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Properties

Label 888.2.bd.a.803.13
Level $888$
Weight $2$
Character 888.803
Analytic conductor $7.091$
Analytic rank $0$
Dimension $296$
Inner twists $8$

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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(491,888)] 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("888.491"); 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([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bd (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: \(296\)
Relative dimension: \(148\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 803.13
Character \(\chi\) \(=\) 888.803
Dual form 888.2.bd.a.491.13

$q$-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+(-1.34708 + 0.430552i) q^{2} +(1.04983 + 1.37763i) q^{3} +(1.62925 - 1.15998i) q^{4} +(-1.48905 + 2.57912i) q^{5} +(-2.00735 - 1.40377i) q^{6} +(-1.43848 - 0.830505i) q^{7} +(-1.69530 + 2.26406i) q^{8} +(-0.795715 + 2.89255i) q^{9} +(0.895430 - 4.11539i) q^{10} +4.72935i q^{11} +(3.30845 + 1.02672i) q^{12} +(3.15343 + 1.82064i) q^{13} +(2.29532 + 0.499418i) q^{14} +(-5.11632 + 0.656273i) q^{15} +(1.30891 - 3.77978i) q^{16} +(5.81067 - 3.35479i) q^{17} +(-0.173502 - 4.23909i) q^{18} +(-2.15641 + 3.73502i) q^{19} +(0.565675 + 5.92929i) q^{20} +(-0.366030 - 2.85358i) q^{21} +(-2.03623 - 6.37082i) q^{22} +1.66390 q^{23} +(-4.89880 + 0.0413856i) q^{24} +(-1.93456 - 3.35076i) q^{25} +(-5.03181 - 1.09482i) q^{26} +(-4.82022 + 1.94049i) q^{27} +(-3.30701 + 0.315500i) q^{28} -9.50892 q^{29} +(6.60953 - 3.08689i) q^{30} -8.60573i q^{31} +(-0.135810 + 5.65522i) q^{32} +(-6.51529 + 4.96501i) q^{33} +(-6.38303 + 7.02097i) q^{34} +(4.28394 - 2.47333i) q^{35} +(2.05887 + 5.63569i) q^{36} +(-5.81839 + 1.77378i) q^{37} +(1.29674 - 5.95981i) q^{38} +(0.802411 + 6.25561i) q^{39} +(-3.31488 - 7.74368i) q^{40} +(2.81760 + 1.62674i) q^{41} +(1.72168 + 3.68640i) q^{42} +1.70602 q^{43} +(5.48594 + 7.70529i) q^{44} +(-6.27536 - 6.35940i) q^{45} +(-2.24140 + 0.716394i) q^{46} -5.83269 q^{47} +(6.58126 - 2.16494i) q^{48} +(-2.12052 - 3.67285i) q^{49} +(4.04869 + 3.68081i) q^{50} +(10.7219 + 4.48298i) q^{51} +(7.24962 - 0.691639i) q^{52} +(-1.04996 - 1.81858i) q^{53} +(5.65774 - 4.68935i) q^{54} +(-12.1976 - 7.04226i) q^{55} +(4.31896 - 1.84884i) q^{56} +(-7.40933 + 0.950398i) q^{57} +(12.8093 - 4.09409i) q^{58} +(2.56985 - 1.48370i) q^{59} +(-7.57449 + 7.00404i) q^{60} +(4.57313 + 2.64030i) q^{61} +(3.70522 + 11.5926i) q^{62} +(3.54689 - 3.50002i) q^{63} +(-2.25192 - 7.67651i) q^{64} +(-9.39126 + 5.42205i) q^{65} +(6.63891 - 9.49344i) q^{66} +(-2.41074 + 4.17552i) q^{67} +(5.57555 - 12.2060i) q^{68} +(1.74681 + 2.29223i) q^{69} +(-4.70591 + 5.17624i) q^{70} +(-2.75475 + 4.77137i) q^{71} +(-5.19993 - 6.70528i) q^{72} +0.868664 q^{73} +(7.07414 - 4.89455i) q^{74} +(2.58514 - 6.18284i) q^{75} +(0.819197 + 8.58666i) q^{76} +(3.92775 - 6.80307i) q^{77} +(-3.77428 - 8.08133i) q^{78} +(6.32966 + 3.65443i) q^{79} +(7.79947 + 9.00413i) q^{80} +(-7.73368 - 4.60329i) q^{81} +(-4.49592 - 0.978226i) q^{82} +(-7.38843 + 4.26571i) q^{83} +(-3.90643 - 4.22460i) q^{84} +19.9819i q^{85} +(-2.29815 + 0.734532i) q^{86} +(-9.98275 - 13.0997i) q^{87} +(-10.7075 - 8.01767i) q^{88} +(4.69970 - 2.71337i) q^{89} +(11.1915 + 5.86475i) q^{90} +(-3.02409 - 5.23789i) q^{91} +(2.71090 - 1.93008i) q^{92} +(11.8555 - 9.03455i) q^{93} +(7.85710 - 2.51128i) q^{94} +(-6.42203 - 11.1233i) q^{95} +(-7.93337 + 5.74993i) q^{96} +8.56691 q^{97} +(4.43787 + 4.03463i) q^{98} +(-13.6799 - 3.76321i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 296 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 10 q^{12} + 6 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{22} + 16 q^{24} - 136 q^{25} - 8 q^{27} - 2 q^{28} - 4 q^{30} + 4 q^{33} - 18 q^{34} + 20 q^{36} + 6 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34708 + 0.430552i −0.952529 + 0.304446i
\(3\) 1.04983 + 1.37763i 0.606120 + 0.795374i
\(4\) 1.62925 − 1.15998i 0.814625 − 0.579988i
\(5\) −1.48905 + 2.57912i −0.665925 + 1.15342i 0.313108 + 0.949717i \(0.398630\pi\)
−0.979034 + 0.203699i \(0.934704\pi\)
\(6\) −2.00735 − 1.40377i −0.819495 − 0.573086i
\(7\) −1.43848 − 0.830505i −0.543693 − 0.313901i 0.202881 − 0.979203i \(-0.434970\pi\)
−0.746574 + 0.665302i \(0.768303\pi\)
\(8\) −1.69530 + 2.26406i −0.599379 + 0.800466i
\(9\) −0.795715 + 2.89255i −0.265238 + 0.964183i
\(10\) 0.895430 − 4.11539i 0.283160 − 1.30140i
\(11\) 4.72935i 1.42595i 0.701188 + 0.712977i \(0.252653\pi\)
−0.701188 + 0.712977i \(0.747347\pi\)
\(12\) 3.30845 + 1.02672i 0.955067 + 0.296389i
\(13\) 3.15343 + 1.82064i 0.874605 + 0.504953i 0.868876 − 0.495030i \(-0.164843\pi\)
0.00572914 + 0.999984i \(0.498176\pi\)
\(14\) 2.29532 + 0.499418i 0.613450 + 0.133475i
\(15\) −5.11632 + 0.656273i −1.32103 + 0.169449i
\(16\) 1.30891 − 3.77978i 0.327227 − 0.944946i
\(17\) 5.81067 − 3.35479i 1.40930 − 0.813657i 0.413975 − 0.910288i \(-0.364140\pi\)
0.995320 + 0.0966312i \(0.0308067\pi\)
\(18\) −0.173502 − 4.23909i −0.0408948 − 0.999163i
\(19\) −2.15641 + 3.73502i −0.494715 + 0.856871i −0.999981 − 0.00609196i \(-0.998061\pi\)
0.505267 + 0.862963i \(0.331394\pi\)
\(20\) 0.565675 + 5.92929i 0.126489 + 1.32583i
\(21\) −0.366030 − 2.85358i −0.0798742 − 0.622701i
\(22\) −2.03623 − 6.37082i −0.434126 − 1.35826i
\(23\) 1.66390 0.346946 0.173473 − 0.984839i \(-0.444501\pi\)
0.173473 + 0.984839i \(0.444501\pi\)
\(24\) −4.89880 + 0.0413856i −0.999964 + 0.00844781i
\(25\) −1.93456 − 3.35076i −0.386913 − 0.670152i
\(26\) −5.03181 − 1.09482i −0.986818 − 0.214713i
\(27\) −4.82022 + 1.94049i −0.927652 + 0.373447i
\(28\) −3.30701 + 0.315500i −0.624965 + 0.0596239i
\(29\) −9.50892 −1.76576 −0.882881 − 0.469597i \(-0.844399\pi\)
−0.882881 + 0.469597i \(0.844399\pi\)
\(30\) 6.60953 − 3.08689i 1.20673 − 0.563587i
\(31\) − 8.60573i − 1.54563i −0.634629 − 0.772817i \(-0.718847\pi\)
0.634629 − 0.772817i \(-0.281153\pi\)
\(32\) −0.135810 + 5.65522i −0.0240081 + 0.999712i
\(33\) −6.51529 + 4.96501i −1.13417 + 0.864298i
\(34\) −6.38303 + 7.02097i −1.09468 + 1.20409i
\(35\) 4.28394 − 2.47333i 0.724118 − 0.418070i
\(36\) 2.05887 + 5.63569i 0.343145 + 0.939282i
\(37\) −5.81839 + 1.77378i −0.956538 + 0.291608i
\(38\) 1.29674 − 5.95981i 0.210359 − 0.966809i
\(39\) 0.802411 + 6.25561i 0.128489 + 1.00170i
\(40\) −3.31488 − 7.74368i −0.524129 − 1.22438i
\(41\) 2.81760 + 1.62674i 0.440034 + 0.254054i 0.703612 − 0.710584i \(-0.251569\pi\)
−0.263578 + 0.964638i \(0.584903\pi\)
\(42\) 1.72168 + 3.68640i 0.265662 + 0.568824i
\(43\) 1.70602 0.260166 0.130083 − 0.991503i \(-0.458476\pi\)
0.130083 + 0.991503i \(0.458476\pi\)
\(44\) 5.48594 + 7.70529i 0.827036 + 1.16162i
\(45\) −6.27536 − 6.35940i −0.935475 − 0.948004i
\(46\) −2.24140 + 0.716394i −0.330477 + 0.105627i
\(47\) −5.83269 −0.850785 −0.425392 − 0.905009i \(-0.639864\pi\)
−0.425392 + 0.905009i \(0.639864\pi\)
\(48\) 6.58126 − 2.16494i 0.949924 − 0.312482i
\(49\) −2.12052 − 3.67285i −0.302932 − 0.524693i
\(50\) 4.04869 + 3.68081i 0.572571 + 0.520546i
\(51\) 10.7219 + 4.48298i 1.50136 + 0.627743i
\(52\) 7.24962 − 0.691639i 1.00534 − 0.0959131i
\(53\) −1.04996 − 1.81858i −0.144223 − 0.249801i 0.784860 − 0.619673i \(-0.212735\pi\)
−0.929083 + 0.369872i \(0.879402\pi\)
\(54\) 5.65774 − 4.68935i 0.769921 − 0.638139i
\(55\) −12.1976 − 7.04226i −1.64472 − 0.949578i
\(56\) 4.31896 − 1.84884i 0.577146 − 0.247062i
\(57\) −7.40933 + 0.950398i −0.981389 + 0.125883i
\(58\) 12.8093 − 4.09409i 1.68194 − 0.537580i
\(59\) 2.56985 − 1.48370i 0.334566 − 0.193162i −0.323300 − 0.946296i \(-0.604792\pi\)
0.657866 + 0.753135i \(0.271459\pi\)
\(60\) −7.57449 + 7.00404i −0.977863 + 0.904218i
\(61\) 4.57313 + 2.64030i 0.585529 + 0.338056i 0.763328 − 0.646011i \(-0.223564\pi\)
−0.177798 + 0.984067i \(0.556897\pi\)
\(62\) 3.70522 + 11.5926i 0.470563 + 1.47226i
\(63\) 3.54689 − 3.50002i 0.446867 − 0.440961i
\(64\) −2.25192 − 7.67651i −0.281490 − 0.959564i
\(65\) −9.39126 + 5.42205i −1.16484 + 0.672522i
\(66\) 6.63891 − 9.49344i 0.817194 − 1.16856i
\(67\) −2.41074 + 4.17552i −0.294518 + 0.510121i −0.974873 − 0.222763i \(-0.928493\pi\)
0.680354 + 0.732883i \(0.261826\pi\)
\(68\) 5.57555 − 12.2060i 0.676135 − 1.48020i
\(69\) 1.74681 + 2.29223i 0.210291 + 0.275952i
\(70\) −4.70591 + 5.17624i −0.562464 + 0.618679i
\(71\) −2.75475 + 4.77137i −0.326929 + 0.566257i −0.981901 − 0.189396i \(-0.939347\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(72\) −5.19993 − 6.70528i −0.612817 − 0.790225i
\(73\) 0.868664 0.101669 0.0508347 − 0.998707i \(-0.483812\pi\)
0.0508347 + 0.998707i \(0.483812\pi\)
\(74\) 7.07414 − 4.89455i 0.822352 − 0.568979i
\(75\) 2.58514 − 6.18284i 0.298506 − 0.713933i
\(76\) 0.819197 + 8.58666i 0.0939684 + 0.984958i
\(77\) 3.92775 − 6.80307i 0.447609 − 0.775281i
\(78\) −3.77428 − 8.08133i −0.427353 − 0.915031i
\(79\) 6.32966 + 3.65443i 0.712142 + 0.411155i 0.811854 − 0.583861i \(-0.198459\pi\)
−0.0997117 + 0.995016i \(0.531792\pi\)
\(80\) 7.79947 + 9.00413i 0.872007 + 1.00669i
\(81\) −7.73368 − 4.60329i −0.859297 − 0.511476i
\(82\) −4.49592 − 0.978226i −0.496492 − 0.108027i
\(83\) −7.38843 + 4.26571i −0.810985 + 0.468223i −0.847298 − 0.531118i \(-0.821772\pi\)
0.0363127 + 0.999340i \(0.488439\pi\)
\(84\) −3.90643 − 4.22460i −0.426227 − 0.460942i
\(85\) 19.9819i 2.16734i
\(86\) −2.29815 + 0.734532i −0.247816 + 0.0792066i
\(87\) −9.98275 − 13.0997i −1.07026 − 1.40444i
\(88\) −10.7075 − 8.01767i −1.14143 − 0.854686i
\(89\) 4.69970 − 2.71337i 0.498167 − 0.287617i −0.229789 − 0.973240i \(-0.573804\pi\)
0.727956 + 0.685624i \(0.240470\pi\)
\(90\) 11.1915 + 5.86475i 1.17968 + 0.618199i
\(91\) −3.02409 − 5.23789i −0.317011 − 0.549080i
\(92\) 2.71090 − 1.93008i 0.282631 − 0.201225i
\(93\) 11.8555 − 9.03455i 1.22936 − 0.936839i
\(94\) 7.85710 − 2.51128i 0.810397 − 0.259018i
\(95\) −6.42203 − 11.1233i −0.658886 − 1.14122i
\(96\) −7.93337 + 5.74993i −0.809696 + 0.586849i
\(97\) 8.56691 0.869838 0.434919 − 0.900470i \(-0.356777\pi\)
0.434919 + 0.900470i \(0.356777\pi\)
\(98\) 4.43787 + 4.03463i 0.448292 + 0.407559i
\(99\) −13.6799 − 3.76321i −1.37488 − 0.378217i
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.bd.a.803.13 yes 296
3.2 odd 2 inner 888.2.bd.a.803.136 yes 296
8.3 odd 2 inner 888.2.bd.a.803.36 yes 296
24.11 even 2 inner 888.2.bd.a.803.113 yes 296
37.10 even 3 inner 888.2.bd.a.491.113 yes 296
111.47 odd 6 inner 888.2.bd.a.491.36 yes 296
296.195 odd 6 inner 888.2.bd.a.491.136 yes 296
888.491 even 6 inner 888.2.bd.a.491.13 ✓ 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.13 ✓ 296 888.491 even 6 inner
888.2.bd.a.491.36 yes 296 111.47 odd 6 inner
888.2.bd.a.491.113 yes 296 37.10 even 3 inner
888.2.bd.a.491.136 yes 296 296.195 odd 6 inner
888.2.bd.a.803.13 yes 296 1.1 even 1 trivial
888.2.bd.a.803.36 yes 296 8.3 odd 2 inner
888.2.bd.a.803.113 yes 296 24.11 even 2 inner
888.2.bd.a.803.136 yes 296 3.2 odd 2 inner