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Properties

Label 6600.2.d.bf
Level $6600$
Weight $2$
Character orbit 6600.d
Analytic conductor $52.701$
Analytic rank $0$
Dimension $6$
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] = [6600,2,Mod(1849,6600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6600.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6600.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: \(52.7012653340\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27206656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 25x^{2} - 30x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{2} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{2} q^{7} - q^{9} - q^{11} + ( - \beta_{5} + \beta_{4}) q^{13} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{17} + (\beta_{3} - 2 \beta_1 + 2) q^{19} + \beta_{3} q^{21} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{23} + \beta_{4} q^{27} + (2 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{3} + \beta_1 - 1) q^{31} + \beta_{4} q^{33} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{37} + (\beta_1 + 1) q^{39} + ( - \beta_{3} + \beta_1 + 3) q^{41} + (3 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{43} + ( - \beta_{5} + 2 \beta_{4}) q^{47} + ( - 2 \beta_{3} + 2 \beta_1) q^{49} + ( - \beta_{3} - \beta_1 - 1) q^{51} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{2}) q^{53} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{57} + (\beta_1 + 10) q^{59} + ( - 2 \beta_{3} - 3 \beta_1 - 3) q^{61} + \beta_{2} q^{63} + ( - \beta_{5} - 5 \beta_{4}) q^{67} + ( - 2 \beta_{3} + \beta_1 + 2) q^{69} + (\beta_1 + 6) q^{71} + ( - 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{2}) q^{73} + \beta_{2} q^{77} + ( - 5 \beta_{3} + 3 \beta_1 + 5) q^{79} + q^{81} + ( - 2 \beta_{5} + 6 \beta_{4} + 4 \beta_{2}) q^{83} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{87} + (2 \beta_{3} + 4 \beta_1) q^{89} + (\beta_1 - 3) q^{91} + (\beta_{5} + \beta_{4} - 2 \beta_{2}) q^{93} + ( - 7 \beta_{4} - 2 \beta_{2}) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} - 6 q^{11} + 14 q^{19} + 2 q^{21} + 4 q^{29} - 10 q^{31} + 6 q^{39} + 16 q^{41} - 4 q^{49} - 8 q^{51} + 60 q^{59} - 22 q^{61} + 8 q^{69} + 36 q^{71} + 20 q^{79} + 6 q^{81} + 4 q^{89} - 18 q^{91} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 25x^{2} - 30x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 35\nu^{4} - 14\nu^{3} - 4\nu^{2} + 12\nu + 726 ) / 213 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} - \nu^{4} + 43\nu^{3} - 79\nu^{2} + 95\nu - 39 ) / 213 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 16\nu^{4} + 49\nu^{3} + 14\nu^{2} - 42\nu + 15 ) / 213 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{5} - 15\nu^{4} + 6\nu^{3} + 93\nu^{2} + 289\nu - 159 ) / 213 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -44\nu^{5} + 60\nu^{4} - 24\nu^{3} - 159\nu^{2} - 1156\nu + 636 ) / 213 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + 7\beta_{4} + 5\beta_{3} + 5\beta_{2} + 2\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 7\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{5} - 59\beta_{4} + 29\beta_{3} - 29\beta_{2} + 18\beta _1 - 59 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(2201\) \(2377\) \(3301\) \(4951\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.90022 + 1.90022i
0.545872 + 0.545872i
−1.44609 − 1.44609i
−1.44609 + 1.44609i
0.545872 − 0.545872i
1.90022 − 1.90022i
0 − 1.00000i 0 0 0 − 2.80044i 0 −1.00000 0
1849.2 0 − 1.00000i 0 0 0 − 0.0917445i 0 −1.00000 0
1849.3 0 − 1.00000i 0 0 0 3.89219i 0 −1.00000 0
1849.4 0 1.00000i 0 0 0 − 3.89219i 0 −1.00000 0
1849.5 0 1.00000i 0 0 0 0.0917445i 0 −1.00000 0
1849.6 0 1.00000i 0 0 0 2.80044i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.6
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 6600.2.d.bf 6
5.b even 2 1 inner 6600.2.d.bf 6
5.c odd 4 1 6600.2.a.bo ✓ 3
5.c odd 4 1 6600.2.a.bv yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6600.2.a.bo ✓ 3 5.c odd 4 1
6600.2.a.bv yes 3 5.c odd 4 1
6600.2.d.bf 6 1.a even 1 1 trivial
6600.2.d.bf 6 5.b 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 \(S_{2}^{\mathrm{new}}(6600, [\chi])\):

\( T_{7}^{6} + 23T_{7}^{4} + 119T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{6} + 25T_{13}^{4} + 112T_{13}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{6} + 66T_{17}^{4} + 1441T_{17}^{2} + 10404 \) Copy content Toggle raw display
\( T_{19}^{3} - 7T_{19}^{2} - 23T_{19} + 173 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 56T_{29} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 23 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 25 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 66 T^{4} + \cdots + 10404 \) Copy content Toggle raw display
$19$ \( (T^{3} - 7 T^{2} + \cdots + 173)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 86 T^{4} + \cdots + 21316 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 56 T + 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 5 T^{2} - 32 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 166 T^{4} + \cdots + 116964 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} + 7 T + 18)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 201 T^{4} + \cdots + 197136 \) Copy content Toggle raw display
$47$ \( T^{6} + 34 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{6} + 380 T^{4} + \cdots + 1915456 \) Copy content Toggle raw display
$59$ \( (T^{3} - 30 T^{2} + \cdots - 892)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 11 T^{2} + \cdots - 1636)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 97 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$71$ \( (T^{3} - 18 T^{2} + \cdots - 152)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 460 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$79$ \( (T^{3} - 10 T^{2} + \cdots + 2196)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 388 T^{4} + \cdots + 1982464 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} + \cdots + 1752)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 211 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
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