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

$q$-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)
 
\(f(q)\) \(=\) \(q-27.1918 q^{3} -170.767i q^{5} -374.384i q^{7} +10.3959 q^{9} -1358.26 q^{11} +2957.84i q^{13} +4643.48i q^{15} -6656.09 q^{17} -9926.98 q^{19} +10180.2i q^{21} -15585.2i q^{23} -13536.5 q^{25} +19540.2 q^{27} +2083.13i q^{29} -28446.6i q^{31} +36933.6 q^{33} -63932.5 q^{35} -3331.58i q^{37} -80429.0i q^{39} +63122.2 q^{41} -88501.2 q^{43} -1775.29i q^{45} -10825.8i q^{47} -22514.1 q^{49} +180991. q^{51} -48049.2i q^{53} +231947. i q^{55} +269933. q^{57} -147032. q^{59} +124334. i q^{61} -3892.07i q^{63} +505102. q^{65} -87535.6 q^{67} +423790. i q^{69} +225599. i q^{71} -475959. q^{73} +368082. q^{75} +508511. i q^{77} -276262. i q^{79} -538912. q^{81} +485661. q^{83} +1.13664e6i q^{85} -56644.0i q^{87} +1.06400e6 q^{89} +1.10737e6 q^{91} +773516. i q^{93} +1.69520e6i q^{95} +1.46796e6 q^{97} -14120.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 48 q^{3} + 3804 q^{9} - 2768 q^{11} - 5304 q^{17} - 20112 q^{19} - 36588 q^{25} + 158112 q^{27} + 71232 q^{33} - 65728 q^{35} + 209848 q^{41} - 116816 q^{43} + 95556 q^{49} + 771936 q^{51} + 526656 q^{57}+ \cdots - 125616 q^{99}+O(q^{100}) \) 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 \(-0.760645\pi\)
0.730353 − 0.683069i \(-0.239355\pi\)
\(6\) 0 0
\(7\) − 374.384i − 1.09150i −0.837949 − 0.545749i \(-0.816245\pi\)
0.837949 − 0.545749i \(-0.183755\pi\)
\(8\) 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 \(0.235061\pi\)
−0.739503 + 0.673154i \(0.764939\pi\)
\(14\) 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 \(-0.778740\pi\)
0.767983 − 0.640470i \(-0.221260\pi\)
\(24\) 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 \(0.0135980\pi\)
−0.999088 + 0.0427063i \(0.986402\pi\)
\(30\) 0 0
\(31\) − 28446.6i − 0.954873i −0.878666 − 0.477437i \(-0.841566\pi\)
0.878666 − 0.477437i \(-0.158434\pi\)
\(32\) 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 \(-0.989530\pi\)
0.999459 − 0.0328863i \(-0.0104699\pi\)
\(38\) 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 \(-0.983397\pi\)
0.998640 − 0.0521357i \(-0.0166028\pi\)
\(48\) 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 \(-0.948408\pi\)
0.986894 − 0.161372i \(-0.0515919\pi\)
\(54\) 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 \(0.0883093\pi\)
−0.961762 + 0.273887i \(0.911691\pi\)
\(62\) 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 \(0.102059\pi\)
−0.949038 + 0.315161i \(0.897941\pi\)
\(72\) 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 \(-0.909612\pi\)
0.959953 − 0.280162i \(-0.0903882\pi\)
\(80\) 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.127.1 4
4.3 odd 2 256.7.d.d.127.3 4
8.3 odd 2 inner 256.7.d.g.127.2 4
8.5 even 2 256.7.d.d.127.4 4
16.3 odd 4 32.7.c.b.31.3 yes 4
16.5 even 4 64.7.c.e.63.3 4
16.11 odd 4 64.7.c.e.63.2 4
16.13 even 4 32.7.c.b.31.2 ✓ 4
48.5 odd 4 576.7.g.l.127.2 4
48.11 even 4 576.7.g.l.127.1 4
48.29 odd 4 288.7.g.b.127.4 4
48.35 even 4 288.7.g.b.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.2 ✓ 4 16.13 even 4
32.7.c.b.31.3 yes 4 16.3 odd 4
64.7.c.e.63.2 4 16.11 odd 4
64.7.c.e.63.3 4 16.5 even 4
256.7.d.d.127.3 4 4.3 odd 2
256.7.d.d.127.4 4 8.5 even 2
256.7.d.g.127.1 4 1.1 even 1 trivial
256.7.d.g.127.2 4 8.3 odd 2 inner
288.7.g.b.127.3 4 48.35 even 4
288.7.g.b.127.4 4 48.29 odd 4
576.7.g.l.127.1 4 48.11 even 4
576.7.g.l.127.2 4 48.5 odd 4