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Properties

Label 1080.4.a.c.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,-24] 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: 3.3.697.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 7x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.782816\) 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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +11.6263 q^{7} +47.4851 q^{11} +84.2727 q^{13} +26.9488 q^{17} -152.060 q^{19} +177.082 q^{23} +25.0000 q^{25} -60.0913 q^{29} -92.1613 q^{31} -58.1315 q^{35} -221.798 q^{37} -115.099 q^{41} +383.827 q^{43} -317.275 q^{47} -207.829 q^{49} +257.738 q^{53} -237.426 q^{55} +642.576 q^{59} -662.092 q^{61} -421.363 q^{65} +597.173 q^{67} +500.152 q^{71} +989.949 q^{73} +552.076 q^{77} -517.554 q^{79} +605.867 q^{83} -134.744 q^{85} +1519.15 q^{89} +979.780 q^{91} +760.301 q^{95} +742.835 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 24 q^{7} - 6 q^{11} + 48 q^{13} + 27 q^{17} - 195 q^{19} + 27 q^{23} + 75 q^{25} - 60 q^{29} - 279 q^{31} + 120 q^{35} - 138 q^{37} - 66 q^{41} + 222 q^{43} + 264 q^{47} - 237 q^{49} + 507 q^{53}+ \cdots - 336 q^{97}+O(q^{100}) \) 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
\(7\) 11.6263 0.627761 0.313881 − 0.949462i \(-0.398371\pi\)
0.313881 + 0.949462i \(0.398371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 47.4851 1.30157 0.650787 − 0.759261i \(-0.274439\pi\)
0.650787 + 0.759261i \(0.274439\pi\)
\(12\) 0 0
\(13\) 84.2727 1.79793 0.898963 − 0.438024i \(-0.144322\pi\)
0.898963 + 0.438024i \(0.144322\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 26.9488 0.384474 0.192237 − 0.981349i \(-0.438426\pi\)
0.192237 + 0.981349i \(0.438426\pi\)
\(18\) 0 0
\(19\) −152.060 −1.83605 −0.918027 − 0.396518i \(-0.870218\pi\)
−0.918027 + 0.396518i \(0.870218\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 177.082 1.60539 0.802697 − 0.596387i \(-0.203397\pi\)
0.802697 + 0.596387i \(0.203397\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −60.0913 −0.384782 −0.192391 − 0.981318i \(-0.561624\pi\)
−0.192391 + 0.981318i \(0.561624\pi\)
\(30\) 0 0
\(31\) −92.1613 −0.533957 −0.266978 − 0.963703i \(-0.586025\pi\)
−0.266978 + 0.963703i \(0.586025\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −58.1315 −0.280743
\(36\) 0 0
\(37\) −221.798 −0.985496 −0.492748 − 0.870172i \(-0.664008\pi\)
−0.492748 + 0.870172i \(0.664008\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −115.099 −0.438426 −0.219213 − 0.975677i \(-0.570349\pi\)
−0.219213 + 0.975677i \(0.570349\pi\)
\(42\) 0 0
\(43\) 383.827 1.36123 0.680617 − 0.732639i \(-0.261712\pi\)
0.680617 + 0.732639i \(0.261712\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −317.275 −0.984666 −0.492333 − 0.870407i \(-0.663856\pi\)
−0.492333 + 0.870407i \(0.663856\pi\)
\(48\) 0 0
\(49\) −207.829 −0.605916
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 257.738 0.667981 0.333991 − 0.942576i \(-0.391605\pi\)
0.333991 + 0.942576i \(0.391605\pi\)
\(54\) 0 0
\(55\) −237.426 −0.582081
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 642.576 1.41790 0.708951 − 0.705257i \(-0.249169\pi\)
0.708951 + 0.705257i \(0.249169\pi\)
\(60\) 0 0
\(61\) −662.092 −1.38971 −0.694854 − 0.719150i \(-0.744531\pi\)
−0.694854 + 0.719150i \(0.744531\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −421.363 −0.804057
\(66\) 0 0
\(67\) 597.173 1.08890 0.544450 − 0.838794i \(-0.316739\pi\)
0.544450 + 0.838794i \(0.316739\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 500.152 0.836016 0.418008 − 0.908443i \(-0.362728\pi\)
0.418008 + 0.908443i \(0.362728\pi\)
\(72\) 0 0
\(73\) 989.949 1.58719 0.793594 − 0.608447i \(-0.208207\pi\)
0.793594 + 0.608447i \(0.208207\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 552.076 0.817077
\(78\) 0 0
\(79\) −517.554 −0.737081 −0.368540 − 0.929612i \(-0.620142\pi\)
−0.368540 + 0.929612i \(0.620142\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 605.867 0.801235 0.400617 − 0.916245i \(-0.368796\pi\)
0.400617 + 0.916245i \(0.368796\pi\)
\(84\) 0 0
\(85\) −134.744 −0.171942
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1519.15 1.80932 0.904658 − 0.426138i \(-0.140126\pi\)
0.904658 + 0.426138i \(0.140126\pi\)
\(90\) 0 0
\(91\) 979.780 1.12867
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 760.301 0.821108
\(96\) 0 0
\(97\) 742.835 0.777561 0.388781 − 0.921330i \(-0.372896\pi\)
0.388781 + 0.921330i \(0.372896\pi\)
\(98\) 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.c.1.3 ✓ 3
3.2 odd 2 1080.4.a.i.1.3 yes 3
4.3 odd 2 2160.4.a.bl.1.1 3
12.11 even 2 2160.4.a.bt.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.3 ✓ 3 1.1 even 1 trivial
1080.4.a.i.1.3 yes 3 3.2 odd 2
2160.4.a.bl.1.1 3 4.3 odd 2
2160.4.a.bt.1.1 3 12.11 even 2