Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,4,Mod(1,735)] 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("735.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-6,17,-10,-3,0,33,18,-5,-22] 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: \(43.3664038542\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 735.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.53113 q^{2} -3.00000 q^{3} +4.46887 q^{4} -5.00000 q^{5} +10.5934 q^{6} +12.4689 q^{8} +9.00000 q^{9} +17.6556 q^{10} -2.93774 q^{11} -13.4066 q^{12} +19.0623 q^{13} +15.0000 q^{15} -79.7802 q^{16} -122.498 q^{17} -31.7802 q^{18} -107.436 q^{19} -22.3444 q^{20} +10.3735 q^{22} +210.623 q^{23} -37.4066 q^{24} +25.0000 q^{25} -67.3113 q^{26} -27.0000 q^{27} +95.4942 q^{29} -52.9669 q^{30} +94.3074 q^{31} +181.963 q^{32} +8.81323 q^{33} +432.556 q^{34} +40.2198 q^{36} +97.1206 q^{37} +379.370 q^{38} -57.1868 q^{39} -62.3444 q^{40} +491.113 q^{41} -43.0039 q^{43} -13.1284 q^{44} -45.0000 q^{45} -743.735 q^{46} -473.494 q^{47} +239.340 q^{48} -88.2782 q^{50} +367.494 q^{51} +85.1868 q^{52} -183.677 q^{53} +95.3405 q^{54} +14.6887 q^{55} +322.307 q^{57} -337.202 q^{58} +760.615 q^{59} +67.0331 q^{60} +198.747 q^{61} -333.012 q^{62} -4.29373 q^{64} -95.3113 q^{65} -31.1206 q^{66} -309.992 q^{67} -547.428 q^{68} -631.868 q^{69} +665.693 q^{71} +112.220 q^{72} -621.288 q^{73} -342.945 q^{74} -75.0000 q^{75} -480.117 q^{76} +201.934 q^{78} -24.7626 q^{79} +398.901 q^{80} +81.0000 q^{81} -1734.18 q^{82} +406.724 q^{83} +612.490 q^{85} +151.852 q^{86} -286.483 q^{87} -36.6303 q^{88} -261.751 q^{89} +158.901 q^{90} +941.245 q^{92} -282.922 q^{93} +1671.97 q^{94} +537.179 q^{95} -545.889 q^{96} +1004.77 q^{97} -26.4397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} + 17 q^{4} - 10 q^{5} - 3 q^{6} + 33 q^{8} + 18 q^{9} - 5 q^{10} - 22 q^{11} - 51 q^{12} + 22 q^{13} + 30 q^{15} - 87 q^{16} - 116 q^{17} + 9 q^{18} - 102 q^{19} - 85 q^{20} - 76 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.53113 −1.24844 −0.624221 − 0.781248i \(-0.714584\pi\)
−0.624221 + 0.781248i \(0.714584\pi\)
\(3\) −3.00000 −0.577350
\(4\) 4.46887 0.558609
\(5\) −5.00000 −0.447214
\(6\) 10.5934 0.720789
\(7\) 0 0
\(8\) 12.4689 0.551051
\(9\) 9.00000 0.333333
\(10\) 17.6556 0.558320
\(11\) −2.93774 −0.0805239 −0.0402619 − 0.999189i \(-0.512819\pi\)
−0.0402619 + 0.999189i \(0.512819\pi\)
\(12\) −13.4066 −0.322513
\(13\) 19.0623 0.406686 0.203343 − 0.979108i \(-0.434819\pi\)
0.203343 + 0.979108i \(0.434819\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) −79.7802 −1.24656
\(17\) −122.498 −1.74766 −0.873828 − 0.486236i \(-0.838370\pi\)
−0.873828 + 0.486236i \(0.838370\pi\)
\(18\) −31.7802 −0.416148
\(19\) −107.436 −1.29723 −0.648617 − 0.761115i \(-0.724652\pi\)
−0.648617 + 0.761115i \(0.724652\pi\)
\(20\) −22.3444 −0.249817
\(21\) 0 0
\(22\) 10.3735 0.100529
\(23\) 210.623 1.90947 0.954736 − 0.297455i \(-0.0961379\pi\)
0.954736 + 0.297455i \(0.0961379\pi\)
\(24\) −37.4066 −0.318150
\(25\) 25.0000 0.200000
\(26\) −67.3113 −0.507724
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 95.4942 0.611477 0.305738 − 0.952116i \(-0.401097\pi\)
0.305738 + 0.952116i \(0.401097\pi\)
\(30\) −52.9669 −0.322346
\(31\) 94.3074 0.546391 0.273195 − 0.961959i \(-0.411919\pi\)
0.273195 + 0.961959i \(0.411919\pi\)
\(32\) 181.963 1.00521
\(33\) 8.81323 0.0464905
\(34\) 432.556 2.18185
\(35\) 0 0
\(36\) 40.2198 0.186203
\(37\) 97.1206 0.431528 0.215764 − 0.976446i \(-0.430776\pi\)
0.215764 + 0.976446i \(0.430776\pi\)
\(38\) 379.370 1.61952
\(39\) −57.1868 −0.234800
\(40\) −62.3444 −0.246438
\(41\) 491.113 1.87071 0.935353 − 0.353716i \(-0.115082\pi\)
0.935353 + 0.353716i \(0.115082\pi\)
\(42\) 0 0
\(43\) −43.0039 −0.152512 −0.0762562 − 0.997088i \(-0.524297\pi\)
−0.0762562 + 0.997088i \(0.524297\pi\)
\(44\) −13.1284 −0.0449814
\(45\) −45.0000 −0.149071
\(46\) −743.735 −2.38387
\(47\) −473.494 −1.46949 −0.734747 − 0.678341i \(-0.762699\pi\)
−0.734747 + 0.678341i \(0.762699\pi\)
\(48\) 239.340 0.719705
\(49\) 0 0
\(50\) −88.2782 −0.249689
\(51\) 367.494 1.00901
\(52\) 85.1868 0.227178
\(53\) −183.677 −0.476038 −0.238019 − 0.971261i \(-0.576498\pi\)
−0.238019 + 0.971261i \(0.576498\pi\)
\(54\) 95.3405 0.240263
\(55\) 14.6887 0.0360114
\(56\) 0 0
\(57\) 322.307 0.748959
\(58\) −337.202 −0.763394
\(59\) 760.615 1.67837 0.839183 − 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(60\) 67.0331 0.144232
\(61\) 198.747 0.417163 0.208582 − 0.978005i \(-0.433115\pi\)
0.208582 + 0.978005i \(0.433115\pi\)
\(62\) −333.012 −0.682137
\(63\) 0 0
\(64\) −4.29373 −0.00838618
\(65\) −95.3113 −0.181876
\(66\) −31.1206 −0.0580407
\(67\) −309.992 −0.565247 −0.282624 − 0.959231i \(-0.591205\pi\)
−0.282624 + 0.959231i \(0.591205\pi\)
\(68\) −547.428 −0.976256
\(69\) −631.868 −1.10243
\(70\) 0 0
\(71\) 665.693 1.11272 0.556360 − 0.830941i \(-0.312197\pi\)
0.556360 + 0.830941i \(0.312197\pi\)
\(72\) 112.220 0.183684
\(73\) −621.288 −0.996113 −0.498057 − 0.867145i \(-0.665953\pi\)
−0.498057 + 0.867145i \(0.665953\pi\)
\(74\) −342.945 −0.538738
\(75\) −75.0000 −0.115470
\(76\) −480.117 −0.724647
\(77\) 0 0
\(78\) 201.934 0.293135
\(79\) −24.7626 −0.0352659 −0.0176330 − 0.999845i \(-0.505613\pi\)
−0.0176330 + 0.999845i \(0.505613\pi\)
\(80\) 398.901 0.557481
\(81\) 81.0000 0.111111
\(82\) −1734.18 −2.33547
\(83\) 406.724 0.537876 0.268938 − 0.963157i \(-0.413327\pi\)
0.268938 + 0.963157i \(0.413327\pi\)
\(84\) 0 0
\(85\) 612.490 0.781575
\(86\) 151.852 0.190403
\(87\) −286.483 −0.353036
\(88\) −36.6303 −0.0443728
\(89\) −261.751 −0.311748 −0.155874 − 0.987777i \(-0.549819\pi\)
−0.155874 + 0.987777i \(0.549819\pi\)
\(90\) 158.901 0.186107
\(91\) 0 0
\(92\) 941.245 1.06665
\(93\) −282.922 −0.315459
\(94\) 1671.97 1.83458
\(95\) 537.179 0.580141
\(96\) −545.889 −0.580360
\(97\) 1004.77 1.05175 0.525873 − 0.850563i \(-0.323739\pi\)
0.525873 + 0.850563i \(0.323739\pi\)
\(98\) 0 0
\(99\) −26.4397 −0.0268413
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 735.4.a.p.1.1 2
3.2 odd 2 2205.4.a.z.1.2 2
7.6 odd 2 105.4.a.f.1.1 ✓ 2
21.20 even 2 315.4.a.i.1.2 2
28.27 even 2 1680.4.a.bg.1.1 2
35.13 even 4 525.4.d.h.274.3 4
35.27 even 4 525.4.d.h.274.2 4
35.34 odd 2 525.4.a.k.1.2 2
105.104 even 2 1575.4.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 ✓ 2 7.6 odd 2
315.4.a.i.1.2 2 21.20 even 2
525.4.a.k.1.2 2 35.34 odd 2
525.4.d.h.274.2 4 35.27 even 4
525.4.d.h.274.3 4 35.13 even 4
735.4.a.p.1.1 2 1.1 even 1 trivial
1575.4.a.w.1.1 2 105.104 even 2
1680.4.a.bg.1.1 2 28.27 even 2
2205.4.a.z.1.2 2 3.2 odd 2