Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,2,Mod(961,1040)] 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("1040.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.1
Root \(-2.60168i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.60168 q^{3} -1.00000i q^{5} -2.76873i q^{7} +3.76873 q^{9} -2.16706i q^{11} +(-0.167055 - 3.60168i) q^{13} +2.60168i q^{15} +5.03630i q^{19} +7.20336i q^{21} +4.93579 q^{23} -1.00000 q^{25} -2.00000 q^{27} -4.43462 q^{29} -3.37041i q^{31} +5.63798i q^{33} -2.76873 q^{35} -3.97209i q^{37} +(0.434624 + 9.37041i) q^{39} +1.66589i q^{41} -9.80504 q^{43} -3.76873i q^{45} -11.6380i q^{47} -0.665890 q^{49} -12.7408 q^{53} -2.16706 q^{55} -13.1028i q^{57} +8.16706i q^{59} -10.3062 q^{61} -10.4346i q^{63} +(-3.60168 + 0.167055i) q^{65} -7.10284i q^{67} -12.8413 q^{69} +16.1112i q^{71} +15.9721i q^{73} +2.60168 q^{75} -6.00000 q^{77} -11.9442 q^{79} -6.10284 q^{81} -3.97209i q^{83} +11.5375 q^{87} -7.94419i q^{89} +(-9.97209 + 0.462531i) q^{91} +8.76873i q^{93} +5.03630 q^{95} -0.462531i q^{97} -8.16706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −2.60168 −1.50208 −0.751040 − 0.660257i \(-0.770448\pi\)
−0.751040 + 0.660257i \(0.770448\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 2.76873i − 1.04648i −0.852184 − 0.523242i \(-0.824723\pi\)
0.852184 − 0.523242i \(-0.175277\pi\)
\(8\) 0 0
\(9\) 3.76873 1.25624
\(10\) 0 0
\(11\) − 2.16706i − 0.653392i −0.945130 − 0.326696i \(-0.894065\pi\)
0.945130 − 0.326696i \(-0.105935\pi\)
\(12\) 0 0
\(13\) −0.167055 − 3.60168i −0.0463328 − 0.998926i
\(14\) 0 0
\(15\) 2.60168i 0.671751i
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.03630i 1.15541i 0.816247 + 0.577704i \(0.196051\pi\)
−0.816247 + 0.577704i \(0.803949\pi\)
\(20\) 0 0
\(21\) 7.20336i 1.57190i
\(22\) 0 0
\(23\) 4.93579 1.02918 0.514592 − 0.857435i \(-0.327944\pi\)
0.514592 + 0.857435i \(0.327944\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) −4.43462 −0.823489 −0.411744 − 0.911299i \(-0.635080\pi\)
−0.411744 + 0.911299i \(0.635080\pi\)
\(30\) 0 0
\(31\) − 3.37041i − 0.605344i −0.953095 − 0.302672i \(-0.902121\pi\)
0.953095 − 0.302672i \(-0.0978787\pi\)
\(32\) 0 0
\(33\) 5.63798i 0.981447i
\(34\) 0 0
\(35\) −2.76873 −0.468002
\(36\) 0 0
\(37\) − 3.97209i − 0.653008i −0.945196 − 0.326504i \(-0.894129\pi\)
0.945196 − 0.326504i \(-0.105871\pi\)
\(38\) 0 0
\(39\) 0.434624 + 9.37041i 0.0695955 + 1.50047i
\(40\) 0 0
\(41\) 1.66589i 0.260168i 0.991503 + 0.130084i \(0.0415247\pi\)
−0.991503 + 0.130084i \(0.958475\pi\)
\(42\) 0 0
\(43\) −9.80504 −1.49525 −0.747627 − 0.664119i \(-0.768807\pi\)
−0.747627 + 0.664119i \(0.768807\pi\)
\(44\) 0 0
\(45\) − 3.76873i − 0.561810i
\(46\) 0 0
\(47\) − 11.6380i − 1.69757i −0.528735 − 0.848787i \(-0.677333\pi\)
0.528735 − 0.848787i \(-0.322667\pi\)
\(48\) 0 0
\(49\) −0.665890 −0.0951271
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.7408 −1.75009 −0.875044 − 0.484044i \(-0.839167\pi\)
−0.875044 + 0.484044i \(0.839167\pi\)
\(54\) 0 0
\(55\) −2.16706 −0.292206
\(56\) 0 0
\(57\) − 13.1028i − 1.73551i
\(58\) 0 0
\(59\) 8.16706i 1.06326i 0.846977 + 0.531630i \(0.178420\pi\)
−0.846977 + 0.531630i \(0.821580\pi\)
\(60\) 0 0
\(61\) −10.3062 −1.31957 −0.659787 − 0.751453i \(-0.729354\pi\)
−0.659787 + 0.751453i \(0.729354\pi\)
\(62\) 0 0
\(63\) − 10.4346i − 1.31464i
\(64\) 0 0
\(65\) −3.60168 + 0.167055i −0.446733 + 0.0207206i
\(66\) 0 0
\(67\) − 7.10284i − 0.867751i −0.900973 − 0.433875i \(-0.857146\pi\)
0.900973 − 0.433875i \(-0.142854\pi\)
\(68\) 0 0
\(69\) −12.8413 −1.54592
\(70\) 0 0
\(71\) 16.1112i 1.91205i 0.293281 + 0.956026i \(0.405253\pi\)
−0.293281 + 0.956026i \(0.594747\pi\)
\(72\) 0 0
\(73\) 15.9721i 1.86939i 0.355448 + 0.934696i \(0.384328\pi\)
−0.355448 + 0.934696i \(0.615672\pi\)
\(74\) 0 0
\(75\) 2.60168 0.300416
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −11.9442 −1.34383 −0.671913 − 0.740630i \(-0.734527\pi\)
−0.671913 + 0.740630i \(0.734527\pi\)
\(80\) 0 0
\(81\) −6.10284 −0.678094
\(82\) 0 0
\(83\) − 3.97209i − 0.435994i −0.975949 − 0.217997i \(-0.930048\pi\)
0.975949 − 0.217997i \(-0.0699523\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.5375 1.23695
\(88\) 0 0
\(89\) − 7.94419i − 0.842082i −0.907042 − 0.421041i \(-0.861665\pi\)
0.907042 − 0.421041i \(-0.138335\pi\)
\(90\) 0 0
\(91\) −9.97209 + 0.462531i −1.04536 + 0.0484865i
\(92\) 0 0
\(93\) 8.76873i 0.909275i
\(94\) 0 0
\(95\) 5.03630 0.516714
\(96\) 0 0
\(97\) − 0.462531i − 0.0469629i −0.999724 − 0.0234815i \(-0.992525\pi\)
0.999724 − 0.0234815i \(-0.00747507\pi\)
\(98\) 0 0
\(99\) − 8.16706i − 0.820820i
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 1040.2.k.c.961.1 6
4.3 odd 2 260.2.f.a.181.5 ✓ 6
12.11 even 2 2340.2.c.d.181.6 6
13.12 even 2 inner 1040.2.k.c.961.2 6
20.3 even 4 1300.2.d.d.649.6 6
20.7 even 4 1300.2.d.c.649.1 6
20.19 odd 2 1300.2.f.e.701.1 6
52.31 even 4 3380.2.a.m.1.3 3
52.47 even 4 3380.2.a.n.1.3 3
52.51 odd 2 260.2.f.a.181.6 yes 6
156.155 even 2 2340.2.c.d.181.1 6
260.103 even 4 1300.2.d.c.649.6 6
260.207 even 4 1300.2.d.d.649.1 6
260.259 odd 2 1300.2.f.e.701.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.5 ✓ 6 4.3 odd 2
260.2.f.a.181.6 yes 6 52.51 odd 2
1040.2.k.c.961.1 6 1.1 even 1 trivial
1040.2.k.c.961.2 6 13.12 even 2 inner
1300.2.d.c.649.1 6 20.7 even 4
1300.2.d.c.649.6 6 260.103 even 4
1300.2.d.d.649.1 6 260.207 even 4
1300.2.d.d.649.6 6 20.3 even 4
1300.2.f.e.701.1 6 20.19 odd 2
1300.2.f.e.701.2 6 260.259 odd 2
2340.2.c.d.181.1 6 156.155 even 2
2340.2.c.d.181.6 6 12.11 even 2
3380.2.a.m.1.3 3 52.31 even 4
3380.2.a.n.1.3 3 52.47 even 4