↑

Properties

Label 112.13.s.c
Level $112$
Weight $13$
Character orbit 112.s
Analytic conductor $102.367$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,13,Mod(17,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.17");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(102.367307535\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{62}\cdot 3^{11}\cdot 7^{7} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{7} - 3 \beta_{5} + \cdots + 756) q^{5}+ \cdots + ( - 4 \beta_{14} - \beta_{13} + \cdots + 295281) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + ( - \beta_{7} - 3 \beta_{5} + \cdots + 756) q^{5}+ \cdots + ( - 909015 \beta_{14} + \cdots - 245073772161) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 18144 q^{5} + 469720 q^{7} + 2362248 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 18144 q^{5} + 469720 q^{7} + 2362248 q^{9} - 2072088 q^{11} + 27163728 q^{15} - 101561040 q^{17} - 174931848 q^{19} - 323731368 q^{21} - 25630560 q^{23} + 521205808 q^{25} + 532360944 q^{29} - 4583818344 q^{31} + 6054957720 q^{33} + 1612540440 q^{35} + 5764524040 q^{37} - 10526083272 q^{39} + 66929432000 q^{43} + 57253352184 q^{45} - 18116171640 q^{47} - 9977452064 q^{49} + 23299256376 q^{51} + 39134161800 q^{53} - 328243960080 q^{57} - 201845459088 q^{59} + 336780254328 q^{61} + 389095094520 q^{63} + 158322703896 q^{65} - 107767119920 q^{67} + 1150259029344 q^{71} - 738414283320 q^{73} - 1537028640000 q^{75} - 321203352960 q^{77} - 227632064768 q^{79} - 391984178400 q^{81} + 710209696080 q^{85} - 1957017683880 q^{87} - 2485007442792 q^{89} - 1803248333904 q^{91} - 458668768680 q^{93} + 186503862960 q^{95} - 3921180354576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 2204556 x^{14} - 87623088 x^{13} + 1948666431190 x^{12} + 195028079162640 x^{11} + \cdots + 24\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!20 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 33\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\!\cdots\!71 \nu^{15} + \cdots + 39\!\cdots\!45 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!82 \nu^{15} + \cdots - 11\!\cdots\!35 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31\!\cdots\!24 \nu^{15} + \cdots + 26\!\cdots\!45 ) / 11\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31\!\cdots\!44 \nu^{15} + \cdots - 26\!\cdots\!45 ) / 11\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!08 \nu^{15} + \cdots - 24\!\cdots\!45 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!33 \nu^{15} + \cdots + 48\!\cdots\!80 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!10 \nu^{15} + \cdots - 27\!\cdots\!35 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67\!\cdots\!17 \nu^{15} + \cdots - 55\!\cdots\!55 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 71\!\cdots\!82 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 84\!\cdots\!07 \nu^{15} + \cdots - 71\!\cdots\!20 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{15} + \cdots - 85\!\cdots\!65 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!77 \nu^{15} + \cdots + 19\!\cdots\!70 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!38 \nu^{15} + \cdots - 13\!\cdots\!05 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 58\!\cdots\!79 \nu^{15} + \cdots - 46\!\cdots\!25 ) / 19\!\cdots\!80 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{14} + \beta_{12} + \beta_{10} + 6 \beta_{9} - 15 \beta_{7} + 11 \beta_{6} + 82 \beta_{5} + \cdots + 826719 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1918 \beta_{14} + 3 \beta_{13} + 295 \beta_{12} + 250 \beta_{11} - 47 \beta_{10} - 4062 \beta_{9} + \cdots + 57968502 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1000 \beta_{15} - 1742034 \beta_{14} - 194 \beta_{13} + 538884 \beta_{12} - 75802 \beta_{11} + \cdots + 361538501343 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 376510 \beta_{15} + 1538420568 \beta_{14} + 2935170 \beta_{13} + 336858376 \beta_{12} + \cdots + 19195727425833 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 957981450 \beta_{15} - 1004134710712 \beta_{14} - 181819605 \beta_{13} + 239576991297 \beta_{12} + \cdots + 18\!\cdots\!80 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 516667029928 \beta_{15} + \cdots + 82\!\cdots\!62 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 685271474358708 \beta_{15} + \cdots + 96\!\cdots\!48 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 51\!\cdots\!00 \beta_{15} + \cdots - 49\!\cdots\!21 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 44\!\cdots\!20 \beta_{15} + \cdots + 53\!\cdots\!07 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 43\!\cdots\!54 \beta_{15} + \cdots - 56\!\cdots\!58 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 28\!\cdots\!98 \beta_{15} + \cdots + 29\!\cdots\!09 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 33\!\cdots\!74 \beta_{15} + \cdots - 46\!\cdots\!34 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 18\!\cdots\!50 \beta_{15} + \cdots + 16\!\cdots\!15 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24\!\cdots\!44 \beta_{15} + \cdots - 34\!\cdots\!12 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
738.105 − 0.866025i
657.612 − 0.866025i
468.905 − 0.866025i
−22.0174 − 0.866025i
−272.613 − 0.866025i
−295.555 − 0.866025i
−496.819 − 0.866025i
−773.618 − 0.866025i
738.105 + 0.866025i
657.612 + 0.866025i
468.905 + 0.866025i
−22.0174 + 0.866025i
−272.613 + 0.866025i
−295.555 + 0.866025i
−496.819 + 0.866025i
−773.618 + 0.866025i
0 −1106.41 − 638.785i 0 24805.4 − 14321.4i 0 66642.6 − 96953.8i 0 550372. + 953273.i 0
17.2 0 −985.668 − 569.076i 0 −12133.2 + 7005.11i 0 103863. − 55261.5i 0 381974. + 661598.i 0
17.3 0 −702.607 − 405.650i 0 −18305.2 + 10568.5i 0 9080.55 + 117298.i 0 63384.0 + 109784.i 0
17.4 0 33.7761 + 19.5006i 0 2481.87 − 1432.91i 0 44556.8 − 108885.i 0 −264960. − 458924.i 0
17.5 0 409.670 + 236.523i 0 18219.6 − 10519.1i 0 −61842.4 + 100084.i 0 −153834. − 266449.i 0
17.6 0 444.082 + 256.391i 0 −12687.9 + 7325.35i 0 −117217. + 10074.6i 0 −134248. − 232524.i 0
17.7 0 745.979 + 430.691i 0 −7532.28 + 4348.76i 0 93774.2 − 71047.1i 0 105269. + 182331.i 0
17.8 0 1161.18 + 670.406i 0 14223.7 − 8212.03i 0 96002.4 + 68006.1i 0 633167. + 1.09668e6i 0
33.1 0 −1106.41 + 638.785i 0 24805.4 + 14321.4i 0 66642.6 + 96953.8i 0 550372. − 953273.i 0
33.2 0 −985.668 + 569.076i 0 −12133.2 − 7005.11i 0 103863. + 55261.5i 0 381974. − 661598.i 0
33.3 0 −702.607 + 405.650i 0 −18305.2 − 10568.5i 0 9080.55 − 117298.i 0 63384.0 − 109784.i 0
33.4 0 33.7761 − 19.5006i 0 2481.87 + 1432.91i 0 44556.8 + 108885.i 0 −264960. + 458924.i 0
33.5 0 409.670 − 236.523i 0 18219.6 + 10519.1i 0 −61842.4 − 100084.i 0 −153834. + 266449.i 0
33.6 0 444.082 − 256.391i 0 −12687.9 − 7325.35i 0 −117217. − 10074.6i 0 −134248. + 232524.i 0
33.7 0 745.979 − 430.691i 0 −7532.28 − 4348.76i 0 93774.2 + 71047.1i 0 105269. − 182331.i 0
33.8 0 1161.18 − 670.406i 0 14223.7 + 8212.03i 0 96002.4 − 68006.1i 0 633167. − 1.09668e6i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.13.s.c 16
4.b odd 2 1 14.13.d.a ✓ 16
7.d odd 6 1 inner 112.13.s.c 16
12.b even 2 1 126.13.n.a 16
28.d even 2 1 98.13.d.a 16
28.f even 6 1 14.13.d.a ✓ 16
28.f even 6 1 98.13.b.c 16
28.g odd 6 1 98.13.b.c 16
28.g odd 6 1 98.13.d.a 16
84.j odd 6 1 126.13.n.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.d.a ✓ 16 4.b odd 2 1
14.13.d.a ✓ 16 28.f even 6 1
98.13.b.c 16 28.f even 6 1
98.13.b.c 16 28.g odd 6 1
98.13.d.a 16 28.d even 2 1
98.13.d.a 16 28.g odd 6 1
112.13.s.c 16 1.a even 1 1 trivial
112.13.s.c 16 7.d odd 6 1 inner
126.13.n.a 16 12.b even 2 1
126.13.n.a 16 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 3306888 T_{3}^{14} + 7637014892322 T_{3}^{12} - 276426941444640 T_{3}^{11} + \cdots + 16\!\cdots\!41 \) acting on \(S_{13}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 29\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 73\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 45\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 24\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 41\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 90\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 53\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 19\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 56\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 97\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 80\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 78\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
show more
show less