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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(26,567)] 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("567.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 458.13
Character \(\chi\) \(=\) 567.458
Dual form 567.2.s.g.26.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.41253 + 0.815523i) q^{2} +(0.330156 + 0.571847i) q^{4} +2.00581 q^{5} +(2.28410 + 1.33524i) q^{7} -2.18509i q^{8} +(2.83327 + 1.63579i) q^{10} +1.31085i q^{11} +(-2.65208 - 1.53118i) q^{13} +(2.13744 + 3.74881i) q^{14} +(2.44231 - 4.23020i) q^{16} +(-0.683325 + 1.18355i) q^{17} +(4.92262 - 2.84207i) q^{19} +(0.662231 + 1.14702i) q^{20} +(-1.06902 + 1.85161i) q^{22} +6.58539i q^{23} -0.976718 q^{25} +(-2.49742 - 4.32566i) q^{26} +(-0.00944366 + 1.74700i) q^{28} +(5.41377 - 3.12564i) q^{29} +(-6.20195 + 3.58070i) q^{31} +(3.11496 - 1.79842i) q^{32} +(-1.93043 + 1.11453i) q^{34} +(4.58148 + 2.67824i) q^{35} +(-2.17992 - 3.77573i) q^{37} +9.27111 q^{38} -4.38288i q^{40} +(-2.79967 + 4.84917i) q^{41} +(-1.56902 - 2.71763i) q^{43} +(-0.749603 + 0.432784i) q^{44} +(-5.37054 + 9.30205i) q^{46} +(-3.86407 + 6.69277i) q^{47} +(3.43426 + 6.09966i) q^{49} +(-1.37964 - 0.796536i) q^{50} -2.02211i q^{52} +(8.54669 + 4.93443i) q^{53} +2.62931i q^{55} +(2.91763 - 4.99098i) q^{56} +10.1961 q^{58} +(-5.78441 - 10.0189i) q^{59} +(-12.5188 - 7.22776i) q^{61} -11.6806 q^{62} -3.90260 q^{64} +(-5.31957 - 3.07126i) q^{65} +(-5.21234 - 9.02803i) q^{67} -0.902416 q^{68} +(4.28730 + 7.51940i) q^{70} +9.13734i q^{71} +(1.16332 + 0.671641i) q^{73} -7.11110i q^{74} +(3.25047 + 1.87666i) q^{76} +(-1.75030 + 2.99411i) q^{77} +(6.78595 - 11.7536i) q^{79} +(4.89881 - 8.48498i) q^{80} +(-7.90922 + 4.56639i) q^{82} +(-0.206808 - 0.358202i) q^{83} +(-1.37062 + 2.37398i) q^{85} -5.11830i q^{86} +2.86432 q^{88} +(-2.39824 - 4.15388i) q^{89} +(-4.01313 - 7.03854i) q^{91} +(-3.76584 + 2.17421i) q^{92} +(-10.9162 + 6.30248i) q^{94} +(9.87384 - 5.70067i) q^{95} +(-15.6273 + 9.02240i) q^{97} +(-0.123433 + 11.4167i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} + 4 q^{7} - 12 q^{13} - 28 q^{16} - 12 q^{22} + 32 q^{25} - 16 q^{28} + 48 q^{31} - 4 q^{37} - 28 q^{43} + 12 q^{46} - 16 q^{49} - 72 q^{58} - 12 q^{61} - 80 q^{64} - 20 q^{67} - 60 q^{70}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41253 + 0.815523i 0.998808 + 0.576662i 0.907895 − 0.419197i \(-0.137688\pi\)
0.0909125 + 0.995859i \(0.471022\pi\)
\(3\) 0 0
\(4\) 0.330156 + 0.571847i 0.165078 + 0.285924i
\(5\) 2.00581 0.897026 0.448513 − 0.893776i \(-0.351954\pi\)
0.448513 + 0.893776i \(0.351954\pi\)
\(6\) 0 0
\(7\) 2.28410 + 1.33524i 0.863310 + 0.504674i
\(8\) − 2.18509i − 0.772547i
\(9\) 0 0
\(10\) 2.83327 + 1.63579i 0.895957 + 0.517281i
\(11\) 1.31085i 0.395235i 0.980279 + 0.197617i \(0.0633203\pi\)
−0.980279 + 0.197617i \(0.936680\pi\)
\(12\) 0 0
\(13\) −2.65208 − 1.53118i −0.735554 − 0.424673i 0.0848963 − 0.996390i \(-0.472944\pi\)
−0.820451 + 0.571717i \(0.806277\pi\)
\(14\) 2.13744 + 3.74881i 0.571254 + 1.00191i
\(15\) 0 0
\(16\) 2.44231 − 4.23020i 0.610577 − 1.05755i
\(17\) −0.683325 + 1.18355i −0.165731 + 0.287054i −0.936914 − 0.349559i \(-0.886332\pi\)
0.771184 + 0.636612i \(0.219665\pi\)
\(18\) 0 0
\(19\) 4.92262 − 2.84207i 1.12933 − 0.652016i 0.185560 − 0.982633i \(-0.440590\pi\)
0.943765 + 0.330616i \(0.107257\pi\)
\(20\) 0.662231 + 1.14702i 0.148079 + 0.256481i
\(21\) 0 0
\(22\) −1.06902 + 1.85161i −0.227917 + 0.394764i
\(23\) 6.58539i 1.37315i 0.727060 + 0.686574i \(0.240886\pi\)
−0.727060 + 0.686574i \(0.759114\pi\)
\(24\) 0 0
\(25\) −0.976718 −0.195344
\(26\) −2.49742 − 4.32566i −0.489785 − 0.848332i
\(27\) 0 0
\(28\) −0.00944366 + 1.74700i −0.00178468 + 0.330151i
\(29\) 5.41377 − 3.12564i 1.00531 − 0.580417i 0.0954960 − 0.995430i \(-0.469556\pi\)
0.909816 + 0.415013i \(0.136223\pi\)
\(30\) 0 0
\(31\) −6.20195 + 3.58070i −1.11390 + 0.643112i −0.939837 − 0.341622i \(-0.889024\pi\)
−0.174065 + 0.984734i \(0.555690\pi\)
\(32\) 3.11496 − 1.79842i 0.550652 − 0.317919i
\(33\) 0 0
\(34\) −1.93043 + 1.11453i −0.331066 + 0.191141i
\(35\) 4.58148 + 2.67824i 0.774412 + 0.452706i
\(36\) 0 0
\(37\) −2.17992 − 3.77573i −0.358377 − 0.620727i 0.629313 − 0.777152i \(-0.283336\pi\)
−0.987690 + 0.156425i \(0.950003\pi\)
\(38\) 9.27111 1.50397
\(39\) 0 0
\(40\) − 4.38288i − 0.692995i
\(41\) −2.79967 + 4.84917i −0.437235 + 0.757313i −0.997475 − 0.0710171i \(-0.977376\pi\)
0.560240 + 0.828330i \(0.310709\pi\)
\(42\) 0 0
\(43\) −1.56902 − 2.71763i −0.239274 − 0.414435i 0.721232 − 0.692693i \(-0.243576\pi\)
−0.960506 + 0.278259i \(0.910243\pi\)
\(44\) −0.749603 + 0.432784i −0.113007 + 0.0652446i
\(45\) 0 0
\(46\) −5.37054 + 9.30205i −0.791843 + 1.37151i
\(47\) −3.86407 + 6.69277i −0.563633 + 0.976241i 0.433543 + 0.901133i \(0.357263\pi\)
−0.997175 + 0.0751077i \(0.976070\pi\)
\(48\) 0 0
\(49\) 3.43426 + 6.09966i 0.490608 + 0.871380i
\(50\) −1.37964 − 0.796536i −0.195111 − 0.112647i
\(51\) 0 0
\(52\) − 2.02211i − 0.280417i
\(53\) 8.54669 + 4.93443i 1.17398 + 0.677796i 0.954614 − 0.297847i \(-0.0962684\pi\)
0.219364 + 0.975643i \(0.429602\pi\)
\(54\) 0 0
\(55\) 2.62931i 0.354536i
\(56\) 2.91763 − 4.99098i 0.389884 − 0.666947i
\(57\) 0 0
\(58\) 10.1961 1.33882
\(59\) −5.78441 − 10.0189i −0.753066 − 1.30435i −0.946330 − 0.323202i \(-0.895241\pi\)
0.193264 − 0.981147i \(-0.438093\pi\)
\(60\) 0 0
\(61\) −12.5188 − 7.22776i −1.60287 − 0.925419i −0.990909 − 0.134534i \(-0.957046\pi\)
−0.611964 − 0.790885i \(-0.709620\pi\)
\(62\) −11.6806 −1.48343
\(63\) 0 0
\(64\) −3.90260 −0.487825
\(65\) −5.31957 − 3.07126i −0.659812 − 0.380942i
\(66\) 0 0
\(67\) −5.21234 − 9.02803i −0.636788 − 1.10295i −0.986133 − 0.165955i \(-0.946929\pi\)
0.349345 − 0.936994i \(-0.386404\pi\)
\(68\) −0.902416 −0.109434
\(69\) 0 0
\(70\) 4.28730 + 7.51940i 0.512430 + 0.898740i
\(71\) 9.13734i 1.08440i 0.840249 + 0.542201i \(0.182409\pi\)
−0.840249 + 0.542201i \(0.817591\pi\)
\(72\) 0 0
\(73\) 1.16332 + 0.671641i 0.136156 + 0.0786097i 0.566531 − 0.824041i \(-0.308285\pi\)
−0.430375 + 0.902650i \(0.641619\pi\)
\(74\) − 7.11110i − 0.826649i
\(75\) 0 0
\(76\) 3.25047 + 1.87666i 0.372854 + 0.215267i
\(77\) −1.75030 + 2.99411i −0.199465 + 0.341210i
\(78\) 0 0
\(79\) 6.78595 − 11.7536i 0.763479 − 1.32238i −0.177569 − 0.984108i \(-0.556823\pi\)
0.941047 − 0.338275i \(-0.109844\pi\)
\(80\) 4.89881 − 8.48498i 0.547703 − 0.948650i
\(81\) 0 0
\(82\) −7.90922 + 4.56639i −0.873427 + 0.504274i
\(83\) −0.206808 − 0.358202i −0.0227001 − 0.0393177i 0.854452 − 0.519530i \(-0.173893\pi\)
−0.877152 + 0.480212i \(0.840560\pi\)
\(84\) 0 0
\(85\) −1.37062 + 2.37398i −0.148665 + 0.257495i
\(86\) − 5.11830i − 0.551921i
\(87\) 0 0
\(88\) 2.86432 0.305337
\(89\) −2.39824 − 4.15388i −0.254213 − 0.440310i 0.710468 − 0.703729i \(-0.248483\pi\)
−0.964681 + 0.263419i \(0.915150\pi\)
\(90\) 0 0
\(91\) −4.01313 − 7.03854i −0.420690 − 0.737839i
\(92\) −3.76584 + 2.17421i −0.392616 + 0.226677i
\(93\) 0 0
\(94\) −10.9162 + 6.30248i −1.12592 + 0.650051i
\(95\) 9.87384 − 5.70067i 1.01303 − 0.584876i
\(96\) 0 0
\(97\) −15.6273 + 9.02240i −1.58671 + 0.916086i −0.592863 + 0.805304i \(0.702002\pi\)
−0.993845 + 0.110782i \(0.964664\pi\)
\(98\) −0.123433 + 11.4167i −0.0124686 + 1.15326i
\(99\) 0 0
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 567.2.s.g.458.13 32
3.2 odd 2 inner 567.2.s.g.458.4 32
7.5 odd 6 567.2.i.g.215.4 32
9.2 odd 6 567.2.i.g.269.13 32
9.4 even 3 567.2.p.e.80.4 ✓ 32
9.5 odd 6 567.2.p.e.80.13 yes 32
9.7 even 3 567.2.i.g.269.4 32
21.5 even 6 567.2.i.g.215.13 32
63.5 even 6 567.2.p.e.404.4 yes 32
63.40 odd 6 567.2.p.e.404.13 yes 32
63.47 even 6 inner 567.2.s.g.26.13 32
63.61 odd 6 inner 567.2.s.g.26.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.4 32 7.5 odd 6
567.2.i.g.215.13 32 21.5 even 6
567.2.i.g.269.4 32 9.7 even 3
567.2.i.g.269.13 32 9.2 odd 6
567.2.p.e.80.4 ✓ 32 9.4 even 3
567.2.p.e.80.13 yes 32 9.5 odd 6
567.2.p.e.404.4 yes 32 63.5 even 6
567.2.p.e.404.13 yes 32 63.40 odd 6
567.2.s.g.26.4 32 63.61 odd 6 inner
567.2.s.g.26.13 32 63.47 even 6 inner
567.2.s.g.458.4 32 3.2 odd 2 inner
567.2.s.g.458.13 32 1.1 even 1 trivial