Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,2,Mod(1,768)] 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("768.1"); 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([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,0,0,0,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.46410 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{15} +6.00000 q^{17} +4.00000 q^{19} +3.46410 q^{21} -6.92820 q^{23} +7.00000 q^{25} +1.00000 q^{27} +3.46410 q^{29} +3.46410 q^{31} -12.0000 q^{35} +6.92820 q^{37} +6.00000 q^{41} +4.00000 q^{43} -3.46410 q^{45} +6.92820 q^{47} +5.00000 q^{49} +6.00000 q^{51} +3.46410 q^{53} +4.00000 q^{57} -12.0000 q^{59} -6.92820 q^{61} +3.46410 q^{63} -4.00000 q^{67} -6.92820 q^{69} +6.92820 q^{71} -2.00000 q^{73} +7.00000 q^{75} -10.3923 q^{79} +1.00000 q^{81} -20.7846 q^{85} +3.46410 q^{87} -6.00000 q^{89} +3.46410 q^{93} -13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} + 8 q^{19} + 14 q^{25} + 2 q^{27} - 24 q^{35} + 12 q^{41} + 8 q^{43} + 10 q^{49} + 12 q^{51} + 8 q^{57} - 24 q^{59} - 8 q^{67} - 4 q^{73} + 14 q^{75} + 2 q^{81} - 12 q^{89}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.46410 −1.54919 −0.774597 − 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 − 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 − 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 − 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) 6.92820 1.13899 0.569495 − 0.821995i \(-0.307139\pi\)
0.569495 + 0.821995i \(0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −3.46410 −0.516398
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 − 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 3.46410 0.475831 0.237915 − 0.971286i \(-0.423536\pi\)
0.237915 + 0.971286i \(0.423536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −6.92820 −0.887066 −0.443533 − 0.896258i \(-0.646275\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 − 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 − 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −20.7846 −2.25441
\(86\) 0 0
\(87\) 3.46410 0.371391
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.46410 0.359211
\(94\) 0 0
\(95\) −13.8564 −1.42164
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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.a.k.1.1 2
3.2 odd 2 2304.2.a.u.1.2 2
4.3 odd 2 768.2.a.j.1.1 2
8.3 odd 2 inner 768.2.a.k.1.2 2
8.5 even 2 768.2.a.j.1.2 2
12.11 even 2 2304.2.a.s.1.2 2
16.3 odd 4 192.2.d.a.97.2 yes 4
16.5 even 4 192.2.d.a.97.1 ✓ 4
16.11 odd 4 192.2.d.a.97.3 yes 4
16.13 even 4 192.2.d.a.97.4 yes 4
24.5 odd 2 2304.2.a.s.1.1 2
24.11 even 2 2304.2.a.u.1.1 2
48.5 odd 4 576.2.d.b.289.3 4
48.11 even 4 576.2.d.b.289.4 4
48.29 odd 4 576.2.d.b.289.1 4
48.35 even 4 576.2.d.b.289.2 4
80.3 even 4 4800.2.d.o.1249.1 4
80.13 odd 4 4800.2.d.j.1249.3 4
80.19 odd 4 4800.2.k.j.2401.3 4
80.27 even 4 4800.2.d.o.1249.4 4
80.29 even 4 4800.2.k.j.2401.2 4
80.37 odd 4 4800.2.d.j.1249.2 4
80.43 even 4 4800.2.d.j.1249.1 4
80.53 odd 4 4800.2.d.o.1249.3 4
80.59 odd 4 4800.2.k.j.2401.1 4
80.67 even 4 4800.2.d.j.1249.4 4
80.69 even 4 4800.2.k.j.2401.4 4
80.77 odd 4 4800.2.d.o.1249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 ✓ 4 16.5 even 4
192.2.d.a.97.2 yes 4 16.3 odd 4
192.2.d.a.97.3 yes 4 16.11 odd 4
192.2.d.a.97.4 yes 4 16.13 even 4
576.2.d.b.289.1 4 48.29 odd 4
576.2.d.b.289.2 4 48.35 even 4
576.2.d.b.289.3 4 48.5 odd 4
576.2.d.b.289.4 4 48.11 even 4
768.2.a.j.1.1 2 4.3 odd 2
768.2.a.j.1.2 2 8.5 even 2
768.2.a.k.1.1 2 1.1 even 1 trivial
768.2.a.k.1.2 2 8.3 odd 2 inner
2304.2.a.s.1.1 2 24.5 odd 2
2304.2.a.s.1.2 2 12.11 even 2
2304.2.a.u.1.1 2 24.11 even 2
2304.2.a.u.1.2 2 3.2 odd 2
4800.2.d.j.1249.1 4 80.43 even 4
4800.2.d.j.1249.2 4 80.37 odd 4
4800.2.d.j.1249.3 4 80.13 odd 4
4800.2.d.j.1249.4 4 80.67 even 4
4800.2.d.o.1249.1 4 80.3 even 4
4800.2.d.o.1249.2 4 80.77 odd 4
4800.2.d.o.1249.3 4 80.53 odd 4
4800.2.d.o.1249.4 4 80.27 even 4
4800.2.k.j.2401.1 4 80.59 odd 4
4800.2.k.j.2401.2 4 80.29 even 4
4800.2.k.j.2401.3 4 80.19 odd 4
4800.2.k.j.2401.4 4 80.69 even 4