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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 565.18
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.18

$q$-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+(-1.08179 + 0.910895i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.340540 - 1.97079i) q^{4} +(-0.713620 + 0.412009i) q^{5} +(-0.481410 + 1.32975i) q^{6} +(0.715669 + 1.23957i) q^{7} +(1.42680 + 2.44218i) q^{8} +(0.500000 - 0.866025i) q^{9} +(0.396690 - 1.09574i) q^{10} +2.82147i q^{11} +(-0.690481 - 1.87703i) q^{12} +(1.59720 - 0.922145i) q^{13} +(-1.90333 - 0.689060i) q^{14} +(-0.412009 + 0.713620i) q^{15} +(-3.76807 - 1.34227i) q^{16} +(-3.08899 + 5.35029i) q^{17} +(0.247963 + 1.39231i) q^{18} +(4.26151 - 2.46038i) q^{19} +(0.568969 + 1.54670i) q^{20} +(1.23957 + 0.715669i) q^{21} +(-2.57007 - 3.05224i) q^{22} +0.650522 q^{23} +(2.45673 + 1.40159i) q^{24} +(-2.16050 + 3.74209i) q^{25} +(-0.887860 + 2.45245i) q^{26} -1.00000i q^{27} +(2.68666 - 0.988312i) q^{28} -5.71190i q^{29} +(-0.204326 - 1.14728i) q^{30} -3.01433 q^{31} +(5.29892 - 1.98026i) q^{32} +(1.41074 + 2.44347i) q^{33} +(-1.53191 - 8.60164i) q^{34} +(-1.02143 - 0.589723i) q^{35} +(-1.53649 - 1.28031i) q^{36} +(1.86624 + 5.78940i) q^{37} +(-2.36891 + 6.54340i) q^{38} +(0.922145 - 1.59720i) q^{39} +(-2.02439 - 1.15494i) q^{40} +(5.74216 + 9.94572i) q^{41} +(-1.99286 + 0.354919i) q^{42} +9.05473i q^{43} +(5.56055 + 0.960824i) q^{44} +0.824018i q^{45} +(-0.703728 + 0.592557i) q^{46} -7.04142 q^{47} +(-3.93437 + 0.721595i) q^{48} +(2.47564 - 4.28793i) q^{49} +(-1.07145 - 6.01614i) q^{50} +6.17798i q^{51} +(-1.27345 - 3.46179i) q^{52} +(0.880186 + 0.508176i) q^{53} +(0.910895 + 1.08179i) q^{54} +(-1.16247 - 2.01346i) q^{55} +(-2.00615 + 3.51641i) q^{56} +(2.46038 - 4.26151i) q^{57} +(5.20294 + 6.17907i) q^{58} +(11.5959 + 6.69490i) q^{59} +(1.26609 + 1.05500i) q^{60} +(11.2885 - 6.51740i) q^{61} +(3.26088 - 2.74574i) q^{62} +1.43134 q^{63} +(-3.92851 + 6.96899i) q^{64} +(-0.759864 + 1.31612i) q^{65} +(-3.75187 - 1.35829i) q^{66} +(2.33209 - 1.34643i) q^{67} +(9.49240 + 7.90976i) q^{68} +(0.563369 - 0.325261i) q^{69} +(1.64215 - 0.292460i) q^{70} +(7.97988 + 13.8216i) q^{71} +(2.82839 - 0.0145498i) q^{72} -1.38365 q^{73} +(-7.29241 - 4.56297i) q^{74} +4.32100i q^{75} +(-3.39770 - 9.23641i) q^{76} +(-3.49743 + 2.01924i) q^{77} +(0.457316 + 2.56782i) q^{78} +(-3.81826 - 6.61342i) q^{79} +(3.24199 - 0.594607i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-15.2713 - 5.52867i) q^{82} +(-2.90046 - 1.67458i) q^{83} +(1.83256 - 2.19923i) q^{84} -5.09077i q^{85} +(-8.24792 - 9.79532i) q^{86} +(-2.85595 - 4.94665i) q^{87} +(-6.89056 + 4.02567i) q^{88} +(-6.73325 + 11.6623i) q^{89} +(-0.750594 - 0.891414i) q^{90} +(2.28613 + 1.31990i) q^{91} +(0.221529 - 1.28205i) q^{92} +(-2.61049 + 1.50717i) q^{93} +(7.61734 - 6.41400i) q^{94} +(-2.02740 + 3.51156i) q^{95} +(3.59887 - 4.36442i) q^{96} +8.82110 q^{97} +(1.22773 + 6.89369i) q^{98} +(2.44347 + 1.41074i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08179 + 0.910895i −0.764941 + 0.644100i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 0.340540 − 1.97079i 0.170270 − 0.985397i
\(5\) −0.713620 + 0.412009i −0.319141 + 0.184256i −0.651009 − 0.759070i \(-0.725654\pi\)
0.331869 + 0.943326i \(0.392321\pi\)
\(6\) −0.481410 + 1.32975i −0.196535 + 0.542870i
\(7\) 0.715669 + 1.23957i 0.270497 + 0.468515i 0.968989 − 0.247103i \(-0.0794785\pi\)
−0.698492 + 0.715618i \(0.746145\pi\)
\(8\) 1.42680 + 2.44218i 0.504448 + 0.863442i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 0.396690 − 1.09574i 0.125445 − 0.346503i
\(11\) 2.82147i 0.850707i 0.905027 + 0.425353i \(0.139850\pi\)
−0.905027 + 0.425353i \(0.860150\pi\)
\(12\) −0.690481 − 1.87703i −0.199325 − 0.541851i
\(13\) 1.59720 − 0.922145i 0.442984 − 0.255757i −0.261878 − 0.965101i \(-0.584342\pi\)
0.704863 + 0.709344i \(0.251009\pi\)
\(14\) −1.90333 − 0.689060i −0.508685 − 0.184159i
\(15\) −0.412009 + 0.713620i −0.106380 + 0.184256i
\(16\) −3.76807 − 1.34227i −0.942016 − 0.335567i
\(17\) −3.08899 + 5.35029i −0.749191 + 1.29764i 0.199021 + 0.979995i \(0.436224\pi\)
−0.948211 + 0.317641i \(0.897109\pi\)
\(18\) 0.247963 + 1.39231i 0.0584455 + 0.328170i
\(19\) 4.26151 − 2.46038i 0.977657 − 0.564450i 0.0760949 − 0.997101i \(-0.475755\pi\)
0.901562 + 0.432650i \(0.142421\pi\)
\(20\) 0.568969 + 1.54670i 0.127225 + 0.345854i
\(21\) 1.23957 + 0.715669i 0.270497 + 0.156172i
\(22\) −2.57007 − 3.05224i −0.547940 − 0.650740i
\(23\) 0.650522 0.135643 0.0678216 − 0.997697i \(-0.478395\pi\)
0.0678216 + 0.997697i \(0.478395\pi\)
\(24\) 2.45673 + 1.40159i 0.501478 + 0.286099i
\(25\) −2.16050 + 3.74209i −0.432100 + 0.748418i
\(26\) −0.887860 + 2.45245i −0.174124 + 0.480965i
\(27\) − 1.00000i − 0.192450i
\(28\) 2.68666 − 0.988312i 0.507731 − 0.186773i
\(29\) − 5.71190i − 1.06067i −0.847787 − 0.530336i \(-0.822066\pi\)
0.847787 − 0.530336i \(-0.177934\pi\)
\(30\) −0.204326 − 1.14728i −0.0373047 − 0.209464i
\(31\) −3.01433 −0.541390 −0.270695 − 0.962665i \(-0.587254\pi\)
−0.270695 + 0.962665i \(0.587254\pi\)
\(32\) 5.29892 − 1.98026i 0.936726 − 0.350064i
\(33\) 1.41074 + 2.44347i 0.245578 + 0.425353i
\(34\) −1.53191 − 8.60164i −0.262721 − 1.47517i
\(35\) −1.02143 − 0.589723i −0.172653 − 0.0996815i
\(36\) −1.53649 − 1.28031i −0.256081 − 0.213386i
\(37\) 1.86624 + 5.78940i 0.306807 + 0.951772i
\(38\) −2.36891 + 6.54340i −0.384287 + 1.06148i
\(39\) 0.922145 − 1.59720i 0.147661 − 0.255757i
\(40\) −2.02439 − 1.15494i −0.320084 − 0.182612i
\(41\) 5.74216 + 9.94572i 0.896775 + 1.55326i 0.831592 + 0.555387i \(0.187430\pi\)
0.0651835 + 0.997873i \(0.479237\pi\)
\(42\) −1.99286 + 0.354919i −0.307505 + 0.0547652i
\(43\) 9.05473i 1.38083i 0.723412 + 0.690417i \(0.242573\pi\)
−0.723412 + 0.690417i \(0.757427\pi\)
\(44\) 5.56055 + 0.960824i 0.838284 + 0.144850i
\(45\) 0.824018i 0.122837i
\(46\) −0.703728 + 0.592557i −0.103759 + 0.0873678i
\(47\) −7.04142 −1.02710 −0.513548 − 0.858061i \(-0.671669\pi\)
−0.513548 + 0.858061i \(0.671669\pi\)
\(48\) −3.93437 + 0.721595i −0.567878 + 0.104153i
\(49\) 2.47564 − 4.28793i 0.353662 − 0.612561i
\(50\) −1.07145 − 6.01614i −0.151526 − 0.850811i
\(51\) 6.17798i 0.865091i
\(52\) −1.27345 − 3.46179i −0.176595 − 0.480063i
\(53\) 0.880186 + 0.508176i 0.120903 + 0.0698033i 0.559232 − 0.829011i \(-0.311096\pi\)
−0.438329 + 0.898815i \(0.644430\pi\)
\(54\) 0.910895 + 1.08179i 0.123957 + 0.147213i
\(55\) −1.16247 − 2.01346i −0.156748 − 0.271495i
\(56\) −2.00615 + 3.51641i −0.268084 + 0.469900i
\(57\) 2.46038 − 4.26151i 0.325886 − 0.564450i
\(58\) 5.20294 + 6.17907i 0.683180 + 0.811352i
\(59\) 11.5959 + 6.69490i 1.50966 + 0.871602i 0.999937 + 0.0112624i \(0.00358501\pi\)
0.509722 + 0.860339i \(0.329748\pi\)
\(60\) 1.26609 + 1.05500i 0.163452 + 0.136200i
\(61\) 11.2885 − 6.51740i 1.44534 − 0.834467i 0.447141 − 0.894464i \(-0.352442\pi\)
0.998199 + 0.0599964i \(0.0191089\pi\)
\(62\) 3.26088 − 2.74574i 0.414132 − 0.348710i
\(63\) 1.43134 0.180332
\(64\) −3.92851 + 6.96899i −0.491064 + 0.871124i
\(65\) −0.759864 + 1.31612i −0.0942495 + 0.163245i
\(66\) −3.75187 − 1.35829i −0.461823 − 0.167193i
\(67\) 2.33209 − 1.34643i 0.284910 − 0.164493i −0.350734 − 0.936475i \(-0.614068\pi\)
0.635644 + 0.771982i \(0.280735\pi\)
\(68\) 9.49240 + 7.90976i 1.15112 + 0.959199i
\(69\) 0.563369 − 0.325261i 0.0678216 − 0.0391568i
\(70\) 1.64215 − 0.292460i 0.196275 − 0.0349556i
\(71\) 7.97988 + 13.8216i 0.947038 + 1.64032i 0.751619 + 0.659598i \(0.229273\pi\)
0.195419 + 0.980720i \(0.437393\pi\)
\(72\) 2.82839 − 0.0145498i 0.333329 − 0.00171471i
\(73\) −1.38365 −0.161943 −0.0809717 − 0.996716i \(-0.525802\pi\)
−0.0809717 + 0.996716i \(0.525802\pi\)
\(74\) −7.29241 − 4.56297i −0.847726 − 0.530435i
\(75\) 4.32100i 0.498946i
\(76\) −3.39770 − 9.23641i −0.389743 − 1.05949i
\(77\) −3.49743 + 2.01924i −0.398569 + 0.230114i
\(78\) 0.457316 + 2.56782i 0.0517809 + 0.290748i
\(79\) −3.81826 − 6.61342i −0.429588 − 0.744068i 0.567249 − 0.823546i \(-0.308008\pi\)
−0.996837 + 0.0794788i \(0.974674\pi\)
\(80\) 3.24199 − 0.594607i 0.362466 − 0.0664791i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −15.2713 − 5.52867i −1.68644 − 0.610540i
\(83\) −2.90046 − 1.67458i −0.318367 − 0.183809i 0.332297 − 0.943175i \(-0.392176\pi\)
−0.650665 + 0.759365i \(0.725510\pi\)
\(84\) 1.83256 − 2.19923i 0.199949 − 0.239956i
\(85\) − 5.09077i − 0.552171i
\(86\) −8.24792 − 9.79532i −0.889396 − 1.05626i
\(87\) −2.85595 − 4.94665i −0.306190 − 0.530336i
\(88\) −6.89056 + 4.02567i −0.734536 + 0.429138i
\(89\) −6.73325 + 11.6623i −0.713723 + 1.23621i 0.249726 + 0.968316i \(0.419659\pi\)
−0.963450 + 0.267889i \(0.913674\pi\)
\(90\) −0.750594 − 0.891414i −0.0791195 − 0.0939633i
\(91\) 2.28613 + 1.31990i 0.239652 + 0.138363i
\(92\) 0.221529 − 1.28205i 0.0230959 − 0.133662i
\(93\) −2.61049 + 1.50717i −0.270695 + 0.156286i
\(94\) 7.61734 − 6.41400i 0.785669 − 0.661553i
\(95\) −2.02740 + 3.51156i −0.208007 + 0.360278i
\(96\) 3.59887 − 4.36442i 0.367308 − 0.445441i
\(97\) 8.82110 0.895647 0.447824 − 0.894122i \(-0.352199\pi\)
0.447824 + 0.894122i \(0.352199\pi\)
\(98\) 1.22773 + 6.89369i 0.124020 + 0.696367i
\(99\) 2.44347 + 1.41074i 0.245578 + 0.141784i
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.565.18 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.35 yes 152
37.26 even 3 inner 888.2.bh.a.877.35 yes 152
296.285 even 6 inner 888.2.bh.a.877.18 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.18 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.35 yes 152 8.5 even 2 inner
888.2.bh.a.877.18 yes 152 296.285 even 6 inner
888.2.bh.a.877.35 yes 152 37.26 even 3 inner