Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3528,2,Mod(361,3528)] 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("3528.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3528.361
Dual form 3528.2.s.be.3313.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41421 - 2.44949i) q^{5} +(-2.00000 + 3.46410i) q^{11} +2.82843 q^{13} +(2.82843 - 4.89898i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(-1.50000 + 2.59808i) q^{25} -2.00000 q^{29} +(-2.82843 + 4.89898i) q^{31} +(-5.00000 - 8.66025i) q^{37} +5.65685 q^{41} -4.00000 q^{43} +(-2.82843 - 4.89898i) q^{47} +(3.00000 - 5.19615i) q^{53} +11.3137 q^{55} +(1.41421 - 2.44949i) q^{59} +(7.07107 + 12.2474i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(-6.00000 + 10.3923i) q^{67} +(-4.00000 - 6.92820i) q^{79} -14.1421 q^{83} -16.0000 q^{85} +(-4.00000 + 6.92820i) q^{95} -5.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{11} - 6 q^{25} - 8 q^{29} - 20 q^{37} - 16 q^{43} + 12 q^{53} - 16 q^{65} - 24 q^{67} - 16 q^{79} - 64 q^{85} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) −1.41421 − 2.44949i −0.632456 − 1.09545i −0.987048 − 0.160424i \(-0.948714\pi\)
0.354593 − 0.935021i \(-0.384620\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 − 4.89898i 0.685994 − 1.18818i −0.287129 − 0.957892i \(-0.592701\pi\)
0.973123 − 0.230285i \(-0.0739659\pi\)
\(18\) 0 0
\(19\) −1.41421 − 2.44949i −0.324443 − 0.561951i 0.656957 − 0.753928i \(-0.271843\pi\)
−0.981399 + 0.191977i \(0.938510\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −1.50000 + 2.59808i −0.300000 + 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −2.82843 + 4.89898i −0.508001 + 0.879883i 0.491957 + 0.870620i \(0.336282\pi\)
−0.999957 + 0.00926296i \(0.997051\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 − 8.66025i −0.821995 − 1.42374i −0.904194 − 0.427121i \(-0.859528\pi\)
0.0821995 − 0.996616i \(-0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 − 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 − 4.89898i −0.412568 − 0.714590i 0.582601 − 0.812758i \(-0.302035\pi\)
−0.995170 + 0.0981685i \(0.968702\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 11.3137 1.52554
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421 − 2.44949i 0.184115 − 0.318896i −0.759163 − 0.650901i \(-0.774391\pi\)
0.943278 + 0.332004i \(0.107725\pi\)
\(60\) 0 0
\(61\) 7.07107 + 12.2474i 0.905357 + 1.56813i 0.820437 + 0.571737i \(0.193730\pi\)
0.0849208 + 0.996388i \(0.472936\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 − 6.92820i −0.496139 − 0.859338i
\(66\) 0 0
\(67\) −6.00000 + 10.3923i −0.733017 + 1.26962i 0.222571 + 0.974916i \(0.428555\pi\)
−0.955588 + 0.294706i \(0.904778\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 − 6.92820i −0.450035 − 0.779484i 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.1421 −1.55230 −0.776151 − 0.630548i \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) −5.65685 −0.574367 −0.287183 − 0.957876i \(-0.592719\pi\)
−0.287183 + 0.957876i \(0.592719\pi\)
\(98\) 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 3528.2.s.be.361.1 4
3.2 odd 2 392.2.i.g.361.2 4
7.2 even 3 inner 3528.2.s.be.3313.1 4
7.3 odd 6 3528.2.a.bj.1.1 2
7.4 even 3 3528.2.a.bj.1.2 2
7.5 odd 6 inner 3528.2.s.be.3313.2 4
7.6 odd 2 inner 3528.2.s.be.361.2 4
12.11 even 2 784.2.i.k.753.1 4
21.2 odd 6 392.2.i.g.177.2 4
21.5 even 6 392.2.i.g.177.1 4
21.11 odd 6 392.2.a.h.1.1 ✓ 2
21.17 even 6 392.2.a.h.1.2 yes 2
21.20 even 2 392.2.i.g.361.1 4
28.3 even 6 7056.2.a.cj.1.1 2
28.11 odd 6 7056.2.a.cj.1.2 2
84.11 even 6 784.2.a.n.1.2 2
84.23 even 6 784.2.i.k.177.1 4
84.47 odd 6 784.2.i.k.177.2 4
84.59 odd 6 784.2.a.n.1.1 2
84.83 odd 2 784.2.i.k.753.2 4
105.59 even 6 9800.2.a.bw.1.1 2
105.74 odd 6 9800.2.a.bw.1.2 2
168.11 even 6 3136.2.a.bq.1.1 2
168.53 odd 6 3136.2.a.bt.1.2 2
168.59 odd 6 3136.2.a.bq.1.2 2
168.101 even 6 3136.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.2.a.h.1.1 ✓ 2 21.11 odd 6
392.2.a.h.1.2 yes 2 21.17 even 6
392.2.i.g.177.1 4 21.5 even 6
392.2.i.g.177.2 4 21.2 odd 6
392.2.i.g.361.1 4 21.20 even 2
392.2.i.g.361.2 4 3.2 odd 2
784.2.a.n.1.1 2 84.59 odd 6
784.2.a.n.1.2 2 84.11 even 6
784.2.i.k.177.1 4 84.23 even 6
784.2.i.k.177.2 4 84.47 odd 6
784.2.i.k.753.1 4 12.11 even 2
784.2.i.k.753.2 4 84.83 odd 2
3136.2.a.bq.1.1 2 168.11 even 6
3136.2.a.bq.1.2 2 168.59 odd 6
3136.2.a.bt.1.1 2 168.101 even 6
3136.2.a.bt.1.2 2 168.53 odd 6
3528.2.a.bj.1.1 2 7.3 odd 6
3528.2.a.bj.1.2 2 7.4 even 3
3528.2.s.be.361.1 4 1.1 even 1 trivial
3528.2.s.be.361.2 4 7.6 odd 2 inner
3528.2.s.be.3313.1 4 7.2 even 3 inner
3528.2.s.be.3313.2 4 7.5 odd 6 inner
7056.2.a.cj.1.1 2 28.3 even 6
7056.2.a.cj.1.2 2 28.11 odd 6
9800.2.a.bw.1.1 2 105.59 even 6
9800.2.a.bw.1.2 2 105.74 odd 6