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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 11.12
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.12

$q$-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+(-0.389211 - 1.35960i) q^{2} +(1.72608 + 0.143748i) q^{3} +(-1.69703 + 1.05834i) 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} +(2.95867 + 0.496241i) q^{9} +(3.49466 - 3.61563i) q^{10} +(0.396703 + 0.229036i) q^{11} +(-3.08134 + 1.58283i) q^{12} +0.799063i q^{13} +(-3.12277 + 2.06114i) q^{14} +(2.62603 + 5.57065i) q^{15} +(1.75982 - 3.59208i) q^{16} +(-5.48456 - 3.16651i) q^{17} +(-0.476858 - 4.21576i) q^{18} +(-2.61696 - 4.53272i) q^{19} +(-6.27598 - 3.34410i) q^{20} +(-1.00675 - 4.47062i) q^{21} +(0.156997 - 0.628501i) q^{22} +(1.55053 + 2.68559i) q^{23} +(3.35131 + 3.57333i) q^{24} +(-3.82137 + 6.61880i) q^{25} +(1.08641 - 0.311004i) q^{26} +(5.03556 + 1.28185i) q^{27} +(4.01774 + 3.44351i) q^{28} -2.60340 q^{29} +(6.55179 - 5.73851i) q^{30} +(-4.42520 - 2.55489i) q^{31} +(-5.56873 - 0.994581i) q^{32} +(0.651815 + 0.452359i) 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} +(-5.14413 + 5.32221i) q^{38} +(-0.114864 + 1.37924i) q^{39} +(-2.10396 + 9.83439i) q^{40} +10.1452i q^{41} +(-5.68643 + 3.10879i) q^{42} +3.56359 q^{43} +(-0.915615 + 0.0311657i) q^{44} +(3.73195 + 9.99285i) q^{45} +(3.04785 - 3.15336i) 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} +(-9.01159 - 6.25404i) q^{51} +(-0.845682 - 1.35603i) q^{52} +(1.16913 - 2.02499i) q^{53} +(-0.217084 - 7.34526i) q^{54} +1.62875i q^{55} +(3.11805 - 6.80278i) q^{56} +(-3.86551 - 8.19999i) q^{57} +(1.01327 + 3.53959i) q^{58} +(1.49741 + 0.864528i) q^{59} +(-10.3521 - 6.67433i) q^{60} +(-4.60841 + 2.66067i) q^{61} +(-1.75130 + 7.01090i) q^{62} +(-1.09508 - 7.86135i) q^{63} +(0.815178 + 7.95836i) q^{64} +(-2.46055 + 1.42060i) q^{65} +(0.361335 - 1.06227i) q^{66} +(0.00979145 - 0.0169593i) q^{67} +(12.6587 - 0.430877i) q^{68} +(2.29028 + 4.85842i) q^{69} +(-11.8986 - 5.95157i) q^{70} -9.04022 q^{71} +(5.27096 + 6.64959i) q^{72} +(4.15674 - 7.19969i) q^{73} +(-3.39835 - 3.28465i) q^{74} +(-7.54741 + 10.8752i) 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} +(14.1897 - 0.967099i) q^{80} +(8.50749 + 2.93643i) q^{81} +(13.7934 - 3.94861i) q^{82} +0.694004i q^{83} +(6.43993 + 6.52130i) q^{84} -22.5181i q^{85} +(-1.38699 - 4.84506i) q^{86} +(-4.49367 - 0.374235i) q^{87} +(0.398740 + 1.23274i) 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} +(-7.27097 - 5.04605i) q^{93} +(-4.24437 + 4.39129i) q^{94} +(9.30504 - 16.1168i) q^{95} +(-9.46909 - 2.51722i) q^{96} +7.05333 q^{97} +(7.68004 + 6.24636i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.389211 − 1.35960i −0.275214 − 0.961383i
\(3\) 1.72608 + 0.143748i 0.996550 + 0.0829931i
\(4\) −1.69703 + 1.05834i −0.848515 + 0.529171i
\(5\) 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
\(9\) 2.95867 + 0.496241i 0.986224 + 0.165414i
\(10\) 3.49466 − 3.61563i 1.10511 − 1.14336i
\(11\) 0.396703 + 0.229036i 0.119610 + 0.0690571i 0.558612 − 0.829429i \(-0.311334\pi\)
−0.439001 + 0.898486i \(0.644668\pi\)
\(12\) −3.08134 + 1.58283i −0.889505 + 0.456925i
\(13\) 0.799063i 0.221620i 0.993842 + 0.110810i \(0.0353445\pi\)
−0.993842 + 0.110810i \(0.964655\pi\)
\(14\) −3.12277 + 2.06114i −0.834596 + 0.550863i
\(15\) 2.62603 + 5.57065i 0.678037 + 1.43834i
\(16\) 1.75982 − 3.59208i 0.439956 − 0.898020i
\(17\) −5.48456 − 3.16651i −1.33020 − 0.767993i −0.344872 − 0.938650i \(-0.612078\pi\)
−0.985330 + 0.170657i \(0.945411\pi\)
\(18\) −0.476858 − 4.21576i −0.112396 − 0.993663i
\(19\) −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
\(21\) −1.00675 − 4.47062i −0.219690 − 0.975570i
\(22\) 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 \(0.728539\pi\)
\(24\) 3.35131 + 3.57333i 0.684084 + 0.729404i
\(25\) −3.82137 + 6.61880i −0.764273 + 1.32376i
\(26\) 1.08641 − 0.311004i 0.213062 − 0.0609928i
\(27\) 5.03556 + 1.28185i 0.969094 + 0.246693i
\(28\) 4.01774 + 3.44351i 0.759282 + 0.650762i
\(29\) −2.60340 −0.483440 −0.241720 − 0.970346i \(-0.577712\pi\)
−0.241720 + 0.970346i \(0.577712\pi\)
\(30\) 6.55179 − 5.73851i 1.19619 − 1.04770i
\(31\) −4.42520 − 2.55489i −0.794790 − 0.458872i 0.0468563 − 0.998902i \(-0.485080\pi\)
−0.841646 + 0.540030i \(0.818413\pi\)
\(32\) −5.56873 − 0.994581i −0.984423 − 0.175819i
\(33\) 0.651815 + 0.452359i 0.113466 + 0.0787457i
\(34\) −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 −0.242859 − 0.970062i \(-0.578085\pi\)
0.718669 + 0.695352i \(0.244752\pi\)
\(38\) −5.14413 + 5.32221i −0.834489 + 0.863376i
\(39\) −0.114864 + 1.37924i −0.0183929 + 0.220856i
\(40\) −2.10396 + 9.83439i −0.332666 + 1.55495i
\(41\) 10.1452i 1.58441i 0.610256 + 0.792205i \(0.291067\pi\)
−0.610256 + 0.792205i \(0.708933\pi\)
\(42\) −5.68643 + 3.10879i −0.877434 + 0.479697i
\(43\) 3.56359 0.543442 0.271721 − 0.962376i \(-0.412407\pi\)
0.271721 + 0.962376i \(0.412407\pi\)
\(44\) −0.915615 + 0.0311657i −0.138034 + 0.00469840i
\(45\) 3.73195 + 9.99285i 0.556326 + 1.48965i
\(46\) 3.04785 − 3.15336i 0.449381 − 0.464937i
\(47\) −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
\(51\) −9.01159 − 6.25404i −1.26188 − 0.875741i
\(52\) −0.845682 − 1.35603i −0.117275 − 0.188048i
\(53\) 1.16913 − 2.02499i 0.160592 − 0.278154i −0.774489 − 0.632587i \(-0.781993\pi\)
0.935081 + 0.354433i \(0.115326\pi\)
\(54\) −0.217084 − 7.34526i −0.0295414 − 0.999564i
\(55\) 1.62875i 0.219621i
\(56\) 3.11805 − 6.80278i 0.416666 − 0.909059i
\(57\) −3.86551 − 8.19999i −0.511999 − 1.08612i
\(58\) 1.01327 + 3.53959i 0.133049 + 0.464771i
\(59\) 1.49741 + 0.864528i 0.194946 + 0.112552i 0.594296 − 0.804247i \(-0.297431\pi\)
−0.399350 + 0.916798i \(0.630764\pi\)
\(60\) −10.3521 − 6.67433i −1.33645 − 0.861652i
\(61\) −4.60841 + 2.66067i −0.590047 + 0.340664i −0.765116 − 0.643892i \(-0.777318\pi\)
0.175069 + 0.984556i \(0.443985\pi\)
\(62\) −1.75130 + 7.01090i −0.222415 + 0.890385i
\(63\) −1.09508 − 7.86135i −0.137967 − 0.990437i
\(64\) 0.815178 + 7.95836i 0.101897 + 0.994795i
\(65\) −2.46055 + 1.42060i −0.305193 + 0.176204i
\(66\) 0.361335 − 1.06227i 0.0444773 − 0.130757i
\(67\) 0.00979145 − 0.0169593i 0.00119622 − 0.00207191i −0.865427 − 0.501036i \(-0.832953\pi\)
0.866623 + 0.498964i \(0.166286\pi\)
\(68\) 12.6587 − 0.430877i 1.53510 − 0.0522515i
\(69\) 2.29028 + 4.85842i 0.275717 + 0.584885i
\(70\) −11.8986 − 5.95157i −1.42216 − 0.711349i
\(71\) −9.04022 −1.07288 −0.536439 − 0.843939i \(-0.680231\pi\)
−0.536439 + 0.843939i \(0.680231\pi\)
\(72\) 5.27096 + 6.64959i 0.621188 + 0.783661i
\(73\) 4.15674 − 7.19969i 0.486510 − 0.842660i −0.513370 − 0.858167i \(-0.671603\pi\)
0.999880 + 0.0155078i \(0.00493650\pi\)
\(74\) −3.39835 − 3.28465i −0.395050 − 0.381832i
\(75\) −7.54741 + 10.8752i −0.871499 + 1.25576i
\(76\) 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 0.258533 − 0.966003i \(-0.416761\pi\)
0.965849 + 0.259105i \(0.0834277\pi\)
\(80\) 14.1897 − 0.967099i 1.58646 − 0.108125i
\(81\) 8.50749 + 2.93643i 0.945277 + 0.326270i
\(82\) 13.7934 − 3.94861i 1.52322 − 0.436051i
\(83\) 0.694004i 0.0761768i 0.999274 + 0.0380884i \(0.0121268\pi\)
−0.999274 + 0.0380884i \(0.987873\pi\)
\(84\) 6.43993 + 6.52130i 0.702654 + 0.711532i
\(85\) − 22.5181i − 2.44243i
\(86\) −1.38699 − 4.84506i −0.149563 − 0.522456i
\(87\) −4.49367 − 0.374235i −0.481772 − 0.0401222i
\(88\) 0.398740 + 1.23274i 0.0425058 + 0.131411i
\(89\) −7.02189 + 4.05409i −0.744319 + 0.429733i −0.823638 − 0.567116i \(-0.808059\pi\)
0.0793185 + 0.996849i \(0.474726\pi\)
\(90\) 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
\(93\) −7.27097 − 5.04605i −0.753965 − 0.523251i
\(94\) −4.24437 + 4.39129i −0.437773 + 0.452927i
\(95\) 9.30504 − 16.1168i 0.954677 − 1.65355i
\(96\) −9.46909 − 2.51722i −0.966435 − 0.256912i
\(97\) 7.05333 0.716157 0.358078 − 0.933692i \(-0.383432\pi\)
0.358078 + 0.933692i \(0.383432\pi\)
\(98\) 7.68004 + 6.24636i 0.775801 + 0.630978i
\(99\) 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.11.12 yes 56
3.2 odd 2 inner 168.2.v.a.11.17 yes 56
4.3 odd 2 672.2.bd.a.431.2 56
7.2 even 3 inner 168.2.v.a.107.7 yes 56
8.3 odd 2 inner 168.2.v.a.11.22 yes 56
8.5 even 2 672.2.bd.a.431.1 56
12.11 even 2 672.2.bd.a.431.19 56
21.2 odd 6 inner 168.2.v.a.107.22 yes 56
24.5 odd 2 672.2.bd.a.431.20 56
24.11 even 2 inner 168.2.v.a.11.7 ✓ 56
28.23 odd 6 672.2.bd.a.527.20 56
56.37 even 6 672.2.bd.a.527.19 56
56.51 odd 6 inner 168.2.v.a.107.17 yes 56
84.23 even 6 672.2.bd.a.527.1 56
168.107 even 6 inner 168.2.v.a.107.12 yes 56
168.149 odd 6 672.2.bd.a.527.2 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.7 ✓ 56 24.11 even 2 inner
168.2.v.a.11.12 yes 56 1.1 even 1 trivial
168.2.v.a.11.17 yes 56 3.2 odd 2 inner
168.2.v.a.11.22 yes 56 8.3 odd 2 inner
168.2.v.a.107.7 yes 56 7.2 even 3 inner
168.2.v.a.107.12 yes 56 168.107 even 6 inner
168.2.v.a.107.17 yes 56 56.51 odd 6 inner
168.2.v.a.107.22 yes 56 21.2 odd 6 inner
672.2.bd.a.431.1 56 8.5 even 2
672.2.bd.a.431.2 56 4.3 odd 2
672.2.bd.a.431.19 56 12.11 even 2
672.2.bd.a.431.20 56 24.5 odd 2
672.2.bd.a.527.1 56 84.23 even 6
672.2.bd.a.527.2 56 168.149 odd 6
672.2.bd.a.527.19 56 56.37 even 6
672.2.bd.a.527.20 56 28.23 odd 6