Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,-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: \(73.4623698596\)
Analytic rank: \(1\)
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 1.3
Root \(-0.568386\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.568386 q^{3} +4.73770 q^{7} -2.67694 q^{9} +0.360532 q^{11} -5.26123 q^{13} -0.370852 q^{17} +4.60586 q^{19} -2.69284 q^{21} -1.00000 q^{23} +3.22669 q^{27} +0.939238 q^{29} -9.66662 q^{31} -0.204921 q^{33} -3.26862 q^{37} +2.99041 q^{39} +5.29977 q^{41} -1.25491 q^{47} +15.4458 q^{49} +0.210787 q^{51} -10.9278 q^{53} -2.61790 q^{57} +9.66955 q^{59} +9.71441 q^{61} -12.6825 q^{63} -7.07001 q^{67} +0.568386 q^{69} -11.3747 q^{71} -0.745086 q^{73} +1.70809 q^{77} -0.415709 q^{79} +6.19680 q^{81} -9.26862 q^{83} -0.533850 q^{87} +12.6122 q^{89} -24.9261 q^{91} +5.49437 q^{93} -14.0404 q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51}+ \cdots + 65 q^{99}+O(q^{100}) \) 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
\(3\) −0.568386 −0.328158 −0.164079 − 0.986447i \(-0.552465\pi\)
−0.164079 + 0.986447i \(0.552465\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.73770 1.79068 0.895341 − 0.445381i \(-0.146932\pi\)
0.895341 + 0.445381i \(0.146932\pi\)
\(8\) 0 0
\(9\) −2.67694 −0.892312
\(10\) 0 0
\(11\) 0.360532 0.108704 0.0543522 − 0.998522i \(-0.482691\pi\)
0.0543522 + 0.998522i \(0.482691\pi\)
\(12\) 0 0
\(13\) −5.26123 −1.45920 −0.729601 − 0.683873i \(-0.760294\pi\)
−0.729601 + 0.683873i \(0.760294\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.370852 −0.0899449 −0.0449724 − 0.998988i \(-0.514320\pi\)
−0.0449724 + 0.998988i \(0.514320\pi\)
\(18\) 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
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.22669 0.620977
\(28\) 0 0
\(29\) 0.939238 0.174412 0.0872061 − 0.996190i \(-0.472206\pi\)
0.0872061 + 0.996190i \(0.472206\pi\)
\(30\) 0 0
\(31\) −9.66662 −1.73618 −0.868088 − 0.496411i \(-0.834651\pi\)
−0.868088 + 0.496411i \(0.834651\pi\)
\(32\) 0 0
\(33\) −0.204921 −0.0356722
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.26862 −0.537357 −0.268679 − 0.963230i \(-0.586587\pi\)
−0.268679 + 0.963230i \(0.586587\pi\)
\(38\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.25491 −0.183048 −0.0915240 − 0.995803i \(-0.529174\pi\)
−0.0915240 + 0.995803i \(0.529174\pi\)
\(48\) 0 0
\(49\) 15.4458 2.20654
\(50\) 0 0
\(51\) 0.210787 0.0295161
\(52\) 0 0
\(53\) −10.9278 −1.50106 −0.750528 − 0.660839i \(-0.770201\pi\)
−0.750528 + 0.660839i \(0.770201\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.61790 −0.346750
\(58\) 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
\(63\) −12.6825 −1.59785
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.07001 −0.863739 −0.431870 − 0.901936i \(-0.642146\pi\)
−0.431870 + 0.901936i \(0.642146\pi\)
\(68\) 0 0
\(69\) 0.568386 0.0684257
\(70\) 0 0
\(71\) −11.3747 −1.34993 −0.674965 − 0.737850i \(-0.735841\pi\)
−0.674965 + 0.737850i \(0.735841\pi\)
\(72\) 0 0
\(73\) −0.745086 −0.0872057 −0.0436029 − 0.999049i \(-0.513884\pi\)
−0.0436029 + 0.999049i \(0.513884\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.70809 0.194655
\(78\) 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
\(83\) −9.26862 −1.01736 −0.508681 − 0.860955i \(-0.669867\pi\)
−0.508681 + 0.860955i \(0.669867\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.533850 −0.0572347
\(88\) 0 0
\(89\) 12.6122 1.33689 0.668444 − 0.743763i \(-0.266961\pi\)
0.668444 + 0.743763i \(0.266961\pi\)
\(90\) 0 0
\(91\) −24.9261 −2.61297
\(92\) 0 0
\(93\) 5.49437 0.569740
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0404 −1.42559 −0.712793 − 0.701374i \(-0.752570\pi\)
−0.712793 + 0.701374i \(0.752570\pi\)
\(98\) 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 9200.2.a.cu.1.3 5
4.3 odd 2 4600.2.a.be.1.3 5
5.4 even 2 1840.2.a.v.1.3 5
20.3 even 4 4600.2.e.u.4049.6 10
20.7 even 4 4600.2.e.u.4049.5 10
20.19 odd 2 920.2.a.j.1.3 ✓ 5
40.19 odd 2 7360.2.a.co.1.3 5
40.29 even 2 7360.2.a.cp.1.3 5
60.59 even 2 8280.2.a.bs.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.3 ✓ 5 20.19 odd 2
1840.2.a.v.1.3 5 5.4 even 2
4600.2.a.be.1.3 5 4.3 odd 2
4600.2.e.u.4049.5 10 20.7 even 4
4600.2.e.u.4049.6 10 20.3 even 4
7360.2.a.co.1.3 5 40.19 odd 2
7360.2.a.cp.1.3 5 40.29 even 2
8280.2.a.bs.1.5 5 60.59 even 2
9200.2.a.cu.1.3 5 1.1 even 1 trivial