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Properties

Label 363.2.a.i.1.2
Level $363$
Weight $2$
Character 363.1
Self dual yes
Analytic conductor $2.899$
Analytic rank $0$
Dimension $2$
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] = [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 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -0.618034 q^{5} -2.61803 q^{6} +1.00000 q^{7} +7.47214 q^{8} +1.00000 q^{9} -1.61803 q^{10} -4.85410 q^{12} -0.236068 q^{13} +2.61803 q^{14} +0.618034 q^{15} +9.85410 q^{16} +1.14590 q^{17} +2.61803 q^{18} -5.85410 q^{19} -3.00000 q^{20} -1.00000 q^{21} +0.236068 q^{23} -7.47214 q^{24} -4.61803 q^{25} -0.618034 q^{26} -1.00000 q^{27} +4.85410 q^{28} +6.00000 q^{29} +1.61803 q^{30} -6.09017 q^{31} +10.8541 q^{32} +3.00000 q^{34} -0.618034 q^{35} +4.85410 q^{36} -6.23607 q^{37} -15.3262 q^{38} +0.236068 q^{39} -4.61803 q^{40} -0.236068 q^{41} -2.61803 q^{42} +6.70820 q^{43} -0.618034 q^{45} +0.618034 q^{46} -10.0902 q^{47} -9.85410 q^{48} -6.00000 q^{49} -12.0902 q^{50} -1.14590 q^{51} -1.14590 q^{52} -0.381966 q^{53} -2.61803 q^{54} +7.47214 q^{56} +5.85410 q^{57} +15.7082 q^{58} +7.38197 q^{59} +3.00000 q^{60} +11.5623 q^{61} -15.9443 q^{62} +1.00000 q^{63} +8.70820 q^{64} +0.145898 q^{65} +1.85410 q^{67} +5.56231 q^{68} -0.236068 q^{69} -1.61803 q^{70} +10.3262 q^{71} +7.47214 q^{72} +5.70820 q^{73} -16.3262 q^{74} +4.61803 q^{75} -28.4164 q^{76} +0.618034 q^{78} -11.0000 q^{79} -6.09017 q^{80} +1.00000 q^{81} -0.618034 q^{82} -1.47214 q^{83} -4.85410 q^{84} -0.708204 q^{85} +17.5623 q^{86} -6.00000 q^{87} -8.23607 q^{89} -1.61803 q^{90} -0.236068 q^{91} +1.14590 q^{92} +6.09017 q^{93} -26.4164 q^{94} +3.61803 q^{95} -10.8541 q^{96} +7.85410 q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - q^{10} - 3 q^{12} + 4 q^{13} + 3 q^{14} - q^{15} + 13 q^{16} + 9 q^{17} + 3 q^{18} - 5 q^{19} - 6 q^{20} - 2 q^{21}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61803 1.85123 0.925615 − 0.378467i \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.85410 2.42705
\(5\) −0.618034 −0.276393 −0.138197 − 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) −2.61803 −1.06881
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 7.47214 2.64180
\(9\) 1.00000 0.333333
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) −4.85410 −1.40126
\(13\) −0.236068 −0.0654735 −0.0327367 − 0.999464i \(-0.510422\pi\)
−0.0327367 + 0.999464i \(0.510422\pi\)
\(14\) 2.61803 0.699699
\(15\) 0.618034 0.159576
\(16\) 9.85410 2.46353
\(17\) 1.14590 0.277921 0.138961 − 0.990298i \(-0.455624\pi\)
0.138961 + 0.990298i \(0.455624\pi\)
\(18\) 2.61803 0.617077
\(19\) −5.85410 −1.34302 −0.671512 − 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 0.236068 0.0492236 0.0246118 − 0.999697i \(-0.492165\pi\)
0.0246118 + 0.999697i \(0.492165\pi\)
\(24\) −7.47214 −1.52524
\(25\) −4.61803 −0.923607
\(26\) −0.618034 −0.121206
\(27\) −1.00000 −0.192450
\(28\) 4.85410 0.917339
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.61803 0.295411
\(31\) −6.09017 −1.09383 −0.546913 − 0.837189i \(-0.684197\pi\)
−0.546913 + 0.837189i \(0.684197\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −0.618034 −0.104467
\(36\) 4.85410 0.809017
\(37\) −6.23607 −1.02520 −0.512602 − 0.858627i \(-0.671318\pi\)
−0.512602 + 0.858627i \(0.671318\pi\)
\(38\) −15.3262 −2.48624
\(39\) 0.236068 0.0378011
\(40\) −4.61803 −0.730175
\(41\) −0.236068 −0.0368676 −0.0184338 − 0.999830i \(-0.505868\pi\)
−0.0184338 + 0.999830i \(0.505868\pi\)
\(42\) −2.61803 −0.403971
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0.618034 0.0911241
\(47\) −10.0902 −1.47180 −0.735901 − 0.677089i \(-0.763241\pi\)
−0.735901 + 0.677089i \(0.763241\pi\)
\(48\) −9.85410 −1.42232
\(49\) −6.00000 −0.857143
\(50\) −12.0902 −1.70981
\(51\) −1.14590 −0.160458
\(52\) −1.14590 −0.158907
\(53\) −0.381966 −0.0524671 −0.0262335 − 0.999656i \(-0.508351\pi\)
−0.0262335 + 0.999656i \(0.508351\pi\)
\(54\) −2.61803 −0.356269
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) 5.85410 0.775395
\(58\) 15.7082 2.06259
\(59\) 7.38197 0.961050 0.480525 − 0.876981i \(-0.340446\pi\)
0.480525 + 0.876981i \(0.340446\pi\)
\(60\) 3.00000 0.387298
\(61\) 11.5623 1.48040 0.740201 − 0.672386i \(-0.234730\pi\)
0.740201 + 0.672386i \(0.234730\pi\)
\(62\) −15.9443 −2.02492
\(63\) 1.00000 0.125988
\(64\) 8.70820 1.08853
\(65\) 0.145898 0.0180964
\(66\) 0 0
\(67\) 1.85410 0.226515 0.113257 − 0.993566i \(-0.463872\pi\)
0.113257 + 0.993566i \(0.463872\pi\)
\(68\) 5.56231 0.674529
\(69\) −0.236068 −0.0284192
\(70\) −1.61803 −0.193392
\(71\) 10.3262 1.22550 0.612749 − 0.790277i \(-0.290063\pi\)
0.612749 + 0.790277i \(0.290063\pi\)
\(72\) 7.47214 0.880600
\(73\) 5.70820 0.668095 0.334047 − 0.942556i \(-0.391585\pi\)
0.334047 + 0.942556i \(0.391585\pi\)
\(74\) −16.3262 −1.89789
\(75\) 4.61803 0.533245
\(76\) −28.4164 −3.25959
\(77\) 0 0
\(78\) 0.618034 0.0699786
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −6.09017 −0.680902
\(81\) 1.00000 0.111111
\(82\) −0.618034 −0.0682504
\(83\) −1.47214 −0.161588 −0.0807940 − 0.996731i \(-0.525746\pi\)
−0.0807940 + 0.996731i \(0.525746\pi\)
\(84\) −4.85410 −0.529626
\(85\) −0.708204 −0.0768155
\(86\) 17.5623 1.89379
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) −1.61803 −0.170556
\(91\) −0.236068 −0.0247466
\(92\) 1.14590 0.119468
\(93\) 6.09017 0.631521
\(94\) −26.4164 −2.72464
\(95\) 3.61803 0.371202
\(96\) −10.8541 −1.10779
\(97\) 7.85410 0.797463 0.398732 − 0.917068i \(-0.369451\pi\)
0.398732 + 0.917068i \(0.369451\pi\)
\(98\) −15.7082 −1.58677
\(99\) 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.1.2 2
3.2 odd 2 1089.2.a.l.1.1 2
4.3 odd 2 5808.2.a.ci.1.1 2
5.4 even 2 9075.2.a.u.1.1 2
11.2 odd 10 363.2.e.k.202.1 4
11.3 even 5 363.2.e.f.130.1 4
11.4 even 5 363.2.e.f.148.1 4
11.5 even 5 363.2.e.b.124.1 4
11.6 odd 10 363.2.e.k.124.1 4
11.7 odd 10 33.2.e.b.16.1 ✓ 4
11.8 odd 10 33.2.e.b.31.1 yes 4
11.9 even 5 363.2.e.b.202.1 4
11.10 odd 2 363.2.a.d.1.1 2
33.8 even 10 99.2.f.a.64.1 4
33.29 even 10 99.2.f.a.82.1 4
33.32 even 2 1089.2.a.t.1.2 2
44.7 even 10 528.2.y.b.49.1 4
44.19 even 10 528.2.y.b.97.1 4
44.43 even 2 5808.2.a.cj.1.1 2
55.7 even 20 825.2.bx.d.49.2 8
55.8 even 20 825.2.bx.d.724.2 8
55.18 even 20 825.2.bx.d.49.1 8
55.19 odd 10 825.2.n.c.526.1 4
55.29 odd 10 825.2.n.c.676.1 4
55.52 even 20 825.2.bx.d.724.1 8
55.54 odd 2 9075.2.a.cb.1.2 2
99.7 odd 30 891.2.n.c.676.1 8
99.29 even 30 891.2.n.b.676.1 8
99.40 odd 30 891.2.n.c.379.1 8
99.41 even 30 891.2.n.b.460.1 8
99.52 odd 30 891.2.n.c.757.1 8
99.74 even 30 891.2.n.b.757.1 8
99.85 odd 30 891.2.n.c.460.1 8
99.95 even 30 891.2.n.b.379.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 11.7 odd 10
33.2.e.b.31.1 yes 4 11.8 odd 10
99.2.f.a.64.1 4 33.8 even 10
99.2.f.a.82.1 4 33.29 even 10
363.2.a.d.1.1 2 11.10 odd 2
363.2.a.i.1.2 2 1.1 even 1 trivial
363.2.e.b.124.1 4 11.5 even 5
363.2.e.b.202.1 4 11.9 even 5
363.2.e.f.130.1 4 11.3 even 5
363.2.e.f.148.1 4 11.4 even 5
363.2.e.k.124.1 4 11.6 odd 10
363.2.e.k.202.1 4 11.2 odd 10
528.2.y.b.49.1 4 44.7 even 10
528.2.y.b.97.1 4 44.19 even 10
825.2.n.c.526.1 4 55.19 odd 10
825.2.n.c.676.1 4 55.29 odd 10
825.2.bx.d.49.1 8 55.18 even 20
825.2.bx.d.49.2 8 55.7 even 20
825.2.bx.d.724.1 8 55.52 even 20
825.2.bx.d.724.2 8 55.8 even 20
891.2.n.b.379.1 8 99.95 even 30
891.2.n.b.460.1 8 99.41 even 30
891.2.n.b.676.1 8 99.29 even 30
891.2.n.b.757.1 8 99.74 even 30
891.2.n.c.379.1 8 99.40 odd 30
891.2.n.c.460.1 8 99.85 odd 30
891.2.n.c.676.1 8 99.7 odd 30
891.2.n.c.757.1 8 99.52 odd 30
1089.2.a.l.1.1 2 3.2 odd 2
1089.2.a.t.1.2 2 33.32 even 2
5808.2.a.ci.1.1 2 4.3 odd 2
5808.2.a.cj.1.1 2 44.43 even 2
9075.2.a.u.1.1 2 5.4 even 2
9075.2.a.cb.1.2 2 55.54 odd 2