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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 569.6
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.56355 + 0.745185i) q^{3} +(0.212050 - 0.791381i) q^{5} +(-0.149206 - 0.258432i) q^{7} +(1.88940 - 2.33027i) q^{9} -1.40612 q^{11} +(2.07555 + 0.556143i) q^{13} +(0.258174 + 1.39538i) q^{15} +(-4.03302 + 1.08064i) q^{17} +(-4.30531 - 1.15360i) q^{19} +(0.425871 + 0.292886i) q^{21} +(-3.40295 - 3.40295i) q^{23} +(3.74881 + 2.16438i) q^{25} +(-1.21769 + 5.05146i) q^{27} +(2.74173 - 2.74173i) q^{29} +(-0.659194 - 0.659194i) q^{31} +(2.19854 - 1.04782i) q^{33} +(-0.236157 + 0.0632781i) q^{35} +(-3.30552 - 5.10622i) q^{37} +(-3.65967 + 0.677114i) q^{39} +(-5.90794 - 10.2328i) q^{41} +(-4.89804 + 4.89804i) q^{43} +(-1.44349 - 1.98937i) q^{45} -10.9310i q^{47} +(3.45548 - 5.98506i) q^{49} +(5.50055 - 4.69499i) q^{51} +(-6.63976 - 3.83347i) q^{53} +(-0.298167 + 1.11277i) q^{55} +(7.59123 - 1.40453i) q^{57} +(1.53365 - 0.410939i) q^{59} +(-0.777010 + 2.89984i) q^{61} +(-0.884126 - 0.140590i) q^{63} +(0.880242 - 1.52462i) q^{65} +(-1.51826 + 0.876567i) q^{67} +(7.85652 + 2.78486i) q^{69} +(2.84275 - 1.64126i) q^{71} +0.542470i q^{73} +(-7.47432 - 0.590559i) q^{75} +(0.209801 + 0.363386i) q^{77} +(4.66772 + 1.25071i) q^{79} +(-1.86036 - 8.80563i) q^{81} +(-7.59339 - 4.38405i) q^{83} +3.42080i q^{85} +(-2.24374 + 6.32994i) q^{87} +(-1.75938 - 6.56608i) q^{89} +(-0.165959 - 0.619369i) q^{91} +(1.52191 + 0.539463i) q^{93} +(-1.82588 + 3.16252i) q^{95} +(7.62300 - 7.62300i) q^{97} +(-2.65672 + 3.27664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −1.56355 + 0.745185i −0.902718 + 0.430233i
\(4\) 0 0
\(5\) 0.212050 − 0.791381i 0.0948316 − 0.353916i −0.902162 − 0.431397i \(-0.858021\pi\)
0.996994 + 0.0774806i \(0.0246876\pi\)
\(6\) 0 0
\(7\) −0.149206 − 0.258432i −0.0563945 − 0.0976781i 0.836450 − 0.548043i \(-0.184627\pi\)
−0.892844 + 0.450365i \(0.851294\pi\)
\(8\) 0 0
\(9\) 1.88940 − 2.33027i 0.629799 − 0.776758i
\(10\) 0 0
\(11\) −1.40612 −0.423961 −0.211980 − 0.977274i \(-0.567991\pi\)
−0.211980 + 0.977274i \(0.567991\pi\)
\(12\) 0 0
\(13\) 2.07555 + 0.556143i 0.575655 + 0.154246i 0.534889 − 0.844922i \(-0.320353\pi\)
0.0407662 + 0.999169i \(0.487020\pi\)
\(14\) 0 0
\(15\) 0.258174 + 1.39538i 0.0666603 + 0.360286i
\(16\) 0 0
\(17\) −4.03302 + 1.08064i −0.978150 + 0.262094i −0.712265 − 0.701910i \(-0.752331\pi\)
−0.265885 + 0.964005i \(0.585664\pi\)
\(18\) 0 0
\(19\) −4.30531 − 1.15360i −0.987706 − 0.264655i −0.271419 − 0.962461i \(-0.587493\pi\)
−0.716287 + 0.697806i \(0.754160\pi\)
\(20\) 0 0
\(21\) 0.425871 + 0.292886i 0.0929326 + 0.0639130i
\(22\) 0 0
\(23\) −3.40295 − 3.40295i −0.709564 − 0.709564i 0.256880 − 0.966443i \(-0.417306\pi\)
−0.966443 + 0.256880i \(0.917306\pi\)
\(24\) 0 0
\(25\) 3.74881 + 2.16438i 0.749762 + 0.432875i
\(26\) 0 0
\(27\) −1.21769 + 5.05146i −0.234344 + 0.972154i
\(28\) 0 0
\(29\) 2.74173 − 2.74173i 0.509126 − 0.509126i −0.405132 − 0.914258i \(-0.632774\pi\)
0.914258 + 0.405132i \(0.132774\pi\)
\(30\) 0 0
\(31\) −0.659194 − 0.659194i −0.118395 − 0.118395i 0.645427 − 0.763822i \(-0.276679\pi\)
−0.763822 + 0.645427i \(0.776679\pi\)
\(32\) 0 0
\(33\) 2.19854 − 1.04782i 0.382717 − 0.182402i
\(34\) 0 0
\(35\) −0.236157 + 0.0632781i −0.0399178 + 0.0106959i
\(36\) 0 0
\(37\) −3.30552 − 5.10622i −0.543425 − 0.839458i
\(38\) 0 0
\(39\) −3.65967 + 0.677114i −0.586016 + 0.108425i
\(40\) 0 0
\(41\) −5.90794 − 10.2328i −0.922665 − 1.59810i −0.795275 − 0.606249i \(-0.792673\pi\)
−0.127390 − 0.991853i \(-0.540660\pi\)
\(42\) 0 0
\(43\) −4.89804 + 4.89804i −0.746945 + 0.746945i −0.973904 − 0.226960i \(-0.927121\pi\)
0.226960 + 0.973904i \(0.427121\pi\)
\(44\) 0 0
\(45\) −1.44349 − 1.98937i −0.215182 − 0.296557i
\(46\) 0 0
\(47\) − 10.9310i − 1.59444i −0.603687 − 0.797222i \(-0.706302\pi\)
0.603687 − 0.797222i \(-0.293698\pi\)
\(48\) 0 0
\(49\) 3.45548 − 5.98506i 0.493639 − 0.855008i
\(50\) 0 0
\(51\) 5.50055 − 4.69499i 0.770232 − 0.657430i
\(52\) 0 0
\(53\) −6.63976 − 3.83347i −0.912041 − 0.526567i −0.0309539 − 0.999521i \(-0.509855\pi\)
−0.881087 + 0.472954i \(0.843188\pi\)
\(54\) 0 0
\(55\) −0.298167 + 1.11277i −0.0402048 + 0.150047i
\(56\) 0 0
\(57\) 7.59123 − 1.40453i 1.00548 − 0.186035i
\(58\) 0 0
\(59\) 1.53365 − 0.410939i 0.199664 − 0.0534997i −0.157601 − 0.987503i \(-0.550376\pi\)
0.357265 + 0.934003i \(0.383709\pi\)
\(60\) 0 0
\(61\) −0.777010 + 2.89984i −0.0994859 + 0.371287i −0.997661 − 0.0683507i \(-0.978226\pi\)
0.898175 + 0.439637i \(0.144893\pi\)
\(62\) 0 0
\(63\) −0.884126 − 0.140590i −0.111389 − 0.0177127i
\(64\) 0 0
\(65\) 0.880242 − 1.52462i 0.109181 − 0.189106i
\(66\) 0 0
\(67\) −1.51826 + 0.876567i −0.185485 + 0.107090i −0.589867 − 0.807500i \(-0.700820\pi\)
0.404382 + 0.914590i \(0.367487\pi\)
\(68\) 0 0
\(69\) 7.85652 + 2.78486i 0.945813 + 0.335258i
\(70\) 0 0
\(71\) 2.84275 − 1.64126i 0.337372 − 0.194782i −0.321737 − 0.946829i \(-0.604267\pi\)
0.659109 + 0.752047i \(0.270933\pi\)
\(72\) 0 0
\(73\) 0.542470i 0.0634913i 0.999496 + 0.0317457i \(0.0101067\pi\)
−0.999496 + 0.0317457i \(0.989893\pi\)
\(74\) 0 0
\(75\) −7.47432 − 0.590559i −0.863061 − 0.0681919i
\(76\) 0 0
\(77\) 0.209801 + 0.363386i 0.0239090 + 0.0414116i
\(78\) 0 0
\(79\) 4.66772 + 1.25071i 0.525159 + 0.140716i 0.511652 − 0.859193i \(-0.329034\pi\)
0.0135071 + 0.999909i \(0.495700\pi\)
\(80\) 0 0
\(81\) −1.86036 − 8.80563i −0.206706 − 0.978403i
\(82\) 0 0
\(83\) −7.59339 − 4.38405i −0.833483 − 0.481212i 0.0215608 − 0.999768i \(-0.493136\pi\)
−0.855044 + 0.518556i \(0.826470\pi\)
\(84\) 0 0
\(85\) 3.42080i 0.371038i
\(86\) 0 0
\(87\) −2.24374 + 6.32994i −0.240554 + 0.678640i
\(88\) 0 0
\(89\) −1.75938 − 6.56608i −0.186494 − 0.696004i −0.994306 − 0.106564i \(-0.966015\pi\)
0.807812 − 0.589440i \(-0.200651\pi\)
\(90\) 0 0
\(91\) −0.165959 − 0.619369i −0.0173973 − 0.0649275i
\(92\) 0 0
\(93\) 1.52191 + 0.539463i 0.157814 + 0.0559398i
\(94\) 0 0
\(95\) −1.82588 + 3.16252i −0.187331 + 0.324467i
\(96\) 0 0
\(97\) 7.62300 − 7.62300i 0.773998 − 0.773998i −0.204805 − 0.978803i \(-0.565656\pi\)
0.978803 + 0.204805i \(0.0656559\pi\)
\(98\) 0 0
\(99\) −2.65672 + 3.27664i −0.267010 + 0.329315i
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.569.6 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.8 yes 152
37.8 odd 12 inner 888.2.br.a.785.8 yes 152
111.8 even 12 inner 888.2.br.a.785.6 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.6 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.8 yes 152 3.2 odd 2 inner
888.2.br.a.785.6 yes 152 111.8 even 12 inner
888.2.br.a.785.8 yes 152 37.8 odd 12 inner