# Properties

 Label 112.10.a.d Level $112$ Weight $10$ Character orbit 112.a Self dual yes Analytic conductor $57.684$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.6840136504$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11209})$$ Defining polynomial: $$x^{2} - x - 2802$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{11209}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 35 - \beta ) q^{3} + ( 777 + 19 \beta ) q^{5} + 2401 q^{7} + ( -7249 - 70 \beta ) q^{9} +O(q^{10})$$ $$q + ( 35 - \beta ) q^{3} + ( 777 + 19 \beta ) q^{5} + 2401 q^{7} + ( -7249 - 70 \beta ) q^{9} + ( -31194 - 182 \beta ) q^{11} + ( 61383 - 87 \beta ) q^{13} + ( -185776 - 112 \beta ) q^{15} + ( 36792 - 2986 \beta ) q^{17} + ( -585599 - 2019 \beta ) q^{19} + ( 84035 - 2401 \beta ) q^{21} + ( -1131192 - 7448 \beta ) q^{23} + ( 2697053 + 29526 \beta ) q^{25} + ( -157990 + 24482 \beta ) q^{27} + ( -961680 - 24878 \beta ) q^{29} + ( -1488942 + 46242 \beta ) q^{31} + ( 948248 + 24824 \beta ) q^{33} + ( 1865577 + 45619 \beta ) q^{35} + ( -6709264 + 136122 \beta ) q^{37} + ( 3123588 - 64428 \beta ) q^{39} + ( -18183900 + 14922 \beta ) q^{41} + ( 10982458 - 52626 \beta ) q^{43} + ( -20540443 - 192121 \beta ) q^{45} + ( 681366 + 167910 \beta ) q^{47} + 5764801 q^{49} + ( 34757794 - 141302 \beta ) q^{51} + ( -8949306 - 702240 \beta ) q^{53} + ( -62998460 - 734100 \beta ) q^{55} + ( 2135006 + 514934 \beta ) q^{57} + ( -112355271 - 312307 \beta ) q^{59} + ( -42923559 - 1314933 \beta ) q^{61} + ( -17404849 - 168070 \beta ) q^{63} + ( 29166114 + 1098678 \beta ) q^{65} + ( -89784436 + 1498728 \beta ) q^{67} + ( 43892912 + 870512 \beta ) q^{69} + ( -115689084 - 30548 \beta ) q^{71} + ( 44049166 + 15252 \beta ) q^{73} + ( -236560079 - 1663643 \beta ) q^{75} + ( -74896794 - 436982 \beta ) q^{77} + ( 92137092 + 2849700 \beta ) q^{79} + ( -137266321 + 2392670 \beta ) q^{81} + ( -312320547 + 3896617 \beta ) q^{83} + ( -607344022 - 1621074 \beta ) q^{85} + ( 245198702 + 90950 \beta ) q^{87} + ( -787388574 + 2093256 \beta ) q^{89} + ( 147380583 - 208887 \beta ) q^{91} + ( -570439548 + 3107412 \beta ) q^{93} + ( -884998872 - 12695144 \beta ) q^{95} + ( 106832992 - 2735970 \beta ) q^{97} + ( 368927966 + 3502898 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 70q^{3} + 1554q^{5} + 4802q^{7} - 14498q^{9} + O(q^{10})$$ $$2q + 70q^{3} + 1554q^{5} + 4802q^{7} - 14498q^{9} - 62388q^{11} + 122766q^{13} - 371552q^{15} + 73584q^{17} - 1171198q^{19} + 168070q^{21} - 2262384q^{23} + 5394106q^{25} - 315980q^{27} - 1923360q^{29} - 2977884q^{31} + 1896496q^{33} + 3731154q^{35} - 13418528q^{37} + 6247176q^{39} - 36367800q^{41} + 21964916q^{43} - 41080886q^{45} + 1362732q^{47} + 11529602q^{49} + 69515588q^{51} - 17898612q^{53} - 125996920q^{55} + 4270012q^{57} - 224710542q^{59} - 85847118q^{61} - 34809698q^{63} + 58332228q^{65} - 179568872q^{67} + 87785824q^{69} - 231378168q^{71} + 88098332q^{73} - 473120158q^{75} - 149793588q^{77} + 184274184q^{79} - 274532642q^{81} - 624641094q^{83} - 1214688044q^{85} + 490397404q^{87} - 1574777148q^{89} + 294761166q^{91} - 1140879096q^{93} - 1769997744q^{95} + 213665984q^{97} + 737855932q^{99} + O(q^{100})$$

## 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
 53.4363 −52.4363
0 −70.8726 0 2788.58 0 2401.00 0 −14660.1 0
1.2 0 140.873 0 −1234.58 0 2401.00 0 162.080 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 112.10.a.d 2
4.b odd 2 1 28.10.a.b 2
12.b even 2 1 252.10.a.b 2
28.d even 2 1 196.10.a.b 2
28.f even 6 2 196.10.e.d 4
28.g odd 6 2 196.10.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.10.a.b 2 4.b odd 2 1
112.10.a.d 2 1.a even 1 1 trivial
196.10.a.b 2 28.d even 2 1
196.10.e.d 4 28.f even 6 2
196.10.e.e 4 28.g odd 6 2
252.10.a.b 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 70 T_{3} - 9984$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-9984 - 70 T + T^{2}$$
$5$ $$-3442720 - 1554 T + T^{2}$$
$7$ $$( -2401 + T )^{2}$$
$11$ $$601778720 + 62388 T + T^{2}$$
$13$ $$3683031768 - 122766 T + T^{2}$$
$17$ $$-98587989700 - 73584 T + T^{2}$$
$19$ $$297234258352 + 1171198 T + T^{2}$$
$23$ $$657801801728 + 2262384 T + T^{2}$$
$29$ $$-6012588512356 + 1923360 T + T^{2}$$
$31$ $$-21751509340512 + 2977884 T + T^{2}$$
$37$ $$-162679566869060 + 13418528 T + T^{2}$$
$41$ $$328158355074444 + 36367800 T + T^{2}$$
$43$ $$89571104447680 - 21964916 T + T^{2}$$
$47$ $$-315559687006944 - 1362732 T + T^{2}$$
$53$ $$-5447527588396764 + 17898612 T + T^{2}$$
$59$ $$11530429683334400 + 224710542 T + T^{2}$$
$61$ $$-17538476020200720 + 85847118 T + T^{2}$$
$67$ $$-17116249644144560 + 179568872 T + T^{2}$$
$71$ $$13373504138731520 + 231378168 T + T^{2}$$
$73$ $$1937721548439220 - 88098332 T + T^{2}$$
$79$ $$-82536692396593536 - 184274184 T + T^{2}$$
$83$ $$-72649117838539792 + 624641094 T + T^{2}$$
$89$ $$570866059346416452 + 1574777148 T + T^{2}$$
$97$ $$-72492038224976036 - 213665984 T + T^{2}$$
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