Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,2,48,-40] 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: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.94485\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.944847 q^{2} +7.13877 q^{3} -7.10726 q^{4} -5.00000 q^{5} -6.74505 q^{6} +21.6633 q^{7} +14.2741 q^{8} +23.9620 q^{9} +4.72424 q^{10} -55.2377 q^{11} -50.7371 q^{12} +24.7185 q^{13} -20.4685 q^{14} -35.6939 q^{15} +43.3713 q^{16} -22.6405 q^{18} +101.114 q^{19} +35.5363 q^{20} +154.649 q^{21} +52.1912 q^{22} -109.317 q^{23} +101.899 q^{24} +25.0000 q^{25} -23.3553 q^{26} -21.6873 q^{27} -153.966 q^{28} +0.675659 q^{29} +33.7252 q^{30} +314.298 q^{31} -155.172 q^{32} -394.330 q^{33} -108.316 q^{35} -170.305 q^{36} -287.411 q^{37} -95.5369 q^{38} +176.460 q^{39} -71.3703 q^{40} +128.033 q^{41} -146.120 q^{42} +118.426 q^{43} +392.589 q^{44} -119.810 q^{45} +103.288 q^{46} -194.700 q^{47} +309.618 q^{48} +126.297 q^{49} -23.6212 q^{50} -175.681 q^{52} +741.714 q^{53} +20.4911 q^{54} +276.189 q^{55} +309.223 q^{56} +721.826 q^{57} -0.638395 q^{58} -187.827 q^{59} +253.686 q^{60} +674.331 q^{61} -296.964 q^{62} +519.096 q^{63} -200.357 q^{64} -123.593 q^{65} +372.581 q^{66} +669.645 q^{67} -780.388 q^{69} +102.342 q^{70} -838.060 q^{71} +342.036 q^{72} -95.8763 q^{73} +271.560 q^{74} +178.469 q^{75} -718.641 q^{76} -1196.63 q^{77} -166.728 q^{78} +743.451 q^{79} -216.857 q^{80} -801.796 q^{81} -120.972 q^{82} -520.180 q^{83} -1099.13 q^{84} -111.894 q^{86} +4.82338 q^{87} -788.467 q^{88} +963.251 q^{89} +113.202 q^{90} +535.484 q^{91} +776.944 q^{92} +2243.70 q^{93} +183.962 q^{94} -505.568 q^{95} -1107.74 q^{96} +969.429 q^{97} -119.331 q^{98} -1323.61 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.944847 −0.334054 −0.167027 − 0.985952i \(-0.553417\pi\)
−0.167027 + 0.985952i \(0.553417\pi\)
\(3\) 7.13877 1.37386 0.686929 − 0.726725i \(-0.258959\pi\)
0.686929 + 0.726725i \(0.258959\pi\)
\(4\) −7.10726 −0.888408
\(5\) −5.00000 −0.447214
\(6\) −6.74505 −0.458942
\(7\) 21.6633 1.16971 0.584853 − 0.811139i \(-0.301152\pi\)
0.584853 + 0.811139i \(0.301152\pi\)
\(8\) 14.2741 0.630830
\(9\) 23.9620 0.887483
\(10\) 4.72424 0.149393
\(11\) −55.2377 −1.51407 −0.757037 − 0.653373i \(-0.773354\pi\)
−0.757037 + 0.653373i \(0.773354\pi\)
\(12\) −50.7371 −1.22055
\(13\) 24.7185 0.527361 0.263680 − 0.964610i \(-0.415064\pi\)
0.263680 + 0.964610i \(0.415064\pi\)
\(14\) −20.4685 −0.390745
\(15\) −35.6939 −0.614408
\(16\) 43.3713 0.677677
\(17\) 0 0
\(18\) −22.6405 −0.296467
\(19\) 101.114 1.22090 0.610448 − 0.792056i \(-0.290989\pi\)
0.610448 + 0.792056i \(0.290989\pi\)
\(20\) 35.5363 0.397308
\(21\) 154.649 1.60701
\(22\) 52.1912 0.505782
\(23\) −109.317 −0.991050 −0.495525 − 0.868594i \(-0.665024\pi\)
−0.495525 + 0.868594i \(0.665024\pi\)
\(24\) 101.899 0.866671
\(25\) 25.0000 0.200000
\(26\) −23.3553 −0.176167
\(27\) −21.6873 −0.154582
\(28\) −153.966 −1.03918
\(29\) 0.675659 0.00432644 0.00216322 − 0.999998i \(-0.499311\pi\)
0.00216322 + 0.999998i \(0.499311\pi\)
\(30\) 33.7252 0.205245
\(31\) 314.298 1.82096 0.910478 − 0.413557i \(-0.135714\pi\)
0.910478 + 0.413557i \(0.135714\pi\)
\(32\) −155.172 −0.857211
\(33\) −394.330 −2.08012
\(34\) 0 0
\(35\) −108.316 −0.523108
\(36\) −170.305 −0.788447
\(37\) −287.411 −1.27703 −0.638515 − 0.769609i \(-0.720451\pi\)
−0.638515 + 0.769609i \(0.720451\pi\)
\(38\) −95.5369 −0.407845
\(39\) 176.460 0.724518
\(40\) −71.3703 −0.282116
\(41\) 128.033 0.487692 0.243846 − 0.969814i \(-0.421591\pi\)
0.243846 + 0.969814i \(0.421591\pi\)
\(42\) −146.120 −0.536828
\(43\) 118.426 0.419995 0.209997 − 0.977702i \(-0.432654\pi\)
0.209997 + 0.977702i \(0.432654\pi\)
\(44\) 392.589 1.34511
\(45\) −119.810 −0.396895
\(46\) 103.288 0.331064
\(47\) −194.700 −0.604253 −0.302126 − 0.953268i \(-0.597696\pi\)
−0.302126 + 0.953268i \(0.597696\pi\)
\(48\) 309.618 0.931031
\(49\) 126.297 0.368212
\(50\) −23.6212 −0.0668108
\(51\) 0 0
\(52\) −175.681 −0.468512
\(53\) 741.714 1.92231 0.961154 − 0.276013i \(-0.0890132\pi\)
0.961154 + 0.276013i \(0.0890132\pi\)
\(54\) 20.4911 0.0516387
\(55\) 276.189 0.677114
\(56\) 309.223 0.737886
\(57\) 721.826 1.67734
\(58\) −0.638395 −0.00144526
\(59\) −187.827 −0.414459 −0.207229 − 0.978292i \(-0.566445\pi\)
−0.207229 + 0.978292i \(0.566445\pi\)
\(60\) 253.686 0.545845
\(61\) 674.331 1.41540 0.707699 − 0.706514i \(-0.249733\pi\)
0.707699 + 0.706514i \(0.249733\pi\)
\(62\) −296.964 −0.608298
\(63\) 519.096 1.03809
\(64\) −200.357 −0.391322
\(65\) −123.593 −0.235843
\(66\) 372.581 0.694872
\(67\) 669.645 1.22105 0.610523 − 0.791998i \(-0.290959\pi\)
0.610523 + 0.791998i \(0.290959\pi\)
\(68\) 0 0
\(69\) −780.388 −1.36156
\(70\) 102.342 0.174746
\(71\) −838.060 −1.40084 −0.700419 − 0.713732i \(-0.747003\pi\)
−0.700419 + 0.713732i \(0.747003\pi\)
\(72\) 342.036 0.559851
\(73\) −95.8763 −0.153719 −0.0768594 − 0.997042i \(-0.524489\pi\)
−0.0768594 + 0.997042i \(0.524489\pi\)
\(74\) 271.560 0.426597
\(75\) 178.469 0.274771
\(76\) −718.641 −1.08465
\(77\) −1196.63 −1.77102
\(78\) −166.728 −0.242028
\(79\) 743.451 1.05880 0.529398 − 0.848374i \(-0.322418\pi\)
0.529398 + 0.848374i \(0.322418\pi\)
\(80\) −216.857 −0.303066
\(81\) −801.796 −1.09986
\(82\) −120.972 −0.162916
\(83\) −520.180 −0.687918 −0.343959 − 0.938985i \(-0.611768\pi\)
−0.343959 + 0.938985i \(0.611768\pi\)
\(84\) −1099.13 −1.42768
\(85\) 0 0
\(86\) −111.894 −0.140301
\(87\) 4.82338 0.00594391
\(88\) −788.467 −0.955123
\(89\) 963.251 1.14724 0.573620 − 0.819121i \(-0.305538\pi\)
0.573620 + 0.819121i \(0.305538\pi\)
\(90\) 113.202 0.132584
\(91\) 535.484 0.616857
\(92\) 776.944 0.880456
\(93\) 2243.70 2.50173
\(94\) 183.962 0.201853
\(95\) −505.568 −0.546002
\(96\) −1107.74 −1.17769
\(97\) 969.429 1.01475 0.507374 − 0.861726i \(-0.330616\pi\)
0.507374 + 0.861726i \(0.330616\pi\)
\(98\) −119.331 −0.123003
\(99\) −1323.61 −1.34371
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 1445.4.a.r.1.3 yes 8
17.16 even 2 1445.4.a.q.1.3 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.3 ✓ 8 17.16 even 2
1445.4.a.r.1.3 yes 8 1.1 even 1 trivial