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Properties

Label 266.2.t.b.103.4
Level $266$
Weight $2$
Character 266.103
Analytic conductor $2.124$
Analytic rank $0$
Dimension $20$
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(31,266)] 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("266.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(266, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} + 10 x^{17} + 28 x^{16} - 146 x^{15} + 410 x^{14} + 628 x^{13} + \cdots + 1521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.4
Root \(-0.355982 - 0.355982i\) of defining polynomial
Character \(\chi\) \(=\) 266.103
Dual form 266.2.t.b.31.4

$q$-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+(-0.866025 + 0.500000i) q^{2} +0.102618 q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.309363 + 0.178611i) q^{5} +(-0.0888701 + 0.0513092i) q^{6} +(-2.28073 + 1.34100i) q^{7} +1.00000i q^{8} -2.98947 q^{9} +(0.178611 - 0.309363i) q^{10} +(2.94165 + 5.09508i) q^{11} +(0.0513092 - 0.0888701i) q^{12} +(2.26521 + 3.92345i) q^{13} +(1.30467 - 2.30170i) q^{14} +(-0.0317463 + 0.0183287i) q^{15} +(-0.500000 - 0.866025i) q^{16} -2.57407i q^{17} +(2.58896 - 1.49473i) q^{18} +(-3.90498 + 1.93678i) q^{19} +0.357221i q^{20} +(-0.234045 + 0.137611i) q^{21} +(-5.09508 - 2.94165i) q^{22} +4.95996 q^{23} +0.102618i q^{24} +(-2.43620 + 4.21962i) q^{25} +(-3.92345 - 2.26521i) q^{26} -0.614630 q^{27} +(0.0209708 + 2.64567i) q^{28} +(-3.07264 + 1.77399i) q^{29} +(0.0183287 - 0.0317463i) q^{30} +(1.61239 + 2.79273i) q^{31} +(0.866025 + 0.500000i) q^{32} +(0.301867 + 0.522849i) q^{33} +(1.28704 + 2.22921i) q^{34} +(0.466057 - 0.822217i) q^{35} +(-1.49473 + 2.58896i) q^{36} +(-5.83284 - 3.36759i) q^{37} +(2.41343 - 3.62979i) q^{38} +(0.232452 + 0.402619i) q^{39} +(-0.178611 - 0.309363i) q^{40} +(1.15983 - 2.00888i) q^{41} +(0.133883 - 0.236197i) q^{42} +(-0.215938 + 0.374016i) q^{43} +5.88329 q^{44} +(0.924830 - 0.533951i) q^{45} +(-4.29545 + 2.47998i) q^{46} -11.3556i q^{47} +(-0.0513092 - 0.0888701i) q^{48} +(3.40346 - 6.11690i) q^{49} -4.87239i q^{50} -0.264147i q^{51} +4.53041 q^{52} +(-2.36090 - 1.36307i) q^{53} +(0.532285 - 0.307315i) q^{54} +(-1.82007 - 1.05082i) q^{55} +(-1.34100 - 2.28073i) q^{56} +(-0.400723 + 0.198749i) q^{57} +(1.77399 - 3.07264i) q^{58} +14.4335 q^{59} +0.0366575i q^{60} +9.09161i q^{61} +(-2.79273 - 1.61239i) q^{62} +(6.81817 - 4.00886i) q^{63} -1.00000 q^{64} +(-1.40154 - 0.809180i) q^{65} +(-0.522849 - 0.301867i) q^{66} +(10.6572 + 6.15291i) q^{67} +(-2.22921 - 1.28704i) q^{68} +0.508983 q^{69} +(0.00749120 + 0.945089i) q^{70} +(6.25104 + 3.60904i) q^{71} -2.98947i q^{72} -16.7575i q^{73} +6.73518 q^{74} +(-0.249999 + 0.433010i) q^{75} +(-0.275192 + 4.35020i) q^{76} +(-13.5416 - 7.67578i) q^{77} +(-0.402619 - 0.232452i) q^{78} +(-9.18010 + 5.30013i) q^{79} +(0.309363 + 0.178611i) q^{80} +8.90534 q^{81} +2.31966i q^{82} -5.16042i q^{83} +(0.00215199 + 0.271494i) q^{84} +(0.459757 + 0.796322i) q^{85} -0.431877i q^{86} +(-0.315310 + 0.182044i) q^{87} +(-5.09508 + 2.94165i) q^{88} -3.10864 q^{89} +(-0.533951 + 0.924830i) q^{90} +(-10.4277 - 5.91071i) q^{91} +(2.47998 - 4.29545i) q^{92} +(0.165460 + 0.286586i) q^{93} +(5.67778 + 9.83421i) q^{94} +(0.862127 - 1.29664i) q^{95} +(0.0888701 + 0.0513092i) q^{96} +(4.57047 - 7.91628i) q^{97} +(0.110963 + 6.99912i) q^{98} +(-8.79396 - 15.2316i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{4} - 6 q^{5} + 6 q^{6} - 2 q^{7} + 16 q^{9} + 4 q^{10} + 4 q^{11} + 12 q^{13} + 16 q^{14} + 18 q^{15} - 10 q^{16} - 18 q^{19} + 24 q^{22} - 28 q^{23} + 8 q^{25} - 12 q^{26} - 4 q^{28} - 12 q^{29}+ \cdots - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(115\) \(211\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0.102618 0.0592468 0.0296234 − 0.999561i \(-0.490569\pi\)
0.0296234 + 0.999561i \(0.490569\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −0.309363 + 0.178611i −0.138351 + 0.0798771i −0.567578 − 0.823320i \(-0.692119\pi\)
0.429227 + 0.903197i \(0.358786\pi\)
\(6\) −0.0888701 + 0.0513092i −0.0362811 + 0.0209469i
\(7\) −2.28073 + 1.34100i −0.862035 + 0.506849i
\(8\) 1.00000i 0.353553i
\(9\) −2.98947 −0.996490
\(10\) 0.178611 − 0.309363i 0.0564817 − 0.0978291i
\(11\) 2.94165 + 5.09508i 0.886940 + 1.53623i 0.843474 + 0.537170i \(0.180507\pi\)
0.0434662 + 0.999055i \(0.486160\pi\)
\(12\) 0.0513092 − 0.0888701i 0.0148117 − 0.0256546i
\(13\) 2.26521 + 3.92345i 0.628256 + 1.08817i 0.987902 + 0.155082i \(0.0495641\pi\)
−0.359646 + 0.933089i \(0.617103\pi\)
\(14\) 1.30467 − 2.30170i 0.348688 − 0.615156i
\(15\) −0.0317463 + 0.0183287i −0.00819686 + 0.00473246i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 2.57407i − 0.624304i −0.950032 − 0.312152i \(-0.898950\pi\)
0.950032 − 0.312152i \(-0.101050\pi\)
\(18\) 2.58896 − 1.49473i 0.610223 − 0.352312i
\(19\) −3.90498 + 1.93678i −0.895864 + 0.444327i
\(20\) 0.357221i 0.0798771i
\(21\) −0.234045 + 0.137611i −0.0510728 + 0.0300291i
\(22\) −5.09508 − 2.94165i −1.08628 − 0.627161i
\(23\) 4.95996 1.03422 0.517112 − 0.855918i \(-0.327007\pi\)
0.517112 + 0.855918i \(0.327007\pi\)
\(24\) 0.102618i 0.0209469i
\(25\) −2.43620 + 4.21962i −0.487239 + 0.843923i
\(26\) −3.92345 − 2.26521i −0.769453 − 0.444244i
\(27\) −0.614630 −0.118286
\(28\) 0.0209708 + 2.64567i 0.00396310 + 0.499984i
\(29\) −3.07264 + 1.77399i −0.570576 + 0.329422i −0.757379 − 0.652975i \(-0.773521\pi\)
0.186803 + 0.982397i \(0.440187\pi\)
\(30\) 0.0183287 − 0.0317463i 0.00334636 − 0.00579606i
\(31\) 1.61239 + 2.79273i 0.289593 + 0.501590i 0.973713 − 0.227780i \(-0.0731467\pi\)
−0.684120 + 0.729370i \(0.739813\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0.301867 + 0.522849i 0.0525483 + 0.0910164i
\(34\) 1.28704 + 2.22921i 0.220725 + 0.382307i
\(35\) 0.466057 − 0.822217i 0.0787780 − 0.138980i
\(36\) −1.49473 + 2.58896i −0.249122 + 0.431493i
\(37\) −5.83284 − 3.36759i −0.958912 − 0.553628i −0.0630742 − 0.998009i \(-0.520090\pi\)
−0.895838 + 0.444381i \(0.853424\pi\)
\(38\) 2.41343 − 3.62979i 0.391509 − 0.588830i
\(39\) 0.232452 + 0.402619i 0.0372221 + 0.0644706i
\(40\) −0.178611 − 0.309363i −0.0282408 − 0.0489145i
\(41\) 1.15983 − 2.00888i 0.181135 − 0.313735i −0.761132 − 0.648596i \(-0.775356\pi\)
0.942267 + 0.334862i \(0.108690\pi\)
\(42\) 0.133883 − 0.236197i 0.0206587 − 0.0364460i
\(43\) −0.215938 + 0.374016i −0.0329303 + 0.0570369i −0.882021 − 0.471210i \(-0.843817\pi\)
0.849091 + 0.528247i \(0.177151\pi\)
\(44\) 5.88329 0.886940
\(45\) 0.924830 − 0.533951i 0.137866 − 0.0795967i
\(46\) −4.29545 + 2.47998i −0.633330 + 0.365653i
\(47\) − 11.3556i − 1.65638i −0.560447 − 0.828190i \(-0.689371\pi\)
0.560447 − 0.828190i \(-0.310629\pi\)
\(48\) −0.0513092 − 0.0888701i −0.00740585 − 0.0128273i
\(49\) 3.40346 − 6.11690i 0.486209 − 0.873843i
\(50\) − 4.87239i − 0.689060i
\(51\) − 0.264147i − 0.0369880i
\(52\) 4.53041 0.628256
\(53\) −2.36090 − 1.36307i −0.324294 − 0.187231i 0.329011 − 0.944326i \(-0.393285\pi\)
−0.653305 + 0.757095i \(0.726618\pi\)
\(54\) 0.532285 − 0.307315i 0.0724348 − 0.0418203i
\(55\) −1.82007 − 1.05082i −0.245418 − 0.141692i
\(56\) −1.34100 − 2.28073i −0.179198 − 0.304775i
\(57\) −0.400723 + 0.198749i −0.0530771 + 0.0263250i
\(58\) 1.77399 − 3.07264i 0.232937 − 0.403458i
\(59\) 14.4335 1.87908 0.939541 − 0.342435i \(-0.111252\pi\)
0.939541 + 0.342435i \(0.111252\pi\)
\(60\) 0.0366575i 0.00473246i
\(61\) 9.09161i 1.16406i 0.813167 + 0.582031i \(0.197742\pi\)
−0.813167 + 0.582031i \(0.802258\pi\)
\(62\) −2.79273 − 1.61239i −0.354678 − 0.204773i
\(63\) 6.81817 − 4.00886i 0.859009 − 0.505069i
\(64\) −1.00000 −0.125000
\(65\) −1.40154 − 0.809180i −0.173840 − 0.100366i
\(66\) −0.522849 − 0.301867i −0.0643583 − 0.0371573i
\(67\) 10.6572 + 6.15291i 1.30198 + 0.751698i 0.980743 − 0.195301i \(-0.0625685\pi\)
0.321236 + 0.946999i \(0.395902\pi\)
\(68\) −2.22921 − 1.28704i −0.270332 − 0.156076i
\(69\) 0.508983 0.0612744
\(70\) 0.00749120 + 0.945089i 0.000895370 + 0.112960i
\(71\) 6.25104 + 3.60904i 0.741862 + 0.428314i 0.822746 − 0.568409i \(-0.192441\pi\)
−0.0808837 + 0.996724i \(0.525774\pi\)
\(72\) − 2.98947i − 0.352312i
\(73\) − 16.7575i − 1.96132i −0.195730 − 0.980658i \(-0.562707\pi\)
0.195730 − 0.980658i \(-0.437293\pi\)
\(74\) 6.73518 0.782949
\(75\) −0.249999 + 0.433010i −0.0288674 + 0.0499997i
\(76\) −0.275192 + 4.35020i −0.0315667 + 0.499003i
\(77\) −13.5416 − 7.67578i −1.54321 − 0.874736i
\(78\) −0.402619 − 0.232452i −0.0455876 − 0.0263200i
\(79\) −9.18010 + 5.30013i −1.03284 + 0.596312i −0.917797 − 0.397049i \(-0.870034\pi\)
−0.115044 + 0.993360i \(0.536701\pi\)
\(80\) 0.309363 + 0.178611i 0.0345878 + 0.0199693i
\(81\) 8.90534 0.989482
\(82\) 2.31966i 0.256163i
\(83\) − 5.16042i − 0.566430i −0.959057 − 0.283215i \(-0.908599\pi\)
0.959057 − 0.283215i \(-0.0914009\pi\)
\(84\) 0.00215199 + 0.271494i 0.000234801 + 0.0296225i
\(85\) 0.459757 + 0.796322i 0.0498676 + 0.0863733i
\(86\) − 0.431877i − 0.0465705i
\(87\) −0.315310 + 0.182044i −0.0338048 + 0.0195172i
\(88\) −5.09508 + 2.94165i −0.543138 + 0.313581i
\(89\) −3.10864 −0.329515 −0.164758 − 0.986334i \(-0.552684\pi\)
−0.164758 + 0.986334i \(0.552684\pi\)
\(90\) −0.533951 + 0.924830i −0.0562834 + 0.0974857i
\(91\) −10.4277 − 5.91071i −1.09312 − 0.619611i
\(92\) 2.47998 − 4.29545i 0.258556 − 0.447832i
\(93\) 0.165460 + 0.286586i 0.0171574 + 0.0297176i
\(94\) 5.67778 + 9.83421i 0.585619 + 1.01432i
\(95\) 0.862127 − 1.29664i 0.0884524 − 0.133032i
\(96\) 0.0888701 + 0.0513092i 0.00907027 + 0.00523672i
\(97\) 4.57047 − 7.91628i 0.464061 − 0.803777i −0.535098 − 0.844790i \(-0.679725\pi\)
0.999159 + 0.0410134i \(0.0130586\pi\)
\(98\) 0.110963 + 6.99912i 0.0112090 + 0.707018i
\(99\) −8.79396 − 15.2316i −0.883827 − 1.53083i
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.t.b.103.4 yes 20
7.3 odd 6 266.2.k.b.255.4 yes 20
19.12 odd 6 266.2.k.b.145.9 ✓ 20
133.31 even 6 inner 266.2.t.b.31.4 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.k.b.145.9 ✓ 20 19.12 odd 6
266.2.k.b.255.4 yes 20 7.3 odd 6
266.2.t.b.31.4 yes 20 133.31 even 6 inner
266.2.t.b.103.4 yes 20 1.1 even 1 trivial