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Properties

Label 4600.2.e.t
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
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] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 38x^{4} + 25x^{2} + 4 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{7} - \beta_1) q^{7} + (\beta_{6} + \beta_{5} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{7} - \beta_1) q^{7} + (\beta_{6} + \beta_{5} + 2 \beta_{2} - 2) q^{9} + ( - \beta_{6} + \beta_{2}) q^{11} + ( - 2 \beta_{7} + \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{6} + 4 \beta_{5} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{9} + 2 q^{11} + 30 q^{19} + 16 q^{21} - 2 q^{29} - 24 q^{31} + 6 q^{39} - 18 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 28 q^{61} - 4 q^{71} + 42 q^{79} + 88 q^{81} + 32 q^{89} + 66 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 13x^{6} + 38x^{4} + 25x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 43\nu^{2} + 14 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} - 89\nu^{5} - 238\nu^{3} - 89\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} + 61\nu^{5} + 143\nu^{3} + 25\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} + 61\nu^{4} + 143\nu^{2} + 34 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{6} + 25\nu^{4} + 65\nu^{2} + 25 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} - 38\nu^{3} - 23\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + \beta_{4} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} + 7\beta_{5} + 13\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{7} - 14\beta_{4} - 47\beta_{3} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 69\beta_{6} - 55\beta_{5} - 130\beta_{2} - 165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -199\beta_{7} + 144\beta_{4} + 459\beta_{3} - 419\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
− 0.774457i
1.73914i
− 3.02228i
− 0.491317i
0.491317i
3.02228i
− 1.73914i
0.774457i
0 − 3.43375i 0 0 0 2.58245i 0 −8.79066 0
4049.2 0 − 2.35292i 0 0 0 − 1.14999i 0 −2.53622 0
4049.3 0 − 0.751385i 0 0 0 0.661751i 0 2.43542 0
4049.4 0 − 0.329452i 0 0 0 4.07069i 0 2.89146 0
4049.5 0 0.329452i 0 0 0 − 4.07069i 0 2.89146 0
4049.6 0 0.751385i 0 0 0 − 0.661751i 0 2.43542 0
4049.7 0 2.35292i 0 0 0 1.14999i 0 −2.53622 0
4049.8 0 3.43375i 0 0 0 − 2.58245i 0 −8.79066 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.8
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 4600.2.e.t 8
5.b even 2 1 inner 4600.2.e.t 8
5.c odd 4 1 4600.2.a.ba ✓ 4
5.c odd 4 1 4600.2.a.bb yes 4
20.e even 4 1 9200.2.a.cn 4
20.e even 4 1 9200.2.a.cp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.ba ✓ 4 5.c odd 4 1
4600.2.a.bb yes 4 5.c odd 4 1
4600.2.e.t 8 1.a even 1 1 trivial
4600.2.e.t 8 5.b even 2 1 inner
9200.2.a.cn 4 20.e even 4 1
9200.2.a.cp 4 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{8} + 18T_{3}^{6} + 77T_{3}^{4} + 45T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 25T_{7}^{6} + 152T_{7}^{4} + 208T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 22T_{11}^{2} - 16T_{11} + 40 \) Copy content Toggle raw display
\( T_{13}^{8} + 71T_{13}^{6} + 1527T_{13}^{4} + 12146T_{13}^{2} + 27889 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 18 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 25 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 22 T^{2} + \cdots + 40)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 71 T^{6} + \cdots + 27889 \) Copy content Toggle raw display
$17$ \( T^{8} + 52 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{4} - 15 T^{3} + \cdots - 184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + T^{3} - 71 T^{2} + \cdots + 893)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots + 46)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 188 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$41$ \( (T^{4} + 9 T^{3} + 17 T^{2} + \cdots - 47)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 161 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{8} + 30 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 288 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$59$ \( (T^{4} - 5 T^{3} + \cdots + 6246)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + \cdots - 2736)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 136 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + \cdots + 522)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 231 T^{6} + \cdots + 120409 \) Copy content Toggle raw display
$79$ \( (T^{4} - 21 T^{3} + \cdots + 9664)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 313 T^{6} + \cdots + 18045504 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + \cdots - 80)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 488 T^{6} + \cdots + 25240576 \) Copy content Toggle raw display
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