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Properties

Label 8001.2.a.s.1.6
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 1.6
Root \(1.17466\) 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)
 
\(f(q)\) \(=\) \(q-1.17466 q^{2} -0.620175 q^{4} -3.75278 q^{5} -1.00000 q^{7} +3.07781 q^{8} +4.40823 q^{10} +0.236817 q^{11} +2.34303 q^{13} +1.17466 q^{14} -2.37503 q^{16} +5.49213 q^{17} +1.29877 q^{19} +2.32738 q^{20} -0.278179 q^{22} -4.33999 q^{23} +9.08332 q^{25} -2.75226 q^{26} +0.620175 q^{28} -3.35099 q^{29} +10.2207 q^{31} -3.36577 q^{32} -6.45138 q^{34} +3.75278 q^{35} +5.67027 q^{37} -1.52562 q^{38} -11.5503 q^{40} -5.91718 q^{41} +7.58953 q^{43} -0.146868 q^{44} +5.09801 q^{46} -6.69813 q^{47} +1.00000 q^{49} -10.6698 q^{50} -1.45309 q^{52} +11.3704 q^{53} -0.888720 q^{55} -3.07781 q^{56} +3.93627 q^{58} +1.67458 q^{59} +6.14626 q^{61} -12.0058 q^{62} +8.70370 q^{64} -8.79286 q^{65} -12.3793 q^{67} -3.40608 q^{68} -4.40823 q^{70} +16.3585 q^{71} -2.42897 q^{73} -6.66064 q^{74} -0.805466 q^{76} -0.236817 q^{77} -12.9909 q^{79} +8.91297 q^{80} +6.95068 q^{82} +1.57938 q^{83} -20.6107 q^{85} -8.91512 q^{86} +0.728878 q^{88} -3.02589 q^{89} -2.34303 q^{91} +2.69155 q^{92} +7.86802 q^{94} -4.87400 q^{95} -1.63077 q^{97} -1.17466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17466 −0.830610 −0.415305 − 0.909682i \(-0.636325\pi\)
−0.415305 + 0.909682i \(0.636325\pi\)
\(3\) 0 0
\(4\) −0.620175 −0.310087
\(5\) −3.75278 −1.67829 −0.839146 − 0.543906i \(-0.816945\pi\)
−0.839146 + 0.543906i \(0.816945\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.07781 1.08817
\(9\) 0 0
\(10\) 4.40823 1.39401
\(11\) 0.236817 0.0714029 0.0357015 − 0.999362i \(-0.488633\pi\)
0.0357015 + 0.999362i \(0.488633\pi\)
\(12\) 0 0
\(13\) 2.34303 0.649839 0.324920 − 0.945742i \(-0.394663\pi\)
0.324920 + 0.945742i \(0.394663\pi\)
\(14\) 1.17466 0.313941
\(15\) 0 0
\(16\) −2.37503 −0.593759
\(17\) 5.49213 1.33204 0.666018 − 0.745935i \(-0.267997\pi\)
0.666018 + 0.745935i \(0.267997\pi\)
\(18\) 0 0
\(19\) 1.29877 0.297959 0.148979 − 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(20\) 2.32738 0.520417
\(21\) 0 0
\(22\) −0.278179 −0.0593080
\(23\) −4.33999 −0.904950 −0.452475 − 0.891777i \(-0.649459\pi\)
−0.452475 + 0.891777i \(0.649459\pi\)
\(24\) 0 0
\(25\) 9.08332 1.81666
\(26\) −2.75226 −0.539763
\(27\) 0 0
\(28\) 0.620175 0.117202
\(29\) −3.35099 −0.622263 −0.311131 − 0.950367i \(-0.600708\pi\)
−0.311131 + 0.950367i \(0.600708\pi\)
\(30\) 0 0
\(31\) 10.2207 1.83569 0.917846 − 0.396937i \(-0.129927\pi\)
0.917846 + 0.396937i \(0.129927\pi\)
\(32\) −3.36577 −0.594990
\(33\) 0 0
\(34\) −6.45138 −1.10640
\(35\) 3.75278 0.634335
\(36\) 0 0
\(37\) 5.67027 0.932186 0.466093 − 0.884736i \(-0.345661\pi\)
0.466093 + 0.884736i \(0.345661\pi\)
\(38\) −1.52562 −0.247488
\(39\) 0 0
\(40\) −11.5503 −1.82627
\(41\) −5.91718 −0.924109 −0.462054 − 0.886852i \(-0.652888\pi\)
−0.462054 + 0.886852i \(0.652888\pi\)
\(42\) 0 0
\(43\) 7.58953 1.15739 0.578697 − 0.815543i \(-0.303562\pi\)
0.578697 + 0.815543i \(0.303562\pi\)
\(44\) −0.146868 −0.0221411
\(45\) 0 0
\(46\) 5.09801 0.751660
\(47\) −6.69813 −0.977022 −0.488511 − 0.872558i \(-0.662460\pi\)
−0.488511 + 0.872558i \(0.662460\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.6698 −1.50894
\(51\) 0 0
\(52\) −1.45309 −0.201507
\(53\) 11.3704 1.56185 0.780926 − 0.624624i \(-0.214748\pi\)
0.780926 + 0.624624i \(0.214748\pi\)
\(54\) 0 0
\(55\) −0.888720 −0.119835
\(56\) −3.07781 −0.411290
\(57\) 0 0
\(58\) 3.93627 0.516858
\(59\) 1.67458 0.218011 0.109006 − 0.994041i \(-0.465233\pi\)
0.109006 + 0.994041i \(0.465233\pi\)
\(60\) 0 0
\(61\) 6.14626 0.786948 0.393474 − 0.919336i \(-0.371273\pi\)
0.393474 + 0.919336i \(0.371273\pi\)
\(62\) −12.0058 −1.52474
\(63\) 0 0
\(64\) 8.70370 1.08796
\(65\) −8.79286 −1.09062
\(66\) 0 0
\(67\) −12.3793 −1.51237 −0.756186 − 0.654357i \(-0.772940\pi\)
−0.756186 + 0.654357i \(0.772940\pi\)
\(68\) −3.40608 −0.413048
\(69\) 0 0
\(70\) −4.40823 −0.526885
\(71\) 16.3585 1.94140 0.970701 − 0.240291i \(-0.0772428\pi\)
0.970701 + 0.240291i \(0.0772428\pi\)
\(72\) 0 0
\(73\) −2.42897 −0.284289 −0.142144 − 0.989846i \(-0.545400\pi\)
−0.142144 + 0.989846i \(0.545400\pi\)
\(74\) −6.66064 −0.774283
\(75\) 0 0
\(76\) −0.805466 −0.0923933
\(77\) −0.236817 −0.0269878
\(78\) 0 0
\(79\) −12.9909 −1.46159 −0.730796 − 0.682596i \(-0.760851\pi\)
−0.730796 + 0.682596i \(0.760851\pi\)
\(80\) 8.91297 0.996500
\(81\) 0 0
\(82\) 6.95068 0.767574
\(83\) 1.57938 0.173359 0.0866795 − 0.996236i \(-0.472374\pi\)
0.0866795 + 0.996236i \(0.472374\pi\)
\(84\) 0 0
\(85\) −20.6107 −2.23555
\(86\) −8.91512 −0.961342
\(87\) 0 0
\(88\) 0.728878 0.0776986
\(89\) −3.02589 −0.320743 −0.160372 − 0.987057i \(-0.551269\pi\)
−0.160372 + 0.987057i \(0.551269\pi\)
\(90\) 0 0
\(91\) −2.34303 −0.245616
\(92\) 2.69155 0.280614
\(93\) 0 0
\(94\) 7.86802 0.811524
\(95\) −4.87400 −0.500062
\(96\) 0 0
\(97\) −1.63077 −0.165580 −0.0827899 − 0.996567i \(-0.526383\pi\)
−0.0827899 + 0.996567i \(0.526383\pi\)
\(98\) −1.17466 −0.118659
\(99\) 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.1.6 16
3.2 odd 2 2667.2.a.n.1.11 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.11 ✓ 16 3.2 odd 2
8001.2.a.s.1.6 16 1.1 even 1 trivial