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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(5,168)] 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("168.5"); 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([0, 3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.12
Character \(\chi\) \(=\) 168.5
Dual form 168.2.ba.c.101.12

$q$-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+(-0.0857242 - 1.41161i) q^{2} +(-0.528554 + 1.64943i) q^{3} +(-1.98530 + 0.242019i) q^{4} +(-2.66818 - 1.54047i) q^{5} +(2.37367 + 0.604718i) q^{6} +(-1.46307 - 2.20441i) q^{7} +(0.511826 + 2.78173i) q^{8} +(-2.44126 - 1.74363i) q^{9} +(-1.94583 + 3.89849i) q^{10} +(0.621780 + 1.07696i) q^{11} +(0.650147 - 3.40254i) q^{12} -5.98957 q^{13} +(-2.98636 + 2.25426i) q^{14} +(3.95119 - 3.58676i) q^{15} +(3.88285 - 0.960961i) q^{16} +(-0.595854 - 1.03205i) q^{17} +(-2.25206 + 3.59559i) q^{18} +(0.614126 - 1.06370i) q^{19} +(5.66997 + 2.41256i) q^{20} +(4.40934 - 1.24808i) q^{21} +(1.46694 - 0.970035i) q^{22} +(2.56956 + 1.48354i) q^{23} +(-4.85881 - 0.626075i) q^{24} +(2.24612 + 3.89040i) q^{25} +(0.513451 + 8.45496i) q^{26} +(4.16634 - 3.10509i) q^{27} +(3.43814 + 4.02234i) q^{28} +3.19900 q^{29} +(-5.40183 - 5.27008i) q^{30} +(1.33987 - 0.773574i) q^{31} +(-1.68936 - 5.39871i) q^{32} +(-2.10501 + 0.456356i) q^{33} +(-1.40578 + 0.929587i) q^{34} +(0.507883 + 8.13559i) q^{35} +(5.26863 + 2.87080i) q^{36} +(-0.334978 - 0.193399i) q^{37} +(-1.55417 - 0.775723i) q^{38} +(3.16581 - 9.87940i) q^{39} +(2.91955 - 8.21062i) q^{40} -9.44060 q^{41} +(-2.13979 - 6.11729i) q^{42} -8.29057i q^{43} +(-1.49507 - 1.98760i) q^{44} +(3.82770 + 8.41302i) q^{45} +(1.87391 - 3.75440i) q^{46} +(-3.34244 + 5.78928i) q^{47} +(-0.467258 + 6.91243i) q^{48} +(-2.71887 + 6.45041i) q^{49} +(5.29919 - 3.50416i) q^{50} +(2.01724 - 0.437327i) q^{51} +(11.8911 - 1.44959i) q^{52} +(-5.25317 - 9.09875i) q^{53} +(-4.74035 - 5.61508i) q^{54} -3.83135i q^{55} +(5.38325 - 5.19813i) q^{56} +(1.42990 + 1.57518i) q^{57} +(-0.274232 - 4.51576i) q^{58} +(3.22898 - 1.86425i) q^{59} +(-6.97624 + 8.07707i) q^{60} +(3.16493 - 5.48181i) q^{61} +(-1.20685 - 1.82506i) q^{62} +(-0.271956 + 7.93259i) q^{63} +(-7.47607 + 2.84752i) q^{64} +(15.9813 + 9.22678i) q^{65} +(0.824648 + 2.93234i) q^{66} +(-10.7324 + 6.19634i) q^{67} +(1.43273 + 1.90472i) q^{68} +(-3.80515 + 3.45419i) q^{69} +(11.4408 - 1.41435i) q^{70} -6.21100i q^{71} +(3.60081 - 7.68337i) q^{72} +(-8.92963 + 5.15552i) q^{73} +(-0.244290 + 0.489438i) q^{74} +(-7.60415 + 1.64854i) q^{75} +(-0.961791 + 2.26039i) q^{76} +(1.46435 - 2.94632i) q^{77} +(-14.2173 - 3.62200i) q^{78} +(-6.41425 + 11.1098i) q^{79} +(-11.8405 - 3.41742i) q^{80} +(2.91950 + 8.51331i) q^{81} +(0.809288 + 13.3265i) q^{82} +5.22882i q^{83} +(-8.45182 + 3.54496i) q^{84} +3.67159i q^{85} +(-11.7031 + 0.710702i) q^{86} +(-1.69085 + 5.27654i) q^{87} +(-2.67756 + 2.28084i) q^{88} +(6.94090 - 12.0220i) q^{89} +(11.5478 - 6.12444i) q^{90} +(8.76314 + 13.2035i) q^{91} +(-5.46040 - 2.32339i) q^{92} +(0.567765 + 2.61890i) q^{93} +(8.45876 + 4.22196i) q^{94} +(-3.27720 + 1.89209i) q^{95} +(9.79773 + 0.0670247i) q^{96} -17.1489i q^{97} +(9.33855 + 3.28504i) q^{98} +(0.359884 - 3.71328i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{4} - 4 q^{7} - 14 q^{9} - 30 q^{10} - 18 q^{12} + 4 q^{15} + 6 q^{16} - 36 q^{22} - 12 q^{24} - 8 q^{25} - 10 q^{28} + 22 q^{30} + 48 q^{31} - 42 q^{33} + 52 q^{36} - 8 q^{39} - 18 q^{40} + 12 q^{42}+ \cdots + 90 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0857242 − 1.41161i −0.0606162 − 0.998161i
\(3\) −0.528554 + 1.64943i −0.305161 + 0.952301i
\(4\) −1.98530 + 0.242019i −0.992651 + 0.121009i
\(5\) −2.66818 − 1.54047i −1.19325 − 0.688921i −0.234205 − 0.972187i \(-0.575249\pi\)
−0.959041 + 0.283266i \(0.908582\pi\)
\(6\) 2.37367 + 0.604718i 0.969047 + 0.246875i
\(7\) −1.46307 − 2.20441i −0.552987 − 0.833190i
\(8\) 0.511826 + 2.78173i 0.180958 + 0.983491i
\(9\) −2.44126 − 1.74363i −0.813753 − 0.581210i
\(10\) −1.94583 + 3.89849i −0.615324 + 1.23281i
\(11\) 0.621780 + 1.07696i 0.187474 + 0.324714i 0.944407 − 0.328778i \(-0.106637\pi\)
−0.756933 + 0.653492i \(0.773303\pi\)
\(12\) 0.650147 − 3.40254i 0.187681 − 0.982230i
\(13\) −5.98957 −1.66121 −0.830604 − 0.556864i \(-0.812005\pi\)
−0.830604 + 0.556864i \(0.812005\pi\)
\(14\) −2.98636 + 2.25426i −0.798138 + 0.602475i
\(15\) 3.95119 − 3.58676i 1.02019 − 0.926098i
\(16\) 3.88285 − 0.960961i 0.970713 − 0.240240i
\(17\) −0.595854 − 1.03205i −0.144516 − 0.250309i 0.784676 − 0.619906i \(-0.212829\pi\)
−0.929192 + 0.369597i \(0.879496\pi\)
\(18\) −2.25206 + 3.59559i −0.530815 + 0.847488i
\(19\) 0.614126 − 1.06370i 0.140890 − 0.244029i −0.786942 − 0.617027i \(-0.788337\pi\)
0.927832 + 0.372998i \(0.121670\pi\)
\(20\) 5.66997 + 2.41256i 1.26784 + 0.539464i
\(21\) 4.40934 − 1.24808i 0.962197 − 0.272353i
\(22\) 1.46694 − 0.970035i 0.312753 − 0.206812i
\(23\) 2.56956 + 1.48354i 0.535791 + 0.309339i 0.743371 − 0.668879i \(-0.233226\pi\)
−0.207581 + 0.978218i \(0.566559\pi\)
\(24\) −4.85881 − 0.626075i −0.991800 − 0.127797i
\(25\) 2.24612 + 3.89040i 0.449225 + 0.778080i
\(26\) 0.513451 + 8.45496i 0.100696 + 1.65815i
\(27\) 4.16634 − 3.10509i 0.801813 − 0.597575i
\(28\) 3.43814 + 4.02234i 0.649747 + 0.760150i
\(29\) 3.19900 0.594040 0.297020 − 0.954871i \(-0.404007\pi\)
0.297020 + 0.954871i \(0.404007\pi\)
\(30\) −5.40183 − 5.27008i −0.986235 − 0.962180i
\(31\) 1.33987 − 0.773574i 0.240648 − 0.138938i −0.374827 − 0.927095i \(-0.622298\pi\)
0.615474 + 0.788157i \(0.288964\pi\)
\(32\) −1.68936 − 5.39871i −0.298640 − 0.954366i
\(33\) −2.10501 + 0.456356i −0.366435 + 0.0794414i
\(34\) −1.40578 + 0.929587i −0.241088 + 0.159423i
\(35\) 0.507883 + 8.13559i 0.0858478 + 1.37517i
\(36\) 5.26863 + 2.87080i 0.878105 + 0.478467i
\(37\) −0.334978 − 0.193399i −0.0550700 − 0.0317947i 0.472212 − 0.881485i \(-0.343456\pi\)
−0.527282 + 0.849690i \(0.676789\pi\)
\(38\) −1.55417 − 0.775723i −0.252120 − 0.125839i
\(39\) 3.16581 − 9.87940i 0.506936 − 1.58197i
\(40\) 2.91955 − 8.21062i 0.461621 − 1.29821i
\(41\) −9.44060 −1.47437 −0.737187 − 0.675689i \(-0.763846\pi\)
−0.737187 + 0.675689i \(0.763846\pi\)
\(42\) −2.13979 − 6.11729i −0.330177 − 0.943919i
\(43\) − 8.29057i − 1.26430i −0.774846 − 0.632150i \(-0.782173\pi\)
0.774846 − 0.632150i \(-0.217827\pi\)
\(44\) −1.49507 − 1.98760i −0.225390 − 0.299642i
\(45\) 3.82770 + 8.41302i 0.570600 + 1.25414i
\(46\) 1.87391 − 3.75440i 0.276292 − 0.553556i
\(47\) −3.34244 + 5.78928i −0.487546 + 0.844454i −0.999897 − 0.0143219i \(-0.995441\pi\)
0.512352 + 0.858776i \(0.328774\pi\)
\(48\) −0.467258 + 6.91243i −0.0674429 + 0.997723i
\(49\) −2.71887 + 6.45041i −0.388410 + 0.921486i
\(50\) 5.29919 − 3.50416i 0.749419 − 0.495563i
\(51\) 2.01724 − 0.437327i 0.282470 − 0.0612380i
\(52\) 11.8911 − 1.44959i 1.64900 − 0.201022i
\(53\) −5.25317 − 9.09875i −0.721578 − 1.24981i −0.960367 − 0.278738i \(-0.910084\pi\)
0.238789 − 0.971071i \(-0.423250\pi\)
\(54\) −4.74035 − 5.61508i −0.645079 − 0.764116i
\(55\) − 3.83135i − 0.516619i
\(56\) 5.38325 − 5.19813i 0.719367 − 0.694630i
\(57\) 1.42990 + 1.57518i 0.189395 + 0.208638i
\(58\) −0.274232 − 4.51576i −0.0360084 − 0.592948i
\(59\) 3.22898 − 1.86425i 0.420377 − 0.242705i −0.274862 − 0.961484i \(-0.588632\pi\)
0.695238 + 0.718779i \(0.255299\pi\)
\(60\) −6.97624 + 8.07707i −0.900629 + 1.04274i
\(61\) 3.16493 − 5.48181i 0.405227 − 0.701874i −0.589121 − 0.808045i \(-0.700526\pi\)
0.994348 + 0.106171i \(0.0338590\pi\)
\(62\) −1.20685 − 1.82506i −0.153270 − 0.231783i
\(63\) −0.271956 + 7.93259i −0.0342632 + 0.999413i
\(64\) −7.47607 + 2.84752i −0.934509 + 0.355940i
\(65\) 15.9813 + 9.22678i 1.98223 + 1.14444i
\(66\) 0.824648 + 2.93234i 0.101507 + 0.360946i
\(67\) −10.7324 + 6.19634i −1.31117 + 0.757003i −0.982290 − 0.187369i \(-0.940004\pi\)
−0.328878 + 0.944372i \(0.606671\pi\)
\(68\) 1.43273 + 1.90472i 0.173744 + 0.230982i
\(69\) −3.80515 + 3.45419i −0.458086 + 0.415836i
\(70\) 11.4408 − 1.41435i 1.36743 − 0.169047i
\(71\) − 6.21100i − 0.737110i −0.929606 − 0.368555i \(-0.879853\pi\)
0.929606 − 0.368555i \(-0.120147\pi\)
\(72\) 3.60081 − 7.68337i 0.424360 − 0.905494i
\(73\) −8.92963 + 5.15552i −1.04513 + 0.603408i −0.921283 − 0.388893i \(-0.872858\pi\)
−0.123851 + 0.992301i \(0.539524\pi\)
\(74\) −0.244290 + 0.489438i −0.0283981 + 0.0568960i
\(75\) −7.60415 + 1.64854i −0.878052 + 0.190357i
\(76\) −0.961791 + 2.26039i −0.110325 + 0.259285i
\(77\) 1.46435 − 2.94632i 0.166878 − 0.335764i
\(78\) −14.2173 − 3.62200i −1.60979 − 0.410111i
\(79\) −6.41425 + 11.1098i −0.721660 + 1.24995i 0.238674 + 0.971100i \(0.423287\pi\)
−0.960334 + 0.278852i \(0.910046\pi\)
\(80\) −11.8405 − 3.41742i −1.32381 − 0.382079i
\(81\) 2.91950 + 8.51331i 0.324389 + 0.945924i
\(82\) 0.809288 + 13.3265i 0.0893709 + 1.47166i
\(83\) 5.22882i 0.573938i 0.957940 + 0.286969i \(0.0926476\pi\)
−0.957940 + 0.286969i \(0.907352\pi\)
\(84\) −8.45182 + 3.54496i −0.922169 + 0.386787i
\(85\) 3.67159i 0.398240i
\(86\) −11.7031 + 0.710702i −1.26197 + 0.0766370i
\(87\) −1.69085 + 5.27654i −0.181278 + 0.565705i
\(88\) −2.67756 + 2.28084i −0.285429 + 0.243138i
\(89\) 6.94090 − 12.0220i 0.735734 − 1.27433i −0.218666 − 0.975800i \(-0.570171\pi\)
0.954400 − 0.298529i \(-0.0964960\pi\)
\(90\) 11.5478 − 6.12444i 1.21725 − 0.645572i
\(91\) 8.76314 + 13.2035i 0.918627 + 1.38410i
\(92\) −5.46040 − 2.32339i −0.569286 − 0.242230i
\(93\) 0.567765 + 2.61890i 0.0588745 + 0.271567i
\(94\) 8.45876 + 4.22196i 0.872454 + 0.435462i
\(95\) −3.27720 + 1.89209i −0.336233 + 0.194124i
\(96\) 9.79773 + 0.0670247i 0.999977 + 0.00684068i
\(97\) − 17.1489i − 1.74120i −0.491989 − 0.870601i \(-0.663730\pi\)
0.491989 − 0.870601i \(-0.336270\pi\)
\(98\) 9.33855 + 3.28504i 0.943336 + 0.331839i
\(99\) 0.359884 − 3.71328i 0.0361697 − 0.373199i
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.ba.c.5.12 yes 48
3.2 odd 2 inner 168.2.ba.c.5.13 yes 48
4.3 odd 2 672.2.bi.c.593.15 48
7.3 odd 6 inner 168.2.ba.c.101.20 yes 48
8.3 odd 2 672.2.bi.c.593.10 48
8.5 even 2 inner 168.2.ba.c.5.5 ✓ 48
12.11 even 2 672.2.bi.c.593.3 48
21.17 even 6 inner 168.2.ba.c.101.5 yes 48
24.5 odd 2 inner 168.2.ba.c.5.20 yes 48
24.11 even 2 672.2.bi.c.593.22 48
28.3 even 6 672.2.bi.c.17.22 48
56.3 even 6 672.2.bi.c.17.3 48
56.45 odd 6 inner 168.2.ba.c.101.13 yes 48
84.59 odd 6 672.2.bi.c.17.10 48
168.59 odd 6 672.2.bi.c.17.15 48
168.101 even 6 inner 168.2.ba.c.101.12 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.5 ✓ 48 8.5 even 2 inner
168.2.ba.c.5.12 yes 48 1.1 even 1 trivial
168.2.ba.c.5.13 yes 48 3.2 odd 2 inner
168.2.ba.c.5.20 yes 48 24.5 odd 2 inner
168.2.ba.c.101.5 yes 48 21.17 even 6 inner
168.2.ba.c.101.12 yes 48 168.101 even 6 inner
168.2.ba.c.101.13 yes 48 56.45 odd 6 inner
168.2.ba.c.101.20 yes 48 7.3 odd 6 inner
672.2.bi.c.17.3 48 56.3 even 6
672.2.bi.c.17.10 48 84.59 odd 6
672.2.bi.c.17.15 48 168.59 odd 6
672.2.bi.c.17.22 48 28.3 even 6
672.2.bi.c.593.3 48 12.11 even 2
672.2.bi.c.593.10 48 8.3 odd 2
672.2.bi.c.593.15 48 4.3 odd 2
672.2.bi.c.593.22 48 24.11 even 2