↑

Properties

Label 18.8.a.b.1.1
Level $18$
Weight $8$
Character 18.1
Self dual yes
Analytic conductor $5.623$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,8,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.62293045871\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +64.0000 q^{4} +210.000 q^{5} +1016.00 q^{7} +512.000 q^{8} +1680.00 q^{10} -1092.00 q^{11} +1382.00 q^{13} +8128.00 q^{14} +4096.00 q^{16} -14706.0 q^{17} -39940.0 q^{19} +13440.0 q^{20} -8736.00 q^{22} -68712.0 q^{23} -34025.0 q^{25} +11056.0 q^{26} +65024.0 q^{28} +102570. q^{29} +227552. q^{31} +32768.0 q^{32} -117648. q^{34} +213360. q^{35} +160526. q^{37} -319520. q^{38} +107520. q^{40} -10842.0 q^{41} -630748. q^{43} -69888.0 q^{44} -549696. q^{46} -472656. q^{47} +208713. q^{49} -272200. q^{50} +88448.0 q^{52} +1.49402e6 q^{53} -229320. q^{55} +520192. q^{56} +820560. q^{58} -2.64066e6 q^{59} +827702. q^{61} +1.82042e6 q^{62} +262144. q^{64} +290220. q^{65} -126004. q^{67} -941184. q^{68} +1.70688e6 q^{70} +1.41473e6 q^{71} +980282. q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} +860160. q^{80} -86736.0 q^{82} -5.67289e6 q^{83} -3.08826e6 q^{85} -5.04598e6 q^{86} -559104. q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} -4.39757e6 q^{92} -3.78125e6 q^{94} -8.38740e6 q^{95} +8.68215e6 q^{97} +1.66970e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) 210.000 0.751319 0.375659 − 0.926758i \(-0.377416\pi\)
0.375659 + 0.926758i \(0.377416\pi\)
\(6\) 0 0
\(7\) 1016.00 1.11957 0.559784 − 0.828638i \(-0.310884\pi\)
0.559784 + 0.828638i \(0.310884\pi\)
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) 1680.00 0.531263
\(11\) −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 \(0.472198\pi\)
\(14\) 8128.00 0.791654
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −14706.0 −0.725978 −0.362989 − 0.931793i \(-0.618244\pi\)
−0.362989 + 0.931793i \(0.618244\pi\)
\(18\) 0 0
\(19\) −39940.0 −1.33589 −0.667945 − 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(20\) 13440.0 0.375659
\(21\) 0 0
\(22\) −8736.00 −0.174917
\(23\) −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
\(26\) 11056.0 0.123365
\(27\) 0 0
\(28\) 65024.0 0.559784
\(29\) 102570. 0.780957 0.390479 − 0.920612i \(-0.372310\pi\)
0.390479 + 0.920612i \(0.372310\pi\)
\(30\) 0 0
\(31\) 227552. 1.37188 0.685938 − 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(32\) 32768.0 0.176777
\(33\) 0 0
\(34\) −117648. −0.513344
\(35\) 213360. 0.841153
\(36\) 0 0
\(37\) 160526. 0.521002 0.260501 − 0.965474i \(-0.416112\pi\)
0.260501 + 0.965474i \(0.416112\pi\)
\(38\) −319520. −0.944616
\(39\) 0 0
\(40\) 107520. 0.265631
\(41\) −10842.0 −0.0245678 −0.0122839 − 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) 0 0
\(43\) −630748. −1.20981 −0.604904 − 0.796299i \(-0.706788\pi\)
−0.604904 + 0.796299i \(0.706788\pi\)
\(44\) −69888.0 −0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) −472656. −0.664053 −0.332026 − 0.943270i \(-0.607732\pi\)
−0.332026 + 0.943270i \(0.607732\pi\)
\(48\) 0 0
\(49\) 208713. 0.253433
\(50\) −272200. −0.307959
\(51\) 0 0
\(52\) 88448.0 0.0872321
\(53\) 1.49402e6 1.37845 0.689224 − 0.724548i \(-0.257952\pi\)
0.689224 + 0.724548i \(0.257952\pi\)
\(54\) 0 0
\(55\) −229320. −0.185854
\(56\) 520192. 0.395827
\(57\) 0 0
\(58\) 820560. 0.552220
\(59\) −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 \(0.424999\pi\)
\(62\) 1.82042e6 0.970063
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 290220. 0.131078
\(66\) 0 0
\(67\) −126004. −0.0511826 −0.0255913 − 0.999672i \(-0.508147\pi\)
−0.0255913 + 0.999672i \(0.508147\pi\)
\(68\) −941184. −0.362989
\(69\) 0 0
\(70\) 1.70688e6 0.594785
\(71\) 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 \(0.452888\pi\)
\(74\) 1.28421e6 0.368404
\(75\) 0 0
\(76\) −2.55616e6 −0.667945
\(77\) −1.10947e6 −0.276948
\(78\) 0 0
\(79\) −3.56680e6 −0.813924 −0.406962 − 0.913445i \(-0.633412\pi\)
−0.406962 + 0.913445i \(0.633412\pi\)
\(80\) 860160. 0.187830
\(81\) 0 0
\(82\) −86736.0 −0.0173720
\(83\) −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
\(86\) −5.04598e6 −0.855463
\(87\) 0 0
\(88\) −559104. −0.0874587
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) 1.40411e6 0.195325
\(92\) −4.39757e6 −0.588783
\(93\) 0 0
\(94\) −3.78125e6 −0.469556
\(95\) −8.38740e6 −1.00368
\(96\) 0 0
\(97\) 8.68215e6 0.965886 0.482943 − 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(98\) 1.66970e6 0.179204
\(99\) 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 18.8.a.b.1.1 1
3.2 odd 2 2.8.a.a.1.1 ✓ 1
4.3 odd 2 144.8.a.i.1.1 1
5.2 odd 4 450.8.c.g.199.2 2
5.3 odd 4 450.8.c.g.199.1 2
5.4 even 2 450.8.a.c.1.1 1
8.3 odd 2 576.8.a.f.1.1 1
8.5 even 2 576.8.a.g.1.1 1
9.2 odd 6 162.8.c.l.109.1 2
9.4 even 3 162.8.c.a.55.1 2
9.5 odd 6 162.8.c.l.55.1 2
9.7 even 3 162.8.c.a.109.1 2
12.11 even 2 16.8.a.b.1.1 1
15.2 even 4 50.8.b.c.49.1 2
15.8 even 4 50.8.b.c.49.2 2
15.14 odd 2 50.8.a.g.1.1 1
21.2 odd 6 98.8.c.d.67.1 2
21.5 even 6 98.8.c.e.67.1 2
21.11 odd 6 98.8.c.d.79.1 2
21.17 even 6 98.8.c.e.79.1 2
21.20 even 2 98.8.a.a.1.1 1
24.5 odd 2 64.8.a.c.1.1 1
24.11 even 2 64.8.a.e.1.1 1
33.32 even 2 242.8.a.e.1.1 1
39.5 even 4 338.8.b.d.337.2 2
39.8 even 4 338.8.b.d.337.1 2
39.38 odd 2 338.8.a.d.1.1 1
48.5 odd 4 256.8.b.b.129.1 2
48.11 even 4 256.8.b.f.129.2 2
48.29 odd 4 256.8.b.b.129.2 2
48.35 even 4 256.8.b.f.129.1 2
51.50 odd 2 578.8.a.b.1.1 1
60.23 odd 4 400.8.c.j.49.1 2
60.47 odd 4 400.8.c.j.49.2 2
60.59 even 2 400.8.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 3.2 odd 2
16.8.a.b.1.1 1 12.11 even 2
18.8.a.b.1.1 1 1.1 even 1 trivial
50.8.a.g.1.1 1 15.14 odd 2
50.8.b.c.49.1 2 15.2 even 4
50.8.b.c.49.2 2 15.8 even 4
64.8.a.c.1.1 1 24.5 odd 2
64.8.a.e.1.1 1 24.11 even 2
98.8.a.a.1.1 1 21.20 even 2
98.8.c.d.67.1 2 21.2 odd 6
98.8.c.d.79.1 2 21.11 odd 6
98.8.c.e.67.1 2 21.5 even 6
98.8.c.e.79.1 2 21.17 even 6
144.8.a.i.1.1 1 4.3 odd 2
162.8.c.a.55.1 2 9.4 even 3
162.8.c.a.109.1 2 9.7 even 3
162.8.c.l.55.1 2 9.5 odd 6
162.8.c.l.109.1 2 9.2 odd 6
242.8.a.e.1.1 1 33.32 even 2
256.8.b.b.129.1 2 48.5 odd 4
256.8.b.b.129.2 2 48.29 odd 4
256.8.b.f.129.1 2 48.35 even 4
256.8.b.f.129.2 2 48.11 even 4
338.8.a.d.1.1 1 39.38 odd 2
338.8.b.d.337.1 2 39.8 even 4
338.8.b.d.337.2 2 39.5 even 4
400.8.a.l.1.1 1 60.59 even 2
400.8.c.j.49.1 2 60.23 odd 4
400.8.c.j.49.2 2 60.47 odd 4
450.8.a.c.1.1 1 5.4 even 2
450.8.c.g.199.1 2 5.3 odd 4
450.8.c.g.199.2 2 5.2 odd 4
576.8.a.f.1.1 1 8.3 odd 2
576.8.a.g.1.1 1 8.5 even 2
578.8.a.b.1.1 1 51.50 odd 2