Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,-2,15,0,-2,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) 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 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.456669\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.456669 q^{2} +0.835165 q^{3} -1.79145 q^{4} +0.381394 q^{6} -4.69915 q^{7} -1.73144 q^{8} -2.30250 q^{9} -1.00000 q^{11} -1.49616 q^{12} -5.89016 q^{13} -2.14596 q^{14} +2.79221 q^{16} -7.06513 q^{17} -1.05148 q^{18} +1.00000 q^{19} -3.92457 q^{21} -0.456669 q^{22} -1.06348 q^{23} -1.44604 q^{24} -2.68985 q^{26} -4.42846 q^{27} +8.41832 q^{28} -7.62662 q^{29} +0.901295 q^{31} +4.73799 q^{32} -0.835165 q^{33} -3.22642 q^{34} +4.12482 q^{36} +2.71758 q^{37} +0.456669 q^{38} -4.91925 q^{39} +0.788714 q^{41} -1.79223 q^{42} -0.714571 q^{43} +1.79145 q^{44} -0.485656 q^{46} -3.96368 q^{47} +2.33196 q^{48} +15.0820 q^{49} -5.90055 q^{51} +10.5519 q^{52} +9.69714 q^{53} -2.02234 q^{54} +8.13629 q^{56} +0.835165 q^{57} -3.48284 q^{58} -7.33476 q^{59} +8.15179 q^{61} +0.411593 q^{62} +10.8198 q^{63} -3.42073 q^{64} -0.381394 q^{66} -7.86697 q^{67} +12.6569 q^{68} -0.888178 q^{69} -3.13400 q^{71} +3.98664 q^{72} +6.49076 q^{73} +1.24103 q^{74} -1.79145 q^{76} +4.69915 q^{77} -2.24647 q^{78} +12.0148 q^{79} +3.20900 q^{81} +0.360181 q^{82} -16.3902 q^{83} +7.03068 q^{84} -0.326322 q^{86} -6.36948 q^{87} +1.73144 q^{88} -12.9487 q^{89} +27.6788 q^{91} +1.90517 q^{92} +0.752730 q^{93} -1.81009 q^{94} +3.95701 q^{96} -8.14414 q^{97} +6.88750 q^{98} +2.30250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} - 7 q^{11} + 16 q^{12} + 4 q^{13} + 6 q^{14} + 27 q^{16} - 2 q^{17} - 9 q^{18} + 7 q^{19} - 14 q^{21} - q^{22} - 10 q^{23} - 2 q^{24}+ \cdots - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.456669 0.322914 0.161457 − 0.986880i \(-0.448381\pi\)
0.161457 + 0.986880i \(0.448381\pi\)
\(3\) 0.835165 0.482183 0.241091 − 0.970502i \(-0.422495\pi\)
0.241091 + 0.970502i \(0.422495\pi\)
\(4\) −1.79145 −0.895727
\(5\) 0 0
\(6\) 0.381394 0.155703
\(7\) −4.69915 −1.77611 −0.888057 − 0.459734i \(-0.847945\pi\)
−0.888057 + 0.459734i \(0.847945\pi\)
\(8\) −1.73144 −0.612156
\(9\) −2.30250 −0.767500
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.49616 −0.431904
\(13\) −5.89016 −1.63364 −0.816818 − 0.576895i \(-0.804264\pi\)
−0.816818 + 0.576895i \(0.804264\pi\)
\(14\) −2.14596 −0.573531
\(15\) 0 0
\(16\) 2.79221 0.698053
\(17\) −7.06513 −1.71355 −0.856773 − 0.515694i \(-0.827534\pi\)
−0.856773 + 0.515694i \(0.827534\pi\)
\(18\) −1.05148 −0.247836
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.92457 −0.856411
\(22\) −0.456669 −0.0973621
\(23\) −1.06348 −0.221750 −0.110875 − 0.993834i \(-0.535365\pi\)
−0.110875 + 0.993834i \(0.535365\pi\)
\(24\) −1.44604 −0.295171
\(25\) 0 0
\(26\) −2.68985 −0.527523
\(27\) −4.42846 −0.852258
\(28\) 8.41832 1.59091
\(29\) −7.62662 −1.41623 −0.708114 − 0.706098i \(-0.750454\pi\)
−0.708114 + 0.706098i \(0.750454\pi\)
\(30\) 0 0
\(31\) 0.901295 0.161877 0.0809387 − 0.996719i \(-0.474208\pi\)
0.0809387 + 0.996719i \(0.474208\pi\)
\(32\) 4.73799 0.837567
\(33\) −0.835165 −0.145384
\(34\) −3.22642 −0.553327
\(35\) 0 0
\(36\) 4.12482 0.687470
\(37\) 2.71758 0.446768 0.223384 − 0.974731i \(-0.428290\pi\)
0.223384 + 0.974731i \(0.428290\pi\)
\(38\) 0.456669 0.0740815
\(39\) −4.91925 −0.787711
\(40\) 0 0
\(41\) 0.788714 0.123176 0.0615882 − 0.998102i \(-0.480383\pi\)
0.0615882 + 0.998102i \(0.480383\pi\)
\(42\) −1.79223 −0.276547
\(43\) −0.714571 −0.108971 −0.0544855 − 0.998515i \(-0.517352\pi\)
−0.0544855 + 0.998515i \(0.517352\pi\)
\(44\) 1.79145 0.270072
\(45\) 0 0
\(46\) −0.485656 −0.0716061
\(47\) −3.96368 −0.578163 −0.289081 − 0.957305i \(-0.593350\pi\)
−0.289081 + 0.957305i \(0.593350\pi\)
\(48\) 2.33196 0.336589
\(49\) 15.0820 2.15458
\(50\) 0 0
\(51\) −5.90055 −0.826242
\(52\) 10.5519 1.46329
\(53\) 9.69714 1.33200 0.666002 − 0.745950i \(-0.268004\pi\)
0.666002 + 0.745950i \(0.268004\pi\)
\(54\) −2.02234 −0.275206
\(55\) 0 0
\(56\) 8.13629 1.08726
\(57\) 0.835165 0.110620
\(58\) −3.48284 −0.457319
\(59\) −7.33476 −0.954904 −0.477452 − 0.878658i \(-0.658440\pi\)
−0.477452 + 0.878658i \(0.658440\pi\)
\(60\) 0 0
\(61\) 8.15179 1.04373 0.521865 − 0.853028i \(-0.325236\pi\)
0.521865 + 0.853028i \(0.325236\pi\)
\(62\) 0.411593 0.0522724
\(63\) 10.8198 1.36317
\(64\) −3.42073 −0.427592
\(65\) 0 0
\(66\) −0.381394 −0.0469463
\(67\) −7.86697 −0.961104 −0.480552 − 0.876966i \(-0.659564\pi\)
−0.480552 + 0.876966i \(0.659564\pi\)
\(68\) 12.6569 1.53487
\(69\) −0.888178 −0.106924
\(70\) 0 0
\(71\) −3.13400 −0.371937 −0.185969 − 0.982556i \(-0.559542\pi\)
−0.185969 + 0.982556i \(0.559542\pi\)
\(72\) 3.98664 0.469830
\(73\) 6.49076 0.759687 0.379843 − 0.925051i \(-0.375978\pi\)
0.379843 + 0.925051i \(0.375978\pi\)
\(74\) 1.24103 0.144267
\(75\) 0 0
\(76\) −1.79145 −0.205494
\(77\) 4.69915 0.535518
\(78\) −2.24647 −0.254363
\(79\) 12.0148 1.35177 0.675884 − 0.737008i \(-0.263762\pi\)
0.675884 + 0.737008i \(0.263762\pi\)
\(80\) 0 0
\(81\) 3.20900 0.356556
\(82\) 0.360181 0.0397753
\(83\) −16.3902 −1.79906 −0.899528 − 0.436863i \(-0.856089\pi\)
−0.899528 + 0.436863i \(0.856089\pi\)
\(84\) 7.03068 0.767110
\(85\) 0 0
\(86\) −0.326322 −0.0351882
\(87\) −6.36948 −0.682880
\(88\) 1.73144 0.184572
\(89\) −12.9487 −1.37256 −0.686281 − 0.727336i \(-0.740758\pi\)
−0.686281 + 0.727336i \(0.740758\pi\)
\(90\) 0 0
\(91\) 27.6788 2.90152
\(92\) 1.90517 0.198628
\(93\) 0.752730 0.0780545
\(94\) −1.81009 −0.186697
\(95\) 0 0
\(96\) 3.95701 0.403860
\(97\) −8.14414 −0.826912 −0.413456 − 0.910524i \(-0.635679\pi\)
−0.413456 + 0.910524i \(0.635679\pi\)
\(98\) 6.88750 0.695742
\(99\) 2.30250 0.231410
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 5225.2.a.n.1.4 7
5.4 even 2 209.2.a.d.1.4 ✓ 7
15.14 odd 2 1881.2.a.p.1.4 7
20.19 odd 2 3344.2.a.ba.1.5 7
55.54 odd 2 2299.2.a.q.1.4 7
95.94 odd 2 3971.2.a.i.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.4 ✓ 7 5.4 even 2
1881.2.a.p.1.4 7 15.14 odd 2
2299.2.a.q.1.4 7 55.54 odd 2
3344.2.a.ba.1.5 7 20.19 odd 2
3971.2.a.i.1.4 7 95.94 odd 2
5225.2.a.n.1.4 7 1.1 even 1 trivial