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Properties

Label 585.2.bt.b.571.8
Level $585$
Weight $2$
Character 585.571
Analytic conductor $4.671$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,2,Mod(376,585)] 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("585.376"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 571.8
Character \(\chi\) \(=\) 585.571
Dual form 585.2.bt.b.376.8

$q$-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+(-1.85413 + 1.07048i) q^{2} +(-1.70628 - 0.297682i) q^{3} +(1.29187 - 2.23759i) q^{4} +(0.866025 + 0.500000i) q^{5} +(3.48233 - 1.27460i) q^{6} +(-4.18669 + 2.41719i) q^{7} +1.24978i q^{8} +(2.82277 + 1.01586i) q^{9} -2.14097 q^{10} +(-0.777530 + 0.448907i) q^{11} +(-2.87039 + 3.43338i) q^{12} +(3.48463 + 0.925930i) q^{13} +(5.17513 - 8.96358i) q^{14} +(-1.32884 - 1.11094i) q^{15} +(1.24587 + 2.15792i) q^{16} -5.40665 q^{17} +(-6.32125 + 1.13820i) q^{18} +6.78820i q^{19} +(2.23759 - 1.29187i) q^{20} +(7.86322 - 2.87809i) q^{21} +(0.961096 - 1.66467i) q^{22} +(0.544448 - 0.943012i) q^{23} +(0.372038 - 2.13248i) q^{24} +(0.500000 + 0.866025i) q^{25} +(-7.45216 + 2.01345i) q^{26} +(-4.51403 - 2.57362i) q^{27} +12.4908i q^{28} +(-1.28370 - 2.22344i) q^{29} +(3.65309 + 0.637329i) q^{30} +(-4.42594 - 2.55532i) q^{31} +(-6.78472 - 3.91716i) q^{32} +(1.46031 - 0.534504i) q^{33} +(10.0246 - 5.78773i) q^{34} -4.83438 q^{35} +(5.91974 - 5.00384i) q^{36} -2.86635i q^{37} +(-7.26667 - 12.5862i) q^{38} +(-5.67012 - 2.61721i) q^{39} +(-0.624892 + 1.08234i) q^{40} +(-3.50683 - 2.02467i) q^{41} +(-11.4985 + 13.7538i) q^{42} +(2.08636 + 3.61369i) q^{43} +2.31972i q^{44} +(1.93666 + 2.29114i) q^{45} +2.33129i q^{46} +(6.64992 - 3.83933i) q^{47} +(-1.48343 - 4.05288i) q^{48} +(8.18560 - 14.1779i) q^{49} +(-1.85413 - 1.07048i) q^{50} +(9.22525 + 1.60946i) q^{51} +(6.57355 - 6.60099i) q^{52} +6.85899 q^{53} +(11.1246 - 0.0603536i) q^{54} -0.897814 q^{55} +(-3.02096 - 5.23246i) q^{56} +(2.02073 - 11.5826i) q^{57} +(4.76031 + 2.74837i) q^{58} +(-13.0778 - 7.55048i) q^{59} +(-4.20252 + 1.53820i) q^{60} +(-3.04265 - 5.27002i) q^{61} +10.9417 q^{62} +(-14.2736 + 2.57008i) q^{63} +11.7895 q^{64} +(2.55481 + 2.54419i) q^{65} +(-2.13544 + 2.55428i) q^{66} +(5.26815 + 3.04157i) q^{67} +(-6.98471 + 12.0979i) q^{68} +(-1.20970 + 1.44697i) q^{69} +(8.96358 - 5.17513i) q^{70} -13.3088i q^{71} +(-1.26960 + 3.52785i) q^{72} -11.8477i q^{73} +(3.06838 + 5.31459i) q^{74} +(-0.595339 - 1.62652i) q^{75} +(15.1892 + 8.76950i) q^{76} +(2.17019 - 3.75887i) q^{77} +(13.3148 - 1.21712i) q^{78} +(-1.01631 - 1.76030i) q^{79} +2.49175i q^{80} +(6.93607 + 5.73507i) q^{81} +8.66951 q^{82} +(-1.21390 + 0.700846i) q^{83} +(3.71829 - 21.3128i) q^{84} +(-4.68230 - 2.70333i) q^{85} +(-7.73679 - 4.46684i) q^{86} +(1.52848 + 4.17594i) q^{87} +(-0.561037 - 0.971744i) q^{88} -3.44366i q^{89} +(-6.04346 - 2.17492i) q^{90} +(-16.8272 + 4.54643i) q^{91} +(-1.40672 - 2.43650i) q^{92} +(6.79122 + 5.67761i) q^{93} +(-8.21989 + 14.2373i) q^{94} +(-3.39410 + 5.87876i) q^{95} +(10.4105 + 8.70345i) q^{96} +(-15.9996 + 9.23740i) q^{97} +35.0502i q^{98} +(-2.65081 + 0.477302i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35}+ \cdots - 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.85413 + 1.07048i −1.31107 + 0.756947i −0.982273 − 0.187454i \(-0.939976\pi\)
−0.328797 + 0.944401i \(0.606643\pi\)
\(3\) −1.70628 − 0.297682i −0.985120 − 0.171867i
\(4\) 1.29187 − 2.23759i 0.645937 − 1.11880i
\(5\) 0.866025 + 0.500000i 0.387298 + 0.223607i
\(6\) 3.48233 − 1.27460i 1.42166 − 0.520354i
\(7\) −4.18669 + 2.41719i −1.58242 + 0.913612i −0.587917 + 0.808921i \(0.700052\pi\)
−0.994505 + 0.104690i \(0.966615\pi\)
\(8\) 1.24978i 0.441865i
\(9\) 2.82277 + 1.01586i 0.940923 + 0.338619i
\(10\) −2.14097 −0.677034
\(11\) −0.777530 + 0.448907i −0.234434 + 0.135351i −0.612616 − 0.790381i \(-0.709883\pi\)
0.378182 + 0.925731i \(0.376549\pi\)
\(12\) −2.87039 + 3.43338i −0.828609 + 0.991132i
\(13\) 3.48463 + 0.925930i 0.966463 + 0.256807i
\(14\) 5.17513 − 8.96358i 1.38311 − 2.39562i
\(15\) −1.32884 − 1.11094i −0.343105 − 0.286843i
\(16\) 1.24587 + 2.15792i 0.311468 + 0.539479i
\(17\) −5.40665 −1.31131 −0.655653 − 0.755063i \(-0.727606\pi\)
−0.655653 + 0.755063i \(0.727606\pi\)
\(18\) −6.32125 + 1.13820i −1.48993 + 0.268275i
\(19\) 6.78820i 1.55732i 0.627446 + 0.778660i \(0.284100\pi\)
−0.627446 + 0.778660i \(0.715900\pi\)
\(20\) 2.23759 − 1.29187i 0.500340 − 0.288872i
\(21\) 7.86322 − 2.87809i 1.71590 − 0.628051i
\(22\) 0.961096 − 1.66467i 0.204906 − 0.354908i
\(23\) 0.544448 − 0.943012i 0.113525 − 0.196632i −0.803664 − 0.595083i \(-0.797119\pi\)
0.917189 + 0.398452i \(0.130452\pi\)
\(24\) 0.372038 − 2.13248i 0.0759420 − 0.435290i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −7.45216 + 2.01345i −1.46149 + 0.394869i
\(27\) −4.51403 − 2.57362i −0.868725 − 0.495294i
\(28\) 12.4908i 2.36054i
\(29\) −1.28370 − 2.22344i −0.238378 − 0.412882i 0.721871 − 0.692027i \(-0.243282\pi\)
−0.960249 + 0.279145i \(0.909949\pi\)
\(30\) 3.65309 + 0.637329i 0.666960 + 0.116360i
\(31\) −4.42594 − 2.55532i −0.794923 − 0.458949i 0.0467698 − 0.998906i \(-0.485107\pi\)
−0.841693 + 0.539957i \(0.818441\pi\)
\(32\) −6.78472 − 3.91716i −1.19938 − 0.692462i
\(33\) 1.46031 − 0.534504i 0.254208 − 0.0930451i
\(34\) 10.0246 − 5.78773i 1.71921 − 0.992588i
\(35\) −4.83438 −0.817159
\(36\) 5.91974 − 5.00384i 0.986623 − 0.833974i
\(37\) − 2.86635i − 0.471224i −0.971847 − 0.235612i \(-0.924290\pi\)
0.971847 − 0.235612i \(-0.0757095\pi\)
\(38\) −7.26667 − 12.5862i −1.17881 − 2.04176i
\(39\) −5.67012 − 2.61721i −0.907945 − 0.419089i
\(40\) −0.624892 + 1.08234i −0.0988041 + 0.171134i
\(41\) −3.50683 − 2.02467i −0.547675 − 0.316200i 0.200509 − 0.979692i \(-0.435741\pi\)
−0.748184 + 0.663492i \(0.769074\pi\)
\(42\) −11.4985 + 13.7538i −1.77426 + 2.12226i
\(43\) 2.08636 + 3.61369i 0.318167 + 0.551082i 0.980106 − 0.198476i \(-0.0635993\pi\)
−0.661938 + 0.749558i \(0.730266\pi\)
\(44\) 2.31972i 0.349712i
\(45\) 1.93666 + 2.29114i 0.288701 + 0.341544i
\(46\) 2.33129i 0.343731i
\(47\) 6.64992 − 3.83933i 0.969991 − 0.560024i 0.0707572 − 0.997494i \(-0.477458\pi\)
0.899233 + 0.437469i \(0.144125\pi\)
\(48\) −1.48343 − 4.05288i −0.214115 − 0.584983i
\(49\) 8.18560 − 14.1779i 1.16937 − 2.02541i
\(50\) −1.85413 − 1.07048i −0.262214 − 0.151389i
\(51\) 9.22525 + 1.60946i 1.29179 + 0.225370i
\(52\) 6.57355 − 6.60099i 0.911588 − 0.915393i
\(53\) 6.85899 0.942155 0.471077 − 0.882092i \(-0.343865\pi\)
0.471077 + 0.882092i \(0.343865\pi\)
\(54\) 11.1246 − 0.0603536i 1.51387 − 0.00821308i
\(55\) −0.897814 −0.121061
\(56\) −3.02096 − 5.23246i −0.403693 − 0.699217i
\(57\) 2.02073 − 11.5826i 0.267652 − 1.53415i
\(58\) 4.76031 + 2.74837i 0.625059 + 0.360878i
\(59\) −13.0778 − 7.55048i −1.70259 − 0.982989i −0.943125 − 0.332438i \(-0.892129\pi\)
−0.759462 − 0.650552i \(-0.774538\pi\)
\(60\) −4.20252 + 1.53820i −0.542543 + 0.198581i
\(61\) −3.04265 − 5.27002i −0.389571 − 0.674757i 0.602821 − 0.797877i \(-0.294043\pi\)
−0.992392 + 0.123120i \(0.960710\pi\)
\(62\) 10.9417 1.38960
\(63\) −14.2736 + 2.57008i −1.79830 + 0.323800i
\(64\) 11.7895 1.47369
\(65\) 2.55481 + 2.54419i 0.316886 + 0.315568i
\(66\) −2.13544 + 2.55428i −0.262854 + 0.314411i
\(67\) 5.26815 + 3.04157i 0.643607 + 0.371587i 0.786003 − 0.618223i \(-0.212147\pi\)
−0.142395 + 0.989810i \(0.545480\pi\)
\(68\) −6.98471 + 12.0979i −0.847020 + 1.46708i
\(69\) −1.20970 + 1.44697i −0.145631 + 0.174195i
\(70\) 8.96358 − 5.17513i 1.07135 − 0.618546i
\(71\) − 13.3088i − 1.57946i −0.613456 − 0.789729i \(-0.710221\pi\)
0.613456 − 0.789729i \(-0.289779\pi\)
\(72\) −1.26960 + 3.52785i −0.149624 + 0.415761i
\(73\) − 11.8477i − 1.38666i −0.720618 − 0.693332i \(-0.756142\pi\)
0.720618 − 0.693332i \(-0.243858\pi\)
\(74\) 3.06838 + 5.31459i 0.356692 + 0.617808i
\(75\) −0.595339 − 1.62652i −0.0687438 − 0.187815i
\(76\) 15.1892 + 8.76950i 1.74232 + 1.00593i
\(77\) 2.17019 − 3.75887i 0.247316 − 0.428363i
\(78\) 13.3148 − 1.21712i 1.50761 − 0.137812i
\(79\) −1.01631 − 1.76030i −0.114344 − 0.198049i 0.803174 − 0.595745i \(-0.203143\pi\)
−0.917517 + 0.397696i \(0.869810\pi\)
\(80\) 2.49175i 0.278586i
\(81\) 6.93607 + 5.73507i 0.770674 + 0.637230i
\(82\) 8.66951 0.957387
\(83\) −1.21390 + 0.700846i −0.133243 + 0.0769279i −0.565140 − 0.824995i \(-0.691178\pi\)
0.431897 + 0.901923i \(0.357844\pi\)
\(84\) 3.71829 − 21.3128i 0.405699 − 2.32542i
\(85\) −4.68230 − 2.70333i −0.507866 − 0.293217i
\(86\) −7.73679 − 4.46684i −0.834279 − 0.481671i
\(87\) 1.52848 + 4.17594i 0.163870 + 0.447708i
\(88\) −0.561037 − 0.971744i −0.0598067 − 0.103588i
\(89\) − 3.44366i − 0.365028i −0.983203 − 0.182514i \(-0.941577\pi\)
0.983203 − 0.182514i \(-0.0584234\pi\)
\(90\) −6.04346 − 2.17492i −0.637037 − 0.229257i
\(91\) −16.8272 + 4.54643i −1.76397 + 0.476595i
\(92\) −1.40672 − 2.43650i −0.146660 − 0.254023i
\(93\) 6.79122 + 5.67761i 0.704216 + 0.588741i
\(94\) −8.21989 + 14.2373i −0.847817 + 1.46846i
\(95\) −3.39410 + 5.87876i −0.348227 + 0.603148i
\(96\) 10.4105 + 8.70345i 1.06252 + 0.888292i
\(97\) −15.9996 + 9.23740i −1.62452 + 0.937916i −0.638828 + 0.769350i \(0.720580\pi\)
−0.985690 + 0.168566i \(0.946086\pi\)
\(98\) 35.0502i 3.54061i
\(99\) −2.65081 + 0.477302i −0.266417 + 0.0479706i
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 585.2.bt.b.571.8 yes 108
9.7 even 3 inner 585.2.bt.b.376.47 yes 108
13.12 even 2 inner 585.2.bt.b.571.47 yes 108
117.25 even 6 inner 585.2.bt.b.376.8 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.bt.b.376.8 ✓ 108 117.25 even 6 inner
585.2.bt.b.376.47 yes 108 9.7 even 3 inner
585.2.bt.b.571.8 yes 108 1.1 even 1 trivial
585.2.bt.b.571.47 yes 108 13.12 even 2 inner