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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 565.14
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.14

$q$-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+(-1.18340 - 0.774312i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.800881 + 1.83265i) q^{4} +(-0.623830 + 0.360168i) q^{5} +(1.41201 + 0.0788731i) q^{6} +(-0.126978 - 0.219933i) q^{7} +(0.471276 - 2.78889i) q^{8} +(0.500000 - 0.866025i) q^{9} +(1.01712 + 0.0568152i) q^{10} +0.302685i q^{11} +(-1.60991 - 1.18668i) q^{12} +(-2.56431 + 1.48051i) q^{13} +(-0.0200304 + 0.358590i) q^{14} +(0.360168 - 0.623830i) q^{15} +(-2.71718 + 2.93546i) q^{16} +(0.295442 - 0.511720i) q^{17} +(-1.26228 + 0.637700i) q^{18} +(-2.05586 + 1.18695i) q^{19} +(-1.15967 - 0.854807i) q^{20} +(0.219933 + 0.126978i) q^{21} +(0.234373 - 0.358198i) q^{22} +8.17803 q^{23} +(0.986307 + 2.65089i) q^{24} +(-2.24056 + 3.88076i) q^{25} +(4.18099 + 0.233544i) q^{26} +1.00000i q^{27} +(0.301365 - 0.408847i) q^{28} -5.86277i q^{29} +(-0.909263 + 0.459359i) q^{30} -7.55655 q^{31} +(5.48848 - 1.36989i) q^{32} +(-0.151342 - 0.262133i) q^{33} +(-0.745858 + 0.376807i) q^{34} +(0.158426 + 0.0914672i) q^{35} +(1.98756 + 0.222740i) q^{36} +(1.90108 - 5.77805i) q^{37} +(3.35198 + 0.187237i) q^{38} +(1.48051 - 2.56431i) q^{39} +(0.710473 + 1.90953i) q^{40} +(-4.67321 - 8.09424i) q^{41} +(-0.161948 - 0.320563i) q^{42} +9.95143i q^{43} +(-0.554714 + 0.242415i) q^{44} +0.720337i q^{45} +(-9.67789 - 6.33235i) q^{46} -12.3290 q^{47} +(0.885416 - 3.90077i) q^{48} +(3.46775 - 6.00632i) q^{49} +(5.65640 - 2.85761i) q^{50} +0.590884i q^{51} +(-4.76695 - 3.51377i) q^{52} +(-7.25251 - 4.18724i) q^{53} +(0.774312 - 1.18340i) q^{54} +(-0.109018 - 0.188824i) q^{55} +(-0.673211 + 0.250480i) q^{56} +(1.18695 - 2.05586i) q^{57} +(-4.53961 + 6.93801i) q^{58} +(-7.43507 - 4.29264i) q^{59} +(1.43171 + 0.160447i) q^{60} +(0.245065 - 0.141488i) q^{61} +(8.94244 + 5.85113i) q^{62} -0.253957 q^{63} +(-7.55580 - 2.62867i) q^{64} +(1.06646 - 1.84717i) q^{65} +(-0.0238737 + 0.427395i) q^{66} +(2.91782 - 1.68461i) q^{67} +(1.17442 + 0.131613i) q^{68} +(-7.08238 + 4.08901i) q^{69} +(-0.116657 - 0.230914i) q^{70} +(-2.72615 - 4.72183i) q^{71} +(-2.17961 - 1.80258i) q^{72} -11.4588 q^{73} +(-6.72376 + 5.36573i) q^{74} -4.48112i q^{75} +(-3.82176 - 2.81706i) q^{76} +(0.0665704 - 0.0384345i) q^{77} +(-3.73761 + 1.88824i) q^{78} +(0.740755 + 1.28303i) q^{79} +(0.637797 - 2.80987i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.737181 + 13.1973i) q^{82} +(11.6520 + 6.72729i) q^{83} +(-0.0565662 + 0.504754i) q^{84} +0.425635i q^{85} +(7.70551 - 11.7765i) q^{86} +(2.93138 + 5.07731i) q^{87} +(0.844154 + 0.142648i) q^{88} +(2.22561 - 3.85487i) q^{89} +(0.557766 - 0.852448i) q^{90} +(0.651225 + 0.375985i) q^{91} +(6.54963 + 14.9874i) q^{92} +(6.54417 - 3.77828i) q^{93} +(14.5902 + 9.54652i) q^{94} +(0.855005 - 1.48091i) q^{95} +(-4.06822 + 3.93060i) q^{96} -11.3104 q^{97} +(-8.75452 + 4.42277i) q^{98} +(0.262133 + 0.151342i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.18340 − 0.774312i −0.836792 − 0.547521i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 0.800881 + 1.83265i 0.400440 + 0.916323i
\(5\) −0.623830 + 0.360168i −0.278985 + 0.161072i −0.632964 − 0.774181i \(-0.718162\pi\)
0.353979 + 0.935253i \(0.384829\pi\)
\(6\) 1.41201 + 0.0788731i 0.576452 + 0.0321998i
\(7\) −0.126978 − 0.219933i −0.0479933 − 0.0831269i 0.841031 − 0.540987i \(-0.181949\pi\)
−0.889024 + 0.457860i \(0.848616\pi\)
\(8\) 0.471276 − 2.78889i 0.166621 − 0.986021i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 1.01712 + 0.0568152i 0.321643 + 0.0179665i
\(11\) 0.302685i 0.0912629i 0.998958 + 0.0456315i \(0.0145300\pi\)
−0.998958 + 0.0456315i \(0.985470\pi\)
\(12\) −1.60991 − 1.18668i −0.464740 − 0.342564i
\(13\) −2.56431 + 1.48051i −0.711213 + 0.410619i −0.811510 − 0.584339i \(-0.801354\pi\)
0.100297 + 0.994958i \(0.468021\pi\)
\(14\) −0.0200304 + 0.358590i −0.00535334 + 0.0958373i
\(15\) 0.360168 − 0.623830i 0.0929951 − 0.161072i
\(16\) −2.71718 + 2.93546i −0.679295 + 0.733865i
\(17\) 0.295442 − 0.511720i 0.0716552 − 0.124110i −0.827972 − 0.560770i \(-0.810505\pi\)
0.899627 + 0.436660i \(0.143839\pi\)
\(18\) −1.26228 + 0.637700i −0.297521 + 0.150307i
\(19\) −2.05586 + 1.18695i −0.471647 + 0.272305i −0.716929 − 0.697146i \(-0.754453\pi\)
0.245282 + 0.969452i \(0.421119\pi\)
\(20\) −1.15967 − 0.854807i −0.259311 − 0.191141i
\(21\) 0.219933 + 0.126978i 0.0479933 + 0.0277090i
\(22\) 0.234373 − 0.358198i 0.0499684 − 0.0763681i
\(23\) 8.17803 1.70524 0.852618 − 0.522534i \(-0.175013\pi\)
0.852618 + 0.522534i \(0.175013\pi\)
\(24\) 0.986307 + 2.65089i 0.201329 + 0.541110i
\(25\) −2.24056 + 3.88076i −0.448112 + 0.776152i
\(26\) 4.18099 + 0.233544i 0.819959 + 0.0458018i
\(27\) 1.00000i 0.192450i
\(28\) 0.301365 − 0.408847i 0.0569526 − 0.0772648i
\(29\) − 5.86277i − 1.08869i −0.838862 − 0.544344i \(-0.816778\pi\)
0.838862 − 0.544344i \(-0.183222\pi\)
\(30\) −0.909263 + 0.459359i −0.166008 + 0.0838671i
\(31\) −7.55655 −1.35720 −0.678598 − 0.734509i \(-0.737412\pi\)
−0.678598 + 0.734509i \(0.737412\pi\)
\(32\) 5.48848 − 1.36989i 0.970235 − 0.242164i
\(33\) −0.151342 − 0.262133i −0.0263453 − 0.0456315i
\(34\) −0.745858 + 0.376807i −0.127914 + 0.0646218i
\(35\) 0.158426 + 0.0914672i 0.0267789 + 0.0154608i
\(36\) 1.98756 + 0.222740i 0.331260 + 0.0371233i
\(37\) 1.90108 − 5.77805i 0.312536 − 0.949906i
\(38\) 3.35198 + 0.187237i 0.543763 + 0.0303739i
\(39\) 1.48051 − 2.56431i 0.237071 − 0.410619i
\(40\) 0.710473 + 1.90953i 0.112336 + 0.301923i
\(41\) −4.67321 − 8.09424i −0.729833 − 1.26411i −0.956954 − 0.290241i \(-0.906265\pi\)
0.227121 − 0.973867i \(-0.427069\pi\)
\(42\) −0.161948 − 0.320563i −0.0249892 − 0.0494640i
\(43\) 9.95143i 1.51758i 0.651336 + 0.758789i \(0.274209\pi\)
−0.651336 + 0.758789i \(0.725791\pi\)
\(44\) −0.554714 + 0.242415i −0.0836263 + 0.0365454i
\(45\) 0.720337i 0.107381i
\(46\) −9.67789 − 6.33235i −1.42693 − 0.933654i
\(47\) −12.3290 −1.79837 −0.899187 − 0.437564i \(-0.855841\pi\)
−0.899187 + 0.437564i \(0.855841\pi\)
\(48\) 0.885416 − 3.90077i 0.127799 − 0.563028i
\(49\) 3.46775 − 6.00632i 0.495393 − 0.858046i
\(50\) 5.65640 − 2.85761i 0.799936 − 0.404127i
\(51\) 0.590884i 0.0827403i
\(52\) −4.76695 − 3.51377i −0.661058 − 0.487272i
\(53\) −7.25251 − 4.18724i −0.996209 − 0.575162i −0.0890848 − 0.996024i \(-0.528394\pi\)
−0.907125 + 0.420862i \(0.861728\pi\)
\(54\) 0.774312 − 1.18340i 0.105371 − 0.161041i
\(55\) −0.109018 − 0.188824i −0.0146999 − 0.0254610i
\(56\) −0.673211 + 0.250480i −0.0899616 + 0.0334717i
\(57\) 1.18695 − 2.05586i 0.157216 − 0.272305i
\(58\) −4.53961 + 6.93801i −0.596081 + 0.911006i
\(59\) −7.43507 − 4.29264i −0.967964 − 0.558854i −0.0693490 − 0.997592i \(-0.522092\pi\)
−0.898615 + 0.438738i \(0.855426\pi\)
\(60\) 1.43171 + 0.160447i 0.184833 + 0.0207137i
\(61\) 0.245065 − 0.141488i 0.0313774 − 0.0181157i −0.484229 − 0.874941i \(-0.660900\pi\)
0.515607 + 0.856825i \(0.327567\pi\)
\(62\) 8.94244 + 5.85113i 1.13569 + 0.743094i
\(63\) −0.253957 −0.0319956
\(64\) −7.55580 − 2.62867i −0.944475 − 0.328584i
\(65\) 1.06646 − 1.84717i 0.132279 − 0.229113i
\(66\) −0.0238737 + 0.427395i −0.00293865 + 0.0526087i
\(67\) 2.91782 − 1.68461i 0.356469 − 0.205807i −0.311062 − 0.950390i \(-0.600685\pi\)
0.667531 + 0.744582i \(0.267351\pi\)
\(68\) 1.17442 + 0.131613i 0.142419 + 0.0159604i
\(69\) −7.08238 + 4.08901i −0.852618 + 0.492259i
\(70\) −0.116657 − 0.230914i −0.0139432 − 0.0275995i
\(71\) −2.72615 − 4.72183i −0.323535 − 0.560379i 0.657680 − 0.753297i \(-0.271538\pi\)
−0.981215 + 0.192919i \(0.938205\pi\)
\(72\) −2.17961 − 1.80258i −0.256870 − 0.212436i
\(73\) −11.4588 −1.34115 −0.670576 − 0.741841i \(-0.733953\pi\)
−0.670576 + 0.741841i \(0.733953\pi\)
\(74\) −6.72376 + 5.36573i −0.781621 + 0.623754i
\(75\) − 4.48112i − 0.517435i
\(76\) −3.82176 − 2.81706i −0.438386 − 0.323139i
\(77\) 0.0665704 − 0.0384345i 0.00758640 − 0.00438001i
\(78\) −3.73761 + 1.88824i −0.423202 + 0.213801i
\(79\) 0.740755 + 1.28303i 0.0833414 + 0.144352i 0.904683 − 0.426085i \(-0.140107\pi\)
−0.821342 + 0.570436i \(0.806774\pi\)
\(80\) 0.637797 − 2.80987i 0.0713079 − 0.314153i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −0.737181 + 13.1973i −0.0814080 + 1.45739i
\(83\) 11.6520 + 6.72729i 1.27897 + 0.738416i 0.976660 − 0.214792i \(-0.0689072\pi\)
0.302315 + 0.953208i \(0.402241\pi\)
\(84\) −0.0565662 + 0.504754i −0.00617188 + 0.0550732i
\(85\) 0.425635i 0.0461666i
\(86\) 7.70551 − 11.7765i 0.830907 − 1.26990i
\(87\) 2.93138 + 5.07731i 0.314277 + 0.544344i
\(88\) 0.844154 + 0.142648i 0.0899872 + 0.0152063i
\(89\) 2.22561 − 3.85487i 0.235914 − 0.408616i −0.723624 − 0.690195i \(-0.757525\pi\)
0.959538 + 0.281579i \(0.0908582\pi\)
\(90\) 0.557766 − 0.852448i 0.0587936 − 0.0898559i
\(91\) 0.651225 + 0.375985i 0.0682669 + 0.0394139i
\(92\) 6.54963 + 14.9874i 0.682846 + 1.56255i
\(93\) 6.54417 − 3.77828i 0.678598 − 0.391789i
\(94\) 14.5902 + 9.54652i 1.50486 + 0.984649i
\(95\) 0.855005 − 1.48091i 0.0877216 − 0.151938i
\(96\) −4.06822 + 3.93060i −0.415211 + 0.401165i
\(97\) −11.3104 −1.14840 −0.574199 − 0.818716i \(-0.694686\pi\)
−0.574199 + 0.818716i \(0.694686\pi\)
\(98\) −8.75452 + 4.42277i −0.884340 + 0.446768i
\(99\) 0.262133 + 0.151342i 0.0263453 + 0.0152105i
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.565.14 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.64 yes 152
37.26 even 3 inner 888.2.bh.a.877.64 yes 152
296.285 even 6 inner 888.2.bh.a.877.14 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.14 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.64 yes 152 8.5 even 2 inner
888.2.bh.a.877.14 yes 152 296.285 even 6 inner
888.2.bh.a.877.64 yes 152 37.26 even 3 inner