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Properties

Label 5610.2.a.i.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)] 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("5610.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-1,1,1,1,2,-1,1,-1,-1,-1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7960755339\)
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 \(\chi\) \(=\) 5610.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +6.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} -2.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -4.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -2.00000 q^{56} +6.00000 q^{57} -2.00000 q^{58} +14.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} +6.00000 q^{69} -2.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -6.00000 q^{76} -2.00000 q^{77} -4.00000 q^{78} +14.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -1.00000 q^{85} +2.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -6.00000 q^{92} -8.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 0.171499
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 6.00000 0.973329
\(39\) 4.00000 0.640513
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −4.00000 −0.554700
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) −2.00000 −0.267261
\(57\) 6.00000 0.794719
\(58\) −2.00000 −0.262613
\(59\) 14.0000 1.82264 0.911322 − 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −1.00000 −0.123091
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.00000 0.722315
\(70\) −2.00000 −0.239046
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −6.00000 −0.688247
\(77\) −2.00000 −0.227921
\(78\) −4.00000 −0.452911
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) −1.00000 −0.108465
\(86\) 2.00000 0.215666
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −6.00000 −0.615587
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 3.00000 0.303046
\(99\) −1.00000 −0.100504
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 5610.2.a.i.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.i.1.1 ✓ 1 1.1 even 1 trivial