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Properties

Label 3360.2.t.n
Level $3360$
Weight $2$
Character orbit 3360.t
Analytic conductor $26.830$
Analytic rank $0$
Dimension $10$
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] = [3360,2,Mod(2689,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, 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("3360.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.t (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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.1016580161536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{5} q^{5} + \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{5} q^{5} + \beta_1 q^{7} - q^{9} + (\beta_{3} + 1) q^{11} + ( - \beta_{4} - \beta_1) q^{13} - \beta_{7} q^{15} + (\beta_{9} - \beta_{4} + 2 \beta_1) q^{17} + ( - \beta_{8} + \beta_{7} - \beta_{3} - 1) q^{19} + q^{21} + (\beta_{9} + \beta_{8} + \cdots - \beta_{4}) q^{23}+ \cdots + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{5} - 10 q^{9} + 12 q^{11} - 12 q^{19} + 10 q^{21} + 2 q^{25} - 4 q^{29} + 12 q^{31} - 12 q^{39} + 12 q^{41} - 2 q^{45} - 10 q^{49} + 16 q^{51} + 4 q^{55} - 32 q^{59} - 4 q^{61} + 12 q^{71} - 8 q^{75} - 8 q^{79} + 10 q^{81} - 8 q^{85} - 20 q^{89} + 12 q^{91} + 4 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 6\nu^{3} + 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 17\nu^{5} + 42\nu^{3} + 30\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 11\nu^{6} + 38\nu^{4} + 42\nu^{2} + 2\nu + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} - 11\nu^{6} - 38\nu^{4} - 42\nu^{2} + 2\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 12\nu^{7} + \nu^{6} + 47\nu^{5} + 6\nu^{4} + 64\nu^{3} + 6\nu^{2} + 20\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 12\nu^{7} - \nu^{6} + 47\nu^{5} - 6\nu^{4} + 64\nu^{3} - 6\nu^{2} + 20\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{9} - 22\nu^{7} - 77\nu^{5} - 90\nu^{3} - 26\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_{7} - 4\beta_{6} - 4\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{3} - 5\beta_{2} + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{9} + 3\beta_{8} + 3\beta_{7} + 9\beta_{6} + 9\beta_{5} - 3\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{8} + 4\beta_{7} - 3\beta_{3} + 12\beta_{2} - 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{9} - 15\beta_{8} - 15\beta_{7} - 42\beta_{6} - 42\beta_{5} + 16\beta_{4} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 25\beta_{8} - 25\beta_{7} - \beta_{6} + \beta_{5} + 14\beta_{3} - 58\beta_{2} + 210 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 71\beta_{9} + 72\beta_{8} + 72\beta_{7} + 199\beta_{6} + 199\beta_{5} - 83\beta_{4} + 110\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
0.815403i
− 2.10518i
2.23025i
− 1.33253i
0.392048i
− 0.815403i
2.10518i
− 2.23025i
1.33253i
− 0.392048i
0 − 1.00000i 0 −2.08209 + 0.815403i 0 1.00000i 0 −1.00000 0
2689.2 0 − 1.00000i 0 −0.753811 − 2.10518i 0 1.00000i 0 −1.00000 0
2689.3 0 − 1.00000i 0 −0.161179 + 2.23025i 0 1.00000i 0 −1.00000 0
2689.4 0 − 1.00000i 0 1.79565 − 1.33253i 0 1.00000i 0 −1.00000 0
2689.5 0 − 1.00000i 0 2.20143 + 0.392048i 0 1.00000i 0 −1.00000 0
2689.6 0 1.00000i 0 −2.08209 − 0.815403i 0 − 1.00000i 0 −1.00000 0
2689.7 0 1.00000i 0 −0.753811 + 2.10518i 0 − 1.00000i 0 −1.00000 0
2689.8 0 1.00000i 0 −0.161179 − 2.23025i 0 − 1.00000i 0 −1.00000 0
2689.9 0 1.00000i 0 1.79565 + 1.33253i 0 − 1.00000i 0 −1.00000 0
2689.10 0 1.00000i 0 2.20143 − 0.392048i 0 − 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2689.10
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 3360.2.t.n yes 10
4.b odd 2 1 3360.2.t.m ✓ 10
5.b even 2 1 inner 3360.2.t.n yes 10
20.d odd 2 1 3360.2.t.m ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.t.m ✓ 10 4.b odd 2 1
3360.2.t.m ✓ 10 20.d odd 2 1
3360.2.t.n yes 10 1.a even 1 1 trivial
3360.2.t.n yes 10 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 \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{5} - 6T_{11}^{4} - 8T_{11}^{3} + 64T_{11}^{2} + 16T_{11} - 96 \) Copy content Toggle raw display
\( T_{19}^{5} + 6T_{19}^{4} - 24T_{19}^{3} - 176T_{19}^{2} - 176T_{19} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( (T^{5} - 6 T^{4} - 8 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 84 T^{8} + \cdots + 541696 \) Copy content Toggle raw display
$17$ \( T^{10} + 96 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{5} + 6 T^{4} - 24 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 144 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{5} + 2 T^{4} - 56 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 6 T^{4} + \cdots - 736)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 256 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( (T^{5} - 6 T^{4} + \cdots - 21024)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 304 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 2466512896 \) Copy content Toggle raw display
$53$ \( T^{10} + 340 T^{8} + \cdots + 63489024 \) Copy content Toggle raw display
$59$ \( (T^{5} + 16 T^{4} + \cdots + 6144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 2 T^{4} - 56 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 192 T^{8} + \cdots + 2359296 \) Copy content Toggle raw display
$71$ \( (T^{5} - 6 T^{4} + \cdots - 2336)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 212 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$79$ \( (T^{5} + 4 T^{4} + \cdots + 44032)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} + \cdots - 18976)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 798401536 \) Copy content Toggle raw display
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