# Properties

 Label 252.4.b.d Level $252$ Weight $4$ Character orbit 252.b Analytic conductor $14.868$ Analytic rank $0$ Dimension $8$ 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 30 x^{6} + 84 x^{5} + 493 x^{4} - 464 x^{3} - 3172 x^{2} + 1072 x + 8978$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + \beta_{7} q^{5} + ( 3 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} + ( 4 + 4 \beta_{2} + 4 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + \beta_{7} q^{5} + ( 3 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} + ( 4 + 4 \beta_{2} + 4 \beta_{5} ) q^{8} + ( -\beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{10} + ( 3 \beta_{2} + \beta_{4} - 5 \beta_{5} ) q^{11} + ( -2 \beta_{1} + 4 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{13} + ( 19 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{14} + ( 8 - 4 \beta_{2} - 12 \beta_{4} + 4 \beta_{5} ) q^{16} + ( -2 \beta_{1} + 4 \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{5} - 2 \beta_{6} ) q^{19} + ( 8 \beta_{1} + 4 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{20} + ( 30 + 2 \beta_{2} + 8 \beta_{4} - 12 \beta_{5} ) q^{22} + ( -21 \beta_{2} - 7 \beta_{4} - 19 \beta_{5} ) q^{23} + ( -59 + 19 \beta_{2} - 19 \beta_{4} ) q^{25} + ( 23 \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{26} + ( -6 - 16 \beta_{1} - 25 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} ) q^{28} + ( 74 + 18 \beta_{2} - 18 \beta_{4} ) q^{29} + ( 6 \beta_{1} - 3 \beta_{5} + 6 \beta_{6} ) q^{31} + ( -144 - 32 \beta_{2} - 16 \beta_{4} ) q^{32} + ( 24 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{34} + ( 7 \beta_{1} - 51 \beta_{2} - 17 \beta_{4} + 28 \beta_{5} + 4 \beta_{6} ) q^{35} + ( 174 + 14 \beta_{2} - 14 \beta_{4} ) q^{37} + ( 7 \beta_{1} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{38} + ( 44 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} - 12 \beta_{6} ) q^{40} + ( -4 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} ) q^{41} + ( 3 \beta_{2} + \beta_{4} + 25 \beta_{5} ) q^{43} + ( 148 - 14 \beta_{2} + 30 \beta_{4} - 18 \beta_{5} ) q^{44} + ( -102 - 14 \beta_{2} + 52 \beta_{4} - 24 \beta_{5} ) q^{46} + ( 6 \beta_{1} + 7 \beta_{5} - 14 \beta_{6} ) q^{47} + ( 185 - 6 \beta_{1} - 26 \beta_{2} + 12 \beta_{3} + 26 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 10 \beta_{7} ) q^{49} + ( -211 + 21 \beta_{2} - 38 \beta_{4} - 38 \beta_{5} ) q^{50} + ( 16 \beta_{1} - 20 \beta_{3} + 3 \beta_{5} - 26 \beta_{6} + 18 \beta_{7} ) q^{52} + ( 146 - 76 \beta_{2} + 76 \beta_{4} ) q^{53} + ( -48 \beta_{1} - 13 \beta_{5} + 26 \beta_{6} ) q^{55} + ( -100 + 4 \beta_{1} + 20 \beta_{3} + 36 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} + 16 \beta_{7} ) q^{56} + ( -70 - 110 \beta_{2} - 36 \beta_{4} - 36 \beta_{5} ) q^{58} + ( 29 \beta_{1} - 15 \beta_{5} + 30 \beta_{6} ) q^{59} + ( 8 \beta_{1} - 16 \beta_{3} - 4 \beta_{5} - 8 \beta_{6} - 39 \beta_{7} ) q^{61} + ( -12 \beta_{1} - 12 \beta_{3} - 6 \beta_{5} + 12 \beta_{7} ) q^{62} + ( -416 + 112 \beta_{2} + 16 \beta_{4} + 16 \beta_{5} ) q^{64} + ( -56 + 127 \beta_{2} - 127 \beta_{4} ) q^{65} + ( -141 \beta_{2} - 47 \beta_{4} - 11 \beta_{5} ) q^{67} + ( 8 \beta_{1} - 24 \beta_{3} - 24 \beta_{6} + 16 \beta_{7} ) q^{68} + ( -400 - 5 \beta_{1} - 34 \beta_{2} - 11 \beta_{3} - 26 \beta_{4} + 90 \beta_{5} - 3 \beta_{6} + 8 \beta_{7} ) q^{70} + ( 126 \beta_{2} + 42 \beta_{4} + 7 \beta_{5} ) q^{71} + ( -6 \beta_{1} + 12 \beta_{3} + 3 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} ) q^{73} + ( 62 - 202 \beta_{2} - 28 \beta_{4} - 28 \beta_{5} ) q^{74} + ( 28 \beta_{1} - 8 \beta_{3} - \beta_{5} - 6 \beta_{6} - 14 \beta_{7} ) q^{76} + ( -46 + 8 \beta_{1} + 9 \beta_{2} - 16 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} - 38 \beta_{7} ) q^{77} + ( 24 \beta_{2} + 8 \beta_{4} - 159 \beta_{5} ) q^{79} + ( 48 \beta_{1} - 32 \beta_{3} + 12 \beta_{5} - 56 \beta_{6} - 24 \beta_{7} ) q^{80} + ( 38 \beta_{1} + 10 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} + 28 \beta_{7} ) q^{82} + ( -119 \beta_{1} - 10 \beta_{5} + 20 \beta_{6} ) q^{83} + ( 128 + 108 \beta_{2} - 108 \beta_{4} ) q^{85} + ( -30 + 2 \beta_{2} - 52 \beta_{4} + 48 \beta_{5} ) q^{86} + ( 472 - 88 \beta_{2} + 80 \beta_{4} + 8 \beta_{5} ) q^{88} + ( -2 \beta_{1} + 4 \beta_{3} + \beta_{5} + 2 \beta_{6} - 70 \beta_{7} ) q^{89} + ( -91 \beta_{1} + 177 \beta_{2} + 59 \beta_{4} + 175 \beta_{5} - 18 \beta_{6} ) q^{91} + ( 476 + 206 \beta_{2} + 114 \beta_{4} + 18 \beta_{5} ) q^{92} + ( 48 \beta_{1} + 8 \beta_{3} + 14 \beta_{5} - 20 \beta_{6} - 28 \beta_{7} ) q^{94} + ( 93 \beta_{2} + 31 \beta_{4} - 74 \beta_{5} ) q^{95} + ( -26 \beta_{1} + 52 \beta_{3} + 13 \beta_{5} + 26 \beta_{6} - 60 \beta_{7} ) q^{97} + ( 393 + 82 \beta_{1} - 133 \beta_{2} - 10 \beta_{3} + 52 \beta_{4} + 36 \beta_{5} + 22 \beta_{6} - 8 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} - 16q^{4} + 32q^{8} + O(q^{10})$$ $$8q + 8q^{2} - 16q^{4} + 32q^{8} + 152q^{14} + 64q^{16} + 240q^{22} - 472q^{25} - 48q^{28} + 592q^{29} - 1152q^{32} + 1392q^{37} + 1184q^{44} - 816q^{46} + 1480q^{49} - 1688q^{50} + 1168q^{53} - 800q^{56} - 560q^{58} - 3328q^{64} - 448q^{65} - 3200q^{70} + 496q^{74} - 368q^{77} + 1024q^{85} - 240q^{86} + 3776q^{88} + 3808q^{92} + 3144q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 30 x^{6} + 84 x^{5} + 493 x^{4} - 464 x^{3} - 3172 x^{2} + 1072 x + 8978$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2107780 \nu^{7} - 28020178 \nu^{6} + 12013496 \nu^{5} + 856630098 \nu^{4} - 1820971716 \nu^{3} - 7812762598 \nu^{2} + 31061346042 \nu + 26653239984$$$$)/ 11759822513$$ $$\beta_{2}$$ $$=$$ $$($$$$-2107780 \nu^{7} + 28020178 \nu^{6} - 12013496 \nu^{5} - 856630098 \nu^{4} + 1820971716 \nu^{3} + 7812762598 \nu^{2} - 7541701016 \nu - 38413062497$$$$)/ 11759822513$$ $$\beta_{3}$$ $$=$$ $$($$$$-5280 \nu^{7} + 27552 \nu^{6} + 648572 \nu^{5} - 3328540 \nu^{4} - 9022980 \nu^{3} + 48793686 \nu^{2} + 78029576 \nu - 204544903$$$$)/7201361$$ $$\beta_{4}$$ $$=$$ $$($$$$37410820 \nu^{7} - 178195062 \nu^{6} - 946363840 \nu^{5} + 4456400722 \nu^{4} + 10555104876 \nu^{3} - 33992527302 \nu^{2} - 34928966104 \nu + 158760439399$$$$)/ 11759822513$$ $$\beta_{5}$$ $$=$$ $$($$$$-71287048 \nu^{7} + 411792664 \nu^{6} + 1067149048 \nu^{5} - 6385173864 \nu^{4} - 14881408792 \nu^{3} + 32020502452 \nu^{2} + 42141065120 \nu - 65434616690$$$$)/ 11759822513$$ $$\beta_{6}$$ $$=$$ $$($$$$85892144 \nu^{7} - 485124166 \nu^{6} - 2515029448 \nu^{5} + 14797753670 \nu^{4} + 29088392180 \nu^{3} - 121788747936 \nu^{2} - 191858039266 \nu + 390648894103$$$$)/ 11759822513$$ $$\beta_{7}$$ $$=$$ $$($$$$-2876 \nu^{7} + 28790 \nu^{6} - 17704 \nu^{5} - 509450 \nu^{4} + 374248 \nu^{3} + 4834142 \nu^{2} - 1479810 \nu - 15131816$$$$)/379019$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 38$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 6 \beta_{6} + 10 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + 79 \beta_{2} + 17 \beta_{1} + 86$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-16 \beta_{7} + 48 \beta_{6} + 83 \beta_{5} + 39 \beta_{4} + 60 \beta_{3} + 305 \beta_{2} + 32 \beta_{1} + 330$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-250 \beta_{7} + 330 \beta_{6} + 965 \beta_{5} + 834 \beta_{4} + 560 \beta_{3} + 3838 \beta_{2} - 68 \beta_{1} + 1764$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-590 \beta_{7} + 754 \beta_{6} + 2877 \beta_{5} + 3229 \beta_{4} + 1576 \beta_{3} + 8337 \beta_{2} - 928 \beta_{1} - 2086$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-6440 \beta_{7} + 4344 \beta_{6} + 28219 \beta_{5} + 39754 \beta_{4} + 15134 \beta_{3} + 73906 \beta_{2} - 20408 \beta_{1} - 63556$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 4.98105 + 1.39897i −2.56684 + 1.39897i −2.56684 − 1.39897i 4.98105 − 1.39897i −2.60656 + 0.736813i 2.19234 + 0.736813i 2.19234 − 0.736813i −2.60656 − 0.736813i
−0.414214 2.79793i 0 −7.65685 + 2.31788i 8.74756i 0 13.8008 + 12.3507i 9.65685 + 20.4633i 0 −24.4751 + 3.62336i
55.2 −0.414214 2.79793i 0 −7.65685 + 2.31788i 8.74756i 0 −13.8008 + 12.3507i 9.65685 + 20.4633i 0 24.4751 3.62336i
55.3 −0.414214 + 2.79793i 0 −7.65685 2.31788i 8.74756i 0 −13.8008 12.3507i 9.65685 20.4633i 0 24.4751 + 3.62336i
55.4 −0.414214 + 2.79793i 0 −7.65685 2.31788i 8.74756i 0 13.8008 12.3507i 9.65685 20.4633i 0 −24.4751 3.62336i
55.5 2.41421 1.47363i 0 3.65685 7.11529i 17.0728i 0 18.3722 + 2.33686i −1.65685 22.5667i 0 −25.1589 41.2174i
55.6 2.41421 1.47363i 0 3.65685 7.11529i 17.0728i 0 −18.3722 + 2.33686i −1.65685 22.5667i 0 25.1589 + 41.2174i
55.7 2.41421 + 1.47363i 0 3.65685 + 7.11529i 17.0728i 0 −18.3722 2.33686i −1.65685 + 22.5667i 0 25.1589 41.2174i
55.8 2.41421 + 1.47363i 0 3.65685 + 7.11529i 17.0728i 0 18.3722 2.33686i −1.65685 + 22.5667i 0 −25.1589 + 41.2174i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.d 8
3.b odd 2 1 28.4.d.b 8
4.b odd 2 1 inner 252.4.b.d 8
7.b odd 2 1 inner 252.4.b.d 8
12.b even 2 1 28.4.d.b 8
21.c even 2 1 28.4.d.b 8
21.g even 6 2 196.4.f.c 16
21.h odd 6 2 196.4.f.c 16
24.f even 2 1 448.4.f.d 8
24.h odd 2 1 448.4.f.d 8
28.d even 2 1 inner 252.4.b.d 8
84.h odd 2 1 28.4.d.b 8
84.j odd 6 2 196.4.f.c 16
84.n even 6 2 196.4.f.c 16
168.e odd 2 1 448.4.f.d 8
168.i even 2 1 448.4.f.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.b 8 3.b odd 2 1
28.4.d.b 8 12.b even 2 1
28.4.d.b 8 21.c even 2 1
28.4.d.b 8 84.h odd 2 1
196.4.f.c 16 21.g even 6 2
196.4.f.c 16 21.h odd 6 2
196.4.f.c 16 84.j odd 6 2
196.4.f.c 16 84.n even 6 2
252.4.b.d 8 1.a even 1 1 trivial
252.4.b.d 8 4.b odd 2 1 inner
252.4.b.d 8 7.b odd 2 1 inner
252.4.b.d 8 28.d even 2 1 inner
448.4.f.d 8 24.f even 2 1
448.4.f.d 8 24.h odd 2 1
448.4.f.d 8 168.e odd 2 1
448.4.f.d 8 168.i even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 368 T_{5}^{2} + 22304$$ $$T_{11}^{4} + 1720 T_{11}^{2} + 272$$ $$T_{19}^{4} - 2128 T_{19}^{2} + 694048$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 64 - 32 T + 12 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 22304 + 368 T^{2} + T^{4} )^{2}$$
$7$ $$13841287201 - 87060260 T^{2} + 285670 T^{4} - 740 T^{6} + T^{8}$$
$11$ $$( 272 + 1720 T^{2} + T^{4} )^{2}$$
$13$ $$( 11798816 + 7792 T^{2} + T^{4} )^{2}$$
$17$ $$( 5709824 + 7936 T^{2} + T^{4} )^{2}$$
$19$ $$( 694048 - 2128 T^{2} + T^{4} )^{2}$$
$23$ $$( 132140048 + 23800 T^{2} + T^{4} )^{2}$$
$29$ $$( -4892 - 148 T + T^{2} )^{4}$$
$31$ $$( 108822528 - 23040 T^{2} + T^{4} )^{2}$$
$37$ $$( 24004 - 348 T + T^{2} )^{4}$$
$41$ $$( 342946304 + 58304 T^{2} + T^{4} )^{2}$$
$43$ $$( 141396752 + 34360 T^{2} + T^{4} )^{2}$$
$47$ $$( 1789861888 - 103680 T^{2} + T^{4} )^{2}$$
$53$ $$( -163516 - 292 T + T^{2} )^{4}$$
$59$ $$( 67995188512 - 570320 T^{2} + T^{4} )^{2}$$
$61$ $$( 6445856 + 606832 T^{2} + T^{4} )^{2}$$
$67$ $$( 12017669648 + 343672 T^{2} + T^{4} )^{2}$$
$71$ $$( 9248152592 + 275576 T^{2} + T^{4} )^{2}$$
$73$ $$( 1798951424 + 92864 T^{2} + T^{4} )^{2}$$
$79$ $$( 108115639568 + 1466680 T^{2} + T^{4} )^{2}$$
$83$ $$( 340971272992 - 1267920 T^{2} + T^{4} )^{2}$$
$89$ $$( 661026681344 + 1846976 T^{2} + T^{4} )^{2}$$
$97$ $$( 1135775350784 + 3065344 T^{2} + T^{4} )^{2}$$