Properties

 Label 112.12.a.d Level $112$ Weight $12$ Character orbit 112.a Self dual yes Analytic conductor $86.054$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$86.0544362227$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3369})$$ Defining polynomial: $$x^{2} - x - 842$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3369}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -60 - 3 \beta ) q^{3} + ( -6750 - 5 \beta ) q^{5} -16807 q^{7} + ( -52263 + 360 \beta ) q^{9} +O(q^{10})$$ $$q + ( -60 - 3 \beta ) q^{3} + ( -6750 - 5 \beta ) q^{5} -16807 q^{7} + ( -52263 + 360 \beta ) q^{9} + ( 375408 + 2080 \beta ) q^{11} + ( -4774 - 14931 \beta ) q^{13} + ( 607140 + 20550 \beta ) q^{15} + ( 2080026 + 1550 \beta ) q^{17} + ( 8999356 - 39069 \beta ) q^{19} + ( 1008420 + 50421 \beta ) q^{21} + ( 33080508 - 19750 \beta ) q^{23} + ( -2928725 + 67500 \beta ) q^{25} + ( -789480 + 666630 \beta ) q^{27} + ( 30757806 + 964026 \beta ) q^{29} + ( 7640776 - 685422 \beta ) q^{31} + ( -106614720 - 1251024 \beta ) q^{33} + ( 113447250 + 84035 \beta ) q^{35} + ( -263609170 + 2849094 \beta ) q^{37} + ( 603916908 + 910182 \beta ) q^{39} + ( -89138070 + 615678 \beta ) q^{41} + ( -913372616 + 4593456 \beta ) q^{43} + ( 328518450 - 2168685 \beta ) q^{45} + ( -284120352 + 19068194 \beta ) q^{47} + 282475249 q^{49} + ( -187464960 - 6333078 \beta ) q^{51} + ( -2092908186 + 32472572 \beta ) q^{53} + ( -2674154400 - 15917040 \beta ) q^{55} + ( 1039520172 - 24653928 \beta ) q^{57} + ( -1555672500 - 52235585 \beta ) q^{59} + ( 7521297530 - 28453437 \beta ) q^{61} + ( 878384241 - 6050520 \beta ) q^{63} + ( 1038275280 + 100808120 \beta ) q^{65} + ( -4928261984 + 101504502 \beta ) q^{67} + ( -1186377480 - 98056524 \beta ) q^{69} + ( 12156005664 + 30465456 \beta ) q^{71} + ( -15445000966 - 99592308 \beta ) q^{73} + ( -2553166500 + 4736175 \beta ) q^{75} + ( -6309482256 - 34958560 \beta ) q^{77} + ( -996402128 - 217493748 \beta ) q^{79} + ( -17644915179 - 101402280 \beta ) q^{81} + ( -2638507284 - 167491737 \beta ) q^{83} + ( -14144614500 - 20862630 \beta ) q^{85} + ( -40819111488 - 150114978 \beta ) q^{87} + ( -50770656414 + 195208036 \beta ) q^{89} + ( 80236618 + 250945317 \beta ) q^{91} + ( 27251794056 + 18202992 \beta ) q^{93} + ( -58113183780 + 218718970 \beta ) q^{95} + ( -96114310558 - 101683134 \beta ) q^{97} + ( -9529119504 + 26439840 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9} + O(q^{10})$$ $$2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9} + 750816 q^{11} - 9548 q^{13} + 1214280 q^{15} + 4160052 q^{17} + 17998712 q^{19} + 2016840 q^{21} + 66161016 q^{23} - 5857450 q^{25} - 1578960 q^{27} + 61515612 q^{29} + 15281552 q^{31} - 213229440 q^{33} + 226894500 q^{35} - 527218340 q^{37} + 1207833816 q^{39} - 178276140 q^{41} - 1826745232 q^{43} + 657036900 q^{45} - 568240704 q^{47} + 564950498 q^{49} - 374929920 q^{51} - 4185816372 q^{53} - 5348308800 q^{55} + 2079040344 q^{57} - 3111345000 q^{59} + 15042595060 q^{61} + 1756768482 q^{63} + 2076550560 q^{65} - 9856523968 q^{67} - 2372754960 q^{69} + 24312011328 q^{71} - 30890001932 q^{73} - 5106333000 q^{75} - 12618964512 q^{77} - 1992804256 q^{79} - 35289830358 q^{81} - 5277014568 q^{83} - 28289229000 q^{85} - 81638222976 q^{87} - 101541312828 q^{89} + 160473236 q^{91} + 54503588112 q^{93} - 116226367560 q^{95} - 192228621116 q^{97} - 19058239008 q^{99} + O(q^{100})$$

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
 29.5215 −28.5215
0 −408.259 0 −7330.43 0 −16807.0 0 −10472.0 0
1.2 0 288.259 0 −6169.57 0 −16807.0 0 −94054.0 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 112.12.a.d 2
4.b odd 2 1 7.12.a.a 2
12.b even 2 1 63.12.a.c 2
20.d odd 2 1 175.12.a.a 2
20.e even 4 2 175.12.b.a 4
28.d even 2 1 49.12.a.c 2
28.f even 6 2 49.12.c.e 4
28.g odd 6 2 49.12.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.a.a 2 4.b odd 2 1
49.12.a.c 2 28.d even 2 1
49.12.c.d 4 28.g odd 6 2
49.12.c.e 4 28.f even 6 2
63.12.a.c 2 12.b even 2 1
112.12.a.d 2 1.a even 1 1 trivial
175.12.a.a 2 20.d odd 2 1
175.12.b.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 120 T_{3} - 117684$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(112))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-117684 + 120 T + T^{2}$$
$5$ $$45225600 + 13500 T + T^{2}$$
$7$ $$( 16807 + T )^{2}$$
$11$ $$82628600064 - 750816 T + T^{2}$$
$13$ $$-3004246048160 + 9548 T + T^{2}$$
$17$ $$4294132070676 - 4160052 T + T^{2}$$
$19$ $$60418820423500 - 17998712 T + T^{2}$$
$23$ $$1089063527288064 - 66161016 T + T^{2}$$
$29$ $$-11577825800104140 - 61515612 T + T^{2}$$
$31$ $$-6272688056617808 - 15281552 T + T^{2}$$
$37$ $$-39899433794297036 + 527218340 T + T^{2}$$
$41$ $$2837391053183316 + 178276140 T + T^{2}$$
$43$ $$549908118448121920 + 1826745232 T + T^{2}$$
$47$ $$-4819095623733362832 + 568240704 T + T^{2}$$
$53$ $$-9829745180584088988 + 4185816372 T + T^{2}$$
$59$ $$-34349908314521774100 + 3111345000 T + T^{2}$$
$61$ $$45659772847609730656 - 15042595060 T + T^{2}$$
$67$ $$-$$$$11\!\cdots\!48$$$$+ 9856523968 T + T^{2}$$
$71$ $$13\!\cdots\!60$$$$- 24312011328 T + T^{2}$$
$73$ $$10\!\cdots\!92$$$$+ 30890001932 T + T^{2}$$
$79$ $$-$$$$63\!\cdots\!20$$$$+ 1992804256 T + T^{2}$$
$83$ $$-$$$$37\!\cdots\!88$$$$+ 5277014568 T + T^{2}$$
$89$ $$20\!\cdots\!00$$$$+ 101541312828 T + T^{2}$$
$97$ $$90\!\cdots\!08$$$$+ 192228621116 T + T^{2}$$