Properties

Label 3720.2.a.l.1.1
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
Dimension $3$
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] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-3,0,-2] 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: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -5.21509 q^{7} +1.00000 q^{9} +4.00000 q^{11} -3.21509 q^{13} +1.00000 q^{15} -5.73240 q^{17} -1.03461 q^{19} +5.21509 q^{21} -2.69779 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.91288 q^{29} -1.00000 q^{31} -4.00000 q^{33} +5.21509 q^{35} +8.24970 q^{37} +3.21509 q^{39} -4.06922 q^{41} -2.96539 q^{43} -1.00000 q^{45} +8.76700 q^{47} +20.1972 q^{49} +5.73240 q^{51} -9.73240 q^{53} -4.00000 q^{55} +1.03461 q^{57} +1.48270 q^{59} -12.4302 q^{61} -5.21509 q^{63} +3.21509 q^{65} +4.18048 q^{67} +2.69779 q^{69} +5.48270 q^{71} -12.2497 q^{73} -1.00000 q^{75} -20.8604 q^{77} +6.69779 q^{79} +1.00000 q^{81} -16.2318 q^{83} +5.73240 q^{85} +9.91288 q^{87} +1.55191 q^{89} +16.7670 q^{91} +1.00000 q^{93} +1.03461 q^{95} +6.36097 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 12 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} + 2 q^{21} + 4 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} - 3 q^{31} - 12 q^{33} + 2 q^{35} + 8 q^{37} - 4 q^{39} - 6 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −5.21509 −1.97112 −0.985560 − 0.169327i \(-0.945840\pi\)
−0.985560 + 0.169327i \(0.945840\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −3.21509 −0.891706 −0.445853 − 0.895106i \(-0.647100\pi\)
−0.445853 + 0.895106i \(0.647100\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −5.73240 −1.39031 −0.695155 − 0.718860i \(-0.744664\pi\)
−0.695155 + 0.718860i \(0.744664\pi\)
\(18\) 0 0
\(19\) −1.03461 −0.237355 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(20\) 0 0
\(21\) 5.21509 1.13803
\(22\) 0 0
\(23\) −2.69779 −0.562528 −0.281264 − 0.959630i \(-0.590754\pi\)
−0.281264 + 0.959630i \(0.590754\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.91288 −1.84078 −0.920388 − 0.391007i \(-0.872127\pi\)
−0.920388 + 0.391007i \(0.872127\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 5.21509 0.881512
\(36\) 0 0
\(37\) 8.24970 1.35624 0.678121 − 0.734950i \(-0.262794\pi\)
0.678121 + 0.734950i \(0.262794\pi\)
\(38\) 0 0
\(39\) 3.21509 0.514827
\(40\) 0 0
\(41\) −4.06922 −0.635505 −0.317752 − 0.948174i \(-0.602928\pi\)
−0.317752 + 0.948174i \(0.602928\pi\)
\(42\) 0 0
\(43\) −2.96539 −0.452218 −0.226109 − 0.974102i \(-0.572601\pi\)
−0.226109 + 0.974102i \(0.572601\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.76700 1.27880 0.639400 − 0.768875i \(-0.279183\pi\)
0.639400 + 0.768875i \(0.279183\pi\)
\(48\) 0 0
\(49\) 20.1972 2.88531
\(50\) 0 0
\(51\) 5.73240 0.802696
\(52\) 0 0
\(53\) −9.73240 −1.33685 −0.668424 − 0.743781i \(-0.733031\pi\)
−0.668424 + 0.743781i \(0.733031\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 1.03461 0.137037
\(58\) 0 0
\(59\) 1.48270 0.193031 0.0965153 − 0.995332i \(-0.469230\pi\)
0.0965153 + 0.995332i \(0.469230\pi\)
\(60\) 0 0
\(61\) −12.4302 −1.59152 −0.795761 − 0.605611i \(-0.792929\pi\)
−0.795761 + 0.605611i \(0.792929\pi\)
\(62\) 0 0
\(63\) −5.21509 −0.657040
\(64\) 0 0
\(65\) 3.21509 0.398783
\(66\) 0 0
\(67\) 4.18048 0.510727 0.255364 − 0.966845i \(-0.417805\pi\)
0.255364 + 0.966845i \(0.417805\pi\)
\(68\) 0 0
\(69\) 2.69779 0.324776
\(70\) 0 0
\(71\) 5.48270 0.650676 0.325338 − 0.945598i \(-0.394522\pi\)
0.325338 + 0.945598i \(0.394522\pi\)
\(72\) 0 0
\(73\) −12.2497 −1.43372 −0.716860 − 0.697218i \(-0.754421\pi\)
−0.716860 + 0.697218i \(0.754421\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −20.8604 −2.37726
\(78\) 0 0
\(79\) 6.69779 0.753560 0.376780 − 0.926303i \(-0.377031\pi\)
0.376780 + 0.926303i \(0.377031\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.2318 −1.78167 −0.890836 − 0.454326i \(-0.849880\pi\)
−0.890836 + 0.454326i \(0.849880\pi\)
\(84\) 0 0
\(85\) 5.73240 0.621766
\(86\) 0 0
\(87\) 9.91288 1.06277
\(88\) 0 0
\(89\) 1.55191 0.164502 0.0822512 − 0.996612i \(-0.473789\pi\)
0.0822512 + 0.996612i \(0.473789\pi\)
\(90\) 0 0
\(91\) 16.7670 1.75766
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 1.03461 0.106149
\(96\) 0 0
\(97\) 6.36097 0.645859 0.322929 − 0.946423i \(-0.395332\pi\)
0.322929 + 0.946423i \(0.395332\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 3720.2.a.l.1.1 ✓ 3
4.3 odd 2 7440.2.a.bu.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.l.1.1 ✓ 3 1.1 even 1 trivial
7440.2.a.bu.1.3 3 4.3 odd 2