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Properties

Label 4001.2.a.b.1.11
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 1.11
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)
 
\(f(q)\) \(=\) \(q-2.64594 q^{2} -1.99415 q^{3} +5.00101 q^{4} -3.20075 q^{5} +5.27640 q^{6} +3.40947 q^{7} -7.94051 q^{8} +0.976617 q^{9} +8.46900 q^{10} +4.31844 q^{11} -9.97275 q^{12} -1.86918 q^{13} -9.02125 q^{14} +6.38276 q^{15} +11.0081 q^{16} -2.01826 q^{17} -2.58407 q^{18} +5.36675 q^{19} -16.0070 q^{20} -6.79897 q^{21} -11.4263 q^{22} +0.806760 q^{23} +15.8345 q^{24} +5.24480 q^{25} +4.94574 q^{26} +4.03492 q^{27} +17.0508 q^{28} -1.06073 q^{29} -16.8884 q^{30} +2.13299 q^{31} -13.2458 q^{32} -8.61160 q^{33} +5.34020 q^{34} -10.9128 q^{35} +4.88407 q^{36} +0.385677 q^{37} -14.2001 q^{38} +3.72741 q^{39} +25.4156 q^{40} +11.8883 q^{41} +17.9897 q^{42} +3.23663 q^{43} +21.5966 q^{44} -3.12591 q^{45} -2.13464 q^{46} +3.16840 q^{47} -21.9518 q^{48} +4.62446 q^{49} -13.8774 q^{50} +4.02471 q^{51} -9.34779 q^{52} +5.24253 q^{53} -10.6762 q^{54} -13.8222 q^{55} -27.0729 q^{56} -10.7021 q^{57} +2.80663 q^{58} +3.68316 q^{59} +31.9203 q^{60} +5.89671 q^{61} -5.64376 q^{62} +3.32974 q^{63} +13.0314 q^{64} +5.98278 q^{65} +22.7858 q^{66} -3.76625 q^{67} -10.0933 q^{68} -1.60880 q^{69} +28.8748 q^{70} -6.14898 q^{71} -7.75483 q^{72} +12.9440 q^{73} -1.02048 q^{74} -10.4589 q^{75} +26.8392 q^{76} +14.7236 q^{77} -9.86253 q^{78} +9.83252 q^{79} -35.2342 q^{80} -10.9761 q^{81} -31.4559 q^{82} -16.4947 q^{83} -34.0017 q^{84} +6.45995 q^{85} -8.56393 q^{86} +2.11525 q^{87} -34.2906 q^{88} +3.74435 q^{89} +8.27097 q^{90} -6.37290 q^{91} +4.03462 q^{92} -4.25349 q^{93} -8.38341 q^{94} -17.1776 q^{95} +26.4141 q^{96} +9.59660 q^{97} -12.2360 q^{98} +4.21746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64594 −1.87096 −0.935482 − 0.353374i \(-0.885034\pi\)
−0.935482 + 0.353374i \(0.885034\pi\)
\(3\) −1.99415 −1.15132 −0.575660 − 0.817689i \(-0.695255\pi\)
−0.575660 + 0.817689i \(0.695255\pi\)
\(4\) 5.00101 2.50051
\(5\) −3.20075 −1.43142 −0.715710 − 0.698398i \(-0.753897\pi\)
−0.715710 + 0.698398i \(0.753897\pi\)
\(6\) 5.27640 2.15408
\(7\) 3.40947 1.28866 0.644328 − 0.764749i \(-0.277137\pi\)
0.644328 + 0.764749i \(0.277137\pi\)
\(8\) −7.94051 −2.80739
\(9\) 0.976617 0.325539
\(10\) 8.46900 2.67813
\(11\) 4.31844 1.30206 0.651029 − 0.759053i \(-0.274338\pi\)
0.651029 + 0.759053i \(0.274338\pi\)
\(12\) −9.97275 −2.87888
\(13\) −1.86918 −0.518417 −0.259208 − 0.965821i \(-0.583462\pi\)
−0.259208 + 0.965821i \(0.583462\pi\)
\(14\) −9.02125 −2.41103
\(15\) 6.38276 1.64802
\(16\) 11.0081 2.75203
\(17\) −2.01826 −0.489500 −0.244750 − 0.969586i \(-0.578706\pi\)
−0.244750 + 0.969586i \(0.578706\pi\)
\(18\) −2.58407 −0.609072
\(19\) 5.36675 1.23122 0.615608 − 0.788052i \(-0.288910\pi\)
0.615608 + 0.788052i \(0.288910\pi\)
\(20\) −16.0070 −3.57927
\(21\) −6.79897 −1.48366
\(22\) −11.4263 −2.43610
\(23\) 0.806760 0.168221 0.0841106 − 0.996456i \(-0.473195\pi\)
0.0841106 + 0.996456i \(0.473195\pi\)
\(24\) 15.8345 3.23221
\(25\) 5.24480 1.04896
\(26\) 4.94574 0.969939
\(27\) 4.03492 0.776521
\(28\) 17.0508 3.22229
\(29\) −1.06073 −0.196973 −0.0984863 − 0.995138i \(-0.531400\pi\)
−0.0984863 + 0.995138i \(0.531400\pi\)
\(30\) −16.8884 −3.08339
\(31\) 2.13299 0.383096 0.191548 − 0.981483i \(-0.438649\pi\)
0.191548 + 0.981483i \(0.438649\pi\)
\(32\) −13.2458 −2.34155
\(33\) −8.61160 −1.49909
\(34\) 5.34020 0.915837
\(35\) −10.9128 −1.84461
\(36\) 4.88407 0.814012
\(37\) 0.385677 0.0634049 0.0317024 − 0.999497i \(-0.489907\pi\)
0.0317024 + 0.999497i \(0.489907\pi\)
\(38\) −14.2001 −2.30356
\(39\) 3.72741 0.596864
\(40\) 25.4156 4.01856
\(41\) 11.8883 1.85665 0.928323 − 0.371775i \(-0.121251\pi\)
0.928323 + 0.371775i \(0.121251\pi\)
\(42\) 17.9897 2.77587
\(43\) 3.23663 0.493581 0.246790 − 0.969069i \(-0.420624\pi\)
0.246790 + 0.969069i \(0.420624\pi\)
\(44\) 21.5966 3.25581
\(45\) −3.12591 −0.465983
\(46\) −2.13464 −0.314736
\(47\) 3.16840 0.462159 0.231079 − 0.972935i \(-0.425774\pi\)
0.231079 + 0.972935i \(0.425774\pi\)
\(48\) −21.9518 −3.16846
\(49\) 4.62446 0.660636
\(50\) −13.8774 −1.96257
\(51\) 4.02471 0.563571
\(52\) −9.34779 −1.29630
\(53\) 5.24253 0.720117 0.360059 − 0.932930i \(-0.382757\pi\)
0.360059 + 0.932930i \(0.382757\pi\)
\(54\) −10.6762 −1.45284
\(55\) −13.8222 −1.86379
\(56\) −27.0729 −3.61777
\(57\) −10.7021 −1.41753
\(58\) 2.80663 0.368529
\(59\) 3.68316 0.479507 0.239753 − 0.970834i \(-0.422933\pi\)
0.239753 + 0.970834i \(0.422933\pi\)
\(60\) 31.9203 4.12089
\(61\) 5.89671 0.754996 0.377498 − 0.926010i \(-0.376784\pi\)
0.377498 + 0.926010i \(0.376784\pi\)
\(62\) −5.64376 −0.716758
\(63\) 3.32974 0.419508
\(64\) 13.0314 1.62893
\(65\) 5.98278 0.742072
\(66\) 22.7858 2.80474
\(67\) −3.76625 −0.460121 −0.230060 − 0.973176i \(-0.573892\pi\)
−0.230060 + 0.973176i \(0.573892\pi\)
\(68\) −10.0933 −1.22400
\(69\) −1.60880 −0.193676
\(70\) 28.8748 3.45120
\(71\) −6.14898 −0.729750 −0.364875 − 0.931057i \(-0.618888\pi\)
−0.364875 + 0.931057i \(0.618888\pi\)
\(72\) −7.75483 −0.913916
\(73\) 12.9440 1.51498 0.757489 − 0.652848i \(-0.226426\pi\)
0.757489 + 0.652848i \(0.226426\pi\)
\(74\) −1.02048 −0.118628
\(75\) −10.4589 −1.20769
\(76\) 26.8392 3.07867
\(77\) 14.7236 1.67791
\(78\) −9.86253 −1.11671
\(79\) 9.83252 1.10625 0.553123 − 0.833100i \(-0.313436\pi\)
0.553123 + 0.833100i \(0.313436\pi\)
\(80\) −35.2342 −3.93930
\(81\) −10.9761 −1.21956
\(82\) −31.4559 −3.47372
\(83\) −16.4947 −1.81053 −0.905266 − 0.424844i \(-0.860329\pi\)
−0.905266 + 0.424844i \(0.860329\pi\)
\(84\) −34.0017 −3.70989
\(85\) 6.45995 0.700680
\(86\) −8.56393 −0.923472
\(87\) 2.11525 0.226779
\(88\) −34.2906 −3.65539
\(89\) 3.74435 0.396900 0.198450 − 0.980111i \(-0.436409\pi\)
0.198450 + 0.980111i \(0.436409\pi\)
\(90\) 8.27097 0.871837
\(91\) −6.37290 −0.668062
\(92\) 4.03462 0.420638
\(93\) −4.25349 −0.441066
\(94\) −8.38341 −0.864683
\(95\) −17.1776 −1.76239
\(96\) 26.4141 2.69587
\(97\) 9.59660 0.974387 0.487194 − 0.873294i \(-0.338021\pi\)
0.487194 + 0.873294i \(0.338021\pi\)
\(98\) −12.2360 −1.23603
\(99\) 4.21746 0.423871
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.1.11 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.11 ✓ 184 1.1 even 1 trivial