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 $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
Defining polynomial: \(x^{2} - x - 576\)
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.

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

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:

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 \( T_{3}^{2} - 14 T_{3} - 57576 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -57576 - 14 T + T^{2} \)
$5$ \( 846720 + 2730 T + T^{2} \)
$7$ \( ( 2401 + T )^{2} \)
$11$ \( -2036361600 + 44940 T + T^{2} \)
$13$ \( 2397429256 - 100282 T + T^{2} \)
$17$ \( 183575251116 + 870408 T + T^{2} \)
$19$ \( -352577596856 + 508774 T + T^{2} \)
$23$ \( -115915968000 + 79800 T + T^{2} \)
$29$ \( -31904129519604 - 2006328 T + T^{2} \)
$31$ \( -2367849772544 + 2188732 T + T^{2} \)
$37$ \( 60225113026444 + 20723576 T + T^{2} \)
$41$ \( -527816477266884 - 19016592 T + T^{2} \)
$43$ \( -271341247682336 + 4193716 T + T^{2} \)
$47$ \( 1296550084878144 - 74542524 T + T^{2} \)
$53$ \( -494746212296124 + 3239748 T + T^{2} \)
$59$ \( 1575046316366136 - 133642362 T + T^{2} \)
$61$ \( 11243934945943024 - 227801686 T + T^{2} \)
$67$ \( 21401918313456496 + 332930272 T + T^{2} \)
$71$ \( -85127259938918400 - 167985720 T + T^{2} \)
$73$ \( -83678371011966956 + 44684276 T + T^{2} \)
$79$ \( 6880003984764544 + 269642776 T + T^{2} \)
$83$ \( -76697718543013464 - 183105762 T + T^{2} \)
$89$ \( 50565747709419876 - 791657748 T + T^{2} \)
$97$ \( -451110491471642900 + 4169480 T + T^{2} \)
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