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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 5.11
Character \(\chi\) \(=\) 168.5
Dual form 168.2.ba.c.101.11

$q$-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+(-0.174432 - 1.40341i) q^{2} +(1.09218 + 1.34430i) q^{3} +(-1.93915 + 0.489600i) q^{4} +(2.46958 + 1.42581i) q^{5} +(1.69610 - 1.76727i) q^{6} +(-1.02032 + 2.44110i) q^{7} +(1.02536 + 2.63603i) q^{8} +(-0.614290 + 2.93643i) q^{9} +(1.57023 - 3.71455i) q^{10} +(-2.42621 - 4.20231i) q^{11} +(-2.77607 - 2.07207i) q^{12} +2.75221 q^{13} +(3.60385 + 1.00612i) q^{14} +(0.780502 + 4.87710i) q^{15} +(3.52058 - 1.89881i) q^{16} +(-1.75366 - 3.03743i) q^{17} +(4.22819 + 0.349896i) q^{18} +(3.14493 - 5.44717i) q^{19} +(-5.48696 - 1.55575i) q^{20} +(-4.39594 + 1.29450i) q^{21} +(-5.47438 + 4.13799i) q^{22} +(-3.15865 - 1.82365i) q^{23} +(-2.42373 + 4.25741i) q^{24} +(1.56588 + 2.71219i) q^{25} +(-0.480073 - 3.86249i) q^{26} +(-4.61837 + 2.38132i) q^{27} +(0.783383 - 5.23319i) q^{28} +3.90427 q^{29} +(6.70845 - 1.94609i) q^{30} +(-0.858051 + 0.495396i) q^{31} +(-3.27893 - 4.60963i) q^{32} +(2.99932 - 7.85123i) q^{33} +(-3.95689 + 2.99094i) q^{34} +(-6.00030 + 4.57370i) q^{35} +(-0.246481 - 5.99494i) q^{36} +(1.06516 + 0.614970i) q^{37} +(-8.19322 - 3.46348i) q^{38} +(3.00591 + 3.69980i) q^{39} +(-1.22627 + 7.97185i) q^{40} +2.10659 q^{41} +(2.58351 + 5.94352i) q^{42} +5.11768i q^{43} +(6.76223 + 6.96103i) q^{44} +(-5.70384 + 6.37590i) q^{45} +(-2.00837 + 4.75100i) q^{46} +(-5.61268 + 9.72145i) q^{47} +(6.39768 + 2.65888i) q^{48} +(-4.91791 - 4.98138i) q^{49} +(3.53319 - 2.67068i) q^{50} +(2.16791 - 5.67487i) q^{51} +(-5.33694 + 1.34748i) q^{52} +(-1.00417 - 1.73927i) q^{53} +(4.14757 + 6.06611i) q^{54} -13.8373i q^{55} +(-7.48099 - 0.186576i) q^{56} +(10.7575 - 1.72156i) q^{57} +(-0.681029 - 5.47932i) q^{58} +(-0.890996 + 0.514417i) q^{59} +(-3.90134 - 9.07528i) q^{60} +(1.24347 - 2.15376i) q^{61} +(0.844918 + 1.11779i) q^{62} +(-6.54135 - 4.49563i) q^{63} +(-5.89727 + 5.40576i) q^{64} +(6.79681 + 3.92414i) q^{65} +(-11.5417 - 2.83979i) q^{66} +(5.02777 - 2.90279i) q^{67} +(4.88774 + 5.03144i) q^{68} +(-0.998281 - 6.23793i) q^{69} +(7.46545 + 7.62311i) q^{70} -9.75277i q^{71} +(-8.37039 + 1.39162i) q^{72} +(0.291019 - 0.168020i) q^{73} +(0.677260 - 1.60213i) q^{74} +(-1.93577 + 5.06722i) q^{75} +(-3.43154 + 12.1026i) q^{76} +(12.7338 - 1.63491i) q^{77} +(4.66803 - 4.86390i) q^{78} +(-2.80082 + 4.85116i) q^{79} +(11.4017 + 0.330420i) q^{80} +(-8.24530 - 3.60764i) q^{81} +(-0.367457 - 2.95643i) q^{82} -0.138115i q^{83} +(7.89058 - 4.66248i) q^{84} -10.0016i q^{85} +(7.18223 - 0.892686i) q^{86} +(4.26417 + 5.24852i) q^{87} +(8.58967 - 10.7044i) q^{88} +(-0.580993 + 1.00631i) q^{89} +(9.94296 + 6.89270i) q^{90} +(-2.80813 + 6.71841i) q^{91} +(7.01795 + 1.98985i) q^{92} +(-1.60311 - 0.612418i) q^{93} +(14.6223 + 6.18119i) q^{94} +(15.5333 - 8.96815i) q^{95} +(2.61555 - 9.44240i) q^{96} -11.0953i q^{97} +(-6.13311 + 7.77078i) q^{98} +(13.8302 - 4.54296i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.174432 − 1.40341i −0.123342 − 0.992364i
\(3\) 1.09218 + 1.34430i 0.630570 + 0.776132i
\(4\) −1.93915 + 0.489600i −0.969574 + 0.244800i
\(5\) 2.46958 + 1.42581i 1.10443 + 0.637643i 0.937381 − 0.348306i \(-0.113243\pi\)
0.167049 + 0.985949i \(0.446576\pi\)
\(6\) 1.69610 − 1.76727i 0.692430 − 0.721485i
\(7\) −1.02032 + 2.44110i −0.385643 + 0.922648i
\(8\) 1.02536 + 2.63603i 0.362520 + 0.931976i
\(9\) −0.614290 + 2.93643i −0.204763 + 0.978812i
\(10\) 1.57023 − 3.71455i 0.496552 − 1.17464i
\(11\) −2.42621 − 4.20231i −0.731529 − 1.26705i −0.956230 − 0.292617i \(-0.905474\pi\)
0.224701 − 0.974428i \(-0.427860\pi\)
\(12\) −2.77607 − 2.07207i −0.801381 − 0.598154i
\(13\) 2.75221 0.763326 0.381663 − 0.924302i \(-0.375351\pi\)
0.381663 + 0.924302i \(0.375351\pi\)
\(14\) 3.60385 + 1.00612i 0.963169 + 0.268898i
\(15\) 0.780502 + 4.87710i 0.201525 + 1.25926i
\(16\) 3.52058 − 1.89881i 0.880146 − 0.474703i
\(17\) −1.75366 − 3.03743i −0.425326 − 0.736686i 0.571125 − 0.820863i \(-0.306507\pi\)
−0.996451 + 0.0841773i \(0.973174\pi\)
\(18\) 4.22819 + 0.349896i 0.996593 + 0.0824713i
\(19\) 3.14493 − 5.44717i 0.721496 − 1.24967i −0.238905 − 0.971043i \(-0.576788\pi\)
0.960400 − 0.278624i \(-0.0898784\pi\)
\(20\) −5.48696 − 1.55575i −1.22692 − 0.347877i
\(21\) −4.39594 + 1.29450i −0.959272 + 0.282484i
\(22\) −5.47438 + 4.13799i −1.16714 + 0.882223i
\(23\) −3.15865 − 1.82365i −0.658624 − 0.380257i 0.133128 − 0.991099i \(-0.457498\pi\)
−0.791753 + 0.610842i \(0.790831\pi\)
\(24\) −2.42373 + 4.25741i −0.494743 + 0.869039i
\(25\) 1.56588 + 2.71219i 0.313177 + 0.542438i
\(26\) −0.480073 − 3.86249i −0.0941501 − 0.757498i
\(27\) −4.61837 + 2.38132i −0.888805 + 0.458286i
\(28\) 0.783383 − 5.23319i 0.148045 − 0.988981i
\(29\) 3.90427 0.725005 0.362503 − 0.931983i \(-0.381922\pi\)
0.362503 + 0.931983i \(0.381922\pi\)
\(30\) 6.70845 − 1.94609i 1.22479 − 0.355306i
\(31\) −0.858051 + 0.495396i −0.154111 + 0.0889758i −0.575072 − 0.818103i \(-0.695026\pi\)
0.420962 + 0.907078i \(0.361693\pi\)
\(32\) −3.27893 − 4.60963i −0.579638 − 0.814874i
\(33\) 2.99932 − 7.85123i 0.522115 − 1.36672i
\(34\) −3.95689 + 2.99094i −0.678600 + 0.512942i
\(35\) −6.00030 + 4.57370i −1.01424 + 0.773097i
\(36\) −0.246481 − 5.99494i −0.0410801 − 0.999156i
\(37\) 1.06516 + 0.614970i 0.175111 + 0.101100i 0.584994 − 0.811038i \(-0.301097\pi\)
−0.409883 + 0.912138i \(0.634430\pi\)
\(38\) −8.19322 − 3.46348i −1.32912 − 0.561850i
\(39\) 3.00591 + 3.69980i 0.481330 + 0.592442i
\(40\) −1.22627 + 7.97185i −0.193890 + 1.26046i
\(41\) 2.10659 0.328995 0.164497 − 0.986378i \(-0.447400\pi\)
0.164497 + 0.986378i \(0.447400\pi\)
\(42\) 2.58351 + 5.94352i 0.398645 + 0.917105i
\(43\) 5.11768i 0.780439i 0.920722 + 0.390220i \(0.127601\pi\)
−0.920722 + 0.390220i \(0.872399\pi\)
\(44\) 6.76223 + 6.96103i 1.01944 + 1.04942i
\(45\) −5.70384 + 6.37590i −0.850279 + 0.950463i
\(46\) −2.00837 + 4.75100i −0.296117 + 0.700497i
\(47\) −5.61268 + 9.72145i −0.818693 + 1.41802i 0.0879518 + 0.996125i \(0.471968\pi\)
−0.906645 + 0.421894i \(0.861365\pi\)
\(48\) 6.39768 + 2.65888i 0.923426 + 0.383776i
\(49\) −4.91791 − 4.98138i −0.702558 − 0.711626i
\(50\) 3.53319 − 2.67068i 0.499668 − 0.377691i
\(51\) 2.16791 − 5.67487i 0.303568 − 0.794641i
\(52\) −5.33694 + 1.34748i −0.740101 + 0.186862i
\(53\) −1.00417 − 1.73927i −0.137933 − 0.238907i 0.788781 − 0.614674i \(-0.210712\pi\)
−0.926714 + 0.375767i \(0.877379\pi\)
\(54\) 4.14757 + 6.06611i 0.564413 + 0.825492i
\(55\) − 13.8373i − 1.86582i
\(56\) −7.48099 − 0.186576i −0.999689 − 0.0249323i
\(57\) 10.7575 − 1.72156i 1.42486 − 0.228026i
\(58\) −0.681029 − 5.47932i −0.0894235 − 0.719469i
\(59\) −0.890996 + 0.514417i −0.115998 + 0.0669714i −0.556876 − 0.830595i \(-0.688000\pi\)
0.440879 + 0.897567i \(0.354667\pi\)
\(60\) −3.90134 − 9.07528i −0.503661 − 1.17161i
\(61\) 1.24347 − 2.15376i 0.159211 − 0.275761i −0.775374 − 0.631503i \(-0.782438\pi\)
0.934584 + 0.355742i \(0.115772\pi\)
\(62\) 0.844918 + 1.11779i 0.107305 + 0.141959i
\(63\) −6.54135 − 4.49563i −0.824133 − 0.566397i
\(64\) −5.89727 + 5.40576i −0.737159 + 0.675720i
\(65\) 6.79681 + 3.92414i 0.843040 + 0.486729i
\(66\) −11.5417 − 2.83979i −1.42069 − 0.349554i
\(67\) 5.02777 − 2.90279i 0.614240 − 0.354632i −0.160383 − 0.987055i \(-0.551273\pi\)
0.774623 + 0.632423i \(0.217940\pi\)
\(68\) 4.88774 + 5.03144i 0.592725 + 0.610151i
\(69\) −0.998281 − 6.23793i −0.120179 − 0.750958i
\(70\) 7.46545 + 7.62311i 0.892292 + 0.911136i
\(71\) − 9.75277i − 1.15744i −0.815526 − 0.578720i \(-0.803552\pi\)
0.815526 − 0.578720i \(-0.196448\pi\)
\(72\) −8.37039 + 1.39162i −0.986460 + 0.164004i
\(73\) 0.291019 − 0.168020i 0.0340612 − 0.0196652i −0.482873 − 0.875691i \(-0.660407\pi\)
0.516934 + 0.856025i \(0.327073\pi\)
\(74\) 0.677260 − 1.60213i 0.0787299 − 0.186244i
\(75\) −1.93577 + 5.06722i −0.223524 + 0.585112i
\(76\) −3.43154 + 12.1026i −0.393624 + 1.38827i
\(77\) 12.7338 − 1.63491i 1.45115 − 0.186316i
\(78\) 4.66803 − 4.86390i 0.528550 − 0.550728i
\(79\) −2.80082 + 4.85116i −0.315117 + 0.545798i −0.979462 − 0.201627i \(-0.935377\pi\)
0.664346 + 0.747426i \(0.268710\pi\)
\(80\) 11.4017 + 0.330420i 1.27475 + 0.0369421i
\(81\) −8.24530 − 3.60764i −0.916144 − 0.400849i
\(82\) −0.367457 − 2.95643i −0.0405788 − 0.326483i
\(83\) − 0.138115i − 0.0151600i −0.999971 − 0.00758002i \(-0.997587\pi\)
0.999971 − 0.00758002i \(-0.00241282\pi\)
\(84\) 7.89058 − 4.66248i 0.860933 − 0.508718i
\(85\) − 10.0016i − 1.08482i
\(86\) 7.18223 − 0.892686i 0.774480 − 0.0962609i
\(87\) 4.26417 + 5.24852i 0.457167 + 0.562700i
\(88\) 8.58967 − 10.7044i 0.915662 − 1.14110i
\(89\) −0.580993 + 1.00631i −0.0615852 + 0.106669i −0.895174 − 0.445717i \(-0.852949\pi\)
0.833589 + 0.552385i \(0.186282\pi\)
\(90\) 9.94296 + 6.89270i 1.04808 + 0.726554i
\(91\) −2.80813 + 6.71841i −0.294372 + 0.704281i
\(92\) 7.01795 + 1.98985i 0.731671 + 0.207456i
\(93\) −1.60311 − 0.612418i −0.166234 − 0.0635048i
\(94\) 14.6223 + 6.18119i 1.50817 + 0.637541i
\(95\) 15.5333 − 8.96815i 1.59368 − 0.920113i
\(96\) 2.61555 − 9.44240i 0.266949 − 0.963711i
\(97\) − 11.0953i − 1.12656i −0.826266 − 0.563280i \(-0.809539\pi\)
0.826266 − 0.563280i \(-0.190461\pi\)
\(98\) −6.13311 + 7.77078i −0.619538 + 0.784967i
\(99\) 13.8302 − 4.54296i 1.38999 − 0.456585i
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.5.11 yes 48
3.2 odd 2 inner 168.2.ba.c.5.14 yes 48
4.3 odd 2 672.2.bi.c.593.7 48
7.3 odd 6 inner 168.2.ba.c.101.21 yes 48
8.3 odd 2 672.2.bi.c.593.18 48
8.5 even 2 inner 168.2.ba.c.5.4 ✓ 48
12.11 even 2 672.2.bi.c.593.8 48
21.17 even 6 inner 168.2.ba.c.101.4 yes 48
24.5 odd 2 inner 168.2.ba.c.5.21 yes 48
24.11 even 2 672.2.bi.c.593.17 48
28.3 even 6 672.2.bi.c.17.17 48
56.3 even 6 672.2.bi.c.17.8 48
56.45 odd 6 inner 168.2.ba.c.101.14 yes 48
84.59 odd 6 672.2.bi.c.17.18 48
168.59 odd 6 672.2.bi.c.17.7 48
168.101 even 6 inner 168.2.ba.c.101.11 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.4 ✓ 48 8.5 even 2 inner
168.2.ba.c.5.11 yes 48 1.1 even 1 trivial
168.2.ba.c.5.14 yes 48 3.2 odd 2 inner
168.2.ba.c.5.21 yes 48 24.5 odd 2 inner
168.2.ba.c.101.4 yes 48 21.17 even 6 inner
168.2.ba.c.101.11 yes 48 168.101 even 6 inner
168.2.ba.c.101.14 yes 48 56.45 odd 6 inner
168.2.ba.c.101.21 yes 48 7.3 odd 6 inner
672.2.bi.c.17.7 48 168.59 odd 6
672.2.bi.c.17.8 48 56.3 even 6
672.2.bi.c.17.17 48 28.3 even 6
672.2.bi.c.17.18 48 84.59 odd 6
672.2.bi.c.593.7 48 4.3 odd 2
672.2.bi.c.593.8 48 12.11 even 2
672.2.bi.c.593.17 48 24.11 even 2
672.2.bi.c.593.18 48 8.3 odd 2