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Properties

Label 160.3.g.a
Level $160$
Weight $3$
Character orbit 160.g
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,3,Mod(111,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.g (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: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.148996000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 7x^{6} - 2x^{5} + 12x^{3} + 47x^{2} + 114x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{5} + \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{2} q^{5} + \beta_{6} q^{7} + (\beta_{5} + \beta_{3} + 3) q^{9} + (\beta_{4} + 4) q^{11} + ( - \beta_{7} + \beta_1) q^{13} - \beta_{7} q^{15} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{17} + (\beta_{4} - 2 \beta_{3} - 4) q^{19} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{2}) q^{21} + ( - \beta_{6} + 4 \beta_{2} + 2 \beta_1) q^{23} - 5 q^{25} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 12) q^{27} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{29} + (2 \beta_{7} - 8 \beta_{2}) q^{31} + ( - \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + 2) q^{33} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{35} + (4 \beta_{7} - 4 \beta_{6} + 2 \beta_{2}) q^{37} + ( - 2 \beta_{6} + 16 \beta_{2} - 4 \beta_1) q^{39} + ( - \beta_{5} - \beta_{3} + 6) q^{41} + ( - 6 \beta_{5} - \beta_{3} - 12) q^{43} + ( - \beta_{7} - 2 \beta_{6} + \cdots - \beta_1) q^{45}+ \cdots + (8 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 32 q^{11} - 32 q^{19} - 40 q^{25} + 96 q^{27} + 16 q^{33} + 48 q^{41} - 96 q^{43} - 88 q^{49} - 64 q^{51} - 176 q^{57} - 224 q^{59} - 160 q^{67} + 160 q^{73} - 56 q^{81} + 480 q^{83} - 48 q^{89} + 224 q^{97} + 352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 7x^{6} - 2x^{5} + 12x^{3} + 47x^{2} + 114x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 37\nu^{7} - 127\nu^{6} - 92\nu^{5} + 1210\nu^{4} - 1350\nu^{3} - 1386\nu^{2} + 4001\nu + 3531 ) / 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\nu^{7} - 193\nu^{6} + 412\nu^{5} - 410\nu^{4} + 390\nu^{3} - 54\nu^{2} + 2639\nu + 3189 ) / 960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{7} - 75\nu^{6} + 132\nu^{5} - 30\nu^{4} - 110\nu^{3} + 494\nu^{2} + 285\nu + 1399 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 65\nu^{6} + 156\nu^{5} - 90\nu^{4} - 250\nu^{3} + 522\nu^{2} + 495\nu + 117 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 23\nu^{6} + 52\nu^{5} - 22\nu^{4} - 70\nu^{3} + 166\nu^{2} + 129\nu + 467 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -47\nu^{7} + 237\nu^{6} - 588\nu^{5} + 690\nu^{4} - 750\nu^{3} - 34\nu^{2} - 2531\nu - 3441 ) / 320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -167\nu^{7} + 917\nu^{6} - 2348\nu^{5} + 3010\nu^{4} - 2910\nu^{3} + 846\nu^{2} - 5371\nu - 10041 ) / 960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 3\beta_{6} - 2\beta_{5} + \beta_{4} + 4\beta_{3} - 4\beta_{2} - \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 10\beta_{6} - 3\beta_{5} - 2\beta_{4} + 7\beta_{3} - 22\beta_{2} + \beta _1 - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} - 25\beta_{6} + 8\beta_{5} - 7\beta_{4} + 2\beta_{3} - 72\beta_{2} + 9\beta _1 - 62 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{7} - 49\beta_{6} + 60\beta_{5} - 25\beta_{4} - 50\beta_{3} - 140\beta_{2} + 13\beta _1 - 206 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 14\beta_{7} - 25\beta_{6} + 106\beta_{5} - 51\beta_{4} - 84\beta_{3} - 32\beta_{2} - 3\beta _1 - 374 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{7} + 155\beta_{6} + 539\beta_{5} - 327\beta_{4} - 253\beta_{3} + 618\beta_{2} - 76\beta _1 - 2684 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
1.01379 − 1.99995i
1.01379 + 1.99995i
−0.528640 − 1.28967i
−0.528640 + 1.28967i
2.49601 + 1.32738i
2.49601 − 1.32738i
−0.981166 − 0.273826i
−0.981166 + 0.273826i
0 −4.13348 0 − 2.23607i 0 2.28473i 0 8.08569 0
111.2 0 −4.13348 0 2.23607i 0 − 2.28473i 0 8.08569 0
111.3 0 −2.08343 0 − 2.23607i 0 1.18656i 0 −4.65931 0
111.4 0 −2.08343 0 2.23607i 0 − 1.18656i 0 −4.65931 0
111.5 0 1.20470 0 − 2.23607i 0 − 12.3974i 0 −7.54870 0
111.6 0 1.20470 0 2.23607i 0 12.3974i 0 −7.54870 0
111.7 0 5.01222 0 − 2.23607i 0 8.92613i 0 16.1223 0
111.8 0 5.01222 0 2.23607i 0 − 8.92613i 0 16.1223 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.g.a 8
3.b odd 2 1 1440.3.g.a 8
4.b odd 2 1 40.3.g.a ✓ 8
5.b even 2 1 800.3.g.g 8
5.c odd 4 2 800.3.e.d 16
8.b even 2 1 40.3.g.a ✓ 8
8.d odd 2 1 inner 160.3.g.a 8
12.b even 2 1 360.3.g.a 8
16.e even 4 2 1280.3.b.i 16
16.f odd 4 2 1280.3.b.i 16
20.d odd 2 1 200.3.g.g 8
20.e even 4 2 200.3.e.d 16
24.f even 2 1 1440.3.g.a 8
24.h odd 2 1 360.3.g.a 8
40.e odd 2 1 800.3.g.g 8
40.f even 2 1 200.3.g.g 8
40.i odd 4 2 200.3.e.d 16
40.k even 4 2 800.3.e.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.g.a ✓ 8 4.b odd 2 1
40.3.g.a ✓ 8 8.b even 2 1
160.3.g.a 8 1.a even 1 1 trivial
160.3.g.a 8 8.d odd 2 1 inner
200.3.e.d 16 20.e even 4 2
200.3.e.d 16 40.i odd 4 2
200.3.g.g 8 20.d odd 2 1
200.3.g.g 8 40.f even 2 1
360.3.g.a 8 12.b even 2 1
360.3.g.a 8 24.h odd 2 1
800.3.e.d 16 5.c odd 4 2
800.3.e.d 16 40.k even 4 2
800.3.g.g 8 5.b even 2 1
800.3.g.g 8 40.e odd 2 1
1280.3.b.i 16 16.e even 4 2
1280.3.b.i 16 16.f odd 4 2
1440.3.g.a 8 3.b odd 2 1
1440.3.g.a 8 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 24 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 240 T^{6} + \cdots + 90000 \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 7792)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 880 T^{6} + \cdots + 138297600 \) Copy content Toggle raw display
$17$ \( (T^{4} - 664 T^{2} + \cdots + 72592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16 T^{3} + \cdots - 2672)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2320 T^{6} + \cdots + 47610000 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 184968806400 \) Copy content Toggle raw display
$31$ \( T^{8} + 2240 T^{6} + \cdots + 23040000 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 134307590400 \) Copy content Toggle raw display
$41$ \( (T^{4} - 24 T^{3} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 48 T^{3} + \cdots + 5598004)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1101366291600 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 3957553209600 \) Copy content Toggle raw display
$59$ \( (T^{4} + 112 T^{3} + \cdots - 4307824)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 39126526214400 \) Copy content Toggle raw display
$67$ \( (T^{4} + 80 T^{3} + \cdots - 22028)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 2536502169600 \) Copy content Toggle raw display
$73$ \( (T^{4} - 80 T^{3} + \cdots - 12225008)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 976317515366400 \) Copy content Toggle raw display
$83$ \( (T^{4} - 240 T^{3} + \cdots - 721708)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 3224432)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 112 T^{3} + \cdots + 83984)^{2} \) Copy content Toggle raw display
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