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Properties

Label 720.4.a.b.1.1
Level $720$
Weight $4$
Character 720.1
Self dual yes
Analytic conductor $42.481$
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] = [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,-32,0,0,0,-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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} -32.0000 q^{7} -60.0000 q^{11} -34.0000 q^{13} -42.0000 q^{17} +76.0000 q^{19} +25.0000 q^{25} -6.00000 q^{29} +232.000 q^{31} +160.000 q^{35} +134.000 q^{37} -234.000 q^{41} +412.000 q^{43} -360.000 q^{47} +681.000 q^{49} -222.000 q^{53} +300.000 q^{55} +660.000 q^{59} -490.000 q^{61} +170.000 q^{65} -812.000 q^{67} +120.000 q^{71} +746.000 q^{73} +1920.00 q^{77} -152.000 q^{79} -804.000 q^{83} +210.000 q^{85} +678.000 q^{89} +1088.00 q^{91} -380.000 q^{95} +194.000 q^{97} +O(q^{100})\)

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
\(7\) −32.0000 −1.72784 −0.863919 − 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −60.0000 −1.64461 −0.822304 − 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 0 0
\(13\) −34.0000 −0.725377 −0.362689 − 0.931910i \(-0.618141\pi\)
−0.362689 + 0.931910i \(0.618141\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −42.0000 −0.599206 −0.299603 − 0.954064i \(-0.596854\pi\)
−0.299603 + 0.954064i \(0.596854\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 − 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) 232.000 1.34414 0.672071 − 0.740486i \(-0.265405\pi\)
0.672071 + 0.740486i \(0.265405\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 160.000 0.772712
\(36\) 0 0
\(37\) 134.000 0.595391 0.297695 − 0.954661i \(-0.403782\pi\)
0.297695 + 0.954661i \(0.403782\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −234.000 −0.891333 −0.445667 − 0.895199i \(-0.647033\pi\)
−0.445667 + 0.895199i \(0.647033\pi\)
\(42\) 0 0
\(43\) 412.000 1.46115 0.730575 − 0.682833i \(-0.239252\pi\)
0.730575 + 0.682833i \(0.239252\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −360.000 −1.11726 −0.558632 − 0.829416i \(-0.688674\pi\)
−0.558632 + 0.829416i \(0.688674\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −222.000 −0.575359 −0.287680 − 0.957727i \(-0.592884\pi\)
−0.287680 + 0.957727i \(0.592884\pi\)
\(54\) 0 0
\(55\) 300.000 0.735491
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 660.000 1.45635 0.728175 − 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −490.000 −1.02849 −0.514246 − 0.857642i \(-0.671928\pi\)
−0.514246 + 0.857642i \(0.671928\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 170.000 0.324399
\(66\) 0 0
\(67\) −812.000 −1.48062 −0.740310 − 0.672265i \(-0.765321\pi\)
−0.740310 + 0.672265i \(0.765321\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 746.000 1.19606 0.598032 − 0.801472i \(-0.295949\pi\)
0.598032 + 0.801472i \(0.295949\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1920.00 2.84161
\(78\) 0 0
\(79\) −152.000 −0.216473 −0.108236 − 0.994125i \(-0.534520\pi\)
−0.108236 + 0.994125i \(0.534520\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −804.000 −1.06326 −0.531629 − 0.846977i \(-0.678420\pi\)
−0.531629 + 0.846977i \(0.678420\pi\)
\(84\) 0 0
\(85\) 210.000 0.267973
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 678.000 0.807504 0.403752 − 0.914868i \(-0.367706\pi\)
0.403752 + 0.914868i \(0.367706\pi\)
\(90\) 0 0
\(91\) 1088.00 1.25333
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −380.000 −0.410391
\(96\) 0 0
\(97\) 194.000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.b.1.1 1
3.2 odd 2 240.4.a.c.1.1 1
4.3 odd 2 90.4.a.d.1.1 1
12.11 even 2 30.4.a.a.1.1 ✓ 1
15.2 even 4 1200.4.f.u.49.2 2
15.8 even 4 1200.4.f.u.49.1 2
15.14 odd 2 1200.4.a.bk.1.1 1
20.3 even 4 450.4.c.k.199.1 2
20.7 even 4 450.4.c.k.199.2 2
20.19 odd 2 450.4.a.b.1.1 1
24.5 odd 2 960.4.a.s.1.1 1
24.11 even 2 960.4.a.j.1.1 1
36.7 odd 6 810.4.e.e.271.1 2
36.11 even 6 810.4.e.m.271.1 2
36.23 even 6 810.4.e.m.541.1 2
36.31 odd 6 810.4.e.e.541.1 2
60.23 odd 4 150.4.c.a.49.2 2
60.47 odd 4 150.4.c.a.49.1 2
60.59 even 2 150.4.a.e.1.1 1
84.83 odd 2 1470.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.a.1.1 ✓ 1 12.11 even 2
90.4.a.d.1.1 1 4.3 odd 2
150.4.a.e.1.1 1 60.59 even 2
150.4.c.a.49.1 2 60.47 odd 4
150.4.c.a.49.2 2 60.23 odd 4
240.4.a.c.1.1 1 3.2 odd 2
450.4.a.b.1.1 1 20.19 odd 2
450.4.c.k.199.1 2 20.3 even 4
450.4.c.k.199.2 2 20.7 even 4
720.4.a.b.1.1 1 1.1 even 1 trivial
810.4.e.e.271.1 2 36.7 odd 6
810.4.e.e.541.1 2 36.31 odd 6
810.4.e.m.271.1 2 36.11 even 6
810.4.e.m.541.1 2 36.23 even 6
960.4.a.j.1.1 1 24.11 even 2
960.4.a.s.1.1 1 24.5 odd 2
1200.4.a.bk.1.1 1 15.14 odd 2
1200.4.f.u.49.1 2 15.8 even 4
1200.4.f.u.49.2 2 15.2 even 4
1470.4.a.a.1.1 1 84.83 odd 2