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Properties

Label 585.2.bm.a.166.15
Level $585$
Weight $2$
Character 585.166
Analytic conductor $4.671$
Analytic rank $0$
Dimension $112$
Inner twists $2$

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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(166,585)] 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("585.166"); 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([2, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 166.15
Character \(\chi\) \(=\) 585.166
Dual form 585.2.bm.a.511.15

$q$-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+(-1.23047 - 0.710412i) q^{2} +(-0.0786278 + 1.73027i) q^{3} +(0.00936902 + 0.0162276i) q^{4} +(0.866025 + 0.500000i) q^{5} +(1.32595 - 2.07318i) q^{6} +4.62258i q^{7} +2.81502i q^{8} +(-2.98764 - 0.272094i) q^{9} +(-0.710412 - 1.23047i) q^{10} +(-0.837582 - 0.483578i) q^{11} +(-0.0288148 + 0.0149350i) q^{12} +(-3.51803 + 0.789610i) q^{13} +(3.28394 - 5.68795i) q^{14} +(-0.933226 + 1.45914i) q^{15} +(2.01856 - 3.49625i) q^{16} +(1.57898 - 2.73488i) q^{17} +(3.48289 + 2.45725i) q^{18} +(-4.10052 - 2.36744i) q^{19} +0.0187380i q^{20} +(-7.99830 - 0.363464i) q^{21} +(0.687079 + 1.19006i) q^{22} +0.286133 q^{23} +(-4.87074 - 0.221339i) q^{24} +(0.500000 + 0.866025i) q^{25} +(4.88977 + 1.52766i) q^{26} +(0.705706 - 5.14801i) q^{27} +(-0.0750136 + 0.0433091i) q^{28} +(1.55192 - 2.68800i) q^{29} +(2.18490 - 1.13245i) q^{30} +(2.73416 + 1.57857i) q^{31} +(-0.0917943 + 0.0529975i) q^{32} +(0.902576 - 1.41122i) q^{33} +(-3.88578 + 2.24346i) q^{34} +(-2.31129 + 4.00328i) q^{35} +(-0.0235758 - 0.0510315i) q^{36} +(-8.86496 + 5.11818i) q^{37} +(3.36371 + 5.82611i) q^{38} +(-1.08962 - 6.14921i) q^{39} +(-1.40751 + 2.43788i) q^{40} -0.396645i q^{41} +(9.58345 + 6.12931i) q^{42} -6.36252 q^{43} -0.0181226i q^{44} +(-2.45132 - 1.72946i) q^{45} +(-0.352077 - 0.203272i) q^{46} +(7.82786 - 4.51941i) q^{47} +(5.89073 + 3.76755i) q^{48} -14.3683 q^{49} -1.42082i q^{50} +(4.60791 + 2.94710i) q^{51} +(-0.0457740 - 0.0496914i) q^{52} -9.55227 q^{53} +(-4.52555 + 5.83312i) q^{54} +(-0.483578 - 0.837582i) q^{55} -13.0127 q^{56} +(4.41871 - 6.90884i) q^{57} +(-3.81918 + 2.20500i) q^{58} +(-10.8517 + 6.26524i) q^{59} +(-0.0324218 - 0.00147333i) q^{60} -2.94833 q^{61} +(-2.24287 - 3.88476i) q^{62} +(1.25778 - 13.8106i) q^{63} -7.92365 q^{64} +(-3.44151 - 1.07519i) q^{65} +(-2.11314 + 1.09526i) q^{66} -1.27150i q^{67} +0.0591741 q^{68} +(-0.0224980 + 0.495085i) q^{69} +(5.68795 - 3.28394i) q^{70} +(6.58608 + 3.80248i) q^{71} +(0.765951 - 8.41026i) q^{72} +6.76186i q^{73} +14.5441 q^{74} +(-1.53777 + 0.797039i) q^{75} -0.0887223i q^{76} +(2.23538 - 3.87179i) q^{77} +(-3.02772 + 8.34048i) q^{78} +(7.85397 + 13.6035i) q^{79} +(3.49625 - 2.01856i) q^{80} +(8.85193 + 1.62584i) q^{81} +(-0.281781 + 0.488059i) q^{82} +(9.64054 - 5.56597i) q^{83} +(-0.0690381 - 0.133199i) q^{84} +(2.73488 - 1.57898i) q^{85} +(7.82888 + 4.52001i) q^{86} +(4.52893 + 2.89658i) q^{87} +(1.36128 - 2.35781i) q^{88} +(-1.26409 + 0.729825i) q^{89} +(1.78765 + 3.86949i) q^{90} +(-3.65004 - 16.2624i) q^{91} +(0.00268078 + 0.00464325i) q^{92} +(-2.94632 + 4.60670i) q^{93} -12.8426 q^{94} +(-2.36744 - 4.10052i) q^{95} +(-0.0844821 - 0.162996i) q^{96} -2.57362i q^{97} +(17.6797 + 10.2074i) q^{98} +(2.37081 + 1.67266i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q + 56 q^{4} - 2 q^{9} + 12 q^{12} + 2 q^{13} - 56 q^{16} - 16 q^{17} + 24 q^{18} + 6 q^{19} - 6 q^{21} + 48 q^{23} - 60 q^{24} + 56 q^{25} - 12 q^{26} - 24 q^{27} + 10 q^{29} + 8 q^{30} - 24 q^{31}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23047 − 0.710412i −0.870073 − 0.502337i −0.00270040 − 0.999996i \(-0.500860\pi\)
−0.867372 + 0.497660i \(0.834193\pi\)
\(3\) −0.0786278 + 1.73027i −0.0453958 + 0.998969i
\(4\) 0.00936902 + 0.0162276i 0.00468451 + 0.00811381i
\(5\) 0.866025 + 0.500000i 0.387298 + 0.223607i
\(6\) 1.32595 − 2.07318i 0.541317 − 0.846372i
\(7\) 4.62258i 1.74717i 0.486669 + 0.873586i \(0.338212\pi\)
−0.486669 + 0.873586i \(0.661788\pi\)
\(8\) 2.81502i 0.995261i
\(9\) −2.98764 − 0.272094i −0.995878 − 0.0906980i
\(10\) −0.710412 − 1.23047i −0.224652 − 0.389108i
\(11\) −0.837582 − 0.483578i −0.252541 − 0.145804i 0.368386 − 0.929673i \(-0.379910\pi\)
−0.620927 + 0.783868i \(0.713244\pi\)
\(12\) −0.0288148 + 0.0149350i −0.00831811 + 0.00431135i
\(13\) −3.51803 + 0.789610i −0.975725 + 0.218998i
\(14\) 3.28394 − 5.68795i 0.877669 − 1.52017i
\(15\) −0.933226 + 1.45914i −0.240958 + 0.376748i
\(16\) 2.01856 − 3.49625i 0.504641 − 0.874063i
\(17\) 1.57898 − 2.73488i 0.382960 − 0.663306i −0.608524 − 0.793535i \(-0.708238\pi\)
0.991484 + 0.130230i \(0.0415715\pi\)
\(18\) 3.48289 + 2.45725i 0.820926 + 0.579180i
\(19\) −4.10052 − 2.36744i −0.940724 − 0.543127i −0.0505365 − 0.998722i \(-0.516093\pi\)
−0.890187 + 0.455595i \(0.849426\pi\)
\(20\) 0.0187380i 0.00418996i
\(21\) −7.99830 − 0.363464i −1.74537 − 0.0793143i
\(22\) 0.687079 + 1.19006i 0.146486 + 0.253721i
\(23\) 0.286133 0.0596628 0.0298314 − 0.999555i \(-0.490503\pi\)
0.0298314 + 0.999555i \(0.490503\pi\)
\(24\) −4.87074 − 0.221339i −0.994235 − 0.0451807i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 4.88977 + 1.52766i 0.958963 + 0.299598i
\(27\) 0.705706 − 5.14801i 0.135813 − 0.990734i
\(28\) −0.0750136 + 0.0433091i −0.0141762 + 0.00818465i
\(29\) 1.55192 − 2.68800i 0.288184 − 0.499150i −0.685192 − 0.728362i \(-0.740282\pi\)
0.973376 + 0.229213i \(0.0736151\pi\)
\(30\) 2.18490 − 1.13245i 0.398906 − 0.206756i
\(31\) 2.73416 + 1.57857i 0.491070 + 0.283519i 0.725018 − 0.688730i \(-0.241831\pi\)
−0.233948 + 0.972249i \(0.575165\pi\)
\(32\) −0.0917943 + 0.0529975i −0.0162271 + 0.00936872i
\(33\) 0.902576 − 1.41122i 0.157118 − 0.245661i
\(34\) −3.88578 + 2.24346i −0.666406 + 0.384750i
\(35\) −2.31129 + 4.00328i −0.390680 + 0.676677i
\(36\) −0.0235758 − 0.0510315i −0.00392930 − 0.00850525i
\(37\) −8.86496 + 5.11818i −1.45739 + 0.841424i −0.998882 − 0.0472661i \(-0.984949\pi\)
−0.458508 + 0.888691i \(0.651616\pi\)
\(38\) 3.36371 + 5.82611i 0.545665 + 0.945120i
\(39\) −1.08962 − 6.14921i −0.174479 − 0.984661i
\(40\) −1.40751 + 2.43788i −0.222547 + 0.385463i
\(41\) − 0.396645i − 0.0619455i −0.999520 − 0.0309728i \(-0.990139\pi\)
0.999520 − 0.0309728i \(-0.00986051\pi\)
\(42\) 9.58345 + 6.12931i 1.47876 + 0.945773i
\(43\) −6.36252 −0.970276 −0.485138 − 0.874438i \(-0.661231\pi\)
−0.485138 + 0.874438i \(0.661231\pi\)
\(44\) − 0.0181226i − 0.00273209i
\(45\) −2.45132 − 1.72946i −0.365421 − 0.257812i
\(46\) −0.352077 − 0.203272i −0.0519110 − 0.0299708i
\(47\) 7.82786 − 4.51941i 1.14181 − 0.659224i 0.194932 − 0.980817i \(-0.437551\pi\)
0.946878 + 0.321592i \(0.104218\pi\)
\(48\) 5.89073 + 3.76755i 0.850254 + 0.543799i
\(49\) −14.3683 −2.05261
\(50\) − 1.42082i − 0.200935i
\(51\) 4.60791 + 2.94710i 0.645237 + 0.412676i
\(52\) −0.0457740 − 0.0496914i −0.00634771 − 0.00689095i
\(53\) −9.55227 −1.31210 −0.656052 − 0.754715i \(-0.727775\pi\)
−0.656052 + 0.754715i \(0.727775\pi\)
\(54\) −4.52555 + 5.83312i −0.615850 + 0.793787i
\(55\) −0.483578 − 0.837582i −0.0652057 − 0.112940i
\(56\) −13.0127 −1.73889
\(57\) 4.41871 − 6.90884i 0.585272 − 0.915098i
\(58\) −3.81918 + 2.20500i −0.501482 + 0.289531i
\(59\) −10.8517 + 6.26524i −1.41277 + 0.815665i −0.995649 − 0.0931838i \(-0.970296\pi\)
−0.417125 + 0.908849i \(0.636962\pi\)
\(60\) −0.0324218 − 0.00147333i −0.00418564 − 0.000190206i
\(61\) −2.94833 −0.377495 −0.188748 − 0.982026i \(-0.560443\pi\)
−0.188748 + 0.982026i \(0.560443\pi\)
\(62\) −2.24287 − 3.88476i −0.284844 − 0.493365i
\(63\) 1.25778 − 13.8106i 0.158465 − 1.73997i
\(64\) −7.92365 −0.990456
\(65\) −3.44151 − 1.07519i −0.426866 − 0.133361i
\(66\) −2.11314 + 1.09526i −0.260109 + 0.134817i
\(67\) − 1.27150i − 0.155339i −0.996979 − 0.0776695i \(-0.975252\pi\)
0.996979 − 0.0776695i \(-0.0247479\pi\)
\(68\) 0.0591741 0.00717592
\(69\) −0.0224980 + 0.495085i −0.00270844 + 0.0596013i
\(70\) 5.68795 − 3.28394i 0.679840 − 0.392506i
\(71\) 6.58608 + 3.80248i 0.781624 + 0.451271i 0.837006 − 0.547194i \(-0.184304\pi\)
−0.0553815 + 0.998465i \(0.517637\pi\)
\(72\) 0.765951 − 8.41026i 0.0902682 − 0.991159i
\(73\) 6.76186i 0.791415i 0.918377 + 0.395708i \(0.129501\pi\)
−0.918377 + 0.395708i \(0.870499\pi\)
\(74\) 14.5441 1.69071
\(75\) −1.53777 + 0.797039i −0.177566 + 0.0920341i
\(76\) − 0.0887223i − 0.0101771i
\(77\) 2.23538 − 3.87179i 0.254745 − 0.441232i
\(78\) −3.02772 + 8.34048i −0.342822 + 0.944374i
\(79\) 7.85397 + 13.6035i 0.883641 + 1.53051i 0.847264 + 0.531173i \(0.178248\pi\)
0.0363774 + 0.999338i \(0.488418\pi\)
\(80\) 3.49625 − 2.01856i 0.390893 − 0.225682i
\(81\) 8.85193 + 1.62584i 0.983548 + 0.180648i
\(82\) −0.281781 + 0.488059i −0.0311175 + 0.0538971i
\(83\) 9.64054 − 5.56597i 1.05819 − 0.610944i 0.133257 − 0.991082i \(-0.457457\pi\)
0.924930 + 0.380137i \(0.124123\pi\)
\(84\) −0.0690381 − 0.133199i −0.00753267 − 0.0145332i
\(85\) 2.73488 − 1.57898i 0.296639 − 0.171265i
\(86\) 7.82888 + 4.52001i 0.844210 + 0.487405i
\(87\) 4.52893 + 2.89658i 0.485553 + 0.310546i
\(88\) 1.36128 − 2.35781i 0.145113 − 0.251344i
\(89\) −1.26409 + 0.729825i −0.133994 + 0.0773613i −0.565498 − 0.824749i \(-0.691316\pi\)
0.431505 + 0.902111i \(0.357983\pi\)
\(90\) 1.78765 + 3.86949i 0.188435 + 0.407880i
\(91\) −3.65004 − 16.2624i −0.382628 − 1.70476i
\(92\) 0.00268078 + 0.00464325i 0.000279491 + 0.000484093i
\(93\) −2.94632 + 4.60670i −0.305520 + 0.477693i
\(94\) −12.8426 −1.32461
\(95\) −2.36744 − 4.10052i −0.242894 − 0.420704i
\(96\) −0.0844821 − 0.162996i −0.00862242 − 0.0166357i
\(97\) − 2.57362i − 0.261311i −0.991428 − 0.130656i \(-0.958292\pi\)
0.991428 − 0.130656i \(-0.0417082\pi\)
\(98\) 17.6797 + 10.2074i 1.78592 + 1.03110i
\(99\) 2.37081 + 1.67266i 0.238276 + 0.168108i
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.bm.a.166.15 yes 112
9.7 even 3 585.2.ba.a.556.42 yes 112
13.4 even 6 585.2.ba.a.121.15 ✓ 112
117.43 even 6 inner 585.2.bm.a.511.15 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.ba.a.121.15 ✓ 112 13.4 even 6
585.2.ba.a.556.42 yes 112 9.7 even 3
585.2.bm.a.166.15 yes 112 1.1 even 1 trivial
585.2.bm.a.511.15 yes 112 117.43 even 6 inner