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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 251.40
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.40

$q$-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 \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

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

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
\(7\) −13757.7 + 23829.0i −0.818569 + 1.41780i 0.0881668 + 0.996106i \(0.471899\pi\)
−0.906736 + 0.421698i \(0.861434\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) −241492. − 139425.i −1.49947 − 0.865722i −0.499475 − 0.866328i \(-0.666474\pi\)
−1.00000 0.000606189i \(0.999807\pi\)
\(12\) 0 0
\(13\) −54722.5 − 94782.1i −0.147384 − 0.255276i 0.782876 − 0.622178i \(-0.213752\pi\)
−0.930260 + 0.366902i \(0.880418\pi\)
\(14\) −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 \(-0.971561\pi\)
0.996011 − 0.0892263i \(-0.0284394\pi\)
\(18\) 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 −0.388130 − 0.921605i \(-0.626879\pi\)
0.604068 + 0.796933i \(0.293545\pi\)
\(24\) 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 0.822653 − 0.568543i \(-0.192493\pi\)
−0.0810463 + 0.996710i \(0.525826\pi\)
\(30\) 0 0
\(31\) −1.21626e7 − 2.10662e7i −0.424833 − 0.735832i 0.571572 − 0.820552i \(-0.306334\pi\)
−0.996405 + 0.0847200i \(0.973000\pi\)
\(32\) −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 −0.00634585 − 0.999980i \(-0.502020\pi\)
0.862835 + 0.505486i \(0.168687\pi\)
\(42\) 0 0
\(43\) 8.76234e7 − 1.51768e8i 0.596043 − 1.03238i −0.397355 − 0.917665i \(-0.630072\pi\)
0.993399 − 0.114713i \(-0.0365947\pi\)
\(44\) − 1.42772e8i − 0.865722i
\(45\) 0 0
\(46\) 3.63139e7 0.176313
\(47\) 2.20818e8 + 1.27489e8i 0.962821 + 0.555885i 0.897040 − 0.441949i \(-0.145713\pi\)
0.0657811 + 0.997834i \(0.479046\pi\)
\(48\) 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 \(0.0902060\pi\)
−0.960113 + 0.279613i \(0.909794\pi\)
\(54\) 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 −0.716313 − 0.697779i \(-0.754172\pi\)
0.246138 + 0.969235i \(0.420838\pi\)
\(60\) 0 0
\(61\) −5.76153e8 + 9.97926e8i −0.682164 + 1.18154i 0.292156 + 0.956371i \(0.405627\pi\)
−0.974319 + 0.225171i \(0.927706\pi\)
\(62\) − 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 0.995040 − 0.0994797i \(-0.0317178\pi\)
−0.583672 + 0.811990i \(0.698385\pi\)
\(68\) 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 \(0.218251\pi\)
−0.774005 + 0.633180i \(0.781749\pi\)
\(72\) 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 −0.0756875 + 0.997132i \(0.524115\pi\)
−0.825698 + 0.564113i \(0.809218\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) 2.59266e9 0.699321
\(83\) 3.73356e9 + 2.15557e9i 0.947836 + 0.547233i 0.892408 − 0.451230i \(-0.149014\pi\)
0.0554278 + 0.998463i \(0.482348\pi\)
\(84\) 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 \(-0.804302\pi\)
0.816886 − 0.576799i \(-0.195698\pi\)
\(90\) 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 −0.997051 − 0.0767421i \(-0.975548\pi\)
0.564986 + 0.825100i \(0.308882\pi\)
\(98\) − 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.251.40 80
3.2 odd 2 90.11.h.a.11.15 ✓ 80
9.4 even 3 90.11.h.a.41.15 yes 80
9.5 odd 6 inner 270.11.h.a.71.40 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.15 ✓ 80 3.2 odd 2
90.11.h.a.41.15 yes 80 9.4 even 3
270.11.h.a.71.40 80 9.5 odd 6 inner
270.11.h.a.251.40 80 1.1 even 1 trivial