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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 11.19
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.19

$q$-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+(0.819527 - 1.15255i) q^{2} +(1.03943 + 1.38549i) q^{3} +(-0.656749 - 1.88910i) q^{4} +(0.692094 + 1.19874i) q^{5} +(2.44869 - 0.0625523i) q^{6} +(2.08863 + 1.62408i) q^{7} +(-2.71550 - 0.791228i) q^{8} +(-0.839162 + 2.88024i) q^{9} +(1.94880 + 0.184728i) q^{10} +(-3.82316 - 2.20731i) q^{11} +(1.93467 - 2.87351i) q^{12} -6.43609i q^{13} +(3.58352 - 1.07627i) q^{14} +(-0.941460 + 2.20490i) q^{15} +(-3.13736 + 2.48132i) q^{16} +(-2.52866 - 1.45992i) q^{17} +(2.63191 + 3.32762i) q^{18} +(1.58928 + 2.75271i) q^{19} +(1.81001 - 2.09470i) q^{20} +(-0.0791630 + 4.58189i) q^{21} +(-5.67722 + 2.59745i) q^{22} +(1.84696 + 3.19902i) q^{23} +(-1.72634 - 4.58473i) q^{24} +(1.54201 - 2.67084i) q^{25} +(-7.41792 - 5.27455i) q^{26} +(-4.86280 + 1.83117i) q^{27} +(1.69634 - 5.01223i) q^{28} -6.67884 q^{29} +(1.76971 + 2.89206i) q^{30} +(2.17580 + 1.25620i) q^{31} +(0.288700 + 5.64948i) q^{32} +(-0.915722 - 7.59130i) q^{33} +(-3.75494 + 1.71796i) q^{34} +(-0.501330 + 3.62774i) q^{35} +(5.99217 - 0.306343i) q^{36} +(-5.00149 + 2.88761i) q^{37} +(4.47510 + 0.424197i) q^{38} +(8.91713 - 6.68987i) q^{39} +(-0.930905 - 3.80279i) q^{40} +0.497427i q^{41} +(5.21599 + 3.84623i) q^{42} +0.865898 q^{43} +(-1.65895 + 8.67197i) q^{44} +(-4.03345 + 0.987462i) q^{45} +(5.20067 + 0.492975i) q^{46} +(1.59001 + 2.75398i) q^{47} +(-6.69892 - 1.76761i) q^{48} +(1.72472 + 6.78420i) q^{49} +(-1.81456 - 3.96608i) q^{50} +(-0.605663 - 5.02092i) q^{51} +(-12.1584 + 4.22690i) q^{52} +(4.12906 - 7.15175i) q^{53} +(-1.87468 + 7.10532i) q^{54} -6.11065i q^{55} +(-4.38665 - 6.06278i) q^{56} +(-2.16191 + 5.06319i) q^{57} +(-5.47350 + 7.69771i) q^{58} +(6.62279 + 3.82367i) q^{59} +(4.78357 + 0.330440i) q^{60} +(2.99321 - 1.72813i) q^{61} +(3.23096 - 1.47823i) q^{62} +(-6.43045 + 4.65289i) q^{63} +(6.74792 + 4.29716i) q^{64} +(7.71521 - 4.45438i) q^{65} +(-9.49982 - 5.16586i) q^{66} +(-3.36806 + 5.83364i) q^{67} +(-1.09724 + 5.73567i) q^{68} +(-2.51243 + 5.88411i) q^{69} +(3.77030 + 3.55084i) q^{70} +1.90437 q^{71} +(4.55768 - 7.15735i) q^{72} +(3.23515 - 5.60345i) q^{73} +(-0.770738 + 8.13095i) q^{74} +(5.30324 - 0.639719i) q^{75} +(4.15638 - 4.81014i) q^{76} +(-4.40032 - 10.8194i) q^{77} +(-0.402592 - 15.7600i) q^{78} +(1.65073 - 0.953048i) q^{79} +(-5.14582 - 2.04358i) q^{80} +(-7.59162 - 4.83398i) q^{81} +(0.573310 + 0.407655i) q^{82} -7.00064i q^{83} +(8.70762 - 2.85961i) q^{84} -4.04161i q^{85} +(0.709627 - 0.997992i) q^{86} +(-6.94220 - 9.25347i) q^{87} +(8.63533 + 9.01894i) q^{88} +(8.22776 - 4.75030i) q^{89} +(-2.16742 + 5.45801i) q^{90} +(10.4527 - 13.4426i) q^{91} +(4.83027 - 5.59004i) q^{92} +(0.521147 + 4.32029i) q^{93} +(4.47717 + 0.424393i) q^{94} +(-2.19986 + 3.81027i) q^{95} +(-7.52721 + 6.27224i) q^{96} +1.19214 q^{97} +(9.23259 + 3.57201i) q^{98} +(9.56583 - 9.15936i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.819527 − 1.15255i 0.579493 − 0.814977i
\(3\) 1.03943 + 1.38549i 0.600116 + 0.799913i
\(4\) −0.656749 − 1.88910i −0.328375 − 0.944548i
\(5\) 0.692094 + 1.19874i 0.309514 + 0.536094i 0.978256 − 0.207401i \(-0.0665004\pi\)
−0.668742 + 0.743494i \(0.733167\pi\)
\(6\) 2.44869 − 0.0625523i 0.999674 − 0.0255369i
\(7\) 2.08863 + 1.62408i 0.789426 + 0.613845i
\(8\) −2.71550 − 0.791228i −0.960075 − 0.279741i
\(9\) −0.839162 + 2.88024i −0.279721 + 0.960081i
\(10\) 1.94880 + 0.184728i 0.616265 + 0.0584162i
\(11\) −3.82316 − 2.20731i −1.15273 − 0.665528i −0.203177 − 0.979142i \(-0.565127\pi\)
−0.949550 + 0.313614i \(0.898460\pi\)
\(12\) 1.93467 − 2.87351i 0.558492 − 0.829510i
\(13\) − 6.43609i − 1.78505i −0.450999 − 0.892525i \(-0.648932\pi\)
0.450999 − 0.892525i \(-0.351068\pi\)
\(14\) 3.58352 − 1.07627i 0.957737 − 0.287645i
\(15\) −0.941460 + 2.20490i −0.243084 + 0.569303i
\(16\) −3.13736 + 2.48132i −0.784340 + 0.620331i
\(17\) −2.52866 − 1.45992i −0.613289 − 0.354083i 0.160962 − 0.986961i \(-0.448540\pi\)
−0.774252 + 0.632878i \(0.781874\pi\)
\(18\) 2.63191 + 3.32762i 0.620348 + 0.784327i
\(19\) 1.58928 + 2.75271i 0.364606 + 0.631515i 0.988713 − 0.149823i \(-0.0478705\pi\)
−0.624107 + 0.781339i \(0.714537\pi\)
\(20\) 1.81001 − 2.09470i 0.404730 − 0.468390i
\(21\) −0.0791630 + 4.58189i −0.0172748 + 0.999851i
\(22\) −5.67722 + 2.59745i −1.21039 + 0.553777i
\(23\) 1.84696 + 3.19902i 0.385117 + 0.667043i 0.991785 − 0.127912i \(-0.0408277\pi\)
−0.606668 + 0.794955i \(0.707494\pi\)
\(24\) −1.72634 − 4.58473i −0.352388 − 0.935854i
\(25\) 1.54201 − 2.67084i 0.308402 − 0.534169i
\(26\) −7.41792 − 5.27455i −1.45477 − 1.03442i
\(27\) −4.86280 + 1.83117i −0.935846 + 0.352409i
\(28\) 1.69634 − 5.01223i 0.320578 − 0.947222i
\(29\) −6.67884 −1.24023 −0.620115 − 0.784511i \(-0.712914\pi\)
−0.620115 + 0.784511i \(0.712914\pi\)
\(30\) 1.76971 + 2.89206i 0.323103 + 0.528015i
\(31\) 2.17580 + 1.25620i 0.390786 + 0.225620i 0.682500 − 0.730885i \(-0.260893\pi\)
−0.291715 + 0.956505i \(0.594226\pi\)
\(32\) 0.288700 + 5.64948i 0.0510355 + 0.998697i
\(33\) −0.915722 − 7.59130i −0.159407 − 1.32148i
\(34\) −3.75494 + 1.71796i −0.643966 + 0.294628i
\(35\) −0.501330 + 3.62774i −0.0847402 + 0.613200i
\(36\) 5.99217 − 0.306343i 0.998696 − 0.0510571i
\(37\) −5.00149 + 2.88761i −0.822240 + 0.474720i −0.851188 − 0.524861i \(-0.824117\pi\)
0.0289485 + 0.999581i \(0.490784\pi\)
\(38\) 4.47510 + 0.424197i 0.725957 + 0.0688139i
\(39\) 8.91713 − 6.68987i 1.42788 − 1.07124i
\(40\) −0.930905 − 3.80279i −0.147189 − 0.601274i
\(41\) 0.497427i 0.0776850i 0.999245 + 0.0388425i \(0.0123671\pi\)
−0.999245 + 0.0388425i \(0.987633\pi\)
\(42\) 5.21599 + 3.84623i 0.804845 + 0.593486i
\(43\) 0.865898 0.132048 0.0660241 − 0.997818i \(-0.478969\pi\)
0.0660241 + 0.997818i \(0.478969\pi\)
\(44\) −1.65895 + 8.67197i −0.250096 + 1.30735i
\(45\) −4.03345 + 0.987462i −0.601271 + 0.147202i
\(46\) 5.20067 + 0.492975i 0.766797 + 0.0726852i
\(47\) 1.59001 + 2.75398i 0.231927 + 0.401710i 0.958375 − 0.285512i \(-0.0921635\pi\)
−0.726448 + 0.687221i \(0.758830\pi\)
\(48\) −6.69892 − 1.76761i −0.966906 − 0.255133i
\(49\) 1.72472 + 6.78420i 0.246388 + 0.969171i
\(50\) −1.81456 − 3.96608i −0.256618 − 0.560888i
\(51\) −0.605663 − 5.02092i −0.0848097 − 0.703069i
\(52\) −12.1584 + 4.22690i −1.68606 + 0.586165i
\(53\) 4.12906 − 7.15175i 0.567171 − 0.982368i −0.429673 − 0.902984i \(-0.641371\pi\)
0.996844 − 0.0793841i \(-0.0252954\pi\)
\(54\) −1.87468 + 7.10532i −0.255112 + 0.966912i
\(55\) − 6.11065i − 0.823960i
\(56\) −4.38665 − 6.06278i −0.586191 − 0.810173i
\(57\) −2.16191 + 5.06319i −0.286351 + 0.670635i
\(58\) −5.47350 + 7.69771i −0.718705 + 1.01076i
\(59\) 6.62279 + 3.82367i 0.862214 + 0.497799i 0.864753 − 0.502197i \(-0.167475\pi\)
−0.00253919 + 0.999997i \(0.500808\pi\)
\(60\) 4.78357 + 0.330440i 0.617556 + 0.0426596i
\(61\) 2.99321 − 1.72813i 0.383241 − 0.221265i −0.295986 − 0.955192i \(-0.595648\pi\)
0.679228 + 0.733928i \(0.262315\pi\)
\(62\) 3.23096 − 1.47823i 0.410333 − 0.187736i
\(63\) −6.43045 + 4.65289i −0.810160 + 0.586209i
\(64\) 6.74792 + 4.29716i 0.843490 + 0.537146i
\(65\) 7.71521 − 4.45438i 0.956954 − 0.552497i
\(66\) −9.49982 − 5.16586i −1.16935 − 0.635873i
\(67\) −3.36806 + 5.83364i −0.411473 + 0.712693i −0.995051 − 0.0993644i \(-0.968319\pi\)
0.583578 + 0.812057i \(0.301652\pi\)
\(68\) −1.09724 + 5.73567i −0.133059 + 0.695553i
\(69\) −2.51243 + 5.88411i −0.302461 + 0.708363i
\(70\) 3.77030 + 3.55084i 0.450638 + 0.424407i
\(71\) 1.90437 0.226007 0.113004 − 0.993595i \(-0.463953\pi\)
0.113004 + 0.993595i \(0.463953\pi\)
\(72\) 4.55768 − 7.15735i 0.537127 − 0.843501i
\(73\) 3.23515 − 5.60345i 0.378646 − 0.655834i −0.612219 − 0.790688i \(-0.709723\pi\)
0.990866 + 0.134854i \(0.0430564\pi\)
\(74\) −0.770738 + 8.13095i −0.0895964 + 0.945204i
\(75\) 5.30324 − 0.639719i 0.612365 − 0.0738684i
\(76\) 4.15638 − 4.81014i 0.476769 − 0.551761i
\(77\) −4.40032 − 10.8194i −0.501463 − 1.23298i
\(78\) −0.402592 − 15.7600i −0.0455846 − 1.78447i
\(79\) 1.65073 − 0.953048i 0.185721 − 0.107226i −0.404257 − 0.914646i \(-0.632470\pi\)
0.589978 + 0.807419i \(0.299136\pi\)
\(80\) −5.14582 − 2.04358i −0.575320 − 0.228479i
\(81\) −7.59162 − 4.83398i −0.843513 − 0.537109i
\(82\) 0.573310 + 0.407655i 0.0633115 + 0.0450179i
\(83\) − 7.00064i − 0.768420i −0.923246 − 0.384210i \(-0.874474\pi\)
0.923246 − 0.384210i \(-0.125526\pi\)
\(84\) 8.70762 − 2.85961i 0.950079 − 0.312009i
\(85\) − 4.04161i − 0.438374i
\(86\) 0.709627 − 0.997992i 0.0765210 − 0.107616i
\(87\) −6.94220 − 9.25347i −0.744282 − 0.992076i
\(88\) 8.63533 + 9.01894i 0.920530 + 0.961422i
\(89\) 8.22776 − 4.75030i 0.872141 − 0.503531i 0.00408165 − 0.999992i \(-0.498701\pi\)
0.868059 + 0.496461i \(0.165367\pi\)
\(90\) −2.16742 + 5.45801i −0.228466 + 0.575325i
\(91\) 10.4527 − 13.4426i 1.09574 − 1.40916i
\(92\) 4.83027 − 5.59004i 0.503591 − 0.582801i
\(93\) 0.521147 + 4.32029i 0.0540404 + 0.447993i
\(94\) 4.47717 + 0.424393i 0.461785 + 0.0437728i
\(95\) −2.19986 + 3.81027i −0.225701 + 0.390926i
\(96\) −7.52721 + 6.27224i −0.768243 + 0.640158i
\(97\) 1.19214 0.121044 0.0605218 − 0.998167i \(-0.480724\pi\)
0.0605218 + 0.998167i \(0.480724\pi\)
\(98\) 9.23259 + 3.57201i 0.932632 + 0.360828i
\(99\) 9.56583 − 9.15936i 0.961402 − 0.920551i
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.11.19 yes 56
3.2 odd 2 inner 168.2.v.a.11.10 yes 56
4.3 odd 2 672.2.bd.a.431.8 56
7.2 even 3 inner 168.2.v.a.107.1 yes 56
8.3 odd 2 inner 168.2.v.a.11.28 yes 56
8.5 even 2 672.2.bd.a.431.7 56
12.11 even 2 672.2.bd.a.431.27 56
21.2 odd 6 inner 168.2.v.a.107.28 yes 56
24.5 odd 2 672.2.bd.a.431.28 56
24.11 even 2 inner 168.2.v.a.11.1 ✓ 56
28.23 odd 6 672.2.bd.a.527.28 56
56.37 even 6 672.2.bd.a.527.27 56
56.51 odd 6 inner 168.2.v.a.107.10 yes 56
84.23 even 6 672.2.bd.a.527.7 56
168.107 even 6 inner 168.2.v.a.107.19 yes 56
168.149 odd 6 672.2.bd.a.527.8 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.1 ✓ 56 24.11 even 2 inner
168.2.v.a.11.10 yes 56 3.2 odd 2 inner
168.2.v.a.11.19 yes 56 1.1 even 1 trivial
168.2.v.a.11.28 yes 56 8.3 odd 2 inner
168.2.v.a.107.1 yes 56 7.2 even 3 inner
168.2.v.a.107.10 yes 56 56.51 odd 6 inner
168.2.v.a.107.19 yes 56 168.107 even 6 inner
168.2.v.a.107.28 yes 56 21.2 odd 6 inner
672.2.bd.a.431.7 56 8.5 even 2
672.2.bd.a.431.8 56 4.3 odd 2
672.2.bd.a.431.27 56 12.11 even 2
672.2.bd.a.431.28 56 24.5 odd 2
672.2.bd.a.527.7 56 84.23 even 6
672.2.bd.a.527.8 56 168.149 odd 6
672.2.bd.a.527.27 56 56.37 even 6
672.2.bd.a.527.28 56 28.23 odd 6