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Properties

Label 168.2.v.a.107.23
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 107.23
Character \(\chi\) \(=\) 168.107
Dual form 168.2.v.a.11.23

$q$-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} +(-1.31637 - 1.12568i) q^{3} +(0.338619 + 1.97113i) q^{4} +(-1.86088 + 3.22314i) q^{5} +(-0.397486 - 2.41702i) q^{6} +(2.46429 + 0.962962i) q^{7} +(-1.43036 + 2.44009i) q^{8} +(0.465685 + 2.96364i) q^{9} +(-4.94989 + 1.78928i) q^{10} +(2.26958 - 1.31034i) q^{11} +(1.77311 - 2.97592i) 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} +(-2.19756 + 3.62915i) q^{18} +(-1.14103 + 1.97631i) q^{19} +(-6.98334 - 2.57661i) q^{20} +(-2.15994 - 4.04162i) q^{21} +(3.64848 + 0.651612i) q^{22} +(2.33586 - 4.04583i) q^{23} +(4.62966 - 1.60195i) 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} +(8.53007 + 3.21664i) q^{30} +(4.26196 - 2.46064i) q^{31} +(-5.29408 - 1.99316i) q^{32} +(-4.46265 - 0.829921i) q^{33} +(-0.299314 - 0.0534570i) q^{34} +(-7.68949 + 6.15077i) q^{35} +(-5.68401 + 1.92147i) q^{36} +(7.31104 + 4.22103i) q^{37} +(-3.03510 + 1.09712i) q^{38} +(-4.02073 + 4.70186i) q^{39} +(-5.20302 - 9.15098i) q^{40} +0.909442i q^{41} +(1.34798 - 6.33900i) q^{42} +3.73541 q^{43} +(3.35137 + 4.02992i) q^{44} +(-10.4188 - 4.01400i) q^{45} +(6.21334 - 2.24599i) q^{46} +(0.586728 - 1.01624i) q^{47} +(6.46632 + 2.48732i) q^{48} +(5.14541 + 4.74603i) q^{49} +(2.20085 - 12.3229i) q^{50} +(0.366107 + 0.0680851i) q^{51} +(7.04051 - 1.20949i) q^{52} +(1.06171 + 1.83894i) q^{53} +(6.97808 - 2.30358i) 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} +(6.29225 + 11.2528i) q^{60} +(-0.301301 - 0.173956i) q^{61} +(6.85134 + 1.22364i) q^{62} +(-1.70629 + 7.75168i) q^{63} +(-3.90812 - 6.98044i) q^{64} +(11.5125 + 6.64673i) q^{65} +(-4.06926 - 4.96479i) 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} +(-7.89765 - 3.10276i) q^{72} +(4.45173 + 7.71062i) q^{73} +(4.05862 + 11.2278i) q^{74} +(-2.80309 + 15.0728i) q^{75} +(-4.28194 - 1.57989i) q^{76} +(6.85470 - 1.04354i) q^{77} +(-8.63318 + 1.41975i) q^{78} +(-9.70950 - 5.60578i) q^{79} +(2.71413 - 14.6375i) q^{80} +(-8.56628 + 2.76024i) q^{81} +(-0.828886 + 0.983422i) q^{82} -1.73937i q^{83} +(7.23514 - 5.62607i) q^{84} -0.800163i q^{85} +(4.03927 + 3.40453i) q^{86} +(3.38855 + 2.89767i) q^{87} +(-0.0489643 + 7.41226i) q^{88} +(-15.4694 - 8.93127i) q^{89} +(-7.60786 - 13.8364i) q^{90} +(3.43953 - 8.80199i) q^{91} +(8.76581 + 3.23428i) q^{92} +(-8.38024 - 1.55848i) q^{93} +(1.56068 - 0.564153i) q^{94} +(-4.24662 - 7.35536i) q^{95} +(4.72533 + 8.58320i) 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 \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.08135 + 0.911422i 0.764627 + 0.644473i
\(3\) −1.31637 − 1.12568i −0.760009 − 0.649912i
\(4\) 0.338619 + 1.97113i 0.169310 + 0.985563i
\(5\) −1.86088 + 3.22314i −0.832210 + 1.44143i 0.0640716 + 0.997945i \(0.479591\pi\)
−0.896282 + 0.443485i \(0.853742\pi\)
\(6\) −0.397486 − 2.41702i −0.162273 − 0.986746i
\(7\) 2.46429 + 0.962962i 0.931412 + 0.363965i
\(8\) −1.43036 + 2.44009i −0.505710 + 0.862704i
\(9\) 0.465685 + 2.96364i 0.155228 + 0.987879i
\(10\) −4.94989 + 1.78928i −1.56529 + 0.565820i
\(11\) 2.26958 − 1.31034i 0.684304 − 0.395083i −0.117171 − 0.993112i \(-0.537382\pi\)
0.801475 + 0.598029i \(0.204049\pi\)
\(12\) 1.77311 − 2.97592i 0.511853 − 0.859073i
\(13\) − 3.57182i − 0.990645i −0.868709 − 0.495323i \(-0.835050\pi\)
0.868709 − 0.495323i \(-0.164950\pi\)
\(14\) 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 \(-0.674967\pi\)
0.477251 + 0.878767i \(0.341633\pi\)
\(18\) −2.19756 + 3.62915i −0.517969 + 0.855399i
\(19\) −1.14103 + 1.97631i −0.261769 + 0.453398i −0.966712 − 0.255866i \(-0.917639\pi\)
0.704943 + 0.709264i \(0.250973\pi\)
\(20\) −6.98334 − 2.57661i −1.56152 − 0.576148i
\(21\) −2.15994 − 4.04162i −0.471337 − 0.881953i
\(22\) 3.64848 + 0.651612i 0.777858 + 0.138924i
\(23\) 2.33586 − 4.04583i 0.487061 − 0.843615i −0.512828 − 0.858491i \(-0.671402\pi\)
0.999889 + 0.0148767i \(0.00473557\pi\)
\(24\) 4.62966 − 1.60195i 0.945026 − 0.326996i
\(25\) −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 \(-0.576821\pi\)
−0.239004 + 0.971019i \(0.576821\pi\)
\(30\) 8.53007 + 3.21664i 1.55737 + 0.587275i
\(31\) 4.26196 − 2.46064i 0.765471 − 0.441945i −0.0657857 − 0.997834i \(-0.520955\pi\)
0.831257 + 0.555889i \(0.187622\pi\)
\(32\) −5.29408 − 1.99316i −0.935870 − 0.352345i
\(33\) −4.46265 − 0.829921i −0.776847 − 0.144471i
\(34\) −0.299314 − 0.0534570i −0.0513319 − 0.00916779i
\(35\) −7.68949 + 6.15077i −1.29976 + 1.03967i
\(36\) −5.68401 + 1.92147i −0.947335 + 0.320244i
\(37\) 7.31104 + 4.22103i 1.20193 + 0.693933i 0.960983 − 0.276606i \(-0.0892098\pi\)
0.240944 + 0.970539i \(0.422543\pi\)
\(38\) −3.03510 + 1.09712i −0.492358 + 0.177977i
\(39\) −4.02073 + 4.70186i −0.643832 + 0.752899i
\(40\) −5.20302 − 9.15098i −0.822670 − 1.44690i
\(41\) 0.909442i 0.142031i 0.997475 + 0.0710155i \(0.0226240\pi\)
−0.997475 + 0.0710155i \(0.977376\pi\)
\(42\) 1.34798 − 6.33900i 0.207998 − 0.978129i
\(43\) 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
\(45\) −10.4188 − 4.01400i −1.55314 − 0.598372i
\(46\) 6.21334 − 2.24599i 0.916107 − 0.331153i
\(47\) 0.586728 − 1.01624i 0.0855831 − 0.148234i −0.820056 − 0.572283i \(-0.806058\pi\)
0.905640 + 0.424048i \(0.139391\pi\)
\(48\) 6.46632 + 2.48732i 0.933332 + 0.359014i
\(49\) 5.14541 + 4.74603i 0.735058 + 0.678004i
\(50\) 2.20085 − 12.3229i 0.311247 − 1.74272i
\(51\) 0.366107 + 0.0680851i 0.0512652 + 0.00953383i
\(52\) 7.04051 − 1.20949i 0.976343 − 0.167726i
\(53\) 1.06171 + 1.83894i 0.145837 + 0.252598i 0.929685 − 0.368355i \(-0.120079\pi\)
−0.783848 + 0.620953i \(0.786746\pi\)
\(54\) 6.97808 − 2.30358i 0.949596 − 0.313477i
\(55\) 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 \(-0.496121\pi\)
0.872055 + 0.489408i \(0.162787\pi\)
\(60\) 6.29225 + 11.2528i 0.812325 + 1.45273i
\(61\) −0.301301 − 0.173956i −0.0385776 − 0.0222728i 0.480587 − 0.876947i \(-0.340424\pi\)
−0.519165 + 0.854674i \(0.673757\pi\)
\(62\) 6.85134 + 1.22364i 0.870121 + 0.155402i
\(63\) −1.70629 + 7.75168i −0.214972 + 0.976620i
\(64\) −3.90812 − 6.98044i −0.488515 − 0.872555i
\(65\) 11.5125 + 6.64673i 1.42795 + 0.824425i
\(66\) −4.06926 − 4.96479i −0.500891 − 0.611123i
\(67\) −4.98736 − 8.63836i −0.609303 − 1.05534i −0.991356 − 0.131203i \(-0.958116\pi\)
0.382053 − 0.924140i \(-0.375217\pi\)
\(68\) −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 \(-0.704111\pi\)
−0.598183 + 0.801359i \(0.704111\pi\)
\(72\) −7.89765 − 3.10276i −0.930747 − 0.365664i
\(73\) 4.45173 + 7.71062i 0.521036 + 0.902460i 0.999701 + 0.0244626i \(0.00778746\pi\)
−0.478665 + 0.877998i \(0.658879\pi\)
\(74\) 4.05862 + 11.2278i 0.471805 + 1.30521i
\(75\) −2.80309 + 15.0728i −0.323673 + 1.74045i
\(76\) −4.28194 − 1.57989i −0.491172 − 0.181226i
\(77\) 6.85470 − 1.04354i 0.781166 − 0.118922i
\(78\) −8.63318 + 1.41975i −0.977515 + 0.160755i
\(79\) −9.70950 − 5.60578i −1.09240 − 0.630700i −0.158189 − 0.987409i \(-0.550566\pi\)
−0.934216 + 0.356709i \(0.883899\pi\)
\(80\) 2.71413 − 14.6375i 0.303449 − 1.63652i
\(81\) −8.56628 + 2.76024i −0.951808 + 0.306693i
\(82\) −0.828886 + 0.983422i −0.0915351 + 0.108601i
\(83\) − 1.73937i − 0.190921i −0.995433 − 0.0954603i \(-0.969568\pi\)
0.995433 − 0.0954603i \(-0.0304323\pi\)
\(84\) 7.23514 − 5.62607i 0.789419 − 0.613855i
\(85\) − 0.800163i − 0.0867898i
\(86\) 4.03927 + 3.40453i 0.435566 + 0.367120i
\(87\) 3.38855 + 2.89767i 0.363290 + 0.310663i
\(88\) −0.0489643 + 7.41226i −0.00521961 + 0.790149i
\(89\) −15.4694 − 8.93127i −1.63975 − 0.946712i −0.980917 − 0.194426i \(-0.937716\pi\)
−0.658837 − 0.752286i \(-0.728951\pi\)
\(90\) −7.60786 − 13.8364i −0.801939 − 1.45849i
\(91\) 3.43953 − 8.80199i 0.360560 − 0.922699i
\(92\) 8.76581 + 3.23428i 0.913899 + 0.337197i
\(93\) −8.38024 − 1.55848i −0.868990 − 0.161607i
\(94\) 1.56068 − 0.564153i 0.160972 − 0.0581880i
\(95\) −4.24662 − 7.35536i −0.435694 − 0.754644i
\(96\) 4.72533 + 8.58320i 0.482277 + 0.876019i
\(97\) −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.107.23 yes 56
3.2 odd 2 inner 168.2.v.a.107.6 yes 56
4.3 odd 2 672.2.bd.a.527.21 56
7.4 even 3 inner 168.2.v.a.11.3 ✓ 56
8.3 odd 2 inner 168.2.v.a.107.26 yes 56
8.5 even 2 672.2.bd.a.527.22 56
12.11 even 2 672.2.bd.a.527.16 56
21.11 odd 6 inner 168.2.v.a.11.26 yes 56
24.5 odd 2 672.2.bd.a.527.15 56
24.11 even 2 inner 168.2.v.a.107.3 yes 56
28.11 odd 6 672.2.bd.a.431.15 56
56.11 odd 6 inner 168.2.v.a.11.6 yes 56
56.53 even 6 672.2.bd.a.431.16 56
84.11 even 6 672.2.bd.a.431.22 56
168.11 even 6 inner 168.2.v.a.11.23 yes 56
168.53 odd 6 672.2.bd.a.431.21 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.3 ✓ 56 7.4 even 3 inner
168.2.v.a.11.6 yes 56 56.11 odd 6 inner
168.2.v.a.11.23 yes 56 168.11 even 6 inner
168.2.v.a.11.26 yes 56 21.11 odd 6 inner
168.2.v.a.107.3 yes 56 24.11 even 2 inner
168.2.v.a.107.6 yes 56 3.2 odd 2 inner
168.2.v.a.107.23 yes 56 1.1 even 1 trivial
168.2.v.a.107.26 yes 56 8.3 odd 2 inner
672.2.bd.a.431.15 56 28.11 odd 6
672.2.bd.a.431.16 56 56.53 even 6
672.2.bd.a.431.21 56 168.53 odd 6
672.2.bd.a.431.22 56 84.11 even 6
672.2.bd.a.527.15 56 24.5 odd 2
672.2.bd.a.527.16 56 12.11 even 2
672.2.bd.a.527.21 56 4.3 odd 2
672.2.bd.a.527.22 56 8.5 even 2