Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1044,4,Mod(1,1044)] 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("1044.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1044, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1044 = 2^{2} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1044.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.5979940460\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148344.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 70x + 238 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.21773\) of defining polynomial
Character \(\chi\) \(=\) 1044.1

$q$-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)
 
\(f(q)\) \(=\) \(q+20.3399 q^{5} +22.2443 q^{7} +10.3330 q^{11} +62.7753 q^{13} -70.8570 q^{17} -13.5507 q^{19} +168.758 q^{23} +288.711 q^{25} -29.0000 q^{29} +116.591 q^{31} +452.447 q^{35} +270.376 q^{37} -307.481 q^{41} -377.951 q^{43} -65.4127 q^{47} +151.810 q^{49} +28.0053 q^{53} +210.171 q^{55} -766.772 q^{59} -565.048 q^{61} +1276.84 q^{65} -220.567 q^{67} -1039.13 q^{71} +815.062 q^{73} +229.850 q^{77} +716.722 q^{79} +785.053 q^{83} -1441.22 q^{85} -802.772 q^{89} +1396.39 q^{91} -275.619 q^{95} +1043.98 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 20 q^{5} + 8 q^{7} - 86 q^{11} + 124 q^{13} - 14 q^{17} + 88 q^{19} - 68 q^{23} + 111 q^{25} - 87 q^{29} + 326 q^{31} + 784 q^{35} + 166 q^{37} - 34 q^{41} - 946 q^{43} - 234 q^{47} + 1067 q^{49}+ \cdots - 406 q^{97}+O(q^{100}) \) 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
\(5\) 20.3399 1.81925 0.909627 − 0.415425i \(-0.136367\pi\)
0.909627 + 0.415425i \(0.136367\pi\)
\(6\) 0 0
\(7\) 22.2443 1.20108 0.600540 − 0.799595i \(-0.294952\pi\)
0.600540 + 0.799595i \(0.294952\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.3330 0.283227 0.141614 − 0.989922i \(-0.454771\pi\)
0.141614 + 0.989922i \(0.454771\pi\)
\(12\) 0 0
\(13\) 62.7753 1.33929 0.669644 − 0.742682i \(-0.266447\pi\)
0.669644 + 0.742682i \(0.266447\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −70.8570 −1.01090 −0.505452 − 0.862855i \(-0.668674\pi\)
−0.505452 + 0.862855i \(0.668674\pi\)
\(18\) 0 0
\(19\) −13.5507 −0.163618 −0.0818089 − 0.996648i \(-0.526070\pi\)
−0.0818089 + 0.996648i \(0.526070\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 168.758 1.52994 0.764969 − 0.644067i \(-0.222754\pi\)
0.764969 + 0.644067i \(0.222754\pi\)
\(24\) 0 0
\(25\) 288.711 2.30969
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 116.591 0.675496 0.337748 − 0.941237i \(-0.390335\pi\)
0.337748 + 0.941237i \(0.390335\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 452.447 2.18507
\(36\) 0 0
\(37\) 270.376 1.20134 0.600670 − 0.799497i \(-0.294901\pi\)
0.600670 + 0.799497i \(0.294901\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −307.481 −1.17123 −0.585616 − 0.810589i \(-0.699147\pi\)
−0.585616 + 0.810589i \(0.699147\pi\)
\(42\) 0 0
\(43\) −377.951 −1.34039 −0.670197 − 0.742183i \(-0.733790\pi\)
−0.670197 + 0.742183i \(0.733790\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −65.4127 −0.203009 −0.101504 − 0.994835i \(-0.532366\pi\)
−0.101504 + 0.994835i \(0.532366\pi\)
\(48\) 0 0
\(49\) 151.810 0.442595
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 28.0053 0.0725817 0.0362908 − 0.999341i \(-0.488446\pi\)
0.0362908 + 0.999341i \(0.488446\pi\)
\(54\) 0 0
\(55\) 210.171 0.515263
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −766.772 −1.69195 −0.845976 − 0.533221i \(-0.820981\pi\)
−0.845976 + 0.533221i \(0.820981\pi\)
\(60\) 0 0
\(61\) −565.048 −1.18602 −0.593008 − 0.805197i \(-0.702060\pi\)
−0.593008 + 0.805197i \(0.702060\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1276.84 2.43651
\(66\) 0 0
\(67\) −220.567 −0.402188 −0.201094 − 0.979572i \(-0.564450\pi\)
−0.201094 + 0.979572i \(0.564450\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1039.13 −1.73693 −0.868467 − 0.495747i \(-0.834894\pi\)
−0.868467 + 0.495747i \(0.834894\pi\)
\(72\) 0 0
\(73\) 815.062 1.30679 0.653396 − 0.757016i \(-0.273344\pi\)
0.653396 + 0.757016i \(0.273344\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 229.850 0.340179
\(78\) 0 0
\(79\) 716.722 1.02073 0.510364 − 0.859958i \(-0.329511\pi\)
0.510364 + 0.859958i \(0.329511\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 785.053 1.03820 0.519101 − 0.854713i \(-0.326267\pi\)
0.519101 + 0.854713i \(0.326267\pi\)
\(84\) 0 0
\(85\) −1441.22 −1.83909
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −802.772 −0.956109 −0.478054 − 0.878330i \(-0.658658\pi\)
−0.478054 + 0.878330i \(0.658658\pi\)
\(90\) 0 0
\(91\) 1396.39 1.60859
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −275.619 −0.297663
\(96\) 0 0
\(97\) 1043.98 1.09279 0.546393 − 0.837529i \(-0.316001\pi\)
0.546393 + 0.837529i \(0.316001\pi\)
\(98\) 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 1044.4.a.f.1.3 3
3.2 odd 2 116.4.a.c.1.2 ✓ 3
12.11 even 2 464.4.a.j.1.2 3
24.5 odd 2 1856.4.a.v.1.2 3
24.11 even 2 1856.4.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
116.4.a.c.1.2 ✓ 3 3.2 odd 2
464.4.a.j.1.2 3 12.11 even 2
1044.4.a.f.1.3 3 1.1 even 1 trivial
1856.4.a.o.1.2 3 24.11 even 2
1856.4.a.v.1.2 3 24.5 odd 2