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Properties

Label 888.2.o.a.517.16
Level $888$
Weight $2$
Character 888.517
Analytic conductor $7.091$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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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] = [888,2,Mod(517,888)] 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("888.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 517.16
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.15

$q$-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+(-1.15841 + 0.811225i) q^{2} -1.00000i q^{3} +(0.683827 - 1.87946i) q^{4} -0.690214 q^{5} +(0.811225 + 1.15841i) q^{6} +3.18564 q^{7} +(0.732517 + 2.73193i) q^{8} -1.00000 q^{9} +(0.799551 - 0.559919i) q^{10} +4.96824i q^{11} +(-1.87946 - 0.683827i) q^{12} -6.93860 q^{13} +(-3.69028 + 2.58428i) q^{14} +0.690214i q^{15} +(-3.06476 - 2.57045i) q^{16} +2.38531i q^{17} +(1.15841 - 0.811225i) q^{18} -1.86165 q^{19} +(-0.471987 + 1.29723i) q^{20} -3.18564i q^{21} +(-4.03036 - 5.75526i) q^{22} +1.90237i q^{23} +(2.73193 - 0.732517i) q^{24} -4.52360 q^{25} +(8.03774 - 5.62877i) q^{26} +1.00000i q^{27} +(2.17843 - 5.98730i) q^{28} +8.83222 q^{29} +(-0.559919 - 0.799551i) q^{30} +1.40457i q^{31} +(5.63547 + 0.491425i) q^{32} +4.96824 q^{33} +(-1.93503 - 2.76317i) q^{34} -2.19878 q^{35} +(-0.683827 + 1.87946i) q^{36} +(1.28333 + 5.94584i) q^{37} +(2.15655 - 1.51021i) q^{38} +6.93860i q^{39} +(-0.505593 - 1.88561i) q^{40} -7.23782 q^{41} +(2.58428 + 3.69028i) q^{42} +1.45646 q^{43} +(9.33763 + 3.39742i) q^{44} +0.690214 q^{45} +(-1.54325 - 2.20372i) q^{46} -7.16139 q^{47} +(-2.57045 + 3.06476i) q^{48} +3.14833 q^{49} +(5.24019 - 3.66966i) q^{50} +2.38531 q^{51} +(-4.74480 + 13.0408i) q^{52} +10.1184i q^{53} +(-0.811225 - 1.15841i) q^{54} -3.42915i q^{55} +(2.33354 + 8.70295i) q^{56} +1.86165i q^{57} +(-10.2313 + 7.16492i) q^{58} -12.8171 q^{59} +(1.29723 + 0.471987i) q^{60} +7.74300 q^{61} +(-1.13943 - 1.62707i) q^{62} -3.18564 q^{63} +(-6.92684 + 4.00236i) q^{64} +4.78912 q^{65} +(-5.75526 + 4.03036i) q^{66} -4.41914i q^{67} +(4.48311 + 1.63114i) q^{68} +1.90237 q^{69} +(2.54708 - 1.78370i) q^{70} +6.10186 q^{71} +(-0.732517 - 2.73193i) q^{72} +9.01569 q^{73} +(-6.31004 - 5.84665i) q^{74} +4.52360i q^{75} +(-1.27304 + 3.49889i) q^{76} +15.8271i q^{77} +(-5.62877 - 8.03774i) q^{78} +7.88044i q^{79} +(2.11534 + 1.77416i) q^{80} +1.00000 q^{81} +(8.38437 - 5.87151i) q^{82} +1.92844i q^{83} +(-5.98730 - 2.17843i) q^{84} -1.64638i q^{85} +(-1.68718 + 1.18152i) q^{86} -8.83222i q^{87} +(-13.5729 + 3.63932i) q^{88} +10.4925i q^{89} +(-0.799551 + 0.559919i) q^{90} -22.1039 q^{91} +(3.57543 + 1.30089i) q^{92} +1.40457 q^{93} +(8.29583 - 5.80950i) q^{94} +1.28493 q^{95} +(0.491425 - 5.63547i) q^{96} +4.29379i q^{97} +(-3.64706 + 2.55401i) q^{98} -4.96824i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15841 + 0.811225i −0.819119 + 0.573623i
\(3\) − 1.00000i − 0.577350i
\(4\) 0.683827 − 1.87946i 0.341913 − 0.939732i
\(5\) −0.690214 −0.308673 −0.154337 − 0.988018i \(-0.549324\pi\)
−0.154337 + 0.988018i \(0.549324\pi\)
\(6\) 0.811225 + 1.15841i 0.331181 + 0.472919i
\(7\) 3.18564 1.20406 0.602030 − 0.798473i \(-0.294359\pi\)
0.602030 + 0.798473i \(0.294359\pi\)
\(8\) 0.732517 + 2.73193i 0.258984 + 0.965882i
\(9\) −1.00000 −0.333333
\(10\) 0.799551 − 0.559919i 0.252840 − 0.177062i
\(11\) 4.96824i 1.49798i 0.662581 + 0.748991i \(0.269461\pi\)
−0.662581 + 0.748991i \(0.730539\pi\)
\(12\) −1.87946 − 0.683827i −0.542554 − 0.197404i
\(13\) −6.93860 −1.92442 −0.962211 − 0.272306i \(-0.912214\pi\)
−0.962211 + 0.272306i \(0.912214\pi\)
\(14\) −3.69028 + 2.58428i −0.986269 + 0.690677i
\(15\) 0.690214i 0.178213i
\(16\) −3.06476 − 2.57045i −0.766191 − 0.642613i
\(17\) 2.38531i 0.578524i 0.957250 + 0.289262i \(0.0934098\pi\)
−0.957250 + 0.289262i \(0.906590\pi\)
\(18\) 1.15841 − 0.811225i 0.273040 − 0.191208i
\(19\) −1.86165 −0.427091 −0.213545 − 0.976933i \(-0.568501\pi\)
−0.213545 + 0.976933i \(0.568501\pi\)
\(20\) −0.471987 + 1.29723i −0.105539 + 0.290070i
\(21\) − 3.18564i − 0.695165i
\(22\) −4.03036 − 5.75526i −0.859276 − 1.22703i
\(23\) 1.90237i 0.396671i 0.980134 + 0.198336i \(0.0635536\pi\)
−0.980134 + 0.198336i \(0.936446\pi\)
\(24\) 2.73193 − 0.732517i 0.557652 − 0.149524i
\(25\) −4.52360 −0.904721
\(26\) 8.03774 − 5.62877i 1.57633 − 1.10389i
\(27\) 1.00000i 0.192450i
\(28\) 2.17843 − 5.98730i 0.411684 − 1.13149i
\(29\) 8.83222 1.64010 0.820051 − 0.572290i \(-0.193945\pi\)
0.820051 + 0.572290i \(0.193945\pi\)
\(30\) −0.559919 − 0.799551i −0.102227 − 0.145977i
\(31\) 1.40457i 0.252269i 0.992013 + 0.126134i \(0.0402571\pi\)
−0.992013 + 0.126134i \(0.959743\pi\)
\(32\) 5.63547 + 0.491425i 0.996219 + 0.0868725i
\(33\) 4.96824 0.864860
\(34\) −1.93503 − 2.76317i −0.331855 − 0.473880i
\(35\) −2.19878 −0.371661
\(36\) −0.683827 + 1.87946i −0.113971 + 0.313244i
\(37\) 1.28333 + 5.94584i 0.210978 + 0.977491i
\(38\) 2.15655 − 1.51021i 0.349838 − 0.244989i
\(39\) 6.93860i 1.11107i
\(40\) −0.505593 − 1.88561i −0.0799413 − 0.298142i
\(41\) −7.23782 −1.13036 −0.565179 − 0.824968i \(-0.691193\pi\)
−0.565179 + 0.824968i \(0.691193\pi\)
\(42\) 2.58428 + 3.69028i 0.398762 + 0.569423i
\(43\) 1.45646 0.222109 0.111054 − 0.993814i \(-0.464577\pi\)
0.111054 + 0.993814i \(0.464577\pi\)
\(44\) 9.33763 + 3.39742i 1.40770 + 0.512180i
\(45\) 0.690214 0.102891
\(46\) −1.54325 − 2.20372i −0.227540 − 0.324921i
\(47\) −7.16139 −1.04460 −0.522298 − 0.852763i \(-0.674925\pi\)
−0.522298 + 0.852763i \(0.674925\pi\)
\(48\) −2.57045 + 3.06476i −0.371013 + 0.442360i
\(49\) 3.14833 0.449762
\(50\) 5.24019 − 3.66966i 0.741074 − 0.518969i
\(51\) 2.38531 0.334011
\(52\) −4.74480 + 13.0408i −0.657985 + 1.80844i
\(53\) 10.1184i 1.38986i 0.719075 + 0.694932i \(0.244566\pi\)
−0.719075 + 0.694932i \(0.755434\pi\)
\(54\) −0.811225 − 1.15841i −0.110394 − 0.157640i
\(55\) − 3.42915i − 0.462387i
\(56\) 2.33354 + 8.70295i 0.311832 + 1.16298i
\(57\) 1.86165i 0.246581i
\(58\) −10.2313 + 7.16492i −1.34344 + 0.940801i
\(59\) −12.8171 −1.66865 −0.834324 − 0.551275i \(-0.814142\pi\)
−0.834324 + 0.551275i \(0.814142\pi\)
\(60\) 1.29723 + 0.471987i 0.167472 + 0.0609332i
\(61\) 7.74300 0.991389 0.495695 − 0.868497i \(-0.334914\pi\)
0.495695 + 0.868497i \(0.334914\pi\)
\(62\) −1.13943 − 1.62707i −0.144707 − 0.206638i
\(63\) −3.18564 −0.401353
\(64\) −6.92684 + 4.00236i −0.865855 + 0.500295i
\(65\) 4.78912 0.594017
\(66\) −5.75526 + 4.03036i −0.708423 + 0.496103i
\(67\) − 4.41914i − 0.539884i −0.962877 − 0.269942i \(-0.912995\pi\)
0.962877 − 0.269942i \(-0.0870045\pi\)
\(68\) 4.48311 + 1.63114i 0.543657 + 0.197805i
\(69\) 1.90237 0.229018
\(70\) 2.54708 − 1.78370i 0.304435 − 0.213193i
\(71\) 6.10186 0.724157 0.362079 − 0.932148i \(-0.382067\pi\)
0.362079 + 0.932148i \(0.382067\pi\)
\(72\) −0.732517 − 2.73193i −0.0863279 − 0.321961i
\(73\) 9.01569 1.05521 0.527604 − 0.849491i \(-0.323091\pi\)
0.527604 + 0.849491i \(0.323091\pi\)
\(74\) −6.31004 − 5.84665i −0.733528 − 0.679660i
\(75\) 4.52360i 0.522341i
\(76\) −1.27304 + 3.49889i −0.146028 + 0.401351i
\(77\) 15.8271i 1.80366i
\(78\) −5.62877 − 8.03774i −0.637333 − 0.910095i
\(79\) 7.88044i 0.886619i 0.896369 + 0.443310i \(0.146196\pi\)
−0.896369 + 0.443310i \(0.853804\pi\)
\(80\) 2.11534 + 1.77416i 0.236502 + 0.198357i
\(81\) 1.00000 0.111111
\(82\) 8.38437 − 5.87151i 0.925898 − 0.648399i
\(83\) 1.92844i 0.211674i 0.994383 + 0.105837i \(0.0337522\pi\)
−0.994383 + 0.105837i \(0.966248\pi\)
\(84\) −5.98730 − 2.17843i −0.653268 − 0.237686i
\(85\) − 1.64638i − 0.178575i
\(86\) −1.68718 + 1.18152i −0.181933 + 0.127407i
\(87\) − 8.83222i − 0.946914i
\(88\) −13.5729 + 3.63932i −1.44687 + 0.387953i
\(89\) 10.4925i 1.11220i 0.831114 + 0.556102i \(0.187704\pi\)
−0.831114 + 0.556102i \(0.812296\pi\)
\(90\) −0.799551 + 0.559919i −0.0842801 + 0.0590207i
\(91\) −22.1039 −2.31712
\(92\) 3.57543 + 1.30089i 0.372764 + 0.135627i
\(93\) 1.40457 0.145648
\(94\) 8.29583 − 5.80950i 0.855649 − 0.599205i
\(95\) 1.28493 0.131831
\(96\) 0.491425 − 5.63547i 0.0501559 − 0.575168i
\(97\) 4.29379i 0.435968i 0.975952 + 0.217984i \(0.0699480\pi\)
−0.975952 + 0.217984i \(0.930052\pi\)
\(98\) −3.64706 + 2.55401i −0.368409 + 0.257994i
\(99\) − 4.96824i − 0.499327i
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 888.2.o.a.517.16 yes 76
4.3 odd 2 3552.2.o.a.2737.66 76
8.3 odd 2 3552.2.o.a.2737.63 76
8.5 even 2 inner 888.2.o.a.517.62 yes 76
37.36 even 2 inner 888.2.o.a.517.61 yes 76
148.147 odd 2 3552.2.o.a.2737.64 76
296.147 odd 2 3552.2.o.a.2737.65 76
296.221 even 2 inner 888.2.o.a.517.15 ✓ 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.15 ✓ 76 296.221 even 2 inner
888.2.o.a.517.16 yes 76 1.1 even 1 trivial
888.2.o.a.517.61 yes 76 37.36 even 2 inner
888.2.o.a.517.62 yes 76 8.5 even 2 inner
3552.2.o.a.2737.63 76 8.3 odd 2
3552.2.o.a.2737.64 76 148.147 odd 2
3552.2.o.a.2737.65 76 296.147 odd 2
3552.2.o.a.2737.66 76 4.3 odd 2