Properties

Label 69.4.a.a.1.2
Level $69$
Weight $4$
Character 69.1
Self dual yes
Analytic conductor $4.071$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [69,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.07113179040\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \(q+0.236068 q^{2} +3.00000 q^{3} -7.94427 q^{4} -15.2361 q^{5} +0.708204 q^{6} -20.6525 q^{7} -3.76393 q^{8} +9.00000 q^{9} -3.59675 q^{10} -7.63932 q^{11} -23.8328 q^{12} +55.0820 q^{13} -4.87539 q^{14} -45.7082 q^{15} +62.6656 q^{16} -72.7639 q^{17} +2.12461 q^{18} -96.7902 q^{19} +121.039 q^{20} -61.9574 q^{21} -1.80340 q^{22} -23.0000 q^{23} -11.2918 q^{24} +107.138 q^{25} +13.0031 q^{26} +27.0000 q^{27} +164.069 q^{28} -228.748 q^{29} -10.7902 q^{30} +336.413 q^{31} +44.9048 q^{32} -22.9180 q^{33} -17.1772 q^{34} +314.663 q^{35} -71.4984 q^{36} +213.108 q^{37} -22.8491 q^{38} +165.246 q^{39} +57.3475 q^{40} -325.108 q^{41} -14.6262 q^{42} -297.787 q^{43} +60.6888 q^{44} -137.125 q^{45} -5.42956 q^{46} -494.545 q^{47} +187.997 q^{48} +83.5248 q^{49} +25.2918 q^{50} -218.292 q^{51} -437.587 q^{52} +220.403 q^{53} +6.37384 q^{54} +116.393 q^{55} +77.7345 q^{56} -290.371 q^{57} -54.0000 q^{58} +502.768 q^{59} +363.118 q^{60} -69.3963 q^{61} +79.4164 q^{62} -185.872 q^{63} -490.724 q^{64} -839.234 q^{65} -5.41020 q^{66} -425.820 q^{67} +578.056 q^{68} -69.0000 q^{69} +74.2817 q^{70} -140.604 q^{71} -33.8754 q^{72} -273.266 q^{73} +50.3081 q^{74} +321.413 q^{75} +768.928 q^{76} +157.771 q^{77} +39.0093 q^{78} -234.521 q^{79} -954.778 q^{80} +81.0000 q^{81} -76.7477 q^{82} -233.358 q^{83} +492.207 q^{84} +1108.64 q^{85} -70.2980 q^{86} -686.243 q^{87} +28.7539 q^{88} +880.581 q^{89} -32.3707 q^{90} -1137.58 q^{91} +182.718 q^{92} +1009.24 q^{93} -116.746 q^{94} +1474.70 q^{95} +134.714 q^{96} +20.4319 q^{97} +19.7175 q^{98} -68.7539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 26 q^{5} - 12 q^{6} - 10 q^{7} - 12 q^{8} + 18 q^{9} + 42 q^{10} - 60 q^{11} + 6 q^{12} - 24 q^{13} - 50 q^{14} - 78 q^{15} + 18 q^{16} - 150 q^{17} - 36 q^{18} - 46 q^{19}+ \cdots - 540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.236068 0.0834626 0.0417313 − 0.999129i \(-0.486713\pi\)
0.0417313 + 0.999129i \(0.486713\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.94427 −0.993034
\(5\) −15.2361 −1.36276 −0.681378 − 0.731932i \(-0.738619\pi\)
−0.681378 + 0.731932i \(0.738619\pi\)
\(6\) 0.708204 0.0481872
\(7\) −20.6525 −1.11513 −0.557564 − 0.830134i \(-0.688264\pi\)
−0.557564 + 0.830134i \(0.688264\pi\)
\(8\) −3.76393 −0.166344
\(9\) 9.00000 0.333333
\(10\) −3.59675 −0.113739
\(11\) −7.63932 −0.209395 −0.104697 − 0.994504i \(-0.533387\pi\)
−0.104697 + 0.994504i \(0.533387\pi\)
\(12\) −23.8328 −0.573328
\(13\) 55.0820 1.17515 0.587577 − 0.809168i \(-0.300082\pi\)
0.587577 + 0.809168i \(0.300082\pi\)
\(14\) −4.87539 −0.0930716
\(15\) −45.7082 −0.786787
\(16\) 62.6656 0.979150
\(17\) −72.7639 −1.03811 −0.519054 − 0.854741i \(-0.673716\pi\)
−0.519054 + 0.854741i \(0.673716\pi\)
\(18\) 2.12461 0.0278209
\(19\) −96.7902 −1.16869 −0.584347 − 0.811504i \(-0.698649\pi\)
−0.584347 + 0.811504i \(0.698649\pi\)
\(20\) 121.039 1.35326
\(21\) −61.9574 −0.643820
\(22\) −1.80340 −0.0174766
\(23\) −23.0000 −0.208514
\(24\) −11.2918 −0.0960387
\(25\) 107.138 0.857102
\(26\) 13.0031 0.0980815
\(27\) 27.0000 0.192450
\(28\) 164.069 1.10736
\(29\) −228.748 −1.46474 −0.732369 − 0.680908i \(-0.761585\pi\)
−0.732369 + 0.680908i \(0.761585\pi\)
\(30\) −10.7902 −0.0656673
\(31\) 336.413 1.94908 0.974542 − 0.224204i \(-0.0719783\pi\)
0.974542 + 0.224204i \(0.0719783\pi\)
\(32\) 44.9048 0.248066
\(33\) −22.9180 −0.120894
\(34\) −17.1772 −0.0866433
\(35\) 314.663 1.51965
\(36\) −71.4984 −0.331011
\(37\) 213.108 0.946886 0.473443 − 0.880824i \(-0.343011\pi\)
0.473443 + 0.880824i \(0.343011\pi\)
\(38\) −22.8491 −0.0975424
\(39\) 165.246 0.678476
\(40\) 57.3475 0.226686
\(41\) −325.108 −1.23838 −0.619188 − 0.785243i \(-0.712538\pi\)
−0.619188 + 0.785243i \(0.712538\pi\)
\(42\) −14.6262 −0.0537349
\(43\) −297.787 −1.05610 −0.528048 − 0.849215i \(-0.677076\pi\)
−0.528048 + 0.849215i \(0.677076\pi\)
\(44\) 60.6888 0.207936
\(45\) −137.125 −0.454252
\(46\) −5.42956 −0.0174032
\(47\) −494.545 −1.53483 −0.767413 − 0.641154i \(-0.778456\pi\)
−0.767413 + 0.641154i \(0.778456\pi\)
\(48\) 187.997 0.565313
\(49\) 83.5248 0.243512
\(50\) 25.2918 0.0715360
\(51\) −218.292 −0.599352
\(52\) −437.587 −1.16697
\(53\) 220.403 0.571221 0.285611 − 0.958346i \(-0.407804\pi\)
0.285611 + 0.958346i \(0.407804\pi\)
\(54\) 6.37384 0.0160624
\(55\) 116.393 0.285354
\(56\) 77.7345 0.185495
\(57\) −290.371 −0.674746
\(58\) −54.0000 −0.122251
\(59\) 502.768 1.10940 0.554702 − 0.832049i \(-0.312832\pi\)
0.554702 + 0.832049i \(0.312832\pi\)
\(60\) 363.118 0.781306
\(61\) −69.3963 −0.145660 −0.0728302 − 0.997344i \(-0.523203\pi\)
−0.0728302 + 0.997344i \(0.523203\pi\)
\(62\) 79.4164 0.162676
\(63\) −185.872 −0.371710
\(64\) −490.724 −0.958446
\(65\) −839.234 −1.60145
\(66\) −5.41020 −0.0100901
\(67\) −425.820 −0.776450 −0.388225 − 0.921565i \(-0.626912\pi\)
−0.388225 + 0.921565i \(0.626912\pi\)
\(68\) 578.056 1.03088
\(69\) −69.0000 −0.120386
\(70\) 74.2817 0.126834
\(71\) −140.604 −0.235022 −0.117511 − 0.993072i \(-0.537492\pi\)
−0.117511 + 0.993072i \(0.537492\pi\)
\(72\) −33.8754 −0.0554480
\(73\) −273.266 −0.438129 −0.219064 − 0.975710i \(-0.570300\pi\)
−0.219064 + 0.975710i \(0.570300\pi\)
\(74\) 50.3081 0.0790296
\(75\) 321.413 0.494848
\(76\) 768.928 1.16055
\(77\) 157.771 0.233502
\(78\) 39.0093 0.0566274
\(79\) −234.521 −0.333996 −0.166998 − 0.985957i \(-0.553407\pi\)
−0.166998 + 0.985957i \(0.553407\pi\)
\(80\) −954.778 −1.33434
\(81\) 81.0000 0.111111
\(82\) −76.7477 −0.103358
\(83\) −233.358 −0.308606 −0.154303 − 0.988024i \(-0.549313\pi\)
−0.154303 + 0.988024i \(0.549313\pi\)
\(84\) 492.207 0.639335
\(85\) 1108.64 1.41469
\(86\) −70.2980 −0.0881445
\(87\) −686.243 −0.845666
\(88\) 28.7539 0.0348315
\(89\) 880.581 1.04878 0.524390 − 0.851478i \(-0.324293\pi\)
0.524390 + 0.851478i \(0.324293\pi\)
\(90\) −32.3707 −0.0379131
\(91\) −1137.58 −1.31045
\(92\) 182.718 0.207062
\(93\) 1009.24 1.12530
\(94\) −116.746 −0.128101
\(95\) 1474.70 1.59265
\(96\) 134.714 0.143221
\(97\) 20.4319 0.0213871 0.0106936 − 0.999943i \(-0.496596\pi\)
0.0106936 + 0.999943i \(0.496596\pi\)
\(98\) 19.7175 0.0203242
\(99\) −68.7539 −0.0697982
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 69.4.a.a.1.2 ✓ 2
3.2 odd 2 207.4.a.c.1.1 2
4.3 odd 2 1104.4.a.h.1.1 2
5.4 even 2 1725.4.a.n.1.1 2
23.22 odd 2 1587.4.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.4.a.a.1.2 ✓ 2 1.1 even 1 trivial
207.4.a.c.1.1 2 3.2 odd 2
1104.4.a.h.1.1 2 4.3 odd 2
1587.4.a.b.1.2 2 23.22 odd 2
1725.4.a.n.1.1 2 5.4 even 2