↑

Properties

Label 3900.2.q.m
Level $3900$
Weight $2$
Character orbit 3900.q
Analytic conductor $31.142$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,2,Mod(601,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.q (of order \(3\), degree \(2\), 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: \(31.1416567883\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.16765488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 13x^{3} + 46x^{2} + 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 780)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{5} - \beta_1) q^{7} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{5} - \beta_1) q^{7} - \beta_{4} q^{9} + ( - \beta_{2} - 1) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{17} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{19} - \beta_1 q^{21} + (2 \beta_{5} + 2 \beta_{4} - 2) q^{23} - q^{27} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{29}+ \cdots + ( - 4 \beta_{5} - 2 \beta_{4} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - q^{7} - 3 q^{9} - 5 q^{13} - 2 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} - 6 q^{27} - 8 q^{29} - 14 q^{31} + 6 q^{37} - 4 q^{39} + 6 q^{41} - 11 q^{43} + 12 q^{47} - 6 q^{49} - 4 q^{51} - 8 q^{53} + 8 q^{57} + 12 q^{59} - 9 q^{61} - q^{63} - 15 q^{67} + 4 q^{69} + 10 q^{71} + 10 q^{73} + 22 q^{79} - 3 q^{81} + 44 q^{83} + 8 q^{87} - 4 q^{89} + 27 q^{91} - 7 q^{93} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 13x^{3} + 46x^{2} + 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{5} + 64\nu^{4} - 123\nu^{3} + 368\nu^{2} + 168\nu + 1768 ) / 389 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 72\nu^{4} - 187\nu^{3} + 414\nu^{2} + 578\nu + 1211 ) / 389 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\nu^{5} - 80\nu^{4} + 251\nu^{3} - 460\nu^{2} + 179\nu - 1043 ) / 389 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -56\nu^{5} + 59\nu^{4} - 472\nu^{3} - 536\nu^{2} - 2714\nu - 72 ) / 1167 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 277\nu^{5} - 271\nu^{4} + 2168\nu^{3} + 3985\nu^{2} + 12466\nu + 5691 ) / 1167 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 15\beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{3} - 8\beta_{2} + 19\beta _1 - 40 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{5} - 43\beta_{4} - 6\beta_{3} - 12\beta_{2} + 14\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -54\beta_{5} - 342\beta_{4} - 154\beta_{3} - 77\beta_{2} - 77\beta _1 + 419 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−0.238226 + 0.412620i
−0.938702 + 1.62588i
1.67693 − 2.90452i
−0.238226 − 0.412620i
−0.938702 − 1.62588i
1.67693 + 2.90452i
0 0.500000 + 0.866025i 0 0 0 −2.14827 + 3.72091i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 0 0 0.201024 − 0.348184i 0 −0.500000 + 0.866025i 0
601.3 0 0.500000 + 0.866025i 0 0 0 1.44725 − 2.50670i 0 −0.500000 + 0.866025i 0
2401.1 0 0.500000 − 0.866025i 0 0 0 −2.14827 − 3.72091i 0 −0.500000 − 0.866025i 0
2401.2 0 0.500000 − 0.866025i 0 0 0 0.201024 + 0.348184i 0 −0.500000 − 0.866025i 0
2401.3 0 0.500000 − 0.866025i 0 0 0 1.44725 + 2.50670i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.q.m 6
5.b even 2 1 780.2.q.d ✓ 6
5.c odd 4 2 3900.2.by.i 12
13.c even 3 1 inner 3900.2.q.m 6
15.d odd 2 1 2340.2.q.i 6
65.n even 6 1 780.2.q.d ✓ 6
65.q odd 12 2 3900.2.by.i 12
195.x odd 6 1 2340.2.q.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.q.d ✓ 6 5.b even 2 1
780.2.q.d ✓ 6 65.n even 6 1
2340.2.q.i 6 15.d odd 2 1
2340.2.q.i 6 195.x odd 6 1
3900.2.q.m 6 1.a even 1 1 trivial
3900.2.q.m 6 13.c even 3 1 inner
3900.2.by.i 12 5.c odd 4 2
3900.2.by.i 12 65.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{6} + T_{7}^{5} + 14T_{7}^{4} - 23T_{7}^{3} + 164T_{7}^{2} - 65T_{7} + 25 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{23}^{6} + 4T_{23}^{5} + 64T_{23}^{4} + 96T_{23}^{3} + 2880T_{23}^{2} + 6912T_{23} + 20736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 5 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 15376 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots + 20736 \) Copy content Toggle raw display
$29$ \( T^{6} + 8 T^{5} + \cdots + 54756 \) Copy content Toggle raw display
$31$ \( (T^{3} + 7 T^{2} + \cdots - 715)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + \cdots + 72900 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} + \cdots + 65025 \) Copy content Toggle raw display
$47$ \( (T^{3} - 6 T^{2} + \cdots + 486)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 4 T^{2} + \cdots - 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 76729 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} + \cdots + 3481 \) Copy content Toggle raw display
$71$ \( T^{6} - 10 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( (T^{3} - 5 T^{2} - 5 T + 27)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 11 T^{2} + \cdots - 97)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 22 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} + \cdots + 93636 \) Copy content Toggle raw display
$97$ \( T^{6} - T^{5} + \cdots + 199809 \) Copy content Toggle raw display
show more
show less