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Properties

Label 585.2.bt.b.571.18
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 571.18
Character \(\chi\) \(=\) 585.571
Dual form 585.2.bt.b.376.18

$q$-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+(-1.07284 + 0.619404i) q^{2} +(-1.48219 - 0.896168i) q^{3} +(-0.232678 + 0.403011i) q^{4} +(0.866025 + 0.500000i) q^{5} +(2.14524 + 0.0433707i) q^{6} +(0.166629 - 0.0962033i) q^{7} -3.05410i q^{8} +(1.39377 + 2.65658i) q^{9} -1.23881 q^{10} +(0.223697 - 0.129152i) q^{11} +(0.706038 - 0.388819i) q^{12} +(-3.45476 - 1.03180i) q^{13} +(-0.119177 + 0.206421i) q^{14} +(-0.835529 - 1.51720i) q^{15} +(1.42637 + 2.47054i) q^{16} +3.42590 q^{17} +(-3.14078 - 1.98678i) q^{18} -2.08354i q^{19} +(-0.403011 + 0.232678i) q^{20} +(-0.333190 - 0.00673617i) q^{21} +(-0.159994 + 0.277118i) q^{22} +(0.614267 - 1.06394i) q^{23} +(-2.73699 + 4.52675i) q^{24} +(0.500000 + 0.866025i) q^{25} +(4.34550 - 1.03294i) q^{26} +(0.314919 - 5.18660i) q^{27} +0.0895377i q^{28} +(4.55796 + 7.89462i) q^{29} +(1.83615 + 1.11018i) q^{30} +(-2.13714 - 1.23388i) q^{31} +(2.22934 + 1.28711i) q^{32} +(-0.447303 - 0.00904322i) q^{33} +(-3.67544 + 2.12202i) q^{34} +0.192407 q^{35} +(-1.39493 - 0.0564262i) q^{36} -1.41990i q^{37} +(1.29055 + 2.23531i) q^{38} +(4.19594 + 4.62537i) q^{39} +(1.52705 - 2.64493i) q^{40} +(9.38294 + 5.41724i) q^{41} +(0.361632 - 0.199152i) q^{42} +(5.21080 + 9.02537i) q^{43} +0.120203i q^{44} +(-0.121254 + 2.99755i) q^{45} +1.52192i q^{46} +(-2.32341 + 1.34142i) q^{47} +(0.0998742 - 4.94006i) q^{48} +(-3.48149 + 6.03012i) q^{49} +(-1.07284 - 0.619404i) q^{50} +(-5.07784 - 3.07019i) q^{51} +(1.21967 - 1.15223i) q^{52} +9.93408 q^{53} +(2.87474 + 5.75945i) q^{54} +0.258303 q^{55} +(-0.293815 - 0.508902i) q^{56} +(-1.86721 + 3.08821i) q^{57} +(-9.77991 - 5.64643i) q^{58} +(10.3335 + 5.96605i) q^{59} +(0.805857 + 0.0162922i) q^{60} +(2.57942 + 4.46768i) q^{61} +3.05707 q^{62} +(0.487814 + 0.308579i) q^{63} -8.89442 q^{64} +(-2.47601 - 2.62095i) q^{65} +(0.485486 - 0.267359i) q^{66} +(8.23135 + 4.75237i) q^{67} +(-0.797133 + 1.38068i) q^{68} +(-1.86393 + 1.02648i) q^{69} +(-0.206421 + 0.119177i) q^{70} -6.11596i q^{71} +(8.11347 - 4.25670i) q^{72} -10.7486i q^{73} +(0.879489 + 1.52332i) q^{74} +(0.0350101 - 1.73170i) q^{75} +(0.839690 + 0.484795i) q^{76} +(0.0248496 - 0.0430409i) q^{77} +(-7.36654 - 2.36329i) q^{78} +(-2.92403 - 5.06458i) q^{79} +2.85273i q^{80} +(-5.11484 + 7.40530i) q^{81} -13.4218 q^{82} +(-3.67679 + 2.12279i) q^{83} +(0.0802408 - 0.132712i) q^{84} +(2.96692 + 1.71295i) q^{85} +(-11.1807 - 6.45518i) q^{86} +(0.319149 - 15.7860i) q^{87} +(-0.394442 - 0.683194i) q^{88} +2.03536i q^{89} +(-1.72661 - 3.29099i) q^{90} +(-0.674927 + 0.160432i) q^{91} +(0.285853 + 0.495112i) q^{92} +(2.06188 + 3.74407i) q^{93} +(1.66176 - 2.87826i) q^{94} +(1.04177 - 1.80440i) q^{95} +(-2.15084 - 3.90560i) q^{96} +(-5.71691 + 3.30066i) q^{97} -8.62579i q^{98} +(0.654883 + 0.414263i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.07284 + 0.619404i −0.758611 + 0.437985i −0.828797 − 0.559549i \(-0.810974\pi\)
0.0701855 + 0.997534i \(0.477641\pi\)
\(3\) −1.48219 − 0.896168i −0.855742 − 0.517403i
\(4\) −0.232678 + 0.403011i −0.116339 + 0.201505i
\(5\) 0.866025 + 0.500000i 0.387298 + 0.223607i
\(6\) 2.14524 + 0.0433707i 0.875790 + 0.0177060i
\(7\) 0.166629 − 0.0962033i 0.0629799 − 0.0363614i −0.468179 − 0.883633i \(-0.655090\pi\)
0.531159 + 0.847272i \(0.321757\pi\)
\(8\) − 3.05410i − 1.07979i
\(9\) 1.39377 + 2.65658i 0.464589 + 0.885527i
\(10\) −1.23881 −0.391745
\(11\) 0.223697 − 0.129152i 0.0674473 − 0.0389407i −0.465897 − 0.884839i \(-0.654268\pi\)
0.533344 + 0.845898i \(0.320935\pi\)
\(12\) 0.706038 − 0.388819i 0.203816 − 0.112242i
\(13\) −3.45476 − 1.03180i −0.958179 − 0.286170i
\(14\) −0.119177 + 0.206421i −0.0318515 + 0.0551684i
\(15\) −0.835529 − 1.51720i −0.215733 − 0.391739i
\(16\) 1.42637 + 2.47054i 0.356591 + 0.617634i
\(17\) 3.42590 0.830904 0.415452 − 0.909615i \(-0.363623\pi\)
0.415452 + 0.909615i \(0.363623\pi\)
\(18\) −3.14078 − 1.98678i −0.740289 − 0.468288i
\(19\) − 2.08354i − 0.477998i −0.971020 − 0.238999i \(-0.923181\pi\)
0.971020 − 0.238999i \(-0.0768192\pi\)
\(20\) −0.403011 + 0.232678i −0.0901159 + 0.0520284i
\(21\) −0.333190 − 0.00673617i −0.0727080 − 0.00146995i
\(22\) −0.159994 + 0.277118i −0.0341108 + 0.0590817i
\(23\) 0.614267 − 1.06394i 0.128083 − 0.221847i −0.794851 − 0.606805i \(-0.792451\pi\)
0.922934 + 0.384958i \(0.125784\pi\)
\(24\) −2.73699 + 4.52675i −0.558685 + 0.924020i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 4.34550 − 1.03294i 0.852223 − 0.202576i
\(27\) 0.314919 − 5.18660i 0.0606062 − 0.998162i
\(28\) 0.0895377i 0.0169210i
\(29\) 4.55796 + 7.89462i 0.846392 + 1.46599i 0.884407 + 0.466717i \(0.154563\pi\)
−0.0380149 + 0.999277i \(0.512103\pi\)
\(30\) 1.83615 + 1.11018i 0.335233 + 0.202690i
\(31\) −2.13714 − 1.23388i −0.383841 − 0.221611i 0.295647 − 0.955297i \(-0.404465\pi\)
−0.679488 + 0.733686i \(0.737798\pi\)
\(32\) 2.22934 + 1.28711i 0.394095 + 0.227531i
\(33\) −0.447303 − 0.00904322i −0.0778655 − 0.00157422i
\(34\) −3.67544 + 2.12202i −0.630333 + 0.363923i
\(35\) 0.192407 0.0325227
\(36\) −1.39493 − 0.0564262i −0.232488 − 0.00940436i
\(37\) − 1.41990i − 0.233430i −0.993165 − 0.116715i \(-0.962764\pi\)
0.993165 − 0.116715i \(-0.0372363\pi\)
\(38\) 1.29055 + 2.23531i 0.209356 + 0.362615i
\(39\) 4.19594 + 4.62537i 0.671889 + 0.740652i
\(40\) 1.52705 − 2.64493i 0.241448 − 0.418200i
\(41\) 9.38294 + 5.41724i 1.46537 + 0.846031i 0.999251 − 0.0386914i \(-0.0123189\pi\)
0.466118 + 0.884723i \(0.345652\pi\)
\(42\) 0.361632 − 0.199152i 0.0558010 − 0.0307299i
\(43\) 5.21080 + 9.02537i 0.794640 + 1.37636i 0.923068 + 0.384637i \(0.125674\pi\)
−0.128428 + 0.991719i \(0.540993\pi\)
\(44\) 0.120203i 0.0181213i
\(45\) −0.121254 + 2.99755i −0.0180754 + 0.446848i
\(46\) 1.52192i 0.224394i
\(47\) −2.32341 + 1.34142i −0.338905 + 0.195667i −0.659788 − 0.751452i \(-0.729354\pi\)
0.320883 + 0.947119i \(0.396020\pi\)
\(48\) 0.0998742 − 4.94006i 0.0144156 − 0.713037i
\(49\) −3.48149 + 6.03012i −0.497356 + 0.861445i
\(50\) −1.07284 − 0.619404i −0.151722 − 0.0875969i
\(51\) −5.07784 − 3.07019i −0.711039 − 0.429912i
\(52\) 1.21967 − 1.15223i 0.169138 − 0.159785i
\(53\) 9.93408 1.36455 0.682276 − 0.731095i \(-0.260990\pi\)
0.682276 + 0.731095i \(0.260990\pi\)
\(54\) 2.87474 + 5.75945i 0.391203 + 0.783761i
\(55\) 0.258303 0.0348296
\(56\) −0.293815 − 0.508902i −0.0392627 − 0.0680049i
\(57\) −1.86721 + 3.08821i −0.247317 + 0.409043i
\(58\) −9.77991 − 5.64643i −1.28417 − 0.741413i
\(59\) 10.3335 + 5.96605i 1.34531 + 0.776714i 0.987581 − 0.157112i \(-0.0502184\pi\)
0.357727 + 0.933826i \(0.383552\pi\)
\(60\) 0.805857 + 0.0162922i 0.104036 + 0.00210331i
\(61\) 2.57942 + 4.46768i 0.330261 + 0.572028i 0.982563 − 0.185931i \(-0.0595300\pi\)
−0.652302 + 0.757959i \(0.726197\pi\)
\(62\) 3.05707 0.388249
\(63\) 0.487814 + 0.308579i 0.0614588 + 0.0388772i
\(64\) −8.89442 −1.11180
\(65\) −2.47601 − 2.62095i −0.307112 − 0.325088i
\(66\) 0.485486 − 0.267359i 0.0597591 − 0.0329097i
\(67\) 8.23135 + 4.75237i 1.00562 + 0.580595i 0.909906 − 0.414814i \(-0.136153\pi\)
0.0957136 + 0.995409i \(0.469487\pi\)
\(68\) −0.797133 + 1.38068i −0.0966666 + 0.167432i
\(69\) −1.86393 + 1.02648i −0.224391 + 0.123573i
\(70\) −0.206421 + 0.119177i −0.0246721 + 0.0142444i
\(71\) − 6.11596i − 0.725831i −0.931822 − 0.362916i \(-0.881781\pi\)
0.931822 − 0.362916i \(-0.118219\pi\)
\(72\) 8.11347 − 4.25670i 0.956181 − 0.501657i
\(73\) − 10.7486i − 1.25803i −0.777394 − 0.629015i \(-0.783459\pi\)
0.777394 − 0.629015i \(-0.216541\pi\)
\(74\) 0.879489 + 1.52332i 0.102239 + 0.177082i
\(75\) 0.0350101 − 1.73170i 0.00404261 − 0.199959i
\(76\) 0.839690 + 0.484795i 0.0963191 + 0.0556098i
\(77\) 0.0248496 − 0.0430409i 0.00283188 − 0.00490496i
\(78\) −7.36654 − 2.36329i −0.834097 − 0.267590i
\(79\) −2.92403 − 5.06458i −0.328980 − 0.569809i 0.653330 − 0.757073i \(-0.273371\pi\)
−0.982310 + 0.187264i \(0.940038\pi\)
\(80\) 2.85273i 0.318945i
\(81\) −5.11484 + 7.40530i −0.568315 + 0.822811i
\(82\) −13.4218 −1.48219
\(83\) −3.67679 + 2.12279i −0.403580 + 0.233007i −0.688027 − 0.725685i \(-0.741523\pi\)
0.284448 + 0.958692i \(0.408190\pi\)
\(84\) 0.0802408 − 0.132712i 0.00875499 − 0.0144800i
\(85\) 2.96692 + 1.71295i 0.321808 + 0.185796i
\(86\) −11.1807 − 6.45518i −1.20565 − 0.696080i
\(87\) 0.319149 − 15.7860i 0.0342164 − 1.69244i
\(88\) −0.394442 − 0.683194i −0.0420477 − 0.0728288i
\(89\) 2.03536i 0.215748i 0.994165 + 0.107874i \(0.0344043\pi\)
−0.994165 + 0.107874i \(0.965596\pi\)
\(90\) −1.72661 − 3.29099i −0.182000 − 0.346901i
\(91\) −0.674927 + 0.160432i −0.0707515 + 0.0168178i
\(92\) 0.285853 + 0.495112i 0.0298022 + 0.0516190i
\(93\) 2.06188 + 3.74407i 0.213807 + 0.388242i
\(94\) 1.66176 − 2.87826i 0.171398 − 0.296870i
\(95\) 1.04177 − 1.80440i 0.106884 − 0.185128i
\(96\) −2.15084 − 3.90560i −0.219519 − 0.398614i
\(97\) −5.71691 + 3.30066i −0.580464 + 0.335131i −0.761318 − 0.648379i \(-0.775447\pi\)
0.180854 + 0.983510i \(0.442114\pi\)
\(98\) − 8.62579i − 0.871336i
\(99\) 0.654883 + 0.414263i 0.0658183 + 0.0416350i
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.571.18 yes 108
9.7 even 3 inner 585.2.bt.b.376.37 yes 108
13.12 even 2 inner 585.2.bt.b.571.37 yes 108
117.25 even 6 inner 585.2.bt.b.376.18 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.bt.b.376.18 ✓ 108 117.25 even 6 inner
585.2.bt.b.376.37 yes 108 9.7 even 3 inner
585.2.bt.b.571.18 yes 108 1.1 even 1 trivial
585.2.bt.b.571.37 yes 108 13.12 even 2 inner