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Properties

Label 888.2.bh.a.565.12
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 565.12
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.12

$q$-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+(-1.23751 - 0.684515i) q^{2} +(0.866025 - 0.500000i) q^{3} +(1.06288 + 1.69419i) q^{4} +(-1.80687 + 1.04320i) q^{5} +(-1.41398 + 0.0259491i) q^{6} +(0.240738 + 0.416970i) q^{7} +(-0.155625 - 2.82414i) q^{8} +(0.500000 - 0.866025i) q^{9} +(2.95011 - 0.0541400i) q^{10} -3.12789i q^{11} +(1.76758 + 0.935775i) q^{12} +(4.51572 - 2.60715i) q^{13} +(-0.0124939 - 0.680795i) q^{14} +(-1.04320 + 1.80687i) q^{15} +(-1.74058 + 3.60144i) q^{16} +(-0.517128 + 0.895691i) q^{17} +(-1.21156 + 0.729460i) q^{18} +(-5.81878 + 3.35948i) q^{19} +(-3.68786 - 1.95240i) q^{20} +(0.416970 + 0.240738i) q^{21} +(-2.14109 + 3.87081i) q^{22} +3.68137 q^{23} +(-1.54685 - 2.36797i) q^{24} +(-0.323480 + 0.560283i) q^{25} +(-7.37290 + 0.135307i) q^{26} -1.00000i q^{27} +(-0.450553 + 0.851044i) q^{28} -6.01256i q^{29} +(2.52780 - 1.52194i) q^{30} +10.4325 q^{31} +(4.61923 - 3.26538i) q^{32} +(-1.56395 - 2.70884i) q^{33} +(1.25307 - 0.754448i) q^{34} +(-0.869964 - 0.502274i) q^{35} +(1.99865 - 0.0733828i) q^{36} +(5.60165 + 2.37097i) q^{37} +(9.50043 - 0.174351i) q^{38} +(2.60715 - 4.51572i) q^{39} +(3.22733 + 4.94051i) q^{40} +(-4.96703 - 8.60314i) q^{41} +(-0.351217 - 0.583338i) q^{42} -10.0225i q^{43} +(5.29926 - 3.32457i) q^{44} +2.08639i q^{45} +(-4.55574 - 2.51995i) q^{46} +3.74815 q^{47} +(0.293333 + 3.98923i) q^{48} +(3.38409 - 5.86142i) q^{49} +(0.783833 - 0.471931i) q^{50} +1.03426i q^{51} +(9.21668 + 4.87942i) q^{52} +(-3.65751 - 2.11167i) q^{53} +(-0.684515 + 1.23751i) q^{54} +(3.26301 + 5.65170i) q^{55} +(1.14012 - 0.744769i) q^{56} +(-3.35948 + 5.81878i) q^{57} +(-4.11569 + 7.44063i) q^{58} +(10.7515 + 6.20736i) q^{59} +(-4.16998 + 0.153105i) q^{60} +(9.47458 - 5.47015i) q^{61} +(-12.9103 - 7.14120i) q^{62} +0.481475 q^{63} +(-7.95156 + 0.879013i) q^{64} +(-5.43955 + 9.42158i) q^{65} +(0.0811660 + 4.42276i) q^{66} +(-5.34946 + 3.08851i) q^{67} +(-2.06712 + 0.0758965i) q^{68} +(3.18816 - 1.84068i) q^{69} +(0.732778 + 1.21707i) q^{70} +(-1.06552 - 1.84554i) q^{71} +(-2.52359 - 1.27730i) q^{72} +2.58154 q^{73} +(-5.30916 - 6.76852i) q^{74} +0.646960i q^{75} +(-11.8763 - 6.28743i) q^{76} +(1.30424 - 0.753002i) q^{77} +(-6.31747 + 3.80363i) q^{78} +(-5.73615 - 9.93531i) q^{79} +(-0.612009 - 8.32311i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.257780 + 14.0465i) q^{82} +(0.969104 + 0.559513i) q^{83} +(0.0353320 + 0.962303i) q^{84} -2.15786i q^{85} +(-6.86057 + 12.4030i) q^{86} +(-3.00628 - 5.20703i) q^{87} +(-8.83362 + 0.486777i) q^{88} +(3.86855 - 6.70053i) q^{89} +(1.42817 - 2.58194i) q^{90} +(2.17421 + 1.25528i) q^{91} +(3.91284 + 6.23695i) q^{92} +(9.03480 - 5.21624i) q^{93} +(-4.63839 - 2.56567i) q^{94} +(7.00919 - 12.1403i) q^{95} +(2.36768 - 5.13752i) q^{96} +9.40788 q^{97} +(-8.20009 + 4.93712i) q^{98} +(-2.70884 - 1.56395i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23751 − 0.684515i −0.875054 − 0.484025i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 1.06288 + 1.69419i 0.531439 + 0.847097i
\(5\) −1.80687 + 1.04320i −0.808057 + 0.466532i −0.846281 − 0.532737i \(-0.821163\pi\)
0.0382237 + 0.999269i \(0.487830\pi\)
\(6\) −1.41398 + 0.0259491i −0.577253 + 0.0105937i
\(7\) 0.240738 + 0.416970i 0.0909903 + 0.157600i 0.907928 − 0.419126i \(-0.137663\pi\)
−0.816938 + 0.576726i \(0.804330\pi\)
\(8\) −0.155625 − 2.82414i −0.0550216 − 0.998485i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 2.95011 − 0.0541400i 0.932907 − 0.0171206i
\(11\) − 3.12789i − 0.943095i −0.881841 − 0.471548i \(-0.843696\pi\)
0.881841 − 0.471548i \(-0.156304\pi\)
\(12\) 1.76758 + 0.935775i 0.510255 + 0.270135i
\(13\) 4.51572 − 2.60715i 1.25244 − 0.723094i 0.280843 − 0.959754i \(-0.409386\pi\)
0.971593 + 0.236659i \(0.0760525\pi\)
\(14\) −0.0124939 − 0.680795i −0.00333912 − 0.181950i
\(15\) −1.04320 + 1.80687i −0.269352 + 0.466532i
\(16\) −1.74058 + 3.60144i −0.435145 + 0.900360i
\(17\) −0.517128 + 0.895691i −0.125422 + 0.217237i −0.921898 − 0.387433i \(-0.873362\pi\)
0.796476 + 0.604670i \(0.206695\pi\)
\(18\) −1.21156 + 0.729460i −0.285568 + 0.171935i
\(19\) −5.81878 + 3.35948i −1.33492 + 0.770716i −0.986049 − 0.166454i \(-0.946768\pi\)
−0.348871 + 0.937171i \(0.613435\pi\)
\(20\) −3.68786 − 1.95240i −0.824631 − 0.436569i
\(21\) 0.416970 + 0.240738i 0.0909903 + 0.0525333i
\(22\) −2.14109 + 3.87081i −0.456482 + 0.825259i
\(23\) 3.68137 0.767618 0.383809 − 0.923413i \(-0.374612\pi\)
0.383809 + 0.923413i \(0.374612\pi\)
\(24\) −1.54685 − 2.36797i −0.315749 − 0.483359i
\(25\) −0.323480 + 0.560283i −0.0646960 + 0.112057i
\(26\) −7.37290 + 0.135307i −1.44595 + 0.0265358i
\(27\) − 1.00000i − 0.192450i
\(28\) −0.450553 + 0.851044i −0.0851465 + 0.160832i
\(29\) − 6.01256i − 1.11651i −0.829671 − 0.558253i \(-0.811472\pi\)
0.829671 − 0.558253i \(-0.188528\pi\)
\(30\) 2.52780 − 1.52194i 0.461511 − 0.277867i
\(31\) 10.4325 1.87373 0.936865 − 0.349691i \(-0.113713\pi\)
0.936865 + 0.349691i \(0.113713\pi\)
\(32\) 4.61923 − 3.26538i 0.816573 − 0.577243i
\(33\) −1.56395 − 2.70884i −0.272248 − 0.471548i
\(34\) 1.25307 − 0.754448i 0.214899 − 0.129387i
\(35\) −0.869964 − 0.502274i −0.147051 − 0.0848998i
\(36\) 1.99865 − 0.0733828i 0.333109 − 0.0122305i
\(37\) 5.60165 + 2.37097i 0.920906 + 0.389784i
\(38\) 9.50043 − 0.174351i 1.54117 − 0.0282834i
\(39\) 2.60715 − 4.51572i 0.417479 − 0.723094i
\(40\) 3.22733 + 4.94051i 0.510286 + 0.781164i
\(41\) −4.96703 − 8.60314i −0.775719 − 1.34358i −0.934389 − 0.356253i \(-0.884054\pi\)
0.158670 − 0.987332i \(-0.449279\pi\)
\(42\) −0.351217 − 0.583338i −0.0541940 − 0.0900111i
\(43\) − 10.0225i − 1.52842i −0.644967 − 0.764211i \(-0.723129\pi\)
0.644967 − 0.764211i \(-0.276871\pi\)
\(44\) 5.29926 − 3.32457i 0.798893 − 0.501198i
\(45\) 2.08639i 0.311021i
\(46\) −4.55574 − 2.51995i −0.671707 − 0.371547i
\(47\) 3.74815 0.546724 0.273362 − 0.961911i \(-0.411864\pi\)
0.273362 + 0.961911i \(0.411864\pi\)
\(48\) 0.293333 + 3.98923i 0.0423390 + 0.575796i
\(49\) 3.38409 − 5.86142i 0.483442 − 0.837345i
\(50\) 0.783833 − 0.471931i 0.110851 − 0.0667412i
\(51\) 1.03426i 0.144825i
\(52\) 9.21668 + 4.87942i 1.27812 + 0.676654i
\(53\) −3.65751 − 2.11167i −0.502398 − 0.290060i 0.227305 − 0.973824i \(-0.427008\pi\)
−0.729703 + 0.683764i \(0.760342\pi\)
\(54\) −0.684515 + 1.23751i −0.0931507 + 0.168404i
\(55\) 3.26301 + 5.65170i 0.439984 + 0.762075i
\(56\) 1.14012 − 0.744769i 0.152355 − 0.0995239i
\(57\) −3.35948 + 5.81878i −0.444973 + 0.770716i
\(58\) −4.11569 + 7.44063i −0.540417 + 0.977002i
\(59\) 10.7515 + 6.20736i 1.39972 + 0.808130i 0.994363 − 0.106028i \(-0.0338132\pi\)
0.405359 + 0.914158i \(0.367147\pi\)
\(60\) −4.16998 + 0.153105i −0.538342 + 0.0197658i
\(61\) 9.47458 − 5.47015i 1.21310 − 0.700381i 0.249663 − 0.968333i \(-0.419680\pi\)
0.963432 + 0.267951i \(0.0863467\pi\)
\(62\) −12.9103 − 7.14120i −1.63962 − 0.906933i
\(63\) 0.481475 0.0606602
\(64\) −7.95156 + 0.879013i −0.993945 + 0.109877i
\(65\) −5.43955 + 9.42158i −0.674693 + 1.16860i
\(66\) 0.0811660 + 4.42276i 0.00999085 + 0.544405i
\(67\) −5.34946 + 3.08851i −0.653540 + 0.377322i −0.789811 − 0.613350i \(-0.789822\pi\)
0.136271 + 0.990672i \(0.456488\pi\)
\(68\) −2.06712 + 0.0758965i −0.250675 + 0.00920381i
\(69\) 3.18816 − 1.84068i 0.383809 − 0.221592i
\(70\) 0.732778 + 1.21707i 0.0875837 + 0.145468i
\(71\) −1.06552 − 1.84554i −0.126454 − 0.219025i 0.795846 − 0.605499i \(-0.207026\pi\)
−0.922300 + 0.386474i \(0.873693\pi\)
\(72\) −2.52359 − 1.27730i −0.297408 − 0.150531i
\(73\) 2.58154 0.302146 0.151073 − 0.988523i \(-0.451727\pi\)
0.151073 + 0.988523i \(0.451727\pi\)
\(74\) −5.30916 − 6.76852i −0.617177 − 0.786824i
\(75\) 0.646960i 0.0747045i
\(76\) −11.8763 − 6.28743i −1.36230 − 0.721217i
\(77\) 1.30424 − 0.753002i 0.148632 − 0.0858125i
\(78\) −6.31747 + 3.80363i −0.715312 + 0.430676i
\(79\) −5.73615 − 9.93531i −0.645368 − 1.11781i −0.984216 − 0.176969i \(-0.943371\pi\)
0.338849 − 0.940841i \(-0.389962\pi\)
\(80\) −0.612009 − 8.32311i −0.0684247 − 0.930552i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0.257780 + 14.0465i 0.0284670 + 1.55118i
\(83\) 0.969104 + 0.559513i 0.106373 + 0.0614145i 0.552243 − 0.833683i \(-0.313772\pi\)
−0.445870 + 0.895098i \(0.647106\pi\)
\(84\) 0.0353320 + 0.962303i 0.00385504 + 0.104996i
\(85\) − 2.15786i − 0.234053i
\(86\) −6.86057 + 12.4030i −0.739795 + 1.33745i
\(87\) −3.00628 − 5.20703i −0.322307 − 0.558253i
\(88\) −8.83362 + 0.486777i −0.941667 + 0.0518907i
\(89\) 3.86855 − 6.70053i 0.410066 − 0.710255i −0.584831 − 0.811155i \(-0.698839\pi\)
0.994897 + 0.100900i \(0.0321724\pi\)
\(90\) 1.42817 − 2.58194i 0.150542 − 0.272160i
\(91\) 2.17421 + 1.25528i 0.227919 + 0.131589i
\(92\) 3.91284 + 6.23695i 0.407942 + 0.650247i
\(93\) 9.03480 − 5.21624i 0.936865 − 0.540899i
\(94\) −4.63839 − 2.56567i −0.478413 − 0.264628i
\(95\) 7.00919 − 12.1403i 0.719128 − 1.24557i
\(96\) 2.36768 − 5.13752i 0.241651 − 0.524346i
\(97\) 9.40788 0.955225 0.477613 − 0.878570i \(-0.341502\pi\)
0.477613 + 0.878570i \(0.341502\pi\)
\(98\) −8.20009 + 4.93712i −0.828334 + 0.498724i
\(99\) −2.70884 − 1.56395i −0.272248 − 0.157183i
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.565.12 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.63 yes 152
37.26 even 3 inner 888.2.bh.a.877.63 yes 152
296.285 even 6 inner 888.2.bh.a.877.12 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.12 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.63 yes 152 8.5 even 2 inner
888.2.bh.a.877.12 yes 152 296.285 even 6 inner
888.2.bh.a.877.63 yes 152 37.26 even 3 inner