Properties

Label 1360.4.a.g.1.1
Level $1360$
Weight $4$
Character 1360.1
Self dual yes
Analytic conductor $80.243$
Analytic rank $0$
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] = [1360,4,Mod(1,1360)] 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("1360.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,5,0,-5,0,22,0,-2,0,-60] 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: \(80.2425976078\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1360.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{3} -5.00000 q^{5} +22.0000 q^{7} -2.00000 q^{9} -60.0000 q^{11} -31.0000 q^{13} -25.0000 q^{15} +17.0000 q^{17} +61.0000 q^{19} +110.000 q^{21} +78.0000 q^{23} +25.0000 q^{25} -145.000 q^{27} +69.0000 q^{29} +31.0000 q^{31} -300.000 q^{33} -110.000 q^{35} +56.0000 q^{37} -155.000 q^{39} -6.00000 q^{41} +538.000 q^{43} +10.0000 q^{45} +465.000 q^{47} +141.000 q^{49} +85.0000 q^{51} +723.000 q^{53} +300.000 q^{55} +305.000 q^{57} +753.000 q^{59} +35.0000 q^{61} -44.0000 q^{63} +155.000 q^{65} +322.000 q^{67} +390.000 q^{69} +99.0000 q^{71} -1123.00 q^{73} +125.000 q^{75} -1320.00 q^{77} -488.000 q^{79} -671.000 q^{81} +852.000 q^{83} -85.0000 q^{85} +345.000 q^{87} +1215.00 q^{89} -682.000 q^{91} +155.000 q^{93} -305.000 q^{95} -601.000 q^{97} +120.000 q^{99} +O(q^{100})\)

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
\(3\) 5.00000 0.962250 0.481125 − 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 22.0000 1.18789 0.593944 − 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) −60.0000 −1.64461 −0.822304 − 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 0 0
\(13\) −31.0000 −0.661373 −0.330687 − 0.943741i \(-0.607280\pi\)
−0.330687 + 0.943741i \(0.607280\pi\)
\(14\) 0 0
\(15\) −25.0000 −0.430331
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) 61.0000 0.736545 0.368273 − 0.929718i \(-0.379949\pi\)
0.368273 + 0.929718i \(0.379949\pi\)
\(20\) 0 0
\(21\) 110.000 1.14305
\(22\) 0 0
\(23\) 78.0000 0.707136 0.353568 − 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −145.000 −1.03353
\(28\) 0 0
\(29\) 69.0000 0.441827 0.220913 − 0.975293i \(-0.429096\pi\)
0.220913 + 0.975293i \(0.429096\pi\)
\(30\) 0 0
\(31\) 31.0000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) −300.000 −1.58252
\(34\) 0 0
\(35\) −110.000 −0.531240
\(36\) 0 0
\(37\) 56.0000 0.248820 0.124410 − 0.992231i \(-0.460296\pi\)
0.124410 + 0.992231i \(0.460296\pi\)
\(38\) 0 0
\(39\) −155.000 −0.636407
\(40\) 0 0
\(41\) −6.00000 −0.0228547 −0.0114273 − 0.999935i \(-0.503638\pi\)
−0.0114273 + 0.999935i \(0.503638\pi\)
\(42\) 0 0
\(43\) 538.000 1.90801 0.954003 − 0.299798i \(-0.0969193\pi\)
0.954003 + 0.299798i \(0.0969193\pi\)
\(44\) 0 0
\(45\) 10.0000 0.0331269
\(46\) 0 0
\(47\) 465.000 1.44313 0.721566 − 0.692345i \(-0.243423\pi\)
0.721566 + 0.692345i \(0.243423\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) 0 0
\(51\) 85.0000 0.233380
\(52\) 0 0
\(53\) 723.000 1.87381 0.936903 − 0.349590i \(-0.113679\pi\)
0.936903 + 0.349590i \(0.113679\pi\)
\(54\) 0 0
\(55\) 300.000 0.735491
\(56\) 0 0
\(57\) 305.000 0.708741
\(58\) 0 0
\(59\) 753.000 1.66156 0.830782 − 0.556598i \(-0.187894\pi\)
0.830782 + 0.556598i \(0.187894\pi\)
\(60\) 0 0
\(61\) 35.0000 0.0734638 0.0367319 − 0.999325i \(-0.488305\pi\)
0.0367319 + 0.999325i \(0.488305\pi\)
\(62\) 0 0
\(63\) −44.0000 −0.0879917
\(64\) 0 0
\(65\) 155.000 0.295775
\(66\) 0 0
\(67\) 322.000 0.587143 0.293571 − 0.955937i \(-0.405156\pi\)
0.293571 + 0.955937i \(0.405156\pi\)
\(68\) 0 0
\(69\) 390.000 0.680442
\(70\) 0 0
\(71\) 99.0000 0.165481 0.0827404 − 0.996571i \(-0.473633\pi\)
0.0827404 + 0.996571i \(0.473633\pi\)
\(72\) 0 0
\(73\) −1123.00 −1.80051 −0.900255 − 0.435363i \(-0.856620\pi\)
−0.900255 + 0.435363i \(0.856620\pi\)
\(74\) 0 0
\(75\) 125.000 0.192450
\(76\) 0 0
\(77\) −1320.00 −1.95361
\(78\) 0 0
\(79\) −488.000 −0.694991 −0.347496 − 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) 852.000 1.12674 0.563368 − 0.826206i \(-0.309505\pi\)
0.563368 + 0.826206i \(0.309505\pi\)
\(84\) 0 0
\(85\) −85.0000 −0.108465
\(86\) 0 0
\(87\) 345.000 0.425148
\(88\) 0 0
\(89\) 1215.00 1.44708 0.723538 − 0.690285i \(-0.242515\pi\)
0.723538 + 0.690285i \(0.242515\pi\)
\(90\) 0 0
\(91\) −682.000 −0.785638
\(92\) 0 0
\(93\) 155.000 0.172825
\(94\) 0 0
\(95\) −305.000 −0.329393
\(96\) 0 0
\(97\) −601.000 −0.629096 −0.314548 − 0.949242i \(-0.601853\pi\)
−0.314548 + 0.949242i \(0.601853\pi\)
\(98\) 0 0
\(99\) 120.000 0.121823
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 1360.4.a.g.1.1 1
4.3 odd 2 85.4.a.b.1.1 ✓ 1
12.11 even 2 765.4.a.c.1.1 1
20.3 even 4 425.4.b.b.324.1 2
20.7 even 4 425.4.b.b.324.2 2
20.19 odd 2 425.4.a.b.1.1 1
68.67 odd 2 1445.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.b.1.1 ✓ 1 4.3 odd 2
425.4.a.b.1.1 1 20.19 odd 2
425.4.b.b.324.1 2 20.3 even 4
425.4.b.b.324.2 2 20.7 even 4
765.4.a.c.1.1 1 12.11 even 2
1360.4.a.g.1.1 1 1.1 even 1 trivial
1445.4.a.g.1.1 1 68.67 odd 2