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Properties

Label 1080.4.a.n.1.3
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [1080,4,Mod(1,1080)] 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("1080.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,15,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.47977.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 60x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-6.83575\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} +23.9261 q^{7} -57.9406 q^{11} -8.16237 q^{13} +50.0884 q^{17} +69.7782 q^{19} -4.92607 q^{23} +25.0000 q^{25} -79.4277 q^{29} +260.294 q^{31} +119.630 q^{35} -223.836 q^{37} +337.939 q^{41} +326.511 q^{43} -89.6851 q^{47} +229.457 q^{49} +543.672 q^{53} -289.703 q^{55} -92.0000 q^{59} +159.129 q^{61} -40.8119 q^{65} -910.561 q^{67} +293.232 q^{71} +142.022 q^{73} -1386.29 q^{77} +1106.50 q^{79} -813.367 q^{83} +250.442 q^{85} -956.666 q^{89} -195.294 q^{91} +348.891 q^{95} +106.363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 9 q^{7} + 18 q^{11} - 21 q^{13} + 84 q^{17} + 21 q^{19} + 48 q^{23} + 75 q^{25} - 36 q^{29} + 324 q^{31} + 45 q^{35} + 33 q^{37} + 114 q^{41} + 282 q^{43} + 282 q^{47} + 228 q^{49} + 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 23.9261 1.29189 0.645943 − 0.763386i \(-0.276464\pi\)
0.645943 + 0.763386i \(0.276464\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −57.9406 −1.58816 −0.794079 − 0.607814i \(-0.792046\pi\)
−0.794079 + 0.607814i \(0.792046\pi\)
\(12\) 0 0
\(13\) −8.16237 −0.174141 −0.0870706 − 0.996202i \(-0.527751\pi\)
−0.0870706 + 0.996202i \(0.527751\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 50.0884 0.714602 0.357301 − 0.933989i \(-0.383697\pi\)
0.357301 + 0.933989i \(0.383697\pi\)
\(18\) 0 0
\(19\) 69.7782 0.842538 0.421269 − 0.906936i \(-0.361585\pi\)
0.421269 + 0.906936i \(0.361585\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.92607 −0.0446590 −0.0223295 − 0.999751i \(-0.507108\pi\)
−0.0223295 + 0.999751i \(0.507108\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −79.4277 −0.508598 −0.254299 − 0.967126i \(-0.581845\pi\)
−0.254299 + 0.967126i \(0.581845\pi\)
\(30\) 0 0
\(31\) 260.294 1.50807 0.754036 − 0.656833i \(-0.228104\pi\)
0.754036 + 0.656833i \(0.228104\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 119.630 0.577749
\(36\) 0 0
\(37\) −223.836 −0.994553 −0.497276 − 0.867592i \(-0.665666\pi\)
−0.497276 + 0.867592i \(0.665666\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 337.939 1.28725 0.643625 − 0.765341i \(-0.277430\pi\)
0.643625 + 0.765341i \(0.277430\pi\)
\(42\) 0 0
\(43\) 326.511 1.15797 0.578983 − 0.815340i \(-0.303450\pi\)
0.578983 + 0.815340i \(0.303450\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −89.6851 −0.278339 −0.139169 − 0.990269i \(-0.544443\pi\)
−0.139169 + 0.990269i \(0.544443\pi\)
\(48\) 0 0
\(49\) 229.457 0.668970
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 543.672 1.40904 0.704520 − 0.709684i \(-0.251162\pi\)
0.704520 + 0.709684i \(0.251162\pi\)
\(54\) 0 0
\(55\) −289.703 −0.710246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −92.0000 −0.203006 −0.101503 − 0.994835i \(-0.532365\pi\)
−0.101503 + 0.994835i \(0.532365\pi\)
\(60\) 0 0
\(61\) 159.129 0.334006 0.167003 − 0.985956i \(-0.446591\pi\)
0.167003 + 0.985956i \(0.446591\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −40.8119 −0.0778783
\(66\) 0 0
\(67\) −910.561 −1.66034 −0.830169 − 0.557511i \(-0.811756\pi\)
−0.830169 + 0.557511i \(0.811756\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 293.232 0.490144 0.245072 − 0.969505i \(-0.421188\pi\)
0.245072 + 0.969505i \(0.421188\pi\)
\(72\) 0 0
\(73\) 142.022 0.227705 0.113853 − 0.993498i \(-0.463681\pi\)
0.113853 + 0.993498i \(0.463681\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1386.29 −2.05172
\(78\) 0 0
\(79\) 1106.50 1.57584 0.787920 − 0.615777i \(-0.211158\pi\)
0.787920 + 0.615777i \(0.211158\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −813.367 −1.07565 −0.537823 − 0.843058i \(-0.680753\pi\)
−0.537823 + 0.843058i \(0.680753\pi\)
\(84\) 0 0
\(85\) 250.442 0.319580
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −956.666 −1.13940 −0.569699 − 0.821853i \(-0.692940\pi\)
−0.569699 + 0.821853i \(0.692940\pi\)
\(90\) 0 0
\(91\) −195.294 −0.224971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 348.891 0.376794
\(96\) 0 0
\(97\) 106.363 0.111335 0.0556677 − 0.998449i \(-0.482271\pi\)
0.0556677 + 0.998449i \(0.482271\pi\)
\(98\) 0 0
\(99\) 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 1080.4.a.n.1.3 yes 3
3.2 odd 2 1080.4.a.h.1.3 ✓ 3
4.3 odd 2 2160.4.a.bn.1.1 3
12.11 even 2 2160.4.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.3 ✓ 3 3.2 odd 2
1080.4.a.n.1.3 yes 3 1.1 even 1 trivial
2160.4.a.bf.1.1 3 12.11 even 2
2160.4.a.bn.1.1 3 4.3 odd 2