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Properties

Label 3600.3.l.w
Level $3600$
Weight $3$
Character orbit 3600.l
Analytic conductor $98.093$
Analytic rank $0$
Dimension $6$
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] = [3600,3,Mod(1601,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.l (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: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1686643200.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} - 44x^{3} + 244x^{2} - 560x + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{7} + (2 \beta_{5} - \beta_{4} + \beta_{2}) q^{11} + (\beta_{3} + 1) q^{13} + (2 \beta_{5} - 3 \beta_{4} + 4 \beta_{2}) q^{17} + ( - 2 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{2}) q^{23} + (4 \beta_{5} + \beta_{4} + 8 \beta_{2}) q^{29} + (3 \beta_{3} + 3 \beta_1 - 6) q^{31} + ( - 4 \beta_{3} - \beta_1 + 33) q^{37} + ( - 4 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{41} + ( - 6 \beta_1 - 18) q^{43} + (8 \beta_{5} - 20 \beta_{2}) q^{47} + ( - 3 \beta_{3} - 3 \beta_1 - 15) q^{49} + ( - 4 \beta_{5} + 3 \beta_{4} + 6 \beta_{2}) q^{53} + ( - 2 \beta_{5} + 3 \beta_{4} + 17 \beta_{2}) q^{59} + (2 \beta_{3} + 10 \beta_1 - 6) q^{61} + ( - 4 \beta_{3} - 4 \beta_1 - 72) q^{67} + ( - 4 \beta_{5} - 4 \beta_{4} + 16 \beta_{2}) q^{71} + (5 \beta_{3} - 9 \beta_1 + 26) q^{73} + (5 \beta_{5} + 3 \beta_{4} + 47 \beta_{2}) q^{77} + ( - 5 \beta_{3} - 13 \beta_1 - 38) q^{79} + ( - 2 \beta_{5} + 6 \beta_{4} - 30 \beta_{2}) q^{83} + ( - 12 \beta_{5} + 6 \beta_{4} + 33 \beta_{2}) q^{89} + ( - \beta_{3} - 5 \beta_1 + 2) q^{91} + (\beta_{3} + 3 \beta_1 + 130) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{7} + 4 q^{13} - 48 q^{31} + 208 q^{37} - 96 q^{43} - 78 q^{49} - 60 q^{61} - 416 q^{67} + 164 q^{73} - 192 q^{79} + 24 q^{91} + 772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} - 44x^{3} + 244x^{2} - 560x + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\nu^{5} + 60\nu^{4} - 319\nu^{3} - 1542\nu^{2} + 2330\nu + 7728 ) / 2748 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{5} - 43\nu^{4} - 50\nu^{3} + 899\nu^{2} - 334\nu + 3530 ) / 2290 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -67\nu^{5} + 6\nu^{4} + 449\nu^{3} + 4380\nu^{2} - 5950\nu + 10116 ) / 2748 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 719\nu^{5} + 1002\nu^{4} - 1045\nu^{3} - 29736\nu^{2} + 51506\nu - 176520 ) / 13740 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 291\nu^{5} + 138\nu^{4} - 665\nu^{3} - 12844\nu^{2} + 40854\nu - 90680 ) / 4580 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 5\beta_{4} + 3\beta_{3} + 17\beta_{2} - \beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} - 13\beta_{4} + 3\beta_{3} - 61\beta_{2} - 19\beta _1 + 108 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 5\beta_{4} + 63\beta_{3} - 71\beta_{2} + 95\beta _1 - 266 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -107\beta_{5} + 329\beta_{4} - 31\beta_{3} + 785\beta_{2} - 273\beta _1 + 1780 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
−3.03182 − 3.08994i
−3.03182 + 3.08994i
3.27692 − 0.290905i
3.27692 + 0.290905i
0.754905 − 1.96663i
0.754905 + 1.96663i
0 0 0 0 0 −8.73967 0 0 0
1601.2 0 0 0 0 0 −8.73967 0 0 0
1601.3 0 0 0 0 0 −0.822805 0 0 0
1601.4 0 0 0 0 0 −0.822805 0 0 0
1601.5 0 0 0 0 0 5.56248 0 0 0
1601.6 0 0 0 0 0 5.56248 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1601.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.l.w 6
3.b odd 2 1 inner 3600.3.l.w 6
4.b odd 2 1 1800.3.l.i 6
5.b even 2 1 3600.3.l.x 6
5.c odd 4 2 720.3.c.d 12
12.b even 2 1 1800.3.l.i 6
15.d odd 2 1 3600.3.l.x 6
15.e even 4 2 720.3.c.d 12
20.d odd 2 1 1800.3.l.h 6
20.e even 4 2 360.3.c.a ✓ 12
40.i odd 4 2 2880.3.c.i 12
40.k even 4 2 2880.3.c.j 12
60.h even 2 1 1800.3.l.h 6
60.l odd 4 2 360.3.c.a ✓ 12
120.q odd 4 2 2880.3.c.j 12
120.w even 4 2 2880.3.c.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.3.c.a ✓ 12 20.e even 4 2
360.3.c.a ✓ 12 60.l odd 4 2
720.3.c.d 12 5.c odd 4 2
720.3.c.d 12 15.e even 4 2
1800.3.l.h 6 20.d odd 2 1
1800.3.l.h 6 60.h even 2 1
1800.3.l.i 6 4.b odd 2 1
1800.3.l.i 6 12.b even 2 1
2880.3.c.i 12 40.i odd 4 2
2880.3.c.i 12 120.w even 4 2
2880.3.c.j 12 40.k even 4 2
2880.3.c.j 12 120.q odd 4 2
3600.3.l.w 6 1.a even 1 1 trivial
3600.3.l.w 6 3.b odd 2 1 inner
3600.3.l.x 6 5.b even 2 1
3600.3.l.x 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{3} + 4T_{7}^{2} - 46T_{7} - 40 \) Copy content Toggle raw display
\( T_{11}^{6} + 516T_{11}^{4} + 57732T_{11}^{2} + 1438208 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 94T_{13} - 292 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + 4 T^{2} - 46 T - 40)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 516 T^{4} + \cdots + 1438208 \) Copy content Toggle raw display
$13$ \( (T^{3} - 2 T^{2} + \cdots - 292)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 1506 T^{4} + \cdots + 83980800 \) Copy content Toggle raw display
$19$ \( (T^{3} - 528 T - 1408)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 1944 T^{4} + \cdots + 90747392 \) Copy content Toggle raw display
$29$ \( T^{6} + 3042 T^{4} + \cdots + 446168192 \) Copy content Toggle raw display
$31$ \( (T^{3} + 24 T^{2} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 104 T^{2} + \cdots + 35120)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3462 T^{4} + \cdots + 483605000 \) Copy content Toggle raw display
$43$ \( (T^{3} + 48 T^{2} + \cdots - 31104)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16564912128 \) Copy content Toggle raw display
$53$ \( T^{6} + 2802 T^{4} + \cdots + 327680000 \) Copy content Toggle raw display
$59$ \( T^{6} + 3396 T^{4} + \cdots + 17428608 \) Copy content Toggle raw display
$61$ \( (T^{3} + 30 T^{2} + \cdots + 82312)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 208 T^{2} + \cdots + 137216)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 7392 T^{4} + \cdots + 99123200 \) Copy content Toggle raw display
$73$ \( (T^{3} - 82 T^{2} + \cdots + 306160)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 96 T^{2} + \cdots - 858320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 8506253312 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 418887045000 \) Copy content Toggle raw display
$97$ \( (T^{3} - 386 T^{2} + \cdots - 2047120)^{2} \) Copy content Toggle raw display
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