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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 71.37
Character \(\chi\) \(=\) 270.71
Dual form 270.11.h.a.251.37

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 - 11.3137i) q^{2} +(256.000 - 443.405i) q^{4} +(-1210.31 - 698.771i) q^{5} +(-10611.8 - 18380.2i) q^{7} -11585.2i q^{8} -31622.8 q^{10} +(232171. - 134044. i) q^{11} +(-266298. + 461242. i) q^{13} +(-415896. - 240118. i) q^{14} +(-131072. - 227023. i) q^{16} +1.68584e6i q^{17} +314597. q^{19} +(-619677. + 357771. i) q^{20} +(3.03307e6 - 5.25343e6i) q^{22} +(7.09048e6 + 4.09369e6i) q^{23} +(976562. + 1.69146e6i) q^{25} +1.20513e7i q^{26} -1.08665e7 q^{28} +(3.12690e7 - 1.80532e7i) q^{29} +(2.27961e7 - 3.94841e7i) q^{31} +(-5.13695e6 - 2.96582e6i) q^{32} +(1.90731e7 + 3.30356e7i) q^{34} +2.96609e7i q^{35} +2.77674e7 q^{37} +(6.16482e6 - 3.55926e6i) q^{38} +(-8.09543e6 + 1.40217e7i) q^{40} +(1.24774e8 + 7.20382e7i) q^{41} +(7.18757e7 + 1.24492e8i) q^{43} -1.37261e8i q^{44} +1.85259e8 q^{46} +(-2.95221e8 + 1.70446e8i) q^{47} +(-8.39830e7 + 1.45463e8i) q^{49} +(3.82733e7 + 2.20971e7i) q^{50} +(1.36345e8 + 2.36156e8i) q^{52} -1.14407e8i q^{53} -3.74664e8 q^{55} +(-2.12939e8 + 1.22940e8i) q^{56} +(4.08496e8 - 7.07536e8i) q^{58} +(-4.76249e7 - 2.74963e7i) q^{59} +(2.78802e8 + 4.82900e8i) q^{61} -1.03164e9i q^{62} -1.34218e8 q^{64} +(6.44605e8 - 3.72163e8i) q^{65} +(8.59232e8 - 1.48823e9i) q^{67} +(7.47509e8 + 4.31575e8i) q^{68} +(3.35575e8 + 5.81232e8i) q^{70} -2.29652e9i q^{71} -2.42924e9 q^{73} +(5.44127e8 - 3.14152e8i) q^{74} +(8.05368e7 - 1.39494e8i) q^{76} +(-4.92750e9 - 2.84490e9i) q^{77} +(1.81753e9 + 3.14806e9i) q^{79} +3.66357e8i q^{80} +3.26008e9 q^{82} +(-4.51472e9 + 2.60657e9i) q^{83} +(1.17802e9 - 2.04038e9i) q^{85} +(2.81694e9 + 1.62636e9i) q^{86} +(-1.55293e9 - 2.68976e9i) q^{88} -6.74728e9i q^{89} +1.13036e10 q^{91} +(3.63033e9 - 2.09597e9i) q^{92} +(-3.85675e9 + 6.68009e9i) q^{94} +(-3.80759e8 - 2.19831e8i) q^{95} +(-1.24083e8 - 2.14918e8i) q^{97} +3.80064e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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
\(7\) −10611.8 − 18380.2i −0.631392 − 1.09360i −0.987267 − 0.159069i \(-0.949151\pi\)
0.355876 − 0.934533i \(-0.384183\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) 232171. − 134044.i 1.44160 − 0.832307i 0.443642 − 0.896204i \(-0.353686\pi\)
0.997957 + 0.0638965i \(0.0203528\pi\)
\(12\) 0 0
\(13\) −266298. + 461242.i −0.717218 + 1.24226i 0.244880 + 0.969553i \(0.421251\pi\)
−0.962098 + 0.272704i \(0.912082\pi\)
\(14\) −415896. − 240118.i −0.773294 − 0.446461i
\(15\) 0 0
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) 1.68584e6i 1.18733i 0.804712 + 0.593665i \(0.202320\pi\)
−0.804712 + 0.593665i \(0.797680\pi\)
\(18\) 0 0
\(19\) 314597. 0.127053 0.0635267 − 0.997980i \(-0.479765\pi\)
0.0635267 + 0.997980i \(0.479765\pi\)
\(20\) −619677. + 357771.i −0.193649 + 0.111803i
\(21\) 0 0
\(22\) 3.03307e6 − 5.25343e6i 0.588530 − 1.01936i
\(23\) 7.09048e6 + 4.09369e6i 1.10163 + 0.636027i 0.936649 − 0.350269i \(-0.113910\pi\)
0.164983 + 0.986296i \(0.447243\pi\)
\(24\) 0 0
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) 1.20513e7i 1.01430i
\(27\) 0 0
\(28\) −1.08665e7 −0.631392
\(29\) 3.12690e7 − 1.80532e7i 1.52449 − 0.880163i 0.524908 − 0.851159i \(-0.324100\pi\)
0.999579 − 0.0290037i \(-0.00923347\pi\)
\(30\) 0 0
\(31\) 2.27961e7 − 3.94841e7i 0.796256 − 1.37916i −0.125782 − 0.992058i \(-0.540144\pi\)
0.922038 − 0.387098i \(-0.126523\pi\)
\(32\) −5.13695e6 − 2.96582e6i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 1.90731e7 + 3.30356e7i 0.419784 + 0.727088i
\(35\) 2.96609e7i 0.564734i
\(36\) 0 0
\(37\) 2.77674e7 0.400430 0.200215 − 0.979752i \(-0.435836\pi\)
0.200215 + 0.979752i \(0.435836\pi\)
\(38\) 6.16482e6 − 3.55926e6i 0.0778041 − 0.0449202i
\(39\) 0 0
\(40\) −8.09543e6 + 1.40217e7i −0.0790569 + 0.136931i
\(41\) 1.24774e8 + 7.20382e7i 1.07697 + 0.621790i 0.930078 − 0.367363i \(-0.119739\pi\)
0.146893 + 0.989152i \(0.453073\pi\)
\(42\) 0 0
\(43\) 7.18757e7 + 1.24492e8i 0.488922 + 0.846838i 0.999919 − 0.0127444i \(-0.00405679\pi\)
−0.510996 + 0.859583i \(0.670723\pi\)
\(44\) − 1.37261e8i − 0.832307i
\(45\) 0 0
\(46\) 1.85259e8 0.899479
\(47\) −2.95221e8 + 1.70446e8i −1.28724 + 0.743186i −0.978160 − 0.207852i \(-0.933353\pi\)
−0.309075 + 0.951038i \(0.600020\pi\)
\(48\) 0 0
\(49\) −8.39830e7 + 1.45463e8i −0.297311 + 0.514958i
\(50\) 3.82733e7 + 2.20971e7i 0.122474 + 0.0707107i
\(51\) 0 0
\(52\) 1.36345e8 + 2.36156e8i 0.358609 + 0.621129i
\(53\) − 1.14407e8i − 0.273574i −0.990600 − 0.136787i \(-0.956322\pi\)
0.990600 − 0.136787i \(-0.0436776\pi\)
\(54\) 0 0
\(55\) −3.74664e8 −0.744438
\(56\) −2.12939e8 + 1.22940e8i −0.386647 + 0.223231i
\(57\) 0 0
\(58\) 4.08496e8 − 7.07536e8i 0.622369 − 1.07798i
\(59\) −4.76249e7 − 2.74963e7i −0.0666154 − 0.0384604i 0.466322 − 0.884615i \(-0.345579\pi\)
−0.532938 + 0.846154i \(0.678912\pi\)
\(60\) 0 0
\(61\) 2.78802e8 + 4.82900e8i 0.330101 + 0.571752i 0.982531 − 0.186097i \(-0.0595838\pi\)
−0.652430 + 0.757849i \(0.726250\pi\)
\(62\) − 1.03164e9i − 1.12608i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 6.44605e8 − 3.72163e8i 0.555554 − 0.320750i
\(66\) 0 0
\(67\) 8.59232e8 − 1.48823e9i 0.636409 − 1.10229i −0.349806 − 0.936822i \(-0.613752\pi\)
0.986215 − 0.165470i \(-0.0529143\pi\)
\(68\) 7.47509e8 + 4.31575e8i 0.514129 + 0.296832i
\(69\) 0 0
\(70\) 3.35575e8 + 5.81232e8i 0.199664 + 0.345827i
\(71\) − 2.29652e9i − 1.27285i −0.771338 − 0.636426i \(-0.780412\pi\)
0.771338 − 0.636426i \(-0.219588\pi\)
\(72\) 0 0
\(73\) −2.42924e9 −1.17181 −0.585904 − 0.810380i \(-0.699260\pi\)
−0.585904 + 0.810380i \(0.699260\pi\)
\(74\) 5.44127e8 − 3.14152e8i 0.245212 − 0.141573i
\(75\) 0 0
\(76\) 8.05368e7 − 1.39494e8i 0.0317634 − 0.0550158i
\(77\) −4.92750e9 − 2.84490e9i −1.82043 − 1.05102i
\(78\) 0 0
\(79\) 1.81753e9 + 3.14806e9i 0.590672 + 1.02307i 0.994142 + 0.108081i \(0.0344706\pi\)
−0.403470 + 0.914993i \(0.632196\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 0 0
\(82\) 3.26008e9 0.879343
\(83\) −4.51472e9 + 2.60657e9i −1.14615 + 0.661728i −0.947945 − 0.318433i \(-0.896843\pi\)
−0.198202 + 0.980161i \(0.563510\pi\)
\(84\) 0 0
\(85\) 1.17802e9 − 2.04038e9i 0.265495 − 0.459851i
\(86\) 2.81694e9 + 1.62636e9i 0.598805 + 0.345720i
\(87\) 0 0
\(88\) −1.55293e9 − 2.68976e9i −0.294265 − 0.509682i
\(89\) − 6.74728e9i − 1.20831i −0.796867 − 0.604155i \(-0.793511\pi\)
0.796867 − 0.604155i \(-0.206489\pi\)
\(90\) 0 0
\(91\) 1.13036e10 1.81138
\(92\) 3.63033e9 − 2.09597e9i 0.550816 − 0.318014i
\(93\) 0 0
\(94\) −3.85675e9 + 6.68009e9i −0.525512 + 0.910213i
\(95\) −3.80759e8 − 2.19831e8i −0.0492076 − 0.0284100i
\(96\) 0 0
\(97\) −1.24083e8 − 2.14918e8i −0.0144495 − 0.0250273i 0.858710 − 0.512461i \(-0.171266\pi\)
−0.873160 + 0.487434i \(0.837933\pi\)
\(98\) 3.80064e9i 0.420461i
\(99\) 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.71.37 80
3.2 odd 2 90.11.h.a.41.6 yes 80
9.2 odd 6 inner 270.11.h.a.251.37 80
9.7 even 3 90.11.h.a.11.6 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.6 ✓ 80 9.7 even 3
90.11.h.a.41.6 yes 80 3.2 odd 2
270.11.h.a.71.37 80 1.1 even 1 trivial
270.11.h.a.251.37 80 9.2 odd 6 inner