Properties

Label 297.3.b.c
Level $297$
Weight $3$
Character orbit 297.b
Analytic conductor $8.093$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,3,Mod(188,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.188");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.b (of order \(2\), degree \(1\), 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: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 30x^{6} + 285x^{4} + 944x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 4) q^{4} + ( - \beta_{4} - \beta_1) q^{5} + (\beta_{6} - \beta_{2}) q^{7} + (\beta_{5} - \beta_{3} - 3 \beta_1) q^{8} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 6) q^{10}+ \cdots + ( - 9 \beta_{5} - 4 \beta_{4} + \cdots + 67 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 28 q^{4} - 8 q^{7} + 36 q^{10} + 28 q^{13} + 68 q^{16} - 80 q^{19} - 124 q^{25} - 104 q^{28} + 64 q^{31} + 132 q^{34} + 76 q^{37} - 84 q^{40} + 16 q^{43} + 324 q^{46} + 276 q^{49} - 548 q^{52} - 360 q^{58}+ \cdots - 488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 30x^{6} + 285x^{4} + 944x^{2} + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 30\nu^{5} + 255\nu^{3} + 494\nu ) / 60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 25\nu^{5} - 170\nu^{3} - 264\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 30\nu^{5} + 315\nu^{3} + 1154\nu ) / 60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{4} + 17\nu^{2} + 44 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 25\nu^{4} + 170\nu^{2} + 260 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 17\beta_{2} + 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{5} + 4\beta_{4} + 29\beta_{3} + 141\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} - 50\beta_{6} + 255\beta_{2} - 1200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 255\beta_{5} - 120\beta_{4} - 555\beta_{3} - 1919\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
3.83254i
3.13595i
1.96614i
1.26955i
1.26955i
1.96614i
3.13595i
3.83254i
3.83254i 0 −10.6883 5.42387i 0 11.7111 25.6333i 0 20.7872
188.2 3.13595i 0 −5.83419 6.09830i 0 −11.4008 5.75194i 0 −19.1240
188.3 1.96614i 0 0.134296 9.56977i 0 −7.52095 8.12860i 0 18.8155
188.4 1.26955i 0 2.38823 1.95240i 0 3.21064 8.11021i 0 −2.47868
188.5 1.26955i 0 2.38823 1.95240i 0 3.21064 8.11021i 0 −2.47868
188.6 1.96614i 0 0.134296 9.56977i 0 −7.52095 8.12860i 0 18.8155
188.7 3.13595i 0 −5.83419 6.09830i 0 −11.4008 5.75194i 0 −19.1240
188.8 3.83254i 0 −10.6883 5.42387i 0 11.7111 25.6333i 0 20.7872
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.b.c 8
3.b odd 2 1 inner 297.3.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.b.c 8 1.a even 1 1 trivial
297.3.b.c 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 30T_{2}^{6} + 285T_{2}^{4} + 944T_{2}^{2} + 900 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 30 T^{6} + \cdots + 900 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 162 T^{6} + \cdots + 381924 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + \cdots + 3224)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 14 T^{3} + \cdots - 21925)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 750 T^{6} + \cdots + 54730404 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{3} + \cdots - 16200)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 7669555776 \) Copy content Toggle raw display
$29$ \( T^{8} + 1632 T^{6} + \cdots + 29160000 \) Copy content Toggle raw display
$31$ \( (T^{4} - 32 T^{3} + \cdots + 451232)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 38 T^{3} + \cdots + 622620)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1171329998400 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2773983518784 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16265008340100 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 7655005165824 \) Copy content Toggle raw display
$61$ \( (T^{4} - 98 T^{3} + \cdots + 104355)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 64 T^{3} + \cdots + 935113)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11293737984 \) Copy content Toggle raw display
$73$ \( (T^{4} + 76 T^{3} + \cdots + 3530000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 74 T^{3} + \cdots + 680095)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 256407458001984 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{4} + 244 T^{3} + \cdots + 5426125)^{2} \) Copy content Toggle raw display
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