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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 11.11
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.11

$q$-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} +(-0.0929747 + 1.72955i) q^{3} +(-1.59413 + 1.20779i) q^{4} +(-0.187787 - 0.325257i) q^{5} +(2.36043 - 0.654497i) q^{6} +(-0.198565 + 2.63829i) q^{7} +(2.33722 + 1.59292i) q^{8} +(-2.98271 - 0.321609i) q^{9} +(-0.351427 + 0.398260i) q^{10} +(4.18624 + 2.41693i) q^{11} +(-1.94072 - 2.86943i) 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} +(0.912528 + 4.14334i) q^{18} +(1.83569 + 3.17951i) q^{19} +(0.692199 + 0.291695i) q^{20} +(-4.54460 - 0.588723i) q^{21} +(1.35417 - 6.70064i) q^{22} +(-3.60943 - 6.25172i) q^{23} +(-2.97234 + 3.89425i) 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} +(-0.656139 - 0.644841i) q^{30} +(2.78588 + 1.60843i) q^{31} +(-5.64974 + 0.283544i) q^{32} +(-4.56942 + 7.01562i) q^{33} +(-0.774447 + 3.83207i) q^{34} +(0.895411 - 0.430853i) q^{35} +(5.14327 - 3.08979i) q^{36} +(-0.459519 + 0.265303i) q^{37} +(3.43534 - 3.89315i) q^{38} +(-7.62216 - 0.409740i) q^{39} +(0.0792077 - 1.05933i) q^{40} -3.88604i q^{41} +(1.25805 + 6.35746i) q^{42} +0.747189 q^{43} +(-9.59255 + 1.20319i) q^{44} +(0.455510 + 1.03054i) q^{45} +(-6.75473 + 7.65490i) q^{46} +(5.20322 + 9.01223i) q^{47} +(6.55941 + 2.23027i) q^{48} +(-6.92114 - 1.04774i) q^{49} +(-6.73541 - 1.36120i) q^{50} +(2.61323 - 4.01220i) q^{51} +(-5.32273 - 7.02535i) q^{52} +(1.40600 - 2.43526i) q^{53} +(-7.25098 + 1.19304i) 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} +(-0.568860 + 1.17007i) q^{60} +(8.43102 - 4.86765i) q^{61} +(0.901184 - 4.45918i) q^{62} +(1.44076 - 7.80540i) q^{63} +(2.92522 + 7.44601i) q^{64} +(1.43341 - 0.827580i) q^{65} +(11.4632 + 2.96511i) 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} +(-6.45896 - 5.50289i) q^{72} +(-0.445737 + 0.772039i) q^{73} +(0.562657 + 0.496492i) q^{74} +(7.05203 + 4.59314i) q^{75} +(-6.76651 - 2.85143i) q^{76} +(-7.20780 + 10.5646i) q^{77} +(2.88438 + 10.4024i) q^{78} +(11.3282 - 6.54032i) q^{79} +(-1.45576 + 0.371028i) q^{80} +(8.79313 + 1.91854i) q^{81} +(-5.20942 + 1.75059i) q^{82} +10.1716i q^{83} +(7.95574 - 4.55041i) q^{84} +1.03826i q^{85} +(-0.336596 - 1.00164i) q^{86} +(0.0900430 - 1.67502i) q^{87} +(5.93421 + 12.3172i) q^{88} +(4.70650 - 2.71730i) q^{89} +(1.17629 - 1.07487i) q^{90} +(-11.6270 - 0.875077i) q^{91} +(13.3046 + 5.60663i) q^{92} +(-3.04088 + 4.66879i) q^{93} +(9.73736 - 11.0350i) q^{94} +(0.689440 - 1.19414i) q^{95} +(0.0348788 - 9.79790i) 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
\(3\) −0.0929747 + 1.72955i −0.0536790 + 0.998558i
\(4\) −1.59413 + 1.20779i −0.797065 + 0.603893i
\(5\) −0.187787 − 0.325257i −0.0839810 − 0.145459i 0.820976 − 0.570963i \(-0.193430\pi\)
−0.904957 + 0.425504i \(0.860097\pi\)
\(6\) 2.36043 − 0.654497i 0.963642 − 0.267197i
\(7\) −0.198565 + 2.63829i −0.0750504 + 0.997180i
\(8\) 2.33722 + 1.59292i 0.826333 + 0.563182i
\(9\) −2.98271 − 0.321609i −0.994237 − 0.107203i
\(10\) −0.351427 + 0.398260i −0.111131 + 0.125941i
\(11\) 4.18624 + 2.41693i 1.26220 + 0.728731i 0.973500 − 0.228688i \(-0.0734437\pi\)
0.288700 + 0.957420i \(0.406777\pi\)
\(12\) −1.94072 − 2.86943i −0.560237 − 0.828332i
\(13\) 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 0.180740 − 0.983531i \(-0.442151\pi\)
−0.761393 + 0.648291i \(0.775484\pi\)
\(18\) 0.912528 + 4.14334i 0.215085 + 0.976595i
\(19\) 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
\(21\) −4.54460 − 0.588723i −0.991713 − 0.128470i
\(22\) 1.35417 − 6.70064i 0.288711 − 1.42858i
\(23\) −3.60943 − 6.25172i −0.752618 − 1.30357i −0.946550 − 0.322558i \(-0.895457\pi\)
0.193932 − 0.981015i \(-0.437876\pi\)
\(24\) −2.97234 + 3.89425i −0.606727 + 0.794910i
\(25\) 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 \(-0.528661\pi\)
−0.0899200 + 0.995949i \(0.528661\pi\)
\(30\) −0.656139 − 0.644841i −0.119794 − 0.117731i
\(31\) 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
\(33\) −4.56942 + 7.01562i −0.795434 + 1.22126i
\(34\) −0.774447 + 3.83207i −0.132817 + 0.657194i
\(35\) 0.895411 − 0.430853i 0.151352 − 0.0728274i
\(36\) 5.14327 − 3.08979i 0.857211 − 0.514965i
\(37\) −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
\(39\) −7.62216 − 0.409740i −1.22052 − 0.0656110i
\(40\) 0.0792077 − 1.05933i 0.0125238 − 0.167495i
\(41\) − 3.88604i − 0.606897i −0.952848 − 0.303449i \(-0.901862\pi\)
0.952848 − 0.303449i \(-0.0981381\pi\)
\(42\) 1.25805 + 6.35746i 0.194122 + 0.980977i
\(43\) 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
\(45\) 0.455510 + 1.03054i 0.0679034 + 0.153624i
\(46\) −6.75473 + 7.65490i −0.995931 + 1.12865i
\(47\) 5.20322 + 9.01223i 0.758967 + 1.31457i 0.943378 + 0.331720i \(0.107629\pi\)
−0.184411 + 0.982849i \(0.559038\pi\)
\(48\) 6.55941 + 2.23027i 0.946770 + 0.321912i
\(49\) −6.92114 − 1.04774i −0.988735 − 0.149678i
\(50\) −6.73541 − 1.36120i −0.952531 − 0.192503i
\(51\) 2.61323 − 4.01220i 0.365926 − 0.561821i
\(52\) −5.32273 − 7.02535i −0.738129 − 0.974240i
\(53\) 1.40600 − 2.43526i 0.193129 − 0.334509i −0.753157 − 0.657841i \(-0.771470\pi\)
0.946285 + 0.323332i \(0.104803\pi\)
\(54\) −7.25098 + 1.19304i −0.986733 + 0.162352i
\(55\) − 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 0.388606 − 0.921404i \(-0.372957\pi\)
−0.603657 + 0.797244i \(0.706290\pi\)
\(60\) −0.568860 + 1.17007i −0.0734395 + 0.151056i
\(61\) 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
\(63\) 1.44076 − 7.80540i 0.181519 − 0.983387i
\(64\) 2.92522 + 7.44601i 0.365652 + 0.930752i
\(65\) 1.43341 − 0.827580i 0.177793 − 0.102649i
\(66\) 11.4632 + 2.96511i 1.41102 + 0.364980i
\(67\) 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 \(-0.392989\pi\)
0.329887 + 0.944020i \(0.392989\pi\)
\(72\) −6.45896 − 5.50289i −0.761196 − 0.648522i
\(73\) −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
\(75\) 7.05203 + 4.59314i 0.814298 + 0.530370i
\(76\) −6.76651 − 2.85143i −0.776172 − 0.327082i
\(77\) −7.20780 + 10.5646i −0.821405 + 1.20395i
\(78\) 2.88438 + 10.4024i 0.326591 + 1.17784i
\(79\) 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
\(81\) 8.79313 + 1.91854i 0.977015 + 0.213171i
\(82\) −5.20942 + 1.75059i −0.575284 + 0.193321i
\(83\) 10.1716i 1.11648i 0.829680 + 0.558239i \(0.188523\pi\)
−0.829680 + 0.558239i \(0.811477\pi\)
\(84\) 7.95574 − 4.55041i 0.868042 − 0.496490i
\(85\) 1.03826i 0.112615i
\(86\) −0.336596 − 1.00164i −0.0362961 − 0.108010i
\(87\) 0.0900430 − 1.67502i 0.00965362 − 0.179581i
\(88\) 5.93421 + 12.3172i 0.632589 + 1.31302i
\(89\) 4.70650 − 2.71730i 0.498888 − 0.288033i −0.229366 − 0.973340i \(-0.573665\pi\)
0.728254 + 0.685307i \(0.240332\pi\)
\(90\) 1.17629 − 1.07487i 0.123992 − 0.113302i
\(91\) −11.6270 − 0.875077i −1.21884 − 0.0917330i
\(92\) 13.3046 + 5.60663i 1.38710 + 0.584532i
\(93\) −3.04088 + 4.66879i −0.315325 + 0.484131i
\(94\) 9.73736 − 11.0350i 1.00433 − 1.13817i
\(95\) 0.689440 − 1.19414i 0.0707350 − 0.122517i
\(96\) 0.0348788 − 9.79790i 0.00355980 − 0.999994i
\(97\) 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.11.11 yes 56
3.2 odd 2 inner 168.2.v.a.11.18 yes 56
4.3 odd 2 672.2.bd.a.431.13 56
7.2 even 3 inner 168.2.v.a.107.8 yes 56
8.3 odd 2 inner 168.2.v.a.11.21 yes 56
8.5 even 2 672.2.bd.a.431.14 56
12.11 even 2 672.2.bd.a.431.24 56
21.2 odd 6 inner 168.2.v.a.107.21 yes 56
24.5 odd 2 672.2.bd.a.431.23 56
24.11 even 2 inner 168.2.v.a.11.8 ✓ 56
28.23 odd 6 672.2.bd.a.527.23 56
56.37 even 6 672.2.bd.a.527.24 56
56.51 odd 6 inner 168.2.v.a.107.18 yes 56
84.23 even 6 672.2.bd.a.527.14 56
168.107 even 6 inner 168.2.v.a.107.11 yes 56
168.149 odd 6 672.2.bd.a.527.13 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.8 ✓ 56 24.11 even 2 inner
168.2.v.a.11.11 yes 56 1.1 even 1 trivial
168.2.v.a.11.18 yes 56 3.2 odd 2 inner
168.2.v.a.11.21 yes 56 8.3 odd 2 inner
168.2.v.a.107.8 yes 56 7.2 even 3 inner
168.2.v.a.107.11 yes 56 168.107 even 6 inner
168.2.v.a.107.18 yes 56 56.51 odd 6 inner
168.2.v.a.107.21 yes 56 21.2 odd 6 inner
672.2.bd.a.431.13 56 4.3 odd 2
672.2.bd.a.431.14 56 8.5 even 2
672.2.bd.a.431.23 56 24.5 odd 2
672.2.bd.a.431.24 56 12.11 even 2
672.2.bd.a.527.13 56 168.149 odd 6
672.2.bd.a.527.14 56 84.23 even 6
672.2.bd.a.527.23 56 28.23 odd 6
672.2.bd.a.527.24 56 56.37 even 6