Properties

Label 480.4.a.h.1.1
Level $480$
Weight $4$
Character 480.1
Self dual yes
Analytic conductor $28.321$
Analytic rank $1$
Dimension $1$
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] = [480,4,Mod(1,480)] 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("480.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,-5,0,12] 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: \(28.3209168028\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 480.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} -20.0000 q^{11} -58.0000 q^{13} -15.0000 q^{15} -70.0000 q^{17} -92.0000 q^{19} +36.0000 q^{21} +112.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +66.0000 q^{29} -108.000 q^{31} -60.0000 q^{33} -60.0000 q^{35} -58.0000 q^{37} -174.000 q^{39} +66.0000 q^{41} -388.000 q^{43} -45.0000 q^{45} -408.000 q^{47} -199.000 q^{49} -210.000 q^{51} +474.000 q^{53} +100.000 q^{55} -276.000 q^{57} -540.000 q^{59} +14.0000 q^{61} +108.000 q^{63} +290.000 q^{65} -276.000 q^{67} +336.000 q^{69} -96.0000 q^{71} -790.000 q^{73} +75.0000 q^{75} -240.000 q^{77} +308.000 q^{79} +81.0000 q^{81} -1036.00 q^{83} +350.000 q^{85} +198.000 q^{87} +1210.00 q^{89} -696.000 q^{91} -324.000 q^{93} +460.000 q^{95} +1426.00 q^{97} -180.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 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
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −20.0000 −0.548202 −0.274101 − 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −70.0000 −0.998676 −0.499338 − 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) −92.0000 −1.11086 −0.555428 − 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(20\) 0 0
\(21\) 36.0000 0.374088
\(22\) 0 0
\(23\) 112.000 1.01537 0.507687 − 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 66.0000 0.422617 0.211308 − 0.977419i \(-0.432228\pi\)
0.211308 + 0.977419i \(0.432228\pi\)
\(30\) 0 0
\(31\) −108.000 −0.625722 −0.312861 − 0.949799i \(-0.601287\pi\)
−0.312861 + 0.949799i \(0.601287\pi\)
\(32\) 0 0
\(33\) −60.0000 −0.316505
\(34\) 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
\(39\) −174.000 −0.714418
\(40\) 0 0
\(41\) 66.0000 0.251402 0.125701 − 0.992068i \(-0.459882\pi\)
0.125701 + 0.992068i \(0.459882\pi\)
\(42\) 0 0
\(43\) −388.000 −1.37603 −0.688017 − 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −408.000 −1.26623 −0.633116 − 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) −210.000 −0.576586
\(52\) 0 0
\(53\) 474.000 1.22847 0.614235 − 0.789123i \(-0.289465\pi\)
0.614235 + 0.789123i \(0.289465\pi\)
\(54\) 0 0
\(55\) 100.000 0.245164
\(56\) 0 0
\(57\) −276.000 −0.641353
\(58\) 0 0
\(59\) −540.000 −1.19156 −0.595780 − 0.803148i \(-0.703157\pi\)
−0.595780 + 0.803148i \(0.703157\pi\)
\(60\) 0 0
\(61\) 14.0000 0.0293855 0.0146928 − 0.999892i \(-0.495323\pi\)
0.0146928 + 0.999892i \(0.495323\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 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
\(69\) 336.000 0.586227
\(70\) 0 0
\(71\) −96.0000 −0.160466 −0.0802331 − 0.996776i \(-0.525566\pi\)
−0.0802331 + 0.996776i \(0.525566\pi\)
\(72\) 0 0
\(73\) −790.000 −1.26661 −0.633305 − 0.773902i \(-0.718302\pi\)
−0.633305 + 0.773902i \(0.718302\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 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
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1036.00 −1.37007 −0.685035 − 0.728510i \(-0.740213\pi\)
−0.685035 + 0.728510i \(0.740213\pi\)
\(84\) 0 0
\(85\) 350.000 0.446622
\(86\) 0 0
\(87\) 198.000 0.243998
\(88\) 0 0
\(89\) 1210.00 1.44112 0.720560 − 0.693392i \(-0.243885\pi\)
0.720560 + 0.693392i \(0.243885\pi\)
\(90\) 0 0
\(91\) −696.000 −0.801765
\(92\) 0 0
\(93\) −324.000 −0.361261
\(94\) 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
\(99\) −180.000 −0.182734
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 480.4.a.h.1.1 yes 1
3.2 odd 2 1440.4.a.q.1.1 1
4.3 odd 2 480.4.a.a.1.1 ✓ 1
5.4 even 2 2400.4.a.b.1.1 1
8.3 odd 2 960.4.a.be.1.1 1
8.5 even 2 960.4.a.p.1.1 1
12.11 even 2 1440.4.a.l.1.1 1
20.19 odd 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 4.3 odd 2
480.4.a.h.1.1 yes 1 1.1 even 1 trivial
960.4.a.p.1.1 1 8.5 even 2
960.4.a.be.1.1 1 8.3 odd 2
1440.4.a.l.1.1 1 12.11 even 2
1440.4.a.q.1.1 1 3.2 odd 2
2400.4.a.b.1.1 1 5.4 even 2
2400.4.a.u.1.1 1 20.19 odd 2