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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,2,Mod(447,448)] 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("448.447"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 447.1
Root \(0.500000 + 1.32288i\) of defining polynomial
Character \(\chi\) \(=\) 448.447
Dual form 448.2.f.b.447.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.64575i q^{7} -3.00000 q^{9} -5.29150i q^{11} -5.29150i q^{23} +5.00000 q^{25} +2.00000 q^{29} -6.00000 q^{37} -5.29150i q^{43} -7.00000 q^{49} +10.0000 q^{53} +7.93725i q^{63} +15.8745i q^{67} -5.29150i q^{71} -14.0000 q^{77} +15.8745i q^{79} +9.00000 q^{81} +15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} + 10 q^{25} + 4 q^{29} - 12 q^{37} - 14 q^{49} + 20 q^{53} - 28 q^{77} + 18 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 1.00000i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 5.29150i − 1.59545i −0.603023 − 0.797724i \(-0.706037\pi\)
0.603023 − 0.797724i \(-0.293963\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5.29150i − 1.10335i −0.834058 − 0.551677i \(-0.813988\pi\)
0.834058 − 0.551677i \(-0.186012\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 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
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 5.29150i − 0.806947i −0.914991 − 0.403473i \(-0.867803\pi\)
0.914991 − 0.403473i \(-0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 7.93725i 1.00000i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8745i 1.93938i 0.244339 + 0.969690i \(0.421429\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.29150i − 0.627986i −0.949425 − 0.313993i \(-0.898333\pi\)
0.949425 − 0.313993i \(-0.101667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.0000 −1.59545
\(78\) 0 0
\(79\) 15.8745i 1.78602i 0.450035 + 0.893011i \(0.351411\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 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
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 15.8745i 1.59545i
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 448.2.f.b.447.1 2
3.2 odd 2 4032.2.b.e.3583.1 2
4.3 odd 2 inner 448.2.f.b.447.2 2
7.6 odd 2 CM 448.2.f.b.447.1 2
8.3 odd 2 28.2.d.a.27.1 ✓ 2
8.5 even 2 28.2.d.a.27.2 yes 2
12.11 even 2 4032.2.b.e.3583.2 2
16.3 odd 4 1792.2.e.b.895.2 4
16.5 even 4 1792.2.e.b.895.4 4
16.11 odd 4 1792.2.e.b.895.1 4
16.13 even 4 1792.2.e.b.895.3 4
21.20 even 2 4032.2.b.e.3583.1 2
24.5 odd 2 252.2.b.a.55.1 2
24.11 even 2 252.2.b.a.55.2 2
28.27 even 2 inner 448.2.f.b.447.2 2
40.3 even 4 700.2.c.d.699.2 4
40.13 odd 4 700.2.c.d.699.4 4
40.19 odd 2 700.2.g.a.251.2 2
40.27 even 4 700.2.c.d.699.3 4
40.29 even 2 700.2.g.a.251.1 2
40.37 odd 4 700.2.c.d.699.1 4
56.3 even 6 196.2.f.b.19.2 4
56.5 odd 6 196.2.f.b.31.2 4
56.11 odd 6 196.2.f.b.19.2 4
56.13 odd 2 28.2.d.a.27.2 yes 2
56.19 even 6 196.2.f.b.31.1 4
56.27 even 2 28.2.d.a.27.1 ✓ 2
56.37 even 6 196.2.f.b.31.2 4
56.45 odd 6 196.2.f.b.19.1 4
56.51 odd 6 196.2.f.b.31.1 4
56.53 even 6 196.2.f.b.19.1 4
84.83 odd 2 4032.2.b.e.3583.2 2
112.13 odd 4 1792.2.e.b.895.3 4
112.27 even 4 1792.2.e.b.895.1 4
112.69 odd 4 1792.2.e.b.895.4 4
112.83 even 4 1792.2.e.b.895.2 4
168.83 odd 2 252.2.b.a.55.2 2
168.125 even 2 252.2.b.a.55.1 2
280.13 even 4 700.2.c.d.699.4 4
280.27 odd 4 700.2.c.d.699.3 4
280.69 odd 2 700.2.g.a.251.1 2
280.83 odd 4 700.2.c.d.699.2 4
280.139 even 2 700.2.g.a.251.2 2
280.237 even 4 700.2.c.d.699.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.d.a.27.1 ✓ 2 8.3 odd 2
28.2.d.a.27.1 ✓ 2 56.27 even 2
28.2.d.a.27.2 yes 2 8.5 even 2
28.2.d.a.27.2 yes 2 56.13 odd 2
196.2.f.b.19.1 4 56.45 odd 6
196.2.f.b.19.1 4 56.53 even 6
196.2.f.b.19.2 4 56.3 even 6
196.2.f.b.19.2 4 56.11 odd 6
196.2.f.b.31.1 4 56.19 even 6
196.2.f.b.31.1 4 56.51 odd 6
196.2.f.b.31.2 4 56.5 odd 6
196.2.f.b.31.2 4 56.37 even 6
252.2.b.a.55.1 2 24.5 odd 2
252.2.b.a.55.1 2 168.125 even 2
252.2.b.a.55.2 2 24.11 even 2
252.2.b.a.55.2 2 168.83 odd 2
448.2.f.b.447.1 2 1.1 even 1 trivial
448.2.f.b.447.1 2 7.6 odd 2 CM
448.2.f.b.447.2 2 4.3 odd 2 inner
448.2.f.b.447.2 2 28.27 even 2 inner
700.2.c.d.699.1 4 40.37 odd 4
700.2.c.d.699.1 4 280.237 even 4
700.2.c.d.699.2 4 40.3 even 4
700.2.c.d.699.2 4 280.83 odd 4
700.2.c.d.699.3 4 40.27 even 4
700.2.c.d.699.3 4 280.27 odd 4
700.2.c.d.699.4 4 40.13 odd 4
700.2.c.d.699.4 4 280.13 even 4
700.2.g.a.251.1 2 40.29 even 2
700.2.g.a.251.1 2 280.69 odd 2
700.2.g.a.251.2 2 40.19 odd 2
700.2.g.a.251.2 2 280.139 even 2
1792.2.e.b.895.1 4 16.11 odd 4
1792.2.e.b.895.1 4 112.27 even 4
1792.2.e.b.895.2 4 16.3 odd 4
1792.2.e.b.895.2 4 112.83 even 4
1792.2.e.b.895.3 4 16.13 even 4
1792.2.e.b.895.3 4 112.13 odd 4
1792.2.e.b.895.4 4 16.5 even 4
1792.2.e.b.895.4 4 112.69 odd 4
4032.2.b.e.3583.1 2 3.2 odd 2
4032.2.b.e.3583.1 2 21.20 even 2
4032.2.b.e.3583.2 2 12.11 even 2
4032.2.b.e.3583.2 2 84.83 odd 2