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 569.38
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.38

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.73067 - 0.0692453i) q^{3} +(-0.476115 + 1.77688i) q^{5} +(-2.08448 - 3.61043i) q^{7} +(2.99041 - 0.239681i) q^{9} -2.49905 q^{11} +(5.42126 + 1.45262i) q^{13} +(-0.700954 + 3.10816i) q^{15} +(4.54452 - 1.21770i) q^{17} +(3.13671 + 0.840480i) q^{19} +(-3.85755 - 6.10411i) q^{21} +(1.79911 + 1.79911i) q^{23} +(1.39950 + 0.808000i) q^{25} +(5.15880 - 0.621880i) q^{27} +(6.12943 - 6.12943i) q^{29} +(-3.95854 - 3.95854i) q^{31} +(-4.32502 + 0.173047i) q^{33} +(7.40777 - 1.98491i) q^{35} +(1.62085 - 5.86283i) q^{37} +(9.48297 + 2.13861i) q^{39} +(2.65064 + 4.59104i) q^{41} +(-5.38702 + 5.38702i) q^{43} +(-0.997892 + 5.42773i) q^{45} +5.49932i q^{47} +(-5.19014 + 8.98958i) q^{49} +(7.78072 - 2.42212i) q^{51} +(-10.7697 - 6.21787i) q^{53} +(1.18983 - 4.44052i) q^{55} +(5.48680 + 1.23739i) q^{57} +(-3.91796 + 1.04981i) q^{59} +(-0.537886 + 2.00742i) q^{61} +(-7.09881 - 10.2971i) q^{63} +(-5.16228 + 8.94133i) q^{65} +(12.0150 - 6.93687i) q^{67} +(3.23824 + 2.98908i) q^{69} +(-13.3444 + 7.70441i) q^{71} +11.3648i q^{73} +(2.47801 + 1.30147i) q^{75} +(5.20922 + 9.02264i) q^{77} +(-9.49928 - 2.54533i) q^{79} +(8.88511 - 1.43349i) q^{81} +(-6.83566 - 3.94657i) q^{83} +8.65484i q^{85} +(10.1836 - 11.0324i) q^{87} +(-2.26255 - 8.44397i) q^{89} +(-6.05593 - 22.6010i) q^{91} +(-7.12502 - 6.57679i) q^{93} +(-2.98687 + 5.17341i) q^{95} +(0.136059 - 0.136059i) q^{97} +(-7.47318 + 0.598974i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.73067 − 0.0692453i 0.999201 − 0.0399788i
\(4\) 0 0
\(5\) −0.476115 + 1.77688i −0.212925 + 0.794647i 0.773962 + 0.633232i \(0.218272\pi\)
−0.986887 + 0.161414i \(0.948394\pi\)
\(6\) 0 0
\(7\) −2.08448 − 3.61043i −0.787860 − 1.36461i −0.927275 − 0.374380i \(-0.877855\pi\)
0.139415 − 0.990234i \(-0.455478\pi\)
\(8\) 0 0
\(9\) 2.99041 − 0.239681i 0.996803 − 0.0798937i
\(10\) 0 0
\(11\) −2.49905 −0.753491 −0.376746 − 0.926317i \(-0.622957\pi\)
−0.376746 + 0.926317i \(0.622957\pi\)
\(12\) 0 0
\(13\) 5.42126 + 1.45262i 1.50359 + 0.402885i 0.914299 − 0.405039i \(-0.132742\pi\)
0.589287 + 0.807924i \(0.299409\pi\)
\(14\) 0 0
\(15\) −0.700954 + 3.10816i −0.180986 + 0.802524i
\(16\) 0 0
\(17\) 4.54452 − 1.21770i 1.10221 − 0.295336i 0.338544 − 0.940951i \(-0.390066\pi\)
0.763663 + 0.645615i \(0.223399\pi\)
\(18\) 0 0
\(19\) 3.13671 + 0.840480i 0.719611 + 0.192819i 0.599998 − 0.800001i \(-0.295168\pi\)
0.119613 + 0.992821i \(0.461835\pi\)
\(20\) 0 0
\(21\) −3.85755 − 6.10411i −0.841786 − 1.33203i
\(22\) 0 0
\(23\) 1.79911 + 1.79911i 0.375140 + 0.375140i 0.869345 − 0.494205i \(-0.164541\pi\)
−0.494205 + 0.869345i \(0.664541\pi\)
\(24\) 0 0
\(25\) 1.39950 + 0.808000i 0.279899 + 0.161600i
\(26\) 0 0
\(27\) 5.15880 − 0.621880i 0.992812 − 0.119681i
\(28\) 0 0
\(29\) 6.12943 − 6.12943i 1.13821 − 1.13821i 0.149435 − 0.988772i \(-0.452255\pi\)
0.988772 − 0.149435i \(-0.0477454\pi\)
\(30\) 0 0
\(31\) −3.95854 − 3.95854i −0.710974 − 0.710974i 0.255765 − 0.966739i \(-0.417673\pi\)
−0.966739 + 0.255765i \(0.917673\pi\)
\(32\) 0 0
\(33\) −4.32502 + 0.173047i −0.752889 + 0.0301237i
\(34\) 0 0
\(35\) 7.40777 − 1.98491i 1.25214 − 0.335510i
\(36\) 0 0
\(37\) 1.62085 − 5.86283i 0.266467 − 0.963844i
\(38\) 0 0
\(39\) 9.48297 + 2.13861i 1.51849 + 0.342451i
\(40\) 0 0
\(41\) 2.65064 + 4.59104i 0.413960 + 0.717000i 0.995319 − 0.0966477i \(-0.0308120\pi\)
−0.581359 + 0.813647i \(0.697479\pi\)
\(42\) 0 0
\(43\) −5.38702 + 5.38702i −0.821513 + 0.821513i −0.986325 − 0.164812i \(-0.947298\pi\)
0.164812 + 0.986325i \(0.447298\pi\)
\(44\) 0 0
\(45\) −0.997892 + 5.42773i −0.148757 + 0.809118i
\(46\) 0 0
\(47\) 5.49932i 0.802158i 0.916043 + 0.401079i \(0.131365\pi\)
−0.916043 + 0.401079i \(0.868635\pi\)
\(48\) 0 0
\(49\) −5.19014 + 8.98958i −0.741448 + 1.28423i
\(50\) 0 0
\(51\) 7.78072 − 2.42212i 1.08952 − 0.339164i
\(52\) 0 0
\(53\) −10.7697 − 6.21787i −1.47933 − 0.854090i −0.479602 − 0.877486i \(-0.659219\pi\)
−0.999726 + 0.0233961i \(0.992552\pi\)
\(54\) 0 0
\(55\) 1.18983 − 4.44052i 0.160437 − 0.598759i
\(56\) 0 0
\(57\) 5.48680 + 1.23739i 0.726745 + 0.163896i
\(58\) 0 0
\(59\) −3.91796 + 1.04981i −0.510075 + 0.136674i −0.504672 − 0.863311i \(-0.668386\pi\)
−0.00540291 + 0.999985i \(0.501720\pi\)
\(60\) 0 0
\(61\) −0.537886 + 2.00742i −0.0688692 + 0.257023i −0.991773 − 0.128006i \(-0.959142\pi\)
0.922904 + 0.385030i \(0.125809\pi\)
\(62\) 0 0
\(63\) −7.09881 − 10.2971i −0.894366 − 1.29731i
\(64\) 0 0
\(65\) −5.16228 + 8.94133i −0.640302 + 1.10904i
\(66\) 0 0
\(67\) 12.0150 − 6.93687i 1.46787 − 0.847474i 0.468515 − 0.883455i \(-0.344789\pi\)
0.999352 + 0.0359815i \(0.0114557\pi\)
\(68\) 0 0
\(69\) 3.23824 + 2.98908i 0.389838 + 0.359843i
\(70\) 0 0
\(71\) −13.3444 + 7.70441i −1.58369 + 0.914346i −0.589379 + 0.807856i \(0.700628\pi\)
−0.994314 + 0.106489i \(0.966039\pi\)
\(72\) 0 0
\(73\) 11.3648i 1.33015i 0.746779 + 0.665073i \(0.231599\pi\)
−0.746779 + 0.665073i \(0.768401\pi\)
\(74\) 0 0
\(75\) 2.47801 + 1.30147i 0.286136 + 0.150281i
\(76\) 0 0
\(77\) 5.20922 + 9.02264i 0.593646 + 1.02822i
\(78\) 0 0
\(79\) −9.49928 − 2.54533i −1.06875 − 0.286371i −0.318772 − 0.947831i \(-0.603271\pi\)
−0.749981 + 0.661460i \(0.769937\pi\)
\(80\) 0 0
\(81\) 8.88511 − 1.43349i 0.987234 − 0.159277i
\(82\) 0 0
\(83\) −6.83566 − 3.94657i −0.750311 − 0.433192i 0.0754955 − 0.997146i \(-0.475946\pi\)
−0.825806 + 0.563954i \(0.809279\pi\)
\(84\) 0 0
\(85\) 8.65484i 0.938749i
\(86\) 0 0
\(87\) 10.1836 − 11.0324i 1.09179 − 1.18280i
\(88\) 0 0
\(89\) −2.26255 − 8.44397i −0.239830 − 0.895059i −0.975912 − 0.218166i \(-0.929993\pi\)
0.736082 − 0.676893i \(-0.236674\pi\)
\(90\) 0 0
\(91\) −6.05593 − 22.6010i −0.634834 − 2.36923i
\(92\) 0 0
\(93\) −7.12502 − 6.57679i −0.738830 − 0.681982i
\(94\) 0 0
\(95\) −2.98687 + 5.17341i −0.306446 + 0.530781i
\(96\) 0 0
\(97\) 0.136059 − 0.136059i 0.0138147 − 0.0138147i −0.700166 − 0.713980i \(-0.746891\pi\)
0.713980 + 0.700166i \(0.246891\pi\)
\(98\) 0 0
\(99\) −7.47318 + 0.598974i −0.751083 + 0.0601992i
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.569.38 yes 152
3.2 odd 2 inner 888.2.br.a.569.26 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.26 yes 152
111.8 even 12 inner 888.2.br.a.785.38 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.26 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.38 yes 152 1.1 even 1 trivial
888.2.br.a.785.26 yes 152 37.8 odd 12 inner
888.2.br.a.785.38 yes 152 111.8 even 12 inner