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Properties

Label 1080.2.m.c.539.21
Level $1080$
Weight $2$
Character 1080.539
Analytic conductor $8.624$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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] = [1080,2,Mod(539,1080)] 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("1080.539"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] 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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 539.21
Character \(\chi\) \(=\) 1080.539
Dual form 1080.2.m.c.539.23

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.341092 - 1.37246i) q^{2} +(-1.76731 + 0.936273i) q^{4} +(-2.13531 + 0.663669i) q^{5} +4.39866 q^{7} +(1.88782 + 2.10622i) q^{8} +(1.63920 + 2.70426i) q^{10} +0.633050i q^{11} -3.40649 q^{13} +(-1.50035 - 6.03700i) q^{14} +(2.24679 - 3.30937i) q^{16} -5.48913 q^{17} +0.323423 q^{19} +(3.15238 - 3.17214i) q^{20} +(0.868838 - 0.215928i) q^{22} +6.26839i q^{23} +(4.11909 - 2.83428i) q^{25} +(1.16193 + 4.67528i) q^{26} +(-7.77381 + 4.11835i) q^{28} -3.77896 q^{29} +7.86723i q^{31} +(-5.30835 - 1.95483i) q^{32} +(1.87230 + 7.53363i) q^{34} +(-9.39250 + 2.91926i) q^{35} -5.90433 q^{37} +(-0.110317 - 0.443887i) q^{38} +(-5.42890 - 3.24454i) q^{40} -0.877482i q^{41} -2.31397i q^{43} +(-0.592707 - 1.11880i) q^{44} +(8.60314 - 2.13810i) q^{46} +8.67090i q^{47} +12.3482 q^{49} +(-5.29493 - 4.68655i) q^{50} +(6.02033 - 3.18940i) q^{52} +7.09416i q^{53} +(-0.420136 - 1.35176i) q^{55} +(8.30386 + 9.26454i) q^{56} +(1.28897 + 5.18649i) q^{58} -10.5321i q^{59} +14.0673i q^{61} +(10.7975 - 2.68345i) q^{62} +(-0.872299 + 7.95230i) q^{64} +(7.27391 - 2.26078i) q^{65} +5.99382i q^{67} +(9.70101 - 5.13933i) q^{68} +(7.21028 + 11.8951i) q^{70} +10.7806 q^{71} +13.9654i q^{73} +(2.01392 + 8.10348i) q^{74} +(-0.571590 + 0.302812i) q^{76} +2.78457i q^{77} -7.33728i q^{79} +(-2.60125 + 8.55766i) q^{80} +(-1.20431 + 0.299302i) q^{82} -1.90716 q^{83} +(11.7210 - 3.64297i) q^{85} +(-3.17585 + 0.789278i) q^{86} +(-1.33334 + 1.19508i) q^{88} -8.16899i q^{89} -14.9840 q^{91} +(-5.86892 - 11.0782i) q^{92} +(11.9005 - 2.95758i) q^{94} +(-0.690609 + 0.214646i) q^{95} +6.27222i q^{97} +(-4.21188 - 16.9475i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{4} - 4 q^{10} + 4 q^{16} - 16 q^{19} - 4 q^{34} + 16 q^{40} + 36 q^{46} + 48 q^{49} + 52 q^{64} + 28 q^{70} - 64 q^{76} + 92 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.341092 − 1.37246i −0.241188 − 0.970478i
\(3\) 0 0
\(4\) −1.76731 + 0.936273i −0.883656 + 0.468136i
\(5\) −2.13531 + 0.663669i −0.954939 + 0.296802i
\(6\) 0 0
\(7\) 4.39866 1.66254 0.831269 − 0.555871i \(-0.187615\pi\)
0.831269 + 0.555871i \(0.187615\pi\)
\(8\) 1.88782 + 2.10622i 0.667444 + 0.744660i
\(9\) 0 0
\(10\) 1.63920 + 2.70426i 0.518360 + 0.855162i
\(11\) 0.633050i 0.190872i 0.995436 + 0.0954359i \(0.0304245\pi\)
−0.995436 + 0.0954359i \(0.969576\pi\)
\(12\) 0 0
\(13\) −3.40649 −0.944790 −0.472395 − 0.881387i \(-0.656610\pi\)
−0.472395 + 0.881387i \(0.656610\pi\)
\(14\) −1.50035 − 6.03700i −0.400985 − 1.61346i
\(15\) 0 0
\(16\) 2.24679 − 3.30937i 0.561697 − 0.827343i
\(17\) −5.48913 −1.33131 −0.665655 − 0.746260i \(-0.731848\pi\)
−0.665655 + 0.746260i \(0.731848\pi\)
\(18\) 0 0
\(19\) 0.323423 0.0741984 0.0370992 − 0.999312i \(-0.488188\pi\)
0.0370992 + 0.999312i \(0.488188\pi\)
\(20\) 3.15238 − 3.17214i 0.704894 − 0.709313i
\(21\) 0 0
\(22\) 0.868838 − 0.215928i 0.185237 − 0.0460361i
\(23\) 6.26839i 1.30705i 0.756905 + 0.653525i \(0.226710\pi\)
−0.756905 + 0.653525i \(0.773290\pi\)
\(24\) 0 0
\(25\) 4.11909 − 2.83428i 0.823817 − 0.566856i
\(26\) 1.16193 + 4.67528i 0.227873 + 0.916899i
\(27\) 0 0
\(28\) −7.77381 + 4.11835i −1.46911 + 0.778294i
\(29\) −3.77896 −0.701736 −0.350868 − 0.936425i \(-0.614113\pi\)
−0.350868 + 0.936425i \(0.614113\pi\)
\(30\) 0 0
\(31\) 7.86723i 1.41300i 0.707715 + 0.706498i \(0.249726\pi\)
−0.707715 + 0.706498i \(0.750274\pi\)
\(32\) −5.30835 − 1.95483i −0.938393 − 0.345569i
\(33\) 0 0
\(34\) 1.87230 + 7.53363i 0.321097 + 1.29201i
\(35\) −9.39250 + 2.91926i −1.58762 + 0.493444i
\(36\) 0 0
\(37\) −5.90433 −0.970666 −0.485333 − 0.874329i \(-0.661302\pi\)
−0.485333 + 0.874329i \(0.661302\pi\)
\(38\) −0.110317 − 0.443887i −0.0178958 − 0.0720079i
\(39\) 0 0
\(40\) −5.42890 − 3.24454i −0.858385 − 0.513006i
\(41\) − 0.877482i − 0.137040i −0.997650 − 0.0685198i \(-0.978172\pi\)
0.997650 − 0.0685198i \(-0.0218276\pi\)
\(42\) 0 0
\(43\) − 2.31397i − 0.352878i −0.984312 − 0.176439i \(-0.943542\pi\)
0.984312 − 0.176439i \(-0.0564578\pi\)
\(44\) −0.592707 − 1.11880i −0.0893540 − 0.168665i
\(45\) 0 0
\(46\) 8.60314 − 2.13810i 1.26846 − 0.315245i
\(47\) 8.67090i 1.26478i 0.774650 + 0.632391i \(0.217926\pi\)
−0.774650 + 0.632391i \(0.782074\pi\)
\(48\) 0 0
\(49\) 12.3482 1.76403
\(50\) −5.29493 − 4.68655i −0.748816 − 0.662778i
\(51\) 0 0
\(52\) 6.02033 − 3.18940i 0.834870 − 0.442291i
\(53\) 7.09416i 0.974457i 0.873274 + 0.487229i \(0.161992\pi\)
−0.873274 + 0.487229i \(0.838008\pi\)
\(54\) 0 0
\(55\) −0.420136 − 1.35176i −0.0566511 − 0.182271i
\(56\) 8.30386 + 9.26454i 1.10965 + 1.23803i
\(57\) 0 0
\(58\) 1.28897 + 5.18649i 0.169251 + 0.681019i
\(59\) − 10.5321i − 1.37117i −0.727993 − 0.685584i \(-0.759547\pi\)
0.727993 − 0.685584i \(-0.240453\pi\)
\(60\) 0 0
\(61\) 14.0673i 1.80113i 0.434717 + 0.900567i \(0.356848\pi\)
−0.434717 + 0.900567i \(0.643152\pi\)
\(62\) 10.7975 − 2.68345i 1.37128 − 0.340798i
\(63\) 0 0
\(64\) −0.872299 + 7.95230i −0.109037 + 0.994038i
\(65\) 7.27391 − 2.26078i 0.902217 − 0.280416i
\(66\) 0 0
\(67\) 5.99382i 0.732262i 0.930563 + 0.366131i \(0.119318\pi\)
−0.930563 + 0.366131i \(0.880682\pi\)
\(68\) 9.70101 − 5.13933i 1.17642 − 0.623235i
\(69\) 0 0
\(70\) 7.21028 + 11.8951i 0.861793 + 1.42174i
\(71\) 10.7806 1.27942 0.639710 − 0.768617i \(-0.279055\pi\)
0.639710 + 0.768617i \(0.279055\pi\)
\(72\) 0 0
\(73\) 13.9654i 1.63453i 0.576264 + 0.817264i \(0.304510\pi\)
−0.576264 + 0.817264i \(0.695490\pi\)
\(74\) 2.01392 + 8.10348i 0.234113 + 0.942010i
\(75\) 0 0
\(76\) −0.571590 + 0.302812i −0.0655659 + 0.0347350i
\(77\) 2.78457i 0.317331i
\(78\) 0 0
\(79\) − 7.33728i − 0.825508i −0.910843 − 0.412754i \(-0.864567\pi\)
0.910843 − 0.412754i \(-0.135433\pi\)
\(80\) −2.60125 + 8.55766i −0.290829 + 0.956775i
\(81\) 0 0
\(82\) −1.20431 + 0.299302i −0.132994 + 0.0330524i
\(83\) −1.90716 −0.209338 −0.104669 − 0.994507i \(-0.533378\pi\)
−0.104669 + 0.994507i \(0.533378\pi\)
\(84\) 0 0
\(85\) 11.7210 − 3.64297i 1.27132 − 0.395135i
\(86\) −3.17585 + 0.789278i −0.342460 + 0.0851101i
\(87\) 0 0
\(88\) −1.33334 + 1.19508i −0.142135 + 0.127396i
\(89\) − 8.16899i − 0.865911i −0.901415 − 0.432956i \(-0.857471\pi\)
0.901415 − 0.432956i \(-0.142529\pi\)
\(90\) 0 0
\(91\) −14.9840 −1.57075
\(92\) −5.86892 − 11.0782i −0.611877 − 1.15498i
\(93\) 0 0
\(94\) 11.9005 − 2.95758i 1.22744 − 0.305051i
\(95\) −0.690609 + 0.214646i −0.0708549 + 0.0220222i
\(96\) 0 0
\(97\) 6.27222i 0.636847i 0.947949 + 0.318424i \(0.103153\pi\)
−0.947949 + 0.318424i \(0.896847\pi\)
\(98\) −4.21188 − 16.9475i −0.425464 − 1.71195i
\(99\) 0 0
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 1080.2.m.c.539.21 ✓ 48
3.2 odd 2 inner 1080.2.m.c.539.28 yes 48
4.3 odd 2 4320.2.m.c.2159.7 48
5.4 even 2 inner 1080.2.m.c.539.27 yes 48
8.3 odd 2 inner 1080.2.m.c.539.24 yes 48
8.5 even 2 4320.2.m.c.2159.42 48
12.11 even 2 4320.2.m.c.2159.41 48
15.14 odd 2 inner 1080.2.m.c.539.22 yes 48
20.19 odd 2 4320.2.m.c.2159.6 48
24.5 odd 2 4320.2.m.c.2159.8 48
24.11 even 2 inner 1080.2.m.c.539.25 yes 48
40.19 odd 2 inner 1080.2.m.c.539.26 yes 48
40.29 even 2 4320.2.m.c.2159.43 48
60.59 even 2 4320.2.m.c.2159.44 48
120.29 odd 2 4320.2.m.c.2159.5 48
120.59 even 2 inner 1080.2.m.c.539.23 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.m.c.539.21 ✓ 48 1.1 even 1 trivial
1080.2.m.c.539.22 yes 48 15.14 odd 2 inner
1080.2.m.c.539.23 yes 48 120.59 even 2 inner
1080.2.m.c.539.24 yes 48 8.3 odd 2 inner
1080.2.m.c.539.25 yes 48 24.11 even 2 inner
1080.2.m.c.539.26 yes 48 40.19 odd 2 inner
1080.2.m.c.539.27 yes 48 5.4 even 2 inner
1080.2.m.c.539.28 yes 48 3.2 odd 2 inner
4320.2.m.c.2159.5 48 120.29 odd 2
4320.2.m.c.2159.6 48 20.19 odd 2
4320.2.m.c.2159.7 48 4.3 odd 2
4320.2.m.c.2159.8 48 24.5 odd 2
4320.2.m.c.2159.41 48 12.11 even 2
4320.2.m.c.2159.42 48 8.5 even 2
4320.2.m.c.2159.43 48 40.29 even 2
4320.2.m.c.2159.44 48 60.59 even 2