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Properties

Label 162.13.d.d
Level $162$
Weight $13$
Character orbit 162.d
Analytic conductor $148.067$
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] = [162,13,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), 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: \(148.066998399\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 1006x^{6} + 3032x^{5} + 379007x^{4} - 763072x^{3} - 63364886x^{2} + 63746928x + 3969381036 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{3}) q^{2} + (2048 \beta_1 + 2048) q^{4} + ( - 200 \beta_{3} + \beta_{2}) q^{5} + ( - 17 \beta_{7} - 17 \beta_{4} + 38270 \beta_1) q^{7} + 2048 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{3}) q^{2} + (2048 \beta_1 + 2048) q^{4} + ( - 200 \beta_{3} + \beta_{2}) q^{5} + ( - 17 \beta_{7} - 17 \beta_{4} + 38270 \beta_1) q^{7} + 2048 \beta_{6} q^{8} + (32 \beta_{4} + 410496) q^{10} + (19492 \beta_{6} + \cdots - 19492 \beta_{3}) q^{11}+ \cdots + ( - 18197979235 \beta_{6} + \cdots + 83275520 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8192 q^{4} - 153080 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8192 q^{4} - 153080 q^{7} + 3283968 q^{10} - 7253000 q^{13} - 16777216 q^{16} - 240536144 q^{19} + 159244800 q^{22} - 435605764 q^{25} - 627015680 q^{28} - 2731727672 q^{31} + 3097810944 q^{34} - 30560240 q^{37} + 3362783232 q^{40} - 1629119960 q^{43} - 59811698688 q^{46} - 72937649100 q^{49} + 14854144000 q^{52} + 12571599744 q^{55} + 62351992320 q^{58} - 45477065096 q^{61} - 68719476736 q^{64} + 213433609960 q^{67} - 293322353664 q^{70} - 508767250160 q^{73} - 246309011456 q^{76} - 308580159032 q^{79} + 469207418880 q^{82} + 18844054272 q^{85} - 326133350400 q^{88} - 6647468692000 q^{91} - 775668172800 q^{94} - 1276228475720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 1006x^{6} + 3032x^{5} + 379007x^{4} - 763072x^{3} - 63364886x^{2} + 63746928x + 3969381036 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 2521\nu^{4} + 5052\nu^{3} + 890081\nu^{2} - 892608\nu - 96448452 ) / 2028066 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 282296 \nu^{7} - 328251688 \nu^{6} + 683166140 \nu^{5} + 413264865476 \nu^{4} + \cdots + 57\!\cdots\!64 ) / 1024247354409 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 645248 \nu^{7} + 2258368 \nu^{6} + 682571008 \nu^{5} - 1712073440 \nu^{4} + \cdots - 19621044296448 ) / 1024247354409 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1451808 \nu^{7} - 5081328 \nu^{6} - 1535784768 \nu^{5} + 3852165240 \nu^{4} + \cdots + 81020254425732 ) / 113805261601 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14110754 \nu^{7} - 82438012030 \nu^{6} + 236505763118 \nu^{5} + 62395616935526 \nu^{4} + \cdots + 12\!\cdots\!48 ) / 1024247354409 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32800 \nu^{7} + 114800 \nu^{6} + 24973616 \nu^{5} - 62721040 \nu^{4} - 6361949200 \nu^{3} + \cdots - 272405933760 ) / 1021183803 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 145294848 \nu^{7} - 508531968 \nu^{6} - 146384520444 \nu^{5} + 367232631030 \nu^{4} + \cdots + 22\!\cdots\!76 ) / 113805261601 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{4} - 81\beta_{3} + 1296 ) / 2592 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - 22\beta_{3} - 4\beta_{2} - 1296\beta _1 + 163620 ) / 648 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{7} - 81\beta_{6} - 518\beta_{4} - 30684\beta_{3} - 12\beta_{2} - 3888\beta _1 + 490536 ) / 1296 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{7} - 88 \beta_{6} - 16 \beta_{5} - 517 \beta_{4} - 31544 \beta_{3} - 2040 \beta_{2} + \cdots + 41474916 ) / 648 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5080 \beta_{7} - 102361 \beta_{6} - 40 \beta_{5} - 69321 \beta_{4} - 6482989 \beta_{3} + \cdots + 103278564 ) / 648 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 15210 \beta_{7} - 315704 \beta_{6} - 20288 \beta_{5} - 206671 \beta_{4} - 19703052 \beta_{3} + \cdots + 10545794676 ) / 648 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2734382 \beta_{7} - 91192556 \beta_{6} - 70868 \beta_{5} - 19084199 \beta_{4} - 2309784560 \beta_{3} + \cdots + 36549092484 ) / 648 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−16.6071 − 0.707107i
15.1576 − 0.707107i
17.6071 + 0.707107i
−14.1576 + 0.707107i
−16.6071 + 0.707107i
15.1576 + 0.707107i
17.6071 − 0.707107i
−14.1576 − 0.707107i
−39.1918 + 22.6274i 0 1024.00 − 1773.62i −14157.9 − 8174.08i 0 −106615. − 184663.i 92681.9i 0 739833.
53.2 −39.1918 + 22.6274i 0 1024.00 − 1773.62i −1553.11 − 896.688i 0 68345.1 + 118377.i 92681.9i 0 81159.0
53.3 39.1918 − 22.6274i 0 1024.00 − 1773.62i 1553.11 + 896.688i 0 68345.1 + 118377.i − 92681.9i 0 81159.0
53.4 39.1918 − 22.6274i 0 1024.00 − 1773.62i 14157.9 + 8174.08i 0 −106615. − 184663.i − 92681.9i 0 739833.
107.1 −39.1918 − 22.6274i 0 1024.00 + 1773.62i −14157.9 + 8174.08i 0 −106615. + 184663.i − 92681.9i 0 739833.
107.2 −39.1918 − 22.6274i 0 1024.00 + 1773.62i −1553.11 + 896.688i 0 68345.1 − 118377.i − 92681.9i 0 81159.0
107.3 39.1918 + 22.6274i 0 1024.00 + 1773.62i 1553.11 − 896.688i 0 68345.1 − 118377.i 92681.9i 0 81159.0
107.4 39.1918 + 22.6274i 0 1024.00 + 1773.62i 14157.9 − 8174.08i 0 −106615. + 184663.i 92681.9i 0 739833.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.13.d.d 8
3.b odd 2 1 inner 162.13.d.d 8
9.c even 3 1 6.13.b.a ✓ 4
9.c even 3 1 inner 162.13.d.d 8
9.d odd 6 1 6.13.b.a ✓ 4
9.d odd 6 1 inner 162.13.d.d 8
36.f odd 6 1 48.13.e.c 4
36.h even 6 1 48.13.e.c 4
45.h odd 6 1 150.13.d.a 4
45.j even 6 1 150.13.d.a 4
45.k odd 12 2 150.13.b.a 8
45.l even 12 2 150.13.b.a 8
72.j odd 6 1 192.13.e.e 4
72.l even 6 1 192.13.e.h 4
72.n even 6 1 192.13.e.e 4
72.p odd 6 1 192.13.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.13.b.a ✓ 4 9.c even 3 1
6.13.b.a ✓ 4 9.d odd 6 1
48.13.e.c 4 36.f odd 6 1
48.13.e.c 4 36.h even 6 1
150.13.b.a 8 45.k odd 12 2
150.13.b.a 8 45.l even 12 2
150.13.d.a 4 45.h odd 6 1
150.13.d.a 4 45.j even 6 1
162.13.d.d 8 1.a even 1 1 trivial
162.13.d.d 8 3.b odd 2 1 inner
162.13.d.d 8 9.c even 3 1 inner
162.13.d.d 8 9.d odd 6 1 inner
192.13.e.e 4 72.j odd 6 1
192.13.e.e 4 72.n even 6 1
192.13.e.h 4 72.l even 6 1
192.13.e.h 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 270478368 T_{5}^{6} + \cdots + 73\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2048 T^{2} + 4194304)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 84\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 36\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 8399894140076)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 72\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 11\!\cdots\!96)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 40\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 79\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 72\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 26\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
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