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Properties

Label 7200.2.k.s
Level $7200$
Weight $2$
Character orbit 7200.k
Analytic conductor $57.492$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 600)
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( q + (\beta_1 + 1) q^{7} - \beta_{5} q^{11} + ( - \beta_{7} + \beta_{6}) q^{13} + ( - \beta_{4} + \beta_1) q^{17} + ( - 2 \beta_{6} + \beta_{5} - \beta_{2}) q^{19} + (2 \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{23} + (2 \beta_{7} - \beta_{6} + \cdots - 3 \beta_{2}) q^{29}+ \cdots + ( - 4 \beta_{4} - 2 \beta_1 + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} - 8 q^{23} - 8 q^{31} - 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} - 2\nu^{4} + 7\nu^{3} + \nu^{2} - 4\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 4\nu^{6} + 8\nu^{5} + 18\nu^{4} - 25\nu^{3} - 18\nu^{2} - 20\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 6\nu^{6} + 4\nu^{5} + 18\nu^{4} - 29\nu^{3} - 8\nu^{2} - 12\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 4\nu^{6} - 8\nu^{5} - 22\nu^{4} + 35\nu^{3} + 30\nu^{2} + 12\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 5\nu^{6} - 6\nu^{5} - 22\nu^{4} + 37\nu^{3} + 17\nu^{2} + 24\nu - 88 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 5\nu^{6} - 6\nu^{5} - 24\nu^{4} + 34\nu^{3} + 15\nu^{2} + 28\nu - 84 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{4} + 4\beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{6} + 3\beta_{5} - 3\beta_{4} + 7\beta_{3} + \beta_{2} - 5\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} + 4\beta_{4} + 2\beta_{3} - 7\beta_{2} + 3\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14\beta_{7} - 7\beta_{6} + 5\beta_{5} + 7\beta_{4} + 7\beta_{3} + 13\beta_{2} + 3\beta _1 + 1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
1.41216 + 0.0762223i
1.41216 − 0.0762223i
1.23291 + 0.692769i
1.23291 − 0.692769i
−1.08003 − 0.912978i
−1.08003 + 0.912978i
−0.565036 − 1.29643i
−0.565036 + 1.29643i
0 0 0 0 0 −1.97676 0 0 0
3601.2 0 0 0 0 0 −1.97676 0 0 0
3601.3 0 0 0 0 0 −0.0802864 0 0 0
3601.4 0 0 0 0 0 −0.0802864 0 0 0
3601.5 0 0 0 0 0 1.33411 0 0 0
3601.6 0 0 0 0 0 1.33411 0 0 0
3601.7 0 0 0 0 0 4.72294 0 0 0
3601.8 0 0 0 0 0 4.72294 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3601.8
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.s 8
3.b odd 2 1 2400.2.k.e 8
4.b odd 2 1 1800.2.k.q 8
5.b even 2 1 7200.2.k.r 8
5.c odd 4 1 7200.2.d.s 8
5.c odd 4 1 7200.2.d.t 8
8.b even 2 1 inner 7200.2.k.s 8
8.d odd 2 1 1800.2.k.q 8
12.b even 2 1 600.2.k.e yes 8
15.d odd 2 1 2400.2.k.d 8
15.e even 4 1 2400.2.d.g 8
15.e even 4 1 2400.2.d.h 8
20.d odd 2 1 1800.2.k.t 8
20.e even 4 1 1800.2.d.s 8
20.e even 4 1 1800.2.d.t 8
24.f even 2 1 600.2.k.e yes 8
24.h odd 2 1 2400.2.k.e 8
40.e odd 2 1 1800.2.k.t 8
40.f even 2 1 7200.2.k.r 8
40.i odd 4 1 7200.2.d.s 8
40.i odd 4 1 7200.2.d.t 8
40.k even 4 1 1800.2.d.s 8
40.k even 4 1 1800.2.d.t 8
60.h even 2 1 600.2.k.d ✓ 8
60.l odd 4 1 600.2.d.g 8
60.l odd 4 1 600.2.d.h 8
120.i odd 2 1 2400.2.k.d 8
120.m even 2 1 600.2.k.d ✓ 8
120.q odd 4 1 600.2.d.g 8
120.q odd 4 1 600.2.d.h 8
120.w even 4 1 2400.2.d.g 8
120.w even 4 1 2400.2.d.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 60.l odd 4 1
600.2.d.g 8 120.q odd 4 1
600.2.d.h 8 60.l odd 4 1
600.2.d.h 8 120.q odd 4 1
600.2.k.d ✓ 8 60.h even 2 1
600.2.k.d ✓ 8 120.m even 2 1
600.2.k.e yes 8 12.b even 2 1
600.2.k.e yes 8 24.f even 2 1
1800.2.d.s 8 20.e even 4 1
1800.2.d.s 8 40.k even 4 1
1800.2.d.t 8 20.e even 4 1
1800.2.d.t 8 40.k even 4 1
1800.2.k.q 8 4.b odd 2 1
1800.2.k.q 8 8.d odd 2 1
1800.2.k.t 8 20.d odd 2 1
1800.2.k.t 8 40.e odd 2 1
2400.2.d.g 8 15.e even 4 1
2400.2.d.g 8 120.w even 4 1
2400.2.d.h 8 15.e even 4 1
2400.2.d.h 8 120.w even 4 1
2400.2.k.d 8 15.d odd 2 1
2400.2.k.d 8 120.i odd 2 1
2400.2.k.e 8 3.b odd 2 1
2400.2.k.e 8 24.h odd 2 1
7200.2.d.s 8 5.c odd 4 1
7200.2.d.s 8 40.i odd 4 1
7200.2.d.t 8 5.c odd 4 1
7200.2.d.t 8 40.i odd 4 1
7200.2.k.r 8 5.b even 2 1
7200.2.k.r 8 40.f even 2 1
7200.2.k.s 8 1.a even 1 1 trivial
7200.2.k.s 8 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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{4} - 4T_{7}^{3} - 6T_{7}^{2} + 12T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + 32T_{11}^{6} + 336T_{11}^{4} + 1344T_{11}^{2} + 1600 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} + 104T_{17} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 116 T^{6} + \cdots + 380689 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 56 T^{2} + \cdots - 88)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 144 T^{6} + \cdots + 627264 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 673)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 128 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{2} + \cdots + 328)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 244 T^{6} + \cdots + 4363921 \) Copy content Toggle raw display
$47$ \( (T^{4} - 72 T^{2} + \cdots - 176)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 256 T^{6} + \cdots + 23104 \) Copy content Toggle raw display
$59$ \( T^{8} + 432 T^{6} + \cdots + 31181056 \) Copy content Toggle raw display
$61$ \( T^{8} + 236 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$67$ \( T^{8} + 372 T^{6} + \cdots + 25979409 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots - 536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} + \cdots - 432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 8080)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 368 T^{6} + \cdots + 3873024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 224 T^{2} + \cdots + 10880)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 8881)^{2} \) Copy content Toggle raw display
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