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Properties

Label 1200.4.a.l.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
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] = [1200,4,Mod(1,1200)] 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("1200.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,4,0,9,0,28] 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: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +28.0000 q^{11} +16.0000 q^{13} -108.000 q^{17} -32.0000 q^{19} -12.0000 q^{21} -28.0000 q^{23} -27.0000 q^{27} -238.000 q^{29} +180.000 q^{31} -84.0000 q^{33} +40.0000 q^{37} -48.0000 q^{39} +422.000 q^{41} +276.000 q^{43} +60.0000 q^{47} -327.000 q^{49} +324.000 q^{51} -220.000 q^{53} +96.0000 q^{57} +804.000 q^{59} -358.000 q^{61} +36.0000 q^{63} -884.000 q^{67} +84.0000 q^{69} +64.0000 q^{71} +152.000 q^{73} +112.000 q^{77} +932.000 q^{79} +81.0000 q^{81} -1292.00 q^{83} +714.000 q^{87} -1146.00 q^{89} +64.0000 q^{91} -540.000 q^{93} -824.000 q^{97} +252.000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 − 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 − 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 16.0000 0.341354 0.170677 − 0.985327i \(-0.445405\pi\)
0.170677 + 0.985327i \(0.445405\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −108.000 −1.54081 −0.770407 − 0.637552i \(-0.779947\pi\)
−0.770407 + 0.637552i \(0.779947\pi\)
\(18\) 0 0
\(19\) −32.0000 −0.386384 −0.193192 − 0.981161i \(-0.561884\pi\)
−0.193192 + 0.981161i \(0.561884\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) −28.0000 −0.253844 −0.126922 − 0.991913i \(-0.540510\pi\)
−0.126922 + 0.991913i \(0.540510\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −238.000 −1.52398 −0.761991 − 0.647587i \(-0.775778\pi\)
−0.761991 + 0.647587i \(0.775778\pi\)
\(30\) 0 0
\(31\) 180.000 1.04287 0.521435 − 0.853291i \(-0.325397\pi\)
0.521435 + 0.853291i \(0.325397\pi\)
\(32\) 0 0
\(33\) −84.0000 −0.443107
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 40.0000 0.177729 0.0888643 − 0.996044i \(-0.471676\pi\)
0.0888643 + 0.996044i \(0.471676\pi\)
\(38\) 0 0
\(39\) −48.0000 −0.197081
\(40\) 0 0
\(41\) 422.000 1.60745 0.803724 − 0.595003i \(-0.202849\pi\)
0.803724 + 0.595003i \(0.202849\pi\)
\(42\) 0 0
\(43\) 276.000 0.978828 0.489414 − 0.872052i \(-0.337211\pi\)
0.489414 + 0.872052i \(0.337211\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 60.0000 0.186211 0.0931053 − 0.995656i \(-0.470321\pi\)
0.0931053 + 0.995656i \(0.470321\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) 324.000 0.889590
\(52\) 0 0
\(53\) −220.000 −0.570176 −0.285088 − 0.958501i \(-0.592023\pi\)
−0.285088 + 0.958501i \(0.592023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 96.0000 0.223079
\(58\) 0 0
\(59\) 804.000 1.77410 0.887050 − 0.461674i \(-0.152751\pi\)
0.887050 + 0.461674i \(0.152751\pi\)
\(60\) 0 0
\(61\) −358.000 −0.751430 −0.375715 − 0.926735i \(-0.622603\pi\)
−0.375715 + 0.926735i \(0.622603\pi\)
\(62\) 0 0
\(63\) 36.0000 0.0719932
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −884.000 −1.61191 −0.805954 − 0.591979i \(-0.798347\pi\)
−0.805954 + 0.591979i \(0.798347\pi\)
\(68\) 0 0
\(69\) 84.0000 0.146557
\(70\) 0 0
\(71\) 64.0000 0.106978 0.0534888 − 0.998568i \(-0.482966\pi\)
0.0534888 + 0.998568i \(0.482966\pi\)
\(72\) 0 0
\(73\) 152.000 0.243702 0.121851 − 0.992548i \(-0.461117\pi\)
0.121851 + 0.992548i \(0.461117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 112.000 0.165761
\(78\) 0 0
\(79\) 932.000 1.32732 0.663659 − 0.748035i \(-0.269002\pi\)
0.663659 + 0.748035i \(0.269002\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1292.00 −1.70862 −0.854310 − 0.519764i \(-0.826020\pi\)
−0.854310 + 0.519764i \(0.826020\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 714.000 0.879872
\(88\) 0 0
\(89\) −1146.00 −1.36490 −0.682448 − 0.730934i \(-0.739085\pi\)
−0.682448 + 0.730934i \(0.739085\pi\)
\(90\) 0 0
\(91\) 64.0000 0.0737255
\(92\) 0 0
\(93\) −540.000 −0.602101
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −824.000 −0.862521 −0.431260 − 0.902227i \(-0.641931\pi\)
−0.431260 + 0.902227i \(0.641931\pi\)
\(98\) 0 0
\(99\) 252.000 0.255828
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 1200.4.a.l.1.1 1
4.3 odd 2 600.4.a.k.1.1 1
5.2 odd 4 240.4.f.e.49.2 2
5.3 odd 4 240.4.f.e.49.1 2
5.4 even 2 1200.4.a.z.1.1 1
12.11 even 2 1800.4.a.o.1.1 1
15.2 even 4 720.4.f.b.289.2 2
15.8 even 4 720.4.f.b.289.1 2
20.3 even 4 120.4.f.c.49.2 yes 2
20.7 even 4 120.4.f.c.49.1 ✓ 2
20.19 odd 2 600.4.a.f.1.1 1
40.3 even 4 960.4.f.b.769.1 2
40.13 odd 4 960.4.f.a.769.2 2
40.27 even 4 960.4.f.b.769.2 2
40.37 odd 4 960.4.f.a.769.1 2
60.23 odd 4 360.4.f.a.289.1 2
60.47 odd 4 360.4.f.a.289.2 2
60.59 even 2 1800.4.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.c.49.1 ✓ 2 20.7 even 4
120.4.f.c.49.2 yes 2 20.3 even 4
240.4.f.e.49.1 2 5.3 odd 4
240.4.f.e.49.2 2 5.2 odd 4
360.4.f.a.289.1 2 60.23 odd 4
360.4.f.a.289.2 2 60.47 odd 4
600.4.a.f.1.1 1 20.19 odd 2
600.4.a.k.1.1 1 4.3 odd 2
720.4.f.b.289.1 2 15.8 even 4
720.4.f.b.289.2 2 15.2 even 4
960.4.f.a.769.1 2 40.37 odd 4
960.4.f.a.769.2 2 40.13 odd 4
960.4.f.b.769.1 2 40.3 even 4
960.4.f.b.769.2 2 40.27 even 4
1200.4.a.l.1.1 1 1.1 even 1 trivial
1200.4.a.z.1.1 1 5.4 even 2
1800.4.a.o.1.1 1 12.11 even 2
1800.4.a.u.1.1 1 60.59 even 2