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Properties

Label 5148.2.e.e
Level $5148$
Weight $2$
Character orbit 5148.e
Analytic conductor $41.107$
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] = [5148,2,Mod(1585,5148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5148.1585");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5148.e (of order \(2\), degree \(1\), 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: \(41.1069869606\)
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: \( x^{10} - 2x^{9} + 2x^{8} + 8x^{7} + 20x^{6} - 10x^{5} + 12x^{4} + 44x^{3} + 64x^{2} + 16x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1716)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + (\beta_{9} + \beta_{8}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + (\beta_{9} + \beta_{8}) q^{7} + \beta_{7} q^{11} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{13} + (\beta_{4} - 2 \beta_{3} + \beta_{2}) q^{17} + (\beta_{8} + \beta_{6} - \beta_1) q^{19} + (\beta_{5} + \beta_{3} + 2 \beta_{2}) q^{23} + ( - \beta_{5} + \beta_{3} + 1) q^{25} + ( - \beta_{5} + \beta_{2} - 2) q^{29} + (2 \beta_{9} + \beta_{8} + 2 \beta_1) q^{31} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 4) q^{35} + ( - 2 \beta_{8} - 2 \beta_{6} - \beta_1) q^{37} + ( - \beta_{8} + 2 \beta_{7} + \cdots - \beta_1) q^{41}+ \cdots + (2 \beta_{9} - 2 \beta_{7} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{13} + 8 q^{17} + 10 q^{25} - 12 q^{29} + 32 q^{35} - 20 q^{43} - 18 q^{49} + 20 q^{53} + 4 q^{55} - 12 q^{61} + 8 q^{79} - 4 q^{91} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} + 8x^{7} + 20x^{6} - 10x^{5} + 12x^{4} + 44x^{3} + 64x^{2} + 16x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18374 \nu^{9} - 46452 \nu^{8} + 73276 \nu^{7} + 95079 \nu^{6} + 326074 \nu^{5} - 267864 \nu^{4} + \cdots + 247726 ) / 860919 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18374 \nu^{9} + 46452 \nu^{8} - 73276 \nu^{7} - 95079 \nu^{6} - 326074 \nu^{5} + 267864 \nu^{4} + \cdots - 247726 ) / 860919 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28078 \nu^{9} + 82980 \nu^{8} - 125189 \nu^{7} - 136485 \nu^{6} - 410198 \nu^{5} + \cdots + 1437364 ) / 860919 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22716 \nu^{9} - 81013 \nu^{8} + 123297 \nu^{7} + 95744 \nu^{6} + 185096 \nu^{5} - 848727 \nu^{4} + \cdots - 927964 ) / 286973 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28914 \nu^{9} + 92278 \nu^{8} - 149889 \nu^{7} - 126567 \nu^{6} - 282594 \nu^{5} + 639835 \nu^{4} + \cdots + 137260 ) / 286973 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41117 \nu^{9} + 77648 \nu^{8} - 40840 \nu^{7} - 438343 \nu^{6} - 689668 \nu^{5} + 458949 \nu^{4} + \cdots - 240520 ) / 286973 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 123863 \nu^{9} + 266100 \nu^{8} - 294178 \nu^{7} - 917628 \nu^{6} - 2382181 \nu^{5} + \cdots - 1016623 ) / 860919 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 229352 \nu^{9} - 485748 \nu^{8} + 515080 \nu^{7} + 1740177 \nu^{6} + 4438288 \nu^{5} + \cdots + 1785520 ) / 860919 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 147773 \nu^{9} - 312335 \nu^{8} + 322310 \nu^{7} + 1181826 \nu^{6} + 2776356 \nu^{5} + \cdots + 1172464 ) / 286973 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 2\beta_{7} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - 5\beta_{2} + 5\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{4} + 7\beta_{3} - 8\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{9} - 11 \beta_{8} - 9 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 11 \beta_{3} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{9} - 52\beta_{8} - 46\beta_{7} + 12\beta_{6} - 63\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{9} - 95 \beta_{8} - 74 \beta_{7} + 30 \beta_{6} - 30 \beta_{5} - 42 \beta_{4} - 95 \beta_{3} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -107\beta_{5} - 167\beta_{4} - 400\beta_{3} + 497\beta_{2} + 322 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 337 \beta_{9} + 771 \beta_{8} + 594 \beta_{7} - 230 \beta_{6} - 230 \beta_{5} - 337 \beta_{4} + \cdots + 594 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5148\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1145\) \(2575\) \(4357\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
−1.17546 − 1.17546i
1.98567 + 1.98567i
1.05917 − 1.05917i
−0.138330 + 0.138330i
−0.731045 − 0.731045i
−0.731045 + 0.731045i
−0.138330 − 0.138330i
1.05917 + 1.05917i
1.98567 − 1.98567i
−1.17546 + 1.17546i
0 0 0 − 3.11435i 0 3.50020i 0 0 0
1585.2 0 0 0 − 1.91444i 0 − 1.46773i 0 0 0
1585.3 0 0 0 − 1.87466i 0 − 0.825804i 0 0 0
1585.4 0 0 0 − 1.68507i 0 4.95245i 0 0 0
1585.5 0 0 0 − 0.530945i 0 2.09419i 0 0 0
1585.6 0 0 0 0.530945i 0 − 2.09419i 0 0 0
1585.7 0 0 0 1.68507i 0 − 4.95245i 0 0 0
1585.8 0 0 0 1.87466i 0 0.825804i 0 0 0
1585.9 0 0 0 1.91444i 0 1.46773i 0 0 0
1585.10 0 0 0 3.11435i 0 − 3.50020i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5148.2.e.e 10
3.b odd 2 1 1716.2.e.a ✓ 10
13.b even 2 1 inner 5148.2.e.e 10
39.d odd 2 1 1716.2.e.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.e.a ✓ 10 3.b odd 2 1
1716.2.e.a ✓ 10 39.d odd 2 1
5148.2.e.e 10 1.a even 1 1 trivial
5148.2.e.e 10 13.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 \(S_{2}^{\mathrm{new}}(5148, [\chi])\):

\( T_{5}^{10} + 20T_{5}^{8} + 136T_{5}^{6} + 396T_{5}^{4} + 456T_{5}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{10} + 44T_{7}^{8} + 580T_{7}^{6} + 2688T_{7}^{4} + 4416T_{7}^{2} + 1936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 20 T^{8} + \cdots + 100 \) Copy content Toggle raw display
$7$ \( T^{10} + 44 T^{8} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 4 T^{4} + \cdots + 162)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 96 T^{8} + \cdots + 32400 \) Copy content Toggle raw display
$23$ \( (T^{5} - 80 T^{3} + \cdots - 1892)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 6 T^{4} - 20 T^{3} + \cdots + 82)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 148 T^{8} + \cdots + 15876 \) Copy content Toggle raw display
$37$ \( T^{10} + 180 T^{8} + \cdots + 627264 \) Copy content Toggle raw display
$41$ \( T^{10} + 68 T^{8} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{5} + 10 T^{4} + \cdots + 2646)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 284 T^{8} + \cdots + 22505536 \) Copy content Toggle raw display
$53$ \( (T^{5} - 10 T^{4} + \cdots - 51440)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 212 T^{8} + \cdots + 8111104 \) Copy content Toggle raw display
$61$ \( (T^{5} + 6 T^{4} + \cdots - 280)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 336 T^{8} + \cdots + 2735716 \) Copy content Toggle raw display
$71$ \( T^{10} + 288 T^{8} + \cdots + 35521600 \) Copy content Toggle raw display
$73$ \( T^{10} + 652 T^{8} + \cdots + 8596624 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} + \cdots - 33786)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 356 T^{8} + \cdots + 63744256 \) Copy content Toggle raw display
$89$ \( T^{10} + 500 T^{8} + \cdots + 6280036 \) Copy content Toggle raw display
$97$ \( T^{10} + 296 T^{8} + \cdots + 26460736 \) Copy content Toggle raw display
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