Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7616,2,Mod(1,7616)] 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("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,-1,0,-3,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.8140661794\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 7616.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.86081 q^{3} +0.462598 q^{5} -1.00000 q^{7} +5.18421 q^{9} +2.00000 q^{11} -6.64681 q^{13} +1.32340 q^{15} -1.00000 q^{17} -4.64681 q^{19} -2.86081 q^{21} -5.72161 q^{23} -4.78600 q^{25} +6.24860 q^{27} -0.796415 q^{29} +0.184210 q^{31} +5.72161 q^{33} -0.462598 q^{35} +7.44322 q^{37} -19.0152 q^{39} -6.71120 q^{41} -5.50761 q^{43} +2.39821 q^{45} -5.72161 q^{47} +1.00000 q^{49} -2.86081 q^{51} +4.33382 q^{53} +0.925197 q^{55} -13.2936 q^{57} -0.646809 q^{59} -4.86081 q^{61} -5.18421 q^{63} -3.07480 q^{65} +15.7562 q^{67} -16.3684 q^{69} -0.556777 q^{71} -4.73202 q^{73} -13.6918 q^{75} -2.00000 q^{77} -13.1648 q^{79} +2.32340 q^{81} +12.2188 q^{83} -0.462598 q^{85} -2.27839 q^{87} -6.36842 q^{89} +6.64681 q^{91} +0.526989 q^{93} -2.14961 q^{95} +14.6814 q^{97} +10.3684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{13} - 4 q^{15} - 3 q^{17} + 2 q^{19} - 3 q^{21} - 6 q^{23} - 4 q^{25} + 6 q^{27} + 4 q^{29} - 13 q^{31} + 6 q^{33} + q^{35} - 14 q^{39} - 5 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) 2.86081 1.65169 0.825844 − 0.563899i \(-0.190699\pi\)
0.825844 + 0.563899i \(0.190699\pi\)
\(4\) 0 0
\(5\) 0.462598 0.206880 0.103440 − 0.994636i \(-0.467015\pi\)
0.103440 + 0.994636i \(0.467015\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 5.18421 1.72807
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −6.64681 −1.84349 −0.921747 − 0.387793i \(-0.873238\pi\)
−0.921747 + 0.387793i \(0.873238\pi\)
\(14\) 0 0
\(15\) 1.32340 0.341702
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −4.64681 −1.06605 −0.533025 − 0.846099i \(-0.678945\pi\)
−0.533025 + 0.846099i \(0.678945\pi\)
\(20\) 0 0
\(21\) −2.86081 −0.624279
\(22\) 0 0
\(23\) −5.72161 −1.19304 −0.596519 − 0.802599i \(-0.703450\pi\)
−0.596519 + 0.802599i \(0.703450\pi\)
\(24\) 0 0
\(25\) −4.78600 −0.957201
\(26\) 0 0
\(27\) 6.24860 1.20254
\(28\) 0 0
\(29\) −0.796415 −0.147891 −0.0739453 − 0.997262i \(-0.523559\pi\)
−0.0739453 + 0.997262i \(0.523559\pi\)
\(30\) 0 0
\(31\) 0.184210 0.0330851 0.0165426 − 0.999863i \(-0.494734\pi\)
0.0165426 + 0.999863i \(0.494734\pi\)
\(32\) 0 0
\(33\) 5.72161 0.996005
\(34\) 0 0
\(35\) −0.462598 −0.0781934
\(36\) 0 0
\(37\) 7.44322 1.22366 0.611829 − 0.790990i \(-0.290434\pi\)
0.611829 + 0.790990i \(0.290434\pi\)
\(38\) 0 0
\(39\) −19.0152 −3.04487
\(40\) 0 0
\(41\) −6.71120 −1.04811 −0.524057 − 0.851683i \(-0.675582\pi\)
−0.524057 + 0.851683i \(0.675582\pi\)
\(42\) 0 0
\(43\) −5.50761 −0.839903 −0.419952 − 0.907546i \(-0.637953\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(44\) 0 0
\(45\) 2.39821 0.357504
\(46\) 0 0
\(47\) −5.72161 −0.834583 −0.417291 − 0.908773i \(-0.637021\pi\)
−0.417291 + 0.908773i \(0.637021\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.86081 −0.400593
\(52\) 0 0
\(53\) 4.33382 0.595295 0.297648 − 0.954676i \(-0.403798\pi\)
0.297648 + 0.954676i \(0.403798\pi\)
\(54\) 0 0
\(55\) 0.925197 0.124754
\(56\) 0 0
\(57\) −13.2936 −1.76078
\(58\) 0 0
\(59\) −0.646809 −0.0842073 −0.0421037 − 0.999113i \(-0.513406\pi\)
−0.0421037 + 0.999113i \(0.513406\pi\)
\(60\) 0 0
\(61\) −4.86081 −0.622362 −0.311181 − 0.950351i \(-0.600725\pi\)
−0.311181 + 0.950351i \(0.600725\pi\)
\(62\) 0 0
\(63\) −5.18421 −0.653149
\(64\) 0 0
\(65\) −3.07480 −0.381382
\(66\) 0 0
\(67\) 15.7562 1.92493 0.962464 − 0.271409i \(-0.0874897\pi\)
0.962464 + 0.271409i \(0.0874897\pi\)
\(68\) 0 0
\(69\) −16.3684 −1.97053
\(70\) 0 0
\(71\) −0.556777 −0.0660772 −0.0330386 − 0.999454i \(-0.510518\pi\)
−0.0330386 + 0.999454i \(0.510518\pi\)
\(72\) 0 0
\(73\) −4.73202 −0.553842 −0.276921 − 0.960893i \(-0.589314\pi\)
−0.276921 + 0.960893i \(0.589314\pi\)
\(74\) 0 0
\(75\) −13.6918 −1.58100
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −13.1648 −1.48116 −0.740580 − 0.671968i \(-0.765449\pi\)
−0.740580 + 0.671968i \(0.765449\pi\)
\(80\) 0 0
\(81\) 2.32340 0.258156
\(82\) 0 0
\(83\) 12.2188 1.34119 0.670595 − 0.741824i \(-0.266039\pi\)
0.670595 + 0.741824i \(0.266039\pi\)
\(84\) 0 0
\(85\) −0.462598 −0.0501758
\(86\) 0 0
\(87\) −2.27839 −0.244269
\(88\) 0 0
\(89\) −6.36842 −0.675051 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(90\) 0 0
\(91\) 6.64681 0.696775
\(92\) 0 0
\(93\) 0.526989 0.0546463
\(94\) 0 0
\(95\) −2.14961 −0.220545
\(96\) 0 0
\(97\) 14.6814 1.49067 0.745336 − 0.666689i \(-0.232289\pi\)
0.745336 + 0.666689i \(0.232289\pi\)
\(98\) 0 0
\(99\) 10.3684 1.04207
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 7616.2.a.bg.1.3 3
4.3 odd 2 7616.2.a.ba.1.1 3
8.3 odd 2 1904.2.a.p.1.3 3
8.5 even 2 952.2.a.d.1.1 ✓ 3
24.5 odd 2 8568.2.a.z.1.2 3
56.13 odd 2 6664.2.a.l.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.d.1.1 ✓ 3 8.5 even 2
1904.2.a.p.1.3 3 8.3 odd 2
6664.2.a.l.1.3 3 56.13 odd 2
7616.2.a.ba.1.1 3 4.3 odd 2
7616.2.a.bg.1.3 3 1.1 even 1 trivial
8568.2.a.z.1.2 3 24.5 odd 2