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Properties

Label 1800.2.k.s
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
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] = [1800,2,Mod(901,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.212336640000.29
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{5} q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{7} + \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{5} q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{7} + \beta_{4} q^{8} + (\beta_{4} + \beta_{3}) q^{11} - \beta_{7} q^{13} + (\beta_{6} + \beta_{4} - \beta_1) q^{14} + ( - \beta_{7} - 1) q^{16} + ( - 2 \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + (2 \beta_{4} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 8 q^{16} - 8 q^{22} + 24 q^{28} + 8 q^{31} - 24 q^{34} - 8 q^{46} + 24 q^{58} + 16 q^{73} + 40 q^{76} - 16 q^{79} + 40 q^{82} - 64 q^{88} + 64 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{4} + 1 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} - 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{6} - 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
1.26979 − 0.622597i
1.26979 + 0.622597i
0.622597 − 1.26979i
0.622597 + 1.26979i
−0.622597 − 1.26979i
−0.622597 + 1.26979i
−1.26979 − 0.622597i
−1.26979 + 0.622597i
−1.26979 − 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 −0.570759 − 2.77024i 0 0
901.2 −1.26979 + 0.622597i 0 1.22474 − 1.58114i 0 0 3.44949 −0.570759 + 2.77024i 0 0
901.3 −0.622597 − 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 2.77024 + 0.570759i 0 0
901.4 −0.622597 + 1.26979i 0 −1.22474 − 1.58114i 0 0 −1.44949 2.77024 − 0.570759i 0 0
901.5 0.622597 − 1.26979i 0 −1.22474 − 1.58114i 0 0 −1.44949 −2.77024 + 0.570759i 0 0
901.6 0.622597 + 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 −2.77024 − 0.570759i 0 0
901.7 1.26979 − 0.622597i 0 1.22474 − 1.58114i 0 0 3.44949 0.570759 − 2.77024i 0 0
901.8 1.26979 + 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 0.570759 + 2.77024i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.s yes 8
3.b odd 2 1 inner 1800.2.k.s yes 8
4.b odd 2 1 7200.2.k.q 8
5.b even 2 1 1800.2.k.r ✓ 8
5.c odd 4 2 1800.2.d.u 16
8.b even 2 1 inner 1800.2.k.s yes 8
8.d odd 2 1 7200.2.k.q 8
12.b even 2 1 7200.2.k.q 8
15.d odd 2 1 1800.2.k.r ✓ 8
15.e even 4 2 1800.2.d.u 16
20.d odd 2 1 7200.2.k.t 8
20.e even 4 2 7200.2.d.u 16
24.f even 2 1 7200.2.k.q 8
24.h odd 2 1 inner 1800.2.k.s yes 8
40.e odd 2 1 7200.2.k.t 8
40.f even 2 1 1800.2.k.r ✓ 8
40.i odd 4 2 1800.2.d.u 16
40.k even 4 2 7200.2.d.u 16
60.h even 2 1 7200.2.k.t 8
60.l odd 4 2 7200.2.d.u 16
120.i odd 2 1 1800.2.k.r ✓ 8
120.m even 2 1 7200.2.k.t 8
120.q odd 4 2 7200.2.d.u 16
120.w even 4 2 1800.2.d.u 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.d.u 16 5.c odd 4 2
1800.2.d.u 16 15.e even 4 2
1800.2.d.u 16 40.i odd 4 2
1800.2.d.u 16 120.w even 4 2
1800.2.k.r ✓ 8 5.b even 2 1
1800.2.k.r ✓ 8 15.d odd 2 1
1800.2.k.r ✓ 8 40.f even 2 1
1800.2.k.r ✓ 8 120.i odd 2 1
1800.2.k.s yes 8 1.a even 1 1 trivial
1800.2.k.s yes 8 3.b odd 2 1 inner
1800.2.k.s yes 8 8.b even 2 1 inner
1800.2.k.s yes 8 24.h odd 2 1 inner
7200.2.d.u 16 20.e even 4 2
7200.2.d.u 16 40.k even 4 2
7200.2.d.u 16 60.l odd 4 2
7200.2.d.u 16 120.q odd 4 2
7200.2.k.q 8 4.b odd 2 1
7200.2.k.q 8 8.d odd 2 1
7200.2.k.q 8 12.b even 2 1
7200.2.k.q 8 24.f even 2 1
7200.2.k.t 8 20.d odd 2 1
7200.2.k.t 8 40.e odd 2 1
7200.2.k.t 8 60.h even 2 1
7200.2.k.t 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} - 2T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{4} + 32T_{11}^{2} + 40 \) Copy content Toggle raw display
\( T_{17}^{4} - 48T_{17}^{2} + 360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 5)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 40)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 15)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 48 T^{2} + 360)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 50 T^{2} + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 48 T^{2} + 360)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 1000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 50 T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 128 T^{2} + 2560)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 80 T^{2} + 1000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 128 T^{2} + 2560)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 110 T^{2} + 625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 210 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 320 T^{2} + 25000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 20)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 192 T^{2} + 5760)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 320 T^{2} + 16000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T - 215)^{4} \) Copy content Toggle raw display
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