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 251.37
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.37

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

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

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
\(7\) −10611.8 + 18380.2i −0.631392 + 1.09360i 0.355876 + 0.934533i \(0.384183\pi\)
−0.987267 + 0.159069i \(0.949151\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) 232171. + 134044.i 1.44160 + 0.832307i 0.997957 − 0.0638965i \(-0.0203528\pi\)
0.443642 + 0.896204i \(0.353686\pi\)
\(12\) 0 0
\(13\) −266298. − 461242.i −0.717218 − 1.24226i −0.962098 − 0.272704i \(-0.912082\pi\)
0.244880 − 0.969553i \(-0.421251\pi\)
\(14\) −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 \(-0.797680\pi\)
0.804712 − 0.593665i \(-0.202320\pi\)
\(18\) 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 0.164983 − 0.986296i \(-0.447243\pi\)
0.936649 + 0.350269i \(0.113910\pi\)
\(24\) 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 0.999579 + 0.0290037i \(0.00923347\pi\)
0.524908 + 0.851159i \(0.324100\pi\)
\(30\) 0 0
\(31\) 2.27961e7 + 3.94841e7i 0.796256 + 1.37916i 0.922038 + 0.387098i \(0.126523\pi\)
−0.125782 + 0.992058i \(0.540144\pi\)
\(32\) −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 0.146893 − 0.989152i \(-0.453073\pi\)
0.930078 + 0.367363i \(0.119739\pi\)
\(42\) 0 0
\(43\) 7.18757e7 − 1.24492e8i 0.488922 − 0.846838i −0.510996 − 0.859583i \(-0.670723\pi\)
0.999919 + 0.0127444i \(0.00405679\pi\)
\(44\) 1.37261e8i 0.832307i
\(45\) 0 0
\(46\) 1.85259e8 0.899479
\(47\) −2.95221e8 − 1.70446e8i −1.28724 − 0.743186i −0.309075 − 0.951038i \(-0.600020\pi\)
−0.978160 + 0.207852i \(0.933353\pi\)
\(48\) 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 \(0.0436776\pi\)
−0.990600 + 0.136787i \(0.956322\pi\)
\(54\) 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 −0.532938 − 0.846154i \(-0.678912\pi\)
0.466322 + 0.884615i \(0.345579\pi\)
\(60\) 0 0
\(61\) 2.78802e8 − 4.82900e8i 0.330101 − 0.571752i −0.652430 − 0.757849i \(-0.726250\pi\)
0.982531 + 0.186097i \(0.0595838\pi\)
\(62\) 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 0.986215 + 0.165470i \(0.0529143\pi\)
−0.349806 + 0.936822i \(0.613752\pi\)
\(68\) 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 \(0.219588\pi\)
−0.771338 + 0.636426i \(0.780412\pi\)
\(72\) 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 −0.403470 − 0.914993i \(-0.632196\pi\)
0.994142 − 0.108081i \(-0.0344706\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) 3.26008e9 0.879343
\(83\) −4.51472e9 − 2.60657e9i −1.14615 − 0.661728i −0.198202 − 0.980161i \(-0.563510\pi\)
−0.947945 + 0.318433i \(0.896843\pi\)
\(84\) 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 \(0.206489\pi\)
−0.796867 + 0.604155i \(0.793511\pi\)
\(90\) 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 −0.873160 − 0.487434i \(-0.837933\pi\)
0.858710 + 0.512461i \(0.171266\pi\)
\(98\) − 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.251.37 80
3.2 odd 2 90.11.h.a.11.6 ✓ 80
9.4 even 3 90.11.h.a.41.6 yes 80
9.5 odd 6 inner 270.11.h.a.71.37 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.6 ✓ 80 3.2 odd 2
90.11.h.a.41.6 yes 80 9.4 even 3
270.11.h.a.71.37 80 9.5 odd 6 inner
270.11.h.a.251.37 80 1.1 even 1 trivial