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Properties

Label 266.2.bd.a.143.14
Level $266$
Weight $2$
Character 266.143
Analytic conductor $2.124$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [266,2,Mod(33,266)] 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("266.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(266, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([15, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.bd (of order \(18\), degree \(6\), 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: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 143.14
Character \(\chi\) \(=\) 266.143
Dual form 266.2.bd.a.173.14

$q$-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+(0.642788 + 0.766044i) q^{2} +(0.552974 + 3.13607i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(-2.41781 + 0.426326i) q^{5} +(-2.04692 + 2.43943i) q^{6} +(1.16653 - 2.37470i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-6.71008 + 2.44227i) q^{9} +(-1.88073 - 1.57812i) q^{10} +3.23570 q^{11} -3.18445 q^{12} +(2.24189 + 1.88117i) q^{13} +(2.56896 - 0.632810i) q^{14} +(-2.67398 - 7.34669i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(0.00506203 - 0.0139078i) q^{17} +(-6.18404 - 3.57036i) q^{18} +(3.48373 + 2.61985i) q^{19} -2.45511i q^{20} +(8.09229 + 2.34519i) q^{21} +(2.07987 + 2.47869i) q^{22} +(-4.66285 - 3.91260i) q^{23} +(-2.04692 - 2.43943i) q^{24} +(0.965607 - 0.351452i) q^{25} +2.92658i q^{26} +(-6.59296 - 11.4193i) q^{27} +(2.13606 + 1.56117i) q^{28} +(-0.993467 - 0.175175i) q^{29} +(3.90909 - 6.77074i) q^{30} +(4.28926 + 7.42921i) q^{31} +(-0.342020 - 0.939693i) q^{32} +(1.78926 + 10.1474i) q^{33} +(0.0139078 - 0.00506203i) q^{34} +(-1.80807 + 6.23890i) q^{35} +(-1.23997 - 7.03224i) q^{36} +(9.19842 - 5.31071i) q^{37} +(0.232382 + 4.35270i) q^{38} +(-4.65977 + 8.07096i) q^{39} +(1.88073 - 1.57812i) q^{40} +(-2.23727 + 1.87729i) q^{41} +(3.40510 + 7.70651i) q^{42} +(-1.85833 - 0.676376i) q^{43} +(-0.561874 + 3.18655i) q^{44} +(15.1825 - 8.76563i) q^{45} -6.08692i q^{46} +(2.03144 + 5.58133i) q^{47} +(0.552974 - 3.13607i) q^{48} +(-4.27839 - 5.54034i) q^{49} +(0.889908 + 0.513789i) q^{50} +(0.0464150 + 0.00818422i) q^{51} +(-2.24189 + 1.88117i) q^{52} +(7.18049 + 1.26611i) q^{53} +(4.50985 - 12.3907i) q^{54} +(-7.82333 + 1.37946i) q^{55} +(0.177101 + 2.63982i) q^{56} +(-6.28961 + 12.3739i) q^{57} +(-0.504397 - 0.873641i) q^{58} +(-7.52606 - 2.73926i) q^{59} +(7.69941 - 1.35761i) q^{60} +(1.69613 - 2.02137i) q^{61} +(-2.93402 + 8.06117i) q^{62} +(-2.02788 + 18.7834i) q^{63} +(0.500000 - 0.866025i) q^{64} +(-6.22246 - 3.59254i) q^{65} +(-6.62324 + 7.89327i) q^{66} +(-2.32842 + 2.77490i) q^{67} +(0.0128175 + 0.00740019i) q^{68} +(9.69175 - 16.7866i) q^{69} +(-5.94148 + 2.62523i) q^{70} +(5.01111 - 13.7679i) q^{71} +(4.58997 - 5.47011i) q^{72} +(11.6551 - 2.05510i) q^{73} +(9.98087 + 3.63274i) q^{74} +(1.63613 + 2.83387i) q^{75} +(-3.18499 + 2.97588i) q^{76} +(3.77456 - 7.68382i) q^{77} +(-9.17796 + 1.61832i) q^{78} +(4.04986 - 11.1269i) q^{79} +(2.41781 + 0.426326i) q^{80} +(15.7558 - 13.2207i) q^{81} +(-2.87618 - 0.507148i) q^{82} +(-2.43571 - 1.40626i) q^{83} +(-3.71477 + 7.56211i) q^{84} +(-0.00630978 + 0.0357846i) q^{85} +(-0.676376 - 1.85833i) q^{86} -3.21245i q^{87} +(-2.80220 + 1.61785i) q^{88} +(1.22753 - 6.96167i) q^{89} +(16.4740 + 5.99605i) q^{90} +(7.08245 - 3.12937i) q^{91} +(4.66285 - 3.91260i) q^{92} +(-20.9267 + 17.5596i) q^{93} +(-2.96977 + 5.14379i) q^{94} +(-9.53993 - 4.84910i) q^{95} +(2.75781 - 1.59222i) q^{96} +(-0.219899 - 1.24711i) q^{97} +(1.49405 - 6.83870i) q^{98} +(-21.7118 + 7.90246i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 12 q^{7} + 12 q^{11} - 12 q^{12} + 6 q^{13} + 6 q^{14} - 30 q^{15} - 54 q^{17} - 36 q^{18} - 42 q^{19} + 12 q^{21} - 18 q^{22} + 12 q^{23} + 36 q^{25} - 24 q^{27} + 12 q^{28} + 36 q^{34} + 6 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.642788 + 0.766044i 0.454519 + 0.541675i
\(3\) 0.552974 + 3.13607i 0.319260 + 1.81061i 0.547274 + 0.836953i \(0.315666\pi\)
−0.228014 + 0.973658i \(0.573223\pi\)
\(4\) −0.173648 + 0.984808i −0.0868241 + 0.492404i
\(5\) −2.41781 + 0.426326i −1.08128 + 0.190659i −0.685781 − 0.727808i \(-0.740539\pi\)
−0.395498 + 0.918467i \(0.629428\pi\)
\(6\) −2.04692 + 2.43943i −0.835654 + 0.995893i
\(7\) 1.16653 − 2.37470i 0.440909 − 0.897552i
\(8\) −0.866025 + 0.500000i −0.306186 + 0.176777i
\(9\) −6.71008 + 2.44227i −2.23669 + 0.814090i
\(10\) −1.88073 − 1.57812i −0.594738 − 0.499044i
\(11\) 3.23570 0.975601 0.487801 − 0.872955i \(-0.337799\pi\)
0.487801 + 0.872955i \(0.337799\pi\)
\(12\) −3.18445 −0.919271
\(13\) 2.24189 + 1.88117i 0.621788 + 0.521742i 0.898365 − 0.439250i \(-0.144756\pi\)
−0.276577 + 0.960992i \(0.589200\pi\)
\(14\) 2.56896 − 0.632810i 0.686583 − 0.169126i
\(15\) −2.67398 − 7.34669i −0.690418 − 1.89691i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) 0.00506203 − 0.0139078i 0.00122772 − 0.00337314i −0.939077 − 0.343706i \(-0.888318\pi\)
0.940305 + 0.340333i \(0.110540\pi\)
\(18\) −6.18404 − 3.57036i −1.45759 − 0.841542i
\(19\) 3.48373 + 2.61985i 0.799223 + 0.601034i
\(20\) − 2.45511i − 0.548980i
\(21\) 8.09229 + 2.34519i 1.76588 + 0.511762i
\(22\) 2.07987 + 2.47869i 0.443430 + 0.528459i
\(23\) −4.66285 − 3.91260i −0.972272 − 0.815833i 0.0106334 − 0.999943i \(-0.496615\pi\)
−0.982906 + 0.184110i \(0.941060\pi\)
\(24\) −2.04692 − 2.43943i −0.417827 − 0.497947i
\(25\) 0.965607 − 0.351452i 0.193121 − 0.0702904i
\(26\) 2.92658i 0.573949i
\(27\) −6.59296 − 11.4193i −1.26881 − 2.19765i
\(28\) 2.13606 + 1.56117i 0.403677 + 0.295034i
\(29\) −0.993467 − 0.175175i −0.184482 − 0.0325292i 0.0806437 − 0.996743i \(-0.474302\pi\)
−0.265126 + 0.964214i \(0.585414\pi\)
\(30\) 3.90909 − 6.77074i 0.713699 − 1.23616i
\(31\) 4.28926 + 7.42921i 0.770373 + 1.33433i 0.937358 + 0.348366i \(0.113264\pi\)
−0.166985 + 0.985959i \(0.553403\pi\)
\(32\) −0.342020 − 0.939693i −0.0604612 − 0.166116i
\(33\) 1.78926 + 10.1474i 0.311470 + 1.76643i
\(34\) 0.0139078 − 0.00506203i 0.00238517 − 0.000868131i
\(35\) −1.80807 + 6.23890i −0.305619 + 1.05457i
\(36\) −1.23997 − 7.03224i −0.206662 − 1.17204i
\(37\) 9.19842 − 5.31071i 1.51221 − 0.873076i 0.512313 − 0.858799i \(-0.328789\pi\)
0.999898 − 0.0142772i \(-0.00454471\pi\)
\(38\) 0.232382 + 4.35270i 0.0376974 + 0.706101i
\(39\) −4.65977 + 8.07096i −0.746161 + 1.29239i
\(40\) 1.88073 − 1.57812i 0.297369 − 0.249522i
\(41\) −2.23727 + 1.87729i −0.349403 + 0.293184i −0.800550 − 0.599266i \(-0.795459\pi\)
0.451147 + 0.892449i \(0.351015\pi\)
\(42\) 3.40510 + 7.70651i 0.525419 + 1.18914i
\(43\) −1.85833 − 0.676376i −0.283392 − 0.103146i 0.196413 − 0.980521i \(-0.437071\pi\)
−0.479806 + 0.877375i \(0.659293\pi\)
\(44\) −0.561874 + 3.18655i −0.0847057 + 0.480390i
\(45\) 15.1825 − 8.76563i 2.26328 − 1.30670i
\(46\) − 6.08692i − 0.897468i
\(47\) 2.03144 + 5.58133i 0.296316 + 0.814121i 0.995108 + 0.0987961i \(0.0314992\pi\)
−0.698792 + 0.715325i \(0.746279\pi\)
\(48\) 0.552974 − 3.13607i 0.0798149 − 0.452653i
\(49\) −4.27839 − 5.54034i −0.611199 − 0.791477i
\(50\) 0.889908 + 0.513789i 0.125852 + 0.0726607i
\(51\) 0.0464150 + 0.00818422i 0.00649940 + 0.00114602i
\(52\) −2.24189 + 1.88117i −0.310894 + 0.260871i
\(53\) 7.18049 + 1.26611i 0.986317 + 0.173914i 0.643465 − 0.765475i \(-0.277496\pi\)
0.342851 + 0.939390i \(0.388607\pi\)
\(54\) 4.50985 − 12.3907i 0.613712 − 1.68616i
\(55\) −7.82333 + 1.37946i −1.05490 + 0.186007i
\(56\) 0.177101 + 2.63982i 0.0236662 + 0.352760i
\(57\) −6.28961 + 12.3739i −0.833079 + 1.63897i
\(58\) −0.504397 − 0.873641i −0.0662305 − 0.114715i
\(59\) −7.52606 − 2.73926i −0.979809 − 0.356621i −0.198043 − 0.980193i \(-0.563459\pi\)
−0.781766 + 0.623572i \(0.785681\pi\)
\(60\) 7.69941 − 1.35761i 0.993989 − 0.175267i
\(61\) 1.69613 − 2.02137i 0.217167 − 0.258810i −0.646452 − 0.762955i \(-0.723748\pi\)
0.863619 + 0.504145i \(0.168192\pi\)
\(62\) −2.93402 + 8.06117i −0.372622 + 1.02377i
\(63\) −2.02788 + 18.7834i −0.255489 + 2.36649i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) −6.22246 − 3.59254i −0.771801 − 0.445600i
\(66\) −6.62324 + 7.89327i −0.815265 + 0.971595i
\(67\) −2.32842 + 2.77490i −0.284462 + 0.339008i −0.889287 − 0.457350i \(-0.848799\pi\)
0.604825 + 0.796358i \(0.293243\pi\)
\(68\) 0.0128175 + 0.00740019i 0.00155435 + 0.000897405i
\(69\) 9.69175 − 16.7866i 1.16675 − 2.02087i
\(70\) −5.94148 + 2.62523i −0.710143 + 0.313775i
\(71\) 5.01111 − 13.7679i 0.594709 − 1.63395i −0.166946 − 0.985966i \(-0.553391\pi\)
0.761655 − 0.647983i \(-0.224387\pi\)
\(72\) 4.58997 − 5.47011i 0.540933 − 0.644658i
\(73\) 11.6551 − 2.05510i 1.36412 − 0.240532i 0.556802 − 0.830645i \(-0.312028\pi\)
0.807320 + 0.590113i \(0.200917\pi\)
\(74\) 9.98087 + 3.63274i 1.16025 + 0.422298i
\(75\) 1.63613 + 2.83387i 0.188924 + 0.327227i
\(76\) −3.18499 + 2.97588i −0.365343 + 0.341356i
\(77\) 3.77456 − 7.68382i 0.430151 − 0.875653i
\(78\) −9.17796 + 1.61832i −1.03920 + 0.183239i
\(79\) 4.04986 − 11.1269i 0.455645 − 1.25188i −0.473051 − 0.881035i \(-0.656847\pi\)
0.928697 − 0.370840i \(-0.120930\pi\)
\(80\) 2.41781 + 0.426326i 0.270320 + 0.0476647i
\(81\) 15.7558 − 13.2207i 1.75064 − 1.46896i
\(82\) −2.87618 − 0.507148i −0.317621 − 0.0560051i
\(83\) −2.43571 − 1.40626i −0.267354 − 0.154357i 0.360331 − 0.932825i \(-0.382664\pi\)
−0.627684 + 0.778468i \(0.715997\pi\)
\(84\) −3.71477 + 7.56211i −0.405315 + 0.825094i
\(85\) −0.00630978 + 0.0357846i −0.000684392 + 0.00388138i
\(86\) −0.676376 − 1.85833i −0.0729355 − 0.200389i
\(87\) − 3.21245i − 0.344411i
\(88\) −2.80220 + 1.61785i −0.298716 + 0.172464i
\(89\) 1.22753 − 6.96167i 0.130118 − 0.737936i −0.848018 − 0.529968i \(-0.822204\pi\)
0.978136 − 0.207968i \(-0.0666850\pi\)
\(90\) 16.4740 + 5.99605i 1.73651 + 0.632039i
\(91\) 7.08245 − 3.12937i 0.742443 − 0.328047i
\(92\) 4.66285 − 3.91260i 0.486136 − 0.407917i
\(93\) −20.9267 + 17.5596i −2.17000 + 1.82084i
\(94\) −2.96977 + 5.14379i −0.306308 + 0.530541i
\(95\) −9.53993 − 4.84910i −0.978776 − 0.497507i
\(96\) 2.75781 − 1.59222i 0.281468 − 0.162506i
\(97\) −0.219899 − 1.24711i −0.0223273 − 0.126625i 0.971607 − 0.236601i \(-0.0760335\pi\)
−0.993934 + 0.109977i \(0.964922\pi\)
\(98\) 1.49405 − 6.83870i 0.150921 − 0.690813i
\(99\) −21.7118 + 7.90246i −2.18212 + 0.794227i
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 266.2.bd.a.143.14 yes 84
7.5 odd 6 266.2.y.a.257.14 yes 84
19.2 odd 18 266.2.y.a.59.14 ✓ 84
133.40 even 18 inner 266.2.bd.a.173.14 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.y.a.59.14 ✓ 84 19.2 odd 18
266.2.y.a.257.14 yes 84 7.5 odd 6
266.2.bd.a.143.14 yes 84 1.1 even 1 trivial
266.2.bd.a.173.14 yes 84 133.40 even 18 inner