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Properties

Label 1764.2.b.j
Level $1764$
Weight $2$
Character orbit 1764.b
Analytic conductor $14.086$
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] = [1764,2,Mod(1567,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.562828176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{6} q^{4} + ( - \beta_{6} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{3} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{6} q^{4} + ( - \beta_{6} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{3} - 1) q^{8} + ( - \beta_{7} - \beta_{3} + \beta_1 + 1) q^{10} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_{3}) q^{13}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{4} - 4 q^{8} + 8 q^{10} + 10 q^{16} - 12 q^{19} + 22 q^{20} - 6 q^{22} - 4 q^{25} - 6 q^{26} + 16 q^{29} + 12 q^{31} + 12 q^{32} + 28 q^{34} - 12 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{44} - 12 q^{46} - 8 q^{47} - 2 q^{50} - 4 q^{52} - 8 q^{53} + 8 q^{55} - 14 q^{58} + 28 q^{59} - 48 q^{62} + 2 q^{64} + 8 q^{65} + 16 q^{68} - 38 q^{74} - 44 q^{76} + 6 q^{80} + 4 q^{82} + 4 q^{83} - 32 q^{85} - 6 q^{86} + 26 q^{88} + 28 q^{92} + 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 2\nu^{4} - 6\nu^{3} + 4\nu^{2} + 4\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{5} + 3\nu^{4} - 2\nu^{3} + 2\nu^{2} + 4\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} - 3\nu^{4} + 4\nu^{3} + 2\nu^{2} - 8\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + \nu^{6} - 3\nu^{4} + 6\nu^{3} + 2\nu^{2} - 4\nu + 20 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta_{3} + 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{7} + 2\beta_{6} + \beta_{5} - 3\beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.33790 + 0.458297i
−1.33790 − 0.458297i
0.0777157 + 1.41208i
0.0777157 − 1.41208i
0.856419 + 1.12541i
0.856419 − 1.12541i
1.40376 + 0.171630i
1.40376 − 0.171630i
−1.33790 − 0.458297i 0 1.57993 + 1.22631i − 2.45262i 0 0 −1.55176 − 2.36475i 0 −1.12403 + 3.28134i
1567.2 −1.33790 + 0.458297i 0 1.57993 − 1.22631i 2.45262i 0 0 −1.55176 + 2.36475i 0 −1.12403 − 3.28134i
1567.3 0.0777157 − 1.41208i 0 −1.98792 − 0.219481i 0.438962i 0 0 −0.464416 + 2.79004i 0 0.619848 + 0.0341142i
1567.4 0.0777157 + 1.41208i 0 −1.98792 + 0.219481i − 0.438962i 0 0 −0.464416 − 2.79004i 0 0.619848 − 0.0341142i
1567.5 0.856419 − 1.12541i 0 −0.533092 − 1.92764i 3.85529i 0 0 −2.62594 − 1.05092i 0 4.33878 + 3.30174i
1567.6 0.856419 + 1.12541i 0 −0.533092 + 1.92764i − 3.85529i 0 0 −2.62594 + 1.05092i 0 4.33878 − 3.30174i
1567.7 1.40376 − 0.171630i 0 1.94109 − 0.481855i 0.963711i 0 0 2.64212 − 1.00956i 0 0.165402 + 1.35282i
1567.8 1.40376 + 0.171630i 0 1.94109 + 0.481855i − 0.963711i 0 0 2.64212 + 1.00956i 0 0.165402 − 1.35282i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.j 8
3.b odd 2 1 588.2.b.a 8
4.b odd 2 1 1764.2.b.i 8
7.b odd 2 1 1764.2.b.i 8
7.c even 3 1 252.2.bf.f 8
7.d odd 6 1 252.2.bf.g 8
12.b even 2 1 588.2.b.b 8
21.c even 2 1 588.2.b.b 8
21.g even 6 1 84.2.o.a ✓ 8
21.g even 6 1 588.2.o.b 8
21.h odd 6 1 84.2.o.b yes 8
21.h odd 6 1 588.2.o.d 8
28.d even 2 1 inner 1764.2.b.j 8
28.f even 6 1 252.2.bf.f 8
28.g odd 6 1 252.2.bf.g 8
84.h odd 2 1 588.2.b.a 8
84.j odd 6 1 84.2.o.b yes 8
84.j odd 6 1 588.2.o.d 8
84.n even 6 1 84.2.o.a ✓ 8
84.n even 6 1 588.2.o.b 8
168.s odd 6 1 1344.2.bl.i 8
168.v even 6 1 1344.2.bl.j 8
168.ba even 6 1 1344.2.bl.j 8
168.be odd 6 1 1344.2.bl.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a ✓ 8 21.g even 6 1
84.2.o.a ✓ 8 84.n even 6 1
84.2.o.b yes 8 21.h odd 6 1
84.2.o.b yes 8 84.j odd 6 1
252.2.bf.f 8 7.c even 3 1
252.2.bf.f 8 28.f even 6 1
252.2.bf.g 8 7.d odd 6 1
252.2.bf.g 8 28.g odd 6 1
588.2.b.a 8 3.b odd 2 1
588.2.b.a 8 84.h odd 2 1
588.2.b.b 8 12.b even 2 1
588.2.b.b 8 21.c even 2 1
588.2.o.b 8 21.g even 6 1
588.2.o.b 8 84.n even 6 1
588.2.o.d 8 21.h odd 6 1
588.2.o.d 8 84.j odd 6 1
1344.2.bl.i 8 168.s odd 6 1
1344.2.bl.i 8 168.be odd 6 1
1344.2.bl.j 8 168.v even 6 1
1344.2.bl.j 8 168.ba even 6 1
1764.2.b.i 8 4.b odd 2 1
1764.2.b.i 8 7.b odd 2 1
1764.2.b.j 8 1.a even 1 1 trivial
1764.2.b.j 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} + 22T_{5}^{6} + 113T_{5}^{4} + 104T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} + 38T_{11}^{6} + 257T_{11}^{4} + 568T_{11}^{2} + 400 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 7T_{19}^{2} - 60T_{19} + 4 \) Copy content Toggle raw display
\( T_{29}^{4} - 8T_{29}^{3} - 45T_{29}^{2} + 352T_{29} - 512 \) Copy content Toggle raw display
\( T_{53}^{4} + 4T_{53}^{3} - 61T_{53}^{2} + 116T_{53} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 22 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 38 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{8} + 38 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 56 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 7 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 80 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots - 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + \cdots + 2043)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6 T^{3} + \cdots - 596)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 208 T^{6} + \cdots + 350464 \) Copy content Toggle raw display
$43$ \( T^{8} + 134 T^{6} + \cdots + 1073296 \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} - 28 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} - 61 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 14 T^{3} + \cdots + 1192)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 272 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$67$ \( T^{8} + 254 T^{6} + \cdots + 4129024 \) Copy content Toggle raw display
$71$ \( T^{8} + 280 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$73$ \( T^{8} + 214 T^{6} + \cdots + 952576 \) Copy content Toggle raw display
$79$ \( T^{8} + 404 T^{6} + \cdots + 241081 \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 184 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{8} + 182 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
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