Properties

Label 95.4.a.a.1.1
Level $95$
Weight $4$
Character 95.1
Self dual yes
Analytic conductor $5.605$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(1,95)] 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("95.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.60518145055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 95.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{3} -8.00000 q^{4} -5.00000 q^{5} -22.0000 q^{7} -11.0000 q^{9} -12.0000 q^{11} -32.0000 q^{12} +8.00000 q^{13} -20.0000 q^{15} +64.0000 q^{16} -66.0000 q^{17} +19.0000 q^{19} +40.0000 q^{20} -88.0000 q^{21} -30.0000 q^{23} +25.0000 q^{25} -152.000 q^{27} +176.000 q^{28} -6.00000 q^{29} -64.0000 q^{31} -48.0000 q^{33} +110.000 q^{35} +88.0000 q^{36} -16.0000 q^{37} +32.0000 q^{39} +54.0000 q^{41} +182.000 q^{43} +96.0000 q^{44} +55.0000 q^{45} +594.000 q^{47} +256.000 q^{48} +141.000 q^{49} -264.000 q^{51} -64.0000 q^{52} +396.000 q^{53} +60.0000 q^{55} +76.0000 q^{57} -564.000 q^{59} +160.000 q^{60} -706.000 q^{61} +242.000 q^{63} -512.000 q^{64} -40.0000 q^{65} -628.000 q^{67} +528.000 q^{68} -120.000 q^{69} -984.000 q^{71} +14.0000 q^{73} +100.000 q^{75} -152.000 q^{76} +264.000 q^{77} -328.000 q^{79} -320.000 q^{80} -311.000 q^{81} -294.000 q^{83} +704.000 q^{84} +330.000 q^{85} -24.0000 q^{87} +918.000 q^{89} -176.000 q^{91} +240.000 q^{92} -256.000 q^{93} -95.0000 q^{95} -1564.00 q^{97} +132.000 q^{99} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 4.00000 0.769800 0.384900 − 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) −8.00000 −1.00000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −22.0000 −1.18789 −0.593944 − 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) −12.0000 −0.328921 −0.164461 − 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) −32.0000 −0.769800
\(13\) 8.00000 0.170677 0.0853385 − 0.996352i \(-0.472803\pi\)
0.0853385 + 0.996352i \(0.472803\pi\)
\(14\) 0 0
\(15\) −20.0000 −0.344265
\(16\) 64.0000 1.00000
\(17\) −66.0000 −0.941609 −0.470804 − 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 40.0000 0.447214
\(21\) −88.0000 −0.914437
\(22\) 0 0
\(23\) −30.0000 −0.271975 −0.135988 − 0.990711i \(-0.543421\pi\)
−0.135988 + 0.990711i \(0.543421\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −152.000 −1.08342
\(28\) 176.000 1.18789
\(29\) −6.00000 −0.0384197 −0.0192099 − 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) −64.0000 −0.370798 −0.185399 − 0.982663i \(-0.559358\pi\)
−0.185399 + 0.982663i \(0.559358\pi\)
\(32\) 0 0
\(33\) −48.0000 −0.253204
\(34\) 0 0
\(35\) 110.000 0.531240
\(36\) 88.0000 0.407407
\(37\) −16.0000 −0.0710915 −0.0355457 − 0.999368i \(-0.511317\pi\)
−0.0355457 + 0.999368i \(0.511317\pi\)
\(38\) 0 0
\(39\) 32.0000 0.131387
\(40\) 0 0
\(41\) 54.0000 0.205692 0.102846 − 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(42\) 0 0
\(43\) 182.000 0.645459 0.322730 − 0.946491i \(-0.395400\pi\)
0.322730 + 0.946491i \(0.395400\pi\)
\(44\) 96.0000 0.328921
\(45\) 55.0000 0.182198
\(46\) 0 0
\(47\) 594.000 1.84349 0.921743 − 0.387802i \(-0.126766\pi\)
0.921743 + 0.387802i \(0.126766\pi\)
\(48\) 256.000 0.769800
\(49\) 141.000 0.411079
\(50\) 0 0
\(51\) −264.000 −0.724851
\(52\) −64.0000 −0.170677
\(53\) 396.000 1.02632 0.513158 − 0.858294i \(-0.328475\pi\)
0.513158 + 0.858294i \(0.328475\pi\)
\(54\) 0 0
\(55\) 60.0000 0.147098
\(56\) 0 0
\(57\) 76.0000 0.176604
\(58\) 0 0
\(59\) −564.000 −1.24452 −0.622259 − 0.782812i \(-0.713785\pi\)
−0.622259 + 0.782812i \(0.713785\pi\)
\(60\) 160.000 0.344265
\(61\) −706.000 −1.48187 −0.740935 − 0.671577i \(-0.765617\pi\)
−0.740935 + 0.671577i \(0.765617\pi\)
\(62\) 0 0
\(63\) 242.000 0.483955
\(64\) −512.000 −1.00000
\(65\) −40.0000 −0.0763291
\(66\) 0 0
\(67\) −628.000 −1.14511 −0.572555 − 0.819866i \(-0.694048\pi\)
−0.572555 + 0.819866i \(0.694048\pi\)
\(68\) 528.000 0.941609
\(69\) −120.000 −0.209367
\(70\) 0 0
\(71\) −984.000 −1.64478 −0.822390 − 0.568925i \(-0.807360\pi\)
−0.822390 + 0.568925i \(0.807360\pi\)
\(72\) 0 0
\(73\) 14.0000 0.0224462 0.0112231 − 0.999937i \(-0.496427\pi\)
0.0112231 + 0.999937i \(0.496427\pi\)
\(74\) 0 0
\(75\) 100.000 0.153960
\(76\) −152.000 −0.229416
\(77\) 264.000 0.390722
\(78\) 0 0
\(79\) −328.000 −0.467125 −0.233563 − 0.972342i \(-0.575038\pi\)
−0.233563 + 0.972342i \(0.575038\pi\)
\(80\) −320.000 −0.447214
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) −294.000 −0.388804 −0.194402 − 0.980922i \(-0.562277\pi\)
−0.194402 + 0.980922i \(0.562277\pi\)
\(84\) 704.000 0.914437
\(85\) 330.000 0.421100
\(86\) 0 0
\(87\) −24.0000 −0.0295755
\(88\) 0 0
\(89\) 918.000 1.09335 0.546673 − 0.837346i \(-0.315894\pi\)
0.546673 + 0.837346i \(0.315894\pi\)
\(90\) 0 0
\(91\) −176.000 −0.202745
\(92\) 240.000 0.271975
\(93\) −256.000 −0.285440
\(94\) 0 0
\(95\) −95.0000 −0.102598
\(96\) 0 0
\(97\) −1564.00 −1.63711 −0.818557 − 0.574425i \(-0.805226\pi\)
−0.818557 + 0.574425i \(0.805226\pi\)
\(98\) 0 0
\(99\) 132.000 0.134005
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 95.4.a.a.1.1 ✓ 1
3.2 odd 2 855.4.a.e.1.1 1
4.3 odd 2 1520.4.a.b.1.1 1
5.2 odd 4 475.4.b.e.324.1 2
5.3 odd 4 475.4.b.e.324.2 2
5.4 even 2 475.4.a.d.1.1 1
19.18 odd 2 1805.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
475.4.a.d.1.1 1 5.4 even 2
475.4.b.e.324.1 2 5.2 odd 4
475.4.b.e.324.2 2 5.3 odd 4
855.4.a.e.1.1 1 3.2 odd 2
1520.4.a.b.1.1 1 4.3 odd 2
1805.4.a.f.1.1 1 19.18 odd 2