Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5290,2,Mod(1,5290)] 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("5290.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,1,3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.68740\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.68740 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.68740 q^{6} +4.59692 q^{7} +1.00000 q^{8} +4.22212 q^{9} +1.00000 q^{10} -5.13163 q^{11} +2.68740 q^{12} -1.22212 q^{13} +4.59692 q^{14} +2.68740 q^{15} +1.00000 q^{16} +4.68740 q^{17} +4.22212 q^{18} +4.59692 q^{19} +1.00000 q^{20} +12.3537 q^{21} -5.13163 q^{22} +2.68740 q^{24} +1.00000 q^{25} -1.22212 q^{26} +3.28432 q^{27} +4.59692 q^{28} +3.37480 q^{29} +2.68740 q^{30} -0.777884 q^{31} +1.00000 q^{32} -13.7907 q^{33} +4.68740 q^{34} +4.59692 q^{35} +4.22212 q^{36} -5.81903 q^{37} +4.59692 q^{38} -3.28432 q^{39} +1.00000 q^{40} -8.50643 q^{41} +12.3537 q^{42} -8.00000 q^{43} -5.13163 q^{44} +4.22212 q^{45} -6.44423 q^{47} +2.68740 q^{48} +14.1316 q^{49} +1.00000 q^{50} +12.5969 q^{51} -1.22212 q^{52} +6.00000 q^{53} +3.28432 q^{54} -5.13163 q^{55} +4.59692 q^{56} +12.3537 q^{57} +3.37480 q^{58} +9.37480 q^{59} +2.68740 q^{60} -10.9507 q^{61} -0.777884 q^{62} +19.4087 q^{63} +1.00000 q^{64} -1.22212 q^{65} -13.7907 q^{66} -15.6381 q^{67} +4.68740 q^{68} +4.59692 q^{70} +1.31260 q^{71} +4.22212 q^{72} -4.44423 q^{73} -5.81903 q^{74} +2.68740 q^{75} +4.59692 q^{76} -23.5897 q^{77} -3.28432 q^{78} +4.88847 q^{79} +1.00000 q^{80} -3.84008 q^{81} -8.50643 q^{82} +3.81903 q^{83} +12.3537 q^{84} +4.68740 q^{85} -8.00000 q^{86} +9.06943 q^{87} -5.13163 q^{88} -8.93057 q^{89} +4.22212 q^{90} -5.61797 q^{91} -2.09048 q^{93} -6.44423 q^{94} +4.59692 q^{95} +2.68740 q^{96} +18.0622 q^{97} +14.1316 q^{98} -21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 10 q^{9} + 3 q^{10} - 3 q^{11} + q^{12} - q^{13} - 3 q^{14} + q^{15} + 3 q^{16} + 7 q^{17} + 10 q^{18} - 3 q^{19} + 3 q^{20}+ \cdots - 57 q^{99}+O(q^{100}) \) 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
\(3\) 2.68740 1.55157 0.775785 − 0.630997i \(-0.217354\pi\)
0.775785 + 0.630997i \(0.217354\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.68740 1.09713
\(7\) 4.59692 1.73747 0.868735 − 0.495277i \(-0.164933\pi\)
0.868735 + 0.495277i \(0.164933\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.22212 1.40737
\(10\) 1.00000 0.316228
\(11\) −5.13163 −1.54725 −0.773623 − 0.633647i \(-0.781557\pi\)
−0.773623 + 0.633647i \(0.781557\pi\)
\(12\) 2.68740 0.775785
\(13\) −1.22212 −0.338954 −0.169477 − 0.985534i \(-0.554208\pi\)
−0.169477 + 0.985534i \(0.554208\pi\)
\(14\) 4.59692 1.22858
\(15\) 2.68740 0.693884
\(16\) 1.00000 0.250000
\(17\) 4.68740 1.13686 0.568431 − 0.822731i \(-0.307551\pi\)
0.568431 + 0.822731i \(0.307551\pi\)
\(18\) 4.22212 0.995162
\(19\) 4.59692 1.05460 0.527302 − 0.849678i \(-0.323204\pi\)
0.527302 + 0.849678i \(0.323204\pi\)
\(20\) 1.00000 0.223607
\(21\) 12.3537 2.69581
\(22\) −5.13163 −1.09407
\(23\) 0 0
\(24\) 2.68740 0.548563
\(25\) 1.00000 0.200000
\(26\) −1.22212 −0.239677
\(27\) 3.28432 0.632067
\(28\) 4.59692 0.868735
\(29\) 3.37480 0.626684 0.313342 − 0.949640i \(-0.398551\pi\)
0.313342 + 0.949640i \(0.398551\pi\)
\(30\) 2.68740 0.490650
\(31\) −0.777884 −0.139712 −0.0698560 − 0.997557i \(-0.522254\pi\)
−0.0698560 + 0.997557i \(0.522254\pi\)
\(32\) 1.00000 0.176777
\(33\) −13.7907 −2.40066
\(34\) 4.68740 0.803882
\(35\) 4.59692 0.777021
\(36\) 4.22212 0.703686
\(37\) −5.81903 −0.956643 −0.478321 − 0.878185i \(-0.658755\pi\)
−0.478321 + 0.878185i \(0.658755\pi\)
\(38\) 4.59692 0.745718
\(39\) −3.28432 −0.525911
\(40\) 1.00000 0.158114
\(41\) −8.50643 −1.32848 −0.664241 − 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(42\) 12.3537 1.90622
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −5.13163 −0.773623
\(45\) 4.22212 0.629396
\(46\) 0 0
\(47\) −6.44423 −0.939988 −0.469994 − 0.882670i \(-0.655744\pi\)
−0.469994 + 0.882670i \(0.655744\pi\)
\(48\) 2.68740 0.387893
\(49\) 14.1316 2.01880
\(50\) 1.00000 0.141421
\(51\) 12.5969 1.76392
\(52\) −1.22212 −0.169477
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 3.28432 0.446939
\(55\) −5.13163 −0.691949
\(56\) 4.59692 0.614289
\(57\) 12.3537 1.63629
\(58\) 3.37480 0.443133
\(59\) 9.37480 1.22049 0.610247 − 0.792211i \(-0.291070\pi\)
0.610247 + 0.792211i \(0.291070\pi\)
\(60\) 2.68740 0.346942
\(61\) −10.9507 −1.40209 −0.701044 − 0.713118i \(-0.747283\pi\)
−0.701044 + 0.713118i \(0.747283\pi\)
\(62\) −0.777884 −0.0987913
\(63\) 19.4087 2.44527
\(64\) 1.00000 0.125000
\(65\) −1.22212 −0.151585
\(66\) −13.7907 −1.69752
\(67\) −15.6381 −1.91049 −0.955247 − 0.295810i \(-0.904410\pi\)
−0.955247 + 0.295810i \(0.904410\pi\)
\(68\) 4.68740 0.568431
\(69\) 0 0
\(70\) 4.59692 0.549436
\(71\) 1.31260 0.155777 0.0778885 − 0.996962i \(-0.475182\pi\)
0.0778885 + 0.996962i \(0.475182\pi\)
\(72\) 4.22212 0.497581
\(73\) −4.44423 −0.520158 −0.260079 − 0.965587i \(-0.583749\pi\)
−0.260079 + 0.965587i \(0.583749\pi\)
\(74\) −5.81903 −0.676449
\(75\) 2.68740 0.310314
\(76\) 4.59692 0.527302
\(77\) −23.5897 −2.68829
\(78\) −3.28432 −0.371875
\(79\) 4.88847 0.549995 0.274998 − 0.961445i \(-0.411323\pi\)
0.274998 + 0.961445i \(0.411323\pi\)
\(80\) 1.00000 0.111803
\(81\) −3.84008 −0.426676
\(82\) −8.50643 −0.939378
\(83\) 3.81903 0.419193 0.209597 − 0.977788i \(-0.432785\pi\)
0.209597 + 0.977788i \(0.432785\pi\)
\(84\) 12.3537 1.34790
\(85\) 4.68740 0.508420
\(86\) −8.00000 −0.862662
\(87\) 9.06943 0.972345
\(88\) −5.13163 −0.547034
\(89\) −8.93057 −0.946638 −0.473319 − 0.880891i \(-0.656944\pi\)
−0.473319 + 0.880891i \(0.656944\pi\)
\(90\) 4.22212 0.445050
\(91\) −5.61797 −0.588923
\(92\) 0 0
\(93\) −2.09048 −0.216773
\(94\) −6.44423 −0.664672
\(95\) 4.59692 0.471634
\(96\) 2.68740 0.274282
\(97\) 18.0622 1.83394 0.916969 − 0.398958i \(-0.130628\pi\)
0.916969 + 0.398958i \(0.130628\pi\)
\(98\) 14.1316 1.42751
\(99\) −21.6663 −2.17755
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 5290.2.a.r.1.3 3
23.22 odd 2 230.2.a.d.1.3 ✓ 3
69.68 even 2 2070.2.a.z.1.1 3
92.91 even 2 1840.2.a.r.1.1 3
115.22 even 4 1150.2.b.j.599.4 6
115.68 even 4 1150.2.b.j.599.3 6
115.114 odd 2 1150.2.a.q.1.1 3
184.45 odd 2 7360.2.a.bz.1.1 3
184.91 even 2 7360.2.a.ce.1.3 3
460.459 even 2 9200.2.a.cf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.3 ✓ 3 23.22 odd 2
1150.2.a.q.1.1 3 115.114 odd 2
1150.2.b.j.599.3 6 115.68 even 4
1150.2.b.j.599.4 6 115.22 even 4
1840.2.a.r.1.1 3 92.91 even 2
2070.2.a.z.1.1 3 69.68 even 2
5290.2.a.r.1.3 3 1.1 even 1 trivial
7360.2.a.bz.1.1 3 184.45 odd 2
7360.2.a.ce.1.3 3 184.91 even 2
9200.2.a.cf.1.3 3 460.459 even 2