.850647 −0.136213
\(40\) −22.6618 −3.58314
\(41\) 4.27416 0.667512 0.333756 − 0.942660i \(-0.391684\pi\)
0.333756 + 0.942660i \(0.391684\pi\)
\(42\) −21.6736 −3.34432
\(43\) −9.28723 −1.41629 −0.708144 − 0.706068i \(-0.750467\pi\)
−0.708144 + 0.706068i \(0.750467\pi\)
\(44\) −23.0784 −3.47920
\(45\) 9.80581 1.46176
\(46\) −23.9067 −3.52486
\(47\) 2.53171 0.369288 0.184644 − 0.982805i \(-0.440887\pi\)
0.184644 + 0.982805i \(0.440887\pi\)
\(48\) 17.8902 2.58223
\(49\) 5.42670 0.775243
\(50\) −18.7303 −2.64886
\(51\) 2.94859 0.412885
\(52\) −1.59973 −0.221843
\(53\) −1.29833 −0.178339 −0.0891694 − 0.996016i \(-0.528421\pi\)
−0.0891694 + 0.996016i \(0.528421\pi\)
\(54\) 1.27676 0.173746
\(55\) −17.9059 −2.41443
\(56\) −22.7487 −3.03992
\(57\) 1.07136 0.141906
\(58\) 16.3982 2.15319
\(59\) −7.77128 −1.01173 −0.505867 − 0.862611i \(-0.668828\pi\)
−0.505867 + 0.862611i \(0.668828\pi\)
\(60\) 38.2531 4.93846
\(61\) 7.07624 0.906020 0.453010 − 0.891505i \(-0.350350\pi\)
0.453010 + 0.891505i \(0.350350\pi\)
\(62\) −11.7463 −1.49178
\(63\) 9.84342 1.24015
\(64\) 0.673206 0.0841507
\(65\) −1.24119 −0.153950
\(66\) 31.3498 3.85889
\(67\) 15.5801 1.90341 0.951704 − 0.307017i \(-0.0993310\pi\)
0.951704 + 0.307017i \(0.0993310\pi\)
\(68\) 5.54514 0.672447
\(69\) 22.5227 2.71141
\(70\) −31.6243 −3.77982
\(71\) 6.33364 0.751665 0.375832 − 0.926688i \(-0.377357\pi\)
0.375832 + 0.926688i \(0.377357\pi\)
\(72\) −18.0197 −2.12364
\(73\) 12.7305 1.49000 0.744999 − 0.667066i \(-0.232450\pi\)
0.744999 + 0.667066i \(0.232450\pi\)
\(74\) 0.211638 0.0246024
\(75\) 17.6459 2.03758
\(76\) 2.01482 0.231115
\(77\) −17.9746 −2.04839
\(78\) 2.17308 0.246053
\(79\) −0.799467 −0.0899471 −0.0449736 − 0.998988i \(-0.514320\pi\)
−0.0449736 + 0.998988i \(0.514320\pi\)
\(80\) 26.1038 2.91850
\(81\) −9.57986 −1.06443
\(82\) −10.9189 −1.20579
\(83\) 2.66341 0.292347 0.146174 − 0.989259i \(-0.453304\pi\)
0.146174 + 0.989259i \(0.453304\pi\)
\(84\) 38.3999 4.18977
\(85\) 4.30232 0.466652
\(86\) 23.7254 2.55837
\(87\) −15.4489 −1.65629
\(88\) 32.9048 3.50767
\(89\) −12.6340 −1.33920 −0.669599 − 0.742723i \(-0.733534\pi\)
−0.669599 + 0.742723i \(0.733534\pi\)
\(90\) −25.0502 −2.64052
\(91\) −1.24595 −0.130611
\(92\) 42.3563 4.41595
\(93\) 11.0663 1.14752
\(94\) −6.46758 −0.667080
\(95\) 1.56324 0.160385
\(96\) −14.6404 −1.49423
\(97\) 7.86963 0.799040 0.399520 − 0.916725i \(-0.369177\pi\)
0.399520 + 0.916725i \(0.369177\pi\)
\(98\) −13.8632 −1.40039
\(99\) −14.2380 −1.43097
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.1.15 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.15 ✓ 184 1.1 even 1 trivial