# Properties

 Label 252.4.k.d 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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{7} + ( 1 - \beta_{1} + 26 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 28 - 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{13} + ( -8 + 8 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} ) q^{17} + ( -85 + 2 \beta_{1} - 84 \beta_{2} - \beta_{3} ) q^{19} + ( -96 - 96 \beta_{2} ) q^{23} + ( 3 - 3 \beta_{1} + 7 \beta_{2} + 6 \beta_{3} ) q^{25} + ( 11 - 17 \beta_{1} - 17 \beta_{2} - 17 \beta_{3} ) q^{29} + ( -10 + 10 \beta_{1} - 51 \beta_{2} - 20 \beta_{3} ) q^{31} + ( -35 - 7 \beta_{1} - 217 \beta_{2} - 7 \beta_{3} ) q^{35} + ( 77 + 38 \beta_{1} + 96 \beta_{2} - 19 \beta_{3} ) q^{37} + ( -104 - 34 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} ) q^{41} + ( -258 - 19 \beta_{1} - 19 \beta_{2} - 19 \beta_{3} ) q^{43} + ( 98 - 32 \beta_{1} + 82 \beta_{2} + 16 \beta_{3} ) q^{47} + ( 117 + 25 \beta_{1} + 312 \beta_{2} + 15 \beta_{3} ) q^{49} + ( -27 + 27 \beta_{1} - 156 \beta_{2} - 54 \beta_{3} ) q^{53} + ( 153 - 27 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} ) q^{55} + ( 3 - 3 \beta_{1} + 636 \beta_{2} + 6 \beta_{3} ) q^{59} + ( -146 - 72 \beta_{1} - 182 \beta_{2} + 36 \beta_{3} ) q^{61} + ( -706 + 76 \beta_{1} - 668 \beta_{2} - 38 \beta_{3} ) q^{65} + ( 9 - 9 \beta_{1} - 433 \beta_{2} + 18 \beta_{3} ) q^{67} + ( 718 + 32 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{71} + ( -21 + 21 \beta_{1} - 691 \beta_{2} - 42 \beta_{3} ) q^{73} + ( -278 + 38 \beta_{1} - 109 \beta_{2} + 41 \beta_{3} ) q^{77} + ( 93 - 8 \beta_{1} + 89 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 221 + 43 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{83} + ( -1032 + 24 \beta_{1} + 24 \beta_{2} + 24 \beta_{3} ) q^{85} + ( 400 + 44 \beta_{1} + 422 \beta_{2} - 22 \beta_{3} ) q^{89} + ( 1039 - 104 \beta_{1} + 1002 \beta_{2} + 93 \beta_{3} ) q^{91} + ( 86 - 86 \beta_{1} + 298 \beta_{2} + 172 \beta_{3} ) q^{95} + ( 431 + 21 \beta_{1} + 21 \beta_{2} + 21 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{5} - 20q^{7} + O(q^{10})$$ $$4q - 3q^{5} - 20q^{7} - 51q^{11} + 122q^{13} + 24q^{17} - 169q^{19} - 192q^{23} - 11q^{25} + 78q^{29} + 92q^{31} + 294q^{35} + 173q^{37} - 348q^{41} - 994q^{43} + 180q^{47} - 146q^{49} + 285q^{53} + 666q^{55} - 1269q^{59} - 328q^{61} - 1374q^{65} + 875q^{67} + 2808q^{71} + 1361q^{73} - 897q^{77} + 182q^{79} + 798q^{83} - 4176q^{85} + 822q^{89} + 1955q^{91} - 510q^{95} + 1682q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} + 56 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 25$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{3} + 2 \nu^{2} + 28 \nu + 35$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 14 \beta_{2} + 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 4 \beta_{1} + 19$$$$)/3$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 0 0 −6.41238 + 11.1066i 0 6.32475 17.4068i 0 0 0
37.2 0 0 0 4.91238 8.50848i 0 −16.3248 + 8.74657i 0 0 0
109.1 0 0 0 −6.41238 11.1066i 0 6.32475 + 17.4068i 0 0 0
109.2 0 0 0 4.91238 + 8.50848i 0 −16.3248 8.74657i 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.d 4
3.b odd 2 1 84.4.i.b 4
7.b odd 2 1 1764.4.k.z 4
7.c even 3 1 inner 252.4.k.d 4
7.c even 3 1 1764.4.a.x 2
7.d odd 6 1 1764.4.a.p 2
7.d odd 6 1 1764.4.k.z 4
12.b even 2 1 336.4.q.h 4
21.c even 2 1 588.4.i.i 4
21.g even 6 1 588.4.a.h 2
21.g even 6 1 588.4.i.i 4
21.h odd 6 1 84.4.i.b 4
21.h odd 6 1 588.4.a.g 2
84.j odd 6 1 2352.4.a.bp 2
84.n even 6 1 336.4.q.h 4
84.n even 6 1 2352.4.a.cb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 3.b odd 2 1
84.4.i.b 4 21.h odd 6 1
252.4.k.d 4 1.a even 1 1 trivial
252.4.k.d 4 7.c even 3 1 inner
336.4.q.h 4 12.b even 2 1
336.4.q.h 4 84.n even 6 1
588.4.a.g 2 21.h odd 6 1
588.4.a.h 2 21.g even 6 1
588.4.i.i 4 21.c even 2 1
588.4.i.i 4 21.g even 6 1
1764.4.a.p 2 7.d odd 6 1
1764.4.a.x 2 7.c even 3 1
1764.4.k.z 4 7.b odd 2 1
1764.4.k.z 4 7.d odd 6 1
2352.4.a.bp 2 84.j odd 6 1
2352.4.a.cb 2 84.n even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 3 T_{5}^{3} + 135 T_{5}^{2} - 378 T_{5} + 15876$$ $$T_{13}^{2} - 61 T_{13} - 2276$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$15876 - 378 T + 135 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$117649 + 6860 T + 273 T^{2} + 20 T^{3} + T^{4}$$
$11$ $$272484 + 26622 T + 2079 T^{2} + 51 T^{3} + T^{4}$$
$13$ $$( -2276 - 61 T + T^{2} )^{2}$$
$17$ $$65028096 + 193536 T + 8640 T^{2} - 24 T^{3} + T^{4}$$
$19$ $$49168144 + 1185028 T + 21549 T^{2} + 169 T^{3} + T^{4}$$
$23$ $$( 9216 + 96 T + T^{2} )^{2}$$
$29$ $$( -36684 - 39 T + T^{2} )^{2}$$
$31$ $$114682681 + 985228 T + 19173 T^{2} - 92 T^{3} + T^{4}$$
$37$ $$1506681856 + 6715168 T + 68745 T^{2} - 173 T^{3} + T^{4}$$
$41$ $$( -140688 + 174 T + T^{2} )^{2}$$
$43$ $$( 15454 + 497 T + T^{2} )^{2}$$
$47$ $$611671824 + 4451760 T + 57132 T^{2} - 180 T^{3} + T^{4}$$
$53$ $$5356483344 + 20858580 T + 154413 T^{2} - 285 T^{3} + T^{4}$$
$59$ $$161150862096 + 509422284 T + 1208925 T^{2} + 1269 T^{3} + T^{4}$$
$61$ $$19408947856 - 45695648 T + 246900 T^{2} + 328 T^{3} + T^{4}$$
$67$ $$32767516324 - 158390750 T + 584607 T^{2} - 875 T^{3} + T^{4}$$
$71$ $$( 361476 - 1404 T + T^{2} )^{2}$$
$73$ $$165260136484 - 553276442 T + 1445799 T^{2} - 1361 T^{3} + T^{4}$$
$79$ $$38800441 - 1133678 T + 26895 T^{2} - 182 T^{3} + T^{4}$$
$83$ $$( -197334 - 399 T + T^{2} )^{2}$$
$89$ $$11416495104 - 87829056 T + 568836 T^{2} - 822 T^{3} + T^{4}$$
$97$ $$( 120262 - 841 T + T^{2} )^{2}$$