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Properties

Label 8100.2.d.q
Level $8100$
Weight $2$
Character orbit 8100.d
Analytic conductor $64.679$
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] = [8100,2,Mod(649,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8100.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), 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: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4057180416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 900)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{7} + ( - \beta_1 - 1) q^{11} - \beta_{2} q^{13} + (\beta_{6} - 2 \beta_{3}) q^{17} + (\beta_{5} + 1) q^{19} + ( - \beta_{7} + \beta_{6} - \beta_{2}) q^{23} + ( - \beta_{5} - 2 \beta_{4} + \beta_1 - 1) q^{29} + \beta_{4} q^{31} + ( - \beta_{7} + \beta_{6}) q^{37} + ( - \beta_{5} + \beta_{4} - \beta_1 - 3) q^{41} + ( - 3 \beta_{3} + 2 \beta_{2}) q^{43} + (\beta_{7} - 2 \beta_{3} - 2 \beta_{2}) q^{47} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_1 - 1) q^{49} + (\beta_{7} + 2 \beta_{6} + \cdots + \beta_{2}) q^{53}+ \cdots + (\beta_{7} - \beta_{6} + 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} + 6\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 7\nu^{5} + 5\nu^{3} - 19\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + 17\nu^{5} + 37\nu^{3} + 13\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} + 9\nu^{4} + 20\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 8\nu^{4} + 17\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} + 37\nu^{5} + 92\nu^{3} + 41\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{7} + 18\nu^{5} + 43\nu^{3} + 21\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} + 3\beta_{6} + 7\beta_{3} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} - 10\beta_{6} - 43\beta_{3} + 12\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 34\beta_{5} - 31\beta_{4} + 7\beta _1 - 111 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -94\beta_{7} + 36\beta_{6} + 247\beta_{3} - 65\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.75080i
− 0.318459i
0.785261i
− 2.28400i
2.28400i
− 0.785261i
0.318459i
− 1.75080i
0 0 0 0 0 − 4.99574i 0 0 0
649.2 0 0 0 0 0 − 3.40179i 0 0 0
649.3 0 0 0 0 0 − 0.680426i 0 0 0
649.4 0 0 0 0 0 − 0.0864793i 0 0 0
649.5 0 0 0 0 0 0.0864793i 0 0 0
649.6 0 0 0 0 0 0.680426i 0 0 0
649.7 0 0 0 0 0 3.40179i 0 0 0
649.8 0 0 0 0 0 4.99574i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.d.q 8
3.b odd 2 1 8100.2.d.s 8
5.b even 2 1 inner 8100.2.d.q 8
5.c odd 4 1 8100.2.a.x 4
5.c odd 4 1 8100.2.a.z 4
9.c even 3 2 900.2.s.d 16
9.d odd 6 2 2700.2.s.d 16
15.d odd 2 1 8100.2.d.s 8
15.e even 4 1 8100.2.a.y 4
15.e even 4 1 8100.2.a.ba 4
45.h odd 6 2 2700.2.s.d 16
45.j even 6 2 900.2.s.d 16
45.k odd 12 2 900.2.i.d ✓ 8
45.k odd 12 2 900.2.i.e yes 8
45.l even 12 2 2700.2.i.d 8
45.l even 12 2 2700.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.i.d ✓ 8 45.k odd 12 2
900.2.i.e yes 8 45.k odd 12 2
900.2.s.d 16 9.c even 3 2
900.2.s.d 16 45.j even 6 2
2700.2.i.d 8 45.l even 12 2
2700.2.i.e 8 45.l even 12 2
2700.2.s.d 16 9.d odd 6 2
2700.2.s.d 16 45.h odd 6 2
8100.2.a.x 4 5.c odd 4 1
8100.2.a.y 4 15.e even 4 1
8100.2.a.z 4 5.c odd 4 1
8100.2.a.ba 4 15.e even 4 1
8100.2.d.q 8 1.a even 1 1 trivial
8100.2.d.q 8 5.b even 2 1 inner
8100.2.d.s 8 3.b odd 2 1
8100.2.d.s 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8100, [\chi])\):

\( T_{7}^{8} + 37T_{7}^{6} + 306T_{7}^{4} + 136T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} - 15T_{11}^{2} - 45T_{11} - 27 \) Copy content Toggle raw display
\( T_{29}^{4} + 9T_{29}^{3} - 60T_{29}^{2} - 666T_{29} - 1161 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 37 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} + 3 T^{3} - 15 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 40 T^{6} + \cdots + 1849 \) Copy content Toggle raw display
$17$ \( T^{8} + 57 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 27 T^{2} + \cdots - 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 147 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( (T^{4} + 9 T^{3} + \cdots - 1161)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 18 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 79 T^{6} + \cdots + 9409 \) Copy content Toggle raw display
$41$ \( (T^{4} + 9 T^{3} - 45 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 172 T^{6} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{8} + 246 T^{6} + \cdots + 998001 \) Copy content Toggle raw display
$53$ \( T^{8} + 300 T^{6} + \cdots + 7733961 \) Copy content Toggle raw display
$59$ \( (T^{4} + 15 T^{3} + \cdots - 10071)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} - 75 T^{2} + \cdots - 317)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 289 T^{6} + \cdots + 8162449 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots - 729)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 124 T^{6} + \cdots + 265225 \) Copy content Toggle raw display
$79$ \( (T^{4} - 7 T^{3} + \cdots + 15493)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 336 T^{6} + \cdots + 13682601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{3} + \cdots + 5913)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 295 T^{6} + \cdots + 4044121 \) Copy content Toggle raw display
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