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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 569.23
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.23

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.595509 + 1.62646i) q^{3} +(0.892825 - 3.33207i) q^{5} +(-1.99103 - 3.44857i) q^{7} +(-2.29074 + 1.93714i) q^{9} -3.36814 q^{11} +(2.63931 + 0.707201i) q^{13} +(5.95116 - 0.532133i) q^{15} +(-0.777980 + 0.208459i) q^{17} +(-7.12280 - 1.90855i) q^{19} +(4.42327 - 5.29198i) q^{21} +(-1.05436 - 1.05436i) q^{23} +(-5.97541 - 3.44990i) q^{25} +(-4.51484 - 2.57221i) q^{27} +(-4.97920 + 4.97920i) q^{29} +(-6.17874 - 6.17874i) q^{31} +(-2.00576 - 5.47815i) q^{33} +(-13.2685 + 3.55528i) q^{35} +(4.71146 - 3.84735i) q^{37} +(0.421499 + 4.71387i) q^{39} +(4.81048 + 8.33200i) q^{41} +(7.91059 - 7.91059i) q^{43} +(4.40946 + 9.36242i) q^{45} -1.70651i q^{47} +(-4.42840 + 7.67022i) q^{49} +(-0.802344 - 1.14121i) q^{51} +(3.10233 + 1.79113i) q^{53} +(-3.00716 + 11.2229i) q^{55} +(-1.13751 - 12.7215i) q^{57} +(7.07465 - 1.89565i) q^{59} +(3.03647 - 11.3323i) q^{61} +(11.2413 + 4.04285i) q^{63} +(4.71288 - 8.16295i) q^{65} +(-7.22475 + 4.17121i) q^{67} +(1.08699 - 2.34275i) q^{69} +(1.70549 - 0.984662i) q^{71} -7.15559i q^{73} +(2.05272 - 11.7732i) q^{75} +(6.70608 + 11.6153i) q^{77} +(16.7356 + 4.48429i) q^{79} +(1.49497 - 8.87497i) q^{81} +(0.914753 + 0.528133i) q^{83} +2.77840i q^{85} +(-11.0636 - 5.13331i) q^{87} +(0.276914 + 1.03346i) q^{89} +(-2.81612 - 10.5099i) q^{91} +(6.36997 - 13.7290i) q^{93} +(-12.7188 + 22.0297i) q^{95} +(2.31047 - 2.31047i) q^{97} +(7.71554 - 6.52457i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 0.595509 + 1.62646i 0.343817 + 0.939037i
\(4\) 0 0
\(5\) 0.892825 − 3.33207i 0.399283 − 1.49015i −0.415077 − 0.909786i \(-0.636245\pi\)
0.814360 − 0.580359i \(-0.197088\pi\)
\(6\) 0 0
\(7\) −1.99103 − 3.44857i −0.752539 − 1.30344i −0.946589 − 0.322443i \(-0.895496\pi\)
0.194050 − 0.980992i \(-0.437838\pi\)
\(8\) 0 0
\(9\) −2.29074 + 1.93714i −0.763579 + 0.645714i
\(10\) 0 0
\(11\) −3.36814 −1.01553 −0.507767 − 0.861495i \(-0.669529\pi\)
−0.507767 + 0.861495i \(0.669529\pi\)
\(12\) 0 0
\(13\) 2.63931 + 0.707201i 0.732013 + 0.196142i 0.605525 − 0.795826i \(-0.292963\pi\)
0.126487 + 0.991968i \(0.459630\pi\)
\(14\) 0 0
\(15\) 5.95116 − 0.532133i 1.53658 − 0.137396i
\(16\) 0 0
\(17\) −0.777980 + 0.208459i −0.188688 + 0.0505587i −0.351925 − 0.936028i \(-0.614473\pi\)
0.163238 + 0.986587i \(0.447806\pi\)
\(18\) 0 0
\(19\) −7.12280 − 1.90855i −1.63408 − 0.437851i −0.678988 − 0.734150i \(-0.737581\pi\)
−0.955095 + 0.296298i \(0.904248\pi\)
\(20\) 0 0
\(21\) 4.42327 − 5.29198i 0.965238 − 1.15480i
\(22\) 0 0
\(23\) −1.05436 − 1.05436i −0.219849 − 0.219849i 0.588586 − 0.808435i \(-0.299685\pi\)
−0.808435 + 0.588586i \(0.799685\pi\)
\(24\) 0 0
\(25\) −5.97541 − 3.44990i −1.19508 − 0.689981i
\(26\) 0 0
\(27\) −4.51484 − 2.57221i −0.868881 − 0.495022i
\(28\) 0 0
\(29\) −4.97920 + 4.97920i −0.924615 + 0.924615i −0.997351 − 0.0727365i \(-0.976827\pi\)
0.0727365 + 0.997351i \(0.476827\pi\)
\(30\) 0 0
\(31\) −6.17874 − 6.17874i −1.10973 − 1.10973i −0.993185 − 0.116549i \(-0.962817\pi\)
−0.116549 − 0.993185i \(-0.537183\pi\)
\(32\) 0 0
\(33\) −2.00576 − 5.47815i −0.349158 − 0.953623i
\(34\) 0 0
\(35\) −13.2685 + 3.55528i −2.24278 + 0.600952i
\(36\) 0 0
\(37\) 4.71146 − 3.84735i 0.774560 − 0.632501i
\(38\) 0 0
\(39\) 0.421499 + 4.71387i 0.0674939 + 0.754824i
\(40\) 0 0
\(41\) 4.81048 + 8.33200i 0.751271 + 1.30124i 0.947207 + 0.320622i \(0.103892\pi\)
−0.195936 + 0.980617i \(0.562775\pi\)
\(42\) 0 0
\(43\) 7.91059 − 7.91059i 1.20635 − 1.20635i 0.234155 − 0.972199i \(-0.424768\pi\)
0.972199 − 0.234155i \(-0.0752324\pi\)
\(44\) 0 0
\(45\) 4.40946 + 9.36242i 0.657323 + 1.39567i
\(46\) 0 0
\(47\) − 1.70651i − 0.248920i −0.992225 − 0.124460i \(-0.960280\pi\)
0.992225 − 0.124460i \(-0.0397198\pi\)
\(48\) 0 0
\(49\) −4.42840 + 7.67022i −0.632629 + 1.09575i
\(50\) 0 0
\(51\) −0.802344 − 1.14121i −0.112351 − 0.159802i
\(52\) 0 0
\(53\) 3.10233 + 1.79113i 0.426138 + 0.246031i 0.697700 − 0.716390i \(-0.254207\pi\)
−0.271562 + 0.962421i \(0.587540\pi\)
\(54\) 0 0
\(55\) −3.00716 + 11.2229i −0.405486 + 1.51329i
\(56\) 0 0
\(57\) −1.13751 − 12.7215i −0.150668 − 1.68500i
\(58\) 0 0
\(59\) 7.07465 − 1.89565i 0.921041 − 0.246792i 0.233011 − 0.972474i \(-0.425142\pi\)
0.688030 + 0.725682i \(0.258476\pi\)
\(60\) 0 0
\(61\) 3.03647 − 11.3323i 0.388781 − 1.45095i −0.443339 − 0.896354i \(-0.646206\pi\)
0.832120 − 0.554596i \(-0.187127\pi\)
\(62\) 0 0
\(63\) 11.2413 + 4.04285i 1.41627 + 0.509352i
\(64\) 0 0
\(65\) 4.71288 − 8.16295i 0.584561 − 1.01249i
\(66\) 0 0
\(67\) −7.22475 + 4.17121i −0.882644 + 0.509595i −0.871529 − 0.490344i \(-0.836871\pi\)
−0.0111147 + 0.999938i \(0.503538\pi\)
\(68\) 0 0
\(69\) 1.08699 − 2.34275i 0.130858 − 0.282034i
\(70\) 0 0
\(71\) 1.70549 − 0.984662i 0.202404 − 0.116858i −0.395372 − 0.918521i \(-0.629384\pi\)
0.597776 + 0.801663i \(0.296051\pi\)
\(72\) 0 0
\(73\) − 7.15559i − 0.837498i −0.908102 − 0.418749i \(-0.862469\pi\)
0.908102 − 0.418749i \(-0.137531\pi\)
\(74\) 0 0
\(75\) 2.05272 − 11.7732i 0.237027 − 1.35945i
\(76\) 0 0
\(77\) 6.70608 + 11.6153i 0.764228 + 1.32368i
\(78\) 0 0
\(79\) 16.7356 + 4.48429i 1.88290 + 0.504522i 0.999345 + 0.0361957i \(0.0115240\pi\)
0.883556 + 0.468326i \(0.155143\pi\)
\(80\) 0 0
\(81\) 1.49497 − 8.87497i 0.166107 − 0.986108i
\(82\) 0 0
\(83\) 0.914753 + 0.528133i 0.100407 + 0.0579701i 0.549363 − 0.835584i \(-0.314871\pi\)
−0.448956 + 0.893554i \(0.648204\pi\)
\(84\) 0 0
\(85\) 2.77840i 0.301360i
\(86\) 0 0
\(87\) −11.0636 − 5.13331i −1.18615 − 0.550349i
\(88\) 0 0
\(89\) 0.276914 + 1.03346i 0.0293528 + 0.109546i 0.979048 − 0.203630i \(-0.0652738\pi\)
−0.949695 + 0.313176i \(0.898607\pi\)
\(90\) 0 0
\(91\) −2.81612 − 10.5099i −0.295209 − 1.10174i
\(92\) 0 0
\(93\) 6.36997 − 13.7290i 0.660535 − 1.42363i
\(94\) 0 0
\(95\) −12.7188 + 22.0297i −1.30492 + 2.26020i
\(96\) 0 0
\(97\) 2.31047 − 2.31047i 0.234593 − 0.234593i −0.580014 − 0.814607i \(-0.696953\pi\)
0.814607 + 0.580014i \(0.196953\pi\)
\(98\) 0 0
\(99\) 7.71554 − 6.52457i 0.775441 − 0.655744i
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.569.23 yes 152
3.2 odd 2 inner 888.2.br.a.569.11 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.11 yes 152
111.8 even 12 inner 888.2.br.a.785.23 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.11 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.23 yes 152 1.1 even 1 trivial
888.2.br.a.785.11 yes 152 37.8 odd 12 inner
888.2.br.a.785.23 yes 152 111.8 even 12 inner