Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,6,Mod(1,880)] 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("880.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 880.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-100,0,90] 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: \(141.137761435\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.50110\) of defining polynomial
Character \(\chi\) \(=\) 880.1

$q$-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)
 
\(f(q)\) \(=\) \(q-22.6701 q^{3} -25.0000 q^{5} +169.118 q^{7} +270.932 q^{9} +121.000 q^{11} +25.3182 q^{13} +566.752 q^{15} -2016.26 q^{17} +773.486 q^{19} -3833.91 q^{21} +541.643 q^{23} +625.000 q^{25} -633.231 q^{27} -5882.28 q^{29} +915.584 q^{31} -2743.08 q^{33} -4227.94 q^{35} +11360.3 q^{37} -573.964 q^{39} -15477.7 q^{41} -6097.77 q^{43} -6773.31 q^{45} -15131.8 q^{47} +11793.8 q^{49} +45708.8 q^{51} +10443.0 q^{53} -3025.00 q^{55} -17535.0 q^{57} +50295.3 q^{59} +45523.0 q^{61} +45819.4 q^{63} -632.954 q^{65} -11285.0 q^{67} -12279.1 q^{69} -64741.3 q^{71} +77769.4 q^{73} -14168.8 q^{75} +20463.2 q^{77} +87890.2 q^{79} -51481.2 q^{81} -18403.3 q^{83} +50406.5 q^{85} +133352. q^{87} +52660.1 q^{89} +4281.74 q^{91} -20756.4 q^{93} -19337.2 q^{95} -38745.0 q^{97} +32782.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 100 q^{5} + 90 q^{7} + 22 q^{9} + 484 q^{11} + 820 q^{13} - 3800 q^{17} + 3394 q^{19} - 4708 q^{21} + 3020 q^{23} + 2500 q^{25} - 5400 q^{27} - 5248 q^{29} - 4732 q^{31} - 2250 q^{35} + 10210 q^{37}+ \cdots + 2662 q^{99}+O(q^{100}) \) 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
\(3\) −22.6701 −1.45429 −0.727143 − 0.686486i \(-0.759152\pi\)
−0.727143 + 0.686486i \(0.759152\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 169.118 1.30450 0.652249 − 0.758004i \(-0.273826\pi\)
0.652249 + 0.758004i \(0.273826\pi\)
\(8\) 0 0
\(9\) 270.932 1.11495
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) 25.3182 0.0415502 0.0207751 − 0.999784i \(-0.493387\pi\)
0.0207751 + 0.999784i \(0.493387\pi\)
\(14\) 0 0
\(15\) 566.752 0.650377
\(16\) 0 0
\(17\) −2016.26 −1.69209 −0.846047 − 0.533108i \(-0.821024\pi\)
−0.846047 + 0.533108i \(0.821024\pi\)
\(18\) 0 0
\(19\) 773.486 0.491551 0.245775 − 0.969327i \(-0.420957\pi\)
0.245775 + 0.969327i \(0.420957\pi\)
\(20\) 0 0
\(21\) −3833.91 −1.89711
\(22\) 0 0
\(23\) 541.643 0.213498 0.106749 − 0.994286i \(-0.465956\pi\)
0.106749 + 0.994286i \(0.465956\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −633.231 −0.167168
\(28\) 0 0
\(29\) −5882.28 −1.29882 −0.649412 − 0.760437i \(-0.724985\pi\)
−0.649412 + 0.760437i \(0.724985\pi\)
\(30\) 0 0
\(31\) 915.584 0.171117 0.0855587 − 0.996333i \(-0.472732\pi\)
0.0855587 + 0.996333i \(0.472732\pi\)
\(32\) 0 0
\(33\) −2743.08 −0.438484
\(34\) 0 0
\(35\) −4227.94 −0.583390
\(36\) 0 0
\(37\) 11360.3 1.36423 0.682115 − 0.731245i \(-0.261061\pi\)
0.682115 + 0.731245i \(0.261061\pi\)
\(38\) 0 0
\(39\) −573.964 −0.0604259
\(40\) 0 0
\(41\) −15477.7 −1.43796 −0.718979 − 0.695031i \(-0.755390\pi\)
−0.718979 + 0.695031i \(0.755390\pi\)
\(42\) 0 0
\(43\) −6097.77 −0.502921 −0.251460 − 0.967868i \(-0.580911\pi\)
−0.251460 + 0.967868i \(0.580911\pi\)
\(44\) 0 0
\(45\) −6773.31 −0.498620
\(46\) 0 0
\(47\) −15131.8 −0.999185 −0.499593 − 0.866260i \(-0.666517\pi\)
−0.499593 + 0.866260i \(0.666517\pi\)
\(48\) 0 0
\(49\) 11793.8 0.701717
\(50\) 0 0
\(51\) 45708.8 2.46079
\(52\) 0 0
\(53\) 10443.0 0.510666 0.255333 − 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) 0 0
\(57\) −17535.0 −0.714856
\(58\) 0 0
\(59\) 50295.3 1.88104 0.940519 − 0.339741i \(-0.110339\pi\)
0.940519 + 0.339741i \(0.110339\pi\)
\(60\) 0 0
\(61\) 45523.0 1.56641 0.783206 − 0.621762i \(-0.213583\pi\)
0.783206 + 0.621762i \(0.213583\pi\)
\(62\) 0 0
\(63\) 45819.4 1.45445
\(64\) 0 0
\(65\) −632.954 −0.0185818
\(66\) 0 0
\(67\) −11285.0 −0.307124 −0.153562 − 0.988139i \(-0.549074\pi\)
−0.153562 + 0.988139i \(0.549074\pi\)
\(68\) 0 0
\(69\) −12279.1 −0.310487
\(70\) 0 0
\(71\) −64741.3 −1.52418 −0.762089 − 0.647473i \(-0.775826\pi\)
−0.762089 + 0.647473i \(0.775826\pi\)
\(72\) 0 0
\(73\) 77769.4 1.70805 0.854027 − 0.520229i \(-0.174153\pi\)
0.854027 + 0.520229i \(0.174153\pi\)
\(74\) 0 0
\(75\) −14168.8 −0.290857
\(76\) 0 0
\(77\) 20463.2 0.393321
\(78\) 0 0
\(79\) 87890.2 1.58443 0.792214 − 0.610243i \(-0.208928\pi\)
0.792214 + 0.610243i \(0.208928\pi\)
\(80\) 0 0
\(81\) −51481.2 −0.871838
\(82\) 0 0
\(83\) −18403.3 −0.293225 −0.146613 − 0.989194i \(-0.546837\pi\)
−0.146613 + 0.989194i \(0.546837\pi\)
\(84\) 0 0
\(85\) 50406.5 0.756728
\(86\) 0 0
\(87\) 133352. 1.88886
\(88\) 0 0
\(89\) 52660.1 0.704704 0.352352 − 0.935868i \(-0.385382\pi\)
0.352352 + 0.935868i \(0.385382\pi\)
\(90\) 0 0
\(91\) 4281.74 0.0542022
\(92\) 0 0
\(93\) −20756.4 −0.248854
\(94\) 0 0
\(95\) −19337.2 −0.219828
\(96\) 0 0
\(97\) −38745.0 −0.418106 −0.209053 − 0.977904i \(-0.567038\pi\)
−0.209053 + 0.977904i \(0.567038\pi\)
\(98\) 0 0
\(99\) 32782.8 0.336170
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 880.6.a.n.1.1 4
4.3 odd 2 55.6.a.b.1.2 ✓ 4
12.11 even 2 495.6.a.g.1.3 4
20.3 even 4 275.6.b.d.199.6 8
20.7 even 4 275.6.b.d.199.3 8
20.19 odd 2 275.6.a.d.1.3 4
44.43 even 2 605.6.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.2 ✓ 4 4.3 odd 2
275.6.a.d.1.3 4 20.19 odd 2
275.6.b.d.199.3 8 20.7 even 4
275.6.b.d.199.6 8 20.3 even 4
495.6.a.g.1.3 4 12.11 even 2
605.6.a.c.1.3 4 44.43 even 2
880.6.a.n.1.1 4 1.1 even 1 trivial