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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 11.4
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.4

$q$-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+(-1.32990 - 0.481016i) q^{2} +(1.28463 + 1.16178i) q^{3} +(1.53725 + 1.27940i) q^{4} +(-0.646402 - 1.11960i) q^{5} +(-1.14959 - 2.16297i) q^{6} +(2.42742 - 1.05244i) q^{7} +(-1.42896 - 2.44091i) q^{8} +(0.300539 + 2.98491i) q^{9} +(0.321101 + 1.79988i) q^{10} +(1.60004 + 0.923782i) q^{11} +(0.488406 + 3.42950i) q^{12} +2.25432i q^{13} +(-3.73446 + 0.232013i) q^{14} +(0.470342 - 2.18925i) q^{15} +(0.726254 + 3.93352i) q^{16} +(3.89580 + 2.24924i) q^{17} +(1.03610 - 4.11418i) q^{18} +(-2.80851 - 4.86448i) q^{19} +(0.438742 - 2.54811i) q^{20} +(4.34104 + 1.46812i) q^{21} +(-1.68353 - 1.99818i) q^{22} +(0.519880 + 0.900459i) q^{23} +(1.00011 - 4.79581i) q^{24} +(1.66433 - 2.88270i) q^{25} +(1.08436 - 2.99801i) q^{26} +(-3.08172 + 4.18366i) q^{27} +(5.07804 + 1.48778i) q^{28} -1.32085 q^{29} +(-1.67857 + 2.68523i) q^{30} +(-3.69700 - 2.13446i) q^{31} +(0.926243 - 5.58051i) q^{32} +(0.982223 + 3.04561i) q^{33} +(-4.09909 - 4.86521i) q^{34} +(-2.74740 - 2.03744i) q^{35} +(-3.35690 + 4.97305i) q^{36} +(-8.18394 + 4.72500i) q^{37} +(1.39513 + 7.82020i) q^{38} +(-2.61902 + 2.89596i) q^{39} +(-1.80916 + 3.17768i) q^{40} -1.39634i q^{41} +(-5.06694 - 4.04056i) q^{42} -6.02578 q^{43} +(1.27776 + 3.46717i) q^{44} +(3.14763 - 2.26593i) q^{45} +(-0.258251 - 1.44759i) q^{46} +(-5.90249 - 10.2234i) q^{47} +(-3.63691 + 5.89685i) q^{48} +(4.78472 - 5.10944i) q^{49} +(-3.60001 + 3.03313i) q^{50} +(2.39154 + 7.41551i) q^{51} +(-2.88418 + 3.46544i) q^{52} +(-6.02901 + 10.4425i) q^{53} +(6.11078 - 4.08147i) q^{54} -2.38854i q^{55} +(-6.03762 - 4.42122i) q^{56} +(2.04356 - 9.51192i) q^{57} +(1.75660 + 0.635353i) q^{58} +(-9.57732 - 5.52947i) q^{59} +(3.52396 - 2.76365i) q^{60} +(7.65220 - 4.41800i) q^{61} +(3.88991 + 4.61693i) q^{62} +(3.87098 + 6.92932i) q^{63} +(-3.91612 + 6.97596i) q^{64} +(2.52393 - 1.45719i) q^{65} +(0.158732 - 4.52281i) q^{66} +(3.05545 - 5.29219i) q^{67} +(3.11112 + 8.44195i) q^{68} +(-0.378281 + 1.76074i) q^{69} +(2.67372 + 4.03113i) q^{70} +14.0121 q^{71} +(6.85644 - 4.99892i) q^{72} +(-4.38664 + 7.59788i) q^{73} +(13.1566 - 2.34715i) q^{74} +(5.48711 - 1.76962i) q^{75} +(1.90626 - 11.0711i) q^{76} +(4.85619 + 0.558456i) q^{77} +(4.87602 - 2.59153i) q^{78} +(2.37061 - 1.36867i) q^{79} +(3.93451 - 3.35575i) q^{80} +(-8.81935 + 1.79416i) q^{81} +(-0.671663 + 1.85699i) q^{82} -4.74366i q^{83} +(4.79492 + 7.81081i) q^{84} -5.81566i q^{85} +(8.01366 + 2.89850i) q^{86} +(-1.69681 - 1.53454i) q^{87} +(-0.0315236 - 5.22561i) q^{88} +(-8.31219 + 4.79904i) q^{89} +(-5.27598 + 1.49939i) q^{90} +(2.37254 + 5.47217i) q^{91} +(-0.352866 + 2.04936i) q^{92} +(-2.26949 - 7.03709i) q^{93} +(2.93207 + 16.4353i) q^{94} +(-3.63085 + 6.28882i) q^{95} +(7.67320 - 6.09279i) q^{96} -8.73466 q^{97} +(-8.82091 + 4.49350i) q^{98} +(-2.27653 + 5.05360i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32990 − 0.481016i −0.940378 − 0.340130i
\(3\) 1.28463 + 1.16178i 0.741680 + 0.670753i
\(4\) 1.53725 + 1.27940i 0.768623 + 0.639702i
\(5\) −0.646402 − 1.11960i −0.289080 − 0.500700i 0.684511 − 0.729003i \(-0.260016\pi\)
−0.973590 + 0.228302i \(0.926683\pi\)
\(6\) −1.14959 − 2.16297i −0.469317 − 0.883030i
\(7\) 2.42742 − 1.05244i 0.917478 − 0.397786i
\(8\) −1.42896 − 2.44091i −0.505215 − 0.862993i
\(9\) 0.300539 + 2.98491i 0.100180 + 0.994969i
\(10\) 0.321101 + 1.79988i 0.101541 + 0.569173i
\(11\) 1.60004 + 0.923782i 0.482430 + 0.278531i 0.721428 − 0.692489i \(-0.243486\pi\)
−0.238999 + 0.971020i \(0.576819\pi\)
\(12\) 0.488406 + 3.42950i 0.140991 + 0.990011i
\(13\) 2.25432i 0.625235i 0.949879 + 0.312617i \(0.101206\pi\)
−0.949879 + 0.312617i \(0.898794\pi\)
\(14\) −3.73446 + 0.232013i −0.998076 + 0.0620081i
\(15\) 0.470342 − 2.18925i 0.121442 − 0.565261i
\(16\) 0.726254 + 3.93352i 0.181564 + 0.983379i
\(17\) 3.89580 + 2.24924i 0.944871 + 0.545522i 0.891484 − 0.453052i \(-0.149665\pi\)
0.0533874 + 0.998574i \(0.482998\pi\)
\(18\) 1.03610 − 4.11418i 0.244212 − 0.969722i
\(19\) −2.80851 − 4.86448i −0.644317 − 1.11599i −0.984459 − 0.175615i \(-0.943809\pi\)
0.340142 − 0.940374i \(-0.389525\pi\)
\(20\) 0.438742 − 2.54811i 0.0981056 − 0.569775i
\(21\) 4.34104 + 1.46812i 0.947292 + 0.320371i
\(22\) −1.68353 − 1.99818i −0.358930 − 0.426013i
\(23\) 0.519880 + 0.900459i 0.108402 + 0.187759i 0.915123 − 0.403174i \(-0.132093\pi\)
−0.806721 + 0.590933i \(0.798760\pi\)
\(24\) 1.00011 − 4.79581i 0.204148 − 0.978940i
\(25\) 1.66433 − 2.88270i 0.332866 − 0.576541i
\(26\) 1.08436 − 2.99801i 0.212661 − 0.587957i
\(27\) −3.08172 + 4.18366i −0.593078 + 0.805145i
\(28\) 5.07804 + 1.48778i 0.959660 + 0.281164i
\(29\) −1.32085 −0.245277 −0.122638 − 0.992451i \(-0.539135\pi\)
−0.122638 + 0.992451i \(0.539135\pi\)
\(30\) −1.67857 + 2.68523i −0.306463 + 0.490253i
\(31\) −3.69700 − 2.13446i −0.664000 − 0.383361i 0.129799 − 0.991540i \(-0.458567\pi\)
−0.793800 + 0.608179i \(0.791900\pi\)
\(32\) 0.926243 − 5.58051i 0.163738 − 0.986504i
\(33\) 0.982223 + 3.04561i 0.170983 + 0.530172i
\(34\) −4.09909 − 4.86521i −0.702988 − 0.834376i
\(35\) −2.74740 − 2.03744i −0.464396 − 0.344390i
\(36\) −3.35690 + 4.97305i −0.559483 + 0.828842i
\(37\) −8.18394 + 4.72500i −1.34543 + 0.776786i −0.987599 − 0.157000i \(-0.949818\pi\)
−0.357834 + 0.933785i \(0.616484\pi\)
\(38\) 1.39513 + 7.82020i 0.226320 + 1.26860i
\(39\) −2.61902 + 2.89596i −0.419378 + 0.463724i
\(40\) −1.80916 + 3.17768i −0.286054 + 0.502435i
\(41\) − 1.39634i − 0.218072i −0.994038 − 0.109036i \(-0.965224\pi\)
0.994038 − 0.109036i \(-0.0347764\pi\)
\(42\) −5.06694 − 4.04056i −0.781845 − 0.623472i
\(43\) −6.02578 −0.918922 −0.459461 − 0.888198i \(-0.651957\pi\)
−0.459461 + 0.888198i \(0.651957\pi\)
\(44\) 1.27776 + 3.46717i 0.192630 + 0.522696i
\(45\) 3.14763 − 2.26593i 0.469222 − 0.337785i
\(46\) −0.258251 − 1.44759i −0.0380770 − 0.213435i
\(47\) −5.90249 − 10.2234i −0.860966 − 1.49124i −0.870997 − 0.491288i \(-0.836526\pi\)
0.0100312 − 0.999950i \(-0.496807\pi\)
\(48\) −3.63691 + 5.89685i −0.524943 + 0.851138i
\(49\) 4.78472 − 5.10944i 0.683532 − 0.729921i
\(50\) −3.60001 + 3.03313i −0.509119 + 0.428949i
\(51\) 2.39154 + 7.41551i 0.334882 + 1.03838i
\(52\) −2.88418 + 3.46544i −0.399964 + 0.480570i
\(53\) −6.02901 + 10.4425i −0.828148 + 1.43439i 0.0713414 + 0.997452i \(0.477272\pi\)
−0.899489 + 0.436943i \(0.856061\pi\)
\(54\) 6.11078 − 4.08147i 0.831571 − 0.555418i
\(55\) − 2.38854i − 0.322070i
\(56\) −6.03762 − 4.42122i −0.806811 − 0.590810i
\(57\) 2.04356 − 9.51192i 0.270676 − 1.25988i
\(58\) 1.75660 + 0.635353i 0.230653 + 0.0834259i
\(59\) −9.57732 − 5.52947i −1.24686 − 0.719875i −0.276378 − 0.961049i \(-0.589134\pi\)
−0.970482 + 0.241174i \(0.922468\pi\)
\(60\) 3.52396 − 2.76365i 0.454941 − 0.356786i
\(61\) 7.65220 − 4.41800i 0.979764 − 0.565667i 0.0775652 − 0.996987i \(-0.475285\pi\)
0.902199 + 0.431320i \(0.141952\pi\)
\(62\) 3.88991 + 4.61693i 0.494019 + 0.586351i
\(63\) 3.87098 + 6.92932i 0.487698 + 0.873012i
\(64\) −3.91612 + 6.97596i −0.489515 + 0.871995i
\(65\) 2.52393 − 1.45719i 0.313055 − 0.180743i
\(66\) 0.158732 − 4.52281i 0.0195386 − 0.556719i
\(67\) 3.05545 − 5.29219i 0.373283 − 0.646544i −0.616786 − 0.787131i \(-0.711566\pi\)
0.990068 + 0.140587i \(0.0448989\pi\)
\(68\) 3.11112 + 8.44195i 0.377279 + 1.02374i
\(69\) −0.378281 + 1.76074i −0.0455397 + 0.211968i
\(70\) 2.67372 + 4.03113i 0.319571 + 0.481812i
\(71\) 14.0121 1.66293 0.831464 − 0.555578i \(-0.187503\pi\)
0.831464 + 0.555578i \(0.187503\pi\)
\(72\) 6.85644 − 4.99892i 0.808040 − 0.589128i
\(73\) −4.38664 + 7.59788i −0.513417 + 0.889264i 0.486462 + 0.873702i \(0.338287\pi\)
−0.999879 + 0.0155626i \(0.995046\pi\)
\(74\) 13.1566 − 2.34715i 1.52942 − 0.272851i
\(75\) 5.48711 − 1.76962i 0.633597 − 0.204338i
\(76\) 1.90626 − 11.0711i 0.218663 − 1.26995i
\(77\) 4.85619 + 0.558456i 0.553414 + 0.0636420i
\(78\) 4.87602 − 2.59153i 0.552101 − 0.293433i
\(79\) 2.37061 − 1.36867i 0.266714 − 0.153987i −0.360679 − 0.932690i \(-0.617455\pi\)
0.627393 + 0.778702i \(0.284122\pi\)
\(80\) 3.93451 − 3.35575i 0.439892 − 0.375184i
\(81\) −8.81935 + 1.79416i −0.979928 + 0.199352i
\(82\) −0.671663 + 1.85699i −0.0741728 + 0.205070i
\(83\) − 4.74366i − 0.520685i −0.965516 − 0.260342i \(-0.916165\pi\)
0.965516 − 0.260342i \(-0.0838354\pi\)
\(84\) 4.79492 + 7.81081i 0.523169 + 0.852229i
\(85\) − 5.81566i − 0.630797i
\(86\) 8.01366 + 2.89850i 0.864135 + 0.312553i
\(87\) −1.69681 − 1.53454i −0.181917 − 0.164520i
\(88\) −0.0315236 − 5.22561i −0.00336043 − 0.557051i
\(89\) −8.31219 + 4.79904i −0.881090 + 0.508697i −0.871018 − 0.491252i \(-0.836539\pi\)
−0.0100723 + 0.999949i \(0.503206\pi\)
\(90\) −5.27598 + 1.49939i −0.556137 + 0.158050i
\(91\) 2.37254 + 5.47217i 0.248710 + 0.573639i
\(92\) −0.352866 + 2.04936i −0.0367888 + 0.213661i
\(93\) −2.26949 − 7.03709i −0.235336 − 0.729712i
\(94\) 2.93207 + 16.4353i 0.302420 + 1.69517i
\(95\) −3.63085 + 6.28882i −0.372517 + 0.645219i
\(96\) 7.67320 − 6.09279i 0.783142 − 0.621843i
\(97\) −8.73466 −0.886871 −0.443435 − 0.896306i \(-0.646240\pi\)
−0.443435 + 0.896306i \(0.646240\pi\)
\(98\) −8.82091 + 4.49350i −0.891046 + 0.453912i
\(99\) −2.27653 + 5.05360i −0.228800 + 0.507906i
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.11.4 ✓ 56
3.2 odd 2 inner 168.2.v.a.11.25 yes 56
4.3 odd 2 672.2.bd.a.431.5 56
7.2 even 3 inner 168.2.v.a.107.16 yes 56
8.3 odd 2 inner 168.2.v.a.11.13 yes 56
8.5 even 2 672.2.bd.a.431.6 56
12.11 even 2 672.2.bd.a.431.26 56
21.2 odd 6 inner 168.2.v.a.107.13 yes 56
24.5 odd 2 672.2.bd.a.431.25 56
24.11 even 2 inner 168.2.v.a.11.16 yes 56
28.23 odd 6 672.2.bd.a.527.25 56
56.37 even 6 672.2.bd.a.527.26 56
56.51 odd 6 inner 168.2.v.a.107.25 yes 56
84.23 even 6 672.2.bd.a.527.6 56
168.107 even 6 inner 168.2.v.a.107.4 yes 56
168.149 odd 6 672.2.bd.a.527.5 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.4 ✓ 56 1.1 even 1 trivial
168.2.v.a.11.13 yes 56 8.3 odd 2 inner
168.2.v.a.11.16 yes 56 24.11 even 2 inner
168.2.v.a.11.25 yes 56 3.2 odd 2 inner
168.2.v.a.107.4 yes 56 168.107 even 6 inner
168.2.v.a.107.13 yes 56 21.2 odd 6 inner
168.2.v.a.107.16 yes 56 7.2 even 3 inner
168.2.v.a.107.25 yes 56 56.51 odd 6 inner
672.2.bd.a.431.5 56 4.3 odd 2
672.2.bd.a.431.6 56 8.5 even 2
672.2.bd.a.431.25 56 24.5 odd 2
672.2.bd.a.431.26 56 12.11 even 2
672.2.bd.a.527.5 56 168.149 odd 6
672.2.bd.a.527.6 56 84.23 even 6
672.2.bd.a.527.25 56 28.23 odd 6
672.2.bd.a.527.26 56 56.37 even 6