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Properties

Label 160.4.n.c
Level $160$
Weight $4$
Character orbit 160.n
Analytic conductor $9.440$
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] = [160,4,Mod(63,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), 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: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 48x^{6} + 628x^{4} + 1556x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1 - 1) q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{5} + (\beta_{7} + \beta_{5} - \beta_{2} + \cdots - 8) q^{7}+ \cdots + (3 \beta_{7} + 3 \beta_{6} + \cdots + 19 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1 - 1) q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{5} + (\beta_{7} + \beta_{5} - \beta_{2} + \cdots - 8) q^{7}+ \cdots + (63 \beta_{7} - 63 \beta_{6} + \cdots - 1000) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 6 q^{5} - 70 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 6 q^{5} - 70 q^{7} + 144 q^{13} + 134 q^{15} - 100 q^{17} - 176 q^{19} - 516 q^{21} + 198 q^{23} - 172 q^{25} + 288 q^{27} + 172 q^{33} - 170 q^{35} + 492 q^{37} - 756 q^{39} - 28 q^{41} - 654 q^{43} + 10 q^{45} + 690 q^{47} + 580 q^{53} + 2392 q^{55} - 248 q^{57} - 1808 q^{59} - 1068 q^{61} - 3106 q^{63} + 2200 q^{65} + 1878 q^{67} + 1952 q^{73} + 2178 q^{75} + 340 q^{77} + 1808 q^{79} - 7844 q^{81} + 506 q^{83} + 184 q^{85} - 784 q^{87} + 3708 q^{93} + 5800 q^{95} + 5576 q^{97} - 8412 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 48x^{6} + 628x^{4} + 1556x^{2} + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{7} - 412\nu^{5} - 4952\nu^{3} - 8604\nu ) / 3880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} - 44\nu^{6} + 412\nu^{5} - 1928\nu^{4} + 4952\nu^{3} - 18864\nu^{2} + 16364\nu + 7600 ) / 3880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} - 44\nu^{6} - 412\nu^{5} - 1928\nu^{4} - 4952\nu^{3} - 18864\nu^{2} - 16364\nu + 7600 ) / 3880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{7} - 46\nu^{6} + 956\nu^{5} - 2192\nu^{4} + 13860\nu^{3} - 26776\nu^{2} + 48428\nu - 31560 ) / 3880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{7} + 46\nu^{6} + 956\nu^{5} + 2192\nu^{4} + 13860\nu^{3} + 26776\nu^{2} + 48428\nu + 31560 ) / 3880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{7} + 20\nu^{6} - 1096\nu^{5} + 700\nu^{4} - 13388\nu^{3} + 5400\nu^{2} - 15780\nu + 5540 ) / 1940 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -23\nu^{7} - 20\nu^{6} - 1096\nu^{5} - 700\nu^{4} - 13388\nu^{3} - 5400\nu^{2} - 15780\nu - 5540 ) / 1940 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 3\beta_{2} - 50 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 11\beta_{3} - 11\beta_{2} - 34\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20\beta_{7} - 20\beta_{6} - 18\beta_{5} + 18\beta_{4} - 37\beta_{3} - 37\beta_{2} + 552 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -79\beta_{7} - 79\beta_{6} - 34\beta_{5} - 34\beta_{4} - 259\beta_{3} + 259\beta_{2} + 1182\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -331\beta_{7} + 331\beta_{6} + 180\beta_{5} - 180\beta_{4} + 445\beta_{3} + 445\beta_{2} - 6562 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1258\beta_{7} + 1258\beta_{6} + 228\beta_{5} + 228\beta_{4} + 3141\beta_{3} - 3141\beta_{2} - 18610\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
0.538834i
− 5.14642i
4.28911i
− 1.68152i
− 0.538834i
5.14642i
− 4.28911i
1.68152i
0 −7.17770 + 7.17770i 0 8.50645 − 7.25537i 0 −7.22872 − 7.22872i 0 − 76.0387i 0
63.2 0 −2.38984 + 2.38984i 0 −6.76288 + 8.90300i 0 9.83542 + 9.83542i 0 15.5773i 0
63.3 0 0.437458 − 0.437458i 0 −8.60288 − 7.14076i 0 −12.8503 − 12.8503i 0 26.6173i 0
63.4 0 6.13008 − 6.13008i 0 3.85931 + 10.4931i 0 −24.7564 − 24.7564i 0 − 48.1559i 0
127.1 0 −7.17770 − 7.17770i 0 8.50645 + 7.25537i 0 −7.22872 + 7.22872i 0 76.0387i 0
127.2 0 −2.38984 − 2.38984i 0 −6.76288 − 8.90300i 0 9.83542 − 9.83542i 0 − 15.5773i 0
127.3 0 0.437458 + 0.437458i 0 −8.60288 + 7.14076i 0 −12.8503 + 12.8503i 0 − 26.6173i 0
127.4 0 6.13008 + 6.13008i 0 3.85931 − 10.4931i 0 −24.7564 + 24.7564i 0 48.1559i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 160.4.n.c ✓ 8
4.b odd 2 1 160.4.n.f yes 8
5.c odd 4 1 160.4.n.f yes 8
8.b even 2 1 320.4.n.i 8
8.d odd 2 1 320.4.n.f 8
20.e even 4 1 inner 160.4.n.c ✓ 8
40.i odd 4 1 320.4.n.f 8
40.k even 4 1 320.4.n.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.n.c ✓ 8 1.a even 1 1 trivial
160.4.n.c ✓ 8 20.e even 4 1 inner
160.4.n.f yes 8 4.b odd 2 1
160.4.n.f yes 8 5.c odd 4 1
320.4.n.f 8 8.d odd 2 1
320.4.n.f 8 40.i odd 4 1
320.4.n.i 8 8.b even 2 1
320.4.n.i 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6T_{3}^{7} + 18T_{3}^{6} - 168T_{3}^{5} + 7028T_{3}^{4} + 28824T_{3}^{3} + 60552T_{3}^{2} - 64032T_{3} + 33856 \) acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots + 33856 \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 8185182784 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 9429321318400 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1659222219664 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 84810469747600 \) Copy content Toggle raw display
$19$ \( (T^{4} + 88 T^{3} + \cdots + 74695424)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 226869868840000 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + 14 T^{3} + \cdots - 111097600)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 904 T^{3} + \cdots - 3059828992)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 534 T^{3} + \cdots + 8377265504)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} - 904 T^{3} + \cdots - 2880741376)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
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