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Properties

Label 888.2.bh.a.565.20
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 565.20
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.20

$q$-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+(-0.989862 - 1.01004i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(-0.0403476 + 1.99959i) q^{4} +(-1.41657 + 0.817855i) q^{5} +(1.36226 + 0.379786i) q^{6} +(2.19997 + 3.81047i) q^{7} +(2.05960 - 1.93857i) q^{8} +(0.500000 - 0.866025i) q^{9} +(2.22827 + 0.621221i) q^{10} +6.07917i q^{11} +(-0.964854 - 1.75187i) q^{12} +(3.64594 - 2.10498i) q^{13} +(1.67104 - 5.99389i) q^{14} +(0.817855 - 1.41657i) q^{15} +(-3.99674 - 0.161357i) q^{16} +(1.03608 - 1.79455i) q^{17} +(-1.36965 + 0.352227i) q^{18} +(2.31643 - 1.33739i) q^{19} +(-1.57822 - 2.86556i) q^{20} +(-3.81047 - 2.19997i) q^{21} +(6.14019 - 6.01754i) q^{22} -8.54149 q^{23} +(-0.814382 + 2.70865i) q^{24} +(-1.16223 + 2.01303i) q^{25} +(-5.73509 - 1.59889i) q^{26} +1.00000i q^{27} +(-7.70815 + 4.24531i) q^{28} +3.90302i q^{29} +(-2.24035 + 0.576142i) q^{30} +3.74280 q^{31} +(3.79325 + 4.19658i) q^{32} +(-3.03959 - 5.26472i) q^{33} +(-2.83814 + 0.729873i) q^{34} +(-6.23282 - 3.59852i) q^{35} +(1.71152 + 1.03474i) q^{36} +(5.58638 - 2.40674i) q^{37} +(-3.64375 - 1.01584i) q^{38} +(-2.10498 + 3.64594i) q^{39} +(-1.33209 + 4.43057i) q^{40} +(-1.83252 - 3.17402i) q^{41} +(1.54978 + 6.02638i) q^{42} +3.95665i q^{43} +(-12.1559 - 0.245280i) q^{44} +1.63571i q^{45} +(8.45489 + 8.62721i) q^{46} +5.58719 q^{47} +(3.54196 - 1.85863i) q^{48} +(-6.17977 + 10.7037i) q^{49} +(3.18368 - 0.818735i) q^{50} +2.07217i q^{51} +(4.06201 + 7.37533i) q^{52} +(-7.50896 - 4.33530i) q^{53} +(1.01004 - 0.989862i) q^{54} +(-4.97188 - 8.61155i) q^{55} +(11.9179 + 3.58324i) q^{56} +(-1.33739 + 2.31643i) q^{57} +(3.94219 - 3.86345i) q^{58} +(6.56890 + 3.79256i) q^{59} +(2.79956 + 1.69253i) q^{60} +(-8.06981 + 4.65911i) q^{61} +(-3.70485 - 3.78036i) q^{62} +4.39995 q^{63} +(0.483908 - 7.98535i) q^{64} +(-3.44315 + 5.96370i) q^{65} +(-2.30879 + 8.28144i) q^{66} +(-11.2653 + 6.50403i) q^{67} +(3.54656 + 2.14415i) q^{68} +(7.39714 - 4.27074i) q^{69} +(2.53499 + 9.85742i) q^{70} +(0.192599 + 0.333592i) q^{71} +(-0.649049 - 2.75295i) q^{72} -3.00807 q^{73} +(-7.96063 - 3.26011i) q^{74} -2.32445i q^{75} +(2.58077 + 4.68587i) q^{76} +(-23.1645 + 13.3740i) q^{77} +(5.76618 - 1.48287i) q^{78} +(-4.37462 - 7.57706i) q^{79} +(5.79362 - 3.04018i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-1.39194 + 4.99276i) q^{82} +(3.78739 + 2.18665i) q^{83} +(4.55280 - 7.53062i) q^{84} +3.38946i q^{85} +(3.99636 - 3.91653i) q^{86} +(-1.95151 - 3.38011i) q^{87} +(11.7849 + 12.5207i) q^{88} +(-5.83444 + 10.1055i) q^{89} +(1.65213 - 1.61913i) q^{90} +(16.0420 + 9.26182i) q^{91} +(0.344628 - 17.0795i) q^{92} +(-3.24136 + 1.87140i) q^{93} +(-5.53054 - 5.64326i) q^{94} +(-2.18758 + 3.78900i) q^{95} +(-5.38334 - 1.73772i) q^{96} +17.6465 q^{97} +(16.9282 - 4.35337i) q^{98} +(5.26472 + 3.03959i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.989862 − 1.01004i −0.699938 − 0.714204i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) −0.0403476 + 1.99959i −0.0201738 + 0.999796i
\(5\) −1.41657 + 0.817855i −0.633508 + 0.365756i −0.782109 − 0.623141i \(-0.785856\pi\)
0.148601 + 0.988897i \(0.452523\pi\)
\(6\) 1.36226 + 0.379786i 0.556142 + 0.155047i
\(7\) 2.19997 + 3.81047i 0.831512 + 1.44022i 0.896839 + 0.442357i \(0.145858\pi\)
−0.0653269 + 0.997864i \(0.520809\pi\)
\(8\) 2.05960 − 1.93857i 0.728179 − 0.685387i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 2.22827 + 0.621221i 0.704641 + 0.196447i
\(11\) 6.07917i 1.83294i 0.400104 + 0.916470i \(0.368974\pi\)
−0.400104 + 0.916470i \(0.631026\pi\)
\(12\) −0.964854 − 1.75187i −0.278529 − 0.505722i
\(13\) 3.64594 − 2.10498i 1.01120 − 0.583818i 0.0996587 − 0.995022i \(-0.468225\pi\)
0.911543 + 0.411204i \(0.134892\pi\)
\(14\) 1.67104 − 5.99389i 0.446604 − 1.60193i
\(15\) 0.817855 − 1.41657i 0.211169 − 0.365756i
\(16\) −3.99674 − 0.161357i −0.999186 − 0.0403394i
\(17\) 1.03608 − 1.79455i 0.251287 − 0.435242i −0.712593 − 0.701577i \(-0.752480\pi\)
0.963880 + 0.266335i \(0.0858129\pi\)
\(18\) −1.36965 + 0.352227i −0.322829 + 0.0830207i
\(19\) 2.31643 − 1.33739i 0.531425 − 0.306818i −0.210172 − 0.977664i \(-0.567402\pi\)
0.741596 + 0.670846i \(0.234069\pi\)
\(20\) −1.57822 − 2.86556i −0.352901 − 0.640758i
\(21\) −3.81047 − 2.19997i −0.831512 − 0.480074i
\(22\) 6.14019 − 6.01754i 1.30909 − 1.28294i
\(23\) −8.54149 −1.78102 −0.890512 − 0.454961i \(-0.849653\pi\)
−0.890512 + 0.454961i \(0.849653\pi\)
\(24\) −0.814382 + 2.70865i −0.166235 + 0.552901i
\(25\) −1.16223 + 2.01303i −0.232445 + 0.402607i
\(26\) −5.73509 − 1.59889i −1.12474 − 0.313568i
\(27\) 1.00000i 0.192450i
\(28\) −7.70815 + 4.24531i −1.45670 + 0.802288i
\(29\) 3.90302i 0.724772i 0.932028 + 0.362386i \(0.118038\pi\)
−0.932028 + 0.362386i \(0.881962\pi\)
\(30\) −2.24035 + 0.576142i −0.409030 + 0.105189i
\(31\) 3.74280 0.672226 0.336113 − 0.941822i \(-0.390888\pi\)
0.336113 + 0.941822i \(0.390888\pi\)
\(32\) 3.79325 + 4.19658i 0.670558 + 0.741857i
\(33\) −3.03959 − 5.26472i −0.529124 − 0.916470i
\(34\) −2.83814 + 0.729873i −0.486737 + 0.125172i
\(35\) −6.23282 − 3.59852i −1.05354 − 0.608261i
\(36\) 1.71152 + 1.03474i 0.285254 + 0.172456i
\(37\) 5.58638 − 2.40674i 0.918395 − 0.395665i
\(38\) −3.64375 − 1.01584i −0.591095 − 0.164792i
\(39\) −2.10498 + 3.64594i −0.337067 + 0.583818i
\(40\) −1.33209 + 4.43057i −0.210623 + 0.700534i
\(41\) −1.83252 − 3.17402i −0.286192 − 0.495699i 0.686705 − 0.726936i \(-0.259056\pi\)
−0.972898 + 0.231237i \(0.925723\pi\)
\(42\) 1.54978 + 6.02638i 0.239136 + 0.929891i
\(43\) 3.95665i 0.603383i 0.953406 + 0.301691i \(0.0975512\pi\)
−0.953406 + 0.301691i \(0.902449\pi\)
\(44\) −12.1559 − 0.245280i −1.83257 − 0.0369773i
\(45\) 1.63571i 0.243837i
\(46\) 8.45489 + 8.62721i 1.24661 + 1.27201i
\(47\) 5.58719 0.814975 0.407488 − 0.913211i \(-0.366405\pi\)
0.407488 + 0.913211i \(0.366405\pi\)
\(48\) 3.54196 − 1.85863i 0.511238 − 0.268270i
\(49\) −6.17977 + 10.7037i −0.882825 + 1.52910i
\(50\) 3.18368 − 0.818735i 0.450240 − 0.115787i
\(51\) 2.07217i 0.290161i
\(52\) 4.06201 + 7.37533i 0.563299 + 1.02277i
\(53\) −7.50896 − 4.33530i −1.03143 − 0.595499i −0.114039 − 0.993476i \(-0.536379\pi\)
−0.917395 + 0.397977i \(0.869712\pi\)
\(54\) 1.01004 − 0.989862i 0.137449 − 0.134703i
\(55\) −4.97188 − 8.61155i −0.670409 − 1.16118i
\(56\) 11.9179 + 3.58324i 1.59260 + 0.478831i
\(57\) −1.33739 + 2.31643i −0.177142 + 0.306818i
\(58\) 3.94219 − 3.86345i 0.517635 − 0.507296i
\(59\) 6.56890 + 3.79256i 0.855198 + 0.493749i 0.862401 − 0.506225i \(-0.168959\pi\)
−0.00720298 + 0.999974i \(0.502293\pi\)
\(60\) 2.79956 + 1.69253i 0.361421 + 0.218505i
\(61\) −8.06981 + 4.65911i −1.03323 + 0.596538i −0.917909 − 0.396790i \(-0.870124\pi\)
−0.115324 + 0.993328i \(0.536791\pi\)
\(62\) −3.70485 − 3.78036i −0.470516 − 0.480106i
\(63\) 4.39995 0.554341
\(64\) 0.483908 − 7.98535i 0.0604885 − 0.998169i
\(65\) −3.44315 + 5.96370i −0.427070 + 0.739706i
\(66\) −2.30879 + 8.28144i −0.284192 + 1.01937i
\(67\) −11.2653 + 6.50403i −1.37628 + 0.794594i −0.991709 − 0.128503i \(-0.958983\pi\)
−0.384568 + 0.923097i \(0.625649\pi\)
\(68\) 3.54656 + 2.14415i 0.430084 + 0.260016i
\(69\) 7.39714 − 4.27074i 0.890512 − 0.514137i
\(70\) 2.53499 + 9.85742i 0.302990 + 1.17819i
\(71\) 0.192599 + 0.333592i 0.0228573 + 0.0395901i 0.877228 − 0.480074i \(-0.159390\pi\)
−0.854370 + 0.519664i \(0.826057\pi\)
\(72\) −0.649049 − 2.75295i −0.0764911 − 0.324438i
\(73\) −3.00807 −0.352068 −0.176034 − 0.984384i \(-0.556327\pi\)
−0.176034 + 0.984384i \(0.556327\pi\)
\(74\) −7.96063 − 3.26011i −0.925405 − 0.378980i
\(75\) − 2.32445i − 0.268404i
\(76\) 2.58077 + 4.68587i 0.296035 + 0.537506i
\(77\) −23.1645 + 13.3740i −2.63984 + 1.52411i
\(78\) 5.76618 − 1.48287i 0.652891 − 0.167901i
\(79\) −4.37462 − 7.57706i −0.492183 − 0.852486i 0.507776 − 0.861489i \(-0.330468\pi\)
−0.999959 + 0.00900285i \(0.997134\pi\)
\(80\) 5.79362 − 3.04018i 0.647747 − 0.339903i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −1.39194 + 4.99276i −0.153714 + 0.551358i
\(83\) 3.78739 + 2.18665i 0.415720 + 0.240016i 0.693244 − 0.720702i \(-0.256181\pi\)
−0.277524 + 0.960719i \(0.589514\pi\)
\(84\) 4.55280 − 7.53062i 0.496751 − 0.821658i
\(85\) 3.38946i 0.367639i
\(86\) 3.99636 − 3.91653i 0.430938 − 0.422330i
\(87\) −1.95151 − 3.38011i −0.209224 − 0.362386i
\(88\) 11.7849 + 12.5207i 1.25627 + 1.33471i
\(89\) −5.83444 + 10.1055i −0.618449 + 1.07119i 0.371320 + 0.928505i \(0.378905\pi\)
−0.989769 + 0.142680i \(0.954428\pi\)
\(90\) 1.65213 − 1.61913i 0.174150 − 0.170671i
\(91\) 16.0420 + 9.26182i 1.68165 + 0.970903i
\(92\) 0.344628 − 17.0795i 0.0359300 − 1.78066i
\(93\) −3.24136 + 1.87140i −0.336113 + 0.194055i
\(94\) −5.53054 − 5.64326i −0.570432 − 0.582058i
\(95\) −2.18758 + 3.78900i −0.224441 + 0.388744i
\(96\) −5.38334 − 1.73772i −0.549435 − 0.177355i
\(97\) 17.6465 1.79173 0.895866 − 0.444324i \(-0.146556\pi\)
0.895866 + 0.444324i \(0.146556\pi\)
\(98\) 16.9282 − 4.35337i 1.71001 − 0.439756i
\(99\) 5.26472 + 3.03959i 0.529124 + 0.305490i
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.565.20 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.70 yes 152
37.26 even 3 inner 888.2.bh.a.877.70 yes 152
296.285 even 6 inner 888.2.bh.a.877.20 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.20 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.70 yes 152 8.5 even 2 inner
888.2.bh.a.877.20 yes 152 296.285 even 6 inner
888.2.bh.a.877.70 yes 152 37.26 even 3 inner