Properties

Label 338.8.a.d.1.1
Level $338$
Weight $8$
Character 338.1
Self dual yes
Analytic conductor $105.586$
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] = [338,8,Mod(1,338)] 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("338.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8,12] 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: \(105.586138614\)
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 \(\chi\) \(=\) 338.1

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

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 \(-0.377416\pi\)
0.375659 + 0.926758i \(0.377416\pi\)
\(6\) 96.0000 0.181444
\(7\) −1016.00 −1.11957 −0.559784 − 0.828638i \(-0.689116\pi\)
−0.559784 + 0.828638i \(0.689116\pi\)
\(8\) 512.000 0.353553
\(9\) −2043.00 −0.934156
\(10\) 1680.00 0.531263
\(11\) −1092.00 −0.247371 −0.123685 − 0.992321i \(-0.539471\pi\)
−0.123685 + 0.992321i \(0.539471\pi\)
\(12\) 768.000 0.128300
\(13\) 0 0
\(14\) −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 \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 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
\(26\) 0 0
\(27\) −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 \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 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 \(-0.583888\pi\)
−0.260501 + 0.965474i \(0.583888\pi\)
\(38\) 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 \(0.503910\pi\)
\(42\) −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 \(-0.607732\pi\)
−0.332026 + 0.943270i \(0.607732\pi\)
\(48\) 49152.0 0.0641500
\(49\) 208713. 0.253433
\(50\) −272200. −0.307959
\(51\) 176472. 0.186286
\(52\) 0 0
\(53\) −1.49402e6 −1.37845 −0.689224 − 0.724548i \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) −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 \(-0.815665\pi\)
−0.836952 + 0.547277i \(0.815665\pi\)
\(60\) 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
\(65\) 0 0
\(66\) −104832. −0.0448838
\(67\) 126004. 0.0511826 0.0255913 − 0.999672i \(-0.491853\pi\)
0.0255913 + 0.999672i \(0.491853\pi\)
\(68\) 941184. 0.362989
\(69\) 824544. 0.302164
\(70\) −1.70688e6 −0.594785
\(71\) 1.41473e6 0.469104 0.234552 − 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) −1.04602e6 −0.330274
\(73\) −980282. −0.294931 −0.147466 − 0.989067i \(-0.547112\pi\)
−0.147466 + 0.989067i \(0.547112\pi\)
\(74\) −1.28421e6 −0.368404
\(75\) −408300. −0.111754
\(76\) 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
\(81\) 3.85892e6 0.806805
\(82\) −86736.0 −0.0173720
\(83\) −5.67289e6 −1.08901 −0.544504 − 0.838758i \(-0.683282\pi\)
−0.544504 + 0.838758i \(0.683282\pi\)
\(84\) −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 \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) −3.43224e6 −0.496282
\(91\) 0 0
\(92\) 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 \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(98\) 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 338.8.a.d.1.1 1
13.5 odd 4 338.8.b.d.337.1 2
13.8 odd 4 338.8.b.d.337.2 2
13.12 even 2 2.8.a.a.1.1 ✓ 1
39.38 odd 2 18.8.a.b.1.1 1
52.51 odd 2 16.8.a.b.1.1 1
65.12 odd 4 50.8.b.c.49.1 2
65.38 odd 4 50.8.b.c.49.2 2
65.64 even 2 50.8.a.g.1.1 1
91.12 odd 6 98.8.c.e.67.1 2
91.25 even 6 98.8.c.d.79.1 2
91.38 odd 6 98.8.c.e.79.1 2
91.51 even 6 98.8.c.d.67.1 2
91.90 odd 2 98.8.a.a.1.1 1
104.51 odd 2 64.8.a.e.1.1 1
104.77 even 2 64.8.a.c.1.1 1
117.25 even 6 162.8.c.l.109.1 2
117.38 odd 6 162.8.c.a.109.1 2
117.77 odd 6 162.8.c.a.55.1 2
117.103 even 6 162.8.c.l.55.1 2
143.142 odd 2 242.8.a.e.1.1 1
156.155 even 2 144.8.a.i.1.1 1
195.38 even 4 450.8.c.g.199.1 2
195.77 even 4 450.8.c.g.199.2 2
195.194 odd 2 450.8.a.c.1.1 1
208.51 odd 4 256.8.b.f.129.1 2
208.77 even 4 256.8.b.b.129.2 2
208.155 odd 4 256.8.b.f.129.2 2
208.181 even 4 256.8.b.b.129.1 2
221.220 even 2 578.8.a.b.1.1 1
260.103 even 4 400.8.c.j.49.1 2
260.207 even 4 400.8.c.j.49.2 2
260.259 odd 2 400.8.a.l.1.1 1
312.77 odd 2 576.8.a.g.1.1 1
312.155 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 13.12 even 2
16.8.a.b.1.1 1 52.51 odd 2
18.8.a.b.1.1 1 39.38 odd 2
50.8.a.g.1.1 1 65.64 even 2
50.8.b.c.49.1 2 65.12 odd 4
50.8.b.c.49.2 2 65.38 odd 4
64.8.a.c.1.1 1 104.77 even 2
64.8.a.e.1.1 1 104.51 odd 2
98.8.a.a.1.1 1 91.90 odd 2
98.8.c.d.67.1 2 91.51 even 6
98.8.c.d.79.1 2 91.25 even 6
98.8.c.e.67.1 2 91.12 odd 6
98.8.c.e.79.1 2 91.38 odd 6
144.8.a.i.1.1 1 156.155 even 2
162.8.c.a.55.1 2 117.77 odd 6
162.8.c.a.109.1 2 117.38 odd 6
162.8.c.l.55.1 2 117.103 even 6
162.8.c.l.109.1 2 117.25 even 6
242.8.a.e.1.1 1 143.142 odd 2
256.8.b.b.129.1 2 208.181 even 4
256.8.b.b.129.2 2 208.77 even 4
256.8.b.f.129.1 2 208.51 odd 4
256.8.b.f.129.2 2 208.155 odd 4
338.8.a.d.1.1 1 1.1 even 1 trivial
338.8.b.d.337.1 2 13.5 odd 4
338.8.b.d.337.2 2 13.8 odd 4
400.8.a.l.1.1 1 260.259 odd 2
400.8.c.j.49.1 2 260.103 even 4
400.8.c.j.49.2 2 260.207 even 4
450.8.a.c.1.1 1 195.194 odd 2
450.8.c.g.199.1 2 195.38 even 4
450.8.c.g.199.2 2 195.77 even 4
576.8.a.f.1.1 1 312.155 even 2
576.8.a.g.1.1 1 312.77 odd 2
578.8.a.b.1.1 1 221.220 even 2