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Properties

Label 888.2.o.a.517.1
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 517.1
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.2

$q$-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+(-1.40736 - 0.139090i) q^{2} +1.00000i q^{3} +(1.96131 + 0.391499i) q^{4} -2.38070 q^{5} +(0.139090 - 1.40736i) q^{6} +1.41343 q^{7} +(-2.70581 - 0.823778i) q^{8} -1.00000 q^{9} +(3.35049 + 0.331132i) q^{10} +0.100135i q^{11} +(-0.391499 + 1.96131i) q^{12} +2.71581 q^{13} +(-1.98920 - 0.196594i) q^{14} -2.38070i q^{15} +(3.69346 + 1.53570i) q^{16} -1.95399i q^{17} +(1.40736 + 0.139090i) q^{18} +0.786700 q^{19} +(-4.66928 - 0.932041i) q^{20} +1.41343i q^{21} +(0.0139278 - 0.140926i) q^{22} +7.52595i q^{23} +(0.823778 - 2.70581i) q^{24} +0.667717 q^{25} +(-3.82211 - 0.377743i) q^{26} -1.00000i q^{27} +(2.77217 + 0.553356i) q^{28} +5.39988 q^{29} +(-0.331132 + 3.35049i) q^{30} +5.05930i q^{31} +(-4.98441 - 2.67500i) q^{32} -0.100135 q^{33} +(-0.271782 + 2.74997i) q^{34} -3.36494 q^{35} +(-1.96131 - 0.391499i) q^{36} +(-5.24981 + 3.07238i) q^{37} +(-1.10717 - 0.109422i) q^{38} +2.71581i q^{39} +(6.44170 + 1.96117i) q^{40} +2.40509 q^{41} +(0.196594 - 1.98920i) q^{42} +6.12142 q^{43} +(-0.0392028 + 0.196395i) q^{44} +2.38070 q^{45} +(1.04679 - 10.5917i) q^{46} -5.12803 q^{47} +(-1.53570 + 3.69346i) q^{48} -5.00222 q^{49} +(-0.939716 - 0.0928730i) q^{50} +1.95399 q^{51} +(5.32654 + 1.06324i) q^{52} +12.9258i q^{53} +(-0.139090 + 1.40736i) q^{54} -0.238391i q^{55} +(-3.82446 - 1.16435i) q^{56} +0.786700i q^{57} +(-7.59956 - 0.751071i) q^{58} +0.981983 q^{59} +(0.932041 - 4.66928i) q^{60} -10.6552 q^{61} +(0.703700 - 7.12025i) q^{62} -1.41343 q^{63} +(6.64278 + 4.45797i) q^{64} -6.46551 q^{65} +(0.140926 + 0.0139278i) q^{66} +5.10005i q^{67} +(0.764988 - 3.83238i) q^{68} -7.52595 q^{69} +(4.73568 + 0.468031i) q^{70} +0.189214 q^{71} +(2.70581 + 0.823778i) q^{72} -2.73557 q^{73} +(7.81569 - 3.59373i) q^{74} +0.667717i q^{75} +(1.54296 + 0.307993i) q^{76} +0.141534i q^{77} +(0.377743 - 3.82211i) q^{78} +3.46459i q^{79} +(-8.79300 - 3.65604i) q^{80} +1.00000 q^{81} +(-3.38483 - 0.334525i) q^{82} -9.45378i q^{83} +(-0.553356 + 2.77217i) q^{84} +4.65187i q^{85} +(-8.61503 - 0.851430i) q^{86} +5.39988i q^{87} +(0.0824890 - 0.270946i) q^{88} +4.92087i q^{89} +(-3.35049 - 0.331132i) q^{90} +3.83860 q^{91} +(-2.94640 + 14.7607i) q^{92} -5.05930 q^{93} +(7.21696 + 0.713259i) q^{94} -1.87290 q^{95} +(2.67500 - 4.98441i) q^{96} +12.7211i q^{97} +(7.03991 + 0.695760i) q^{98} -0.100135i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40736 − 0.139090i −0.995152 − 0.0983517i
\(3\) 1.00000i 0.577350i
\(4\) 1.96131 + 0.391499i 0.980654 + 0.195750i
\(5\) −2.38070 −1.06468 −0.532340 − 0.846531i \(-0.678687\pi\)
−0.532340 + 0.846531i \(0.678687\pi\)
\(6\) 0.139090 − 1.40736i 0.0567834 − 0.574551i
\(7\) 1.41343 0.534226 0.267113 − 0.963665i \(-0.413930\pi\)
0.267113 + 0.963665i \(0.413930\pi\)
\(8\) −2.70581 − 0.823778i −0.956647 − 0.291250i
\(9\) −1.00000 −0.333333
\(10\) 3.35049 + 0.331132i 1.05952 + 0.104713i
\(11\) 0.100135i 0.0301918i 0.999886 + 0.0150959i \(0.00480536\pi\)
−0.999886 + 0.0150959i \(0.995195\pi\)
\(12\) −0.391499 + 1.96131i −0.113016 + 0.566181i
\(13\) 2.71581 0.753230 0.376615 − 0.926370i \(-0.377088\pi\)
0.376615 + 0.926370i \(0.377088\pi\)
\(14\) −1.98920 − 0.196594i −0.531636 − 0.0525420i
\(15\) − 2.38070i − 0.614693i
\(16\) 3.69346 + 1.53570i 0.923364 + 0.383925i
\(17\) − 1.95399i − 0.473913i −0.971520 − 0.236957i \(-0.923850\pi\)
0.971520 − 0.236957i \(-0.0761499\pi\)
\(18\) 1.40736 + 0.139090i 0.331717 + 0.0327839i
\(19\) 0.786700 0.180481 0.0902407 − 0.995920i \(-0.471236\pi\)
0.0902407 + 0.995920i \(0.471236\pi\)
\(20\) −4.66928 − 0.932041i −1.04408 − 0.208411i
\(21\) 1.41343i 0.308435i
\(22\) 0.0139278 − 0.140926i 0.00296942 − 0.0300454i
\(23\) 7.52595i 1.56927i 0.619959 + 0.784634i \(0.287149\pi\)
−0.619959 + 0.784634i \(0.712851\pi\)
\(24\) 0.823778 − 2.70581i 0.168153 − 0.552320i
\(25\) 0.667717 0.133543
\(26\) −3.82211 − 0.377743i −0.749578 − 0.0740814i
\(27\) − 1.00000i − 0.192450i
\(28\) 2.77217 + 0.553356i 0.523891 + 0.104575i
\(29\) 5.39988 1.00273 0.501367 − 0.865235i \(-0.332831\pi\)
0.501367 + 0.865235i \(0.332831\pi\)
\(30\) −0.331132 + 3.35049i −0.0604561 + 0.611713i
\(31\) 5.05930i 0.908678i 0.890829 + 0.454339i \(0.150124\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(32\) −4.98441 − 2.67500i −0.881128 − 0.472878i
\(33\) −0.100135 −0.0174312
\(34\) −0.271782 + 2.74997i −0.0466102 + 0.471616i
\(35\) −3.36494 −0.568779
\(36\) −1.96131 − 0.391499i −0.326885 − 0.0652499i
\(37\) −5.24981 + 3.07238i −0.863063 + 0.505096i
\(38\) −1.10717 − 0.109422i −0.179606 − 0.0177507i
\(39\) 2.71581i 0.434877i
\(40\) 6.44170 + 1.96117i 1.01852 + 0.310088i
\(41\) 2.40509 0.375613 0.187806 − 0.982206i \(-0.439862\pi\)
0.187806 + 0.982206i \(0.439862\pi\)
\(42\) 0.196594 − 1.98920i 0.0303351 − 0.306940i
\(43\) 6.12142 0.933508 0.466754 − 0.884387i \(-0.345423\pi\)
0.466754 + 0.884387i \(0.345423\pi\)
\(44\) −0.0392028 + 0.196395i −0.00591004 + 0.0296077i
\(45\) 2.38070 0.354893
\(46\) 1.04679 − 10.5917i 0.154340 − 1.56166i
\(47\) −5.12803 −0.747999 −0.374000 − 0.927429i \(-0.622014\pi\)
−0.374000 + 0.927429i \(0.622014\pi\)
\(48\) −1.53570 + 3.69346i −0.221659 + 0.533105i
\(49\) −5.00222 −0.714603
\(50\) −0.939716 − 0.0928730i −0.132896 − 0.0131342i
\(51\) 1.95399 0.273614
\(52\) 5.32654 + 1.06324i 0.738658 + 0.147444i
\(53\) 12.9258i 1.77549i 0.460337 + 0.887744i \(0.347729\pi\)
−0.460337 + 0.887744i \(0.652271\pi\)
\(54\) −0.139090 + 1.40736i −0.0189278 + 0.191517i
\(55\) − 0.238391i − 0.0321446i
\(56\) −3.82446 − 1.16435i −0.511065 − 0.155593i
\(57\) 0.786700i 0.104201i
\(58\) −7.59956 − 0.751071i −0.997872 − 0.0986205i
\(59\) 0.981983 0.127843 0.0639216 − 0.997955i \(-0.479639\pi\)
0.0639216 + 0.997955i \(0.479639\pi\)
\(60\) 0.932041 − 4.66928i 0.120326 − 0.602801i
\(61\) −10.6552 −1.36426 −0.682132 − 0.731229i \(-0.738947\pi\)
−0.682132 + 0.731229i \(0.738947\pi\)
\(62\) 0.703700 − 7.12025i 0.0893700 − 0.904272i
\(63\) −1.41343 −0.178075
\(64\) 6.64278 + 4.45797i 0.830347 + 0.557246i
\(65\) −6.46551 −0.801948
\(66\) 0.140926 + 0.0139278i 0.0173467 + 0.00171439i
\(67\) 5.10005i 0.623070i 0.950235 + 0.311535i \(0.100843\pi\)
−0.950235 + 0.311535i \(0.899157\pi\)
\(68\) 0.764988 − 3.83238i 0.0927684 − 0.464745i
\(69\) −7.52595 −0.906017
\(70\) 4.73568 + 0.468031i 0.566022 + 0.0559404i
\(71\) 0.189214 0.0224556 0.0112278 − 0.999937i \(-0.496426\pi\)
0.0112278 + 0.999937i \(0.496426\pi\)
\(72\) 2.70581 + 0.823778i 0.318882 + 0.0970832i
\(73\) −2.73557 −0.320174 −0.160087 − 0.987103i \(-0.551177\pi\)
−0.160087 + 0.987103i \(0.551177\pi\)
\(74\) 7.81569 − 3.59373i 0.908556 − 0.417763i
\(75\) 0.667717i 0.0771013i
\(76\) 1.54296 + 0.307993i 0.176990 + 0.0353292i
\(77\) 0.141534i 0.0161292i
\(78\) 0.377743 − 3.82211i 0.0427709 − 0.432769i
\(79\) 3.46459i 0.389797i 0.980823 + 0.194898i \(0.0624377\pi\)
−0.980823 + 0.194898i \(0.937562\pi\)
\(80\) −8.79300 − 3.65604i −0.983087 − 0.408758i
\(81\) 1.00000 0.111111
\(82\) −3.38483 − 0.334525i −0.373791 − 0.0369421i
\(83\) − 9.45378i − 1.03769i −0.854869 − 0.518844i \(-0.826363\pi\)
0.854869 − 0.518844i \(-0.173637\pi\)
\(84\) −0.553356 + 2.77217i −0.0603761 + 0.302468i
\(85\) 4.65187i 0.504566i
\(86\) −8.61503 − 0.851430i −0.928982 − 0.0918121i
\(87\) 5.39988i 0.578928i
\(88\) 0.0824890 − 0.270946i 0.00879335 − 0.0288829i
\(89\) 4.92087i 0.521611i 0.965391 + 0.260805i \(0.0839881\pi\)
−0.965391 + 0.260805i \(0.916012\pi\)
\(90\) −3.35049 − 0.331132i −0.353173 − 0.0349044i
\(91\) 3.83860 0.402395
\(92\) −2.94640 + 14.7607i −0.307184 + 1.53891i
\(93\) −5.05930 −0.524625
\(94\) 7.21696 + 0.713259i 0.744373 + 0.0735670i
\(95\) −1.87290 −0.192155
\(96\) 2.67500 − 4.98441i 0.273017 − 0.508719i
\(97\) 12.7211i 1.29164i 0.763491 + 0.645818i \(0.223484\pi\)
−0.763491 + 0.645818i \(0.776516\pi\)
\(98\) 7.03991 + 0.695760i 0.711138 + 0.0702824i
\(99\) − 0.100135i − 0.0100639i
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.517.1 ✓ 76
4.3 odd 2 3552.2.o.a.2737.49 76
8.3 odd 2 3552.2.o.a.2737.38 76
8.5 even 2 inner 888.2.o.a.517.75 yes 76
37.36 even 2 inner 888.2.o.a.517.76 yes 76
148.147 odd 2 3552.2.o.a.2737.37 76
296.147 odd 2 3552.2.o.a.2737.50 76
296.221 even 2 inner 888.2.o.a.517.2 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.1 ✓ 76 1.1 even 1 trivial
888.2.o.a.517.2 yes 76 296.221 even 2 inner
888.2.o.a.517.75 yes 76 8.5 even 2 inner
888.2.o.a.517.76 yes 76 37.36 even 2 inner
3552.2.o.a.2737.37 76 148.147 odd 2
3552.2.o.a.2737.38 76 8.3 odd 2
3552.2.o.a.2737.49 76 4.3 odd 2
3552.2.o.a.2737.50 76 296.147 odd 2