Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1760,2,Mod(1,1760)] 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("1760.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1760.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,5,0,2,0,11,0,5] 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: \(14.0536707557\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.792644.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 4x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.34129\) of defining polynomial
Character \(\chi\) \(=\) 1760.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.18777 q^{3} +1.00000 q^{5} -2.49480 q^{7} +1.78634 q^{9} +1.00000 q^{11} +3.56183 q^{13} +2.18777 q^{15} +5.89623 q^{17} +0.786343 q^{19} -5.45806 q^{21} +7.64583 q^{23} +1.00000 q^{25} -2.65520 q^{27} -5.03075 q^{29} -9.45806 q^{31} +2.18777 q^{33} -2.49480 q^{35} -3.16189 q^{37} +7.79247 q^{39} +11.3652 q^{41} +7.65669 q^{43} +1.78634 q^{45} +4.84298 q^{47} -0.775951 q^{49} +12.8996 q^{51} -3.16189 q^{53} +1.00000 q^{55} +1.72034 q^{57} +8.98961 q^{59} +9.03075 q^{61} -4.45658 q^{63} +3.56183 q^{65} -6.09486 q^{67} +16.7273 q^{69} -10.4167 q^{71} +12.9270 q^{73} +2.18777 q^{75} -2.49480 q^{77} -4.37554 q^{79} -11.1680 q^{81} -6.40480 q^{83} +5.89623 q^{85} -11.0061 q^{87} +13.9647 q^{89} -8.88606 q^{91} -20.6921 q^{93} +0.786343 q^{95} -5.79247 q^{97} +1.78634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} + 2 q^{7} + 11 q^{9} + 5 q^{11} + 6 q^{13} + 2 q^{17} + 6 q^{19} + 12 q^{21} - 12 q^{23} + 5 q^{25} + 10 q^{29} - 8 q^{31} + 2 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 4 q^{43} + 11 q^{45}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
\(3\) 2.18777 1.26311 0.631555 − 0.775331i \(-0.282417\pi\)
0.631555 + 0.775331i \(0.282417\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.49480 −0.942947 −0.471474 − 0.881880i \(-0.656278\pi\)
−0.471474 + 0.881880i \(0.656278\pi\)
\(8\) 0 0
\(9\) 1.78634 0.595448
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 3.56183 0.987873 0.493937 − 0.869498i \(-0.335557\pi\)
0.493937 + 0.869498i \(0.335557\pi\)
\(14\) 0 0
\(15\) 2.18777 0.564880
\(16\) 0 0
\(17\) 5.89623 1.43005 0.715023 − 0.699101i \(-0.246416\pi\)
0.715023 + 0.699101i \(0.246416\pi\)
\(18\) 0 0
\(19\) 0.786343 0.180399 0.0901997 − 0.995924i \(-0.471249\pi\)
0.0901997 + 0.995924i \(0.471249\pi\)
\(20\) 0 0
\(21\) −5.45806 −1.19105
\(22\) 0 0
\(23\) 7.64583 1.59427 0.797133 − 0.603804i \(-0.206349\pi\)
0.797133 + 0.603804i \(0.206349\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.65520 −0.510994
\(28\) 0 0
\(29\) −5.03075 −0.934186 −0.467093 − 0.884208i \(-0.654699\pi\)
−0.467093 + 0.884208i \(0.654699\pi\)
\(30\) 0 0
\(31\) −9.45806 −1.69872 −0.849359 − 0.527816i \(-0.823011\pi\)
−0.849359 + 0.527816i \(0.823011\pi\)
\(32\) 0 0
\(33\) 2.18777 0.380842
\(34\) 0 0
\(35\) −2.49480 −0.421699
\(36\) 0 0
\(37\) −3.16189 −0.519811 −0.259905 − 0.965634i \(-0.583691\pi\)
−0.259905 + 0.965634i \(0.583691\pi\)
\(38\) 0 0
\(39\) 7.79247 1.24779
\(40\) 0 0
\(41\) 11.3652 1.77494 0.887469 − 0.460867i \(-0.152462\pi\)
0.887469 + 0.460867i \(0.152462\pi\)
\(42\) 0 0
\(43\) 7.65669 1.16763 0.583817 − 0.811885i \(-0.301559\pi\)
0.583817 + 0.811885i \(0.301559\pi\)
\(44\) 0 0
\(45\) 1.78634 0.266292
\(46\) 0 0
\(47\) 4.84298 0.706421 0.353210 − 0.935544i \(-0.385090\pi\)
0.353210 + 0.935544i \(0.385090\pi\)
\(48\) 0 0
\(49\) −0.775951 −0.110850
\(50\) 0 0
\(51\) 12.8996 1.80631
\(52\) 0 0
\(53\) −3.16189 −0.434318 −0.217159 − 0.976136i \(-0.569679\pi\)
−0.217159 + 0.976136i \(0.569679\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 1.72034 0.227864
\(58\) 0 0
\(59\) 8.98961 1.17035 0.585174 − 0.810908i \(-0.301026\pi\)
0.585174 + 0.810908i \(0.301026\pi\)
\(60\) 0 0
\(61\) 9.03075 1.15627 0.578134 − 0.815942i \(-0.303781\pi\)
0.578134 + 0.815942i \(0.303781\pi\)
\(62\) 0 0
\(63\) −4.45658 −0.561476
\(64\) 0 0
\(65\) 3.56183 0.441790
\(66\) 0 0
\(67\) −6.09486 −0.744606 −0.372303 − 0.928111i \(-0.621432\pi\)
−0.372303 + 0.928111i \(0.621432\pi\)
\(68\) 0 0
\(69\) 16.7273 2.01373
\(70\) 0 0
\(71\) −10.4167 −1.23623 −0.618116 − 0.786087i \(-0.712104\pi\)
−0.618116 + 0.786087i \(0.712104\pi\)
\(72\) 0 0
\(73\) 12.9270 1.51299 0.756494 − 0.654000i \(-0.226911\pi\)
0.756494 + 0.654000i \(0.226911\pi\)
\(74\) 0 0
\(75\) 2.18777 0.252622
\(76\) 0 0
\(77\) −2.49480 −0.284309
\(78\) 0 0
\(79\) −4.37554 −0.492287 −0.246143 − 0.969233i \(-0.579163\pi\)
−0.246143 + 0.969233i \(0.579163\pi\)
\(80\) 0 0
\(81\) −11.1680 −1.24089
\(82\) 0 0
\(83\) −6.40480 −0.703019 −0.351509 − 0.936184i \(-0.614331\pi\)
−0.351509 + 0.936184i \(0.614331\pi\)
\(84\) 0 0
\(85\) 5.89623 0.639536
\(86\) 0 0
\(87\) −11.0061 −1.17998
\(88\) 0 0
\(89\) 13.9647 1.48026 0.740130 − 0.672464i \(-0.234764\pi\)
0.740130 + 0.672464i \(0.234764\pi\)
\(90\) 0 0
\(91\) −8.88606 −0.931513
\(92\) 0 0
\(93\) −20.6921 −2.14567
\(94\) 0 0
\(95\) 0.786343 0.0806771
\(96\) 0 0
\(97\) −5.79247 −0.588136 −0.294068 − 0.955785i \(-0.595009\pi\)
−0.294068 + 0.955785i \(0.595009\pi\)
\(98\) 0 0
\(99\) 1.78634 0.179534
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 1760.2.a.v.1.4 yes 5
4.3 odd 2 1760.2.a.u.1.2 ✓ 5
5.4 even 2 8800.2.a.bu.1.2 5
8.3 odd 2 3520.2.a.by.1.4 5
8.5 even 2 3520.2.a.bz.1.2 5
20.19 odd 2 8800.2.a.bv.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1760.2.a.u.1.2 ✓ 5 4.3 odd 2
1760.2.a.v.1.4 yes 5 1.1 even 1 trivial
3520.2.a.by.1.4 5 8.3 odd 2
3520.2.a.bz.1.2 5 8.5 even 2
8800.2.a.bu.1.2 5 5.4 even 2
8800.2.a.bv.1.4 5 20.19 odd 2