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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 11.13
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.13

$q$-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+(-0.248376 + 1.39223i) q^{2} +(1.28463 + 1.16178i) q^{3} +(-1.87662 - 0.691593i) q^{4} +(0.646402 + 1.11960i) q^{5} +(-1.93654 + 1.49994i) q^{6} +(-2.42742 + 1.05244i) q^{7} +(1.42896 - 2.44091i) q^{8} +(0.300539 + 2.98491i) q^{9} +(-1.71929 + 0.621859i) q^{10} +(1.60004 + 0.923782i) q^{11} +(-1.60728 - 3.06866i) q^{12} -2.25432i q^{13} +(-0.862334 - 3.64093i) q^{14} +(-0.470342 + 2.18925i) q^{15} +(3.04340 + 2.59571i) q^{16} +(3.89580 + 2.24924i) q^{17} +(-4.23033 - 0.322958i) 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} +(4.67149 - 1.47553i) q^{24} +(1.66433 - 2.88270i) q^{25} +(3.13853 + 0.559917i) q^{26} +(-3.08172 + 4.18366i) q^{27} +(5.28320 - 0.296251i) q^{28} +1.32085 q^{29} +(-2.93112 - 1.19858i) q^{30} +(3.69700 + 2.13446i) q^{31} +(-4.36974 + 3.59240i) q^{32} +(0.982223 + 3.04561i) q^{33} +(-4.09909 + 4.86521i) q^{34} +(-2.74740 - 2.03744i) q^{35} +(1.50034 - 5.80939i) q^{36} +(8.18394 - 4.72500i) q^{37} +(7.47005 - 2.70188i) q^{38} +(2.61902 - 2.89596i) q^{39} +(3.65653 + 0.0220581i) q^{40} -1.39634i q^{41} +(3.12218 - 5.67908i) q^{42} -6.02578 q^{43} +(-2.36378 - 2.84016i) q^{44} +(-3.14763 + 2.26593i) q^{45} +(1.38277 - 0.500142i) q^{46} +(5.90249 + 10.2234i) q^{47} +(0.893991 + 6.87028i) q^{48} +(4.78472 - 5.10944i) q^{49} +(3.60001 + 3.03313i) q^{50} +(2.39154 + 7.41551i) q^{51} +(-1.55907 + 4.23049i) q^{52} +(6.02901 - 10.4425i) q^{53} +(-5.05920 - 5.32959i) q^{54} +2.38854i q^{55} +(-0.899769 + 7.42903i) q^{56} +(2.04356 - 9.51192i) q^{57} +(-0.328068 + 1.83894i) q^{58} +(-9.57732 - 5.52947i) q^{59} +(2.39672 - 3.78309i) 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} +(-4.48415 + 0.611027i) q^{66} +(3.05545 - 5.29219i) q^{67} +(-5.75538 - 6.91528i) q^{68} +(0.378281 - 1.76074i) q^{69} +(3.51897 - 3.31897i) q^{70} -14.0121 q^{71} +(7.71536 + 3.53174i) q^{72} +(-4.38664 + 7.59788i) q^{73} +(4.54561 + 12.5675i) q^{74} +(5.48711 - 1.76962i) q^{75} +(1.90626 + 11.0711i) q^{76} +(-4.85619 - 0.558456i) q^{77} +(3.38135 + 4.36556i) q^{78} +(-2.37061 + 1.36867i) q^{79} +(-0.938904 + 5.08526i) q^{80} +(-8.81935 + 1.79416i) q^{81} +(1.94403 + 0.346817i) q^{82} -4.74366i q^{83} +(7.13113 + 5.75734i) q^{84} +5.81566i q^{85} +(1.49666 - 8.38928i) q^{86} +(1.69681 + 1.53454i) q^{87} +(4.54127 - 2.58550i) q^{88} +(-8.31219 + 4.79904i) q^{89} +(-2.37291 - 4.94504i) q^{90} +(2.37254 + 5.47217i) q^{91} +(0.352866 + 2.04936i) q^{92} +(2.26949 + 7.03709i) q^{93} +(-15.6994 + 5.67839i) q^{94} +(3.63085 - 6.28882i) q^{95} +(-9.78707 - 0.461768i) q^{96} -8.73466 q^{97} +(5.92512 + 7.93051i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.248376 + 1.39223i −0.175628 + 0.984457i
\(3\) 1.28463 + 1.16178i 0.741680 + 0.670753i
\(4\) −1.87662 − 0.691593i −0.938310 − 0.345796i
\(5\) 0.646402 + 1.11960i 0.289080 + 0.500700i 0.973590 − 0.228302i \(-0.0733175\pi\)
−0.684511 + 0.729003i \(0.739984\pi\)
\(6\) −1.93654 + 1.49994i −0.790588 + 0.612349i
\(7\) −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
\(10\) −1.71929 + 0.621859i −0.543688 + 0.196649i
\(11\) 1.60004 + 0.923782i 0.482430 + 0.278531i 0.721428 − 0.692489i \(-0.243486\pi\)
−0.238999 + 0.971020i \(0.576819\pi\)
\(12\) −1.60728 − 3.06866i −0.463982 − 0.885845i
\(13\) − 2.25432i − 0.625235i −0.949879 − 0.312617i \(-0.898794\pi\)
0.949879 − 0.312617i \(-0.101206\pi\)
\(14\) −0.862334 − 3.64093i −0.230469 − 0.973080i
\(15\) −0.470342 + 2.18925i −0.121442 + 0.565261i
\(16\) 3.04340 + 2.59571i 0.760850 + 0.648928i
\(17\) 3.89580 + 2.24924i 0.944871 + 0.545522i 0.891484 − 0.453052i \(-0.149665\pi\)
0.0533874 + 0.998574i \(0.482998\pi\)
\(18\) −4.23033 − 0.322958i −0.997099 − 0.0761219i
\(19\) −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 0.806721 − 0.590933i \(-0.201240\pi\)
−0.915123 + 0.403174i \(0.867907\pi\)
\(24\) 4.67149 − 1.47553i 0.953564 − 0.301191i
\(25\) 1.66433 − 2.88270i 0.332866 − 0.576541i
\(26\) 3.13853 + 0.559917i 0.615517 + 0.109809i
\(27\) −3.08172 + 4.18366i −0.593078 + 0.805145i
\(28\) 5.28320 − 0.296251i 0.998432 − 0.0559862i
\(29\) 1.32085 0.245277 0.122638 − 0.992451i \(-0.460865\pi\)
0.122638 + 0.992451i \(0.460865\pi\)
\(30\) −2.93112 − 1.19858i −0.535146 − 0.218830i
\(31\) 3.69700 + 2.13446i 0.664000 + 0.383361i 0.793800 − 0.608179i \(-0.208100\pi\)
−0.129799 + 0.991540i \(0.541433\pi\)
\(32\) −4.36974 + 3.59240i −0.772468 + 0.635053i
\(33\) 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
\(36\) 1.50034 − 5.80939i 0.250057 − 0.968231i
\(37\) 8.18394 − 4.72500i 1.34543 − 0.776786i 0.357834 − 0.933785i \(-0.383516\pi\)
0.987599 + 0.157000i \(0.0501822\pi\)
\(38\) 7.47005 − 2.70188i 1.21180 − 0.438303i
\(39\) 2.61902 − 2.89596i 0.419378 − 0.463724i
\(40\) 3.65653 + 0.0220581i 0.578149 + 0.00348770i
\(41\) − 1.39634i − 0.218072i −0.994038 − 0.109036i \(-0.965224\pi\)
0.994038 − 0.109036i \(-0.0347764\pi\)
\(42\) 3.12218 − 5.67908i 0.481763 − 0.876302i
\(43\) −6.02578 −0.918922 −0.459461 − 0.888198i \(-0.651957\pi\)
−0.459461 + 0.888198i \(0.651957\pi\)
\(44\) −2.36378 − 2.84016i −0.356353 − 0.428171i
\(45\) −3.14763 + 2.26593i −0.469222 + 0.337785i
\(46\) 1.38277 − 0.500142i 0.203879 − 0.0737418i
\(47\) 5.90249 + 10.2234i 0.860966 + 1.49124i 0.870997 + 0.491288i \(0.163474\pi\)
−0.0100312 + 0.999950i \(0.503193\pi\)
\(48\) 0.893991 + 6.87028i 0.129036 + 0.991640i
\(49\) 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
\(52\) −1.55907 + 4.23049i −0.216204 + 0.586664i
\(53\) 6.02901 − 10.4425i 0.828148 − 1.43439i −0.0713414 − 0.997452i \(-0.522728\pi\)
0.899489 − 0.436943i \(-0.143939\pi\)
\(54\) −5.05920 − 5.32959i −0.688469 − 0.725265i
\(55\) 2.38854i 0.322070i
\(56\) −0.899769 + 7.42903i −0.120237 + 0.992745i
\(57\) 2.04356 − 9.51192i 0.270676 − 1.25988i
\(58\) −0.328068 + 1.83894i −0.0430775 + 0.241464i
\(59\) −9.57732 − 5.52947i −1.24686 − 0.719875i −0.276378 − 0.961049i \(-0.589134\pi\)
−0.970482 + 0.241174i \(0.922468\pi\)
\(60\) 2.39672 − 3.78309i 0.309415 − 0.488395i
\(61\) −7.65220 + 4.41800i −0.979764 + 0.565667i −0.902199 − 0.431320i \(-0.858048\pi\)
−0.0775652 + 0.996987i \(0.524715\pi\)
\(62\) −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
\(66\) −4.48415 + 0.611027i −0.551961 + 0.0752123i
\(67\) 3.05545 − 5.29219i 0.373283 − 0.646544i −0.616786 − 0.787131i \(-0.711566\pi\)
0.990068 + 0.140587i \(0.0448989\pi\)
\(68\) −5.75538 − 6.91528i −0.697942 − 0.838601i
\(69\) 0.378281 − 1.76074i 0.0455397 − 0.211968i
\(70\) 3.51897 − 3.31897i 0.420598 − 0.396693i
\(71\) −14.0121 −1.66293 −0.831464 − 0.555578i \(-0.812497\pi\)
−0.831464 + 0.555578i \(0.812497\pi\)
\(72\) 7.71536 + 3.53174i 0.909264 + 0.416219i
\(73\) −4.38664 + 7.59788i −0.513417 + 0.889264i 0.486462 + 0.873702i \(0.338287\pi\)
−0.999879 + 0.0155626i \(0.995046\pi\)
\(74\) 4.54561 + 12.5675i 0.528416 + 1.46095i
\(75\) 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
\(78\) 3.38135 + 4.36556i 0.382862 + 0.494303i
\(79\) −2.37061 + 1.36867i −0.266714 + 0.153987i −0.627393 − 0.778702i \(-0.715878\pi\)
0.360679 + 0.932690i \(0.382545\pi\)
\(80\) −0.938904 + 5.08526i −0.104973 + 0.568550i
\(81\) −8.81935 + 1.79416i −0.979928 + 0.199352i
\(82\) 1.94403 + 0.346817i 0.214682 + 0.0382996i
\(83\) − 4.74366i − 0.520685i −0.965516 − 0.260342i \(-0.916165\pi\)
0.965516 − 0.260342i \(-0.0838354\pi\)
\(84\) 7.13113 + 5.75734i 0.778070 + 0.628178i
\(85\) 5.81566i 0.630797i
\(86\) 1.49666 − 8.38928i 0.161389 − 0.904639i
\(87\) 1.69681 + 1.53454i 0.181917 + 0.164520i
\(88\) 4.54127 − 2.58550i 0.484101 − 0.275616i
\(89\) −8.31219 + 4.79904i −0.881090 + 0.508697i −0.871018 − 0.491252i \(-0.836539\pi\)
−0.0100723 + 0.999949i \(0.503206\pi\)
\(90\) −2.37291 − 4.94504i −0.250126 − 0.521253i
\(91\) 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
\(94\) −15.6994 + 5.67839i −1.61927 + 0.585681i
\(95\) 3.63085 − 6.28882i 0.372517 − 0.645219i
\(96\) −9.78707 − 0.461768i −0.998889 − 0.0471290i
\(97\) −8.73466 −0.886871 −0.443435 − 0.896306i \(-0.646240\pi\)
−0.443435 + 0.896306i \(0.646240\pi\)
\(98\) 5.92512 + 7.93051i 0.598528 + 0.801102i
\(99\) −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.11.13 yes 56
3.2 odd 2 inner 168.2.v.a.11.16 yes 56
4.3 odd 2 672.2.bd.a.431.6 56
7.2 even 3 inner 168.2.v.a.107.25 yes 56
8.3 odd 2 inner 168.2.v.a.11.4 ✓ 56
8.5 even 2 672.2.bd.a.431.5 56
12.11 even 2 672.2.bd.a.431.25 56
21.2 odd 6 inner 168.2.v.a.107.4 yes 56
24.5 odd 2 672.2.bd.a.431.26 56
24.11 even 2 inner 168.2.v.a.11.25 yes 56
28.23 odd 6 672.2.bd.a.527.26 56
56.37 even 6 672.2.bd.a.527.25 56
56.51 odd 6 inner 168.2.v.a.107.16 yes 56
84.23 even 6 672.2.bd.a.527.5 56
168.107 even 6 inner 168.2.v.a.107.13 yes 56
168.149 odd 6 672.2.bd.a.527.6 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.4 ✓ 56 8.3 odd 2 inner
168.2.v.a.11.13 yes 56 1.1 even 1 trivial
168.2.v.a.11.16 yes 56 3.2 odd 2 inner
168.2.v.a.11.25 yes 56 24.11 even 2 inner
168.2.v.a.107.4 yes 56 21.2 odd 6 inner
168.2.v.a.107.13 yes 56 168.107 even 6 inner
168.2.v.a.107.16 yes 56 56.51 odd 6 inner
168.2.v.a.107.25 yes 56 7.2 even 3 inner
672.2.bd.a.431.5 56 8.5 even 2
672.2.bd.a.431.6 56 4.3 odd 2
672.2.bd.a.431.25 56 12.11 even 2
672.2.bd.a.431.26 56 24.5 odd 2
672.2.bd.a.527.5 56 84.23 even 6
672.2.bd.a.527.6 56 168.149 odd 6
672.2.bd.a.527.25 56 56.37 even 6
672.2.bd.a.527.26 56 28.23 odd 6