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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 11.7
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.7

$q$-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+(-0.982844 - 1.01687i) q^{2} +(-0.987527 - 1.42295i) q^{3} +(-0.0680365 + 1.99884i) q^{4} +(1.77783 + 3.07929i) q^{5} +(-0.476367 + 2.40272i) q^{6} +(0.793456 + 2.52397i) q^{7} +(2.09943 - 1.89537i) q^{8} +(-1.04958 + 2.81041i) q^{9} +(1.38390 - 4.83428i) q^{10} +(-0.396703 - 0.229036i) q^{11} +(2.91144 - 1.87710i) q^{12} -0.799063i q^{13} +(1.78670 - 3.28751i) q^{14} +(2.62603 - 5.57065i) q^{15} +(-3.99074 - 0.271989i) q^{16} +(5.48456 + 3.16651i) q^{17} +(3.88938 - 1.69491i) q^{18} +(-2.61696 - 4.53272i) q^{19} +(-6.27598 + 3.34410i) q^{20} +(2.80793 - 3.62154i) q^{21} +(0.156997 + 0.628501i) q^{22} +(1.55053 + 2.68559i) q^{23} +(-4.77025 - 1.11565i) q^{24} +(-3.82137 + 6.61880i) q^{25} +(-0.812540 + 0.785354i) q^{26} +(5.03556 - 1.28185i) q^{27} +(-5.09900 + 1.41427i) q^{28} -2.60340 q^{29} +(-8.24559 + 2.80476i) q^{30} +(4.42520 + 2.55489i) q^{31} +(3.64570 + 4.32538i) q^{32} +(0.0658472 + 0.790668i) q^{33} +(-2.17055 - 8.68926i) q^{34} +(-6.36142 + 6.93048i) q^{35} +(-5.54615 - 2.28915i) q^{36} +(-2.89424 + 1.67099i) q^{37} +(-2.03710 + 7.11606i) q^{38} +(-1.13703 + 0.789096i) q^{39} +(9.56881 + 3.09511i) q^{40} -10.1452i q^{41} +(-6.44238 + 0.704118i) q^{42} +3.56359 q^{43} +(0.484798 - 0.777363i) q^{44} +(-10.5200 + 1.76446i) q^{45} +(1.20696 - 4.21620i) q^{46} +(-2.15923 - 3.73989i) q^{47} +(3.55394 + 5.94723i) q^{48} +(-5.74086 + 4.00532i) q^{49} +(10.4862 - 2.61943i) q^{50} +(-0.910362 - 10.9313i) q^{51} +(1.59720 + 0.0543654i) q^{52} +(1.16913 - 2.02499i) q^{53} +(-6.25264 - 3.86063i) q^{54} -1.62875i q^{55} +(6.44965 + 3.79500i) q^{56} +(-3.86551 + 8.19999i) q^{57} +(2.55874 + 2.64731i) q^{58} +(-1.49741 - 0.864528i) q^{59} +(10.9562 + 5.62802i) q^{60} +(4.60841 - 2.66067i) q^{61} +(-1.75130 - 7.01090i) q^{62} +(-7.92618 - 0.419175i) q^{63} +(0.815178 - 7.95836i) q^{64} +(2.46055 - 1.42060i) q^{65} +(0.739287 - 0.844061i) q^{66} +(0.00979145 - 0.0169593i) q^{67} +(-6.70251 + 10.7473i) q^{68} +(2.29028 - 4.85842i) q^{69} +(13.2997 - 0.342860i) q^{70} -9.04022 q^{71} +(3.12323 + 7.88958i) q^{72} +(4.15674 - 7.19969i) q^{73} +(4.54376 + 1.30074i) q^{74} +(13.1919 - 1.09863i) q^{75} +(9.23823 - 4.92251i) q^{76} +(0.263315 - 1.18300i) q^{77} +(1.91993 + 0.380647i) q^{78} +(-10.8825 + 6.28304i) q^{79} +(-6.25733 - 12.7722i) q^{80} +(-6.79677 - 5.89949i) q^{81} +(-10.3163 + 9.97112i) q^{82} -0.694004i q^{83} +(7.04784 + 5.85900i) q^{84} +22.5181i q^{85} +(-3.50245 - 3.62369i) q^{86} +(2.57093 + 3.70452i) q^{87} +(-1.26696 + 0.271052i) q^{88} +(7.02189 - 4.05409i) q^{89} +(12.1338 + 8.96329i) q^{90} +(2.01681 - 0.634021i) q^{91} +(-5.47357 + 2.91654i) q^{92} +(-0.734523 - 8.81987i) q^{93} +(-1.68079 + 5.87138i) q^{94} +(9.30504 - 16.1168i) q^{95} +(2.55457 - 9.45908i) q^{96} +7.05333 q^{97} +(9.71524 + 1.90109i) q^{98} +(1.06006 - 0.874504i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.982844 − 1.01687i −0.694975 − 0.719033i
\(3\) −0.987527 − 1.42295i −0.570149 − 0.821541i
\(4\) −0.0680365 + 1.99884i −0.0340183 + 0.999421i
\(5\) 1.77783 + 3.07929i 0.795070 + 1.37710i 0.922795 + 0.385292i \(0.125899\pi\)
−0.127724 + 0.991810i \(0.540767\pi\)
\(6\) −0.476367 + 2.40272i −0.194476 + 0.980907i
\(7\) 0.793456 + 2.52397i 0.299898 + 0.953971i
\(8\) 2.09943 − 1.89537i 0.742259 − 0.670113i
\(9\) −1.04958 + 2.81041i −0.349860 + 0.936802i
\(10\) 1.38390 − 4.83428i 0.437628 − 1.52873i
\(11\) −0.396703 − 0.229036i −0.119610 − 0.0690571i 0.439001 − 0.898486i \(-0.355332\pi\)
−0.558612 + 0.829429i \(0.688666\pi\)
\(12\) 2.91144 − 1.87710i 0.840461 − 0.541872i
\(13\) − 0.799063i − 0.221620i −0.993842 − 0.110810i \(-0.964655\pi\)
0.993842 − 0.110810i \(-0.0353445\pi\)
\(14\) 1.78670 − 3.28751i 0.477516 − 0.878623i
\(15\) 2.62603 − 5.57065i 0.678037 − 1.43834i
\(16\) −3.99074 − 0.271989i −0.997686 − 0.0679971i
\(17\) 5.48456 + 3.16651i 1.33020 + 0.767993i 0.985330 − 0.170657i \(-0.0545891\pi\)
0.344872 + 0.938650i \(0.387922\pi\)
\(18\) 3.88938 − 1.69491i 0.916736 − 0.399494i
\(19\) −2.61696 − 4.53272i −0.600373 − 1.03988i −0.992764 − 0.120078i \(-0.961686\pi\)
0.392392 − 0.919798i \(-0.371648\pi\)
\(20\) −6.27598 + 3.34410i −1.40335 + 0.747763i
\(21\) 2.80793 − 3.62154i 0.612740 − 0.790285i
\(22\) 0.156997 + 0.628501i 0.0334719 + 0.133997i
\(23\) 1.55053 + 2.68559i 0.323307 + 0.559985i 0.981168 − 0.193155i \(-0.0618719\pi\)
−0.657861 + 0.753139i \(0.728539\pi\)
\(24\) −4.77025 − 1.11565i −0.973724 − 0.227732i
\(25\) −3.82137 + 6.61880i −0.764273 + 1.32376i
\(26\) −0.812540 + 0.785354i −0.159352 + 0.154021i
\(27\) 5.03556 − 1.28185i 0.969094 − 0.246693i
\(28\) −5.09900 + 1.41427i −0.963621 + 0.267272i
\(29\) −2.60340 −0.483440 −0.241720 − 0.970346i \(-0.577712\pi\)
−0.241720 + 0.970346i \(0.577712\pi\)
\(30\) −8.24559 + 2.80476i −1.50543 + 0.512077i
\(31\) 4.42520 + 2.55489i 0.794790 + 0.458872i 0.841646 − 0.540030i \(-0.181587\pi\)
−0.0468563 + 0.998902i \(0.514920\pi\)
\(32\) 3.64570 + 4.32538i 0.644475 + 0.764626i
\(33\) 0.0658472 + 0.790668i 0.0114625 + 0.137638i
\(34\) −2.17055 − 8.68926i −0.372245 − 1.49020i
\(35\) −6.36142 + 6.93048i −1.07528 + 1.17146i
\(36\) −5.54615 − 2.28915i −0.924358 − 0.381526i
\(37\) −2.89424 + 1.67099i −0.475810 + 0.274709i −0.718669 − 0.695352i \(-0.755248\pi\)
0.242859 + 0.970062i \(0.421915\pi\)
\(38\) −2.03710 + 7.11606i −0.330461 + 1.15438i
\(39\) −1.13703 + 0.789096i −0.182070 + 0.126357i
\(40\) 9.56881 + 3.09511i 1.51296 + 0.489380i
\(41\) − 10.1452i − 1.58441i −0.610256 − 0.792205i \(-0.708933\pi\)
0.610256 − 0.792205i \(-0.291067\pi\)
\(42\) −6.44238 + 0.704118i −0.994080 + 0.108648i
\(43\) 3.56359 0.543442 0.271721 − 0.962376i \(-0.412407\pi\)
0.271721 + 0.962376i \(0.412407\pi\)
\(44\) 0.484798 − 0.777363i 0.0730860 − 0.117192i
\(45\) −10.5200 + 1.76446i −1.56824 + 0.263031i
\(46\) 1.20696 − 4.21620i 0.177957 − 0.621644i
\(47\) −2.15923 − 3.73989i −0.314956 − 0.545519i 0.664472 − 0.747313i \(-0.268656\pi\)
−0.979428 + 0.201794i \(0.935323\pi\)
\(48\) 3.55394 + 5.94723i 0.512967 + 0.858408i
\(49\) −5.74086 + 4.00532i −0.820122 + 0.572188i
\(50\) 10.4862 − 2.61943i 1.48298 − 0.370443i
\(51\) −0.910362 − 10.9313i −0.127476 − 1.53069i
\(52\) 1.59720 + 0.0543654i 0.221492 + 0.00753913i
\(53\) 1.16913 − 2.02499i 0.160592 − 0.278154i −0.774489 − 0.632587i \(-0.781993\pi\)
0.935081 + 0.354433i \(0.115326\pi\)
\(54\) −6.25264 − 3.86063i −0.850877 − 0.525365i
\(55\) − 1.62875i − 0.219621i
\(56\) 6.44965 + 3.79500i 0.861871 + 0.507128i
\(57\) −3.86551 + 8.19999i −0.511999 + 1.08612i
\(58\) 2.55874 + 2.64731i 0.335979 + 0.347609i
\(59\) −1.49741 − 0.864528i −0.194946 − 0.112552i 0.399350 − 0.916798i \(-0.369236\pi\)
−0.594296 + 0.804247i \(0.702569\pi\)
\(60\) 10.9562 + 5.62802i 1.41444 + 0.726575i
\(61\) 4.60841 − 2.66067i 0.590047 − 0.340664i −0.175069 − 0.984556i \(-0.556015\pi\)
0.765116 + 0.643892i \(0.222682\pi\)
\(62\) −1.75130 − 7.01090i −0.222415 − 0.890385i
\(63\) −7.92618 − 0.419175i −0.998605 − 0.0528111i
\(64\) 0.815178 − 7.95836i 0.101897 − 0.994795i
\(65\) 2.46055 − 1.42060i 0.305193 − 0.176204i
\(66\) 0.739287 − 0.844061i 0.0909999 − 0.103897i
\(67\) 0.00979145 − 0.0169593i 0.00119622 − 0.00207191i −0.865427 − 0.501036i \(-0.832953\pi\)
0.866623 + 0.498964i \(0.166286\pi\)
\(68\) −6.70251 + 10.7473i −0.812799 + 1.30331i
\(69\) 2.29028 − 4.85842i 0.275717 − 0.584885i
\(70\) 13.2997 − 0.342860i 1.58961 − 0.0409796i
\(71\) −9.04022 −1.07288 −0.536439 − 0.843939i \(-0.680231\pi\)
−0.536439 + 0.843939i \(0.680231\pi\)
\(72\) 3.12323 + 7.88958i 0.368077 + 0.929795i
\(73\) 4.15674 − 7.19969i 0.486510 − 0.842660i −0.513370 − 0.858167i \(-0.671603\pi\)
0.999880 + 0.0155078i \(0.00493650\pi\)
\(74\) 4.54376 + 1.30074i 0.528202 + 0.151207i
\(75\) 13.1919 − 1.09863i 1.52327 − 0.126859i
\(76\) 9.23823 − 4.92251i 1.05970 − 0.564651i
\(77\) 0.263315 − 1.18300i 0.0300076 − 0.134815i
\(78\) 1.91993 + 0.380647i 0.217389 + 0.0430998i
\(79\) −10.8825 + 6.28304i −1.22438 + 0.706897i −0.965849 − 0.259105i \(-0.916572\pi\)
−0.258533 + 0.966003i \(0.583239\pi\)
\(80\) −6.25733 − 12.7722i −0.699591 − 1.42798i
\(81\) −6.79677 − 5.89949i −0.755196 − 0.655499i
\(82\) −10.3163 + 9.97112i −1.13924 + 1.10113i
\(83\) − 0.694004i − 0.0761768i −0.999274 − 0.0380884i \(-0.987873\pi\)
0.999274 − 0.0380884i \(-0.0121268\pi\)
\(84\) 7.04784 + 5.85900i 0.768983 + 0.639270i
\(85\) 22.5181i 2.44243i
\(86\) −3.50245 − 3.62369i −0.377679 − 0.390753i
\(87\) 2.57093 + 3.70452i 0.275633 + 0.397166i
\(88\) −1.26696 + 0.271052i −0.135058 + 0.0288942i
\(89\) 7.02189 − 4.05409i 0.744319 − 0.429733i −0.0793185 − 0.996849i \(-0.525274\pi\)
0.823638 + 0.567116i \(0.191941\pi\)
\(90\) 12.1338 + 8.96329i 1.27901 + 0.944814i
\(91\) 2.01681 − 0.634021i 0.211419 − 0.0664634i
\(92\) −5.47357 + 2.91654i −0.570659 + 0.304070i
\(93\) −0.734523 − 8.81987i −0.0761664 − 0.914578i
\(94\) −1.68079 + 5.87138i −0.173360 + 0.605586i
\(95\) 9.30504 − 16.1168i 0.954677 − 1.65355i
\(96\) 2.55457 − 9.45908i 0.260725 − 0.965413i
\(97\) 7.05333 0.716157 0.358078 − 0.933692i \(-0.383432\pi\)
0.358078 + 0.933692i \(0.383432\pi\)
\(98\) 9.71524 + 1.90109i 0.981387 + 0.192039i
\(99\) 1.06006 − 0.874504i 0.106540 − 0.0878909i
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.11.7 ✓ 56
3.2 odd 2 inner 168.2.v.a.11.22 yes 56
4.3 odd 2 672.2.bd.a.431.20 56
7.2 even 3 inner 168.2.v.a.107.12 yes 56
8.3 odd 2 inner 168.2.v.a.11.17 yes 56
8.5 even 2 672.2.bd.a.431.19 56
12.11 even 2 672.2.bd.a.431.1 56
21.2 odd 6 inner 168.2.v.a.107.17 yes 56
24.5 odd 2 672.2.bd.a.431.2 56
24.11 even 2 inner 168.2.v.a.11.12 yes 56
28.23 odd 6 672.2.bd.a.527.2 56
56.37 even 6 672.2.bd.a.527.1 56
56.51 odd 6 inner 168.2.v.a.107.22 yes 56
84.23 even 6 672.2.bd.a.527.19 56
168.107 even 6 inner 168.2.v.a.107.7 yes 56
168.149 odd 6 672.2.bd.a.527.20 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.7 ✓ 56 1.1 even 1 trivial
168.2.v.a.11.12 yes 56 24.11 even 2 inner
168.2.v.a.11.17 yes 56 8.3 odd 2 inner
168.2.v.a.11.22 yes 56 3.2 odd 2 inner
168.2.v.a.107.7 yes 56 168.107 even 6 inner
168.2.v.a.107.12 yes 56 7.2 even 3 inner
168.2.v.a.107.17 yes 56 21.2 odd 6 inner
168.2.v.a.107.22 yes 56 56.51 odd 6 inner
672.2.bd.a.431.1 56 12.11 even 2
672.2.bd.a.431.2 56 24.5 odd 2
672.2.bd.a.431.19 56 8.5 even 2
672.2.bd.a.431.20 56 4.3 odd 2
672.2.bd.a.527.1 56 56.37 even 6
672.2.bd.a.527.2 56 28.23 odd 6
672.2.bd.a.527.19 56 84.23 even 6
672.2.bd.a.527.20 56 168.149 odd 6