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Properties

Label 140.2.w.b.123.10
Level $140$
Weight $2$
Character 140.123
Analytic conductor $1.118$
Analytic rank $0$
Dimension $72$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [140,2,Mod(23,140)] 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("140.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(140, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 123.10
Character \(\chi\) \(=\) 140.123
Dual form 140.2.w.b.107.10

$q$-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+(0.157385 + 1.40543i) q^{2} +(0.290169 + 1.08292i) q^{3} +(-1.95046 + 0.442386i) q^{4} +(-2.11448 + 0.727309i) q^{5} +(-1.47630 + 0.578247i) q^{6} +(-2.16744 + 1.51730i) q^{7} +(-0.928715 - 2.67161i) q^{8} +(1.50955 - 0.871538i) q^{9} +(-1.35497 - 2.85728i) q^{10} +(2.58391 + 1.49182i) q^{11} +(-1.04503 - 1.98383i) q^{12} +(4.05418 + 4.05418i) q^{13} +(-2.47357 - 2.80739i) q^{14} +(-1.40118 - 2.07878i) q^{15} +(3.60859 - 1.72571i) q^{16} +(0.617565 + 2.30478i) q^{17} +(1.46247 + 1.98440i) q^{18} +(-1.25694 - 2.17708i) q^{19} +(3.80246 - 2.35400i) q^{20} +(-2.27204 - 1.90691i) q^{21} +(-1.68998 + 3.86629i) q^{22} +(-5.79414 - 1.55254i) q^{23} +(2.62367 - 1.78095i) q^{24} +(3.94204 - 3.07576i) q^{25} +(-5.05979 + 6.33592i) q^{26} +(3.76010 + 3.76010i) q^{27} +(3.55628 - 3.91827i) q^{28} -2.55433i q^{29} +(2.70105 - 2.29642i) q^{30} +(3.12248 + 1.80277i) q^{31} +(2.99330 + 4.80001i) q^{32} +(-0.865758 + 3.23105i) q^{33} +(-3.14202 + 1.23068i) q^{34} +(3.47947 - 4.78469i) q^{35} +(-2.55876 + 2.36770i) q^{36} +(4.61014 + 1.23528i) q^{37} +(2.86191 - 2.10918i) q^{38} +(-3.21397 + 5.56676i) q^{39} +(3.90683 + 4.97360i) q^{40} -7.93727 q^{41} +(2.32244 - 3.49331i) q^{42} +(7.62646 - 7.62646i) q^{43} +(-5.69977 - 1.76665i) q^{44} +(-2.55803 + 2.94076i) q^{45} +(1.27007 - 8.38760i) q^{46} +(0.765987 - 2.85870i) q^{47} +(2.91592 + 3.40708i) q^{48} +(2.39563 - 6.57731i) q^{49} +(4.94318 + 5.05618i) q^{50} +(-2.31671 + 1.33755i) q^{51} +(-9.70103 - 6.11400i) q^{52} +(-2.47243 + 0.662485i) q^{53} +(-4.69277 + 5.87634i) q^{54} +(-6.54863 - 1.27512i) q^{55} +(6.06656 + 4.38142i) q^{56} +(1.99289 - 1.99289i) q^{57} +(3.58993 - 0.402012i) q^{58} +(1.04694 - 1.81336i) q^{59} +(3.65256 + 3.43471i) q^{60} +(0.950478 + 1.64628i) q^{61} +(-2.04223 + 4.67216i) q^{62} +(-1.94948 + 4.17944i) q^{63} +(-6.27498 + 4.96233i) q^{64} +(-11.5211 - 5.62384i) q^{65} +(-4.67727 - 0.708243i) q^{66} +(3.00512 - 0.805219i) q^{67} +(-2.22414 - 4.22219i) q^{68} -6.72512i q^{69} +(7.27216 + 4.13711i) q^{70} -8.09721i q^{71} +(-3.73035 - 3.22351i) q^{72} +(-9.41954 + 2.52396i) q^{73} +(-1.01054 + 6.67364i) q^{74} +(4.47467 + 3.37645i) q^{75} +(3.41472 + 3.69026i) q^{76} +(-7.86400 + 0.687116i) q^{77} +(-8.32952 - 3.64089i) q^{78} +(4.03058 + 6.98117i) q^{79} +(-6.37516 + 6.27354i) q^{80} +(-0.366226 + 0.634322i) q^{81} +(-1.24921 - 11.1553i) q^{82} +(-5.99790 + 5.99790i) q^{83} +(5.27511 + 2.71422i) q^{84} +(-2.98212 - 4.42426i) q^{85} +(11.9187 + 9.51816i) q^{86} +(2.76614 - 0.741186i) q^{87} +(1.58584 - 8.28866i) q^{88} +(-1.77990 + 1.02763i) q^{89} +(-4.53562 - 3.13230i) q^{90} +(-14.9386 - 2.63582i) q^{91} +(11.9881 + 0.464910i) q^{92} +(-1.04621 + 3.90452i) q^{93} +(4.13826 + 0.626624i) q^{94} +(4.24118 + 3.68921i) q^{95} +(-4.32949 + 4.63434i) q^{96} +(6.63160 - 6.63160i) q^{97} +(9.62097 + 2.33171i) q^{98} +5.20071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8} + 2 q^{10} + 10 q^{12} - 28 q^{16} + 4 q^{17} - 20 q^{18} - 56 q^{20} + 4 q^{21} - 16 q^{22} - 16 q^{25} - 4 q^{26} + 42 q^{28} - 32 q^{30} - 38 q^{32} - 64 q^{33}+ \cdots - 90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.157385 + 1.40543i 0.111288 + 0.993788i
\(3\) 0.290169 + 1.08292i 0.167529 + 0.625227i 0.997704 + 0.0677240i \(0.0215737\pi\)
−0.830175 + 0.557503i \(0.811760\pi\)
\(4\) −1.95046 + 0.442386i −0.975230 + 0.221193i
\(5\) −2.11448 + 0.727309i −0.945624 + 0.325262i
\(6\) −1.47630 + 0.578247i −0.602699 + 0.236068i
\(7\) −2.16744 + 1.51730i −0.819217 + 0.573484i
\(8\) −0.928715 − 2.67161i −0.328350 − 0.944556i
\(9\) 1.50955 − 0.871538i 0.503183 − 0.290513i
\(10\) −1.35497 − 2.85728i −0.428478 − 0.903552i
\(11\) 2.58391 + 1.49182i 0.779077 + 0.449800i 0.836103 − 0.548572i \(-0.184828\pi\)
−0.0570261 + 0.998373i \(0.518162\pi\)
\(12\) −1.04503 − 1.98383i −0.301675 − 0.572684i
\(13\) 4.05418 + 4.05418i 1.12443 + 1.12443i 0.991068 + 0.133359i \(0.0425763\pi\)
0.133359 + 0.991068i \(0.457424\pi\)
\(14\) −2.47357 − 2.80739i −0.661090 − 0.750306i
\(15\) −1.40118 − 2.07878i −0.361782 − 0.536738i
\(16\) 3.60859 − 1.72571i 0.902147 − 0.431428i
\(17\) 0.617565 + 2.30478i 0.149782 + 0.558992i 0.999496 + 0.0317490i \(0.0101077\pi\)
−0.849714 + 0.527243i \(0.823226\pi\)
\(18\) 1.46247 + 1.98440i 0.344706 + 0.467727i
\(19\) −1.25694 − 2.17708i −0.288362 − 0.499457i 0.685057 − 0.728489i \(-0.259777\pi\)
−0.973419 + 0.229032i \(0.926444\pi\)
\(20\) 3.80246 − 2.35400i 0.850255 − 0.526371i
\(21\) −2.27204 − 1.90691i −0.495800 − 0.416121i
\(22\) −1.68998 + 3.86629i −0.360304 + 0.824295i
\(23\) −5.79414 − 1.55254i −1.20816 − 0.323726i −0.402122 − 0.915586i \(-0.631727\pi\)
−0.806041 + 0.591860i \(0.798394\pi\)
\(24\) 2.62367 − 1.78095i 0.535553 − 0.363534i
\(25\) 3.94204 − 3.07576i 0.788409 − 0.615152i
\(26\) −5.05979 + 6.33592i −0.992307 + 1.24258i
\(27\) 3.76010 + 3.76010i 0.723632 + 0.723632i
\(28\) 3.55628 − 3.91827i 0.672074 − 0.740484i
\(29\) − 2.55433i − 0.474327i −0.971470 − 0.237163i \(-0.923782\pi\)
0.971470 − 0.237163i \(-0.0762177\pi\)
\(30\) 2.70105 − 2.29642i 0.493142 − 0.419267i
\(31\) 3.12248 + 1.80277i 0.560815 + 0.323787i 0.753472 − 0.657479i \(-0.228377\pi\)
−0.192658 + 0.981266i \(0.561711\pi\)
\(32\) 2.99330 + 4.80001i 0.529147 + 0.848530i
\(33\) −0.865758 + 3.23105i −0.150709 + 0.562454i
\(34\) −3.14202 + 1.23068i −0.538851 + 0.211060i
\(35\) 3.47947 − 4.78469i 0.588138 − 0.808760i
\(36\) −2.55876 + 2.36770i −0.426460 + 0.394617i
\(37\) 4.61014 + 1.23528i 0.757903 + 0.203079i 0.617021 − 0.786947i \(-0.288339\pi\)
0.140882 + 0.990026i \(0.455006\pi\)
\(38\) 2.86191 − 2.10918i 0.464263 − 0.342154i
\(39\) −3.21397 + 5.56676i −0.514648 + 0.891396i
\(40\) 3.90683 + 4.97360i 0.617724 + 0.786395i
\(41\) −7.93727 −1.23959 −0.619797 − 0.784763i \(-0.712785\pi\)
−0.619797 + 0.784763i \(0.712785\pi\)
\(42\) 2.32244 − 3.49331i 0.358360 − 0.539029i
\(43\) 7.62646 − 7.62646i 1.16302 − 1.16302i 0.179215 − 0.983810i \(-0.442644\pi\)
0.983810 − 0.179215i \(-0.0573557\pi\)
\(44\) −5.69977 − 1.76665i −0.859272 − 0.266332i
\(45\) −2.55803 + 2.94076i −0.381329 + 0.438382i
\(46\) 1.27007 − 8.38760i 0.187261 − 1.23668i
\(47\) 0.765987 − 2.85870i 0.111731 − 0.416985i −0.887291 − 0.461210i \(-0.847415\pi\)
0.999022 + 0.0442256i \(0.0140821\pi\)
\(48\) 2.91592 + 3.40708i 0.420876 + 0.491770i
\(49\) 2.39563 − 6.57731i 0.342232 − 0.939615i
\(50\) 4.94318 + 5.05618i 0.699071 + 0.715053i
\(51\) −2.31671 + 1.33755i −0.324404 + 0.187295i
\(52\) −9.70103 − 6.11400i −1.34529 − 0.847859i
\(53\) −2.47243 + 0.662485i −0.339614 + 0.0909993i −0.424595 − 0.905383i \(-0.639584\pi\)
0.0849813 + 0.996383i \(0.472917\pi\)
\(54\) −4.69277 + 5.87634i −0.638605 + 0.799668i
\(55\) −6.54863 − 1.27512i −0.883017 − 0.171938i
\(56\) 6.06656 + 4.38142i 0.810678 + 0.585492i
\(57\) 1.99289 − 1.99289i 0.263965 − 0.263965i
\(58\) 3.58993 − 0.402012i 0.471380 − 0.0527868i
\(59\) 1.04694 − 1.81336i 0.136301 − 0.236079i −0.789793 − 0.613374i \(-0.789812\pi\)
0.926094 + 0.377294i \(0.123145\pi\)
\(60\) 3.65256 + 3.43471i 0.471544 + 0.443420i
\(61\) 0.950478 + 1.64628i 0.121696 + 0.210784i 0.920437 − 0.390892i \(-0.127833\pi\)
−0.798740 + 0.601676i \(0.794500\pi\)
\(62\) −2.04223 + 4.67216i −0.259363 + 0.593365i
\(63\) −1.94948 + 4.17944i −0.245612 + 0.526560i
\(64\) −6.27498 + 4.96233i −0.784372 + 0.620291i
\(65\) −11.5211 − 5.62384i −1.42902 − 0.697551i
\(66\) −4.67727 − 0.708243i −0.575733 − 0.0871787i
\(67\) 3.00512 − 0.805219i 0.367134 − 0.0983732i −0.0705354 − 0.997509i \(-0.522471\pi\)
0.437669 + 0.899136i \(0.355804\pi\)
\(68\) −2.22414 − 4.22219i −0.269717 − 0.512016i
\(69\) − 6.72512i − 0.809609i
\(70\) 7.27216 + 4.13711i 0.869189 + 0.494480i
\(71\) − 8.09721i − 0.960962i −0.877005 − 0.480481i \(-0.840462\pi\)
0.877005 − 0.480481i \(-0.159538\pi\)
\(72\) −3.73035 − 3.22351i −0.439626 − 0.379894i
\(73\) −9.41954 + 2.52396i −1.10247 + 0.295407i −0.763771 − 0.645487i \(-0.776655\pi\)
−0.338702 + 0.940894i \(0.609988\pi\)
\(74\) −1.01054 + 6.67364i −0.117473 + 0.775795i
\(75\) 4.47467 + 3.37645i 0.516691 + 0.389879i
\(76\) 3.41472 + 3.69026i 0.391695 + 0.423302i
\(77\) −7.86400 + 0.687116i −0.896186 + 0.0783041i
\(78\) −8.32952 − 3.64089i −0.943133 − 0.412249i
\(79\) 4.03058 + 6.98117i 0.453476 + 0.785443i 0.998599 − 0.0529126i \(-0.0168505\pi\)
−0.545123 + 0.838356i \(0.683517\pi\)
\(80\) −6.37516 + 6.27354i −0.712764 + 0.701403i
\(81\) −0.366226 + 0.634322i −0.0406918 + 0.0704802i
\(82\) −1.24921 − 11.1553i −0.137952 − 1.23189i
\(83\) −5.99790 + 5.99790i −0.658356 + 0.658356i −0.954991 − 0.296635i \(-0.904135\pi\)
0.296635 + 0.954991i \(0.404135\pi\)
\(84\) 5.27511 + 2.71422i 0.575562 + 0.296146i
\(85\) −2.98212 − 4.42426i −0.323456 − 0.479878i
\(86\) 11.9187 + 9.51816i 1.28523 + 1.02637i
\(87\) 2.76614 − 0.741186i 0.296562 − 0.0794635i
\(88\) 1.58584 − 8.28866i 0.169051 − 0.883574i
\(89\) −1.77990 + 1.02763i −0.188669 + 0.108928i −0.591359 − 0.806408i \(-0.701408\pi\)
0.402690 + 0.915336i \(0.368075\pi\)
\(90\) −4.53562 − 3.13230i −0.478096 − 0.330174i
\(91\) −14.9386 − 2.63582i −1.56599 − 0.276309i
\(92\) 11.9881 + 0.464910i 1.24984 + 0.0484702i
\(93\) −1.04621 + 3.90452i −0.108487 + 0.404880i
\(94\) 4.13826 + 0.626624i 0.426829 + 0.0646313i
\(95\) 4.24118 + 3.68921i 0.435136 + 0.378505i
\(96\) −4.32949 + 4.63434i −0.441877 + 0.472990i
\(97\) 6.63160 − 6.63160i 0.673337 − 0.673337i −0.285147 − 0.958484i \(-0.592042\pi\)
0.958484 + 0.285147i \(0.0920425\pi\)
\(98\) 9.62097 + 2.33171i 0.971865 + 0.235539i
\(99\) 5.20071 0.522691
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 140.2.w.b.123.10 yes 72
4.3 odd 2 inner 140.2.w.b.123.13 yes 72
5.2 odd 4 inner 140.2.w.b.67.2 yes 72
5.3 odd 4 700.2.be.e.207.17 72
5.4 even 2 700.2.be.e.543.9 72
7.2 even 3 inner 140.2.w.b.23.14 yes 72
7.3 odd 6 980.2.k.j.883.3 36
7.4 even 3 980.2.k.k.883.3 36
7.5 odd 6 980.2.x.m.863.14 72
7.6 odd 2 980.2.x.m.263.10 72
20.3 even 4 700.2.be.e.207.5 72
20.7 even 4 inner 140.2.w.b.67.14 yes 72
20.19 odd 2 700.2.be.e.543.6 72
28.3 even 6 980.2.k.j.883.13 36
28.11 odd 6 980.2.k.k.883.13 36
28.19 even 6 980.2.x.m.863.2 72
28.23 odd 6 inner 140.2.w.b.23.2 ✓ 72
28.27 even 2 980.2.x.m.263.13 72
35.2 odd 12 inner 140.2.w.b.107.13 yes 72
35.9 even 6 700.2.be.e.443.5 72
35.12 even 12 980.2.x.m.667.13 72
35.17 even 12 980.2.k.j.687.13 36
35.23 odd 12 700.2.be.e.107.6 72
35.27 even 4 980.2.x.m.67.2 72
35.32 odd 12 980.2.k.k.687.13 36
140.23 even 12 700.2.be.e.107.9 72
140.27 odd 4 980.2.x.m.67.14 72
140.47 odd 12 980.2.x.m.667.10 72
140.67 even 12 980.2.k.k.687.3 36
140.79 odd 6 700.2.be.e.443.17 72
140.87 odd 12 980.2.k.j.687.3 36
140.107 even 12 inner 140.2.w.b.107.10 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.w.b.23.2 ✓ 72 28.23 odd 6 inner
140.2.w.b.23.14 yes 72 7.2 even 3 inner
140.2.w.b.67.2 yes 72 5.2 odd 4 inner
140.2.w.b.67.14 yes 72 20.7 even 4 inner
140.2.w.b.107.10 yes 72 140.107 even 12 inner
140.2.w.b.107.13 yes 72 35.2 odd 12 inner
140.2.w.b.123.10 yes 72 1.1 even 1 trivial
140.2.w.b.123.13 yes 72 4.3 odd 2 inner
700.2.be.e.107.6 72 35.23 odd 12
700.2.be.e.107.9 72 140.23 even 12
700.2.be.e.207.5 72 20.3 even 4
700.2.be.e.207.17 72 5.3 odd 4
700.2.be.e.443.5 72 35.9 even 6
700.2.be.e.443.17 72 140.79 odd 6
700.2.be.e.543.6 72 20.19 odd 2
700.2.be.e.543.9 72 5.4 even 2
980.2.k.j.687.3 36 140.87 odd 12
980.2.k.j.687.13 36 35.17 even 12
980.2.k.j.883.3 36 7.3 odd 6
980.2.k.j.883.13 36 28.3 even 6
980.2.k.k.687.3 36 140.67 even 12
980.2.k.k.687.13 36 35.32 odd 12
980.2.k.k.883.3 36 7.4 even 3
980.2.k.k.883.13 36 28.11 odd 6
980.2.x.m.67.2 72 35.27 even 4
980.2.x.m.67.14 72 140.27 odd 4
980.2.x.m.263.10 72 7.6 odd 2
980.2.x.m.263.13 72 28.27 even 2
980.2.x.m.667.10 72 140.47 odd 12
980.2.x.m.667.13 72 35.12 even 12
980.2.x.m.863.2 72 28.19 even 6
980.2.x.m.863.14 72 7.5 odd 6