Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,11,Mod(71,270)] 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("270.71"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 270.h (of order \(6\), degree \(2\), not 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: \(171.546458222\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 251.21
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.21

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 + 11.3137i) q^{2} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(15545.1 - 26924.9i) q^{7} +11585.2i q^{8} -31622.8 q^{10} +(-234772. - 135546. i) q^{11} +(276994. + 479767. i) q^{13} +(609241. - 351745. i) q^{14} +(-131072. + 227023. i) q^{16} +534391. i q^{17} +167539. q^{19} +(-619677. - 357771. i) q^{20} +(-3.06705e6 - 5.31228e6i) q^{22} +(747130. - 431356. i) q^{23} +(976562. - 1.69146e6i) q^{25} +1.25353e7i q^{26} +1.59182e7 q^{28} +(2.04189e7 + 1.17889e7i) q^{29} +(-2.24824e6 - 3.89407e6i) q^{31} +(-5.13695e6 + 2.96582e6i) q^{32} +(-6.04595e6 + 1.04719e7i) q^{34} +4.34499e7i q^{35} +5.34610e7 q^{37} +(3.28307e6 + 1.89548e6i) q^{38} +(-8.09543e6 - 1.40217e7i) q^{40} +(5.45882e7 - 3.15165e7i) q^{41} +(2.64717e7 - 4.58503e7i) q^{43} -1.38799e8i q^{44} +1.95209e7 q^{46} +(3.24688e7 + 1.87459e7i) q^{47} +(-3.42062e8 - 5.92469e8i) q^{49} +(3.82733e7 - 2.20971e7i) q^{50} +(-1.41821e8 + 2.45641e8i) q^{52} -7.23937e8i q^{53} +3.78862e8 q^{55} +(3.11931e8 + 1.80094e8i) q^{56} +(2.66751e8 + 4.62027e8i) q^{58} +(-9.77489e8 + 5.64354e8i) q^{59} +(-6.92108e8 + 1.19877e9i) q^{61} -1.01744e8i q^{62} -1.34218e8 q^{64} +(-6.70495e8 - 3.87111e8i) q^{65} +(-5.79876e8 - 1.00437e9i) q^{67} +(-2.36952e8 + 1.36804e8i) q^{68} +(-4.91579e8 + 8.51440e8i) q^{70} -9.14733e8i q^{71} +4.31063e8 q^{73} +(1.04762e9 + 6.04842e8i) q^{74} +(4.28899e7 + 7.42875e7i) q^{76} +(-7.29910e9 + 4.21414e9i) q^{77} +(2.08726e9 - 3.61524e9i) q^{79} -3.66357e8i q^{80} +1.42628e9 q^{82} +(-5.62364e9 - 3.24681e9i) q^{83} +(-3.73417e8 - 6.46777e8i) q^{85} +(1.03747e9 - 5.98986e8i) q^{86} +(1.57033e9 - 2.71989e9i) q^{88} -4.48046e9i q^{89} +1.72236e10 q^{91} +(3.82530e8 + 2.20854e8i) q^{92} +(4.24171e8 + 7.34685e8i) q^{94} +(-2.02773e8 + 1.17071e8i) q^{95} +(4.77216e9 - 8.26562e9i) q^{97} -1.54800e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 19.5959 + 11.3137i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) 0 0
\(7\) 15545.1 − 26924.9i 0.924918 − 1.60200i 0.133224 − 0.991086i \(-0.457467\pi\)
0.791693 − 0.610919i \(-0.209200\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) −234772. − 135546.i −1.45775 − 0.841632i −0.458848 − 0.888515i \(-0.651738\pi\)
−0.998900 + 0.0468829i \(0.985071\pi\)
\(12\) 0 0
\(13\) 276994. + 479767.i 0.746025 + 1.29215i 0.949714 + 0.313117i \(0.101373\pi\)
−0.203690 + 0.979035i \(0.565293\pi\)
\(14\) 609241. − 351745.i 1.13279 − 0.654016i
\(15\) 0 0
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 534391.i 0.376370i 0.982134 + 0.188185i \(0.0602604\pi\)
−0.982134 + 0.188185i \(0.939740\pi\)
\(18\) 0 0
\(19\) 167539. 0.0676624 0.0338312 − 0.999428i \(-0.489229\pi\)
0.0338312 + 0.999428i \(0.489229\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 0 0
\(22\) −3.06705e6 − 5.31228e6i −0.595123 − 1.03078i
\(23\) 747130. − 431356.i 0.116080 − 0.0670187i −0.440836 − 0.897588i \(-0.645318\pi\)
0.556916 + 0.830569i \(0.311985\pi\)
\(24\) 0 0
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 1.25353e7i 1.05504i
\(27\) 0 0
\(28\) 1.59182e7 0.924918
\(29\) 2.04189e7 + 1.17889e7i 0.995503 + 0.574754i 0.906915 − 0.421315i \(-0.138431\pi\)
0.0885882 + 0.996068i \(0.471765\pi\)
\(30\) 0 0
\(31\) −2.24824e6 − 3.89407e6i −0.0785297 − 0.136018i 0.824086 − 0.566465i \(-0.191689\pi\)
−0.902616 + 0.430447i \(0.858356\pi\)
\(32\) −5.13695e6 + 2.96582e6i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −6.04595e6 + 1.04719e7i −0.133067 + 0.230478i
\(35\) 4.34499e7i 0.827272i
\(36\) 0 0
\(37\) 5.34610e7 0.770954 0.385477 − 0.922718i \(-0.374037\pi\)
0.385477 + 0.922718i \(0.374037\pi\)
\(38\) 3.28307e6 + 1.89548e6i 0.0414346 + 0.0239223i
\(39\) 0 0
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) 5.45882e7 − 3.15165e7i 0.471172 − 0.272031i −0.245558 − 0.969382i \(-0.578971\pi\)
0.716730 + 0.697350i \(0.245638\pi\)
\(42\) 0 0
\(43\) 2.64717e7 − 4.58503e7i 0.180069 − 0.311889i −0.761835 − 0.647771i \(-0.775701\pi\)
0.941904 + 0.335883i \(0.109035\pi\)
\(44\) − 1.38799e8i − 0.841632i
\(45\) 0 0
\(46\) 1.95209e7 0.0947788
\(47\) 3.24688e7 + 1.87459e7i 0.141572 + 0.0817366i 0.569113 − 0.822259i \(-0.307287\pi\)
−0.427541 + 0.903996i \(0.640620\pi\)
\(48\) 0 0
\(49\) −3.42062e8 − 5.92469e8i −1.21095 − 2.09742i
\(50\) 3.82733e7 − 2.20971e7i 0.122474 − 0.0707107i
\(51\) 0 0
\(52\) −1.41821e8 + 2.45641e8i −0.373012 + 0.646076i
\(53\) − 7.23937e8i − 1.73110i −0.500825 − 0.865549i \(-0.666970\pi\)
0.500825 − 0.865549i \(-0.333030\pi\)
\(54\) 0 0
\(55\) 3.78862e8 0.752778
\(56\) 3.11931e8 + 1.80094e8i 0.566394 + 0.327008i
\(57\) 0 0
\(58\) 2.66751e8 + 4.62027e8i 0.406412 + 0.703927i
\(59\) −9.77489e8 + 5.64354e8i −1.36726 + 0.789390i −0.990578 − 0.136952i \(-0.956269\pi\)
−0.376685 + 0.926341i \(0.622936\pi\)
\(60\) 0 0
\(61\) −6.92108e8 + 1.19877e9i −0.819454 + 1.41934i 0.0866315 + 0.996240i \(0.472390\pi\)
−0.906085 + 0.423095i \(0.860944\pi\)
\(62\) − 1.01744e8i − 0.111058i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) −6.70495e8 − 3.87111e8i −0.577868 − 0.333632i
\(66\) 0 0
\(67\) −5.79876e8 − 1.00437e9i −0.429498 − 0.743912i 0.567331 − 0.823490i \(-0.307976\pi\)
−0.996829 + 0.0795780i \(0.974643\pi\)
\(68\) −2.36952e8 + 1.36804e8i −0.162973 + 0.0940924i
\(69\) 0 0
\(70\) −4.91579e8 + 8.51440e8i −0.292485 + 0.506598i
\(71\) − 9.14733e8i − 0.506994i −0.967336 − 0.253497i \(-0.918419\pi\)
0.967336 − 0.253497i \(-0.0815808\pi\)
\(72\) 0 0
\(73\) 4.31063e8 0.207934 0.103967 − 0.994581i \(-0.466846\pi\)
0.103967 + 0.994581i \(0.466846\pi\)
\(74\) 1.04762e9 + 6.04842e8i 0.472111 + 0.272573i
\(75\) 0 0
\(76\) 4.28899e7 + 7.42875e7i 0.0169156 + 0.0292987i
\(77\) −7.29910e9 + 4.21414e9i −2.69660 + 1.55688i
\(78\) 0 0
\(79\) 2.08726e9 − 3.61524e9i 0.678331 − 1.17490i −0.297153 − 0.954830i \(-0.596037\pi\)
0.975483 − 0.220073i \(-0.0706296\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) 1.42628e9 0.384711
\(83\) −5.62364e9 − 3.24681e9i −1.42767 − 0.824264i −0.430730 − 0.902481i \(-0.641744\pi\)
−0.996936 + 0.0782172i \(0.975077\pi\)
\(84\) 0 0
\(85\) −3.73417e8 − 6.46777e8i −0.0841588 − 0.145767i
\(86\) 1.03747e9 − 5.98986e8i 0.220539 − 0.127328i
\(87\) 0 0
\(88\) 1.57033e9 − 2.71989e9i 0.297562 − 0.515392i
\(89\) − 4.48046e9i − 0.802365i −0.915998 − 0.401183i \(-0.868599\pi\)
0.915998 − 0.401183i \(-0.131401\pi\)
\(90\) 0 0
\(91\) 1.72236e10 2.76005
\(92\) 3.82530e8 + 2.20854e8i 0.0580399 + 0.0335094i
\(93\) 0 0
\(94\) 4.24171e8 + 7.34685e8i 0.0577965 + 0.100106i
\(95\) −2.02773e8 + 1.17071e8i −0.0262055 + 0.0151298i
\(96\) 0 0
\(97\) 4.77216e9 − 8.26562e9i 0.555720 − 0.962536i −0.442127 − 0.896953i \(-0.645776\pi\)
0.997847 − 0.0655833i \(-0.0208908\pi\)
\(98\) − 1.54800e10i − 1.71254i
\(99\) 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 270.11.h.a.251.21 80
3.2 odd 2 90.11.h.a.11.17 ✓ 80
9.4 even 3 90.11.h.a.41.17 yes 80
9.5 odd 6 inner 270.11.h.a.71.21 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.17 ✓ 80 3.2 odd 2
90.11.h.a.41.17 yes 80 9.4 even 3
270.11.h.a.71.21 80 9.5 odd 6 inner
270.11.h.a.251.21 80 1.1 even 1 trivial