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Properties

Label 322.2.a.b.1.1
Level $322$
Weight $2$
Character 322.1
Self dual yes
Analytic conductor $2.571$
Analytic rank $0$
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] = [322,2,Mod(1,322)] 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("322.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,2] 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: \(2.57118294509\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 322.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +2.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} -2.00000 q^{24} -5.00000 q^{25} -4.00000 q^{27} +1.00000 q^{28} +10.0000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +8.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} -10.0000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{46} +12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +12.0000 q^{51} -6.00000 q^{53} +4.00000 q^{54} -1.00000 q^{56} -12.0000 q^{57} -10.0000 q^{58} -2.00000 q^{59} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{66} +6.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -10.0000 q^{75} -6.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} +10.0000 q^{82} -14.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +20.0000 q^{87} -4.00000 q^{88} -14.0000 q^{89} -1.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} -2.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) −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 \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −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
\(28\) 1.00000 0.188982
\(29\) 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 \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −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 \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −2.00000 −0.308607
\(43\) −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 \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) −1.00000 −0.133631
\(57\) −12.0000 −1.58944
\(58\) −10.0000 −1.31306
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 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 \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −10.0000 −1.15470
\(76\) −6.00000 −0.688247
\(77\) 4.00000 0.455842
\(78\) 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 \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 2.00000 0.218218
\(85\) 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 \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 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 \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 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 322.2.a.b.1.1 ✓ 1
3.2 odd 2 2898.2.a.p.1.1 1
4.3 odd 2 2576.2.a.c.1.1 1
5.4 even 2 8050.2.a.n.1.1 1
7.6 odd 2 2254.2.a.a.1.1 1
23.22 odd 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 1.1 even 1 trivial
2254.2.a.a.1.1 1 7.6 odd 2
2576.2.a.c.1.1 1 4.3 odd 2
2898.2.a.p.1.1 1 3.2 odd 2
7406.2.a.e.1.1 1 23.22 odd 2
8050.2.a.n.1.1 1 5.4 even 2