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Properties

Label 888.2.bh.a.565.3
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 565.3
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.3

$q$-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+(-1.41004 - 0.108511i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(1.97645 + 0.306011i) q^{4} +(-2.48652 + 1.43559i) q^{5} +(1.27539 - 0.611049i) q^{6} +(1.49987 + 2.59785i) q^{7} +(-2.75368 - 0.645957i) q^{8} +(0.500000 - 0.866025i) q^{9} +(3.66188 - 1.75443i) q^{10} -3.66220i q^{11} +(-1.86466 + 0.723212i) q^{12} +(3.25869 - 1.88141i) q^{13} +(-1.83299 - 3.82584i) q^{14} +(1.43559 - 2.48652i) q^{15} +(3.81271 + 1.20963i) q^{16} +(1.33253 - 2.30801i) q^{17} +(-0.798996 + 1.16688i) q^{18} +(1.35324 - 0.781296i) q^{19} +(-5.35379 + 2.07647i) q^{20} +(-2.59785 - 1.49987i) q^{21} +(-0.397390 + 5.16386i) q^{22} +5.66795 q^{23} +(2.70773 - 0.817424i) q^{24} +(1.62185 - 2.80912i) q^{25} +(-4.79905 + 2.29926i) q^{26} +1.00000i q^{27} +(2.16945 + 5.59350i) q^{28} +3.18557i q^{29} +(-2.29406 + 3.35032i) q^{30} -1.13123 q^{31} +(-5.24484 - 2.11936i) q^{32} +(1.83110 + 3.17156i) q^{33} +(-2.12937 + 3.10980i) q^{34} +(-7.45891 - 4.30640i) q^{35} +(1.25324 - 1.55865i) q^{36} +(-5.20959 + 3.14009i) q^{37} +(-1.99291 + 0.954819i) q^{38} +(-1.88141 + 3.25869i) q^{39} +(7.77440 - 2.34697i) q^{40} +(2.29487 + 3.97484i) q^{41} +(3.50033 + 2.39678i) q^{42} +3.81843i q^{43} +(1.12067 - 7.23816i) q^{44} +2.87118i q^{45} +(-7.99207 - 0.615037i) q^{46} +3.40946 q^{47} +(-3.90672 + 0.858784i) q^{48} +(-0.999224 + 1.73071i) q^{49} +(-2.59170 + 3.78500i) q^{50} +2.66506i q^{51} +(7.01637 - 2.72131i) q^{52} +(11.7199 + 6.76649i) q^{53} +(0.108511 - 1.41004i) q^{54} +(5.25742 + 9.10612i) q^{55} +(-2.45206 - 8.12250i) q^{56} +(-0.781296 + 1.35324i) q^{57} +(0.345670 - 4.49179i) q^{58} +(-5.44940 - 3.14621i) q^{59} +(3.59828 - 4.47517i) q^{60} +(3.19381 - 1.84395i) q^{61} +(1.59508 + 0.122751i) q^{62} +2.99974 q^{63} +(7.16548 + 3.55751i) q^{64} +(-5.40186 + 9.35630i) q^{65} +(-2.23778 - 4.67073i) q^{66} +(-3.26220 + 1.88343i) q^{67} +(3.33996 - 4.15390i) q^{68} +(-4.90859 + 2.83398i) q^{69} +(10.0501 + 6.88160i) q^{70} +(-0.397991 - 0.689341i) q^{71} +(-1.93625 + 2.06178i) q^{72} -4.14236 q^{73} +(7.68649 - 3.86237i) q^{74} +3.24369i q^{75} +(2.91370 - 1.13008i) q^{76} +(9.51385 - 5.49282i) q^{77} +(3.00647 - 4.39074i) q^{78} +(-3.75182 - 6.49835i) q^{79} +(-11.2169 + 2.46573i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-2.80456 - 5.85371i) q^{82} +(11.1852 + 6.45775i) q^{83} +(-4.67555 - 3.75939i) q^{84} +7.65187i q^{85} +(0.414343 - 5.38416i) q^{86} +(-1.59278 - 2.75878i) q^{87} +(-2.36562 + 10.0845i) q^{88} +(-6.66193 + 11.5388i) q^{89} +(0.311556 - 4.04850i) q^{90} +(9.77523 + 5.64373i) q^{91} +(11.2024 + 1.73446i) q^{92} +(0.979671 - 0.565613i) q^{93} +(-4.80748 - 0.369964i) q^{94} +(-2.24324 + 3.88541i) q^{95} +(5.60184 - 0.787000i) q^{96} +12.1761 q^{97} +(1.59675 - 2.33195i) q^{98} +(-3.17156 - 1.83110i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41004 − 0.108511i −0.997052 − 0.0767291i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 1.97645 + 0.306011i 0.988225 + 0.153006i
\(5\) −2.48652 + 1.43559i −1.11200 + 0.642016i −0.939348 − 0.342966i \(-0.888568\pi\)
−0.172657 + 0.984982i \(0.555235\pi\)
\(6\) 1.27539 − 0.611049i 0.520676 − 0.249460i
\(7\) 1.49987 + 2.59785i 0.566898 + 0.981896i 0.996870 + 0.0790538i \(0.0251899\pi\)
−0.429973 + 0.902842i \(0.641477\pi\)
\(8\) −2.75368 − 0.645957i −0.973572 − 0.228380i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 3.66188 − 1.75443i 1.15799 − 0.554800i
\(11\) − 3.66220i − 1.10419i −0.833780 − 0.552097i \(-0.813828\pi\)
0.833780 − 0.552097i \(-0.186172\pi\)
\(12\) −1.86466 + 0.723212i −0.538282 + 0.208773i
\(13\) 3.25869 − 1.88141i 0.903798 − 0.521808i 0.0253677 − 0.999678i \(-0.491924\pi\)
0.878431 + 0.477870i \(0.158591\pi\)
\(14\) −1.83299 − 3.82584i −0.489887 − 1.02250i
\(15\) 1.43559 − 2.48652i 0.370668 − 0.642016i
\(16\) 3.81271 + 1.20963i 0.953178 + 0.302408i
\(17\) 1.33253 − 2.30801i 0.323186 − 0.559774i −0.657958 − 0.753055i \(-0.728579\pi\)
0.981144 + 0.193281i \(0.0619127\pi\)
\(18\) −0.798996 + 1.16688i −0.188325 + 0.275036i
\(19\) 1.35324 − 0.781296i 0.310455 − 0.179242i −0.336675 − 0.941621i \(-0.609302\pi\)
0.647130 + 0.762379i \(0.275969\pi\)
\(20\) −5.35379 + 2.07647i −1.19714 + 0.464313i
\(21\) −2.59785 − 1.49987i −0.566898 − 0.327299i
\(22\) −0.397390 + 5.16386i −0.0847238 + 1.10094i
\(23\) 5.66795 1.18185 0.590925 − 0.806726i \(-0.298763\pi\)
0.590925 + 0.806726i \(0.298763\pi\)
\(24\) 2.70773 − 0.817424i 0.552714 − 0.166856i
\(25\) 1.62185 − 2.80912i 0.324369 − 0.561824i
\(26\) −4.79905 + 2.29926i −0.941172 + 0.450922i
\(27\) 1.00000i 0.192450i
\(28\) 2.16945 + 5.59350i 0.409987 + 1.05707i
\(29\) 3.18557i 0.591545i 0.955258 + 0.295772i \(0.0955770\pi\)
−0.955258 + 0.295772i \(0.904423\pi\)
\(30\) −2.29406 + 3.35032i −0.418837 + 0.611682i
\(31\) −1.13123 −0.203174 −0.101587 − 0.994827i \(-0.532392\pi\)
−0.101587 + 0.994827i \(0.532392\pi\)
\(32\) −5.24484 − 2.11936i −0.927165 − 0.374653i
\(33\) 1.83110 + 3.17156i 0.318754 + 0.552097i
\(34\) −2.12937 + 3.10980i −0.365184 + 0.533326i
\(35\) −7.45891 − 4.30640i −1.26079 − 0.727915i
\(36\) 1.25324 − 1.55865i 0.208873 − 0.259775i
\(37\) −5.20959 + 3.14009i −0.856451 + 0.516228i
\(38\) −1.99291 + 0.954819i −0.323293 + 0.154892i
\(39\) −1.88141 + 3.25869i −0.301266 + 0.521808i
\(40\) 7.77440 − 2.34697i 1.22924 − 0.371089i
\(41\) 2.29487 + 3.97484i 0.358399 + 0.620765i 0.987694 − 0.156402i \(-0.0499895\pi\)
−0.629295 + 0.777167i \(0.716656\pi\)
\(42\) 3.50033 + 2.39678i 0.540113 + 0.369831i
\(43\) 3.81843i 0.582306i 0.956677 + 0.291153i \(0.0940388\pi\)
−0.956677 + 0.291153i \(0.905961\pi\)
\(44\) 1.12067 − 7.23816i 0.168948 − 1.09119i
\(45\) 2.87118i 0.428011i
\(46\) −7.99207 − 0.615037i −1.17837 − 0.0906822i
\(47\) 3.40946 0.497320 0.248660 − 0.968591i \(-0.420010\pi\)
0.248660 + 0.968591i \(0.420010\pi\)
\(48\) −3.90672 + 0.858784i −0.563887 + 0.123955i
\(49\) −0.999224 + 1.73071i −0.142746 + 0.247244i
\(50\) −2.59170 + 3.78500i −0.366521 + 0.535279i
\(51\) 2.66506i 0.373183i
\(52\) 7.01637 − 2.72131i 0.972996 − 0.377378i
\(53\) 11.7199 + 6.76649i 1.60985 + 0.929449i 0.989402 + 0.145205i \(0.0463841\pi\)
0.620452 + 0.784245i \(0.286949\pi\)
\(54\) 0.108511 − 1.41004i 0.0147665 − 0.191883i
\(55\) 5.25742 + 9.10612i 0.708911 + 1.22787i
\(56\) −2.45206 − 8.12250i −0.327670 − 1.08541i
\(57\) −0.781296 + 1.35324i −0.103485 + 0.179242i
\(58\) 0.345670 − 4.49179i 0.0453887 − 0.589801i
\(59\) −5.44940 − 3.14621i −0.709451 − 0.409602i 0.101407 − 0.994845i \(-0.467666\pi\)
−0.810858 + 0.585243i \(0.800999\pi\)
\(60\) 3.59828 − 4.47517i 0.464536 − 0.577742i
\(61\) 3.19381 − 1.84395i 0.408926 − 0.236093i −0.281402 − 0.959590i \(-0.590800\pi\)
0.690328 + 0.723497i \(0.257466\pi\)
\(62\) 1.59508 + 0.122751i 0.202575 + 0.0155894i
\(63\) 2.99974 0.377932
\(64\) 7.16548 + 3.55751i 0.895685 + 0.444689i
\(65\) −5.40186 + 9.35630i −0.670018 + 1.16051i
\(66\) −2.23778 − 4.67073i −0.275452 − 0.574927i
\(67\) −3.26220 + 1.88343i −0.398541 + 0.230098i −0.685854 − 0.727739i \(-0.740571\pi\)
0.287313 + 0.957837i \(0.407238\pi\)
\(68\) 3.33996 − 4.15390i 0.405029 − 0.503734i
\(69\) −4.90859 + 2.83398i −0.590925 + 0.341171i
\(70\) 10.0501 + 6.88160i 1.20122 + 0.822508i
\(71\) −0.397991 − 0.689341i −0.0472329 − 0.0818097i 0.841442 − 0.540347i \(-0.181707\pi\)
−0.888675 + 0.458537i \(0.848374\pi\)
\(72\) −1.93625 + 2.06178i −0.228190 + 0.242983i
\(73\) −4.14236 −0.484826 −0.242413 − 0.970173i \(-0.577939\pi\)
−0.242413 + 0.970173i \(0.577939\pi\)
\(74\) 7.68649 − 3.86237i 0.893536 − 0.448992i
\(75\) 3.24369i 0.374549i
\(76\) 2.91370 − 1.13008i 0.334225 − 0.129630i
\(77\) 9.51385 − 5.49282i 1.08420 − 0.625965i
\(78\) 3.00647 − 4.39074i 0.340416 − 0.497154i
\(79\) −3.75182 − 6.49835i −0.422113 − 0.731121i 0.574033 − 0.818832i \(-0.305378\pi\)
−0.996146 + 0.0877110i \(0.972045\pi\)
\(80\) −11.2169 + 2.46573i −1.25409 + 0.275677i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −2.80456 − 5.85371i −0.309712 − 0.646435i
\(83\) 11.1852 + 6.45775i 1.22773 + 0.708831i 0.966555 − 0.256461i \(-0.0825564\pi\)
0.261176 + 0.965291i \(0.415890\pi\)
\(84\) −4.67555 − 3.75939i −0.510144 − 0.410183i
\(85\) 7.65187i 0.829962i
\(86\) 0.414343 − 5.38416i 0.0446798 − 0.580589i
\(87\) −1.59278 − 2.75878i −0.170764 − 0.295772i
\(88\) −2.36562 + 10.0845i −0.252176 + 1.07501i
\(89\) −6.66193 + 11.5388i −0.706164 + 1.22311i 0.260106 + 0.965580i \(0.416242\pi\)
−0.966270 + 0.257531i \(0.917091\pi\)
\(90\) 0.311556 − 4.04850i 0.0328409 − 0.426749i
\(91\) 9.77523 + 5.64373i 1.02472 + 0.591624i
\(92\) 11.2024 + 1.73446i 1.16793 + 0.180830i
\(93\) 0.979671 − 0.565613i 0.101587 − 0.0586513i
\(94\) −4.80748 − 0.369964i −0.495854 − 0.0381589i
\(95\) −2.24324 + 3.88541i −0.230152 + 0.398635i
\(96\) 5.60184 − 0.787000i 0.571736 − 0.0803228i
\(97\) 12.1761 1.23629 0.618147 − 0.786063i \(-0.287884\pi\)
0.618147 + 0.786063i \(0.287884\pi\)
\(98\) 1.59675 − 2.33195i 0.161296 − 0.235562i
\(99\) −3.17156 − 1.83110i −0.318754 − 0.184032i
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.565.3 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.53 yes 152
37.26 even 3 inner 888.2.bh.a.877.53 yes 152
296.285 even 6 inner 888.2.bh.a.877.3 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.3 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.53 yes 152 8.5 even 2 inner
888.2.bh.a.877.3 yes 152 296.285 even 6 inner
888.2.bh.a.877.53 yes 152 37.26 even 3 inner