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Properties

Label 888.2.bh.a.565.17
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 565.17
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.17

$q$-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+(-1.08808 - 0.903371i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.367840 + 1.96588i) q^{4} +(-1.28795 + 0.743600i) q^{5} +(-1.39399 - 0.238302i) q^{6} +(-2.36259 - 4.09212i) q^{7} +(1.37568 - 2.47134i) q^{8} +(0.500000 - 0.866025i) q^{9} +(2.07314 + 0.354403i) q^{10} +2.68067i q^{11} +(1.30150 + 1.51858i) q^{12} +(-2.62862 + 1.51764i) q^{13} +(-1.12602 + 6.58685i) q^{14} +(-0.743600 + 1.28795i) q^{15} +(-3.72939 + 1.44626i) q^{16} +(1.31986 - 2.28607i) q^{17} +(-1.32638 + 0.490620i) q^{18} +(-1.48362 + 0.856569i) q^{19} +(-1.93559 - 2.25844i) q^{20} +(-4.09212 - 2.36259i) q^{21} +(2.42164 - 2.91679i) q^{22} +1.08203 q^{23} +(-0.0442918 - 2.82808i) q^{24} +(-1.39412 + 2.41468i) q^{25} +(4.23114 + 0.723312i) q^{26} -1.00000i q^{27} +(7.17557 - 6.14981i) q^{28} +7.94145i q^{29} +(1.97260 - 0.729650i) q^{30} -0.642621 q^{31} +(5.36439 + 1.79537i) q^{32} +(1.34034 + 2.32153i) q^{33} +(-3.50128 + 1.29510i) q^{34} +(6.08580 + 3.51364i) q^{35} +(1.88642 + 0.664382i) q^{36} +(-6.07556 + 0.295871i) q^{37} +(2.38810 + 0.408245i) q^{38} +(-1.51764 + 2.62862i) q^{39} +(0.0658707 + 4.20592i) q^{40} +(3.79677 + 6.57619i) q^{41} +(2.31826 + 6.26739i) q^{42} -5.62167i q^{43} +(-5.26989 + 0.986059i) q^{44} +1.48720i q^{45} +(-1.17734 - 0.977477i) q^{46} -10.2321 q^{47} +(-2.50661 + 3.11719i) q^{48} +(-7.66363 + 13.2738i) q^{49} +(3.69827 - 1.36796i) q^{50} -2.63972i q^{51} +(-3.95041 - 4.60932i) q^{52} +(-4.76853 - 2.75311i) q^{53} +(-0.903371 + 1.08808i) q^{54} +(-1.99335 - 3.45258i) q^{55} +(-13.3632 + 0.209286i) q^{56} +(-0.856569 + 1.48362i) q^{57} +(7.17408 - 8.64094i) q^{58} +(2.14528 + 1.23858i) q^{59} +(-2.80549 - 0.988070i) q^{60} +(-2.95293 + 1.70487i) q^{61} +(0.699223 + 0.580525i) q^{62} -4.72517 q^{63} +(-4.21500 - 6.79954i) q^{64} +(2.25703 - 3.90929i) q^{65} +(0.638810 - 3.73683i) q^{66} +(-10.5804 + 6.10860i) q^{67} +(4.97963 + 1.75378i) q^{68} +(0.937067 - 0.541016i) q^{69} +(-3.44772 - 9.32086i) q^{70} +(7.71919 + 13.3700i) q^{71} +(-1.45240 - 2.42704i) q^{72} -10.6753 q^{73} +(6.87798 + 5.16656i) q^{74} +2.78823i q^{75} +(-2.22965 - 2.60155i) q^{76} +(10.9696 - 6.33332i) q^{77} +(4.02593 - 1.48917i) q^{78} +(-6.98127 - 12.0919i) q^{79} +(3.72784 - 4.63589i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(1.80956 - 10.5853i) q^{82} +(5.98889 + 3.45769i) q^{83} +(3.13932 - 8.91368i) q^{84} +3.92580i q^{85} +(-5.07845 + 6.11683i) q^{86} +(3.97073 + 6.87750i) q^{87} +(6.62484 + 3.68775i) q^{88} +(-2.91925 + 5.05630i) q^{89} +(1.34349 - 1.61819i) q^{90} +(12.4207 + 7.17109i) q^{91} +(0.398015 + 2.12715i) q^{92} +(-0.556526 + 0.321310i) q^{93} +(11.1333 + 9.24335i) q^{94} +(1.27389 - 2.20644i) q^{95} +(5.54338 - 1.12735i) q^{96} +3.65462 q^{97} +(20.3298 - 7.51986i) q^{98} +(2.32153 + 1.34034i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08808 − 0.903371i −0.769389 − 0.638780i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 0.367840 + 1.96588i 0.183920 + 0.982941i
\(5\) −1.28795 + 0.743600i −0.575990 + 0.332548i −0.759538 − 0.650463i \(-0.774575\pi\)
0.183548 + 0.983011i \(0.441242\pi\)
\(6\) −1.39399 − 0.238302i −0.569095 − 0.0972865i
\(7\) −2.36259 − 4.09212i −0.892974 − 1.54668i −0.836293 − 0.548283i \(-0.815282\pi\)
−0.0566809 − 0.998392i \(-0.518052\pi\)
\(8\) 1.37568 − 2.47134i 0.486377 − 0.873749i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 2.07314 + 0.354403i 0.655586 + 0.112072i
\(11\) 2.68067i 0.808253i 0.914703 + 0.404127i \(0.132424\pi\)
−0.914703 + 0.404127i \(0.867576\pi\)
\(12\) 1.30150 + 1.51858i 0.375711 + 0.438377i
\(13\) −2.62862 + 1.51764i −0.729049 + 0.420917i −0.818074 − 0.575113i \(-0.804958\pi\)
0.0890252 + 0.996029i \(0.471625\pi\)
\(14\) −1.12602 + 6.58685i −0.300941 + 1.76041i
\(15\) −0.743600 + 1.28795i −0.191997 + 0.332548i
\(16\) −3.72939 + 1.44626i −0.932347 + 0.361565i
\(17\) 1.31986 − 2.28607i 0.320113 − 0.554452i −0.660398 − 0.750916i \(-0.729612\pi\)
0.980511 + 0.196463i \(0.0629457\pi\)
\(18\) −1.32638 + 0.490620i −0.312631 + 0.115640i
\(19\) −1.48362 + 0.856569i −0.340366 + 0.196510i −0.660434 − 0.750884i \(-0.729628\pi\)
0.320068 + 0.947395i \(0.396294\pi\)
\(20\) −1.93559 − 2.25844i −0.432811 − 0.505002i
\(21\) −4.09212 − 2.36259i −0.892974 − 0.515559i
\(22\) 2.42164 − 2.91679i 0.516296 − 0.621861i
\(23\) 1.08203 0.225619 0.112810 − 0.993617i \(-0.464015\pi\)
0.112810 + 0.993617i \(0.464015\pi\)
\(24\) −0.0442918 − 2.82808i −0.00904102 − 0.577279i
\(25\) −1.39412 + 2.41468i −0.278823 + 0.482936i
\(26\) 4.23114 + 0.723312i 0.829796 + 0.141853i
\(27\) − 1.00000i − 0.192450i
\(28\) 7.17557 − 6.14981i 1.35606 − 1.16221i
\(29\) 7.94145i 1.47469i 0.675516 + 0.737345i \(0.263921\pi\)
−0.675516 + 0.737345i \(0.736079\pi\)
\(30\) 1.97260 − 0.729650i 0.360145 − 0.133215i
\(31\) −0.642621 −0.115418 −0.0577090 − 0.998333i \(-0.518380\pi\)
−0.0577090 + 0.998333i \(0.518380\pi\)
\(32\) 5.36439 + 1.79537i 0.948298 + 0.317380i
\(33\) 1.34034 + 2.32153i 0.233323 + 0.404127i
\(34\) −3.50128 + 1.29510i −0.600465 + 0.222108i
\(35\) 6.08580 + 3.51364i 1.02869 + 0.593913i
\(36\) 1.88642 + 0.664382i 0.314404 + 0.110730i
\(37\) −6.07556 + 0.295871i −0.998816 + 0.0486409i
\(38\) 2.38810 + 0.408245i 0.387401 + 0.0662260i
\(39\) −1.51764 + 2.62862i −0.243016 + 0.420917i
\(40\) 0.0658707 + 4.20592i 0.0104151 + 0.665015i
\(41\) 3.79677 + 6.57619i 0.592955 + 1.02703i 0.993832 + 0.110896i \(0.0353722\pi\)
−0.400877 + 0.916132i \(0.631294\pi\)
\(42\) 2.31826 + 6.26739i 0.357716 + 0.967079i
\(43\) − 5.62167i − 0.857296i −0.903472 − 0.428648i \(-0.858990\pi\)
0.903472 − 0.428648i \(-0.141010\pi\)
\(44\) −5.26989 + 0.986059i −0.794465 + 0.148654i
\(45\) 1.48720i 0.221699i
\(46\) −1.17734 − 0.977477i −0.173589 − 0.144121i
\(47\) −10.2321 −1.49250 −0.746250 − 0.665666i \(-0.768147\pi\)
−0.746250 + 0.665666i \(0.768147\pi\)
\(48\) −2.50661 + 3.11719i −0.361799 + 0.449928i
\(49\) −7.66363 + 13.2738i −1.09480 + 1.89626i
\(50\) 3.69827 − 1.36796i 0.523014 − 0.193459i
\(51\) − 2.63972i − 0.369635i
\(52\) −3.95041 − 4.60932i −0.547823 − 0.639197i
\(53\) −4.76853 − 2.75311i −0.655009 − 0.378169i 0.135364 − 0.990796i \(-0.456780\pi\)
−0.790372 + 0.612627i \(0.790113\pi\)
\(54\) −0.903371 + 1.08808i −0.122933 + 0.148069i
\(55\) −1.99335 − 3.45258i −0.268783 − 0.465546i
\(56\) −13.3632 + 0.209286i −1.78573 + 0.0279670i
\(57\) −0.856569 + 1.48362i −0.113455 + 0.196510i
\(58\) 7.17408 − 8.64094i 0.942003 − 1.13461i
\(59\) 2.14528 + 1.23858i 0.279292 + 0.161249i 0.633103 − 0.774068i \(-0.281781\pi\)
−0.353811 + 0.935317i \(0.615114\pi\)
\(60\) −2.80549 − 0.988070i −0.362187 − 0.127559i
\(61\) −2.95293 + 1.70487i −0.378084 + 0.218287i −0.676984 − 0.735998i \(-0.736713\pi\)
0.298900 + 0.954284i \(0.403380\pi\)
\(62\) 0.699223 + 0.580525i 0.0888015 + 0.0737268i
\(63\) −4.72517 −0.595316
\(64\) −4.21500 − 6.79954i −0.526875 − 0.849943i
\(65\) 2.25703 − 3.90929i 0.279950 − 0.484888i
\(66\) 0.638810 − 3.73683i 0.0786321 − 0.459973i
\(67\) −10.5804 + 6.10860i −1.29260 + 0.746284i −0.979115 − 0.203308i \(-0.934831\pi\)
−0.313487 + 0.949592i \(0.601497\pi\)
\(68\) 4.97963 + 1.75378i 0.603869 + 0.212678i
\(69\) 0.937067 − 0.541016i 0.112810 − 0.0651307i
\(70\) −3.44772 − 9.32086i −0.412082 − 1.11406i
\(71\) 7.71919 + 13.3700i 0.916100 + 1.58673i 0.805283 + 0.592890i \(0.202013\pi\)
0.110816 + 0.993841i \(0.464653\pi\)
\(72\) −1.45240 − 2.42704i −0.171167 − 0.286030i
\(73\) −10.6753 −1.24944 −0.624722 − 0.780847i \(-0.714788\pi\)
−0.624722 + 0.780847i \(0.714788\pi\)
\(74\) 6.87798 + 5.16656i 0.799549 + 0.600600i
\(75\) 2.78823i 0.321958i
\(76\) −2.22965 − 2.60155i −0.255758 − 0.298418i
\(77\) 10.9696 − 6.33332i 1.25011 − 0.721749i
\(78\) 4.02593 − 1.48917i 0.455847 − 0.168615i
\(79\) −6.98127 − 12.0919i −0.785454 − 1.36045i −0.928727 − 0.370763i \(-0.879096\pi\)
0.143273 − 0.989683i \(-0.454237\pi\)
\(80\) 3.72784 − 4.63589i 0.416785 − 0.518308i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.80956 − 10.5853i 0.199832 − 1.16895i
\(83\) 5.98889 + 3.45769i 0.657366 + 0.379531i 0.791273 − 0.611463i \(-0.209419\pi\)
−0.133906 + 0.990994i \(0.542752\pi\)
\(84\) 3.13932 − 8.91368i 0.342528 − 0.972562i
\(85\) 3.92580i 0.425812i
\(86\) −5.07845 + 6.11683i −0.547624 + 0.659594i
\(87\) 3.97073 + 6.87750i 0.425706 + 0.737345i
\(88\) 6.62484 + 3.68775i 0.706210 + 0.393116i
\(89\) −2.91925 + 5.05630i −0.309440 + 0.535966i −0.978240 − 0.207476i \(-0.933475\pi\)
0.668800 + 0.743443i \(0.266808\pi\)
\(90\) 1.34349 − 1.61819i 0.141617 − 0.170573i
\(91\) 12.4207 + 7.17109i 1.30204 + 0.751735i
\(92\) 0.398015 + 2.12715i 0.0414959 + 0.221770i
\(93\) −0.556526 + 0.321310i −0.0577090 + 0.0333183i
\(94\) 11.1333 + 9.24335i 1.14831 + 0.953379i
\(95\) 1.27389 − 2.20644i 0.130698 − 0.226376i
\(96\) 5.54338 − 1.12735i 0.565769 − 0.115060i
\(97\) 3.65462 0.371070 0.185535 − 0.982638i \(-0.440598\pi\)
0.185535 + 0.982638i \(0.440598\pi\)
\(98\) 20.3298 − 7.51986i 2.05362 − 0.759620i
\(99\) 2.32153 + 1.34034i 0.233323 + 0.134709i
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.565.17 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.68 yes 152
37.26 even 3 inner 888.2.bh.a.877.68 yes 152
296.285 even 6 inner 888.2.bh.a.877.17 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.17 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.68 yes 152 8.5 even 2 inner
888.2.bh.a.877.17 yes 152 296.285 even 6 inner
888.2.bh.a.877.68 yes 152 37.26 even 3 inner