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Properties

Label 585.2.bm.a.166.2
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 166.2
Character \(\chi\) \(=\) 585.166
Dual form 585.2.bm.a.511.2

$q$-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+(-2.39541 - 1.38299i) q^{2} +(-0.747450 - 1.56247i) q^{3} +(2.82531 + 4.89358i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-0.370435 + 4.77647i) q^{6} +1.27008i q^{7} -10.0975i q^{8} +(-1.88264 + 2.33574i) q^{9} +(1.38299 + 2.39541i) q^{10} +(-3.05130 - 1.76167i) q^{11} +(5.53431 - 8.07218i) q^{12} +(-2.29817 - 2.77820i) q^{13} +(1.75651 - 3.04236i) q^{14} +(-0.133926 + 1.72687i) q^{15} +(-8.31415 + 14.4005i) q^{16} +(-2.16857 + 3.75607i) q^{17} +(7.73998 - 2.99137i) q^{18} +(-0.126666 - 0.0731304i) q^{19} -5.65062i q^{20} +(1.98447 - 0.949322i) q^{21} +(4.87274 + 8.43983i) q^{22} +7.72264 q^{23} +(-15.7771 + 7.54740i) q^{24} +(0.500000 + 0.866025i) q^{25} +(1.66283 + 9.83326i) q^{26} +(5.05670 + 1.19572i) q^{27} +(-6.21525 + 3.58838i) q^{28} +(-3.22500 + 5.58586i) q^{29} +(2.70904 - 3.95132i) q^{30} +(5.73819 + 3.31294i) q^{31} +(22.3421 - 12.8992i) q^{32} +(-0.471866 + 6.08434i) q^{33} +(10.3892 - 5.99821i) q^{34} +(0.635041 - 1.09992i) q^{35} +(-16.7492 - 2.61366i) q^{36} +(3.44613 - 1.98963i) q^{37} +(0.202277 + 0.350354i) q^{38} +(-2.62310 + 5.66740i) q^{39} +(-5.04877 + 8.74472i) q^{40} +3.47204i q^{41} +(-6.06650 - 0.470483i) q^{42} -10.0878 q^{43} -19.9091i q^{44} +(2.79828 - 1.08149i) q^{45} +(-18.4989 - 10.6803i) q^{46} +(-3.42004 + 1.97456i) q^{47} +(28.7148 + 2.22696i) q^{48} +5.38689 q^{49} -2.76598i q^{50} +(7.48966 + 0.580855i) q^{51} +(7.10231 - 19.0956i) q^{52} +2.25211 q^{53} +(-10.4592 - 9.85760i) q^{54} +(1.76167 + 3.05130i) q^{55} +12.8247 q^{56} +(-0.0195881 + 0.252573i) q^{57} +(15.4504 - 8.92027i) q^{58} +(5.65865 - 3.26702i) q^{59} +(-8.82894 + 4.22356i) q^{60} +10.2798 q^{61} +(-9.16352 - 15.8717i) q^{62} +(-2.96658 - 2.39110i) q^{63} -38.1011 q^{64} +(0.601175 + 3.55508i) q^{65} +(9.54487 - 13.9219i) q^{66} -3.26284i q^{67} -24.5075 q^{68} +(-5.77228 - 12.0664i) q^{69} +(-3.04236 + 1.75651i) q^{70} +(8.16850 + 4.71608i) q^{71} +(23.5852 + 19.0100i) q^{72} +1.86545i q^{73} -11.0065 q^{74} +(0.979416 - 1.42855i) q^{75} -0.826465i q^{76} +(2.23747 - 3.87540i) q^{77} +(14.1213 - 9.94800i) q^{78} +(2.06157 + 3.57075i) q^{79} +(14.4005 - 8.31415i) q^{80} +(-1.91135 - 8.79470i) q^{81} +(4.80178 - 8.31694i) q^{82} +(-5.74724 + 3.31817i) q^{83} +(10.2523 + 7.02902i) q^{84} +(3.75607 - 2.16857i) q^{85} +(24.1644 + 13.9513i) q^{86} +(11.1383 + 0.863821i) q^{87} +(-17.7885 + 30.8106i) q^{88} +(-0.249767 + 0.144203i) q^{89} +(-8.19871 - 1.27938i) q^{90} +(3.52854 - 2.91887i) q^{91} +(21.8189 + 37.7914i) q^{92} +(0.887377 - 11.4420i) q^{93} +10.9232 q^{94} +(0.0731304 + 0.126666i) q^{95} +(-36.8542 - 25.2673i) q^{96} -1.65184i q^{97} +(-12.9038 - 7.45001i) q^{98} +(9.85930 - 3.81046i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39541 − 1.38299i −1.69381 − 0.977920i −0.951393 − 0.307979i \(-0.900347\pi\)
−0.742414 − 0.669941i \(-0.766319\pi\)
\(3\) −0.747450 − 1.56247i −0.431540 − 0.902094i
\(4\) 2.82531 + 4.89358i 1.41266 + 2.44679i
\(5\) −0.866025 − 0.500000i −0.387298 − 0.223607i
\(6\) −0.370435 + 4.77647i −0.151230 + 1.94998i
\(7\) 1.27008i 0.480046i 0.970767 + 0.240023i \(0.0771549\pi\)
−0.970767 + 0.240023i \(0.922845\pi\)
\(8\) − 10.0975i − 3.57002i
\(9\) −1.88264 + 2.33574i −0.627546 + 0.778579i
\(10\) 1.38299 + 2.39541i 0.437339 + 0.757494i
\(11\) −3.05130 − 1.76167i −0.920003 − 0.531164i −0.0363666 − 0.999339i \(-0.511578\pi\)
−0.883636 + 0.468175i \(0.844912\pi\)
\(12\) 5.53431 − 8.07218i 1.59762 − 2.33024i
\(13\) −2.29817 − 2.77820i −0.637398 − 0.770535i
\(14\) 1.75651 − 3.04236i 0.469446 − 0.813105i
\(15\) −0.133926 + 1.72687i −0.0345795 + 0.445875i
\(16\) −8.31415 + 14.4005i −2.07854 + 3.60013i
\(17\) −2.16857 + 3.75607i −0.525955 + 0.910981i 0.473587 + 0.880747i \(0.342959\pi\)
−0.999543 + 0.0302347i \(0.990375\pi\)
\(18\) 7.73998 − 2.99137i 1.82433 − 0.705074i
\(19\) −0.126666 − 0.0731304i −0.0290591 − 0.0167773i 0.485400 − 0.874292i \(-0.338674\pi\)
−0.514459 + 0.857515i \(0.672007\pi\)
\(20\) − 5.65062i − 1.26352i
\(21\) 1.98447 − 0.949322i 0.433046 − 0.207159i
\(22\) 4.87274 + 8.43983i 1.03887 + 1.79938i
\(23\) 7.72264 1.61028 0.805141 − 0.593084i \(-0.202090\pi\)
0.805141 + 0.593084i \(0.202090\pi\)
\(24\) −15.7771 + 7.54740i −3.22049 + 1.54061i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 1.66283 + 9.83326i 0.326109 + 1.92846i
\(27\) 5.05670 + 1.19572i 0.973163 + 0.230117i
\(28\) −6.21525 + 3.58838i −1.17457 + 0.678139i
\(29\) −3.22500 + 5.58586i −0.598867 + 1.03727i 0.394121 + 0.919058i \(0.371049\pi\)
−0.992989 + 0.118210i \(0.962284\pi\)
\(30\) 2.70904 − 3.95132i 0.494601 − 0.721410i
\(31\) 5.73819 + 3.31294i 1.03061 + 0.595022i 0.917159 − 0.398521i \(-0.130477\pi\)
0.113450 + 0.993544i \(0.463810\pi\)
\(32\) 22.3421 − 12.8992i 3.94955 − 2.28028i
\(33\) −0.471866 + 6.08434i −0.0821414 + 1.05915i
\(34\) 10.3892 − 5.99821i 1.78173 − 1.02868i
\(35\) 0.635041 − 1.09992i 0.107341 − 0.185921i
\(36\) −16.7492 − 2.61366i −2.79153 − 0.435610i
\(37\) 3.44613 − 1.98963i 0.566541 − 0.327093i −0.189226 − 0.981934i \(-0.560598\pi\)
0.755767 + 0.654841i \(0.227264\pi\)
\(38\) 0.202277 + 0.350354i 0.0328137 + 0.0568349i
\(39\) −2.62310 + 5.66740i −0.420031 + 0.907510i
\(40\) −5.04877 + 8.74472i −0.798280 + 1.38266i
\(41\) 3.47204i 0.542241i 0.962545 + 0.271121i \(0.0873942\pi\)
−0.962545 + 0.271121i \(0.912606\pi\)
\(42\) −6.06650 − 0.470483i −0.936082 − 0.0725972i
\(43\) −10.0878 −1.53838 −0.769190 − 0.639021i \(-0.779340\pi\)
−0.769190 + 0.639021i \(0.779340\pi\)
\(44\) − 19.9091i − 3.00141i
\(45\) 2.79828 − 1.08149i 0.417143 − 0.161219i
\(46\) −18.4989 − 10.6803i −2.72751 − 1.57473i
\(47\) −3.42004 + 1.97456i −0.498864 + 0.288019i −0.728244 − 0.685318i \(-0.759663\pi\)
0.229381 + 0.973337i \(0.426330\pi\)
\(48\) 28.7148 + 2.22696i 4.14463 + 0.321434i
\(49\) 5.38689 0.769556
\(50\) − 2.76598i − 0.391168i
\(51\) 7.48966 + 0.580855i 1.04876 + 0.0813359i
\(52\) 7.10231 − 19.0956i 0.984913 − 2.64808i
\(53\) 2.25211 0.309351 0.154675 − 0.987965i \(-0.450567\pi\)
0.154675 + 0.987965i \(0.450567\pi\)
\(54\) −10.4592 − 9.85760i −1.42331 − 1.34145i
\(55\) 1.76167 + 3.05130i 0.237544 + 0.411438i
\(56\) 12.8247 1.71377
\(57\) −0.0195881 + 0.252573i −0.00259451 + 0.0334541i
\(58\) 15.4504 − 8.92027i 2.02873 − 1.17129i
\(59\) 5.65865 − 3.26702i 0.736693 − 0.425330i −0.0841727 − 0.996451i \(-0.526825\pi\)
0.820866 + 0.571121i \(0.193491\pi\)
\(60\) −8.82894 + 4.22356i −1.13981 + 0.545259i
\(61\) 10.2798 1.31619 0.658097 − 0.752933i \(-0.271361\pi\)
0.658097 + 0.752933i \(0.271361\pi\)
\(62\) −9.16352 − 15.8717i −1.16377 − 2.01571i
\(63\) −2.96658 − 2.39110i −0.373754 − 0.301251i
\(64\) −38.1011 −4.76264
\(65\) 0.601175 + 3.55508i 0.0745665 + 0.440953i
\(66\) 9.54487 − 13.9219i 1.17489 − 1.71366i
\(67\) − 3.26284i − 0.398619i −0.979937 − 0.199309i \(-0.936130\pi\)
0.979937 − 0.199309i \(-0.0638699\pi\)
\(68\) −24.5075 −2.97198
\(69\) −5.77228 − 12.0664i −0.694901 − 1.45263i
\(70\) −3.04236 + 1.75651i −0.363632 + 0.209943i
\(71\) 8.16850 + 4.71608i 0.969422 + 0.559696i 0.899060 − 0.437825i \(-0.144251\pi\)
0.0703621 + 0.997522i \(0.477585\pi\)
\(72\) 23.5852 + 19.0100i 2.77954 + 2.24035i
\(73\) 1.86545i 0.218334i 0.994023 + 0.109167i \(0.0348184\pi\)
−0.994023 + 0.109167i \(0.965182\pi\)
\(74\) −11.0065 −1.27948
\(75\) 0.979416 − 1.42855i 0.113093 − 0.164954i
\(76\) − 0.826465i − 0.0948020i
\(77\) 2.23747 − 3.87540i 0.254983 − 0.441643i
\(78\) 14.1213 − 9.94800i 1.59892 − 1.12639i
\(79\) 2.06157 + 3.57075i 0.231945 + 0.401741i 0.958380 − 0.285494i \(-0.0921577\pi\)
−0.726435 + 0.687235i \(0.758824\pi\)
\(80\) 14.4005 − 8.31415i 1.61003 − 0.929550i
\(81\) −1.91135 − 8.79470i −0.212372 − 0.977189i
\(82\) 4.80178 − 8.31694i 0.530268 − 0.918452i
\(83\) −5.74724 + 3.31817i −0.630842 + 0.364217i −0.781078 − 0.624434i \(-0.785330\pi\)
0.150236 + 0.988650i \(0.451997\pi\)
\(84\) 10.2523 + 7.02902i 1.11862 + 0.766929i
\(85\) 3.75607 − 2.16857i 0.407403 − 0.235214i
\(86\) 24.1644 + 13.9513i 2.60572 + 1.50441i
\(87\) 11.1383 + 0.863821i 1.19415 + 0.0926113i
\(88\) −17.7885 + 30.8106i −1.89626 + 3.28443i
\(89\) −0.249767 + 0.144203i −0.0264753 + 0.0152855i −0.513179 − 0.858281i \(-0.671532\pi\)
0.486704 + 0.873567i \(0.338199\pi\)
\(90\) −8.19871 − 1.27938i −0.864219 − 0.134859i
\(91\) 3.52854 − 2.91887i 0.369892 − 0.305980i
\(92\) 21.8189 + 37.7914i 2.27477 + 3.94002i
\(93\) 0.887377 − 11.4420i 0.0920167 − 1.18648i
\(94\) 10.9232 1.12664
\(95\) 0.0731304 + 0.126666i 0.00750302 + 0.0129956i
\(96\) −36.8542 − 25.2673i −3.76141 − 2.57884i
\(97\) − 1.65184i − 0.167719i −0.996478 − 0.0838594i \(-0.973275\pi\)
0.996478 − 0.0838594i \(-0.0267247\pi\)
\(98\) −12.9038 − 7.45001i −1.30348 − 0.752564i
\(99\) 9.85930 − 3.81046i 0.990897 − 0.382965i
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.166.2 yes 112
9.7 even 3 585.2.ba.a.556.55 yes 112
13.4 even 6 585.2.ba.a.121.2 ✓ 112
117.43 even 6 inner 585.2.bm.a.511.2 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.ba.a.121.2 ✓ 112 13.4 even 6
585.2.ba.a.556.55 yes 112 9.7 even 3
585.2.bm.a.166.2 yes 112 1.1 even 1 trivial
585.2.bm.a.511.2 yes 112 117.43 even 6 inner