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Properties

Label 198.3.j.a
Level $198$
Weight $3$
Character orbit 198.j
Analytic conductor $5.395$
Analytic rank $0$
Dimension $8$
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] = [198,3,Mod(19,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 198.j (of order \(10\), degree \(4\), 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: \(5.39510923433\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{5}+ \cdots + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{5}+ \cdots + ( - 25 \beta_{7} - 48 \beta_{6} + \cdots + 20) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 2 q^{5} - 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 2 q^{5} - 30 q^{7} + 4 q^{11} + 30 q^{13} - 16 q^{14} - 8 q^{16} - 30 q^{17} - 30 q^{19} + 4 q^{20} + 24 q^{22} + 104 q^{23} - 12 q^{25} + 96 q^{26} - 40 q^{28} + 10 q^{29} + 46 q^{31} + 112 q^{34} - 70 q^{35} + 6 q^{37} - 108 q^{38} + 80 q^{40} - 250 q^{41} + 12 q^{44} - 160 q^{46} + 54 q^{47} - 144 q^{49} + 80 q^{50} - 40 q^{52} + 274 q^{53} - 26 q^{55} - 48 q^{56} + 64 q^{58} - 50 q^{59} + 50 q^{61} - 20 q^{62} + 16 q^{64} + 112 q^{67} - 60 q^{68} + 4 q^{70} - 54 q^{71} - 70 q^{73} + 40 q^{74} - 266 q^{77} + 370 q^{79} - 48 q^{80} - 96 q^{82} + 150 q^{83} - 330 q^{85} + 72 q^{86} + 72 q^{88} - 24 q^{89} - 294 q^{91} + 112 q^{92} - 20 q^{94} + 330 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(155\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.831254 − 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i
0.831254 − 1.14412i
−1.34500 + 0.437016i
1.34500 − 0.437016i
−1.34500 − 0.437016i
1.34500 + 0.437016i
−0.831254 − 1.14412i 0 −0.618034 + 1.90211i 2.42545 + 1.76219i 0 −3.70473 − 1.20374i 2.68999 − 0.874032i 0 − 4.23984i
19.2 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −6.27955 − 4.56236i 0 −2.67724 − 0.869888i −2.68999 + 0.874032i 0 − 10.9771i
73.1 −0.831254 + 1.14412i 0 −0.618034 − 1.90211i 2.42545 − 1.76219i 0 −3.70473 + 1.20374i 2.68999 + 0.874032i 0 4.23984i
73.2 0.831254 − 1.14412i 0 −0.618034 − 1.90211i −6.27955 + 4.56236i 0 −2.67724 + 0.869888i −2.68999 − 0.874032i 0 10.9771i
127.1 −1.34500 + 0.437016i 0 1.61803 − 1.17557i 2.45454 − 7.55429i 0 −2.13277 − 2.93550i −1.66251 + 2.28825i 0 11.2332i
127.2 1.34500 − 0.437016i 0 1.61803 − 1.17557i 0.399565 − 1.22973i 0 −6.48527 − 8.92621i 1.66251 − 2.28825i 0 − 1.82860i
145.1 −1.34500 − 0.437016i 0 1.61803 + 1.17557i 2.45454 + 7.55429i 0 −2.13277 + 2.93550i −1.66251 − 2.28825i 0 − 11.2332i
145.2 1.34500 + 0.437016i 0 1.61803 + 1.17557i 0.399565 + 1.22973i 0 −6.48527 + 8.92621i 1.66251 + 2.28825i 0 1.82860i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.3.j.a 8
3.b odd 2 1 22.3.d.a ✓ 8
11.c even 5 1 2178.3.d.l 8
11.d odd 10 1 inner 198.3.j.a 8
11.d odd 10 1 2178.3.d.l 8
12.b even 2 1 176.3.n.b 8
33.d even 2 1 242.3.d.c 8
33.f even 10 1 22.3.d.a ✓ 8
33.f even 10 1 242.3.b.d 8
33.f even 10 1 242.3.d.d 8
33.f even 10 1 242.3.d.e 8
33.h odd 10 1 242.3.b.d 8
33.h odd 10 1 242.3.d.c 8
33.h odd 10 1 242.3.d.d 8
33.h odd 10 1 242.3.d.e 8
132.n odd 10 1 176.3.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a ✓ 8 3.b odd 2 1
22.3.d.a ✓ 8 33.f even 10 1
176.3.n.b 8 12.b even 2 1
176.3.n.b 8 132.n odd 10 1
198.3.j.a 8 1.a even 1 1 trivial
198.3.j.a 8 11.d odd 10 1 inner
242.3.b.d 8 33.f even 10 1
242.3.b.d 8 33.h odd 10 1
242.3.d.c 8 33.d even 2 1
242.3.d.c 8 33.h odd 10 1
242.3.d.d 8 33.f even 10 1
242.3.d.d 8 33.h odd 10 1
242.3.d.e 8 33.f even 10 1
242.3.d.e 8 33.h odd 10 1
2178.3.d.l 8 11.c even 5 1
2178.3.d.l 8 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2T_{5}^{7} + 33T_{5}^{6} + 244T_{5}^{5} + 1790T_{5}^{4} - 15086T_{5}^{3} + 48588T_{5}^{2} - 50668T_{5} + 57121 \) acting on \(S_{3}^{\mathrm{new}}(198, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$7$ \( T^{8} + 30 T^{7} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2268521641 \) Copy content Toggle raw display
$17$ \( T^{8} + 30 T^{7} + \cdots + 22934521 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3189877441 \) Copy content Toggle raw display
$23$ \( (T^{4} - 52 T^{3} + \cdots - 90224)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 206760274681 \) Copy content Toggle raw display
$31$ \( T^{8} - 46 T^{7} + \cdots + 996728041 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 20079361 \) Copy content Toggle raw display
$41$ \( T^{8} + 250 T^{7} + \cdots + 405257161 \) Copy content Toggle raw display
$43$ \( T^{8} + 3632 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 7428543721 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 189900554154481 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 47196831300025 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 4097365398025 \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{3} + \cdots + 112576)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14204072031241 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36635874025 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 4768279282321 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots - 22808304)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 185279454481 \) Copy content Toggle raw display
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