Properties

Label 3720.2.a.l.1.3
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 1.3
Root \(-0.210756\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.95558 q^{7} +1.00000 q^{9} +4.00000 q^{11} +4.95558 q^{13} +1.00000 q^{15} +0.421512 q^{17} -5.06814 q^{19} -2.95558 q^{21} +7.48965 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.44523 q^{29} -1.00000 q^{31} -4.00000 q^{33} -2.95558 q^{35} +4.11256 q^{37} -4.95558 q^{39} -12.1363 q^{41} +1.06814 q^{43} -1.00000 q^{45} +6.64663 q^{47} +1.73546 q^{49} -0.421512 q^{51} -3.57849 q^{53} -4.00000 q^{55} +5.06814 q^{57} -0.534070 q^{59} +3.91116 q^{61} +2.95558 q^{63} -4.95558 q^{65} -8.02372 q^{67} -7.48965 q^{69} +3.46593 q^{71} -8.11256 q^{73} -1.00000 q^{75} +11.8223 q^{77} -3.48965 q^{79} +1.00000 q^{81} -1.80361 q^{83} -0.421512 q^{85} -8.44523 q^{87} +7.60221 q^{89} +14.6466 q^{91} +1.00000 q^{93} +5.06814 q^{95} -18.0474 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 12 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} + 2 q^{21} + 4 q^{23} + 3 q^{25} - 3 q^{27} - 4 q^{29} - 3 q^{31} - 12 q^{33} + 2 q^{35} + 8 q^{37} - 4 q^{39} - 6 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(7\) 2.95558 1.11710 0.558552 − 0.829469i \(-0.311357\pi\)
0.558552 + 0.829469i \(0.311357\pi\)
\(8\) 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
\(13\) 4.95558 1.37443 0.687216 − 0.726454i \(-0.258833\pi\)
0.687216 + 0.726454i \(0.258833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.421512 0.102232 0.0511158 − 0.998693i \(-0.483722\pi\)
0.0511158 + 0.998693i \(0.483722\pi\)
\(18\) 0 0
\(19\) −5.06814 −1.16271 −0.581356 − 0.813650i \(-0.697477\pi\)
−0.581356 + 0.813650i \(0.697477\pi\)
\(20\) 0 0
\(21\) −2.95558 −0.644961
\(22\) 0 0
\(23\) 7.48965 1.56170 0.780850 − 0.624718i \(-0.214786\pi\)
0.780850 + 0.624718i \(0.214786\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.44523 1.56824 0.784120 − 0.620609i \(-0.213114\pi\)
0.784120 + 0.620609i \(0.213114\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −2.95558 −0.499585
\(36\) 0 0
\(37\) 4.11256 0.676100 0.338050 − 0.941128i \(-0.390233\pi\)
0.338050 + 0.941128i \(0.390233\pi\)
\(38\) 0 0
\(39\) −4.95558 −0.793528
\(40\) 0 0
\(41\) −12.1363 −1.89537 −0.947684 − 0.319209i \(-0.896583\pi\)
−0.947684 + 0.319209i \(0.896583\pi\)
\(42\) 0 0
\(43\) 1.06814 0.162890 0.0814449 − 0.996678i \(-0.474047\pi\)
0.0814449 + 0.996678i \(0.474047\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.64663 0.969510 0.484755 − 0.874650i \(-0.338909\pi\)
0.484755 + 0.874650i \(0.338909\pi\)
\(48\) 0 0
\(49\) 1.73546 0.247924
\(50\) 0 0
\(51\) −0.421512 −0.0590235
\(52\) 0 0
\(53\) −3.57849 −0.491543 −0.245772 − 0.969328i \(-0.579041\pi\)
−0.245772 + 0.969328i \(0.579041\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 5.06814 0.671292
\(58\) 0 0
\(59\) −0.534070 −0.0695300 −0.0347650 − 0.999396i \(-0.511068\pi\)
−0.0347650 + 0.999396i \(0.511068\pi\)
\(60\) 0 0
\(61\) 3.91116 0.500773 0.250387 − 0.968146i \(-0.419442\pi\)
0.250387 + 0.968146i \(0.419442\pi\)
\(62\) 0 0
\(63\) 2.95558 0.372368
\(64\) 0 0
\(65\) −4.95558 −0.614664
\(66\) 0 0
\(67\) −8.02372 −0.980254 −0.490127 − 0.871651i \(-0.663050\pi\)
−0.490127 + 0.871651i \(0.663050\pi\)
\(68\) 0 0
\(69\) −7.48965 −0.901648
\(70\) 0 0
\(71\) 3.46593 0.411330 0.205665 − 0.978622i \(-0.434064\pi\)
0.205665 + 0.978622i \(0.434064\pi\)
\(72\) 0 0
\(73\) −8.11256 −0.949503 −0.474752 − 0.880120i \(-0.657462\pi\)
−0.474752 + 0.880120i \(0.657462\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 11.8223 1.34728
\(78\) 0 0
\(79\) −3.48965 −0.392617 −0.196308 − 0.980542i \(-0.562895\pi\)
−0.196308 + 0.980542i \(0.562895\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.80361 −0.197971 −0.0989857 − 0.995089i \(-0.531560\pi\)
−0.0989857 + 0.995089i \(0.531560\pi\)
\(84\) 0 0
\(85\) −0.421512 −0.0457194
\(86\) 0 0
\(87\) −8.44523 −0.905424
\(88\) 0 0
\(89\) 7.60221 0.805833 0.402916 − 0.915237i \(-0.367997\pi\)
0.402916 + 0.915237i \(0.367997\pi\)
\(90\) 0 0
\(91\) 14.6466 1.53538
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 5.06814 0.519980
\(96\) 0 0
\(97\) −18.0474 −1.83244 −0.916220 − 0.400675i \(-0.868776\pi\)
−0.916220 + 0.400675i \(0.868776\pi\)
\(98\) 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.1.3 ✓ 3
4.3 odd 2 7440.2.a.bu.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.l.1.3 ✓ 3 1.1 even 1 trivial
7440.2.a.bu.1.1 3 4.3 odd 2