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Properties

Label 2.8.a.a.1.1
Level $2$
Weight $8$
Character 2.1
Self dual yes
Analytic conductor $0.625$
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] = [2,8,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.624770050968\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +12.0000 q^{3} +64.0000 q^{4} -210.000 q^{5} -96.0000 q^{6} +1016.00 q^{7} -512.000 q^{8} -2043.00 q^{9} +1680.00 q^{10} +1092.00 q^{11} +768.000 q^{12} +1382.00 q^{13} -8128.00 q^{14} -2520.00 q^{15} +4096.00 q^{16} +14706.0 q^{17} +16344.0 q^{18} -39940.0 q^{19} -13440.0 q^{20} +12192.0 q^{21} -8736.00 q^{22} +68712.0 q^{23} -6144.00 q^{24} -34025.0 q^{25} -11056.0 q^{26} -50760.0 q^{27} +65024.0 q^{28} -102570. q^{29} +20160.0 q^{30} +227552. q^{31} -32768.0 q^{32} +13104.0 q^{33} -117648. q^{34} -213360. q^{35} -130752. q^{36} +160526. q^{37} +319520. q^{38} +16584.0 q^{39} +107520. q^{40} +10842.0 q^{41} -97536.0 q^{42} -630748. q^{43} +69888.0 q^{44} +429030. q^{45} -549696. q^{46} +472656. q^{47} +49152.0 q^{48} +208713. q^{49} +272200. q^{50} +176472. q^{51} +88448.0 q^{52} -1.49402e6 q^{53} +406080. q^{54} -229320. q^{55} -520192. q^{56} -479280. q^{57} +820560. q^{58} +2.64066e6 q^{59} -161280. q^{60} +827702. q^{61} -1.82042e6 q^{62} -2.07569e6 q^{63} +262144. q^{64} -290220. q^{65} -104832. q^{66} -126004. q^{67} +941184. q^{68} +824544. q^{69} +1.70688e6 q^{70} -1.41473e6 q^{71} +1.04602e6 q^{72} +980282. q^{73} -1.28421e6 q^{74} -408300. q^{75} -2.55616e6 q^{76} +1.10947e6 q^{77} -132672. q^{78} -3.56680e6 q^{79} -860160. q^{80} +3.85892e6 q^{81} -86736.0 q^{82} +5.67289e6 q^{83} +780288. q^{84} -3.08826e6 q^{85} +5.04598e6 q^{86} -1.23084e6 q^{87} -559104. q^{88} -1.19512e7 q^{89} -3.43224e6 q^{90} +1.40411e6 q^{91} +4.39757e6 q^{92} +2.73062e6 q^{93} -3.78125e6 q^{94} +8.38740e6 q^{95} -393216. q^{96} +8.68215e6 q^{97} -1.66970e6 q^{98} -2.23096e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 12.0000 0.256600 0.128300 − 0.991735i \(-0.459048\pi\)
0.128300 + 0.991735i \(0.459048\pi\)
\(4\) 64.0000 0.500000
\(5\) −210.000 −0.751319 −0.375659 − 0.926758i \(-0.622584\pi\)
−0.375659 + 0.926758i \(0.622584\pi\)
\(6\) −96.0000 −0.181444
\(7\) 1016.00 1.11957 0.559784 − 0.828638i \(-0.310884\pi\)
0.559784 + 0.828638i \(0.310884\pi\)
\(8\) −512.000 −0.353553
\(9\) −2043.00 −0.934156
\(10\) 1680.00 0.531263
\(11\) 1092.00 0.247371 0.123685 − 0.992321i \(-0.460529\pi\)
0.123685 + 0.992321i \(0.460529\pi\)
\(12\) 768.000 0.128300
\(13\) 1382.00 0.174464 0.0872321 − 0.996188i \(-0.472198\pi\)
0.0872321 + 0.996188i \(0.472198\pi\)
\(14\) −8128.00 −0.791654
\(15\) −2520.00 −0.192789
\(16\) 4096.00 0.250000
\(17\) 14706.0 0.725978 0.362989 − 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) 16344.0 0.660548
\(19\) −39940.0 −1.33589 −0.667945 − 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(20\) −13440.0 −0.375659
\(21\) 12192.0 0.287281
\(22\) −8736.00 −0.174917
\(23\) 68712.0 1.17757 0.588783 − 0.808291i \(-0.299607\pi\)
0.588783 + 0.808291i \(0.299607\pi\)
\(24\) −6144.00 −0.0907218
\(25\) −34025.0 −0.435520
\(26\) −11056.0 −0.123365
\(27\) −50760.0 −0.496305
\(28\) 65024.0 0.559784
\(29\) −102570. −0.780957 −0.390479 − 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) 20160.0 0.136322
\(31\) 227552. 1.37188 0.685938 − 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(32\) −32768.0 −0.176777
\(33\) 13104.0 0.0634753
\(34\) −117648. −0.513344
\(35\) −213360. −0.841153
\(36\) −130752. −0.467078
\(37\) 160526. 0.521002 0.260501 − 0.965474i \(-0.416112\pi\)
0.260501 + 0.965474i \(0.416112\pi\)
\(38\) 319520. 0.944616
\(39\) 16584.0 0.0447675
\(40\) 107520. 0.265631
\(41\) 10842.0 0.0245678 0.0122839 − 0.999925i \(-0.496090\pi\)
0.0122839 + 0.999925i \(0.496090\pi\)
\(42\) −97536.0 −0.203139
\(43\) −630748. −1.20981 −0.604904 − 0.796299i \(-0.706788\pi\)
−0.604904 + 0.796299i \(0.706788\pi\)
\(44\) 69888.0 0.123685
\(45\) 429030. 0.701849
\(46\) −549696. −0.832665
\(47\) 472656. 0.664053 0.332026 − 0.943270i \(-0.392268\pi\)
0.332026 + 0.943270i \(0.392268\pi\)
\(48\) 49152.0 0.0641500
\(49\) 208713. 0.253433
\(50\) 272200. 0.307959
\(51\) 176472. 0.186286
\(52\) 88448.0 0.0872321
\(53\) −1.49402e6 −1.37845 −0.689224 − 0.724548i \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) 406080. 0.350940
\(55\) −229320. −0.185854
\(56\) −520192. −0.395827
\(57\) −479280. −0.342789
\(58\) 820560. 0.552220
\(59\) 2.64066e6 1.67390 0.836952 − 0.547277i \(-0.184335\pi\)
0.836952 + 0.547277i \(0.184335\pi\)
\(60\) −161280. −0.0963943
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) −1.82042e6 −0.970063
\(63\) −2.07569e6 −1.04585
\(64\) 262144. 0.125000
\(65\) −290220. −0.131078
\(66\) −104832. −0.0448838
\(67\) −126004. −0.0511826 −0.0255913 − 0.999672i \(-0.508147\pi\)
−0.0255913 + 0.999672i \(0.508147\pi\)
\(68\) 941184. 0.362989
\(69\) 824544. 0.302164
\(70\) 1.70688e6 0.594785
\(71\) −1.41473e6 −0.469104 −0.234552 − 0.972104i \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) 1.04602e6 0.330274
\(73\) 980282. 0.294931 0.147466 − 0.989067i \(-0.452888\pi\)
0.147466 + 0.989067i \(0.452888\pi\)
\(74\) −1.28421e6 −0.368404
\(75\) −408300. −0.111754
\(76\) −2.55616e6 −0.667945
\(77\) 1.10947e6 0.276948
\(78\) −132672. −0.0316554
\(79\) −3.56680e6 −0.813924 −0.406962 − 0.913445i \(-0.633412\pi\)
−0.406962 + 0.913445i \(0.633412\pi\)
\(80\) −860160. −0.187830
\(81\) 3.85892e6 0.806805
\(82\) −86736.0 −0.0173720
\(83\) 5.67289e6 1.08901 0.544504 − 0.838758i \(-0.316718\pi\)
0.544504 + 0.838758i \(0.316718\pi\)
\(84\) 780288. 0.143641
\(85\) −3.08826e6 −0.545441
\(86\) 5.04598e6 0.855463
\(87\) −1.23084e6 −0.200394
\(88\) −559104. −0.0874587
\(89\) −1.19512e7 −1.79699 −0.898496 − 0.438982i \(-0.855339\pi\)
−0.898496 + 0.438982i \(0.855339\pi\)
\(90\) −3.43224e6 −0.496282
\(91\) 1.40411e6 0.195325
\(92\) 4.39757e6 0.588783
\(93\) 2.73062e6 0.352023
\(94\) −3.78125e6 −0.469556
\(95\) 8.38740e6 1.00368
\(96\) −393216. −0.0453609
\(97\) 8.68215e6 0.965886 0.482943 − 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(98\) −1.66970e6 −0.179204
\(99\) −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 2.8.a.a.1.1 ✓ 1
3.2 odd 2 18.8.a.b.1.1 1
4.3 odd 2 16.8.a.b.1.1 1
5.2 odd 4 50.8.b.c.49.1 2
5.3 odd 4 50.8.b.c.49.2 2
5.4 even 2 50.8.a.g.1.1 1
7.2 even 3 98.8.c.d.67.1 2
7.3 odd 6 98.8.c.e.79.1 2
7.4 even 3 98.8.c.d.79.1 2
7.5 odd 6 98.8.c.e.67.1 2
7.6 odd 2 98.8.a.a.1.1 1
8.3 odd 2 64.8.a.e.1.1 1
8.5 even 2 64.8.a.c.1.1 1
9.2 odd 6 162.8.c.a.109.1 2
9.4 even 3 162.8.c.l.55.1 2
9.5 odd 6 162.8.c.a.55.1 2
9.7 even 3 162.8.c.l.109.1 2
11.10 odd 2 242.8.a.e.1.1 1
12.11 even 2 144.8.a.i.1.1 1
13.5 odd 4 338.8.b.d.337.2 2
13.8 odd 4 338.8.b.d.337.1 2
13.12 even 2 338.8.a.d.1.1 1
15.2 even 4 450.8.c.g.199.2 2
15.8 even 4 450.8.c.g.199.1 2
15.14 odd 2 450.8.a.c.1.1 1
16.3 odd 4 256.8.b.f.129.1 2
16.5 even 4 256.8.b.b.129.1 2
16.11 odd 4 256.8.b.f.129.2 2
16.13 even 4 256.8.b.b.129.2 2
17.16 even 2 578.8.a.b.1.1 1
20.3 even 4 400.8.c.j.49.1 2
20.7 even 4 400.8.c.j.49.2 2
20.19 odd 2 400.8.a.l.1.1 1
24.5 odd 2 576.8.a.g.1.1 1
24.11 even 2 576.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
16.8.a.b.1.1 1 4.3 odd 2
18.8.a.b.1.1 1 3.2 odd 2
50.8.a.g.1.1 1 5.4 even 2
50.8.b.c.49.1 2 5.2 odd 4
50.8.b.c.49.2 2 5.3 odd 4
64.8.a.c.1.1 1 8.5 even 2
64.8.a.e.1.1 1 8.3 odd 2
98.8.a.a.1.1 1 7.6 odd 2
98.8.c.d.67.1 2 7.2 even 3
98.8.c.d.79.1 2 7.4 even 3
98.8.c.e.67.1 2 7.5 odd 6
98.8.c.e.79.1 2 7.3 odd 6
144.8.a.i.1.1 1 12.11 even 2
162.8.c.a.55.1 2 9.5 odd 6
162.8.c.a.109.1 2 9.2 odd 6
162.8.c.l.55.1 2 9.4 even 3
162.8.c.l.109.1 2 9.7 even 3
242.8.a.e.1.1 1 11.10 odd 2
256.8.b.b.129.1 2 16.5 even 4
256.8.b.b.129.2 2 16.13 even 4
256.8.b.f.129.1 2 16.3 odd 4
256.8.b.f.129.2 2 16.11 odd 4
338.8.a.d.1.1 1 13.12 even 2
338.8.b.d.337.1 2 13.8 odd 4
338.8.b.d.337.2 2 13.5 odd 4
400.8.a.l.1.1 1 20.19 odd 2
400.8.c.j.49.1 2 20.3 even 4
400.8.c.j.49.2 2 20.7 even 4
450.8.a.c.1.1 1 15.14 odd 2
450.8.c.g.199.1 2 15.8 even 4
450.8.c.g.199.2 2 15.2 even 4
576.8.a.f.1.1 1 24.11 even 2
576.8.a.g.1.1 1 24.5 odd 2
578.8.a.b.1.1 1 17.16 even 2