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Properties

Label 168.2.v.a.107.19
Level $168$
Weight $2$
Character 168.107
Analytic conductor $1.341$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

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 107.19
Character \(\chi\) \(=\) 168.107
Dual form 168.2.v.a.11.19

$q$-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 \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(-1\)

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 −0.668742 − 0.743494i \(-0.733167\pi\)
0.978256 + 0.207401i \(0.0665004\pi\)
\(6\) 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 −0.949550 − 0.313614i \(-0.898460\pi\)
−0.203177 + 0.979142i \(0.565127\pi\)
\(12\) 1.93467 + 2.87351i 0.558492 + 0.829510i
\(13\) 6.43609i 1.78505i 0.450999 + 0.892525i \(0.351068\pi\)
−0.450999 + 0.892525i \(0.648932\pi\)
\(14\) 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 −0.774252 − 0.632878i \(-0.781874\pi\)
0.160962 + 0.986961i \(0.448540\pi\)
\(18\) 2.63191 − 3.32762i 0.620348 − 0.784327i
\(19\) 1.58928 − 2.75271i 0.364606 − 0.631515i −0.624107 − 0.781339i \(-0.714537\pi\)
0.988713 + 0.149823i \(0.0478705\pi\)
\(20\) 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 −0.606668 − 0.794955i \(-0.707494\pi\)
0.991785 + 0.127912i \(0.0408277\pi\)
\(24\) −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 −0.291715 − 0.956505i \(-0.594226\pi\)
0.682500 + 0.730885i \(0.260893\pi\)
\(32\) 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 0.0289485 − 0.999581i \(-0.490784\pi\)
−0.851188 + 0.524861i \(0.824117\pi\)
\(38\) 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 \(-0.987633\pi\)
0.999245 − 0.0388425i \(-0.0123671\pi\)
\(42\) 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 −0.726448 − 0.687221i \(-0.758830\pi\)
0.958375 + 0.285512i \(0.0921635\pi\)
\(48\) −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 0.996844 + 0.0793841i \(0.0252954\pi\)
−0.429673 + 0.902984i \(0.641371\pi\)
\(54\) −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 −0.00253919 − 0.999997i \(-0.500808\pi\)
0.864753 + 0.502197i \(0.167475\pi\)
\(60\) 4.78357 − 0.330440i 0.617556 − 0.0426596i
\(61\) 2.99321 + 1.72813i 0.383241 + 0.221265i 0.679228 − 0.733928i \(-0.262315\pi\)
−0.295986 + 0.955192i \(0.595648\pi\)
\(62\) 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 0.583578 − 0.812057i \(-0.301652\pi\)
−0.995051 + 0.0993644i \(0.968319\pi\)
\(68\) −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 0.990866 − 0.134854i \(-0.0430564\pi\)
−0.612219 + 0.790688i \(0.709723\pi\)
\(74\) −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 0.589978 − 0.807419i \(-0.299136\pi\)
−0.404257 + 0.914646i \(0.632470\pi\)
\(80\) −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 \(0.125526\pi\)
−0.923246 + 0.384210i \(0.874474\pi\)
\(84\) 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 0.868059 − 0.496461i \(-0.165367\pi\)
0.00408165 + 0.999992i \(0.498701\pi\)
\(90\) −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.107.19 yes 56
3.2 odd 2 inner 168.2.v.a.107.10 yes 56
4.3 odd 2 672.2.bd.a.527.8 56
7.4 even 3 inner 168.2.v.a.11.1 ✓ 56
8.3 odd 2 inner 168.2.v.a.107.28 yes 56
8.5 even 2 672.2.bd.a.527.7 56
12.11 even 2 672.2.bd.a.527.27 56
21.11 odd 6 inner 168.2.v.a.11.28 yes 56
24.5 odd 2 672.2.bd.a.527.28 56
24.11 even 2 inner 168.2.v.a.107.1 yes 56
28.11 odd 6 672.2.bd.a.431.28 56
56.11 odd 6 inner 168.2.v.a.11.10 yes 56
56.53 even 6 672.2.bd.a.431.27 56
84.11 even 6 672.2.bd.a.431.7 56
168.11 even 6 inner 168.2.v.a.11.19 yes 56
168.53 odd 6 672.2.bd.a.431.8 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.1 ✓ 56 7.4 even 3 inner
168.2.v.a.11.10 yes 56 56.11 odd 6 inner
168.2.v.a.11.19 yes 56 168.11 even 6 inner
168.2.v.a.11.28 yes 56 21.11 odd 6 inner
168.2.v.a.107.1 yes 56 24.11 even 2 inner
168.2.v.a.107.10 yes 56 3.2 odd 2 inner
168.2.v.a.107.19 yes 56 1.1 even 1 trivial
168.2.v.a.107.28 yes 56 8.3 odd 2 inner
672.2.bd.a.431.7 56 84.11 even 6
672.2.bd.a.431.8 56 168.53 odd 6
672.2.bd.a.431.27 56 56.53 even 6
672.2.bd.a.431.28 56 28.11 odd 6
672.2.bd.a.527.7 56 8.5 even 2
672.2.bd.a.527.8 56 4.3 odd 2
672.2.bd.a.527.27 56 12.11 even 2
672.2.bd.a.527.28 56 24.5 odd 2