Properties

Label 736.2.a.b.1.2
Level $736$
Weight $2$
Character 736.1
Self dual yes
Analytic conductor $5.877$
Analytic rank $1$
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] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-6] 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: \(5.87698958877\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} -0.732051 q^{5} -4.73205 q^{7} -1.26795 q^{11} -6.46410 q^{13} -1.26795 q^{15} +4.19615 q^{17} -3.46410 q^{19} -8.19615 q^{21} +1.00000 q^{23} -4.46410 q^{25} -5.19615 q^{27} +8.46410 q^{29} -4.26795 q^{31} -2.19615 q^{33} +3.46410 q^{35} +6.19615 q^{37} -11.1962 q^{39} -1.53590 q^{41} +6.92820 q^{43} -11.1962 q^{47} +15.3923 q^{49} +7.26795 q^{51} +8.92820 q^{53} +0.928203 q^{55} -6.00000 q^{57} -9.46410 q^{59} +2.00000 q^{61} +4.73205 q^{65} +3.80385 q^{67} +1.73205 q^{69} -10.2679 q^{71} -0.464102 q^{73} -7.73205 q^{75} +6.00000 q^{77} +10.3923 q^{79} -9.00000 q^{81} -2.19615 q^{83} -3.07180 q^{85} +14.6603 q^{87} -9.85641 q^{89} +30.5885 q^{91} -7.39230 q^{93} +2.53590 q^{95} -1.26795 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 6 q^{7} - 6 q^{11} - 6 q^{13} - 6 q^{15} - 2 q^{17} - 6 q^{21} + 2 q^{23} - 2 q^{25} + 10 q^{29} - 12 q^{31} + 6 q^{33} + 2 q^{37} - 12 q^{39} - 10 q^{41} - 12 q^{47} + 10 q^{49} + 18 q^{51}+ \cdots - 6 q^{97}+O(q^{100}) \) 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 0
\(3\) 1.73205 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) −0.732051 −0.327383 −0.163692 − 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 0 0
\(7\) −4.73205 −1.78855 −0.894274 − 0.447521i \(-0.852307\pi\)
−0.894274 + 0.447521i \(0.852307\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.26795 −0.382301 −0.191151 − 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) 0 0
\(13\) −6.46410 −1.79282 −0.896410 − 0.443227i \(-0.853834\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 0 0
\(15\) −1.26795 −0.327383
\(16\) 0 0
\(17\) 4.19615 1.01772 0.508858 − 0.860850i \(-0.330068\pi\)
0.508858 + 0.860850i \(0.330068\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 − 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) −8.19615 −1.78855
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 8.46410 1.57174 0.785872 − 0.618389i \(-0.212214\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 0 0
\(31\) −4.26795 −0.766546 −0.383273 − 0.923635i \(-0.625203\pi\)
−0.383273 + 0.923635i \(0.625203\pi\)
\(32\) 0 0
\(33\) −2.19615 −0.382301
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) 6.19615 1.01864 0.509321 − 0.860577i \(-0.329897\pi\)
0.509321 + 0.860577i \(0.329897\pi\)
\(38\) 0 0
\(39\) −11.1962 −1.79282
\(40\) 0 0
\(41\) −1.53590 −0.239867 −0.119934 − 0.992782i \(-0.538268\pi\)
−0.119934 + 0.992782i \(0.538268\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 − 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1962 −1.63313 −0.816563 − 0.577256i \(-0.804124\pi\)
−0.816563 + 0.577256i \(0.804124\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 0 0
\(51\) 7.26795 1.01772
\(52\) 0 0
\(53\) 8.92820 1.22638 0.613192 − 0.789934i \(-0.289885\pi\)
0.613192 + 0.789934i \(0.289885\pi\)
\(54\) 0 0
\(55\) 0.928203 0.125159
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −9.46410 −1.23212 −0.616061 − 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.73205 0.586939
\(66\) 0 0
\(67\) 3.80385 0.464714 0.232357 − 0.972631i \(-0.425356\pi\)
0.232357 + 0.972631i \(0.425356\pi\)
\(68\) 0 0
\(69\) 1.73205 0.208514
\(70\) 0 0
\(71\) −10.2679 −1.21858 −0.609291 − 0.792947i \(-0.708546\pi\)
−0.609291 + 0.792947i \(0.708546\pi\)
\(72\) 0 0
\(73\) −0.464102 −0.0543190 −0.0271595 − 0.999631i \(-0.508646\pi\)
−0.0271595 + 0.999631i \(0.508646\pi\)
\(74\) 0 0
\(75\) −7.73205 −0.892820
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −2.19615 −0.241059 −0.120530 − 0.992710i \(-0.538459\pi\)
−0.120530 + 0.992710i \(0.538459\pi\)
\(84\) 0 0
\(85\) −3.07180 −0.333183
\(86\) 0 0
\(87\) 14.6603 1.57174
\(88\) 0 0
\(89\) −9.85641 −1.04478 −0.522388 − 0.852708i \(-0.674959\pi\)
−0.522388 + 0.852708i \(0.674959\pi\)
\(90\) 0 0
\(91\) 30.5885 3.20654
\(92\) 0 0
\(93\) −7.39230 −0.766546
\(94\) 0 0
\(95\) 2.53590 0.260178
\(96\) 0 0
\(97\) −1.26795 −0.128741 −0.0643704 − 0.997926i \(-0.520504\pi\)
−0.0643704 + 0.997926i \(0.520504\pi\)
\(98\) 0 0
\(99\) 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 736.2.a.b.1.2 ✓ 2
3.2 odd 2 6624.2.a.k.1.2 2
4.3 odd 2 736.2.a.c.1.1 yes 2
8.3 odd 2 1472.2.a.r.1.2 2
8.5 even 2 1472.2.a.q.1.1 2
12.11 even 2 6624.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.b.1.2 ✓ 2 1.1 even 1 trivial
736.2.a.c.1.1 yes 2 4.3 odd 2
1472.2.a.q.1.1 2 8.5 even 2
1472.2.a.r.1.2 2 8.3 odd 2
6624.2.a.k.1.2 2 3.2 odd 2
6624.2.a.n.1.2 2 12.11 even 2