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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,2,Mod(767,768)] 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("768.767"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20] 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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 767.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 768.767
Dual form 768.2.c.g.767.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +4.00000i q^{7} -3.00000 q^{9} -6.92820 q^{13} -3.46410i q^{19} -6.92820 q^{21} +5.00000 q^{25} -5.19615i q^{27} +4.00000i q^{31} -6.92820 q^{37} -12.0000i q^{39} +10.3923i q^{43} -9.00000 q^{49} +6.00000 q^{57} -6.92820 q^{61} -12.0000i q^{63} +3.46410i q^{67} -10.0000 q^{73} +8.66025i q^{75} +4.00000i q^{79} +9.00000 q^{81} -27.7128i q^{91} -6.92820 q^{93} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} - 36 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.92820 −1.92154 −0.960769 − 0.277350i \(-0.910544\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 − 0.397360i \(-0.869927\pi\)
0.917663 − 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) −6.92820 −1.51186
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.92820 −1.13899 −0.569495 − 0.821995i \(-0.692861\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) − 12.0000i − 1.92154i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.92820 −0.887066 −0.443533 − 0.896258i \(-0.646275\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) − 12.0000i − 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 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
\(91\) − 27.7128i − 2.90509i
\(92\) 0 0
\(93\) −6.92820 −0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 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 768.2.c.g.767.4 4
3.2 odd 2 CM 768.2.c.g.767.4 4
4.3 odd 2 inner 768.2.c.g.767.1 4
8.3 odd 2 inner 768.2.c.g.767.3 4
8.5 even 2 inner 768.2.c.g.767.2 4
12.11 even 2 inner 768.2.c.g.767.1 4
16.3 odd 4 192.2.f.b.95.4 yes 4
16.5 even 4 192.2.f.b.95.3 yes 4
16.11 odd 4 192.2.f.b.95.2 yes 4
16.13 even 4 192.2.f.b.95.1 ✓ 4
24.5 odd 2 inner 768.2.c.g.767.2 4
24.11 even 2 inner 768.2.c.g.767.3 4
48.5 odd 4 192.2.f.b.95.3 yes 4
48.11 even 4 192.2.f.b.95.2 yes 4
48.29 odd 4 192.2.f.b.95.1 ✓ 4
48.35 even 4 192.2.f.b.95.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.f.b.95.1 ✓ 4 16.13 even 4
192.2.f.b.95.1 ✓ 4 48.29 odd 4
192.2.f.b.95.2 yes 4 16.11 odd 4
192.2.f.b.95.2 yes 4 48.11 even 4
192.2.f.b.95.3 yes 4 16.5 even 4
192.2.f.b.95.3 yes 4 48.5 odd 4
192.2.f.b.95.4 yes 4 16.3 odd 4
192.2.f.b.95.4 yes 4 48.35 even 4
768.2.c.g.767.1 4 4.3 odd 2 inner
768.2.c.g.767.1 4 12.11 even 2 inner
768.2.c.g.767.2 4 8.5 even 2 inner
768.2.c.g.767.2 4 24.5 odd 2 inner
768.2.c.g.767.3 4 8.3 odd 2 inner
768.2.c.g.767.3 4 24.11 even 2 inner
768.2.c.g.767.4 4 1.1 even 1 trivial
768.2.c.g.767.4 4 3.2 odd 2 CM