↑

Properties

Label 8001.2.a.ba.1.29
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $40$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [40,0,0,54,0] 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: \(40\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.29
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)
 
\(f(q)\) \(=\) \(q+1.42650 q^{2} +0.0348943 q^{4} -4.07722 q^{5} +1.00000 q^{7} -2.80322 q^{8} -5.81614 q^{10} -3.50251 q^{11} -5.98419 q^{13} +1.42650 q^{14} -4.06857 q^{16} -7.04872 q^{17} +3.99853 q^{19} -0.142272 q^{20} -4.99633 q^{22} -7.32524 q^{23} +11.6237 q^{25} -8.53642 q^{26} +0.0348943 q^{28} -7.77438 q^{29} +2.84394 q^{31} -0.197369 q^{32} -10.0550 q^{34} -4.07722 q^{35} -1.23220 q^{37} +5.70390 q^{38} +11.4293 q^{40} -1.75959 q^{41} -10.6816 q^{43} -0.122218 q^{44} -10.4494 q^{46} -4.77897 q^{47} +1.00000 q^{49} +16.5812 q^{50} -0.208814 q^{52} +4.77146 q^{53} +14.2805 q^{55} -2.80322 q^{56} -11.0901 q^{58} -13.2639 q^{59} -2.14883 q^{61} +4.05688 q^{62} +7.85559 q^{64} +24.3988 q^{65} -5.59570 q^{67} -0.245960 q^{68} -5.81614 q^{70} +11.6689 q^{71} +5.10805 q^{73} -1.75772 q^{74} +0.139526 q^{76} -3.50251 q^{77} +8.07023 q^{79} +16.5885 q^{80} -2.51006 q^{82} +5.09750 q^{83} +28.7392 q^{85} -15.2373 q^{86} +9.81831 q^{88} -0.679380 q^{89} -5.98419 q^{91} -0.255609 q^{92} -6.81718 q^{94} -16.3029 q^{95} -18.1740 q^{97} +1.42650 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52}+ \cdots + 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.42650 1.00869 0.504343 − 0.863503i \(-0.331735\pi\)
0.504343 + 0.863503i \(0.331735\pi\)
\(3\) 0 0
\(4\) 0.0348943 0.0174471
\(5\) −4.07722 −1.82339 −0.911694 − 0.410870i \(-0.865225\pi\)
−0.911694 + 0.410870i \(0.865225\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.80322 −0.991087
\(9\) 0 0
\(10\) −5.81614 −1.83923
\(11\) −3.50251 −1.05605 −0.528024 − 0.849229i \(-0.677067\pi\)
−0.528024 + 0.849229i \(0.677067\pi\)
\(12\) 0 0
\(13\) −5.98419 −1.65971 −0.829857 − 0.557976i \(-0.811578\pi\)
−0.829857 + 0.557976i \(0.811578\pi\)
\(14\) 1.42650 0.381247
\(15\) 0 0
\(16\) −4.06857 −1.01714
\(17\) −7.04872 −1.70956 −0.854782 − 0.518987i \(-0.826309\pi\)
−0.854782 + 0.518987i \(0.826309\pi\)
\(18\) 0 0
\(19\) 3.99853 0.917326 0.458663 − 0.888610i \(-0.348328\pi\)
0.458663 + 0.888610i \(0.348328\pi\)
\(20\) −0.142272 −0.0318129
\(21\) 0 0
\(22\) −4.99633 −1.06522
\(23\) −7.32524 −1.52742 −0.763709 − 0.645561i \(-0.776624\pi\)
−0.763709 + 0.645561i \(0.776624\pi\)
\(24\) 0 0
\(25\) 11.6237 2.32474
\(26\) −8.53642 −1.67413
\(27\) 0 0
\(28\) 0.0348943 0.00659440
\(29\) −7.77438 −1.44367 −0.721833 − 0.692068i \(-0.756700\pi\)
−0.721833 + 0.692068i \(0.756700\pi\)
\(30\) 0 0
\(31\) 2.84394 0.510787 0.255394 − 0.966837i \(-0.417795\pi\)
0.255394 + 0.966837i \(0.417795\pi\)
\(32\) −0.197369 −0.0348903
\(33\) 0 0
\(34\) −10.0550 −1.72441
\(35\) −4.07722 −0.689176
\(36\) 0 0
\(37\) −1.23220 −0.202572 −0.101286 − 0.994857i \(-0.532296\pi\)
−0.101286 + 0.994857i \(0.532296\pi\)
\(38\) 5.70390 0.925294
\(39\) 0 0
\(40\) 11.4293 1.80714
\(41\) −1.75959 −0.274802 −0.137401 − 0.990515i \(-0.543875\pi\)
−0.137401 + 0.990515i \(0.543875\pi\)
\(42\) 0 0
\(43\) −10.6816 −1.62894 −0.814468 − 0.580209i \(-0.802971\pi\)
−0.814468 + 0.580209i \(0.802971\pi\)
\(44\) −0.122218 −0.0184250
\(45\) 0 0
\(46\) −10.4494 −1.54068
\(47\) −4.77897 −0.697084 −0.348542 − 0.937293i \(-0.613323\pi\)
−0.348542 + 0.937293i \(0.613323\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 16.5812 2.34494
\(51\) 0 0
\(52\) −0.208814 −0.0289573
\(53\) 4.77146 0.655410 0.327705 − 0.944780i \(-0.393725\pi\)
0.327705 + 0.944780i \(0.393725\pi\)
\(54\) 0 0
\(55\) 14.2805 1.92559
\(56\) −2.80322 −0.374596
\(57\) 0 0
\(58\) −11.0901 −1.45620
\(59\) −13.2639 −1.72681 −0.863403 − 0.504514i \(-0.831672\pi\)
−0.863403 + 0.504514i \(0.831672\pi\)
\(60\) 0 0
\(61\) −2.14883 −0.275130 −0.137565 − 0.990493i \(-0.543928\pi\)
−0.137565 + 0.990493i \(0.543928\pi\)
\(62\) 4.05688 0.515224
\(63\) 0 0
\(64\) 7.85559 0.981949
\(65\) 24.3988 3.02630
\(66\) 0 0
\(67\) −5.59570 −0.683623 −0.341812 − 0.939769i \(-0.611040\pi\)
−0.341812 + 0.939769i \(0.611040\pi\)
\(68\) −0.245960 −0.0298270
\(69\) 0 0
\(70\) −5.81614 −0.695162
\(71\) 11.6689 1.38485 0.692424 − 0.721490i \(-0.256543\pi\)
0.692424 + 0.721490i \(0.256543\pi\)
\(72\) 0 0
\(73\) 5.10805 0.597852 0.298926 − 0.954276i \(-0.403372\pi\)
0.298926 + 0.954276i \(0.403372\pi\)
\(74\) −1.75772 −0.204331
\(75\) 0 0
\(76\) 0.139526 0.0160047
\(77\) −3.50251 −0.399149
\(78\) 0 0
\(79\) 8.07023 0.907972 0.453986 − 0.891009i \(-0.350002\pi\)
0.453986 + 0.891009i \(0.350002\pi\)
\(80\) 16.5885 1.85465
\(81\) 0 0
\(82\) −2.51006 −0.277189
\(83\) 5.09750 0.559524 0.279762 − 0.960069i \(-0.409744\pi\)
0.279762 + 0.960069i \(0.409744\pi\)
\(84\) 0 0
\(85\) 28.7392 3.11720
\(86\) −15.2373 −1.64308
\(87\) 0 0
\(88\) 9.81831 1.04664
\(89\) −0.679380 −0.0720141 −0.0360071 − 0.999352i \(-0.511464\pi\)
−0.0360071 + 0.999352i \(0.511464\pi\)
\(90\) 0 0
\(91\) −5.98419 −0.627313
\(92\) −0.255609 −0.0266491
\(93\) 0 0
\(94\) −6.81718 −0.703139
\(95\) −16.3029 −1.67264
\(96\) 0 0
\(97\) −18.1740 −1.84529 −0.922643 − 0.385654i \(-0.873976\pi\)
−0.922643 + 0.385654i \(0.873976\pi\)
\(98\) 1.42650 0.144098
\(99\) 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.ba.1.29 yes 40
3.2 odd 2 inner 8001.2.a.ba.1.12 ✓ 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.ba.1.12 ✓ 40 3.2 odd 2 inner
8001.2.a.ba.1.29 yes 40 1.1 even 1 trivial