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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3200,2,Mod(449,3200)] 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("3200.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-6,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,24,0,0,0,20,0,0,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.c.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{9} -4.00000 q^{13} +2.00000i q^{17} -4.00000i q^{29} +12.0000 q^{37} +10.0000 q^{41} +7.00000 q^{49} +4.00000 q^{53} -12.0000i q^{61} +6.00000i q^{73} +9.00000 q^{81} +10.0000 q^{89} +18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} - 8 q^{13} + 24 q^{37} + 20 q^{41} + 14 q^{49} + 8 q^{53} + 18 q^{81} + 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.00000i − 0.742781i −0.928477 − 0.371391i \(-0.878881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0000 1.97279 0.986394 − 0.164399i \(-0.0525685\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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 3200.2.f.c.449.2 2
4.3 odd 2 CM 3200.2.f.c.449.2 2
5.2 odd 4 128.2.b.b.65.1 ✓ 2
5.3 odd 4 3200.2.d.e.1601.1 2
5.4 even 2 3200.2.f.d.449.1 2
8.3 odd 2 3200.2.f.d.449.2 2
8.5 even 2 3200.2.f.d.449.2 2
15.2 even 4 1152.2.d.d.577.2 2
20.3 even 4 3200.2.d.e.1601.1 2
20.7 even 4 128.2.b.b.65.1 ✓ 2
20.19 odd 2 3200.2.f.d.449.1 2
40.3 even 4 3200.2.d.e.1601.2 2
40.13 odd 4 3200.2.d.e.1601.2 2
40.19 odd 2 inner 3200.2.f.c.449.1 2
40.27 even 4 128.2.b.b.65.2 yes 2
40.29 even 2 inner 3200.2.f.c.449.1 2
40.37 odd 4 128.2.b.b.65.2 yes 2
60.47 odd 4 1152.2.d.d.577.2 2
80.3 even 4 6400.2.a.m.1.1 1
80.13 odd 4 6400.2.a.m.1.1 1
80.27 even 4 256.2.a.c.1.1 1
80.37 odd 4 256.2.a.c.1.1 1
80.43 even 4 6400.2.a.l.1.1 1
80.53 odd 4 6400.2.a.l.1.1 1
80.67 even 4 256.2.a.b.1.1 1
80.77 odd 4 256.2.a.b.1.1 1
120.77 even 4 1152.2.d.d.577.1 2
120.107 odd 4 1152.2.d.d.577.1 2
160.27 even 8 1024.2.e.k.257.2 4
160.37 odd 8 1024.2.e.k.257.2 4
160.67 even 8 1024.2.e.k.769.2 4
160.77 odd 8 1024.2.e.k.769.1 4
160.107 even 8 1024.2.e.k.257.1 4
160.117 odd 8 1024.2.e.k.257.1 4
160.147 even 8 1024.2.e.k.769.1 4
160.157 odd 8 1024.2.e.k.769.2 4
240.77 even 4 2304.2.a.p.1.1 1
240.107 odd 4 2304.2.a.a.1.1 1
240.197 even 4 2304.2.a.a.1.1 1
240.227 odd 4 2304.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.b.65.1 ✓ 2 5.2 odd 4
128.2.b.b.65.1 ✓ 2 20.7 even 4
128.2.b.b.65.2 yes 2 40.27 even 4
128.2.b.b.65.2 yes 2 40.37 odd 4
256.2.a.b.1.1 1 80.67 even 4
256.2.a.b.1.1 1 80.77 odd 4
256.2.a.c.1.1 1 80.27 even 4
256.2.a.c.1.1 1 80.37 odd 4
1024.2.e.k.257.1 4 160.107 even 8
1024.2.e.k.257.1 4 160.117 odd 8
1024.2.e.k.257.2 4 160.27 even 8
1024.2.e.k.257.2 4 160.37 odd 8
1024.2.e.k.769.1 4 160.77 odd 8
1024.2.e.k.769.1 4 160.147 even 8
1024.2.e.k.769.2 4 160.67 even 8
1024.2.e.k.769.2 4 160.157 odd 8
1152.2.d.d.577.1 2 120.77 even 4
1152.2.d.d.577.1 2 120.107 odd 4
1152.2.d.d.577.2 2 15.2 even 4
1152.2.d.d.577.2 2 60.47 odd 4
2304.2.a.a.1.1 1 240.107 odd 4
2304.2.a.a.1.1 1 240.197 even 4
2304.2.a.p.1.1 1 240.77 even 4
2304.2.a.p.1.1 1 240.227 odd 4
3200.2.d.e.1601.1 2 5.3 odd 4
3200.2.d.e.1601.1 2 20.3 even 4
3200.2.d.e.1601.2 2 40.3 even 4
3200.2.d.e.1601.2 2 40.13 odd 4
3200.2.f.c.449.1 2 40.19 odd 2 inner
3200.2.f.c.449.1 2 40.29 even 2 inner
3200.2.f.c.449.2 2 1.1 even 1 trivial
3200.2.f.c.449.2 2 4.3 odd 2 CM
3200.2.f.d.449.1 2 5.4 even 2
3200.2.f.d.449.1 2 20.19 odd 2
3200.2.f.d.449.2 2 8.3 odd 2
3200.2.f.d.449.2 2 8.5 even 2
6400.2.a.l.1.1 1 80.43 even 4
6400.2.a.l.1.1 1 80.53 odd 4
6400.2.a.m.1.1 1 80.3 even 4
6400.2.a.m.1.1 1 80.13 odd 4