Properties

Label 736.2.a.g.1.2
Level $736$
Weight $2$
Character 736.1
Self dual yes
Analytic conductor $5.877$
Analytic rank $0$
Dimension $4$
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 = [4,0,-2,0,6,0,4] 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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.489088\) 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)
 
\(f(q)\) \(=\) \(q-2.27170 q^{3} -2.08924 q^{5} -1.11106 q^{7} +2.16064 q^{9} -4.08924 q^{11} +0.839360 q^{13} +4.74614 q^{15} +3.11106 q^{17} -2.97818 q^{19} +2.52401 q^{21} +1.00000 q^{23} -0.635073 q^{25} +1.90678 q^{27} +9.01784 q^{29} +0.315351 q^{31} +9.28955 q^{33} +2.32128 q^{35} -2.08924 q^{37} -1.90678 q^{39} +11.3391 q^{41} -0.478415 q^{43} -4.51410 q^{45} +8.22806 q^{47} -5.76554 q^{49} -7.06742 q^{51} +0.434768 q^{53} +8.54341 q^{55} +6.76554 q^{57} +2.86469 q^{59} +3.56523 q^{61} -2.40061 q^{63} -1.75362 q^{65} +6.31137 q^{67} -2.27170 q^{69} +8.12891 q^{71} -11.8825 q^{73} +1.44270 q^{75} +4.54341 q^{77} +10.4998 q^{79} -10.8136 q^{81} -0.454168 q^{83} -6.49976 q^{85} -20.4859 q^{87} +4.32128 q^{89} -0.932583 q^{91} -0.716384 q^{93} +6.22213 q^{95} -11.5553 q^{97} -8.83538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 10 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{15} + 4 q^{17} - 6 q^{19} + 4 q^{21} + 4 q^{23} + 12 q^{25} - 14 q^{27} + 6 q^{29} + 6 q^{31} + 12 q^{35} + 6 q^{37} + 14 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.27170 −1.31157 −0.655785 − 0.754948i \(-0.727662\pi\)
−0.655785 + 0.754948i \(0.727662\pi\)
\(4\) 0 0
\(5\) −2.08924 −0.934337 −0.467168 − 0.884168i \(-0.654726\pi\)
−0.467168 + 0.884168i \(0.654726\pi\)
\(6\) 0 0
\(7\) −1.11106 −0.419943 −0.209971 − 0.977708i \(-0.567337\pi\)
−0.209971 + 0.977708i \(0.567337\pi\)
\(8\) 0 0
\(9\) 2.16064 0.720213
\(10\) 0 0
\(11\) −4.08924 −1.23295 −0.616476 − 0.787374i \(-0.711440\pi\)
−0.616476 + 0.787374i \(0.711440\pi\)
\(12\) 0 0
\(13\) 0.839360 0.232797 0.116398 − 0.993203i \(-0.462865\pi\)
0.116398 + 0.993203i \(0.462865\pi\)
\(14\) 0 0
\(15\) 4.74614 1.22545
\(16\) 0 0
\(17\) 3.11106 0.754544 0.377272 − 0.926103i \(-0.376862\pi\)
0.377272 + 0.926103i \(0.376862\pi\)
\(18\) 0 0
\(19\) −2.97818 −0.683241 −0.341620 − 0.939838i \(-0.610976\pi\)
−0.341620 + 0.939838i \(0.610976\pi\)
\(20\) 0 0
\(21\) 2.52401 0.550784
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −0.635073 −0.127015
\(26\) 0 0
\(27\) 1.90678 0.366959
\(28\) 0 0
\(29\) 9.01784 1.67457 0.837286 − 0.546766i \(-0.184141\pi\)
0.837286 + 0.546766i \(0.184141\pi\)
\(30\) 0 0
\(31\) 0.315351 0.0566387 0.0283193 − 0.999599i \(-0.490984\pi\)
0.0283193 + 0.999599i \(0.490984\pi\)
\(32\) 0 0
\(33\) 9.28955 1.61710
\(34\) 0 0
\(35\) 2.32128 0.392368
\(36\) 0 0
\(37\) −2.08924 −0.343469 −0.171735 − 0.985143i \(-0.554937\pi\)
−0.171735 + 0.985143i \(0.554937\pi\)
\(38\) 0 0
\(39\) −1.90678 −0.305329
\(40\) 0 0
\(41\) 11.3391 1.77087 0.885437 − 0.464760i \(-0.153859\pi\)
0.885437 + 0.464760i \(0.153859\pi\)
\(42\) 0 0
\(43\) −0.478415 −0.0729576 −0.0364788 − 0.999334i \(-0.511614\pi\)
−0.0364788 + 0.999334i \(0.511614\pi\)
\(44\) 0 0
\(45\) −4.51410 −0.672922
\(46\) 0 0
\(47\) 8.22806 1.20019 0.600093 − 0.799930i \(-0.295130\pi\)
0.600093 + 0.799930i \(0.295130\pi\)
\(48\) 0 0
\(49\) −5.76554 −0.823648
\(50\) 0 0
\(51\) −7.06742 −0.989636
\(52\) 0 0
\(53\) 0.434768 0.0597200 0.0298600 − 0.999554i \(-0.490494\pi\)
0.0298600 + 0.999554i \(0.490494\pi\)
\(54\) 0 0
\(55\) 8.54341 1.15199
\(56\) 0 0
\(57\) 6.76554 0.896117
\(58\) 0 0
\(59\) 2.86469 0.372951 0.186475 − 0.982460i \(-0.440294\pi\)
0.186475 + 0.982460i \(0.440294\pi\)
\(60\) 0 0
\(61\) 3.56523 0.456481 0.228241 − 0.973605i \(-0.426703\pi\)
0.228241 + 0.973605i \(0.426703\pi\)
\(62\) 0 0
\(63\) −2.40061 −0.302448
\(64\) 0 0
\(65\) −1.75362 −0.217510
\(66\) 0 0
\(67\) 6.31137 0.771056 0.385528 − 0.922696i \(-0.374019\pi\)
0.385528 + 0.922696i \(0.374019\pi\)
\(68\) 0 0
\(69\) −2.27170 −0.273481
\(70\) 0 0
\(71\) 8.12891 0.964724 0.482362 − 0.875972i \(-0.339779\pi\)
0.482362 + 0.875972i \(0.339779\pi\)
\(72\) 0 0
\(73\) −11.8825 −1.39074 −0.695372 − 0.718650i \(-0.744761\pi\)
−0.695372 + 0.718650i \(0.744761\pi\)
\(74\) 0 0
\(75\) 1.44270 0.166588
\(76\) 0 0
\(77\) 4.54341 0.517769
\(78\) 0 0
\(79\) 10.4998 1.18132 0.590658 − 0.806922i \(-0.298868\pi\)
0.590658 + 0.806922i \(0.298868\pi\)
\(80\) 0 0
\(81\) −10.8136 −1.20151
\(82\) 0 0
\(83\) −0.454168 −0.0498514 −0.0249257 − 0.999689i \(-0.507935\pi\)
−0.0249257 + 0.999689i \(0.507935\pi\)
\(84\) 0 0
\(85\) −6.49976 −0.704998
\(86\) 0 0
\(87\) −20.4859 −2.19632
\(88\) 0 0
\(89\) 4.32128 0.458055 0.229027 − 0.973420i \(-0.426445\pi\)
0.229027 + 0.973420i \(0.426445\pi\)
\(90\) 0 0
\(91\) −0.932583 −0.0977612
\(92\) 0 0
\(93\) −0.716384 −0.0742855
\(94\) 0 0
\(95\) 6.22213 0.638377
\(96\) 0 0
\(97\) −11.5553 −1.17327 −0.586633 − 0.809853i \(-0.699547\pi\)
−0.586633 + 0.809853i \(0.699547\pi\)
\(98\) 0 0
\(99\) −8.83538 −0.887989
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.g.1.2 ✓ 4
3.2 odd 2 6624.2.a.bf.1.4 4
4.3 odd 2 736.2.a.h.1.3 yes 4
8.3 odd 2 1472.2.a.y.1.2 4
8.5 even 2 1472.2.a.z.1.3 4
12.11 even 2 6624.2.a.be.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.g.1.2 ✓ 4 1.1 even 1 trivial
736.2.a.h.1.3 yes 4 4.3 odd 2
1472.2.a.y.1.2 4 8.3 odd 2
1472.2.a.z.1.3 4 8.5 even 2
6624.2.a.be.1.4 4 12.11 even 2
6624.2.a.bf.1.4 4 3.2 odd 2