Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(1,1776)] 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("1776.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,2,0,5,0,3,0,3] 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: \(14.1814313990\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 1776.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.52444 q^{5} -2.91638 q^{7} +1.00000 q^{9} +1.28917 q^{11} +3.28917 q^{13} +2.52444 q^{15} +1.23527 q^{17} -1.28917 q^{19} -2.91638 q^{21} +4.01916 q^{23} +1.37279 q^{25} +1.00000 q^{27} +1.47556 q^{29} +2.57834 q^{31} +1.28917 q^{33} -7.36222 q^{35} -1.00000 q^{37} +3.28917 q^{39} +10.4111 q^{41} -4.00000 q^{43} +2.52444 q^{45} +5.83276 q^{47} +1.50528 q^{49} +1.23527 q^{51} -2.54359 q^{53} +3.25443 q^{55} -1.28917 q^{57} +2.05390 q^{59} +14.4111 q^{61} -2.91638 q^{63} +8.30330 q^{65} -13.2544 q^{67} +4.01916 q^{69} -3.25443 q^{71} +3.86751 q^{73} +1.37279 q^{75} -3.75971 q^{77} +4.78389 q^{79} +1.00000 q^{81} +4.54359 q^{83} +3.11836 q^{85} +1.47556 q^{87} -6.76473 q^{89} -9.59247 q^{91} +2.57834 q^{93} -3.25443 q^{95} +11.0489 q^{97} +1.28917 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{5} + 5 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{13} + 2 q^{15} - q^{17} - 3 q^{19} + 5 q^{21} - 9 q^{23} + 17 q^{25} + 3 q^{27} + 10 q^{29} + 6 q^{31} + 3 q^{33} - 4 q^{35} - 3 q^{37} + 9 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.52444 1.12896 0.564481 − 0.825446i \(-0.309076\pi\)
0.564481 + 0.825446i \(0.309076\pi\)
\(6\) 0 0
\(7\) −2.91638 −1.10229 −0.551144 − 0.834410i \(-0.685809\pi\)
−0.551144 + 0.834410i \(0.685809\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.28917 0.388699 0.194349 − 0.980932i \(-0.437740\pi\)
0.194349 + 0.980932i \(0.437740\pi\)
\(12\) 0 0
\(13\) 3.28917 0.912251 0.456126 − 0.889915i \(-0.349237\pi\)
0.456126 + 0.889915i \(0.349237\pi\)
\(14\) 0 0
\(15\) 2.52444 0.651807
\(16\) 0 0
\(17\) 1.23527 0.299597 0.149798 − 0.988717i \(-0.452138\pi\)
0.149798 + 0.988717i \(0.452138\pi\)
\(18\) 0 0
\(19\) −1.28917 −0.295756 −0.147878 − 0.989006i \(-0.547244\pi\)
−0.147878 + 0.989006i \(0.547244\pi\)
\(20\) 0 0
\(21\) −2.91638 −0.636407
\(22\) 0 0
\(23\) 4.01916 0.838052 0.419026 − 0.907974i \(-0.362372\pi\)
0.419026 + 0.907974i \(0.362372\pi\)
\(24\) 0 0
\(25\) 1.37279 0.274557
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.47556 0.274005 0.137002 − 0.990571i \(-0.456253\pi\)
0.137002 + 0.990571i \(0.456253\pi\)
\(30\) 0 0
\(31\) 2.57834 0.463083 0.231542 − 0.972825i \(-0.425623\pi\)
0.231542 + 0.972825i \(0.425623\pi\)
\(32\) 0 0
\(33\) 1.28917 0.224415
\(34\) 0 0
\(35\) −7.36222 −1.24444
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 3.28917 0.526688
\(40\) 0 0
\(41\) 10.4111 1.62594 0.812970 − 0.582305i \(-0.197849\pi\)
0.812970 + 0.582305i \(0.197849\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.52444 0.376321
\(46\) 0 0
\(47\) 5.83276 0.850796 0.425398 − 0.905006i \(-0.360134\pi\)
0.425398 + 0.905006i \(0.360134\pi\)
\(48\) 0 0
\(49\) 1.50528 0.215040
\(50\) 0 0
\(51\) 1.23527 0.172972
\(52\) 0 0
\(53\) −2.54359 −0.349390 −0.174695 − 0.984623i \(-0.555894\pi\)
−0.174695 + 0.984623i \(0.555894\pi\)
\(54\) 0 0
\(55\) 3.25443 0.438827
\(56\) 0 0
\(57\) −1.28917 −0.170755
\(58\) 0 0
\(59\) 2.05390 0.267395 0.133697 − 0.991022i \(-0.457315\pi\)
0.133697 + 0.991022i \(0.457315\pi\)
\(60\) 0 0
\(61\) 14.4111 1.84515 0.922576 − 0.385815i \(-0.126080\pi\)
0.922576 + 0.385815i \(0.126080\pi\)
\(62\) 0 0
\(63\) −2.91638 −0.367430
\(64\) 0 0
\(65\) 8.30330 1.02990
\(66\) 0 0
\(67\) −13.2544 −1.61929 −0.809643 − 0.586923i \(-0.800339\pi\)
−0.809643 + 0.586923i \(0.800339\pi\)
\(68\) 0 0
\(69\) 4.01916 0.483850
\(70\) 0 0
\(71\) −3.25443 −0.386229 −0.193115 − 0.981176i \(-0.561859\pi\)
−0.193115 + 0.981176i \(0.561859\pi\)
\(72\) 0 0
\(73\) 3.86751 0.452657 0.226329 − 0.974051i \(-0.427328\pi\)
0.226329 + 0.974051i \(0.427328\pi\)
\(74\) 0 0
\(75\) 1.37279 0.158516
\(76\) 0 0
\(77\) −3.75971 −0.428458
\(78\) 0 0
\(79\) 4.78389 0.538229 0.269115 − 0.963108i \(-0.413269\pi\)
0.269115 + 0.963108i \(0.413269\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.54359 0.498724 0.249362 − 0.968410i \(-0.419779\pi\)
0.249362 + 0.968410i \(0.419779\pi\)
\(84\) 0 0
\(85\) 3.11836 0.338234
\(86\) 0 0
\(87\) 1.47556 0.158197
\(88\) 0 0
\(89\) −6.76473 −0.717060 −0.358530 − 0.933518i \(-0.616722\pi\)
−0.358530 + 0.933518i \(0.616722\pi\)
\(90\) 0 0
\(91\) −9.59247 −1.00556
\(92\) 0 0
\(93\) 2.57834 0.267361
\(94\) 0 0
\(95\) −3.25443 −0.333897
\(96\) 0 0
\(97\) 11.0489 1.12184 0.560922 − 0.827869i \(-0.310447\pi\)
0.560922 + 0.827869i \(0.310447\pi\)
\(98\) 0 0
\(99\) 1.28917 0.129566
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 1776.2.a.s.1.2 3
3.2 odd 2 5328.2.a.bl.1.2 3
4.3 odd 2 888.2.a.i.1.2 ✓ 3
8.3 odd 2 7104.2.a.bw.1.2 3
8.5 even 2 7104.2.a.bq.1.2 3
12.11 even 2 2664.2.a.m.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.i.1.2 ✓ 3 4.3 odd 2
1776.2.a.s.1.2 3 1.1 even 1 trivial
2664.2.a.m.1.2 3 12.11 even 2
5328.2.a.bl.1.2 3 3.2 odd 2
7104.2.a.bq.1.2 3 8.5 even 2
7104.2.a.bw.1.2 3 8.3 odd 2