Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-1,0,-2,0,-3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
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 \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{3} +2.85410 q^{5} +1.23607 q^{7} -2.61803 q^{9} +3.61803 q^{11} -3.85410 q^{13} -1.76393 q^{15} +4.47214 q^{17} +4.47214 q^{19} -0.763932 q^{21} -3.85410 q^{23} +3.14590 q^{25} +3.47214 q^{27} -6.32624 q^{29} +9.61803 q^{31} -2.23607 q^{33} +3.52786 q^{35} +1.00000 q^{37} +2.38197 q^{39} +7.38197 q^{41} +0.763932 q^{43} -7.47214 q^{45} +3.23607 q^{47} -5.47214 q^{49} -2.76393 q^{51} +8.47214 q^{53} +10.3262 q^{55} -2.76393 q^{57} +9.23607 q^{59} -8.38197 q^{61} -3.23607 q^{63} -11.0000 q^{65} +10.0902 q^{67} +2.38197 q^{69} -14.9443 q^{71} -4.09017 q^{73} -1.94427 q^{75} +4.47214 q^{77} +11.5623 q^{79} +5.70820 q^{81} +5.52786 q^{83} +12.7639 q^{85} +3.90983 q^{87} -10.4721 q^{89} -4.76393 q^{91} -5.94427 q^{93} +12.7639 q^{95} +8.47214 q^{97} -9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 0 0
\(5\) 2.85410 1.27639 0.638197 − 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) 3.61803 1.09088 0.545439 − 0.838150i \(-0.316363\pi\)
0.545439 + 0.838150i \(0.316363\pi\)
\(12\) 0 0
\(13\) −3.85410 −1.06894 −0.534468 − 0.845189i \(-0.679488\pi\)
−0.534468 + 0.845189i \(0.679488\pi\)
\(14\) 0 0
\(15\) −1.76393 −0.455445
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) −0.763932 −0.166704
\(22\) 0 0
\(23\) −3.85410 −0.803636 −0.401818 − 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(24\) 0 0
\(25\) 3.14590 0.629180
\(26\) 0 0
\(27\) 3.47214 0.668213
\(28\) 0 0
\(29\) −6.32624 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(30\) 0 0
\(31\) 9.61803 1.72745 0.863725 − 0.503964i \(-0.168125\pi\)
0.863725 + 0.503964i \(0.168125\pi\)
\(32\) 0 0
\(33\) −2.23607 −0.389249
\(34\) 0 0
\(35\) 3.52786 0.596318
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 2.38197 0.381420
\(40\) 0 0
\(41\) 7.38197 1.15287 0.576435 − 0.817143i \(-0.304444\pi\)
0.576435 + 0.817143i \(0.304444\pi\)
\(42\) 0 0
\(43\) 0.763932 0.116499 0.0582493 − 0.998302i \(-0.481448\pi\)
0.0582493 + 0.998302i \(0.481448\pi\)
\(44\) 0 0
\(45\) −7.47214 −1.11388
\(46\) 0 0
\(47\) 3.23607 0.472029 0.236015 − 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) −2.76393 −0.387028
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 10.3262 1.39239
\(56\) 0 0
\(57\) −2.76393 −0.366092
\(58\) 0 0
\(59\) 9.23607 1.20243 0.601217 − 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) −8.38197 −1.07320 −0.536600 − 0.843836i \(-0.680292\pi\)
−0.536600 + 0.843836i \(0.680292\pi\)
\(62\) 0 0
\(63\) −3.23607 −0.407706
\(64\) 0 0
\(65\) −11.0000 −1.36438
\(66\) 0 0
\(67\) 10.0902 1.23271 0.616355 − 0.787468i \(-0.288609\pi\)
0.616355 + 0.787468i \(0.288609\pi\)
\(68\) 0 0
\(69\) 2.38197 0.286755
\(70\) 0 0
\(71\) −14.9443 −1.77356 −0.886779 − 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) 0 0
\(73\) −4.09017 −0.478718 −0.239359 − 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(74\) 0 0
\(75\) −1.94427 −0.224505
\(76\) 0 0
\(77\) 4.47214 0.509647
\(78\) 0 0
\(79\) 11.5623 1.30086 0.650431 − 0.759566i \(-0.274589\pi\)
0.650431 + 0.759566i \(0.274589\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) 5.52786 0.606762 0.303381 − 0.952869i \(-0.401885\pi\)
0.303381 + 0.952869i \(0.401885\pi\)
\(84\) 0 0
\(85\) 12.7639 1.38444
\(86\) 0 0
\(87\) 3.90983 0.419178
\(88\) 0 0
\(89\) −10.4721 −1.11004 −0.555022 − 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) −4.76393 −0.499396
\(92\) 0 0
\(93\) −5.94427 −0.616392
\(94\) 0 0
\(95\) 12.7639 1.30955
\(96\) 0 0
\(97\) 8.47214 0.860215 0.430108 − 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 0 0
\(99\) −9.47214 −0.951985
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 2368.2.a.y.1.1 2
4.3 odd 2 2368.2.a.u.1.2 2
8.3 odd 2 592.2.a.g.1.1 2
8.5 even 2 74.2.a.b.1.2 ✓ 2
24.5 odd 2 666.2.a.i.1.2 2
24.11 even 2 5328.2.a.bc.1.2 2
40.13 odd 4 1850.2.b.j.149.2 4
40.29 even 2 1850.2.a.t.1.1 2
40.37 odd 4 1850.2.b.j.149.3 4
56.13 odd 2 3626.2.a.s.1.1 2
88.21 odd 2 8954.2.a.j.1.2 2
296.221 even 2 2738.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 ✓ 2 8.5 even 2
592.2.a.g.1.1 2 8.3 odd 2
666.2.a.i.1.2 2 24.5 odd 2
1850.2.a.t.1.1 2 40.29 even 2
1850.2.b.j.149.2 4 40.13 odd 4
1850.2.b.j.149.3 4 40.37 odd 4
2368.2.a.u.1.2 2 4.3 odd 2
2368.2.a.y.1.1 2 1.1 even 1 trivial
2738.2.a.g.1.2 2 296.221 even 2
3626.2.a.s.1.1 2 56.13 odd 2
5328.2.a.bc.1.2 2 24.11 even 2
8954.2.a.j.1.2 2 88.21 odd 2