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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 11.5
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.5

$q$-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+(-1.18883 - 0.765954i) q^{2} +(0.728136 - 1.57157i) q^{3} +(0.826627 + 1.82118i) 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} +(-1.93964 - 2.28863i) q^{9} +(-0.137955 + 2.82736i) q^{10} +(3.33711 + 1.92668i) q^{11} +(3.46400 + 0.0269657i) q^{12} -2.05924i q^{13} +(2.34268 + 2.91751i) q^{14} +(-3.45297 + 0.310649i) q^{15} +(-2.63337 + 3.01087i) q^{16} +(-4.92484 - 2.84336i) q^{17} +(0.552911 + 4.20646i) 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} +(-4.09744 - 2.68532i) q^{24} +(0.496746 - 0.860389i) q^{25} +(-1.57728 + 2.44808i) q^{26} +(-5.00905 + 1.38183i) q^{27} +(-0.550373 - 5.26280i) q^{28} +9.25829 q^{29} +(4.34294 + 2.27551i) q^{30} +(4.99699 + 2.88502i) q^{31} +(5.43682 - 1.56236i) q^{32} +(5.45777 - 3.84160i) 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} +(-0.0320962 + 0.657805i) q^{38} +(-3.23622 - 1.49940i) q^{39} +(-5.26317 + 2.08593i) q^{40} -8.42924i q^{41} +(6.29084 - 1.55734i) q^{42} +6.77074 q^{43} +(-0.750281 + 7.67011i) q^{44} +(-2.02603 + 5.65277i) q^{45} +(0.0980225 - 2.00895i) 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} +(-8.05448 + 5.66936i) q^{51} +(3.75023 - 1.70222i) q^{52} +(1.50566 - 2.60789i) q^{53} +(7.01332 + 2.19394i) q^{54} -7.71298i q^{55} +(-3.37677 + 6.67813i) q^{56} +(-0.803357 + 0.0722746i) q^{57} +(-11.0065 - 7.09143i) q^{58} +(5.72041 + 3.30268i) q^{59} +(-3.42007 - 6.03168i) 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} +(-9.43084 + 0.386597i) q^{66} +(-3.78271 + 6.55184i) q^{67} +(1.10725 - 11.3194i) q^{68} +(2.45347 - 0.220728i) q^{69} +(2.71279 - 6.98082i) q^{70} -14.0426 q^{71} +(-7.20366 + 4.48412i) q^{72} +(-2.02113 + 3.50071i) q^{73} +(0.928223 + 0.0452908i) q^{74} +(-0.990459 - 1.40715i) 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} +(7.85474 + 1.55153i) q^{80} +(-1.47562 + 8.87821i) q^{81} +(-6.45642 + 10.0209i) q^{82} -8.56466i q^{83} +(-8.67158 - 2.96709i) q^{84} +11.3827i q^{85} +(-8.04925 - 5.18608i) q^{86} +(6.74130 - 14.5500i) q^{87} +(6.76691 - 8.54376i) 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} +(8.17248 - 5.75242i) q^{93} +(0.247479 - 5.07202i) q^{94} +(-0.466070 + 0.807257i) q^{95} +(1.50339 - 9.68193i) q^{96} -4.17304 q^{97} +(-3.43782 - 9.28339i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.18883 − 0.765954i −0.840629 − 0.541612i
\(3\) 0.728136 − 1.57157i 0.420390 − 0.907344i
\(4\) 0.826627 + 1.82118i 0.413314 + 0.910589i
\(5\) −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
\(9\) −1.93964 − 2.28863i −0.646545 − 0.762876i
\(10\) −0.137955 + 2.82736i −0.0436254 + 0.894091i
\(11\) 3.33711 + 1.92668i 1.00618 + 0.580916i 0.910070 − 0.414455i \(-0.136028\pi\)
0.0961058 + 0.995371i \(0.469361\pi\)
\(12\) 3.46400 + 0.0269657i 0.999970 + 0.00778433i
\(13\) − 2.05924i − 0.571129i −0.958359 − 0.285565i \(-0.907819\pi\)
0.958359 − 0.285565i \(-0.0921811\pi\)
\(14\) 2.34268 + 2.91751i 0.626108 + 0.779736i
\(15\) −3.45297 + 0.310649i −0.891554 + 0.0802093i
\(16\) −2.63337 + 3.01087i −0.658344 + 0.752718i
\(17\) −4.92484 − 2.84336i −1.19445 − 0.689616i −0.235137 − 0.971962i \(-0.575554\pi\)
−0.959313 + 0.282346i \(0.908887\pi\)
\(18\) 0.552911 + 4.20646i 0.130322 + 0.991472i
\(19\) −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 0.930591 − 0.366061i \(-0.119294\pi\)
−0.782313 + 0.622885i \(0.785960\pi\)
\(24\) −4.09744 − 2.68532i −0.836387 − 0.548139i
\(25\) 0.496746 − 0.860389i 0.0993492 − 0.172078i
\(26\) −1.57728 + 2.44808i −0.309330 + 0.480108i
\(27\) −5.00905 + 1.38183i −0.963991 + 0.265934i
\(28\) −0.550373 − 5.26280i −0.104011 − 0.994576i
\(29\) 9.25829 1.71922 0.859611 − 0.510949i \(-0.170706\pi\)
0.859611 + 0.510949i \(0.170706\pi\)
\(30\) 4.34294 + 2.27551i 0.792908 + 0.415450i
\(31\) 4.99699 + 2.88502i 0.897486 + 0.518164i 0.876384 − 0.481613i \(-0.159949\pi\)
0.0211026 + 0.999777i \(0.493282\pi\)
\(32\) 5.43682 − 1.56236i 0.961103 − 0.276190i
\(33\) 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
\(36\) 2.56464 − 5.42426i 0.427440 − 0.904044i
\(37\) −0.569094 + 0.328567i −0.0935586 + 0.0540161i −0.546049 − 0.837753i \(-0.683869\pi\)
0.452491 + 0.891769i \(0.350536\pi\)
\(38\) −0.0320962 + 0.657805i −0.00520670 + 0.106710i
\(39\) −3.23622 − 1.49940i −0.518211 − 0.240097i
\(40\) −5.26317 + 2.08593i −0.832180 + 0.329815i
\(41\) − 8.42924i − 1.31643i −0.752832 − 0.658213i \(-0.771313\pi\)
0.752832 − 0.658213i \(-0.228687\pi\)
\(42\) 6.29084 − 1.55734i 0.970698 − 0.240303i
\(43\) 6.77074 1.03253 0.516264 − 0.856430i \(-0.327322\pi\)
0.516264 + 0.856430i \(0.327322\pi\)
\(44\) −0.750281 + 7.67011i −0.113109 + 1.15631i
\(45\) −2.02603 + 5.65277i −0.302022 + 0.842665i
\(46\) 0.0980225 − 2.00895i 0.0144526 − 0.296203i
\(47\) 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
\(51\) −8.05448 + 5.66936i −1.12785 + 0.793869i
\(52\) 3.75023 − 1.70222i 0.520064 − 0.236056i
\(53\) 1.50566 − 2.60789i 0.206819 − 0.358221i −0.743892 − 0.668300i \(-0.767022\pi\)
0.950711 + 0.310079i \(0.100356\pi\)
\(54\) 7.01332 + 2.19394i 0.954392 + 0.298557i
\(55\) − 7.71298i − 1.04002i
\(56\) −3.37677 + 6.67813i −0.451240 + 0.892403i
\(57\) −0.803357 + 0.0722746i −0.106407 + 0.00957300i
\(58\) −11.0065 − 7.09143i −1.44523 − 0.931151i
\(59\) 5.72041 + 3.30268i 0.744735 + 0.429973i 0.823788 − 0.566898i \(-0.191857\pi\)
−0.0790537 + 0.996870i \(0.525190\pi\)
\(60\) −3.42007 − 6.03168i −0.441529 − 0.778687i
\(61\) 8.50530 − 4.91054i 1.08899 − 0.628730i 0.155684 − 0.987807i \(-0.450242\pi\)
0.933308 + 0.359077i \(0.116908\pi\)
\(62\) −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
\(66\) −9.43084 + 0.386597i −1.16086 + 0.0475868i
\(67\) −3.78271 + 6.55184i −0.462131 + 0.800435i −0.999067 − 0.0431883i \(-0.986248\pi\)
0.536936 + 0.843623i \(0.319582\pi\)
\(68\) 1.10725 − 11.3194i 0.134274 − 1.37268i
\(69\) 2.45347 − 0.220728i 0.295363 − 0.0265725i
\(70\) 2.71279 − 6.98082i 0.324240 − 0.834368i
\(71\) −14.0426 −1.66655 −0.833274 − 0.552861i \(-0.813536\pi\)
−0.833274 + 0.552861i \(0.813536\pi\)
\(72\) −7.20366 + 4.48412i −0.848959 + 0.528459i
\(73\) −2.02113 + 3.50071i −0.236556 + 0.409727i −0.959724 − 0.280946i \(-0.909352\pi\)
0.723168 + 0.690672i \(0.242685\pi\)
\(74\) 0.928223 + 0.0452908i 0.107904 + 0.00526494i
\(75\) −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
\(78\) 2.69884 + 4.26133i 0.305584 + 0.482501i
\(79\) 0.00488142 − 0.00281829i 0.000549203 − 0.000317082i −0.499725 − 0.866184i \(-0.666566\pi\)
0.500275 + 0.865867i \(0.333232\pi\)
\(80\) 7.85474 + 1.55153i 0.878186 + 0.173466i
\(81\) −1.47562 + 8.87821i −0.163958 + 0.986467i
\(82\) −6.45642 + 10.0209i −0.712992 + 1.10663i
\(83\) − 8.56466i − 0.940093i −0.882642 − 0.470047i \(-0.844237\pi\)
0.882642 − 0.470047i \(-0.155763\pi\)
\(84\) −8.67158 − 2.96709i −0.946147 − 0.323736i
\(85\) 11.3827i 1.23463i
\(86\) −8.04925 − 5.18608i −0.867973 − 0.559229i
\(87\) 6.74130 − 14.5500i 0.722743 − 1.55993i
\(88\) 6.76691 − 8.54376i 0.721355 − 0.910768i
\(89\) −8.14017 + 4.69973i −0.862857 + 0.498170i −0.864968 − 0.501827i \(-0.832661\pi\)
0.00211125 + 0.999998i \(0.499328\pi\)
\(90\) 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
\(93\) 8.17248 − 5.75242i 0.847447 − 0.596498i
\(94\) 0.247479 − 5.07202i 0.0255255 − 0.523139i
\(95\) −0.466070 + 0.807257i −0.0478178 + 0.0828228i
\(96\) 1.50339 − 9.68193i 0.153439 − 0.988158i
\(97\) −4.17304 −0.423708 −0.211854 − 0.977301i \(-0.567950\pi\)
−0.211854 + 0.977301i \(0.567950\pi\)
\(98\) −3.43782 − 9.28339i −0.347273 − 0.937764i
\(99\) −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.11.5 ✓ 56
3.2 odd 2 inner 168.2.v.a.11.24 yes 56
4.3 odd 2 672.2.bd.a.431.11 56
7.2 even 3 inner 168.2.v.a.107.14 yes 56
8.3 odd 2 inner 168.2.v.a.11.15 yes 56
8.5 even 2 672.2.bd.a.431.12 56
12.11 even 2 672.2.bd.a.431.10 56
21.2 odd 6 inner 168.2.v.a.107.15 yes 56
24.5 odd 2 672.2.bd.a.431.9 56
24.11 even 2 inner 168.2.v.a.11.14 yes 56
28.23 odd 6 672.2.bd.a.527.9 56
56.37 even 6 672.2.bd.a.527.10 56
56.51 odd 6 inner 168.2.v.a.107.24 yes 56
84.23 even 6 672.2.bd.a.527.12 56
168.107 even 6 inner 168.2.v.a.107.5 yes 56
168.149 odd 6 672.2.bd.a.527.11 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.5 ✓ 56 1.1 even 1 trivial
168.2.v.a.11.14 yes 56 24.11 even 2 inner
168.2.v.a.11.15 yes 56 8.3 odd 2 inner
168.2.v.a.11.24 yes 56 3.2 odd 2 inner
168.2.v.a.107.5 yes 56 168.107 even 6 inner
168.2.v.a.107.14 yes 56 7.2 even 3 inner
168.2.v.a.107.15 yes 56 21.2 odd 6 inner
168.2.v.a.107.24 yes 56 56.51 odd 6 inner
672.2.bd.a.431.9 56 24.5 odd 2
672.2.bd.a.431.10 56 12.11 even 2
672.2.bd.a.431.11 56 4.3 odd 2
672.2.bd.a.431.12 56 8.5 even 2
672.2.bd.a.527.9 56 28.23 odd 6
672.2.bd.a.527.10 56 56.37 even 6
672.2.bd.a.527.11 56 168.149 odd 6
672.2.bd.a.527.12 56 84.23 even 6