Properties

Label 3380.2.f.h
Level $3380$
Weight $2$
Character orbit 3380.f
Analytic conductor $26.989$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (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: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{2} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{2} + 4) q^{9} + ( - \beta_{5} + \beta_1) q^{11} + (\beta_{4} + \beta_1) q^{15} + 2 \beta_{3} q^{17} + ( - \beta_{5} + 2 \beta_{4} - \beta_1) q^{19} + (2 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{21} + ( - \beta_{3} + 3) q^{23} - q^{25} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{27} + (\beta_{2} + 3) q^{29} + (\beta_{5} - 4 \beta_{4} - \beta_1) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{33} + (\beta_{2} - 1) q^{35} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{37} + 2 \beta_1 q^{41} + (\beta_{3} + 1) q^{43} + (\beta_{5} + 4 \beta_{4}) q^{45} + ( - \beta_{5} + 3 \beta_{4}) q^{47} + ( - 4 \beta_{3} - 9) q^{49} + ( - 2 \beta_{3} - 2 \beta_{2} + 12) q^{51} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{53} + ( - \beta_{3} - \beta_{2}) q^{55} + ( - 3 \beta_{5} - 7 \beta_{4} + 2 \beta_1) q^{57} + ( - \beta_{5} + 6 \beta_{4} + 3 \beta_1) q^{59} + ( - \beta_{2} + 5) q^{61} + (3 \beta_{5} + 19 \beta_{4} + 4 \beta_1) q^{63} + ( - \beta_{5} + 5 \beta_{4} + 2 \beta_1) q^{67} + (4 \beta_{3} + \beta_{2} - 3) q^{69} + (\beta_{5} - \beta_1) q^{71} + ( - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_1) q^{73} + ( - \beta_{3} - 1) q^{75} + (2 \beta_{3} + 2 \beta_{2} + 12) q^{77} + ( - 2 \beta_{3} + 2) q^{79} + (4 \beta_{3} - 3 \beta_{2} + 10) q^{81} + (\beta_{5} - 3 \beta_{4} - 4 \beta_1) q^{83} + 2 \beta_1 q^{85} + (2 \beta_{3} + 2 \beta_{2}) q^{87} + (2 \beta_{5} - 4 \beta_1) q^{89} + (\beta_{5} - 7 \beta_{4} - 2 \beta_1) q^{93} + (\beta_{3} - \beta_{2} - 2) q^{95} + ( - 2 \beta_{5} + 8 \beta_{4}) q^{97} + ( - \beta_{5} - 12 \beta_{4} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 22 q^{9} - 4 q^{17} + 20 q^{23} - 6 q^{25} + 16 q^{27} + 20 q^{29} - 4 q^{35} + 4 q^{43} - 46 q^{49} + 72 q^{51} - 36 q^{53} + 28 q^{61} - 24 q^{69} - 4 q^{75} + 72 q^{77} + 16 q^{79} + 46 q^{81} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 13\nu^{4} - 36\nu^{3} - 6\nu^{2} + 36\nu + 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -161\nu^{5} + 196\nu^{4} - 49\nu^{3} - 292\nu^{2} - 2702\nu + 1869 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 5\beta_{4} - 4\beta_{3} + \beta_{2} + 4\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} + 3\beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{5} - 31\beta_{4} - 18\beta_{3} + 7\beta_{2} - 18\beta _1 - 31 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
−1.33641 + 1.33641i
−1.33641 1.33641i
0.675970 0.675970i
0.675970 + 0.675970i
1.66044 1.66044i
1.66044 + 1.66044i
0 −2.67282 0 1.00000i 0 1.14399i 0 4.14399 0
3041.2 0 −2.67282 0 1.00000i 0 1.14399i 0 4.14399 0
3041.3 0 1.35194 0 1.00000i 0 4.17226i 0 −1.17226 0
3041.4 0 1.35194 0 1.00000i 0 4.17226i 0 −1.17226 0
3041.5 0 3.32088 0 1.00000i 0 5.02827i 0 8.02827 0
3041.6 0 3.32088 0 1.00000i 0 5.02827i 0 8.02827 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3041.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.h 6
13.b even 2 1 inner 3380.2.f.h 6
13.d odd 4 1 260.2.a.b 3
13.d odd 4 1 3380.2.a.o 3
39.f even 4 1 2340.2.a.n 3
52.f even 4 1 1040.2.a.o 3
65.f even 4 1 1300.2.c.f 6
65.g odd 4 1 1300.2.a.i 3
65.k even 4 1 1300.2.c.f 6
104.j odd 4 1 4160.2.a.bo 3
104.m even 4 1 4160.2.a.br 3
156.l odd 4 1 9360.2.a.da 3
260.u even 4 1 5200.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.b 3 13.d odd 4 1
1040.2.a.o 3 52.f even 4 1
1300.2.a.i 3 65.g odd 4 1
1300.2.c.f 6 65.f even 4 1
1300.2.c.f 6 65.k even 4 1
2340.2.a.n 3 39.f even 4 1
3380.2.a.o 3 13.d odd 4 1
3380.2.f.h 6 1.a even 1 1 trivial
3380.2.f.h 6 13.b even 2 1 inner
4160.2.a.bo 3 104.j odd 4 1
4160.2.a.br 3 104.m even 4 1
5200.2.a.ci 3 260.u even 4 1
9360.2.a.da 3 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3}^{3} - 2T_{3}^{2} - 8T_{3} + 12 \) Copy content Toggle raw display
\( T_{19}^{6} + 96T_{19}^{4} + 2880T_{19}^{2} + 26896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 44 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{6} + 48 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 36 T + 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 96 T^{4} + \cdots + 26896 \) Copy content Toggle raw display
$23$ \( (T^{3} - 10 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 10 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 96 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{6} + 92 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( T^{6} + 76 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 76 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} + 18 T^{2} + \cdots - 648)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 256 T^{4} + \cdots + 318096 \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + 44 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 156 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$71$ \( T^{6} + 48 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{6} + 444 T^{4} + \cdots + 3182656 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 300 T^{4} + \cdots + 876096 \) Copy content Toggle raw display
$89$ \( T^{6} + 364 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$97$ \( T^{6} + 396 T^{4} + \cdots + 64 \) Copy content Toggle raw display
show more
show less