Properties

Label 252.4.k.e
Level $252$
Weight $4$
Character orbit 252.k
Analytic conductor $14.868$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{385})\)
Defining polynomial: \(x^{4} - x^{3} + 97 x^{2} + 96 x + 9216\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -7 + 21 \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -7 + 21 \beta_{1} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{11} -54 q^{13} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{17} -2 \beta_{1} q^{19} + 10 \beta_{2} q^{23} + ( -260 + 260 \beta_{1} ) q^{25} + 5 \beta_{3} q^{29} + ( -201 + 201 \beta_{1} ) q^{31} + ( -14 \beta_{2} - 21 \beta_{3} ) q^{35} + 202 \beta_{1} q^{37} + 24 \beta_{3} q^{41} -244 q^{43} -12 \beta_{2} q^{47} + ( -392 + 147 \beta_{1} ) q^{49} + ( -33 \beta_{2} - 33 \beta_{3} ) q^{53} -385 q^{55} + ( 31 \beta_{2} + 31 \beta_{3} ) q^{59} + 588 \beta_{1} q^{61} + 54 \beta_{2} q^{65} + ( 302 - 302 \beta_{1} ) q^{67} -8 \beta_{3} q^{71} + ( 446 - 446 \beta_{1} ) q^{73} + ( 7 \beta_{2} - 14 \beta_{3} ) q^{77} + 267 \beta_{1} q^{79} + 37 \beta_{3} q^{83} + 770 q^{85} -6 \beta_{2} q^{89} + ( 378 - 1134 \beta_{1} ) q^{91} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{95} + 595 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{7} + O(q^{10}) \) \( 4q + 14q^{7} - 216q^{13} - 4q^{19} - 520q^{25} - 402q^{31} + 404q^{37} - 976q^{43} - 1274q^{49} - 1540q^{55} + 1176q^{61} + 604q^{67} + 892q^{73} + 534q^{79} + 3080q^{85} - 756q^{91} + 2380q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 97 x^{2} + 96 x + 9216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 97 \nu^{2} - 97 \nu + 9216 \)\()/9312\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 97 \nu^{2} + 18721 \nu - 9216 \)\()/9312\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + 289 \)\()/97\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 193 \beta_{1} - 193\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(97 \beta_{3} - 289\)\()/2\)

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
5.15535 8.92934i
−4.65535 + 8.06331i
5.15535 + 8.92934i
−4.65535 8.06331i
0 0 0 −9.81071 + 16.9926i 0 3.50000 18.1865i 0 0 0
37.2 0 0 0 9.81071 16.9926i 0 3.50000 18.1865i 0 0 0
109.1 0 0 0 −9.81071 16.9926i 0 3.50000 + 18.1865i 0 0 0
109.2 0 0 0 9.81071 + 16.9926i 0 3.50000 + 18.1865i 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 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.k.e 4
3.b odd 2 1 inner 252.4.k.e 4
7.b odd 2 1 1764.4.k.x 4
7.c even 3 1 inner 252.4.k.e 4
7.c even 3 1 1764.4.a.r 2
7.d odd 6 1 1764.4.a.u 2
7.d odd 6 1 1764.4.k.x 4
21.c even 2 1 1764.4.k.x 4
21.g even 6 1 1764.4.a.u 2
21.g even 6 1 1764.4.k.x 4
21.h odd 6 1 inner 252.4.k.e 4
21.h odd 6 1 1764.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.k.e 4 1.a even 1 1 trivial
252.4.k.e 4 3.b odd 2 1 inner
252.4.k.e 4 7.c even 3 1 inner
252.4.k.e 4 21.h odd 6 1 inner
1764.4.a.r 2 7.c even 3 1
1764.4.a.r 2 21.h odd 6 1
1764.4.a.u 2 7.d odd 6 1
1764.4.a.u 2 21.g even 6 1
1764.4.k.x 4 7.b odd 2 1
1764.4.k.x 4 7.d odd 6 1
1764.4.k.x 4 21.c even 2 1
1764.4.k.x 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} + 385 T_{5}^{2} + 148225 \)
\( T_{13} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 148225 + 385 T^{2} + T^{4} \)
$7$ \( ( 343 - 7 T + T^{2} )^{2} \)
$11$ \( 148225 + 385 T^{2} + T^{4} \)
$13$ \( ( 54 + T )^{4} \)
$17$ \( 2371600 + 1540 T^{2} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 1482250000 + 38500 T^{2} + T^{4} \)
$29$ \( ( -9625 + T^{2} )^{2} \)
$31$ \( ( 40401 + 201 T + T^{2} )^{2} \)
$37$ \( ( 40804 - 202 T + T^{2} )^{2} \)
$41$ \( ( -221760 + T^{2} )^{2} \)
$43$ \( ( 244 + T )^{4} \)
$47$ \( 3073593600 + 55440 T^{2} + T^{4} \)
$53$ \( 175783140225 + 419265 T^{2} + T^{4} \)
$59$ \( 136888900225 + 369985 T^{2} + T^{4} \)
$61$ \( ( 345744 - 588 T + T^{2} )^{2} \)
$67$ \( ( 91204 - 302 T + T^{2} )^{2} \)
$71$ \( ( -24640 + T^{2} )^{2} \)
$73$ \( ( 198916 - 446 T + T^{2} )^{2} \)
$79$ \( ( 71289 - 267 T + T^{2} )^{2} \)
$83$ \( ( -527065 + T^{2} )^{2} \)
$89$ \( 192099600 + 13860 T^{2} + T^{4} \)
$97$ \( ( -595 + T )^{4} \)
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