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Properties

Label 8001.2.a.m.1.5
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $11$
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 = [11,2,0,12,-1] 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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.910036\) 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)
 
\(f(q)\) \(=\) \(q-0.910036 q^{2} -1.17183 q^{4} -2.84869 q^{5} -1.00000 q^{7} +2.88648 q^{8} +2.59241 q^{10} +1.86466 q^{11} -6.19675 q^{13} +0.910036 q^{14} -0.283136 q^{16} -2.49732 q^{17} -6.22887 q^{19} +3.33819 q^{20} -1.69691 q^{22} +5.12980 q^{23} +3.11503 q^{25} +5.63926 q^{26} +1.17183 q^{28} +7.68531 q^{29} +4.32504 q^{31} -5.51530 q^{32} +2.27265 q^{34} +2.84869 q^{35} +2.90361 q^{37} +5.66849 q^{38} -8.22269 q^{40} -4.60795 q^{41} -9.98321 q^{43} -2.18507 q^{44} -4.66830 q^{46} +10.9278 q^{47} +1.00000 q^{49} -2.83479 q^{50} +7.26156 q^{52} +10.7010 q^{53} -5.31184 q^{55} -2.88648 q^{56} -6.99391 q^{58} -11.4224 q^{59} +1.39989 q^{61} -3.93594 q^{62} +5.58540 q^{64} +17.6526 q^{65} +14.3600 q^{67} +2.92644 q^{68} -2.59241 q^{70} -9.49609 q^{71} -16.1291 q^{73} -2.64239 q^{74} +7.29920 q^{76} -1.86466 q^{77} +12.5570 q^{79} +0.806567 q^{80} +4.19340 q^{82} +6.86150 q^{83} +7.11407 q^{85} +9.08509 q^{86} +5.38232 q^{88} +8.88528 q^{89} +6.19675 q^{91} -6.01128 q^{92} -9.94466 q^{94} +17.7441 q^{95} +1.23342 q^{97} -0.910036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.910036 −0.643493 −0.321746 − 0.946826i \(-0.604270\pi\)
−0.321746 + 0.946826i \(0.604270\pi\)
\(3\) 0 0
\(4\) −1.17183 −0.585917
\(5\) −2.84869 −1.27397 −0.636986 − 0.770875i \(-0.719819\pi\)
−0.636986 + 0.770875i \(0.719819\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.88648 1.02053
\(9\) 0 0
\(10\) 2.59241 0.819792
\(11\) 1.86466 0.562217 0.281108 − 0.959676i \(-0.409298\pi\)
0.281108 + 0.959676i \(0.409298\pi\)
\(12\) 0 0
\(13\) −6.19675 −1.71867 −0.859334 − 0.511415i \(-0.829122\pi\)
−0.859334 + 0.511415i \(0.829122\pi\)
\(14\) 0.910036 0.243217
\(15\) 0 0
\(16\) −0.283136 −0.0707841
\(17\) −2.49732 −0.605688 −0.302844 − 0.953040i \(-0.597936\pi\)
−0.302844 + 0.953040i \(0.597936\pi\)
\(18\) 0 0
\(19\) −6.22887 −1.42900 −0.714500 − 0.699635i \(-0.753346\pi\)
−0.714500 + 0.699635i \(0.753346\pi\)
\(20\) 3.33819 0.746442
\(21\) 0 0
\(22\) −1.69691 −0.361782
\(23\) 5.12980 1.06964 0.534819 − 0.844967i \(-0.320380\pi\)
0.534819 + 0.844967i \(0.320380\pi\)
\(24\) 0 0
\(25\) 3.11503 0.623006
\(26\) 5.63926 1.10595
\(27\) 0 0
\(28\) 1.17183 0.221456
\(29\) 7.68531 1.42713 0.713563 − 0.700591i \(-0.247080\pi\)
0.713563 + 0.700591i \(0.247080\pi\)
\(30\) 0 0
\(31\) 4.32504 0.776800 0.388400 − 0.921491i \(-0.373028\pi\)
0.388400 + 0.921491i \(0.373028\pi\)
\(32\) −5.51530 −0.974977
\(33\) 0 0
\(34\) 2.27265 0.389756
\(35\) 2.84869 0.481516
\(36\) 0 0
\(37\) 2.90361 0.477350 0.238675 − 0.971100i \(-0.423287\pi\)
0.238675 + 0.971100i \(0.423287\pi\)
\(38\) 5.66849 0.919551
\(39\) 0 0
\(40\) −8.22269 −1.30012
\(41\) −4.60795 −0.719640 −0.359820 − 0.933022i \(-0.617162\pi\)
−0.359820 + 0.933022i \(0.617162\pi\)
\(42\) 0 0
\(43\) −9.98321 −1.52243 −0.761213 − 0.648502i \(-0.775396\pi\)
−0.761213 + 0.648502i \(0.775396\pi\)
\(44\) −2.18507 −0.329412
\(45\) 0 0
\(46\) −4.66830 −0.688304
\(47\) 10.9278 1.59398 0.796989 − 0.603994i \(-0.206425\pi\)
0.796989 + 0.603994i \(0.206425\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.83479 −0.400900
\(51\) 0 0
\(52\) 7.26156 1.00700
\(53\) 10.7010 1.46990 0.734948 − 0.678123i \(-0.237207\pi\)
0.734948 + 0.678123i \(0.237207\pi\)
\(54\) 0 0
\(55\) −5.31184 −0.716249
\(56\) −2.88648 −0.385723
\(57\) 0 0
\(58\) −6.99391 −0.918345
\(59\) −11.4224 −1.48707 −0.743533 − 0.668699i \(-0.766851\pi\)
−0.743533 + 0.668699i \(0.766851\pi\)
\(60\) 0 0
\(61\) 1.39989 0.179237 0.0896185 − 0.995976i \(-0.471435\pi\)
0.0896185 + 0.995976i \(0.471435\pi\)
\(62\) −3.93594 −0.499865
\(63\) 0 0
\(64\) 5.58540 0.698175
\(65\) 17.6526 2.18954
\(66\) 0 0
\(67\) 14.3600 1.75436 0.877178 − 0.480164i \(-0.159423\pi\)
0.877178 + 0.480164i \(0.159423\pi\)
\(68\) 2.92644 0.354883
\(69\) 0 0
\(70\) −2.59241 −0.309852
\(71\) −9.49609 −1.12698 −0.563489 − 0.826123i \(-0.690541\pi\)
−0.563489 + 0.826123i \(0.690541\pi\)
\(72\) 0 0
\(73\) −16.1291 −1.88777 −0.943887 − 0.330269i \(-0.892861\pi\)
−0.943887 + 0.330269i \(0.892861\pi\)
\(74\) −2.64239 −0.307171
\(75\) 0 0
\(76\) 7.29920 0.837276
\(77\) −1.86466 −0.212498
\(78\) 0 0
\(79\) 12.5570 1.41277 0.706384 − 0.707829i \(-0.250325\pi\)
0.706384 + 0.707829i \(0.250325\pi\)
\(80\) 0.806567 0.0901770
\(81\) 0 0
\(82\) 4.19340 0.463083
\(83\) 6.86150 0.753148 0.376574 − 0.926387i \(-0.377102\pi\)
0.376574 + 0.926387i \(0.377102\pi\)
\(84\) 0 0
\(85\) 7.11407 0.771630
\(86\) 9.08509 0.979670
\(87\) 0 0
\(88\) 5.38232 0.573757
\(89\) 8.88528 0.941838 0.470919 − 0.882176i \(-0.343922\pi\)
0.470919 + 0.882176i \(0.343922\pi\)
\(90\) 0 0
\(91\) 6.19675 0.649596
\(92\) −6.01128 −0.626719
\(93\) 0 0
\(94\) −9.94466 −1.02571
\(95\) 17.7441 1.82051
\(96\) 0 0
\(97\) 1.23342 0.125235 0.0626176 − 0.998038i \(-0.480055\pi\)
0.0626176 + 0.998038i \(0.480055\pi\)
\(98\) −0.910036 −0.0919275
\(99\) 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.m.1.5 11
3.2 odd 2 2667.2.a.k.1.7 ✓ 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.k.1.7 ✓ 11 3.2 odd 2
8001.2.a.m.1.5 11 1.1 even 1 trivial