Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,0,2,0,0,-4,0,-3,0,-1,0,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -4.00000 q^{7} -3.00000 q^{9} -1.00000 q^{11} -2.00000 q^{13} -8.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -6.00000 q^{18} -2.00000 q^{22} +6.00000 q^{23} -4.00000 q^{26} -8.00000 q^{28} +9.00000 q^{29} -7.00000 q^{31} -8.00000 q^{32} -4.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} +2.00000 q^{41} +2.00000 q^{43} -2.00000 q^{44} +12.0000 q^{46} +6.00000 q^{47} +9.00000 q^{49} -4.00000 q^{52} +4.00000 q^{53} +18.0000 q^{58} +9.00000 q^{59} -7.00000 q^{61} -14.0000 q^{62} +12.0000 q^{63} -8.00000 q^{64} -10.0000 q^{67} -4.00000 q^{68} +1.00000 q^{71} +10.0000 q^{73} +4.00000 q^{74} +4.00000 q^{77} +1.00000 q^{79} +9.00000 q^{81} +4.00000 q^{82} -6.00000 q^{83} +4.00000 q^{86} -11.0000 q^{89} +8.00000 q^{91} +12.0000 q^{92} +12.0000 q^{94} -6.00000 q^{97} +18.0000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −8.00000 −2.13809
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −6.00000 −1.41421
\(19\) 0 0
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −8.00000 −1.51186
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 18.0000 2.36352
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −14.0000 −1.77800
\(63\) 12.0000 1.51186
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 4.00000 0.441726
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 − 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 18.0000 1.81827
\(99\) 3.00000 0.301511
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 9025.2.a.i.1.1 1
5.2 odd 4 1805.2.b.b.1084.2 2
5.3 odd 4 1805.2.b.b.1084.1 2
5.4 even 2 9025.2.a.b.1.1 1
19.7 even 3 475.2.e.a.201.1 2
19.11 even 3 475.2.e.a.26.1 2
19.18 odd 2 9025.2.a.a.1.1 1
95.7 odd 12 95.2.i.a.49.1 ✓ 4
95.18 even 4 1805.2.b.a.1084.2 2
95.37 even 4 1805.2.b.a.1084.1 2
95.49 even 6 475.2.e.c.26.1 2
95.64 even 6 475.2.e.c.201.1 2
95.68 odd 12 95.2.i.a.64.1 yes 4
95.83 odd 12 95.2.i.a.49.2 yes 4
95.87 odd 12 95.2.i.a.64.2 yes 4
95.94 odd 2 9025.2.a.j.1.1 1
285.68 even 12 855.2.be.a.64.2 4
285.83 even 12 855.2.be.a.334.1 4
285.182 even 12 855.2.be.a.64.1 4
285.197 even 12 855.2.be.a.334.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.a.49.1 ✓ 4 95.7 odd 12
95.2.i.a.49.2 yes 4 95.83 odd 12
95.2.i.a.64.1 yes 4 95.68 odd 12
95.2.i.a.64.2 yes 4 95.87 odd 12
475.2.e.a.26.1 2 19.11 even 3
475.2.e.a.201.1 2 19.7 even 3
475.2.e.c.26.1 2 95.49 even 6
475.2.e.c.201.1 2 95.64 even 6
855.2.be.a.64.1 4 285.182 even 12
855.2.be.a.64.2 4 285.68 even 12
855.2.be.a.334.1 4 285.83 even 12
855.2.be.a.334.2 4 285.197 even 12
1805.2.b.a.1084.1 2 95.37 even 4
1805.2.b.a.1084.2 2 95.18 even 4
1805.2.b.b.1084.1 2 5.3 odd 4
1805.2.b.b.1084.2 2 5.2 odd 4
9025.2.a.a.1.1 1 19.18 odd 2
9025.2.a.b.1.1 1 5.4 even 2
9025.2.a.i.1.1 1 1.1 even 1 trivial
9025.2.a.j.1.1 1 95.94 odd 2