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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,2,Mod(383,768)] 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("768.383"); 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, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 383.4
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 768.383
Dual form 768.2.f.d.383.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +3.46410i q^{7} +3.00000 q^{9} +2.00000i q^{13} +3.46410 q^{19} +6.00000i q^{21} -5.00000 q^{25} +5.19615 q^{27} +10.3923i q^{31} -10.0000i q^{37} +3.46410i q^{39} +10.3923 q^{43} -5.00000 q^{49} +6.00000 q^{57} -14.0000i q^{61} +10.3923i q^{63} +3.46410 q^{67} -10.0000 q^{73} -8.66025 q^{75} -17.3205i q^{79} +9.00000 q^{81} -6.92820 q^{91} +18.0000i q^{93} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} - 20 q^{25} - 20 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} - 56 q^{97}+O(q^{100}) \) 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
\(3\) 1.73205 1.00000
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−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
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 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
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 10.3923i 1.86651i 0.359211 + 0.933257i \(0.383046\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 3.46410i 0.554700i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 10.3923 1.58481 0.792406 − 0.609994i \(-0.208828\pi\)
0.792406 + 0.609994i \(0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 14.0000i − 1.79252i −0.443533 − 0.896258i \(-0.646275\pi\)
0.443533 − 0.896258i \(-0.353725\pi\)
\(62\) 0 0
\(63\) 10.3923i 1.30931i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 0.423207 0.211604 − 0.977356i \(-0.432131\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 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
\(75\) −8.66025 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 17.3205i − 1.94871i −0.225018 − 0.974355i \(-0.572244\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 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
\(91\) −6.92820 −0.726273
\(92\) 0 0
\(93\) 18.0000i 1.86651i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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.f.d.383.4 4
3.2 odd 2 CM 768.2.f.d.383.4 4
4.3 odd 2 inner 768.2.f.d.383.1 4
8.3 odd 2 inner 768.2.f.d.383.3 4
8.5 even 2 inner 768.2.f.d.383.2 4
12.11 even 2 inner 768.2.f.d.383.1 4
16.3 odd 4 48.2.c.a.47.1 ✓ 2
16.5 even 4 192.2.c.a.191.1 2
16.11 odd 4 192.2.c.a.191.2 2
16.13 even 4 48.2.c.a.47.2 yes 2
24.5 odd 2 inner 768.2.f.d.383.2 4
24.11 even 2 inner 768.2.f.d.383.3 4
48.5 odd 4 192.2.c.a.191.1 2
48.11 even 4 192.2.c.a.191.2 2
48.29 odd 4 48.2.c.a.47.2 yes 2
48.35 even 4 48.2.c.a.47.1 ✓ 2
80.3 even 4 1200.2.o.i.1199.3 4
80.13 odd 4 1200.2.o.i.1199.1 4
80.19 odd 4 1200.2.h.e.1151.2 2
80.29 even 4 1200.2.h.e.1151.1 2
80.67 even 4 1200.2.o.i.1199.2 4
80.77 odd 4 1200.2.o.i.1199.4 4
112.13 odd 4 2352.2.h.c.2255.1 2
112.83 even 4 2352.2.h.c.2255.2 2
144.13 even 12 1296.2.s.e.431.1 2
144.29 odd 12 1296.2.s.b.863.1 2
144.61 even 12 1296.2.s.b.863.1 2
144.67 odd 12 1296.2.s.b.431.1 2
144.77 odd 12 1296.2.s.e.431.1 2
144.83 even 12 1296.2.s.e.863.1 2
144.115 odd 12 1296.2.s.e.863.1 2
144.131 even 12 1296.2.s.b.431.1 2
240.29 odd 4 1200.2.h.e.1151.1 2
240.77 even 4 1200.2.o.i.1199.4 4
240.83 odd 4 1200.2.o.i.1199.3 4
240.173 even 4 1200.2.o.i.1199.1 4
240.179 even 4 1200.2.h.e.1151.2 2
240.227 odd 4 1200.2.o.i.1199.2 4
336.83 odd 4 2352.2.h.c.2255.2 2
336.125 even 4 2352.2.h.c.2255.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.c.a.47.1 ✓ 2 16.3 odd 4
48.2.c.a.47.1 ✓ 2 48.35 even 4
48.2.c.a.47.2 yes 2 16.13 even 4
48.2.c.a.47.2 yes 2 48.29 odd 4
192.2.c.a.191.1 2 16.5 even 4
192.2.c.a.191.1 2 48.5 odd 4
192.2.c.a.191.2 2 16.11 odd 4
192.2.c.a.191.2 2 48.11 even 4
768.2.f.d.383.1 4 4.3 odd 2 inner
768.2.f.d.383.1 4 12.11 even 2 inner
768.2.f.d.383.2 4 8.5 even 2 inner
768.2.f.d.383.2 4 24.5 odd 2 inner
768.2.f.d.383.3 4 8.3 odd 2 inner
768.2.f.d.383.3 4 24.11 even 2 inner
768.2.f.d.383.4 4 1.1 even 1 trivial
768.2.f.d.383.4 4 3.2 odd 2 CM
1200.2.h.e.1151.1 2 80.29 even 4
1200.2.h.e.1151.1 2 240.29 odd 4
1200.2.h.e.1151.2 2 80.19 odd 4
1200.2.h.e.1151.2 2 240.179 even 4
1200.2.o.i.1199.1 4 80.13 odd 4
1200.2.o.i.1199.1 4 240.173 even 4
1200.2.o.i.1199.2 4 80.67 even 4
1200.2.o.i.1199.2 4 240.227 odd 4
1200.2.o.i.1199.3 4 80.3 even 4
1200.2.o.i.1199.3 4 240.83 odd 4
1200.2.o.i.1199.4 4 80.77 odd 4
1200.2.o.i.1199.4 4 240.77 even 4
1296.2.s.b.431.1 2 144.67 odd 12
1296.2.s.b.431.1 2 144.131 even 12
1296.2.s.b.863.1 2 144.29 odd 12
1296.2.s.b.863.1 2 144.61 even 12
1296.2.s.e.431.1 2 144.13 even 12
1296.2.s.e.431.1 2 144.77 odd 12
1296.2.s.e.863.1 2 144.83 even 12
1296.2.s.e.863.1 2 144.115 odd 12
2352.2.h.c.2255.1 2 112.13 odd 4
2352.2.h.c.2255.1 2 336.125 even 4
2352.2.h.c.2255.2 2 112.83 even 4
2352.2.h.c.2255.2 2 336.83 odd 4