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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 11.15
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.15

$q$-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} +(0.728136 - 1.57157i) q^{3} +(-1.99050 + 0.194708i) q^{4} +(1.00081 + 1.73346i) q^{5} +(2.27007 + 0.920201i) q^{6} +(2.50986 + 0.837013i) q^{7} +(-0.412221 - 2.79823i) q^{8} +(-1.93964 - 2.28863i) q^{9} +(-2.37959 + 1.53315i) q^{10} +(3.33711 + 1.92668i) q^{11} +(-1.14336 + 3.26997i) 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} +(3.09908 - 2.89754i) q^{18} +(-0.232846 - 0.403301i) q^{19} +(-2.32964 - 3.25558i) q^{20} +(3.14294 - 3.33495i) q^{21} +(-2.49150 + 4.84656i) q^{22} +(-0.711115 - 1.23169i) q^{23} +(-4.69775 - 1.38966i) 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} +(0.676787 + 4.85603i) q^{30} +(-4.99699 - 2.88502i) q^{31} +(1.36536 + 5.48961i) q^{32} +(5.45777 - 3.84160i) q^{33} +(3.67691 - 7.15247i) q^{34} +(1.06098 + 5.18844i) q^{35} +(4.30646 + 4.17785i) q^{36} +(0.569094 - 0.328567i) q^{37} +(0.553627 - 0.356698i) q^{38} +(3.23622 + 1.49940i) q^{39} +(4.43806 - 3.51507i) q^{40} -8.42924i q^{41} +(4.92735 + 4.20966i) q^{42} +6.77074 q^{43} +(-7.01765 - 3.18529i) q^{44} +(2.02603 - 5.65277i) q^{45} +(1.69079 - 1.08936i) q^{46} +(-1.79536 - 3.10966i) q^{47} +(1.63916 - 6.73150i) q^{48} +(5.59882 + 4.20157i) q^{49} +(1.24956 + 0.642371i) q^{50} +(-8.05448 + 5.66936i) q^{51} +(-0.400950 - 4.09891i) q^{52} +(-1.50566 + 2.60789i) q^{53} +(-2.29712 - 6.98020i) 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} +(-6.81266 + 1.29067i) q^{60} +(-8.50530 + 4.91054i) q^{61} +(3.73078 - 7.25726i) q^{62} +(-2.95261 - 7.36764i) q^{63} +(-7.66015 + 2.30697i) q^{64} +(-3.56960 + 2.06091i) q^{65} +(5.80254 + 7.44451i) 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} +(-5.60454 + 6.37096i) q^{72} +(-2.02113 + 3.50071i) q^{73} +(0.503334 + 0.781219i) q^{74} +(-0.990459 - 1.40715i) q^{75} +(0.542005 + 0.757433i) q^{76} +(6.76302 + 7.62890i) q^{77} +(-1.89491 + 4.67462i) q^{78} +(-0.00488142 + 0.00281829i) q^{79} +(5.27103 + 6.02664i) q^{80} +(-1.47562 + 8.87821i) q^{81} +(11.9066 - 0.580958i) q^{82} -8.56466i q^{83} +(-5.60668 + 7.25018i) q^{84} -11.3827i q^{85} +(0.466651 + 9.56389i) q^{86} +(-6.74130 + 14.5500i) q^{87} +(4.01566 - 10.1322i) q^{88} +(-8.14017 + 4.69973i) q^{89} +(8.12436 + 2.47223i) q^{90} +(-1.72361 + 5.16840i) q^{91} +(1.65529 + 2.31321i) q^{92} +(-8.17248 + 5.75242i) q^{93} +(4.26876 - 2.75033i) q^{94} +(0.466070 - 0.807257i) q^{95} +(9.62145 + 1.85142i) 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
\(3\) 0.728136 − 1.57157i 0.420390 − 0.907344i
\(4\) −1.99050 + 0.194708i −0.995250 + 0.0973542i
\(5\) 1.00081 + 1.73346i 0.447577 + 0.775227i 0.998228 − 0.0595094i \(-0.0189536\pi\)
−0.550650 + 0.834736i \(0.685620\pi\)
\(6\) 2.27007 + 0.920201i 0.926753 + 0.375671i
\(7\) 2.50986 + 0.837013i 0.948639 + 0.316361i
\(8\) −0.412221 − 2.79823i −0.145742 − 0.989323i
\(9\) −1.93964 − 2.28863i −0.646545 − 0.762876i
\(10\) −2.37959 + 1.53315i −0.752493 + 0.484826i
\(11\) 3.33711 + 1.92668i 1.00618 + 0.580916i 0.910070 − 0.414455i \(-0.136028\pi\)
0.0961058 + 0.995371i \(0.469361\pi\)
\(12\) −1.14336 + 3.26997i −0.330059 + 0.943960i
\(13\) 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 −0.235137 − 0.971962i \(-0.575554\pi\)
−0.959313 + 0.282346i \(0.908887\pi\)
\(18\) 3.09908 − 2.89754i 0.730460 − 0.682956i
\(19\) −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
\(21\) 3.14294 − 3.33495i 0.685846 − 0.727747i
\(22\) −2.49150 + 4.84656i −0.531189 + 1.03329i
\(23\) −0.711115 − 1.23169i −0.148278 − 0.256825i 0.782313 − 0.622885i \(-0.214040\pi\)
−0.930591 + 0.366061i \(0.880706\pi\)
\(24\) −4.69775 − 1.38966i −0.958924 − 0.283663i
\(25\) 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 \(-0.829294\pi\)
−0.859611 + 0.510949i \(0.829294\pi\)
\(30\) 0.676787 + 4.85603i 0.123564 + 0.886585i
\(31\) −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
\(33\) 5.45777 − 3.84160i 0.950076 − 0.668736i
\(34\) 3.67691 − 7.15247i 0.630585 − 1.22664i
\(35\) 1.06098 + 5.18844i 0.179338 + 0.877006i
\(36\) 4.30646 + 4.17785i 0.717743 + 0.696308i
\(37\) 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
\(39\) 3.23622 + 1.49940i 0.518211 + 0.240097i
\(40\) 4.43806 − 3.51507i 0.701718 − 0.555781i
\(41\) − 8.42924i − 1.31643i −0.752832 − 0.658213i \(-0.771313\pi\)
0.752832 − 0.658213i \(-0.228687\pi\)
\(42\) 4.92735 + 4.20966i 0.760307 + 0.649564i
\(43\) 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
\(45\) 2.02603 − 5.65277i 0.302022 − 0.842665i
\(46\) 1.69079 − 1.08936i 0.249293 − 0.160618i
\(47\) −1.79536 − 3.10966i −0.261881 − 0.453590i 0.704861 − 0.709345i \(-0.251009\pi\)
−0.966742 + 0.255755i \(0.917676\pi\)
\(48\) 1.63916 − 6.73150i 0.236593 − 0.971609i
\(49\) 5.59882 + 4.20157i 0.799831 + 0.600225i
\(50\) 1.24956 + 0.642371i 0.176715 + 0.0908449i
\(51\) −8.05448 + 5.66936i −1.12785 + 0.793869i
\(52\) −0.400950 − 4.09891i −0.0556018 − 0.568416i
\(53\) −1.50566 + 2.60789i −0.206819 + 0.358221i −0.950711 − 0.310079i \(-0.899644\pi\)
0.743892 + 0.668300i \(0.232978\pi\)
\(54\) −2.29712 − 6.98020i −0.312598 − 0.949885i
\(55\) 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 0.823788 − 0.566898i \(-0.191857\pi\)
−0.0790537 + 0.996870i \(0.525190\pi\)
\(60\) −6.81266 + 1.29067i −0.879510 + 0.166625i
\(61\) −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
\(63\) −2.95261 − 7.36764i −0.371994 − 0.928235i
\(64\) −7.66015 + 2.30697i −0.957519 + 0.288372i
\(65\) −3.56960 + 2.06091i −0.442755 + 0.255624i
\(66\) 5.80254 + 7.44451i 0.714243 + 0.916356i
\(67\) −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 \(-0.186464\pi\)
0.833274 + 0.552861i \(0.186464\pi\)
\(72\) −5.60454 + 6.37096i −0.660501 + 0.750825i
\(73\) −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
\(75\) −0.990459 − 1.40715i −0.114368 − 0.162484i
\(76\) 0.542005 + 0.757433i 0.0621723 + 0.0868835i
\(77\) 6.76302 + 7.62890i 0.770718 + 0.869394i
\(78\) −1.89491 + 4.67462i −0.214556 + 0.529296i
\(79\) −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
\(81\) −1.47562 + 8.87821i −0.163958 + 0.986467i
\(82\) 11.9066 − 0.580958i 1.31486 − 0.0641560i
\(83\) − 8.56466i − 0.940093i −0.882642 − 0.470047i \(-0.844237\pi\)
0.882642 − 0.470047i \(-0.155763\pi\)
\(84\) −5.60668 + 7.25018i −0.611739 + 0.791060i
\(85\) − 11.3827i − 1.23463i
\(86\) 0.466651 + 9.56389i 0.0503202 + 1.03130i
\(87\) −6.74130 + 14.5500i −0.722743 + 1.55993i
\(88\) 4.01566 − 10.1322i 0.428071 − 1.08010i
\(89\) −8.14017 + 4.69973i −0.862857 + 0.498170i −0.864968 − 0.501827i \(-0.832661\pi\)
0.00211125 + 0.999998i \(0.499328\pi\)
\(90\) 8.12436 + 2.47223i 0.856383 + 0.260596i
\(91\) −1.72361 + 5.16840i −0.180683 + 0.541795i
\(92\) 1.65529 + 2.31321i 0.172576 + 0.241169i
\(93\) −8.17248 + 5.75242i −0.847447 + 0.596498i
\(94\) 4.26876 − 2.75033i 0.440289 − 0.283675i
\(95\) 0.466070 − 0.807257i 0.0478178 − 0.0828228i
\(96\) 9.62145 + 1.85142i 0.981985 + 0.188960i
\(97\) −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.11.15 yes 56
3.2 odd 2 inner 168.2.v.a.11.14 yes 56
4.3 odd 2 672.2.bd.a.431.12 56
7.2 even 3 inner 168.2.v.a.107.24 yes 56
8.3 odd 2 inner 168.2.v.a.11.5 ✓ 56
8.5 even 2 672.2.bd.a.431.11 56
12.11 even 2 672.2.bd.a.431.9 56
21.2 odd 6 inner 168.2.v.a.107.5 yes 56
24.5 odd 2 672.2.bd.a.431.10 56
24.11 even 2 inner 168.2.v.a.11.24 yes 56
28.23 odd 6 672.2.bd.a.527.10 56
56.37 even 6 672.2.bd.a.527.9 56
56.51 odd 6 inner 168.2.v.a.107.14 yes 56
84.23 even 6 672.2.bd.a.527.11 56
168.107 even 6 inner 168.2.v.a.107.15 yes 56
168.149 odd 6 672.2.bd.a.527.12 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.5 ✓ 56 8.3 odd 2 inner
168.2.v.a.11.14 yes 56 3.2 odd 2 inner
168.2.v.a.11.15 yes 56 1.1 even 1 trivial
168.2.v.a.11.24 yes 56 24.11 even 2 inner
168.2.v.a.107.5 yes 56 21.2 odd 6 inner
168.2.v.a.107.14 yes 56 56.51 odd 6 inner
168.2.v.a.107.15 yes 56 168.107 even 6 inner
168.2.v.a.107.24 yes 56 7.2 even 3 inner
672.2.bd.a.431.9 56 12.11 even 2
672.2.bd.a.431.10 56 24.5 odd 2
672.2.bd.a.431.11 56 8.5 even 2
672.2.bd.a.431.12 56 4.3 odd 2
672.2.bd.a.527.9 56 56.37 even 6
672.2.bd.a.527.10 56 28.23 odd 6
672.2.bd.a.527.11 56 84.23 even 6
672.2.bd.a.527.12 56 168.149 odd 6