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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 251.39
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.39

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 + 11.3137i) q^{2} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(-12940.6 + 22413.8i) q^{7} +11585.2i q^{8} +31622.8 q^{10} +(-53638.1 - 30967.9i) q^{11} +(37959.2 + 65747.3i) q^{13} +(-507166. + 292813. i) q^{14} +(-131072. + 227023. i) q^{16} -866449. i q^{17} +3.81151e6 q^{19} +(619677. + 357771. i) q^{20} +(-700725. - 1.21369e6i) q^{22} +(1.05354e7 - 6.08261e6i) q^{23} +(976562. - 1.69146e6i) q^{25} +1.71784e6i q^{26} -1.32512e7 q^{28} +(1.23594e7 + 7.13569e6i) q^{29} +(8.75637e6 + 1.51665e7i) q^{31} +(-5.13695e6 + 2.96582e6i) q^{32} +(9.80275e6 - 1.69789e7i) q^{34} +3.61701e7i q^{35} +1.16632e7 q^{37} +(7.46900e7 + 4.31223e7i) q^{38} +(8.09543e6 + 1.40217e7i) q^{40} +(-1.28946e8 + 7.44471e7i) q^{41} +(4.99447e7 - 8.65067e7i) q^{43} -3.17112e7i q^{44} +2.75268e8 q^{46} +(3.16872e8 + 1.82946e8i) q^{47} +(-1.93681e8 - 3.35466e8i) q^{49} +(3.82733e7 - 2.20971e7i) q^{50} +(-1.94351e7 + 3.36626e7i) q^{52} -9.47585e6i q^{53} -8.65580e7 q^{55} +(-2.59669e8 - 1.49920e8i) q^{56} +(1.61462e8 + 2.79661e8i) q^{58} +(-5.16273e8 + 2.98070e8i) q^{59} +(-6.45295e8 + 1.11768e9i) q^{61} +3.96268e8i q^{62} -1.34218e8 q^{64} +(9.18846e7 + 5.30496e7i) q^{65} +(-1.95210e8 - 3.38114e8i) q^{67} +(3.84188e8 - 2.21811e8i) q^{68} +(-4.09218e8 + 7.08787e8i) q^{70} +1.89256e8i q^{71} -3.42543e9 q^{73} +(2.28552e8 + 1.31954e8i) q^{74} +(9.75746e8 + 1.69004e9i) q^{76} +(1.38822e9 - 8.01488e8i) q^{77} +(-1.01333e9 + 1.75514e9i) q^{79} +3.66357e8i q^{80} -3.36909e9 q^{82} +(3.87487e9 + 2.23716e9i) q^{83} +(-6.05450e8 - 1.04867e9i) q^{85} +(1.95742e9 - 1.13012e9i) q^{86} +(3.58771e8 - 6.21410e8i) q^{88} -3.09752e9i q^{89} -1.96486e9 q^{91} +(5.39412e9 + 3.11430e9i) q^{92} +(4.13959e9 + 7.16998e9i) q^{94} +(4.61310e9 - 2.66337e9i) q^{95} +(1.68792e9 - 2.92356e9i) q^{97} -8.76502e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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
\(7\) −12940.6 + 22413.8i −0.769954 + 1.33360i 0.167633 + 0.985849i \(0.446387\pi\)
−0.937587 + 0.347750i \(0.886946\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) 31622.8 0.316228
\(11\) −53638.1 − 30967.9i −0.333050 − 0.192287i 0.324144 − 0.946008i \(-0.394924\pi\)
−0.657194 + 0.753721i \(0.728257\pi\)
\(12\) 0 0
\(13\) 37959.2 + 65747.3i 0.102235 + 0.177077i 0.912605 − 0.408842i \(-0.134067\pi\)
−0.810370 + 0.585918i \(0.800734\pi\)
\(14\) −507166. + 292813.i −0.942997 + 0.544440i
\(15\) 0 0
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 866449.i − 0.610237i −0.952314 − 0.305118i \(-0.901304\pi\)
0.952314 − 0.305118i \(-0.0986960\pi\)
\(18\) 0 0
\(19\) 3.81151e6 1.53932 0.769660 − 0.638454i \(-0.220426\pi\)
0.769660 + 0.638454i \(0.220426\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) 0 0
\(22\) −700725. − 1.21369e6i −0.135967 − 0.235502i
\(23\) 1.05354e7 − 6.08261e6i 1.63686 − 0.945042i 0.654956 − 0.755667i \(-0.272687\pi\)
0.981905 − 0.189375i \(-0.0606461\pi\)
\(24\) 0 0
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 1.71784e6i 0.144582i
\(27\) 0 0
\(28\) −1.32512e7 −0.769954
\(29\) 1.23594e7 + 7.13569e6i 0.602569 + 0.347893i 0.770052 − 0.637982i \(-0.220230\pi\)
−0.167483 + 0.985875i \(0.553564\pi\)
\(30\) 0 0
\(31\) 8.75637e6 + 1.51665e7i 0.305855 + 0.529757i 0.977451 − 0.211161i \(-0.0677243\pi\)
−0.671596 + 0.740917i \(0.734391\pi\)
\(32\) −5.13695e6 + 2.96582e6i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 9.80275e6 − 1.69789e7i 0.215751 − 0.373692i
\(35\) 3.61701e7i 0.688668i
\(36\) 0 0
\(37\) 1.16632e7 0.168194 0.0840969 − 0.996458i \(-0.473199\pi\)
0.0840969 + 0.996458i \(0.473199\pi\)
\(38\) 7.46900e7 + 4.31223e7i 0.942637 + 0.544232i
\(39\) 0 0
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) −1.28946e8 + 7.44471e7i −1.11299 + 0.642582i −0.939601 − 0.342272i \(-0.888804\pi\)
−0.173384 + 0.984854i \(0.555470\pi\)
\(42\) 0 0
\(43\) 4.99447e7 − 8.65067e7i 0.339740 − 0.588447i −0.644644 − 0.764483i \(-0.722994\pi\)
0.984384 + 0.176036i \(0.0563275\pi\)
\(44\) − 3.17112e7i − 0.192287i
\(45\) 0 0
\(46\) 2.75268e8 1.33649
\(47\) 3.16872e8 + 1.82946e8i 1.38164 + 0.797689i 0.992353 − 0.123429i \(-0.0393893\pi\)
0.389284 + 0.921118i \(0.372723\pi\)
\(48\) 0 0
\(49\) −1.93681e8 − 3.35466e8i −0.685658 − 1.18759i
\(50\) 3.82733e7 − 2.20971e7i 0.122474 − 0.0707107i
\(51\) 0 0
\(52\) −1.94351e7 + 3.36626e7i −0.0511176 + 0.0885383i
\(53\) − 9.47585e6i − 0.0226589i −0.999936 − 0.0113295i \(-0.996394\pi\)
0.999936 − 0.0113295i \(-0.00360636\pi\)
\(54\) 0 0
\(55\) −8.65580e7 −0.171986
\(56\) −2.59669e8 − 1.49920e8i −0.471499 − 0.272220i
\(57\) 0 0
\(58\) 1.61462e8 + 2.79661e8i 0.245998 + 0.426081i
\(59\) −5.16273e8 + 2.98070e8i −0.722136 + 0.416926i −0.815538 − 0.578703i \(-0.803559\pi\)
0.0934022 + 0.995628i \(0.470226\pi\)
\(60\) 0 0
\(61\) −6.45295e8 + 1.11768e9i −0.764028 + 1.32334i 0.176730 + 0.984259i \(0.443448\pi\)
−0.940759 + 0.339077i \(0.889885\pi\)
\(62\) 3.96268e8i 0.432544i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 9.18846e7 + 5.30496e7i 0.0791910 + 0.0457210i
\(66\) 0 0
\(67\) −1.95210e8 − 3.38114e8i −0.144587 − 0.250431i 0.784632 − 0.619962i \(-0.212852\pi\)
−0.929219 + 0.369530i \(0.879519\pi\)
\(68\) 3.84188e8 − 2.21811e8i 0.264240 − 0.152559i
\(69\) 0 0
\(70\) −4.09218e8 + 7.08787e8i −0.243481 + 0.421721i
\(71\) 1.89256e8i 0.104896i 0.998624 + 0.0524479i \(0.0167024\pi\)
−0.998624 + 0.0524479i \(0.983298\pi\)
\(72\) 0 0
\(73\) −3.42543e9 −1.65235 −0.826173 − 0.563416i \(-0.809487\pi\)
−0.826173 + 0.563416i \(0.809487\pi\)
\(74\) 2.28552e8 + 1.31954e8i 0.102997 + 0.0594655i
\(75\) 0 0
\(76\) 9.75746e8 + 1.69004e9i 0.384830 + 0.666545i
\(77\) 1.38822e9 − 8.01488e8i 0.512866 − 0.296104i
\(78\) 0 0
\(79\) −1.01333e9 + 1.75514e9i −0.329318 + 0.570396i −0.982377 − 0.186911i \(-0.940152\pi\)
0.653058 + 0.757308i \(0.273486\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 0 0
\(82\) −3.36909e9 −0.908749
\(83\) 3.87487e9 + 2.23716e9i 0.983709 + 0.567945i 0.903388 − 0.428825i \(-0.141072\pi\)
0.0803210 + 0.996769i \(0.474405\pi\)
\(84\) 0 0
\(85\) −6.05450e8 − 1.04867e9i −0.136453 − 0.236344i
\(86\) 1.95742e9 − 1.13012e9i 0.416095 − 0.240233i
\(87\) 0 0
\(88\) 3.58771e8 − 6.21410e8i 0.0679836 − 0.117751i
\(89\) − 3.09752e9i − 0.554708i −0.960768 − 0.277354i \(-0.910542\pi\)
0.960768 − 0.277354i \(-0.0894575\pi\)
\(90\) 0 0
\(91\) −1.96486e9 −0.314866
\(92\) 5.39412e9 + 3.11430e9i 0.818430 + 0.472521i
\(93\) 0 0
\(94\) 4.13959e9 + 7.16998e9i 0.564051 + 0.976965i
\(95\) 4.61310e9 − 2.66337e9i 0.596176 − 0.344202i
\(96\) 0 0
\(97\) 1.68792e9 − 2.92356e9i 0.196559 − 0.340451i −0.750851 − 0.660471i \(-0.770357\pi\)
0.947411 + 0.320021i \(0.103690\pi\)
\(98\) − 8.76502e9i − 0.969667i
\(99\) 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.251.39 80
3.2 odd 2 90.11.h.a.11.16 ✓ 80
9.4 even 3 90.11.h.a.41.16 yes 80
9.5 odd 6 inner 270.11.h.a.71.39 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.16 ✓ 80 3.2 odd 2
90.11.h.a.41.16 yes 80 9.4 even 3
270.11.h.a.71.39 80 9.5 odd 6 inner
270.11.h.a.251.39 80 1.1 even 1 trivial