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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 569.15
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.546883 + 1.64345i) q^{3} +(0.245456 - 0.916055i) q^{5} +(-0.268154 - 0.464456i) q^{7} +(-2.40184 - 1.79755i) q^{9} +2.15491 q^{11} +(-4.73322 - 1.26826i) q^{13} +(1.37125 + 0.904369i) q^{15} +(-7.20293 + 1.93002i) q^{17} +(-0.618377 - 0.165694i) q^{19} +(0.909958 - 0.186694i) q^{21} +(-3.45779 - 3.45779i) q^{23} +(3.55122 + 2.05030i) q^{25} +(4.26770 - 2.96425i) q^{27} +(-4.81676 + 4.81676i) q^{29} +(-5.47926 - 5.47926i) q^{31} +(-1.17849 + 3.54149i) q^{33} +(-0.491287 + 0.131640i) q^{35} +(-3.08394 + 5.24303i) q^{37} +(4.67284 - 7.08521i) q^{39} +(-2.52574 - 4.37472i) q^{41} +(5.21814 - 5.21814i) q^{43} +(-2.23620 + 1.75900i) q^{45} +9.16245i q^{47} +(3.35619 - 5.81309i) q^{49} +(0.767275 - 12.8931i) q^{51} +(-6.26907 - 3.61945i) q^{53} +(0.528937 - 1.97402i) q^{55} +(0.610489 - 0.925655i) q^{57} +(-9.47946 + 2.54001i) q^{59} +(3.71091 - 13.8493i) q^{61} +(-0.190819 + 1.59757i) q^{63} +(-2.32360 + 4.02459i) q^{65} +(8.60720 - 4.96937i) q^{67} +(7.57371 - 3.79169i) q^{69} +(-4.24180 + 2.44900i) q^{71} +0.462418i q^{73} +(-5.31166 + 4.71497i) q^{75} +(-0.577848 - 1.00086i) q^{77} +(6.71860 + 1.80024i) q^{79} +(2.53765 + 8.63483i) q^{81} +(1.63527 + 0.944121i) q^{83} +7.07201i q^{85} +(-5.28189 - 10.5503i) q^{87} +(-0.448156 - 1.67254i) q^{89} +(0.680179 + 2.53846i) q^{91} +(12.0014 - 6.00837i) q^{93} +(-0.303569 + 0.525797i) q^{95} +(-9.43307 + 9.43307i) q^{97} +(-5.17575 - 3.87356i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −0.546883 + 1.64345i −0.315743 + 0.948845i
\(4\) 0 0
\(5\) 0.245456 − 0.916055i 0.109771 − 0.409672i −0.889071 − 0.457769i \(-0.848649\pi\)
0.998843 + 0.0480964i \(0.0153155\pi\)
\(6\) 0 0
\(7\) −0.268154 − 0.464456i −0.101353 − 0.175548i 0.810890 − 0.585199i \(-0.198984\pi\)
−0.912242 + 0.409651i \(0.865650\pi\)
\(8\) 0 0
\(9\) −2.40184 − 1.79755i −0.800613 − 0.599182i
\(10\) 0 0
\(11\) 2.15491 0.649731 0.324865 − 0.945760i \(-0.394681\pi\)
0.324865 + 0.945760i \(0.394681\pi\)
\(12\) 0 0
\(13\) −4.73322 − 1.26826i −1.31276 − 0.351753i −0.466499 − 0.884522i \(-0.654485\pi\)
−0.846261 + 0.532769i \(0.821151\pi\)
\(14\) 0 0
\(15\) 1.37125 + 0.904369i 0.354056 + 0.233507i
\(16\) 0 0
\(17\) −7.20293 + 1.93002i −1.74697 + 0.468098i −0.983974 − 0.178310i \(-0.942937\pi\)
−0.762992 + 0.646408i \(0.776270\pi\)
\(18\) 0 0
\(19\) −0.618377 − 0.165694i −0.141865 − 0.0380127i 0.187188 − 0.982324i \(-0.440063\pi\)
−0.329053 + 0.944311i \(0.606729\pi\)
\(20\) 0 0
\(21\) 0.909958 − 0.186694i 0.198569 − 0.0407399i
\(22\) 0 0
\(23\) −3.45779 − 3.45779i −0.721000 − 0.721000i 0.247809 − 0.968809i \(-0.420289\pi\)
−0.968809 + 0.247809i \(0.920289\pi\)
\(24\) 0 0
\(25\) 3.55122 + 2.05030i 0.710244 + 0.410059i
\(26\) 0 0
\(27\) 4.26770 − 2.96425i 0.821319 − 0.570469i
\(28\) 0 0
\(29\) −4.81676 + 4.81676i −0.894450 + 0.894450i −0.994938 − 0.100488i \(-0.967960\pi\)
0.100488 + 0.994938i \(0.467960\pi\)
\(30\) 0 0
\(31\) −5.47926 − 5.47926i −0.984105 − 0.984105i 0.0157708 − 0.999876i \(-0.494980\pi\)
−0.999876 + 0.0157708i \(0.994980\pi\)
\(32\) 0 0
\(33\) −1.17849 + 3.54149i −0.205148 + 0.616494i
\(34\) 0 0
\(35\) −0.491287 + 0.131640i −0.0830427 + 0.0222512i
\(36\) 0 0
\(37\) −3.08394 + 5.24303i −0.506997 + 0.861948i
\(38\) 0 0
\(39\) 4.67284 − 7.08521i 0.748253 − 1.13454i
\(40\) 0 0
\(41\) −2.52574 − 4.37472i −0.394455 − 0.683216i 0.598577 − 0.801066i \(-0.295733\pi\)
−0.993031 + 0.117850i \(0.962400\pi\)
\(42\) 0 0
\(43\) 5.21814 − 5.21814i 0.795759 − 0.795759i −0.186664 − 0.982424i \(-0.559768\pi\)
0.982424 + 0.186664i \(0.0597677\pi\)
\(44\) 0 0
\(45\) −2.23620 + 1.75900i −0.333353 + 0.262216i
\(46\) 0 0
\(47\) 9.16245i 1.33648i 0.743946 + 0.668240i \(0.232952\pi\)
−0.743946 + 0.668240i \(0.767048\pi\)
\(48\) 0 0
\(49\) 3.35619 − 5.81309i 0.479455 − 0.830441i
\(50\) 0 0
\(51\) 0.767275 − 12.8931i 0.107440 − 1.80540i
\(52\) 0 0
\(53\) −6.26907 − 3.61945i −0.861123 − 0.497170i 0.00326508 − 0.999995i \(-0.498961\pi\)
−0.864388 + 0.502825i \(0.832294\pi\)
\(54\) 0 0
\(55\) 0.528937 − 1.97402i 0.0713218 − 0.266177i
\(56\) 0 0
\(57\) 0.610489 − 0.925655i 0.0808612 − 0.122606i
\(58\) 0 0
\(59\) −9.47946 + 2.54001i −1.23412 + 0.330682i −0.816183 − 0.577794i \(-0.803914\pi\)
−0.417938 + 0.908476i \(0.637247\pi\)
\(60\) 0 0
\(61\) 3.71091 − 13.8493i 0.475133 − 1.77322i −0.145754 − 0.989321i \(-0.546561\pi\)
0.620887 − 0.783900i \(-0.286773\pi\)
\(62\) 0 0
\(63\) −0.190819 + 1.59757i −0.0240410 + 0.201275i
\(64\) 0 0
\(65\) −2.32360 + 4.02459i −0.288207 + 0.499189i
\(66\) 0 0
\(67\) 8.60720 − 4.96937i 1.05154 − 0.607105i 0.128458 − 0.991715i \(-0.458997\pi\)
0.923079 + 0.384610i \(0.125664\pi\)
\(68\) 0 0
\(69\) 7.57371 − 3.79169i 0.911768 − 0.456466i
\(70\) 0 0
\(71\) −4.24180 + 2.44900i −0.503409 + 0.290643i −0.730120 − 0.683319i \(-0.760536\pi\)
0.226712 + 0.973962i \(0.427203\pi\)
\(72\) 0 0
\(73\) 0.462418i 0.0541219i 0.999634 + 0.0270610i \(0.00861483\pi\)
−0.999634 + 0.0270610i \(0.991385\pi\)
\(74\) 0 0
\(75\) −5.31166 + 4.71497i −0.613337 + 0.544438i
\(76\) 0 0
\(77\) −0.577848 − 1.00086i −0.0658519 − 0.114059i
\(78\) 0 0
\(79\) 6.71860 + 1.80024i 0.755901 + 0.202543i 0.616134 − 0.787641i \(-0.288698\pi\)
0.139767 + 0.990184i \(0.455365\pi\)
\(80\) 0 0
\(81\) 2.53765 + 8.63483i 0.281961 + 0.959426i
\(82\) 0 0
\(83\) 1.63527 + 0.944121i 0.179494 + 0.103631i 0.587055 − 0.809547i \(-0.300287\pi\)
−0.407561 + 0.913178i \(0.633621\pi\)
\(84\) 0 0
\(85\) 7.07201i 0.767067i
\(86\) 0 0
\(87\) −5.28189 − 10.5503i −0.566278 − 1.13111i
\(88\) 0 0
\(89\) −0.448156 − 1.67254i −0.0475045 − 0.177289i 0.938097 − 0.346371i \(-0.112586\pi\)
−0.985602 + 0.169082i \(0.945920\pi\)
\(90\) 0 0
\(91\) 0.680179 + 2.53846i 0.0713021 + 0.266103i
\(92\) 0 0
\(93\) 12.0014 − 6.00837i 1.24449 − 0.623038i
\(94\) 0 0
\(95\) −0.303569 + 0.525797i −0.0311455 + 0.0539456i
\(96\) 0 0
\(97\) −9.43307 + 9.43307i −0.957783 + 0.957783i −0.999144 − 0.0413609i \(-0.986831\pi\)
0.0413609 + 0.999144i \(0.486831\pi\)
\(98\) 0 0
\(99\) −5.17575 − 3.87356i −0.520183 − 0.389307i
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.569.15 yes 152
3.2 odd 2 inner 888.2.br.a.569.2 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.2 yes 152
111.8 even 12 inner 888.2.br.a.785.15 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.2 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.15 yes 152 1.1 even 1 trivial
888.2.br.a.785.2 yes 152 37.8 odd 12 inner
888.2.br.a.785.15 yes 152 111.8 even 12 inner