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Properties

Label 2106.2.b.c.649.5
Level $2106$
Weight $2$
Character 2106.649
Analytic conductor $16.816$
Analytic rank $0$
Dimension $14$
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] = [2106,2,Mod(649,2106)] 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("2106.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2106.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,-14,0,0,0,0,0,0,0,0,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.8164946657\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.5
Root \(3.32820i\) of defining polynomial
Character \(\chi\) \(=\) 2106.649
Dual form 2106.2.b.c.649.10

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +0.594758i q^{5} +1.67682i q^{7} +1.00000i q^{8} +0.594758 q^{10} +0.480744i q^{11} +(-1.28905 - 3.36725i) q^{13} +1.67682 q^{14} +1.00000 q^{16} -2.09349 q^{17} +0.480744i q^{19} -0.594758i q^{20} +0.480744 q^{22} +3.66679 q^{23} +4.64626 q^{25} +(-3.36725 + 1.28905i) q^{26} -1.67682i q^{28} -2.46678 q^{29} -1.14753i q^{31} -1.00000i q^{32} +2.09349i q^{34} -0.997301 q^{35} +3.65012i q^{37} +0.480744 q^{38} -0.594758 q^{40} +9.91005i q^{41} +6.91644 q^{43} -0.480744i q^{44} -3.66679i q^{46} +6.24102i q^{47} +4.18828 q^{49} -4.64626i q^{50} +(1.28905 + 3.36725i) q^{52} -5.08592 q^{53} -0.285927 q^{55} -1.67682 q^{56} +2.46678i q^{58} +9.38985i q^{59} +7.81270 q^{61} -1.14753 q^{62} -1.00000 q^{64} +(2.00270 - 0.766671i) q^{65} +14.3819i q^{67} +2.09349 q^{68} +0.997301i q^{70} -6.51028i q^{71} +5.91514i q^{73} +3.65012 q^{74} -0.480744i q^{76} -0.806121 q^{77} -2.05790 q^{79} +0.594758i q^{80} +9.91005 q^{82} +11.0601i q^{83} -1.24512i q^{85} -6.91644i q^{86} -0.480744 q^{88} +9.48720i q^{89} +(5.64626 - 2.16150i) q^{91} -3.66679 q^{92} +6.24102 q^{94} -0.285927 q^{95} -9.71535i q^{97} -4.18828i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times\).

\(n\) \(1379\) \(1783\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0.594758i 0.265984i 0.991117 + 0.132992i \(0.0424585\pi\)
−0.991117 + 0.132992i \(0.957542\pi\)
\(6\) 0 0
\(7\) 1.67682i 0.633778i 0.948463 + 0.316889i \(0.102638\pi\)
−0.948463 + 0.316889i \(0.897362\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.594758 0.188079
\(11\) 0.480744i 0.144950i 0.997370 + 0.0724750i \(0.0230897\pi\)
−0.997370 + 0.0724750i \(0.976910\pi\)
\(12\) 0 0
\(13\) −1.28905 − 3.36725i −0.357517 − 0.933907i
\(14\) 1.67682 0.448149
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.09349 −0.507746 −0.253873 − 0.967238i \(-0.581705\pi\)
−0.253873 + 0.967238i \(0.581705\pi\)
\(18\) 0 0
\(19\) 0.480744i 0.110290i 0.998478 + 0.0551452i \(0.0175622\pi\)
−0.998478 + 0.0551452i \(0.982438\pi\)
\(20\) − 0.594758i − 0.132992i
\(21\) 0 0
\(22\) 0.480744 0.102495
\(23\) 3.66679 0.764578 0.382289 − 0.924043i \(-0.375136\pi\)
0.382289 + 0.924043i \(0.375136\pi\)
\(24\) 0 0
\(25\) 4.64626 0.929253
\(26\) −3.36725 + 1.28905i −0.660372 + 0.252803i
\(27\) 0 0
\(28\) − 1.67682i − 0.316889i
\(29\) −2.46678 −0.458069 −0.229035 − 0.973418i \(-0.573557\pi\)
−0.229035 + 0.973418i \(0.573557\pi\)
\(30\) 0 0
\(31\) − 1.14753i − 0.206103i −0.994676 − 0.103051i \(-0.967139\pi\)
0.994676 − 0.103051i \(-0.0328606\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 2.09349i 0.359031i
\(35\) −0.997301 −0.168575
\(36\) 0 0
\(37\) 3.65012i 0.600076i 0.953927 + 0.300038i \(0.0969994\pi\)
−0.953927 + 0.300038i \(0.903001\pi\)
\(38\) 0.480744 0.0779870
\(39\) 0 0
\(40\) −0.594758 −0.0940395
\(41\) 9.91005i 1.54769i 0.633376 + 0.773845i \(0.281669\pi\)
−0.633376 + 0.773845i \(0.718331\pi\)
\(42\) 0 0
\(43\) 6.91644 1.05475 0.527374 − 0.849633i \(-0.323177\pi\)
0.527374 + 0.849633i \(0.323177\pi\)
\(44\) − 0.480744i − 0.0724750i
\(45\) 0 0
\(46\) − 3.66679i − 0.540638i
\(47\) 6.24102i 0.910346i 0.890403 + 0.455173i \(0.150423\pi\)
−0.890403 + 0.455173i \(0.849577\pi\)
\(48\) 0 0
\(49\) 4.18828 0.598326
\(50\) − 4.64626i − 0.657081i
\(51\) 0 0
\(52\) 1.28905 + 3.36725i 0.178759 + 0.466953i
\(53\) −5.08592 −0.698605 −0.349302 − 0.937010i \(-0.613581\pi\)
−0.349302 + 0.937010i \(0.613581\pi\)
\(54\) 0 0
\(55\) −0.285927 −0.0385544
\(56\) −1.67682 −0.224074
\(57\) 0 0
\(58\) 2.46678i 0.323904i
\(59\) 9.38985i 1.22245i 0.791455 + 0.611227i \(0.209324\pi\)
−0.791455 + 0.611227i \(0.790676\pi\)
\(60\) 0 0
\(61\) 7.81270 1.00031 0.500157 − 0.865935i \(-0.333276\pi\)
0.500157 + 0.865935i \(0.333276\pi\)
\(62\) −1.14753 −0.145737
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.00270 − 0.766671i 0.248404 − 0.0950939i
\(66\) 0 0
\(67\) 14.3819i 1.75703i 0.477714 + 0.878516i \(0.341466\pi\)
−0.477714 + 0.878516i \(0.658534\pi\)
\(68\) 2.09349 0.253873
\(69\) 0 0
\(70\) 0.997301i 0.119200i
\(71\) − 6.51028i − 0.772628i −0.922367 − 0.386314i \(-0.873748\pi\)
0.922367 − 0.386314i \(-0.126252\pi\)
\(72\) 0 0
\(73\) 5.91514i 0.692315i 0.938176 + 0.346157i \(0.112514\pi\)
−0.938176 + 0.346157i \(0.887486\pi\)
\(74\) 3.65012 0.424318
\(75\) 0 0
\(76\) − 0.480744i − 0.0551452i
\(77\) −0.806121 −0.0918660
\(78\) 0 0
\(79\) −2.05790 −0.231532 −0.115766 − 0.993277i \(-0.536932\pi\)
−0.115766 + 0.993277i \(0.536932\pi\)
\(80\) 0.594758i 0.0664960i
\(81\) 0 0
\(82\) 9.91005 1.09438
\(83\) 11.0601i 1.21401i 0.794700 + 0.607003i \(0.207628\pi\)
−0.794700 + 0.607003i \(0.792372\pi\)
\(84\) 0 0
\(85\) − 1.24512i − 0.135052i
\(86\) − 6.91644i − 0.745819i
\(87\) 0 0
\(88\) −0.480744 −0.0512475
\(89\) 9.48720i 1.00564i 0.864391 + 0.502821i \(0.167704\pi\)
−0.864391 + 0.502821i \(0.832296\pi\)
\(90\) 0 0
\(91\) 5.64626 − 2.16150i 0.591889 − 0.226586i
\(92\) −3.66679 −0.382289
\(93\) 0 0
\(94\) 6.24102 0.643712
\(95\) −0.285927 −0.0293355
\(96\) 0 0
\(97\) − 9.71535i − 0.986444i −0.869903 − 0.493222i \(-0.835819\pi\)
0.869903 − 0.493222i \(-0.164181\pi\)
\(98\) − 4.18828i − 0.423080i
\(99\) 0 0
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 2106.2.b.c.649.5 14
3.2 odd 2 2106.2.b.d.649.10 14
9.2 odd 6 702.2.t.a.415.12 28
9.4 even 3 234.2.t.a.25.14 yes 28
9.5 odd 6 702.2.t.a.181.3 28
9.7 even 3 234.2.t.a.103.7 yes 28
13.12 even 2 inner 2106.2.b.c.649.10 14
39.38 odd 2 2106.2.b.d.649.5 14
117.25 even 6 234.2.t.a.103.14 yes 28
117.38 odd 6 702.2.t.a.415.3 28
117.77 odd 6 702.2.t.a.181.12 28
117.103 even 6 234.2.t.a.25.7 ✓ 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.7 ✓ 28 117.103 even 6
234.2.t.a.25.14 yes 28 9.4 even 3
234.2.t.a.103.7 yes 28 9.7 even 3
234.2.t.a.103.14 yes 28 117.25 even 6
702.2.t.a.181.3 28 9.5 odd 6
702.2.t.a.181.12 28 117.77 odd 6
702.2.t.a.415.3 28 117.38 odd 6
702.2.t.a.415.12 28 9.2 odd 6
2106.2.b.c.649.5 14 1.1 even 1 trivial
2106.2.b.c.649.10 14 13.12 even 2 inner
2106.2.b.d.649.5 14 39.38 odd 2
2106.2.b.d.649.10 14 3.2 odd 2