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Properties

Label 888.2.r.e.43.10
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
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(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.10
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.10

$q$-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+(-0.851073 + 1.12946i) q^{2} -1.00000i q^{3} +(-0.551349 - 1.92250i) q^{4} +(1.41934 - 1.41934i) q^{5} +(1.12946 + 0.851073i) q^{6} +1.22240i q^{7} +(2.64062 + 1.01347i) q^{8} -1.00000 q^{9} +(0.395121 + 2.81104i) q^{10} -3.17135i q^{11} +(-1.92250 + 0.551349i) q^{12} +(3.40871 - 3.40871i) q^{13} +(-1.38065 - 1.04035i) q^{14} +(-1.41934 - 1.41934i) q^{15} +(-3.39203 + 2.11994i) q^{16} +(5.11017 + 5.11017i) q^{17} +(0.851073 - 1.12946i) q^{18} +(-2.54661 - 2.54661i) q^{19} +(-3.51122 - 1.94613i) q^{20} +1.22240 q^{21} +(3.58190 + 2.69905i) q^{22} +(-0.0251759 + 0.0251759i) q^{23} +(1.01347 - 2.64062i) q^{24} +0.970974i q^{25} +(0.948931 + 6.75106i) q^{26} +1.00000i q^{27} +(2.35007 - 0.673969i) q^{28} +(-6.48955 - 6.48955i) q^{29} +(2.81104 - 0.395121i) q^{30} +(-0.109600 - 0.109600i) q^{31} +(0.492486 - 5.63538i) q^{32} -3.17135 q^{33} +(-10.1208 + 1.42259i) q^{34} +(1.73500 + 1.73500i) q^{35} +(0.551349 + 1.92250i) q^{36} +(2.21402 + 5.66552i) q^{37} +(5.04364 - 0.708936i) q^{38} +(-3.40871 - 3.40871i) q^{39} +(5.18638 - 2.30948i) q^{40} -4.43641i q^{41} +(-1.04035 + 1.38065i) q^{42} +(0.554403 + 0.554403i) q^{43} +(-6.09692 + 1.74852i) q^{44} +(-1.41934 + 1.41934i) q^{45} +(-0.00700859 - 0.0498617i) q^{46} -9.19271i q^{47} +(2.11994 + 3.39203i) q^{48} +5.50574 q^{49} +(-1.09667 - 0.826370i) q^{50} +(5.11017 - 5.11017i) q^{51} +(-8.43264 - 4.67387i) q^{52} -11.6007i q^{53} +(-1.12946 - 0.851073i) q^{54} +(-4.50121 - 4.50121i) q^{55} +(-1.23886 + 3.22790i) q^{56} +(-2.54661 + 2.54661i) q^{57} +(12.8528 - 1.80659i) q^{58} +(-0.00863022 - 0.00863022i) q^{59} +(-1.94613 + 3.51122i) q^{60} +(-5.10610 - 5.10610i) q^{61} +(0.217065 - 0.0305108i) q^{62} -1.22240i q^{63} +(5.94578 + 5.35236i) q^{64} -9.67621i q^{65} +(2.69905 - 3.58190i) q^{66} +1.59286i q^{67} +(7.00683 - 12.6418i) q^{68} +(0.0251759 + 0.0251759i) q^{69} +(-3.43621 + 0.482996i) q^{70} -15.5936i q^{71} +(-2.64062 - 1.01347i) q^{72} +7.80454i q^{73} +(-8.28326 - 2.32114i) q^{74} +0.970974 q^{75} +(-3.49179 + 6.29994i) q^{76} +3.87666 q^{77} +(6.75106 - 0.948931i) q^{78} +(-8.42132 + 8.42132i) q^{79} +(-1.80553 + 7.82333i) q^{80} +1.00000 q^{81} +(5.01074 + 3.77571i) q^{82} +14.2778 q^{83} +(-0.673969 - 2.35007i) q^{84} +14.5061 q^{85} +(-1.09801 + 0.154337i) q^{86} +(-6.48955 + 6.48955i) q^{87} +(3.21405 - 8.37433i) q^{88} +(-1.42694 + 1.42694i) q^{89} +(-0.395121 - 2.81104i) q^{90} +(4.16681 + 4.16681i) q^{91} +(0.0622815 + 0.0345201i) q^{92} +(-0.109600 + 0.109600i) q^{93} +(10.3828 + 7.82367i) q^{94} -7.22899 q^{95} +(-5.63538 - 0.492486i) q^{96} +(6.08152 + 6.08152i) q^{97} +(-4.68579 + 6.21850i) q^{98} +3.17135i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.851073 + 1.12946i −0.601800 + 0.798647i
\(3\) − 1.00000i − 0.577350i
\(4\) −0.551349 − 1.92250i −0.275674 − 0.961251i
\(5\) 1.41934 − 1.41934i 0.634746 − 0.634746i −0.314509 − 0.949255i \(-0.601840\pi\)
0.949255 + 0.314509i \(0.101840\pi\)
\(6\) 1.12946 + 0.851073i 0.461099 + 0.347449i
\(7\) 1.22240i 0.462024i 0.972951 + 0.231012i \(0.0742036\pi\)
−0.972951 + 0.231012i \(0.925796\pi\)
\(8\) 2.64062 + 1.01347i 0.933601 + 0.358314i
\(9\) −1.00000 −0.333333
\(10\) 0.395121 + 2.81104i 0.124948 + 0.888928i
\(11\) − 3.17135i − 0.956198i −0.878306 − 0.478099i \(-0.841326\pi\)
0.878306 − 0.478099i \(-0.158674\pi\)
\(12\) −1.92250 + 0.551349i −0.554979 + 0.159161i
\(13\) 3.40871 − 3.40871i 0.945406 − 0.945406i −0.0531789 − 0.998585i \(-0.516935\pi\)
0.998585 + 0.0531789i \(0.0169354\pi\)
\(14\) −1.38065 − 1.04035i −0.368994 − 0.278046i
\(15\) −1.41934 − 1.41934i −0.366471 − 0.366471i
\(16\) −3.39203 + 2.11994i −0.848007 + 0.529984i
\(17\) 5.11017 + 5.11017i 1.23940 + 1.23940i 0.960247 + 0.279150i \(0.0900528\pi\)
0.279150 + 0.960247i \(0.409947\pi\)
\(18\) 0.851073 − 1.12946i 0.200600 − 0.266216i
\(19\) −2.54661 − 2.54661i −0.584233 − 0.584233i 0.351831 − 0.936064i \(-0.385559\pi\)
−0.936064 + 0.351831i \(0.885559\pi\)
\(20\) −3.51122 − 1.94613i −0.785134 − 0.435167i
\(21\) 1.22240 0.266750
\(22\) 3.58190 + 2.69905i 0.763664 + 0.575439i
\(23\) −0.0251759 + 0.0251759i −0.00524955 + 0.00524955i −0.709727 − 0.704477i \(-0.751182\pi\)
0.704477 + 0.709727i \(0.251182\pi\)
\(24\) 1.01347 − 2.64062i 0.206873 − 0.539015i
\(25\) 0.970974i 0.194195i
\(26\) 0.948931 + 6.75106i 0.186101 + 1.32399i
\(27\) 1.00000i 0.192450i
\(28\) 2.35007 − 0.673969i 0.444121 − 0.127368i
\(29\) −6.48955 − 6.48955i −1.20508 − 1.20508i −0.972602 − 0.232477i \(-0.925317\pi\)
−0.232477 − 0.972602i \(-0.574683\pi\)
\(30\) 2.81104 − 0.395121i 0.513223 − 0.0721388i
\(31\) −0.109600 − 0.109600i −0.0196847 − 0.0196847i 0.697196 − 0.716881i \(-0.254431\pi\)
−0.716881 + 0.697196i \(0.754431\pi\)
\(32\) 0.492486 − 5.63538i 0.0870601 − 0.996203i
\(33\) −3.17135 −0.552061
\(34\) −10.1208 + 1.42259i −1.73571 + 0.243972i
\(35\) 1.73500 + 1.73500i 0.293268 + 0.293268i
\(36\) 0.551349 + 1.92250i 0.0918914 + 0.320417i
\(37\) 2.21402 + 5.66552i 0.363982 + 0.931406i
\(38\) 5.04364 − 0.708936i 0.818187 − 0.115005i
\(39\) −3.40871 − 3.40871i −0.545830 − 0.545830i
\(40\) 5.18638 − 2.30948i 0.820038 − 0.365161i
\(41\) − 4.43641i − 0.692851i −0.938077 − 0.346425i \(-0.887395\pi\)
0.938077 − 0.346425i \(-0.112605\pi\)
\(42\) −1.04035 + 1.38065i −0.160530 + 0.213039i
\(43\) 0.554403 + 0.554403i 0.0845457 + 0.0845457i 0.748115 − 0.663569i \(-0.230959\pi\)
−0.663569 + 0.748115i \(0.730959\pi\)
\(44\) −6.09692 + 1.74852i −0.919146 + 0.263599i
\(45\) −1.41934 + 1.41934i −0.211582 + 0.211582i
\(46\) −0.00700859 − 0.0498617i −0.00103336 − 0.00735171i
\(47\) − 9.19271i − 1.34090i −0.741957 − 0.670448i \(-0.766102\pi\)
0.741957 − 0.670448i \(-0.233898\pi\)
\(48\) 2.11994 + 3.39203i 0.305987 + 0.489597i
\(49\) 5.50574 0.786534
\(50\) −1.09667 − 0.826370i −0.155093 − 0.116866i
\(51\) 5.11017 − 5.11017i 0.715567 − 0.715567i
\(52\) −8.43264 − 4.67387i −1.16940 − 0.648148i
\(53\) − 11.6007i − 1.59348i −0.604324 − 0.796739i \(-0.706557\pi\)
0.604324 − 0.796739i \(-0.293443\pi\)
\(54\) −1.12946 − 0.851073i −0.153700 − 0.115816i
\(55\) −4.50121 − 4.50121i −0.606943 − 0.606943i
\(56\) −1.23886 + 3.22790i −0.165550 + 0.431346i
\(57\) −2.54661 + 2.54661i −0.337307 + 0.337307i
\(58\) 12.8528 − 1.80659i 1.68765 − 0.237217i
\(59\) −0.00863022 − 0.00863022i −0.00112356 − 0.00112356i 0.706545 − 0.707668i \(-0.250253\pi\)
−0.707668 + 0.706545i \(0.750253\pi\)
\(60\) −1.94613 + 3.51122i −0.251244 + 0.453297i
\(61\) −5.10610 − 5.10610i −0.653769 − 0.653769i 0.300129 − 0.953899i \(-0.402970\pi\)
−0.953899 + 0.300129i \(0.902970\pi\)
\(62\) 0.217065 − 0.0305108i 0.0275673 − 0.00387488i
\(63\) − 1.22240i − 0.154008i
\(64\) 5.94578 + 5.35236i 0.743222 + 0.669045i
\(65\) − 9.67621i − 1.20019i
\(66\) 2.69905 − 3.58190i 0.332230 − 0.440902i
\(67\) 1.59286i 0.194598i 0.995255 + 0.0972991i \(0.0310203\pi\)
−0.995255 + 0.0972991i \(0.968980\pi\)
\(68\) 7.00683 − 12.6418i 0.849702 − 1.53304i
\(69\) 0.0251759 + 0.0251759i 0.00303083 + 0.00303083i
\(70\) −3.43621 + 0.482996i −0.410706 + 0.0577290i
\(71\) − 15.5936i − 1.85062i −0.379206 − 0.925312i \(-0.623803\pi\)
0.379206 − 0.925312i \(-0.376197\pi\)
\(72\) −2.64062 − 1.01347i −0.311200 − 0.119438i
\(73\) 7.80454i 0.913452i 0.889607 + 0.456726i \(0.150978\pi\)
−0.889607 + 0.456726i \(0.849022\pi\)
\(74\) −8.28326 − 2.32114i −0.962909 − 0.269827i
\(75\) 0.970974 0.112118
\(76\) −3.49179 + 6.29994i −0.400536 + 0.722652i
\(77\) 3.87666 0.441786
\(78\) 6.75106 − 0.948931i 0.764406 − 0.107445i
\(79\) −8.42132 + 8.42132i −0.947473 + 0.947473i −0.998688 − 0.0512148i \(-0.983691\pi\)
0.0512148 + 0.998688i \(0.483691\pi\)
\(80\) −1.80553 + 7.82333i −0.201864 + 0.874675i
\(81\) 1.00000 0.111111
\(82\) 5.01074 + 3.77571i 0.553343 + 0.416957i
\(83\) 14.2778 1.56719 0.783596 − 0.621271i \(-0.213383\pi\)
0.783596 + 0.621271i \(0.213383\pi\)
\(84\) −0.673969 − 2.35007i −0.0735360 − 0.256413i
\(85\) 14.5061 1.57341
\(86\) −1.09801 + 0.154337i −0.118402 + 0.0166426i
\(87\) −6.48955 + 6.48955i −0.695753 + 0.695753i
\(88\) 3.21405 − 8.37433i 0.342619 − 0.892707i
\(89\) −1.42694 + 1.42694i −0.151256 + 0.151256i −0.778679 − 0.627423i \(-0.784110\pi\)
0.627423 + 0.778679i \(0.284110\pi\)
\(90\) −0.395121 − 2.81104i −0.0416494 − 0.296309i
\(91\) 4.16681 + 4.16681i 0.436800 + 0.436800i
\(92\) 0.0622815 + 0.0345201i 0.00649330 + 0.00359897i
\(93\) −0.109600 + 0.109600i −0.0113650 + 0.0113650i
\(94\) 10.3828 + 7.82367i 1.07090 + 0.806950i
\(95\) −7.22899 −0.741679
\(96\) −5.63538 − 0.492486i −0.575158 − 0.0502642i
\(97\) 6.08152 + 6.08152i 0.617485 + 0.617485i 0.944886 − 0.327401i \(-0.106173\pi\)
−0.327401 + 0.944886i \(0.606173\pi\)
\(98\) −4.68579 + 6.21850i −0.473336 + 0.628163i
\(99\) 3.17135i 0.318733i
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.r.e.43.10 yes 72
8.3 odd 2 inner 888.2.r.e.43.8 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.8 yes 72
296.179 even 4 inner 888.2.r.e.475.10 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.8 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.10 yes 72 1.1 even 1 trivial
888.2.r.e.475.8 yes 72 37.31 odd 4 inner
888.2.r.e.475.10 yes 72 296.179 even 4 inner