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Properties

Label 4001.2.a.b.1.3
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 1.3
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)
 
\(f(q)\) \(=\) \(q-2.77961 q^{2} -0.970175 q^{3} +5.72623 q^{4} +0.973772 q^{5} +2.69671 q^{6} +0.392591 q^{7} -10.3575 q^{8} -2.05876 q^{9} -2.70671 q^{10} -4.34690 q^{11} -5.55544 q^{12} +2.35614 q^{13} -1.09125 q^{14} -0.944729 q^{15} +17.3373 q^{16} +6.89369 q^{17} +5.72255 q^{18} +6.69926 q^{19} +5.57604 q^{20} -0.380881 q^{21} +12.0827 q^{22} -2.69910 q^{23} +10.0486 q^{24} -4.05177 q^{25} -6.54914 q^{26} +4.90788 q^{27} +2.24806 q^{28} +0.233462 q^{29} +2.62598 q^{30} +9.20330 q^{31} -27.4759 q^{32} +4.21725 q^{33} -19.1618 q^{34} +0.382294 q^{35} -11.7889 q^{36} -6.23130 q^{37} -18.6213 q^{38} -2.28586 q^{39} -10.0858 q^{40} +11.9724 q^{41} +1.05870 q^{42} +5.31449 q^{43} -24.8913 q^{44} -2.00476 q^{45} +7.50244 q^{46} -5.36754 q^{47} -16.8202 q^{48} -6.84587 q^{49} +11.2623 q^{50} -6.68808 q^{51} +13.4918 q^{52} -5.62281 q^{53} -13.6420 q^{54} -4.23289 q^{55} -4.06624 q^{56} -6.49946 q^{57} -0.648933 q^{58} -7.14186 q^{59} -5.40974 q^{60} +9.12946 q^{61} -25.5816 q^{62} -0.808250 q^{63} +41.6977 q^{64} +2.29434 q^{65} -11.7223 q^{66} -2.47276 q^{67} +39.4749 q^{68} +2.61860 q^{69} -1.06263 q^{70} -7.84018 q^{71} +21.3236 q^{72} -12.2002 q^{73} +17.3206 q^{74} +3.93092 q^{75} +38.3615 q^{76} -1.70655 q^{77} +6.35381 q^{78} -9.57153 q^{79} +16.8825 q^{80} +1.41478 q^{81} -33.2787 q^{82} +13.8667 q^{83} -2.18101 q^{84} +6.71288 q^{85} -14.7722 q^{86} -0.226499 q^{87} +45.0229 q^{88} -9.52619 q^{89} +5.57246 q^{90} +0.924997 q^{91} -15.4557 q^{92} -8.92880 q^{93} +14.9197 q^{94} +6.52355 q^{95} +26.6564 q^{96} +15.4817 q^{97} +19.0289 q^{98} +8.94923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77961 −1.96548 −0.982741 − 0.184989i \(-0.940775\pi\)
−0.982741 + 0.184989i \(0.940775\pi\)
\(3\) −0.970175 −0.560131 −0.280065 − 0.959981i \(-0.590356\pi\)
−0.280065 + 0.959981i \(0.590356\pi\)
\(4\) 5.72623 2.86312
\(5\) 0.973772 0.435484 0.217742 − 0.976006i \(-0.430131\pi\)
0.217742 + 0.976006i \(0.430131\pi\)
\(6\) 2.69671 1.10093
\(7\) 0.392591 0.148385 0.0741926 − 0.997244i \(-0.476362\pi\)
0.0741926 + 0.997244i \(0.476362\pi\)
\(8\) −10.3575 −3.66192
\(9\) −2.05876 −0.686254
\(10\) −2.70671 −0.855936
\(11\) −4.34690 −1.31064 −0.655320 − 0.755352i \(-0.727466\pi\)
−0.655320 + 0.755352i \(0.727466\pi\)
\(12\) −5.55544 −1.60372
\(13\) 2.35614 0.653475 0.326737 − 0.945115i \(-0.394051\pi\)
0.326737 + 0.945115i \(0.394051\pi\)
\(14\) −1.09125 −0.291648
\(15\) −0.944729 −0.243928
\(16\) 17.3373 4.33432
\(17\) 6.89369 1.67196 0.835982 − 0.548756i \(-0.184898\pi\)
0.835982 + 0.548756i \(0.184898\pi\)
\(18\) 5.72255 1.34882
\(19\) 6.69926 1.53692 0.768458 − 0.639900i \(-0.221024\pi\)
0.768458 + 0.639900i \(0.221024\pi\)
\(20\) 5.57604 1.24684
\(21\) −0.380881 −0.0831151
\(22\) 12.0827 2.57604
\(23\) −2.69910 −0.562801 −0.281400 − 0.959590i \(-0.590799\pi\)
−0.281400 + 0.959590i \(0.590799\pi\)
\(24\) 10.0486 2.05115
\(25\) −4.05177 −0.810354
\(26\) −6.54914 −1.28439
\(27\) 4.90788 0.944522
\(28\) 2.24806 0.424844
\(29\) 0.233462 0.0433528 0.0216764 − 0.999765i \(-0.493100\pi\)
0.0216764 + 0.999765i \(0.493100\pi\)
\(30\) 2.62598 0.479436
\(31\) 9.20330 1.65296 0.826480 − 0.562966i \(-0.190340\pi\)
0.826480 + 0.562966i \(0.190340\pi\)
\(32\) −27.4759 −4.85710
\(33\) 4.21725 0.734129
\(34\) −19.1618 −3.28622
\(35\) 0.382294 0.0646194
\(36\) −11.7889 −1.96482
\(37\) −6.23130 −1.02442 −0.512210 − 0.858860i \(-0.671173\pi\)
−0.512210 + 0.858860i \(0.671173\pi\)
\(38\) −18.6213 −3.02078
\(39\) −2.28586 −0.366031
\(40\) −10.0858 −1.59471
\(41\) 11.9724 1.86978 0.934890 − 0.354937i \(-0.115498\pi\)
0.934890 + 0.354937i \(0.115498\pi\)
\(42\) 1.05870 0.163361
\(43\) 5.31449 0.810452 0.405226 − 0.914217i \(-0.367193\pi\)
0.405226 + 0.914217i \(0.367193\pi\)
\(44\) −24.8913 −3.75251
\(45\) −2.00476 −0.298852
\(46\) 7.50244 1.10617
\(47\) −5.36754 −0.782937 −0.391468 − 0.920192i \(-0.628033\pi\)
−0.391468 + 0.920192i \(0.628033\pi\)
\(48\) −16.8202 −2.42778
\(49\) −6.84587 −0.977982
\(50\) 11.2623 1.59273
\(51\) −6.68808 −0.936519
\(52\) 13.4918 1.87097
\(53\) −5.62281 −0.772352 −0.386176 − 0.922425i \(-0.626204\pi\)
−0.386176 + 0.922425i \(0.626204\pi\)
\(54\) −13.6420 −1.85644
\(55\) −4.23289 −0.570762
\(56\) −4.06624 −0.543375
\(57\) −6.49946 −0.860874
\(58\) −0.648933 −0.0852091
\(59\) −7.14186 −0.929791 −0.464896 − 0.885366i \(-0.653908\pi\)
−0.464896 + 0.885366i \(0.653908\pi\)
\(60\) −5.40974 −0.698394
\(61\) 9.12946 1.16891 0.584454 − 0.811427i \(-0.301309\pi\)
0.584454 + 0.811427i \(0.301309\pi\)
\(62\) −25.5816 −3.24886
\(63\) −0.808250 −0.101830
\(64\) 41.6977 5.21222
\(65\) 2.29434 0.284578
\(66\) −11.7223 −1.44292
\(67\) −2.47276 −0.302096 −0.151048 − 0.988526i \(-0.548265\pi\)
−0.151048 + 0.988526i \(0.548265\pi\)
\(68\) 39.4749 4.78703
\(69\) 2.61860 0.315242
\(70\) −1.06263 −0.127008
\(71\) −7.84018 −0.930458 −0.465229 − 0.885190i \(-0.654028\pi\)
−0.465229 + 0.885190i \(0.654028\pi\)
\(72\) 21.3236 2.51301
\(73\) −12.2002 −1.42792 −0.713962 − 0.700185i \(-0.753101\pi\)
−0.713962 + 0.700185i \(0.753101\pi\)
\(74\) 17.3206 2.01348
\(75\) 3.93092 0.453904
\(76\) 38.3615 4.40037
\(77\) −1.70655 −0.194480
\(78\) 6.35381 0.719427
\(79\) −9.57153 −1.07688 −0.538441 − 0.842663i \(-0.680986\pi\)
−0.538441 + 0.842663i \(0.680986\pi\)
\(80\) 16.8825 1.88753
\(81\) 1.41478 0.157198
\(82\) −33.2787 −3.67502
\(83\) 13.8667 1.52207 0.761037 − 0.648709i \(-0.224691\pi\)
0.761037 + 0.648709i \(0.224691\pi\)
\(84\) −2.18101 −0.237968
\(85\) 6.71288 0.728114
\(86\) −14.7722 −1.59293
\(87\) −0.226499 −0.0242832
\(88\) 45.0229 4.79945
\(89\) −9.52619 −1.00977 −0.504887 − 0.863185i \(-0.668466\pi\)
−0.504887 + 0.863185i \(0.668466\pi\)
\(90\) 5.57246 0.587389
\(91\) 0.924997 0.0969660
\(92\) −15.4557 −1.61136
\(93\) −8.92880 −0.925874
\(94\) 14.9197 1.53885
\(95\) 6.52355 0.669302
\(96\) 26.6564 2.72061
\(97\) 15.4817 1.57193 0.785964 − 0.618273i \(-0.212167\pi\)
0.785964 + 0.618273i \(0.212167\pi\)
\(98\) 19.0289 1.92220
\(99\) 8.94923 0.899431
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.1.3 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.3 ✓ 184 1.1 even 1 trivial