Properties

Label 736.2.a.a.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,-2,0,-4,0,4] 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: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(1.41421\) 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)
 
\(f(q)\) \(=\) \(q+0.414214 q^{3} -0.585786 q^{5} +0.585786 q^{7} -2.82843 q^{9} -4.24264 q^{11} -3.82843 q^{13} -0.242641 q^{15} -2.58579 q^{17} +0.828427 q^{19} +0.242641 q^{21} +1.00000 q^{23} -4.65685 q^{25} -2.41421 q^{27} -5.00000 q^{29} +9.24264 q^{31} -1.75736 q^{33} -0.343146 q^{35} +5.07107 q^{37} -1.58579 q^{39} -1.34315 q^{41} -4.00000 q^{43} +1.65685 q^{45} +5.24264 q^{47} -6.65685 q^{49} -1.07107 q^{51} -9.31371 q^{53} +2.48528 q^{55} +0.343146 q^{57} -1.17157 q^{59} -3.65685 q^{61} -1.65685 q^{63} +2.24264 q^{65} -13.8995 q^{67} +0.414214 q^{69} -3.58579 q^{71} +7.82843 q^{73} -1.92893 q^{75} -2.48528 q^{77} +14.4853 q^{79} +7.48528 q^{81} +10.7279 q^{83} +1.51472 q^{85} -2.07107 q^{87} -8.00000 q^{89} -2.24264 q^{91} +3.82843 q^{93} -0.485281 q^{95} +9.89949 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} - 2 q^{13} + 8 q^{15} - 8 q^{17} - 4 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 10 q^{31} - 12 q^{33} - 12 q^{35} - 4 q^{37} - 6 q^{39} - 14 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) \) 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 0
\(3\) 0.414214 0.239146 0.119573 − 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 0 0
\(5\) −0.585786 −0.261972 −0.130986 − 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) 0.585786 0.221406 0.110703 − 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −4.24264 −1.27920 −0.639602 − 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 − 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) −0.242641 −0.0626496
\(16\) 0 0
\(17\) −2.58579 −0.627145 −0.313573 − 0.949564i \(-0.601526\pi\)
−0.313573 + 0.949564i \(0.601526\pi\)
\(18\) 0 0
\(19\) 0.828427 0.190054 0.0950271 − 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 0 0
\(21\) 0.242641 0.0529485
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 9.24264 1.66003 0.830014 − 0.557743i \(-0.188333\pi\)
0.830014 + 0.557743i \(0.188333\pi\)
\(32\) 0 0
\(33\) −1.75736 −0.305917
\(34\) 0 0
\(35\) −0.343146 −0.0580022
\(36\) 0 0
\(37\) 5.07107 0.833678 0.416839 − 0.908980i \(-0.363138\pi\)
0.416839 + 0.908980i \(0.363138\pi\)
\(38\) 0 0
\(39\) −1.58579 −0.253929
\(40\) 0 0
\(41\) −1.34315 −0.209764 −0.104882 − 0.994485i \(-0.533447\pi\)
−0.104882 + 0.994485i \(0.533447\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.65685 0.246989
\(46\) 0 0
\(47\) 5.24264 0.764718 0.382359 − 0.924014i \(-0.375112\pi\)
0.382359 + 0.924014i \(0.375112\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) −1.07107 −0.149979
\(52\) 0 0
\(53\) −9.31371 −1.27934 −0.639668 − 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) 0 0
\(55\) 2.48528 0.335115
\(56\) 0 0
\(57\) 0.343146 0.0454508
\(58\) 0 0
\(59\) −1.17157 −0.152526 −0.0762629 − 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 − 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 0 0
\(63\) −1.65685 −0.208744
\(64\) 0 0
\(65\) 2.24264 0.278165
\(66\) 0 0
\(67\) −13.8995 −1.69809 −0.849047 − 0.528318i \(-0.822823\pi\)
−0.849047 + 0.528318i \(0.822823\pi\)
\(68\) 0 0
\(69\) 0.414214 0.0498655
\(70\) 0 0
\(71\) −3.58579 −0.425555 −0.212777 − 0.977101i \(-0.568251\pi\)
−0.212777 + 0.977101i \(0.568251\pi\)
\(72\) 0 0
\(73\) 7.82843 0.916248 0.458124 − 0.888888i \(-0.348522\pi\)
0.458124 + 0.888888i \(0.348522\pi\)
\(74\) 0 0
\(75\) −1.92893 −0.222734
\(76\) 0 0
\(77\) −2.48528 −0.283224
\(78\) 0 0
\(79\) 14.4853 1.62972 0.814861 − 0.579657i \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 10.7279 1.17754 0.588771 − 0.808300i \(-0.299612\pi\)
0.588771 + 0.808300i \(0.299612\pi\)
\(84\) 0 0
\(85\) 1.51472 0.164294
\(86\) 0 0
\(87\) −2.07107 −0.222042
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −2.24264 −0.235093
\(92\) 0 0
\(93\) 3.82843 0.396989
\(94\) 0 0
\(95\) −0.485281 −0.0497888
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 − 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
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.a.1.2 ✓ 2
3.2 odd 2 6624.2.a.t.1.1 2
4.3 odd 2 736.2.a.d.1.1 yes 2
8.3 odd 2 1472.2.a.o.1.2 2
8.5 even 2 1472.2.a.v.1.1 2
12.11 even 2 6624.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.a.1.2 ✓ 2 1.1 even 1 trivial
736.2.a.d.1.1 yes 2 4.3 odd 2
1472.2.a.o.1.2 2 8.3 odd 2
1472.2.a.v.1.1 2 8.5 even 2
6624.2.a.s.1.1 2 12.11 even 2
6624.2.a.t.1.1 2 3.2 odd 2