↑

Properties

Label 162.7.b.b
Level $162$
Weight $7$
Character orbit 162.b
Analytic conductor $37.269$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,7,Mod(161,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), 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: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 1382x^{6} - 288x^{5} + 716245x^{4} + 201312x^{3} - 164876604x^{2} - 33576768x + 14252103396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{16} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 4 \beta_1 q^{2} - 32 q^{4} + (\beta_{3} + 12 \beta_1) q^{5} + (\beta_{6} + \beta_{4} + 121) q^{7} - 128 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_1 q^{2} - 32 q^{4} + (\beta_{3} + 12 \beta_1) q^{5} + (\beta_{6} + \beta_{4} + 121) q^{7} - 128 \beta_1 q^{8} + ( - 4 \beta_{7} - 96) q^{10} + (\beta_{5} + 11 \beta_{3} + \cdots - 421 \beta_1) q^{11}+ \cdots + ( - 776 \beta_{5} + \cdots + 647860 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 256 q^{4} + 964 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 256 q^{4} + 964 q^{7} - 768 q^{10} + 4540 q^{13} + 8192 q^{16} - 23684 q^{19} + 27072 q^{22} - 32392 q^{25} - 30848 q^{28} + 77056 q^{31} - 52608 q^{34} - 11348 q^{37} + 24576 q^{40} - 226604 q^{43} - 162720 q^{46} + 1298088 q^{49} - 145280 q^{52} - 1460916 q^{55} + 867456 q^{58} - 327476 q^{61} - 262144 q^{64} + 1713292 q^{67} - 176352 q^{70} - 2189216 q^{73} + 757888 q^{76} - 1326884 q^{79} - 1158816 q^{82} + 3483180 q^{85} - 866304 q^{88} + 1130324 q^{91} - 26400 q^{94} - 2200064 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 1382x^{6} - 288x^{5} + 716245x^{4} + 201312x^{3} - 164876604x^{2} - 33576768x + 14252103396 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1757827 \nu^{7} + 15550472 \nu^{6} + 1837618022 \nu^{5} - 15612889936 \nu^{4} + \cdots - 641041597968432 ) / 10048369201740 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4113462533 \nu^{7} - 29950787701 \nu^{6} - 5668141371706 \nu^{5} + 50235543376490 \nu^{4} + \cdots + 35\!\cdots\!98 ) / 462224983280040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1723203713 \nu^{7} + 17183136963 \nu^{6} + 2405144883478 \nu^{5} + \cdots - 23\!\cdots\!98 ) / 154074994426680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 287061637 \nu^{7} + 5164927080 \nu^{6} + 295082651276 \nu^{5} - 5260447677576 \nu^{4} + \cdots - 20\!\cdots\!00 ) / 6698912801160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 501851953 \nu^{7} - 16874291267 \nu^{6} - 550388138492 \nu^{5} + 18049866731152 \nu^{4} + \cdots + 80\!\cdots\!98 ) / 10048369201740 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48249 \nu^{7} - 989496 \nu^{6} - 49735080 \nu^{5} + 1009495872 \nu^{4} + 17248065129 \nu^{3} + \cdots + 39500788589136 ) / 722837560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18088455757 \nu^{7} + 320767116792 \nu^{6} + 18629076849992 \nu^{5} + \cdots - 12\!\cdots\!52 ) / 154074994426680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + \beta_{6} - 6\beta_{4} - 162\beta _1 - 3 ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 27\beta_{7} + 55\beta_{6} - 12\beta_{5} - 12\beta_{3} - 4\beta_{2} + 4\beta _1 + 55971 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 559\beta_{7} + 401\beta_{6} - 686\beta_{4} - 54\beta_{3} - 110\beta_{2} - 56242\beta _1 + 5489 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18873 \beta_{7} + 37709 \beta_{6} - 8328 \beta_{5} - 864 \beta_{4} - 13512 \beta_{3} - 9656 \beta_{2} + \cdots + 19335645 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 794109 \beta_{7} + 706027 \beta_{6} - 4320 \beta_{5} - 705642 \beta_{4} - 190890 \beta_{3} + \cdots + 19563459 ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3275343 \beta_{7} + 6401147 \beta_{6} - 1466932 \beta_{5} - 292032 \beta_{4} - 3266644 \beta_{3} + \cdots + 2223539235 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 343662537 \beta_{7} + 339858959 \beta_{6} - 6253632 \beta_{5} - 241403874 \beta_{4} + \cdots + 13932837591 ) / 162 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−18.3354 + 1.41421i
20.5718 + 1.41421i
16.6034 + 1.41421i
−18.8398 + 1.41421i
−18.8398 − 1.41421i
16.6034 − 1.41421i
20.5718 − 1.41421i
−18.3354 − 1.41421i
− 5.65685i 0 −32.0000 − 171.064i 0 560.839 181.019i 0 −967.683
161.2 − 5.65685i 0 −32.0000 − 136.059i 0 −187.053 181.019i 0 −769.663
161.3 − 5.65685i 0 −32.0000 85.6833i 0 −564.058 181.019i 0 484.698
161.4 − 5.65685i 0 −32.0000 153.557i 0 672.272 181.019i 0 868.648
161.5 5.65685i 0 −32.0000 − 153.557i 0 672.272 − 181.019i 0 868.648
161.6 5.65685i 0 −32.0000 − 85.6833i 0 −564.058 − 181.019i 0 484.698
161.7 5.65685i 0 −32.0000 136.059i 0 −187.053 − 181.019i 0 −769.663
161.8 5.65685i 0 −32.0000 171.064i 0 560.839 − 181.019i 0 −967.683
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.7.b.b ✓ 8
3.b odd 2 1 inner 162.7.b.b ✓ 8
9.c even 3 2 162.7.d.g 16
9.d odd 6 2 162.7.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.7.b.b ✓ 8 1.a even 1 1 trivial
162.7.b.b ✓ 8 3.b odd 2 1 inner
162.7.d.g 16 9.c even 3 2
162.7.d.g 16 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 78696T_{5}^{6} + 2192079186T_{5}^{4} + 25020814545000T_{5}^{2} + 93777195687890625 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} - 482 T^{3} + \cdots + 39780617104)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 3271697830489)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 89\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 41\!\cdots\!44)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 79\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 68\!\cdots\!73)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 95\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 26\!\cdots\!57)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 93\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 40\!\cdots\!71)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
show more
show less