Properties

Label 1280.2.n.r
Level $1280$
Weight $2$
Character orbit 1280.n
Analytic conductor $10.221$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(767,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.767");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 27x^{8} + 107x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{7} q^{5} + \beta_{6} q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{7} q^{5} + \beta_{6} q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{9} + (\beta_{9} - \beta_{6} - \beta_1) q^{11} + (\beta_{8} - 2 \beta_{5} + \beta_{4} + 2) q^{13} + (\beta_{11} + \beta_{9} - \beta_{6} - \beta_{2}) q^{15} + ( - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{17} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{2} - \beta_1) q^{19} + (\beta_{4} - \beta_{3}) q^{21} + (\beta_{10} - \beta_{9} + 2 \beta_{2} + \beta_1) q^{23} + ( - \beta_{8} - \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 1) q^{25} + (\beta_{10} + \beta_{9} - 2 \beta_{6}) q^{27} + ( - \beta_{8} + \beta_{7} + 4 \beta_{5} - \beta_{4} - \beta_{3}) q^{29} + ( - 2 \beta_{11} - \beta_{6} - 2 \beta_{2} - \beta_1) q^{31} + ( - 3 \beta_{8} + \beta_{7} + 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2) q^{33} + (\beta_{9} + \beta_{6} + \beta_{2} + \beta_1) q^{35} + (\beta_{8} - 2 \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{37} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{6} + 2 \beta_{2} - \beta_1) q^{39} + ( - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} + \beta_{3}) q^{41} - \beta_{2} q^{43} + ( - \beta_{8} - \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 6) q^{45} + (2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{6}) q^{47} + ( - \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{49} + (\beta_{11} - 2 \beta_{9} + \beta_{2}) q^{51} + ( - 2 \beta_{8} - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2) q^{53} + (3 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{2}) q^{55} + ( - 2 \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + 2) q^{57} + (\beta_{11} - 3 \beta_{10} + \beta_{6} - \beta_{2} - \beta_1) q^{59} + ( - \beta_{8} - \beta_{7} - 4) q^{61} + ( - \beta_{10} + \beta_{9} + 2 \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 3) q^{65} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{9}) q^{67} + (\beta_{8} - \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{69} + ( - 4 \beta_{11} + 2 \beta_{9} + \beta_{6} - 4 \beta_{2} + \beta_1) q^{71} + (2 \beta_{7} - 3 \beta_{5} + 2 \beta_{3} + 3) q^{73} + (\beta_{10} - 3 \beta_{9} - \beta_{2} + 2 \beta_1) q^{75} + (\beta_{8} + \beta_{7} + 6 \beta_{5} - \beta_{4} - \beta_{3} + 6) q^{77} + ( - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{2}) q^{79} + ( - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} + \beta_{3} - 1) q^{81} + ( - \beta_{10} + \beta_{9} - 3 \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{8} - 2 \beta_{5} - 3 \beta_{3} - 6) q^{85} + (4 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{6}) q^{87} + ( - 2 \beta_{8} + 2 \beta_{7}) q^{89} + ( - \beta_{11} + 2 \beta_{6} - \beta_{2} + 2 \beta_1) q^{91} + (\beta_{8} - \beta_{7} - 8 \beta_{5} + \beta_{4} - \beta_{3} + 8) q^{93} + (3 \beta_{11} - \beta_{10} + 2 \beta_{9} + 3 \beta_{6} - \beta_{2} - \beta_1) q^{95} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{3} + 1) q^{97} + (4 \beta_{11} + 3 \beta_{10} - \beta_{6} - 4 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} + 20 q^{13} - 4 q^{17} - 4 q^{25} - 16 q^{33} + 4 q^{37} + 16 q^{41} - 64 q^{45} + 36 q^{53} + 32 q^{57} - 40 q^{61} + 36 q^{65} + 28 q^{73} + 64 q^{77} + 4 q^{81} - 68 q^{85} + 96 q^{93} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 27x^{8} + 107x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} + 38\nu^{5} + 373\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{9} - 190\nu^{5} - 711\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{10} - 3\nu^{8} + 76\nu^{6} - 76\nu^{4} + 245\nu^{2} - 169 ) / 76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} + 3\nu^{8} + 76\nu^{6} + 76\nu^{4} + 245\nu^{2} + 169 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{10} + 190\nu^{6} + 787\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\nu^{11} + 950\nu^{7} + 3859\nu^{3} ) / 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{10} - 2\nu^{8} + 456\nu^{6} - 38\nu^{4} + 1743\nu^{2} - 24 ) / 76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -17\nu^{10} - 2\nu^{8} - 456\nu^{6} - 38\nu^{4} - 1743\nu^{2} - 24 ) / 76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -24\nu^{11} - 3\nu^{9} - 646\nu^{7} - 76\nu^{5} - 2530\nu^{3} - 245\nu ) / 38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -24\nu^{11} + 3\nu^{9} - 646\nu^{7} + 76\nu^{5} - 2530\nu^{3} + 245\nu ) / 38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -69\nu^{11} - 1862\nu^{7} - 7345\nu^{3} ) / 76 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} + 2\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + 4\beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{11} - 5\beta_{10} - 5\beta_{9} + 6\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + 3\beta_{7} + 2\beta_{4} - 2\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{10} + 25\beta_{9} - 46\beta_{2} - 22\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -17\beta_{8} + 17\beta_{7} - 56\beta_{5} - 31\beta_{4} - 31\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -214\beta_{11} + 121\beta_{10} + 121\beta_{9} - 90\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -76\beta_{8} - 76\beta_{7} - 38\beta_{4} + 38\beta_{3} + 121 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 577\beta_{10} - 577\beta_{9} + 1002\beta_{2} + 394\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 349\beta_{8} - 349\beta_{7} + 1092\beta_{5} + 729\beta_{4} + 729\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4706\beta_{11} - 2733\beta_{10} - 2733\beta_{9} + 1790\beta_{6} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
0.219986 + 0.219986i
−1.53448 1.53448i
−1.04736 1.04736i
1.04736 + 1.04736i
1.53448 + 1.53448i
−0.219986 0.219986i
0.219986 0.219986i
−1.53448 + 1.53448i
−1.04736 + 1.04736i
1.04736 1.04736i
1.53448 1.53448i
−0.219986 + 0.219986i
0 −2.05288 2.05288i 0 −0.311108 + 2.21432i 0 −1.07864 + 1.07864i 0 5.42864i 0
767.2 0 −1.20864 1.20864i 0 −2.17009 0.539189i 0 0.446112 0.446112i 0 0.0783777i 0
767.3 0 −0.569973 0.569973i 0 1.48119 1.67513i 0 2.93897 2.93897i 0 2.35026i 0
767.4 0 0.569973 + 0.569973i 0 1.48119 1.67513i 0 −2.93897 + 2.93897i 0 2.35026i 0
767.5 0 1.20864 + 1.20864i 0 −2.17009 0.539189i 0 −0.446112 + 0.446112i 0 0.0783777i 0
767.6 0 2.05288 + 2.05288i 0 −0.311108 + 2.21432i 0 1.07864 1.07864i 0 5.42864i 0
1023.1 0 −2.05288 + 2.05288i 0 −0.311108 2.21432i 0 −1.07864 1.07864i 0 5.42864i 0
1023.2 0 −1.20864 + 1.20864i 0 −2.17009 + 0.539189i 0 0.446112 + 0.446112i 0 0.0783777i 0
1023.3 0 −0.569973 + 0.569973i 0 1.48119 + 1.67513i 0 2.93897 + 2.93897i 0 2.35026i 0
1023.4 0 0.569973 0.569973i 0 1.48119 + 1.67513i 0 −2.93897 2.93897i 0 2.35026i 0
1023.5 0 1.20864 1.20864i 0 −2.17009 + 0.539189i 0 −0.446112 0.446112i 0 0.0783777i 0
1023.6 0 2.05288 2.05288i 0 −0.311108 2.21432i 0 1.07864 + 1.07864i 0 5.42864i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 767.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.n.r 12
4.b odd 2 1 inner 1280.2.n.r 12
5.c odd 4 1 inner 1280.2.n.r 12
8.b even 2 1 1280.2.n.s 12
8.d odd 2 1 1280.2.n.s 12
16.e even 4 1 640.2.o.j 12
16.e even 4 1 640.2.o.k yes 12
16.f odd 4 1 640.2.o.j 12
16.f odd 4 1 640.2.o.k yes 12
20.e even 4 1 inner 1280.2.n.r 12
40.i odd 4 1 1280.2.n.s 12
40.k even 4 1 1280.2.n.s 12
80.i odd 4 1 640.2.o.j 12
80.j even 4 1 640.2.o.k yes 12
80.s even 4 1 640.2.o.j 12
80.t odd 4 1 640.2.o.k yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.o.j 12 16.e even 4 1
640.2.o.j 12 16.f odd 4 1
640.2.o.j 12 80.i odd 4 1
640.2.o.j 12 80.s even 4 1
640.2.o.k yes 12 16.e even 4 1
640.2.o.k yes 12 16.f odd 4 1
640.2.o.k yes 12 80.j even 4 1
640.2.o.k yes 12 80.t odd 4 1
1280.2.n.r 12 1.a even 1 1 trivial
1280.2.n.r 12 4.b odd 2 1 inner
1280.2.n.r 12 5.c odd 4 1 inner
1280.2.n.r 12 20.e even 4 1 inner
1280.2.n.s 12 8.b even 2 1
1280.2.n.s 12 8.d odd 2 1
1280.2.n.s 12 40.i odd 4 1
1280.2.n.s 12 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{12} + 80T_{3}^{8} + 640T_{3}^{4} + 256 \) Copy content Toggle raw display
\( T_{7}^{12} + 304T_{7}^{8} + 1664T_{7}^{4} + 256 \) Copy content Toggle raw display
\( T_{13}^{6} - 10T_{13}^{5} + 50T_{13}^{4} - 80T_{13}^{3} + 36T_{13}^{2} + 120T_{13} + 200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 80 T^{8} + 640 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 304 T^{8} + 1664 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{6} + 48 T^{4} + 704 T^{2} + 3200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 10 T^{5} + 50 T^{4} - 80 T^{3} + \cdots + 200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 88 T^{4} + 1984 T^{2} + \cdots - 12800)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 4208 T^{8} + 41088 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{6} + 76 T^{4} + 944 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 136 T^{4} + 5824 T^{2} + \cdots + 80000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 2 T^{5} + 2 T^{4} - 208 T^{3} + \cdots + 57800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 4 T^{2} - 48 T - 80)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + 80 T^{8} + 640 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{12} + 4080 T^{8} + \cdots + 479785216 \) Copy content Toggle raw display
$53$ \( (T^{6} - 18 T^{5} + 162 T^{4} + 16 T^{3} + \cdots + 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 184 T^{4} + 6336 T^{2} + \cdots - 12800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 10 T^{2} + 20 T - 8)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + 21712 T^{8} + \cdots + 133090713856 \) Copy content Toggle raw display
$71$ \( (T^{6} + 424 T^{4} + 53056 T^{2} + \cdots + 1692800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 14 T^{5} + 98 T^{4} + 608 T^{3} + \cdots + 200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 352 T^{4} + 24576 T^{2} + \cdots - 204800)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 17040 T^{8} + \cdots + 133090713856 \) Copy content Toggle raw display
$89$ \( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 10 T^{5} + 50 T^{4} + 160 T^{3} + \cdots + 5000)^{2} \) Copy content Toggle raw display
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