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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 71.40
Character \(\chi\) \(=\) 270.71
Dual form 270.11.h.a.251.40

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 - 11.3137i) q^{2} +(256.000 - 443.405i) q^{4} +(-1210.31 - 698.771i) q^{5} +(-13757.7 - 23829.0i) q^{7} -11585.2i q^{8} -31622.8 q^{10} +(-241492. + 139425. i) q^{11} +(-54722.5 + 94782.1i) q^{13} +(-539189. - 311301. i) q^{14} +(-131072. - 227023. i) q^{16} +253377. i q^{17} -4.61535e6 q^{19} +(-619677. + 357771. i) q^{20} +(-3.15484e6 + 5.46434e6i) q^{22} +(1.38985e6 + 802432. i) q^{23} +(976562. + 1.69146e6i) q^{25} +2.47646e6i q^{26} -1.40879e7 q^{28} +(1.52112e7 - 8.78220e6i) q^{29} +(-1.21626e7 + 2.10662e7i) q^{31} +(-5.13695e6 - 2.96582e6i) q^{32} +(2.86664e6 + 4.96516e6i) q^{34} +3.84539e7i q^{35} -2.74413e7 q^{37} +(-9.04421e7 + 5.22168e7i) q^{38} +(-8.09543e6 + 1.40217e7i) q^{40} +(9.92296e7 + 5.72902e7i) q^{41} +(8.76234e7 + 1.51768e8i) q^{43} +1.42772e8i q^{44} +3.63139e7 q^{46} +(2.20818e8 - 1.27489e8i) q^{47} +(-2.37311e8 + 4.11034e8i) q^{49} +(3.82733e7 + 2.20971e7i) q^{50} +(2.80179e7 + 4.85284e7i) q^{52} -2.33866e8i q^{53} +3.89706e8 q^{55} +(-2.76065e8 + 1.59386e8i) q^{56} +(1.98718e8 - 3.44190e8i) q^{58} +(-3.36140e8 - 1.94070e8i) q^{59} +(-5.76153e8 - 9.97926e8i) q^{61} +5.50416e8i q^{62} -1.34218e8 q^{64} +(1.32462e8 - 7.64770e7i) q^{65} +(5.55398e8 - 9.61978e8i) q^{67} +(1.12349e8 + 6.48645e7i) q^{68} +(4.35057e8 + 7.53540e8i) q^{70} -2.28480e9i q^{71} -1.52546e9 q^{73} +(-5.37738e8 + 3.10463e8i) q^{74} +(-1.18153e9 + 2.04647e9i) q^{76} +(6.64474e9 + 3.83635e9i) q^{77} +(-2.77361e9 - 4.80404e9i) q^{79} +3.66357e8i q^{80} +2.59266e9 q^{82} +(3.73356e9 - 2.15557e9i) q^{83} +(1.77053e8 - 3.06664e8i) q^{85} +(3.43412e9 + 1.98269e9i) q^{86} +(1.61528e9 + 2.79774e9i) q^{88} +6.44176e9i q^{89} +3.01142e9 q^{91} +(7.11604e8 - 4.10845e8i) q^{92} +(2.88476e9 - 4.99655e9i) q^{94} +(5.58600e9 + 3.22508e9i) q^{95} +(-3.71029e9 - 6.42641e9i) q^{97} +1.07395e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(7\) −13757.7 − 23829.0i −0.818569 − 1.41780i −0.906736 − 0.421698i \(-0.861434\pi\)
0.0881668 − 0.996106i \(-0.471899\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) −241492. + 139425.i −1.49947 + 0.865722i −1.00000 0.000606189i \(-0.999807\pi\)
−0.499475 + 0.866328i \(0.666474\pi\)
\(12\) 0 0
\(13\) −54722.5 + 94782.1i −0.147384 + 0.255276i −0.930260 − 0.366902i \(-0.880418\pi\)
0.782876 + 0.622178i \(0.213752\pi\)
\(14\) −539189. − 311301.i −1.00254 − 0.578816i
\(15\) 0 0
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) 253377.i 0.178453i 0.996011 + 0.0892263i \(0.0284394\pi\)
−0.996011 + 0.0892263i \(0.971561\pi\)
\(18\) 0 0
\(19\) −4.61535e6 −1.86396 −0.931981 − 0.362507i \(-0.881921\pi\)
−0.931981 + 0.362507i \(0.881921\pi\)
\(20\) −619677. + 357771.i −0.193649 + 0.111803i
\(21\) 0 0
\(22\) −3.15484e6 + 5.46434e6i −0.612158 + 1.06029i
\(23\) 1.38985e6 + 802432.i 0.215938 + 0.124672i 0.604068 − 0.796933i \(-0.293545\pi\)
−0.388130 + 0.921605i \(0.626879\pi\)
\(24\) 0 0
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) 2.47646e6i 0.208432i
\(27\) 0 0
\(28\) −1.40879e7 −0.818569
\(29\) 1.52112e7 − 8.78220e6i 0.741607 − 0.428167i −0.0810463 − 0.996710i \(-0.525826\pi\)
0.822653 + 0.568543i \(0.192493\pi\)
\(30\) 0 0
\(31\) −1.21626e7 + 2.10662e7i −0.424833 + 0.735832i −0.996405 − 0.0847200i \(-0.973000\pi\)
0.571572 + 0.820552i \(0.306334\pi\)
\(32\) −5.13695e6 − 2.96582e6i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 2.86664e6 + 4.96516e6i 0.0630925 + 0.109279i
\(35\) 3.84539e7i 0.732151i
\(36\) 0 0
\(37\) −2.74413e7 −0.395728 −0.197864 − 0.980229i \(-0.563400\pi\)
−0.197864 + 0.980229i \(0.563400\pi\)
\(38\) −9.04421e7 + 5.22168e7i −1.14144 + 0.659010i
\(39\) 0 0
\(40\) −8.09543e6 + 1.40217e7i −0.0790569 + 0.136931i
\(41\) 9.92296e7 + 5.72902e7i 0.856489 + 0.494494i 0.862835 − 0.505486i \(-0.168687\pi\)
−0.00634585 + 0.999980i \(0.502020\pi\)
\(42\) 0 0
\(43\) 8.76234e7 + 1.51768e8i 0.596043 + 1.03238i 0.993399 + 0.114713i \(0.0365947\pi\)
−0.397355 + 0.917665i \(0.630072\pi\)
\(44\) 1.42772e8i 0.865722i
\(45\) 0 0
\(46\) 3.63139e7 0.176313
\(47\) 2.20818e8 − 1.27489e8i 0.962821 − 0.555885i 0.0657811 − 0.997834i \(-0.479046\pi\)
0.897040 + 0.441949i \(0.145713\pi\)
\(48\) 0 0
\(49\) −2.37311e8 + 4.11034e8i −0.840112 + 1.45512i
\(50\) 3.82733e7 + 2.20971e7i 0.122474 + 0.0707107i
\(51\) 0 0
\(52\) 2.80179e7 + 4.85284e7i 0.0736918 + 0.127638i
\(53\) − 2.33866e8i − 0.559225i −0.960113 − 0.279613i \(-0.909794\pi\)
0.960113 − 0.279613i \(-0.0902060\pi\)
\(54\) 0 0
\(55\) 3.89706e8 0.774325
\(56\) −2.76065e8 + 1.59386e8i −0.501269 + 0.289408i
\(57\) 0 0
\(58\) 1.98718e8 − 3.44190e8i 0.302760 − 0.524395i
\(59\) −3.36140e8 − 1.94070e8i −0.470176 − 0.271456i 0.246138 − 0.969235i \(-0.420838\pi\)
−0.716313 + 0.697779i \(0.754172\pi\)
\(60\) 0 0
\(61\) −5.76153e8 − 9.97926e8i −0.682164 − 1.18154i −0.974319 − 0.225171i \(-0.927706\pi\)
0.292156 − 0.956371i \(-0.405627\pi\)
\(62\) 5.50416e8i 0.600804i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 1.32462e8 − 7.64770e7i 0.114163 − 0.0659119i
\(66\) 0 0
\(67\) 5.55398e8 − 9.61978e8i 0.411368 − 0.712510i −0.583672 − 0.811990i \(-0.698385\pi\)
0.995040 + 0.0994797i \(0.0317178\pi\)
\(68\) 1.12349e8 + 6.48645e7i 0.0772722 + 0.0446131i
\(69\) 0 0
\(70\) 4.35057e8 + 7.53540e8i 0.258854 + 0.448349i
\(71\) − 2.28480e9i − 1.26636i −0.774005 − 0.633180i \(-0.781749\pi\)
0.774005 − 0.633180i \(-0.218251\pi\)
\(72\) 0 0
\(73\) −1.52546e9 −0.735846 −0.367923 − 0.929856i \(-0.619931\pi\)
−0.367923 + 0.929856i \(0.619931\pi\)
\(74\) −5.37738e8 + 3.10463e8i −0.242333 + 0.139911i
\(75\) 0 0
\(76\) −1.18153e9 + 2.04647e9i −0.465990 + 0.807119i
\(77\) 6.64474e9 + 3.83635e9i 2.45485 + 1.41731i
\(78\) 0 0
\(79\) −2.77361e9 − 4.80404e9i −0.901385 − 1.56124i −0.825698 − 0.564113i \(-0.809218\pi\)
−0.0756875 − 0.997132i \(-0.524115\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 0 0
\(82\) 2.59266e9 0.699321
\(83\) 3.73356e9 − 2.15557e9i 0.947836 − 0.547233i 0.0554278 − 0.998463i \(-0.482348\pi\)
0.892408 + 0.451230i \(0.149014\pi\)
\(84\) 0 0
\(85\) 1.77053e8 − 3.06664e8i 0.0399032 − 0.0691144i
\(86\) 3.43412e9 + 1.98269e9i 0.730001 + 0.421466i
\(87\) 0 0
\(88\) 1.61528e9 + 2.79774e9i 0.306079 + 0.530144i
\(89\) 6.44176e9i 1.15360i 0.816886 + 0.576799i \(0.195698\pi\)
−0.816886 + 0.576799i \(0.804302\pi\)
\(90\) 0 0
\(91\) 3.01142e9 0.482575
\(92\) 7.11604e8 − 4.10845e8i 0.107969 − 0.0623360i
\(93\) 0 0
\(94\) 2.88476e9 − 4.99655e9i 0.393070 − 0.680817i
\(95\) 5.58600e9 + 3.22508e9i 0.721909 + 0.416795i
\(96\) 0 0
\(97\) −3.71029e9 − 6.42641e9i −0.432065 − 0.748358i 0.564986 − 0.825100i \(-0.308882\pi\)
−0.997051 + 0.0767421i \(0.975548\pi\)
\(98\) 1.07395e10i 1.18810i
\(99\) 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.71.40 80
3.2 odd 2 90.11.h.a.41.15 yes 80
9.2 odd 6 inner 270.11.h.a.251.40 80
9.7 even 3 90.11.h.a.11.15 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.15 ✓ 80 9.7 even 3
90.11.h.a.41.15 yes 80 3.2 odd 2
270.11.h.a.71.40 80 1.1 even 1 trivial
270.11.h.a.251.40 80 9.2 odd 6 inner