Properties

Label 1440.4.a.q.1.1
Level $1440$
Weight $4$
Character 1440.1
Self dual yes
Analytic conductor $84.963$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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] = [1440,4,Mod(1,1440)] 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("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,5,0,12,0,0,0,20,0,-58] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(84.9627504083\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1440.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} +12.0000 q^{7} +20.0000 q^{11} -58.0000 q^{13} +70.0000 q^{17} -92.0000 q^{19} -112.000 q^{23} +25.0000 q^{25} -66.0000 q^{29} -108.000 q^{31} +60.0000 q^{35} -58.0000 q^{37} -66.0000 q^{41} -388.000 q^{43} +408.000 q^{47} -199.000 q^{49} -474.000 q^{53} +100.000 q^{55} +540.000 q^{59} +14.0000 q^{61} -290.000 q^{65} -276.000 q^{67} +96.0000 q^{71} -790.000 q^{73} +240.000 q^{77} +308.000 q^{79} +1036.00 q^{83} +350.000 q^{85} -1210.00 q^{89} -696.000 q^{91} -460.000 q^{95} +1426.00 q^{97} +O(q^{100})\)

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
\(7\) 12.0000 0.647939 0.323970 − 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 20.0000 0.548202 0.274101 − 0.961701i \(-0.411620\pi\)
0.274101 + 0.961701i \(0.411620\pi\)
\(12\) 0 0
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 70.0000 0.998676 0.499338 − 0.866407i \(-0.333577\pi\)
0.499338 + 0.866407i \(0.333577\pi\)
\(18\) 0 0
\(19\) −92.0000 −1.11086 −0.555428 − 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −112.000 −1.01537 −0.507687 − 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −66.0000 −0.422617 −0.211308 − 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(30\) 0 0
\(31\) −108.000 −0.625722 −0.312861 − 0.949799i \(-0.601287\pi\)
−0.312861 + 0.949799i \(0.601287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 60.0000 0.289767
\(36\) 0 0
\(37\) −58.0000 −0.257707 −0.128853 − 0.991664i \(-0.541130\pi\)
−0.128853 + 0.991664i \(0.541130\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −66.0000 −0.251402 −0.125701 − 0.992068i \(-0.540118\pi\)
−0.125701 + 0.992068i \(0.540118\pi\)
\(42\) 0 0
\(43\) −388.000 −1.37603 −0.688017 − 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 408.000 1.26623 0.633116 − 0.774057i \(-0.281776\pi\)
0.633116 + 0.774057i \(0.281776\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −474.000 −1.22847 −0.614235 − 0.789123i \(-0.710535\pi\)
−0.614235 + 0.789123i \(0.710535\pi\)
\(54\) 0 0
\(55\) 100.000 0.245164
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 540.000 1.19156 0.595780 − 0.803148i \(-0.296843\pi\)
0.595780 + 0.803148i \(0.296843\pi\)
\(60\) 0 0
\(61\) 14.0000 0.0293855 0.0146928 − 0.999892i \(-0.495323\pi\)
0.0146928 + 0.999892i \(0.495323\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −290.000 −0.553386
\(66\) 0 0
\(67\) −276.000 −0.503265 −0.251633 − 0.967823i \(-0.580967\pi\)
−0.251633 + 0.967823i \(0.580967\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 96.0000 0.160466 0.0802331 − 0.996776i \(-0.474434\pi\)
0.0802331 + 0.996776i \(0.474434\pi\)
\(72\) 0 0
\(73\) −790.000 −1.26661 −0.633305 − 0.773902i \(-0.718302\pi\)
−0.633305 + 0.773902i \(0.718302\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 240.000 0.355202
\(78\) 0 0
\(79\) 308.000 0.438642 0.219321 − 0.975653i \(-0.429616\pi\)
0.219321 + 0.975653i \(0.429616\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1036.00 1.37007 0.685035 − 0.728510i \(-0.259787\pi\)
0.685035 + 0.728510i \(0.259787\pi\)
\(84\) 0 0
\(85\) 350.000 0.446622
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1210.00 −1.44112 −0.720560 − 0.693392i \(-0.756115\pi\)
−0.720560 + 0.693392i \(0.756115\pi\)
\(90\) 0 0
\(91\) −696.000 −0.801765
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −460.000 −0.496790
\(96\) 0 0
\(97\) 1426.00 1.49266 0.746332 − 0.665574i \(-0.231813\pi\)
0.746332 + 0.665574i \(0.231813\pi\)
\(98\) 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 1440.4.a.q.1.1 1
3.2 odd 2 480.4.a.h.1.1 yes 1
4.3 odd 2 1440.4.a.l.1.1 1
12.11 even 2 480.4.a.a.1.1 ✓ 1
15.14 odd 2 2400.4.a.b.1.1 1
24.5 odd 2 960.4.a.p.1.1 1
24.11 even 2 960.4.a.be.1.1 1
60.59 even 2 2400.4.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.a.1.1 ✓ 1 12.11 even 2
480.4.a.h.1.1 yes 1 3.2 odd 2
960.4.a.p.1.1 1 24.5 odd 2
960.4.a.be.1.1 1 24.11 even 2
1440.4.a.l.1.1 1 4.3 odd 2
1440.4.a.q.1.1 1 1.1 even 1 trivial
2400.4.a.b.1.1 1 15.14 odd 2
2400.4.a.u.1.1 1 60.59 even 2