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Properties

Label 888.2.bh.a.565.19
Level $888$
Weight $2$
Character 888.565
Analytic conductor $7.091$
Analytic rank $0$
Dimension $152$
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(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), 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: \(152\)
Relative dimension: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 565.19
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.19

$q$-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+(-1.06938 + 0.925428i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.287166 - 1.97928i) q^{4} +(-2.81225 + 1.62365i) q^{5} +(0.463400 - 1.33614i) q^{6} +(0.617779 + 1.07002i) q^{7} +(1.52459 + 2.38236i) q^{8} +(0.500000 - 0.866025i) q^{9} +(1.50480 - 4.33884i) q^{10} +3.14154i q^{11} +(0.740945 + 1.85769i) q^{12} +(1.67404 - 0.966510i) q^{13} +(-1.65087 - 0.572557i) q^{14} +(1.62365 - 2.81225i) q^{15} +(-3.83507 - 1.13676i) q^{16} +(-2.17609 + 3.76911i) q^{17} +(0.266752 + 1.38883i) q^{18} +(-5.07639 + 2.93086i) q^{19} +(2.40607 + 6.03247i) q^{20} +(-1.07002 - 0.617779i) q^{21} +(-2.90727 - 3.35952i) q^{22} +3.62594 q^{23} +(-2.51151 - 1.30089i) q^{24} +(2.77249 - 4.80210i) q^{25} +(-0.895762 + 2.58278i) q^{26} +1.00000i q^{27} +(2.29528 - 0.915480i) q^{28} +4.87092i q^{29} +(0.866225 + 4.50995i) q^{30} -1.80997 q^{31} +(5.15316 - 2.33345i) q^{32} +(-1.57077 - 2.72066i) q^{33} +(-1.16095 - 6.04444i) q^{34} +(-3.47469 - 2.00612i) q^{35} +(-1.57052 - 1.23833i) q^{36} +(0.322857 + 6.07419i) q^{37} +(2.71632 - 7.83205i) q^{38} +(-0.966510 + 1.67404i) q^{39} +(-8.15564 - 4.22439i) q^{40} +(-6.02714 - 10.4393i) q^{41} +(1.71598 - 0.329587i) q^{42} -9.17544i q^{43} +(6.21798 + 0.902145i) q^{44} +3.24730i q^{45} +(-3.87753 + 3.35555i) q^{46} +5.92595 q^{47} +(3.88965 - 0.933071i) q^{48} +(2.73670 - 4.74010i) q^{49} +(1.47914 + 7.70103i) q^{50} -4.35219i q^{51} +(-1.43226 - 3.59095i) q^{52} +(-10.9646 - 6.33042i) q^{53} +(-0.925428 - 1.06938i) q^{54} +(-5.10077 - 8.83480i) q^{55} +(-1.60732 + 3.10312i) q^{56} +(2.93086 - 5.07639i) q^{57} +(-4.50769 - 5.20889i) q^{58} +(-5.39595 - 3.11535i) q^{59} +(-5.09996 - 4.02124i) q^{60} +(2.06087 - 1.18984i) q^{61} +(1.93556 - 1.67500i) q^{62} +1.23556 q^{63} +(-3.35127 + 7.26423i) q^{64} +(-3.13855 + 5.43613i) q^{65} +(4.19753 + 1.45579i) q^{66} +(7.48437 - 4.32110i) q^{67} +(6.83520 + 5.38945i) q^{68} +(-3.14016 + 1.81297i) q^{69} +(5.57230 - 1.07027i) q^{70} +(4.97558 + 8.61796i) q^{71} +(2.82548 - 0.129152i) q^{72} -10.4800 q^{73} +(-5.96648 - 6.19686i) q^{74} +5.54499i q^{75} +(4.34321 + 10.8892i) q^{76} +(-3.36153 + 1.94078i) q^{77} +(-0.515637 - 2.68463i) q^{78} +(-0.158267 - 0.274127i) q^{79} +(12.6309 - 3.02997i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(16.1062 + 5.58596i) q^{82} +(-13.9318 - 8.04354i) q^{83} +(-1.53003 + 1.94047i) q^{84} -14.1329i q^{85} +(8.49120 + 9.81207i) q^{86} +(-2.43546 - 4.21834i) q^{87} +(-7.48428 + 4.78956i) q^{88} +(-3.05437 + 5.29032i) q^{89} +(-3.00515 - 3.47262i) q^{90} +(2.06838 + 1.19418i) q^{91} +(1.04125 - 7.17675i) q^{92} +(1.56748 - 0.904987i) q^{93} +(-6.33711 + 5.48404i) q^{94} +(9.51739 - 16.4846i) q^{95} +(-3.29604 + 4.59740i) q^{96} -14.8277 q^{97} +(1.46004 + 7.60161i) q^{98} +(2.72066 + 1.57077i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06938 + 0.925428i −0.756169 + 0.654376i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 0.287166 − 1.97928i 0.143583 − 0.989638i
\(5\) −2.81225 + 1.62365i −1.25768 + 0.726119i −0.972622 − 0.232392i \(-0.925345\pi\)
−0.285054 + 0.958512i \(0.592011\pi\)
\(6\) 0.463400 − 1.33614i 0.189182 − 0.545475i
\(7\) 0.617779 + 1.07002i 0.233498 + 0.404431i 0.958835 − 0.283963i \(-0.0916493\pi\)
−0.725337 + 0.688394i \(0.758316\pi\)
\(8\) 1.52459 + 2.38236i 0.539023 + 0.842291i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 1.50480 − 4.33884i 0.475860 − 1.37206i
\(11\) 3.14154i 0.947211i 0.880737 + 0.473606i \(0.157048\pi\)
−0.880737 + 0.473606i \(0.842952\pi\)
\(12\) 0.740945 + 1.85769i 0.213892 + 0.536268i
\(13\) 1.67404 − 0.966510i 0.464296 − 0.268062i −0.249553 − 0.968361i \(-0.580284\pi\)
0.713849 + 0.700300i \(0.246950\pi\)
\(14\) −1.65087 − 0.572557i −0.441214 − 0.153022i
\(15\) 1.62365 − 2.81225i 0.419225 − 0.726119i
\(16\) −3.83507 − 1.13676i −0.958768 − 0.284191i
\(17\) −2.17609 + 3.76911i −0.527780 + 0.914143i 0.471695 + 0.881762i \(0.343642\pi\)
−0.999476 + 0.0323808i \(0.989691\pi\)
\(18\) 0.266752 + 1.38883i 0.0628740 + 0.327350i
\(19\) −5.07639 + 2.93086i −1.16460 + 0.672385i −0.952403 − 0.304841i \(-0.901397\pi\)
−0.212202 + 0.977226i \(0.568063\pi\)
\(20\) 2.40607 + 6.03247i 0.538015 + 1.34890i
\(21\) −1.07002 − 0.617779i −0.233498 − 0.134810i
\(22\) −2.90727 − 3.35952i −0.619833 − 0.716252i
\(23\) 3.62594 0.756062 0.378031 − 0.925793i \(-0.376601\pi\)
0.378031 + 0.925793i \(0.376601\pi\)
\(24\) −2.51151 − 1.30089i −0.512660 − 0.265543i
\(25\) 2.77249 − 4.80210i 0.554499 − 0.960420i
\(26\) −0.895762 + 2.58278i −0.175673 + 0.506524i
\(27\) 1.00000i 0.192450i
\(28\) 2.29528 − 0.915480i 0.433767 − 0.173010i
\(29\) 4.87092i 0.904508i 0.891889 + 0.452254i \(0.149380\pi\)
−0.891889 + 0.452254i \(0.850620\pi\)
\(30\) 0.866225 + 4.50995i 0.158150 + 0.823400i
\(31\) −1.80997 −0.325081 −0.162541 − 0.986702i \(-0.551969\pi\)
−0.162541 + 0.986702i \(0.551969\pi\)
\(32\) 5.15316 − 2.33345i 0.910958 − 0.412499i
\(33\) −1.57077 − 2.72066i −0.273436 − 0.473606i
\(34\) −1.16095 − 6.04444i −0.199102 − 1.03661i
\(35\) −3.47469 − 2.00612i −0.587331 − 0.339095i
\(36\) −1.57052 − 1.23833i −0.261753 − 0.206389i
\(37\) 0.322857 + 6.07419i 0.0530773 + 0.998590i
\(38\) 2.71632 − 7.83205i 0.440645 − 1.27053i
\(39\) −0.966510 + 1.67404i −0.154765 + 0.268062i
\(40\) −8.15564 − 4.22439i −1.28952 − 0.667934i
\(41\) −6.02714 − 10.4393i −0.941281 − 1.63035i −0.763031 − 0.646362i \(-0.776290\pi\)
−0.178251 − 0.983985i \(-0.557044\pi\)
\(42\) 1.71598 − 0.329587i 0.264781 − 0.0508564i
\(43\) − 9.17544i − 1.39924i −0.714515 − 0.699620i \(-0.753352\pi\)
0.714515 − 0.699620i \(-0.246648\pi\)
\(44\) 6.21798 + 0.902145i 0.937396 + 0.136003i
\(45\) 3.24730i 0.484080i
\(46\) −3.87753 + 3.35555i −0.571710 + 0.494749i
\(47\) 5.92595 0.864388 0.432194 − 0.901781i \(-0.357740\pi\)
0.432194 + 0.901781i \(0.357740\pi\)
\(48\) 3.88965 − 0.933071i 0.561423 − 0.134677i
\(49\) 2.73670 − 4.74010i 0.390957 − 0.677157i
\(50\) 1.47914 + 7.70103i 0.209181 + 1.08909i
\(51\) − 4.35219i − 0.609428i
\(52\) −1.43226 − 3.59095i −0.198619 − 0.497975i
\(53\) −10.9646 − 6.33042i −1.50610 − 0.869550i −0.999975 − 0.00709222i \(-0.997742\pi\)
−0.506129 − 0.862458i \(-0.668924\pi\)
\(54\) −0.925428 − 1.06938i −0.125935 − 0.145525i
\(55\) −5.10077 − 8.83480i −0.687788 − 1.19128i
\(56\) −1.60732 + 3.10312i −0.214788 + 0.414671i
\(57\) 2.93086 − 5.07639i 0.388202 − 0.672385i
\(58\) −4.50769 − 5.20889i −0.591889 − 0.683961i
\(59\) −5.39595 − 3.11535i −0.702493 − 0.405585i 0.105782 − 0.994389i \(-0.466265\pi\)
−0.808275 + 0.588805i \(0.799599\pi\)
\(60\) −5.09996 − 4.02124i −0.658402 − 0.519140i
\(61\) 2.06087 − 1.18984i 0.263867 − 0.152344i −0.362230 − 0.932089i \(-0.617985\pi\)
0.626097 + 0.779745i \(0.284651\pi\)
\(62\) 1.93556 − 1.67500i 0.245816 − 0.212725i
\(63\) 1.23556 0.155666
\(64\) −3.35127 + 7.26423i −0.418909 + 0.908028i
\(65\) −3.13855 + 5.43613i −0.389289 + 0.674269i
\(66\) 4.19753 + 1.45579i 0.516680 + 0.179196i
\(67\) 7.48437 − 4.32110i 0.914361 − 0.527907i 0.0325296 − 0.999471i \(-0.489644\pi\)
0.881832 + 0.471564i \(0.156310\pi\)
\(68\) 6.83520 + 5.38945i 0.828890 + 0.653567i
\(69\) −3.14016 + 1.81297i −0.378031 + 0.218256i
\(70\) 5.57230 − 1.07027i 0.666017 − 0.127922i
\(71\) 4.97558 + 8.61796i 0.590493 + 1.02276i 0.994166 + 0.107860i \(0.0343999\pi\)
−0.403673 + 0.914903i \(0.632267\pi\)
\(72\) 2.82548 − 0.129152i 0.332986 − 0.0152207i
\(73\) −10.4800 −1.22659 −0.613296 − 0.789853i \(-0.710157\pi\)
−0.613296 + 0.789853i \(0.710157\pi\)
\(74\) −5.96648 − 6.19686i −0.693589 − 0.720371i
\(75\) 5.54499i 0.640280i
\(76\) 4.34321 + 10.8892i 0.498200 + 1.24908i
\(77\) −3.36153 + 1.94078i −0.383082 + 0.221172i
\(78\) −0.515637 − 2.68463i −0.0583844 − 0.303975i
\(79\) −0.158267 − 0.274127i −0.0178064 − 0.0308416i 0.856985 − 0.515342i \(-0.172335\pi\)
−0.874791 + 0.484500i \(0.839002\pi\)
\(80\) 12.6309 − 3.02997i 1.41218 − 0.338760i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 16.1062 + 5.58596i 1.77863 + 0.616866i
\(83\) −13.9318 − 8.04354i −1.52922 − 0.882893i −0.999395 − 0.0347832i \(-0.988926\pi\)
−0.529821 − 0.848110i \(-0.677741\pi\)
\(84\) −1.53003 + 1.94047i −0.166940 + 0.211723i
\(85\) − 14.1329i − 1.53293i
\(86\) 8.49120 + 9.81207i 0.915630 + 1.05806i
\(87\) −2.43546 − 4.21834i −0.261109 − 0.452254i
\(88\) −7.48428 + 4.78956i −0.797827 + 0.510568i
\(89\) −3.05437 + 5.29032i −0.323762 + 0.560773i −0.981261 − 0.192683i \(-0.938281\pi\)
0.657499 + 0.753456i \(0.271615\pi\)
\(90\) −3.00515 − 3.47262i −0.316770 − 0.366046i
\(91\) 2.06838 + 1.19418i 0.216825 + 0.125184i
\(92\) 1.04125 − 7.17675i 0.108558 − 0.748228i
\(93\) 1.56748 − 0.904987i 0.162541 − 0.0938428i
\(94\) −6.33711 + 5.48404i −0.653623 + 0.565635i
\(95\) 9.51739 − 16.4846i 0.976463 − 1.69128i
\(96\) −3.29604 + 4.59740i −0.336401 + 0.469220i
\(97\) −14.8277 −1.50553 −0.752763 − 0.658292i \(-0.771279\pi\)
−0.752763 + 0.658292i \(0.771279\pi\)
\(98\) 1.46004 + 7.60161i 0.147486 + 0.767878i
\(99\) 2.72066 + 1.57077i 0.273436 + 0.157869i
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.bh.a.565.19 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.34 yes 152
37.26 even 3 inner 888.2.bh.a.877.34 yes 152
296.285 even 6 inner 888.2.bh.a.877.19 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.19 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.34 yes 152 8.5 even 2 inner
888.2.bh.a.877.19 yes 152 296.285 even 6 inner
888.2.bh.a.877.34 yes 152 37.26 even 3 inner