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Properties

Label 1080.4.a.i.1.1
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $1$
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: \(1\)
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 1.1
Root \(-2.16601\) 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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -21.1457 q^{7} +67.4294 q^{11} +2.55493 q^{13} -126.114 q^{17} -103.539 q^{19} +200.434 q^{23} +25.0000 q^{25} -71.4215 q^{29} -158.130 q^{31} -105.729 q^{35} +7.18163 q^{37} +347.052 q^{41} +189.318 q^{43} -585.443 q^{47} +104.142 q^{49} -77.2791 q^{53} +337.147 q^{55} -200.790 q^{59} +681.137 q^{61} +12.7746 q^{65} -810.751 q^{67} -515.476 q^{71} +385.577 q^{73} -1425.84 q^{77} -209.405 q^{79} -887.189 q^{83} -630.572 q^{85} +548.890 q^{89} -54.0258 q^{91} -517.696 q^{95} -1523.96 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} - 24 q^{7} + 6 q^{11} + 48 q^{13} - 27 q^{17} - 195 q^{19} - 27 q^{23} + 75 q^{25} + 60 q^{29} - 279 q^{31} - 120 q^{35} - 138 q^{37} + 66 q^{41} + 222 q^{43} - 264 q^{47} - 237 q^{49} - 507 q^{53}+ \cdots - 336 q^{97}+O(q^{100}) \) 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
\(7\) −21.1457 −1.14176 −0.570881 − 0.821033i \(-0.693398\pi\)
−0.570881 + 0.821033i \(0.693398\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 67.4294 1.84825 0.924124 − 0.382093i \(-0.124797\pi\)
0.924124 + 0.382093i \(0.124797\pi\)
\(12\) 0 0
\(13\) 2.55493 0.0545084 0.0272542 − 0.999629i \(-0.491324\pi\)
0.0272542 + 0.999629i \(0.491324\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −126.114 −1.79925 −0.899624 − 0.436665i \(-0.856160\pi\)
−0.899624 + 0.436665i \(0.856160\pi\)
\(18\) 0 0
\(19\) −103.539 −1.25019 −0.625093 − 0.780550i \(-0.714939\pi\)
−0.625093 + 0.780550i \(0.714939\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 200.434 1.81710 0.908551 − 0.417774i \(-0.137190\pi\)
0.908551 + 0.417774i \(0.137190\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −71.4215 −0.457333 −0.228666 − 0.973505i \(-0.573436\pi\)
−0.228666 + 0.973505i \(0.573436\pi\)
\(30\) 0 0
\(31\) −158.130 −0.916161 −0.458081 − 0.888911i \(-0.651463\pi\)
−0.458081 + 0.888911i \(0.651463\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −105.729 −0.510612
\(36\) 0 0
\(37\) 7.18163 0.0319095 0.0159548 − 0.999873i \(-0.494921\pi\)
0.0159548 + 0.999873i \(0.494921\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 347.052 1.32196 0.660980 − 0.750404i \(-0.270141\pi\)
0.660980 + 0.750404i \(0.270141\pi\)
\(42\) 0 0
\(43\) 189.318 0.671413 0.335707 − 0.941967i \(-0.391025\pi\)
0.335707 + 0.941967i \(0.391025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −585.443 −1.81693 −0.908464 − 0.417963i \(-0.862744\pi\)
−0.908464 + 0.417963i \(0.862744\pi\)
\(48\) 0 0
\(49\) 104.142 0.303622
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −77.2791 −0.200285 −0.100143 − 0.994973i \(-0.531930\pi\)
−0.100143 + 0.994973i \(0.531930\pi\)
\(54\) 0 0
\(55\) 337.147 0.826561
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −200.790 −0.443062 −0.221531 − 0.975153i \(-0.571105\pi\)
−0.221531 + 0.975153i \(0.571105\pi\)
\(60\) 0 0
\(61\) 681.137 1.42968 0.714841 − 0.699287i \(-0.246499\pi\)
0.714841 + 0.699287i \(0.246499\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.7746 0.0243769
\(66\) 0 0
\(67\) −810.751 −1.47834 −0.739172 − 0.673517i \(-0.764783\pi\)
−0.739172 + 0.673517i \(0.764783\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −515.476 −0.861631 −0.430816 − 0.902440i \(-0.641774\pi\)
−0.430816 + 0.902440i \(0.641774\pi\)
\(72\) 0 0
\(73\) 385.577 0.618197 0.309099 − 0.951030i \(-0.399973\pi\)
0.309099 + 0.951030i \(0.399973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1425.84 −2.11026
\(78\) 0 0
\(79\) −209.405 −0.298226 −0.149113 − 0.988820i \(-0.547642\pi\)
−0.149113 + 0.988820i \(0.547642\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −887.189 −1.17327 −0.586637 − 0.809850i \(-0.699548\pi\)
−0.586637 + 0.809850i \(0.699548\pi\)
\(84\) 0 0
\(85\) −630.572 −0.804648
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 548.890 0.653733 0.326867 − 0.945071i \(-0.394007\pi\)
0.326867 + 0.945071i \(0.394007\pi\)
\(90\) 0 0
\(91\) −54.0258 −0.0622357
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −517.696 −0.559100
\(96\) 0 0
\(97\) −1523.96 −1.59520 −0.797600 − 0.603186i \(-0.793897\pi\)
−0.797600 + 0.603186i \(0.793897\pi\)
\(98\) 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.i.1.1 yes 3
3.2 odd 2 1080.4.a.c.1.1 ✓ 3
4.3 odd 2 2160.4.a.bt.1.3 3
12.11 even 2 2160.4.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.1 ✓ 3 3.2 odd 2
1080.4.a.i.1.1 yes 3 1.1 even 1 trivial
2160.4.a.bl.1.3 3 12.11 even 2
2160.4.a.bt.1.3 3 4.3 odd 2