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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,2,Mod(69,544)] 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("544.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.bd (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,20] 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: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 69.1
Character \(\chi\) \(=\) 544.69
Dual form 544.2.bd.b.205.1

$q$-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+(-1.41395 - 0.0273756i) q^{2} +(-0.343657 - 0.829662i) q^{3} +(1.99850 + 0.0774154i) q^{4} +(2.82705 + 1.17100i) q^{5} +(0.463201 + 1.18251i) q^{6} +(-0.551065 - 0.551065i) q^{7} +(-2.82366 - 0.164172i) q^{8} +(1.55108 - 1.55108i) q^{9} +(-3.96525 - 1.73313i) q^{10} +(0.660143 - 1.59373i) q^{11} +(-0.622571 - 1.68469i) q^{12} +(1.61856 - 0.670429i) q^{13} +(0.764092 + 0.794263i) q^{14} -2.74792i q^{15} +(3.98801 + 0.309429i) q^{16} +1.00000i q^{17} +(-2.23561 + 2.15069i) q^{18} +(0.212318 - 0.0879452i) q^{19} +(5.55921 + 2.55911i) q^{20} +(-0.267820 + 0.646575i) q^{21} +(-0.977037 + 2.23537i) q^{22} +(-1.41166 + 1.41166i) q^{23} +(0.834164 + 2.39910i) q^{24} +(3.08544 + 3.08544i) q^{25} +(-2.30691 + 0.903643i) q^{26} +(-4.30890 - 1.78481i) q^{27} +(-1.05864 - 1.14396i) q^{28} +(0.673845 + 1.62681i) q^{29} +(-0.0752260 + 3.88542i) q^{30} +0.625716 q^{31} +(-5.63038 - 0.546692i) q^{32} -1.54912 q^{33} +(0.0273756 - 1.41395i) q^{34} +(-0.912590 - 2.20319i) q^{35} +(3.21992 - 2.97976i) q^{36} +(7.15251 + 2.96266i) q^{37} +(-0.302615 + 0.118538i) q^{38} +(-1.11246 - 1.11246i) q^{39} +(-7.79038 - 3.77063i) q^{40} +(2.46134 - 2.46134i) q^{41} +(0.396385 - 0.906893i) q^{42} +(1.92010 - 4.63552i) q^{43} +(1.44268 - 3.13396i) q^{44} +(6.20131 - 2.56867i) q^{45} +(2.03467 - 1.95738i) q^{46} -1.60171i q^{47} +(-1.11379 - 3.41504i) q^{48} -6.39265i q^{49} +(-4.27819 - 4.44712i) q^{50} +(0.829662 - 0.343657i) q^{51} +(3.28659 - 1.21455i) q^{52} +(-0.903832 + 2.18204i) q^{53} +(6.04370 + 2.64158i) q^{54} +(3.73252 - 3.73252i) q^{55} +(1.46555 + 1.64649i) q^{56} +(-0.145930 - 0.145930i) q^{57} +(-0.908247 - 2.31867i) q^{58} +(8.34042 + 3.45471i) q^{59} +(0.212731 - 5.49173i) q^{60} +(-2.50218 - 6.04080i) q^{61} +(-0.884731 - 0.0171294i) q^{62} -1.70949 q^{63} +(7.94610 + 0.927129i) q^{64} +5.36082 q^{65} +(2.19037 + 0.0424080i) q^{66} +(-5.05744 - 12.2097i) q^{67} +(-0.0774154 + 1.99850i) q^{68} +(1.65633 + 0.686076i) q^{69} +(1.23004 + 3.14018i) q^{70} +(9.28897 + 9.28897i) q^{71} +(-4.63437 + 4.12508i) q^{72} +(0.376770 - 0.376770i) q^{73} +(-10.0322 - 4.38486i) q^{74} +(1.49954 - 3.62021i) q^{75} +(0.431127 - 0.159322i) q^{76} +(-1.24203 + 0.514465i) q^{77} +(1.54251 + 1.60341i) q^{78} -4.87827i q^{79} +(10.9120 + 5.54475i) q^{80} -2.39239i q^{81} +(-3.54758 + 3.41282i) q^{82} +(-13.0073 + 5.38778i) q^{83} +(-0.585294 + 1.27145i) q^{84} +(-1.17100 + 2.82705i) q^{85} +(-2.84182 + 6.50183i) q^{86} +(1.11813 - 1.11813i) q^{87} +(-2.12566 + 4.39176i) q^{88} +(11.3333 + 11.3333i) q^{89} +(-8.83865 + 3.46220i) q^{90} +(-1.26138 - 0.522481i) q^{91} +(-2.93050 + 2.71193i) q^{92} +(-0.215032 - 0.519133i) q^{93} +(-0.0438478 + 2.26474i) q^{94} +0.703219 q^{95} +(1.48135 + 4.85919i) q^{96} -12.8535 q^{97} +(-0.175003 + 9.03889i) q^{98} +(-1.44806 - 3.49593i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 20 q^{6} - 24 q^{8} - 16 q^{12} - 20 q^{16} - 20 q^{18} - 48 q^{22} - 8 q^{23} + 68 q^{24} + 20 q^{26} - 24 q^{27} + 20 q^{30} + 48 q^{31} - 24 q^{35} + 8 q^{36} + 68 q^{38} - 24 q^{39} - 36 q^{40}+ \cdots - 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/544\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41395 − 0.0273756i −0.999813 − 0.0193575i
\(3\) −0.343657 − 0.829662i −0.198411 − 0.479006i 0.793090 − 0.609104i \(-0.208471\pi\)
−0.991501 + 0.130098i \(0.958471\pi\)
\(4\) 1.99850 + 0.0774154i 0.999251 + 0.0387077i
\(5\) 2.82705 + 1.17100i 1.26430 + 0.523689i 0.911225 − 0.411908i \(-0.135138\pi\)
0.353071 + 0.935597i \(0.385138\pi\)
\(6\) 0.463201 + 1.18251i 0.189101 + 0.482757i
\(7\) −0.551065 − 0.551065i −0.208283 − 0.208283i 0.595254 − 0.803537i \(-0.297051\pi\)
−0.803537 + 0.595254i \(0.797051\pi\)
\(8\) −2.82366 − 0.164172i −0.998314 − 0.0580434i
\(9\) 1.55108 − 1.55108i 0.517027 − 0.517027i
\(10\) −3.96525 − 1.73313i −1.25392 − 0.548064i
\(11\) 0.660143 − 1.59373i 0.199041 − 0.480526i −0.792571 − 0.609780i \(-0.791258\pi\)
0.991612 + 0.129253i \(0.0412580\pi\)
\(12\) −0.622571 − 1.68469i −0.179721 − 0.486327i
\(13\) 1.61856 − 0.670429i 0.448907 − 0.185944i −0.146764 − 0.989171i \(-0.546886\pi\)
0.595672 + 0.803228i \(0.296886\pi\)
\(14\) 0.764092 + 0.794263i 0.204212 + 0.212276i
\(15\) − 2.74792i − 0.709510i
\(16\) 3.98801 + 0.309429i 0.997003 + 0.0773574i
\(17\) 1.00000i 0.242536i
\(18\) −2.23561 + 2.15069i −0.526939 + 0.506922i
\(19\) 0.212318 − 0.0879452i 0.0487092 − 0.0201760i −0.358196 − 0.933646i \(-0.616608\pi\)
0.406905 + 0.913470i \(0.366608\pi\)
\(20\) 5.55921 + 2.55911i 1.24308 + 0.572234i
\(21\) −0.267820 + 0.646575i −0.0584432 + 0.141094i
\(22\) −0.977037 + 2.23537i −0.208305 + 0.476583i
\(23\) −1.41166 + 1.41166i −0.294352 + 0.294352i −0.838797 − 0.544445i \(-0.816740\pi\)
0.544445 + 0.838797i \(0.316740\pi\)
\(24\) 0.834164 + 2.39910i 0.170273 + 0.489715i
\(25\) 3.08544 + 3.08544i 0.617088 + 0.617088i
\(26\) −2.30691 + 0.903643i −0.452423 + 0.177219i
\(27\) −4.30890 − 1.78481i −0.829248 − 0.343486i
\(28\) −1.05864 − 1.14396i −0.200065 − 0.216189i
\(29\) 0.673845 + 1.62681i 0.125130 + 0.302090i 0.974014 − 0.226489i \(-0.0727248\pi\)
−0.848884 + 0.528580i \(0.822725\pi\)
\(30\) −0.0752260 + 3.88542i −0.0137343 + 0.709378i
\(31\) 0.625716 0.112382 0.0561910 − 0.998420i \(-0.482104\pi\)
0.0561910 + 0.998420i \(0.482104\pi\)
\(32\) −5.63038 − 0.546692i −0.995319 − 0.0966423i
\(33\) −1.54912 −0.269667
\(34\) 0.0273756 − 1.41395i 0.00469488 − 0.242490i
\(35\) −0.912590 − 2.20319i −0.154256 − 0.372407i
\(36\) 3.21992 − 2.97976i 0.536653 − 0.496627i
\(37\) 7.15251 + 2.96266i 1.17586 + 0.487059i 0.883127 − 0.469134i \(-0.155434\pi\)
0.292737 + 0.956193i \(0.405434\pi\)
\(38\) −0.302615 + 0.118538i −0.0490906 + 0.0192293i
\(39\) −1.11246 − 1.11246i −0.178136 − 0.178136i
\(40\) −7.79038 − 3.77063i −1.23177 − 0.596190i
\(41\) 2.46134 − 2.46134i 0.384396 − 0.384396i −0.488287 − 0.872683i \(-0.662378\pi\)
0.872683 + 0.488287i \(0.162378\pi\)
\(42\) 0.396385 − 0.906893i 0.0611635 − 0.139937i
\(43\) 1.92010 − 4.63552i 0.292812 − 0.706910i −0.707188 − 0.707025i \(-0.750037\pi\)
1.00000 0.000114977i \(3.65984e-5\pi\)
\(44\) 1.44268 − 3.13396i 0.217491 − 0.472462i
\(45\) 6.20131 − 2.56867i 0.924436 − 0.382914i
\(46\) 2.03467 − 1.95738i 0.299995 − 0.288599i
\(47\) − 1.60171i − 0.233634i −0.993153 − 0.116817i \(-0.962731\pi\)
0.993153 − 0.116817i \(-0.0372691\pi\)
\(48\) −1.11379 − 3.41504i −0.160762 − 0.492919i
\(49\) − 6.39265i − 0.913236i
\(50\) −4.27819 − 4.44712i −0.605027 − 0.628917i
\(51\) 0.829662 − 0.343657i 0.116176 − 0.0481217i
\(52\) 3.28659 − 1.21455i 0.455768 − 0.168428i
\(53\) −0.903832 + 2.18204i −0.124151 + 0.299727i −0.973719 − 0.227752i \(-0.926862\pi\)
0.849568 + 0.527479i \(0.176862\pi\)
\(54\) 6.04370 + 2.64158i 0.822444 + 0.359474i
\(55\) 3.73252 − 3.73252i 0.503292 − 0.503292i
\(56\) 1.46555 + 1.64649i 0.195842 + 0.220021i
\(57\) −0.145930 − 0.145930i −0.0193288 − 0.0193288i
\(58\) −0.908247 − 2.31867i −0.119259 − 0.304456i
\(59\) 8.34042 + 3.45471i 1.08583 + 0.449765i 0.852551 − 0.522644i \(-0.175054\pi\)
0.233279 + 0.972410i \(0.425054\pi\)
\(60\) 0.212731 − 5.49173i 0.0274635 − 0.708979i
\(61\) −2.50218 − 6.04080i −0.320371 − 0.773445i −0.999232 − 0.0391781i \(-0.987526\pi\)
0.678861 − 0.734267i \(-0.262474\pi\)
\(62\) −0.884731 − 0.0171294i −0.112361 − 0.00217543i
\(63\) −1.70949 −0.215376
\(64\) 7.94610 + 0.927129i 0.993262 + 0.115891i
\(65\) 5.36082 0.664928
\(66\) 2.19037 + 0.0424080i 0.269616 + 0.00522007i
\(67\) −5.05744 − 12.2097i −0.617865 − 1.49166i −0.854178 − 0.519980i \(-0.825939\pi\)
0.236313 − 0.971677i \(-0.424061\pi\)
\(68\) −0.0774154 + 1.99850i −0.00938799 + 0.242354i
\(69\) 1.65633 + 0.686076i 0.199399 + 0.0825938i
\(70\) 1.23004 + 3.14018i 0.147018 + 0.375323i
\(71\) 9.28897 + 9.28897i 1.10240 + 1.10240i 0.994121 + 0.108277i \(0.0345334\pi\)
0.108277 + 0.994121i \(0.465467\pi\)
\(72\) −4.63437 + 4.12508i −0.546165 + 0.486145i
\(73\) 0.376770 − 0.376770i 0.0440977 − 0.0440977i −0.684714 − 0.728812i \(-0.740073\pi\)
0.728812 + 0.684714i \(0.240073\pi\)
\(74\) −10.0322 − 4.38486i −1.16622 − 0.509730i
\(75\) 1.49954 − 3.62021i 0.173152 − 0.418025i
\(76\) 0.431127 − 0.159322i 0.0494536 − 0.0182755i
\(77\) −1.24203 + 0.514465i −0.141542 + 0.0586287i
\(78\) 1.54251 + 1.60341i 0.174654 + 0.181551i
\(79\) − 4.87827i − 0.548848i −0.961609 − 0.274424i \(-0.911513\pi\)
0.961609 − 0.274424i \(-0.0884872\pi\)
\(80\) 10.9120 + 5.54475i 1.22000 + 0.619922i
\(81\) − 2.39239i − 0.265821i
\(82\) −3.54758 + 3.41282i −0.391765 + 0.376883i
\(83\) −13.0073 + 5.38778i −1.42773 + 0.591386i −0.956790 − 0.290778i \(-0.906086\pi\)
−0.470942 + 0.882164i \(0.656086\pi\)
\(84\) −0.585294 + 1.27145i −0.0638608 + 0.138726i
\(85\) −1.17100 + 2.82705i −0.127013 + 0.306637i
\(86\) −2.84182 + 6.50183i −0.306441 + 0.701110i
\(87\) 1.11813 − 1.11813i 0.119876 − 0.119876i
\(88\) −2.12566 + 4.39176i −0.226596 + 0.468163i
\(89\) 11.3333 + 11.3333i 1.20133 + 1.20133i 0.973762 + 0.227567i \(0.0730771\pi\)
0.227567 + 0.973762i \(0.426923\pi\)
\(90\) −8.83865 + 3.46220i −0.931675 + 0.364948i
\(91\) −1.26138 − 0.522481i −0.132229 − 0.0547709i
\(92\) −2.93050 + 2.71193i −0.305525 + 0.282738i
\(93\) −0.215032 − 0.519133i −0.0222978 − 0.0538316i
\(94\) −0.0438478 + 2.26474i −0.00452256 + 0.233590i
\(95\) 0.703219 0.0721488
\(96\) 1.48135 + 4.85919i 0.151190 + 0.495939i
\(97\) −12.8535 −1.30508 −0.652539 − 0.757755i \(-0.726296\pi\)
−0.652539 + 0.757755i \(0.726296\pi\)
\(98\) −0.175003 + 9.03889i −0.0176779 + 0.913065i
\(99\) −1.44806 − 3.49593i −0.145536 − 0.351354i
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 544.2.bd.b.69.1 ✓ 128
32.13 even 8 inner 544.2.bd.b.205.1 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
544.2.bd.b.69.1 ✓ 128 1.1 even 1 trivial
544.2.bd.b.205.1 yes 128 32.13 even 8 inner