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Properties

Label 16.8.a.b.1.1
Level $16$
Weight $8$
Character 16.1
Self dual yes
Analytic conductor $4.998$
Analytic rank $1$
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] = [16,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-12] 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: \(4.99816040775\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \(q-12.0000 q^{3} -210.000 q^{5} -1016.00 q^{7} -2043.00 q^{9} -1092.00 q^{11} +1382.00 q^{13} +2520.00 q^{15} +14706.0 q^{17} +39940.0 q^{19} +12192.0 q^{21} -68712.0 q^{23} -34025.0 q^{25} +50760.0 q^{27} -102570. q^{29} -227552. q^{31} +13104.0 q^{33} +213360. q^{35} +160526. q^{37} -16584.0 q^{39} +10842.0 q^{41} +630748. q^{43} +429030. q^{45} -472656. q^{47} +208713. q^{49} -176472. q^{51} -1.49402e6 q^{53} +229320. q^{55} -479280. q^{57} -2.64066e6 q^{59} +827702. q^{61} +2.07569e6 q^{63} -290220. q^{65} +126004. q^{67} +824544. q^{69} +1.41473e6 q^{71} +980282. q^{73} +408300. q^{75} +1.10947e6 q^{77} +3.56680e6 q^{79} +3.85892e6 q^{81} -5.67289e6 q^{83} -3.08826e6 q^{85} +1.23084e6 q^{87} -1.19512e7 q^{89} -1.40411e6 q^{91} +2.73062e6 q^{93} -8.38740e6 q^{95} +8.68215e6 q^{97} +2.23096e6 q^{99} +O(q^{100})\)

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
\(3\) −12.0000 −0.256600 −0.128300 − 0.991735i \(-0.540952\pi\)
−0.128300 + 0.991735i \(0.540952\pi\)
\(4\) 0 0
\(5\) −210.000 −0.751319 −0.375659 − 0.926758i \(-0.622584\pi\)
−0.375659 + 0.926758i \(0.622584\pi\)
\(6\) 0 0
\(7\) −1016.00 −1.11957 −0.559784 − 0.828638i \(-0.689116\pi\)
−0.559784 + 0.828638i \(0.689116\pi\)
\(8\) 0 0
\(9\) −2043.00 −0.934156
\(10\) 0 0
\(11\) −1092.00 −0.247371 −0.123685 − 0.992321i \(-0.539471\pi\)
−0.123685 + 0.992321i \(0.539471\pi\)
\(12\) 0 0
\(13\) 1382.00 0.174464 0.0872321 − 0.996188i \(-0.472198\pi\)
0.0872321 + 0.996188i \(0.472198\pi\)
\(14\) 0 0
\(15\) 2520.00 0.192789
\(16\) 0 0
\(17\) 14706.0 0.725978 0.362989 − 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) 0 0
\(19\) 39940.0 1.33589 0.667945 − 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 0 0
\(21\) 12192.0 0.287281
\(22\) 0 0
\(23\) −68712.0 −1.17757 −0.588783 − 0.808291i \(-0.700393\pi\)
−0.588783 + 0.808291i \(0.700393\pi\)
\(24\) 0 0
\(25\) −34025.0 −0.435520
\(26\) 0 0
\(27\) 50760.0 0.496305
\(28\) 0 0
\(29\) −102570. −0.780957 −0.390479 − 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) 0 0
\(31\) −227552. −1.37188 −0.685938 − 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 0 0
\(33\) 13104.0 0.0634753
\(34\) 0 0
\(35\) 213360. 0.841153
\(36\) 0 0
\(37\) 160526. 0.521002 0.260501 − 0.965474i \(-0.416112\pi\)
0.260501 + 0.965474i \(0.416112\pi\)
\(38\) 0 0
\(39\) −16584.0 −0.0447675
\(40\) 0 0
\(41\) 10842.0 0.0245678 0.0122839 − 0.999925i \(-0.496090\pi\)
0.0122839 + 0.999925i \(0.496090\pi\)
\(42\) 0 0
\(43\) 630748. 1.20981 0.604904 − 0.796299i \(-0.293212\pi\)
0.604904 + 0.796299i \(0.293212\pi\)
\(44\) 0 0
\(45\) 429030. 0.701849
\(46\) 0 0
\(47\) −472656. −0.664053 −0.332026 − 0.943270i \(-0.607732\pi\)
−0.332026 + 0.943270i \(0.607732\pi\)
\(48\) 0 0
\(49\) 208713. 0.253433
\(50\) 0 0
\(51\) −176472. −0.186286
\(52\) 0 0
\(53\) −1.49402e6 −1.37845 −0.689224 − 0.724548i \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) 0 0
\(55\) 229320. 0.185854
\(56\) 0 0
\(57\) −479280. −0.342789
\(58\) 0 0
\(59\) −2.64066e6 −1.67390 −0.836952 − 0.547277i \(-0.815665\pi\)
−0.836952 + 0.547277i \(0.815665\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 0 0
\(63\) 2.07569e6 1.04585
\(64\) 0 0
\(65\) −290220. −0.131078
\(66\) 0 0
\(67\) 126004. 0.0511826 0.0255913 − 0.999672i \(-0.491853\pi\)
0.0255913 + 0.999672i \(0.491853\pi\)
\(68\) 0 0
\(69\) 824544. 0.302164
\(70\) 0 0
\(71\) 1.41473e6 0.469104 0.234552 − 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) 0 0
\(73\) 980282. 0.294931 0.147466 − 0.989067i \(-0.452888\pi\)
0.147466 + 0.989067i \(0.452888\pi\)
\(74\) 0 0
\(75\) 408300. 0.111754
\(76\) 0 0
\(77\) 1.10947e6 0.276948
\(78\) 0 0
\(79\) 3.56680e6 0.813924 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(80\) 0 0
\(81\) 3.85892e6 0.806805
\(82\) 0 0
\(83\) −5.67289e6 −1.08901 −0.544504 − 0.838758i \(-0.683282\pi\)
−0.544504 + 0.838758i \(0.683282\pi\)
\(84\) 0 0
\(85\) −3.08826e6 −0.545441
\(86\) 0 0
\(87\) 1.23084e6 0.200394
\(88\) 0 0
\(89\) −1.19512e7 −1.79699 −0.898496 − 0.438982i \(-0.855339\pi\)
−0.898496 + 0.438982i \(0.855339\pi\)
\(90\) 0 0
\(91\) −1.40411e6 −0.195325
\(92\) 0 0
\(93\) 2.73062e6 0.352023
\(94\) 0 0
\(95\) −8.38740e6 −1.00368
\(96\) 0 0
\(97\) 8.68215e6 0.965886 0.482943 − 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(98\) 0 0
\(99\) 2.23096e6 0.231083
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 16.8.a.b.1.1 1
3.2 odd 2 144.8.a.i.1.1 1
4.3 odd 2 2.8.a.a.1.1 ✓ 1
5.2 odd 4 400.8.c.j.49.2 2
5.3 odd 4 400.8.c.j.49.1 2
5.4 even 2 400.8.a.l.1.1 1
8.3 odd 2 64.8.a.c.1.1 1
8.5 even 2 64.8.a.e.1.1 1
12.11 even 2 18.8.a.b.1.1 1
16.3 odd 4 256.8.b.b.129.2 2
16.5 even 4 256.8.b.f.129.2 2
16.11 odd 4 256.8.b.b.129.1 2
16.13 even 4 256.8.b.f.129.1 2
20.3 even 4 50.8.b.c.49.2 2
20.7 even 4 50.8.b.c.49.1 2
20.19 odd 2 50.8.a.g.1.1 1
24.5 odd 2 576.8.a.f.1.1 1
24.11 even 2 576.8.a.g.1.1 1
28.3 even 6 98.8.c.e.79.1 2
28.11 odd 6 98.8.c.d.79.1 2
28.19 even 6 98.8.c.e.67.1 2
28.23 odd 6 98.8.c.d.67.1 2
28.27 even 2 98.8.a.a.1.1 1
36.7 odd 6 162.8.c.l.109.1 2
36.11 even 6 162.8.c.a.109.1 2
36.23 even 6 162.8.c.a.55.1 2
36.31 odd 6 162.8.c.l.55.1 2
44.43 even 2 242.8.a.e.1.1 1
52.31 even 4 338.8.b.d.337.2 2
52.47 even 4 338.8.b.d.337.1 2
52.51 odd 2 338.8.a.d.1.1 1
60.23 odd 4 450.8.c.g.199.1 2
60.47 odd 4 450.8.c.g.199.2 2
60.59 even 2 450.8.a.c.1.1 1
68.67 odd 2 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 4.3 odd 2
16.8.a.b.1.1 1 1.1 even 1 trivial
18.8.a.b.1.1 1 12.11 even 2
50.8.a.g.1.1 1 20.19 odd 2
50.8.b.c.49.1 2 20.7 even 4
50.8.b.c.49.2 2 20.3 even 4
64.8.a.c.1.1 1 8.3 odd 2
64.8.a.e.1.1 1 8.5 even 2
98.8.a.a.1.1 1 28.27 even 2
98.8.c.d.67.1 2 28.23 odd 6
98.8.c.d.79.1 2 28.11 odd 6
98.8.c.e.67.1 2 28.19 even 6
98.8.c.e.79.1 2 28.3 even 6
144.8.a.i.1.1 1 3.2 odd 2
162.8.c.a.55.1 2 36.23 even 6
162.8.c.a.109.1 2 36.11 even 6
162.8.c.l.55.1 2 36.31 odd 6
162.8.c.l.109.1 2 36.7 odd 6
242.8.a.e.1.1 1 44.43 even 2
256.8.b.b.129.1 2 16.11 odd 4
256.8.b.b.129.2 2 16.3 odd 4
256.8.b.f.129.1 2 16.13 even 4
256.8.b.f.129.2 2 16.5 even 4
338.8.a.d.1.1 1 52.51 odd 2
338.8.b.d.337.1 2 52.47 even 4
338.8.b.d.337.2 2 52.31 even 4
400.8.a.l.1.1 1 5.4 even 2
400.8.c.j.49.1 2 5.3 odd 4
400.8.c.j.49.2 2 5.2 odd 4
450.8.a.c.1.1 1 60.59 even 2
450.8.c.g.199.1 2 60.23 odd 4
450.8.c.g.199.2 2 60.47 odd 4
576.8.a.f.1.1 1 24.5 odd 2
576.8.a.g.1.1 1 24.11 even 2
578.8.a.b.1.1 1 68.67 odd 2