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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 569.33
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.33

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.63979 - 0.557758i) q^{3} +(-0.799283 + 2.98296i) q^{5} +(-0.335390 - 0.580913i) q^{7} +(2.37781 - 1.82921i) q^{9} +4.05514 q^{11} +(-0.293096 - 0.0785350i) q^{13} +(0.353119 + 5.33724i) q^{15} +(-0.650482 + 0.174296i) q^{17} +(0.739950 + 0.198269i) q^{19} +(-0.873979 - 0.765508i) q^{21} +(2.25350 + 2.25350i) q^{23} +(-3.92910 - 2.26847i) q^{25} +(2.87885 - 4.32576i) q^{27} +(-1.54903 + 1.54903i) q^{29} +(6.22364 + 6.22364i) q^{31} +(6.64956 - 2.26179i) q^{33} +(2.00092 - 0.536144i) q^{35} +(0.816062 + 6.02777i) q^{37} +(-0.524420 + 0.0346963i) q^{39} +(3.27826 + 5.67812i) q^{41} +(-3.22323 + 3.22323i) q^{43} +(3.55593 + 8.55499i) q^{45} -0.410860i q^{47} +(3.27503 - 5.67251i) q^{49} +(-0.969438 + 0.648621i) q^{51} +(-5.53742 - 3.19703i) q^{53} +(-3.24120 + 12.0963i) q^{55} +(1.32395 - 0.0875940i) q^{57} +(-9.82950 + 2.63381i) q^{59} +(1.33086 - 4.96682i) q^{61} +(-1.86011 - 0.767802i) q^{63} +(0.468534 - 0.811525i) q^{65} +(3.94446 - 2.27734i) q^{67} +(4.95217 + 2.43835i) q^{69} +(9.45301 - 5.45770i) q^{71} -13.9639i q^{73} +(-7.70815 - 1.52832i) q^{75} +(-1.36005 - 2.35568i) q^{77} +(-6.36126 - 1.70449i) q^{79} +(2.30797 - 8.69904i) q^{81} +(-8.30167 - 4.79297i) q^{83} -2.07968i q^{85} +(-1.67610 + 3.40407i) q^{87} +(-0.326263 - 1.21763i) q^{89} +(0.0526798 + 0.196604i) q^{91} +(13.6767 + 6.73417i) q^{93} +(-1.18286 + 2.04877i) q^{95} +(10.9399 - 10.9399i) q^{97} +(9.64235 - 7.41770i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.63979 − 0.557758i 0.946732 − 0.322022i
\(4\) 0 0
\(5\) −0.799283 + 2.98296i −0.357450 + 1.33402i 0.519923 + 0.854213i \(0.325961\pi\)
−0.877373 + 0.479809i \(0.840706\pi\)
\(6\) 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
\(9\) 2.37781 − 1.82921i 0.792604 − 0.609737i
\(10\) 0 0
\(11\) 4.05514 1.22267 0.611335 − 0.791372i \(-0.290633\pi\)
0.611335 + 0.791372i \(0.290633\pi\)
\(12\) 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
\(15\) 0.353119 + 5.33724i 0.0911748 + 1.37807i
\(16\) 0 0
\(17\) −0.650482 + 0.174296i −0.157765 + 0.0422730i −0.336837 − 0.941563i \(-0.609357\pi\)
0.179072 + 0.983836i \(0.442690\pi\)
\(18\) 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
\(21\) −0.873979 − 0.765508i −0.190718 − 0.167048i
\(22\) 0 0
\(23\) 2.25350 + 2.25350i 0.469887 + 0.469887i 0.901878 − 0.431991i \(-0.142189\pi\)
−0.431991 + 0.901878i \(0.642189\pi\)
\(24\) 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 −0.836150 − 0.548501i \(-0.815199\pi\)
0.548501 + 0.836150i \(0.315199\pi\)
\(30\) 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
\(33\) 6.64956 − 2.26179i 1.15754 − 0.393726i
\(34\) 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
\(39\) −0.524420 + 0.0346963i −0.0839744 + 0.00555585i
\(40\) 0 0
\(41\) 3.27826 + 5.67812i 0.511979 + 0.886773i 0.999904 + 0.0138873i \(0.00442061\pi\)
−0.487925 + 0.872886i \(0.662246\pi\)
\(42\) 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
\(45\) 3.55593 + 8.55499i 0.530087 + 1.27530i
\(46\) 0 0
\(47\) − 0.410860i − 0.0599301i −0.999551 − 0.0299651i \(-0.990460\pi\)
0.999551 − 0.0299651i \(-0.00953960\pi\)
\(48\) 0 0
\(49\) 3.27503 − 5.67251i 0.467861 − 0.810359i
\(50\) 0 0
\(51\) −0.969438 + 0.648621i −0.135748 + 0.0908251i
\(52\) 0 0
\(53\) −5.53742 − 3.19703i −0.760623 − 0.439146i 0.0688965 − 0.997624i \(-0.478052\pi\)
−0.829519 + 0.558478i \(0.811385\pi\)
\(54\) 0 0
\(55\) −3.24120 + 12.0963i −0.437044 + 1.63107i
\(56\) 0 0
\(57\) 1.32395 − 0.0875940i 0.175361 − 0.0116021i
\(58\) 0 0
\(59\) −9.82950 + 2.63381i −1.27969 + 0.342892i −0.833736 − 0.552163i \(-0.813803\pi\)
−0.445955 + 0.895055i \(0.647136\pi\)
\(60\) 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
\(69\) 4.95217 + 2.43835i 0.596171 + 0.293543i
\(70\) 0 0
\(71\) 9.45301 − 5.45770i 1.12187 − 0.647710i 0.179990 − 0.983668i \(-0.442394\pi\)
0.941877 + 0.335959i \(0.109060\pi\)
\(72\) 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
\(81\) 2.30797 − 8.69904i 0.256441 − 0.966560i
\(82\) 0 0
\(83\) −8.30167 − 4.79297i −0.911226 − 0.526097i −0.0304007 − 0.999538i \(-0.509678\pi\)
−0.880825 + 0.473441i \(0.843012\pi\)
\(84\) 0 0
\(85\) − 2.07968i − 0.225573i
\(86\) 0 0
\(87\) −1.67610 + 3.40407i −0.179697 + 0.364955i
\(88\) 0 0
\(89\) −0.326263 − 1.21763i −0.0345838 − 0.129068i 0.946476 − 0.322775i \(-0.104616\pi\)
−0.981060 + 0.193707i \(0.937949\pi\)
\(90\) 0 0
\(91\) 0.0526798 + 0.196604i 0.00552234 + 0.0206097i
\(92\) 0 0
\(93\) 13.6767 + 6.73417i 1.41821 + 0.698301i
\(94\) 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
\(99\) 9.64235 − 7.41770i 0.969092 − 0.745507i
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.569.33 yes 152
3.2 odd 2 inner 888.2.br.a.569.30 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.30 yes 152
111.8 even 12 inner 888.2.br.a.785.33 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.30 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.33 yes 152 1.1 even 1 trivial
888.2.br.a.785.30 yes 152 37.8 odd 12 inner
888.2.br.a.785.33 yes 152 111.8 even 12 inner