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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(473,888)] 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("888.473"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.br (of order \(12\), degree \(4\), 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: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 569.30
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.30

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.30293 - 1.14122i) q^{3} +(0.799283 - 2.98296i) q^{5} +(-0.335390 - 0.580913i) q^{7} +(0.395238 - 2.97385i) q^{9} -4.05514 q^{11} +(-0.293096 - 0.0785350i) q^{13} +(-2.36281 - 4.79874i) q^{15} +(0.650482 - 0.174296i) q^{17} +(0.739950 + 0.198269i) q^{19} +(-1.09994 - 0.374134i) q^{21} +(-2.25350 - 2.25350i) q^{23} +(-3.92910 - 2.26847i) q^{25} +(-2.87885 - 4.32576i) q^{27} +(1.54903 - 1.54903i) q^{29} +(6.22364 + 6.22364i) q^{31} +(-5.28355 + 4.62780i) q^{33} +(-2.00092 + 0.536144i) q^{35} +(0.816062 + 6.02777i) q^{37} +(-0.471509 + 0.232162i) q^{39} +(-3.27826 - 5.67812i) q^{41} +(-3.22323 + 3.22323i) q^{43} +(-8.55499 - 3.55593i) q^{45} +0.410860i q^{47} +(3.27503 - 5.67251i) q^{49} +(0.648621 - 0.969438i) q^{51} +(5.53742 + 3.19703i) q^{53} +(-3.24120 + 12.0963i) q^{55} +(1.19037 - 0.586115i) q^{57} +(9.82950 - 2.63381i) q^{59} +(1.33086 - 4.96682i) q^{61} +(-1.86011 + 0.767802i) q^{63} +(-0.468534 + 0.811525i) q^{65} +(3.94446 - 2.27734i) q^{67} +(-5.50788 - 0.364409i) q^{69} +(-9.45301 + 5.45770i) q^{71} -13.9639i q^{73} +(-7.70815 + 1.52832i) q^{75} +(1.36005 + 2.35568i) q^{77} +(-6.36126 - 1.70449i) q^{79} +(-8.68757 - 2.35076i) q^{81} +(8.30167 + 4.79297i) q^{83} -2.07968i q^{85} +(0.250491 - 3.78607i) q^{87} +(0.326263 + 1.21763i) q^{89} +(0.0526798 + 0.196604i) q^{91} +(15.2115 + 1.00641i) q^{93} +(1.18286 - 2.04877i) q^{95} +(10.9399 - 10.9399i) q^{97} +(-1.60274 + 12.0594i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\)

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 0
\(3\) 1.30293 − 1.14122i 0.752245 − 0.658883i
\(4\) 0 0
\(5\) 0.799283 − 2.98296i 0.357450 − 1.33402i −0.519923 − 0.854213i \(-0.674039\pi\)
0.877373 − 0.479809i \(-0.159294\pi\)
\(6\) 0 0
\(7\) −0.335390 − 0.580913i −0.126766 − 0.219565i 0.795656 − 0.605749i \(-0.207126\pi\)
−0.922422 + 0.386184i \(0.873793\pi\)
\(8\) 0 0
\(9\) 0.395238 − 2.97385i 0.131746 − 0.991284i
\(10\) 0 0
\(11\) −4.05514 −1.22267 −0.611335 − 0.791372i \(-0.709367\pi\)
−0.611335 + 0.791372i \(0.709367\pi\)
\(12\) 0 0
\(13\) −0.293096 − 0.0785350i −0.0812903 − 0.0217817i 0.217945 − 0.975961i \(-0.430065\pi\)
−0.299235 + 0.954179i \(0.596731\pi\)
\(14\) 0 0
\(15\) −2.36281 − 4.79874i −0.610075 − 1.23903i
\(16\) 0 0
\(17\) 0.650482 − 0.174296i 0.157765 − 0.0422730i −0.179072 − 0.983836i \(-0.557310\pi\)
0.336837 + 0.941563i \(0.390643\pi\)
\(18\) 0 0
\(19\) 0.739950 + 0.198269i 0.169756 + 0.0454860i 0.342696 − 0.939446i \(-0.388660\pi\)
−0.172940 + 0.984932i \(0.555327\pi\)
\(20\) 0 0
\(21\) −1.09994 − 0.374134i −0.240026 − 0.0816427i
\(22\) 0 0
\(23\) −2.25350 − 2.25350i −0.469887 − 0.469887i 0.431991 − 0.901878i \(-0.357811\pi\)
−0.901878 + 0.431991i \(0.857811\pi\)
\(24\) 0 0
\(25\) −3.92910 − 2.26847i −0.785820 − 0.453693i
\(26\) 0 0
\(27\) −2.87885 − 4.32576i −0.554035 − 0.832493i
\(28\) 0 0
\(29\) 1.54903 − 1.54903i 0.287648 − 0.287648i −0.548501 − 0.836150i \(-0.684801\pi\)
0.836150 + 0.548501i \(0.184801\pi\)
\(30\) 0 0
\(31\) 6.22364 + 6.22364i 1.11780 + 1.11780i 0.992064 + 0.125735i \(0.0401290\pi\)
0.125735 + 0.992064i \(0.459871\pi\)
\(32\) 0 0
\(33\) −5.28355 + 4.62780i −0.919747 + 0.805596i
\(34\) 0 0
\(35\) −2.00092 + 0.536144i −0.338217 + 0.0906249i
\(36\) 0 0
\(37\) 0.816062 + 6.02777i 0.134160 + 0.990960i
\(38\) 0 0
\(39\) −0.471509 + 0.232162i −0.0755019 + 0.0371757i
\(40\) 0 0
\(41\) −3.27826 − 5.67812i −0.511979 − 0.886773i −0.999904 − 0.0138873i \(-0.995579\pi\)
0.487925 − 0.872886i \(-0.337754\pi\)
\(42\) 0 0
\(43\) −3.22323 + 3.22323i −0.491538 + 0.491538i −0.908790 − 0.417253i \(-0.862993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(44\) 0 0
\(45\) −8.55499 − 3.55593i −1.27530 − 0.530087i
\(46\) 0 0
\(47\) 0.410860i 0.0599301i 0.999551 + 0.0299651i \(0.00953960\pi\)
−0.999551 + 0.0299651i \(0.990460\pi\)
\(48\) 0 0
\(49\) 3.27503 − 5.67251i 0.467861 − 0.810359i
\(50\) 0 0
\(51\) 0.648621 − 0.969438i 0.0908251 − 0.135748i
\(52\) 0 0
\(53\) 5.53742 + 3.19703i 0.760623 + 0.439146i 0.829519 − 0.558478i \(-0.188615\pi\)
−0.0688965 + 0.997624i \(0.521948\pi\)
\(54\) 0 0
\(55\) −3.24120 + 12.0963i −0.437044 + 1.63107i
\(56\) 0 0
\(57\) 1.19037 − 0.586115i 0.157668 − 0.0776328i
\(58\) 0 0
\(59\) 9.82950 − 2.63381i 1.27969 − 0.342892i 0.445955 − 0.895055i \(-0.352864\pi\)
0.833736 + 0.552163i \(0.186197\pi\)
\(60\) 0 0
\(61\) 1.33086 − 4.96682i 0.170399 − 0.635936i −0.826891 − 0.562362i \(-0.809893\pi\)
0.997290 − 0.0735743i \(-0.0234406\pi\)
\(62\) 0 0
\(63\) −1.86011 + 0.767802i −0.234352 + 0.0967340i
\(64\) 0 0
\(65\) −0.468534 + 0.811525i −0.0581145 + 0.100657i
\(66\) 0 0
\(67\) 3.94446 − 2.27734i 0.481893 − 0.278221i −0.239312 − 0.970943i \(-0.576922\pi\)
0.721205 + 0.692722i \(0.243589\pi\)
\(68\) 0 0
\(69\) −5.50788 − 0.364409i −0.663071 − 0.0438696i
\(70\) 0 0
\(71\) −9.45301 + 5.45770i −1.12187 + 0.647710i −0.941877 − 0.335959i \(-0.890940\pi\)
−0.179990 + 0.983668i \(0.557606\pi\)
\(72\) 0 0
\(73\) − 13.9639i − 1.63435i −0.576391 − 0.817174i \(-0.695539\pi\)
0.576391 − 0.817174i \(-0.304461\pi\)
\(74\) 0 0
\(75\) −7.70815 + 1.52832i −0.890060 + 0.176475i
\(76\) 0 0
\(77\) 1.36005 + 2.35568i 0.154993 + 0.268455i
\(78\) 0 0
\(79\) −6.36126 − 1.70449i −0.715698 − 0.191771i −0.117446 − 0.993079i \(-0.537471\pi\)
−0.598251 + 0.801309i \(0.704138\pi\)
\(80\) 0 0
\(81\) −8.68757 − 2.35076i −0.965286 − 0.261195i
\(82\) 0 0
\(83\) 8.30167 + 4.79297i 0.911226 + 0.526097i 0.880825 − 0.473441i \(-0.156988\pi\)
0.0304007 + 0.999538i \(0.490322\pi\)
\(84\) 0 0
\(85\) − 2.07968i − 0.225573i
\(86\) 0 0
\(87\) 0.250491 − 3.78607i 0.0268555 − 0.405909i
\(88\) 0 0
\(89\) 0.326263 + 1.21763i 0.0345838 + 0.129068i 0.981060 − 0.193707i \(-0.0620510\pi\)
−0.946476 + 0.322775i \(0.895384\pi\)
\(90\) 0 0
\(91\) 0.0526798 + 0.196604i 0.00552234 + 0.0206097i
\(92\) 0 0
\(93\) 15.2115 + 1.00641i 1.57736 + 0.104360i
\(94\) 0 0
\(95\) 1.18286 − 2.04877i 0.121359 − 0.210199i
\(96\) 0 0
\(97\) 10.9399 − 10.9399i 1.11078 − 1.11078i 0.117731 − 0.993046i \(-0.462438\pi\)
0.993046 − 0.117731i \(-0.0375619\pi\)
\(98\) 0 0
\(99\) −1.60274 + 12.0594i −0.161082 + 1.21201i
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.br.a.569.30 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.33 yes 152
37.8 odd 12 inner 888.2.br.a.785.33 yes 152
111.8 even 12 inner 888.2.br.a.785.30 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.30 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.33 yes 152 3.2 odd 2 inner
888.2.br.a.785.30 yes 152 111.8 even 12 inner
888.2.br.a.785.33 yes 152 37.8 odd 12 inner