# Properties

 Label 252.4.k.c Level $252$ Weight $4$ Character orbit 252.k Analytic conductor $14.868$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{-3}, \sqrt{37})$$ Defining polynomial: $$x^{4} - x^{3} + 10 x^{2} + 9 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 28) 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.

 $$f(q)$$ $$=$$ $$q + ( -7 \beta_{1} - 2 \beta_{2} ) q^{5} + ( 10 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -7 \beta_{1} - 2 \beta_{2} ) q^{5} + ( 10 - 8 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} + ( 16 - 16 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{11} + ( -14 - 4 \beta_{3} ) q^{13} + ( -77 + 77 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 112 \beta_{1} - 3 \beta_{2} ) q^{19} + ( 34 \beta_{1} - 7 \beta_{2} ) q^{23} + ( -72 + 72 \beta_{1} + 28 \beta_{2} + 28 \beta_{3} ) q^{25} + ( 118 + 28 \beta_{3} ) q^{29} + ( 98 - 98 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{31} + ( -204 + 208 \beta_{1} + 17 \beta_{2} + 30 \beta_{3} ) q^{35} + ( -173 \beta_{1} - 42 \beta_{2} ) q^{37} + ( 210 + 12 \beta_{3} ) q^{41} -172 q^{43} + ( 42 \beta_{1} - 15 \beta_{2} ) q^{47} + ( -75 + 200 \beta_{1} + 8 \beta_{2} + 52 \beta_{3} ) q^{49} + ( -219 + 219 \beta_{1} + 42 \beta_{2} + 42 \beta_{3} ) q^{53} + ( 406 - 17 \beta_{3} ) q^{55} + ( -28 + 28 \beta_{1} - 37 \beta_{2} - 37 \beta_{3} ) q^{59} + ( -49 \beta_{1} + 74 \beta_{2} ) q^{61} -198 \beta_{1} q^{65} + ( -168 + 168 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{67} + ( -448 + 56 \beta_{3} ) q^{71} + ( -483 + 483 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} ) q^{73} + ( 809 - 419 \beta_{1} + 86 \beta_{2} + 62 \beta_{3} ) q^{77} + ( 26 \beta_{1} - 133 \beta_{2} ) q^{79} + ( -196 - 88 \beta_{3} ) q^{83} + ( 835 - 182 \beta_{3} ) q^{85} + ( 147 \beta_{1} + 60 \beta_{2} ) q^{89} + ( -288 - 184 \beta_{1} - 4 \beta_{2} - 54 \beta_{3} ) q^{91} + ( 562 - 562 \beta_{1} - 203 \beta_{2} - 203 \beta_{3} ) q^{95} + ( -210 + 76 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 14q^{5} + 24q^{7} + O(q^{10})$$ $$4q - 14q^{5} + 24q^{7} + 32q^{11} - 56q^{13} - 154q^{17} + 224q^{19} + 68q^{23} - 144q^{25} + 472q^{29} + 196q^{31} - 400q^{35} - 346q^{37} + 840q^{41} - 688q^{43} + 84q^{47} + 100q^{49} - 438q^{53} + 1624q^{55} - 56q^{59} - 98q^{61} - 396q^{65} - 336q^{67} - 1792q^{71} - 966q^{73} + 2398q^{77} + 52q^{79} - 784q^{83} + 3340q^{85} + 294q^{89} - 1520q^{91} + 1124q^{95} - 840q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 10 x^{2} + 9 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu^{2} - 10 \nu + 81$$$$)/90$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 10 \nu^{2} + 190 \nu - 81$$$$)/90$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 19 \beta_{1} - 19$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 14$$

## 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
 1.77069 − 3.06693i −1.27069 + 2.20090i 1.77069 + 3.06693i −1.27069 − 2.20090i
0 0 0 −9.58276 + 16.5978i 0 −6.16553 + 17.4639i 0 0 0
37.2 0 0 0 2.58276 4.47348i 0 18.1655 3.60745i 0 0 0
109.1 0 0 0 −9.58276 16.5978i 0 −6.16553 17.4639i 0 0 0
109.2 0 0 0 2.58276 + 4.47348i 0 18.1655 + 3.60745i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.k.c 4
3.b odd 2 1 28.4.e.a 4
7.b odd 2 1 1764.4.k.ba 4
7.c even 3 1 inner 252.4.k.c 4
7.c even 3 1 1764.4.a.z 2
7.d odd 6 1 1764.4.a.n 2
7.d odd 6 1 1764.4.k.ba 4
12.b even 2 1 112.4.i.d 4
15.d odd 2 1 700.4.i.g 4
15.e even 4 2 700.4.r.d 8
21.c even 2 1 196.4.e.g 4
21.g even 6 1 196.4.a.g 2
21.g even 6 1 196.4.e.g 4
21.h odd 6 1 28.4.e.a 4
21.h odd 6 1 196.4.a.e 2
24.f even 2 1 448.4.i.g 4
24.h odd 2 1 448.4.i.h 4
84.j odd 6 1 784.4.a.ba 2
84.n even 6 1 112.4.i.d 4
84.n even 6 1 784.4.a.u 2
105.o odd 6 1 700.4.i.g 4
105.x even 12 2 700.4.r.d 8
168.s odd 6 1 448.4.i.h 4
168.v even 6 1 448.4.i.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.e.a 4 3.b odd 2 1
28.4.e.a 4 21.h odd 6 1
112.4.i.d 4 12.b even 2 1
112.4.i.d 4 84.n even 6 1
196.4.a.e 2 21.h odd 6 1
196.4.a.g 2 21.g even 6 1
196.4.e.g 4 21.c even 2 1
196.4.e.g 4 21.g even 6 1
252.4.k.c 4 1.a even 1 1 trivial
252.4.k.c 4 7.c even 3 1 inner
448.4.i.g 4 24.f even 2 1
448.4.i.g 4 168.v even 6 1
448.4.i.h 4 24.h odd 2 1
448.4.i.h 4 168.s odd 6 1
700.4.i.g 4 15.d odd 2 1
700.4.i.g 4 105.o odd 6 1
700.4.r.d 8 15.e even 4 2
700.4.r.d 8 105.x even 12 2
784.4.a.u 2 84.n even 6 1
784.4.a.ba 2 84.j odd 6 1
1764.4.a.n 2 7.d odd 6 1
1764.4.a.z 2 7.c even 3 1
1764.4.k.ba 4 7.b odd 2 1
1764.4.k.ba 4 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{3} + 295 T_{5}^{2} - 1386 T_{5} + 9801$$ $$T_{13}^{2} + 28 T_{13} - 396$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$9801 - 1386 T + 295 T^{2} + 14 T^{3} + T^{4}$$
$7$ $$117649 - 8232 T + 238 T^{2} - 24 T^{3} + T^{4}$$
$11$ $$2424249 + 49824 T + 2581 T^{2} - 32 T^{3} + T^{4}$$
$13$ $$( -396 + 28 T + T^{2} )^{2}$$
$17$ $$28483569 + 821898 T + 18379 T^{2} + 154 T^{3} + T^{4}$$
$19$ $$149108521 - 2735264 T + 37965 T^{2} - 224 T^{3} + T^{4}$$
$23$ $$431649 + 44676 T + 5281 T^{2} - 68 T^{3} + T^{4}$$
$29$ $$( -15084 - 236 T + T^{2} )^{2}$$
$31$ $$60699681 - 1527036 T + 30625 T^{2} - 196 T^{3} + T^{4}$$
$37$ $$1248844921 - 12227294 T + 155055 T^{2} + 346 T^{3} + T^{4}$$
$41$ $$( 38772 - 420 T + T^{2} )^{2}$$
$43$ $$( 172 + T )^{4}$$
$47$ $$43046721 + 551124 T + 13617 T^{2} - 84 T^{3} + T^{4}$$
$53$ $$299532249 - 7580466 T + 209151 T^{2} + 438 T^{3} + T^{4}$$
$59$ $$2486917161 - 2792664 T + 53005 T^{2} + 56 T^{3} + T^{4}$$
$61$ $$40084444521 - 19620678 T + 209815 T^{2} + 98 T^{3} + T^{4}$$
$67$ $$697540921 + 8874096 T + 86485 T^{2} + 336 T^{3} + T^{4}$$
$71$ $$( 84672 + 896 T + T^{2} )^{2}$$
$73$ $$47737443121 + 211060374 T + 714667 T^{2} + 966 T^{3} + T^{4}$$
$79$ $$427476669489 + 33998484 T + 656521 T^{2} - 52 T^{3} + T^{4}$$
$83$ $$( -248112 + 392 T + T^{2} )^{2}$$
$89$ $$12452551281 + 32807754 T + 198027 T^{2} - 294 T^{3} + T^{4}$$
$97$ $$( -169612 + 420 T + T^{2} )^{2}$$