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Properties

Label 37.2.a.a.1.1
Level $37$
Weight $2$
Character 37.1
Self dual yes
Analytic conductor $0.295$
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] = [37,2,Mod(1,37)] 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("37.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(37, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2] 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: \(0.295446487479\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 37.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} -5.00000 q^{11} -6.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +6.00000 q^{15} -4.00000 q^{16} -12.0000 q^{18} -4.00000 q^{20} +3.00000 q^{21} +10.0000 q^{22} +2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} +8.00000 q^{32} +15.0000 q^{33} +2.00000 q^{35} +12.0000 q^{36} -1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -12.0000 q^{45} -4.00000 q^{46} -9.00000 q^{47} +12.0000 q^{48} -6.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +1.00000 q^{53} +18.0000 q^{54} +10.0000 q^{55} -12.0000 q^{58} +8.00000 q^{59} +12.0000 q^{60} -8.00000 q^{61} +8.00000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -30.0000 q^{66} +8.00000 q^{67} -6.00000 q^{69} -4.00000 q^{70} +9.00000 q^{71} -1.00000 q^{73} +2.00000 q^{74} +3.00000 q^{75} +5.00000 q^{77} -12.0000 q^{78} +4.00000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +18.0000 q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} -18.0000 q^{87} +4.00000 q^{89} +24.0000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +12.0000 q^{93} +18.0000 q^{94} -24.0000 q^{96} +4.00000 q^{97} +12.0000 q^{98} -30.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 2.00000 1.00000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 6.00000 2.44949
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 4.00000 1.26491
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −6.00000 −1.73205
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 6.00000 1.54919
\(16\) −4.00000 −1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −12.0000 −2.82843
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.00000 −0.894427
\(21\) 3.00000 0.654654
\(22\) 10.0000 2.13201
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −9.00000 −1.73205
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −12.0000 −2.19089
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 8.00000 1.41421
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 12.0000 2.00000
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) −6.00000 −0.925820
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −10.0000 −1.50756
\(45\) −12.0000 −1.78885
\(46\) −4.00000 −0.589768
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 12.0000 1.73205
\(49\) −6.00000 −0.857143
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 18.0000 2.44949
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) −12.0000 −1.57568
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 12.0000 1.54919
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 8.00000 1.01600
\(63\) −6.00000 −0.755929
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) −30.0000 −3.69274
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) −4.00000 −0.478091
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 2.00000 0.232495
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) −12.0000 −1.35873
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 8.00000 0.894427
\(81\) 9.00000 1.00000
\(82\) 18.0000 1.98777
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 24.0000 2.52982
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 12.0000 1.24434
\(94\) 18.0000 1.85656
\(95\) 0 0
\(96\) −24.0000 −2.44949
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 12.0000 1.21218
\(99\) −30.0000 −3.01511
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 37.2.a.a.1.1 ✓ 1
3.2 odd 2 333.2.a.d.1.1 1
4.3 odd 2 592.2.a.e.1.1 1
5.2 odd 4 925.2.b.b.149.1 2
5.3 odd 4 925.2.b.b.149.2 2
5.4 even 2 925.2.a.e.1.1 1
7.6 odd 2 1813.2.a.a.1.1 1
8.3 odd 2 2368.2.a.b.1.1 1
8.5 even 2 2368.2.a.q.1.1 1
11.10 odd 2 4477.2.a.b.1.1 1
12.11 even 2 5328.2.a.r.1.1 1
13.12 even 2 6253.2.a.c.1.1 1
15.14 odd 2 8325.2.a.e.1.1 1
37.6 odd 4 1369.2.b.c.1368.2 2
37.31 odd 4 1369.2.b.c.1368.1 2
37.36 even 2 1369.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
333.2.a.d.1.1 1 3.2 odd 2
592.2.a.e.1.1 1 4.3 odd 2
925.2.a.e.1.1 1 5.4 even 2
925.2.b.b.149.1 2 5.2 odd 4
925.2.b.b.149.2 2 5.3 odd 4
1369.2.a.e.1.1 1 37.36 even 2
1369.2.b.c.1368.1 2 37.31 odd 4
1369.2.b.c.1368.2 2 37.6 odd 4
1813.2.a.a.1.1 1 7.6 odd 2
2368.2.a.b.1.1 1 8.3 odd 2
2368.2.a.q.1.1 1 8.5 even 2
4477.2.a.b.1.1 1 11.10 odd 2
5328.2.a.r.1.1 1 12.11 even 2
6253.2.a.c.1.1 1 13.12 even 2
8325.2.a.e.1.1 1 15.14 odd 2