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Properties

Label 140.2.w.b.123.13
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 123.13
Character \(\chi\) \(=\) 140.123
Dual form 140.2.w.b.107.13

$q$-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+(0.839014 - 1.13844i) q^{2} +(-0.290169 - 1.08292i) q^{3} +(-0.592112 - 1.91034i) q^{4} +(-2.11448 + 0.727309i) q^{5} +(-1.47630 - 0.578247i) q^{6} +(2.16744 - 1.51730i) q^{7} +(-2.67161 - 0.928715i) q^{8} +(1.50955 - 0.871538i) q^{9} +(-0.946076 + 3.01744i) q^{10} +(-2.58391 - 1.49182i) q^{11} +(-1.89694 + 1.19553i) q^{12} +(4.05418 + 4.05418i) q^{13} +(0.0911579 - 3.74055i) q^{14} +(1.40118 + 2.07878i) q^{15} +(-3.29881 + 2.26227i) q^{16} +(0.617565 + 2.30478i) q^{17} +(0.274334 - 2.44977i) q^{18} +(1.25694 + 2.17708i) q^{19} +(2.64142 + 3.60873i) q^{20} +(-2.27204 - 1.90691i) q^{21} +(-3.86629 + 1.68998i) q^{22} +(5.79414 + 1.55254i) q^{23} +(-0.230511 + 3.16263i) q^{24} +(3.94204 - 3.07576i) q^{25} +(8.01697 - 1.21395i) q^{26} +(-3.76010 - 3.76010i) q^{27} +(-4.18192 - 3.24215i) q^{28} -2.55433i q^{29} +(3.54218 + 0.148963i) q^{30} +(-3.12248 - 1.80277i) q^{31} +(-0.192271 + 5.65359i) 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} +(3.53308 + 0.395646i) q^{38} +(3.21397 - 5.56676i) q^{39} +(6.32452 + 0.0206657i) q^{40} -7.93727 q^{41} +(-4.07718 + 0.986673i) q^{42} +(-7.62646 + 7.62646i) q^{43} +(-1.31992 + 5.81947i) q^{44} +(-2.55803 + 2.94076i) q^{45} +(6.62884 - 5.29371i) q^{46} +(-0.765987 + 2.85870i) q^{47} +(3.40708 + 2.91592i) q^{48} +(2.39563 - 6.57731i) q^{49} +(-0.194151 - 7.06840i) q^{50} +(2.31671 - 1.33755i) q^{51} +(5.34433 - 10.1454i) q^{52} +(-2.47243 + 0.662485i) q^{53} +(-7.43544 + 1.12589i) q^{54} +(6.54863 + 1.27512i) q^{55} +(-7.19970 + 2.04068i) q^{56} +(1.99289 - 1.99289i) q^{57} +(-2.90796 - 2.14312i) q^{58} +(-1.04694 + 1.81336i) q^{59} +(3.14152 - 3.90759i) q^{60} +(0.950478 + 1.64628i) q^{61} +(-4.67216 + 2.04223i) q^{62} +(1.94948 - 4.17944i) q^{63} +(6.27498 + 4.96233i) q^{64} +(-11.5211 - 5.62384i) q^{65} +(2.95199 + 3.69652i) q^{66} +(-3.00512 + 0.805219i) q^{67} +(4.03726 - 2.54445i) q^{68} -6.72512i q^{69} +(2.52778 + 7.97561i) q^{70} +8.09721i q^{71} +(-4.84233 + 0.926468i) q^{72} +(-9.41954 + 2.52396i) q^{73} +(5.27428 - 4.21197i) q^{74} +(-4.47467 - 3.37645i) q^{75} +(3.41472 - 3.69026i) q^{76} +(-7.86400 + 0.687116i) q^{77} +(-3.64089 - 8.32952i) q^{78} +(-4.03058 - 6.98117i) q^{79} +(5.32989 - 7.18278i) q^{80} +(-0.366226 + 0.634322i) q^{81} +(-6.65948 + 9.03614i) q^{82} +(5.99790 - 5.99790i) q^{83} +(-2.29754 + 5.46948i) q^{84} +(-2.98212 - 4.42426i) q^{85} +(2.28360 + 15.0810i) q^{86} +(-2.76614 + 0.741186i) q^{87} +(5.51771 + 6.38527i) q^{88} +(-1.77990 + 1.02763i) q^{89} +(1.20167 + 5.37951i) q^{90} +(14.9386 + 2.63582i) q^{91} +(-0.464910 - 11.9881i) q^{92} +(-1.04621 + 3.90452i) q^{93} +(2.61180 + 3.27052i) q^{94} +(-4.24118 - 3.68921i) q^{95} +(6.17820 - 1.43228i) q^{96} +(6.63160 - 6.63160i) q^{97} +(-5.47794 - 8.24574i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.839014 − 1.13844i 0.593272 − 0.805002i
\(3\) −0.290169 − 1.08292i −0.167529 − 0.625227i −0.997704 − 0.0677240i \(-0.978426\pi\)
0.830175 − 0.557503i \(-0.188240\pi\)
\(4\) −0.592112 − 1.91034i −0.296056 − 0.955171i
\(5\) −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
\(8\) −2.67161 − 0.928715i −0.944556 − 0.328350i
\(9\) 1.50955 − 0.871538i 0.503183 − 0.290513i
\(10\) −0.946076 + 3.01744i −0.299176 + 0.954198i
\(11\) −2.58391 − 1.49182i −0.779077 − 0.449800i 0.0570261 − 0.998373i \(-0.481838\pi\)
−0.836103 + 0.548572i \(0.815172\pi\)
\(12\) −1.89694 + 1.19553i −0.547600 + 0.345121i
\(13\) 4.05418 + 4.05418i 1.12443 + 1.12443i 0.991068 + 0.133359i \(0.0425763\pi\)
0.133359 + 0.991068i \(0.457424\pi\)
\(14\) 0.0911579 − 3.74055i 0.0243630 − 0.999703i
\(15\) 1.40118 + 2.07878i 0.361782 + 0.536738i
\(16\) −3.29881 + 2.26227i −0.824702 + 0.565568i
\(17\) 0.617565 + 2.30478i 0.149782 + 0.558992i 0.999496 + 0.0317490i \(0.0101077\pi\)
−0.849714 + 0.527243i \(0.823226\pi\)
\(18\) 0.274334 − 2.44977i 0.0646611 − 0.577416i
\(19\) 1.25694 + 2.17708i 0.288362 + 0.499457i 0.973419 − 0.229032i \(-0.0735561\pi\)
−0.685057 + 0.728489i \(0.740223\pi\)
\(20\) 2.64142 + 3.60873i 0.590639 + 0.806936i
\(21\) −2.27204 − 1.90691i −0.495800 − 0.416121i
\(22\) −3.86629 + 1.68998i −0.824295 + 0.360304i
\(23\) 5.79414 + 1.55254i 1.20816 + 0.323726i 0.806041 − 0.591860i \(-0.201606\pi\)
0.402122 + 0.915586i \(0.368273\pi\)
\(24\) −0.230511 + 3.16263i −0.0470529 + 0.645570i
\(25\) 3.94204 − 3.07576i 0.788409 − 0.615152i
\(26\) 8.01697 − 1.21395i 1.57226 − 0.238075i
\(27\) −3.76010 − 3.76010i −0.723632 − 0.723632i
\(28\) −4.18192 − 3.24215i −0.790309 − 0.612708i
\(29\) − 2.55433i − 0.474327i −0.971470 − 0.237163i \(-0.923782\pi\)
0.971470 − 0.237163i \(-0.0762177\pi\)
\(30\) 3.54218 + 0.148963i 0.646711 + 0.0271967i
\(31\) −3.12248 − 1.80277i −0.560815 − 0.323787i 0.192658 − 0.981266i \(-0.438289\pi\)
−0.753472 + 0.657479i \(0.771623\pi\)
\(32\) −0.192271 + 5.65359i −0.0339891 + 0.999422i
\(33\) −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 \(0.455006\pi\)
\(38\) 3.53308 + 0.395646i 0.573141 + 0.0641823i
\(39\) 3.21397 − 5.56676i 0.514648 − 0.891396i
\(40\) 6.32452 + 0.0206657i 0.999995 + 0.00326753i
\(41\) −7.93727 −1.23959 −0.619797 − 0.784763i \(-0.712785\pi\)
−0.619797 + 0.784763i \(0.712785\pi\)
\(42\) −4.07718 + 0.986673i −0.629123 + 0.152247i
\(43\) −7.62646 + 7.62646i −1.16302 + 1.16302i −0.179215 + 0.983810i \(0.557356\pi\)
−0.983810 + 0.179215i \(0.942644\pi\)
\(44\) −1.31992 + 5.81947i −0.198986 + 0.877318i
\(45\) −2.55803 + 2.94076i −0.381329 + 0.438382i
\(46\) 6.62884 − 5.29371i 0.977369 − 0.780515i
\(47\) −0.765987 + 2.85870i −0.111731 + 0.416985i −0.999022 − 0.0442256i \(-0.985918\pi\)
0.887291 + 0.461210i \(0.152585\pi\)
\(48\) 3.40708 + 2.91592i 0.491770 + 0.420876i
\(49\) 2.39563 − 6.57731i 0.342232 − 0.939615i
\(50\) −0.194151 − 7.06840i −0.0274571 − 0.999623i
\(51\) 2.31671 − 1.33755i 0.324404 − 0.187295i
\(52\) 5.34433 − 10.1454i 0.741126 − 1.40691i
\(53\) −2.47243 + 0.662485i −0.339614 + 0.0909993i −0.424595 − 0.905383i \(-0.639584\pi\)
0.0849813 + 0.996383i \(0.472917\pi\)
\(54\) −7.43544 + 1.12589i −1.01184 + 0.153214i
\(55\) 6.54863 + 1.27512i 0.883017 + 0.171938i
\(56\) −7.19970 + 2.04068i −0.962100 + 0.272697i
\(57\) 1.99289 − 1.99289i 0.263965 − 0.263965i
\(58\) −2.90796 − 2.14312i −0.381834 − 0.281405i
\(59\) −1.04694 + 1.81336i −0.136301 + 0.236079i −0.926094 − 0.377294i \(-0.876855\pi\)
0.789793 + 0.613374i \(0.210188\pi\)
\(60\) 3.14152 − 3.90759i 0.405569 − 0.504468i
\(61\) 0.950478 + 1.64628i 0.121696 + 0.210784i 0.920437 − 0.390892i \(-0.127833\pi\)
−0.798740 + 0.601676i \(0.794500\pi\)
\(62\) −4.67216 + 2.04223i −0.593365 + 0.259363i
\(63\) 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
\(66\) 2.95199 + 3.69652i 0.363365 + 0.455010i
\(67\) −3.00512 + 0.805219i −0.367134 + 0.0983732i −0.437669 − 0.899136i \(-0.644196\pi\)
0.0705354 + 0.997509i \(0.477529\pi\)
\(68\) 4.03726 − 2.54445i 0.489589 − 0.308560i
\(69\) − 6.72512i − 0.809609i
\(70\) 2.52778 + 7.97561i 0.302128 + 0.953268i
\(71\) 8.09721i 0.960962i 0.877005 + 0.480481i \(0.159538\pi\)
−0.877005 + 0.480481i \(0.840462\pi\)
\(72\) −4.84233 + 0.926468i −0.570674 + 0.109185i
\(73\) −9.41954 + 2.52396i −1.10247 + 0.295407i −0.763771 − 0.645487i \(-0.776655\pi\)
−0.338702 + 0.940894i \(0.609988\pi\)
\(74\) 5.27428 − 4.21197i 0.613122 − 0.489632i
\(75\) −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
\(78\) −3.64089 − 8.32952i −0.412249 − 0.943133i
\(79\) −4.03058 − 6.98117i −0.453476 − 0.785443i 0.545123 − 0.838356i \(-0.316483\pi\)
−0.998599 + 0.0529126i \(0.983150\pi\)
\(80\) 5.32989 − 7.18278i 0.595899 − 0.803059i
\(81\) −0.366226 + 0.634322i −0.0406918 + 0.0704802i
\(82\) −6.65948 + 9.03614i −0.735416 + 0.997875i
\(83\) 5.99790 − 5.99790i 0.658356 − 0.658356i −0.296635 − 0.954991i \(-0.595865\pi\)
0.954991 + 0.296635i \(0.0958646\pi\)
\(84\) −2.29754 + 5.46948i −0.250682 + 0.596769i
\(85\) −2.98212 − 4.42426i −0.323456 − 0.479878i
\(86\) 2.28360 + 15.0810i 0.246247 + 1.62623i
\(87\) −2.76614 + 0.741186i −0.296562 + 0.0794635i
\(88\) 5.51771 + 6.38527i 0.588190 + 0.680672i
\(89\) −1.77990 + 1.02763i −0.188669 + 0.108928i −0.591359 − 0.806408i \(-0.701408\pi\)
0.402690 + 0.915336i \(0.368075\pi\)
\(90\) 1.20167 + 5.37951i 0.126667 + 0.567051i
\(91\) 14.9386 + 2.63582i 1.56599 + 0.276309i
\(92\) −0.464910 − 11.9881i −0.0484702 − 1.24984i
\(93\) −1.04621 + 3.90452i −0.108487 + 0.404880i
\(94\) 2.61180 + 3.27052i 0.269387 + 0.337329i
\(95\) −4.24118 − 3.68921i −0.435136 − 0.378505i
\(96\) 6.17820 − 1.43228i 0.630560 − 0.146181i
\(97\) 6.63160 − 6.63160i 0.673337 − 0.673337i −0.285147 − 0.958484i \(-0.592042\pi\)
0.958484 + 0.285147i \(0.0920425\pi\)
\(98\) −5.47794 − 8.24574i −0.553355 − 0.832945i
\(99\) −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.123.13 yes 72
4.3 odd 2 inner 140.2.w.b.123.10 yes 72
5.2 odd 4 inner 140.2.w.b.67.14 yes 72
5.3 odd 4 700.2.be.e.207.5 72
5.4 even 2 700.2.be.e.543.6 72
7.2 even 3 inner 140.2.w.b.23.2 ✓ 72
7.3 odd 6 980.2.k.j.883.13 36
7.4 even 3 980.2.k.k.883.13 36
7.5 odd 6 980.2.x.m.863.2 72
7.6 odd 2 980.2.x.m.263.13 72
20.3 even 4 700.2.be.e.207.17 72
20.7 even 4 inner 140.2.w.b.67.2 yes 72
20.19 odd 2 700.2.be.e.543.9 72
28.3 even 6 980.2.k.j.883.3 36
28.11 odd 6 980.2.k.k.883.3 36
28.19 even 6 980.2.x.m.863.14 72
28.23 odd 6 inner 140.2.w.b.23.14 yes 72
28.27 even 2 980.2.x.m.263.10 72
35.2 odd 12 inner 140.2.w.b.107.10 yes 72
35.9 even 6 700.2.be.e.443.17 72
35.12 even 12 980.2.x.m.667.10 72
35.17 even 12 980.2.k.j.687.3 36
35.23 odd 12 700.2.be.e.107.9 72
35.27 even 4 980.2.x.m.67.14 72
35.32 odd 12 980.2.k.k.687.3 36
140.23 even 12 700.2.be.e.107.6 72
140.27 odd 4 980.2.x.m.67.2 72
140.47 odd 12 980.2.x.m.667.13 72
140.67 even 12 980.2.k.k.687.13 36
140.79 odd 6 700.2.be.e.443.5 72
140.87 odd 12 980.2.k.j.687.13 36
140.107 even 12 inner 140.2.w.b.107.13 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.w.b.23.2 ✓ 72 7.2 even 3 inner
140.2.w.b.23.14 yes 72 28.23 odd 6 inner
140.2.w.b.67.2 yes 72 20.7 even 4 inner
140.2.w.b.67.14 yes 72 5.2 odd 4 inner
140.2.w.b.107.10 yes 72 35.2 odd 12 inner
140.2.w.b.107.13 yes 72 140.107 even 12 inner
140.2.w.b.123.10 yes 72 4.3 odd 2 inner
140.2.w.b.123.13 yes 72 1.1 even 1 trivial
700.2.be.e.107.6 72 140.23 even 12
700.2.be.e.107.9 72 35.23 odd 12
700.2.be.e.207.5 72 5.3 odd 4
700.2.be.e.207.17 72 20.3 even 4
700.2.be.e.443.5 72 140.79 odd 6
700.2.be.e.443.17 72 35.9 even 6
700.2.be.e.543.6 72 5.4 even 2
700.2.be.e.543.9 72 20.19 odd 2
980.2.k.j.687.3 36 35.17 even 12
980.2.k.j.687.13 36 140.87 odd 12
980.2.k.j.883.3 36 28.3 even 6
980.2.k.j.883.13 36 7.3 odd 6
980.2.k.k.687.3 36 35.32 odd 12
980.2.k.k.687.13 36 140.67 even 12
980.2.k.k.883.3 36 28.11 odd 6
980.2.k.k.883.13 36 7.4 even 3
980.2.x.m.67.2 72 140.27 odd 4
980.2.x.m.67.14 72 35.27 even 4
980.2.x.m.263.10 72 28.27 even 2
980.2.x.m.263.13 72 7.6 odd 2
980.2.x.m.667.10 72 35.12 even 12
980.2.x.m.667.13 72 140.47 odd 12
980.2.x.m.863.2 72 7.5 odd 6
980.2.x.m.863.14 72 28.19 even 6