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Properties

Label 8001.2.a.ba.1.35
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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 = [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 1.35
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)
 
\(f(q)\) \(=\) \(q+2.31568 q^{2} +3.36240 q^{4} -1.76797 q^{5} +1.00000 q^{7} +3.15488 q^{8} -4.09407 q^{10} +5.91269 q^{11} -6.94886 q^{13} +2.31568 q^{14} +0.580914 q^{16} -4.59921 q^{17} +2.11313 q^{19} -5.94463 q^{20} +13.6919 q^{22} +7.36361 q^{23} -1.87427 q^{25} -16.0914 q^{26} +3.36240 q^{28} -1.96837 q^{29} +8.41994 q^{31} -4.96455 q^{32} -10.6503 q^{34} -1.76797 q^{35} +10.4399 q^{37} +4.89334 q^{38} -5.57774 q^{40} +8.05785 q^{41} +2.89613 q^{43} +19.8808 q^{44} +17.0518 q^{46} +6.04059 q^{47} +1.00000 q^{49} -4.34022 q^{50} -23.3648 q^{52} +0.912285 q^{53} -10.4535 q^{55} +3.15488 q^{56} -4.55812 q^{58} +5.80425 q^{59} +0.182737 q^{61} +19.4979 q^{62} -12.6581 q^{64} +12.2854 q^{65} +6.40252 q^{67} -15.4644 q^{68} -4.09407 q^{70} +2.69489 q^{71} -3.40449 q^{73} +24.1756 q^{74} +7.10518 q^{76} +5.91269 q^{77} +1.85374 q^{79} -1.02704 q^{80} +18.6594 q^{82} -12.7542 q^{83} +8.13128 q^{85} +6.70653 q^{86} +18.6538 q^{88} +6.46307 q^{89} -6.94886 q^{91} +24.7594 q^{92} +13.9881 q^{94} -3.73596 q^{95} -11.2916 q^{97} +2.31568 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52}+ \cdots + 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31568 1.63744 0.818718 − 0.574196i \(-0.194685\pi\)
0.818718 + 0.574196i \(0.194685\pi\)
\(3\) 0 0
\(4\) 3.36240 1.68120
\(5\) −1.76797 −0.790662 −0.395331 − 0.918539i \(-0.629370\pi\)
−0.395331 + 0.918539i \(0.629370\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.15488 1.11542
\(9\) 0 0
\(10\) −4.09407 −1.29466
\(11\) 5.91269 1.78274 0.891371 − 0.453274i \(-0.149744\pi\)
0.891371 + 0.453274i \(0.149744\pi\)
\(12\) 0 0
\(13\) −6.94886 −1.92727 −0.963633 − 0.267227i \(-0.913893\pi\)
−0.963633 + 0.267227i \(0.913893\pi\)
\(14\) 2.31568 0.618893
\(15\) 0 0
\(16\) 0.580914 0.145228
\(17\) −4.59921 −1.11547 −0.557736 − 0.830018i \(-0.688330\pi\)
−0.557736 + 0.830018i \(0.688330\pi\)
\(18\) 0 0
\(19\) 2.11313 0.484785 0.242393 − 0.970178i \(-0.422068\pi\)
0.242393 + 0.970178i \(0.422068\pi\)
\(20\) −5.94463 −1.32926
\(21\) 0 0
\(22\) 13.6919 2.91913
\(23\) 7.36361 1.53542 0.767710 − 0.640798i \(-0.221396\pi\)
0.767710 + 0.640798i \(0.221396\pi\)
\(24\) 0 0
\(25\) −1.87427 −0.374854
\(26\) −16.0914 −3.15578
\(27\) 0 0
\(28\) 3.36240 0.635433
\(29\) −1.96837 −0.365517 −0.182759 − 0.983158i \(-0.558503\pi\)
−0.182759 + 0.983158i \(0.558503\pi\)
\(30\) 0 0
\(31\) 8.41994 1.51227 0.756133 − 0.654418i \(-0.227086\pi\)
0.756133 + 0.654418i \(0.227086\pi\)
\(32\) −4.96455 −0.877616
\(33\) 0 0
\(34\) −10.6503 −1.82652
\(35\) −1.76797 −0.298842
\(36\) 0 0
\(37\) 10.4399 1.71631 0.858157 − 0.513387i \(-0.171609\pi\)
0.858157 + 0.513387i \(0.171609\pi\)
\(38\) 4.89334 0.793805
\(39\) 0 0
\(40\) −5.57774 −0.881919
\(41\) 8.05785 1.25842 0.629212 − 0.777234i \(-0.283378\pi\)
0.629212 + 0.777234i \(0.283378\pi\)
\(42\) 0 0
\(43\) 2.89613 0.441656 0.220828 − 0.975313i \(-0.429124\pi\)
0.220828 + 0.975313i \(0.429124\pi\)
\(44\) 19.8808 2.99714
\(45\) 0 0
\(46\) 17.0518 2.51415
\(47\) 6.04059 0.881111 0.440555 − 0.897725i \(-0.354782\pi\)
0.440555 + 0.897725i \(0.354782\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.34022 −0.613799
\(51\) 0 0
\(52\) −23.3648 −3.24012
\(53\) 0.912285 0.125312 0.0626560 − 0.998035i \(-0.480043\pi\)
0.0626560 + 0.998035i \(0.480043\pi\)
\(54\) 0 0
\(55\) −10.4535 −1.40955
\(56\) 3.15488 0.421588
\(57\) 0 0
\(58\) −4.55812 −0.598511
\(59\) 5.80425 0.755649 0.377824 − 0.925877i \(-0.376672\pi\)
0.377824 + 0.925877i \(0.376672\pi\)
\(60\) 0 0
\(61\) 0.182737 0.0233971 0.0116986 − 0.999932i \(-0.496276\pi\)
0.0116986 + 0.999932i \(0.496276\pi\)
\(62\) 19.4979 2.47624
\(63\) 0 0
\(64\) −12.6581 −1.58227
\(65\) 12.2854 1.52382
\(66\) 0 0
\(67\) 6.40252 0.782192 0.391096 − 0.920350i \(-0.372096\pi\)
0.391096 + 0.920350i \(0.372096\pi\)
\(68\) −15.4644 −1.87533
\(69\) 0 0
\(70\) −4.09407 −0.489335
\(71\) 2.69489 0.319824 0.159912 − 0.987131i \(-0.448879\pi\)
0.159912 + 0.987131i \(0.448879\pi\)
\(72\) 0 0
\(73\) −3.40449 −0.398465 −0.199232 − 0.979952i \(-0.563845\pi\)
−0.199232 + 0.979952i \(0.563845\pi\)
\(74\) 24.1756 2.81036
\(75\) 0 0
\(76\) 7.10518 0.815020
\(77\) 5.91269 0.673814
\(78\) 0 0
\(79\) 1.85374 0.208562 0.104281 − 0.994548i \(-0.466746\pi\)
0.104281 + 0.994548i \(0.466746\pi\)
\(80\) −1.02704 −0.114827
\(81\) 0 0
\(82\) 18.6594 2.06059
\(83\) −12.7542 −1.39996 −0.699979 − 0.714163i \(-0.746807\pi\)
−0.699979 + 0.714163i \(0.746807\pi\)
\(84\) 0 0
\(85\) 8.13128 0.881962
\(86\) 6.70653 0.723184
\(87\) 0 0
\(88\) 18.6538 1.98850
\(89\) 6.46307 0.685085 0.342542 − 0.939502i \(-0.388712\pi\)
0.342542 + 0.939502i \(0.388712\pi\)
\(90\) 0 0
\(91\) −6.94886 −0.728438
\(92\) 24.7594 2.58134
\(93\) 0 0
\(94\) 13.9881 1.44276
\(95\) −3.73596 −0.383301
\(96\) 0 0
\(97\) −11.2916 −1.14649 −0.573243 − 0.819386i \(-0.694315\pi\)
−0.573243 + 0.819386i \(0.694315\pi\)
\(98\) 2.31568 0.233919
\(99\) 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.1.35 yes 40
3.2 odd 2 inner 8001.2.a.ba.1.6 ✓ 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.ba.1.6 ✓ 40 3.2 odd 2 inner
8001.2.a.ba.1.35 yes 40 1.1 even 1 trivial