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 1.1
Root \(0.162962\) 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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} -26.8184 q^{7} +30.8629 q^{11} +32.8629 q^{13} -71.5442 q^{17} +49.3219 q^{19} +72.5035 q^{23} +25.0000 q^{25} -54.4628 q^{29} -146.736 q^{31} +134.092 q^{35} -65.6294 q^{37} -148.626 q^{41} +453.009 q^{43} -171.139 q^{47} +376.228 q^{49} +440.506 q^{53} -154.314 q^{55} -128.143 q^{59} +395.450 q^{61} -164.314 q^{65} +380.925 q^{67} +490.158 q^{71} -288.396 q^{73} -827.694 q^{77} +395.847 q^{79} +845.940 q^{83} +357.721 q^{85} +743.631 q^{89} -881.331 q^{91} -246.610 q^{95} -1840.93 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 6 q^{7} + 12 q^{11} + 18 q^{13} + 21 q^{17} - 57 q^{19} + 87 q^{23} + 75 q^{25} - 138 q^{29} - 117 q^{31} + 30 q^{35} + 150 q^{37} - 180 q^{43} + 684 q^{47} - 81 q^{49} - 87 q^{53} - 60 q^{55}+ \cdots - 1080 q^{97}+O(q^{100}) \) 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
\(7\) −26.8184 −1.44806 −0.724030 − 0.689769i \(-0.757712\pi\)
−0.724030 + 0.689769i \(0.757712\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 30.8629 0.845956 0.422978 − 0.906140i \(-0.360985\pi\)
0.422978 + 0.906140i \(0.360985\pi\)
\(12\) 0 0
\(13\) 32.8629 0.701117 0.350559 − 0.936541i \(-0.385992\pi\)
0.350559 + 0.936541i \(0.385992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −71.5442 −1.02071 −0.510354 − 0.859965i \(-0.670485\pi\)
−0.510354 + 0.859965i \(0.670485\pi\)
\(18\) 0 0
\(19\) 49.3219 0.595538 0.297769 − 0.954638i \(-0.403757\pi\)
0.297769 + 0.954638i \(0.403757\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 72.5035 0.657305 0.328653 − 0.944451i \(-0.393405\pi\)
0.328653 + 0.944451i \(0.393405\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −54.4628 −0.348741 −0.174370 − 0.984680i \(-0.555789\pi\)
−0.174370 + 0.984680i \(0.555789\pi\)
\(30\) 0 0
\(31\) −146.736 −0.850150 −0.425075 − 0.905158i \(-0.639752\pi\)
−0.425075 + 0.905158i \(0.639752\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 134.092 0.647592
\(36\) 0 0
\(37\) −65.6294 −0.291606 −0.145803 − 0.989314i \(-0.546576\pi\)
−0.145803 + 0.989314i \(0.546576\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −148.626 −0.566132 −0.283066 − 0.959100i \(-0.591352\pi\)
−0.283066 + 0.959100i \(0.591352\pi\)
\(42\) 0 0
\(43\) 453.009 1.60659 0.803294 − 0.595583i \(-0.203079\pi\)
0.803294 + 0.595583i \(0.203079\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −171.139 −0.531133 −0.265566 − 0.964093i \(-0.585559\pi\)
−0.265566 + 0.964093i \(0.585559\pi\)
\(48\) 0 0
\(49\) 376.228 1.09688
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 440.506 1.14166 0.570831 − 0.821067i \(-0.306621\pi\)
0.570831 + 0.821067i \(0.306621\pi\)
\(54\) 0 0
\(55\) −154.314 −0.378323
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −128.143 −0.282760 −0.141380 − 0.989955i \(-0.545154\pi\)
−0.141380 + 0.989955i \(0.545154\pi\)
\(60\) 0 0
\(61\) 395.450 0.830036 0.415018 − 0.909813i \(-0.363775\pi\)
0.415018 + 0.909813i \(0.363775\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −164.314 −0.313549
\(66\) 0 0
\(67\) 380.925 0.694587 0.347294 − 0.937756i \(-0.387101\pi\)
0.347294 + 0.937756i \(0.387101\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 490.158 0.819311 0.409655 − 0.912240i \(-0.365649\pi\)
0.409655 + 0.912240i \(0.365649\pi\)
\(72\) 0 0
\(73\) −288.396 −0.462386 −0.231193 − 0.972908i \(-0.574263\pi\)
−0.231193 + 0.972908i \(0.574263\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −827.694 −1.22499
\(78\) 0 0
\(79\) 395.847 0.563750 0.281875 − 0.959451i \(-0.409044\pi\)
0.281875 + 0.959451i \(0.409044\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 845.940 1.11872 0.559361 − 0.828924i \(-0.311046\pi\)
0.559361 + 0.828924i \(0.311046\pi\)
\(84\) 0 0
\(85\) 357.721 0.456474
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 743.631 0.885671 0.442835 − 0.896603i \(-0.353973\pi\)
0.442835 + 0.896603i \(0.353973\pi\)
\(90\) 0 0
\(91\) −881.331 −1.01526
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −246.610 −0.266333
\(96\) 0 0
\(97\) −1840.93 −1.92699 −0.963494 − 0.267728i \(-0.913727\pi\)
−0.963494 + 0.267728i \(0.913727\pi\)
\(98\) 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.1.1 3
3.2 odd 2 2160.4.a.bo.1.1 3
4.3 odd 2 1080.4.a.g.1.3 ✓ 3
12.11 even 2 1080.4.a.m.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.g.1.3 ✓ 3 4.3 odd 2
1080.4.a.m.1.3 yes 3 12.11 even 2
2160.4.a.bg.1.1 3 1.1 even 1 trivial
2160.4.a.bo.1.1 3 3.2 odd 2