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Properties

Label 768.3.g.a.511.1
Level $768$
Weight $3$
Character 768.511
Analytic conductor $20.926$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,3,Mod(511,768)] 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("768.511"); S:= CuspForms(chi, 3); 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, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-8,0,0,0,-6,0,0,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 511.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 768.511
Dual form 768.3.g.a.511.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} -4.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +6.92820i q^{11} +6.92820i q^{15} -18.0000 q^{17} +20.7846i q^{19} -12.0000 q^{21} -41.5692i q^{23} -9.00000 q^{25} +5.19615i q^{27} +4.00000 q^{29} +48.4974i q^{31} +12.0000 q^{33} +27.7128i q^{35} +72.0000 q^{37} -18.0000 q^{41} +62.3538i q^{43} +12.0000 q^{45} +41.5692i q^{47} +1.00000 q^{49} +31.1769i q^{51} -44.0000 q^{53} -27.7128i q^{55} +36.0000 q^{57} +62.3538i q^{59} -72.0000 q^{61} +20.7846i q^{63} +20.7846i q^{67} -72.0000 q^{69} +41.5692i q^{71} +82.0000 q^{73} +15.5885i q^{75} +48.0000 q^{77} -62.3538i q^{79} +9.00000 q^{81} +131.636i q^{83} +72.0000 q^{85} -6.92820i q^{87} -126.000 q^{89} +84.0000 q^{93} -83.1384i q^{95} +110.000 q^{97} -20.7846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{5} - 6 q^{9} - 36 q^{17} - 24 q^{21} - 18 q^{25} + 8 q^{29} + 24 q^{33} + 144 q^{37} - 36 q^{41} + 24 q^{45} + 2 q^{49} - 88 q^{53} + 72 q^{57} - 144 q^{61} - 144 q^{69} + 164 q^{73} + 96 q^{77}+ \cdots + 220 q^{97}+O(q^{100}) \) 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 − 0.577350i
\(4\) 0 0
\(5\) −4.00000 −0.800000 −0.400000 − 0.916515i \(-0.630990\pi\)
−0.400000 + 0.916515i \(0.630990\pi\)
\(6\) 0 0
\(7\) − 6.92820i − 0.989743i −0.868966 − 0.494872i \(-0.835215\pi\)
0.868966 − 0.494872i \(-0.164785\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 6.92820i 0.629837i 0.949119 + 0.314918i \(0.101977\pi\)
−0.949119 + 0.314918i \(0.898023\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 6.92820i 0.461880i
\(16\) 0 0
\(17\) −18.0000 −1.05882 −0.529412 − 0.848365i \(-0.677587\pi\)
−0.529412 + 0.848365i \(0.677587\pi\)
\(18\) 0 0
\(19\) 20.7846i 1.09393i 0.837157 + 0.546963i \(0.184216\pi\)
−0.837157 + 0.546963i \(0.815784\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.571429
\(22\) 0 0
\(23\) − 41.5692i − 1.80736i −0.428211 − 0.903679i \(-0.640856\pi\)
0.428211 − 0.903679i \(-0.359144\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 4.00000 0.137931 0.0689655 − 0.997619i \(-0.478030\pi\)
0.0689655 + 0.997619i \(0.478030\pi\)
\(30\) 0 0
\(31\) 48.4974i 1.56443i 0.623007 + 0.782216i \(0.285911\pi\)
−0.623007 + 0.782216i \(0.714089\pi\)
\(32\) 0 0
\(33\) 12.0000 0.363636
\(34\) 0 0
\(35\) 27.7128i 0.791795i
\(36\) 0 0
\(37\) 72.0000 1.94595 0.972973 − 0.230919i \(-0.0741732\pi\)
0.972973 + 0.230919i \(0.0741732\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −18.0000 −0.439024 −0.219512 − 0.975610i \(-0.570447\pi\)
−0.219512 + 0.975610i \(0.570447\pi\)
\(42\) 0 0
\(43\) 62.3538i 1.45009i 0.688702 + 0.725045i \(0.258181\pi\)
−0.688702 + 0.725045i \(0.741819\pi\)
\(44\) 0 0
\(45\) 12.0000 0.266667
\(46\) 0 0
\(47\) 41.5692i 0.884451i 0.896904 + 0.442226i \(0.145811\pi\)
−0.896904 + 0.442226i \(0.854189\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) 0 0
\(51\) 31.1769i 0.611312i
\(52\) 0 0
\(53\) −44.0000 −0.830189 −0.415094 − 0.909778i \(-0.636251\pi\)
−0.415094 + 0.909778i \(0.636251\pi\)
\(54\) 0 0
\(55\) − 27.7128i − 0.503869i
\(56\) 0 0
\(57\) 36.0000 0.631579
\(58\) 0 0
\(59\) 62.3538i 1.05684i 0.848982 + 0.528422i \(0.177216\pi\)
−0.848982 + 0.528422i \(0.822784\pi\)
\(60\) 0 0
\(61\) −72.0000 −1.18033 −0.590164 − 0.807283i \(-0.700937\pi\)
−0.590164 + 0.807283i \(0.700937\pi\)
\(62\) 0 0
\(63\) 20.7846i 0.329914i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 20.7846i 0.310218i 0.987897 + 0.155109i \(0.0495729\pi\)
−0.987897 + 0.155109i \(0.950427\pi\)
\(68\) 0 0
\(69\) −72.0000 −1.04348
\(70\) 0 0
\(71\) 41.5692i 0.585482i 0.956192 + 0.292741i \(0.0945674\pi\)
−0.956192 + 0.292741i \(0.905433\pi\)
\(72\) 0 0
\(73\) 82.0000 1.12329 0.561644 − 0.827379i \(-0.310169\pi\)
0.561644 + 0.827379i \(0.310169\pi\)
\(74\) 0 0
\(75\) 15.5885i 0.207846i
\(76\) 0 0
\(77\) 48.0000 0.623377
\(78\) 0 0
\(79\) − 62.3538i − 0.789289i −0.918834 − 0.394644i \(-0.870868\pi\)
0.918834 − 0.394644i \(-0.129132\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 131.636i 1.58597i 0.609238 + 0.792987i \(0.291475\pi\)
−0.609238 + 0.792987i \(0.708525\pi\)
\(84\) 0 0
\(85\) 72.0000 0.847059
\(86\) 0 0
\(87\) − 6.92820i − 0.0796345i
\(88\) 0 0
\(89\) −126.000 −1.41573 −0.707865 − 0.706348i \(-0.750342\pi\)
−0.707865 + 0.706348i \(0.750342\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 84.0000 0.903226
\(94\) 0 0
\(95\) − 83.1384i − 0.875141i
\(96\) 0 0
\(97\) 110.000 1.13402 0.567010 − 0.823711i \(-0.308100\pi\)
0.567010 + 0.823711i \(0.308100\pi\)
\(98\) 0 0
\(99\) − 20.7846i − 0.209946i
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 768.3.g.a.511.1 2
3.2 odd 2 2304.3.g.n.1279.1 2
4.3 odd 2 inner 768.3.g.a.511.2 2
8.3 odd 2 768.3.g.b.511.1 2
8.5 even 2 768.3.g.b.511.2 2
12.11 even 2 2304.3.g.n.1279.2 2
16.3 odd 4 384.3.b.a.319.2 yes 4
16.5 even 4 384.3.b.a.319.1 ✓ 4
16.11 odd 4 384.3.b.a.319.3 yes 4
16.13 even 4 384.3.b.a.319.4 yes 4
24.5 odd 2 2304.3.g.g.1279.1 2
24.11 even 2 2304.3.g.g.1279.2 2
48.5 odd 4 1152.3.b.h.703.4 4
48.11 even 4 1152.3.b.h.703.3 4
48.29 odd 4 1152.3.b.h.703.2 4
48.35 even 4 1152.3.b.h.703.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.a.319.1 ✓ 4 16.5 even 4
384.3.b.a.319.2 yes 4 16.3 odd 4
384.3.b.a.319.3 yes 4 16.11 odd 4
384.3.b.a.319.4 yes 4 16.13 even 4
768.3.g.a.511.1 2 1.1 even 1 trivial
768.3.g.a.511.2 2 4.3 odd 2 inner
768.3.g.b.511.1 2 8.3 odd 2
768.3.g.b.511.2 2 8.5 even 2
1152.3.b.h.703.1 4 48.35 even 4
1152.3.b.h.703.2 4 48.29 odd 4
1152.3.b.h.703.3 4 48.11 even 4
1152.3.b.h.703.4 4 48.5 odd 4
2304.3.g.g.1279.1 2 24.5 odd 2
2304.3.g.g.1279.2 2 24.11 even 2
2304.3.g.n.1279.1 2 3.2 odd 2
2304.3.g.n.1279.2 2 12.11 even 2