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 251.35
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.35

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 + 11.3137i) q^{2} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(-8869.13 + 15361.8i) q^{7} +11585.2i q^{8} -31622.8 q^{10} +(15696.6 + 9062.45i) q^{11} +(-40262.4 - 69736.6i) q^{13} +(-347598. + 200686. i) q^{14} +(-131072. + 227023. i) q^{16} +2.53262e6i q^{17} -2.06614e6 q^{19} +(-619677. - 357771. i) q^{20} +(205060. + 355174. i) q^{22} +(-62543.2 + 36109.3i) q^{23} +(976562. - 1.69146e6i) q^{25} -1.82207e6i q^{26} -9.08199e6 q^{28} +(1.16643e7 + 6.73441e6i) q^{29} +(1.82535e6 + 3.16160e6i) q^{31} +(-5.13695e6 + 2.96582e6i) q^{32} +(-2.86533e7 + 4.96290e7i) q^{34} -2.47900e7i q^{35} +2.49721e7 q^{37} +(-4.04879e7 - 2.33757e7i) q^{38} +(-8.09543e6 - 1.40217e7i) q^{40} +(-1.76629e8 + 1.01977e8i) q^{41} +(-8.73241e7 + 1.51250e8i) q^{43} +9.27995e6i q^{44} -1.63412e6 q^{46} +(-1.02658e8 - 5.92693e7i) q^{47} +(-1.60853e7 - 2.78606e7i) q^{49} +(3.82733e7 - 2.20971e7i) q^{50} +(2.06144e7 - 3.57051e7i) q^{52} +1.02801e8i q^{53} -2.53303e7 q^{55} +(-1.77970e8 - 1.02751e8i) q^{56} +(1.52382e8 + 2.63934e8i) q^{58} +(8.09584e8 - 4.67413e8i) q^{59} +(2.40616e8 - 4.16759e8i) q^{61} +8.26058e7i q^{62} -1.34218e8 q^{64} +(9.74598e7 + 5.62685e7i) q^{65} +(-8.24447e8 - 1.42798e9i) q^{67} +(-1.12298e9 + 6.48351e8i) q^{68} +(2.80467e8 - 4.85782e8i) q^{70} +1.63717e9i q^{71} -6.40707e7 q^{73} +(4.89352e8 + 2.82527e8i) q^{74} +(-5.28931e8 - 9.16136e8i) q^{76} +(-2.78431e8 + 1.60752e8i) q^{77} +(2.96682e9 - 5.13868e9i) q^{79} -3.66357e8i q^{80} -4.61495e9 q^{82} +(-1.30648e9 - 7.54297e8i) q^{83} +(-1.76972e9 - 3.06525e9i) q^{85} +(-3.42239e9 + 1.97592e9i) q^{86} +(-1.04991e8 + 1.81849e8i) q^{88} +6.86120e9i q^{89} +1.42837e9 q^{91} +(-3.20221e7 - 1.84880e7i) q^{92} +(-1.34111e9 - 2.32287e9i) q^{94} +(2.50066e9 - 1.44376e9i) q^{95} +(3.09809e9 - 5.36605e9i) q^{97} -7.27939e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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
\(7\) −8869.13 + 15361.8i −0.527705 + 0.914011i 0.471774 + 0.881720i \(0.343614\pi\)
−0.999478 + 0.0322916i \(0.989719\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) 15696.6 + 9062.45i 0.0974637 + 0.0562707i 0.547940 − 0.836518i \(-0.315412\pi\)
−0.450476 + 0.892789i \(0.648746\pi\)
\(12\) 0 0
\(13\) −40262.4 − 69736.6i −0.108438 − 0.187821i 0.806699 − 0.590962i \(-0.201252\pi\)
−0.915138 + 0.403141i \(0.867918\pi\)
\(14\) −347598. + 200686.i −0.646303 + 0.373143i
\(15\) 0 0
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 2.53262e6i 1.78372i 0.452316 + 0.891858i \(0.350598\pi\)
−0.452316 + 0.891858i \(0.649402\pi\)
\(18\) 0 0
\(19\) −2.06614e6 −0.834433 −0.417216 − 0.908807i \(-0.636994\pi\)
−0.417216 + 0.908807i \(0.636994\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 0 0
\(22\) 205060. + 355174.i 0.0397894 + 0.0689172i
\(23\) −62543.2 + 36109.3i −0.00971719 + 0.00561022i −0.504851 − 0.863207i \(-0.668452\pi\)
0.495134 + 0.868817i \(0.335119\pi\)
\(24\) 0 0
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.82207e6i − 0.153355i
\(27\) 0 0
\(28\) −9.08199e6 −0.527705
\(29\) 1.16643e7 + 6.73441e6i 0.568683 + 0.328329i 0.756623 − 0.653851i \(-0.226848\pi\)
−0.187940 + 0.982180i \(0.560181\pi\)
\(30\) 0 0
\(31\) 1.82535e6 + 3.16160e6i 0.0637584 + 0.110433i 0.896143 − 0.443766i \(-0.146358\pi\)
−0.832384 + 0.554199i \(0.813025\pi\)
\(32\) −5.13695e6 + 2.96582e6i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −2.86533e7 + 4.96290e7i −0.630639 + 1.09230i
\(35\) − 2.47900e7i − 0.471993i
\(36\) 0 0
\(37\) 2.49721e7 0.360120 0.180060 − 0.983656i \(-0.442371\pi\)
0.180060 + 0.983656i \(0.442371\pi\)
\(38\) −4.04879e7 − 2.33757e7i −0.510984 − 0.295017i
\(39\) 0 0
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −1.76629e8 + 1.01977e8i −1.52455 + 0.880202i −0.524977 + 0.851116i \(0.675926\pi\)
−0.999577 + 0.0290858i \(0.990740\pi\)
\(42\) 0 0
\(43\) −8.73241e7 + 1.51250e8i −0.594007 + 1.02885i 0.399679 + 0.916655i \(0.369122\pi\)
−0.993686 + 0.112195i \(0.964212\pi\)
\(44\) 9.27995e6i 0.0562707i
\(45\) 0 0
\(46\) −1.63412e6 −0.00793405
\(47\) −1.02658e8 − 5.92693e7i −0.447612 − 0.258429i 0.259209 − 0.965821i \(-0.416538\pi\)
−0.706821 + 0.707392i \(0.749871\pi\)
\(48\) 0 0
\(49\) −1.60853e7 − 2.78606e7i −0.0569442 − 0.0986303i
\(50\) 3.82733e7 − 2.20971e7i 0.122474 − 0.0707107i
\(51\) 0 0
\(52\) 2.06144e7 − 3.57051e7i 0.0542192 − 0.0939104i
\(53\) 1.02801e8i 0.245820i 0.992418 + 0.122910i \(0.0392227\pi\)
−0.992418 + 0.122910i \(0.960777\pi\)
\(54\) 0 0
\(55\) −2.53303e7 −0.0503300
\(56\) −1.77970e8 − 1.02751e8i −0.323152 − 0.186572i
\(57\) 0 0
\(58\) 1.52382e8 + 2.63934e8i 0.232164 + 0.402120i
\(59\) 8.09584e8 − 4.67413e8i 1.13240 − 0.653794i 0.187866 − 0.982195i \(-0.439843\pi\)
0.944539 + 0.328400i \(0.106510\pi\)
\(60\) 0 0
\(61\) 2.40616e8 − 4.16759e8i 0.284889 − 0.493442i −0.687693 − 0.726001i \(-0.741377\pi\)
0.972582 + 0.232559i \(0.0747100\pi\)
\(62\) 8.26058e7i 0.0901680i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 9.74598e7 + 5.62685e7i 0.0839961 + 0.0484951i
\(66\) 0 0
\(67\) −8.24447e8 − 1.42798e9i −0.610645 − 1.05767i −0.991132 − 0.132882i \(-0.957577\pi\)
0.380487 − 0.924786i \(-0.375756\pi\)
\(68\) −1.12298e9 + 6.48351e8i −0.772371 + 0.445929i
\(69\) 0 0
\(70\) 2.80467e8 − 4.85782e8i 0.166875 − 0.289036i
\(71\) 1.63717e9i 0.907405i 0.891153 + 0.453703i \(0.149897\pi\)
−0.891153 + 0.453703i \(0.850103\pi\)
\(72\) 0 0
\(73\) −6.40707e7 −0.0309062 −0.0154531 − 0.999881i \(-0.504919\pi\)
−0.0154531 + 0.999881i \(0.504919\pi\)
\(74\) 4.89352e8 + 2.82527e8i 0.220527 + 0.127321i
\(75\) 0 0
\(76\) −5.28931e8 − 9.16136e8i −0.208608 − 0.361320i
\(77\) −2.78431e8 + 1.60752e8i −0.102864 + 0.0593886i
\(78\) 0 0
\(79\) 2.96682e9 − 5.13868e9i 0.964175 − 1.67000i 0.252359 − 0.967634i \(-0.418794\pi\)
0.711816 − 0.702366i \(-0.247873\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) −4.61495e9 −1.24479
\(83\) −1.30648e9 − 7.54297e8i −0.331675 − 0.191493i 0.324910 − 0.945745i \(-0.394666\pi\)
−0.656584 + 0.754253i \(0.727999\pi\)
\(84\) 0 0
\(85\) −1.76972e9 − 3.06525e9i −0.398851 − 0.690830i
\(86\) −3.42239e9 + 1.97592e9i −0.727507 + 0.420026i
\(87\) 0 0
\(88\) −1.04991e8 + 1.81849e8i −0.0198947 + 0.0344586i
\(89\) 6.86120e9i 1.22871i 0.789029 + 0.614356i \(0.210584\pi\)
−0.789029 + 0.614356i \(0.789416\pi\)
\(90\) 0 0
\(91\) 1.42837e9 0.228894
\(92\) −3.20221e7 − 1.84880e7i −0.00485860 − 0.00280511i
\(93\) 0 0
\(94\) −1.34111e9 − 2.32287e9i −0.182737 − 0.316509i
\(95\) 2.50066e9 − 1.44376e9i 0.323174 − 0.186585i
\(96\) 0 0
\(97\) 3.09809e9 − 5.36605e9i 0.360774 − 0.624879i −0.627314 − 0.778766i \(-0.715846\pi\)
0.988088 + 0.153887i \(0.0491792\pi\)
\(98\) − 7.27939e8i − 0.0805313i
\(99\) 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.251.35 80
3.2 odd 2 90.11.h.a.11.13 ✓ 80
9.4 even 3 90.11.h.a.41.13 yes 80
9.5 odd 6 inner 270.11.h.a.71.35 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.13 ✓ 80 3.2 odd 2
90.11.h.a.41.13 yes 80 9.4 even 3
270.11.h.a.71.35 80 9.5 odd 6 inner
270.11.h.a.251.35 80 1.1 even 1 trivial