Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,4,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.1044889615\)
Analytic rank: \(1\)
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: \( 2^{3} \)
Twist minimal: no (minimal twist has level 64)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 256.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +13.8564 q^{5} -27.7128 q^{7} -23.0000 q^{9} -42.0000 q^{11} -41.5692 q^{13} +27.7128 q^{15} -6.00000 q^{17} -94.0000 q^{19} -55.4256 q^{21} +138.564 q^{23} +67.0000 q^{25} -100.000 q^{27} +235.559 q^{29} -110.851 q^{31} -84.0000 q^{33} -384.000 q^{35} +13.8564 q^{37} -83.1384 q^{39} +54.0000 q^{41} -442.000 q^{43} -318.697 q^{45} -55.4256 q^{47} +425.000 q^{49} -12.0000 q^{51} +69.2820 q^{53} -581.969 q^{55} -188.000 q^{57} -138.000 q^{59} -429.549 q^{61} +637.395 q^{63} -576.000 q^{65} +178.000 q^{67} +277.128 q^{69} +859.097 q^{71} -434.000 q^{73} +134.000 q^{75} +1163.94 q^{77} -166.277 q^{79} +421.000 q^{81} -270.000 q^{83} -83.1384 q^{85} +471.118 q^{87} +1182.00 q^{89} +1152.00 q^{91} -221.703 q^{93} -1302.50 q^{95} -1238.00 q^{97} +966.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 46 q^{9} - 84 q^{11} - 12 q^{17} - 188 q^{19} + 134 q^{25} - 200 q^{27} - 168 q^{33} - 768 q^{35} + 108 q^{41} - 884 q^{43} + 850 q^{49} - 24 q^{51} - 376 q^{57} - 276 q^{59} - 1152 q^{65}+ \cdots + 1932 q^{99}+O(q^{100}) \) 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
\(3\) 2.00000 0.384900 0.192450 − 0.981307i \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) 0 0
\(5\) 13.8564 1.23935 0.619677 − 0.784857i \(-0.287263\pi\)
0.619677 + 0.784857i \(0.287263\pi\)
\(6\) 0 0
\(7\) −27.7128 −1.49635 −0.748176 − 0.663501i \(-0.769070\pi\)
−0.748176 + 0.663501i \(0.769070\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −42.0000 −1.15123 −0.575613 − 0.817723i \(-0.695236\pi\)
−0.575613 + 0.817723i \(0.695236\pi\)
\(12\) 0 0
\(13\) −41.5692 −0.886864 −0.443432 − 0.896308i \(-0.646239\pi\)
−0.443432 + 0.896308i \(0.646239\pi\)
\(14\) 0 0
\(15\) 27.7128 0.477028
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 − 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) −94.0000 −1.13500 −0.567502 − 0.823372i \(-0.692090\pi\)
−0.567502 + 0.823372i \(0.692090\pi\)
\(20\) 0 0
\(21\) −55.4256 −0.575946
\(22\) 0 0
\(23\) 138.564 1.25620 0.628100 − 0.778133i \(-0.283833\pi\)
0.628100 + 0.778133i \(0.283833\pi\)
\(24\) 0 0
\(25\) 67.0000 0.536000
\(26\) 0 0
\(27\) −100.000 −0.712778
\(28\) 0 0
\(29\) 235.559 1.50835 0.754176 − 0.656673i \(-0.228037\pi\)
0.754176 + 0.656673i \(0.228037\pi\)
\(30\) 0 0
\(31\) −110.851 −0.642241 −0.321121 − 0.947038i \(-0.604059\pi\)
−0.321121 + 0.947038i \(0.604059\pi\)
\(32\) 0 0
\(33\) −84.0000 −0.443107
\(34\) 0 0
\(35\) −384.000 −1.85451
\(36\) 0 0
\(37\) 13.8564 0.0615670 0.0307835 − 0.999526i \(-0.490200\pi\)
0.0307835 + 0.999526i \(0.490200\pi\)
\(38\) 0 0
\(39\) −83.1384 −0.341354
\(40\) 0 0
\(41\) 54.0000 0.205692 0.102846 − 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(42\) 0 0
\(43\) −442.000 −1.56754 −0.783772 − 0.621049i \(-0.786707\pi\)
−0.783772 + 0.621049i \(0.786707\pi\)
\(44\) 0 0
\(45\) −318.697 −1.05575
\(46\) 0 0
\(47\) −55.4256 −0.172014 −0.0860070 − 0.996295i \(-0.527411\pi\)
−0.0860070 + 0.996295i \(0.527411\pi\)
\(48\) 0 0
\(49\) 425.000 1.23907
\(50\) 0 0
\(51\) −12.0000 −0.0329478
\(52\) 0 0
\(53\) 69.2820 0.179559 0.0897794 − 0.995962i \(-0.471384\pi\)
0.0897794 + 0.995962i \(0.471384\pi\)
\(54\) 0 0
\(55\) −581.969 −1.42678
\(56\) 0 0
\(57\) −188.000 −0.436863
\(58\) 0 0
\(59\) −138.000 −0.304510 −0.152255 − 0.988341i \(-0.548653\pi\)
−0.152255 + 0.988341i \(0.548653\pi\)
\(60\) 0 0
\(61\) −429.549 −0.901608 −0.450804 − 0.892623i \(-0.648863\pi\)
−0.450804 + 0.892623i \(0.648863\pi\)
\(62\) 0 0
\(63\) 637.395 1.27467
\(64\) 0 0
\(65\) −576.000 −1.09914
\(66\) 0 0
\(67\) 178.000 0.324570 0.162285 − 0.986744i \(-0.448114\pi\)
0.162285 + 0.986744i \(0.448114\pi\)
\(68\) 0 0
\(69\) 277.128 0.483512
\(70\) 0 0
\(71\) 859.097 1.43600 0.718001 − 0.696043i \(-0.245058\pi\)
0.718001 + 0.696043i \(0.245058\pi\)
\(72\) 0 0
\(73\) −434.000 −0.695834 −0.347917 − 0.937525i \(-0.613111\pi\)
−0.347917 + 0.937525i \(0.613111\pi\)
\(74\) 0 0
\(75\) 134.000 0.206306
\(76\) 0 0
\(77\) 1163.94 1.72264
\(78\) 0 0
\(79\) −166.277 −0.236805 −0.118403 − 0.992966i \(-0.537777\pi\)
−0.118403 + 0.992966i \(0.537777\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) −270.000 −0.357064 −0.178532 − 0.983934i \(-0.557135\pi\)
−0.178532 + 0.983934i \(0.557135\pi\)
\(84\) 0 0
\(85\) −83.1384 −0.106090
\(86\) 0 0
\(87\) 471.118 0.580565
\(88\) 0 0
\(89\) 1182.00 1.40777 0.703886 − 0.710313i \(-0.251446\pi\)
0.703886 + 0.710313i \(0.251446\pi\)
\(90\) 0 0
\(91\) 1152.00 1.32706
\(92\) 0 0
\(93\) −221.703 −0.247199
\(94\) 0 0
\(95\) −1302.50 −1.40667
\(96\) 0 0
\(97\) −1238.00 −1.29587 −0.647937 − 0.761694i \(-0.724368\pi\)
−0.647937 + 0.761694i \(0.724368\pi\)
\(98\) 0 0
\(99\) 966.000 0.980673
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 256.4.a.m.1.2 2
3.2 odd 2 2304.4.a.bi.1.1 2
4.3 odd 2 256.4.a.i.1.2 2
8.3 odd 2 inner 256.4.a.m.1.1 2
8.5 even 2 256.4.a.i.1.1 2
12.11 even 2 2304.4.a.ba.1.1 2
16.3 odd 4 64.4.b.b.33.1 ✓ 4
16.5 even 4 64.4.b.b.33.2 yes 4
16.11 odd 4 64.4.b.b.33.4 yes 4
16.13 even 4 64.4.b.b.33.3 yes 4
24.5 odd 2 2304.4.a.ba.1.2 2
24.11 even 2 2304.4.a.bi.1.2 2
48.5 odd 4 576.4.d.e.289.2 4
48.11 even 4 576.4.d.e.289.1 4
48.29 odd 4 576.4.d.e.289.4 4
48.35 even 4 576.4.d.e.289.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
64.4.b.b.33.1 ✓ 4 16.3 odd 4
64.4.b.b.33.2 yes 4 16.5 even 4
64.4.b.b.33.3 yes 4 16.13 even 4
64.4.b.b.33.4 yes 4 16.11 odd 4
256.4.a.i.1.1 2 8.5 even 2
256.4.a.i.1.2 2 4.3 odd 2
256.4.a.m.1.1 2 8.3 odd 2 inner
256.4.a.m.1.2 2 1.1 even 1 trivial
576.4.d.e.289.1 4 48.11 even 4
576.4.d.e.289.2 4 48.5 odd 4
576.4.d.e.289.3 4 48.35 even 4
576.4.d.e.289.4 4 48.29 odd 4
2304.4.a.ba.1.1 2 12.11 even 2
2304.4.a.ba.1.2 2 24.5 odd 2
2304.4.a.bi.1.1 2 3.2 odd 2
2304.4.a.bi.1.2 2 24.11 even 2