# Properties

 Label 112.10.a.c Level $112$ Weight $10$ Character orbit 112.a Self dual yes Analytic conductor $57.684$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2305})$$ Defining polynomial: $$x^{2} - x - 576$$ x^2 - x - 576 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{2305}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (5 \beta + 7) q^{3} + ( - 21 \beta - 1365) q^{5} - 2401 q^{7} + (70 \beta + 37991) q^{9}+O(q^{10})$$ q + (5*b + 7) * q^3 + (-21*b - 1365) * q^5 - 2401 * q^7 + (70*b + 37991) * q^9 $$q + (5 \beta + 7) q^{3} + ( - 21 \beta - 1365) q^{5} - 2401 q^{7} + (70 \beta + 37991) q^{9} + (1050 \beta - 22470) q^{11} + (225 \beta + 50141) q^{13} + ( - 6972 \beta - 251580) q^{15} + (1590 \beta - 435204) q^{17} + (13455 \beta - 254387) q^{19} + ( - 12005 \beta - 16807) q^{21} + (7140 \beta - 39900) q^{23} + (57330 \beta + 926605) q^{25} + (92030 \beta + 934906) q^{27} + (119490 \beta + 1003164) q^{29} + ( - 39330 \beta - 1094366) q^{31} + ( - 105000 \beta + 11943960) q^{33} + (50421 \beta + 3277365) q^{35} + (143010 \beta - 10361788) q^{37} + (252280 \beta + 2944112) q^{39} + (517890 \beta + 9508296) q^{41} + (345870 \beta - 2096858) q^{43} + ( - 893361 \beta - 55246065) q^{45} + ( - 200430 \beta + 37271262) q^{47} + 5764801 q^{49} + ( - 2164890 \beta + 15278322) q^{51} + ( - 464520 \beta - 1619874) q^{53} + ( - 961380 \beta - 20153700) q^{55} + ( - 1177750 \beta + 153288166) q^{57} + (1119735 \beta + 66821181) q^{59} + ( - 866205 \beta + 113900843) q^{61} + ( - 168070 \beta - 91216391) q^{63} + ( - 1360086 \beta - 79333590) q^{65} + (1654380 \beta - 166465136) q^{67} + ( - 149520 \beta + 82009200) q^{69} + ( - 6323940 \beta + 83992860) q^{71} + (6043140 \beta - 22342138) q^{73} + (5034335 \beta + 667214485) q^{75} + ( - 2521050 \beta + 53950470) q^{77} + (2213820 \beta - 134821388) q^{79} + (3940930 \beta + 319413239) q^{81} + (6075435 \beta + 91552881) q^{83} + (6968934 \beta + 517089510) q^{85} + (5852250 \beta + 1384144398) q^{87} + (6785040 \beta + 395828874) q^{89} + ( - 540225 \beta - 120388541) q^{91} + ( - 5747140 \beta - 460938812) q^{93} + ( - 13023948 \beta - 304051020) q^{95} + ( - 13989690 \beta - 2084740) q^{97} + (38317650 \beta - 684240270) q^{99}+O(q^{100})$$ q + (5*b + 7) * q^3 + (-21*b - 1365) * q^5 - 2401 * q^7 + (70*b + 37991) * q^9 + (1050*b - 22470) * q^11 + (225*b + 50141) * q^13 + (-6972*b - 251580) * q^15 + (1590*b - 435204) * q^17 + (13455*b - 254387) * q^19 + (-12005*b - 16807) * q^21 + (7140*b - 39900) * q^23 + (57330*b + 926605) * q^25 + (92030*b + 934906) * q^27 + (119490*b + 1003164) * q^29 + (-39330*b - 1094366) * q^31 + (-105000*b + 11943960) * q^33 + (50421*b + 3277365) * q^35 + (143010*b - 10361788) * q^37 + (252280*b + 2944112) * q^39 + (517890*b + 9508296) * q^41 + (345870*b - 2096858) * q^43 + (-893361*b - 55246065) * q^45 + (-200430*b + 37271262) * q^47 + 5764801 * q^49 + (-2164890*b + 15278322) * q^51 + (-464520*b - 1619874) * q^53 + (-961380*b - 20153700) * q^55 + (-1177750*b + 153288166) * q^57 + (1119735*b + 66821181) * q^59 + (-866205*b + 113900843) * q^61 + (-168070*b - 91216391) * q^63 + (-1360086*b - 79333590) * q^65 + (1654380*b - 166465136) * q^67 + (-149520*b + 82009200) * q^69 + (-6323940*b + 83992860) * q^71 + (6043140*b - 22342138) * q^73 + (5034335*b + 667214485) * q^75 + (-2521050*b + 53950470) * q^77 + (2213820*b - 134821388) * q^79 + (3940930*b + 319413239) * q^81 + (6075435*b + 91552881) * q^83 + (6968934*b + 517089510) * q^85 + (5852250*b + 1384144398) * q^87 + (6785040*b + 395828874) * q^89 + (-540225*b - 120388541) * q^91 + (-5747140*b - 460938812) * q^93 + (-13023948*b - 304051020) * q^95 + (-13989690*b - 2084740) * q^97 + (38317650*b - 684240270) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9}+O(q^{10})$$ 2 * q + 14 * q^3 - 2730 * q^5 - 4802 * q^7 + 75982 * q^9 $$2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9} - 44940 q^{11} + 100282 q^{13} - 503160 q^{15} - 870408 q^{17} - 508774 q^{19} - 33614 q^{21} - 79800 q^{23} + 1853210 q^{25} + 1869812 q^{27} + 2006328 q^{29} - 2188732 q^{31} + 23887920 q^{33} + 6554730 q^{35} - 20723576 q^{37} + 5888224 q^{39} + 19016592 q^{41} - 4193716 q^{43} - 110492130 q^{45} + 74542524 q^{47} + 11529602 q^{49} + 30556644 q^{51} - 3239748 q^{53} - 40307400 q^{55} + 306576332 q^{57} + 133642362 q^{59} + 227801686 q^{61} - 182432782 q^{63} - 158667180 q^{65} - 332930272 q^{67} + 164018400 q^{69} + 167985720 q^{71} - 44684276 q^{73} + 1334428970 q^{75} + 107900940 q^{77} - 269642776 q^{79} + 638826478 q^{81} + 183105762 q^{83} + 1034179020 q^{85} + 2768288796 q^{87} + 791657748 q^{89} - 240777082 q^{91} - 921877624 q^{93} - 608102040 q^{95} - 4169480 q^{97} - 1368480540 q^{99}+O(q^{100})$$ 2 * q + 14 * q^3 - 2730 * q^5 - 4802 * q^7 + 75982 * q^9 - 44940 * q^11 + 100282 * q^13 - 503160 * q^15 - 870408 * q^17 - 508774 * q^19 - 33614 * q^21 - 79800 * q^23 + 1853210 * q^25 + 1869812 * q^27 + 2006328 * q^29 - 2188732 * q^31 + 23887920 * q^33 + 6554730 * q^35 - 20723576 * q^37 + 5888224 * q^39 + 19016592 * q^41 - 4193716 * q^43 - 110492130 * q^45 + 74542524 * q^47 + 11529602 * q^49 + 30556644 * q^51 - 3239748 * q^53 - 40307400 * q^55 + 306576332 * q^57 + 133642362 * q^59 + 227801686 * q^61 - 182432782 * q^63 - 158667180 * q^65 - 332930272 * q^67 + 164018400 * q^69 + 167985720 * q^71 - 44684276 * q^73 + 1334428970 * q^75 + 107900940 * q^77 - 269642776 * q^79 + 638826478 * q^81 + 183105762 * q^83 + 1034179020 * q^85 + 2768288796 * q^87 + 791657748 * q^89 - 240777082 * q^91 - 921877624 * q^93 - 608102040 * q^95 - 4169480 * q^97 - 1368480540 * q^99

## 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
 −23.5052 24.5052
0 −233.052 0 −356.781 0 −2401.00 0 34630.3 0
1.2 0 247.052 0 −2373.22 0 −2401.00 0 41351.7 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.c 2
4.b odd 2 1 14.10.a.c 2
12.b even 2 1 126.10.a.o 2
20.d odd 2 1 350.10.a.j 2
20.e even 4 2 350.10.c.j 4
28.d even 2 1 98.10.a.e 2
28.f even 6 2 98.10.c.h 4
28.g odd 6 2 98.10.c.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 4.b odd 2 1
98.10.a.e 2 28.d even 2 1
98.10.c.h 4 28.f even 6 2
98.10.c.j 4 28.g odd 6 2
112.10.a.c 2 1.a even 1 1 trivial
126.10.a.o 2 12.b even 2 1
350.10.a.j 2 20.d odd 2 1
350.10.c.j 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 14T_{3} - 57576$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 14T - 57576$$
$5$ $$T^{2} + 2730 T + 846720$$
$7$ $$(T + 2401)^{2}$$
$11$ $$T^{2} + 44940 T - 2036361600$$
$13$ $$T^{2} - 100282 T + 2397429256$$
$17$ $$T^{2} + 870408 T + 183575251116$$
$19$ $$T^{2} + 508774 T - 352577596856$$
$23$ $$T^{2} + 79800 T - 115915968000$$
$29$ $$T^{2} - 2006328 T - 31904129519604$$
$31$ $$T^{2} + 2188732 T - 2367849772544$$
$37$ $$T^{2} + 20723576 T + 60225113026444$$
$41$ $$T^{2} + \cdots - 527816477266884$$
$43$ $$T^{2} + \cdots - 271341247682336$$
$47$ $$T^{2} - 74542524 T + 12\!\cdots\!44$$
$53$ $$T^{2} + \cdots - 494746212296124$$
$59$ $$T^{2} - 133642362 T + 15\!\cdots\!36$$
$61$ $$T^{2} - 227801686 T + 11\!\cdots\!24$$
$67$ $$T^{2} + 332930272 T + 21\!\cdots\!96$$
$71$ $$T^{2} - 167985720 T - 85\!\cdots\!00$$
$73$ $$T^{2} + 44684276 T - 83\!\cdots\!56$$
$79$ $$T^{2} + 269642776 T + 68\!\cdots\!44$$
$83$ $$T^{2} - 183105762 T - 76\!\cdots\!64$$
$89$ $$T^{2} - 791657748 T + 50\!\cdots\!76$$
$97$ $$T^{2} + 4169480 T - 45\!\cdots\!00$$