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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 4049.6
Root \(0.568386i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.5

$q$-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)
 
\(f(q)\) \(=\) \(q+0.568386i q^{3} +4.73770i q^{7} +2.67694 q^{9} -0.360532 q^{11} -5.26123i q^{13} +0.370852i q^{17} +4.60586 q^{19} -2.69284 q^{21} +1.00000i q^{23} +3.22669i q^{27} -0.939238 q^{29} +9.66662 q^{31} -0.204921i q^{33} +3.26862i q^{37} +2.99041 q^{39} +5.29977 q^{41} -1.25491i q^{47} -15.4458 q^{49} -0.210787 q^{51} -10.9278i q^{53} +2.61790i q^{57} +9.66955 q^{59} +9.71441 q^{61} +12.6825i q^{63} -7.07001i q^{67} -0.568386 q^{69} +11.3747 q^{71} -0.745086i q^{73} -1.70809i q^{77} -0.415709 q^{79} +6.19680 q^{81} +9.26862i q^{83} -0.533850i q^{87} -12.6122 q^{89} +24.9261 q^{91} +5.49437i q^{93} +14.0404i q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91}+ \cdots + 130 q^{99}+O(q^{100}) \) 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
\(3\) 0.568386i 0.328158i 0.986447 + 0.164079i \(0.0524652\pi\)
−0.986447 + 0.164079i \(0.947535\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.73770i 1.79068i 0.445381 + 0.895341i \(0.353068\pi\)
−0.445381 + 0.895341i \(0.646932\pi\)
\(8\) 0 0
\(9\) 2.67694 0.892312
\(10\) 0 0
\(11\) −0.360532 −0.108704 −0.0543522 − 0.998522i \(-0.517309\pi\)
−0.0543522 + 0.998522i \(0.517309\pi\)
\(12\) 0 0
\(13\) − 5.26123i − 1.45920i −0.683873 − 0.729601i \(-0.739706\pi\)
0.683873 − 0.729601i \(-0.260294\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.370852i 0.0899449i 0.998988 + 0.0449724i \(0.0143200\pi\)
−0.998988 + 0.0449724i \(0.985680\pi\)
\(18\) 0 0
\(19\) 4.60586 1.05666 0.528328 − 0.849040i \(-0.322819\pi\)
0.528328 + 0.849040i \(0.322819\pi\)
\(20\) 0 0
\(21\) −2.69284 −0.587626
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.22669i 0.620977i
\(28\) 0 0
\(29\) −0.939238 −0.174412 −0.0872061 − 0.996190i \(-0.527794\pi\)
−0.0872061 + 0.996190i \(0.527794\pi\)
\(30\) 0 0
\(31\) 9.66662 1.73618 0.868088 − 0.496411i \(-0.165349\pi\)
0.868088 + 0.496411i \(0.165349\pi\)
\(32\) 0 0
\(33\) − 0.204921i − 0.0356722i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.26862i 0.537357i 0.963230 + 0.268679i \(0.0865869\pi\)
−0.963230 + 0.268679i \(0.913413\pi\)
\(38\) 0 0
\(39\) 2.99041 0.478849
\(40\) 0 0
\(41\) 5.29977 0.827685 0.413843 − 0.910348i \(-0.364186\pi\)
0.413843 + 0.910348i \(0.364186\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.25491i − 0.183048i −0.995803 − 0.0915240i \(-0.970826\pi\)
0.995803 − 0.0915240i \(-0.0291738\pi\)
\(48\) 0 0
\(49\) −15.4458 −2.20654
\(50\) 0 0
\(51\) −0.210787 −0.0295161
\(52\) 0 0
\(53\) − 10.9278i − 1.50106i −0.660839 − 0.750528i \(-0.729799\pi\)
0.660839 − 0.750528i \(-0.270201\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.61790i 0.346750i
\(58\) 0 0
\(59\) 9.66955 1.25887 0.629434 − 0.777054i \(-0.283287\pi\)
0.629434 + 0.777054i \(0.283287\pi\)
\(60\) 0 0
\(61\) 9.71441 1.24380 0.621901 − 0.783096i \(-0.286361\pi\)
0.621901 + 0.783096i \(0.286361\pi\)
\(62\) 0 0
\(63\) 12.6825i 1.59785i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.07001i − 0.863739i −0.901936 − 0.431870i \(-0.857854\pi\)
0.901936 − 0.431870i \(-0.142146\pi\)
\(68\) 0 0
\(69\) −0.568386 −0.0684257
\(70\) 0 0
\(71\) 11.3747 1.34993 0.674965 − 0.737850i \(-0.264159\pi\)
0.674965 + 0.737850i \(0.264159\pi\)
\(72\) 0 0
\(73\) − 0.745086i − 0.0872057i −0.999049 − 0.0436029i \(-0.986116\pi\)
0.999049 − 0.0436029i \(-0.0138836\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.70809i − 0.194655i
\(78\) 0 0
\(79\) −0.415709 −0.0467709 −0.0233854 − 0.999727i \(-0.507444\pi\)
−0.0233854 + 0.999727i \(0.507444\pi\)
\(80\) 0 0
\(81\) 6.19680 0.688534
\(82\) 0 0
\(83\) 9.26862i 1.01736i 0.860955 + 0.508681i \(0.169867\pi\)
−0.860955 + 0.508681i \(0.830133\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.533850i − 0.0572347i
\(88\) 0 0
\(89\) −12.6122 −1.33689 −0.668444 − 0.743763i \(-0.733039\pi\)
−0.668444 + 0.743763i \(0.733039\pi\)
\(90\) 0 0
\(91\) 24.9261 2.61297
\(92\) 0 0
\(93\) 5.49437i 0.569740i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0404i 1.42559i 0.701374 + 0.712793i \(0.252570\pi\)
−0.701374 + 0.712793i \(0.747430\pi\)
\(98\) 0 0
\(99\) −0.965121 −0.0969983
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.4049.6 10
5.2 odd 4 4600.2.a.be.1.3 5
5.3 odd 4 920.2.a.j.1.3 ✓ 5
5.4 even 2 inner 4600.2.e.u.4049.5 10
15.8 even 4 8280.2.a.bs.1.5 5
20.3 even 4 1840.2.a.v.1.3 5
20.7 even 4 9200.2.a.cu.1.3 5
40.3 even 4 7360.2.a.cp.1.3 5
40.13 odd 4 7360.2.a.co.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.3 ✓ 5 5.3 odd 4
1840.2.a.v.1.3 5 20.3 even 4
4600.2.a.be.1.3 5 5.2 odd 4
4600.2.e.u.4049.5 10 5.4 even 2 inner
4600.2.e.u.4049.6 10 1.1 even 1 trivial
7360.2.a.co.1.3 5 40.13 odd 4
7360.2.a.cp.1.3 5 40.3 even 4
8280.2.a.bs.1.5 5 15.8 even 4
9200.2.a.cu.1.3 5 20.7 even 4