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Properties

Label 4600.2.e.w
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} - \beta_{4}) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} - \beta_{4}) q^{7} + (\beta_{2} - 1) q^{9} - \beta_{5} q^{11} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{13} + (2 \beta_{9} - 2 \beta_{4} - 2 \beta_{3}) q^{17} + \beta_{5} q^{19} + (\beta_{8} + \beta_{5} - 2 \beta_{2} + 1) q^{21} - \beta_{4} q^{23} + (\beta_{6} - 2 \beta_{4} + \beta_1) q^{27} + ( - 3 \beta_{7} - \beta_{5} - \beta_{2} - 2) q^{29} + (\beta_{8} - \beta_{7} - \beta_{5} + \cdots - 3) q^{31}+ \cdots + ( - 2 \beta_{7} - \beta_{5} - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 15\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 61\nu^{5} + 81\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 12\nu^{4} + 39\nu^{2} + 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 61\nu^{5} + 93\nu^{3} + 60\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 14\nu^{6} + 63\nu^{4} + 99\nu^{2} + 36 ) / 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} - 16\nu^{6} - 81\nu^{4} - 135\nu^{2} - 48 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{9} - 50\nu^{7} - 279\nu^{5} - 579\nu^{3} - 336\nu ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{5} - 7\beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{6} + 18\beta_{4} + 3\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{8} - 12\beta_{7} - 9\beta_{5} + 45\beta_{2} - 123 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{9} + 66\beta_{6} - 141\beta_{4} - 36\beta_{3} - 192\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105\beta_{8} + 111\beta_{7} + 63\beta_{5} - 288\beta_{2} + 759 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 42\beta_{9} - 456\beta_{6} + 1062\beta_{4} + 321\beta_{3} + 1263\beta_1 \) 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
− 2.61696i
− 2.29544i
− 1.83957i
− 1.36629i
− 0.794805i
0.794805i
1.36629i
1.83957i
2.29544i
2.61696i
0 − 2.61696i 0 0 0 3.83744i 0 −3.84849 0
4049.2 0 − 2.29544i 0 0 0 1.61758i 0 −2.26905 0
4049.3 0 − 1.83957i 0 0 0 − 3.97272i 0 −0.384010 0
4049.4 0 − 1.36629i 0 0 0 − 3.28093i 0 1.13327 0
4049.5 0 − 0.794805i 0 0 0 − 2.47193i 0 2.36829 0
4049.6 0 0.794805i 0 0 0 2.47193i 0 2.36829 0
4049.7 0 1.36629i 0 0 0 3.28093i 0 1.13327 0
4049.8 0 1.83957i 0 0 0 3.97272i 0 −0.384010 0
4049.9 0 2.29544i 0 0 0 − 1.61758i 0 −2.26905 0
4049.10 0 2.61696i 0 0 0 − 3.83744i 0 −3.84849 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.10
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.w 10
5.b even 2 1 inner 4600.2.e.w 10
5.c odd 4 1 4600.2.a.bd ✓ 5
5.c odd 4 1 4600.2.a.bf yes 5
20.e even 4 1 9200.2.a.ct 5
20.e even 4 1 9200.2.a.cv 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bd ✓ 5 5.c odd 4 1
4600.2.a.bf yes 5 5.c odd 4 1
4600.2.e.w 10 1.a even 1 1 trivial
4600.2.e.w 10 5.b even 2 1 inner
9200.2.a.ct 5 20.e even 4 1
9200.2.a.cv 5 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{10} + 18T_{3}^{8} + 117T_{3}^{6} + 333T_{3}^{4} + 396T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{7}^{10} + 50T_{7}^{8} + 937T_{7}^{6} + 8056T_{7}^{4} + 30800T_{7}^{2} + 40000 \) Copy content Toggle raw display
\( T_{11}^{5} - 15T_{11}^{3} - 6T_{11}^{2} + 48T_{11} + 24 \) Copy content Toggle raw display
\( T_{13}^{10} + 80T_{13}^{8} + 2326T_{13}^{6} + 29425T_{13}^{4} + 145523T_{13}^{2} + 120409 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 18 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 50 T^{8} + \cdots + 40000 \) Copy content Toggle raw display
$11$ \( (T^{5} - 15 T^{3} + \cdots + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 80 T^{8} + \cdots + 120409 \) Copy content Toggle raw display
$17$ \( T^{10} + 132 T^{8} + \cdots + 147456 \) Copy content Toggle raw display
$19$ \( (T^{5} - 15 T^{3} + \cdots - 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots + 2787)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 18 T^{4} + \cdots - 228)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 164 T^{8} + \cdots + 369664 \) Copy content Toggle raw display
$41$ \( (T^{5} + 6 T^{4} + \cdots + 453)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 146 T^{8} + \cdots + 2166784 \) Copy content Toggle raw display
$47$ \( T^{10} + 290 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{10} + 284 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{5} - T^{4} - 83 T^{3} + \cdots + 3004)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots + 7648)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 440 T^{8} + \cdots + 94633984 \) Copy content Toggle raw display
$71$ \( (T^{5} - 8 T^{4} + \cdots + 8084)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 816187761 \) Copy content Toggle raw display
$79$ \( (T^{5} - 153 T^{3} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 386 T^{8} + \cdots + 5721664 \) Copy content Toggle raw display
$89$ \( (T^{5} + 14 T^{4} + \cdots - 3104)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 324 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
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