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 11.6
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.6

$q$-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+(-1.08135 + 0.911422i) q^{2} +(-0.316681 + 1.70285i) q^{3} +(0.338619 - 1.97113i) q^{4} +(1.86088 + 3.22314i) q^{5} +(-1.20958 - 2.13001i) q^{6} +(2.46429 - 0.962962i) q^{7} +(1.43036 + 2.44009i) q^{8} +(-2.79943 - 1.07852i) q^{9} +(-4.94989 - 1.78928i) q^{10} +(-2.26958 - 1.31034i) q^{11} +(3.24931 + 1.20084i) q^{12} +3.57182i q^{13} +(-1.78708 + 3.28730i) q^{14} +(-6.07784 + 2.14810i) q^{15} +(-3.77067 - 1.33492i) q^{16} +(0.186192 + 0.107498i) q^{17} +(4.01014 - 1.38520i) q^{18} +(-1.14103 - 1.97631i) q^{19} +(6.98334 - 2.57661i) q^{20} +(0.859391 + 4.50127i) q^{21} +(3.64848 - 0.651612i) q^{22} +(-2.33586 - 4.04583i) q^{23} +(-4.60809 + 1.66297i) q^{24} +(-4.42574 + 7.66560i) q^{25} +(-3.25544 - 3.86238i) q^{26} +(2.72309 - 4.42547i) q^{27} +(-1.06366 - 5.18349i) q^{28} +2.57415 q^{29} +(4.61442 - 7.86231i) q^{30} +(4.26196 + 2.46064i) q^{31} +(5.29408 - 1.99316i) q^{32} +(2.95006 - 3.44980i) q^{33} +(-0.299314 + 0.0534570i) q^{34} +(7.68949 + 6.15077i) q^{35} +(-3.07384 + 5.15281i) q^{36} +(7.31104 - 4.22103i) q^{37} +(3.03510 + 1.09712i) q^{38} +(-6.08229 - 1.13113i) q^{39} +(-5.20302 + 9.15098i) q^{40} +0.909442i q^{41} +(-5.03186 - 4.08416i) q^{42} +3.73541 q^{43} +(-3.35137 + 4.02992i) q^{44} +(-1.73317 - 11.0299i) q^{45} +(6.21334 + 2.24599i) q^{46} +(-0.586728 - 1.01624i) q^{47} +(3.46728 - 5.99816i) q^{48} +(5.14541 - 4.74603i) q^{49} +(-2.20085 - 12.3229i) q^{50} +(-0.242017 + 0.283015i) q^{51} +(7.04051 + 1.20949i) q^{52} +(-1.06171 + 1.83894i) q^{53} +(1.08886 + 7.26735i) q^{54} -9.75355i q^{55} +(5.87454 + 4.63570i) q^{56} +(3.72672 - 1.31714i) q^{57} +(-2.78355 + 2.34614i) q^{58} +(-6.79199 - 3.92136i) q^{59} +(2.17610 + 12.7076i) q^{60} +(-0.301301 + 0.173956i) q^{61} +(-6.85134 + 1.22364i) q^{62} +(-7.93716 + 0.0379510i) q^{63} +(-3.90812 + 6.98044i) q^{64} +(-11.5125 + 6.64673i) q^{65} +(-0.0458036 + 6.41918i) q^{66} +(-4.98736 + 8.63836i) q^{67} +(0.274940 - 0.330607i) q^{68} +(7.62919 - 2.69640i) q^{69} +(-13.9210 + 0.357259i) q^{70} +10.0808 q^{71} +(-1.37250 - 8.37354i) q^{72} +(4.45173 - 7.71062i) q^{73} +(-4.05862 + 11.2278i) q^{74} +(-11.6519 - 9.96394i) q^{75} +(-4.28194 + 1.57989i) q^{76} +(-6.85470 - 1.04354i) q^{77} +(7.60800 - 4.32040i) q^{78} +(-9.70950 + 5.60578i) q^{79} +(-2.71413 - 14.6375i) q^{80} +(6.67358 + 6.03849i) q^{81} +(-0.828886 - 0.983422i) q^{82} -1.73937i q^{83} +(9.16358 - 0.169751i) q^{84} +0.800163i q^{85} +(-4.03927 + 3.40453i) q^{86} +(-0.815185 + 4.38341i) q^{87} +(-0.0489643 - 7.41226i) q^{88} +(15.4694 - 8.93127i) q^{89} +(11.9271 + 10.3475i) q^{90} +(3.43953 + 8.80199i) q^{91} +(-8.76581 + 3.23428i) q^{92} +(-5.53980 + 6.47826i) q^{93} +(1.56068 + 0.564153i) q^{94} +(4.24662 - 7.35536i) q^{95} +(1.71753 + 9.64625i) q^{96} -8.88553 q^{97} +(-1.23834 + 9.82174i) q^{98} +(4.94029 + 6.11600i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08135 + 0.911422i −0.764627 + 0.644473i
\(3\) −0.316681 + 1.70285i −0.182836 + 0.983143i
\(4\) 0.338619 − 1.97113i 0.169310 − 0.985563i
\(5\) 1.86088 + 3.22314i 0.832210 + 1.44143i 0.896282 + 0.443485i \(0.146258\pi\)
−0.0640716 + 0.997945i \(0.520409\pi\)
\(6\) −1.20958 − 2.13001i −0.493808 − 0.869571i
\(7\) 2.46429 − 0.962962i 0.931412 − 0.363965i
\(8\) 1.43036 + 2.44009i 0.505710 + 0.862704i
\(9\) −2.79943 − 1.07852i −0.933142 − 0.359508i
\(10\) −4.94989 − 1.78928i −1.56529 − 0.565820i
\(11\) −2.26958 − 1.31034i −0.684304 − 0.395083i 0.117171 − 0.993112i \(-0.462618\pi\)
−0.801475 + 0.598029i \(0.795951\pi\)
\(12\) 3.24931 + 1.20084i 0.937994 + 0.346652i
\(13\) 3.57182i 0.990645i 0.868709 + 0.495323i \(0.164950\pi\)
−0.868709 + 0.495323i \(0.835050\pi\)
\(14\) −1.78708 + 3.28730i −0.477618 + 0.878568i
\(15\) −6.07784 + 2.14810i −1.56929 + 0.554637i
\(16\) −3.77067 − 1.33492i −0.942669 − 0.333730i
\(17\) 0.186192 + 0.107498i 0.0451582 + 0.0260721i 0.522409 − 0.852695i \(-0.325033\pi\)
−0.477251 + 0.878767i \(0.658367\pi\)
\(18\) 4.01014 − 1.38520i 0.945199 − 0.326495i
\(19\) −1.14103 − 1.97631i −0.261769 − 0.453398i 0.704943 − 0.709264i \(-0.250973\pi\)
−0.966712 + 0.255866i \(0.917639\pi\)
\(20\) 6.98334 − 2.57661i 1.56152 − 0.576148i
\(21\) 0.859391 + 4.50127i 0.187535 + 0.982258i
\(22\) 3.64848 − 0.651612i 0.777858 − 0.138924i
\(23\) −2.33586 − 4.04583i −0.487061 − 0.843615i 0.512828 − 0.858491i \(-0.328598\pi\)
−0.999889 + 0.0148767i \(0.995264\pi\)
\(24\) −4.60809 + 1.66297i −0.940623 + 0.339452i
\(25\) −4.42574 + 7.66560i −0.885148 + 1.53312i
\(26\) −3.25544 − 3.86238i −0.638444 − 0.757474i
\(27\) 2.72309 − 4.42547i 0.524060 − 0.851682i
\(28\) −1.06366 − 5.18349i −0.201014 − 0.979588i
\(29\) 2.57415 0.478008 0.239004 − 0.971019i \(-0.423179\pi\)
0.239004 + 0.971019i \(0.423179\pi\)
\(30\) 4.61442 − 7.86231i 0.842474 − 1.43546i
\(31\) 4.26196 + 2.46064i 0.765471 + 0.441945i 0.831257 − 0.555889i \(-0.187622\pi\)
−0.0657857 + 0.997834i \(0.520955\pi\)
\(32\) 5.29408 − 1.99316i 0.935870 − 0.352345i
\(33\) 2.95006 − 3.44980i 0.513539 − 0.600534i
\(34\) −0.299314 + 0.0534570i −0.0513319 + 0.00916779i
\(35\) 7.68949 + 6.15077i 1.29976 + 1.03967i
\(36\) −3.07384 + 5.15281i −0.512307 + 0.858802i
\(37\) 7.31104 − 4.22103i 1.20193 − 0.693933i 0.240944 − 0.970539i \(-0.422543\pi\)
0.960983 + 0.276606i \(0.0892098\pi\)
\(38\) 3.03510 + 1.09712i 0.492358 + 0.177977i
\(39\) −6.08229 − 1.13113i −0.973946 − 0.181125i
\(40\) −5.20302 + 9.15098i −0.822670 + 1.44690i
\(41\) 0.909442i 0.142031i 0.997475 + 0.0710155i \(0.0226240\pi\)
−0.997475 + 0.0710155i \(0.977376\pi\)
\(42\) −5.03186 − 4.08416i −0.776433 − 0.630200i
\(43\) 3.73541 0.569644 0.284822 − 0.958580i \(-0.408065\pi\)
0.284822 + 0.958580i \(0.408065\pi\)
\(44\) −3.35137 + 4.02992i −0.505239 + 0.607533i
\(45\) −1.73317 − 11.0299i −0.258365 − 1.64425i
\(46\) 6.21334 + 2.24599i 0.916107 + 0.331153i
\(47\) −0.586728 − 1.01624i −0.0855831 − 0.148234i 0.820056 − 0.572283i \(-0.193942\pi\)
−0.905640 + 0.424048i \(0.860609\pi\)
\(48\) 3.46728 − 5.99816i 0.500459 − 0.865761i
\(49\) 5.14541 − 4.74603i 0.735058 − 0.678004i
\(50\) −2.20085 − 12.3229i −0.311247 − 1.74272i
\(51\) −0.242017 + 0.283015i −0.0338891 + 0.0396301i
\(52\) 7.04051 + 1.20949i 0.976343 + 0.167726i
\(53\) −1.06171 + 1.83894i −0.145837 + 0.252598i −0.929685 − 0.368355i \(-0.879921\pi\)
0.783848 + 0.620953i \(0.213254\pi\)
\(54\) 1.08886 + 7.26735i 0.148176 + 0.988961i
\(55\) − 9.75355i − 1.31517i
\(56\) 5.87454 + 4.63570i 0.785019 + 0.619472i
\(57\) 3.72672 − 1.31714i 0.493616 − 0.174459i
\(58\) −2.78355 + 2.34614i −0.365498 + 0.308063i
\(59\) −6.79199 − 3.92136i −0.884242 − 0.510517i −0.0121872 − 0.999926i \(-0.503879\pi\)
−0.872055 + 0.489408i \(0.837213\pi\)
\(60\) 2.17610 + 12.7076i 0.280934 + 1.64054i
\(61\) −0.301301 + 0.173956i −0.0385776 + 0.0222728i −0.519165 − 0.854674i \(-0.673757\pi\)
0.480587 + 0.876947i \(0.340424\pi\)
\(62\) −6.85134 + 1.22364i −0.870121 + 0.155402i
\(63\) −7.93716 + 0.0379510i −0.999989 + 0.00478137i
\(64\) −3.90812 + 6.98044i −0.488515 + 0.872555i
\(65\) −11.5125 + 6.64673i −1.42795 + 0.824425i
\(66\) −0.0458036 + 6.41918i −0.00563803 + 0.790146i
\(67\) −4.98736 + 8.63836i −0.609303 + 1.05534i 0.382053 + 0.924140i \(0.375217\pi\)
−0.991356 + 0.131203i \(0.958116\pi\)
\(68\) 0.274940 − 0.330607i 0.0333414 − 0.0400920i
\(69\) 7.62919 − 2.69640i 0.918446 − 0.324608i
\(70\) −13.9210 + 0.357259i −1.66387 + 0.0427006i
\(71\) 10.0808 1.19637 0.598183 − 0.801359i \(-0.295889\pi\)
0.598183 + 0.801359i \(0.295889\pi\)
\(72\) −1.37250 − 8.37354i −0.161750 − 0.986832i
\(73\) 4.45173 − 7.71062i 0.521036 − 0.902460i −0.478665 − 0.877998i \(-0.658879\pi\)
0.999701 − 0.0244626i \(-0.00778746\pi\)
\(74\) −4.05862 + 11.2278i −0.471805 + 1.30521i
\(75\) −11.6519 − 9.96394i −1.34544 − 1.15054i
\(76\) −4.28194 + 1.57989i −0.491172 + 0.181226i
\(77\) −6.85470 − 1.04354i −0.781166 − 0.118922i
\(78\) 7.60800 − 4.32040i 0.861436 − 0.489188i
\(79\) −9.70950 + 5.60578i −1.09240 + 0.630700i −0.934216 − 0.356709i \(-0.883899\pi\)
−0.158189 + 0.987409i \(0.550566\pi\)
\(80\) −2.71413 − 14.6375i −0.303449 − 1.63652i
\(81\) 6.67358 + 6.03849i 0.741508 + 0.670944i
\(82\) −0.828886 − 0.983422i −0.0915351 − 0.108601i
\(83\) − 1.73937i − 0.190921i −0.995433 − 0.0954603i \(-0.969568\pi\)
0.995433 − 0.0954603i \(-0.0304323\pi\)
\(84\) 9.16358 − 0.169751i 0.999828 − 0.0185214i
\(85\) 0.800163i 0.0867898i
\(86\) −4.03927 + 3.40453i −0.435566 + 0.367120i
\(87\) −0.815185 + 4.38341i −0.0873970 + 0.469950i
\(88\) −0.0489643 − 7.41226i −0.00521961 − 0.790149i
\(89\) 15.4694 − 8.93127i 1.63975 − 0.946712i 0.658837 − 0.752286i \(-0.271049\pi\)
0.980917 − 0.194426i \(-0.0622845\pi\)
\(90\) 11.9271 + 10.3475i 1.25722 + 1.09073i
\(91\) 3.43953 + 8.80199i 0.360560 + 0.922699i
\(92\) −8.76581 + 3.23428i −0.913899 + 0.337197i
\(93\) −5.53980 + 6.47826i −0.574451 + 0.671764i
\(94\) 1.56068 + 0.564153i 0.160972 + 0.0581880i
\(95\) 4.24662 − 7.35536i 0.435694 − 0.754644i
\(96\) 1.71753 + 9.64625i 0.175295 + 0.984516i
\(97\) −8.88553 −0.902189 −0.451094 − 0.892476i \(-0.648966\pi\)
−0.451094 + 0.892476i \(0.648966\pi\)
\(98\) −1.23834 + 9.82174i −0.125091 + 0.992145i
\(99\) 4.94029 + 6.11600i 0.496518 + 0.614681i
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.11.6 yes 56
3.2 odd 2 inner 168.2.v.a.11.23 yes 56
4.3 odd 2 672.2.bd.a.431.16 56
7.2 even 3 inner 168.2.v.a.107.26 yes 56
8.3 odd 2 inner 168.2.v.a.11.3 ✓ 56
8.5 even 2 672.2.bd.a.431.15 56
12.11 even 2 672.2.bd.a.431.21 56
21.2 odd 6 inner 168.2.v.a.107.3 yes 56
24.5 odd 2 672.2.bd.a.431.22 56
24.11 even 2 inner 168.2.v.a.11.26 yes 56
28.23 odd 6 672.2.bd.a.527.22 56
56.37 even 6 672.2.bd.a.527.21 56
56.51 odd 6 inner 168.2.v.a.107.23 yes 56
84.23 even 6 672.2.bd.a.527.15 56
168.107 even 6 inner 168.2.v.a.107.6 yes 56
168.149 odd 6 672.2.bd.a.527.16 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.3 ✓ 56 8.3 odd 2 inner
168.2.v.a.11.6 yes 56 1.1 even 1 trivial
168.2.v.a.11.23 yes 56 3.2 odd 2 inner
168.2.v.a.11.26 yes 56 24.11 even 2 inner
168.2.v.a.107.3 yes 56 21.2 odd 6 inner
168.2.v.a.107.6 yes 56 168.107 even 6 inner
168.2.v.a.107.23 yes 56 56.51 odd 6 inner
168.2.v.a.107.26 yes 56 7.2 even 3 inner
672.2.bd.a.431.15 56 8.5 even 2
672.2.bd.a.431.16 56 4.3 odd 2
672.2.bd.a.431.21 56 12.11 even 2
672.2.bd.a.431.22 56 24.5 odd 2
672.2.bd.a.527.15 56 84.23 even 6
672.2.bd.a.527.16 56 168.149 odd 6
672.2.bd.a.527.21 56 56.37 even 6
672.2.bd.a.527.22 56 28.23 odd 6