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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,3,Mod(257,768)] 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("768.257"); 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([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,20,0,-18,0,0,0,0,0,12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == 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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 257.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.257
Dual form 768.3.e.d.257.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +2.00000i q^{5} +10.0000 q^{7} -9.00000 q^{9} -10.0000i q^{11} +6.00000 q^{15} -30.0000i q^{21} +21.0000 q^{25} +27.0000i q^{27} -50.0000i q^{29} +38.0000 q^{31} -30.0000 q^{33} +20.0000i q^{35} -18.0000i q^{45} +51.0000 q^{49} -94.0000i q^{53} +20.0000 q^{55} -10.0000i q^{59} -90.0000 q^{63} -50.0000 q^{73} -63.0000i q^{75} -100.000i q^{77} -58.0000 q^{79} +81.0000 q^{81} -134.000i q^{83} -150.000 q^{87} -114.000i q^{93} -190.000 q^{97} +90.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{7} - 18 q^{9} + 12 q^{15} + 42 q^{25} + 76 q^{31} - 60 q^{33} + 102 q^{49} + 40 q^{55} - 180 q^{63} - 100 q^{73} - 116 q^{79} + 162 q^{81} - 300 q^{87} - 380 q^{97}+O(q^{100}) \) 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
\(3\) − 3.00000i − 1.00000i
\(4\) 0 0
\(5\) 2.00000i 0.400000i 0.979796 + 0.200000i \(0.0640942\pi\)
−0.979796 + 0.200000i \(0.935906\pi\)
\(6\) 0 0
\(7\) 10.0000 1.42857 0.714286 − 0.699854i \(-0.246752\pi\)
0.714286 + 0.699854i \(0.246752\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) − 10.0000i − 0.909091i −0.890724 − 0.454545i \(-0.849802\pi\)
0.890724 − 0.454545i \(-0.150198\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 6.00000 0.400000
\(16\) 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
\(21\) − 30.0000i − 1.42857i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 21.0000 0.840000
\(26\) 0 0
\(27\) 27.0000i 1.00000i
\(28\) 0 0
\(29\) − 50.0000i − 1.72414i −0.506791 − 0.862069i \(-0.669168\pi\)
0.506791 − 0.862069i \(-0.330832\pi\)
\(30\) 0 0
\(31\) 38.0000 1.22581 0.612903 − 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(32\) 0 0
\(33\) −30.0000 −0.909091
\(34\) 0 0
\(35\) 20.0000i 0.571429i
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) − 18.0000i − 0.400000i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 51.0000 1.04082
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 94.0000i − 1.77358i −0.462168 − 0.886792i \(-0.652928\pi\)
0.462168 − 0.886792i \(-0.347072\pi\)
\(54\) 0 0
\(55\) 20.0000 0.363636
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 10.0000i − 0.169492i −0.996403 − 0.0847458i \(-0.972992\pi\)
0.996403 − 0.0847458i \(-0.0270078\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −90.0000 −1.42857
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −50.0000 −0.684932 −0.342466 − 0.939530i \(-0.611262\pi\)
−0.342466 + 0.939530i \(0.611262\pi\)
\(74\) 0 0
\(75\) − 63.0000i − 0.840000i
\(76\) 0 0
\(77\) − 100.000i − 1.29870i
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 − 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) − 134.000i − 1.61446i −0.590238 − 0.807229i \(-0.700966\pi\)
0.590238 − 0.807229i \(-0.299034\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −150.000 −1.72414
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 114.000i − 1.22581i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −190.000 −1.95876 −0.979381 − 0.202020i \(-0.935249\pi\)
−0.979381 + 0.202020i \(0.935249\pi\)
\(98\) 0 0
\(99\) 90.0000i 0.909091i
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.e.d.257.1 2
3.2 odd 2 inner 768.3.e.d.257.2 2
4.3 odd 2 768.3.e.c.257.2 2
8.3 odd 2 768.3.e.c.257.1 2
8.5 even 2 inner 768.3.e.d.257.2 2
12.11 even 2 768.3.e.c.257.1 2
16.3 odd 4 96.3.h.a.17.1 1
16.5 even 4 24.3.h.b.5.1 yes 1
16.11 odd 4 96.3.h.b.17.1 1
16.13 even 4 24.3.h.a.5.1 ✓ 1
24.5 odd 2 CM 768.3.e.d.257.1 2
24.11 even 2 768.3.e.c.257.2 2
48.5 odd 4 24.3.h.a.5.1 ✓ 1
48.11 even 4 96.3.h.a.17.1 1
48.29 odd 4 24.3.h.b.5.1 yes 1
48.35 even 4 96.3.h.b.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.h.a.5.1 ✓ 1 16.13 even 4
24.3.h.a.5.1 ✓ 1 48.5 odd 4
24.3.h.b.5.1 yes 1 16.5 even 4
24.3.h.b.5.1 yes 1 48.29 odd 4
96.3.h.a.17.1 1 16.3 odd 4
96.3.h.a.17.1 1 48.11 even 4
96.3.h.b.17.1 1 16.11 odd 4
96.3.h.b.17.1 1 48.35 even 4
768.3.e.c.257.1 2 8.3 odd 2
768.3.e.c.257.1 2 12.11 even 2
768.3.e.c.257.2 2 4.3 odd 2
768.3.e.c.257.2 2 24.11 even 2
768.3.e.d.257.1 2 1.1 even 1 trivial
768.3.e.d.257.1 2 24.5 odd 2 CM
768.3.e.d.257.2 2 3.2 odd 2 inner
768.3.e.d.257.2 2 8.5 even 2 inner