Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(11,288)] 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("288.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([12, 15, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.bf (of order \(24\), degree \(8\), 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: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(368\)
Relative dimension: \(46\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label 11.14
Character \(\chi\) \(=\) 288.11
Dual form 288.2.bf.a.131.14

$q$-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+(-0.931130 - 1.06442i) q^{2} +(-1.32517 + 1.11531i) q^{3} +(-0.265992 + 1.98223i) q^{4} +(0.240456 + 1.82644i) q^{5} +(2.42107 + 0.372039i) q^{6} +(-0.327296 - 1.22149i) q^{7} +(2.35761 - 1.56259i) q^{8} +(0.512151 - 2.95596i) q^{9} +(1.72021 - 1.95660i) q^{10} +(-3.90217 + 2.99424i) q^{11} +(-1.85833 - 2.92346i) q^{12} +(0.296220 - 0.386041i) q^{13} +(-0.995423 + 1.48574i) q^{14} +(-2.35570 - 2.15216i) q^{15} +(-3.85850 - 1.05452i) q^{16} -5.61635 q^{17} +(-3.62327 + 2.20724i) q^{18} +(-0.888394 + 0.367985i) q^{19} +(-3.68439 - 0.00918028i) q^{20} +(1.79606 + 1.25364i) q^{21} +(6.82056 + 1.36553i) q^{22} +(-7.36969 - 1.97470i) q^{23} +(-1.38145 + 4.70017i) q^{24} +(1.55156 - 0.415740i) q^{25} +(-0.686730 + 0.0441515i) q^{26} +(2.61814 + 4.48836i) q^{27} +(2.50833 - 0.323872i) q^{28} +(-0.0749405 - 0.00986610i) q^{29} +(-0.0973469 + 4.51140i) q^{30} +(-3.52968 + 2.03786i) q^{31} +(2.47031 + 5.08896i) q^{32} +(1.83152 - 8.32001i) q^{33} +(5.22955 + 5.97817i) q^{34} +(2.15227 - 0.891501i) q^{35} +(5.72317 + 1.80147i) q^{36} +(-1.40911 + 3.40189i) q^{37} +(1.21890 + 0.602985i) q^{38} +(0.0380154 + 0.841948i) q^{39} +(3.42088 + 3.93030i) q^{40} +(-0.814802 + 3.04088i) q^{41} +(-0.337967 - 3.07907i) q^{42} +(-5.44883 - 7.10106i) q^{43} +(-4.89733 - 8.53144i) q^{44} +(5.52204 + 0.224636i) q^{45} +(4.76022 + 9.68317i) q^{46} +(2.04433 + 1.18029i) q^{47} +(6.28928 - 2.90602i) q^{48} +(4.67727 - 2.70042i) q^{49} +(-1.88723 - 1.26441i) q^{50} +(7.44261 - 6.26399i) q^{51} +(0.686431 + 0.689860i) q^{52} +(1.85750 - 4.48441i) q^{53} +(2.33969 - 6.96605i) q^{54} +(-6.40709 - 6.40709i) q^{55} +(-2.68032 - 2.36836i) q^{56} +(0.766854 - 1.47848i) q^{57} +(0.0592776 + 0.0889550i) q^{58} +(-12.3026 + 1.61967i) q^{59} +(4.89268 - 4.09709i) q^{60} +(-1.32434 + 10.0594i) q^{61} +(5.45574 + 1.85956i) q^{62} +(-3.77829 + 0.341890i) q^{63} +(3.11663 - 7.36795i) q^{64} +(0.776309 + 0.448202i) q^{65} +(-10.5614 + 5.79750i) q^{66} +(3.05893 - 3.98648i) q^{67} +(1.49390 - 11.1329i) q^{68} +(11.9685 - 5.60270i) q^{69} +(-2.95298 - 1.46083i) q^{70} +(8.83986 + 8.83986i) q^{71} +(-3.41150 - 7.76928i) q^{72} +(-2.27334 + 2.27334i) q^{73} +(4.93312 - 1.66772i) q^{74} +(-1.59240 + 2.28140i) q^{75} +(-0.493126 - 1.85888i) q^{76} +(4.93458 + 3.78644i) q^{77} +(0.860791 - 0.824428i) q^{78} +(-3.84864 + 6.66604i) q^{79} +(0.998217 - 7.30088i) q^{80} +(-8.47540 - 3.02780i) q^{81} +(3.99547 - 1.96416i) q^{82} +(16.6163 + 2.18758i) q^{83} +(-2.96274 + 3.22676i) q^{84} +(-1.35048 - 10.2579i) q^{85} +(-2.48495 + 12.4119i) q^{86} +(0.110313 - 0.0705079i) q^{87} +(-4.52101 + 13.1567i) q^{88} +(2.24981 - 2.24981i) q^{89} +(-4.90263 - 6.08695i) q^{90} +(-0.568495 - 0.235479i) q^{91} +(5.87460 - 14.0832i) q^{92} +(2.40457 - 6.63722i) q^{93} +(-0.647205 - 3.27504i) q^{94} +(-0.885722 - 1.53411i) q^{95} +(-8.94937 - 3.98857i) q^{96} +(0.186438 - 0.322920i) q^{97} +(-7.22954 - 2.46415i) q^{98} +(6.85235 + 13.0681i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 368 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9} - 16 q^{10} - 12 q^{11} - 8 q^{12} - 4 q^{13} - 12 q^{14} - 16 q^{15} - 4 q^{16} - 8 q^{18} - 16 q^{19} - 12 q^{20} - 8 q^{21}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.931130 − 1.06442i −0.658409 − 0.752661i
\(3\) −1.32517 + 1.11531i −0.765087 + 0.643927i
\(4\) −0.265992 + 1.98223i −0.132996 + 0.991117i
\(5\) 0.240456 + 1.82644i 0.107535 + 0.816809i 0.957188 + 0.289467i \(0.0934781\pi\)
−0.849653 + 0.527342i \(0.823189\pi\)
\(6\) 2.42107 + 0.372039i 0.988398 + 0.151884i
\(7\) −0.327296 − 1.22149i −0.123706 − 0.461678i 0.876084 − 0.482159i \(-0.160147\pi\)
−0.999790 + 0.0204802i \(0.993480\pi\)
\(8\) 2.35761 − 1.56259i 0.833540 − 0.552459i
\(9\) 0.512151 − 2.95596i 0.170717 − 0.985320i
\(10\) 1.72021 − 1.95660i 0.543978 − 0.618732i
\(11\) −3.90217 + 2.99424i −1.17655 + 0.902796i −0.996722 − 0.0809023i \(-0.974220\pi\)
−0.179825 + 0.983699i \(0.557553\pi\)
\(12\) −1.85833 − 2.92346i −0.536453 − 0.843930i
\(13\) 0.296220 − 0.386041i 0.0821566 − 0.107069i −0.750467 − 0.660908i \(-0.770171\pi\)
0.832623 + 0.553840i \(0.186838\pi\)
\(14\) −0.995423 + 1.48574i −0.266038 + 0.397082i
\(15\) −2.35570 − 2.15216i −0.608239 − 0.555686i
\(16\) −3.85850 − 1.05452i −0.964624 − 0.263629i
\(17\) −5.61635 −1.36216 −0.681082 − 0.732207i \(-0.738490\pi\)
−0.681082 + 0.732207i \(0.738490\pi\)
\(18\) −3.62327 + 2.20724i −0.854013 + 0.520251i
\(19\) −0.888394 + 0.367985i −0.203811 + 0.0844215i −0.482254 − 0.876031i \(-0.660182\pi\)
0.278443 + 0.960453i \(0.410182\pi\)
\(20\) −3.68439 − 0.00918028i −0.823855 − 0.00205277i
\(21\) 1.79606 + 1.25364i 0.391933 + 0.273566i
\(22\) 6.82056 + 1.36553i 1.45415 + 0.291132i
\(23\) −7.36969 − 1.97470i −1.53669 − 0.411754i −0.611493 − 0.791250i \(-0.709431\pi\)
−0.925193 + 0.379496i \(0.876097\pi\)
\(24\) −1.38145 + 4.70017i −0.281988 + 0.959418i
\(25\) 1.55156 − 0.415740i 0.310312 − 0.0831479i
\(26\) −0.686730 + 0.0441515i −0.134679 + 0.00865882i
\(27\) 2.61814 + 4.48836i 0.503860 + 0.863785i
\(28\) 2.50833 − 0.323872i 0.474030 − 0.0612060i
\(29\) −0.0749405 − 0.00986610i −0.0139161 − 0.00183209i 0.123565 − 0.992336i \(-0.460567\pi\)
−0.137481 + 0.990504i \(0.543901\pi\)
\(30\) −0.0973469 + 4.51140i −0.0177730 + 0.823666i
\(31\) −3.52968 + 2.03786i −0.633950 + 0.366011i −0.782280 − 0.622927i \(-0.785943\pi\)
0.148330 + 0.988938i \(0.452610\pi\)
\(32\) 2.47031 + 5.08896i 0.436693 + 0.899610i
\(33\) 1.83152 − 8.32001i 0.318827 − 1.44833i
\(34\) 5.22955 + 5.97817i 0.896861 + 1.02525i
\(35\) 2.15227 − 0.891501i 0.363800 − 0.150691i
\(36\) 5.72317 + 1.80147i 0.953862 + 0.300244i
\(37\) −1.40911 + 3.40189i −0.231656 + 0.559268i −0.996372 − 0.0850996i \(-0.972879\pi\)
0.764716 + 0.644367i \(0.222879\pi\)
\(38\) 1.21890 + 0.602985i 0.197732 + 0.0978171i
\(39\) 0.0380154 + 0.841948i 0.00608733 + 0.134820i
\(40\) 3.42088 + 3.93030i 0.540888 + 0.621435i
\(41\) −0.814802 + 3.04088i −0.127251 + 0.474906i −0.999910 − 0.0134238i \(-0.995727\pi\)
0.872659 + 0.488330i \(0.162394\pi\)
\(42\) −0.337967 − 3.07907i −0.0521495 − 0.475111i
\(43\) −5.44883 − 7.10106i −0.830939 − 1.08290i −0.995332 − 0.0965063i \(-0.969233\pi\)
0.164393 − 0.986395i \(-0.447433\pi\)
\(44\) −4.89733 − 8.53144i −0.738300 − 1.28616i
\(45\) 5.52204 + 0.224636i 0.823177 + 0.0334868i
\(46\) 4.76022 + 9.68317i 0.701856 + 1.42771i
\(47\) 2.04433 + 1.18029i 0.298196 + 0.172163i 0.641632 − 0.767013i \(-0.278258\pi\)
−0.343436 + 0.939176i \(0.611591\pi\)
\(48\) 6.28928 − 2.90602i 0.907780 − 0.419448i
\(49\) 4.67727 − 2.70042i 0.668182 − 0.385775i
\(50\) −1.88723 − 1.26441i −0.266894 − 0.178815i
\(51\) 7.44261 − 6.26399i 1.04217 − 0.877134i
\(52\) 0.686431 + 0.689860i 0.0951909 + 0.0956664i
\(53\) 1.85750 − 4.48441i 0.255148 − 0.615981i −0.743457 − 0.668783i \(-0.766815\pi\)
0.998605 + 0.0528024i \(0.0168153\pi\)
\(54\) 2.33969 − 6.96605i 0.318391 − 0.947960i
\(55\) −6.40709 − 6.40709i −0.863932 − 0.863932i
\(56\) −2.68032 − 2.36836i −0.358173 − 0.316485i
\(57\) 0.766854 − 1.47848i 0.101572 − 0.195829i
\(58\) 0.0592776 + 0.0889550i 0.00778354 + 0.0116804i
\(59\) −12.3026 + 1.61967i −1.60166 + 0.210863i −0.877781 − 0.479061i \(-0.840977\pi\)
−0.723882 + 0.689924i \(0.757644\pi\)
\(60\) 4.89268 − 4.09709i 0.631643 − 0.528932i
\(61\) −1.32434 + 10.0594i −0.169564 + 1.28797i 0.669039 + 0.743227i \(0.266706\pi\)
−0.838604 + 0.544742i \(0.816628\pi\)
\(62\) 5.45574 + 1.85956i 0.692880 + 0.236164i
\(63\) −3.77829 + 0.341890i −0.476020 + 0.0430740i
\(64\) 3.11663 − 7.36795i 0.389579 − 0.920993i
\(65\) 0.776309 + 0.448202i 0.0962893 + 0.0555926i
\(66\) −10.5614 + 5.79750i −1.30002 + 0.713623i
\(67\) 3.05893 − 3.98648i 0.373708 − 0.487026i −0.568203 − 0.822888i \(-0.692361\pi\)
0.941911 + 0.335863i \(0.109028\pi\)
\(68\) 1.49390 − 11.1329i 0.181163 − 1.35006i
\(69\) 11.9685 − 5.60270i 1.44084 − 0.674486i
\(70\) −2.95298 − 1.46083i −0.352949 − 0.174602i
\(71\) 8.83986 + 8.83986i 1.04910 + 1.04910i 0.998731 + 0.0503682i \(0.0160395\pi\)
0.0503682 + 0.998731i \(0.483961\pi\)
\(72\) −3.41150 − 7.76928i −0.402049 − 0.915618i
\(73\) −2.27334 + 2.27334i −0.266075 + 0.266075i −0.827516 − 0.561442i \(-0.810247\pi\)
0.561442 + 0.827516i \(0.310247\pi\)
\(74\) 4.93312 − 1.66772i 0.573463 − 0.193868i
\(75\) −1.59240 + 2.28140i −0.183875 + 0.263434i
\(76\) −0.493126 − 1.85888i −0.0565654 − 0.213229i
\(77\) 4.93458 + 3.78644i 0.562348 + 0.431505i
\(78\) 0.860791 − 0.824428i 0.0974654 − 0.0933481i
\(79\) −3.84864 + 6.66604i −0.433006 + 0.749988i −0.997131 − 0.0757014i \(-0.975880\pi\)
0.564125 + 0.825690i \(0.309214\pi\)
\(80\) 0.998217 − 7.30088i 0.111604 − 0.816263i
\(81\) −8.47540 − 3.02780i −0.941711 − 0.336422i
\(82\) 3.99547 − 1.96416i 0.441226 − 0.216906i
\(83\) 16.6163 + 2.18758i 1.82388 + 0.240118i 0.963154 − 0.268952i \(-0.0866773\pi\)
0.860725 + 0.509070i \(0.170011\pi\)
\(84\) −2.96274 + 3.22676i −0.323262 + 0.352068i
\(85\) −1.35048 − 10.2579i −0.146480 − 1.11263i
\(86\) −2.48495 + 12.4119i −0.267960 + 1.33841i
\(87\) 0.110313 − 0.0705079i 0.0118268 − 0.00755924i
\(88\) −4.52101 + 13.1567i −0.481942 + 1.40251i
\(89\) 2.24981 − 2.24981i 0.238479 − 0.238479i −0.577741 − 0.816220i \(-0.696066\pi\)
0.816220 + 0.577741i \(0.196066\pi\)
\(90\) −4.90263 − 6.08695i −0.516782 − 0.641621i
\(91\) −0.568495 − 0.235479i −0.0595945 − 0.0246849i
\(92\) 5.87460 − 14.0832i 0.612469 − 1.46827i
\(93\) 2.40457 − 6.63722i 0.249343 − 0.688247i
\(94\) −0.647205 − 3.27504i −0.0667541 − 0.337794i
\(95\) −0.885722 − 1.53411i −0.0908731 − 0.157397i
\(96\) −8.94937 − 3.98857i −0.913392 − 0.407082i
\(97\) 0.186438 − 0.322920i 0.0189299 − 0.0327876i −0.856405 − 0.516304i \(-0.827307\pi\)
0.875335 + 0.483517i \(0.160641\pi\)
\(98\) −7.22954 − 2.46415i −0.730294 − 0.248917i
\(99\) 6.85235 + 13.0681i 0.688687 + 1.31340i
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 288.2.bf.a.11.14 ✓ 368
3.2 odd 2 864.2.bn.a.683.33 368
9.4 even 3 864.2.bn.a.395.44 368
9.5 odd 6 inner 288.2.bf.a.203.3 yes 368
32.3 odd 8 inner 288.2.bf.a.227.3 yes 368
96.35 even 8 864.2.bn.a.35.44 368
288.67 odd 24 864.2.bn.a.611.33 368
288.131 even 24 inner 288.2.bf.a.131.14 yes 368
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.bf.a.11.14 ✓ 368 1.1 even 1 trivial
288.2.bf.a.131.14 yes 368 288.131 even 24 inner
288.2.bf.a.203.3 yes 368 9.5 odd 6 inner
288.2.bf.a.227.3 yes 368 32.3 odd 8 inner
864.2.bn.a.35.44 368 96.35 even 8
864.2.bn.a.395.44 368 9.4 even 3
864.2.bn.a.611.33 368 288.67 odd 24
864.2.bn.a.683.33 368 3.2 odd 2