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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 11.16
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.16

$q$-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+(0.248376 - 1.39223i) q^{2} +(-1.64844 - 0.531631i) q^{3} +(-1.87662 - 0.691593i) q^{4} +(-0.646402 - 1.11960i) q^{5} +(-1.14959 + 2.16297i) q^{6} +(-2.42742 + 1.05244i) q^{7} +(-1.42896 + 2.44091i) q^{8} +(2.43474 + 1.75273i) q^{9} +(-1.71929 + 0.621859i) q^{10} +(-1.60004 - 0.923782i) q^{11} +(2.72583 + 2.13772i) q^{12} -2.25432i q^{13} +(0.862334 + 3.64093i) q^{14} +(0.470342 + 2.18925i) q^{15} +(3.04340 + 2.59571i) q^{16} +(-3.89580 - 2.24924i) q^{17} +(3.04493 - 2.95438i) q^{18} +(-2.80851 - 4.86448i) q^{19} +(0.438742 + 2.54811i) q^{20} +(4.56098 - 0.444404i) q^{21} +(-1.68353 + 1.99818i) q^{22} +(0.519880 + 0.900459i) q^{23} +(3.65323 - 3.26403i) q^{24} +(1.66433 - 2.88270i) q^{25} +(-3.13853 - 0.559917i) q^{26} +(-3.08172 - 4.18366i) q^{27} +(5.28320 - 0.296251i) q^{28} -1.32085 q^{29} +(3.16476 - 0.111070i) q^{30} +(3.69700 + 2.13446i) q^{31} +(4.36974 - 3.59240i) q^{32} +(2.14646 + 2.37343i) q^{33} +(-4.09909 + 4.86521i) q^{34} +(2.74740 + 2.03744i) q^{35} +(-3.35690 - 4.97305i) q^{36} +(8.18394 - 4.72500i) q^{37} +(-7.47005 + 2.70188i) q^{38} +(-1.19846 + 3.71611i) q^{39} +(3.65653 + 0.0220581i) q^{40} +1.39634i q^{41} +(0.514122 - 6.46032i) q^{42} -6.02578 q^{43} +(2.36378 + 2.84016i) q^{44} +(0.388538 - 3.85890i) q^{45} +(1.38277 - 0.500142i) q^{46} +(-5.90249 - 10.2234i) q^{47} +(-3.63691 - 5.89685i) q^{48} +(4.78472 - 5.10944i) q^{49} +(-3.60001 - 3.03313i) q^{50} +(5.22625 + 5.77888i) q^{51} +(-1.55907 + 4.23049i) q^{52} +(-6.02901 + 10.4425i) q^{53} +(-6.59005 + 3.25135i) q^{54} +2.38854i q^{55} +(0.899769 - 7.42903i) q^{56} +(2.04356 + 9.51192i) q^{57} +(-0.328068 + 1.83894i) q^{58} +(9.57732 + 5.52947i) q^{59} +(0.631413 - 4.43367i) q^{60} +(-7.65220 + 4.41800i) q^{61} +(3.88991 - 4.61693i) q^{62} +(-7.75477 - 1.69218i) q^{63} +(-3.91612 - 6.97596i) q^{64} +(-2.52393 + 1.45719i) q^{65} +(3.83750 - 2.39887i) q^{66} +(3.05545 - 5.29219i) q^{67} +(5.75538 + 6.91528i) q^{68} +(-0.378281 - 1.76074i) q^{69} +(3.51897 - 3.31897i) q^{70} +14.0121 q^{71} +(-7.75741 + 3.43840i) q^{72} +(-4.38664 + 7.59788i) q^{73} +(-4.54561 - 12.5675i) q^{74} +(-4.27609 + 3.86717i) q^{75} +(1.90626 + 11.0711i) q^{76} +(4.85619 + 0.558456i) q^{77} +(4.87602 + 2.59153i) q^{78} +(-2.37061 + 1.36867i) q^{79} +(0.938904 - 5.08526i) q^{80} +(2.85588 + 8.53487i) q^{81} +(1.94403 + 0.346817i) q^{82} +4.74366i q^{83} +(-8.86656 - 2.32036i) q^{84} +5.81566i q^{85} +(-1.49666 + 8.38928i) q^{86} +(2.17736 + 0.702207i) q^{87} +(4.54127 - 2.58550i) q^{88} +(8.31219 - 4.79904i) q^{89} +(-5.27598 - 1.49939i) q^{90} +(2.37254 + 5.47217i) q^{91} +(-0.352866 - 2.04936i) q^{92} +(-4.95955 - 5.48398i) q^{93} +(-15.6994 + 5.67839i) q^{94} +(-3.63085 + 6.28882i) q^{95} +(-9.11311 + 3.59879i) q^{96} -8.73466 q^{97} +(-5.92512 - 7.93051i) q^{98} +(-2.27653 - 5.05360i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.248376 − 1.39223i 0.175628 − 0.984457i
\(3\) −1.64844 − 0.531631i −0.951730 − 0.306937i
\(4\) −1.87662 − 0.691593i −0.938310 − 0.345796i
\(5\) −0.646402 − 1.11960i −0.289080 − 0.500700i 0.684511 − 0.729003i \(-0.260016\pi\)
−0.973590 + 0.228302i \(0.926683\pi\)
\(6\) −1.14959 + 2.16297i −0.469317 + 0.883030i
\(7\) −2.42742 + 1.05244i −0.917478 + 0.397786i
\(8\) −1.42896 + 2.44091i −0.505215 + 0.862993i
\(9\) 2.43474 + 1.75273i 0.811579 + 0.584243i
\(10\) −1.71929 + 0.621859i −0.543688 + 0.196649i
\(11\) −1.60004 − 0.923782i −0.482430 − 0.278531i 0.238999 − 0.971020i \(-0.423181\pi\)
−0.721428 + 0.692489i \(0.756514\pi\)
\(12\) 2.72583 + 2.13772i 0.786879 + 0.617107i
\(13\) − 2.25432i − 0.625235i −0.949879 − 0.312617i \(-0.898794\pi\)
0.949879 − 0.312617i \(-0.101206\pi\)
\(14\) 0.862334 + 3.64093i 0.230469 + 0.973080i
\(15\) 0.470342 + 2.18925i 0.121442 + 0.565261i
\(16\) 3.04340 + 2.59571i 0.760850 + 0.648928i
\(17\) −3.89580 − 2.24924i −0.944871 − 0.545522i −0.0533874 − 0.998574i \(-0.517002\pi\)
−0.891484 + 0.453052i \(0.850335\pi\)
\(18\) 3.04493 − 2.95438i 0.717698 − 0.696355i
\(19\) −2.80851 − 4.86448i −0.644317 − 1.11599i −0.984459 − 0.175615i \(-0.943809\pi\)
0.340142 − 0.940374i \(-0.389525\pi\)
\(20\) 0.438742 + 2.54811i 0.0981056 + 0.569775i
\(21\) 4.56098 − 0.444404i 0.995287 − 0.0969769i
\(22\) −1.68353 + 1.99818i −0.358930 + 0.426013i
\(23\) 0.519880 + 0.900459i 0.108402 + 0.187759i 0.915123 − 0.403174i \(-0.132093\pi\)
−0.806721 + 0.590933i \(0.798760\pi\)
\(24\) 3.65323 − 3.26403i 0.745713 − 0.666267i
\(25\) 1.66433 − 2.88270i 0.332866 − 0.576541i
\(26\) −3.13853 − 0.559917i −0.615517 − 0.109809i
\(27\) −3.08172 − 4.18366i −0.593078 − 0.805145i
\(28\) 5.28320 − 0.296251i 0.998432 − 0.0559862i
\(29\) −1.32085 −0.245277 −0.122638 − 0.992451i \(-0.539135\pi\)
−0.122638 + 0.992451i \(0.539135\pi\)
\(30\) 3.16476 − 0.111070i 0.577803 − 0.0202786i
\(31\) 3.69700 + 2.13446i 0.664000 + 0.383361i 0.793800 − 0.608179i \(-0.208100\pi\)
−0.129799 + 0.991540i \(0.541433\pi\)
\(32\) 4.36974 − 3.59240i 0.772468 − 0.635053i
\(33\) 2.14646 + 2.37343i 0.373651 + 0.413162i
\(34\) −4.09909 + 4.86521i −0.702988 + 0.834376i
\(35\) 2.74740 + 2.03744i 0.464396 + 0.344390i
\(36\) −3.35690 − 4.97305i −0.559483 − 0.828842i
\(37\) 8.18394 − 4.72500i 1.34543 − 0.776786i 0.357834 − 0.933785i \(-0.383516\pi\)
0.987599 + 0.157000i \(0.0501822\pi\)
\(38\) −7.47005 + 2.70188i −1.21180 + 0.438303i
\(39\) −1.19846 + 3.71611i −0.191908 + 0.595055i
\(40\) 3.65653 + 0.0220581i 0.578149 + 0.00348770i
\(41\) 1.39634i 0.218072i 0.994038 + 0.109036i \(0.0347764\pi\)
−0.994038 + 0.109036i \(0.965224\pi\)
\(42\) 0.514122 − 6.46032i 0.0793308 − 0.996848i
\(43\) −6.02578 −0.918922 −0.459461 − 0.888198i \(-0.651957\pi\)
−0.459461 + 0.888198i \(0.651957\pi\)
\(44\) 2.36378 + 2.84016i 0.356353 + 0.428171i
\(45\) 0.388538 − 3.85890i 0.0579198 − 0.575251i
\(46\) 1.38277 − 0.500142i 0.203879 − 0.0737418i
\(47\) −5.90249 − 10.2234i −0.860966 − 1.49124i −0.870997 − 0.491288i \(-0.836526\pi\)
0.0100312 − 0.999950i \(-0.496807\pi\)
\(48\) −3.63691 − 5.89685i −0.524943 − 0.851138i
\(49\) 4.78472 − 5.10944i 0.683532 − 0.729921i
\(50\) −3.60001 − 3.03313i −0.509119 − 0.428949i
\(51\) 5.22625 + 5.77888i 0.731821 + 0.809206i
\(52\) −1.55907 + 4.23049i −0.216204 + 0.586664i
\(53\) −6.02901 + 10.4425i −0.828148 + 1.43439i 0.0713414 + 0.997452i \(0.477272\pi\)
−0.899489 + 0.436943i \(0.856061\pi\)
\(54\) −6.59005 + 3.25135i −0.896792 + 0.442453i
\(55\) 2.38854i 0.322070i
\(56\) 0.899769 − 7.42903i 0.120237 − 0.992745i
\(57\) 2.04356 + 9.51192i 0.270676 + 1.25988i
\(58\) −0.328068 + 1.83894i −0.0430775 + 0.241464i
\(59\) 9.57732 + 5.52947i 1.24686 + 0.719875i 0.970482 − 0.241174i \(-0.0775324\pi\)
0.276378 + 0.961049i \(0.410866\pi\)
\(60\) 0.631413 − 4.43367i 0.0815151 − 0.572384i
\(61\) −7.65220 + 4.41800i −0.979764 + 0.565667i −0.902199 − 0.431320i \(-0.858048\pi\)
−0.0775652 + 0.996987i \(0.524715\pi\)
\(62\) 3.88991 − 4.61693i 0.494019 − 0.586351i
\(63\) −7.75477 − 1.69218i −0.977010 − 0.213195i
\(64\) −3.91612 − 6.97596i −0.489515 − 0.871995i
\(65\) −2.52393 + 1.45719i −0.313055 + 0.180743i
\(66\) 3.83750 − 2.39887i 0.472363 − 0.295280i
\(67\) 3.05545 − 5.29219i 0.373283 − 0.646544i −0.616786 − 0.787131i \(-0.711566\pi\)
0.990068 + 0.140587i \(0.0448989\pi\)
\(68\) 5.75538 + 6.91528i 0.697942 + 0.838601i
\(69\) −0.378281 − 1.76074i −0.0455397 − 0.211968i
\(70\) 3.51897 − 3.31897i 0.420598 − 0.396693i
\(71\) 14.0121 1.66293 0.831464 − 0.555578i \(-0.187503\pi\)
0.831464 + 0.555578i \(0.187503\pi\)
\(72\) −7.75741 + 3.43840i −0.914220 + 0.405219i
\(73\) −4.38664 + 7.59788i −0.513417 + 0.889264i 0.486462 + 0.873702i \(0.338287\pi\)
−0.999879 + 0.0155626i \(0.995046\pi\)
\(74\) −4.54561 − 12.5675i −0.528416 − 1.46095i
\(75\) −4.27609 + 3.86717i −0.493760 + 0.446542i
\(76\) 1.90626 + 11.0711i 0.218663 + 1.26995i
\(77\) 4.85619 + 0.558456i 0.553414 + 0.0636420i
\(78\) 4.87602 + 2.59153i 0.552101 + 0.293433i
\(79\) −2.37061 + 1.36867i −0.266714 + 0.153987i −0.627393 − 0.778702i \(-0.715878\pi\)
0.360679 + 0.932690i \(0.382545\pi\)
\(80\) 0.938904 − 5.08526i 0.104973 − 0.568550i
\(81\) 2.85588 + 8.53487i 0.317320 + 0.948318i
\(82\) 1.94403 + 0.346817i 0.214682 + 0.0382996i
\(83\) 4.74366i 0.520685i 0.965516 + 0.260342i \(0.0838354\pi\)
−0.965516 + 0.260342i \(0.916165\pi\)
\(84\) −8.86656 − 2.32036i −0.967421 − 0.253172i
\(85\) 5.81566i 0.630797i
\(86\) −1.49666 + 8.38928i −0.161389 + 0.904639i
\(87\) 2.17736 + 0.702207i 0.233437 + 0.0752845i
\(88\) 4.54127 − 2.58550i 0.484101 − 0.275616i
\(89\) 8.31219 − 4.79904i 0.881090 − 0.508697i 0.0100723 − 0.999949i \(-0.496794\pi\)
0.871018 + 0.491252i \(0.163461\pi\)
\(90\) −5.27598 − 1.49939i −0.556137 − 0.158050i
\(91\) 2.37254 + 5.47217i 0.248710 + 0.573639i
\(92\) −0.352866 − 2.04936i −0.0367888 − 0.213661i
\(93\) −4.95955 − 5.48398i −0.514281 − 0.568662i
\(94\) −15.6994 + 5.67839i −1.61927 + 0.585681i
\(95\) −3.63085 + 6.28882i −0.372517 + 0.645219i
\(96\) −9.11311 + 3.59879i −0.930103 + 0.367300i
\(97\) −8.73466 −0.886871 −0.443435 − 0.896306i \(-0.646240\pi\)
−0.443435 + 0.896306i \(0.646240\pi\)
\(98\) −5.92512 − 7.93051i −0.598528 − 0.801102i
\(99\) −2.27653 − 5.05360i −0.228800 − 0.507906i
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.11.16 yes 56
3.2 odd 2 inner 168.2.v.a.11.13 yes 56
4.3 odd 2 672.2.bd.a.431.25 56
7.2 even 3 inner 168.2.v.a.107.4 yes 56
8.3 odd 2 inner 168.2.v.a.11.25 yes 56
8.5 even 2 672.2.bd.a.431.26 56
12.11 even 2 672.2.bd.a.431.6 56
21.2 odd 6 inner 168.2.v.a.107.25 yes 56
24.5 odd 2 672.2.bd.a.431.5 56
24.11 even 2 inner 168.2.v.a.11.4 ✓ 56
28.23 odd 6 672.2.bd.a.527.5 56
56.37 even 6 672.2.bd.a.527.6 56
56.51 odd 6 inner 168.2.v.a.107.13 yes 56
84.23 even 6 672.2.bd.a.527.26 56
168.107 even 6 inner 168.2.v.a.107.16 yes 56
168.149 odd 6 672.2.bd.a.527.25 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.4 ✓ 56 24.11 even 2 inner
168.2.v.a.11.13 yes 56 3.2 odd 2 inner
168.2.v.a.11.16 yes 56 1.1 even 1 trivial
168.2.v.a.11.25 yes 56 8.3 odd 2 inner
168.2.v.a.107.4 yes 56 7.2 even 3 inner
168.2.v.a.107.13 yes 56 56.51 odd 6 inner
168.2.v.a.107.16 yes 56 168.107 even 6 inner
168.2.v.a.107.25 yes 56 21.2 odd 6 inner
672.2.bd.a.431.5 56 24.5 odd 2
672.2.bd.a.431.6 56 12.11 even 2
672.2.bd.a.431.25 56 4.3 odd 2
672.2.bd.a.431.26 56 8.5 even 2
672.2.bd.a.527.5 56 28.23 odd 6
672.2.bd.a.527.6 56 56.37 even 6
672.2.bd.a.527.25 56 168.149 odd 6
672.2.bd.a.527.26 56 84.23 even 6