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Properties

Label 720.4.a.r.1.1
Level $720$
Weight $4$
Character 720.1
Self dual yes
Analytic conductor $42.481$
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] = [720,4,Mod(1,720)] 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("720.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,5,0,-20,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.4813752041\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 720.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -20.0000 q^{7} -24.0000 q^{11} +74.0000 q^{13} -54.0000 q^{17} +124.000 q^{19} -120.000 q^{23} +25.0000 q^{25} +78.0000 q^{29} -200.000 q^{31} -100.000 q^{35} -70.0000 q^{37} -330.000 q^{41} -92.0000 q^{43} -24.0000 q^{47} +57.0000 q^{49} -450.000 q^{53} -120.000 q^{55} +24.0000 q^{59} -322.000 q^{61} +370.000 q^{65} +196.000 q^{67} -288.000 q^{71} -430.000 q^{73} +480.000 q^{77} +520.000 q^{79} +156.000 q^{83} -270.000 q^{85} -1026.00 q^{89} -1480.00 q^{91} +620.000 q^{95} -286.000 q^{97} +O(q^{100})\)

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
\(7\) −20.0000 −1.07990 −0.539949 − 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 − 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 0 0
\(13\) 74.0000 1.57876 0.789381 − 0.613904i \(-0.210402\pi\)
0.789381 + 0.613904i \(0.210402\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −54.0000 −0.770407 −0.385204 − 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 − 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −120.000 −1.08790 −0.543951 − 0.839117i \(-0.683072\pi\)
−0.543951 + 0.839117i \(0.683072\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 78.0000 0.499456 0.249728 − 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) −200.000 −1.15874 −0.579372 − 0.815063i \(-0.696702\pi\)
−0.579372 + 0.815063i \(0.696702\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −100.000 −0.482945
\(36\) 0 0
\(37\) −70.0000 −0.311025 −0.155513 − 0.987834i \(-0.549703\pi\)
−0.155513 + 0.987834i \(0.549703\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −330.000 −1.25701 −0.628504 − 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 0 0
\(43\) −92.0000 −0.326276 −0.163138 − 0.986603i \(-0.552162\pi\)
−0.163138 + 0.986603i \(0.552162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −24.0000 −0.0744843 −0.0372421 − 0.999306i \(-0.511857\pi\)
−0.0372421 + 0.999306i \(0.511857\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −450.000 −1.16627 −0.583134 − 0.812376i \(-0.698174\pi\)
−0.583134 + 0.812376i \(0.698174\pi\)
\(54\) 0 0
\(55\) −120.000 −0.294196
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 24.0000 0.0529582 0.0264791 − 0.999649i \(-0.491570\pi\)
0.0264791 + 0.999649i \(0.491570\pi\)
\(60\) 0 0
\(61\) −322.000 −0.675867 −0.337933 − 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 370.000 0.706044
\(66\) 0 0
\(67\) 196.000 0.357391 0.178696 − 0.983904i \(-0.442812\pi\)
0.178696 + 0.983904i \(0.442812\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −288.000 −0.481399 −0.240699 − 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) 0 0
\(73\) −430.000 −0.689420 −0.344710 − 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 480.000 0.710404
\(78\) 0 0
\(79\) 520.000 0.740564 0.370282 − 0.928919i \(-0.379261\pi\)
0.370282 + 0.928919i \(0.379261\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 156.000 0.206304 0.103152 − 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(84\) 0 0
\(85\) −270.000 −0.344537
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1026.00 −1.22198 −0.610988 − 0.791640i \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) −1480.00 −1.70490
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 620.000 0.669586
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 − 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 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 720.4.a.r.1.1 1
3.2 odd 2 240.4.a.f.1.1 1
4.3 odd 2 45.4.a.b.1.1 1
12.11 even 2 15.4.a.b.1.1 ✓ 1
15.2 even 4 1200.4.f.m.49.1 2
15.8 even 4 1200.4.f.m.49.2 2
15.14 odd 2 1200.4.a.o.1.1 1
20.3 even 4 225.4.b.d.199.2 2
20.7 even 4 225.4.b.d.199.1 2
20.19 odd 2 225.4.a.g.1.1 1
24.5 odd 2 960.4.a.l.1.1 1
24.11 even 2 960.4.a.bi.1.1 1
28.27 even 2 2205.4.a.c.1.1 1
36.7 odd 6 405.4.e.k.271.1 2
36.11 even 6 405.4.e.d.271.1 2
36.23 even 6 405.4.e.d.136.1 2
36.31 odd 6 405.4.e.k.136.1 2
60.23 odd 4 75.4.b.a.49.1 2
60.47 odd 4 75.4.b.a.49.2 2
60.59 even 2 75.4.a.a.1.1 1
84.83 odd 2 735.4.a.i.1.1 1
132.131 odd 2 1815.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 ✓ 1 12.11 even 2
45.4.a.b.1.1 1 4.3 odd 2
75.4.a.a.1.1 1 60.59 even 2
75.4.b.a.49.1 2 60.23 odd 4
75.4.b.a.49.2 2 60.47 odd 4
225.4.a.g.1.1 1 20.19 odd 2
225.4.b.d.199.1 2 20.7 even 4
225.4.b.d.199.2 2 20.3 even 4
240.4.a.f.1.1 1 3.2 odd 2
405.4.e.d.136.1 2 36.23 even 6
405.4.e.d.271.1 2 36.11 even 6
405.4.e.k.136.1 2 36.31 odd 6
405.4.e.k.271.1 2 36.7 odd 6
720.4.a.r.1.1 1 1.1 even 1 trivial
735.4.a.i.1.1 1 84.83 odd 2
960.4.a.l.1.1 1 24.5 odd 2
960.4.a.bi.1.1 1 24.11 even 2
1200.4.a.o.1.1 1 15.14 odd 2
1200.4.f.m.49.1 2 15.2 even 4
1200.4.f.m.49.2 2 15.8 even 4
1815.4.a.a.1.1 1 132.131 odd 2
2205.4.a.c.1.1 1 28.27 even 2