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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,2,Mod(23,270)] 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("270.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([22, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.r (of order \(36\), degree \(12\), 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 113.10
Character \(\chi\) \(=\) 270.113
Dual form 270.2.r.a.227.10

$q$-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+(0.996195 + 0.0871557i) q^{2} +(-1.71950 + 0.208167i) q^{3} +(0.984808 + 0.173648i) q^{4} +(-2.00125 + 0.997503i) q^{5} +(-1.73110 + 0.0575113i) q^{6} +(2.07619 + 1.45376i) q^{7} +(0.965926 + 0.258819i) q^{8} +(2.91333 - 0.715886i) q^{9} +(-2.08057 + 0.819287i) q^{10} +(-1.56319 + 4.29482i) q^{11} +(-1.72952 - 0.0935825i) q^{12} +(0.388268 + 4.43792i) q^{13} +(1.94158 + 1.62918i) q^{14} +(3.23349 - 2.13180i) q^{15} +(0.939693 + 0.342020i) q^{16} +(-3.33214 + 0.892843i) q^{17} +(2.96464 - 0.459248i) q^{18} +(0.649453 - 0.374962i) q^{19} +(-2.14406 + 0.634835i) q^{20} +(-3.87262 - 2.06754i) q^{21} +(-1.93156 + 4.14224i) q^{22} +(-0.184128 - 0.262962i) q^{23} +(-1.71478 - 0.243964i) q^{24} +(3.00998 - 3.99250i) q^{25} +4.45487i q^{26} +(-4.86044 + 1.83742i) q^{27} +(1.79220 + 1.79220i) q^{28} +(7.59485 - 6.37284i) q^{29} +(3.40698 - 1.84187i) q^{30} +(0.402651 - 2.28354i) q^{31} +(0.906308 + 0.422618i) q^{32} +(1.79385 - 7.71033i) q^{33} +(-3.39727 + 0.599031i) q^{34} +(-5.60510 - 0.838335i) q^{35} +(2.99338 - 0.199115i) q^{36} +(-2.33615 - 8.71863i) q^{37} +(0.679661 - 0.316931i) q^{38} +(-1.59145 - 7.55016i) q^{39} +(-2.19123 + 0.445553i) q^{40} +(5.59486 - 6.66770i) q^{41} +(-3.67769 - 2.39720i) q^{42} +(5.41671 + 11.6162i) q^{43} +(-2.28523 + 3.95813i) q^{44} +(-5.11620 + 4.33872i) q^{45} +(-0.160508 - 0.278009i) q^{46} +(-2.83852 + 4.05383i) q^{47} +(-1.68700 - 0.392489i) q^{48} +(-0.197010 - 0.541282i) q^{49} +(3.34649 - 3.71497i) q^{50} +(5.54373 - 2.22888i) q^{51} +(-0.388268 + 4.43792i) q^{52} +(-6.98671 + 6.98671i) q^{53} +(-5.00209 + 1.40682i) q^{54} +(-1.15577 - 10.1543i) q^{55} +(1.62918 + 1.94158i) q^{56} +(-1.03868 + 0.779940i) q^{57} +(8.12138 - 5.68665i) q^{58} +(-4.25217 + 1.54766i) q^{59} +(3.55455 - 1.53792i) q^{60} +(-0.193936 - 1.09986i) q^{61} +(0.600142 - 2.23976i) q^{62} +(7.08935 + 2.74898i) q^{63} +(0.866025 + 0.500000i) q^{64} +(-5.20385 - 8.49407i) q^{65} +(2.45902 - 7.52464i) q^{66} +(8.09502 - 0.708223i) q^{67} +(-3.43655 + 0.300659i) q^{68} +(0.371347 + 0.413832i) q^{69} +(-5.51070 - 1.32366i) q^{70} +(1.12339 + 0.648590i) q^{71} +(2.99935 + 0.0625334i) q^{72} +(0.981783 - 3.66406i) q^{73} +(-1.56738 - 8.88907i) q^{74} +(-4.34454 + 7.49166i) q^{75} +(0.704697 - 0.256489i) q^{76} +(-9.48911 + 6.64435i) q^{77} +(-0.927359 - 7.66013i) q^{78} +(3.09680 + 3.69062i) q^{79} +(-2.22172 + 0.252879i) q^{80} +(7.97501 - 4.17123i) q^{81} +(6.15470 - 6.15470i) q^{82} +(0.504523 - 5.76673i) q^{83} +(-3.45476 - 2.70861i) q^{84} +(5.77781 - 5.11061i) q^{85} +(4.38368 + 12.0441i) q^{86} +(-11.7327 + 12.5391i) q^{87} +(-2.62150 + 3.74389i) q^{88} +(3.81802 + 6.61301i) q^{89} +(-5.47488 + 3.87630i) q^{90} +(-5.64556 + 9.77840i) q^{91} +(-0.135668 - 0.290940i) q^{92} +(-0.216997 + 4.01036i) q^{93} +(-3.18104 + 3.79101i) q^{94} +(-0.925690 + 1.39822i) q^{95} +(-1.64637 - 0.538027i) q^{96} +(2.43551 - 1.13570i) q^{97} +(-0.149085 - 0.556392i) q^{98} +(-1.47948 + 13.6313i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} + 24 q^{11} - 24 q^{20} + 24 q^{23} + 36 q^{25} + 12 q^{27} - 72 q^{30} - 12 q^{33} - 108 q^{35} - 12 q^{36} - 36 q^{38} - 72 q^{41} - 48 q^{42} - 60 q^{45} - 48 q^{47} - 12 q^{48} - 48 q^{50}+ \cdots - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.996195 + 0.0871557i 0.704416 + 0.0616284i
\(3\) −1.71950 + 0.208167i −0.992751 + 0.120185i
\(4\) 0.984808 + 0.173648i 0.492404 + 0.0868241i
\(5\) −2.00125 + 0.997503i −0.894985 + 0.446097i
\(6\) −1.73110 + 0.0575113i −0.706717 + 0.0234789i
\(7\) 2.07619 + 1.45376i 0.784725 + 0.549470i 0.895861 − 0.444335i \(-0.146560\pi\)
−0.111136 + 0.993805i \(0.535449\pi\)
\(8\) 0.965926 + 0.258819i 0.341506 + 0.0915064i
\(9\) 2.91333 − 0.715886i 0.971111 − 0.238629i
\(10\) −2.08057 + 0.819287i −0.657934 + 0.259081i
\(11\) −1.56319 + 4.29482i −0.471318 + 1.29494i 0.445375 + 0.895344i \(0.353071\pi\)
−0.916693 + 0.399592i \(0.869152\pi\)
\(12\) −1.72952 − 0.0935825i −0.499270 − 0.0270149i
\(13\) 0.388268 + 4.43792i 0.107686 + 1.23086i 0.837202 + 0.546894i \(0.184190\pi\)
−0.729516 + 0.683964i \(0.760255\pi\)
\(14\) 1.94158 + 1.62918i 0.518910 + 0.435417i
\(15\) 3.23349 − 2.13180i 0.834883 − 0.550427i
\(16\) 0.939693 + 0.342020i 0.234923 + 0.0855050i
\(17\) −3.33214 + 0.892843i −0.808162 + 0.216546i −0.639164 − 0.769070i \(-0.720720\pi\)
−0.168997 + 0.985617i \(0.554053\pi\)
\(18\) 2.96464 − 0.459248i 0.698772 − 0.108246i
\(19\) 0.649453 − 0.374962i 0.148995 − 0.0860221i −0.423649 − 0.905826i \(-0.639251\pi\)
0.572644 + 0.819804i \(0.305918\pi\)
\(20\) −2.14406 + 0.634835i −0.479426 + 0.141954i
\(21\) −3.87262 − 2.06754i −0.845075 − 0.451175i
\(22\) −1.93156 + 4.14224i −0.411809 + 0.883127i
\(23\) −0.184128 − 0.262962i −0.0383933 − 0.0548313i 0.799496 − 0.600671i \(-0.205100\pi\)
−0.837890 + 0.545839i \(0.816211\pi\)
\(24\) −1.71478 − 0.243964i −0.350029 − 0.0497990i
\(25\) 3.00998 − 3.99250i 0.601995 − 0.798499i
\(26\) 4.45487i 0.873672i
\(27\) −4.86044 + 1.83742i −0.935392 + 0.353612i
\(28\) 1.79220 + 1.79220i 0.338694 + 0.338694i
\(29\) 7.59485 − 6.37284i 1.41033 − 1.18341i 0.454040 − 0.890982i \(-0.349982\pi\)
0.956289 − 0.292424i \(-0.0944620\pi\)
\(30\) 3.40698 − 1.84187i 0.622027 − 0.336277i
\(31\) 0.402651 − 2.28354i 0.0723182 − 0.410137i −0.927061 − 0.374910i \(-0.877674\pi\)
0.999379 − 0.0352266i \(-0.0112153\pi\)
\(32\) 0.906308 + 0.422618i 0.160214 + 0.0747091i
\(33\) 1.79385 − 7.71033i 0.312269 − 1.34220i
\(34\) −3.39727 + 0.599031i −0.582627 + 0.102733i
\(35\) −5.60510 − 0.838335i −0.947434 − 0.141704i
\(36\) 2.99338 − 0.199115i 0.498897 − 0.0331858i
\(37\) −2.33615 − 8.71863i −0.384061 − 1.43333i −0.839642 − 0.543140i \(-0.817235\pi\)
0.455581 − 0.890194i \(-0.349432\pi\)
\(38\) 0.679661 − 0.316931i 0.110256 − 0.0514130i
\(39\) −1.59145 − 7.55016i −0.254837 − 1.20899i
\(40\) −2.19123 + 0.445553i −0.346464 + 0.0704481i
\(41\) 5.59486 − 6.66770i 0.873771 − 1.04132i −0.125020 − 0.992154i \(-0.539899\pi\)
0.998791 − 0.0491653i \(-0.0156561\pi\)
\(42\) −3.67769 − 2.39720i −0.567479 − 0.369896i
\(43\) 5.41671 + 11.6162i 0.826041 + 1.77145i 0.605073 + 0.796170i \(0.293144\pi\)
0.220968 + 0.975281i \(0.429078\pi\)
\(44\) −2.28523 + 3.95813i −0.344511 + 0.596710i
\(45\) −5.11620 + 4.33872i −0.762678 + 0.646778i
\(46\) −0.160508 − 0.278009i −0.0236657 − 0.0409902i
\(47\) −2.83852 + 4.05383i −0.414041 + 0.591312i −0.970956 − 0.239257i \(-0.923096\pi\)
0.556915 + 0.830569i \(0.311985\pi\)
\(48\) −1.68700 − 0.392489i −0.243497 − 0.0566509i
\(49\) −0.197010 − 0.541282i −0.0281443 − 0.0773259i
\(50\) 3.34649 − 3.71497i 0.473266 − 0.525376i
\(51\) 5.54373 − 2.22888i 0.776278 − 0.312106i
\(52\) −0.388268 + 4.43792i −0.0538430 + 0.615429i
\(53\) −6.98671 + 6.98671i −0.959699 + 0.959699i −0.999219 − 0.0395200i \(-0.987417\pi\)
0.0395200 + 0.999219i \(0.487417\pi\)
\(54\) −5.00209 + 1.40682i −0.680698 + 0.191443i
\(55\) −1.15577 − 10.1543i −0.155844 − 1.36920i
\(56\) 1.62918 + 1.94158i 0.217709 + 0.259455i
\(57\) −1.03868 + 0.779940i −0.137576 + 0.103306i
\(58\) 8.12138 − 5.68665i 1.06639 − 0.746694i
\(59\) −4.25217 + 1.54766i −0.553586 + 0.201489i −0.603639 − 0.797258i \(-0.706283\pi\)
0.0500531 + 0.998747i \(0.484061\pi\)
\(60\) 3.55455 − 1.53792i 0.458890 − 0.198545i
\(61\) −0.193936 − 1.09986i −0.0248309 − 0.140823i 0.969872 − 0.243615i \(-0.0783334\pi\)
−0.994703 + 0.102792i \(0.967222\pi\)
\(62\) 0.600142 − 2.23976i 0.0762182 − 0.284450i
\(63\) 7.08935 + 2.74898i 0.893175 + 0.346339i
\(64\) 0.866025 + 0.500000i 0.108253 + 0.0625000i
\(65\) −5.20385 − 8.49407i −0.645459 − 1.05356i
\(66\) 2.45902 − 7.52464i 0.302685 − 0.926220i
\(67\) 8.09502 − 0.708223i 0.988965 − 0.0865232i 0.418821 − 0.908069i \(-0.362443\pi\)
0.570143 + 0.821545i \(0.306888\pi\)
\(68\) −3.43655 + 0.300659i −0.416743 + 0.0364603i
\(69\) 0.371347 + 0.413832i 0.0447049 + 0.0498195i
\(70\) −5.51070 − 1.32366i −0.658655 − 0.158208i
\(71\) 1.12339 + 0.648590i 0.133322 + 0.0769735i 0.565178 − 0.824969i \(-0.308808\pi\)
−0.431856 + 0.901943i \(0.642141\pi\)
\(72\) 2.99935 + 0.0625334i 0.353477 + 0.00736963i
\(73\) 0.981783 − 3.66406i 0.114909 − 0.428846i −0.884371 − 0.466785i \(-0.845412\pi\)
0.999280 + 0.0379383i \(0.0120790\pi\)
\(74\) −1.56738 − 8.88907i −0.182204 − 1.03333i
\(75\) −4.34454 + 7.49166i −0.501664 + 0.865063i
\(76\) 0.704697 − 0.256489i 0.0808343 − 0.0294213i
\(77\) −9.48911 + 6.64435i −1.08138 + 0.757194i
\(78\) −0.927359 − 7.66013i −0.105003 − 0.867339i
\(79\) 3.09680 + 3.69062i 0.348417 + 0.415228i 0.911583 − 0.411117i \(-0.134861\pi\)
−0.563166 + 0.826344i \(0.690417\pi\)
\(80\) −2.22172 + 0.252879i −0.248396 + 0.0282727i
\(81\) 7.97501 − 4.17123i 0.886113 − 0.463470i
\(82\) 6.15470 − 6.15470i 0.679673 − 0.679673i
\(83\) 0.504523 − 5.76673i 0.0553786 − 0.632980i −0.916894 − 0.399131i \(-0.869312\pi\)
0.972273 − 0.233850i \(-0.0751324\pi\)
\(84\) −3.45476 − 2.70861i −0.376946 − 0.295533i
\(85\) 5.77781 − 5.11061i 0.626692 − 0.554324i
\(86\) 4.38368 + 12.0441i 0.472705 + 1.29875i
\(87\) −11.7327 + 12.5391i −1.25788 + 1.34433i
\(88\) −2.62150 + 3.74389i −0.279453 + 0.399100i
\(89\) 3.81802 + 6.61301i 0.404709 + 0.700977i 0.994288 − 0.106734i \(-0.0340393\pi\)
−0.589578 + 0.807711i \(0.700706\pi\)
\(90\) −5.47488 + 3.87630i −0.577103 + 0.408598i
\(91\) −5.64556 + 9.77840i −0.591816 + 1.02506i
\(92\) −0.135668 − 0.290940i −0.0141443 − 0.0303326i
\(93\) −0.216997 + 4.01036i −0.0225015 + 0.415855i
\(94\) −3.18104 + 3.79101i −0.328099 + 0.391013i
\(95\) −0.925690 + 1.39822i −0.0949738 + 0.143454i
\(96\) −1.64637 − 0.538027i −0.168032 − 0.0549121i
\(97\) 2.43551 − 1.13570i 0.247289 − 0.115313i −0.295022 − 0.955490i \(-0.595327\pi\)
0.542311 + 0.840178i \(0.317549\pi\)
\(98\) −0.149085 − 0.556392i −0.0150598 − 0.0562041i
\(99\) −1.47948 + 13.6313i −0.148693 + 1.37000i
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 270.2.r.a.113.10 ✓ 216
3.2 odd 2 810.2.s.a.773.8 216
5.2 odd 4 inner 270.2.r.a.167.6 yes 216
15.2 even 4 810.2.s.a.287.11 216
27.11 odd 18 inner 270.2.r.a.173.6 yes 216
27.16 even 9 810.2.s.a.683.11 216
135.92 even 36 inner 270.2.r.a.227.10 yes 216
135.97 odd 36 810.2.s.a.197.8 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.r.a.113.10 ✓ 216 1.1 even 1 trivial
270.2.r.a.167.6 yes 216 5.2 odd 4 inner
270.2.r.a.173.6 yes 216 27.11 odd 18 inner
270.2.r.a.227.10 yes 216 135.92 even 36 inner
810.2.s.a.197.8 216 135.97 odd 36
810.2.s.a.287.11 216 15.2 even 4
810.2.s.a.683.11 216 27.16 even 9
810.2.s.a.773.8 216 3.2 odd 2