Properties

Label 69.2.a.b.1.2
Level $69$
Weight $2$
Character 69.1
Self dual yes
Analytic conductor $0.551$
Analytic rank $0$
Dimension $2$
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] = [69,2,Mod(1,69)] 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("69.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(69, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -3.23607 q^{5} -2.23607 q^{6} -1.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -7.23607 q^{10} +4.00000 q^{11} -3.00000 q^{12} +4.47214 q^{13} -2.76393 q^{14} +3.23607 q^{15} -1.00000 q^{16} -7.23607 q^{17} +2.23607 q^{18} +2.76393 q^{19} -9.70820 q^{20} +1.23607 q^{21} +8.94427 q^{22} +1.00000 q^{23} -2.23607 q^{24} +5.47214 q^{25} +10.0000 q^{26} -1.00000 q^{27} -3.70820 q^{28} -4.47214 q^{29} +7.23607 q^{30} +2.47214 q^{31} -6.70820 q^{32} -4.00000 q^{33} -16.1803 q^{34} +4.00000 q^{35} +3.00000 q^{36} -4.47214 q^{37} +6.18034 q^{38} -4.47214 q^{39} -7.23607 q^{40} +6.94427 q^{41} +2.76393 q^{42} +7.70820 q^{43} +12.0000 q^{44} -3.23607 q^{45} +2.23607 q^{46} -4.00000 q^{47} +1.00000 q^{48} -5.47214 q^{49} +12.2361 q^{50} +7.23607 q^{51} +13.4164 q^{52} -0.763932 q^{53} -2.23607 q^{54} -12.9443 q^{55} -2.76393 q^{56} -2.76393 q^{57} -10.0000 q^{58} +12.9443 q^{59} +9.70820 q^{60} -4.47214 q^{61} +5.52786 q^{62} -1.23607 q^{63} -13.0000 q^{64} -14.4721 q^{65} -8.94427 q^{66} +5.23607 q^{67} -21.7082 q^{68} -1.00000 q^{69} +8.94427 q^{70} -8.00000 q^{71} +2.23607 q^{72} -10.9443 q^{73} -10.0000 q^{74} -5.47214 q^{75} +8.29180 q^{76} -4.94427 q^{77} -10.0000 q^{78} -3.70820 q^{79} +3.23607 q^{80} +1.00000 q^{81} +15.5279 q^{82} +4.00000 q^{83} +3.70820 q^{84} +23.4164 q^{85} +17.2361 q^{86} +4.47214 q^{87} +8.94427 q^{88} +3.23607 q^{89} -7.23607 q^{90} -5.52786 q^{91} +3.00000 q^{92} -2.47214 q^{93} -8.94427 q^{94} -8.94427 q^{95} +6.70820 q^{96} -0.472136 q^{97} -12.2361 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 10 q^{10} + 8 q^{11} - 6 q^{12} - 10 q^{14} + 2 q^{15} - 2 q^{16} - 10 q^{17} + 10 q^{19} - 6 q^{20} - 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{26}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 − 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) −3.23607 −1.44721 −0.723607 − 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) −2.23607 −0.912871
\(7\) −1.23607 −0.467190 −0.233595 − 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −7.23607 −2.28825
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −3.00000 −0.866025
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) −2.76393 −0.738692
\(15\) 3.23607 0.835549
\(16\) −1.00000 −0.250000
\(17\) −7.23607 −1.75500 −0.877502 − 0.479573i \(-0.840792\pi\)
−0.877502 + 0.479573i \(0.840792\pi\)
\(18\) 2.23607 0.527046
\(19\) 2.76393 0.634089 0.317045 − 0.948411i \(-0.397309\pi\)
0.317045 + 0.948411i \(0.397309\pi\)
\(20\) −9.70820 −2.17082
\(21\) 1.23607 0.269732
\(22\) 8.94427 1.90693
\(23\) 1.00000 0.208514
\(24\) −2.23607 −0.456435
\(25\) 5.47214 1.09443
\(26\) 10.0000 1.96116
\(27\) −1.00000 −0.192450
\(28\) −3.70820 −0.700785
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 7.23607 1.32112
\(31\) 2.47214 0.444009 0.222004 − 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) −6.70820 −1.18585
\(33\) −4.00000 −0.696311
\(34\) −16.1803 −2.77491
\(35\) 4.00000 0.676123
\(36\) 3.00000 0.500000
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 6.18034 1.00258
\(39\) −4.47214 −0.716115
\(40\) −7.23607 −1.14412
\(41\) 6.94427 1.08451 0.542257 − 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 2.76393 0.426484
\(43\) 7.70820 1.17549 0.587745 − 0.809046i \(-0.300016\pi\)
0.587745 + 0.809046i \(0.300016\pi\)
\(44\) 12.0000 1.80907
\(45\) −3.23607 −0.482405
\(46\) 2.23607 0.329690
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.47214 −0.781734
\(50\) 12.2361 1.73044
\(51\) 7.23607 1.01325
\(52\) 13.4164 1.86052
\(53\) −0.763932 −0.104934 −0.0524671 − 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) −2.23607 −0.304290
\(55\) −12.9443 −1.74541
\(56\) −2.76393 −0.369346
\(57\) −2.76393 −0.366092
\(58\) −10.0000 −1.31306
\(59\) 12.9443 1.68520 0.842600 − 0.538539i \(-0.181024\pi\)
0.842600 + 0.538539i \(0.181024\pi\)
\(60\) 9.70820 1.25332
\(61\) −4.47214 −0.572598 −0.286299 − 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 5.52786 0.702039
\(63\) −1.23607 −0.155730
\(64\) −13.0000 −1.62500
\(65\) −14.4721 −1.79505
\(66\) −8.94427 −1.10096
\(67\) 5.23607 0.639688 0.319844 − 0.947470i \(-0.396370\pi\)
0.319844 + 0.947470i \(0.396370\pi\)
\(68\) −21.7082 −2.63251
\(69\) −1.00000 −0.120386
\(70\) 8.94427 1.06904
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 2.23607 0.263523
\(73\) −10.9443 −1.28093 −0.640465 − 0.767987i \(-0.721258\pi\)
−0.640465 + 0.767987i \(0.721258\pi\)
\(74\) −10.0000 −1.16248
\(75\) −5.47214 −0.631868
\(76\) 8.29180 0.951134
\(77\) −4.94427 −0.563452
\(78\) −10.0000 −1.13228
\(79\) −3.70820 −0.417206 −0.208603 − 0.978000i \(-0.566892\pi\)
−0.208603 + 0.978000i \(0.566892\pi\)
\(80\) 3.23607 0.361803
\(81\) 1.00000 0.111111
\(82\) 15.5279 1.71477
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 3.70820 0.404598
\(85\) 23.4164 2.53987
\(86\) 17.2361 1.85861
\(87\) 4.47214 0.479463
\(88\) 8.94427 0.953463
\(89\) 3.23607 0.343023 0.171511 − 0.985182i \(-0.445135\pi\)
0.171511 + 0.985182i \(0.445135\pi\)
\(90\) −7.23607 −0.762749
\(91\) −5.52786 −0.579478
\(92\) 3.00000 0.312772
\(93\) −2.47214 −0.256349
\(94\) −8.94427 −0.922531
\(95\) −8.94427 −0.917663
\(96\) 6.70820 0.684653
\(97\) −0.472136 −0.0479381 −0.0239691 − 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) −12.2361 −1.23603
\(99\) 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 69.2.a.b.1.2 ✓ 2
3.2 odd 2 207.2.a.c.1.1 2
4.3 odd 2 1104.2.a.m.1.1 2
5.2 odd 4 1725.2.b.o.1174.4 4
5.3 odd 4 1725.2.b.o.1174.1 4
5.4 even 2 1725.2.a.ba.1.1 2
7.6 odd 2 3381.2.a.t.1.2 2
8.3 odd 2 4416.2.a.bg.1.2 2
8.5 even 2 4416.2.a.bm.1.2 2
11.10 odd 2 8349.2.a.i.1.1 2
12.11 even 2 3312.2.a.bb.1.2 2
15.14 odd 2 5175.2.a.bk.1.2 2
23.22 odd 2 1587.2.a.i.1.2 2
69.68 even 2 4761.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 ✓ 2 1.1 even 1 trivial
207.2.a.c.1.1 2 3.2 odd 2
1104.2.a.m.1.1 2 4.3 odd 2
1587.2.a.i.1.2 2 23.22 odd 2
1725.2.a.ba.1.1 2 5.4 even 2
1725.2.b.o.1174.1 4 5.3 odd 4
1725.2.b.o.1174.4 4 5.2 odd 4
3312.2.a.bb.1.2 2 12.11 even 2
3381.2.a.t.1.2 2 7.6 odd 2
4416.2.a.bg.1.2 2 8.3 odd 2
4416.2.a.bm.1.2 2 8.5 even 2
4761.2.a.v.1.1 2 69.68 even 2
5175.2.a.bk.1.2 2 15.14 odd 2
8349.2.a.i.1.1 2 11.10 odd 2