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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5760,2,Mod(2881,5760)] 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("5760.2881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5760.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,-4,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.9938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2881.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 5760.2881
Dual form 5760.2.k.o.2881.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{5} -4.24264 q^{7} +5.65685i q^{11} -2.00000i q^{13} -6.00000 q^{17} -2.82843i q^{19} -7.07107 q^{23} -1.00000 q^{25} +4.00000i q^{29} -2.82843 q^{31} -4.24264i q^{35} -2.00000i q^{37} +8.00000 q^{41} +1.41421i q^{43} +1.41421 q^{47} +11.0000 q^{49} -2.00000i q^{53} -5.65685 q^{55} +2.82843i q^{59} -14.0000i q^{61} +2.00000 q^{65} -4.24264i q^{67} -2.82843 q^{71} -6.00000 q^{73} -24.0000i q^{77} +16.9706 q^{79} +12.7279i q^{83} -6.00000i q^{85} +6.00000 q^{89} +8.48528i q^{91} +2.82843 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{17} - 4 q^{25} + 32 q^{41} + 44 q^{49} + 8 q^{65} - 24 q^{73} + 24 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5760\mathbb{Z}\right)^\times\).

\(n\) \(641\) \(901\) \(2431\) \(3457\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 − 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) − 2.82843i − 0.648886i −0.945905 − 0.324443i \(-0.894823\pi\)
0.945905 − 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.07107 −1.47442 −0.737210 − 0.675664i \(-0.763857\pi\)
−0.737210 + 0.675664i \(0.763857\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 − 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.24264i − 0.717137i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 1.41421i 0.215666i 0.994169 + 0.107833i \(0.0343911\pi\)
−0.994169 + 0.107833i \(0.965609\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41421 0.206284 0.103142 − 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.82843i 0.368230i 0.982905 + 0.184115i \(0.0589419\pi\)
−0.982905 + 0.184115i \(0.941058\pi\)
\(60\) 0 0
\(61\) − 14.0000i − 1.79252i −0.443533 − 0.896258i \(-0.646275\pi\)
0.443533 − 0.896258i \(-0.353725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) − 4.24264i − 0.518321i −0.965834 − 0.259161i \(-0.916554\pi\)
0.965834 − 0.259161i \(-0.0834459\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 − 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 24.0000i − 2.73505i
\(78\) 0 0
\(79\) 16.9706 1.90934 0.954669 − 0.297670i \(-0.0962096\pi\)
0.954669 + 0.297670i \(0.0962096\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.7279i 1.39707i 0.715575 + 0.698535i \(0.246165\pi\)
−0.715575 + 0.698535i \(0.753835\pi\)
\(84\) 0 0
\(85\) − 6.00000i − 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.48528i 0.889499i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 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 5760.2.k.o.2881.3 4
3.2 odd 2 640.2.d.d.321.3 yes 4
4.3 odd 2 inner 5760.2.k.o.2881.4 4
8.3 odd 2 inner 5760.2.k.o.2881.2 4
8.5 even 2 inner 5760.2.k.o.2881.1 4
12.11 even 2 640.2.d.d.321.1 ✓ 4
15.2 even 4 3200.2.f.i.449.3 4
15.8 even 4 3200.2.f.j.449.2 4
15.14 odd 2 3200.2.d.s.1601.2 4
24.5 odd 2 640.2.d.d.321.2 yes 4
24.11 even 2 640.2.d.d.321.4 yes 4
48.5 odd 4 1280.2.a.j.1.2 2
48.11 even 4 1280.2.a.j.1.1 2
48.29 odd 4 1280.2.a.f.1.1 2
48.35 even 4 1280.2.a.f.1.2 2
60.23 odd 4 3200.2.f.j.449.3 4
60.47 odd 4 3200.2.f.i.449.2 4
60.59 even 2 3200.2.d.s.1601.3 4
120.29 odd 2 3200.2.d.s.1601.4 4
120.53 even 4 3200.2.f.i.449.4 4
120.59 even 2 3200.2.d.s.1601.1 4
120.77 even 4 3200.2.f.j.449.1 4
120.83 odd 4 3200.2.f.i.449.1 4
120.107 odd 4 3200.2.f.j.449.4 4
240.29 odd 4 6400.2.a.bl.1.2 2
240.59 even 4 6400.2.a.bn.1.2 2
240.149 odd 4 6400.2.a.bn.1.1 2
240.179 even 4 6400.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.d.321.1 ✓ 4 12.11 even 2
640.2.d.d.321.2 yes 4 24.5 odd 2
640.2.d.d.321.3 yes 4 3.2 odd 2
640.2.d.d.321.4 yes 4 24.11 even 2
1280.2.a.f.1.1 2 48.29 odd 4
1280.2.a.f.1.2 2 48.35 even 4
1280.2.a.j.1.1 2 48.11 even 4
1280.2.a.j.1.2 2 48.5 odd 4
3200.2.d.s.1601.1 4 120.59 even 2
3200.2.d.s.1601.2 4 15.14 odd 2
3200.2.d.s.1601.3 4 60.59 even 2
3200.2.d.s.1601.4 4 120.29 odd 2
3200.2.f.i.449.1 4 120.83 odd 4
3200.2.f.i.449.2 4 60.47 odd 4
3200.2.f.i.449.3 4 15.2 even 4
3200.2.f.i.449.4 4 120.53 even 4
3200.2.f.j.449.1 4 120.77 even 4
3200.2.f.j.449.2 4 15.8 even 4
3200.2.f.j.449.3 4 60.23 odd 4
3200.2.f.j.449.4 4 120.107 odd 4
5760.2.k.o.2881.1 4 8.5 even 2 inner
5760.2.k.o.2881.2 4 8.3 odd 2 inner
5760.2.k.o.2881.3 4 1.1 even 1 trivial
5760.2.k.o.2881.4 4 4.3 odd 2 inner
6400.2.a.bl.1.1 2 240.179 even 4
6400.2.a.bl.1.2 2 240.29 odd 4
6400.2.a.bn.1.1 2 240.149 odd 4
6400.2.a.bn.1.2 2 240.59 even 4