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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 69.5
Character \(\chi\) \(=\) 544.69
Dual form 544.2.bd.b.205.5

$q$-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+(-1.26627 - 0.629734i) q^{2} +(-0.330180 - 0.797125i) q^{3} +(1.20687 + 1.59482i) q^{4} +(-1.39863 - 0.579332i) q^{5} +(-0.0838800 + 1.21730i) q^{6} +(-0.237221 - 0.237221i) q^{7} +(-0.523909 - 2.77948i) q^{8} +(1.59493 - 1.59493i) q^{9} +(1.40622 + 1.61435i) q^{10} +(0.156876 - 0.378732i) q^{11} +(0.872789 - 1.48861i) q^{12} +(1.26763 - 0.525070i) q^{13} +(0.150999 + 0.449772i) q^{14} +1.30617i q^{15} +(-1.08692 + 3.84949i) q^{16} +1.00000i q^{17} +(-3.02399 + 1.01523i) q^{18} +(-4.96174 + 2.05522i) q^{19} +(-0.764035 - 2.92975i) q^{20} +(-0.110769 + 0.267421i) q^{21} +(-0.437147 + 0.380786i) q^{22} +(6.02313 - 6.02313i) q^{23} +(-2.04261 + 1.33535i) q^{24} +(-1.91499 - 1.91499i) q^{25} +(-1.93582 - 0.133390i) q^{26} +(-4.18935 - 1.73529i) q^{27} +(0.0920305 - 0.664621i) q^{28} +(-2.60399 - 6.28660i) q^{29} +(0.822538 - 1.65396i) q^{30} -6.74619 q^{31} +(3.80049 - 4.19002i) q^{32} -0.353694 q^{33} +(0.629734 - 1.26627i) q^{34} +(0.194355 + 0.469215i) q^{35} +(4.46851 + 0.618757i) q^{36} +(-1.55421 - 0.643773i) q^{37} +(7.57713 + 0.522114i) q^{38} +(-0.837093 - 0.837093i) q^{39} +(-0.877488 + 4.19099i) q^{40} +(-4.35369 + 4.35369i) q^{41} +(0.308667 - 0.268871i) q^{42} +(-4.71721 + 11.3884i) q^{43} +(0.793340 - 0.206891i) q^{44} +(-3.15471 + 1.30673i) q^{45} +(-11.4199 + 3.83393i) q^{46} -1.91108i q^{47} +(3.42741 - 0.404611i) q^{48} -6.88745i q^{49} +(1.21896 + 3.63083i) q^{50} +(0.797125 - 0.330180i) q^{51} +(2.36726 + 1.38796i) q^{52} +(3.54600 - 8.56079i) q^{53} +(4.21207 + 4.83551i) q^{54} +(-0.438823 + 0.438823i) q^{55} +(-0.535070 + 0.783634i) q^{56} +(3.27653 + 3.27653i) q^{57} +(-0.661527 + 9.60034i) q^{58} +(-11.0324 - 4.56976i) q^{59} +(-2.08311 + 1.57638i) q^{60} +(0.326785 + 0.788928i) q^{61} +(8.54249 + 4.24831i) q^{62} -0.756702 q^{63} +(-7.45104 + 2.91239i) q^{64} -2.07714 q^{65} +(0.447872 + 0.222733i) q^{66} +(2.20713 + 5.32849i) q^{67} +(-1.59482 + 1.20687i) q^{68} +(-6.78991 - 2.81247i) q^{69} +(0.0493746 - 0.716544i) q^{70} +(-6.93872 - 6.93872i) q^{71} +(-5.26868 - 3.59748i) q^{72} +(4.45843 - 4.45843i) q^{73} +(1.56264 + 1.79393i) q^{74} +(-0.894195 + 2.15878i) q^{75} +(-9.26589 - 5.43271i) q^{76} +(-0.127057 + 0.0526289i) q^{77} +(0.532839 + 1.58713i) q^{78} +0.0645555i q^{79} +(3.75034 - 4.75433i) q^{80} -2.85432i q^{81} +(8.25460 - 2.77127i) q^{82} +(-10.8380 + 4.48925i) q^{83} +(-0.560173 + 0.146085i) q^{84} +(0.579332 - 1.39863i) q^{85} +(13.1449 - 11.4501i) q^{86} +(-4.15142 + 4.15142i) q^{87} +(-1.13487 - 0.237613i) q^{88} +(11.7403 + 11.7403i) q^{89} +(4.81760 + 0.331965i) q^{90} +(-0.425266 - 0.176151i) q^{91} +(16.8750 + 2.33669i) q^{92} +(2.22746 + 5.37756i) q^{93} +(-1.20347 + 2.41994i) q^{94} +8.13030 q^{95} +(-4.59482 - 1.64601i) q^{96} +9.59336 q^{97} +(-4.33726 + 8.72136i) q^{98} +(-0.353845 - 0.854257i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26627 − 0.629734i −0.895387 − 0.445289i
\(3\) −0.330180 − 0.797125i −0.190630 − 0.460221i 0.799449 − 0.600734i \(-0.205125\pi\)
−0.990079 + 0.140513i \(0.955125\pi\)
\(4\) 1.20687 + 1.59482i 0.603436 + 0.797412i
\(5\) −1.39863 − 0.579332i −0.625487 − 0.259085i 0.0473473 − 0.998878i \(-0.484923\pi\)
−0.672834 + 0.739793i \(0.734923\pi\)
\(6\) −0.0838800 + 1.21730i −0.0342439 + 0.496961i
\(7\) −0.237221 − 0.237221i −0.0896611 − 0.0896611i 0.660854 − 0.750515i \(-0.270194\pi\)
−0.750515 + 0.660854i \(0.770194\pi\)
\(8\) −0.523909 − 2.77948i −0.185230 − 0.982695i
\(9\) 1.59493 − 1.59493i 0.531644 − 0.531644i
\(10\) 1.40622 + 1.61435i 0.444685 + 0.510504i
\(11\) 0.156876 − 0.378732i 0.0472999 − 0.114192i −0.898464 − 0.439048i \(-0.855316\pi\)
0.945764 + 0.324856i \(0.105316\pi\)
\(12\) 0.872789 − 1.48861i 0.251953 − 0.429724i
\(13\) 1.26763 − 0.525070i 0.351578 − 0.145628i −0.199904 − 0.979816i \(-0.564063\pi\)
0.551481 + 0.834187i \(0.314063\pi\)
\(14\) 0.150999 + 0.449772i 0.0403563 + 0.120207i
\(15\) 1.30617i 0.337251i
\(16\) −1.08692 + 3.84949i −0.271731 + 0.962373i
\(17\) 1.00000i 0.242536i
\(18\) −3.02399 + 1.01523i −0.712762 + 0.239292i
\(19\) −4.96174 + 2.05522i −1.13830 + 0.471500i −0.870595 − 0.492001i \(-0.836266\pi\)
−0.267706 + 0.963501i \(0.586266\pi\)
\(20\) −0.764035 − 2.92975i −0.170843 − 0.655112i
\(21\) −0.110769 + 0.267421i −0.0241718 + 0.0583560i
\(22\) −0.437147 + 0.380786i −0.0932001 + 0.0811839i
\(23\) 6.02313 − 6.02313i 1.25591 − 1.25591i 0.302881 − 0.953028i \(-0.402051\pi\)
0.953028 − 0.302881i \(-0.0979486\pi\)
\(24\) −2.04261 + 1.33535i −0.416946 + 0.272577i
\(25\) −1.91499 − 1.91499i −0.382998 − 0.382998i
\(26\) −1.93582 − 0.133390i −0.379645 − 0.0261600i
\(27\) −4.18935 − 1.73529i −0.806241 − 0.333956i
\(28\) 0.0920305 − 0.664621i 0.0173921 − 0.125602i
\(29\) −2.60399 − 6.28660i −0.483550 − 1.16739i −0.957912 − 0.287062i \(-0.907321\pi\)
0.474362 − 0.880330i \(-0.342679\pi\)
\(30\) 0.822538 − 1.65396i 0.150174 − 0.301970i
\(31\) −6.74619 −1.21165 −0.605826 − 0.795597i \(-0.707157\pi\)
−0.605826 + 0.795597i \(0.707157\pi\)
\(32\) 3.80049 − 4.19002i 0.671839 − 0.740698i
\(33\) −0.353694 −0.0615702
\(34\) 0.629734 − 1.26627i 0.107998 − 0.217163i
\(35\) 0.194355 + 0.469215i 0.0328520 + 0.0793117i
\(36\) 4.46851 + 0.618757i 0.744751 + 0.103126i
\(37\) −1.55421 − 0.643773i −0.255510 − 0.105836i 0.251252 − 0.967922i \(-0.419158\pi\)
−0.506762 + 0.862086i \(0.669158\pi\)
\(38\) 7.57713 + 0.522114i 1.22917 + 0.0846981i
\(39\) −0.837093 − 0.837093i −0.134042 − 0.134042i
\(40\) −0.877488 + 4.19099i −0.138743 + 0.662653i
\(41\) −4.35369 + 4.35369i −0.679932 + 0.679932i −0.959985 − 0.280053i \(-0.909648\pi\)
0.280053 + 0.959985i \(0.409648\pi\)
\(42\) 0.308667 − 0.268871i 0.0476284 − 0.0414877i
\(43\) −4.71721 + 11.3884i −0.719368 + 1.73671i −0.0442238 + 0.999022i \(0.514081\pi\)
−0.675144 + 0.737686i \(0.735919\pi\)
\(44\) 0.793340 − 0.206891i 0.119600 − 0.0311900i
\(45\) −3.15471 + 1.30673i −0.470277 + 0.194795i
\(46\) −11.4199 + 3.83393i −1.68377 + 0.565282i
\(47\) − 1.91108i − 0.278759i −0.990239 − 0.139380i \(-0.955489\pi\)
0.990239 − 0.139380i \(-0.0445108\pi\)
\(48\) 3.42741 − 0.404611i 0.494704 − 0.0584006i
\(49\) − 6.88745i − 0.983922i
\(50\) 1.21896 + 3.63083i 0.172387 + 0.513476i
\(51\) 0.797125 − 0.330180i 0.111620 − 0.0462345i
\(52\) 2.36726 + 1.38796i 0.328280 + 0.192475i
\(53\) 3.54600 − 8.56079i 0.487080 − 1.17592i −0.469103 − 0.883144i \(-0.655423\pi\)
0.956182 − 0.292771i \(-0.0945775\pi\)
\(54\) 4.21207 + 4.83551i 0.573191 + 0.658030i
\(55\) −0.438823 + 0.438823i −0.0591709 + 0.0591709i
\(56\) −0.535070 + 0.783634i −0.0715017 + 0.104717i
\(57\) 3.27653 + 3.27653i 0.433988 + 0.433988i
\(58\) −0.661527 + 9.60034i −0.0868627 + 1.26059i
\(59\) −11.0324 − 4.56976i −1.43629 − 0.594932i −0.477396 − 0.878688i \(-0.658419\pi\)
−0.958896 + 0.283756i \(0.908419\pi\)
\(60\) −2.08311 + 1.57638i −0.268928 + 0.203509i
\(61\) 0.326785 + 0.788928i 0.0418405 + 0.101012i 0.943418 − 0.331605i \(-0.107590\pi\)
−0.901578 + 0.432617i \(0.857590\pi\)
\(62\) 8.54249 + 4.24831i 1.08490 + 0.539535i
\(63\) −0.756702 −0.0953355
\(64\) −7.45104 + 2.91239i −0.931380 + 0.364049i
\(65\) −2.07714 −0.257637
\(66\) 0.447872 + 0.222733i 0.0551292 + 0.0274165i
\(67\) 2.20713 + 5.32849i 0.269644 + 0.650979i 0.999467 − 0.0326579i \(-0.0103972\pi\)
−0.729822 + 0.683637i \(0.760397\pi\)
\(68\) −1.59482 + 1.20687i −0.193401 + 0.146355i
\(69\) −6.78991 − 2.81247i −0.817409 − 0.338582i
\(70\) 0.0493746 − 0.716544i 0.00590139 − 0.0856433i
\(71\) −6.93872 − 6.93872i −0.823475 − 0.823475i 0.163130 − 0.986605i \(-0.447841\pi\)
−0.986605 + 0.163130i \(0.947841\pi\)
\(72\) −5.26868 − 3.59748i −0.620920 − 0.423967i
\(73\) 4.45843 − 4.45843i 0.521820 − 0.521820i −0.396301 − 0.918121i \(-0.629706\pi\)
0.918121 + 0.396301i \(0.129706\pi\)
\(74\) 1.56264 + 1.79393i 0.181653 + 0.208540i
\(75\) −0.894195 + 2.15878i −0.103253 + 0.249274i
\(76\) −9.26589 − 5.43271i −1.06287 − 0.623175i
\(77\) −0.127057 + 0.0526289i −0.0144795 + 0.00599762i
\(78\) 0.532839 + 1.58713i 0.0603321 + 0.179707i
\(79\) 0.0645555i 0.00726307i 0.999993 + 0.00363153i \(0.00115596\pi\)
−0.999993 + 0.00363153i \(0.998844\pi\)
\(80\) 3.75034 − 4.75433i 0.419301 − 0.531550i
\(81\) − 2.85432i − 0.317147i
\(82\) 8.25460 − 2.77127i 0.911568 − 0.306036i
\(83\) −10.8380 + 4.48925i −1.18963 + 0.492759i −0.887633 − 0.460551i \(-0.847652\pi\)
−0.301993 + 0.953310i \(0.597652\pi\)
\(84\) −0.560173 + 0.146085i −0.0611199 + 0.0159392i
\(85\) 0.579332 − 1.39863i 0.0628374 − 0.151703i
\(86\) 13.1449 − 11.4501i 1.41745 − 1.23470i
\(87\) −4.15142 + 4.15142i −0.445079 + 0.445079i
\(88\) −1.13487 − 0.237613i −0.120977 − 0.0253296i
\(89\) 11.7403 + 11.7403i 1.24447 + 1.24447i 0.958127 + 0.286343i \(0.0924398\pi\)
0.286343 + 0.958127i \(0.407560\pi\)
\(90\) 4.81760 + 0.331965i 0.507820 + 0.0349921i
\(91\) −0.425266 − 0.176151i −0.0445800 − 0.0184657i
\(92\) 16.8750 + 2.33669i 1.75934 + 0.243617i
\(93\) 2.22746 + 5.37756i 0.230977 + 0.557627i
\(94\) −1.20347 + 2.41994i −0.124128 + 0.249597i
\(95\) 8.13030 0.834151
\(96\) −4.59482 − 1.64601i −0.468957 − 0.167995i
\(97\) 9.59336 0.974058 0.487029 − 0.873386i \(-0.338081\pi\)
0.487029 + 0.873386i \(0.338081\pi\)
\(98\) −4.33726 + 8.72136i −0.438129 + 0.880991i
\(99\) −0.353845 − 0.854257i −0.0355628 − 0.0858561i
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.69.5 ✓ 128
32.13 even 8 inner 544.2.bd.b.205.5 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
544.2.bd.b.69.5 ✓ 128 1.1 even 1 trivial
544.2.bd.b.205.5 yes 128 32.13 even 8 inner