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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 569.20
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0117331 + 1.73201i) q^{3} +(-1.03088 + 3.84731i) q^{5} +(-0.379322 - 0.657004i) q^{7} +(-2.99972 - 0.0406437i) q^{9} -3.35502 q^{11} +(6.41138 + 1.71792i) q^{13} +(-6.65149 - 1.83064i) q^{15} +(-5.90896 + 1.58330i) q^{17} +(2.47784 + 0.663934i) q^{19} +(1.14239 - 0.649280i) q^{21} +(2.09499 + 2.09499i) q^{23} +(-9.40897 - 5.43227i) q^{25} +(0.105591 - 5.19508i) q^{27} +(-0.264527 + 0.264527i) q^{29} +(-2.04469 - 2.04469i) q^{31} +(0.0393648 - 5.81093i) q^{33} +(2.91874 - 0.782074i) q^{35} +(-5.73981 + 2.01360i) q^{37} +(-3.05069 + 11.0844i) q^{39} +(2.58103 + 4.47048i) q^{41} +(0.470320 - 0.470320i) q^{43} +(3.24874 - 11.4990i) q^{45} -13.4220i q^{47} +(3.21223 - 5.56375i) q^{49} +(-2.67297 - 10.2530i) q^{51} +(-4.60988 - 2.66152i) q^{53} +(3.45864 - 12.9078i) q^{55} +(-1.17901 + 4.28385i) q^{57} +(13.7858 - 3.69391i) q^{59} +(-1.70253 + 6.35391i) q^{61} +(1.11116 + 1.98625i) q^{63} +(-13.2188 + 22.8956i) q^{65} +(-4.28393 + 2.47333i) q^{67} +(-3.65313 + 3.60397i) q^{69} +(-7.31708 + 4.22452i) q^{71} +6.99903i q^{73} +(9.51916 - 16.2327i) q^{75} +(1.27263 + 2.20426i) q^{77} +(-2.73710 - 0.733404i) q^{79} +(8.99670 + 0.243840i) q^{81} +(-2.14535 - 1.23862i) q^{83} -24.3658i q^{85} +(-0.455061 - 0.461268i) q^{87} +(3.69427 + 13.7872i) q^{89} +(-1.30329 - 4.86395i) q^{91} +(3.56542 - 3.51743i) q^{93} +(-5.10873 + 8.84858i) q^{95} +(-7.60305 + 7.60305i) q^{97} +(10.0641 + 0.136360i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −0.0117331 + 1.73201i −0.00677411 + 0.999977i
\(4\) 0 0
\(5\) −1.03088 + 3.84731i −0.461026 + 1.72057i 0.208715 + 0.977976i \(0.433072\pi\)
−0.669741 + 0.742595i \(0.733595\pi\)
\(6\) 0 0
\(7\) −0.379322 − 0.657004i −0.143370 − 0.248324i 0.785394 − 0.618997i \(-0.212461\pi\)
−0.928764 + 0.370672i \(0.879127\pi\)
\(8\) 0 0
\(9\) −2.99972 − 0.0406437i −0.999908 − 0.0135479i
\(10\) 0 0
\(11\) −3.35502 −1.01158 −0.505788 − 0.862658i \(-0.668798\pi\)
−0.505788 + 0.862658i \(0.668798\pi\)
\(12\) 0 0
\(13\) 6.41138 + 1.71792i 1.77820 + 0.476467i 0.990253 − 0.139280i \(-0.0444788\pi\)
0.787944 + 0.615746i \(0.211145\pi\)
\(14\) 0 0
\(15\) −6.65149 − 1.83064i −1.71741 − 0.472670i
\(16\) 0 0
\(17\) −5.90896 + 1.58330i −1.43313 + 0.384007i −0.890124 − 0.455719i \(-0.849382\pi\)
−0.543010 + 0.839726i \(0.682716\pi\)
\(18\) 0 0
\(19\) 2.47784 + 0.663934i 0.568455 + 0.152317i 0.531588 − 0.847003i \(-0.321596\pi\)
0.0368670 + 0.999320i \(0.488262\pi\)
\(20\) 0 0
\(21\) 1.14239 − 0.649280i 0.249290 − 0.141685i
\(22\) 0 0
\(23\) 2.09499 + 2.09499i 0.436836 + 0.436836i 0.890946 − 0.454110i \(-0.150043\pi\)
−0.454110 + 0.890946i \(0.650043\pi\)
\(24\) 0 0
\(25\) −9.40897 − 5.43227i −1.88179 − 1.08645i
\(26\) 0 0
\(27\) 0.105591 − 5.19508i 0.0203211 − 0.999794i
\(28\) 0 0
\(29\) −0.264527 + 0.264527i −0.0491215 + 0.0491215i −0.731241 − 0.682119i \(-0.761058\pi\)
0.682119 + 0.731241i \(0.261058\pi\)
\(30\) 0 0
\(31\) −2.04469 − 2.04469i −0.367237 − 0.367237i 0.499232 − 0.866469i \(-0.333616\pi\)
−0.866469 + 0.499232i \(0.833616\pi\)
\(32\) 0 0
\(33\) 0.0393648 − 5.81093i 0.00685253 − 1.01155i
\(34\) 0 0
\(35\) 2.91874 − 0.782074i 0.493357 − 0.132195i
\(36\) 0 0
\(37\) −5.73981 + 2.01360i −0.943619 + 0.331034i
\(38\) 0 0
\(39\) −3.05069 + 11.0844i −0.488501 + 1.77493i
\(40\) 0 0
\(41\) 2.58103 + 4.47048i 0.403089 + 0.698171i 0.994097 − 0.108495i \(-0.0346031\pi\)
−0.591008 + 0.806666i \(0.701270\pi\)
\(42\) 0 0
\(43\) 0.470320 − 0.470320i 0.0717231 − 0.0717231i −0.670335 − 0.742058i \(-0.733850\pi\)
0.742058 + 0.670335i \(0.233850\pi\)
\(44\) 0 0
\(45\) 3.24874 − 11.4990i 0.484293 − 1.71417i
\(46\) 0 0
\(47\) − 13.4220i − 1.95780i −0.204343 − 0.978899i \(-0.565506\pi\)
0.204343 − 0.978899i \(-0.434494\pi\)
\(48\) 0 0
\(49\) 3.21223 − 5.56375i 0.458890 − 0.794821i
\(50\) 0 0
\(51\) −2.67297 − 10.2530i −0.374290 − 1.43570i
\(52\) 0 0
\(53\) −4.60988 − 2.66152i −0.633216 − 0.365588i 0.148780 − 0.988870i \(-0.452465\pi\)
−0.781997 + 0.623283i \(0.785799\pi\)
\(54\) 0 0
\(55\) 3.45864 − 12.9078i 0.466363 − 1.74049i
\(56\) 0 0
\(57\) −1.17901 + 4.28385i −0.156164 + 0.567410i
\(58\) 0 0
\(59\) 13.7858 − 3.69391i 1.79476 − 0.480906i 0.801624 − 0.597829i \(-0.203970\pi\)
0.993141 + 0.116924i \(0.0373032\pi\)
\(60\) 0 0
\(61\) −1.70253 + 6.35391i −0.217986 + 0.813535i 0.767108 + 0.641518i \(0.221695\pi\)
−0.985094 + 0.172017i \(0.944972\pi\)
\(62\) 0 0
\(63\) 1.11116 + 1.98625i 0.139993 + 0.250244i
\(64\) 0 0
\(65\) −13.2188 + 22.8956i −1.63959 + 2.83985i
\(66\) 0 0
\(67\) −4.28393 + 2.47333i −0.523366 + 0.302165i −0.738311 − 0.674461i \(-0.764376\pi\)
0.214945 + 0.976626i \(0.431043\pi\)
\(68\) 0 0
\(69\) −3.65313 + 3.60397i −0.439785 + 0.433867i
\(70\) 0 0
\(71\) −7.31708 + 4.22452i −0.868378 + 0.501358i −0.866809 − 0.498641i \(-0.833833\pi\)
−0.00156885 + 0.999999i \(0.500499\pi\)
\(72\) 0 0
\(73\) 6.99903i 0.819174i 0.912271 + 0.409587i \(0.134327\pi\)
−0.912271 + 0.409587i \(0.865673\pi\)
\(74\) 0 0
\(75\) 9.51916 − 16.2327i 1.09918 − 1.87439i
\(76\) 0 0
\(77\) 1.27263 + 2.20426i 0.145030 + 0.251199i
\(78\) 0 0
\(79\) −2.73710 − 0.733404i −0.307948 − 0.0825144i 0.101536 − 0.994832i \(-0.467624\pi\)
−0.409484 + 0.912318i \(0.634291\pi\)
\(80\) 0 0
\(81\) 8.99670 + 0.243840i 0.999633 + 0.0270933i
\(82\) 0 0
\(83\) −2.14535 − 1.23862i −0.235483 − 0.135956i 0.377616 − 0.925962i \(-0.376744\pi\)
−0.613099 + 0.790006i \(0.710077\pi\)
\(84\) 0 0
\(85\) − 24.3658i − 2.64285i
\(86\) 0 0
\(87\) −0.455061 − 0.461268i −0.0487876 − 0.0494531i
\(88\) 0 0
\(89\) 3.69427 + 13.7872i 0.391592 + 1.46144i 0.827509 + 0.561453i \(0.189757\pi\)
−0.435917 + 0.899987i \(0.643576\pi\)
\(90\) 0 0
\(91\) −1.30329 − 4.86395i −0.136622 − 0.509881i
\(92\) 0 0
\(93\) 3.56542 − 3.51743i 0.369716 − 0.364741i
\(94\) 0 0
\(95\) −5.10873 + 8.84858i −0.524144 + 0.907845i
\(96\) 0 0
\(97\) −7.60305 + 7.60305i −0.771973 + 0.771973i −0.978451 − 0.206478i \(-0.933800\pi\)
0.206478 + 0.978451i \(0.433800\pi\)
\(98\) 0 0
\(99\) 10.0641 + 0.136360i 1.01148 + 0.0137047i
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.569.20 yes 152
3.2 odd 2 inner 888.2.br.a.569.7 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.7 yes 152
111.8 even 12 inner 888.2.br.a.785.20 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.7 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.20 yes 152 1.1 even 1 trivial
888.2.br.a.785.7 yes 152 37.8 odd 12 inner
888.2.br.a.785.20 yes 152 111.8 even 12 inner