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(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 569.11
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.11080 - 1.32896i) q^{3} +(-0.892825 + 3.33207i) q^{5} +(-1.99103 - 3.44857i) q^{7} +(-0.532245 + 2.95241i) q^{9} +3.36814 q^{11} +(2.63931 + 0.707201i) q^{13} +(5.41992 - 2.51474i) q^{15} +(0.777980 - 0.208459i) q^{17} +(-7.12280 - 1.90855i) q^{19} +(-2.37135 + 6.47666i) q^{21} +(1.05436 + 1.05436i) q^{23} +(-5.97541 - 3.44990i) q^{25} +(4.51484 - 2.57221i) q^{27} +(4.97920 - 4.97920i) q^{29} +(-6.17874 - 6.17874i) q^{31} +(-3.74134 - 4.47611i) q^{33} +(13.2685 - 3.55528i) q^{35} +(4.71146 - 3.84735i) q^{37} +(-1.99191 - 4.29308i) q^{39} +(-4.81048 - 8.33200i) q^{41} +(7.91059 - 7.91059i) q^{43} +(-9.36242 - 4.40946i) q^{45} +1.70651i q^{47} +(-4.42840 + 7.67022i) q^{49} +(-1.14121 - 0.802344i) q^{51} +(-3.10233 - 1.79113i) q^{53} +(-3.00716 + 11.2229i) q^{55} +(5.37564 + 11.5859i) q^{57} +(-7.07465 + 1.89565i) q^{59} +(3.03647 - 11.3323i) q^{61} +(11.2413 - 4.04285i) q^{63} +(-4.71288 + 8.16295i) q^{65} +(-7.22475 + 4.17121i) q^{67} +(0.230014 - 2.57238i) q^{69} +(-1.70549 + 0.984662i) q^{71} -7.15559i q^{73} +(2.05272 + 11.7732i) q^{75} +(-6.70608 - 11.6153i) q^{77} +(16.7356 + 4.48429i) q^{79} +(-8.43343 - 3.14281i) q^{81} +(-0.914753 - 0.528133i) q^{83} +2.77840i q^{85} +(-12.1480 - 1.08624i) q^{87} +(-0.276914 - 1.03346i) q^{89} +(-2.81612 - 10.5099i) q^{91} +(-1.34792 + 15.0746i) q^{93} +(12.7188 - 22.0297i) q^{95} +(2.31047 - 2.31047i) q^{97} +(-1.79268 + 9.94414i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −1.11080 − 1.32896i −0.641321 − 0.767273i
\(4\) 0 0
\(5\) −0.892825 + 3.33207i −0.399283 + 1.49015i 0.415077 + 0.909786i \(0.363755\pi\)
−0.814360 + 0.580359i \(0.802912\pi\)
\(6\) 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
\(9\) −0.532245 + 2.95241i −0.177415 + 0.984136i
\(10\) 0 0
\(11\) 3.36814 1.01553 0.507767 − 0.861495i \(-0.330471\pi\)
0.507767 + 0.861495i \(0.330471\pi\)
\(12\) 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
\(15\) 5.41992 − 2.51474i 1.39942 − 0.649302i
\(16\) 0 0
\(17\) 0.777980 − 0.208459i 0.188688 − 0.0505587i −0.163238 − 0.986587i \(-0.552194\pi\)
0.351925 + 0.936028i \(0.385527\pi\)
\(18\) 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
\(21\) −2.37135 + 6.47666i −0.517471 + 1.41332i
\(22\) 0 0
\(23\) 1.05436 + 1.05436i 0.219849 + 0.219849i 0.808435 − 0.588586i \(-0.200315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(24\) 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 −0.0727365 − 0.997351i \(-0.523173\pi\)
0.997351 + 0.0727365i \(0.0231732\pi\)
\(30\) 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
\(33\) −3.74134 − 4.47611i −0.651283 − 0.779191i
\(34\) 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
\(39\) −1.99191 − 4.29308i −0.318961 − 0.687444i
\(40\) 0 0
\(41\) −4.81048 − 8.33200i −0.751271 − 1.30124i −0.947207 − 0.320622i \(-0.896108\pi\)
0.195936 − 0.980617i \(-0.437225\pi\)
\(42\) 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
\(45\) −9.36242 − 4.40946i −1.39567 − 0.657323i
\(46\) 0 0
\(47\) 1.70651i 0.248920i 0.992225 + 0.124460i \(0.0397198\pi\)
−0.992225 + 0.124460i \(0.960280\pi\)
\(48\) 0 0
\(49\) −4.42840 + 7.67022i −0.632629 + 1.09575i
\(50\) 0 0
\(51\) −1.14121 − 0.802344i −0.159802 − 0.112351i
\(52\) 0 0
\(53\) −3.10233 − 1.79113i −0.426138 − 0.246031i 0.271562 − 0.962421i \(-0.412460\pi\)
−0.697700 + 0.716390i \(0.745793\pi\)
\(54\) 0 0
\(55\) −3.00716 + 11.2229i −0.405486 + 1.51329i
\(56\) 0 0
\(57\) 5.37564 + 11.5859i 0.712021 + 1.53459i
\(58\) 0 0
\(59\) −7.07465 + 1.89565i −0.921041 + 0.246792i −0.688030 − 0.725682i \(-0.741524\pi\)
−0.233011 + 0.972474i \(0.574858\pi\)
\(60\) 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
\(69\) 0.230014 − 2.57238i 0.0276904 − 0.309678i
\(70\) 0 0
\(71\) −1.70549 + 0.984662i −0.202404 + 0.116858i −0.597776 − 0.801663i \(-0.703949\pi\)
0.395372 + 0.918521i \(0.370616\pi\)
\(72\) 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
\(81\) −8.43343 − 3.14281i −0.937048 − 0.349201i
\(82\) 0 0
\(83\) −0.914753 − 0.528133i −0.100407 − 0.0579701i 0.448956 − 0.893554i \(-0.351796\pi\)
−0.549363 + 0.835584i \(0.685129\pi\)
\(84\) 0 0
\(85\) 2.77840i 0.301360i
\(86\) 0 0
\(87\) −12.1480 − 1.08624i −1.30241 − 0.116457i
\(88\) 0 0
\(89\) −0.276914 − 1.03346i −0.0293528 − 0.109546i 0.949695 − 0.313176i \(-0.101393\pi\)
−0.979048 + 0.203630i \(0.934726\pi\)
\(90\) 0 0
\(91\) −2.81612 − 10.5099i −0.295209 − 1.10174i
\(92\) 0 0
\(93\) −1.34792 + 15.0746i −0.139773 + 1.56316i
\(94\) 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
\(99\) −1.79268 + 9.94414i −0.180171 + 0.999424i
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.569.11 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.23 yes 152
37.8 odd 12 inner 888.2.br.a.785.23 yes 152
111.8 even 12 inner 888.2.br.a.785.11 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.11 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.23 yes 152 3.2 odd 2 inner
888.2.br.a.785.11 yes 152 111.8 even 12 inner
888.2.br.a.785.23 yes 152 37.8 odd 12 inner