Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 736.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,14,0,30,0,-28] 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: \(43.4254057642\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 155x^{7} + 342x^{6} + 7139x^{5} - 8520x^{4} - 113229x^{3} + 12582x^{2} + 530388x + 301320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-6.97389\) 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)
 
\(f(q)\) \(=\) \(q+8.97389 q^{3} -6.92259 q^{5} -34.1625 q^{7} +53.5307 q^{9} -32.0243 q^{11} +61.5703 q^{13} -62.1226 q^{15} +105.992 q^{17} +132.294 q^{19} -306.571 q^{21} +23.0000 q^{23} -77.0777 q^{25} +238.084 q^{27} -25.6713 q^{29} +121.679 q^{31} -287.382 q^{33} +236.493 q^{35} +197.441 q^{37} +552.525 q^{39} +488.821 q^{41} -434.961 q^{43} -370.571 q^{45} -54.5330 q^{47} +824.079 q^{49} +951.159 q^{51} -106.629 q^{53} +221.691 q^{55} +1187.19 q^{57} +638.303 q^{59} -122.412 q^{61} -1828.75 q^{63} -426.226 q^{65} +126.987 q^{67} +206.400 q^{69} +98.7902 q^{71} +213.071 q^{73} -691.687 q^{75} +1094.03 q^{77} +1367.97 q^{79} +691.209 q^{81} +271.826 q^{83} -733.738 q^{85} -230.372 q^{87} +692.216 q^{89} -2103.40 q^{91} +1091.93 q^{93} -915.818 q^{95} +61.5121 q^{97} -1714.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 14 q^{3} + 30 q^{5} - 28 q^{7} + 103 q^{9} - 40 q^{11} + 76 q^{13} - 72 q^{15} + 166 q^{17} + 148 q^{19} + 96 q^{21} + 207 q^{23} + 339 q^{25} + 326 q^{27} + 252 q^{29} + 258 q^{31} + 460 q^{33} + 276 q^{35}+ \cdots + 2024 q^{99}+O(q^{100}) \) 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
\(3\) 8.97389 1.72703 0.863513 − 0.504326i \(-0.168259\pi\)
0.863513 + 0.504326i \(0.168259\pi\)
\(4\) 0 0
\(5\) −6.92259 −0.619175 −0.309588 − 0.950871i \(-0.600191\pi\)
−0.309588 + 0.950871i \(0.600191\pi\)
\(6\) 0 0
\(7\) −34.1625 −1.84460 −0.922302 − 0.386471i \(-0.873694\pi\)
−0.922302 + 0.386471i \(0.873694\pi\)
\(8\) 0 0
\(9\) 53.5307 1.98262
\(10\) 0 0
\(11\) −32.0243 −0.877789 −0.438895 − 0.898538i \(-0.644630\pi\)
−0.438895 + 0.898538i \(0.644630\pi\)
\(12\) 0 0
\(13\) 61.5703 1.31358 0.656789 − 0.754074i \(-0.271914\pi\)
0.656789 + 0.754074i \(0.271914\pi\)
\(14\) 0 0
\(15\) −62.1226 −1.06933
\(16\) 0 0
\(17\) 105.992 1.51216 0.756082 − 0.654477i \(-0.227111\pi\)
0.756082 + 0.654477i \(0.227111\pi\)
\(18\) 0 0
\(19\) 132.294 1.59739 0.798693 − 0.601739i \(-0.205525\pi\)
0.798693 + 0.601739i \(0.205525\pi\)
\(20\) 0 0
\(21\) −306.571 −3.18568
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) −77.0777 −0.616622
\(26\) 0 0
\(27\) 238.084 1.69701
\(28\) 0 0
\(29\) −25.6713 −0.164381 −0.0821905 − 0.996617i \(-0.526192\pi\)
−0.0821905 + 0.996617i \(0.526192\pi\)
\(30\) 0 0
\(31\) 121.679 0.704974 0.352487 − 0.935817i \(-0.385336\pi\)
0.352487 + 0.935817i \(0.385336\pi\)
\(32\) 0 0
\(33\) −287.382 −1.51597
\(34\) 0 0
\(35\) 236.493 1.14213
\(36\) 0 0
\(37\) 197.441 0.877274 0.438637 − 0.898664i \(-0.355461\pi\)
0.438637 + 0.898664i \(0.355461\pi\)
\(38\) 0 0
\(39\) 552.525 2.26858
\(40\) 0 0
\(41\) 488.821 1.86198 0.930989 − 0.365048i \(-0.118948\pi\)
0.930989 + 0.365048i \(0.118948\pi\)
\(42\) 0 0
\(43\) −434.961 −1.54258 −0.771290 − 0.636484i \(-0.780388\pi\)
−0.771290 + 0.636484i \(0.780388\pi\)
\(44\) 0 0
\(45\) −370.571 −1.22759
\(46\) 0 0
\(47\) −54.5330 −0.169244 −0.0846219 − 0.996413i \(-0.526968\pi\)
−0.0846219 + 0.996413i \(0.526968\pi\)
\(48\) 0 0
\(49\) 824.079 2.40256
\(50\) 0 0
\(51\) 951.159 2.61155
\(52\) 0 0
\(53\) −106.629 −0.276350 −0.138175 − 0.990408i \(-0.544124\pi\)
−0.138175 + 0.990408i \(0.544124\pi\)
\(54\) 0 0
\(55\) 221.691 0.543506
\(56\) 0 0
\(57\) 1187.19 2.75873
\(58\) 0 0
\(59\) 638.303 1.40847 0.704237 − 0.709965i \(-0.251289\pi\)
0.704237 + 0.709965i \(0.251289\pi\)
\(60\) 0 0
\(61\) −122.412 −0.256938 −0.128469 − 0.991714i \(-0.541006\pi\)
−0.128469 + 0.991714i \(0.541006\pi\)
\(62\) 0 0
\(63\) −1828.75 −3.65715
\(64\) 0 0
\(65\) −426.226 −0.813336
\(66\) 0 0
\(67\) 126.987 0.231551 0.115776 − 0.993275i \(-0.463065\pi\)
0.115776 + 0.993275i \(0.463065\pi\)
\(68\) 0 0
\(69\) 206.400 0.360110
\(70\) 0 0
\(71\) 98.7902 0.165130 0.0825651 − 0.996586i \(-0.473689\pi\)
0.0825651 + 0.996586i \(0.473689\pi\)
\(72\) 0 0
\(73\) 213.071 0.341618 0.170809 − 0.985304i \(-0.445362\pi\)
0.170809 + 0.985304i \(0.445362\pi\)
\(74\) 0 0
\(75\) −691.687 −1.06492
\(76\) 0 0
\(77\) 1094.03 1.61917
\(78\) 0 0
\(79\) 1367.97 1.94821 0.974105 − 0.226095i \(-0.0725959\pi\)
0.974105 + 0.226095i \(0.0725959\pi\)
\(80\) 0 0
\(81\) 691.209 0.948161
\(82\) 0 0
\(83\) 271.826 0.359479 0.179740 − 0.983714i \(-0.442474\pi\)
0.179740 + 0.983714i \(0.442474\pi\)
\(84\) 0 0
\(85\) −733.738 −0.936295
\(86\) 0 0
\(87\) −230.372 −0.283890
\(88\) 0 0
\(89\) 692.216 0.824435 0.412218 − 0.911085i \(-0.364754\pi\)
0.412218 + 0.911085i \(0.364754\pi\)
\(90\) 0 0
\(91\) −2103.40 −2.42303
\(92\) 0 0
\(93\) 1091.93 1.21751
\(94\) 0 0
\(95\) −915.818 −0.989062
\(96\) 0 0
\(97\) 61.5121 0.0643878 0.0321939 − 0.999482i \(-0.489751\pi\)
0.0321939 + 0.999482i \(0.489751\pi\)
\(98\) 0 0
\(99\) −1714.28 −1.74032
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 736.4.a.j.1.8 yes 9
4.3 odd 2 736.4.a.g.1.2 ✓ 9
8.3 odd 2 1472.4.a.bl.1.8 9
8.5 even 2 1472.4.a.bi.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.4.a.g.1.2 ✓ 9 4.3 odd 2
736.4.a.j.1.8 yes 9 1.1 even 1 trivial
1472.4.a.bi.1.2 9 8.5 even 2
1472.4.a.bl.1.8 9 8.3 odd 2