Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(1,575)] 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("575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-6,-4] 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: \(33.9260982533\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) 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 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.41740\) of defining polynomial
Character \(\chi\) \(=\) 575.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.41740 q^{2} -7.84147 q^{3} +11.5134 q^{4} +34.6389 q^{6} +8.97260 q^{7} -15.5200 q^{8} +34.4886 q^{9} -28.9011 q^{11} -90.2818 q^{12} -16.0879 q^{13} -39.6355 q^{14} -23.5490 q^{16} -25.1771 q^{17} -152.350 q^{18} -35.6298 q^{19} -70.3583 q^{21} +127.668 q^{22} +23.0000 q^{23} +121.700 q^{24} +71.0668 q^{26} -58.7218 q^{27} +103.305 q^{28} +138.272 q^{29} +40.1277 q^{31} +228.186 q^{32} +226.627 q^{33} +111.217 q^{34} +397.081 q^{36} -379.745 q^{37} +157.391 q^{38} +126.153 q^{39} +412.514 q^{41} +310.801 q^{42} +402.095 q^{43} -332.750 q^{44} -101.600 q^{46} -110.070 q^{47} +184.659 q^{48} -262.492 q^{49} +197.426 q^{51} -185.227 q^{52} +421.300 q^{53} +259.397 q^{54} -139.255 q^{56} +279.390 q^{57} -610.802 q^{58} +755.913 q^{59} -307.032 q^{61} -177.260 q^{62} +309.453 q^{63} -819.594 q^{64} -1001.10 q^{66} -319.974 q^{67} -289.874 q^{68} -180.354 q^{69} -554.138 q^{71} -535.264 q^{72} +705.131 q^{73} +1677.48 q^{74} -410.219 q^{76} -259.318 q^{77} -557.268 q^{78} +1170.51 q^{79} -470.728 q^{81} -1822.24 q^{82} +455.978 q^{83} -810.063 q^{84} -1776.21 q^{86} -1084.26 q^{87} +448.546 q^{88} -1495.57 q^{89} -144.351 q^{91} +264.808 q^{92} -314.660 q^{93} +486.222 q^{94} -1789.31 q^{96} -1041.24 q^{97} +1159.53 q^{98} -996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9} + 23 q^{11} - 47 q^{12} - 132 q^{13} + 93 q^{14} + 282 q^{16} - 23 q^{17} + 15 q^{18} - 161 q^{19} - 60 q^{21} - 193 q^{22}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.41740 −1.56179 −0.780893 − 0.624665i \(-0.785235\pi\)
−0.780893 + 0.624665i \(0.785235\pi\)
\(3\) −7.84147 −1.50909 −0.754546 − 0.656248i \(-0.772143\pi\)
−0.754546 + 0.656248i \(0.772143\pi\)
\(4\) 11.5134 1.43917
\(5\) 0 0
\(6\) 34.6389 2.35688
\(7\) 8.97260 0.484475 0.242237 − 0.970217i \(-0.422119\pi\)
0.242237 + 0.970217i \(0.422119\pi\)
\(8\) −15.5200 −0.685894
\(9\) 34.4886 1.27736
\(10\) 0 0
\(11\) −28.9011 −0.792183 −0.396092 − 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(12\) −90.2818 −2.17184
\(13\) −16.0879 −0.343230 −0.171615 − 0.985164i \(-0.554898\pi\)
−0.171615 + 0.985164i \(0.554898\pi\)
\(14\) −39.6355 −0.756646
\(15\) 0 0
\(16\) −23.5490 −0.367954
\(17\) −25.1771 −0.359197 −0.179599 − 0.983740i \(-0.557480\pi\)
−0.179599 + 0.983740i \(0.557480\pi\)
\(18\) −152.350 −1.99496
\(19\) −35.6298 −0.430212 −0.215106 − 0.976591i \(-0.569010\pi\)
−0.215106 + 0.976591i \(0.569010\pi\)
\(20\) 0 0
\(21\) −70.3583 −0.731117
\(22\) 127.668 1.23722
\(23\) 23.0000 0.208514
\(24\) 121.700 1.03508
\(25\) 0 0
\(26\) 71.0668 0.536051
\(27\) −58.7218 −0.418556
\(28\) 103.305 0.697243
\(29\) 138.272 0.885395 0.442698 − 0.896671i \(-0.354022\pi\)
0.442698 + 0.896671i \(0.354022\pi\)
\(30\) 0 0
\(31\) 40.1277 0.232489 0.116244 − 0.993221i \(-0.462914\pi\)
0.116244 + 0.993221i \(0.462914\pi\)
\(32\) 228.186 1.26056
\(33\) 226.627 1.19548
\(34\) 111.217 0.560989
\(35\) 0 0
\(36\) 397.081 1.83834
\(37\) −379.745 −1.68729 −0.843645 − 0.536902i \(-0.819595\pi\)
−0.843645 + 0.536902i \(0.819595\pi\)
\(38\) 157.391 0.671899
\(39\) 126.153 0.517965
\(40\) 0 0
\(41\) 412.514 1.57132 0.785658 − 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) 310.801 1.14185
\(43\) 402.095 1.42602 0.713011 − 0.701153i \(-0.247331\pi\)
0.713011 + 0.701153i \(0.247331\pi\)
\(44\) −332.750 −1.14009
\(45\) 0 0
\(46\) −101.600 −0.325655
\(47\) −110.070 −0.341603 −0.170801 − 0.985305i \(-0.554636\pi\)
−0.170801 + 0.985305i \(0.554636\pi\)
\(48\) 184.659 0.555276
\(49\) −262.492 −0.765284
\(50\) 0 0
\(51\) 197.426 0.542061
\(52\) −185.227 −0.493967
\(53\) 421.300 1.09189 0.545943 − 0.837822i \(-0.316171\pi\)
0.545943 + 0.837822i \(0.316171\pi\)
\(54\) 259.397 0.653695
\(55\) 0 0
\(56\) −139.255 −0.332298
\(57\) 279.390 0.649229
\(58\) −610.802 −1.38280
\(59\) 755.913 1.66799 0.833996 − 0.551770i \(-0.186048\pi\)
0.833996 + 0.551770i \(0.186048\pi\)
\(60\) 0 0
\(61\) −307.032 −0.644450 −0.322225 − 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) −177.260 −0.363098
\(63\) 309.453 0.618847
\(64\) −819.594 −1.60077
\(65\) 0 0
\(66\) −1001.10 −1.86708
\(67\) −319.974 −0.583448 −0.291724 − 0.956502i \(-0.594229\pi\)
−0.291724 + 0.956502i \(0.594229\pi\)
\(68\) −289.874 −0.516947
\(69\) −180.354 −0.314667
\(70\) 0 0
\(71\) −554.138 −0.926254 −0.463127 − 0.886292i \(-0.653273\pi\)
−0.463127 + 0.886292i \(0.653273\pi\)
\(72\) −535.264 −0.876131
\(73\) 705.131 1.13054 0.565269 − 0.824907i \(-0.308772\pi\)
0.565269 + 0.824907i \(0.308772\pi\)
\(74\) 1677.48 2.63518
\(75\) 0 0
\(76\) −410.219 −0.619149
\(77\) −259.318 −0.383793
\(78\) −557.268 −0.808950
\(79\) 1170.51 1.66699 0.833495 − 0.552526i \(-0.186336\pi\)
0.833495 + 0.552526i \(0.186336\pi\)
\(80\) 0 0
\(81\) −470.728 −0.645717
\(82\) −1822.24 −2.45406
\(83\) 455.978 0.603014 0.301507 − 0.953464i \(-0.402510\pi\)
0.301507 + 0.953464i \(0.402510\pi\)
\(84\) −810.063 −1.05220
\(85\) 0 0
\(86\) −1776.21 −2.22714
\(87\) −1084.26 −1.33614
\(88\) 448.546 0.543354
\(89\) −1495.57 −1.78124 −0.890621 − 0.454746i \(-0.849730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(90\) 0 0
\(91\) −144.351 −0.166286
\(92\) 264.808 0.300088
\(93\) −314.660 −0.350847
\(94\) 486.222 0.533510
\(95\) 0 0
\(96\) −1789.31 −1.90230
\(97\) −1041.24 −1.08992 −0.544958 − 0.838463i \(-0.683455\pi\)
−0.544958 + 0.838463i \(0.683455\pi\)
\(98\) 1159.53 1.19521
\(99\) −996.760 −1.01190
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 575.4.a.j.1.2 5
5.2 odd 4 575.4.b.i.24.2 10
5.3 odd 4 575.4.b.i.24.9 10
5.4 even 2 115.4.a.e.1.4 ✓ 5
15.14 odd 2 1035.4.a.k.1.2 5
20.19 odd 2 1840.4.a.n.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 ✓ 5 5.4 even 2
575.4.a.j.1.2 5 1.1 even 1 trivial
575.4.b.i.24.2 10 5.2 odd 4
575.4.b.i.24.9 10 5.3 odd 4
1035.4.a.k.1.2 5 15.14 odd 2
1840.4.a.n.1.2 5 20.19 odd 2