Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,6,Mod(1,1008)] 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("1008.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,76,0,-98,0,0,0,-564,0,516] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(161.666890371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 504)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-10.2956\) of defining polynomial
Character \(\chi\) \(=\) 1008.1

$q$-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)
 
\(f(q)\) \(=\) \(q-23.7738 q^{5} -49.0000 q^{7} -590.869 q^{11} +505.095 q^{13} +517.964 q^{17} -932.905 q^{19} +3550.73 q^{23} -2559.81 q^{25} +5395.48 q^{29} +8329.00 q^{31} +1164.92 q^{35} -4936.43 q^{37} +5880.27 q^{41} -13540.8 q^{43} +7357.78 q^{47} +2401.00 q^{49} -3341.88 q^{53} +14047.2 q^{55} -42675.2 q^{59} +51327.6 q^{61} -12008.0 q^{65} +33755.0 q^{67} +17848.0 q^{71} -20628.2 q^{73} +28952.6 q^{77} -99202.0 q^{79} -61202.1 q^{83} -12314.0 q^{85} +90153.0 q^{89} -24749.7 q^{91} +22178.7 q^{95} -155739. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 76 q^{5} - 98 q^{7} - 564 q^{11} + 516 q^{13} - 76 q^{17} - 2360 q^{19} + 2036 q^{23} + 4270 q^{25} + 8320 q^{29} + 6280 q^{31} - 3724 q^{35} - 2460 q^{37} - 14308 q^{41} - 23128 q^{43} - 12712 q^{47}+ \cdots - 272932 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) −23.7738 −0.425278 −0.212639 − 0.977131i \(-0.568206\pi\)
−0.212639 + 0.977131i \(0.568206\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −590.869 −1.47234 −0.736172 − 0.676794i \(-0.763369\pi\)
−0.736172 + 0.676794i \(0.763369\pi\)
\(12\) 0 0
\(13\) 505.095 0.828924 0.414462 − 0.910067i \(-0.363970\pi\)
0.414462 + 0.910067i \(0.363970\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 517.964 0.434688 0.217344 − 0.976095i \(-0.430261\pi\)
0.217344 + 0.976095i \(0.430261\pi\)
\(18\) 0 0
\(19\) −932.905 −0.592862 −0.296431 − 0.955054i \(-0.595796\pi\)
−0.296431 + 0.955054i \(0.595796\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3550.73 1.39958 0.699790 − 0.714349i \(-0.253277\pi\)
0.699790 + 0.714349i \(0.253277\pi\)
\(24\) 0 0
\(25\) −2559.81 −0.819138
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5395.48 1.19134 0.595669 − 0.803230i \(-0.296887\pi\)
0.595669 + 0.803230i \(0.296887\pi\)
\(30\) 0 0
\(31\) 8329.00 1.55664 0.778321 − 0.627867i \(-0.216072\pi\)
0.778321 + 0.627867i \(0.216072\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1164.92 0.160740
\(36\) 0 0
\(37\) −4936.43 −0.592800 −0.296400 − 0.955064i \(-0.595786\pi\)
−0.296400 + 0.955064i \(0.595786\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5880.27 0.546308 0.273154 − 0.961970i \(-0.411933\pi\)
0.273154 + 0.961970i \(0.411933\pi\)
\(42\) 0 0
\(43\) −13540.8 −1.11679 −0.558396 − 0.829575i \(-0.688583\pi\)
−0.558396 + 0.829575i \(0.688583\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7357.78 0.485850 0.242925 − 0.970045i \(-0.421893\pi\)
0.242925 + 0.970045i \(0.421893\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3341.88 −0.163419 −0.0817093 − 0.996656i \(-0.526038\pi\)
−0.0817093 + 0.996656i \(0.526038\pi\)
\(54\) 0 0
\(55\) 14047.2 0.626156
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −42675.2 −1.59605 −0.798023 − 0.602626i \(-0.794121\pi\)
−0.798023 + 0.602626i \(0.794121\pi\)
\(60\) 0 0
\(61\) 51327.6 1.76615 0.883073 − 0.469235i \(-0.155470\pi\)
0.883073 + 0.469235i \(0.155470\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12008.0 −0.352523
\(66\) 0 0
\(67\) 33755.0 0.918651 0.459325 − 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 17848.0 0.420188 0.210094 − 0.977681i \(-0.432623\pi\)
0.210094 + 0.977681i \(0.432623\pi\)
\(72\) 0 0
\(73\) −20628.2 −0.453058 −0.226529 − 0.974004i \(-0.572738\pi\)
−0.226529 + 0.974004i \(0.572738\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 28952.6 0.556494
\(78\) 0 0
\(79\) −99202.0 −1.78835 −0.894175 − 0.447718i \(-0.852237\pi\)
−0.894175 + 0.447718i \(0.852237\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −61202.1 −0.975149 −0.487575 − 0.873081i \(-0.662118\pi\)
−0.487575 + 0.873081i \(0.662118\pi\)
\(84\) 0 0
\(85\) −12314.0 −0.184863
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 90153.0 1.20644 0.603219 − 0.797576i \(-0.293884\pi\)
0.603219 + 0.797576i \(0.293884\pi\)
\(90\) 0 0
\(91\) −24749.7 −0.313304
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 22178.7 0.252131
\(96\) 0 0
\(97\) −155739. −1.68062 −0.840309 − 0.542107i \(-0.817627\pi\)
−0.840309 + 0.542107i \(0.817627\pi\)
\(98\) 0 0
\(99\) 0 0
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 1008.6.a.bv.1.1 2
3.2 odd 2 1008.6.a.bg.1.2 2
4.3 odd 2 504.6.a.u.1.1 yes 2
12.11 even 2 504.6.a.k.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.k.1.2 ✓ 2 12.11 even 2
504.6.a.u.1.1 yes 2 4.3 odd 2
1008.6.a.bg.1.2 2 3.2 odd 2
1008.6.a.bv.1.1 2 1.1 even 1 trivial