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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-26,0,-2,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(-3.30649i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.9

$q$-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)
 
\(f(q)\) \(=\) \(q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} -2.72314 q^{11} -7.12637i q^{13} +0.924010i q^{17} -7.51623 q^{19} -8.43286 q^{21} +1.00000i q^{23} +16.3105i q^{27} +2.38248 q^{29} +0.866248 q^{31} +9.00402i q^{33} +0.352855i q^{37} -23.5633 q^{39} +4.34066 q^{41} -13.3239i q^{47} +0.495474 q^{49} +3.05523 q^{51} -3.99262i q^{53} +24.8523i q^{57} +3.84064 q^{59} -9.14262 q^{61} +20.2320i q^{63} -3.15933i q^{67} +3.30649 q^{69} -6.07883 q^{71} +11.3239i q^{73} +6.94508i q^{77} +12.0593 q^{79} +30.1319 q^{81} +6.35285i q^{83} -7.87765i q^{87} +9.71377 q^{89} -18.1751 q^{91} -2.86424i q^{93} +8.76465i q^{97} +21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91}+ \cdots + 130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.30649i − 1.90900i −0.298206 − 0.954501i \(-0.596388\pi\)
0.298206 − 0.954501i \(-0.403612\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.55040i − 0.963960i −0.876182 − 0.481980i \(-0.839918\pi\)
0.876182 − 0.481980i \(-0.160082\pi\)
\(8\) 0 0
\(9\) −7.93288 −2.64429
\(10\) 0 0
\(11\) −2.72314 −0.821056 −0.410528 − 0.911848i \(-0.634656\pi\)
−0.410528 + 0.911848i \(0.634656\pi\)
\(12\) 0 0
\(13\) − 7.12637i − 1.97650i −0.152845 − 0.988250i \(-0.548844\pi\)
0.152845 − 0.988250i \(-0.451156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.924010i 0.224105i 0.993702 + 0.112053i \(0.0357426\pi\)
−0.993702 + 0.112053i \(0.964257\pi\)
\(18\) 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
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.3105i 3.13896i
\(28\) 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
\(33\) 9.00402i 1.56740i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.352855i 0.0580089i 0.999579 + 0.0290045i \(0.00923370\pi\)
−0.999579 + 0.0290045i \(0.990766\pi\)
\(38\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13.3239i − 1.94349i −0.236027 − 0.971746i \(-0.575845\pi\)
0.236027 − 0.971746i \(-0.424155\pi\)
\(48\) 0 0
\(49\) 0.495474 0.0707819
\(50\) 0 0
\(51\) 3.05523 0.427818
\(52\) 0 0
\(53\) − 3.99262i − 0.548429i −0.961669 − 0.274214i \(-0.911582\pi\)
0.961669 − 0.274214i \(-0.0884178\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.8523i 3.29177i
\(58\) 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 \(-0.699022\pi\)
−0.585296 + 0.810820i \(0.699022\pi\)
\(62\) 0 0
\(63\) 20.2320i 2.54899i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.15933i − 0.385974i −0.981201 − 0.192987i \(-0.938183\pi\)
0.981201 − 0.192987i \(-0.0618175\pi\)
\(68\) 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
\(73\) 11.3239i 1.32536i 0.748901 + 0.662682i \(0.230582\pi\)
−0.748901 + 0.662682i \(0.769418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.94508i 0.791465i
\(78\) 0 0
\(79\) 12.0593 1.35677 0.678386 − 0.734706i \(-0.262680\pi\)
0.678386 + 0.734706i \(0.262680\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) 6.35285i 0.697316i 0.937250 + 0.348658i \(0.113363\pi\)
−0.937250 + 0.348658i \(0.886637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.87765i − 0.844572i
\(88\) 0 0
\(89\) 9.71377 1.02966 0.514829 − 0.857293i \(-0.327855\pi\)
0.514829 + 0.857293i \(0.327855\pi\)
\(90\) 0 0
\(91\) −18.1751 −1.90527
\(92\) 0 0
\(93\) − 2.86424i − 0.297008i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.76465i 0.889916i 0.895552 + 0.444958i \(0.146781\pi\)
−0.895552 + 0.444958i \(0.853219\pi\)
\(98\) 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 4600.2.e.u.4049.2 10
5.2 odd 4 4600.2.a.be.1.1 5
5.3 odd 4 920.2.a.j.1.5 ✓ 5
5.4 even 2 inner 4600.2.e.u.4049.9 10
15.8 even 4 8280.2.a.bs.1.2 5
20.3 even 4 1840.2.a.v.1.1 5
20.7 even 4 9200.2.a.cu.1.5 5
40.3 even 4 7360.2.a.cp.1.5 5
40.13 odd 4 7360.2.a.co.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 ✓ 5 5.3 odd 4
1840.2.a.v.1.1 5 20.3 even 4
4600.2.a.be.1.1 5 5.2 odd 4
4600.2.e.u.4049.2 10 1.1 even 1 trivial
4600.2.e.u.4049.9 10 5.4 even 2 inner
7360.2.a.co.1.1 5 40.13 odd 4
7360.2.a.cp.1.5 5 40.3 even 4
8280.2.a.bs.1.2 5 15.8 even 4
9200.2.a.cu.1.5 5 20.7 even 4