Properties

Label 112.6.a.i
Level $112$
Weight $6$
Character orbit 112.a
Self dual yes
Analytic conductor $17.963$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.9629878191\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
Defining polynomial: \(x^{2} - x - 86\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 + \beta ) q^{3} + ( 41 - 3 \beta ) q^{5} + 49 q^{7} + ( 111 + 6 \beta ) q^{9} +O(q^{10})\) \( q + ( 3 + \beta ) q^{3} + ( 41 - 3 \beta ) q^{5} + 49 q^{7} + ( 111 + 6 \beta ) q^{9} + ( -170 + 6 \beta ) q^{11} + ( 455 + 39 \beta ) q^{13} + ( -912 + 32 \beta ) q^{15} + ( 1608 - 6 \beta ) q^{17} + ( 337 - 45 \beta ) q^{19} + ( 147 + 49 \beta ) q^{21} + ( 552 + 72 \beta ) q^{23} + ( 1661 - 246 \beta ) q^{25} + ( 1674 - 114 \beta ) q^{27} + ( 4032 - 114 \beta ) q^{29} + ( 3106 - 210 \beta ) q^{31} + ( 1560 - 152 \beta ) q^{33} + ( 2009 - 147 \beta ) q^{35} + ( -4256 + 390 \beta ) q^{37} + ( 14820 + 572 \beta ) q^{39} + ( -652 + 678 \beta ) q^{41} + ( 5002 - 798 \beta ) q^{43} + ( -1659 - 87 \beta ) q^{45} + ( 6374 + 714 \beta ) q^{47} + 2401 q^{49} + ( 2754 + 1590 \beta ) q^{51} + ( -5610 - 768 \beta ) q^{53} + ( -13180 + 756 \beta ) q^{55} + ( -14514 + 202 \beta ) q^{57} + ( 6009 - 2253 \beta ) q^{59} + ( 51369 - 75 \beta ) q^{61} + ( 5439 + 294 \beta ) q^{63} + ( -21710 + 234 \beta ) q^{65} + ( -12068 - 1944 \beta ) q^{67} + ( 26496 + 768 \beta ) q^{69} + ( -44860 - 1740 \beta ) q^{71} + ( -27794 - 1716 \beta ) q^{73} + ( -79887 + 923 \beta ) q^{75} + ( -8330 + 294 \beta ) q^{77} + ( -24412 - 1956 \beta ) q^{79} + ( -61281 - 126 \beta ) q^{81} + ( -17891 + 3447 \beta ) q^{83} + ( 72138 - 5070 \beta ) q^{85} + ( -27234 + 3690 \beta ) q^{87} + ( -9150 + 4728 \beta ) q^{89} + ( 22295 + 1911 \beta ) q^{91} + ( -63132 + 2476 \beta ) q^{93} + ( 60392 - 2856 \beta ) q^{95} + ( -34992 - 462 \beta ) q^{97} + ( -6450 - 354 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 82q^{5} + 98q^{7} + 222q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 82q^{5} + 98q^{7} + 222q^{9} - 340q^{11} + 910q^{13} - 1824q^{15} + 3216q^{17} + 674q^{19} + 294q^{21} + 1104q^{23} + 3322q^{25} + 3348q^{27} + 8064q^{29} + 6212q^{31} + 3120q^{33} + 4018q^{35} - 8512q^{37} + 29640q^{39} - 1304q^{41} + 10004q^{43} - 3318q^{45} + 12748q^{47} + 4802q^{49} + 5508q^{51} - 11220q^{53} - 26360q^{55} - 29028q^{57} + 12018q^{59} + 102738q^{61} + 10878q^{63} - 43420q^{65} - 24136q^{67} + 52992q^{69} - 89720q^{71} - 55588q^{73} - 159774q^{75} - 16660q^{77} - 48824q^{79} - 122562q^{81} - 35782q^{83} + 144276q^{85} - 54468q^{87} - 18300q^{89} + 44590q^{91} - 126264q^{93} + 120784q^{95} - 69984q^{97} - 12900q^{99} + O(q^{100}) \)

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
−8.78709
9.78709
0 −15.5742 0 96.7225 0 49.0000 0 −0.445054 0
1.2 0 21.5742 0 −14.7225 0 49.0000 0 222.445 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 112.6.a.i 2
3.b odd 2 1 1008.6.a.bd 2
4.b odd 2 1 56.6.a.e 2
7.b odd 2 1 784.6.a.u 2
8.b even 2 1 448.6.a.t 2
8.d odd 2 1 448.6.a.v 2
12.b even 2 1 504.6.a.i 2
28.d even 2 1 392.6.a.d 2
28.f even 6 2 392.6.i.i 4
28.g odd 6 2 392.6.i.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 4.b odd 2 1
112.6.a.i 2 1.a even 1 1 trivial
392.6.a.d 2 28.d even 2 1
392.6.i.i 4 28.f even 6 2
392.6.i.j 4 28.g odd 6 2
448.6.a.t 2 8.b even 2 1
448.6.a.v 2 8.d odd 2 1
504.6.a.i 2 12.b even 2 1
784.6.a.u 2 7.b odd 2 1
1008.6.a.bd 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6 T_{3} - 336 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(112))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -336 - 6 T + T^{2} \)
$5$ \( -1424 - 82 T + T^{2} \)
$7$ \( ( -49 + T )^{2} \)
$11$ \( 16480 + 340 T + T^{2} \)
$13$ \( -317720 - 910 T + T^{2} \)
$17$ \( 2573244 - 3216 T + T^{2} \)
$19$ \( -585056 - 674 T + T^{2} \)
$23$ \( -1483776 - 1104 T + T^{2} \)
$29$ \( 11773404 - 8064 T + T^{2} \)
$31$ \( -5567264 - 6212 T + T^{2} \)
$37$ \( -34360964 + 8512 T + T^{2} \)
$41$ \( -158165876 + 1304 T + T^{2} \)
$43$ \( -194677376 - 10004 T + T^{2} \)
$47$ \( -135251744 - 12748 T + T^{2} \)
$53$ \( -172017180 + 11220 T + T^{2} \)
$59$ \( -1715115024 - 12018 T + T^{2} \)
$61$ \( 2636833536 - 102738 T + T^{2} \)
$67$ \( -1158165296 + 24136 T + T^{2} \)
$71$ \( 967897600 + 89720 T + T^{2} \)
$73$ \( -243399884 + 55588 T + T^{2} \)
$79$ \( -724002176 + 48824 T + T^{2} \)
$83$ \( -3779136224 + 35782 T + T^{2} \)
$89$ \( -7628401980 + 18300 T + T^{2} \)
$97$ \( 1150801884 + 69984 T + T^{2} \)
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