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 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.61803 q^{3} -3.85410 q^{5} -3.23607 q^{7} -0.381966 q^{9} +1.38197 q^{11} +2.85410 q^{13} -6.23607 q^{15} -4.47214 q^{17} -4.47214 q^{19} -5.23607 q^{21} +2.85410 q^{23} +9.85410 q^{25} -5.47214 q^{27} +9.32624 q^{29} +7.38197 q^{31} +2.23607 q^{33} +12.4721 q^{35} +1.00000 q^{37} +4.61803 q^{39} +9.61803 q^{41} +5.23607 q^{43} +1.47214 q^{45} -1.23607 q^{47} +3.47214 q^{49} -7.23607 q^{51} -0.472136 q^{53} -5.32624 q^{55} -7.23607 q^{57} +4.76393 q^{59} -10.6180 q^{61} +1.23607 q^{63} -11.0000 q^{65} -1.09017 q^{67} +4.61803 q^{69} +2.94427 q^{71} +7.09017 q^{73} +15.9443 q^{75} -4.47214 q^{77} -8.56231 q^{79} -7.70820 q^{81} +14.4721 q^{83} +17.2361 q^{85} +15.0902 q^{87} -1.52786 q^{89} -9.23607 q^{91} +11.9443 q^{93} +17.2361 q^{95} -0.472136 q^{97} -0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45}+ \cdots - 10 q^{99}+O(q^{100}) \) 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
\(3\) 1.61803 0.934172 0.467086 − 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 0 0
\(5\) −3.85410 −1.72361 −0.861803 − 0.507242i \(-0.830665\pi\)
−0.861803 + 0.507242i \(0.830665\pi\)
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 − 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) 1.38197 0.416678 0.208339 − 0.978057i \(-0.433194\pi\)
0.208339 + 0.978057i \(0.433194\pi\)
\(12\) 0 0
\(13\) 2.85410 0.791585 0.395793 − 0.918340i \(-0.370470\pi\)
0.395793 + 0.918340i \(0.370470\pi\)
\(14\) 0 0
\(15\) −6.23607 −1.61015
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 − 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 − 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) −5.23607 −1.14260
\(22\) 0 0
\(23\) 2.85410 0.595121 0.297561 − 0.954703i \(-0.403827\pi\)
0.297561 + 0.954703i \(0.403827\pi\)
\(24\) 0 0
\(25\) 9.85410 1.97082
\(26\) 0 0
\(27\) −5.47214 −1.05311
\(28\) 0 0
\(29\) 9.32624 1.73184 0.865919 − 0.500183i \(-0.166734\pi\)
0.865919 + 0.500183i \(0.166734\pi\)
\(30\) 0 0
\(31\) 7.38197 1.32584 0.662920 − 0.748690i \(-0.269317\pi\)
0.662920 + 0.748690i \(0.269317\pi\)
\(32\) 0 0
\(33\) 2.23607 0.389249
\(34\) 0 0
\(35\) 12.4721 2.10818
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 4.61803 0.739477
\(40\) 0 0
\(41\) 9.61803 1.50208 0.751042 − 0.660254i \(-0.229551\pi\)
0.751042 + 0.660254i \(0.229551\pi\)
\(42\) 0 0
\(43\) 5.23607 0.798493 0.399246 − 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) 1.47214 0.219453
\(46\) 0 0
\(47\) −1.23607 −0.180299 −0.0901495 − 0.995928i \(-0.528734\pi\)
−0.0901495 + 0.995928i \(0.528734\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) −7.23607 −1.01325
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −5.32624 −0.718190
\(56\) 0 0
\(57\) −7.23607 −0.958441
\(58\) 0 0
\(59\) 4.76393 0.620211 0.310106 − 0.950702i \(-0.399636\pi\)
0.310106 + 0.950702i \(0.399636\pi\)
\(60\) 0 0
\(61\) −10.6180 −1.35950 −0.679750 − 0.733444i \(-0.737912\pi\)
−0.679750 + 0.733444i \(0.737912\pi\)
\(62\) 0 0
\(63\) 1.23607 0.155730
\(64\) 0 0
\(65\) −11.0000 −1.36438
\(66\) 0 0
\(67\) −1.09017 −0.133185 −0.0665927 − 0.997780i \(-0.521213\pi\)
−0.0665927 + 0.997780i \(0.521213\pi\)
\(68\) 0 0
\(69\) 4.61803 0.555946
\(70\) 0 0
\(71\) 2.94427 0.349421 0.174710 − 0.984620i \(-0.444101\pi\)
0.174710 + 0.984620i \(0.444101\pi\)
\(72\) 0 0
\(73\) 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
\(76\) 0 0
\(77\) −4.47214 −0.509647
\(78\) 0 0
\(79\) −8.56231 −0.963335 −0.481667 − 0.876354i \(-0.659969\pi\)
−0.481667 + 0.876354i \(0.659969\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) 14.4721 1.58852 0.794262 − 0.607576i \(-0.207858\pi\)
0.794262 + 0.607576i \(0.207858\pi\)
\(84\) 0 0
\(85\) 17.2361 1.86951
\(86\) 0 0
\(87\) 15.0902 1.61784
\(88\) 0 0
\(89\) −1.52786 −0.161953 −0.0809766 − 0.996716i \(-0.525804\pi\)
−0.0809766 + 0.996716i \(0.525804\pi\)
\(90\) 0 0
\(91\) −9.23607 −0.968203
\(92\) 0 0
\(93\) 11.9443 1.23856
\(94\) 0 0
\(95\) 17.2361 1.76838
\(96\) 0 0
\(97\) −0.472136 −0.0479381 −0.0239691 − 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 0 0
\(99\) −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 2368.2.a.y.1.2 2
4.3 odd 2 2368.2.a.u.1.1 2
8.3 odd 2 592.2.a.g.1.2 2
8.5 even 2 74.2.a.b.1.1 ✓ 2
24.5 odd 2 666.2.a.i.1.1 2
24.11 even 2 5328.2.a.bc.1.1 2
40.13 odd 4 1850.2.b.j.149.1 4
40.29 even 2 1850.2.a.t.1.2 2
40.37 odd 4 1850.2.b.j.149.4 4
56.13 odd 2 3626.2.a.s.1.2 2
88.21 odd 2 8954.2.a.j.1.1 2
296.221 even 2 2738.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 ✓ 2 8.5 even 2
592.2.a.g.1.2 2 8.3 odd 2
666.2.a.i.1.1 2 24.5 odd 2
1850.2.a.t.1.2 2 40.29 even 2
1850.2.b.j.149.1 4 40.13 odd 4
1850.2.b.j.149.4 4 40.37 odd 4
2368.2.a.u.1.1 2 4.3 odd 2
2368.2.a.y.1.2 2 1.1 even 1 trivial
2738.2.a.g.1.1 2 296.221 even 2
3626.2.a.s.1.2 2 56.13 odd 2
5328.2.a.bc.1.1 2 24.11 even 2
8954.2.a.j.1.1 2 88.21 odd 2