Properties

Label 16.22.a.c.1.1
Level $16$
Weight $22$
Character 16.1
Self dual yes
Analytic conductor $44.716$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [16,22,Mod(1,16)] 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("16.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,128844] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7163750859\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 16.1

$q$-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)
 
\(f(q)\) \(=\) \(q+128844. q^{3} +2.16410e7 q^{5} +7.68079e8 q^{7} +6.14042e9 q^{9} +9.47249e10 q^{11} -8.06218e10 q^{13} +2.78831e12 q^{15} +3.05228e12 q^{17} +7.92079e12 q^{19} +9.89623e13 q^{21} +7.38454e13 q^{23} -8.50644e12 q^{25} -5.56597e14 q^{27} -4.25303e15 q^{29} -1.90054e15 q^{31} +1.22047e16 q^{33} +1.66220e16 q^{35} +2.21914e16 q^{37} -1.03876e16 q^{39} -2.06228e16 q^{41} +1.93606e17 q^{43} +1.32885e17 q^{45} -1.46961e17 q^{47} +3.13992e16 q^{49} +3.93268e17 q^{51} +2.03827e18 q^{53} +2.04994e18 q^{55} +1.02055e18 q^{57} +5.97588e18 q^{59} +6.19062e18 q^{61} +4.71633e18 q^{63} -1.74473e18 q^{65} -1.69613e19 q^{67} +9.51454e18 q^{69} +5.63276e18 q^{71} -4.32848e19 q^{73} -1.09600e18 q^{75} +7.27562e19 q^{77} +5.12649e19 q^{79} -1.35945e20 q^{81} -4.89119e19 q^{83} +6.60543e19 q^{85} -5.47978e20 q^{87} -5.04303e20 q^{89} -6.19239e19 q^{91} -2.44873e20 q^{93} +1.71413e20 q^{95} +8.08275e20 q^{97} +5.81651e20 q^{99} +O(q^{100})\)

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 0
\(3\) 128844. 1.25977 0.629885 − 0.776689i \(-0.283102\pi\)
0.629885 + 0.776689i \(0.283102\pi\)
\(4\) 0 0
\(5\) 2.16410e7 0.991040 0.495520 − 0.868596i \(-0.334977\pi\)
0.495520 + 0.868596i \(0.334977\pi\)
\(6\) 0 0
\(7\) 7.68079e8 1.02772 0.513862 − 0.857873i \(-0.328214\pi\)
0.513862 + 0.857873i \(0.328214\pi\)
\(8\) 0 0
\(9\) 6.14042e9 0.587019
\(10\) 0 0
\(11\) 9.47249e10 1.10114 0.550568 − 0.834790i \(-0.314411\pi\)
0.550568 + 0.834790i \(0.314411\pi\)
\(12\) 0 0
\(13\) −8.06218e10 −0.162199 −0.0810993 − 0.996706i \(-0.525843\pi\)
−0.0810993 + 0.996706i \(0.525843\pi\)
\(14\) 0 0
\(15\) 2.78831e12 1.24848
\(16\) 0 0
\(17\) 3.05228e12 0.367207 0.183604 − 0.983000i \(-0.441224\pi\)
0.183604 + 0.983000i \(0.441224\pi\)
\(18\) 0 0
\(19\) 7.92079e12 0.296385 0.148192 − 0.988959i \(-0.452655\pi\)
0.148192 + 0.988959i \(0.452655\pi\)
\(20\) 0 0
\(21\) 9.89623e13 1.29469
\(22\) 0 0
\(23\) 7.38454e13 0.371690 0.185845 − 0.982579i \(-0.440498\pi\)
0.185845 + 0.982579i \(0.440498\pi\)
\(24\) 0 0
\(25\) −8.50644e12 −0.0178393
\(26\) 0 0
\(27\) −5.56597e14 −0.520261
\(28\) 0 0
\(29\) −4.25303e15 −1.87724 −0.938620 − 0.344954i \(-0.887895\pi\)
−0.938620 + 0.344954i \(0.887895\pi\)
\(30\) 0 0
\(31\) −1.90054e15 −0.416466 −0.208233 − 0.978079i \(-0.566771\pi\)
−0.208233 + 0.978079i \(0.566771\pi\)
\(32\) 0 0
\(33\) 1.22047e16 1.38718
\(34\) 0 0
\(35\) 1.66220e16 1.01852
\(36\) 0 0
\(37\) 2.21914e16 0.758695 0.379347 − 0.925254i \(-0.376149\pi\)
0.379347 + 0.925254i \(0.376149\pi\)
\(38\) 0 0
\(39\) −1.03876e16 −0.204333
\(40\) 0 0
\(41\) −2.06228e16 −0.239948 −0.119974 − 0.992777i \(-0.538281\pi\)
−0.119974 + 0.992777i \(0.538281\pi\)
\(42\) 0 0
\(43\) 1.93606e17 1.36615 0.683077 − 0.730346i \(-0.260641\pi\)
0.683077 + 0.730346i \(0.260641\pi\)
\(44\) 0 0
\(45\) 1.32885e17 0.581759
\(46\) 0 0
\(47\) −1.46961e17 −0.407543 −0.203771 − 0.979019i \(-0.565320\pi\)
−0.203771 + 0.979019i \(0.565320\pi\)
\(48\) 0 0
\(49\) 3.13992e16 0.0562160
\(50\) 0 0
\(51\) 3.93268e17 0.462596
\(52\) 0 0
\(53\) 2.03827e18 1.60090 0.800450 − 0.599399i \(-0.204594\pi\)
0.800450 + 0.599399i \(0.204594\pi\)
\(54\) 0 0
\(55\) 2.04994e18 1.09127
\(56\) 0 0
\(57\) 1.02055e18 0.373376
\(58\) 0 0
\(59\) 5.97588e18 1.52214 0.761072 − 0.648667i \(-0.224673\pi\)
0.761072 + 0.648667i \(0.224673\pi\)
\(60\) 0 0
\(61\) 6.19062e18 1.11114 0.555572 − 0.831468i \(-0.312499\pi\)
0.555572 + 0.831468i \(0.312499\pi\)
\(62\) 0 0
\(63\) 4.71633e18 0.603293
\(64\) 0 0
\(65\) −1.74473e18 −0.160745
\(66\) 0 0
\(67\) −1.69613e19 −1.13677 −0.568387 − 0.822761i \(-0.692432\pi\)
−0.568387 + 0.822761i \(0.692432\pi\)
\(68\) 0 0
\(69\) 9.51454e18 0.468244
\(70\) 0 0
\(71\) 5.63276e18 0.205357 0.102678 − 0.994715i \(-0.467259\pi\)
0.102678 + 0.994715i \(0.467259\pi\)
\(72\) 0 0
\(73\) −4.32848e19 −1.17881 −0.589407 − 0.807837i \(-0.700638\pi\)
−0.589407 + 0.807837i \(0.700638\pi\)
\(74\) 0 0
\(75\) −1.09600e18 −0.0224734
\(76\) 0 0
\(77\) 7.27562e19 1.13166
\(78\) 0 0
\(79\) 5.12649e19 0.609166 0.304583 − 0.952486i \(-0.401483\pi\)
0.304583 + 0.952486i \(0.401483\pi\)
\(80\) 0 0
\(81\) −1.35945e20 −1.24243
\(82\) 0 0
\(83\) −4.89119e19 −0.346014 −0.173007 − 0.984921i \(-0.555348\pi\)
−0.173007 + 0.984921i \(0.555348\pi\)
\(84\) 0 0
\(85\) 6.60543e19 0.363917
\(86\) 0 0
\(87\) −5.47978e20 −2.36489
\(88\) 0 0
\(89\) −5.04303e20 −1.71434 −0.857170 − 0.515034i \(-0.827779\pi\)
−0.857170 + 0.515034i \(0.827779\pi\)
\(90\) 0 0
\(91\) −6.19239e19 −0.166695
\(92\) 0 0
\(93\) −2.44873e20 −0.524651
\(94\) 0 0
\(95\) 1.71413e20 0.293729
\(96\) 0 0
\(97\) 8.08275e20 1.11290 0.556450 − 0.830881i \(-0.312163\pi\)
0.556450 + 0.830881i \(0.312163\pi\)
\(98\) 0 0
\(99\) 5.81651e20 0.646388
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 16.22.a.c.1.1 1
4.3 odd 2 1.22.a.a.1.1 ✓ 1
8.3 odd 2 64.22.a.g.1.1 1
8.5 even 2 64.22.a.a.1.1 1
12.11 even 2 9.22.a.c.1.1 1
20.3 even 4 25.22.b.a.24.2 2
20.7 even 4 25.22.b.a.24.1 2
20.19 odd 2 25.22.a.a.1.1 1
28.27 even 2 49.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 ✓ 1 4.3 odd 2
9.22.a.c.1.1 1 12.11 even 2
16.22.a.c.1.1 1 1.1 even 1 trivial
25.22.a.a.1.1 1 20.19 odd 2
25.22.b.a.24.1 2 20.7 even 4
25.22.b.a.24.2 2 20.3 even 4
49.22.a.a.1.1 1 28.27 even 2
64.22.a.a.1.1 1 8.5 even 2
64.22.a.g.1.1 1 8.3 odd 2