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Properties

Label 256.3.d.b.127.2
Level $256$
Weight $3$
Character 256.127
Analytic conductor $6.975$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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] = [256,3,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, 3); 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, 3, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.97549476762\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 127.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.3.d.b.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)
 
\(f(q)\) \(=\) \(q+6.00000i q^{5} -9.00000 q^{9} +10.0000i q^{13} -30.0000 q^{17} -11.0000 q^{25} +42.0000i q^{29} +70.0000i q^{37} -18.0000 q^{41} -54.0000i q^{45} +49.0000 q^{49} -90.0000i q^{53} -22.0000i q^{61} -60.0000 q^{65} +110.000 q^{73} +81.0000 q^{81} -180.000i q^{85} +78.0000 q^{89} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 60 q^{17} - 22 q^{25} - 36 q^{41} + 98 q^{49} - 120 q^{65} + 220 q^{73} + 162 q^{81} + 156 q^{89} + 260 q^{97}+O(q^{100}) \) 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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 6.00000i 1.20000i 0.800000 + 0.600000i \(0.204833\pi\)
−0.800000 + 0.600000i \(0.795167\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 10.0000i 0.769231i 0.923077 + 0.384615i \(0.125666\pi\)
−0.923077 + 0.384615i \(0.874334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.0000 −1.76471 −0.882353 − 0.470588i \(-0.844042\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −11.0000 −0.440000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 42.0000i 1.44828i 0.689655 + 0.724138i \(0.257762\pi\)
−0.689655 + 0.724138i \(0.742238\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 70.0000i 1.89189i 0.324324 + 0.945946i \(0.394863\pi\)
−0.324324 + 0.945946i \(0.605137\pi\)
\(38\) 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
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) − 54.0000i − 1.20000i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 90.0000i − 1.69811i −0.528302 − 0.849057i \(-0.677171\pi\)
0.528302 − 0.849057i \(-0.322829\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) − 22.0000i − 0.360656i −0.983607 − 0.180328i \(-0.942284\pi\)
0.983607 − 0.180328i \(-0.0577159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −60.0000 −0.923077
\(66\) 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
\(73\) 110.000 1.50685 0.753425 − 0.657534i \(-0.228401\pi\)
0.753425 + 0.657534i \(0.228401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) − 180.000i − 2.11765i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 78.0000 0.876404 0.438202 − 0.898876i \(-0.355615\pi\)
0.438202 + 0.898876i \(0.355615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 130.000 1.34021 0.670103 − 0.742268i \(-0.266250\pi\)
0.670103 + 0.742268i \(0.266250\pi\)
\(98\) 0 0
\(99\) 0 0
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 256.3.d.b.127.2 2
3.2 odd 2 2304.3.b.f.127.1 2
4.3 odd 2 CM 256.3.d.b.127.2 2
8.3 odd 2 inner 256.3.d.b.127.1 2
8.5 even 2 inner 256.3.d.b.127.1 2
12.11 even 2 2304.3.b.f.127.1 2
16.3 odd 4 64.3.c.a.63.1 1
16.5 even 4 16.3.c.a.15.1 ✓ 1
16.11 odd 4 16.3.c.a.15.1 ✓ 1
16.13 even 4 64.3.c.a.63.1 1
24.5 odd 2 2304.3.b.f.127.2 2
24.11 even 2 2304.3.b.f.127.2 2
48.5 odd 4 144.3.g.a.127.1 1
48.11 even 4 144.3.g.a.127.1 1
48.29 odd 4 576.3.g.b.127.1 1
48.35 even 4 576.3.g.b.127.1 1
80.3 even 4 1600.3.h.b.1599.1 2
80.13 odd 4 1600.3.h.b.1599.1 2
80.19 odd 4 1600.3.b.b.1151.1 1
80.27 even 4 400.3.h.a.399.1 2
80.29 even 4 1600.3.b.b.1151.1 1
80.37 odd 4 400.3.h.a.399.1 2
80.43 even 4 400.3.h.a.399.2 2
80.53 odd 4 400.3.h.a.399.2 2
80.59 odd 4 400.3.b.a.351.1 1
80.67 even 4 1600.3.h.b.1599.2 2
80.69 even 4 400.3.b.a.351.1 1
80.77 odd 4 1600.3.h.b.1599.2 2
112.5 odd 12 784.3.r.d.655.1 2
112.11 odd 12 784.3.r.e.79.1 2
112.27 even 4 784.3.d.b.687.1 1
112.37 even 12 784.3.r.e.655.1 2
112.53 even 12 784.3.r.e.79.1 2
112.59 even 12 784.3.r.d.79.1 2
112.69 odd 4 784.3.d.b.687.1 1
112.75 even 12 784.3.r.d.655.1 2
112.101 odd 12 784.3.r.d.79.1 2
112.107 odd 12 784.3.r.e.655.1 2
144.5 odd 12 1296.3.o.b.703.1 2
144.11 even 12 1296.3.o.b.271.1 2
144.43 odd 12 1296.3.o.o.271.1 2
144.59 even 12 1296.3.o.b.703.1 2
144.85 even 12 1296.3.o.o.703.1 2
144.101 odd 12 1296.3.o.b.271.1 2
144.133 even 12 1296.3.o.o.271.1 2
144.139 odd 12 1296.3.o.o.703.1 2
240.53 even 4 3600.3.j.a.1999.2 2
240.59 even 4 3600.3.e.c.3151.1 1
240.107 odd 4 3600.3.j.a.1999.1 2
240.149 odd 4 3600.3.e.c.3151.1 1
240.197 even 4 3600.3.j.a.1999.1 2
240.203 odd 4 3600.3.j.a.1999.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.c.a.15.1 ✓ 1 16.5 even 4
16.3.c.a.15.1 ✓ 1 16.11 odd 4
64.3.c.a.63.1 1 16.3 odd 4
64.3.c.a.63.1 1 16.13 even 4
144.3.g.a.127.1 1 48.5 odd 4
144.3.g.a.127.1 1 48.11 even 4
256.3.d.b.127.1 2 8.3 odd 2 inner
256.3.d.b.127.1 2 8.5 even 2 inner
256.3.d.b.127.2 2 1.1 even 1 trivial
256.3.d.b.127.2 2 4.3 odd 2 CM
400.3.b.a.351.1 1 80.59 odd 4
400.3.b.a.351.1 1 80.69 even 4
400.3.h.a.399.1 2 80.27 even 4
400.3.h.a.399.1 2 80.37 odd 4
400.3.h.a.399.2 2 80.43 even 4
400.3.h.a.399.2 2 80.53 odd 4
576.3.g.b.127.1 1 48.29 odd 4
576.3.g.b.127.1 1 48.35 even 4
784.3.d.b.687.1 1 112.27 even 4
784.3.d.b.687.1 1 112.69 odd 4
784.3.r.d.79.1 2 112.59 even 12
784.3.r.d.79.1 2 112.101 odd 12
784.3.r.d.655.1 2 112.5 odd 12
784.3.r.d.655.1 2 112.75 even 12
784.3.r.e.79.1 2 112.11 odd 12
784.3.r.e.79.1 2 112.53 even 12
784.3.r.e.655.1 2 112.37 even 12
784.3.r.e.655.1 2 112.107 odd 12
1296.3.o.b.271.1 2 144.11 even 12
1296.3.o.b.271.1 2 144.101 odd 12
1296.3.o.b.703.1 2 144.5 odd 12
1296.3.o.b.703.1 2 144.59 even 12
1296.3.o.o.271.1 2 144.43 odd 12
1296.3.o.o.271.1 2 144.133 even 12
1296.3.o.o.703.1 2 144.85 even 12
1296.3.o.o.703.1 2 144.139 odd 12
1600.3.b.b.1151.1 1 80.19 odd 4
1600.3.b.b.1151.1 1 80.29 even 4
1600.3.h.b.1599.1 2 80.3 even 4
1600.3.h.b.1599.1 2 80.13 odd 4
1600.3.h.b.1599.2 2 80.67 even 4
1600.3.h.b.1599.2 2 80.77 odd 4
2304.3.b.f.127.1 2 3.2 odd 2
2304.3.b.f.127.1 2 12.11 even 2
2304.3.b.f.127.2 2 24.5 odd 2
2304.3.b.f.127.2 2 24.11 even 2
3600.3.e.c.3151.1 1 240.59 even 4
3600.3.e.c.3151.1 1 240.149 odd 4
3600.3.j.a.1999.1 2 240.107 odd 4
3600.3.j.a.1999.1 2 240.197 even 4
3600.3.j.a.1999.2 2 240.53 even 4
3600.3.j.a.1999.2 2 240.203 odd 4