Properties

Label 162.10.c.e
Level $162$
Weight $10$
Character orbit 162.c
Analytic conductor $83.436$
Analytic rank $0$
Dimension $2$
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] = [162,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(83.4358054585\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (16 \zeta_{6} - 16) q^{2} - 256 \zeta_{6} q^{4} + 2694 \zeta_{6} q^{5} + ( - 3544 \zeta_{6} + 3544) q^{7} + 4096 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (16 \zeta_{6} - 16) q^{2} - 256 \zeta_{6} q^{4} + 2694 \zeta_{6} q^{5} + ( - 3544 \zeta_{6} + 3544) q^{7} + 4096 q^{8} - 43104 q^{10} + ( - 29580 \zeta_{6} + 29580) q^{11} + 44818 \zeta_{6} q^{13} + 56704 \zeta_{6} q^{14} + (65536 \zeta_{6} - 65536) q^{16} + 101934 q^{17} - 895084 q^{19} + ( - 689664 \zeta_{6} + 689664) q^{20} + 473280 \zeta_{6} q^{22} - 1113000 \zeta_{6} q^{23} + (5304511 \zeta_{6} - 5304511) q^{25} - 717088 q^{26} - 907264 q^{28} + (2357346 \zeta_{6} - 2357346) q^{29} - 175808 \zeta_{6} q^{31} - 1048576 \zeta_{6} q^{32} + (1630944 \zeta_{6} - 1630944) q^{34} + 9547536 q^{35} - 2919418 q^{37} + ( - 14321344 \zeta_{6} + 14321344) q^{38} + 11034624 \zeta_{6} q^{40} + 26218794 \zeta_{6} q^{41} + ( - 18762964 \zeta_{6} + 18762964) q^{43} - 7572480 q^{44} + 17808000 q^{46} + (20966160 \zeta_{6} - 20966160) q^{47} + 27793671 \zeta_{6} q^{49} - 84872176 \zeta_{6} q^{50} + ( - 11473408 \zeta_{6} + 11473408) q^{52} - 57251574 q^{53} + 79688520 q^{55} + ( - 14516224 \zeta_{6} + 14516224) q^{56} - 37717536 \zeta_{6} q^{58} + 33587580 \zeta_{6} q^{59} + (82260830 \zeta_{6} - 82260830) q^{61} + 2812928 q^{62} + 16777216 q^{64} + (120739692 \zeta_{6} - 120739692) q^{65} + 188455804 \zeta_{6} q^{67} - 26095104 \zeta_{6} q^{68} + (152760576 \zeta_{6} - 152760576) q^{70} - 80924040 q^{71} - 236140918 q^{73} + ( - 46710688 \zeta_{6} + 46710688) q^{74} + 229141504 \zeta_{6} q^{76} - 104831520 \zeta_{6} q^{77} + (526909808 \zeta_{6} - 526909808) q^{79} - 176553984 q^{80} - 419500704 q^{82} + ( - 18346452 \zeta_{6} + 18346452) q^{83} + 274610196 \zeta_{6} q^{85} + 300207424 \zeta_{6} q^{86} + ( - 121159680 \zeta_{6} + 121159680) q^{88} - 690643098 q^{89} + 158834992 q^{91} + (284928000 \zeta_{6} - 284928000) q^{92} - 335458560 \zeta_{6} q^{94} - 2411356296 \zeta_{6} q^{95} + ( - 438251038 \zeta_{6} + 438251038) q^{97} - 444698736 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} - 256 q^{4} + 2694 q^{5} + 3544 q^{7} + 8192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{2} - 256 q^{4} + 2694 q^{5} + 3544 q^{7} + 8192 q^{8} - 86208 q^{10} + 29580 q^{11} + 44818 q^{13} + 56704 q^{14} - 65536 q^{16} + 203868 q^{17} - 1790168 q^{19} + 689664 q^{20} + 473280 q^{22} - 1113000 q^{23} - 5304511 q^{25} - 1434176 q^{26} - 1814528 q^{28} - 2357346 q^{29} - 175808 q^{31} - 1048576 q^{32} - 1630944 q^{34} + 19095072 q^{35} - 5838836 q^{37} + 14321344 q^{38} + 11034624 q^{40} + 26218794 q^{41} + 18762964 q^{43} - 15144960 q^{44} + 35616000 q^{46} - 20966160 q^{47} + 27793671 q^{49} - 84872176 q^{50} + 11473408 q^{52} - 114503148 q^{53} + 159377040 q^{55} + 14516224 q^{56} - 37717536 q^{58} + 33587580 q^{59} - 82260830 q^{61} + 5625856 q^{62} + 33554432 q^{64} - 120739692 q^{65} + 188455804 q^{67} - 26095104 q^{68} - 152760576 q^{70} - 161848080 q^{71} - 472281836 q^{73} + 46710688 q^{74} + 229141504 q^{76} - 104831520 q^{77} - 526909808 q^{79} - 353107968 q^{80} - 839001408 q^{82} + 18346452 q^{83} + 274610196 q^{85} + 300207424 q^{86} + 121159680 q^{88} - 1381286196 q^{89} + 317669984 q^{91} - 284928000 q^{92} - 335458560 q^{94} - 2411356296 q^{95} + 438251038 q^{97} - 889397472 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−8.00000 + 13.8564i 0 −128.000 221.703i 1347.00 + 2333.07i 0 1772.00 3069.19i 4096.00 0 −43104.0
109.1 −8.00000 13.8564i 0 −128.000 + 221.703i 1347.00 2333.07i 0 1772.00 + 3069.19i 4096.00 0 −43104.0
\(n\): 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 162.10.c.e 2
3.b odd 2 1 162.10.c.f 2
9.c even 3 1 18.10.a.c 1
9.c even 3 1 inner 162.10.c.e 2
9.d odd 6 1 6.10.a.a 1
9.d odd 6 1 162.10.c.f 2
36.f odd 6 1 144.10.a.a 1
36.h even 6 1 48.10.a.d 1
45.h odd 6 1 150.10.a.h 1
45.l even 12 2 150.10.c.d 2
63.o even 6 1 294.10.a.a 1
72.j odd 6 1 192.10.a.a 1
72.l even 6 1 192.10.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.10.a.a 1 9.d odd 6 1
18.10.a.c 1 9.c even 3 1
48.10.a.d 1 36.h even 6 1
144.10.a.a 1 36.f odd 6 1
150.10.a.h 1 45.h odd 6 1
150.10.c.d 2 45.l even 12 2
162.10.c.e 2 1.a even 1 1 trivial
162.10.c.e 2 9.c even 3 1 inner
162.10.c.f 2 3.b odd 2 1
162.10.c.f 2 9.d odd 6 1
192.10.a.a 1 72.j odd 6 1
192.10.a.h 1 72.l even 6 1
294.10.a.a 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2694T_{5} + 7257636 \) acting on \(S_{10}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2694 T + 7257636 \) Copy content Toggle raw display
$7$ \( T^{2} - 3544 T + 12559936 \) Copy content Toggle raw display
$11$ \( T^{2} - 29580 T + 874976400 \) Copy content Toggle raw display
$13$ \( T^{2} - 44818 T + 2008653124 \) Copy content Toggle raw display
$17$ \( (T - 101934)^{2} \) Copy content Toggle raw display
$19$ \( (T + 895084)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1113000 T + 1238769000000 \) Copy content Toggle raw display
$29$ \( T^{2} + 2357346 T + 5557080163716 \) Copy content Toggle raw display
$31$ \( T^{2} + 175808 T + 30908452864 \) Copy content Toggle raw display
$37$ \( (T + 2919418)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 687425158814436 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 352048818065296 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 439579865145600 \) Copy content Toggle raw display
$53$ \( (T + 57251574)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 33587580 T + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 82260830 T + 67\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} - 188455804 T + 35\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T + 80924040)^{2} \) Copy content Toggle raw display
$73$ \( (T + 236140918)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 526909808 T + 27\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 336592300988304 \) Copy content Toggle raw display
$89$ \( (T + 690643098)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 438251038 T + 19\!\cdots\!44 \) Copy content Toggle raw display
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