Properties

Label 7200.2.k.p
Level $7200$
Weight $2$
Character orbit 7200.k
Analytic conductor $57.492$
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] = [7200,2,Mod(3601,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.3601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.k (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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{11} - \beta_{4} q^{13} + ( - \beta_1 + 2) q^{17} + ( - \beta_{5} + \beta_{2}) q^{19} + (\beta_{3} + 1) q^{23} - \beta_{2} q^{29} + (\beta_1 + 2) q^{31} + ( - 2 \beta_{5} - \beta_{4}) q^{37} + ( - 2 \beta_{3} + 4) q^{41} + 2 \beta_{5} q^{43} + ( - \beta_{3} - 2 \beta_1 - 1) q^{47} + (2 \beta_1 + 5) q^{49} - \beta_{2} q^{53} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{59} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{61} + 2 \beta_{2} q^{67} + (2 \beta_{3} + 2 \beta_1 - 2) q^{71} + 6 q^{73} + (4 \beta_{5} - 4 \beta_{2}) q^{77} + (\beta_1 - 6) q^{79} + (2 \beta_{5} + 2 \beta_{2}) q^{83} + (2 \beta_{3} + 4) q^{89} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{91} + (2 \beta_1 - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{7} + 12 q^{17} + 8 q^{23} + 12 q^{31} + 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{71} + 36 q^{73} - 36 q^{79} + 28 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 7 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3601.1
1.40680 + 0.144584i
1.40680 0.144584i
0.264658 + 1.38923i
0.264658 1.38923i
−0.671462 + 1.24464i
−0.671462 1.24464i
0 0 0 0 0 −3.62721 0 0 0
3601.2 0 0 0 0 0 −3.62721 0 0 0
3601.3 0 0 0 0 0 0.941367 0 0 0
3601.4 0 0 0 0 0 0.941367 0 0 0
3601.5 0 0 0 0 0 4.68585 0 0 0
3601.6 0 0 0 0 0 4.68585 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3601.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.k.p 6
3.b odd 2 1 2400.2.k.c 6
4.b odd 2 1 1800.2.k.p 6
5.b even 2 1 1440.2.k.f 6
5.c odd 4 1 7200.2.d.q 6
5.c odd 4 1 7200.2.d.r 6
8.b even 2 1 inner 7200.2.k.p 6
8.d odd 2 1 1800.2.k.p 6
12.b even 2 1 600.2.k.c 6
15.d odd 2 1 480.2.k.b 6
15.e even 4 1 2400.2.d.e 6
15.e even 4 1 2400.2.d.f 6
20.d odd 2 1 360.2.k.f 6
20.e even 4 1 1800.2.d.q 6
20.e even 4 1 1800.2.d.r 6
24.f even 2 1 600.2.k.c 6
24.h odd 2 1 2400.2.k.c 6
40.e odd 2 1 360.2.k.f 6
40.f even 2 1 1440.2.k.f 6
40.i odd 4 1 7200.2.d.q 6
40.i odd 4 1 7200.2.d.r 6
40.k even 4 1 1800.2.d.q 6
40.k even 4 1 1800.2.d.r 6
60.h even 2 1 120.2.k.b 6
60.l odd 4 1 600.2.d.e 6
60.l odd 4 1 600.2.d.f 6
120.i odd 2 1 480.2.k.b 6
120.m even 2 1 120.2.k.b 6
120.q odd 4 1 600.2.d.e 6
120.q odd 4 1 600.2.d.f 6
120.w even 4 1 2400.2.d.e 6
120.w even 4 1 2400.2.d.f 6
240.t even 4 1 3840.2.a.bp 3
240.t even 4 1 3840.2.a.bq 3
240.bm odd 4 1 3840.2.a.bo 3
240.bm odd 4 1 3840.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 60.h even 2 1
120.2.k.b 6 120.m even 2 1
360.2.k.f 6 20.d odd 2 1
360.2.k.f 6 40.e odd 2 1
480.2.k.b 6 15.d odd 2 1
480.2.k.b 6 120.i odd 2 1
600.2.d.e 6 60.l odd 4 1
600.2.d.e 6 120.q odd 4 1
600.2.d.f 6 60.l odd 4 1
600.2.d.f 6 120.q odd 4 1
600.2.k.c 6 12.b even 2 1
600.2.k.c 6 24.f even 2 1
1440.2.k.f 6 5.b even 2 1
1440.2.k.f 6 40.f even 2 1
1800.2.d.q 6 20.e even 4 1
1800.2.d.q 6 40.k even 4 1
1800.2.d.r 6 20.e even 4 1
1800.2.d.r 6 40.k even 4 1
1800.2.k.p 6 4.b odd 2 1
1800.2.k.p 6 8.d odd 2 1
2400.2.d.e 6 15.e even 4 1
2400.2.d.e 6 120.w even 4 1
2400.2.d.f 6 15.e even 4 1
2400.2.d.f 6 120.w even 4 1
2400.2.k.c 6 3.b odd 2 1
2400.2.k.c 6 24.h odd 2 1
3840.2.a.bo 3 240.bm odd 4 1
3840.2.a.bp 3 240.t even 4 1
3840.2.a.bq 3 240.t even 4 1
3840.2.a.br 3 240.bm odd 4 1
7200.2.d.q 6 5.c odd 4 1
7200.2.d.q 6 40.i odd 4 1
7200.2.d.r 6 5.c odd 4 1
7200.2.d.r 6 40.i odd 4 1
7200.2.k.p 6 1.a even 1 1 trivial
7200.2.k.p 6 8.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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{3} - 2T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 64T_{11}^{4} + 1088T_{11}^{2} + 4096 \) Copy content Toggle raw display
\( T_{17}^{3} - 6T_{17}^{2} - 16T_{17} + 32 \) 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
$7$ \( (T^{3} - 2 T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{3} - 4 T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 16 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 136 T^{4} + 5648 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} - 36 T + 232)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( (T^{3} + 4 T^{2} - 92 T - 496)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$59$ \( T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{6} + 256 T^{4} + 17408 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 112 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{6} \) Copy content Toggle raw display
$79$ \( (T^{3} + 18 T^{2} + 80 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 224 T^{4} + 8448 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{3} - 14 T^{2} - 4 T + 184)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 18 T^{2} - 4 T - 328)^{2} \) Copy content Toggle raw display
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