Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2160,4,Mod(1,2160)] 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-15,0,-6,0,0,0,12] 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: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.985.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.93080\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +7.95278 q^{7} -37.1224 q^{11} -35.1224 q^{13} +99.1976 q^{17} +44.6505 q^{19} +102.603 q^{23} +25.0000 q^{25} -285.404 q^{29} -238.593 q^{31} -39.7639 q^{35} +339.168 q^{37} +423.799 q^{41} -144.330 q^{43} +418.414 q^{47} -279.753 q^{49} -186.933 q^{53} +185.612 q^{55} +293.783 q^{59} -701.657 q^{61} +175.612 q^{65} +292.697 q^{67} +738.742 q^{71} +453.214 q^{73} -295.226 q^{77} -892.759 q^{79} -66.4113 q^{83} -495.988 q^{85} -868.816 q^{89} -279.321 q^{91} -223.253 q^{95} +112.488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 6 q^{7} + 12 q^{11} + 18 q^{13} + 21 q^{17} - 57 q^{19} + 87 q^{23} + 75 q^{25} - 138 q^{29} - 117 q^{31} + 30 q^{35} + 150 q^{37} - 180 q^{43} + 684 q^{47} - 81 q^{49} - 87 q^{53} - 60 q^{55}+ \cdots - 1080 q^{97}+O(q^{100}) \) 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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 7.95278 0.429410 0.214705 − 0.976679i \(-0.431121\pi\)
0.214705 + 0.976679i \(0.431121\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −37.1224 −1.01753 −0.508765 − 0.860906i \(-0.669898\pi\)
−0.508765 + 0.860906i \(0.669898\pi\)
\(12\) 0 0
\(13\) −35.1224 −0.749323 −0.374662 − 0.927162i \(-0.622241\pi\)
−0.374662 + 0.927162i \(0.622241\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 99.1976 1.41523 0.707616 − 0.706597i \(-0.249771\pi\)
0.707616 + 0.706597i \(0.249771\pi\)
\(18\) 0 0
\(19\) 44.6505 0.539133 0.269567 − 0.962982i \(-0.413120\pi\)
0.269567 + 0.962982i \(0.413120\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 102.603 0.930186 0.465093 − 0.885262i \(-0.346021\pi\)
0.465093 + 0.885262i \(0.346021\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −285.404 −1.82752 −0.913762 − 0.406249i \(-0.866837\pi\)
−0.913762 + 0.406249i \(0.866837\pi\)
\(30\) 0 0
\(31\) −238.593 −1.38234 −0.691171 − 0.722692i \(-0.742905\pi\)
−0.691171 + 0.722692i \(0.742905\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −39.7639 −0.192038
\(36\) 0 0
\(37\) 339.168 1.50700 0.753498 − 0.657450i \(-0.228365\pi\)
0.753498 + 0.657450i \(0.228365\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 423.799 1.61430 0.807150 − 0.590346i \(-0.201009\pi\)
0.807150 + 0.590346i \(0.201009\pi\)
\(42\) 0 0
\(43\) −144.330 −0.511863 −0.255931 − 0.966695i \(-0.582382\pi\)
−0.255931 + 0.966695i \(0.582382\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 418.414 1.29855 0.649276 − 0.760553i \(-0.275072\pi\)
0.649276 + 0.760553i \(0.275072\pi\)
\(48\) 0 0
\(49\) −279.753 −0.815607
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −186.933 −0.484476 −0.242238 − 0.970217i \(-0.577882\pi\)
−0.242238 + 0.970217i \(0.577882\pi\)
\(54\) 0 0
\(55\) 185.612 0.455053
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 293.783 0.648259 0.324129 − 0.946013i \(-0.394929\pi\)
0.324129 + 0.946013i \(0.394929\pi\)
\(60\) 0 0
\(61\) −701.657 −1.47275 −0.736377 − 0.676571i \(-0.763465\pi\)
−0.736377 + 0.676571i \(0.763465\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 175.612 0.335107
\(66\) 0 0
\(67\) 292.697 0.533710 0.266855 − 0.963737i \(-0.414016\pi\)
0.266855 + 0.963737i \(0.414016\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 738.742 1.23482 0.617412 − 0.786640i \(-0.288181\pi\)
0.617412 + 0.786640i \(0.288181\pi\)
\(72\) 0 0
\(73\) 453.214 0.726639 0.363320 − 0.931665i \(-0.381643\pi\)
0.363320 + 0.931665i \(0.381643\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −295.226 −0.436937
\(78\) 0 0
\(79\) −892.759 −1.27143 −0.635717 − 0.771922i \(-0.719295\pi\)
−0.635717 + 0.771922i \(0.719295\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −66.4113 −0.0878264 −0.0439132 − 0.999035i \(-0.513982\pi\)
−0.0439132 + 0.999035i \(0.513982\pi\)
\(84\) 0 0
\(85\) −495.988 −0.632911
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −868.816 −1.03477 −0.517384 − 0.855753i \(-0.673094\pi\)
−0.517384 + 0.855753i \(0.673094\pi\)
\(90\) 0 0
\(91\) −279.321 −0.321767
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −223.253 −0.241108
\(96\) 0 0
\(97\) 112.488 0.117747 0.0588735 − 0.998265i \(-0.481249\pi\)
0.0588735 + 0.998265i \(0.481249\pi\)
\(98\) 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 2160.4.a.bg.1.2 3
3.2 odd 2 2160.4.a.bo.1.2 3
4.3 odd 2 1080.4.a.g.1.2 ✓ 3
12.11 even 2 1080.4.a.m.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.g.1.2 ✓ 3 4.3 odd 2
1080.4.a.m.1.2 yes 3 12.11 even 2
2160.4.a.bg.1.2 3 1.1 even 1 trivial
2160.4.a.bo.1.2 3 3.2 odd 2