Properties

Label 112.6.a.h
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 + 3 \beta ) q^{3} + ( -9 + 5 \beta ) q^{5} -49 q^{7} + ( 279 + 18 \beta ) q^{9} +O(q^{10})\) \( q + ( 3 + 3 \beta ) q^{3} + ( -9 + 5 \beta ) q^{5} -49 q^{7} + ( 279 + 18 \beta ) q^{9} + ( -198 + 62 \beta ) q^{11} + ( -175 + 63 \beta ) q^{13} + ( 828 - 12 \beta ) q^{15} + ( 900 - 38 \beta ) q^{17} + ( 1633 + 9 \beta ) q^{19} + ( -147 - 147 \beta ) q^{21} + ( -1044 - 284 \beta ) q^{23} + ( -1619 - 90 \beta ) q^{25} + ( 3186 + 162 \beta ) q^{27} + ( 3348 + 126 \beta ) q^{29} + ( 10 - 270 \beta ) q^{31} + ( 10008 - 408 \beta ) q^{33} + ( 441 - 245 \beta ) q^{35} + ( 3116 + 270 \beta ) q^{37} + ( 10248 - 336 \beta ) q^{39} + ( -3024 - 546 \beta ) q^{41} + ( 1510 + 2394 \beta ) q^{43} + ( 2619 + 1233 \beta ) q^{45} + ( -5850 - 1874 \beta ) q^{47} + 2401 q^{49} + ( -3798 + 2586 \beta ) q^{51} + ( 4734 - 104 \beta ) q^{53} + ( 19452 - 1548 \beta ) q^{55} + ( 6438 + 4926 \beta ) q^{57} + ( 21969 + 1025 \beta ) q^{59} + ( -32377 - 2403 \beta ) q^{61} + ( -13671 - 882 \beta ) q^{63} + ( 19530 - 1442 \beta ) q^{65} + ( -12392 + 972 \beta ) q^{67} + ( -51696 - 3984 \beta ) q^{69} + ( -48708 + 2100 \beta ) q^{71} + ( 8726 - 2628 \beta ) q^{73} + ( -20247 - 5127 \beta ) q^{75} + ( 9702 - 3038 \beta ) q^{77} + ( -25628 - 7452 \beta ) q^{79} + ( -30537 + 5670 \beta ) q^{81} + ( -58779 - 7875 \beta ) q^{83} + ( -18930 + 4842 \beta ) q^{85} + ( 31590 + 10422 \beta ) q^{87} + ( 42138 - 11104 \beta ) q^{89} + ( 8575 - 3087 \beta ) q^{91} + ( -46140 - 780 \beta ) q^{93} + ( -12132 + 8084 \beta ) q^{95} + ( 10388 + 4410 \beta ) q^{97} + ( 8370 + 13734 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 18q^{5} - 98q^{7} + 558q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 18q^{5} - 98q^{7} + 558q^{9} - 396q^{11} - 350q^{13} + 1656q^{15} + 1800q^{17} + 3266q^{19} - 294q^{21} - 2088q^{23} - 3238q^{25} + 6372q^{27} + 6696q^{29} + 20q^{31} + 20016q^{33} + 882q^{35} + 6232q^{37} + 20496q^{39} - 6048q^{41} + 3020q^{43} + 5238q^{45} - 11700q^{47} + 4802q^{49} - 7596q^{51} + 9468q^{53} + 38904q^{55} + 12876q^{57} + 43938q^{59} - 64754q^{61} - 27342q^{63} + 39060q^{65} - 24784q^{67} - 103392q^{69} - 97416q^{71} + 17452q^{73} - 40494q^{75} + 19404q^{77} - 51256q^{79} - 61074q^{81} - 117558q^{83} - 37860q^{85} + 63180q^{87} + 84276q^{89} + 17150q^{91} - 92280q^{93} - 24264q^{95} + 20776q^{97} + 16740q^{99} + O(q^{100}) \)

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
−3.27492
4.27492
0 −19.6495 0 −46.7492 0 −49.0000 0 143.103 0
1.2 0 25.6495 0 28.7492 0 −49.0000 0 414.897 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.h 2
3.b odd 2 1 1008.6.a.bq 2
4.b odd 2 1 7.6.a.b 2
7.b odd 2 1 784.6.a.v 2
8.b even 2 1 448.6.a.u 2
8.d odd 2 1 448.6.a.w 2
12.b even 2 1 63.6.a.f 2
20.d odd 2 1 175.6.a.c 2
20.e even 4 2 175.6.b.c 4
28.d even 2 1 49.6.a.f 2
28.f even 6 2 49.6.c.d 4
28.g odd 6 2 49.6.c.e 4
44.c even 2 1 847.6.a.c 2
84.h odd 2 1 441.6.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 4.b odd 2 1
49.6.a.f 2 28.d even 2 1
49.6.c.d 4 28.f even 6 2
49.6.c.e 4 28.g odd 6 2
63.6.a.f 2 12.b even 2 1
112.6.a.h 2 1.a even 1 1 trivial
175.6.a.c 2 20.d odd 2 1
175.6.b.c 4 20.e even 4 2
441.6.a.l 2 84.h odd 2 1
448.6.a.u 2 8.b even 2 1
448.6.a.w 2 8.d odd 2 1
784.6.a.v 2 7.b odd 2 1
847.6.a.c 2 44.c even 2 1
1008.6.a.bq 2 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6 T_{3} - 504 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(112))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 6 T - 18 T^{2} - 1458 T^{3} + 59049 T^{4} \)
$5$ \( 1 + 18 T + 4906 T^{2} + 56250 T^{3} + 9765625 T^{4} \)
$7$ \( ( 1 + 49 T )^{2} \)
$11$ \( 1 + 396 T + 142198 T^{2} + 63776196 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 + 350 T + 546978 T^{2} + 129952550 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 - 1800 T + 3567406 T^{2} - 2555742600 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 - 3266 T + 7614270 T^{2} - 8086939334 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 + 2088 T + 9365230 T^{2} + 13439084184 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 - 20 T + 53103102 T^{2} - 572583020 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 - 3020 T - 30383466 T^{2} - 443965497860 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 + 11700 T + 292735582 T^{2} + 2683336581900 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 - 9468 T + 858185230 T^{2} - 3959474927724 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 - 43938 T + 1852599934 T^{2} - 31412343849462 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 + 24784 T + 2799959190 T^{2} + 33461500651888 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 + 97416 T + 5729557966 T^{2} + 175760806457016 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 - 17452 T + 3828622374 T^{2} - 36179245441036 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 + 51256 T + 3645565854 T^{2} + 157717602787144 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 + 117558 T + 7798161502 T^{2} + 463065739909794 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 - 84276 T + 5915697430 T^{2} - 470602194123924 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 - 20776 T + 16174049358 T^{2} - 178410581179432 T^{3} + 73742412689492826049 T^{4} \)
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