Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,7,Mod(127,256)] 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("256.127"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,48,0,0,0,0,0,3804] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.8938454067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.7.d.g.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q-27.1918 q^{3} +170.767i q^{5} +374.384i q^{7} +10.3959 q^{9} -1358.26 q^{11} -2957.84i q^{13} -4643.48i q^{15} -6656.09 q^{17} -9926.98 q^{19} -10180.2i q^{21} +15585.2i q^{23} -13536.5 q^{25} +19540.2 q^{27} -2083.13i q^{29} +28446.6i q^{31} +36933.6 q^{33} -63932.5 q^{35} +3331.58i q^{37} +80429.0i q^{39} +63122.2 q^{41} -88501.2 q^{43} +1775.29i q^{45} +10825.8i q^{47} -22514.1 q^{49} +180991. q^{51} +48049.2i q^{53} -231947. i q^{55} +269933. q^{57} -147032. q^{59} -124334. i q^{61} +3892.07i q^{63} +505102. q^{65} -87535.6 q^{67} -423790. i q^{69} -225599. i q^{71} -475959. q^{73} +368082. q^{75} -508511. i q^{77} +276262. i q^{79} -538912. q^{81} +485661. q^{83} -1.13664e6i q^{85} +56644.0i q^{87} +1.06400e6 q^{89} +1.10737e6 q^{91} -773516. i q^{93} -1.69520e6i q^{95} +1.46796e6 q^{97} -14120.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 48 q^{3} + 3804 q^{9} - 2768 q^{11} - 5304 q^{17} - 20112 q^{19} - 36588 q^{25} + 158112 q^{27} + 71232 q^{33} - 65728 q^{35} + 209848 q^{41} - 116816 q^{43} + 95556 q^{49} + 771936 q^{51} + 526656 q^{57}+ \cdots - 125616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −27.1918 −1.00711 −0.503553 − 0.863965i \(-0.667974\pi\)
−0.503553 + 0.863965i \(0.667974\pi\)
\(4\) 0 0
\(5\) 170.767i 1.36614i 0.730353 + 0.683069i \(0.239355\pi\)
−0.730353 + 0.683069i \(0.760645\pi\)
\(6\) 0 0
\(7\) 374.384i 1.09150i 0.837949 + 0.545749i \(0.183755\pi\)
−0.837949 + 0.545749i \(0.816245\pi\)
\(8\) 0 0
\(9\) 10.3959 0.0142605
\(10\) 0 0
\(11\) −1358.26 −1.02048 −0.510241 − 0.860032i \(-0.670444\pi\)
−0.510241 + 0.860032i \(0.670444\pi\)
\(12\) 0 0
\(13\) − 2957.84i − 1.34631i −0.739503 − 0.673154i \(-0.764939\pi\)
0.739503 − 0.673154i \(-0.235061\pi\)
\(14\) 0 0
\(15\) − 4643.48i − 1.37585i
\(16\) 0 0
\(17\) −6656.09 −1.35479 −0.677396 − 0.735619i \(-0.736891\pi\)
−0.677396 + 0.735619i \(0.736891\pi\)
\(18\) 0 0
\(19\) −9926.98 −1.44729 −0.723646 − 0.690171i \(-0.757535\pi\)
−0.723646 + 0.690171i \(0.757535\pi\)
\(20\) 0 0
\(21\) − 10180.2i − 1.09925i
\(22\) 0 0
\(23\) 15585.2i 1.28094i 0.767983 + 0.640470i \(0.221260\pi\)
−0.767983 + 0.640470i \(0.778740\pi\)
\(24\) 0 0
\(25\) −13536.5 −0.866335
\(26\) 0 0
\(27\) 19540.2 0.992743
\(28\) 0 0
\(29\) − 2083.13i − 0.0854125i −0.999088 − 0.0427063i \(-0.986402\pi\)
0.999088 − 0.0427063i \(-0.0135980\pi\)
\(30\) 0 0
\(31\) 28446.6i 0.954873i 0.878666 + 0.477437i \(0.158434\pi\)
−0.878666 + 0.477437i \(0.841566\pi\)
\(32\) 0 0
\(33\) 36933.6 1.02773
\(34\) 0 0
\(35\) −63932.5 −1.49114
\(36\) 0 0
\(37\) 3331.58i 0.0657727i 0.999459 + 0.0328863i \(0.0104699\pi\)
−0.999459 + 0.0328863i \(0.989530\pi\)
\(38\) 0 0
\(39\) 80429.0i 1.35587i
\(40\) 0 0
\(41\) 63122.2 0.915863 0.457931 − 0.888988i \(-0.348591\pi\)
0.457931 + 0.888988i \(0.348591\pi\)
\(42\) 0 0
\(43\) −88501.2 −1.11313 −0.556563 − 0.830806i \(-0.687880\pi\)
−0.556563 + 0.830806i \(0.687880\pi\)
\(44\) 0 0
\(45\) 1775.29i 0.0194819i
\(46\) 0 0
\(47\) 10825.8i 0.104271i 0.998640 + 0.0521357i \(0.0166028\pi\)
−0.998640 + 0.0521357i \(0.983397\pi\)
\(48\) 0 0
\(49\) −22514.1 −0.191367
\(50\) 0 0
\(51\) 180991. 1.36442
\(52\) 0 0
\(53\) 48049.2i 0.322744i 0.986894 + 0.161372i \(0.0515919\pi\)
−0.986894 + 0.161372i \(0.948408\pi\)
\(54\) 0 0
\(55\) − 231947.i − 1.39412i
\(56\) 0 0
\(57\) 269933. 1.45758
\(58\) 0 0
\(59\) −147032. −0.715906 −0.357953 − 0.933740i \(-0.616525\pi\)
−0.357953 + 0.933740i \(0.616525\pi\)
\(60\) 0 0
\(61\) − 124334.i − 0.547773i −0.961762 − 0.273887i \(-0.911691\pi\)
0.961762 − 0.273887i \(-0.0883093\pi\)
\(62\) 0 0
\(63\) 3892.07i 0.0155654i
\(64\) 0 0
\(65\) 505102. 1.83924
\(66\) 0 0
\(67\) −87535.6 −0.291045 −0.145523 − 0.989355i \(-0.546486\pi\)
−0.145523 + 0.989355i \(0.546486\pi\)
\(68\) 0 0
\(69\) − 423790.i − 1.29004i
\(70\) 0 0
\(71\) − 225599.i − 0.630322i −0.949038 − 0.315161i \(-0.897941\pi\)
0.949038 − 0.315161i \(-0.102059\pi\)
\(72\) 0 0
\(73\) −475959. −1.22349 −0.611746 − 0.791054i \(-0.709532\pi\)
−0.611746 + 0.791054i \(0.709532\pi\)
\(74\) 0 0
\(75\) 368082. 0.872490
\(76\) 0 0
\(77\) − 508511.i − 1.11385i
\(78\) 0 0
\(79\) 276262.i 0.560324i 0.959953 + 0.280162i \(0.0903882\pi\)
−0.959953 + 0.280162i \(0.909612\pi\)
\(80\) 0 0
\(81\) −538912. −1.01406
\(82\) 0 0
\(83\) 485661. 0.849374 0.424687 − 0.905340i \(-0.360384\pi\)
0.424687 + 0.905340i \(0.360384\pi\)
\(84\) 0 0
\(85\) − 1.13664e6i − 1.85083i
\(86\) 0 0
\(87\) 56644.0i 0.0860194i
\(88\) 0 0
\(89\) 1.06400e6 1.50929 0.754645 − 0.656134i \(-0.227809\pi\)
0.754645 + 0.656134i \(0.227809\pi\)
\(90\) 0 0
\(91\) 1.10737e6 1.46949
\(92\) 0 0
\(93\) − 773516.i − 0.961658i
\(94\) 0 0
\(95\) − 1.69520e6i − 1.97720i
\(96\) 0 0
\(97\) 1.46796e6 1.60842 0.804209 − 0.594346i \(-0.202589\pi\)
0.804209 + 0.594346i \(0.202589\pi\)
\(98\) 0 0
\(99\) −14120.4 −0.0145526
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 256.7.d.g.127.2 4
4.3 odd 2 256.7.d.d.127.4 4
8.3 odd 2 inner 256.7.d.g.127.1 4
8.5 even 2 256.7.d.d.127.3 4
16.3 odd 4 64.7.c.e.63.3 4
16.5 even 4 32.7.c.b.31.3 yes 4
16.11 odd 4 32.7.c.b.31.2 ✓ 4
16.13 even 4 64.7.c.e.63.2 4
48.5 odd 4 288.7.g.b.127.3 4
48.11 even 4 288.7.g.b.127.4 4
48.29 odd 4 576.7.g.l.127.1 4
48.35 even 4 576.7.g.l.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.2 ✓ 4 16.11 odd 4
32.7.c.b.31.3 yes 4 16.5 even 4
64.7.c.e.63.2 4 16.13 even 4
64.7.c.e.63.3 4 16.3 odd 4
256.7.d.d.127.3 4 8.5 even 2
256.7.d.d.127.4 4 4.3 odd 2
256.7.d.g.127.1 4 8.3 odd 2 inner
256.7.d.g.127.2 4 1.1 even 1 trivial
288.7.g.b.127.3 4 48.5 odd 4
288.7.g.b.127.4 4 48.11 even 4
576.7.g.l.127.1 4 48.29 odd 4
576.7.g.l.127.2 4 48.35 even 4