Show commands: Magma / Pari/GP / SageMath

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 257.17
Character \(\chi\) \(=\) 270.257
Dual form 270.2.r.a.83.17

$q$-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+(0.819152 + 0.573576i) q^{2} +(1.61695 - 0.620860i) q^{3} +(0.342020 + 0.939693i) q^{4} +(-0.730092 - 2.11352i) q^{5} +(1.68064 + 0.418867i) q^{6} +(0.512056 + 1.09811i) q^{7} +(-0.258819 + 0.965926i) q^{8} +(2.22907 - 2.00780i) q^{9} +(0.614209 - 2.15006i) q^{10} +(-0.131652 - 0.156897i) q^{11} +(1.13645 + 1.30709i) q^{12} +(1.17604 + 1.67956i) q^{13} +(-0.210397 + 1.19322i) q^{14} +(-2.49272 - 2.96418i) q^{15} +(-0.766044 + 0.642788i) q^{16} +(0.232092 + 0.866178i) q^{17} +(2.97757 - 0.366155i) q^{18} +(-0.760464 + 0.439054i) q^{19} +(1.73635 - 1.40893i) q^{20} +(1.50974 + 1.45767i) q^{21} +(-0.0178508 - 0.204035i) q^{22} +(-5.70966 - 2.66246i) q^{23} +(0.181207 + 1.72255i) q^{24} +(-3.93393 + 3.08613i) q^{25} +2.05036i q^{26} +(2.35773 - 4.63046i) q^{27} +(-0.856750 + 0.856750i) q^{28} +(0.321053 + 1.82078i) q^{29} +(-0.341738 - 3.85788i) q^{30} +(-7.59799 + 2.76544i) q^{31} +(-0.996195 + 0.0871557i) q^{32} +(-0.310287 - 0.171957i) q^{33} +(-0.306701 + 0.842654i) q^{34} +(1.94702 - 1.88396i) q^{35} +(2.64910 + 1.40793i) q^{36} +(-2.05840 + 0.551547i) q^{37} +(-0.874767 - 0.0765322i) q^{38} +(2.94437 + 1.98561i) q^{39} +(2.23047 - 0.158195i) q^{40} +(-0.641136 - 0.113050i) q^{41} +(0.400621 + 2.06001i) q^{42} +(-1.12279 + 12.8336i) q^{43} +(0.102407 - 0.177375i) q^{44} +(-5.87095 - 3.24530i) q^{45} +(-3.14995 - 5.45588i) q^{46} +(8.41431 - 3.92366i) q^{47} +(-0.839576 + 1.51496i) q^{48} +(3.55587 - 4.23773i) q^{49} +(-4.99262 + 0.271595i) q^{50} +(0.913056 + 1.25647i) q^{51} +(-1.17604 + 1.67956i) q^{52} +(-4.82344 - 4.82344i) q^{53} +(4.58726 - 2.44071i) q^{54} +(-0.235487 + 0.392799i) q^{55} +(-1.19322 + 0.210397i) q^{56} +(-0.957043 + 1.18207i) q^{57} +(-0.781366 + 1.67564i) q^{58} +(-3.62459 - 3.04139i) q^{59} +(1.93285 - 3.35620i) q^{60} +(-1.71537 - 0.624344i) q^{61} +(-7.81010 - 2.09271i) q^{62} +(3.34619 + 1.41965i) q^{63} +(-0.866025 - 0.500000i) q^{64} +(2.69117 - 3.71182i) q^{65} +(-0.155541 - 0.318832i) q^{66} +(6.81694 - 4.77327i) q^{67} +(-0.734561 + 0.514345i) q^{68} +(-10.8853 - 0.760167i) q^{69} +(2.67550 - 0.426482i) q^{70} +(2.96211 + 1.71017i) q^{71} +(1.36246 + 2.67277i) q^{72} +(12.5459 + 3.36167i) q^{73} +(-2.00250 - 0.728850i) q^{74} +(-4.44493 + 7.43254i) q^{75} +(-0.672670 - 0.564437i) q^{76} +(0.104877 - 0.224909i) q^{77} +(1.27299 + 3.31534i) q^{78} +(-13.9602 + 2.46157i) q^{79} +(1.91783 + 1.14976i) q^{80} +(0.937470 - 8.95104i) q^{81} +(-0.460346 - 0.460346i) q^{82} +(-6.90097 + 9.85560i) q^{83} +(-0.853402 + 1.91725i) q^{84} +(1.66124 - 1.12292i) q^{85} +(-8.28078 + 9.86864i) q^{86} +(1.64958 + 2.74479i) q^{87} +(0.185625 - 0.0865584i) q^{88} +(-8.58613 - 14.8716i) q^{89} +(-2.94777 - 6.02583i) q^{90} +(-1.24214 + 2.15145i) q^{91} +(0.549073 - 6.27594i) q^{92} +(-10.5686 + 9.18887i) q^{93} +(9.14312 + 1.61218i) q^{94} +(1.48316 + 1.28671i) q^{95} +(-1.55669 + 0.759424i) q^{96} +(10.1255 + 0.885866i) q^{97} +(5.34346 - 1.43178i) q^{98} +(-0.608480 - 0.0854024i) 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 \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.819152 + 0.573576i 0.579228 + 0.405580i
\(3\) 1.61695 − 0.620860i 0.933548 − 0.358454i
\(4\) 0.342020 + 0.939693i 0.171010 + 0.469846i
\(5\) −0.730092 − 2.11352i −0.326507 − 0.945195i
\(6\) 1.68064 + 0.418867i 0.686118 + 0.171002i
\(7\) 0.512056 + 1.09811i 0.193539 + 0.415046i 0.978999 − 0.203867i \(-0.0653511\pi\)
−0.785460 + 0.618913i \(0.787573\pi\)
\(8\) −0.258819 + 0.965926i −0.0915064 + 0.341506i
\(9\) 2.22907 − 2.00780i 0.743022 − 0.669267i
\(10\) 0.614209 − 2.15006i 0.194230 − 0.679908i
\(11\) −0.131652 − 0.156897i −0.0396947 − 0.0473063i 0.745832 − 0.666134i \(-0.232052\pi\)
−0.785527 + 0.618828i \(0.787608\pi\)
\(12\) 1.13645 + 1.30709i 0.328064 + 0.377325i
\(13\) 1.17604 + 1.67956i 0.326175 + 0.465826i 0.948429 − 0.316989i \(-0.102672\pi\)
−0.622254 + 0.782815i \(0.713783\pi\)
\(14\) −0.210397 + 1.19322i −0.0562310 + 0.318902i
\(15\) −2.49272 − 2.96418i −0.643618 − 0.765347i
\(16\) −0.766044 + 0.642788i −0.191511 + 0.160697i
\(17\) 0.232092 + 0.866178i 0.0562905 + 0.210079i 0.988343 − 0.152245i \(-0.0486502\pi\)
−0.932052 + 0.362324i \(0.881984\pi\)
\(18\) 2.97757 − 0.366155i 0.701820 − 0.0863035i
\(19\) −0.760464 + 0.439054i −0.174462 + 0.100726i −0.584688 − 0.811258i \(-0.698783\pi\)
0.410226 + 0.911984i \(0.365450\pi\)
\(20\) 1.73635 − 1.40893i 0.388260 − 0.315046i
\(21\) 1.50974 + 1.45767i 0.329452 + 0.318090i
\(22\) −0.0178508 − 0.204035i −0.00380580 − 0.0435005i
\(23\) −5.70966 − 2.66246i −1.19055 − 0.555161i −0.276595 − 0.960986i \(-0.589206\pi\)
−0.913950 + 0.405826i \(0.866984\pi\)
\(24\) 0.181207 + 1.72255i 0.0369887 + 0.351613i
\(25\) −3.93393 + 3.08613i −0.786787 + 0.617225i
\(26\) 2.05036i 0.402109i
\(27\) 2.35773 − 4.63046i 0.453745 − 0.891132i
\(28\) −0.856750 + 0.856750i −0.161911 + 0.161911i
\(29\) 0.321053 + 1.82078i 0.0596180 + 0.338110i 0.999998 − 0.00201529i \(-0.000641486\pi\)
−0.940380 + 0.340126i \(0.889530\pi\)
\(30\) −0.341738 − 3.85788i −0.0623925 − 0.704349i
\(31\) −7.59799 + 2.76544i −1.36464 + 0.496688i −0.917486 − 0.397769i \(-0.869785\pi\)
−0.447154 + 0.894457i \(0.647562\pi\)
\(32\) −0.996195 + 0.0871557i −0.176104 + 0.0154071i
\(33\) −0.310287 − 0.171957i −0.0540140 − 0.0299339i
\(34\) −0.306701 + 0.842654i −0.0525988 + 0.144514i
\(35\) 1.94702 − 1.88396i 0.329107 − 0.318447i
\(36\) 2.64910 + 1.40793i 0.441517 + 0.234655i
\(37\) −2.05840 + 0.551547i −0.338399 + 0.0906738i −0.424017 − 0.905654i \(-0.639380\pi\)
0.0856178 + 0.996328i \(0.472714\pi\)
\(38\) −0.874767 − 0.0765322i −0.141906 − 0.0124152i
\(39\) 2.94437 + 1.98561i 0.471477 + 0.317952i
\(40\) 2.23047 − 0.158195i 0.352667 − 0.0250128i
\(41\) −0.641136 − 0.113050i −0.100129 − 0.0176554i 0.123359 − 0.992362i \(-0.460633\pi\)
−0.223488 + 0.974707i \(0.571744\pi\)
\(42\) 0.400621 + 2.06001i 0.0618172 + 0.317866i
\(43\) −1.12279 + 12.8336i −0.171224 + 1.95710i 0.101046 + 0.994882i \(0.467781\pi\)
−0.272271 + 0.962221i \(0.587775\pi\)
\(44\) 0.102407 − 0.177375i 0.0154385 − 0.0267402i
\(45\) −5.87095 − 3.24530i −0.875190 − 0.483780i
\(46\) −3.14995 − 5.45588i −0.464435 − 0.804426i
\(47\) 8.41431 − 3.92366i 1.22735 − 0.572324i 0.302683 − 0.953091i \(-0.402118\pi\)
0.924671 + 0.380767i \(0.124340\pi\)
\(48\) −0.839576 + 1.51496i −0.121182 + 0.218666i
\(49\) 3.55587 − 4.23773i 0.507982 − 0.605389i
\(50\) −4.99262 + 0.271595i −0.706063 + 0.0384094i
\(51\) 0.913056 + 1.25647i 0.127853 + 0.175941i
\(52\) −1.17604 + 1.67956i −0.163087 + 0.232913i
\(53\) −4.82344 − 4.82344i −0.662550 − 0.662550i 0.293430 − 0.955980i \(-0.405203\pi\)
−0.955980 + 0.293430i \(0.905203\pi\)
\(54\) 4.58726 − 2.44071i 0.624247 − 0.332138i
\(55\) −0.235487 + 0.392799i −0.0317531 + 0.0529650i
\(56\) −1.19322 + 0.210397i −0.159451 + 0.0281155i
\(57\) −0.957043 + 1.18207i −0.126763 + 0.156569i
\(58\) −0.781366 + 1.67564i −0.102598 + 0.220023i
\(59\) −3.62459 − 3.04139i −0.471882 − 0.395956i 0.375599 − 0.926782i \(-0.377437\pi\)
−0.847480 + 0.530827i \(0.821882\pi\)
\(60\) 1.93285 − 3.35620i 0.249530 − 0.433284i
\(61\) −1.71537 − 0.624344i −0.219631 − 0.0799391i 0.229861 − 0.973223i \(-0.426173\pi\)
−0.449492 + 0.893284i \(0.648395\pi\)
\(62\) −7.81010 − 2.09271i −0.991884 − 0.265774i
\(63\) 3.34619 + 1.41965i 0.421580 + 0.178859i
\(64\) −0.866025 − 0.500000i −0.108253 − 0.0625000i
\(65\) 2.69117 − 3.71182i 0.333798 − 0.460394i
\(66\) −0.155541 − 0.318832i −0.0191458 − 0.0392456i
\(67\) 6.81694 − 4.77327i 0.832822 − 0.583148i −0.0775268 − 0.996990i \(-0.524702\pi\)
0.910349 + 0.413842i \(0.135813\pi\)
\(68\) −0.734561 + 0.514345i −0.0890786 + 0.0623735i
\(69\) −10.8853 − 0.760167i −1.31043 − 0.0915133i
\(70\) 2.67550 − 0.426482i 0.319784 − 0.0509743i
\(71\) 2.96211 + 1.71017i 0.351537 + 0.202960i 0.665362 − 0.746521i \(-0.268277\pi\)
−0.313825 + 0.949481i \(0.601610\pi\)
\(72\) 1.36246 + 2.67277i 0.160568 + 0.314989i
\(73\) 12.5459 + 3.36167i 1.46839 + 0.393454i 0.902379 − 0.430943i \(-0.141819\pi\)
0.566011 + 0.824397i \(0.308486\pi\)
\(74\) −2.00250 − 0.728850i −0.232786 − 0.0847271i
\(75\) −4.44493 + 7.43254i −0.513256 + 0.858236i
\(76\) −0.672670 − 0.564437i −0.0771606 − 0.0647454i
\(77\) 0.104877 − 0.224909i 0.0119518 − 0.0256307i
\(78\) 1.27299 + 3.31534i 0.144138 + 0.375388i
\(79\) −13.9602 + 2.46157i −1.57065 + 0.276948i −0.890104 − 0.455757i \(-0.849369\pi\)
−0.680546 + 0.732705i \(0.738257\pi\)
\(80\) 1.91783 + 1.14976i 0.214420 + 0.128547i
\(81\) 0.937470 − 8.95104i 0.104163 − 0.994560i
\(82\) −0.460346 − 0.460346i −0.0508367 − 0.0508367i
\(83\) −6.90097 + 9.85560i −0.757479 + 1.08179i 0.236436 + 0.971647i \(0.424021\pi\)
−0.993915 + 0.110146i \(0.964868\pi\)
\(84\) −0.853402 + 1.91725i −0.0931138 + 0.209189i
\(85\) 1.66124 − 1.12292i 0.180186 − 0.121798i
\(86\) −8.28078 + 9.86864i −0.892939 + 1.06416i
\(87\) 1.64958 + 2.74479i 0.176853 + 0.294272i
\(88\) 0.185625 − 0.0865584i 0.0197877 − 0.00922716i
\(89\) −8.58613 − 14.8716i −0.910128 − 1.57639i −0.813882 − 0.581031i \(-0.802650\pi\)
−0.0962464 − 0.995358i \(-0.530684\pi\)
\(90\) −2.94777 − 6.02583i −0.310723 − 0.635178i
\(91\) −1.24214 + 2.15145i −0.130212 + 0.225533i
\(92\) 0.549073 − 6.27594i 0.0572448 − 0.654312i
\(93\) −10.5686 + 9.18887i −1.09592 + 0.952842i
\(94\) 9.14312 + 1.61218i 0.943041 + 0.166284i
\(95\) 1.48316 + 1.28671i 0.152169 + 0.132013i
\(96\) −1.55669 + 0.759424i −0.158879 + 0.0775084i
\(97\) 10.1255 + 0.885866i 1.02809 + 0.0899460i 0.588700 − 0.808351i \(-0.299640\pi\)
0.439388 + 0.898297i \(0.355195\pi\)
\(98\) 5.34346 − 1.43178i 0.539771 − 0.144631i
\(99\) −0.608480 − 0.0854024i −0.0611545 − 0.00858326i
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.257.17 yes 216
3.2 odd 2 810.2.s.a.557.7 216
5.3 odd 4 inner 270.2.r.a.203.15 yes 216
15.8 even 4 810.2.s.a.233.2 216
27.2 odd 18 inner 270.2.r.a.137.15 yes 216
27.25 even 9 810.2.s.a.737.2 216
135.83 even 36 inner 270.2.r.a.83.17 ✓ 216
135.133 odd 36 810.2.s.a.413.7 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.r.a.83.17 ✓ 216 135.83 even 36 inner
270.2.r.a.137.15 yes 216 27.2 odd 18 inner
270.2.r.a.203.15 yes 216 5.3 odd 4 inner
270.2.r.a.257.17 yes 216 1.1 even 1 trivial
810.2.s.a.233.2 216 15.8 even 4
810.2.s.a.413.7 216 135.133 odd 36
810.2.s.a.557.7 216 3.2 odd 2
810.2.s.a.737.2 216 27.25 even 9