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Properties

Label 3744.2.a.be.1.3
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [3744,2,Mod(1,3744)] 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("3744.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,-14,0,0,0,0,0,0,0,22,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.546295\) of defining polynomial
Character \(\chi\) \(=\) 3744.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.70156 q^{5} -4.20732 q^{7} -1.09259 q^{11} +1.00000 q^{13} -0.298438 q^{17} -1.09259 q^{19} +8.70156 q^{25} +2.00000 q^{29} +5.13688 q^{31} -15.5737 q^{35} -3.70156 q^{37} +9.40312 q^{41} +5.29991 q^{43} +4.20732 q^{47} +10.7016 q^{49} -1.40312 q^{53} -4.04429 q^{55} -13.5515 q^{59} +9.40312 q^{61} +3.70156 q^{65} +11.3663 q^{67} +8.25161 q^{71} -6.00000 q^{73} +4.59688 q^{77} +14.6441 q^{79} +7.32206 q^{83} -1.10469 q^{85} +6.00000 q^{89} -4.20732 q^{91} -4.04429 q^{95} +8.80625 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 4 q^{13} - 14 q^{17} + 22 q^{25} + 8 q^{29} - 2 q^{37} + 12 q^{41} + 30 q^{49} + 20 q^{53} + 12 q^{61} + 2 q^{65} - 24 q^{73} + 44 q^{77} + 34 q^{85} + 24 q^{89} - 16 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) 3.70156 1.65539 0.827694 − 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 0 0
\(7\) −4.20732 −1.59022 −0.795109 − 0.606466i \(-0.792587\pi\)
−0.795109 + 0.606466i \(0.792587\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.09259 −0.329428 −0.164714 − 0.986341i \(-0.552670\pi\)
−0.164714 + 0.986341i \(0.552670\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.298438 −0.0723818 −0.0361909 − 0.999345i \(-0.511522\pi\)
−0.0361909 + 0.999345i \(0.511522\pi\)
\(18\) 0 0
\(19\) −1.09259 −0.250657 −0.125329 − 0.992115i \(-0.539999\pi\)
−0.125329 + 0.992115i \(0.539999\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 8.70156 1.74031
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 5.13688 0.922610 0.461305 − 0.887242i \(-0.347381\pi\)
0.461305 + 0.887242i \(0.347381\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −15.5737 −2.63243
\(36\) 0 0
\(37\) −3.70156 −0.608533 −0.304267 − 0.952587i \(-0.598411\pi\)
−0.304267 + 0.952587i \(0.598411\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.40312 1.46852 0.734261 − 0.678868i \(-0.237529\pi\)
0.734261 + 0.678868i \(0.237529\pi\)
\(42\) 0 0
\(43\) 5.29991 0.808229 0.404114 − 0.914708i \(-0.367580\pi\)
0.404114 + 0.914708i \(0.367580\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.20732 0.613701 0.306851 − 0.951758i \(-0.400725\pi\)
0.306851 + 0.951758i \(0.400725\pi\)
\(48\) 0 0
\(49\) 10.7016 1.52879
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.40312 −0.192734 −0.0963670 − 0.995346i \(-0.530722\pi\)
−0.0963670 + 0.995346i \(0.530722\pi\)
\(54\) 0 0
\(55\) −4.04429 −0.545332
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.5515 −1.76426 −0.882129 − 0.471008i \(-0.843890\pi\)
−0.882129 + 0.471008i \(0.843890\pi\)
\(60\) 0 0
\(61\) 9.40312 1.20395 0.601973 − 0.798516i \(-0.294381\pi\)
0.601973 + 0.798516i \(0.294381\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.70156 0.459122
\(66\) 0 0
\(67\) 11.3663 1.38862 0.694310 − 0.719676i \(-0.255710\pi\)
0.694310 + 0.719676i \(0.255710\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.25161 0.979286 0.489643 − 0.871923i \(-0.337127\pi\)
0.489643 + 0.871923i \(0.337127\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.59688 0.523863
\(78\) 0 0
\(79\) 14.6441 1.64759 0.823796 − 0.566887i \(-0.191852\pi\)
0.823796 + 0.566887i \(0.191852\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.32206 0.803700 0.401850 − 0.915706i \(-0.368367\pi\)
0.401850 + 0.915706i \(0.368367\pi\)
\(84\) 0 0
\(85\) −1.10469 −0.119820
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.20732 −0.441047
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.04429 −0.414935
\(96\) 0 0
\(97\) 8.80625 0.894139 0.447070 − 0.894499i \(-0.352468\pi\)
0.447070 + 0.894499i \(0.352468\pi\)
\(98\) 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 3744.2.a.be.1.3 4
3.2 odd 2 416.2.a.f.1.1 ✓ 4
4.3 odd 2 inner 3744.2.a.be.1.4 4
8.3 odd 2 7488.2.a.da.1.2 4
8.5 even 2 7488.2.a.da.1.1 4
12.11 even 2 416.2.a.f.1.4 yes 4
24.5 odd 2 832.2.a.p.1.4 4
24.11 even 2 832.2.a.p.1.1 4
39.38 odd 2 5408.2.a.bj.1.1 4
48.5 odd 4 3328.2.b.bb.1665.7 8
48.11 even 4 3328.2.b.bb.1665.1 8
48.29 odd 4 3328.2.b.bb.1665.2 8
48.35 even 4 3328.2.b.bb.1665.8 8
156.155 even 2 5408.2.a.bj.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.f.1.1 ✓ 4 3.2 odd 2
416.2.a.f.1.4 yes 4 12.11 even 2
832.2.a.p.1.1 4 24.11 even 2
832.2.a.p.1.4 4 24.5 odd 2
3328.2.b.bb.1665.1 8 48.11 even 4
3328.2.b.bb.1665.2 8 48.29 odd 4
3328.2.b.bb.1665.7 8 48.5 odd 4
3328.2.b.bb.1665.8 8 48.35 even 4
3744.2.a.be.1.3 4 1.1 even 1 trivial
3744.2.a.be.1.4 4 4.3 odd 2 inner
5408.2.a.bj.1.1 4 39.38 odd 2
5408.2.a.bj.1.4 4 156.155 even 2
7488.2.a.da.1.1 4 8.5 even 2
7488.2.a.da.1.2 4 8.3 odd 2