Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7360,2,Mod(1,7360)] 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("7360.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,5,0,-2,0,13,0,1] 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: \(58.7698958877\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.30649\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.30649 q^{3} +1.00000 q^{5} -2.55040 q^{7} +7.93288 q^{9} +2.72314 q^{11} -7.12637 q^{13} -3.30649 q^{15} +0.924010 q^{17} -7.51623 q^{19} +8.43286 q^{21} -1.00000 q^{23} +1.00000 q^{25} -16.3105 q^{27} +2.38248 q^{29} +0.866248 q^{31} -9.00402 q^{33} -2.55040 q^{35} -0.352855 q^{37} +23.5633 q^{39} +4.34066 q^{41} +7.93288 q^{45} -13.3239 q^{47} -0.495474 q^{49} -3.05523 q^{51} -3.99262 q^{53} +2.72314 q^{55} +24.8523 q^{57} +3.84064 q^{59} +9.14262 q^{61} -20.2320 q^{63} -7.12637 q^{65} +3.15933 q^{67} +3.30649 q^{69} -6.07883 q^{71} -11.3239 q^{73} -3.30649 q^{75} -6.94508 q^{77} -12.0593 q^{79} +30.1319 q^{81} +6.35285 q^{83} +0.924010 q^{85} -7.87765 q^{87} -9.71377 q^{89} +18.1751 q^{91} -2.86424 q^{93} -7.51623 q^{95} +8.76465 q^{97} +21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} + 4 q^{17} - 7 q^{19} - 6 q^{21} - 5 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} + 19 q^{31} + 17 q^{33} - 2 q^{35} - 15 q^{37} + 19 q^{39} + 25 q^{41}+ \cdots + 65 q^{99}+O(q^{100}) \) 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
\(3\) −3.30649 −1.90900 −0.954501 − 0.298206i \(-0.903612\pi\)
−0.954501 + 0.298206i \(0.903612\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.55040 −0.963960 −0.481980 − 0.876182i \(-0.660082\pi\)
−0.481980 + 0.876182i \(0.660082\pi\)
\(8\) 0 0
\(9\) 7.93288 2.64429
\(10\) 0 0
\(11\) 2.72314 0.821056 0.410528 − 0.911848i \(-0.365344\pi\)
0.410528 + 0.911848i \(0.365344\pi\)
\(12\) 0 0
\(13\) −7.12637 −1.97650 −0.988250 − 0.152845i \(-0.951156\pi\)
−0.988250 + 0.152845i \(0.951156\pi\)
\(14\) 0 0
\(15\) −3.30649 −0.853732
\(16\) 0 0
\(17\) 0.924010 0.224105 0.112053 − 0.993702i \(-0.464257\pi\)
0.112053 + 0.993702i \(0.464257\pi\)
\(18\) 0 0
\(19\) −7.51623 −1.72434 −0.862171 − 0.506617i \(-0.830896\pi\)
−0.862171 + 0.506617i \(0.830896\pi\)
\(20\) 0 0
\(21\) 8.43286 1.84020
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −16.3105 −3.13896
\(28\) 0 0
\(29\) 2.38248 0.442415 0.221208 − 0.975227i \(-0.429000\pi\)
0.221208 + 0.975227i \(0.429000\pi\)
\(30\) 0 0
\(31\) 0.866248 0.155583 0.0777913 − 0.996970i \(-0.475213\pi\)
0.0777913 + 0.996970i \(0.475213\pi\)
\(32\) 0 0
\(33\) −9.00402 −1.56740
\(34\) 0 0
\(35\) −2.55040 −0.431096
\(36\) 0 0
\(37\) −0.352855 −0.0580089 −0.0290045 − 0.999579i \(-0.509234\pi\)
−0.0290045 + 0.999579i \(0.509234\pi\)
\(38\) 0 0
\(39\) 23.5633 3.77315
\(40\) 0 0
\(41\) 4.34066 0.677896 0.338948 − 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 7.93288 1.18256
\(46\) 0 0
\(47\) −13.3239 −1.94349 −0.971746 − 0.236027i \(-0.924155\pi\)
−0.971746 + 0.236027i \(0.924155\pi\)
\(48\) 0 0
\(49\) −0.495474 −0.0707819
\(50\) 0 0
\(51\) −3.05523 −0.427818
\(52\) 0 0
\(53\) −3.99262 −0.548429 −0.274214 − 0.961669i \(-0.588418\pi\)
−0.274214 + 0.961669i \(0.588418\pi\)
\(54\) 0 0
\(55\) 2.72314 0.367187
\(56\) 0 0
\(57\) 24.8523 3.29177
\(58\) 0 0
\(59\) 3.84064 0.500009 0.250004 − 0.968245i \(-0.419568\pi\)
0.250004 + 0.968245i \(0.419568\pi\)
\(60\) 0 0
\(61\) 9.14262 1.17059 0.585296 − 0.810820i \(-0.300978\pi\)
0.585296 + 0.810820i \(0.300978\pi\)
\(62\) 0 0
\(63\) −20.2320 −2.54899
\(64\) 0 0
\(65\) −7.12637 −0.883918
\(66\) 0 0
\(67\) 3.15933 0.385974 0.192987 − 0.981201i \(-0.438183\pi\)
0.192987 + 0.981201i \(0.438183\pi\)
\(68\) 0 0
\(69\) 3.30649 0.398055
\(70\) 0 0
\(71\) −6.07883 −0.721424 −0.360712 − 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(72\) 0 0
\(73\) −11.3239 −1.32536 −0.662682 − 0.748901i \(-0.730582\pi\)
−0.662682 + 0.748901i \(0.730582\pi\)
\(74\) 0 0
\(75\) −3.30649 −0.381801
\(76\) 0 0
\(77\) −6.94508 −0.791465
\(78\) 0 0
\(79\) −12.0593 −1.35677 −0.678386 − 0.734706i \(-0.737320\pi\)
−0.678386 + 0.734706i \(0.737320\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) 6.35285 0.697316 0.348658 − 0.937250i \(-0.386637\pi\)
0.348658 + 0.937250i \(0.386637\pi\)
\(84\) 0 0
\(85\) 0.924010 0.100223
\(86\) 0 0
\(87\) −7.87765 −0.844572
\(88\) 0 0
\(89\) −9.71377 −1.02966 −0.514829 − 0.857293i \(-0.672145\pi\)
−0.514829 + 0.857293i \(0.672145\pi\)
\(90\) 0 0
\(91\) 18.1751 1.90527
\(92\) 0 0
\(93\) −2.86424 −0.297008
\(94\) 0 0
\(95\) −7.51623 −0.771149
\(96\) 0 0
\(97\) 8.76465 0.889916 0.444958 − 0.895552i \(-0.353219\pi\)
0.444958 + 0.895552i \(0.353219\pi\)
\(98\) 0 0
\(99\) 21.6023 2.17111
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 7360.2.a.co.1.1 5
4.3 odd 2 7360.2.a.cp.1.5 5
8.3 odd 2 1840.2.a.v.1.1 5
8.5 even 2 920.2.a.j.1.5 ✓ 5
24.5 odd 2 8280.2.a.bs.1.2 5
40.13 odd 4 4600.2.e.u.4049.9 10
40.19 odd 2 9200.2.a.cu.1.5 5
40.29 even 2 4600.2.a.be.1.1 5
40.37 odd 4 4600.2.e.u.4049.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 ✓ 5 8.5 even 2
1840.2.a.v.1.1 5 8.3 odd 2
4600.2.a.be.1.1 5 40.29 even 2
4600.2.e.u.4049.2 10 40.37 odd 4
4600.2.e.u.4049.9 10 40.13 odd 4
7360.2.a.co.1.1 5 1.1 even 1 trivial
7360.2.a.cp.1.5 5 4.3 odd 2
8280.2.a.bs.1.2 5 24.5 odd 2
9200.2.a.cu.1.5 5 40.19 odd 2