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Properties

Label 8001.2.a.o.1.6
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $13$
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 = [13,-4,0,10,-12] 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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 \) 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 \(1.18273\) 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)
 
\(f(q)\) \(=\) \(q-1.18273 q^{2} -0.601156 q^{4} -4.39766 q^{5} +1.00000 q^{7} +3.07646 q^{8} +5.20123 q^{10} +1.66461 q^{11} +1.04052 q^{13} -1.18273 q^{14} -2.43630 q^{16} -2.31427 q^{17} -1.96198 q^{19} +2.64368 q^{20} -1.96878 q^{22} -2.53372 q^{23} +14.3394 q^{25} -1.23065 q^{26} -0.601156 q^{28} -3.94733 q^{29} +2.35543 q^{31} -3.27144 q^{32} +2.73716 q^{34} -4.39766 q^{35} -3.81874 q^{37} +2.32049 q^{38} -13.5292 q^{40} -5.25377 q^{41} +1.58885 q^{43} -1.00069 q^{44} +2.99670 q^{46} +5.23626 q^{47} +1.00000 q^{49} -16.9596 q^{50} -0.625517 q^{52} -6.13449 q^{53} -7.32038 q^{55} +3.07646 q^{56} +4.66862 q^{58} +1.48950 q^{59} +0.959646 q^{61} -2.78583 q^{62} +8.74182 q^{64} -4.57586 q^{65} +4.40683 q^{67} +1.39124 q^{68} +5.20123 q^{70} -4.45038 q^{71} +1.66690 q^{73} +4.51652 q^{74} +1.17946 q^{76} +1.66461 q^{77} +0.363151 q^{79} +10.7140 q^{80} +6.21377 q^{82} +9.34690 q^{83} +10.1774 q^{85} -1.87918 q^{86} +5.12110 q^{88} +3.44222 q^{89} +1.04052 q^{91} +1.52316 q^{92} -6.19307 q^{94} +8.62813 q^{95} +13.6324 q^{97} -1.18273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.18273 −0.836314 −0.418157 − 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(3\) 0 0
\(4\) −0.601156 −0.300578
\(5\) −4.39766 −1.96669 −0.983347 − 0.181739i \(-0.941827\pi\)
−0.983347 + 0.181739i \(0.941827\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.07646 1.08769
\(9\) 0 0
\(10\) 5.20123 1.64477
\(11\) 1.66461 0.501898 0.250949 − 0.968000i \(-0.419257\pi\)
0.250949 + 0.968000i \(0.419257\pi\)
\(12\) 0 0
\(13\) 1.04052 0.288589 0.144294 − 0.989535i \(-0.453909\pi\)
0.144294 + 0.989535i \(0.453909\pi\)
\(14\) −1.18273 −0.316097
\(15\) 0 0
\(16\) −2.43630 −0.609075
\(17\) −2.31427 −0.561294 −0.280647 − 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(18\) 0 0
\(19\) −1.96198 −0.450109 −0.225055 − 0.974346i \(-0.572256\pi\)
−0.225055 + 0.974346i \(0.572256\pi\)
\(20\) 2.64368 0.591145
\(21\) 0 0
\(22\) −1.96878 −0.419745
\(23\) −2.53372 −0.528318 −0.264159 − 0.964479i \(-0.585094\pi\)
−0.264159 + 0.964479i \(0.585094\pi\)
\(24\) 0 0
\(25\) 14.3394 2.86788
\(26\) −1.23065 −0.241351
\(27\) 0 0
\(28\) −0.601156 −0.113608
\(29\) −3.94733 −0.733001 −0.366501 − 0.930418i \(-0.619444\pi\)
−0.366501 + 0.930418i \(0.619444\pi\)
\(30\) 0 0
\(31\) 2.35543 0.423048 0.211524 − 0.977373i \(-0.432157\pi\)
0.211524 + 0.977373i \(0.432157\pi\)
\(32\) −3.27144 −0.578314
\(33\) 0 0
\(34\) 2.73716 0.469418
\(35\) −4.39766 −0.743340
\(36\) 0 0
\(37\) −3.81874 −0.627797 −0.313898 − 0.949457i \(-0.601635\pi\)
−0.313898 + 0.949457i \(0.601635\pi\)
\(38\) 2.32049 0.376433
\(39\) 0 0
\(40\) −13.5292 −2.13916
\(41\) −5.25377 −0.820500 −0.410250 − 0.911973i \(-0.634559\pi\)
−0.410250 + 0.911973i \(0.634559\pi\)
\(42\) 0 0
\(43\) 1.58885 0.242298 0.121149 − 0.992634i \(-0.461342\pi\)
0.121149 + 0.992634i \(0.461342\pi\)
\(44\) −1.00069 −0.150860
\(45\) 0 0
\(46\) 2.99670 0.441840
\(47\) 5.23626 0.763787 0.381893 − 0.924206i \(-0.375272\pi\)
0.381893 + 0.924206i \(0.375272\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −16.9596 −2.39845
\(51\) 0 0
\(52\) −0.625517 −0.0867435
\(53\) −6.13449 −0.842637 −0.421318 − 0.906913i \(-0.638432\pi\)
−0.421318 + 0.906913i \(0.638432\pi\)
\(54\) 0 0
\(55\) −7.32038 −0.987080
\(56\) 3.07646 0.411109
\(57\) 0 0
\(58\) 4.66862 0.613020
\(59\) 1.48950 0.193916 0.0969580 − 0.995288i \(-0.469089\pi\)
0.0969580 + 0.995288i \(0.469089\pi\)
\(60\) 0 0
\(61\) 0.959646 0.122870 0.0614350 − 0.998111i \(-0.480432\pi\)
0.0614350 + 0.998111i \(0.480432\pi\)
\(62\) −2.78583 −0.353801
\(63\) 0 0
\(64\) 8.74182 1.09273
\(65\) −4.57586 −0.567566
\(66\) 0 0
\(67\) 4.40683 0.538380 0.269190 − 0.963087i \(-0.413244\pi\)
0.269190 + 0.963087i \(0.413244\pi\)
\(68\) 1.39124 0.168713
\(69\) 0 0
\(70\) 5.20123 0.621666
\(71\) −4.45038 −0.528163 −0.264082 − 0.964500i \(-0.585069\pi\)
−0.264082 + 0.964500i \(0.585069\pi\)
\(72\) 0 0
\(73\) 1.66690 0.195096 0.0975479 − 0.995231i \(-0.468900\pi\)
0.0975479 + 0.995231i \(0.468900\pi\)
\(74\) 4.51652 0.525035
\(75\) 0 0
\(76\) 1.17946 0.135293
\(77\) 1.66461 0.189700
\(78\) 0 0
\(79\) 0.363151 0.0408577 0.0204288 − 0.999791i \(-0.493497\pi\)
0.0204288 + 0.999791i \(0.493497\pi\)
\(80\) 10.7140 1.19786
\(81\) 0 0
\(82\) 6.21377 0.686196
\(83\) 9.34690 1.02596 0.512978 − 0.858402i \(-0.328542\pi\)
0.512978 + 0.858402i \(0.328542\pi\)
\(84\) 0 0
\(85\) 10.1774 1.10389
\(86\) −1.87918 −0.202637
\(87\) 0 0
\(88\) 5.12110 0.545911
\(89\) 3.44222 0.364874 0.182437 − 0.983218i \(-0.441601\pi\)
0.182437 + 0.983218i \(0.441601\pi\)
\(90\) 0 0
\(91\) 1.04052 0.109076
\(92\) 1.52316 0.158801
\(93\) 0 0
\(94\) −6.19307 −0.638766
\(95\) 8.62813 0.885227
\(96\) 0 0
\(97\) 13.6324 1.38416 0.692082 − 0.721819i \(-0.256694\pi\)
0.692082 + 0.721819i \(0.256694\pi\)
\(98\) −1.18273 −0.119473
\(99\) 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.o.1.6 13
3.2 odd 2 2667.2.a.l.1.8 ✓ 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.l.1.8 ✓ 13 3.2 odd 2
8001.2.a.o.1.6 13 1.1 even 1 trivial