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Properties

Label 8001.2.a.s.1.9
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 1.9
Root \(0.284398\) 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)
 
\(f(q)\) \(=\) \(q-0.284398 q^{2} -1.91912 q^{4} +3.66581 q^{5} -1.00000 q^{7} +1.11459 q^{8} -1.04255 q^{10} +2.89343 q^{11} +1.88240 q^{13} +0.284398 q^{14} +3.52125 q^{16} +5.41792 q^{17} -4.15704 q^{19} -7.03512 q^{20} -0.822888 q^{22} +4.94630 q^{23} +8.43817 q^{25} -0.535352 q^{26} +1.91912 q^{28} +6.42248 q^{29} +8.67066 q^{31} -3.23062 q^{32} -1.54085 q^{34} -3.66581 q^{35} -7.13997 q^{37} +1.18226 q^{38} +4.08588 q^{40} -3.80151 q^{41} +6.40053 q^{43} -5.55284 q^{44} -1.40672 q^{46} -6.07023 q^{47} +1.00000 q^{49} -2.39980 q^{50} -3.61255 q^{52} -3.63552 q^{53} +10.6068 q^{55} -1.11459 q^{56} -1.82654 q^{58} +7.78156 q^{59} +6.41002 q^{61} -2.46592 q^{62} -6.12371 q^{64} +6.90052 q^{65} +13.3619 q^{67} -10.3976 q^{68} +1.04255 q^{70} -8.96501 q^{71} -5.77303 q^{73} +2.03060 q^{74} +7.97785 q^{76} -2.89343 q^{77} -1.95064 q^{79} +12.9082 q^{80} +1.08114 q^{82} +8.20767 q^{83} +19.8611 q^{85} -1.82030 q^{86} +3.22499 q^{88} +0.196140 q^{89} -1.88240 q^{91} -9.49253 q^{92} +1.72637 q^{94} -15.2389 q^{95} -4.42233 q^{97} -0.284398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.284398 −0.201100 −0.100550 − 0.994932i \(-0.532060\pi\)
−0.100550 + 0.994932i \(0.532060\pi\)
\(3\) 0 0
\(4\) −1.91912 −0.959559
\(5\) 3.66581 1.63940 0.819700 − 0.572793i \(-0.194140\pi\)
0.819700 + 0.572793i \(0.194140\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.11459 0.394067
\(9\) 0 0
\(10\) −1.04255 −0.329684
\(11\) 2.89343 0.872403 0.436201 − 0.899849i \(-0.356324\pi\)
0.436201 + 0.899849i \(0.356324\pi\)
\(12\) 0 0
\(13\) 1.88240 0.522084 0.261042 − 0.965327i \(-0.415934\pi\)
0.261042 + 0.965327i \(0.415934\pi\)
\(14\) 0.284398 0.0760087
\(15\) 0 0
\(16\) 3.52125 0.880312
\(17\) 5.41792 1.31404 0.657019 − 0.753874i \(-0.271817\pi\)
0.657019 + 0.753874i \(0.271817\pi\)
\(18\) 0 0
\(19\) −4.15704 −0.953691 −0.476845 − 0.878987i \(-0.658220\pi\)
−0.476845 + 0.878987i \(0.658220\pi\)
\(20\) −7.03512 −1.57310
\(21\) 0 0
\(22\) −0.822888 −0.175440
\(23\) 4.94630 1.03137 0.515687 − 0.856777i \(-0.327537\pi\)
0.515687 + 0.856777i \(0.327537\pi\)
\(24\) 0 0
\(25\) 8.43817 1.68763
\(26\) −0.535352 −0.104991
\(27\) 0 0
\(28\) 1.91912 0.362679
\(29\) 6.42248 1.19262 0.596312 − 0.802753i \(-0.296632\pi\)
0.596312 + 0.802753i \(0.296632\pi\)
\(30\) 0 0
\(31\) 8.67066 1.55730 0.778648 − 0.627461i \(-0.215906\pi\)
0.778648 + 0.627461i \(0.215906\pi\)
\(32\) −3.23062 −0.571098
\(33\) 0 0
\(34\) −1.54085 −0.264253
\(35\) −3.66581 −0.619635
\(36\) 0 0
\(37\) −7.13997 −1.17380 −0.586902 − 0.809658i \(-0.699653\pi\)
−0.586902 + 0.809658i \(0.699653\pi\)
\(38\) 1.18226 0.191787
\(39\) 0 0
\(40\) 4.08588 0.646034
\(41\) −3.80151 −0.593697 −0.296848 − 0.954925i \(-0.595936\pi\)
−0.296848 + 0.954925i \(0.595936\pi\)
\(42\) 0 0
\(43\) 6.40053 0.976072 0.488036 − 0.872823i \(-0.337713\pi\)
0.488036 + 0.872823i \(0.337713\pi\)
\(44\) −5.55284 −0.837122
\(45\) 0 0
\(46\) −1.40672 −0.207410
\(47\) −6.07023 −0.885435 −0.442717 − 0.896661i \(-0.645986\pi\)
−0.442717 + 0.896661i \(0.645986\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.39980 −0.339383
\(51\) 0 0
\(52\) −3.61255 −0.500970
\(53\) −3.63552 −0.499377 −0.249689 − 0.968326i \(-0.580328\pi\)
−0.249689 + 0.968326i \(0.580328\pi\)
\(54\) 0 0
\(55\) 10.6068 1.43022
\(56\) −1.11459 −0.148943
\(57\) 0 0
\(58\) −1.82654 −0.239837
\(59\) 7.78156 1.01307 0.506536 − 0.862219i \(-0.330926\pi\)
0.506536 + 0.862219i \(0.330926\pi\)
\(60\) 0 0
\(61\) 6.41002 0.820719 0.410360 − 0.911924i \(-0.365403\pi\)
0.410360 + 0.911924i \(0.365403\pi\)
\(62\) −2.46592 −0.313172
\(63\) 0 0
\(64\) −6.12371 −0.765464
\(65\) 6.90052 0.855904
\(66\) 0 0
\(67\) 13.3619 1.63242 0.816208 − 0.577758i \(-0.196072\pi\)
0.816208 + 0.577758i \(0.196072\pi\)
\(68\) −10.3976 −1.26090
\(69\) 0 0
\(70\) 1.04255 0.124609
\(71\) −8.96501 −1.06395 −0.531975 − 0.846760i \(-0.678550\pi\)
−0.531975 + 0.846760i \(0.678550\pi\)
\(72\) 0 0
\(73\) −5.77303 −0.675682 −0.337841 − 0.941203i \(-0.609697\pi\)
−0.337841 + 0.941203i \(0.609697\pi\)
\(74\) 2.03060 0.236052
\(75\) 0 0
\(76\) 7.97785 0.915123
\(77\) −2.89343 −0.329737
\(78\) 0 0
\(79\) −1.95064 −0.219464 −0.109732 − 0.993961i \(-0.534999\pi\)
−0.109732 + 0.993961i \(0.534999\pi\)
\(80\) 12.9082 1.44318
\(81\) 0 0
\(82\) 1.08114 0.119392
\(83\) 8.20767 0.900909 0.450454 − 0.892799i \(-0.351262\pi\)
0.450454 + 0.892799i \(0.351262\pi\)
\(84\) 0 0
\(85\) 19.8611 2.15424
\(86\) −1.82030 −0.196288
\(87\) 0 0
\(88\) 3.22499 0.343785
\(89\) 0.196140 0.0207908 0.0103954 − 0.999946i \(-0.496691\pi\)
0.0103954 + 0.999946i \(0.496691\pi\)
\(90\) 0 0
\(91\) −1.88240 −0.197329
\(92\) −9.49253 −0.989665
\(93\) 0 0
\(94\) 1.72637 0.178061
\(95\) −15.2389 −1.56348
\(96\) 0 0
\(97\) −4.42233 −0.449020 −0.224510 − 0.974472i \(-0.572078\pi\)
−0.224510 + 0.974472i \(0.572078\pi\)
\(98\) −0.284398 −0.0287286
\(99\) 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.1.9 16
3.2 odd 2 2667.2.a.n.1.8 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.8 ✓ 16 3.2 odd 2
8001.2.a.s.1.9 16 1.1 even 1 trivial