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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 569.26
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.26

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.925301 - 1.46418i) q^{3} +(0.476115 - 1.77688i) q^{5} +(-2.08448 - 3.61043i) q^{7} +(-1.28764 - 2.70961i) q^{9} +2.49905 q^{11} +(5.42126 + 1.45262i) q^{13} +(-2.16112 - 2.34127i) q^{15} +(-4.54452 + 1.21770i) q^{17} +(3.13671 + 0.840480i) q^{19} +(-7.21509 - 0.288681i) q^{21} +(-1.79911 - 1.79911i) q^{23} +(1.39950 + 0.808000i) q^{25} +(-5.15880 - 0.621880i) q^{27} +(-6.12943 + 6.12943i) q^{29} +(-3.95854 - 3.95854i) q^{31} +(2.31237 - 3.65905i) q^{33} +(-7.40777 + 1.98491i) q^{35} +(1.62085 - 5.86283i) q^{37} +(7.14319 - 6.59357i) q^{39} +(-2.65064 - 4.59104i) q^{41} +(-5.38702 + 5.38702i) q^{43} +(-5.42773 + 0.997892i) q^{45} -5.49932i q^{47} +(-5.19014 + 8.98958i) q^{49} +(-2.42212 + 7.78072i) q^{51} +(10.7697 + 6.21787i) q^{53} +(1.18983 - 4.44052i) q^{55} +(4.13302 - 3.81501i) q^{57} +(3.91796 - 1.04981i) q^{59} +(-0.537886 + 2.00742i) q^{61} +(-7.09881 + 10.2971i) q^{63} +(5.16228 - 8.94133i) q^{65} +(12.0150 - 6.93687i) q^{67} +(-4.29894 + 0.969499i) q^{69} +(13.3444 - 7.70441i) q^{71} +11.3648i q^{73} +(2.47801 - 1.30147i) q^{75} +(-5.20922 - 9.02264i) q^{77} +(-9.49928 - 2.54533i) q^{79} +(-5.68399 + 6.97798i) q^{81} +(6.83566 + 3.94657i) q^{83} +8.65484i q^{85} +(3.30301 + 14.6461i) q^{87} +(2.26255 + 8.44397i) q^{89} +(-6.05593 - 22.6010i) q^{91} +(-9.45884 + 2.13316i) q^{93} +(2.98687 - 5.17341i) q^{95} +(0.136059 - 0.136059i) q^{97} +(-3.21786 - 6.77145i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 0.925301 − 1.46418i 0.534223 − 0.845344i
\(4\) 0 0
\(5\) 0.476115 − 1.77688i 0.212925 − 0.794647i −0.773962 − 0.633232i \(-0.781728\pi\)
0.986887 − 0.161414i \(-0.0516055\pi\)
\(6\) 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
\(9\) −1.28764 − 2.70961i −0.429212 − 0.903204i
\(10\) 0 0
\(11\) 2.49905 0.753491 0.376746 − 0.926317i \(-0.377043\pi\)
0.376746 + 0.926317i \(0.377043\pi\)
\(12\) 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
\(15\) −2.16112 − 2.34127i −0.558000 − 0.604513i
\(16\) 0 0
\(17\) −4.54452 + 1.21770i −1.10221 + 0.295336i −0.763663 − 0.645615i \(-0.776601\pi\)
−0.338544 + 0.940951i \(0.609934\pi\)
\(18\) 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
\(21\) −7.21509 − 0.288681i −1.57446 − 0.0629954i
\(22\) 0 0
\(23\) −1.79911 − 1.79911i −0.375140 − 0.375140i 0.494205 − 0.869345i \(-0.335459\pi\)
−0.869345 + 0.494205i \(0.835459\pi\)
\(24\) 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 \(0.547745\pi\)
−0.988772 + 0.149435i \(0.952255\pi\)
\(30\) 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
\(33\) 2.31237 − 3.65905i 0.402532 − 0.636959i
\(34\) 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
\(39\) 7.14319 − 6.59357i 1.14383 − 1.05582i
\(40\) 0 0
\(41\) −2.65064 − 4.59104i −0.413960 − 0.717000i 0.581359 − 0.813647i \(-0.302521\pi\)
−0.995319 + 0.0966477i \(0.969188\pi\)
\(42\) 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
\(45\) −5.42773 + 0.997892i −0.809118 + 0.148757i
\(46\) 0 0
\(47\) − 5.49932i − 0.802158i −0.916043 − 0.401079i \(-0.868635\pi\)
0.916043 − 0.401079i \(-0.131365\pi\)
\(48\) 0 0
\(49\) −5.19014 + 8.98958i −0.741448 + 1.28423i
\(50\) 0 0
\(51\) −2.42212 + 7.78072i −0.339164 + 1.08952i
\(52\) 0 0
\(53\) 10.7697 + 6.21787i 1.47933 + 0.854090i 0.999726 − 0.0233961i \(-0.00744790\pi\)
0.479602 + 0.877486i \(0.340781\pi\)
\(54\) 0 0
\(55\) 1.18983 − 4.44052i 0.160437 − 0.598759i
\(56\) 0 0
\(57\) 4.13302 − 3.81501i 0.547431 − 0.505310i
\(58\) 0 0
\(59\) 3.91796 − 1.04981i 0.510075 − 0.136674i 0.00540291 − 0.999985i \(-0.498280\pi\)
0.504672 + 0.863311i \(0.331614\pi\)
\(60\) 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
\(69\) −4.29894 + 0.969499i −0.517531 + 0.116714i
\(70\) 0 0
\(71\) 13.3444 − 7.70441i 1.58369 − 0.914346i 0.589379 − 0.807856i \(-0.299372\pi\)
0.994314 − 0.106489i \(-0.0339610\pi\)
\(72\) 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
\(81\) −5.68399 + 6.97798i −0.631555 + 0.775331i
\(82\) 0 0
\(83\) 6.83566 + 3.94657i 0.750311 + 0.433192i 0.825806 − 0.563954i \(-0.190721\pi\)
−0.0754955 + 0.997146i \(0.524054\pi\)
\(84\) 0 0
\(85\) 8.65484i 0.938749i
\(86\) 0 0
\(87\) 3.30301 + 14.6461i 0.354120 + 1.57023i
\(88\) 0 0
\(89\) 2.26255 + 8.44397i 0.239830 + 0.895059i 0.975912 + 0.218166i \(0.0700073\pi\)
−0.736082 + 0.676893i \(0.763326\pi\)
\(90\) 0 0
\(91\) −6.05593 − 22.6010i −0.634834 − 2.36923i
\(92\) 0 0
\(93\) −9.45884 + 2.13316i −0.980836 + 0.221199i
\(94\) 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
\(99\) −3.21786 − 6.77145i −0.323407 − 0.680556i
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.569.26 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.38 yes 152
37.8 odd 12 inner 888.2.br.a.785.38 yes 152
111.8 even 12 inner 888.2.br.a.785.26 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.26 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.38 yes 152 3.2 odd 2 inner
888.2.br.a.785.26 yes 152 111.8 even 12 inner
888.2.br.a.785.38 yes 152 37.8 odd 12 inner