Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,-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: \(2.89856959337\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +1.61803 q^{5} -0.381966 q^{6} +1.00000 q^{7} -1.47214 q^{8} +1.00000 q^{9} +0.618034 q^{10} +1.85410 q^{12} +4.23607 q^{13} +0.381966 q^{14} -1.61803 q^{15} +3.14590 q^{16} +7.85410 q^{17} +0.381966 q^{18} +0.854102 q^{19} -3.00000 q^{20} -1.00000 q^{21} -4.23607 q^{23} +1.47214 q^{24} -2.38197 q^{25} +1.61803 q^{26} -1.00000 q^{27} -1.85410 q^{28} +6.00000 q^{29} -0.618034 q^{30} +5.09017 q^{31} +4.14590 q^{32} +3.00000 q^{34} +1.61803 q^{35} -1.85410 q^{36} -1.76393 q^{37} +0.326238 q^{38} -4.23607 q^{39} -2.38197 q^{40} +4.23607 q^{41} -0.381966 q^{42} -6.70820 q^{43} +1.61803 q^{45} -1.61803 q^{46} +1.09017 q^{47} -3.14590 q^{48} -6.00000 q^{49} -0.909830 q^{50} -7.85410 q^{51} -7.85410 q^{52} -2.61803 q^{53} -0.381966 q^{54} -1.47214 q^{56} -0.854102 q^{57} +2.29180 q^{58} +9.61803 q^{59} +3.00000 q^{60} -8.56231 q^{61} +1.94427 q^{62} +1.00000 q^{63} -4.70820 q^{64} +6.85410 q^{65} -4.85410 q^{67} -14.5623 q^{68} +4.23607 q^{69} +0.618034 q^{70} -5.32624 q^{71} -1.47214 q^{72} -7.70820 q^{73} -0.673762 q^{74} +2.38197 q^{75} -1.58359 q^{76} -1.61803 q^{78} -11.0000 q^{79} +5.09017 q^{80} +1.00000 q^{81} +1.61803 q^{82} +7.47214 q^{83} +1.85410 q^{84} +12.7082 q^{85} -2.56231 q^{86} -6.00000 q^{87} -3.76393 q^{89} +0.618034 q^{90} +4.23607 q^{91} +7.85410 q^{92} -5.09017 q^{93} +0.416408 q^{94} +1.38197 q^{95} -4.14590 q^{96} +1.14590 q^{97} -2.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - q^{10} - 3 q^{12} + 4 q^{13} + 3 q^{14} - q^{15} + 13 q^{16} + 9 q^{17} + 3 q^{18} - 5 q^{19} - 6 q^{20} - 2 q^{21}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 − 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.85410 −0.927051
\(5\) 1.61803 0.723607 0.361803 − 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) −0.381966 −0.155937
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.47214 −0.520479
\(9\) 1.00000 0.333333
\(10\) 0.618034 0.195440
\(11\) 0 0
\(12\) 1.85410 0.535233
\(13\) 4.23607 1.17487 0.587437 − 0.809270i \(-0.300137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(14\) 0.381966 0.102085
\(15\) −1.61803 −0.417775
\(16\) 3.14590 0.786475
\(17\) 7.85410 1.90490 0.952450 − 0.304696i \(-0.0985548\pi\)
0.952450 + 0.304696i \(0.0985548\pi\)
\(18\) 0.381966 0.0900303
\(19\) 0.854102 0.195944 0.0979722 − 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.23607 −0.883281 −0.441641 − 0.897192i \(-0.645603\pi\)
−0.441641 + 0.897192i \(0.645603\pi\)
\(24\) 1.47214 0.300498
\(25\) −2.38197 −0.476393
\(26\) 1.61803 0.317323
\(27\) −1.00000 −0.192450
\(28\) −1.85410 −0.350392
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −0.618034 −0.112837
\(31\) 5.09017 0.914222 0.457111 − 0.889410i \(-0.348884\pi\)
0.457111 + 0.889410i \(0.348884\pi\)
\(32\) 4.14590 0.732898
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 1.61803 0.273498
\(36\) −1.85410 −0.309017
\(37\) −1.76393 −0.289989 −0.144994 − 0.989432i \(-0.546316\pi\)
−0.144994 + 0.989432i \(0.546316\pi\)
\(38\) 0.326238 0.0529228
\(39\) −4.23607 −0.678314
\(40\) −2.38197 −0.376622
\(41\) 4.23607 0.661563 0.330781 − 0.943707i \(-0.392688\pi\)
0.330781 + 0.943707i \(0.392688\pi\)
\(42\) −0.381966 −0.0589386
\(43\) −6.70820 −1.02299 −0.511496 − 0.859286i \(-0.670908\pi\)
−0.511496 + 0.859286i \(0.670908\pi\)
\(44\) 0 0
\(45\) 1.61803 0.241202
\(46\) −1.61803 −0.238566
\(47\) 1.09017 0.159018 0.0795088 − 0.996834i \(-0.474665\pi\)
0.0795088 + 0.996834i \(0.474665\pi\)
\(48\) −3.14590 −0.454071
\(49\) −6.00000 −0.857143
\(50\) −0.909830 −0.128669
\(51\) −7.85410 −1.09979
\(52\) −7.85410 −1.08917
\(53\) −2.61803 −0.359615 −0.179807 − 0.983702i \(-0.557547\pi\)
−0.179807 + 0.983702i \(0.557547\pi\)
\(54\) −0.381966 −0.0519790
\(55\) 0 0
\(56\) −1.47214 −0.196722
\(57\) −0.854102 −0.113129
\(58\) 2.29180 0.300928
\(59\) 9.61803 1.25216 0.626081 − 0.779758i \(-0.284658\pi\)
0.626081 + 0.779758i \(0.284658\pi\)
\(60\) 3.00000 0.387298
\(61\) −8.56231 −1.09629 −0.548145 − 0.836383i \(-0.684666\pi\)
−0.548145 + 0.836383i \(0.684666\pi\)
\(62\) 1.94427 0.246923
\(63\) 1.00000 0.125988
\(64\) −4.70820 −0.588525
\(65\) 6.85410 0.850147
\(66\) 0 0
\(67\) −4.85410 −0.593023 −0.296511 − 0.955029i \(-0.595823\pi\)
−0.296511 + 0.955029i \(0.595823\pi\)
\(68\) −14.5623 −1.76594
\(69\) 4.23607 0.509963
\(70\) 0.618034 0.0738692
\(71\) −5.32624 −0.632108 −0.316054 − 0.948741i \(-0.602358\pi\)
−0.316054 + 0.948741i \(0.602358\pi\)
\(72\) −1.47214 −0.173493
\(73\) −7.70820 −0.902177 −0.451089 − 0.892479i \(-0.648964\pi\)
−0.451089 + 0.892479i \(0.648964\pi\)
\(74\) −0.673762 −0.0783233
\(75\) 2.38197 0.275046
\(76\) −1.58359 −0.181650
\(77\) 0 0
\(78\) −1.61803 −0.183206
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 5.09017 0.569098
\(81\) 1.00000 0.111111
\(82\) 1.61803 0.178682
\(83\) 7.47214 0.820173 0.410087 − 0.912047i \(-0.365498\pi\)
0.410087 + 0.912047i \(0.365498\pi\)
\(84\) 1.85410 0.202299
\(85\) 12.7082 1.37840
\(86\) −2.56231 −0.276301
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −3.76393 −0.398976 −0.199488 − 0.979900i \(-0.563928\pi\)
−0.199488 + 0.979900i \(0.563928\pi\)
\(90\) 0.618034 0.0651465
\(91\) 4.23607 0.444061
\(92\) 7.85410 0.818847
\(93\) −5.09017 −0.527826
\(94\) 0.416408 0.0429492
\(95\) 1.38197 0.141787
\(96\) −4.14590 −0.423139
\(97\) 1.14590 0.116348 0.0581742 − 0.998306i \(-0.481472\pi\)
0.0581742 + 0.998306i \(0.481472\pi\)
\(98\) −2.29180 −0.231506
\(99\) 0 0
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 363.2.a.i.1.1 2
3.2 odd 2 1089.2.a.l.1.2 2
4.3 odd 2 5808.2.a.ci.1.2 2
5.4 even 2 9075.2.a.u.1.2 2
11.2 odd 10 33.2.e.b.4.1 ✓ 4
11.3 even 5 363.2.e.b.130.1 4
11.4 even 5 363.2.e.b.148.1 4
11.5 even 5 363.2.e.f.124.1 4
11.6 odd 10 33.2.e.b.25.1 yes 4
11.7 odd 10 363.2.e.k.148.1 4
11.8 odd 10 363.2.e.k.130.1 4
11.9 even 5 363.2.e.f.202.1 4
11.10 odd 2 363.2.a.d.1.2 2
33.2 even 10 99.2.f.a.37.1 4
33.17 even 10 99.2.f.a.91.1 4
33.32 even 2 1089.2.a.t.1.1 2
44.35 even 10 528.2.y.b.433.1 4
44.39 even 10 528.2.y.b.289.1 4
44.43 even 2 5808.2.a.cj.1.2 2
55.2 even 20 825.2.bx.d.499.1 8
55.13 even 20 825.2.bx.d.499.2 8
55.17 even 20 825.2.bx.d.124.2 8
55.24 odd 10 825.2.n.c.301.1 4
55.28 even 20 825.2.bx.d.124.1 8
55.39 odd 10 825.2.n.c.751.1 4
55.54 odd 2 9075.2.a.cb.1.1 2
99.2 even 30 891.2.n.b.433.1 8
99.13 odd 30 891.2.n.c.136.1 8
99.50 even 30 891.2.n.b.784.1 8
99.61 odd 30 891.2.n.c.190.1 8
99.68 even 30 891.2.n.b.136.1 8
99.79 odd 30 891.2.n.c.433.1 8
99.83 even 30 891.2.n.b.190.1 8
99.94 odd 30 891.2.n.c.784.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.4.1 ✓ 4 11.2 odd 10
33.2.e.b.25.1 yes 4 11.6 odd 10
99.2.f.a.37.1 4 33.2 even 10
99.2.f.a.91.1 4 33.17 even 10
363.2.a.d.1.2 2 11.10 odd 2
363.2.a.i.1.1 2 1.1 even 1 trivial
363.2.e.b.130.1 4 11.3 even 5
363.2.e.b.148.1 4 11.4 even 5
363.2.e.f.124.1 4 11.5 even 5
363.2.e.f.202.1 4 11.9 even 5
363.2.e.k.130.1 4 11.8 odd 10
363.2.e.k.148.1 4 11.7 odd 10
528.2.y.b.289.1 4 44.39 even 10
528.2.y.b.433.1 4 44.35 even 10
825.2.n.c.301.1 4 55.24 odd 10
825.2.n.c.751.1 4 55.39 odd 10
825.2.bx.d.124.1 8 55.28 even 20
825.2.bx.d.124.2 8 55.17 even 20
825.2.bx.d.499.1 8 55.2 even 20
825.2.bx.d.499.2 8 55.13 even 20
891.2.n.b.136.1 8 99.68 even 30
891.2.n.b.190.1 8 99.83 even 30
891.2.n.b.433.1 8 99.2 even 30
891.2.n.b.784.1 8 99.50 even 30
891.2.n.c.136.1 8 99.13 odd 30
891.2.n.c.190.1 8 99.61 odd 30
891.2.n.c.433.1 8 99.79 odd 30
891.2.n.c.784.1 8 99.94 odd 30
1089.2.a.l.1.2 2 3.2 odd 2
1089.2.a.t.1.1 2 33.32 even 2
5808.2.a.ci.1.2 2 4.3 odd 2
5808.2.a.cj.1.2 2 44.43 even 2
9075.2.a.u.1.2 2 5.4 even 2
9075.2.a.cb.1.1 2 55.54 odd 2