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Properties

Label 160.6.n.d
Level $160$
Weight $6$
Character orbit 160.n
Analytic conductor $25.661$
Analytic rank $0$
Dimension $16$
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] = [160,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1375 x^{14} + 743087 x^{12} + 198706725 x^{10} + 26872635188 x^{8} + 1612811892960 x^{6} + \cdots + 177426662425600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{41}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{8} + \beta_{4} - 3) q^{5} + (\beta_{10} - \beta_{8} + 6 \beta_{4} + \cdots - 6) q^{7}+ \cdots + ( - \beta_{7} + 103 \beta_{4} + \cdots - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{8} + \beta_{4} - 3) q^{5} + (\beta_{10} - \beta_{8} + 6 \beta_{4} + \cdots - 6) q^{7}+ \cdots + ( - 69 \beta_{15} - 69 \beta_{13} + \cdots + 8707) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{3} - 42 q^{5} - 86 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{3} - 42 q^{5} - 86 q^{7} + 536 q^{13} - 698 q^{15} - 1828 q^{17} + 2512 q^{19} - 4284 q^{21} - 7642 q^{23} + 9140 q^{25} - 12272 q^{27} + 11876 q^{33} + 10518 q^{35} - 7620 q^{37} + 11244 q^{39} - 21284 q^{41} + 20002 q^{43} + 686 q^{45} + 25298 q^{47} + 12852 q^{53} - 10584 q^{55} + 55848 q^{57} - 142704 q^{59} - 20564 q^{61} - 115282 q^{63} - 38256 q^{65} - 10506 q^{67} + 15432 q^{73} + 256226 q^{75} + 133852 q^{77} - 159344 q^{79} - 236116 q^{81} - 61222 q^{83} + 7056 q^{85} + 162176 q^{87} + 122180 q^{93} + 267512 q^{95} - 17344 q^{97} + 107332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1375 x^{14} + 743087 x^{12} + 198706725 x^{10} + 26872635188 x^{8} + 1612811892960 x^{6} + \cdots + 177426662425600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} + 344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11825110207571 \nu^{14} + \cdots + 19\!\cdots\!80 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11825110207571 \nu^{14} + \cdots - 19\!\cdots\!80 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!91 \nu^{15} + \cdots - 37\!\cdots\!20 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\!\cdots\!15 \nu^{15} + \cdots - 28\!\cdots\!40 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!62 \nu^{15} + \cdots - 46\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{15} + \cdots - 26\!\cdots\!60 \nu ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55\!\cdots\!67 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55\!\cdots\!67 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!59 \nu^{15} + \cdots - 69\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!03 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!01 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!43 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!84 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!53 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 - 344 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 3 \beta_{11} + 4 \beta_{10} - 3 \beta_{7} + 25 \beta_{6} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 49 \beta_{15} - 49 \beta_{13} - 3 \beta_{12} + 25 \beta_{11} - 35 \beta_{10} - 56 \beta_{9} + \cdots + 203177 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 423 \beta_{15} - 663 \beta_{14} + 423 \beta_{13} + 321 \beta_{12} + 1797 \beta_{11} - 2202 \beta_{10} + \cdots - 321 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 28051 \beta_{15} + 28051 \beta_{13} - 4959 \beta_{12} - 14515 \beta_{11} + 20789 \beta_{10} + \cdots - 65791835 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 144878 \beta_{15} + 328013 \beta_{14} - 144878 \beta_{13} - 85532 \beta_{12} - 817110 \beta_{11} + \cdots + 85532 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12331845 \beta_{15} - 12331845 \beta_{13} + 4799217 \beta_{12} + 6328509 \beta_{11} + \cdots + 22568502013 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 44989245 \beta_{15} - 146692755 \beta_{14} + 44989245 \beta_{13} + 19862799 \beta_{12} + \cdots - 19862799 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4950034887 \beta_{15} + 4950034887 \beta_{13} - 2926608051 \beta_{12} - 2531335191 \beta_{11} + \cdots - 8047494130639 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12950771260 \beta_{15} + 62493690265 \beta_{14} - 12950771260 \beta_{13} - 3616179250 \beta_{12} + \cdots + 3616179250 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1909566697801 \beta_{15} - 1909566697801 \beta_{13} + 1502887482645 \beta_{12} + \cdots + 29\!\cdots\!61 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3397983388347 \beta_{15} - 25911317704287 \beta_{14} + 3397983388347 \beta_{13} + 180343826229 \beta_{12} + \cdots - 180343826229 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 722959127959243 \beta_{15} + 722959127959243 \beta_{13} - 707341430714391 \beta_{12} + \cdots - 10\!\cdots\!75 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 752369968134842 \beta_{15} + \cdots - 298058147893480 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{4}\) \(1\)

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
− 18.6675i
− 14.0500i
− 13.0932i
1.56776i
2.29308i
3.47866i
15.6412i
19.8299i
18.6675i
14.0500i
13.0932i
− 1.56776i
− 2.29308i
− 3.47866i
− 15.6412i
− 19.8299i
0 −17.6675 + 17.6675i 0 55.7953 − 3.44692i 0 52.6478 + 52.6478i 0 − 381.282i 0
63.2 0 −13.0500 + 13.0500i 0 −30.1056 − 47.1026i 0 −150.919 − 150.919i 0 − 97.6048i 0
63.3 0 −12.0932 + 12.0932i 0 −34.4070 + 44.0586i 0 74.8536 + 74.8536i 0 − 49.4886i 0
63.4 0 2.56776 − 2.56776i 0 3.91778 + 55.7642i 0 −86.5899 − 86.5899i 0 229.813i 0
63.5 0 3.29308 − 3.29308i 0 35.3847 − 43.2773i 0 −5.96093 − 5.96093i 0 221.311i 0
63.6 0 4.47866 − 4.47866i 0 −49.0494 − 26.8170i 0 132.222 + 132.222i 0 202.883i 0
63.7 0 16.6412 − 16.6412i 0 53.0700 + 17.5663i 0 76.6521 + 76.6521i 0 − 310.861i 0
63.8 0 20.8299 − 20.8299i 0 −55.6057 − 5.74531i 0 −135.906 − 135.906i 0 − 624.772i 0
127.1 0 −17.6675 − 17.6675i 0 55.7953 + 3.44692i 0 52.6478 − 52.6478i 0 381.282i 0
127.2 0 −13.0500 − 13.0500i 0 −30.1056 + 47.1026i 0 −150.919 + 150.919i 0 97.6048i 0
127.3 0 −12.0932 − 12.0932i 0 −34.4070 − 44.0586i 0 74.8536 − 74.8536i 0 49.4886i 0
127.4 0 2.56776 + 2.56776i 0 3.91778 − 55.7642i 0 −86.5899 + 86.5899i 0 − 229.813i 0
127.5 0 3.29308 + 3.29308i 0 35.3847 + 43.2773i 0 −5.96093 + 5.96093i 0 − 221.311i 0
127.6 0 4.47866 + 4.47866i 0 −49.0494 + 26.8170i 0 132.222 − 132.222i 0 − 202.883i 0
127.7 0 16.6412 + 16.6412i 0 53.0700 − 17.5663i 0 76.6521 − 76.6521i 0 310.861i 0
127.8 0 20.8299 + 20.8299i 0 −55.6057 + 5.74531i 0 −135.906 + 135.906i 0 624.772i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.8
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 160.6.n.d yes 16
4.b odd 2 1 160.6.n.c ✓ 16
5.c odd 4 1 160.6.n.c ✓ 16
20.e even 4 1 inner 160.6.n.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.n.c ✓ 16 4.b odd 2 1
160.6.n.c ✓ 16 5.c odd 4 1
160.6.n.d yes 16 1.a even 1 1 trivial
160.6.n.d yes 16 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 10 T_{3}^{15} + 50 T_{3}^{14} + 5544 T_{3}^{13} + 769976 T_{3}^{12} - 3435728 T_{3}^{11} + \cdots + 34\!\cdots\!24 \) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 53\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 62\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 40\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 21\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
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