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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3528,3,Mod(1961,3528)] 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("3528.1961"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3528.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,32,0,0,0,0,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(96.1310372663\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1961.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 3528.1961
Dual form 3528.3.d.a.1961.2

$q$-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)
 
\(f(q)\) \(=\) \(q-7.07107i q^{5} -5.65685i q^{11} +8.00000 q^{13} -9.89949i q^{17} +16.0000 q^{19} -39.5980i q^{23} -25.0000 q^{25} -29.6985i q^{29} +4.00000 q^{31} +30.0000 q^{37} +21.2132i q^{41} -8.00000 q^{43} +16.9706i q^{47} +49.4975i q^{53} -40.0000 q^{55} -79.1960i q^{59} +14.0000 q^{61} -56.5685i q^{65} -88.0000 q^{67} -28.2843i q^{71} +80.0000 q^{73} +100.000 q^{79} +130.108i q^{83} -70.0000 q^{85} -148.492i q^{89} -113.137i q^{95} +112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{13} + 32 q^{19} - 50 q^{25} + 8 q^{31} + 60 q^{37} - 16 q^{43} - 80 q^{55} + 28 q^{61} - 176 q^{67} + 160 q^{73} + 200 q^{79} - 140 q^{85} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) − 7.07107i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.65685i − 0.514259i −0.966377 − 0.257130i \(-0.917223\pi\)
0.966377 − 0.257130i \(-0.0827768\pi\)
\(12\) 0 0
\(13\) 8.00000 0.615385 0.307692 − 0.951486i \(-0.400443\pi\)
0.307692 + 0.951486i \(0.400443\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 9.89949i − 0.582323i −0.956674 − 0.291162i \(-0.905958\pi\)
0.956674 − 0.291162i \(-0.0940417\pi\)
\(18\) 0 0
\(19\) 16.0000 0.842105 0.421053 − 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 39.5980i − 1.72165i −0.508900 − 0.860826i \(-0.669948\pi\)
0.508900 − 0.860826i \(-0.330052\pi\)
\(24\) 0 0
\(25\) −25.0000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 29.6985i − 1.02409i −0.858960 − 0.512043i \(-0.828889\pi\)
0.858960 − 0.512043i \(-0.171111\pi\)
\(30\) 0 0
\(31\) 4.00000 0.129032 0.0645161 − 0.997917i \(-0.479450\pi\)
0.0645161 + 0.997917i \(0.479450\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 30.0000 0.810811 0.405405 − 0.914137i \(-0.367130\pi\)
0.405405 + 0.914137i \(0.367130\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 21.2132i 0.517395i 0.965958 + 0.258698i \(0.0832933\pi\)
−0.965958 + 0.258698i \(0.916707\pi\)
\(42\) 0 0
\(43\) −8.00000 −0.186047 −0.0930233 − 0.995664i \(-0.529653\pi\)
−0.0930233 + 0.995664i \(0.529653\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 49.4975i 0.933915i 0.884280 + 0.466957i \(0.154650\pi\)
−0.884280 + 0.466957i \(0.845350\pi\)
\(54\) 0 0
\(55\) −40.0000 −0.727273
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 79.1960i − 1.34230i −0.741320 − 0.671152i \(-0.765800\pi\)
0.741320 − 0.671152i \(-0.234200\pi\)
\(60\) 0 0
\(61\) 14.0000 0.229508 0.114754 − 0.993394i \(-0.463392\pi\)
0.114754 + 0.993394i \(0.463392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 56.5685i − 0.870285i
\(66\) 0 0
\(67\) −88.0000 −1.31343 −0.656716 − 0.754138i \(-0.728055\pi\)
−0.656716 + 0.754138i \(0.728055\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 28.2843i − 0.398370i −0.979962 − 0.199185i \(-0.936171\pi\)
0.979962 − 0.199185i \(-0.0638295\pi\)
\(72\) 0 0
\(73\) 80.0000 1.09589 0.547945 − 0.836514i \(-0.315410\pi\)
0.547945 + 0.836514i \(0.315410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 100.000 1.26582 0.632911 − 0.774224i \(-0.281860\pi\)
0.632911 + 0.774224i \(0.281860\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 130.108i 1.56756i 0.621037 + 0.783781i \(0.286712\pi\)
−0.621037 + 0.783781i \(0.713288\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.823529
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 148.492i − 1.66845i −0.551421 − 0.834227i \(-0.685914\pi\)
0.551421 − 0.834227i \(-0.314086\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 113.137i − 1.19092i
\(96\) 0 0
\(97\) 112.000 1.15464 0.577320 − 0.816518i \(-0.304099\pi\)
0.577320 + 0.816518i \(0.304099\pi\)
\(98\) 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 3528.3.d.a.1961.1 2
3.2 odd 2 inner 3528.3.d.a.1961.2 2
7.6 odd 2 72.3.e.a.17.2 yes 2
21.20 even 2 72.3.e.a.17.1 ✓ 2
28.27 even 2 144.3.e.a.17.2 2
35.13 even 4 1800.3.c.a.449.1 4
35.27 even 4 1800.3.c.a.449.3 4
35.34 odd 2 1800.3.l.a.1601.1 2
56.13 odd 2 576.3.e.h.449.1 2
56.27 even 2 576.3.e.a.449.1 2
63.13 odd 6 648.3.m.a.593.2 4
63.20 even 6 648.3.m.a.377.2 4
63.34 odd 6 648.3.m.a.377.1 4
63.41 even 6 648.3.m.a.593.1 4
84.83 odd 2 144.3.e.a.17.1 2
105.62 odd 4 1800.3.c.a.449.4 4
105.83 odd 4 1800.3.c.a.449.2 4
105.104 even 2 1800.3.l.a.1601.2 2
112.13 odd 4 2304.3.h.a.2177.3 4
112.27 even 4 2304.3.h.h.2177.2 4
112.69 odd 4 2304.3.h.a.2177.2 4
112.83 even 4 2304.3.h.h.2177.3 4
140.27 odd 4 3600.3.c.c.449.2 4
140.83 odd 4 3600.3.c.c.449.4 4
140.139 even 2 3600.3.l.l.1601.2 2
168.83 odd 2 576.3.e.a.449.2 2
168.125 even 2 576.3.e.h.449.2 2
252.83 odd 6 1296.3.q.k.1025.2 4
252.139 even 6 1296.3.q.k.593.2 4
252.167 odd 6 1296.3.q.k.593.1 4
252.223 even 6 1296.3.q.k.1025.1 4
336.83 odd 4 2304.3.h.h.2177.1 4
336.125 even 4 2304.3.h.a.2177.1 4
336.251 odd 4 2304.3.h.h.2177.4 4
336.293 even 4 2304.3.h.a.2177.4 4
420.83 even 4 3600.3.c.c.449.3 4
420.167 even 4 3600.3.c.c.449.1 4
420.419 odd 2 3600.3.l.l.1601.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.e.a.17.1 ✓ 2 21.20 even 2
72.3.e.a.17.2 yes 2 7.6 odd 2
144.3.e.a.17.1 2 84.83 odd 2
144.3.e.a.17.2 2 28.27 even 2
576.3.e.a.449.1 2 56.27 even 2
576.3.e.a.449.2 2 168.83 odd 2
576.3.e.h.449.1 2 56.13 odd 2
576.3.e.h.449.2 2 168.125 even 2
648.3.m.a.377.1 4 63.34 odd 6
648.3.m.a.377.2 4 63.20 even 6
648.3.m.a.593.1 4 63.41 even 6
648.3.m.a.593.2 4 63.13 odd 6
1296.3.q.k.593.1 4 252.167 odd 6
1296.3.q.k.593.2 4 252.139 even 6
1296.3.q.k.1025.1 4 252.223 even 6
1296.3.q.k.1025.2 4 252.83 odd 6
1800.3.c.a.449.1 4 35.13 even 4
1800.3.c.a.449.2 4 105.83 odd 4
1800.3.c.a.449.3 4 35.27 even 4
1800.3.c.a.449.4 4 105.62 odd 4
1800.3.l.a.1601.1 2 35.34 odd 2
1800.3.l.a.1601.2 2 105.104 even 2
2304.3.h.a.2177.1 4 336.125 even 4
2304.3.h.a.2177.2 4 112.69 odd 4
2304.3.h.a.2177.3 4 112.13 odd 4
2304.3.h.a.2177.4 4 336.293 even 4
2304.3.h.h.2177.1 4 336.83 odd 4
2304.3.h.h.2177.2 4 112.27 even 4
2304.3.h.h.2177.3 4 112.83 even 4
2304.3.h.h.2177.4 4 336.251 odd 4
3528.3.d.a.1961.1 2 1.1 even 1 trivial
3528.3.d.a.1961.2 2 3.2 odd 2 inner
3600.3.c.c.449.1 4 420.167 even 4
3600.3.c.c.449.2 4 140.27 odd 4
3600.3.c.c.449.3 4 420.83 even 4
3600.3.c.c.449.4 4 140.83 odd 4
3600.3.l.l.1601.1 2 420.419 odd 2
3600.3.l.l.1601.2 2 140.139 even 2