Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(1,522)] 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("522.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,1,3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 174)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 522.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +5.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} -4.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +4.00000 q^{25} -4.00000 q^{26} +5.00000 q^{28} +1.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +15.0000 q^{35} -1.00000 q^{37} -1.00000 q^{38} +3.00000 q^{40} +9.00000 q^{41} -7.00000 q^{43} -6.00000 q^{44} +3.00000 q^{47} +18.0000 q^{49} +4.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} -18.0000 q^{55} +5.00000 q^{56} +1.00000 q^{58} -3.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} -3.00000 q^{68} +15.0000 q^{70} -12.0000 q^{71} +2.00000 q^{73} -1.00000 q^{74} -1.00000 q^{76} -30.0000 q^{77} +14.0000 q^{79} +3.00000 q^{80} +9.00000 q^{82} -9.00000 q^{85} -7.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -20.0000 q^{91} +3.00000 q^{94} -3.00000 q^{95} +8.00000 q^{97} +18.0000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 − 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 5.00000 0.944911
\(29\) 1.00000 0.185695
\(30\) 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
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 15.0000 2.53546
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 − 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 5.00000 0.668153
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 15.0000 1.79284
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −30.0000 −3.41882
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −7.00000 −0.754829
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 18.0000 1.81827
\(99\) 0 0
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 522.2.a.m.1.1 1
3.2 odd 2 174.2.a.b.1.1 ✓ 1
4.3 odd 2 4176.2.a.bd.1.1 1
12.11 even 2 1392.2.a.b.1.1 1
15.2 even 4 4350.2.e.j.349.1 2
15.8 even 4 4350.2.e.j.349.2 2
15.14 odd 2 4350.2.a.o.1.1 1
21.20 even 2 8526.2.a.f.1.1 1
24.5 odd 2 5568.2.a.q.1.1 1
24.11 even 2 5568.2.a.bh.1.1 1
87.86 odd 2 5046.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
174.2.a.b.1.1 ✓ 1 3.2 odd 2
522.2.a.m.1.1 1 1.1 even 1 trivial
1392.2.a.b.1.1 1 12.11 even 2
4176.2.a.bd.1.1 1 4.3 odd 2
4350.2.a.o.1.1 1 15.14 odd 2
4350.2.e.j.349.1 2 15.2 even 4
4350.2.e.j.349.2 2 15.8 even 4
5046.2.a.i.1.1 1 87.86 odd 2
5568.2.a.q.1.1 1 24.5 odd 2
5568.2.a.bh.1.1 1 24.11 even 2
8526.2.a.f.1.1 1 21.20 even 2