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Properties

Label 160.3.e.c
Level $160$
Weight $3$
Character orbit 160.e
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
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] = [160,3,Mod(79,160)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.e (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: 8.0.53824000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 36x^{4} + 96x^{2} + 256 \) 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{7} + (2 \beta_{5} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{7} + (2 \beta_{5} - 5) q^{9} + ( - \beta_{5} - 4) q^{11} - 4 \beta_1 q^{13} + (\beta_{7} + 2 \beta_{4} - 3 \beta_1) q^{15} + (\beta_{6} - \beta_{2}) q^{17} + (5 \beta_{5} - 4) q^{19} + (2 \beta_{7} + \beta_{4} + \beta_{3}) q^{21} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{23} + ( - \beta_{6} - 4 \beta_{5} - 3 \beta_{2} - 5) q^{25} + (2 \beta_{6} + 2 \beta_{2}) q^{27} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_{3}) q^{29} + ( - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3}) q^{31} + ( - \beta_{6} + \beta_{2}) q^{33} + ( - 2 \beta_{6} - 3 \beta_{5} + \cdots + 20) q^{35}+ \cdots + ( - 3 \beta_{5} - 20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{9} - 32 q^{11} - 32 q^{19} - 40 q^{25} + 160 q^{35} + 192 q^{41} + 8 q^{49} - 128 q^{51} + 224 q^{59} - 320 q^{75} - 168 q^{81} + 112 q^{89} + 320 q^{91} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 36x^{4} + 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 20\nu^{3} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 10\nu^{5} - 28\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 14\nu^{5} + 8\nu^{4} + 52\nu^{3} - 16\nu^{2} + 128\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 4\nu^{6} - 14\nu^{5} + 8\nu^{4} - 52\nu^{3} - 16\nu^{2} - 128\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 6\nu^{4} + 20\nu^{2} + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} + 34\nu^{5} + 172\nu^{3} + 640\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 6\nu^{4} + 52\nu^{2} + 96 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + \beta_{3} + 2\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 3\beta_{5} + 2\beta_{4} + 2\beta_{3} - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{6} - 5\beta_{4} + 5\beta_{3} - 7\beta_{2} - 10\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} + 4\beta_{5} - 6\beta_{4} - 6\beta_{3} + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5\beta_{6} + 13\beta_{4} - 13\beta_{3} - 19\beta_{2} - 2\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.34500 + 1.48020i
−1.34500 + 1.48020i
−0.831254 − 1.81907i
0.831254 − 1.81907i
−0.831254 + 1.81907i
0.831254 + 1.81907i
1.34500 − 1.48020i
−1.34500 − 1.48020i
0 − 4.79002i 0 −4.35250 + 2.46084i 0 −7.67752 0 −13.9443 0
79.2 0 − 4.79002i 0 4.35250 − 2.46084i 0 7.67752 0 −13.9443 0
79.3 0 − 2.24849i 0 −1.02749 − 4.89329i 0 −6.40747 0 3.94427 0
79.4 0 − 2.24849i 0 1.02749 + 4.89329i 0 6.40747 0 3.94427 0
79.5 0 2.24849i 0 −1.02749 + 4.89329i 0 −6.40747 0 3.94427 0
79.6 0 2.24849i 0 1.02749 − 4.89329i 0 6.40747 0 3.94427 0
79.7 0 4.79002i 0 −4.35250 − 2.46084i 0 −7.67752 0 −13.9443 0
79.8 0 4.79002i 0 4.35250 + 2.46084i 0 7.67752 0 −13.9443 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e 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.e.c 8
3.b odd 2 1 1440.3.p.g 8
4.b odd 2 1 40.3.e.c ✓ 8
5.b even 2 1 inner 160.3.e.c 8
5.c odd 4 2 800.3.g.h 8
8.b even 2 1 40.3.e.c ✓ 8
8.d odd 2 1 inner 160.3.e.c 8
12.b even 2 1 360.3.p.g 8
15.d odd 2 1 1440.3.p.g 8
16.e even 4 2 1280.3.h.m 16
16.f odd 4 2 1280.3.h.m 16
20.d odd 2 1 40.3.e.c ✓ 8
20.e even 4 2 200.3.g.h 8
24.f even 2 1 1440.3.p.g 8
24.h odd 2 1 360.3.p.g 8
40.e odd 2 1 inner 160.3.e.c 8
40.f even 2 1 40.3.e.c ✓ 8
40.i odd 4 2 200.3.g.h 8
40.k even 4 2 800.3.g.h 8
60.h even 2 1 360.3.p.g 8
80.k odd 4 2 1280.3.h.m 16
80.q even 4 2 1280.3.h.m 16
120.i odd 2 1 360.3.p.g 8
120.m even 2 1 1440.3.p.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.e.c ✓ 8 4.b odd 2 1
40.3.e.c ✓ 8 8.b even 2 1
40.3.e.c ✓ 8 20.d odd 2 1
40.3.e.c ✓ 8 40.f even 2 1
160.3.e.c 8 1.a even 1 1 trivial
160.3.e.c 8 5.b even 2 1 inner
160.3.e.c 8 8.d odd 2 1 inner
160.3.e.c 8 40.e odd 2 1 inner
200.3.g.h 8 20.e even 4 2
200.3.g.h 8 40.i odd 4 2
360.3.p.g 8 12.b even 2 1
360.3.p.g 8 24.h odd 2 1
360.3.p.g 8 60.h even 2 1
360.3.p.g 8 120.i odd 2 1
800.3.g.h 8 5.c odd 4 2
800.3.g.h 8 40.k even 4 2
1280.3.h.m 16 16.e even 4 2
1280.3.h.m 16 16.f odd 4 2
1280.3.h.m 16 80.k odd 4 2
1280.3.h.m 16 80.q even 4 2
1440.3.p.g 8 3.b odd 2 1
1440.3.p.g 8 15.d odd 2 1
1440.3.p.g 8 24.f even 2 1
1440.3.p.g 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(160, [\chi])\):

\( T_{3}^{4} + 28T_{3}^{2} + 116 \) Copy content Toggle raw display
\( T_{7}^{4} - 100T_{7}^{2} + 2420 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 28 T^{2} + 116)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 20 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 100 T^{2} + 2420)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 320 T^{2} + 5120)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 368 T^{2} + 1856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T - 484)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 100 T^{2} + 500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1760 T^{2} + 37120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2400 T^{2} + 928000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3280 T^{2} + 1613120)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48 T + 496)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1372 T^{2} + 111476)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 5860 T^{2} + 444020)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 5760 T^{2} + 6635520)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 56 T - 1636)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1080 T^{2} + 187920)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 12988 T^{2} + 41620916)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12000 T^{2} + 35672320)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 7408 T^{2} + 3119936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 22880 T^{2} + 129214720)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1628 T^{2} + 116)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 28 T - 4924)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 44912 T^{2} + 344772416)^{2} \) Copy content Toggle raw display
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