Properties

Label 238.2.a.e.1.1
Level $238$
Weight $2$
Character 238.1
Self dual yes
Analytic conductor $1.900$
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] = [238,2,Mod(1,238)] 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("238.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,2] 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: \(1.90043956811\)
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 \(\chi\) \(=\) 238.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.00000 q^{21} -2.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{39} -2.00000 q^{41} -2.00000 q^{42} -2.00000 q^{44} +4.00000 q^{46} +2.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{58} +4.00000 q^{59} -12.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -8.00000 q^{67} -1.00000 q^{68} +8.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -10.0000 q^{75} +2.00000 q^{77} -4.00000 q^{78} +12.0000 q^{79} -11.0000 q^{81} -2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +8.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} +2.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −2.00000 −0.426401
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.00000 −0.308607
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −1.00000 −0.121268
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 8.00000 0.929981
\(75\) −10.0000 −1.15470
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) −4.00000 −0.452911
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −2.00000 −0.220863
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000 0.857690
\(88\) −2.00000 −0.213201
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.00000 −0.201008
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 238.2.a.e.1.1 ✓ 1
3.2 odd 2 2142.2.a.d.1.1 1
4.3 odd 2 1904.2.a.a.1.1 1
5.4 even 2 5950.2.a.b.1.1 1
7.6 odd 2 1666.2.a.i.1.1 1
8.3 odd 2 7616.2.a.j.1.1 1
8.5 even 2 7616.2.a.c.1.1 1
17.16 even 2 4046.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
238.2.a.e.1.1 ✓ 1 1.1 even 1 trivial
1666.2.a.i.1.1 1 7.6 odd 2
1904.2.a.a.1.1 1 4.3 odd 2
2142.2.a.d.1.1 1 3.2 odd 2
4046.2.a.i.1.1 1 17.16 even 2
5950.2.a.b.1.1 1 5.4 even 2
7616.2.a.c.1.1 1 8.5 even 2
7616.2.a.j.1.1 1 8.3 odd 2