Properties

Label 162.8.c.o
Level $162$
Weight $8$
Character orbit 162.c
Analytic conductor $50.606$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,8,Mod(55,162)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), 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: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-643})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 160x^{2} - 161x + 25921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta_1 q^{2} + ( - 64 \beta_1 - 64) q^{4} + ( - \beta_{3} - 57 \beta_1 - 57) q^{5} + (4 \beta_{3} - 4 \beta_{2} + 140 \beta_1) q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta_1 q^{2} + ( - 64 \beta_1 - 64) q^{4} + ( - \beta_{3} - 57 \beta_1 - 57) q^{5} + (4 \beta_{3} - 4 \beta_{2} + 140 \beta_1) q^{7} - 512 q^{8} + ( - 8 \beta_{2} - 456) q^{10} + (16 \beta_{3} - 16 \beta_{2} + 2592 \beta_1) q^{11} + ( - 41 \beta_{3} + 3331 \beta_1 + 3331) q^{13} + (32 \beta_{3} + 1120 \beta_1 + 1120) q^{14} + 4096 \beta_1 q^{16} + (62 \beta_{2} + 18255) q^{17} + (56 \beta_{2} + 3236) q^{19} + (64 \beta_{3} - 64 \beta_{2} + 3648 \beta_1) q^{20} + (128 \beta_{3} + 20736 \beta_1 + 20736) q^{22} + (40 \beta_{3} + 6480 \beta_1 + 6480) q^{23} + (114 \beta_{3} - 114 \beta_{2} - 5432 \beta_1) q^{25} + ( - 328 \beta_{2} + 26648) q^{26} + (256 \beta_{2} + 8960) q^{28} + ( - 311 \beta_{3} + 311 \beta_{2} + 8037 \beta_1) q^{29} + ( - 108 \beta_{3} + 80068 \beta_1 + 80068) q^{31} + (32768 \beta_1 + 32768) q^{32} + ( - 496 \beta_{3} + \cdots - 146040 \beta_1) q^{34}+ \cdots + ( - 8960 \beta_{2} - 2457288) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 128 q^{4} - 114 q^{5} - 280 q^{7} - 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{2} - 128 q^{4} - 114 q^{5} - 280 q^{7} - 2048 q^{8} - 1824 q^{10} - 5184 q^{11} + 6662 q^{13} + 2240 q^{14} - 8192 q^{16} + 73020 q^{17} + 12944 q^{19} - 7296 q^{20} + 41472 q^{22} + 12960 q^{23} + 10864 q^{25} + 106592 q^{26} + 35840 q^{28} - 16074 q^{29} + 160136 q^{31} + 65536 q^{32} + 292080 q^{34} + 1143024 q^{35} - 573916 q^{37} + 51776 q^{38} + 58368 q^{40} + 538284 q^{41} + 1770896 q^{43} + 663552 q^{44} + 207360 q^{46} - 279960 q^{47} - 614322 q^{49} - 86912 q^{50} + 426368 q^{52} + 4747704 q^{53} + 5035392 q^{55} + 143360 q^{56} + 128592 q^{58} - 690336 q^{59} + 3688886 q^{61} + 2562176 q^{62} + 1048576 q^{64} - 5314674 q^{65} + 983096 q^{67} - 2336640 q^{68} + 4572096 q^{70} + 5553840 q^{71} + 19782188 q^{73} - 2295664 q^{74} - 414208 q^{76} - 9614592 q^{77} + 5078336 q^{79} + 933888 q^{80} + 8612544 q^{82} - 8946312 q^{83} - 10692126 q^{85} - 14167168 q^{86} + 2654208 q^{88} + 1365132 q^{89} + 43689904 q^{91} + 829440 q^{92} + 2239680 q^{94} - 8146632 q^{95} + 18464252 q^{97} - 9829152 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 160x^{2} - 161x + 25921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 160\nu^{2} - 160\nu - 25921 ) / 25760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{3} + 12\nu^{2} + 3852\nu + 966 ) / 161 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 969\nu^{3} + 480\nu^{2} + 154080\nu - 310569 ) / 12880 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 18\beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 5778\beta _1 + 5778 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 80\beta_{3} - 40\beta_{2} + 2169 ) / 9 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
11.2301 + 5.90635i
−10.7301 6.77237i
11.2301 5.90635i
−10.7301 + 6.77237i
4.00000 6.92820i 0 −32.0000 55.4256i −160.261 277.580i 0 −597.045 + 1034.11i −512.000 0 −2564.18
55.2 4.00000 6.92820i 0 −32.0000 55.4256i 103.261 + 178.854i 0 457.045 791.624i −512.000 0 1652.18
109.1 4.00000 + 6.92820i 0 −32.0000 + 55.4256i −160.261 + 277.580i 0 −597.045 1034.11i −512.000 0 −2564.18
109.2 4.00000 + 6.92820i 0 −32.0000 + 55.4256i 103.261 178.854i 0 457.045 + 791.624i −512.000 0 1652.18
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.8.c.o 4
3.b odd 2 1 162.8.c.n 4
9.c even 3 1 162.8.a.c 2
9.c even 3 1 inner 162.8.c.o 4
9.d odd 6 1 162.8.a.d yes 2
9.d odd 6 1 162.8.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.c 2 9.c even 3 1
162.8.a.d yes 2 9.d odd 6 1
162.8.c.n 4 3.b odd 2 1
162.8.c.n 4 9.d odd 6 1
162.8.c.o 4 1.a even 1 1 trivial
162.8.c.o 4 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 114T_{5}^{3} + 79191T_{5}^{2} - 7546230T_{5} + 4381778025 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 4381778025 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 1191380982016 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 122305904640000 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( (T^{2} - 36510 T + 66302289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6472 T - 207304688)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 44\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{2} + 286958 T - 119637948563)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2373852 T + 947978500932)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2776920 T - 608963703984)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 24356425782073)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 24042892404087)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
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