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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 569.36
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.36

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70811 - 0.287007i) q^{3} +(-0.721322 + 2.69201i) q^{5} +(2.47898 + 4.29371i) q^{7} +(2.83525 - 0.980478i) q^{9} -1.92072 q^{11} +(1.17839 + 0.315747i) q^{13} +(-0.459467 + 4.80526i) q^{15} +(-5.26023 + 1.40947i) q^{17} +(-4.56995 - 1.22452i) q^{19} +(5.46668 + 6.62264i) q^{21} +(-2.41576 - 2.41576i) q^{23} +(-2.39648 - 1.38361i) q^{25} +(4.56151 - 2.48850i) q^{27} +(3.35285 - 3.35285i) q^{29} +(2.96309 + 2.96309i) q^{31} +(-3.28079 + 0.551260i) q^{33} +(-13.3469 + 3.57628i) q^{35} +(-5.71600 - 2.08022i) q^{37} +(2.10343 + 0.201125i) q^{39} +(2.27871 + 3.94685i) q^{41} +(3.86120 - 3.86120i) q^{43} +(0.594327 + 8.33977i) q^{45} -1.47732i q^{47} +(-8.79066 + 15.2259i) q^{49} +(-8.58050 + 3.91726i) q^{51} +(11.0443 + 6.37642i) q^{53} +(1.38545 - 5.17059i) q^{55} +(-8.15741 - 0.779992i) q^{57} +(10.7362 - 2.87675i) q^{59} +(2.65045 - 9.89160i) q^{61} +(11.2384 + 9.74319i) q^{63} +(-1.69999 + 2.94447i) q^{65} +(8.89621 - 5.13623i) q^{67} +(-4.81971 - 3.43303i) q^{69} +(1.72584 - 0.996416i) q^{71} -0.224313i q^{73} +(-4.49054 - 1.67554i) q^{75} +(-4.76141 - 8.24701i) q^{77} +(7.22868 + 1.93692i) q^{79} +(7.07732 - 5.55981i) q^{81} +(13.1112 + 7.56973i) q^{83} -15.1773i q^{85} +(4.76473 - 6.68931i) q^{87} +(-1.85313 - 6.91598i) q^{89} +(1.56546 + 5.84238i) q^{91} +(5.91169 + 4.21084i) q^{93} +(6.59281 - 11.4191i) q^{95} +(-10.0950 + 10.0950i) q^{97} +(-5.44572 + 1.88322i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.70811 − 0.287007i 0.986176 − 0.165704i
\(4\) 0 0
\(5\) −0.721322 + 2.69201i −0.322585 + 1.20390i 0.594133 + 0.804367i \(0.297495\pi\)
−0.916718 + 0.399536i \(0.869171\pi\)
\(6\) 0 0
\(7\) 2.47898 + 4.29371i 0.936965 + 1.62287i 0.771091 + 0.636725i \(0.219711\pi\)
0.165874 + 0.986147i \(0.446955\pi\)
\(8\) 0 0
\(9\) 2.83525 − 0.980478i 0.945084 − 0.326826i
\(10\) 0 0
\(11\) −1.92072 −0.579118 −0.289559 − 0.957160i \(-0.593509\pi\)
−0.289559 + 0.957160i \(0.593509\pi\)
\(12\) 0 0
\(13\) 1.17839 + 0.315747i 0.326825 + 0.0875726i 0.418501 − 0.908216i \(-0.362556\pi\)
−0.0916757 + 0.995789i \(0.529222\pi\)
\(14\) 0 0
\(15\) −0.459467 + 4.80526i −0.118634 + 1.24071i
\(16\) 0 0
\(17\) −5.26023 + 1.40947i −1.27579 + 0.341848i −0.832247 − 0.554404i \(-0.812946\pi\)
−0.443545 + 0.896252i \(0.646279\pi\)
\(18\) 0 0
\(19\) −4.56995 − 1.22452i −1.04842 − 0.280923i −0.306821 − 0.951767i \(-0.599265\pi\)
−0.741598 + 0.670844i \(0.765932\pi\)
\(20\) 0 0
\(21\) 5.46668 + 6.62264i 1.19293 + 1.44518i
\(22\) 0 0
\(23\) −2.41576 − 2.41576i −0.503720 − 0.503720i 0.408872 − 0.912592i \(-0.365922\pi\)
−0.912592 + 0.408872i \(0.865922\pi\)
\(24\) 0 0
\(25\) −2.39648 − 1.38361i −0.479296 − 0.276721i
\(26\) 0 0
\(27\) 4.56151 − 2.48850i 0.877863 − 0.478912i
\(28\) 0 0
\(29\) 3.35285 − 3.35285i 0.622608 − 0.622608i −0.323589 − 0.946198i \(-0.604890\pi\)
0.946198 + 0.323589i \(0.104890\pi\)
\(30\) 0 0
\(31\) 2.96309 + 2.96309i 0.532186 + 0.532186i 0.921222 − 0.389037i \(-0.127192\pi\)
−0.389037 + 0.921222i \(0.627192\pi\)
\(32\) 0 0
\(33\) −3.28079 + 0.551260i −0.571112 + 0.0959621i
\(34\) 0 0
\(35\) −13.3469 + 3.57628i −2.25603 + 0.604502i
\(36\) 0 0
\(37\) −5.71600 − 2.08022i −0.939705 − 0.341986i
\(38\) 0 0
\(39\) 2.10343 + 0.201125i 0.336818 + 0.0322057i
\(40\) 0 0
\(41\) 2.27871 + 3.94685i 0.355875 + 0.616394i 0.987267 − 0.159070i \(-0.0508494\pi\)
−0.631392 + 0.775464i \(0.717516\pi\)
\(42\) 0 0
\(43\) 3.86120 − 3.86120i 0.588827 − 0.588827i −0.348486 − 0.937314i \(-0.613304\pi\)
0.937314 + 0.348486i \(0.113304\pi\)
\(44\) 0 0
\(45\) 0.594327 + 8.33977i 0.0885971 + 1.24322i
\(46\) 0 0
\(47\) − 1.47732i − 0.215489i −0.994179 − 0.107745i \(-0.965637\pi\)
0.994179 − 0.107745i \(-0.0343629\pi\)
\(48\) 0 0
\(49\) −8.79066 + 15.2259i −1.25581 + 2.17512i
\(50\) 0 0
\(51\) −8.58050 + 3.91726i −1.20151 + 0.548525i
\(52\) 0 0
\(53\) 11.0443 + 6.37642i 1.51705 + 0.875869i 0.999799 + 0.0200345i \(0.00637760\pi\)
0.517250 + 0.855834i \(0.326956\pi\)
\(54\) 0 0
\(55\) 1.38545 − 5.17059i 0.186815 − 0.697202i
\(56\) 0 0
\(57\) −8.15741 − 0.779992i −1.08048 − 0.103312i
\(58\) 0 0
\(59\) 10.7362 − 2.87675i 1.39773 − 0.374521i 0.520201 − 0.854044i \(-0.325857\pi\)
0.877529 + 0.479523i \(0.159190\pi\)
\(60\) 0 0
\(61\) 2.65045 − 9.89160i 0.339355 − 1.26649i −0.559715 − 0.828685i \(-0.689090\pi\)
0.899070 − 0.437804i \(-0.144244\pi\)
\(62\) 0 0
\(63\) 11.2384 + 9.74319i 1.41591 + 1.22753i
\(64\) 0 0
\(65\) −1.69999 + 2.94447i −0.210858 + 0.365216i
\(66\) 0 0
\(67\) 8.89621 − 5.13623i 1.08684 − 0.627490i 0.154110 − 0.988054i \(-0.450749\pi\)
0.932735 + 0.360564i \(0.117416\pi\)
\(68\) 0 0
\(69\) −4.81971 − 3.43303i −0.580224 − 0.413288i
\(70\) 0 0
\(71\) 1.72584 − 0.996416i 0.204820 − 0.118253i −0.394082 − 0.919075i \(-0.628937\pi\)
0.598902 + 0.800823i \(0.295604\pi\)
\(72\) 0 0
\(73\) − 0.224313i − 0.0262539i −0.999914 − 0.0131269i \(-0.995821\pi\)
0.999914 − 0.0131269i \(-0.00417855\pi\)
\(74\) 0 0
\(75\) −4.49054 − 1.67554i −0.518523 − 0.193475i
\(76\) 0 0
\(77\) −4.76141 − 8.24701i −0.542614 − 0.939834i
\(78\) 0 0
\(79\) 7.22868 + 1.93692i 0.813290 + 0.217921i 0.641411 − 0.767197i \(-0.278349\pi\)
0.171879 + 0.985118i \(0.445016\pi\)
\(80\) 0 0
\(81\) 7.07732 − 5.55981i 0.786369 − 0.617757i
\(82\) 0 0
\(83\) 13.1112 + 7.56973i 1.43914 + 0.830886i 0.997790 − 0.0664499i \(-0.0211673\pi\)
0.441348 + 0.897336i \(0.354501\pi\)
\(84\) 0 0
\(85\) − 15.1773i − 1.64621i
\(86\) 0 0
\(87\) 4.76473 − 6.68931i 0.510832 − 0.717170i
\(88\) 0 0
\(89\) −1.85313 − 6.91598i −0.196432 − 0.733093i −0.991892 − 0.127087i \(-0.959437\pi\)
0.795460 − 0.606006i \(-0.207229\pi\)
\(90\) 0 0
\(91\) 1.56546 + 5.84238i 0.164105 + 0.612448i
\(92\) 0 0
\(93\) 5.91169 + 4.21084i 0.613014 + 0.436643i
\(94\) 0 0
\(95\) 6.59281 − 11.4191i 0.676408 − 1.17157i
\(96\) 0 0
\(97\) −10.0950 + 10.0950i −1.02499 + 1.02499i −0.0253127 + 0.999680i \(0.508058\pi\)
−0.999680 + 0.0253127i \(0.991942\pi\)
\(98\) 0 0
\(99\) −5.44572 + 1.88322i −0.547315 + 0.189271i
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.569.36 yes 152
3.2 odd 2 inner 888.2.br.a.569.29 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.29 yes 152
111.8 even 12 inner 888.2.br.a.785.36 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.29 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.36 yes 152 1.1 even 1 trivial
888.2.br.a.785.29 yes 152 37.8 odd 12 inner
888.2.br.a.785.36 yes 152 111.8 even 12 inner