Properties

Label 345.2.a.f.1.1
Level $345$
Weight $2$
Character 345.1
Self dual yes
Analytic conductor $2.755$
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] = [345,2,Mod(1,345)] 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("345.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,-1] 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: \(2.75483886973\)
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 \(\chi\) \(=\) 345.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} +6.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} -5.00000 q^{29} -2.00000 q^{30} +3.00000 q^{31} -8.00000 q^{32} -2.00000 q^{33} +10.0000 q^{34} +3.00000 q^{35} +2.00000 q^{36} -7.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -11.0000 q^{41} -6.00000 q^{42} -8.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +8.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} -5.00000 q^{51} -4.00000 q^{52} +5.00000 q^{53} -2.00000 q^{54} +2.00000 q^{55} +2.00000 q^{57} -10.0000 q^{58} -1.00000 q^{59} -2.00000 q^{60} -8.00000 q^{61} +6.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -9.00000 q^{67} +10.0000 q^{68} +1.00000 q^{69} +6.00000 q^{70} +1.00000 q^{71} +10.0000 q^{73} -14.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{77} +4.00000 q^{78} -4.00000 q^{80} +1.00000 q^{81} -22.0000 q^{82} +15.0000 q^{83} -6.00000 q^{84} +5.00000 q^{85} -16.0000 q^{86} +5.00000 q^{87} +2.00000 q^{90} -6.00000 q^{91} -2.00000 q^{92} -3.00000 q^{93} +16.0000 q^{94} -2.00000 q^{95} +8.00000 q^{96} -10.0000 q^{97} +4.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(0.250000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) −2.00000 −0.816497
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 6.00000 1.60357
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 2.00000 0.471405
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) −3.00000 −0.654654
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 6.00000 1.13389
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −2.00000 −0.365148
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −8.00000 −1.41421
\(33\) −2.00000 −0.348155
\(34\) 10.0000 1.71499
\(35\) 3.00000 0.507093
\(36\) 2.00000 0.333333
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 − 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) −6.00000 −0.925820
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −2.00000 −0.294884
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 4.00000 0.577350
\(49\) 2.00000 0.285714
\(50\) 2.00000 0.282843
\(51\) −5.00000 −0.700140
\(52\) −4.00000 −0.554700
\(53\) 5.00000 0.686803 0.343401 − 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −2.00000 −0.272166
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −10.0000 −1.31306
\(59\) −1.00000 −0.130189 −0.0650945 − 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) −2.00000 −0.258199
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 6.00000 0.762001
\(63\) 3.00000 0.377964
\(64\) −8.00000 −1.00000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −9.00000 −1.09952 −0.549762 − 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 10.0000 1.21268
\(69\) 1.00000 0.120386
\(70\) 6.00000 0.717137
\(71\) 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 \(0.301014\pi\)
\(74\) −14.0000 −1.62747
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 6.00000 0.683763
\(78\) 4.00000 0.452911
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) −22.0000 −2.42949
\(83\) 15.0000 1.64646 0.823232 − 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) −6.00000 −0.654654
\(85\) 5.00000 0.542326
\(86\) −16.0000 −1.72532
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.00000 0.210819
\(91\) −6.00000 −0.628971
\(92\) −2.00000 −0.208514
\(93\) −3.00000 −0.311086
\(94\) 16.0000 1.65027
\(95\) −2.00000 −0.205196
\(96\) 8.00000 0.816497
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 4.00000 0.404061
\(99\) 2.00000 0.201008
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 345.2.a.f.1.1 ✓ 1
3.2 odd 2 1035.2.a.a.1.1 1
4.3 odd 2 5520.2.a.bc.1.1 1
5.2 odd 4 1725.2.b.b.1174.2 2
5.3 odd 4 1725.2.b.b.1174.1 2
5.4 even 2 1725.2.a.a.1.1 1
15.14 odd 2 5175.2.a.x.1.1 1
23.22 odd 2 7935.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
1035.2.a.a.1.1 1 3.2 odd 2
1725.2.a.a.1.1 1 5.4 even 2
1725.2.b.b.1174.1 2 5.3 odd 4
1725.2.b.b.1174.2 2 5.2 odd 4
5175.2.a.x.1.1 1 15.14 odd 2
5520.2.a.bc.1.1 1 4.3 odd 2
7935.2.a.k.1.1 1 23.22 odd 2