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: \( x^{2} - x - 576 \) Copy content Toggle raw display
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.

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

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

Hecke characteristic polynomials

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