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Properties

Label 162.13.d.e
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: \( x^{8} - 1052x^{6} + 831079x^{4} - 289957500x^{2} + 75969140625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - 2048 \beta_1 + 2048) q^{4} + (5 \beta_{4} + 60 \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{6} + 17071 \beta_1) q^{7} + (2048 \beta_{3} + 2048 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - 2048 \beta_1 + 2048) q^{4} + (5 \beta_{4} + 60 \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{6} + 17071 \beta_1) q^{7} + (2048 \beta_{3} + 2048 \beta_{2}) q^{8} + ( - 5 \beta_{6} - 122880) q^{10} + (407 \beta_{5} - 9636 \beta_{3}) q^{11} + (59 \beta_{7} + 1835759 \beta_1 - 1835759) q^{13} + (2048 \beta_{4} - 17071 \beta_{2}) q^{14} - 4194304 \beta_1 q^{16} + (12071 \beta_{5} + 12071 \beta_{4} + 295596 \beta_{3} + 295596 \beta_{2}) q^{17} + (62 \beta_{6} + 3007649) q^{19} + ( - 10240 \beta_{5} - 122880 \beta_{3}) q^{20} + ( - 407 \beta_{7} + 19734528 \beta_1 - 19734528) q^{22} + (49033 \beta_{4} - 370452 \beta_{2}) q^{23} + (600 \beta_{7} + 600 \beta_{6} - 83532025 \beta_1) q^{25} + ( - 120832 \beta_{5} - 120832 \beta_{4} + \cdots - 1835759 \beta_{2}) q^{26}+ \cdots + (69922816 \beta_{5} + 69922816 \beta_{4} + \cdots + 996791424 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8192 q^{4} + 68284 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8192 q^{4} + 68284 q^{7} - 983040 q^{10} - 7343036 q^{13} - 16777216 q^{16} + 24061192 q^{19} - 78938112 q^{22} - 334128100 q^{25} + 279691264 q^{28} - 3973833944 q^{31} - 2421522432 q^{34} + 10859552440 q^{37} - 1006632960 q^{40} - 11025548504 q^{43} + 6069485568 q^{46} + 3987165696 q^{49} + 15038537728 q^{52} - 90314579520 q^{55} + 26310672384 q^{58} + 175535338564 q^{61} - 68719476736 q^{64} - 88085449700 q^{67} + 242670796800 q^{70} + 378811048696 q^{73} + 24638660608 q^{76} + 416321284636 q^{79} - 72114634752 q^{82} - 1625058819360 q^{85} + 161665253376 q^{88} + 5674346284280 q^{91} + 311943266304 q^{94} + 352304406244 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 1052x^{6} + 831079x^{4} - 289957500x^{2} + 75969140625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1052\nu^{6} - 831079\nu^{4} + 874295108\nu^{2} - 75969140625 ) / 229066149375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32\nu^{7} + 4736676256\nu ) / 436316475 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8920928\nu^{7} - 14015316256\nu^{5} + 7413995921312\nu^{3} - 2586689980560000\nu ) / 120259728421875 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\nu^{7} + 18371773494\nu ) / 145438825 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2766758\nu^{7} - 2621223166\nu^{5} + 2299394471882\nu^{3} - 802242232785000\nu ) / 4454064015625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3456\nu^{6} + 508688826624 ) / 831079 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 106647168\nu^{6} - 167864660736\nu^{4} + 88632221734272\nu^{2} - 30923146215360000 ) / 25451794375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 16\beta_{4} - 27\beta_{2} ) / 1728 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 1817856\beta_1 ) / 3456 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8432\beta_{5} + 8432\beta_{4} - 42579\beta_{3} - 42579\beta_{2} ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -263\beta_{7} + 239956128\beta _1 - 239956128 ) / 864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4460464\beta_{5} - 37351233\beta_{3} ) / 1728 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 831079\beta_{6} - 508688826624 ) / 3456 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2368338128\beta_{4} + 27557660241\beta_{2} ) / 1728 \) 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
−20.4650 − 11.8154i
19.2402 + 11.1083i
−19.2402 − 11.1083i
20.4650 + 11.8154i
−20.4650 + 11.8154i
19.2402 − 11.1083i
−19.2402 + 11.1083i
20.4650 − 11.8154i
−39.1918 + 22.6274i 0 1024.00 − 1773.62i −8368.88 − 4831.78i 0 64555.8 + 111814.i 92681.9i 0 437323.
53.2 −39.1918 + 22.6274i 0 1024.00 − 1773.62i 13071.9 + 7547.07i 0 −47484.8 − 82246.0i 92681.9i 0 −683083.
53.3 39.1918 − 22.6274i 0 1024.00 − 1773.62i −13071.9 − 7547.07i 0 −47484.8 − 82246.0i − 92681.9i 0 −683083.
53.4 39.1918 − 22.6274i 0 1024.00 − 1773.62i 8368.88 + 4831.78i 0 64555.8 + 111814.i − 92681.9i 0 437323.
107.1 −39.1918 − 22.6274i 0 1024.00 + 1773.62i −8368.88 + 4831.78i 0 64555.8 − 111814.i − 92681.9i 0 437323.
107.2 −39.1918 − 22.6274i 0 1024.00 + 1773.62i 13071.9 − 7547.07i 0 −47484.8 + 82246.0i − 92681.9i 0 −683083.
107.3 39.1918 + 22.6274i 0 1024.00 + 1773.62i −13071.9 + 7547.07i 0 −47484.8 + 82246.0i 92681.9i 0 −683083.
107.4 39.1918 + 22.6274i 0 1024.00 + 1773.62i 8368.88 − 4831.78i 0 64555.8 − 111814.i 92681.9i 0 437323.
\(n\): 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.e 8
3.b odd 2 1 inner 162.13.d.e 8
9.c even 3 1 54.13.b.a ✓ 4
9.c even 3 1 inner 162.13.d.e 8
9.d odd 6 1 54.13.b.a ✓ 4
9.d odd 6 1 inner 162.13.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.13.b.a ✓ 4 9.c even 3 1
54.13.b.a ✓ 4 9.d odd 6 1
162.13.d.e 8 1.a even 1 1 trivial
162.13.d.e 8 3.b odd 2 1 inner
162.13.d.e 8 9.c even 3 1 inner
162.13.d.e 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 321217200 T_{5}^{6} + \cdots + 45\!\cdots\!00 \) 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
$5$ \( T^{8} - 321217200 T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} - 34142 T^{3} + \cdots + 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 2410992386352 T^{6} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3671518 T^{3} + \cdots + 16\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6015298 T - 39208074465983)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1986916972 T^{3} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2714888110 T - 11\!\cdots\!75)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + 5512774252 T^{3} + \cdots + 92\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} - 87767669282 T^{3} + \cdots + 54\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 44042724850 T^{3} + \cdots + 61\!\cdots\!21)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 78\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 94702762174 T - 11\!\cdots\!67)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 208160642318 T^{3} + \cdots + 49\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 98\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 176152203122 T^{3} + \cdots + 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
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