Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2738,2,Mod(1,2738)] 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("2738.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2738, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-1,2,-1,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.8630400734\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -3.85410 q^{5} +1.61803 q^{6} -3.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +3.85410 q^{10} -1.38197 q^{11} -1.61803 q^{12} +2.85410 q^{13} +3.23607 q^{14} +6.23607 q^{15} +1.00000 q^{16} +4.47214 q^{17} +0.381966 q^{18} -4.47214 q^{19} -3.85410 q^{20} +5.23607 q^{21} +1.38197 q^{22} -2.85410 q^{23} +1.61803 q^{24} +9.85410 q^{25} -2.85410 q^{26} +5.47214 q^{27} -3.23607 q^{28} +9.32624 q^{29} -6.23607 q^{30} -7.38197 q^{31} -1.00000 q^{32} +2.23607 q^{33} -4.47214 q^{34} +12.4721 q^{35} -0.381966 q^{36} +4.47214 q^{38} -4.61803 q^{39} +3.85410 q^{40} +9.61803 q^{41} -5.23607 q^{42} +5.23607 q^{43} -1.38197 q^{44} +1.47214 q^{45} +2.85410 q^{46} -1.23607 q^{47} -1.61803 q^{48} +3.47214 q^{49} -9.85410 q^{50} -7.23607 q^{51} +2.85410 q^{52} +0.472136 q^{53} -5.47214 q^{54} +5.32624 q^{55} +3.23607 q^{56} +7.23607 q^{57} -9.32624 q^{58} +4.76393 q^{59} +6.23607 q^{60} -10.6180 q^{61} +7.38197 q^{62} +1.23607 q^{63} +1.00000 q^{64} -11.0000 q^{65} -2.23607 q^{66} +1.09017 q^{67} +4.47214 q^{68} +4.61803 q^{69} -12.4721 q^{70} +2.94427 q^{71} +0.381966 q^{72} +7.09017 q^{73} -15.9443 q^{75} -4.47214 q^{76} +4.47214 q^{77} +4.61803 q^{78} +8.56231 q^{79} -3.85410 q^{80} -7.70820 q^{81} -9.61803 q^{82} -14.4721 q^{83} +5.23607 q^{84} -17.2361 q^{85} -5.23607 q^{86} -15.0902 q^{87} +1.38197 q^{88} +1.52786 q^{89} -1.47214 q^{90} -9.23607 q^{91} -2.85410 q^{92} +11.9443 q^{93} +1.23607 q^{94} +17.2361 q^{95} +1.61803 q^{96} +0.472136 q^{97} -3.47214 q^{98} +0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} + q^{10} - 5 q^{11} - q^{12} - q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 3 q^{18} - q^{20} + 6 q^{21} + 5 q^{22} + q^{23}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.61803 −0.934172 −0.467086 − 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.85410 −1.72361 −0.861803 − 0.507242i \(-0.830665\pi\)
−0.861803 + 0.507242i \(0.830665\pi\)
\(6\) 1.61803 0.660560
\(7\) −3.23607 −1.22312 −0.611559 − 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.381966 −0.127322
\(10\) 3.85410 1.21877
\(11\) −1.38197 −0.416678 −0.208339 − 0.978057i \(-0.566806\pi\)
−0.208339 + 0.978057i \(0.566806\pi\)
\(12\) −1.61803 −0.467086
\(13\) 2.85410 0.791585 0.395793 − 0.918340i \(-0.370470\pi\)
0.395793 + 0.918340i \(0.370470\pi\)
\(14\) 3.23607 0.864876
\(15\) 6.23607 1.61015
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0.381966 0.0900303
\(19\) −4.47214 −1.02598 −0.512989 − 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) −3.85410 −0.861803
\(21\) 5.23607 1.14260
\(22\) 1.38197 0.294636
\(23\) −2.85410 −0.595121 −0.297561 − 0.954703i \(-0.596173\pi\)
−0.297561 + 0.954703i \(0.596173\pi\)
\(24\) 1.61803 0.330280
\(25\) 9.85410 1.97082
\(26\) −2.85410 −0.559735
\(27\) 5.47214 1.05311
\(28\) −3.23607 −0.611559
\(29\) 9.32624 1.73184 0.865919 − 0.500183i \(-0.166734\pi\)
0.865919 + 0.500183i \(0.166734\pi\)
\(30\) −6.23607 −1.13855
\(31\) −7.38197 −1.32584 −0.662920 − 0.748690i \(-0.730683\pi\)
−0.662920 + 0.748690i \(0.730683\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.23607 0.389249
\(34\) −4.47214 −0.766965
\(35\) 12.4721 2.10818
\(36\) −0.381966 −0.0636610
\(37\) 0 0
\(38\) 4.47214 0.725476
\(39\) −4.61803 −0.739477
\(40\) 3.85410 0.609387
\(41\) 9.61803 1.50208 0.751042 − 0.660254i \(-0.229551\pi\)
0.751042 + 0.660254i \(0.229551\pi\)
\(42\) −5.23607 −0.807943
\(43\) 5.23607 0.798493 0.399246 − 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) −1.38197 −0.208339
\(45\) 1.47214 0.219453
\(46\) 2.85410 0.420814
\(47\) −1.23607 −0.180299 −0.0901495 − 0.995928i \(-0.528734\pi\)
−0.0901495 + 0.995928i \(0.528734\pi\)
\(48\) −1.61803 −0.233543
\(49\) 3.47214 0.496019
\(50\) −9.85410 −1.39358
\(51\) −7.23607 −1.01325
\(52\) 2.85410 0.395793
\(53\) 0.472136 0.0648529 0.0324264 − 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) −5.47214 −0.744663
\(55\) 5.32624 0.718190
\(56\) 3.23607 0.432438
\(57\) 7.23607 0.958441
\(58\) −9.32624 −1.22460
\(59\) 4.76393 0.620211 0.310106 − 0.950702i \(-0.399636\pi\)
0.310106 + 0.950702i \(0.399636\pi\)
\(60\) 6.23607 0.805073
\(61\) −10.6180 −1.35950 −0.679750 − 0.733444i \(-0.737912\pi\)
−0.679750 + 0.733444i \(0.737912\pi\)
\(62\) 7.38197 0.937511
\(63\) 1.23607 0.155730
\(64\) 1.00000 0.125000
\(65\) −11.0000 −1.36438
\(66\) −2.23607 −0.275241
\(67\) 1.09017 0.133185 0.0665927 − 0.997780i \(-0.478787\pi\)
0.0665927 + 0.997780i \(0.478787\pi\)
\(68\) 4.47214 0.542326
\(69\) 4.61803 0.555946
\(70\) −12.4721 −1.49071
\(71\) 2.94427 0.349421 0.174710 − 0.984620i \(-0.444101\pi\)
0.174710 + 0.984620i \(0.444101\pi\)
\(72\) 0.381966 0.0450151
\(73\) 7.09017 0.829842 0.414921 − 0.909858i \(-0.363809\pi\)
0.414921 + 0.909858i \(0.363809\pi\)
\(74\) 0 0
\(75\) −15.9443 −1.84109
\(76\) −4.47214 −0.512989
\(77\) 4.47214 0.509647
\(78\) 4.61803 0.522889
\(79\) 8.56231 0.963335 0.481667 − 0.876354i \(-0.340031\pi\)
0.481667 + 0.876354i \(0.340031\pi\)
\(80\) −3.85410 −0.430902
\(81\) −7.70820 −0.856467
\(82\) −9.61803 −1.06213
\(83\) −14.4721 −1.58852 −0.794262 − 0.607576i \(-0.792142\pi\)
−0.794262 + 0.607576i \(0.792142\pi\)
\(84\) 5.23607 0.571302
\(85\) −17.2361 −1.86951
\(86\) −5.23607 −0.564620
\(87\) −15.0902 −1.61784
\(88\) 1.38197 0.147318
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) −1.47214 −0.155177
\(91\) −9.23607 −0.968203
\(92\) −2.85410 −0.297561
\(93\) 11.9443 1.23856
\(94\) 1.23607 0.127491
\(95\) 17.2361 1.76838
\(96\) 1.61803 0.165140
\(97\) 0.472136 0.0479381 0.0239691 − 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) −3.47214 −0.350739
\(99\) 0.527864 0.0530523
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 2738.2.a.g.1.1 2
37.36 even 2 74.2.a.b.1.1 ✓ 2
111.110 odd 2 666.2.a.i.1.1 2
148.147 odd 2 592.2.a.g.1.2 2
185.73 odd 4 1850.2.b.j.149.1 4
185.147 odd 4 1850.2.b.j.149.4 4
185.184 even 2 1850.2.a.t.1.2 2
259.258 odd 2 3626.2.a.s.1.2 2
296.147 odd 2 2368.2.a.u.1.1 2
296.221 even 2 2368.2.a.y.1.2 2
407.406 odd 2 8954.2.a.j.1.1 2
444.443 even 2 5328.2.a.bc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 ✓ 2 37.36 even 2
592.2.a.g.1.2 2 148.147 odd 2
666.2.a.i.1.1 2 111.110 odd 2
1850.2.a.t.1.2 2 185.184 even 2
1850.2.b.j.149.1 4 185.73 odd 4
1850.2.b.j.149.4 4 185.147 odd 4
2368.2.a.u.1.1 2 296.147 odd 2
2368.2.a.y.1.2 2 296.221 even 2
2738.2.a.g.1.1 2 1.1 even 1 trivial
3626.2.a.s.1.2 2 259.258 odd 2
5328.2.a.bc.1.1 2 444.443 even 2
8954.2.a.j.1.1 2 407.406 odd 2