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Properties

Label 297.2.d.b
Level $297$
Weight $2$
Character orbit 297.d
Analytic conductor $2.372$
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] = [297,2,Mod(296,297)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.296");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.d (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: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.764411904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{5} + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{6} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{5} + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{6} + \beta_1) q^{8} - \beta_{7} q^{10} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{11} + (\beta_{7} + 2 \beta_{4}) q^{13} + (\beta_{3} + 3 \beta_{2}) q^{14} + q^{16} + (\beta_{6} + \beta_1) q^{17} + ( - \beta_{7} + 2 \beta_{4}) q^{19} + (2 \beta_{3} + \beta_{2}) q^{20} + (2 \beta_{7} - \beta_{5} + \beta_{4}) q^{22} + (\beta_{3} - 5 \beta_{2}) q^{23} + 3 q^{25} + ( - 4 \beta_{3} - 3 \beta_{2}) q^{26} + ( - 2 \beta_{7} - 3 \beta_{4}) q^{28} + (3 \beta_{6} - 2 \beta_1) q^{29} + (3 \beta_{5} + 2) q^{31} + (\beta_{6} - 2 \beta_1) q^{32} - 3 q^{34} + (2 \beta_{6} + \beta_1) q^{35} + ( - \beta_{5} - 1) q^{37} + 3 \beta_{2} q^{38} + ( - \beta_{7} - 2 \beta_{4}) q^{40} + \beta_{6} q^{41} + ( - 2 \beta_{7} - 3 \beta_{4}) q^{43} + (2 \beta_{6} - 3 \beta_{3} + \cdots + \beta_1) q^{44}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 8 q^{16} + 24 q^{25} + 16 q^{31} - 24 q^{34} - 8 q^{37} - 16 q^{49} + 16 q^{55} - 72 q^{58} - 40 q^{64} + 16 q^{67} - 48 q^{70} - 24 q^{82} - 48 q^{88} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 3\nu^{5} - 39\nu^{3} + 189\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 18\nu^{2} + 33 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 21\nu^{3} - 27\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 33\nu^{5} + 66\nu^{3} - 216\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - 3\nu^{5} + 15\nu^{3} - 27\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 4\beta_{6} + 4\beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{5} - 3\beta_{3} - 6\beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - 9\beta_{6} - 3\beta_{4} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{5} - 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 33\beta_{7} + 3\beta_{6} - 21\beta_{4} + 12\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
296.1
−1.69185 − 0.370982i
−1.69185 + 0.370982i
1.27970 + 1.16721i
1.27970 − 1.16721i
−1.27970 − 1.16721i
−1.27970 + 1.16721i
1.69185 + 0.370982i
1.69185 − 0.370982i
−2.33441 0 3.44949 − 1.41421i 0 2.55940i −3.38371 0 3.30136i
296.2 −2.33441 0 3.44949 1.41421i 0 − 2.55940i −3.38371 0 − 3.30136i
296.3 −0.741964 0 −1.44949 − 1.41421i 0 3.38371i 2.55940 0 1.04930i
296.4 −0.741964 0 −1.44949 1.41421i 0 − 3.38371i 2.55940 0 − 1.04930i
296.5 0.741964 0 −1.44949 − 1.41421i 0 − 3.38371i −2.55940 0 − 1.04930i
296.6 0.741964 0 −1.44949 1.41421i 0 3.38371i −2.55940 0 1.04930i
296.7 2.33441 0 3.44949 − 1.41421i 0 − 2.55940i 3.38371 0 − 3.30136i
296.8 2.33441 0 3.44949 1.41421i 0 2.55940i 3.38371 0 3.30136i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 296.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.d.b ✓ 8
3.b odd 2 1 inner 297.2.d.b ✓ 8
4.b odd 2 1 4752.2.b.h 8
9.c even 3 1 891.2.g.b 8
9.c even 3 1 891.2.g.d 8
9.d odd 6 1 891.2.g.b 8
9.d odd 6 1 891.2.g.d 8
11.b odd 2 1 inner 297.2.d.b ✓ 8
12.b even 2 1 4752.2.b.h 8
33.d even 2 1 inner 297.2.d.b ✓ 8
44.c even 2 1 4752.2.b.h 8
99.g even 6 1 891.2.g.b 8
99.g even 6 1 891.2.g.d 8
99.h odd 6 1 891.2.g.b 8
99.h odd 6 1 891.2.g.d 8
132.d odd 2 1 4752.2.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.d.b ✓ 8 1.a even 1 1 trivial
297.2.d.b ✓ 8 3.b odd 2 1 inner
297.2.d.b ✓ 8 11.b odd 2 1 inner
297.2.d.b ✓ 8 33.d even 2 1 inner
891.2.g.b 8 9.c even 3 1
891.2.g.b 8 9.d odd 6 1
891.2.g.b 8 99.g even 6 1
891.2.g.b 8 99.h odd 6 1
891.2.g.d 8 9.c even 3 1
891.2.g.d 8 9.d odd 6 1
891.2.g.d 8 99.g even 6 1
891.2.g.d 8 99.h odd 6 1
4752.2.b.h 8 4.b odd 2 1
4752.2.b.h 8 12.b even 2 1
4752.2.b.h 8 44.c even 2 1
4752.2.b.h 8 132.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 6 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 18 T^{2} + 75)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 300)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 18 T^{2} + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 106 T^{2} + 2209)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 102 T^{2} + 1587)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 50)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{2} + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 102 T^{2} + 1875)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 118 T^{2} + 25)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 58 T^{2} + 625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 204 T^{2} + 6348)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 50)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 160 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 216 T^{2} + 10800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 342 T^{2} + 28227)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 204 T^{2} + 6348)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 232 T^{2} + 10000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 53)^{4} \) Copy content Toggle raw display
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