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Properties

Label 168.2.v.a.11.26
Level $168$
Weight $2$
Character 168.11
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 11.26
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.26

$q$-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+(1.32999 - 0.480762i) q^{2} +(-1.31637 + 1.12568i) q^{3} +(1.53774 - 1.27882i) 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} +(0.465685 - 2.96364i) q^{9} +(4.02451 + 3.39209i) q^{10} +(2.26958 + 1.31034i) q^{11} +(-0.584698 + 3.41440i) q^{12} -3.57182i q^{13} +(-2.81451 + 2.46546i) q^{14} +(-6.07784 - 2.14810i) q^{15} +(0.729261 - 3.93296i) q^{16} +(-0.186192 - 0.107498i) q^{17} +(-0.805449 - 4.16548i) 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} +(0.863874 + 4.82221i) q^{24} +(-4.42574 + 7.66560i) q^{25} +(-1.71720 - 4.75048i) q^{26} +(2.72309 + 4.42547i) q^{27} +(-2.55797 + 4.63215i) q^{28} +2.57415 q^{29} +(-9.11617 + 0.0650478i) q^{30} +(-4.26196 - 2.46064i) q^{31} +(-0.920911 - 5.58139i) q^{32} +(-4.46265 + 0.829921i) 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} +(-2.46769 - 2.07991i) q^{38} +(4.02073 + 4.70186i) q^{39} +(10.5265 + 0.0695365i) q^{40} -0.909442i q^{41} +(0.929631 - 6.41372i) q^{42} +3.73541 q^{43} +(5.16570 - 0.887414i) q^{44} +(10.4188 - 4.01400i) q^{45} +(-5.05175 - 4.25791i) 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} +(0.366107 - 0.0680851i) q^{51} +(-4.56770 - 5.49252i) q^{52} +(-1.06171 + 1.83894i) q^{53} +(5.74928 + 4.57666i) q^{54} +9.75355i q^{55} +(-1.17511 + 7.39048i) q^{56} +(3.72672 + 1.31714i) q^{57} +(3.42359 - 1.23755i) q^{58} +(6.79199 + 3.92136i) q^{59} +(-12.0931 + 4.46922i) 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} +(-5.53627 + 3.24926i) q^{66} +(-4.98736 + 8.63836i) q^{67} +(-0.423784 + 0.0728018i) q^{68} +(7.62919 + 2.69640i) q^{69} +(-13.1840 - 4.48364i) q^{70} +10.0808 q^{71} +(-6.56545 - 5.37539i) q^{72} +(4.45173 - 7.71062i) q^{73} +(-7.69428 + 9.12879i) q^{74} +(-2.80309 - 15.0728i) 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} +(14.0335 - 4.96826i) q^{80} +(-8.56628 - 2.76024i) q^{81} +(-0.437225 - 1.20955i) q^{82} +1.73937i q^{83} +(-1.84707 - 8.97710i) q^{84} -0.800163i q^{85} +(4.96805 - 1.79584i) q^{86} +(-3.38855 + 2.89767i) q^{87} +(6.44368 - 3.66372i) 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} +(8.38024 - 1.55848i) q^{93} +(-1.26891 - 1.06951i) q^{94} +(4.24662 - 7.35536i) q^{95} +(7.49513 + 6.31055i) q^{96} -8.88553 q^{97} +(4.56162 - 8.78588i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32999 − 0.480762i 0.940443 − 0.339950i
\(3\) −1.31637 + 1.12568i −0.760009 + 0.649912i
\(4\) 1.53774 − 1.27882i 0.768868 − 0.639408i
\(5\) 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
\(9\) 0.465685 − 2.96364i 0.155228 − 0.987879i
\(10\) 4.02451 + 3.39209i 1.27266 + 1.07267i
\(11\) 2.26958 + 1.31034i 0.684304 + 0.395083i 0.801475 − 0.598029i \(-0.204049\pi\)
−0.117171 + 0.993112i \(0.537382\pi\)
\(12\) −0.584698 + 3.41440i −0.168788 + 0.985652i
\(13\) − 3.57182i − 0.990645i −0.868709 − 0.495323i \(-0.835050\pi\)
0.868709 − 0.495323i \(-0.164950\pi\)
\(14\) −2.81451 + 2.46546i −0.752211 + 0.658923i
\(15\) −6.07784 − 2.14810i −1.56929 − 0.554637i
\(16\) 0.729261 − 3.93296i 0.182315 − 0.983240i
\(17\) −0.186192 − 0.107498i −0.0451582 − 0.0260721i 0.477251 − 0.878767i \(-0.341633\pi\)
−0.522409 + 0.852695i \(0.674967\pi\)
\(18\) −0.805449 − 4.16548i −0.189846 − 0.981814i
\(19\) −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
\(21\) 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 \(-0.328598\pi\)
−0.999889 + 0.0148767i \(0.995264\pi\)
\(24\) 0.863874 + 4.82221i 0.176337 + 0.984330i
\(25\) −4.42574 + 7.66560i −0.885148 + 1.53312i
\(26\) −1.71720 − 4.75048i −0.336770 − 0.931646i
\(27\) 2.72309 + 4.42547i 0.524060 + 0.851682i
\(28\) −2.55797 + 4.63215i −0.483411 + 0.875394i
\(29\) 2.57415 0.478008 0.239004 − 0.971019i \(-0.423179\pi\)
0.239004 + 0.971019i \(0.423179\pi\)
\(30\) −9.11617 + 0.0650478i −1.66438 + 0.0118760i
\(31\) −4.26196 − 2.46064i −0.765471 − 0.441945i 0.0657857 − 0.997834i \(-0.479045\pi\)
−0.831257 + 0.555889i \(0.812378\pi\)
\(32\) −0.920911 − 5.58139i −0.162796 − 0.986660i
\(33\) −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
\(36\) −3.07384 − 5.15281i −0.512307 − 0.858802i
\(37\) −7.31104 + 4.22103i −1.20193 + 0.693933i −0.960983 − 0.276606i \(-0.910790\pi\)
−0.240944 + 0.970539i \(0.577457\pi\)
\(38\) −2.46769 − 2.07991i −0.400312 − 0.337406i
\(39\) 4.02073 + 4.70186i 0.643832 + 0.752899i
\(40\) 10.5265 + 0.0695365i 1.66438 + 0.0109947i
\(41\) − 0.909442i − 0.142031i −0.997475 − 0.0710155i \(-0.977376\pi\)
0.997475 − 0.0710155i \(-0.0226240\pi\)
\(42\) 0.929631 − 6.41372i 0.143445 − 0.989658i
\(43\) 3.73541 0.569644 0.284822 − 0.958580i \(-0.408065\pi\)
0.284822 + 0.958580i \(0.408065\pi\)
\(44\) 5.16570 − 0.887414i 0.778759 − 0.133783i
\(45\) 10.4188 − 4.01400i 1.55314 − 0.598372i
\(46\) −5.05175 − 4.25791i −0.744840 − 0.627795i
\(47\) −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
\(51\) 0.366107 − 0.0680851i 0.0512652 − 0.00953383i
\(52\) −4.56770 − 5.49252i −0.633426 − 0.761675i
\(53\) −1.06171 + 1.83894i −0.145837 + 0.252598i −0.929685 − 0.368355i \(-0.879921\pi\)
0.783848 + 0.620953i \(0.213254\pi\)
\(54\) 5.74928 + 4.57666i 0.782378 + 0.622804i
\(55\) 9.75355i 1.31517i
\(56\) −1.17511 + 7.39048i −0.157030 + 0.987594i
\(57\) 3.72672 + 1.31714i 0.493616 + 0.174459i
\(58\) 3.42359 − 1.23755i 0.449539 − 0.162499i
\(59\) 6.79199 + 3.92136i 0.884242 + 0.510517i 0.872055 − 0.489408i \(-0.162787\pi\)
0.0121872 + 0.999926i \(0.496121\pi\)
\(60\) −12.0931 + 4.46922i −1.56122 + 0.576974i
\(61\) 0.301301 − 0.173956i 0.0385776 − 0.0222728i −0.480587 − 0.876947i \(-0.659576\pi\)
0.519165 + 0.854674i \(0.326243\pi\)
\(62\) −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
\(66\) −5.53627 + 3.24926i −0.681468 + 0.399956i
\(67\) −4.98736 + 8.63836i −0.609303 + 1.05534i 0.382053 + 0.924140i \(0.375217\pi\)
−0.991356 + 0.131203i \(0.958116\pi\)
\(68\) −0.423784 + 0.0728018i −0.0513914 + 0.00882851i
\(69\) 7.62919 + 2.69640i 0.918446 + 0.324608i
\(70\) −13.1840 − 4.48364i −1.57579 − 0.535897i
\(71\) 10.0808 1.19637 0.598183 − 0.801359i \(-0.295889\pi\)
0.598183 + 0.801359i \(0.295889\pi\)
\(72\) −6.56545 − 5.37539i −0.773746 − 0.633496i
\(73\) 4.45173 − 7.71062i 0.521036 − 0.902460i −0.478665 − 0.877998i \(-0.658879\pi\)
0.999701 − 0.0244626i \(-0.00778746\pi\)
\(74\) −7.69428 + 9.12879i −0.894442 + 1.06120i
\(75\) −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
\(78\) 7.60800 + 4.32040i 0.861436 + 0.489188i
\(79\) 9.70950 − 5.60578i 1.09240 − 0.630700i 0.158189 − 0.987409i \(-0.449434\pi\)
0.934216 + 0.356709i \(0.116101\pi\)
\(80\) 14.0335 − 4.96826i 1.56900 − 0.555468i
\(81\) −8.56628 − 2.76024i −0.951808 − 0.306693i
\(82\) −0.437225 − 1.20955i −0.0482835 − 0.133572i
\(83\) 1.73937i 0.190921i 0.995433 + 0.0954603i \(0.0304323\pi\)
−0.995433 + 0.0954603i \(0.969568\pi\)
\(84\) −1.84707 − 8.97710i −0.201532 − 0.979482i
\(85\) − 0.800163i − 0.0867898i
\(86\) 4.96805 − 1.79584i 0.535718 − 0.193651i
\(87\) −3.38855 + 2.89767i −0.363290 + 0.310663i
\(88\) 6.44368 − 3.66372i 0.686899 − 0.390554i
\(89\) −15.4694 + 8.93127i −1.63975 + 0.946712i −0.658837 + 0.752286i \(0.728951\pi\)
−0.980917 + 0.194426i \(0.937716\pi\)
\(90\) 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
\(93\) 8.38024 − 1.55848i 0.868990 − 0.161607i
\(94\) −1.26891 − 1.06951i −0.130878 − 0.110312i
\(95\) 4.24662 − 7.35536i 0.435694 − 0.754644i
\(96\) 7.49513 + 6.31055i 0.764968 + 0.644068i
\(97\) −8.88553 −0.902189 −0.451094 − 0.892476i \(-0.648966\pi\)
−0.451094 + 0.892476i \(0.648966\pi\)
\(98\) 4.56162 − 8.78588i 0.460793 − 0.887507i
\(99\) 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.11.26 yes 56
3.2 odd 2 inner 168.2.v.a.11.3 ✓ 56
4.3 odd 2 672.2.bd.a.431.22 56
7.2 even 3 inner 168.2.v.a.107.6 yes 56
8.3 odd 2 inner 168.2.v.a.11.23 yes 56
8.5 even 2 672.2.bd.a.431.21 56
12.11 even 2 672.2.bd.a.431.15 56
21.2 odd 6 inner 168.2.v.a.107.23 yes 56
24.5 odd 2 672.2.bd.a.431.16 56
24.11 even 2 inner 168.2.v.a.11.6 yes 56
28.23 odd 6 672.2.bd.a.527.16 56
56.37 even 6 672.2.bd.a.527.15 56
56.51 odd 6 inner 168.2.v.a.107.3 yes 56
84.23 even 6 672.2.bd.a.527.21 56
168.107 even 6 inner 168.2.v.a.107.26 yes 56
168.149 odd 6 672.2.bd.a.527.22 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.3 ✓ 56 3.2 odd 2 inner
168.2.v.a.11.6 yes 56 24.11 even 2 inner
168.2.v.a.11.23 yes 56 8.3 odd 2 inner
168.2.v.a.11.26 yes 56 1.1 even 1 trivial
168.2.v.a.107.3 yes 56 56.51 odd 6 inner
168.2.v.a.107.6 yes 56 7.2 even 3 inner
168.2.v.a.107.23 yes 56 21.2 odd 6 inner
168.2.v.a.107.26 yes 56 168.107 even 6 inner
672.2.bd.a.431.15 56 12.11 even 2
672.2.bd.a.431.16 56 24.5 odd 2
672.2.bd.a.431.21 56 8.5 even 2
672.2.bd.a.431.22 56 4.3 odd 2
672.2.bd.a.527.15 56 56.37 even 6
672.2.bd.a.527.16 56 28.23 odd 6
672.2.bd.a.527.21 56 84.23 even 6
672.2.bd.a.527.22 56 168.149 odd 6