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Properties

Label 3040.2.a.r.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $4$
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] = [3040,2,Mod(1,3040)] 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("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-1,0,4,0,-5,0,1,0,-6] 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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.787711\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.787711 q^{3} +1.00000 q^{5} +1.37951 q^{7} -2.37951 q^{9} -6.04159 q^{11} +3.25388 q^{13} +0.787711 q^{15} -4.23750 q^{17} -1.00000 q^{19} +1.08666 q^{21} -0.0553572 q^{23} +1.00000 q^{25} -4.23750 q^{27} +2.66208 q^{29} -0.816397 q^{31} -4.75902 q^{33} +1.37951 q^{35} -6.30777 q^{37} +2.56311 q^{39} +1.57542 q^{41} -0.717435 q^{43} -2.37951 q^{45} -7.22519 q^{47} -5.09695 q^{49} -3.33792 q^{51} -10.7719 q^{53} -6.04159 q^{55} -0.787711 q^{57} +3.84568 q^{59} +8.66006 q^{61} -3.28257 q^{63} +3.25388 q^{65} -5.40618 q^{67} -0.0436054 q^{69} -12.5078 q^{71} +4.48876 q^{73} +0.787711 q^{75} -8.33445 q^{77} -7.43487 q^{79} +3.80061 q^{81} -3.28257 q^{83} -4.23750 q^{85} +2.09695 q^{87} -1.74873 q^{89} +4.48876 q^{91} -0.643085 q^{93} -1.00000 q^{95} +4.59180 q^{97} +14.3760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.787711 0.454785 0.227392 − 0.973803i \(-0.426980\pi\)
0.227392 + 0.973803i \(0.426980\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.37951 0.521407 0.260703 − 0.965419i \(-0.416046\pi\)
0.260703 + 0.965419i \(0.416046\pi\)
\(8\) 0 0
\(9\) −2.37951 −0.793171
\(10\) 0 0
\(11\) −6.04159 −1.82161 −0.910804 − 0.412839i \(-0.864537\pi\)
−0.910804 + 0.412839i \(0.864537\pi\)
\(12\) 0 0
\(13\) 3.25388 0.902464 0.451232 − 0.892407i \(-0.350985\pi\)
0.451232 + 0.892407i \(0.350985\pi\)
\(14\) 0 0
\(15\) 0.787711 0.203386
\(16\) 0 0
\(17\) −4.23750 −1.02774 −0.513872 − 0.857867i \(-0.671789\pi\)
−0.513872 + 0.857867i \(0.671789\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.08666 0.237128
\(22\) 0 0
\(23\) −0.0553572 −0.0115428 −0.00577138 − 0.999983i \(-0.501837\pi\)
−0.00577138 + 0.999983i \(0.501837\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.23750 −0.815507
\(28\) 0 0
\(29\) 2.66208 0.494335 0.247168 − 0.968973i \(-0.420500\pi\)
0.247168 + 0.968973i \(0.420500\pi\)
\(30\) 0 0
\(31\) −0.816397 −0.146629 −0.0733146 − 0.997309i \(-0.523358\pi\)
−0.0733146 + 0.997309i \(0.523358\pi\)
\(32\) 0 0
\(33\) −4.75902 −0.828440
\(34\) 0 0
\(35\) 1.37951 0.233180
\(36\) 0 0
\(37\) −6.30777 −1.03699 −0.518496 − 0.855080i \(-0.673508\pi\)
−0.518496 + 0.855080i \(0.673508\pi\)
\(38\) 0 0
\(39\) 2.56311 0.410427
\(40\) 0 0
\(41\) 1.57542 0.246039 0.123020 − 0.992404i \(-0.460742\pi\)
0.123020 + 0.992404i \(0.460742\pi\)
\(42\) 0 0
\(43\) −0.717435 −0.109408 −0.0547039 − 0.998503i \(-0.517421\pi\)
−0.0547039 + 0.998503i \(0.517421\pi\)
\(44\) 0 0
\(45\) −2.37951 −0.354717
\(46\) 0 0
\(47\) −7.22519 −1.05390 −0.526951 − 0.849895i \(-0.676665\pi\)
−0.526951 + 0.849895i \(0.676665\pi\)
\(48\) 0 0
\(49\) −5.09695 −0.728135
\(50\) 0 0
\(51\) −3.33792 −0.467403
\(52\) 0 0
\(53\) −10.7719 −1.47964 −0.739819 − 0.672806i \(-0.765089\pi\)
−0.739819 + 0.672806i \(0.765089\pi\)
\(54\) 0 0
\(55\) −6.04159 −0.814648
\(56\) 0 0
\(57\) −0.787711 −0.104335
\(58\) 0 0
\(59\) 3.84568 0.500665 0.250332 − 0.968160i \(-0.419460\pi\)
0.250332 + 0.968160i \(0.419460\pi\)
\(60\) 0 0
\(61\) 8.66006 1.10881 0.554404 − 0.832248i \(-0.312946\pi\)
0.554404 + 0.832248i \(0.312946\pi\)
\(62\) 0 0
\(63\) −3.28257 −0.413564
\(64\) 0 0
\(65\) 3.25388 0.403594
\(66\) 0 0
\(67\) −5.40618 −0.660470 −0.330235 − 0.943899i \(-0.607128\pi\)
−0.330235 + 0.943899i \(0.607128\pi\)
\(68\) 0 0
\(69\) −0.0436054 −0.00524948
\(70\) 0 0
\(71\) −12.5078 −1.48440 −0.742199 − 0.670180i \(-0.766217\pi\)
−0.742199 + 0.670180i \(0.766217\pi\)
\(72\) 0 0
\(73\) 4.48876 0.525370 0.262685 − 0.964882i \(-0.415392\pi\)
0.262685 + 0.964882i \(0.415392\pi\)
\(74\) 0 0
\(75\) 0.787711 0.0909570
\(76\) 0 0
\(77\) −8.33445 −0.949798
\(78\) 0 0
\(79\) −7.43487 −0.836488 −0.418244 − 0.908335i \(-0.637354\pi\)
−0.418244 + 0.908335i \(0.637354\pi\)
\(80\) 0 0
\(81\) 3.80061 0.422290
\(82\) 0 0
\(83\) −3.28257 −0.360308 −0.180154 − 0.983638i \(-0.557660\pi\)
−0.180154 + 0.983638i \(0.557660\pi\)
\(84\) 0 0
\(85\) −4.23750 −0.459621
\(86\) 0 0
\(87\) 2.09695 0.224816
\(88\) 0 0
\(89\) −1.74873 −0.185365 −0.0926827 − 0.995696i \(-0.529544\pi\)
−0.0926827 + 0.995696i \(0.529544\pi\)
\(90\) 0 0
\(91\) 4.48876 0.470550
\(92\) 0 0
\(93\) −0.643085 −0.0666848
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 4.59180 0.466227 0.233113 − 0.972450i \(-0.425109\pi\)
0.233113 + 0.972450i \(0.425109\pi\)
\(98\) 0 0
\(99\) 14.3760 1.44485
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 3040.2.a.r.1.3 ✓ 4
4.3 odd 2 3040.2.a.t.1.2 yes 4
8.3 odd 2 6080.2.a.cd.1.3 4
8.5 even 2 6080.2.a.cf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.r.1.3 ✓ 4 1.1 even 1 trivial
3040.2.a.t.1.2 yes 4 4.3 odd 2
6080.2.a.cd.1.3 4 8.3 odd 2
6080.2.a.cf.1.2 4 8.5 even 2