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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 11.14
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.14

$q$-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+(-0.0689217 - 1.41253i) q^{2} +(0.996948 - 1.41637i) q^{3} +(-1.99050 + 0.194708i) q^{4} +(-1.00081 - 1.73346i) q^{5} +(-2.06938 - 1.31060i) q^{6} +(2.50986 + 0.837013i) q^{7} +(0.412221 + 2.79823i) q^{8} +(-1.01219 - 2.82409i) q^{9} +(-2.37959 + 1.53315i) q^{10} +(-3.33711 - 1.92668i) q^{11} +(-1.70865 + 3.01339i) q^{12} +2.05924i q^{13} +(1.00932 - 3.60295i) q^{14} +(-3.45297 - 0.310649i) q^{15} +(3.92418 - 0.775134i) q^{16} +(4.92484 + 2.84336i) q^{17} +(-3.91936 + 1.62439i) q^{18} +(-0.232846 - 0.403301i) q^{19} +(2.32964 + 3.25558i) q^{20} +(3.68772 - 2.72043i) q^{21} +(-2.49150 + 4.84656i) q^{22} +(0.711115 + 1.23169i) q^{23} +(4.37428 + 2.20583i) q^{24} +(0.496746 - 0.860389i) q^{25} +(2.90874 - 0.141926i) q^{26} +(-5.00905 - 1.38183i) q^{27} +(-5.15885 - 1.17738i) q^{28} +9.25829 q^{29} +(-0.200818 + 4.89885i) q^{30} +(-4.99699 - 2.88502i) q^{31} +(-1.36536 - 5.48961i) q^{32} +(-6.05580 + 2.80577i) q^{33} +(3.67691 - 7.15247i) q^{34} +(-1.06098 - 5.18844i) q^{35} +(2.56464 + 5.42426i) q^{36} +(0.569094 - 0.328567i) q^{37} +(-0.553627 + 0.356698i) q^{38} +(2.91663 + 2.05295i) q^{39} +(4.43806 - 3.51507i) q^{40} +8.42924i q^{41} +(-4.09686 - 5.02153i) q^{42} +6.77074 q^{43} +(7.01765 + 3.18529i) q^{44} +(-3.88243 + 4.58098i) q^{45} +(1.69079 - 1.08936i) q^{46} +(1.79536 + 3.10966i) q^{47} +(2.81433 - 6.33084i) q^{48} +(5.59882 + 4.20157i) q^{49} +(-1.24956 - 0.642371i) q^{50} +(8.93705 - 4.14070i) q^{51} +(-0.400950 - 4.09891i) q^{52} +(1.50566 - 2.60789i) q^{53} +(-1.60665 + 7.17068i) q^{54} +7.71298i q^{55} +(-1.30753 + 7.36820i) q^{56} +(-0.803357 - 0.0722746i) q^{57} +(-0.638097 - 13.0776i) q^{58} +(-5.72041 - 3.30268i) q^{59} +(6.93363 - 0.0539753i) q^{60} +(-8.50530 + 4.91054i) q^{61} +(-3.73078 + 7.25726i) q^{62} +(-0.176663 - 7.93529i) q^{63} +(-7.66015 + 2.30697i) q^{64} +(3.56960 - 2.06091i) q^{65} +(4.38062 + 8.36065i) q^{66} +(-3.78271 + 6.55184i) q^{67} +(-10.3565 - 4.70080i) q^{68} +(2.45347 + 0.220728i) q^{69} +(-7.25572 + 1.85626i) q^{70} -14.0426 q^{71} +(7.48519 - 3.99649i) q^{72} +(-2.02113 + 3.50071i) q^{73} +(-0.503334 - 0.781219i) q^{74} +(-0.723397 - 1.56134i) q^{75} +(0.542005 + 0.757433i) q^{76} +(-6.76302 - 7.62890i) q^{77} +(2.69884 - 4.26133i) q^{78} +(-0.00488142 + 0.00281829i) q^{79} +(-5.27103 - 6.02664i) q^{80} +(-6.95094 + 5.71703i) q^{81} +(11.9066 - 0.580958i) q^{82} +8.56466i q^{83} +(-6.81071 + 6.13304i) q^{84} -11.3827i q^{85} +(-0.466651 - 9.56389i) q^{86} +(9.23003 - 13.1131i) q^{87} +(4.01566 - 10.1322i) q^{88} +(8.14017 - 4.69973i) q^{89} +(6.73836 + 5.16833i) q^{90} +(-1.72361 + 5.16840i) q^{91} +(-1.65529 - 2.31321i) q^{92} +(-9.06798 + 4.20137i) q^{93} +(4.26876 - 2.75033i) q^{94} +(-0.466070 + 0.807257i) q^{95} +(-9.13649 - 3.53900i) q^{96} -4.17304 q^{97} +(5.54898 - 8.19810i) q^{98} +(-2.06332 + 11.3744i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0689217 − 1.41253i −0.0487350 − 0.998812i
\(3\) 0.996948 − 1.41637i 0.575588 − 0.817740i
\(4\) −1.99050 + 0.194708i −0.995250 + 0.0973542i
\(5\) −1.00081 − 1.73346i −0.447577 − 0.775227i 0.550650 − 0.834736i \(-0.314380\pi\)
−0.998228 + 0.0595094i \(0.981046\pi\)
\(6\) −2.06938 − 1.31060i −0.844819 − 0.535051i
\(7\) 2.50986 + 0.837013i 0.948639 + 0.316361i
\(8\) 0.412221 + 2.79823i 0.145742 + 0.989323i
\(9\) −1.01219 − 2.82409i −0.337397 − 0.941362i
\(10\) −2.37959 + 1.53315i −0.752493 + 0.484826i
\(11\) −3.33711 − 1.92668i −1.00618 − 0.580916i −0.0961058 − 0.995371i \(-0.530639\pi\)
−0.910070 + 0.414455i \(0.863972\pi\)
\(12\) −1.70865 + 3.01339i −0.493243 + 0.869891i
\(13\) 2.05924i 0.571129i 0.958359 + 0.285565i \(0.0921811\pi\)
−0.958359 + 0.285565i \(0.907819\pi\)
\(14\) 1.00932 − 3.60295i 0.269753 − 0.962929i
\(15\) −3.45297 − 0.310649i −0.891554 − 0.0802093i
\(16\) 3.92418 − 0.775134i 0.981044 − 0.193783i
\(17\) 4.92484 + 2.84336i 1.19445 + 0.689616i 0.959313 − 0.282346i \(-0.0911127\pi\)
0.235137 + 0.971962i \(0.424446\pi\)
\(18\) −3.91936 + 1.62439i −0.923801 + 0.382873i
\(19\) −0.232846 − 0.403301i −0.0534185 − 0.0925235i 0.838080 − 0.545548i \(-0.183678\pi\)
−0.891498 + 0.453024i \(0.850345\pi\)
\(20\) 2.32964 + 3.25558i 0.520923 + 0.727971i
\(21\) 3.68772 − 2.72043i 0.804726 − 0.593646i
\(22\) −2.49150 + 4.84656i −0.531189 + 1.03329i
\(23\) 0.711115 + 1.23169i 0.148278 + 0.256825i 0.930591 − 0.366061i \(-0.119294\pi\)
−0.782313 + 0.622885i \(0.785960\pi\)
\(24\) 4.37428 + 2.20583i 0.892896 + 0.450263i
\(25\) 0.496746 − 0.860389i 0.0993492 − 0.172078i
\(26\) 2.90874 − 0.141926i 0.570451 − 0.0278340i
\(27\) −5.00905 − 1.38183i −0.963991 − 0.265934i
\(28\) −5.15885 − 1.17738i −0.974932 − 0.222504i
\(29\) 9.25829 1.71922 0.859611 − 0.510949i \(-0.170706\pi\)
0.859611 + 0.510949i \(0.170706\pi\)
\(30\) −0.200818 + 4.89885i −0.0366641 + 0.894403i
\(31\) −4.99699 − 2.88502i −0.897486 − 0.518164i −0.0211026 − 0.999777i \(-0.506718\pi\)
−0.876384 + 0.481613i \(0.840051\pi\)
\(32\) −1.36536 − 5.48961i −0.241364 − 0.970435i
\(33\) −6.05580 + 2.80577i −1.05418 + 0.488422i
\(34\) 3.67691 − 7.15247i 0.630585 − 1.22664i
\(35\) −1.06098 − 5.18844i −0.179338 − 0.877006i
\(36\) 2.56464 + 5.42426i 0.427440 + 0.904044i
\(37\) 0.569094 − 0.328567i 0.0935586 − 0.0540161i −0.452491 − 0.891769i \(-0.649464\pi\)
0.546049 + 0.837753i \(0.316131\pi\)
\(38\) −0.553627 + 0.356698i −0.0898102 + 0.0578641i
\(39\) 2.91663 + 2.05295i 0.467035 + 0.328735i
\(40\) 4.43806 − 3.51507i 0.701718 − 0.555781i
\(41\) 8.42924i 1.31643i 0.752832 + 0.658213i \(0.228687\pi\)
−0.752832 + 0.658213i \(0.771313\pi\)
\(42\) −4.09686 − 5.02153i −0.632159 − 0.774839i
\(43\) 6.77074 1.03253 0.516264 − 0.856430i \(-0.327322\pi\)
0.516264 + 0.856430i \(0.327322\pi\)
\(44\) 7.01765 + 3.18529i 1.05795 + 0.480201i
\(45\) −3.88243 + 4.58098i −0.578758 + 0.682892i
\(46\) 1.69079 − 1.08936i 0.249293 − 0.160618i
\(47\) 1.79536 + 3.10966i 0.261881 + 0.453590i 0.966742 − 0.255755i \(-0.0823240\pi\)
−0.704861 + 0.709345i \(0.748991\pi\)
\(48\) 2.81433 − 6.33084i 0.406213 − 0.913778i
\(49\) 5.59882 + 4.20157i 0.799831 + 0.600225i
\(50\) −1.24956 − 0.642371i −0.176715 − 0.0908449i
\(51\) 8.93705 − 4.14070i 1.25144 − 0.579815i
\(52\) −0.400950 − 4.09891i −0.0556018 − 0.568416i
\(53\) 1.50566 − 2.60789i 0.206819 − 0.358221i −0.743892 − 0.668300i \(-0.767022\pi\)
0.950711 + 0.310079i \(0.100356\pi\)
\(54\) −1.60665 + 7.17068i −0.218638 + 0.975806i
\(55\) 7.71298i 1.04002i
\(56\) −1.30753 + 7.36820i −0.174727 + 0.984617i
\(57\) −0.803357 − 0.0722746i −0.106407 − 0.00957300i
\(58\) −0.638097 − 13.0776i −0.0837863 − 1.71718i
\(59\) −5.72041 − 3.30268i −0.744735 − 0.429973i 0.0790537 − 0.996870i \(-0.474810\pi\)
−0.823788 + 0.566898i \(0.808143\pi\)
\(60\) 6.93363 − 0.0539753i 0.895127 − 0.00696818i
\(61\) −8.50530 + 4.91054i −1.08899 + 0.628730i −0.933308 − 0.359077i \(-0.883092\pi\)
−0.155684 + 0.987807i \(0.549758\pi\)
\(62\) −3.73078 + 7.25726i −0.473809 + 0.921673i
\(63\) −0.176663 − 7.93529i −0.0222575 − 0.999752i
\(64\) −7.66015 + 2.30697i −0.957519 + 0.288372i
\(65\) 3.56960 − 2.06091i 0.442755 − 0.255624i
\(66\) 4.38062 + 8.36065i 0.539217 + 1.02912i
\(67\) −3.78271 + 6.55184i −0.462131 + 0.800435i −0.999067 − 0.0431883i \(-0.986248\pi\)
0.536936 + 0.843623i \(0.319582\pi\)
\(68\) −10.3565 − 4.70080i −1.25591 − 0.570055i
\(69\) 2.45347 + 0.220728i 0.295363 + 0.0265725i
\(70\) −7.25572 + 1.85626i −0.867224 + 0.221865i
\(71\) −14.0426 −1.66655 −0.833274 − 0.552861i \(-0.813536\pi\)
−0.833274 + 0.552861i \(0.813536\pi\)
\(72\) 7.48519 − 3.99649i 0.882138 − 0.470991i
\(73\) −2.02113 + 3.50071i −0.236556 + 0.409727i −0.959724 − 0.280946i \(-0.909352\pi\)
0.723168 + 0.690672i \(0.242685\pi\)
\(74\) −0.503334 − 0.781219i −0.0585114 − 0.0908149i
\(75\) −0.723397 − 1.56134i −0.0835307 − 0.180288i
\(76\) 0.542005 + 0.757433i 0.0621723 + 0.0868835i
\(77\) −6.76302 − 7.62890i −0.770718 − 0.869394i
\(78\) 2.69884 − 4.26133i 0.305584 − 0.482501i
\(79\) −0.00488142 + 0.00281829i −0.000549203 + 0.000317082i −0.500275 − 0.865867i \(-0.666768\pi\)
0.499725 + 0.866184i \(0.333434\pi\)
\(80\) −5.27103 − 6.02664i −0.589319 − 0.673799i
\(81\) −6.95094 + 5.71703i −0.772327 + 0.635226i
\(82\) 11.9066 − 0.580958i 1.31486 − 0.0641560i
\(83\) 8.56466i 0.940093i 0.882642 + 0.470047i \(0.155763\pi\)
−0.882642 + 0.470047i \(0.844237\pi\)
\(84\) −6.81071 + 6.13304i −0.743110 + 0.669170i
\(85\) − 11.3827i − 1.23463i
\(86\) −0.466651 − 9.56389i −0.0503202 − 1.03130i
\(87\) 9.23003 − 13.1131i 0.989564 − 1.40588i
\(88\) 4.01566 − 10.1322i 0.428071 − 1.08010i
\(89\) 8.14017 − 4.69973i 0.862857 − 0.498170i −0.00211125 − 0.999998i \(-0.500672\pi\)
0.864968 + 0.501827i \(0.167339\pi\)
\(90\) 6.73836 + 5.16833i 0.710286 + 0.544790i
\(91\) −1.72361 + 5.16840i −0.180683 + 0.541795i
\(92\) −1.65529 − 2.31321i −0.172576 − 0.241169i
\(93\) −9.06798 + 4.20137i −0.940306 + 0.435661i
\(94\) 4.26876 − 2.75033i 0.440289 − 0.283675i
\(95\) −0.466070 + 0.807257i −0.0478178 + 0.0828228i
\(96\) −9.13649 − 3.53900i −0.932489 − 0.361197i
\(97\) −4.17304 −0.423708 −0.211854 − 0.977301i \(-0.567950\pi\)
−0.211854 + 0.977301i \(0.567950\pi\)
\(98\) 5.54898 − 8.19810i 0.560532 − 0.828133i
\(99\) −2.06332 + 11.3744i −0.207372 + 1.14317i
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.11.14 yes 56
3.2 odd 2 inner 168.2.v.a.11.15 yes 56
4.3 odd 2 672.2.bd.a.431.9 56
7.2 even 3 inner 168.2.v.a.107.5 yes 56
8.3 odd 2 inner 168.2.v.a.11.24 yes 56
8.5 even 2 672.2.bd.a.431.10 56
12.11 even 2 672.2.bd.a.431.12 56
21.2 odd 6 inner 168.2.v.a.107.24 yes 56
24.5 odd 2 672.2.bd.a.431.11 56
24.11 even 2 inner 168.2.v.a.11.5 ✓ 56
28.23 odd 6 672.2.bd.a.527.11 56
56.37 even 6 672.2.bd.a.527.12 56
56.51 odd 6 inner 168.2.v.a.107.15 yes 56
84.23 even 6 672.2.bd.a.527.10 56
168.107 even 6 inner 168.2.v.a.107.14 yes 56
168.149 odd 6 672.2.bd.a.527.9 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.5 ✓ 56 24.11 even 2 inner
168.2.v.a.11.14 yes 56 1.1 even 1 trivial
168.2.v.a.11.15 yes 56 3.2 odd 2 inner
168.2.v.a.11.24 yes 56 8.3 odd 2 inner
168.2.v.a.107.5 yes 56 7.2 even 3 inner
168.2.v.a.107.14 yes 56 168.107 even 6 inner
168.2.v.a.107.15 yes 56 56.51 odd 6 inner
168.2.v.a.107.24 yes 56 21.2 odd 6 inner
672.2.bd.a.431.9 56 4.3 odd 2
672.2.bd.a.431.10 56 8.5 even 2
672.2.bd.a.431.11 56 24.5 odd 2
672.2.bd.a.431.12 56 12.11 even 2
672.2.bd.a.527.9 56 168.149 odd 6
672.2.bd.a.527.10 56 84.23 even 6
672.2.bd.a.527.11 56 28.23 odd 6
672.2.bd.a.527.12 56 56.37 even 6