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Properties

Label 2700.2.i.e
Level $2700$
Weight $2$
Character orbit 2700.i
Analytic conductor $21.560$
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] = [2700,2,Mod(901,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2700.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.i (of order \(3\), degree \(2\), 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: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.142635249.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} + 2) q^{17} + (\beta_{7} - 1) q^{19} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{23}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{7} - 3 q^{11} - 2 q^{13} + 18 q^{17} - 8 q^{19} + 3 q^{23} + 9 q^{29} - 2 q^{31} - 2 q^{37} - 9 q^{41} - 8 q^{43} - 12 q^{47} - 9 q^{49} + 24 q^{53} + 15 q^{59} + q^{61} - 11 q^{67} + 24 q^{71} - 20 q^{73} - 36 q^{77} + 7 q^{79} - 12 q^{83} - 6 q^{89} - 22 q^{91} - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - \nu^{5} + 7\nu^{4} - 7\nu^{3} - 6\nu^{2} + 16\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - \nu^{4} + 3\nu^{3} - 3\nu^{2} - 5\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 2\nu^{3} + 3\nu^{2} + 10\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 9\nu^{6} + 17\nu^{5} + 9\nu^{4} - 41\nu^{3} + 30\nu^{2} + 52\nu - 80 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 6\nu^{6} - 6\nu^{5} - 8\nu^{4} + 22\nu^{3} - 9\nu^{2} - 36\nu + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} - \nu^{5} + 9\nu^{4} - 5\nu^{3} - 9\nu^{2} + 16\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 15\nu^{6} - 11\nu^{5} - 27\nu^{4} + 59\nu^{3} + 6\nu^{2} - 88\nu + 96 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3\beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{5} + 3\beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} - 11\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} + 3\beta_{5} - \beta_{4} + \beta_{3} - 6\beta_{2} - 3\beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - 7\beta_{5} - 6\beta_{4} - 3\beta_{3} - \beta_{2} - 23\beta _1 + 6 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.620769 − 1.27069i
−1.32841 − 0.485097i
1.38941 − 0.263711i
0.818235 + 1.15347i
0.620769 + 1.27069i
−1.32841 + 0.485097i
1.38941 + 0.263711i
0.818235 − 1.15347i
0 0 0 0 0 −1.70089 − 2.94604i 0 0 0
901.2 0 0 0 0 0 −0.340213 − 0.589266i 0 0 0
901.3 0 0 0 0 0 0.0432397 + 0.0748933i 0 0 0
901.4 0 0 0 0 0 2.49787 + 4.32643i 0 0 0
1801.1 0 0 0 0 0 −1.70089 + 2.94604i 0 0 0
1801.2 0 0 0 0 0 −0.340213 + 0.589266i 0 0 0
1801.3 0 0 0 0 0 0.0432397 − 0.0748933i 0 0 0
1801.4 0 0 0 0 0 2.49787 − 4.32643i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7}^{8} - T_{7}^{7} + 19T_{7}^{6} + 38T_{7}^{5} + 313T_{7}^{4} + 182T_{7}^{3} + 118T_{7}^{2} - 10T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + 3T_{11}^{7} + 24T_{11}^{6} + 45T_{11}^{5} + 387T_{11}^{4} + 837T_{11}^{3} + 1620T_{11}^{2} + 1215T_{11} + 729 \) 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
$7$ \( T^{8} - T^{7} + 19 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 1849 \) Copy content Toggle raw display
$17$ \( (T^{4} - 9 T^{3} + 12 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} - 27 T^{2} + \cdots - 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{8} - 9 T^{7} + \cdots + 1347921 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} - 39 T^{2} + \cdots + 97)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 998001 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 2781)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 15 T^{7} + \cdots + 101425041 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + \cdots + 100489 \) Copy content Toggle raw display
$67$ \( T^{8} + 11 T^{7} + \cdots + 8162449 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 729)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 10 T^{3} + \cdots - 515)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 7 T^{7} + \cdots + 240033049 \) Copy content Toggle raw display
$83$ \( T^{8} + 12 T^{7} + \cdots + 13682601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{3} + \cdots + 5913)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 5 T^{7} + \cdots + 4044121 \) Copy content Toggle raw display
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