↑

Properties

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

Related objects

Downloads

Learn more

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 = [4,0,0,0,0,0,0,0,0,0,8,0,-4,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0] 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: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) 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)
 
\(f(q)\) \(=\) \(q+1.03528 q^{5} +3.86370 q^{7} -1.46410 q^{11} -1.00000 q^{13} +5.65685 q^{17} -1.79315 q^{19} +6.92820 q^{23} -3.92820 q^{25} +2.07055 q^{29} +3.86370 q^{31} +4.00000 q^{35} -2.00000 q^{37} -1.03528 q^{41} -5.65685 q^{43} +2.53590 q^{47} +7.92820 q^{49} +5.65685 q^{53} -1.51575 q^{55} +9.46410 q^{59} -8.92820 q^{61} -1.03528 q^{65} +1.79315 q^{67} +2.53590 q^{71} +11.8564 q^{73} -5.65685 q^{77} +6.53590 q^{83} +5.85641 q^{85} -14.4195 q^{89} -3.86370 q^{91} -1.85641 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{11} - 4 q^{13} + 12 q^{25} + 16 q^{35} - 8 q^{37} + 24 q^{47} + 4 q^{49} + 24 q^{59} - 8 q^{61} + 24 q^{71} - 8 q^{73} + 40 q^{83} - 32 q^{85} + 48 q^{95} + 24 q^{97}+O(q^{100}) \) 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
\(5\) 1.03528 0.462990 0.231495 − 0.972836i \(-0.425638\pi\)
0.231495 + 0.972836i \(0.425638\pi\)
\(6\) 0 0
\(7\) 3.86370 1.46034 0.730171 − 0.683264i \(-0.239440\pi\)
0.730171 + 0.683264i \(0.239440\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.46410 −0.441443 −0.220722 − 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 − 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) −1.79315 −0.411377 −0.205689 − 0.978618i \(-0.565943\pi\)
−0.205689 + 0.978618i \(0.565943\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 − 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) −3.92820 −0.785641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.07055 0.384492 0.192246 − 0.981347i \(-0.438423\pi\)
0.192246 + 0.981347i \(0.438423\pi\)
\(30\) 0 0
\(31\) 3.86370 0.693942 0.346971 − 0.937876i \(-0.387210\pi\)
0.346971 + 0.937876i \(0.387210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.03528 −0.161683 −0.0808415 − 0.996727i \(-0.525761\pi\)
−0.0808415 + 0.996727i \(0.525761\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.53590 0.369899 0.184949 − 0.982748i \(-0.440788\pi\)
0.184949 + 0.982748i \(0.440788\pi\)
\(48\) 0 0
\(49\) 7.92820 1.13260
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 − 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) −1.51575 −0.204384
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.46410 1.23212 0.616061 − 0.787699i \(-0.288728\pi\)
0.616061 + 0.787699i \(0.288728\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 − 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.03528 −0.128410
\(66\) 0 0
\(67\) 1.79315 0.219068 0.109534 − 0.993983i \(-0.465064\pi\)
0.109534 + 0.993983i \(0.465064\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) 11.8564 1.38769 0.693844 − 0.720126i \(-0.255916\pi\)
0.693844 + 0.720126i \(0.255916\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.53590 0.717408 0.358704 − 0.933451i \(-0.383219\pi\)
0.358704 + 0.933451i \(0.383219\pi\)
\(84\) 0 0
\(85\) 5.85641 0.635216
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.4195 −1.52847 −0.764234 − 0.644939i \(-0.776883\pi\)
−0.764234 + 0.644939i \(0.776883\pi\)
\(90\) 0 0
\(91\) −3.86370 −0.405026
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.85641 −0.190463
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 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.bd.1.3 yes 4
3.2 odd 2 3744.2.a.bc.1.2 ✓ 4
4.3 odd 2 3744.2.a.bc.1.3 yes 4
8.3 odd 2 7488.2.a.dc.1.2 4
8.5 even 2 7488.2.a.db.1.2 4
12.11 even 2 inner 3744.2.a.bd.1.2 yes 4
24.5 odd 2 7488.2.a.dc.1.3 4
24.11 even 2 7488.2.a.db.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3744.2.a.bc.1.2 ✓ 4 3.2 odd 2
3744.2.a.bc.1.3 yes 4 4.3 odd 2
3744.2.a.bd.1.2 yes 4 12.11 even 2 inner
3744.2.a.bd.1.3 yes 4 1.1 even 1 trivial
7488.2.a.db.1.2 4 8.5 even 2
7488.2.a.db.1.3 4 24.11 even 2
7488.2.a.dc.1.2 4 8.3 odd 2
7488.2.a.dc.1.3 4 24.5 odd 2