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Properties

Label 4851.2.a.cg
Level $4851$
Weight $2$
Character orbit 4851.a
Self dual yes
Analytic conductor $38.735$
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] = [4851,2,Mod(1,4851)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4851.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

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: yes
Analytic conductor: \(38.7354300205\)
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: \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 539)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{9} - \beta_{8} - 2 \beta_{5} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{9} - \beta_{8} - 2 \beta_{5} + \cdots + 1) q^{8}+ \cdots + (\beta_{7} - 2 \beta_{3} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 18 q^{4} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + 18 q^{4} + 6 q^{8} - 10 q^{11} + 42 q^{16} + 2 q^{22} - 4 q^{23} + 18 q^{25} - 12 q^{29} + 30 q^{32} + 40 q^{37} - 8 q^{43} - 18 q^{44} + 44 q^{46} + 62 q^{50} - 16 q^{53} - 28 q^{58} + 106 q^{64} + 32 q^{65} - 4 q^{67} - 36 q^{71} + 28 q^{74} + 8 q^{79} + 88 q^{85} - 32 q^{86} - 6 q^{88} + 52 q^{92} + 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{9} - 160\nu^{7} + 1187\nu^{5} - 3108\nu^{3} + 1572\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{9} - 324\nu^{7} + 2267\nu^{5} - 5912\nu^{3} + 3972\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 22\nu^{6} + 157\nu^{4} - 410\nu^{2} + 252 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{9} - 540\nu^{7} + 3749\nu^{5} - 9472\nu^{3} + 5652\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -71\nu^{9} + 1516\nu^{7} - 10355\nu^{5} + 25672\nu^{3} - 14876\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{8} - 106\nu^{6} + 717\nu^{4} - 1762\nu^{2} + 1024 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{8} + 108\nu^{6} - 749\nu^{4} + 1880\nu^{2} - 1080 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 4\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + \beta_{8} + 5\beta_{5} + 13\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} - 55\beta_{6} + 18\beta_{4} + 26\beta_{3} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{9} + 18\beta_{8} + 80\beta_{5} + 149\beta_{2} + 277 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -151\beta_{7} - 648\beta_{6} + 229\beta_{4} + 292\beta_{3} + 541\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 434\beta_{9} + 239\beta_{8} + 979\beta_{5} + 1647\beta_{2} + 2554 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1691\beta_{7} - 7261\beta_{6} + 2626\beta_{4} + 3166\beta_{3} + 5414\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.903205
−0.903205
3.27614
−3.27614
−2.32267
2.32267
2.15293
−2.15293
−2.10267
2.10267
−2.62810 0 4.90690 −0.337987 0 0 −7.63960 0 0.888264
1.2 −2.62810 0 4.90690 0.337987 0 0 −7.63960 0 −0.888264
1.3 −1.70296 0 0.900071 −0.246676 0 0 1.87313 0 0.420079
1.4 −1.70296 0 0.900071 0.246676 0 0 1.87313 0 −0.420079
1.5 −0.566092 0 −1.67954 −3.58219 0 0 2.08296 0 2.02785
1.6 −0.566092 0 −1.67954 3.58219 0 0 2.08296 0 −2.02785
1.7 1.14898 0 −0.679834 −3.87589 0 0 −3.07909 0 −4.45334
1.8 1.14898 0 −0.679834 3.87589 0 0 −3.07909 0 4.45334
1.9 2.74816 0 5.55241 −2.44342 0 0 9.76260 0 −6.71492
1.10 2.74816 0 5.55241 2.44342 0 0 9.76260 0 6.71492
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4851.2.a.cg 10
3.b odd 2 1 539.2.a.l ✓ 10
7.b odd 2 1 inner 4851.2.a.cg 10
12.b even 2 1 8624.2.a.df 10
21.c even 2 1 539.2.a.l ✓ 10
21.g even 6 2 539.2.e.o 20
21.h odd 6 2 539.2.e.o 20
33.d even 2 1 5929.2.a.bv 10
84.h odd 2 1 8624.2.a.df 10
231.h odd 2 1 5929.2.a.bv 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.l ✓ 10 3.b odd 2 1
539.2.a.l ✓ 10 21.c even 2 1
539.2.e.o 20 21.g even 6 2
539.2.e.o 20 21.h odd 6 2
4851.2.a.cg 10 1.a even 1 1 trivial
4851.2.a.cg 10 7.b odd 2 1 inner
5929.2.a.bv 10 33.d even 2 1
5929.2.a.bv 10 231.h odd 2 1
8624.2.a.df 10 12.b even 2 1
8624.2.a.df 10 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):

\( T_{2}^{5} + T_{2}^{4} - 9T_{2}^{3} - 9T_{2}^{2} + 12T_{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{10} - 34T_{5}^{8} + 365T_{5}^{6} - 1214T_{5}^{4} + 204T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{10} - 72T_{13}^{8} + 1696T_{13}^{6} - 16672T_{13}^{4} + 69120T_{13}^{2} - 100352 \) Copy content Toggle raw display
\( T_{17}^{10} - 136T_{17}^{8} + 6192T_{17}^{6} - 110752T_{17}^{4} + 705536T_{17}^{2} - 430592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + T^{4} - 9 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 34 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 72 T^{8} + \cdots - 100352 \) Copy content Toggle raw display
$17$ \( T^{10} - 136 T^{8} + \cdots - 430592 \) Copy content Toggle raw display
$19$ \( T^{10} - 144 T^{8} + \cdots - 6422528 \) Copy content Toggle raw display
$23$ \( (T^{5} + 2 T^{4} + \cdots + 232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 6 T^{4} + \cdots - 224)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 154 T^{8} + \cdots - 3998792 \) Copy content Toggle raw display
$37$ \( (T^{5} - 20 T^{4} + \cdots - 472)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 168 T^{8} + \cdots - 25088 \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 1158537248 \) Copy content Toggle raw display
$53$ \( (T^{5} + 8 T^{4} + \cdots + 11264)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 108162632 \) Copy content Toggle raw display
$61$ \( T^{10} - 144 T^{8} + \cdots - 3527168 \) Copy content Toggle raw display
$67$ \( (T^{5} + 2 T^{4} + \cdots + 4648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 18 T^{4} + \cdots - 88)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 400 T^{8} + \cdots - 65987072 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} + \cdots - 704)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 102760448 \) Copy content Toggle raw display
$89$ \( T^{10} - 34 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$97$ \( T^{10} - 226 T^{8} + \cdots - 19208 \) Copy content Toggle raw display
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