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Properties

Label 2205.2.d.p
Level $2205$
Weight $2$
Character orbit 2205.d
Analytic conductor $17.607$
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] = [2205,2,Mod(1324,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2205.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4441101041664.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 38x^{4} - 200x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{2}) q^{8} + ( - \beta_{5} - \beta_{4} - 1) q^{10} + \beta_{7} q^{11} + \beta_{5} q^{13} + 2 q^{16} + ( - \beta_{6} - 2 \beta_{2}) q^{17} + ( - \beta_{3} - 2) q^{19} + ( - \beta_{7} + \beta_{6} - 2 \beta_1) q^{20} + ( - 2 \beta_{5} - \beta_{4}) q^{22} + ( - \beta_{6} + 2 \beta_{2}) q^{23} + ( - \beta_{4} + \beta_{3} + 2) q^{25} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{26}+ \cdots + (2 \beta_{5} + 3 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{10} + 16 q^{16} - 16 q^{19} + 16 q^{25} + 40 q^{31} - 40 q^{34} - 8 q^{40} - 8 q^{46} + 24 q^{61} + 48 q^{64} + 72 q^{76} + 16 q^{79} + 24 q^{85} - 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 38x^{4} - 200x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 17\nu^{5} - 37\nu^{3} - 125\nu ) / 500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} - 13\nu^{2} + 100 ) / 50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 8\nu^{4} + 63\nu^{2} - 200 ) / 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} + \nu^{4} - 11\nu^{2} - 125 ) / 100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{5} + 23\nu^{3} - 85\nu ) / 100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 31\nu^{5} + 59\nu^{3} + 325\nu ) / 500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{6} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} + 2\beta_{4} + 8\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} + 4\beta_{6} + 22\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{5} + 3\beta_{4} + \beta_{3} + 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -28\beta_{7} + 43\beta_{6} + 89\beta_{2} + 77\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
−2.19610 + 0.420861i
2.19610 + 0.420861i
−1.47551 + 1.68014i
1.47551 + 1.68014i
−1.47551 − 1.68014i
1.47551 − 1.68014i
−2.19610 − 0.420861i
2.19610 − 0.420861i
− 2.37608i 0 −3.64575 −2.19610 − 0.420861i 0 0 3.91044i 0 −1.00000 + 5.21812i
1324.2 − 2.37608i 0 −3.64575 2.19610 − 0.420861i 0 0 3.91044i 0 −1.00000 − 5.21812i
1324.3 − 0.595188i 0 1.64575 −1.47551 − 1.68014i 0 0 − 2.16991i 0 −1.00000 + 0.878205i
1324.4 − 0.595188i 0 1.64575 1.47551 − 1.68014i 0 0 − 2.16991i 0 −1.00000 − 0.878205i
1324.5 0.595188i 0 1.64575 −1.47551 + 1.68014i 0 0 2.16991i 0 −1.00000 − 0.878205i
1324.6 0.595188i 0 1.64575 1.47551 + 1.68014i 0 0 2.16991i 0 −1.00000 + 0.878205i
1324.7 2.37608i 0 −3.64575 −2.19610 + 0.420861i 0 0 − 3.91044i 0 −1.00000 − 5.21812i
1324.8 2.37608i 0 −3.64575 2.19610 + 0.420861i 0 0 − 3.91044i 0 −1.00000 + 5.21812i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1324.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.2.d.p 8
3.b odd 2 1 inner 2205.2.d.p 8
5.b even 2 1 inner 2205.2.d.p 8
7.b odd 2 1 2205.2.d.r 8
7.c even 3 2 315.2.bf.c ✓ 16
15.d odd 2 1 inner 2205.2.d.p 8
21.c even 2 1 2205.2.d.r 8
21.h odd 6 2 315.2.bf.c ✓ 16
35.c odd 2 1 2205.2.d.r 8
35.j even 6 2 315.2.bf.c ✓ 16
105.g even 2 1 2205.2.d.r 8
105.o odd 6 2 315.2.bf.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bf.c ✓ 16 7.c even 3 2
315.2.bf.c ✓ 16 21.h odd 6 2
315.2.bf.c ✓ 16 35.j even 6 2
315.2.bf.c ✓ 16 105.o odd 6 2
2205.2.d.p 8 1.a even 1 1 trivial
2205.2.d.p 8 3.b odd 2 1 inner
2205.2.d.p 8 5.b even 2 1 inner
2205.2.d.p 8 15.d odd 2 1 inner
2205.2.d.r 8 7.b odd 2 1
2205.2.d.r 8 21.c even 2 1
2205.2.d.r 8 35.c odd 2 1
2205.2.d.r 8 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} - 42T_{11}^{2} + 378 \) Copy content Toggle raw display
\( T_{13}^{4} + 28T_{13}^{2} + 189 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 3 \) Copy content Toggle raw display
\( T_{29}^{4} - 70T_{29}^{2} + 378 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 6 T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 42 T^{2} + 378)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{2} + 189)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 54 T^{2} + 722)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 38 T^{2} + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 70 T^{2} + 378)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 28 T^{2} + 21)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 154 T^{2} + 3402)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 196 T^{2} + 9261)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 104 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 70 T^{2} + 42)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 196 T^{2} + 1029)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 70 T^{2} + 378)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 84 T^{2} + 189)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 59)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 150 T^{2} + 1922)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 42 T^{2} + 378)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 196 T^{2} + 6804)^{2} \) Copy content Toggle raw display
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