Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2254,2,Mod(1,2254)] 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("2254.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-2,1,0,2,0,-1,1,0,4] 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: \(17.9982806156\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2254.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -2.00000 q^{12} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -4.00000 q^{22} -1.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{27} +10.0000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{46} -12.0000 q^{47} -2.00000 q^{48} +5.00000 q^{50} +12.0000 q^{51} -6.00000 q^{53} -4.00000 q^{54} -12.0000 q^{57} -10.0000 q^{58} +2.00000 q^{59} +4.00000 q^{62} +1.00000 q^{64} +8.00000 q^{66} -6.00000 q^{68} +2.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +10.0000 q^{75} +6.00000 q^{76} -8.00000 q^{79} -11.0000 q^{81} -10.0000 q^{82} +14.0000 q^{83} +4.00000 q^{86} -20.0000 q^{87} -4.00000 q^{88} +14.0000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +12.0000 q^{94} +2.00000 q^{96} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) 2.00000 0.408248
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −8.00000 −1.39262
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −2.00000 −0.288675
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) −10.0000 −1.31306
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 8.00000 0.984732
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −6.00000 −0.727607
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) 10.0000 1.15470
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −10.0000 −1.10432
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −20.0000 −2.14423
\(88\) −4.00000 −0.426401
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 2254.2.a.a.1.1 1
7.6 odd 2 322.2.a.b.1.1 ✓ 1
21.20 even 2 2898.2.a.p.1.1 1
28.27 even 2 2576.2.a.c.1.1 1
35.34 odd 2 8050.2.a.n.1.1 1
161.160 even 2 7406.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.b.1.1 ✓ 1 7.6 odd 2
2254.2.a.a.1.1 1 1.1 even 1 trivial
2576.2.a.c.1.1 1 28.27 even 2
2898.2.a.p.1.1 1 21.20 even 2
7406.2.a.e.1.1 1 161.160 even 2
8050.2.a.n.1.1 1 35.34 odd 2