Properties

Label 1900.2.c.f
Level $1900$
Weight $2$
Character orbit 1900.c
Analytic conductor $15.172$
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] = [1900,2,Mod(1749,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, 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("1900.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.6594624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 18x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{3} + ( - \beta_{5} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{3} + ( - \beta_{5} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{2} - 2) q^{9} - \beta_{3} q^{11} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{13} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{17} - q^{19} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{21} + (5 \beta_{4} + \beta_1) q^{23} + ( - \beta_{5} + 2 \beta_1) q^{27} + (\beta_{3} + 2 \beta_{2} - 2) q^{29} + ( - \beta_{3} + 3 \beta_{2} - 1) q^{31} + (\beta_{5} + 4 \beta_{4} - \beta_1) q^{33} + (\beta_{5} + 4 \beta_{4}) q^{37} + ( - \beta_{2} - 4) q^{39} - 3 \beta_{2} q^{41} + ( - 2 \beta_{5} + 5 \beta_{4}) q^{43} + ( - \beta_{5} - \beta_{4} - 5 \beta_1) q^{47} + ( - 3 \beta_{2} - 6) q^{49} + ( - 2 \beta_{3} + 5 \beta_{2} - 5) q^{51} + (\beta_{4} - 4 \beta_1) q^{53} + (\beta_{5} - \beta_{4}) q^{57} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{59} + ( - \beta_{3} + 5 \beta_{2} - 3) q^{61} + (7 \beta_{4} - 2 \beta_1) q^{63} + ( - 3 \beta_{5} - 4 \beta_{4} - 4 \beta_1) q^{67} + (5 \beta_{3} - 2 \beta_{2} - 4) q^{69} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{71} + (2 \beta_{5} + 4 \beta_{4} + \beta_1) q^{73} + (3 \beta_{5} + 2 \beta_{4} + \beta_1) q^{77} + ( - 5 \beta_{3} + 1) q^{79} + ( - \beta_{3} - 8) q^{81} + (10 \beta_{4} - \beta_1) q^{83} + (\beta_{5} - 8 \beta_{4} + 5 \beta_1) q^{87} + ( - 2 \beta_{3} - 3 \beta_{2} - 3) q^{89} + ( - 5 \beta_{2} - 6) q^{91} + (2 \beta_{5} + 5 \beta_1) q^{93} + (5 \beta_{5} - \beta_{4} - \beta_1) q^{97} + (2 \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 18x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 5\nu^{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} + 19\nu^{3} + 41\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{3} + 8\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 19\beta_{4} + 27\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
1.69963i
0.239123i
2.46050i
2.46050i
0.239123i
1.69963i
0 2.58836i 0 0 0 2.81089i 0 −3.69963 0
1749.2 0 2.18194i 0 0 0 3.70370i 0 −1.76088 0
1749.3 0 1.59358i 0 0 0 4.51459i 0 0.460505 0
1749.4 0 1.59358i 0 0 0 4.51459i 0 0.460505 0
1749.5 0 2.18194i 0 0 0 3.70370i 0 −1.76088 0
1749.6 0 2.58836i 0 0 0 2.81089i 0 −3.69963 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.6
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 1900.2.c.f 6
5.b even 2 1 inner 1900.2.c.f 6
5.c odd 4 1 1900.2.a.g 3
5.c odd 4 1 1900.2.a.i yes 3
20.e even 4 1 7600.2.a.bl 3
20.e even 4 1 7600.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 5.c odd 4 1
1900.2.a.i yes 3 5.c odd 4 1
1900.2.c.f 6 1.a even 1 1 trivial
1900.2.c.f 6 5.b even 2 1 inner
7600.2.a.bl 3 20.e even 4 1
7600.2.a.ca 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{6} + 14T_{3}^{4} + 61T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{6} + 42T_{7}^{4} + 549T_{7}^{2} + 2209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + 61 T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 42 T^{4} + 549 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 21 T^{4} + 78 T^{2} + 49 \) Copy content Toggle raw display
$17$ \( T^{6} + 66 T^{4} + 1413 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 94 T^{4} + 2433 T^{2} + \cdots + 16641 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 24 T + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 40 T - 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 69 T^{4} + 966 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$41$ \( (T^{3} + 3 T^{2} - 36 T - 27)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 107 T^{4} + 2755 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$47$ \( T^{6} + 273 T^{4} + 20394 T^{2} + \cdots + 385641 \) Copy content Toggle raw display
$53$ \( T^{6} + 139 T^{4} + 4755 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} - 24 T + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 5 T^{2} - 98 T - 489)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 413 T^{4} + 50950 T^{2} + \cdots + 1750329 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} - 33 T - 27)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 141 T^{4} + 3234 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( (T^{3} + 2 T^{2} - 157 T + 529)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 289 T^{4} + 26238 T^{2} + \cdots + 741321 \) Copy content Toggle raw display
$89$ \( (T^{3} + 14 T^{2} - 9 T - 489)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 309 T^{4} + 22350 T^{2} + \cdots + 466489 \) Copy content Toggle raw display
show more
show less