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Properties

Label 168.2.v.a.107.26
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 107.26
Character \(\chi\) \(=\) 168.107
Dual form 168.2.v.a.11.26

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32999 + 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 −0.0640716 − 0.997945i \(-0.520409\pi\)
0.896282 − 0.443485i \(-0.146258\pi\)
\(6\) −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 −0.117171 − 0.993112i \(-0.537382\pi\)
0.801475 + 0.598029i \(0.204049\pi\)
\(12\) −0.584698 − 3.41440i −0.168788 − 0.985652i
\(13\) 3.57182i 0.990645i 0.868709 + 0.495323i \(0.164950\pi\)
−0.868709 + 0.495323i \(0.835050\pi\)
\(14\) −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 −0.522409 − 0.852695i \(-0.674967\pi\)
0.477251 + 0.878767i \(0.341633\pi\)
\(18\) −0.805449 + 4.16548i −0.189846 + 0.981814i
\(19\) −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 −0.999889 − 0.0148767i \(-0.995264\pi\)
0.512828 + 0.858491i \(0.328598\pi\)
\(24\) 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 −0.831257 − 0.555889i \(-0.812378\pi\)
0.0657857 + 0.997834i \(0.479045\pi\)
\(32\) −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 −0.240944 − 0.970539i \(-0.577457\pi\)
−0.960983 + 0.276606i \(0.910790\pi\)
\(38\) −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 \(0.0226240\pi\)
−0.997475 + 0.0710155i \(0.977376\pi\)
\(42\) 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 −0.905640 − 0.424048i \(-0.860609\pi\)
0.820056 + 0.572283i \(0.193942\pi\)
\(48\) 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 0.783848 − 0.620953i \(-0.213254\pi\)
−0.929685 + 0.368355i \(0.879921\pi\)
\(54\) 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 0.0121872 − 0.999926i \(-0.496121\pi\)
0.872055 + 0.489408i \(0.162787\pi\)
\(60\) −12.0931 − 4.46922i −1.56122 − 0.576974i
\(61\) 0.301301 + 0.173956i 0.0385776 + 0.0222728i 0.519165 − 0.854674i \(-0.326243\pi\)
−0.480587 + 0.876947i \(0.659576\pi\)
\(62\) −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 −0.991356 − 0.131203i \(-0.958116\pi\)
0.382053 − 0.924140i \(-0.375217\pi\)
\(68\) −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 0.999701 + 0.0244626i \(0.00778746\pi\)
−0.478665 + 0.877998i \(0.658879\pi\)
\(74\) −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 0.934216 − 0.356709i \(-0.116101\pi\)
0.158189 + 0.987409i \(0.449434\pi\)
\(80\) 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 \(-0.969568\pi\)
0.995433 − 0.0954603i \(-0.0304323\pi\)
\(84\) −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 −0.980917 − 0.194426i \(-0.937716\pi\)
−0.658837 − 0.752286i \(-0.728951\pi\)
\(90\) 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.107.26 yes 56
3.2 odd 2 inner 168.2.v.a.107.3 yes 56
4.3 odd 2 672.2.bd.a.527.22 56
7.4 even 3 inner 168.2.v.a.11.6 yes 56
8.3 odd 2 inner 168.2.v.a.107.23 yes 56
8.5 even 2 672.2.bd.a.527.21 56
12.11 even 2 672.2.bd.a.527.15 56
21.11 odd 6 inner 168.2.v.a.11.23 yes 56
24.5 odd 2 672.2.bd.a.527.16 56
24.11 even 2 inner 168.2.v.a.107.6 yes 56
28.11 odd 6 672.2.bd.a.431.16 56
56.11 odd 6 inner 168.2.v.a.11.3 ✓ 56
56.53 even 6 672.2.bd.a.431.15 56
84.11 even 6 672.2.bd.a.431.21 56
168.11 even 6 inner 168.2.v.a.11.26 yes 56
168.53 odd 6 672.2.bd.a.431.22 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.3 ✓ 56 56.11 odd 6 inner
168.2.v.a.11.6 yes 56 7.4 even 3 inner
168.2.v.a.11.23 yes 56 21.11 odd 6 inner
168.2.v.a.11.26 yes 56 168.11 even 6 inner
168.2.v.a.107.3 yes 56 3.2 odd 2 inner
168.2.v.a.107.6 yes 56 24.11 even 2 inner
168.2.v.a.107.23 yes 56 8.3 odd 2 inner
168.2.v.a.107.26 yes 56 1.1 even 1 trivial
672.2.bd.a.431.15 56 56.53 even 6
672.2.bd.a.431.16 56 28.11 odd 6
672.2.bd.a.431.21 56 84.11 even 6
672.2.bd.a.431.22 56 168.53 odd 6
672.2.bd.a.527.15 56 12.11 even 2
672.2.bd.a.527.16 56 24.5 odd 2
672.2.bd.a.527.21 56 8.5 even 2
672.2.bd.a.527.22 56 4.3 odd 2