# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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