↑

Properties

Label 1080.2.m.c.539.16
Level $1080$
Weight $2$
Character 1080.539
Analytic conductor $8.624$
Analytic rank $0$
Dimension $48$
Inner twists $8$

Related objects

Downloads

Learn more

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 539.16
Character \(\chi\) \(=\) 1080.539
Dual form 1080.2.m.c.539.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.957850 + 1.04044i) q^{2} +(-0.165046 - 1.99318i) q^{4} +(0.214370 - 2.22577i) q^{5} -2.80642 q^{7} +(2.23188 + 1.73744i) q^{8} +(2.11045 + 2.35499i) q^{10} +2.27115i q^{11} -3.91324 q^{13} +(2.68813 - 2.91992i) q^{14} +(-3.94552 + 0.657933i) q^{16} +1.00493 q^{17} +1.82621 q^{19} +(-4.47173 - 0.0599231i) q^{20} +(-2.36300 - 2.17542i) q^{22} -1.93778i q^{23} +(-4.90809 - 0.954277i) q^{25} +(3.74830 - 4.07151i) q^{26} +(0.463189 + 5.59370i) q^{28} +7.74448 q^{29} +3.94024i q^{31} +(3.09467 - 4.73529i) q^{32} +(-0.962571 + 1.04557i) q^{34} +(-0.601613 + 6.24644i) q^{35} -8.82415 q^{37} +(-1.74923 + 1.90007i) q^{38} +(4.34560 - 4.59519i) q^{40} +2.23967i q^{41} +11.7263i q^{43} +(4.52680 - 0.374844i) q^{44} +(2.01615 + 1.85610i) q^{46} +6.26994i q^{47} +0.875992 q^{49} +(5.69409 - 4.19254i) q^{50} +(0.645866 + 7.79979i) q^{52} +6.73830i q^{53} +(5.05505 + 0.486866i) q^{55} +(-6.26359 - 4.87600i) q^{56} +(-7.41805 + 8.05769i) q^{58} +11.4611i q^{59} -4.08558i q^{61} +(-4.09960 - 3.77416i) q^{62} +(1.96257 + 7.75553i) q^{64} +(-0.838883 + 8.70998i) q^{65} +6.94219i q^{67} +(-0.165860 - 2.00300i) q^{68} +(-5.92282 - 6.60910i) q^{70} -7.36200 q^{71} +5.13736i q^{73} +(8.45222 - 9.18104i) q^{74} +(-0.301409 - 3.63996i) q^{76} -6.37379i q^{77} -4.14620i q^{79} +(0.618605 + 8.92285i) q^{80} +(-2.33025 - 2.14527i) q^{82} +14.8166 q^{83} +(0.215427 - 2.23674i) q^{85} +(-12.2005 - 11.2320i) q^{86} +(-3.94599 + 5.06893i) q^{88} +4.82493i q^{89} +10.9822 q^{91} +(-3.86233 + 0.319823i) q^{92} +(-6.52352 - 6.00566i) q^{94} +(0.391485 - 4.06472i) q^{95} -14.9168i q^{97} +(-0.839069 + 0.911421i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{4} - 4 q^{10} + 4 q^{16} - 16 q^{19} - 4 q^{34} + 16 q^{40} + 36 q^{46} + 48 q^{49} + 52 q^{64} + 28 q^{70} - 64 q^{76} + 92 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.957850 + 1.04044i −0.677302 + 0.735705i
\(3\) 0 0
\(4\) −0.165046 − 1.99318i −0.0825231 − 0.996589i
\(5\) 0.214370 − 2.22577i 0.0958692 − 0.995394i
\(6\) 0 0
\(7\) −2.80642 −1.06073 −0.530363 − 0.847770i \(-0.677945\pi\)
−0.530363 + 0.847770i \(0.677945\pi\)
\(8\) 2.23188 + 1.73744i 0.789089 + 0.614279i
\(9\) 0 0
\(10\) 2.11045 + 2.35499i 0.667384 + 0.744714i
\(11\) 2.27115i 0.684777i 0.939559 + 0.342388i \(0.111236\pi\)
−0.939559 + 0.342388i \(0.888764\pi\)
\(12\) 0 0
\(13\) −3.91324 −1.08534 −0.542669 − 0.839946i \(-0.682586\pi\)
−0.542669 + 0.839946i \(0.682586\pi\)
\(14\) 2.68813 − 2.91992i 0.718433 − 0.780382i
\(15\) 0 0
\(16\) −3.94552 + 0.657933i −0.986380 + 0.164483i
\(17\) 1.00493 0.243731 0.121866 − 0.992547i \(-0.461112\pi\)
0.121866 + 0.992547i \(0.461112\pi\)
\(18\) 0 0
\(19\) 1.82621 0.418961 0.209481 − 0.977813i \(-0.432823\pi\)
0.209481 + 0.977813i \(0.432823\pi\)
\(20\) −4.47173 − 0.0599231i −0.999910 − 0.0133992i
\(21\) 0 0
\(22\) −2.36300 − 2.17542i −0.503794 − 0.463801i
\(23\) − 1.93778i − 0.404054i −0.979380 − 0.202027i \(-0.935247\pi\)
0.979380 − 0.202027i \(-0.0647529\pi\)
\(24\) 0 0
\(25\) −4.90809 − 0.954277i −0.981618 − 0.190855i
\(26\) 3.74830 − 4.07151i 0.735102 − 0.798489i
\(27\) 0 0
\(28\) 0.463189 + 5.59370i 0.0875345 + 1.05711i
\(29\) 7.74448 1.43811 0.719057 − 0.694951i \(-0.244574\pi\)
0.719057 + 0.694951i \(0.244574\pi\)
\(30\) 0 0
\(31\) 3.94024i 0.707689i 0.935304 + 0.353844i \(0.115126\pi\)
−0.935304 + 0.353844i \(0.884874\pi\)
\(32\) 3.09467 − 4.73529i 0.547066 − 0.837089i
\(33\) 0 0
\(34\) −0.962571 + 1.04557i −0.165080 + 0.179314i
\(35\) −0.601613 + 6.24644i −0.101691 + 1.05584i
\(36\) 0 0
\(37\) −8.82415 −1.45068 −0.725341 − 0.688390i \(-0.758318\pi\)
−0.725341 + 0.688390i \(0.758318\pi\)
\(38\) −1.74923 + 1.90007i −0.283763 + 0.308232i
\(39\) 0 0
\(40\) 4.34560 − 4.59519i 0.687099 − 0.726563i
\(41\) 2.23967i 0.349778i 0.984588 + 0.174889i \(0.0559566\pi\)
−0.984588 + 0.174889i \(0.944043\pi\)
\(42\) 0 0
\(43\) 11.7263i 1.78824i 0.447829 + 0.894119i \(0.352197\pi\)
−0.447829 + 0.894119i \(0.647803\pi\)
\(44\) 4.52680 − 0.374844i 0.682441 − 0.0565099i
\(45\) 0 0
\(46\) 2.01615 + 1.85610i 0.297265 + 0.273667i
\(47\) 6.26994i 0.914564i 0.889322 + 0.457282i \(0.151177\pi\)
−0.889322 + 0.457282i \(0.848823\pi\)
\(48\) 0 0
\(49\) 0.875992 0.125142
\(50\) 5.69409 − 4.19254i 0.805265 − 0.592914i
\(51\) 0 0
\(52\) 0.645866 + 7.79979i 0.0895655 + 1.08164i
\(53\) 6.73830i 0.925576i 0.886469 + 0.462788i \(0.153151\pi\)
−0.886469 + 0.462788i \(0.846849\pi\)
\(54\) 0 0
\(55\) 5.05505 + 0.486866i 0.681623 + 0.0656490i
\(56\) −6.26359 − 4.87600i −0.837008 − 0.651583i
\(57\) 0 0
\(58\) −7.41805 + 8.05769i −0.974038 + 1.05803i
\(59\) 11.4611i 1.49211i 0.665885 + 0.746055i \(0.268054\pi\)
−0.665885 + 0.746055i \(0.731946\pi\)
\(60\) 0 0
\(61\) − 4.08558i − 0.523105i −0.965189 − 0.261553i \(-0.915765\pi\)
0.965189 − 0.261553i \(-0.0842345\pi\)
\(62\) −4.09960 − 3.77416i −0.520650 − 0.479319i
\(63\) 0 0
\(64\) 1.96257 + 7.75553i 0.245321 + 0.969442i
\(65\) −0.838883 + 8.70998i −0.104051 + 1.08034i
\(66\) 0 0
\(67\) 6.94219i 0.848123i 0.905633 + 0.424062i \(0.139396\pi\)
−0.905633 + 0.424062i \(0.860604\pi\)
\(68\) −0.165860 − 2.00300i −0.0201135 − 0.242900i
\(69\) 0 0
\(70\) −5.92282 − 6.60910i −0.707912 − 0.789938i
\(71\) −7.36200 −0.873709 −0.436854 − 0.899532i \(-0.643908\pi\)
−0.436854 + 0.899532i \(0.643908\pi\)
\(72\) 0 0
\(73\) 5.13736i 0.601283i 0.953737 + 0.300641i \(0.0972007\pi\)
−0.953737 + 0.300641i \(0.902799\pi\)
\(74\) 8.45222 − 9.18104i 0.982550 − 1.06727i
\(75\) 0 0
\(76\) −0.301409 − 3.63996i −0.0345740 − 0.417532i
\(77\) − 6.37379i − 0.726361i
\(78\) 0 0
\(79\) − 4.14620i − 0.466484i −0.972419 − 0.233242i \(-0.925066\pi\)
0.972419 − 0.233242i \(-0.0749335\pi\)
\(80\) 0.618605 + 8.92285i 0.0691622 + 0.997605i
\(81\) 0 0
\(82\) −2.33025 − 2.14527i −0.257333 − 0.236905i
\(83\) 14.8166 1.62633 0.813167 − 0.582031i \(-0.197742\pi\)
0.813167 + 0.582031i \(0.197742\pi\)
\(84\) 0 0
\(85\) 0.215427 − 2.23674i 0.0233663 − 0.242608i
\(86\) −12.2005 − 11.2320i −1.31562 − 1.21118i
\(87\) 0 0
\(88\) −3.94599 + 5.06893i −0.420644 + 0.540350i
\(89\) 4.82493i 0.511442i 0.966751 + 0.255721i \(0.0823128\pi\)
−0.966751 + 0.255721i \(0.917687\pi\)
\(90\) 0 0
\(91\) 10.9822 1.15125
\(92\) −3.86233 + 0.319823i −0.402676 + 0.0333438i
\(93\) 0 0
\(94\) −6.52352 − 6.00566i −0.672849 − 0.619437i
\(95\) 0.391485 − 4.06472i 0.0401655 − 0.417031i
\(96\) 0 0
\(97\) − 14.9168i − 1.51457i −0.653084 − 0.757285i \(-0.726525\pi\)
0.653084 − 0.757285i \(-0.273475\pi\)
\(98\) −0.839069 + 0.911421i −0.0847588 + 0.0920674i
\(99\) 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.539.16 yes 48
3.2 odd 2 inner 1080.2.m.c.539.33 yes 48
4.3 odd 2 4320.2.m.c.2159.26 48
5.4 even 2 inner 1080.2.m.c.539.34 yes 48
8.3 odd 2 inner 1080.2.m.c.539.13 ✓ 48
8.5 even 2 4320.2.m.c.2159.23 48
12.11 even 2 4320.2.m.c.2159.24 48
15.14 odd 2 inner 1080.2.m.c.539.15 yes 48
20.19 odd 2 4320.2.m.c.2159.27 48
24.5 odd 2 4320.2.m.c.2159.25 48
24.11 even 2 inner 1080.2.m.c.539.36 yes 48
40.19 odd 2 inner 1080.2.m.c.539.35 yes 48
40.29 even 2 4320.2.m.c.2159.22 48
60.59 even 2 4320.2.m.c.2159.21 48
120.29 odd 2 4320.2.m.c.2159.28 48
120.59 even 2 inner 1080.2.m.c.539.14 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.m.c.539.13 ✓ 48 8.3 odd 2 inner
1080.2.m.c.539.14 yes 48 120.59 even 2 inner
1080.2.m.c.539.15 yes 48 15.14 odd 2 inner
1080.2.m.c.539.16 yes 48 1.1 even 1 trivial
1080.2.m.c.539.33 yes 48 3.2 odd 2 inner
1080.2.m.c.539.34 yes 48 5.4 even 2 inner
1080.2.m.c.539.35 yes 48 40.19 odd 2 inner
1080.2.m.c.539.36 yes 48 24.11 even 2 inner
4320.2.m.c.2159.21 48 60.59 even 2
4320.2.m.c.2159.22 48 40.29 even 2
4320.2.m.c.2159.23 48 8.5 even 2
4320.2.m.c.2159.24 48 12.11 even 2
4320.2.m.c.2159.25 48 24.5 odd 2
4320.2.m.c.2159.26 48 4.3 odd 2
4320.2.m.c.2159.27 48 20.19 odd 2
4320.2.m.c.2159.28 48 120.29 odd 2