Properties

Label 825.4.c.i
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + (8 \beta_{2} + 7 \beta_1) q^{8} - 9 q^{9} + 11 q^{11} + 3 \beta_1 q^{12}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44}+ \cdots - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 17x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
3.37228i
2.37228i
2.37228i
3.37228i
3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
199.2 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.3 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.4 3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 825.4.c.i 4
5.b even 2 1 inner 825.4.c.i 4
5.c odd 4 1 33.4.a.d 2
5.c odd 4 1 825.4.a.k 2
15.e even 4 1 99.4.a.e 2
15.e even 4 1 2475.4.a.o 2
20.e even 4 1 528.4.a.o 2
35.f even 4 1 1617.4.a.j 2
40.i odd 4 1 2112.4.a.ba 2
40.k even 4 1 2112.4.a.bh 2
55.e even 4 1 363.4.a.j 2
60.l odd 4 1 1584.4.a.x 2
165.l odd 4 1 1089.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 5.c odd 4 1
99.4.a.e 2 15.e even 4 1
363.4.a.j 2 55.e even 4 1
528.4.a.o 2 20.e even 4 1
825.4.a.k 2 5.c odd 4 1
825.4.c.i 4 1.a even 1 1 trivial
825.4.c.i 4 5.b even 2 1 inner
1089.4.a.t 2 165.l odd 4 1
1584.4.a.x 2 60.l odd 4 1
1617.4.a.j 2 35.f even 4 1
2112.4.a.ba 2 40.i odd 4 1
2112.4.a.bh 2 40.k even 4 1
2475.4.a.o 2 15.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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 17T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} + 68T_{7}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 17T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 68T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 3944 T^{2} + 839056 \) Copy content Toggle raw display
$17$ \( T^{4} + 15188 T^{2} + 52649536 \) Copy content Toggle raw display
$19$ \( (T^{2} - 54 T - 1944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 222 T - 5136)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T - 88832)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 33096 T^{2} + 237036816 \) Copy content Toggle raw display
$41$ \( (T^{2} + 494 T + 60976)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 3591365184 \) Copy content Toggle raw display
$47$ \( T^{4} + 40064 T^{2} + 323424256 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 17796627216 \) Copy content Toggle raw display
$59$ \( (T + 196)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1104 T + 282396)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 811808 T^{2} + 609497344 \) Copy content Toggle raw display
$71$ \( (T^{2} - 456 T - 227328)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 190350709264 \) Copy content Toggle raw display
$79$ \( (T^{2} - 230 T - 31952)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 698521522176 \) Copy content Toggle raw display
$89$ \( (T^{2} + 972 T + 235668)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1220662586896 \) Copy content Toggle raw display
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