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Properties

Label 1280.3.h.i
Level $1280$
Weight $3$
Character orbit 1280.h
Analytic conductor $34.877$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,3,Mod(1279,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.h (of order \(2\), degree \(1\), 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: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.691798081536.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{6} q^{5} - \beta_{2} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_{6} q^{5} - \beta_{2} q^{7} - 3 q^{9} - \beta_{7} q^{11} - 4 \beta_{5} q^{13} + ( - \beta_{2} + 3 \beta_1) q^{15} + \beta_{3} q^{17} + 3 \beta_{7} q^{19} + (6 \beta_{6} + 3 \beta_{5}) q^{21} + \beta_{2} q^{23} + (\beta_{3} + 13) q^{25} + 12 \beta_{4} q^{27} + ( - 8 \beta_{6} - 4 \beta_{5}) q^{29} + 2 \beta_1 q^{31} - \beta_{3} q^{33} + (3 \beta_{7} - 19 \beta_{4}) q^{35} + 3 \beta_{5} q^{37} + 24 \beta_1 q^{39} + 24 q^{41} - 23 \beta_{4} q^{43} - 3 \beta_{6} q^{45} - \beta_{2} q^{47} + 65 q^{49} + 6 \beta_{7} q^{51} + 10 \beta_{5} q^{53} + ( - 2 \beta_{2} - 19 \beta_1) q^{55} + 3 \beta_{3} q^{57} + 5 \beta_{7} q^{59} + (6 \beta_{6} + 3 \beta_{5}) q^{61} + 3 \beta_{2} q^{63} + (4 \beta_{3} - 48) q^{65} - 3 \beta_{4} q^{67} + ( - 6 \beta_{6} - 3 \beta_{5}) q^{69} - 42 \beta_1 q^{71} - 5 \beta_{3} q^{73} + (6 \beta_{7} - 13 \beta_{4}) q^{75} + 19 \beta_{5} q^{77} - 50 \beta_1 q^{79} - 45 q^{81} - 7 \beta_{4} q^{83} + ( - 12 \beta_{6} - 25 \beta_{5}) q^{85} + 8 \beta_{2} q^{87} - 150 q^{89} + 24 \beta_{7} q^{91} - 2 \beta_{5} q^{93} + (6 \beta_{2} + 57 \beta_1) q^{95} - \beta_{3} q^{97} + 3 \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 104 q^{25} + 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} - 1200 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{6} + 12\nu^{5} - 110\nu^{4} + 200\nu^{3} - 886\nu^{2} + 788\nu - 280 ) / 975 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} - 15\nu^{4} + 25\nu^{3} - 9\nu^{2} - 3\nu + 730 ) / 75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 42\nu^{4} - 74\nu^{3} + 378\nu^{2} - 342\nu + 88 ) / 39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\nu^{7} - 77\nu^{6} + 483\nu^{5} - 1015\nu^{4} + 3773\nu^{3} - 4683\nu^{2} - 6763\nu + 4130 ) / 7575 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\nu^{7} - 238\nu^{6} + 2154\nu^{5} - 4790\nu^{4} + 23782\nu^{3} - 31002\nu^{2} + 51706\nu - 20840 ) / 19695 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\nu^{7} - 49\nu^{6} + 289\nu^{5} - 600\nu^{4} + 1189\nu^{3} - 1208\nu^{2} - 11135\nu + 5750 ) / 3939 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 304\nu^{7} - 1064\nu^{6} + 7776\nu^{5} - 16780\nu^{4} + 72336\nu^{3} - 92256\nu^{2} + 158864\nu - 64590 ) / 32825 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - 2\beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} - 36\beta_{6} + 3\beta_{5} + 66\beta_{4} + 3\beta_{3} + 6\beta_{2} + 18\beta _1 - 56 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} - 36\beta_{6} + 4\beta_{5} + 68\beta_{4} - 14\beta_{3} - 4\beta_{2} - 162\beta _1 + 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -68\beta_{7} + 480\beta_{6} + 373\beta_{5} - 814\beta_{4} - 75\beta_{3} - 30\beta_{2} - 840\beta _1 + 224 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -79\beta_{7} + 810\beta_{6} + 549\beta_{5} - 1392\beta_{4} + 126\beta_{3} - 228\beta_{2} + 1341\beta _1 + 2322 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1261 \beta_{7} - 1008 \beta_{6} - 2766 \beta_{5} + 1878 \beta_{4} + 574 \beta_{3} - 742 \beta_{2} + \cdots + 7702 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
−0.724745 − 3.40419i
−0.724745 + 3.40419i
−0.724745 + 0.954705i
−0.724745 − 0.954705i
1.72474 + 0.954705i
1.72474 − 0.954705i
1.72474 − 3.40419i
1.72474 + 3.40419i
0 −2.44949 0 −4.35890 − 2.44949i 0 10.6771 0 −3.00000 0
1279.2 0 −2.44949 0 −4.35890 + 2.44949i 0 10.6771 0 −3.00000 0
1279.3 0 −2.44949 0 4.35890 − 2.44949i 0 −10.6771 0 −3.00000 0
1279.4 0 −2.44949 0 4.35890 + 2.44949i 0 −10.6771 0 −3.00000 0
1279.5 0 2.44949 0 −4.35890 − 2.44949i 0 −10.6771 0 −3.00000 0
1279.6 0 2.44949 0 −4.35890 + 2.44949i 0 −10.6771 0 −3.00000 0
1279.7 0 2.44949 0 4.35890 − 2.44949i 0 10.6771 0 −3.00000 0
1279.8 0 2.44949 0 4.35890 + 2.44949i 0 10.6771 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.h.i 8
4.b odd 2 1 inner 1280.3.h.i 8
5.b even 2 1 inner 1280.3.h.i 8
8.b even 2 1 inner 1280.3.h.i 8
8.d odd 2 1 inner 1280.3.h.i 8
16.e even 4 2 320.3.e.a ✓ 8
16.f odd 4 2 320.3.e.a ✓ 8
20.d odd 2 1 inner 1280.3.h.i 8
40.e odd 2 1 inner 1280.3.h.i 8
40.f even 2 1 inner 1280.3.h.i 8
80.i odd 4 2 1600.3.g.h 8
80.j even 4 2 1600.3.g.h 8
80.k odd 4 2 320.3.e.a ✓ 8
80.q even 4 2 320.3.e.a ✓ 8
80.s even 4 2 1600.3.g.h 8
80.t odd 4 2 1600.3.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.e.a ✓ 8 16.e even 4 2
320.3.e.a ✓ 8 16.f odd 4 2
320.3.e.a ✓ 8 80.k odd 4 2
320.3.e.a ✓ 8 80.q even 4 2
1280.3.h.i 8 1.a even 1 1 trivial
1280.3.h.i 8 4.b odd 2 1 inner
1280.3.h.i 8 5.b even 2 1 inner
1280.3.h.i 8 8.b even 2 1 inner
1280.3.h.i 8 8.d odd 2 1 inner
1280.3.h.i 8 20.d odd 2 1 inner
1280.3.h.i 8 40.e odd 2 1 inner
1280.3.h.i 8 40.f even 2 1 inner
1600.3.g.h 8 80.i odd 4 2
1600.3.g.h 8 80.j even 4 2
1600.3.g.h 8 80.s even 4 2
1600.3.g.h 8 80.t odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 114 \) Copy content Toggle raw display
\( T_{29}^{2} - 1216 \) Copy content Toggle raw display
\( T_{61}^{2} - 684 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 26 T^{2} + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 76)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 384)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 456)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 684)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1216)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$41$ \( (T - 24)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} - 3174)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 114)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2400)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1900)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 684)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7056)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 10000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 294)^{4} \) Copy content Toggle raw display
$89$ \( (T + 150)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 456)^{4} \) Copy content Toggle raw display
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