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Properties

Label 338.8.b.d.337.1
Level $338$
Weight $8$
Character 338.337
Analytic conductor $105.586$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(337,338)] 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("338.337"); 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([1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,24] 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: no
Analytic conductor: \(105.586138614\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.8.b.d.337.2

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000i q^{2} +12.0000 q^{3} -64.0000 q^{4} -210.000i q^{5} -96.0000i q^{6} -1016.00i q^{7} +512.000i q^{8} -2043.00 q^{9} -1680.00 q^{10} -1092.00i q^{11} -768.000 q^{12} -8128.00 q^{14} -2520.00i q^{15} +4096.00 q^{16} -14706.0 q^{17} +16344.0i q^{18} -39940.0i q^{19} +13440.0i q^{20} -12192.0i q^{21} -8736.00 q^{22} -68712.0 q^{23} +6144.00i q^{24} +34025.0 q^{25} -50760.0 q^{27} +65024.0i q^{28} -102570. q^{29} -20160.0 q^{30} +227552. i q^{31} -32768.0i q^{32} -13104.0i q^{33} +117648. i q^{34} -213360. q^{35} +130752. q^{36} -160526. i q^{37} -319520. q^{38} +107520. q^{40} +10842.0i q^{41} -97536.0 q^{42} +630748. q^{43} +69888.0i q^{44} +429030. i q^{45} +549696. i q^{46} -472656. i q^{47} +49152.0 q^{48} -208713. q^{49} -272200. i q^{50} -176472. q^{51} -1.49402e6 q^{53} +406080. i q^{54} -229320. q^{55} +520192. q^{56} -479280. i q^{57} +820560. i q^{58} -2.64066e6i q^{59} +161280. i q^{60} +827702. q^{61} +1.82042e6 q^{62} +2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} -126004. i q^{67} +941184. q^{68} -824544. q^{69} +1.70688e6i q^{70} -1.41473e6i q^{71} -1.04602e6i q^{72} -980282. i q^{73} -1.28421e6 q^{74} +408300. q^{75} +2.55616e6i q^{76} -1.10947e6 q^{77} -3.56680e6 q^{79} -860160. i q^{80} +3.85892e6 q^{81} +86736.0 q^{82} +5.67289e6i q^{83} +780288. i q^{84} +3.08826e6i q^{85} -5.04598e6i q^{86} -1.23084e6 q^{87} +559104. q^{88} +1.19512e7i q^{89} +3.43224e6 q^{90} +4.39757e6 q^{92} +2.73062e6i q^{93} -3.78125e6 q^{94} -8.38740e6 q^{95} -393216. i q^{96} +8.68215e6i q^{97} +1.66970e6i q^{98} +2.23096e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{3} - 128 q^{4} - 4086 q^{9} - 3360 q^{10} - 1536 q^{12} - 16256 q^{14} + 8192 q^{16} - 29412 q^{17} - 17472 q^{22} - 137424 q^{23} + 68050 q^{25} - 101520 q^{27} - 205140 q^{29} - 40320 q^{30}+ \cdots - 16774800 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 8.00000i − 0.707107i
\(3\) 12.0000 0.256600 0.128300 − 0.991735i \(-0.459048\pi\)
0.128300 + 0.991735i \(0.459048\pi\)
\(4\) −64.0000 −0.500000
\(5\) − 210.000i − 0.751319i −0.926758 − 0.375659i \(-0.877416\pi\)
0.926758 − 0.375659i \(-0.122584\pi\)
\(6\) − 96.0000i − 0.181444i
\(7\) − 1016.00i − 1.11957i −0.828638 − 0.559784i \(-0.810884\pi\)
0.828638 − 0.559784i \(-0.189116\pi\)
\(8\) 512.000i 0.353553i
\(9\) −2043.00 −0.934156
\(10\) −1680.00 −0.531263
\(11\) − 1092.00i − 0.247371i −0.992321 − 0.123685i \(-0.960529\pi\)
0.992321 − 0.123685i \(-0.0394713\pi\)
\(12\) −768.000 −0.128300
\(13\) 0 0
\(14\) −8128.00 −0.791654
\(15\) − 2520.00i − 0.192789i
\(16\) 4096.00 0.250000
\(17\) −14706.0 −0.725978 −0.362989 − 0.931793i \(-0.618244\pi\)
−0.362989 + 0.931793i \(0.618244\pi\)
\(18\) 16344.0i 0.660548i
\(19\) − 39940.0i − 1.33589i −0.744211 − 0.667945i \(-0.767174\pi\)
0.744211 − 0.667945i \(-0.232826\pi\)
\(20\) 13440.0i 0.375659i
\(21\) − 12192.0i − 0.287281i
\(22\) −8736.00 −0.174917
\(23\) −68712.0 −1.17757 −0.588783 − 0.808291i \(-0.700393\pi\)
−0.588783 + 0.808291i \(0.700393\pi\)
\(24\) 6144.00i 0.0907218i
\(25\) 34025.0 0.435520
\(26\) 0 0
\(27\) −50760.0 −0.496305
\(28\) 65024.0i 0.559784i
\(29\) −102570. −0.780957 −0.390479 − 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) −20160.0 −0.136322
\(31\) 227552.i 1.37188i 0.727660 + 0.685938i \(0.240608\pi\)
−0.727660 + 0.685938i \(0.759392\pi\)
\(32\) − 32768.0i − 0.176777i
\(33\) − 13104.0i − 0.0634753i
\(34\) 117648.i 0.513344i
\(35\) −213360. −0.841153
\(36\) 130752. 0.467078
\(37\) − 160526.i − 0.521002i −0.965474 − 0.260501i \(-0.916112\pi\)
0.965474 − 0.260501i \(-0.0838877\pi\)
\(38\) −319520. −0.944616
\(39\) 0 0
\(40\) 107520. 0.265631
\(41\) 10842.0i 0.0245678i 0.999925 + 0.0122839i \(0.00391018\pi\)
−0.999925 + 0.0122839i \(0.996090\pi\)
\(42\) −97536.0 −0.203139
\(43\) 630748. 1.20981 0.604904 − 0.796299i \(-0.293212\pi\)
0.604904 + 0.796299i \(0.293212\pi\)
\(44\) 69888.0i 0.123685i
\(45\) 429030.i 0.701849i
\(46\) 549696.i 0.832665i
\(47\) − 472656.i − 0.664053i −0.943270 − 0.332026i \(-0.892268\pi\)
0.943270 − 0.332026i \(-0.107732\pi\)
\(48\) 49152.0 0.0641500
\(49\) −208713. −0.253433
\(50\) − 272200.i − 0.307959i
\(51\) −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.i 0.350940i
\(55\) −229320. −0.185854
\(56\) 520192. 0.395827
\(57\) − 479280.i − 0.342789i
\(58\) 820560.i 0.552220i
\(59\) − 2.64066e6i − 1.67390i −0.547277 − 0.836952i \(-0.684335\pi\)
0.547277 − 0.836952i \(-0.315665\pi\)
\(60\) 161280.i 0.0963943i
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 1.82042e6 0.970063
\(63\) 2.07569e6i 1.04585i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) −104832. −0.0448838
\(67\) − 126004.i − 0.0511826i −0.999672 − 0.0255913i \(-0.991853\pi\)
0.999672 − 0.0255913i \(-0.00814686\pi\)
\(68\) 941184. 0.362989
\(69\) −824544. −0.302164
\(70\) 1.70688e6i 0.594785i
\(71\) − 1.41473e6i − 0.469104i −0.972104 − 0.234552i \(-0.924638\pi\)
0.972104 − 0.234552i \(-0.0753622\pi\)
\(72\) − 1.04602e6i − 0.330274i
\(73\) − 980282.i − 0.294931i −0.989067 − 0.147466i \(-0.952888\pi\)
0.989067 − 0.147466i \(-0.0471116\pi\)
\(74\) −1.28421e6 −0.368404
\(75\) 408300. 0.111754
\(76\) 2.55616e6i 0.667945i
\(77\) −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.i − 0.187830i
\(81\) 3.85892e6 0.806805
\(82\) 86736.0 0.0173720
\(83\) 5.67289e6i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(84\) 780288.i 0.143641i
\(85\) 3.08826e6i 0.545441i
\(86\) − 5.04598e6i − 0.855463i
\(87\) −1.23084e6 −0.200394
\(88\) 559104. 0.0874587
\(89\) 1.19512e7i 1.79699i 0.438982 + 0.898496i \(0.355339\pi\)
−0.438982 + 0.898496i \(0.644661\pi\)
\(90\) 3.43224e6 0.496282
\(91\) 0 0
\(92\) 4.39757e6 0.588783
\(93\) 2.73062e6i 0.352023i
\(94\) −3.78125e6 −0.469556
\(95\) −8.38740e6 −1.00368
\(96\) − 393216.i − 0.0453609i
\(97\) 8.68215e6i 0.965886i 0.875652 + 0.482943i \(0.160432\pi\)
−0.875652 + 0.482943i \(0.839568\pi\)
\(98\) 1.66970e6i 0.179204i
\(99\) 2.23096e6i 0.231083i
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.b.d.337.1 2
13.5 odd 4 2.8.a.a.1.1 ✓ 1
13.8 odd 4 338.8.a.d.1.1 1
13.12 even 2 inner 338.8.b.d.337.2 2
39.5 even 4 18.8.a.b.1.1 1
52.31 even 4 16.8.a.b.1.1 1
65.18 even 4 50.8.b.c.49.2 2
65.44 odd 4 50.8.a.g.1.1 1
65.57 even 4 50.8.b.c.49.1 2
91.5 even 12 98.8.c.e.67.1 2
91.18 odd 12 98.8.c.d.79.1 2
91.31 even 12 98.8.c.e.79.1 2
91.44 odd 12 98.8.c.d.67.1 2
91.83 even 4 98.8.a.a.1.1 1
104.5 odd 4 64.8.a.c.1.1 1
104.83 even 4 64.8.a.e.1.1 1
117.5 even 12 162.8.c.a.55.1 2
117.31 odd 12 162.8.c.l.55.1 2
117.70 odd 12 162.8.c.l.109.1 2
117.83 even 12 162.8.c.a.109.1 2
143.109 even 4 242.8.a.e.1.1 1
156.83 odd 4 144.8.a.i.1.1 1
195.44 even 4 450.8.a.c.1.1 1
195.83 odd 4 450.8.c.g.199.1 2
195.122 odd 4 450.8.c.g.199.2 2
208.5 odd 4 256.8.b.b.129.1 2
208.83 even 4 256.8.b.f.129.1 2
208.109 odd 4 256.8.b.b.129.2 2
208.187 even 4 256.8.b.f.129.2 2
221.135 odd 4 578.8.a.b.1.1 1
260.83 odd 4 400.8.c.j.49.1 2
260.187 odd 4 400.8.c.j.49.2 2
260.239 even 4 400.8.a.l.1.1 1
312.5 even 4 576.8.a.g.1.1 1
312.83 odd 4 576.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 13.5 odd 4
16.8.a.b.1.1 1 52.31 even 4
18.8.a.b.1.1 1 39.5 even 4
50.8.a.g.1.1 1 65.44 odd 4
50.8.b.c.49.1 2 65.57 even 4
50.8.b.c.49.2 2 65.18 even 4
64.8.a.c.1.1 1 104.5 odd 4
64.8.a.e.1.1 1 104.83 even 4
98.8.a.a.1.1 1 91.83 even 4
98.8.c.d.67.1 2 91.44 odd 12
98.8.c.d.79.1 2 91.18 odd 12
98.8.c.e.67.1 2 91.5 even 12
98.8.c.e.79.1 2 91.31 even 12
144.8.a.i.1.1 1 156.83 odd 4
162.8.c.a.55.1 2 117.5 even 12
162.8.c.a.109.1 2 117.83 even 12
162.8.c.l.55.1 2 117.31 odd 12
162.8.c.l.109.1 2 117.70 odd 12
242.8.a.e.1.1 1 143.109 even 4
256.8.b.b.129.1 2 208.5 odd 4
256.8.b.b.129.2 2 208.109 odd 4
256.8.b.f.129.1 2 208.83 even 4
256.8.b.f.129.2 2 208.187 even 4
338.8.a.d.1.1 1 13.8 odd 4
338.8.b.d.337.1 2 1.1 even 1 trivial
338.8.b.d.337.2 2 13.12 even 2 inner
400.8.a.l.1.1 1 260.239 even 4
400.8.c.j.49.1 2 260.83 odd 4
400.8.c.j.49.2 2 260.187 odd 4
450.8.a.c.1.1 1 195.44 even 4
450.8.c.g.199.1 2 195.83 odd 4
450.8.c.g.199.2 2 195.122 odd 4
576.8.a.f.1.1 1 312.83 odd 4
576.8.a.g.1.1 1 312.5 even 4
578.8.a.b.1.1 1 221.135 odd 4