Properties

Label 252.9.d.a
Level $252$
Weight $9$
Character orbit 252.d
Analytic conductor $102.659$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1265 - 1504 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -1265 - 1504 \zeta_{6} ) q^{7} + ( 25696 - 51392 \zeta_{6} ) q^{13} + ( 19136 - 38272 \zeta_{6} ) q^{19} + 390625 q^{25} + ( 214176 - 428352 \zeta_{6} ) q^{31} -503522 q^{37} -3492194 q^{43} + ( -661791 + 6067136 \zeta_{6} ) q^{49} + ( -8147104 + 16294208 \zeta_{6} ) q^{61} -5421406 q^{67} + ( 31434560 - 62869120 \zeta_{6} ) q^{73} -18887038 q^{79} + ( -109799008 + 103657664 \zeta_{6} ) q^{91} + ( -4214336 + 8428672 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4034 q^{7} + O(q^{10}) \) \( 2 q - 4034 q^{7} + 781250 q^{25} - 1007044 q^{37} - 6984388 q^{43} + 4743554 q^{49} - 10842812 q^{67} - 37774076 q^{79} - 115940352 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −2017.00 1302.50i 0 0 0
181.2 0 0 0 0 0 −2017.00 + 1302.50i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.d.a 2
3.b odd 2 1 CM 252.9.d.a 2
7.b odd 2 1 inner 252.9.d.a 2
21.c even 2 1 inner 252.9.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.d.a 2 1.a even 1 1 trivial
252.9.d.a 2 3.b odd 2 1 CM
252.9.d.a 2 7.b odd 2 1 inner
252.9.d.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 5764801 + 4034 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1980853248 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1098559488 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 137614076928 + T^{2} \)
$37$ \( ( 503522 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 3492194 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 199125910760448 + T^{2} \)
$67$ \( ( 5421406 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 2964394687180800 + T^{2} \)
$79$ \( ( 18887038 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 53281883762688 + T^{2} \)
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