Properties

Label 4140.2.f.a
Level $4140$
Weight $2$
Character orbit 4140.f
Analytic conductor $33.058$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{5} - \beta_{3} q^{7} + \beta_1 q^{11} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{13} + ( - 2 \beta_{5} + \beta_{4}) q^{17} + (\beta_{2} - 2 \beta_1 + 1) q^{19} + \beta_{3} q^{23} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (3 \beta_{2} - 2) q^{29} + (\beta_{2} - \beta_1 - 4) q^{31} + (\beta_{5} + \beta_{2}) q^{35} + (\beta_{5} - 3 \beta_{4}) q^{37} + (4 \beta_{2} + \beta_1 + 3) q^{41} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{47} + 6 q^{49} + (\beta_{5} + 2 \beta_{4} + \beta_{3}) q^{53} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{55} + (\beta_{2} + 2) q^{59} + ( - \beta_{2} + 4 \beta_1 - 7) q^{61} + ( - 3 \beta_{5} + \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - \beta_1) q^{65} + ( - \beta_{5} - 5 \beta_{4} + 2 \beta_{3}) q^{67} + ( - 4 \beta_{2} + 3 \beta_1 + 1) q^{71} + ( - 2 \beta_{5} - \beta_{4} - 7 \beta_{3}) q^{73} + \beta_{5} q^{77} + (2 \beta_{2} - 6 \beta_1 + 2) q^{79} + ( - 2 \beta_{5} + 3 \beta_{4} - 8 \beta_{3}) q^{83} + (2 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1) q^{85} + (4 \beta_{2} + 2) q^{89} + (\beta_{2} - 2 \beta_1 + 1) q^{91} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 3 \beta_1 - 5) q^{95} + ( - 5 \beta_{5} + 7 \beta_{4} - 3 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{19} + 2 q^{25} - 18 q^{29} - 26 q^{31} - 2 q^{35} + 10 q^{41} + 36 q^{49} + 16 q^{55} + 10 q^{59} - 40 q^{61} - 4 q^{65} + 14 q^{71} + 8 q^{79} + 6 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
1.45161 + 1.45161i
1.45161 1.45161i
−0.854638 0.854638i
−0.854638 + 0.854638i
0.403032 0.403032i
0.403032 + 0.403032i
0 0 0 −2.21432 0.311108i 0 1.00000i 0 0 0
829.2 0 0 0 −2.21432 + 0.311108i 0 1.00000i 0 0 0
829.3 0 0 0 0.539189 2.17009i 0 1.00000i 0 0 0
829.4 0 0 0 0.539189 + 2.17009i 0 1.00000i 0 0 0
829.5 0 0 0 1.67513 1.48119i 0 1.00000i 0 0 0
829.6 0 0 0 1.67513 + 1.48119i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.6
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 4140.2.f.a 6
3.b odd 2 1 1380.2.f.a 6
5.b even 2 1 inner 4140.2.f.a 6
15.d odd 2 1 1380.2.f.a 6
15.e even 4 1 6900.2.a.y 3
15.e even 4 1 6900.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.a 6 3.b odd 2 1
1380.2.f.a 6 15.d odd 2 1
4140.2.f.a 6 1.a even 1 1 trivial
4140.2.f.a 6 5.b even 2 1 inner
6900.2.a.y 3 15.e even 4 1
6900.2.a.z 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 32 T^{4} + 156 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$17$ \( T^{6} + 31 T^{4} + 275 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 14 T - 10)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} + 9 T^{2} - 3 T - 61)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 13 T^{2} + 51 T + 59)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 59 T^{4} + 475 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( (T^{3} - 5 T^{2} - 57 T - 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$47$ \( T^{6} + 92 T^{4} + 1080 T^{2} + \cdots + 3364 \) Copy content Toggle raw display
$53$ \( T^{6} + 51 T^{4} + 839 T^{2} + \cdots + 4489 \) Copy content Toggle raw display
$59$ \( (T^{3} - 5 T^{2} + 5 T + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 20 T^{2} + 74 T + 74)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 195 T^{4} + 9083 T^{2} + \cdots + 26569 \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} - 49 T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 208 T^{4} + 9980 T^{2} + \cdots + 42436 \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} - 128 T - 416)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 215 T^{4} + 10963 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 388 T^{4} + 40400 T^{2} + \cdots + 781456 \) Copy content Toggle raw display
show more
show less