Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(11,168)] 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("168.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.v (of order \(6\), degree \(2\), 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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.18
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.18

$q$-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+(0.450483 + 1.34055i) q^{2} +(-1.45135 + 0.945295i) q^{3} +(-1.59413 + 1.20779i) q^{4} +(0.187787 + 0.325257i) q^{5} +(-1.92102 - 1.51976i) q^{6} +(-0.198565 + 2.63829i) q^{7} +(-2.33722 - 1.59292i) q^{8} +(1.21283 - 2.74391i) q^{9} +(-0.351427 + 0.398260i) q^{10} +(-4.18624 - 2.41693i) q^{11} +(1.17193 - 3.25984i) q^{12} +4.40701i q^{13} +(-3.62620 + 0.922319i) q^{14} +(-0.580009 - 0.294548i) q^{15} +(1.08250 - 3.85074i) q^{16} +(2.39410 + 1.38223i) q^{17} +(4.22470 + 0.389777i) q^{18} +(1.83569 + 3.17951i) q^{19} +(-0.692199 - 0.291695i) q^{20} +(-2.20578 - 4.01678i) q^{21} +(1.35417 - 6.70064i) q^{22} +(3.60943 + 6.25172i) q^{23} +(4.89791 + 0.102518i) q^{24} +(2.42947 - 4.20797i) q^{25} +(-5.90780 + 1.98528i) q^{26} +(0.833557 + 5.12886i) q^{27} +(-2.86995 - 4.44560i) q^{28} +0.968468 q^{29} +(0.133571 - 0.910218i) q^{30} +(2.78588 + 1.60843i) q^{31} +(5.64974 - 0.283544i) q^{32} +(8.36041 - 0.449426i) q^{33} +(-0.774447 + 3.83207i) q^{34} +(-0.895411 + 0.430853i) q^{35} +(1.38064 + 5.83899i) q^{36} +(-0.459519 + 0.265303i) q^{37} +(-3.43534 + 3.89315i) q^{38} +(-4.16592 - 6.39611i) q^{39} +(0.0792077 - 1.05933i) q^{40} +3.88604i q^{41} +(4.39102 - 4.76644i) q^{42} +0.747189 q^{43} +(9.59255 - 1.20319i) q^{44} +(1.12023 - 0.120788i) q^{45} +(-6.75473 + 7.65490i) q^{46} +(-5.20322 - 9.01223i) q^{47} +(2.06899 + 6.61206i) q^{48} +(-6.92114 - 1.04774i) q^{49} +(6.73541 + 1.36120i) q^{50} +(-4.78129 + 0.257025i) q^{51} +(-5.32273 - 7.02535i) q^{52} +(-1.40600 + 2.43526i) q^{53} +(-6.49997 + 3.42788i) q^{54} -1.81547i q^{55} +(4.66667 - 5.84997i) q^{56} +(-5.66981 - 2.87932i) q^{57} +(0.436278 + 1.29828i) q^{58} +(1.65184 + 0.953689i) q^{59} +(1.28036 - 0.230980i) q^{60} +(8.43102 - 4.86765i) q^{61} +(-0.901184 + 4.45918i) q^{62} +(6.99840 + 3.74465i) q^{63} +(2.92522 + 7.44601i) q^{64} +(-1.43341 + 0.827580i) q^{65} +(4.36870 + 11.0051i) q^{66} +(3.85681 - 6.68019i) q^{67} +(-5.48594 + 0.688098i) q^{68} +(-11.1483 - 5.66145i) q^{69} +(-0.980945 - 1.00625i) q^{70} -5.55936 q^{71} +(-7.20549 + 4.48118i) q^{72} +(-0.445737 + 0.772039i) q^{73} +(-0.562657 - 0.496492i) q^{74} +(0.451759 + 8.40380i) q^{75} +(-6.76651 - 2.85143i) q^{76} +(7.20780 - 10.5646i) q^{77} +(6.69761 - 8.46595i) q^{78} +(11.3282 - 6.54032i) q^{79} +(1.45576 - 0.371028i) q^{80} +(-6.05807 - 6.65581i) q^{81} +(-5.20942 + 1.75059i) q^{82} -10.1716i q^{83} +(8.36771 + 3.73917i) q^{84} +1.03826i q^{85} +(0.336596 + 1.00164i) q^{86} +(-1.40559 + 0.915488i) q^{87} +(5.93421 + 12.3172i) q^{88} +(-4.70650 + 2.71730i) q^{89} +(0.666567 + 1.44731i) q^{90} +(-11.6270 - 0.875077i) q^{91} +(-13.3046 - 5.60663i) q^{92} +(-5.56373 + 0.299087i) q^{93} +(9.73736 - 11.0350i) q^{94} +(-0.689440 + 1.19414i) q^{95} +(-7.93172 + 5.75220i) q^{96} +5.31623 q^{97} +(-1.71331 - 9.75011i) q^{98} +(-11.7090 + 8.55533i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 6 q^{10} + 10 q^{12} - 10 q^{16} - 10 q^{18} - 4 q^{19} - 20 q^{22} - 8 q^{24} - 16 q^{25} - 8 q^{27} - 22 q^{28} - 12 q^{30} - 14 q^{33} - 56 q^{34} + 4 q^{36}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.450483 + 1.34055i 0.318539 + 0.947910i
\(3\) −1.45135 + 0.945295i −0.837937 + 0.545766i
\(4\) −1.59413 + 1.20779i −0.797065 + 0.603893i
\(5\) 0.187787 + 0.325257i 0.0839810 + 0.145459i 0.904957 − 0.425504i \(-0.139903\pi\)
−0.820976 + 0.570963i \(0.806570\pi\)
\(6\) −1.92102 − 1.51976i −0.784253 − 0.620441i
\(7\) −0.198565 + 2.63829i −0.0750504 + 0.997180i
\(8\) −2.33722 − 1.59292i −0.826333 − 0.563182i
\(9\) 1.21283 − 2.74391i 0.404278 − 0.914636i
\(10\) −0.351427 + 0.398260i −0.111131 + 0.125941i
\(11\) −4.18624 − 2.41693i −1.26220 − 0.728731i −0.288700 − 0.957420i \(-0.593223\pi\)
−0.973500 + 0.228688i \(0.926556\pi\)
\(12\) 1.17193 − 3.25984i 0.338306 − 0.941036i
\(13\) 4.40701i 1.22228i 0.791521 + 0.611142i \(0.209290\pi\)
−0.791521 + 0.611142i \(0.790710\pi\)
\(14\) −3.62620 + 0.922319i −0.969143 + 0.246500i
\(15\) −0.580009 − 0.294548i −0.149758 − 0.0760518i
\(16\) 1.08250 − 3.85074i 0.270626 − 0.962685i
\(17\) 2.39410 + 1.38223i 0.580653 + 0.335240i 0.761393 − 0.648291i \(-0.224516\pi\)
−0.180740 + 0.983531i \(0.557849\pi\)
\(18\) 4.22470 + 0.389777i 0.995771 + 0.0918712i
\(19\) 1.83569 + 3.17951i 0.421137 + 0.729431i 0.996051 − 0.0887833i \(-0.0282979\pi\)
−0.574914 + 0.818214i \(0.694965\pi\)
\(20\) −0.692199 − 0.291695i −0.154780 − 0.0652251i
\(21\) −2.20578 − 4.01678i −0.481340 − 0.876534i
\(22\) 1.35417 − 6.70064i 0.288711 − 1.42858i
\(23\) 3.60943 + 6.25172i 0.752618 + 1.30357i 0.946550 + 0.322558i \(0.104543\pi\)
−0.193932 + 0.981015i \(0.562124\pi\)
\(24\) 4.89791 + 0.102518i 0.999781 + 0.0209265i
\(25\) 2.42947 − 4.20797i 0.485894 − 0.841594i
\(26\) −5.90780 + 1.98528i −1.15862 + 0.389346i
\(27\) 0.833557 + 5.12886i 0.160418 + 0.987049i
\(28\) −2.86995 − 4.44560i −0.542370 − 0.840140i
\(29\) 0.968468 0.179840 0.0899200 − 0.995949i \(-0.471339\pi\)
0.0899200 + 0.995949i \(0.471339\pi\)
\(30\) 0.133571 − 0.910218i 0.0243865 − 0.166182i
\(31\) 2.78588 + 1.60843i 0.500359 + 0.288883i 0.728862 − 0.684661i \(-0.240050\pi\)
−0.228503 + 0.973543i \(0.573383\pi\)
\(32\) 5.64974 − 0.283544i 0.998743 − 0.0501240i
\(33\) 8.36041 − 0.449426i 1.45536 − 0.0782351i
\(34\) −0.774447 + 3.83207i −0.132817 + 0.657194i
\(35\) −0.895411 + 0.430853i −0.151352 + 0.0728274i
\(36\) 1.38064 + 5.83899i 0.230107 + 0.973165i
\(37\) −0.459519 + 0.265303i −0.0755445 + 0.0436156i −0.537296 − 0.843393i \(-0.680554\pi\)
0.461752 + 0.887009i \(0.347221\pi\)
\(38\) −3.43534 + 3.89315i −0.557285 + 0.631552i
\(39\) −4.16592 − 6.39611i −0.667082 − 1.02420i
\(40\) 0.0792077 − 1.05933i 0.0125238 − 0.167495i
\(41\) 3.88604i 0.606897i 0.952848 + 0.303449i \(0.0981381\pi\)
−0.952848 + 0.303449i \(0.901862\pi\)
\(42\) 4.39102 − 4.76644i 0.677549 − 0.735477i
\(43\) 0.747189 0.113945 0.0569726 − 0.998376i \(-0.481855\pi\)
0.0569726 + 0.998376i \(0.481855\pi\)
\(44\) 9.59255 − 1.20319i 1.44613 − 0.181387i
\(45\) 1.12023 − 0.120788i 0.166994 − 0.0180061i
\(46\) −6.75473 + 7.65490i −0.995931 + 1.12865i
\(47\) −5.20322 − 9.01223i −0.758967 − 1.31457i −0.943378 − 0.331720i \(-0.892371\pi\)
0.184411 − 0.982849i \(-0.440962\pi\)
\(48\) 2.06899 + 6.61206i 0.298633 + 0.954368i
\(49\) −6.92114 − 1.04774i −0.988735 − 0.149678i
\(50\) 6.73541 + 1.36120i 0.952531 + 0.192503i
\(51\) −4.78129 + 0.257025i −0.669514 + 0.0359907i
\(52\) −5.32273 − 7.02535i −0.738129 − 0.974240i
\(53\) −1.40600 + 2.43526i −0.193129 + 0.334509i −0.946285 − 0.323332i \(-0.895197\pi\)
0.753157 + 0.657841i \(0.228530\pi\)
\(54\) −6.49997 + 3.42788i −0.884534 + 0.466476i
\(55\) − 1.81547i − 0.244798i
\(56\) 4.66667 − 5.84997i 0.623610 − 0.781735i
\(57\) −5.66981 − 2.87932i −0.750985 − 0.381375i
\(58\) 0.436278 + 1.29828i 0.0572861 + 0.170472i
\(59\) 1.65184 + 0.953689i 0.215051 + 0.124160i 0.603657 − 0.797244i \(-0.293710\pi\)
−0.388606 + 0.921404i \(0.627043\pi\)
\(60\) 1.28036 − 0.230980i 0.165294 − 0.0298194i
\(61\) 8.43102 − 4.86765i 1.07948 − 0.623239i 0.148725 − 0.988879i \(-0.452483\pi\)
0.930756 + 0.365640i \(0.119150\pi\)
\(62\) −0.901184 + 4.45918i −0.114450 + 0.566316i
\(63\) 6.99840 + 3.74465i 0.881715 + 0.471782i
\(64\) 2.92522 + 7.44601i 0.365652 + 0.930752i
\(65\) −1.43341 + 0.827580i −0.177793 + 0.102649i
\(66\) 4.36870 + 11.0051i 0.537750 + 1.35463i
\(67\) 3.85681 − 6.68019i 0.471184 − 0.816115i −0.528273 − 0.849075i \(-0.677160\pi\)
0.999457 + 0.0329601i \(0.0104934\pi\)
\(68\) −5.48594 + 0.688098i −0.665268 + 0.0834441i
\(69\) −11.1483 − 5.66145i −1.34209 − 0.681559i
\(70\) −0.980945 − 1.00625i −0.117245 − 0.120270i
\(71\) −5.55936 −0.659774 −0.329887 − 0.944020i \(-0.607011\pi\)
−0.329887 + 0.944020i \(0.607011\pi\)
\(72\) −7.20549 + 4.48118i −0.849175 + 0.528112i
\(73\) −0.445737 + 0.772039i −0.0521696 + 0.0903603i −0.890931 − 0.454139i \(-0.849947\pi\)
0.838761 + 0.544499i \(0.183280\pi\)
\(74\) −0.562657 − 0.496492i −0.0654076 − 0.0577160i
\(75\) 0.451759 + 8.40380i 0.0521646 + 0.970388i
\(76\) −6.76651 − 2.85143i −0.776172 − 0.327082i
\(77\) 7.20780 − 10.5646i 0.821405 − 1.20395i
\(78\) 6.69761 − 8.46595i 0.758355 − 0.958581i
\(79\) 11.3282 − 6.54032i 1.27452 − 0.735844i 0.298684 − 0.954352i \(-0.403453\pi\)
0.975835 + 0.218509i \(0.0701192\pi\)
\(80\) 1.45576 − 0.371028i 0.162759 − 0.0414821i
\(81\) −6.05807 − 6.65581i −0.673119 − 0.739534i
\(82\) −5.20942 + 1.75059i −0.575284 + 0.193321i
\(83\) − 10.1716i − 1.11648i −0.829680 − 0.558239i \(-0.811477\pi\)
0.829680 − 0.558239i \(-0.188523\pi\)
\(84\) 8.36771 + 3.73917i 0.912992 + 0.407977i
\(85\) 1.03826i 0.112615i
\(86\) 0.336596 + 1.00164i 0.0362961 + 0.108010i
\(87\) −1.40559 + 0.915488i −0.150695 + 0.0981506i
\(88\) 5.93421 + 12.3172i 0.632589 + 1.31302i
\(89\) −4.70650 + 2.71730i −0.498888 + 0.288033i −0.728254 − 0.685307i \(-0.759668\pi\)
0.229366 + 0.973340i \(0.426335\pi\)
\(90\) 0.666567 + 1.44731i 0.0702623 + 0.152560i
\(91\) −11.6270 − 0.875077i −1.21884 − 0.0917330i
\(92\) −13.3046 − 5.60663i −1.38710 − 0.584532i
\(93\) −5.56373 + 0.299087i −0.576932 + 0.0310138i
\(94\) 9.73736 − 11.0350i 1.00433 − 1.13817i
\(95\) −0.689440 + 1.19414i −0.0707350 + 0.122517i
\(96\) −7.93172 + 5.75220i −0.809528 + 0.587081i
\(97\) 5.31623 0.539781 0.269891 − 0.962891i \(-0.413012\pi\)
0.269891 + 0.962891i \(0.413012\pi\)
\(98\) −1.71331 − 9.75011i −0.173070 − 0.984909i
\(99\) −11.7090 + 8.55533i −1.17680 + 0.859843i
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 168.2.v.a.11.18 yes 56
3.2 odd 2 inner 168.2.v.a.11.11 yes 56
4.3 odd 2 672.2.bd.a.431.24 56
7.2 even 3 inner 168.2.v.a.107.21 yes 56
8.3 odd 2 inner 168.2.v.a.11.8 ✓ 56
8.5 even 2 672.2.bd.a.431.23 56
12.11 even 2 672.2.bd.a.431.13 56
21.2 odd 6 inner 168.2.v.a.107.8 yes 56
24.5 odd 2 672.2.bd.a.431.14 56
24.11 even 2 inner 168.2.v.a.11.21 yes 56
28.23 odd 6 672.2.bd.a.527.14 56
56.37 even 6 672.2.bd.a.527.13 56
56.51 odd 6 inner 168.2.v.a.107.11 yes 56
84.23 even 6 672.2.bd.a.527.23 56
168.107 even 6 inner 168.2.v.a.107.18 yes 56
168.149 odd 6 672.2.bd.a.527.24 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.8 ✓ 56 8.3 odd 2 inner
168.2.v.a.11.11 yes 56 3.2 odd 2 inner
168.2.v.a.11.18 yes 56 1.1 even 1 trivial
168.2.v.a.11.21 yes 56 24.11 even 2 inner
168.2.v.a.107.8 yes 56 21.2 odd 6 inner
168.2.v.a.107.11 yes 56 56.51 odd 6 inner
168.2.v.a.107.18 yes 56 168.107 even 6 inner
168.2.v.a.107.21 yes 56 7.2 even 3 inner
672.2.bd.a.431.13 56 12.11 even 2
672.2.bd.a.431.14 56 24.5 odd 2
672.2.bd.a.431.23 56 8.5 even 2
672.2.bd.a.431.24 56 4.3 odd 2
672.2.bd.a.527.13 56 56.37 even 6
672.2.bd.a.527.14 56 28.23 odd 6
672.2.bd.a.527.23 56 84.23 even 6
672.2.bd.a.527.24 56 168.149 odd 6