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Properties

Label 888.2.bd.a.491.13
Level $888$
Weight $2$
Character 888.491
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 491.13
Character \(\chi\) \(=\) 888.491
Dual form 888.2.bd.a.803.13

$q$-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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34708 − 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 −0.979034 − 0.203699i \(-0.934704\pi\)
0.313108 − 0.949717i \(-0.398630\pi\)
\(6\) −2.00735 + 1.40377i −0.819495 + 0.573086i
\(7\) −1.43848 + 0.830505i −0.543693 + 0.313901i −0.746574 − 0.665302i \(-0.768303\pi\)
0.202881 + 0.979203i \(0.434970\pi\)
\(8\) −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 \(-0.747347\pi\)
0.701188 − 0.712977i \(-0.252653\pi\)
\(12\) 3.30845 − 1.02672i 0.955067 − 0.296389i
\(13\) 3.15343 − 1.82064i 0.874605 − 0.504953i 0.00572914 − 0.999984i \(-0.498176\pi\)
0.868876 + 0.495030i \(0.164843\pi\)
\(14\) 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 0.995320 − 0.0966312i \(-0.0308067\pi\)
0.413975 + 0.910288i \(0.364140\pi\)
\(18\) −0.173502 + 4.23909i −0.0408948 + 0.999163i
\(19\) −2.15641 − 3.73502i −0.494715 − 0.856871i 0.505267 − 0.862963i \(-0.331394\pi\)
−0.999981 + 0.00609196i \(0.998061\pi\)
\(20\) 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 \(0.281153\pi\)
−0.634629 + 0.772817i \(0.718847\pi\)
\(32\) −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 −0.263578 − 0.964638i \(-0.584903\pi\)
0.703612 + 0.710584i \(0.251569\pi\)
\(42\) 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 −0.929083 − 0.369872i \(-0.879402\pi\)
0.784860 + 0.619673i \(0.212735\pi\)
\(54\) 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 0.657866 − 0.753135i \(-0.271459\pi\)
−0.323300 + 0.946296i \(0.604792\pi\)
\(60\) −7.57449 − 7.00404i −0.977863 − 0.904218i
\(61\) 4.57313 − 2.64030i 0.585529 − 0.338056i −0.177798 − 0.984067i \(-0.556897\pi\)
0.763328 + 0.646011i \(0.223564\pi\)
\(62\) 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 0.680354 − 0.732883i \(-0.261826\pi\)
−0.974873 + 0.222763i \(0.928493\pi\)
\(68\) 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 0.654972 − 0.755653i \(-0.272680\pi\)
−0.981901 + 0.189396i \(0.939347\pi\)
\(72\) −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 −0.0997117 − 0.995016i \(-0.531792\pi\)
0.811854 + 0.583861i \(0.198459\pi\)
\(80\) 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 0.0363127 − 0.999340i \(-0.488439\pi\)
−0.847298 + 0.531118i \(0.821772\pi\)
\(84\) −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 0.727956 − 0.685624i \(-0.240470\pi\)
−0.229789 + 0.973240i \(0.573804\pi\)
\(90\) 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.491.13 ✓ 296
3.2 odd 2 inner 888.2.bd.a.491.136 yes 296
8.3 odd 2 inner 888.2.bd.a.491.36 yes 296
24.11 even 2 inner 888.2.bd.a.491.113 yes 296
37.26 even 3 inner 888.2.bd.a.803.113 yes 296
111.26 odd 6 inner 888.2.bd.a.803.36 yes 296
296.211 odd 6 inner 888.2.bd.a.803.136 yes 296
888.803 even 6 inner 888.2.bd.a.803.13 yes 296
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bd.a.491.13 ✓ 296 1.1 even 1 trivial
888.2.bd.a.491.36 yes 296 8.3 odd 2 inner
888.2.bd.a.491.113 yes 296 24.11 even 2 inner
888.2.bd.a.491.136 yes 296 3.2 odd 2 inner
888.2.bd.a.803.13 yes 296 888.803 even 6 inner
888.2.bd.a.803.36 yes 296 111.26 odd 6 inner
888.2.bd.a.803.113 yes 296 37.26 even 3 inner
888.2.bd.a.803.136 yes 296 296.211 odd 6 inner