# Properties

 Label 112.14.a.d Level $112$ Weight $14$ Character orbit 112.a Self dual yes Analytic conductor $120.099$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$120.098640426$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{100129})$$ Defining polynomial: $$x^{2} - x - 25032$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{100129}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -476 - \beta ) q^{3} + ( 16002 + 63 \beta ) q^{5} -117649 q^{7} + ( -967231 + 952 \beta ) q^{9} +O(q^{10})$$ $$q + ( -476 - \beta ) q^{3} + ( 16002 + 63 \beta ) q^{5} -117649 q^{7} + ( -967231 + 952 \beta ) q^{9} + ( 676368 - 11088 \beta ) q^{11} + ( 1755194 - 50895 \beta ) q^{13} + ( -32849460 - 45990 \beta ) q^{15} + ( 108855978 + 128286 \beta ) q^{17} + ( -295667876 + 114129 \beta ) q^{19} + ( 56000924 + 117649 \beta ) q^{21} + ( -420367500 + 1099350 \beta ) q^{23} + ( 625008883 + 2016252 \beta ) q^{25} + ( 838008472 + 2108402 \beta ) q^{27} + ( -243811770 - 2389086 \beta ) q^{29} + ( -1096538072 - 3506274 \beta ) q^{31} + ( 4118970240 + 4601520 \beta ) q^{33} + ( -1882619298 - 7411887 \beta ) q^{35} + ( 202530134 - 9857106 \beta ) q^{37} + ( 19548789476 + 22470826 \beta ) q^{39} + ( 4259086314 + 51860862 \beta ) q^{41} + ( -13112522648 + 36481536 \beta ) q^{43} + ( 8543717154 - 45701649 \beta ) q^{45} + ( -77524424880 - 23291442 \beta ) q^{47} + 13841287201 q^{49} + ( -103196041104 - 169920114 \beta ) q^{51} + ( 33003525246 - 339839892 \beta ) q^{53} + ( -268954807968 - 134818992 \beta ) q^{55} + ( 95027418412 + 241342472 \beta ) q^{57} + ( 238181148492 - 225693891 \beta ) q^{59} + ( 98689425002 - 401532849 \beta ) q^{61} + ( 113793759919 - 112001848 \beta ) q^{63} + ( -1256121880272 - 703844568 \beta ) q^{65} + ( 859366429744 - 840313278 \beta ) q^{67} + ( -240212334600 - 102923100 \beta ) q^{69} + ( 347771739168 + 1823602032 \beta ) q^{71} + ( -233042619670 + 1429760268 \beta ) q^{73} + ( -1105045414340 - 1584744835 \beta ) q^{75} + ( -79574018832 + 1304492112 \beta ) q^{77} + ( 1216008287920 + 2547097812 \beta ) q^{79} + ( 298737861509 - 3359403320 \beta ) q^{81} + ( 871992247308 + 1600755741 \beta ) q^{83} + ( 4978890881244 + 8910759186 \beta ) q^{85} + ( 1072921570896 + 1381016706 \beta ) q^{87} + ( 1511290120242 - 2399150268 \beta ) q^{89} + ( -206496818906 + 5987745855 \beta ) q^{91} + ( 1926270959656 + 2765524496 \beta ) q^{93} + ( -1851516446220 - 16800783930 \beta ) q^{95} + ( 3880031330546 - 4519008846 \beta ) q^{97} + ( -4881961277424 + 11368559664 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 952 q^{3} + 32004 q^{5} - 235298 q^{7} - 1934462 q^{9} + O(q^{10})$$ $$2 q - 952 q^{3} + 32004 q^{5} - 235298 q^{7} - 1934462 q^{9} + 1352736 q^{11} + 3510388 q^{13} - 65698920 q^{15} + 217711956 q^{17} - 591335752 q^{19} + 112001848 q^{21} - 840735000 q^{23} + 1250017766 q^{25} + 1676016944 q^{27} - 487623540 q^{29} - 2193076144 q^{31} + 8237940480 q^{33} - 3765238596 q^{35} + 405060268 q^{37} + 39097578952 q^{39} + 8518172628 q^{41} - 26225045296 q^{43} + 17087434308 q^{45} - 155048849760 q^{47} + 27682574402 q^{49} - 206392082208 q^{51} + 66007050492 q^{53} - 537909615936 q^{55} + 190054836824 q^{57} + 476362296984 q^{59} + 197378850004 q^{61} + 227587519838 q^{63} - 2512243760544 q^{65} + 1718732859488 q^{67} - 480424669200 q^{69} + 695543478336 q^{71} - 466085239340 q^{73} - 2210090828680 q^{75} - 159148037664 q^{77} + 2432016575840 q^{79} + 597475723018 q^{81} + 1743984494616 q^{83} + 9957781762488 q^{85} + 2145843141792 q^{87} + 3022580240484 q^{89} - 412993637812 q^{91} + 3852541919312 q^{93} - 3703032892440 q^{95} + 7760062661092 q^{97} - 9763922554848 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 158.716 −157.716
0 −1108.86 0 55872.4 0 −117649. 0 −364745. 0
1.2 0 156.863 0 −23868.4 0 −117649. 0 −1.56972e6 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.14.a.d 2
4.b odd 2 1 14.14.a.c 2
12.b even 2 1 126.14.a.l 2
28.d even 2 1 98.14.a.e 2
28.f even 6 2 98.14.c.m 4
28.g odd 6 2 98.14.c.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.c 2 4.b odd 2 1
98.14.a.e 2 28.d even 2 1
98.14.c.l 4 28.g odd 6 2
98.14.c.m 4 28.f even 6 2
112.14.a.d 2 1.a even 1 1 trivial
126.14.a.l 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 952 T_{3} - 173940$$ acting on $$S_{14}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-173940 + 952 T + T^{2}$$
$5$ $$-1333584000 - 32004 T + T^{2}$$
$7$ $$( 117649 + T )^{2}$$
$11$ $$-48783462900480 - 1352736 T + T^{2}$$
$13$ $$-1034376299351264 - 3510388 T + T^{2}$$
$17$ $$5258212862273748 - 217711956 T + T^{2}$$
$19$ $$82202600320772620 + 591335752 T + T^{2}$$
$23$ $$-307342956281760000 + 840735000 T + T^{2}$$
$29$ $$-2226593776636211436 + 487623540 T + T^{2}$$
$31$ $$-3721530883884270032 + 2193076144 T + T^{2}$$
$37$ $$-38874132892883083820 - 405060268 T + T^{2}$$
$41$ $$-$$$$10\!\cdots\!08$$$$- 8518172628 T + T^{2}$$
$43$ $$-$$$$36\!\cdots\!32$$$$+ 26225045296 T + T^{2}$$
$47$ $$57\!\cdots\!76$$$$+ 155048849760 T + T^{2}$$
$53$ $$-$$$$45\!\cdots\!08$$$$- 66007050492 T + T^{2}$$
$59$ $$36\!\cdots\!68$$$$- 476362296984 T + T^{2}$$
$61$ $$-$$$$54\!\cdots\!12$$$$- 197378850004 T + T^{2}$$
$67$ $$45\!\cdots\!92$$$$- 1718732859488 T + T^{2}$$
$71$ $$-$$$$12\!\cdots\!60$$$$- 695543478336 T + T^{2}$$
$73$ $$-$$$$76\!\cdots\!84$$$$+ 466085239340 T + T^{2}$$
$79$ $$-$$$$11\!\cdots\!04$$$$- 2432016575840 T + T^{2}$$
$83$ $$-$$$$26\!\cdots\!32$$$$- 1743984494616 T + T^{2}$$
$89$ $$-$$$$21\!\cdots\!20$$$$- 3022580240484 T + T^{2}$$
$97$ $$68\!\cdots\!60$$$$- 7760062661092 T + T^{2}$$