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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 71.35
Character \(\chi\) \(=\) 270.71
Dual form 270.11.h.a.251.35

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(7\) −8869.13 − 15361.8i −0.527705 − 0.914011i −0.999478 − 0.0322916i \(-0.989719\pi\)
0.471774 − 0.881720i \(-0.343614\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) 15696.6 − 9062.45i 0.0974637 − 0.0562707i −0.450476 − 0.892789i \(-0.648746\pi\)
0.547940 + 0.836518i \(0.315412\pi\)
\(12\) 0 0
\(13\) −40262.4 + 69736.6i −0.108438 + 0.187821i −0.915138 − 0.403141i \(-0.867918\pi\)
0.806699 + 0.590962i \(0.201252\pi\)
\(14\) −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 \(-0.649402\pi\)
0.452316 − 0.891858i \(-0.350598\pi\)
\(18\) 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 0.495134 − 0.868817i \(-0.335119\pi\)
−0.504851 + 0.863207i \(0.668452\pi\)
\(24\) 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 −0.187940 − 0.982180i \(-0.560181\pi\)
0.756623 + 0.653851i \(0.226848\pi\)
\(30\) 0 0
\(31\) 1.82535e6 − 3.16160e6i 0.0637584 − 0.110433i −0.832384 − 0.554199i \(-0.813025\pi\)
0.896143 + 0.443766i \(0.146358\pi\)
\(32\) −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 −0.999577 − 0.0290858i \(-0.990740\pi\)
−0.524977 − 0.851116i \(-0.675926\pi\)
\(42\) 0 0
\(43\) −8.73241e7 − 1.51250e8i −0.594007 − 1.02885i −0.993686 − 0.112195i \(-0.964212\pi\)
0.399679 − 0.916655i \(-0.369122\pi\)
\(44\) − 9.27995e6i − 0.0562707i
\(45\) 0 0
\(46\) −1.63412e6 −0.00793405
\(47\) −1.02658e8 + 5.92693e7i −0.447612 + 0.258429i −0.706821 − 0.707392i \(-0.749871\pi\)
0.259209 + 0.965821i \(0.416538\pi\)
\(48\) 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 \(-0.960777\pi\)
0.992418 − 0.122910i \(-0.0392227\pi\)
\(54\) 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 0.944539 − 0.328400i \(-0.106510\pi\)
0.187866 + 0.982195i \(0.439843\pi\)
\(60\) 0 0
\(61\) 2.40616e8 + 4.16759e8i 0.284889 + 0.493442i 0.972582 − 0.232559i \(-0.0747100\pi\)
−0.687693 + 0.726001i \(0.741377\pi\)
\(62\) − 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 0.380487 + 0.924786i \(0.375756\pi\)
−0.991132 + 0.132882i \(0.957577\pi\)
\(68\) −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 \(-0.850103\pi\)
0.891153 − 0.453703i \(-0.149897\pi\)
\(72\) 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 0.711816 + 0.702366i \(0.247873\pi\)
0.252359 + 0.967634i \(0.418794\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 0 0
\(82\) −4.61495e9 −1.24479
\(83\) −1.30648e9 + 7.54297e8i −0.331675 + 0.191493i −0.656584 − 0.754253i \(-0.727999\pi\)
0.324910 + 0.945745i \(0.394666\pi\)
\(84\) 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 \(-0.789416\pi\)
0.789029 − 0.614356i \(-0.210584\pi\)
\(90\) 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 0.988088 − 0.153887i \(-0.0491792\pi\)
−0.627314 + 0.778766i \(0.715846\pi\)
\(98\) 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.71.35 80
3.2 odd 2 90.11.h.a.41.13 yes 80
9.2 odd 6 inner 270.11.h.a.251.35 80
9.7 even 3 90.11.h.a.11.13 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.13 ✓ 80 9.7 even 3
90.11.h.a.41.13 yes 80 3.2 odd 2
270.11.h.a.71.35 80 1.1 even 1 trivial
270.11.h.a.251.35 80 9.2 odd 6 inner