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Properties

Label 8001.2.a.s.1.8
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
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] = [8001,2,Mod(1,8001)] 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("8001.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8001, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-4,0,20,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.716430\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.716430 q^{2} -1.48673 q^{4} -0.617976 q^{5} -1.00000 q^{7} +2.49800 q^{8} +0.442737 q^{10} -2.03917 q^{11} -2.26367 q^{13} +0.716430 q^{14} +1.18382 q^{16} +1.11795 q^{17} +7.92719 q^{19} +0.918763 q^{20} +1.46092 q^{22} -6.62633 q^{23} -4.61811 q^{25} +1.62176 q^{26} +1.48673 q^{28} +3.70380 q^{29} -1.50424 q^{31} -5.84411 q^{32} -0.800934 q^{34} +0.617976 q^{35} +4.29904 q^{37} -5.67928 q^{38} -1.54370 q^{40} +2.12378 q^{41} -5.68717 q^{43} +3.03169 q^{44} +4.74730 q^{46} -7.25675 q^{47} +1.00000 q^{49} +3.30855 q^{50} +3.36546 q^{52} -7.02300 q^{53} +1.26016 q^{55} -2.49800 q^{56} -2.65351 q^{58} +1.93188 q^{59} +7.43893 q^{61} +1.07768 q^{62} +1.81927 q^{64} +1.39890 q^{65} +10.4720 q^{67} -1.66209 q^{68} -0.442737 q^{70} -9.88028 q^{71} +5.42411 q^{73} -3.07996 q^{74} -11.7856 q^{76} +2.03917 q^{77} +15.1144 q^{79} -0.731571 q^{80} -1.52154 q^{82} -8.98829 q^{83} -0.690868 q^{85} +4.07446 q^{86} -5.09384 q^{88} -10.5005 q^{89} +2.26367 q^{91} +9.85155 q^{92} +5.19896 q^{94} -4.89882 q^{95} -14.4370 q^{97} -0.716430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.716430 −0.506592 −0.253296 − 0.967389i \(-0.581515\pi\)
−0.253296 + 0.967389i \(0.581515\pi\)
\(3\) 0 0
\(4\) −1.48673 −0.743364
\(5\) −0.617976 −0.276367 −0.138184 − 0.990407i \(-0.544126\pi\)
−0.138184 + 0.990407i \(0.544126\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.49800 0.883175
\(9\) 0 0
\(10\) 0.442737 0.140006
\(11\) −2.03917 −0.614833 −0.307416 − 0.951575i \(-0.599464\pi\)
−0.307416 + 0.951575i \(0.599464\pi\)
\(12\) 0 0
\(13\) −2.26367 −0.627829 −0.313915 − 0.949451i \(-0.601641\pi\)
−0.313915 + 0.949451i \(0.601641\pi\)
\(14\) 0.716430 0.191474
\(15\) 0 0
\(16\) 1.18382 0.295954
\(17\) 1.11795 0.271143 0.135572 − 0.990768i \(-0.456713\pi\)
0.135572 + 0.990768i \(0.456713\pi\)
\(18\) 0 0
\(19\) 7.92719 1.81862 0.909311 − 0.416117i \(-0.136609\pi\)
0.909311 + 0.416117i \(0.136609\pi\)
\(20\) 0.918763 0.205442
\(21\) 0 0
\(22\) 1.46092 0.311470
\(23\) −6.62633 −1.38168 −0.690842 − 0.723005i \(-0.742760\pi\)
−0.690842 + 0.723005i \(0.742760\pi\)
\(24\) 0 0
\(25\) −4.61811 −0.923621
\(26\) 1.62176 0.318054
\(27\) 0 0
\(28\) 1.48673 0.280965
\(29\) 3.70380 0.687778 0.343889 − 0.939010i \(-0.388256\pi\)
0.343889 + 0.939010i \(0.388256\pi\)
\(30\) 0 0
\(31\) −1.50424 −0.270170 −0.135085 − 0.990834i \(-0.543131\pi\)
−0.135085 + 0.990834i \(0.543131\pi\)
\(32\) −5.84411 −1.03310
\(33\) 0 0
\(34\) −0.800934 −0.137359
\(35\) 0.617976 0.104457
\(36\) 0 0
\(37\) 4.29904 0.706757 0.353379 − 0.935480i \(-0.385033\pi\)
0.353379 + 0.935480i \(0.385033\pi\)
\(38\) −5.67928 −0.921300
\(39\) 0 0
\(40\) −1.54370 −0.244081
\(41\) 2.12378 0.331679 0.165839 − 0.986153i \(-0.446967\pi\)
0.165839 + 0.986153i \(0.446967\pi\)
\(42\) 0 0
\(43\) −5.68717 −0.867285 −0.433642 − 0.901085i \(-0.642772\pi\)
−0.433642 + 0.901085i \(0.642772\pi\)
\(44\) 3.03169 0.457044
\(45\) 0 0
\(46\) 4.74730 0.699951
\(47\) −7.25675 −1.05851 −0.529253 − 0.848464i \(-0.677528\pi\)
−0.529253 + 0.848464i \(0.677528\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.30855 0.467899
\(51\) 0 0
\(52\) 3.36546 0.466706
\(53\) −7.02300 −0.964683 −0.482341 − 0.875983i \(-0.660214\pi\)
−0.482341 + 0.875983i \(0.660214\pi\)
\(54\) 0 0
\(55\) 1.26016 0.169920
\(56\) −2.49800 −0.333809
\(57\) 0 0
\(58\) −2.65351 −0.348423
\(59\) 1.93188 0.251509 0.125754 − 0.992061i \(-0.459865\pi\)
0.125754 + 0.992061i \(0.459865\pi\)
\(60\) 0 0
\(61\) 7.43893 0.952457 0.476229 − 0.879322i \(-0.342003\pi\)
0.476229 + 0.879322i \(0.342003\pi\)
\(62\) 1.07768 0.136866
\(63\) 0 0
\(64\) 1.81927 0.227408
\(65\) 1.39890 0.173512
\(66\) 0 0
\(67\) 10.4720 1.27936 0.639682 − 0.768640i \(-0.279066\pi\)
0.639682 + 0.768640i \(0.279066\pi\)
\(68\) −1.66209 −0.201558
\(69\) 0 0
\(70\) −0.442737 −0.0529172
\(71\) −9.88028 −1.17257 −0.586287 − 0.810104i \(-0.699411\pi\)
−0.586287 + 0.810104i \(0.699411\pi\)
\(72\) 0 0
\(73\) 5.42411 0.634845 0.317422 − 0.948284i \(-0.397183\pi\)
0.317422 + 0.948284i \(0.397183\pi\)
\(74\) −3.07996 −0.358038
\(75\) 0 0
\(76\) −11.7856 −1.35190
\(77\) 2.03917 0.232385
\(78\) 0 0
\(79\) 15.1144 1.70050 0.850250 − 0.526378i \(-0.176450\pi\)
0.850250 + 0.526378i \(0.176450\pi\)
\(80\) −0.731571 −0.0817921
\(81\) 0 0
\(82\) −1.52154 −0.168026
\(83\) −8.98829 −0.986593 −0.493296 − 0.869861i \(-0.664208\pi\)
−0.493296 + 0.869861i \(0.664208\pi\)
\(84\) 0 0
\(85\) −0.690868 −0.0749351
\(86\) 4.07446 0.439360
\(87\) 0 0
\(88\) −5.09384 −0.543005
\(89\) −10.5005 −1.11305 −0.556523 − 0.830832i \(-0.687865\pi\)
−0.556523 + 0.830832i \(0.687865\pi\)
\(90\) 0 0
\(91\) 2.26367 0.237297
\(92\) 9.85155 1.02709
\(93\) 0 0
\(94\) 5.19896 0.536231
\(95\) −4.89882 −0.502608
\(96\) 0 0
\(97\) −14.4370 −1.46585 −0.732925 − 0.680309i \(-0.761845\pi\)
−0.732925 + 0.680309i \(0.761845\pi\)
\(98\) −0.716430 −0.0723704
\(99\) 0 0
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 8001.2.a.s.1.8 16
3.2 odd 2 2667.2.a.n.1.9 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.9 ✓ 16 3.2 odd 2
8001.2.a.s.1.8 16 1.1 even 1 trivial