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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 569.1
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.72309 - 0.175939i) q^{3} +(0.440252 - 1.64304i) q^{5} +(-2.41379 - 4.18081i) q^{7} +(2.93809 + 0.606319i) q^{9} -5.79998 q^{11} +(0.132470 + 0.0354952i) q^{13} +(-1.04767 + 2.75366i) q^{15} +(-1.95366 + 0.523480i) q^{17} +(2.26131 + 0.605916i) q^{19} +(3.42361 + 7.62860i) q^{21} +(2.57074 + 2.57074i) q^{23} +(1.82436 + 1.05329i) q^{25} +(-4.95592 - 1.56167i) q^{27} +(-1.61116 + 1.61116i) q^{29} +(2.85614 + 2.85614i) q^{31} +(9.99390 + 1.02044i) q^{33} +(-7.93193 + 2.12535i) q^{35} +(-4.69397 + 3.86868i) q^{37} +(-0.222013 - 0.0844681i) q^{39} +(-2.81750 - 4.88006i) q^{41} +(7.06828 - 7.06828i) q^{43} +(2.28971 - 4.56048i) q^{45} +7.92849i q^{47} +(-8.15277 + 14.1210i) q^{49} +(3.45843 - 0.558280i) q^{51} +(-4.24708 - 2.45205i) q^{53} +(-2.55345 + 9.52962i) q^{55} +(-3.78984 - 1.44190i) q^{57} +(-4.36135 + 1.16862i) q^{59} +(-3.81348 + 14.2321i) q^{61} +(-4.55703 - 13.7471i) q^{63} +(0.116640 - 0.202027i) q^{65} +(-1.58274 + 0.913795i) q^{67} +(-3.97733 - 4.88191i) q^{69} +(-10.1420 + 5.85550i) q^{71} -13.5662i q^{73} +(-2.95822 - 2.13590i) q^{75} +(13.9999 + 24.2486i) q^{77} +(-6.87939 - 1.84333i) q^{79} +(8.26475 + 3.56284i) q^{81} +(-3.79517 - 2.19114i) q^{83} +3.44040i q^{85} +(3.05963 - 2.49270i) q^{87} +(-1.55789 - 5.81411i) q^{89} +(-0.171356 - 0.639508i) q^{91} +(-4.41888 - 5.42389i) q^{93} +(1.99109 - 3.44867i) q^{95} +(12.1216 - 12.1216i) q^{97} +(-17.0409 - 3.51664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −1.72309 − 0.175939i −0.994828 − 0.101579i
\(4\) 0 0
\(5\) 0.440252 − 1.64304i 0.196887 − 0.734791i −0.794884 − 0.606762i \(-0.792468\pi\)
0.991770 − 0.128029i \(-0.0408652\pi\)
\(6\) 0 0
\(7\) −2.41379 − 4.18081i −0.912327 − 1.58020i −0.810768 − 0.585367i \(-0.800950\pi\)
−0.101559 − 0.994830i \(-0.532383\pi\)
\(8\) 0 0
\(9\) 2.93809 + 0.606319i 0.979364 + 0.202106i
\(10\) 0 0
\(11\) −5.79998 −1.74876 −0.874380 − 0.485242i \(-0.838731\pi\)
−0.874380 + 0.485242i \(0.838731\pi\)
\(12\) 0 0
\(13\) 0.132470 + 0.0354952i 0.0367405 + 0.00984459i 0.277142 − 0.960829i \(-0.410613\pi\)
−0.240402 + 0.970673i \(0.577279\pi\)
\(14\) 0 0
\(15\) −1.04767 + 2.75366i −0.270507 + 0.710991i
\(16\) 0 0
\(17\) −1.95366 + 0.523480i −0.473831 + 0.126963i −0.487829 − 0.872939i \(-0.662211\pi\)
0.0139976 + 0.999902i \(0.495544\pi\)
\(18\) 0 0
\(19\) 2.26131 + 0.605916i 0.518780 + 0.139007i 0.508702 − 0.860943i \(-0.330125\pi\)
0.0100776 + 0.999949i \(0.496792\pi\)
\(20\) 0 0
\(21\) 3.42361 + 7.62860i 0.747094 + 1.66470i
\(22\) 0 0
\(23\) 2.57074 + 2.57074i 0.536036 + 0.536036i 0.922362 − 0.386326i \(-0.126256\pi\)
−0.386326 + 0.922362i \(0.626256\pi\)
\(24\) 0 0
\(25\) 1.82436 + 1.05329i 0.364871 + 0.210659i
\(26\) 0 0
\(27\) −4.95592 − 1.56167i −0.953768 − 0.300543i
\(28\) 0 0
\(29\) −1.61116 + 1.61116i −0.299184 + 0.299184i −0.840694 − 0.541510i \(-0.817853\pi\)
0.541510 + 0.840694i \(0.317853\pi\)
\(30\) 0 0
\(31\) 2.85614 + 2.85614i 0.512977 + 0.512977i 0.915437 − 0.402460i \(-0.131845\pi\)
−0.402460 + 0.915437i \(0.631845\pi\)
\(32\) 0 0
\(33\) 9.99390 + 1.02044i 1.73971 + 0.177636i
\(34\) 0 0
\(35\) −7.93193 + 2.12535i −1.34074 + 0.359250i
\(36\) 0 0
\(37\) −4.69397 + 3.86868i −0.771684 + 0.636007i
\(38\) 0 0
\(39\) −0.222013 − 0.0844681i −0.0355505 − 0.0135257i
\(40\) 0 0
\(41\) −2.81750 − 4.88006i −0.440020 − 0.762137i 0.557670 − 0.830063i \(-0.311695\pi\)
−0.997690 + 0.0679253i \(0.978362\pi\)
\(42\) 0 0
\(43\) 7.06828 − 7.06828i 1.07790 − 1.07790i 0.0812059 − 0.996697i \(-0.474123\pi\)
0.996697 − 0.0812059i \(-0.0258771\pi\)
\(44\) 0 0
\(45\) 2.28971 − 4.56048i 0.341330 − 0.679836i
\(46\) 0 0
\(47\) 7.92849i 1.15649i 0.815864 + 0.578244i \(0.196262\pi\)
−0.815864 + 0.578244i \(0.803738\pi\)
\(48\) 0 0
\(49\) −8.15277 + 14.1210i −1.16468 + 2.01729i
\(50\) 0 0
\(51\) 3.45843 − 0.558280i 0.484277 − 0.0781749i
\(52\) 0 0
\(53\) −4.24708 − 2.45205i −0.583382 − 0.336816i 0.179094 − 0.983832i \(-0.442683\pi\)
−0.762476 + 0.647016i \(0.776017\pi\)
\(54\) 0 0
\(55\) −2.55345 + 9.52962i −0.344308 + 1.28497i
\(56\) 0 0
\(57\) −3.78984 − 1.44190i −0.501976 − 0.190984i
\(58\) 0 0
\(59\) −4.36135 + 1.16862i −0.567799 + 0.152141i −0.531287 − 0.847192i \(-0.678291\pi\)
−0.0365121 + 0.999333i \(0.511625\pi\)
\(60\) 0 0
\(61\) −3.81348 + 14.2321i −0.488267 + 1.82224i 0.0766114 + 0.997061i \(0.475590\pi\)
−0.564878 + 0.825174i \(0.691077\pi\)
\(62\) 0 0
\(63\) −4.55703 − 13.7471i −0.574132 − 1.73197i
\(64\) 0 0
\(65\) 0.116640 − 0.202027i 0.0144674 − 0.0250583i
\(66\) 0 0
\(67\) −1.58274 + 0.913795i −0.193362 + 0.111638i −0.593556 − 0.804793i \(-0.702276\pi\)
0.400193 + 0.916431i \(0.368943\pi\)
\(68\) 0 0
\(69\) −3.97733 − 4.88191i −0.478814 − 0.587713i
\(70\) 0 0
\(71\) −10.1420 + 5.85550i −1.20364 + 0.694920i −0.961362 − 0.275288i \(-0.911227\pi\)
−0.242274 + 0.970208i \(0.577893\pi\)
\(72\) 0 0
\(73\) − 13.5662i − 1.58780i −0.608048 − 0.793900i \(-0.708047\pi\)
0.608048 − 0.793900i \(-0.291953\pi\)
\(74\) 0 0
\(75\) −2.95822 − 2.13590i −0.341586 − 0.246632i
\(76\) 0 0
\(77\) 13.9999 + 24.2486i 1.59544 + 2.76338i
\(78\) 0 0
\(79\) −6.87939 − 1.84333i −0.773992 − 0.207391i −0.149858 − 0.988708i \(-0.547882\pi\)
−0.624135 + 0.781317i \(0.714548\pi\)
\(80\) 0 0
\(81\) 8.26475 + 3.56284i 0.918306 + 0.395871i
\(82\) 0 0
\(83\) −3.79517 − 2.19114i −0.416574 − 0.240509i 0.277036 − 0.960859i \(-0.410648\pi\)
−0.693611 + 0.720350i \(0.743981\pi\)
\(84\) 0 0
\(85\) 3.44040i 0.373164i
\(86\) 0 0
\(87\) 3.05963 − 2.49270i 0.328027 − 0.267246i
\(88\) 0 0
\(89\) −1.55789 − 5.81411i −0.165136 − 0.616295i −0.998023 − 0.0628518i \(-0.979980\pi\)
0.832887 − 0.553443i \(-0.186686\pi\)
\(90\) 0 0
\(91\) −0.171356 − 0.639508i −0.0179630 − 0.0670387i
\(92\) 0 0
\(93\) −4.41888 − 5.42389i −0.458216 − 0.562431i
\(94\) 0 0
\(95\) 1.99109 − 3.44867i 0.204282 − 0.353826i
\(96\) 0 0
\(97\) 12.1216 − 12.1216i 1.23076 − 1.23076i 0.267094 − 0.963670i \(-0.413936\pi\)
0.963670 − 0.267094i \(-0.0860635\pi\)
\(98\) 0 0
\(99\) −17.0409 − 3.51664i −1.71267 − 0.353435i
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.569.1 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.14 yes 152
37.8 odd 12 inner 888.2.br.a.785.14 yes 152
111.8 even 12 inner 888.2.br.a.785.1 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.1 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.14 yes 152 3.2 odd 2 inner
888.2.br.a.785.1 yes 152 111.8 even 12 inner
888.2.br.a.785.14 yes 152 37.8 odd 12 inner