Properties

Label 736.2.a.e.1.1
Level $736$
Weight $2$
Character 736.1
Self dual yes
Analytic conductor $5.877$
Analytic rank $1$
Dimension $3$
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 = [3,0,-4,0,-2,0,2] 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: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 1.1
Root \(2.34292\) 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)
 
\(f(q)\) \(=\) \(q-3.34292 q^{3} -1.14637 q^{5} +1.14637 q^{7} +8.17513 q^{9} -3.14637 q^{11} +2.48929 q^{13} +3.83221 q^{15} +0.853635 q^{17} +5.66442 q^{19} -3.83221 q^{21} -1.00000 q^{23} -3.68585 q^{25} -17.3001 q^{27} -6.88240 q^{29} -8.32150 q^{31} +10.5181 q^{33} -1.31415 q^{35} +8.81079 q^{37} -8.32150 q^{39} -6.48929 q^{41} +2.97858 q^{43} -9.37169 q^{45} -2.94981 q^{47} -5.68585 q^{49} -2.85363 q^{51} +0.393115 q^{53} +3.60688 q^{55} -18.9357 q^{57} -5.70727 q^{59} -14.3503 q^{61} +9.37169 q^{63} -2.85363 q^{65} -7.93260 q^{67} +3.34292 q^{69} +0.657077 q^{71} -1.90383 q^{73} +12.3215 q^{75} -3.60688 q^{77} +16.0575 q^{79} +33.3074 q^{81} -2.75325 q^{83} -0.978577 q^{85} +23.0073 q^{87} -15.7648 q^{89} +2.85363 q^{91} +27.8181 q^{93} -6.49350 q^{95} -14.8108 q^{97} -25.7220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 2 q^{5} + 2 q^{7} + 5 q^{9} - 8 q^{11} - 2 q^{15} + 4 q^{17} - 10 q^{19} + 2 q^{21} - 3 q^{23} + q^{25} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 6 q^{33} - 16 q^{35} - 2 q^{37} - 4 q^{39} - 12 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) \) 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
\(3\) −3.34292 −1.93004 −0.965019 − 0.262181i \(-0.915558\pi\)
−0.965019 + 0.262181i \(0.915558\pi\)
\(4\) 0 0
\(5\) −1.14637 −0.512670 −0.256335 − 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(6\) 0 0
\(7\) 1.14637 0.433285 0.216643 − 0.976251i \(-0.430489\pi\)
0.216643 + 0.976251i \(0.430489\pi\)
\(8\) 0 0
\(9\) 8.17513 2.72504
\(10\) 0 0
\(11\) −3.14637 −0.948665 −0.474332 − 0.880346i \(-0.657311\pi\)
−0.474332 + 0.880346i \(0.657311\pi\)
\(12\) 0 0
\(13\) 2.48929 0.690404 0.345202 − 0.938528i \(-0.387810\pi\)
0.345202 + 0.938528i \(0.387810\pi\)
\(14\) 0 0
\(15\) 3.83221 0.989473
\(16\) 0 0
\(17\) 0.853635 0.207037 0.103518 − 0.994628i \(-0.466990\pi\)
0.103518 + 0.994628i \(0.466990\pi\)
\(18\) 0 0
\(19\) 5.66442 1.29951 0.649754 − 0.760145i \(-0.274872\pi\)
0.649754 + 0.760145i \(0.274872\pi\)
\(20\) 0 0
\(21\) −3.83221 −0.836257
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.68585 −0.737169
\(26\) 0 0
\(27\) −17.3001 −3.32940
\(28\) 0 0
\(29\) −6.88240 −1.27803 −0.639015 − 0.769194i \(-0.720658\pi\)
−0.639015 + 0.769194i \(0.720658\pi\)
\(30\) 0 0
\(31\) −8.32150 −1.49459 −0.747293 − 0.664495i \(-0.768647\pi\)
−0.747293 + 0.664495i \(0.768647\pi\)
\(32\) 0 0
\(33\) 10.5181 1.83096
\(34\) 0 0
\(35\) −1.31415 −0.222133
\(36\) 0 0
\(37\) 8.81079 1.44848 0.724242 − 0.689545i \(-0.242190\pi\)
0.724242 + 0.689545i \(0.242190\pi\)
\(38\) 0 0
\(39\) −8.32150 −1.33251
\(40\) 0 0
\(41\) −6.48929 −1.01346 −0.506728 − 0.862106i \(-0.669145\pi\)
−0.506728 + 0.862106i \(0.669145\pi\)
\(42\) 0 0
\(43\) 2.97858 0.454229 0.227114 − 0.973868i \(-0.427071\pi\)
0.227114 + 0.973868i \(0.427071\pi\)
\(44\) 0 0
\(45\) −9.37169 −1.39705
\(46\) 0 0
\(47\) −2.94981 −0.430274 −0.215137 − 0.976584i \(-0.569020\pi\)
−0.215137 + 0.976584i \(0.569020\pi\)
\(48\) 0 0
\(49\) −5.68585 −0.812264
\(50\) 0 0
\(51\) −2.85363 −0.399589
\(52\) 0 0
\(53\) 0.393115 0.0539985 0.0269993 − 0.999635i \(-0.491405\pi\)
0.0269993 + 0.999635i \(0.491405\pi\)
\(54\) 0 0
\(55\) 3.60688 0.486352
\(56\) 0 0
\(57\) −18.9357 −2.50810
\(58\) 0 0
\(59\) −5.70727 −0.743023 −0.371512 − 0.928428i \(-0.621160\pi\)
−0.371512 + 0.928428i \(0.621160\pi\)
\(60\) 0 0
\(61\) −14.3503 −1.83736 −0.918682 − 0.394998i \(-0.870745\pi\)
−0.918682 + 0.394998i \(0.870745\pi\)
\(62\) 0 0
\(63\) 9.37169 1.18072
\(64\) 0 0
\(65\) −2.85363 −0.353950
\(66\) 0 0
\(67\) −7.93260 −0.969121 −0.484560 − 0.874758i \(-0.661020\pi\)
−0.484560 + 0.874758i \(0.661020\pi\)
\(68\) 0 0
\(69\) 3.34292 0.402441
\(70\) 0 0
\(71\) 0.657077 0.0779807 0.0389903 − 0.999240i \(-0.487586\pi\)
0.0389903 + 0.999240i \(0.487586\pi\)
\(72\) 0 0
\(73\) −1.90383 −0.222826 −0.111413 − 0.993774i \(-0.535538\pi\)
−0.111413 + 0.993774i \(0.535538\pi\)
\(74\) 0 0
\(75\) 12.3215 1.42276
\(76\) 0 0
\(77\) −3.60688 −0.411043
\(78\) 0 0
\(79\) 16.0575 1.80661 0.903307 − 0.428994i \(-0.141132\pi\)
0.903307 + 0.428994i \(0.141132\pi\)
\(80\) 0 0
\(81\) 33.3074 3.70082
\(82\) 0 0
\(83\) −2.75325 −0.302208 −0.151104 − 0.988518i \(-0.548283\pi\)
−0.151104 + 0.988518i \(0.548283\pi\)
\(84\) 0 0
\(85\) −0.978577 −0.106142
\(86\) 0 0
\(87\) 23.0073 2.46665
\(88\) 0 0
\(89\) −15.7648 −1.67107 −0.835533 − 0.549440i \(-0.814841\pi\)
−0.835533 + 0.549440i \(0.814841\pi\)
\(90\) 0 0
\(91\) 2.85363 0.299142
\(92\) 0 0
\(93\) 27.8181 2.88461
\(94\) 0 0
\(95\) −6.49350 −0.666219
\(96\) 0 0
\(97\) −14.8108 −1.50381 −0.751904 − 0.659273i \(-0.770864\pi\)
−0.751904 + 0.659273i \(0.770864\pi\)
\(98\) 0 0
\(99\) −25.7220 −2.58515
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.e.1.1 ✓ 3
3.2 odd 2 6624.2.a.z.1.2 3
4.3 odd 2 736.2.a.f.1.3 yes 3
8.3 odd 2 1472.2.a.w.1.1 3
8.5 even 2 1472.2.a.x.1.3 3
12.11 even 2 6624.2.a.y.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.e.1.1 ✓ 3 1.1 even 1 trivial
736.2.a.f.1.3 yes 3 4.3 odd 2
1472.2.a.w.1.1 3 8.3 odd 2
1472.2.a.x.1.3 3 8.5 even 2
6624.2.a.y.1.2 3 12.11 even 2
6624.2.a.z.1.2 3 3.2 odd 2