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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(24,575)] 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("575.24"); 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([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-44,0,38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.9
Root \(3.41740i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.4.b.i.24.2

$q$-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)
 
\(f(q)\) \(=\) \(q+4.41740i q^{2} -7.84147i q^{3} -11.5134 q^{4} +34.6389 q^{6} -8.97260i q^{7} -15.5200i q^{8} -34.4886 q^{9} -28.9011 q^{11} +90.2818i q^{12} -16.0879i q^{13} +39.6355 q^{14} -23.5490 q^{16} +25.1771i q^{17} -152.350i q^{18} +35.6298 q^{19} -70.3583 q^{21} -127.668i q^{22} +23.0000i q^{23} -121.700 q^{24} +71.0668 q^{26} +58.7218i q^{27} +103.305i q^{28} -138.272 q^{29} +40.1277 q^{31} -228.186i q^{32} +226.627i q^{33} -111.217 q^{34} +397.081 q^{36} +379.745i q^{37} +157.391i q^{38} -126.153 q^{39} +412.514 q^{41} -310.801i q^{42} +402.095i q^{43} +332.750 q^{44} -101.600 q^{46} +110.070i q^{47} +184.659i q^{48} +262.492 q^{49} +197.426 q^{51} +185.227i q^{52} +421.300i q^{53} -259.397 q^{54} -139.255 q^{56} -279.390i q^{57} -610.802i q^{58} -755.913 q^{59} -307.032 q^{61} +177.260i q^{62} +309.453i q^{63} +819.594 q^{64} -1001.10 q^{66} +319.974i q^{67} -289.874i q^{68} +180.354 q^{69} -554.138 q^{71} +535.264i q^{72} +705.131i q^{73} -1677.48 q^{74} -410.219 q^{76} +259.318i q^{77} -557.268i q^{78} -1170.51 q^{79} -470.728 q^{81} +1822.24i q^{82} +455.978i q^{83} +810.063 q^{84} -1776.21 q^{86} +1084.26i q^{87} +448.546i q^{88} +1495.57 q^{89} -144.351 q^{91} -264.808i q^{92} -314.660i q^{93} -486.222 q^{94} -1789.31 q^{96} +1041.24i q^{97} +1159.53i q^{98} +996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41}+ \cdots - 4042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.41740i 1.56179i 0.624665 + 0.780893i \(0.285235\pi\)
−0.624665 + 0.780893i \(0.714765\pi\)
\(3\) − 7.84147i − 1.50909i −0.656248 − 0.754546i \(-0.727857\pi\)
0.656248 − 0.754546i \(-0.272143\pi\)
\(4\) −11.5134 −1.43917
\(5\) 0 0
\(6\) 34.6389 2.35688
\(7\) − 8.97260i − 0.484475i −0.970217 − 0.242237i \(-0.922119\pi\)
0.970217 − 0.242237i \(-0.0778813\pi\)
\(8\) − 15.5200i − 0.685894i
\(9\) −34.4886 −1.27736
\(10\) 0 0
\(11\) −28.9011 −0.792183 −0.396092 − 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(12\) 90.2818i 2.17184i
\(13\) − 16.0879i − 0.343230i −0.985164 − 0.171615i \(-0.945102\pi\)
0.985164 − 0.171615i \(-0.0548985\pi\)
\(14\) 39.6355 0.756646
\(15\) 0 0
\(16\) −23.5490 −0.367954
\(17\) 25.1771i 0.359197i 0.983740 + 0.179599i \(0.0574799\pi\)
−0.983740 + 0.179599i \(0.942520\pi\)
\(18\) − 152.350i − 1.99496i
\(19\) 35.6298 0.430212 0.215106 − 0.976591i \(-0.430990\pi\)
0.215106 + 0.976591i \(0.430990\pi\)
\(20\) 0 0
\(21\) −70.3583 −0.731117
\(22\) − 127.668i − 1.23722i
\(23\) 23.0000i 0.208514i
\(24\) −121.700 −1.03508
\(25\) 0 0
\(26\) 71.0668 0.536051
\(27\) 58.7218i 0.418556i
\(28\) 103.305i 0.697243i
\(29\) −138.272 −0.885395 −0.442698 − 0.896671i \(-0.645978\pi\)
−0.442698 + 0.896671i \(0.645978\pi\)
\(30\) 0 0
\(31\) 40.1277 0.232489 0.116244 − 0.993221i \(-0.462914\pi\)
0.116244 + 0.993221i \(0.462914\pi\)
\(32\) − 228.186i − 1.26056i
\(33\) 226.627i 1.19548i
\(34\) −111.217 −0.560989
\(35\) 0 0
\(36\) 397.081 1.83834
\(37\) 379.745i 1.68729i 0.536902 + 0.843645i \(0.319595\pi\)
−0.536902 + 0.843645i \(0.680405\pi\)
\(38\) 157.391i 0.671899i
\(39\) −126.153 −0.517965
\(40\) 0 0
\(41\) 412.514 1.57132 0.785658 − 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) − 310.801i − 1.14185i
\(43\) 402.095i 1.42602i 0.701153 + 0.713011i \(0.252669\pi\)
−0.701153 + 0.713011i \(0.747331\pi\)
\(44\) 332.750 1.14009
\(45\) 0 0
\(46\) −101.600 −0.325655
\(47\) 110.070i 0.341603i 0.985305 + 0.170801i \(0.0546357\pi\)
−0.985305 + 0.170801i \(0.945364\pi\)
\(48\) 184.659i 0.555276i
\(49\) 262.492 0.765284
\(50\) 0 0
\(51\) 197.426 0.542061
\(52\) 185.227i 0.493967i
\(53\) 421.300i 1.09189i 0.837822 + 0.545943i \(0.183829\pi\)
−0.837822 + 0.545943i \(0.816171\pi\)
\(54\) −259.397 −0.653695
\(55\) 0 0
\(56\) −139.255 −0.332298
\(57\) − 279.390i − 0.649229i
\(58\) − 610.802i − 1.38280i
\(59\) −755.913 −1.66799 −0.833996 − 0.551770i \(-0.813952\pi\)
−0.833996 + 0.551770i \(0.813952\pi\)
\(60\) 0 0
\(61\) −307.032 −0.644450 −0.322225 − 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) 177.260i 0.363098i
\(63\) 309.453i 0.618847i
\(64\) 819.594 1.60077
\(65\) 0 0
\(66\) −1001.10 −1.86708
\(67\) 319.974i 0.583448i 0.956502 + 0.291724i \(0.0942290\pi\)
−0.956502 + 0.291724i \(0.905771\pi\)
\(68\) − 289.874i − 0.516947i
\(69\) 180.354 0.314667
\(70\) 0 0
\(71\) −554.138 −0.926254 −0.463127 − 0.886292i \(-0.653273\pi\)
−0.463127 + 0.886292i \(0.653273\pi\)
\(72\) 535.264i 0.876131i
\(73\) 705.131i 1.13054i 0.824907 + 0.565269i \(0.191228\pi\)
−0.824907 + 0.565269i \(0.808772\pi\)
\(74\) −1677.48 −2.63518
\(75\) 0 0
\(76\) −410.219 −0.619149
\(77\) 259.318i 0.383793i
\(78\) − 557.268i − 0.808950i
\(79\) −1170.51 −1.66699 −0.833495 − 0.552526i \(-0.813664\pi\)
−0.833495 + 0.552526i \(0.813664\pi\)
\(80\) 0 0
\(81\) −470.728 −0.645717
\(82\) 1822.24i 2.45406i
\(83\) 455.978i 0.603014i 0.953464 + 0.301507i \(0.0974896\pi\)
−0.953464 + 0.301507i \(0.902510\pi\)
\(84\) 810.063 1.05220
\(85\) 0 0
\(86\) −1776.21 −2.22714
\(87\) 1084.26i 1.33614i
\(88\) 448.546i 0.543354i
\(89\) 1495.57 1.78124 0.890621 − 0.454746i \(-0.150270\pi\)
0.890621 + 0.454746i \(0.150270\pi\)
\(90\) 0 0
\(91\) −144.351 −0.166286
\(92\) − 264.808i − 0.300088i
\(93\) − 314.660i − 0.350847i
\(94\) −486.222 −0.533510
\(95\) 0 0
\(96\) −1789.31 −1.90230
\(97\) 1041.24i 1.08992i 0.838463 + 0.544958i \(0.183455\pi\)
−0.838463 + 0.544958i \(0.816545\pi\)
\(98\) 1159.53i 1.19521i
\(99\) 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.b.i.24.9 10
5.2 odd 4 575.4.a.j.1.2 5
5.3 odd 4 115.4.a.e.1.4 ✓ 5
5.4 even 2 inner 575.4.b.i.24.2 10
15.8 even 4 1035.4.a.k.1.2 5
20.3 even 4 1840.4.a.n.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 ✓ 5 5.3 odd 4
575.4.a.j.1.2 5 5.2 odd 4
575.4.b.i.24.2 10 5.4 even 2 inner
575.4.b.i.24.9 10 1.1 even 1 trivial
1035.4.a.k.1.2 5 15.8 even 4
1840.4.a.n.1.2 5 20.3 even 4