↑

Properties

Label 112.10.i.f
Level $112$
Weight $10$
Character orbit 112.i
Analytic conductor $57.684$
Analytic rank $0$
Dimension $18$
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] = [112,10,Mod(65,112)]
 
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, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.65");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), 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: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 58021 x^{16} + 299572 x^{15} + 1344281856 x^{14} - 13223849184 x^{13} + \cdots + 45\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{64}\cdot 3^{9}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + 10 \beta_1 + 10) q^{3} + ( - \beta_{8} + \beta_{5} + \cdots - 89 \beta_1) q^{5}+ \cdots + (\beta_{10} - \beta_{7} + 8 \beta_{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} + 10 \beta_1 + 10) q^{3} + ( - \beta_{8} + \beta_{5} + \cdots - 89 \beta_1) q^{5}+ \cdots + (9795 \beta_{17} - 9795 \beta_{16} + \cdots - 35689165) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 91 q^{3} + 801 q^{5} + 4820 q^{7} - 55886 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 91 q^{3} + 801 q^{5} + 4820 q^{7} - 55886 q^{9} + 42719 q^{11} + 301620 q^{13} - 318842 q^{15} + 11357 q^{17} + 1237749 q^{19} - 352965 q^{21} - 95019 q^{23} - 848460 q^{25} - 983570 q^{27} + 12010116 q^{29} + 10454399 q^{31} - 22110949 q^{33} + 14069675 q^{35} + 18487597 q^{37} - 26537570 q^{39} - 9446020 q^{41} - 15588088 q^{43} + 16445950 q^{45} - 6482527 q^{47} - 97119278 q^{49} + 91171769 q^{51} + 15314401 q^{53} + 27790478 q^{55} - 86196434 q^{57} + 84965083 q^{59} + 122850341 q^{61} - 682061514 q^{63} + 148004010 q^{65} + 27388339 q^{67} - 717062290 q^{69} + 473284416 q^{71} + 217201553 q^{73} - 86436700 q^{75} + 112464495 q^{77} + 326283565 q^{79} - 592248617 q^{81} - 1411349592 q^{83} + 1568978282 q^{85} + 787721718 q^{87} + 153096753 q^{89} + 878501576 q^{91} - 158673269 q^{93} - 1281844653 q^{95} + 919981372 q^{97} - 654760036 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 58021 x^{16} + 299572 x^{15} + 1344281856 x^{14} - 13223849184 x^{13} + \cdots + 45\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16\!\cdots\!12 \nu^{17} + \cdots + 35\!\cdots\!24 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!24 \nu^{17} + \cdots + 70\!\cdots\!23 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 68\!\cdots\!76 \nu^{17} + \cdots + 14\!\cdots\!02 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 87\!\cdots\!32 \nu^{17} + \cdots + 12\!\cdots\!39 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!56 \nu^{17} + \cdots - 16\!\cdots\!49 ) / 34\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50\!\cdots\!68 \nu^{17} + \cdots - 98\!\cdots\!26 ) / 24\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!36 \nu^{17} + \cdots + 66\!\cdots\!22 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!76 \nu^{17} + \cdots + 94\!\cdots\!17 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81\!\cdots\!36 \nu^{17} + \cdots + 20\!\cdots\!32 ) / 40\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!92 \nu^{17} + \cdots - 62\!\cdots\!34 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80\!\cdots\!08 \nu^{17} + \cdots - 15\!\cdots\!11 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!48 \nu^{17} + \cdots + 68\!\cdots\!41 ) / 40\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20\!\cdots\!92 \nu^{17} + \cdots - 47\!\cdots\!94 ) / 24\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!04 \nu^{17} + \cdots + 20\!\cdots\!53 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!84 \nu^{17} + \cdots + 26\!\cdots\!23 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 12\!\cdots\!48 \nu^{17} + \cdots + 29\!\cdots\!96 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 14\!\cdots\!72 \nu^{17} + \cdots - 29\!\cdots\!89 ) / 12\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{4} - 4\beta_{3} - 15\beta_{2} + \beta _1 + 25790 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{17} + 18 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 45 \beta_{13} + 14 \beta_{12} + \cdots - 404521 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 999 \beta_{17} - 855 \beta_{16} + 37 \beta_{15} + 21 \beta_{14} + 1755 \beta_{13} + 5617 \beta_{12} + \cdots + 1205765406 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 859518 \beta_{17} + 854883 \beta_{16} + 66432 \beta_{15} + 66287 \beta_{14} + 1890504 \beta_{13} + \cdots - 12643722776 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 57665025 \beta_{17} - 47376459 \beta_{16} + 10807644 \beta_{15} + 10011090 \beta_{14} + \cdots + 33565268678462 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 63576071865 \beta_{17} + 62840586807 \beta_{16} + 3069578238 \beta_{15} + 2923841010 \beta_{14} + \cdots - 711161351912546 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1211985465174 \beta_{17} - 959008101858 \beta_{16} + 299814304041 \beta_{15} + 287816313117 \beta_{14} + \cdots + 50\!\cdots\!20 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10\!\cdots\!15 \beta_{17} + \cdots - 10\!\cdots\!14 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 22\!\cdots\!88 \beta_{17} + \cdots + 77\!\cdots\!01 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18\!\cdots\!86 \beta_{17} + \cdots - 15\!\cdots\!83 ) / 64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 39\!\cdots\!15 \beta_{17} + \cdots + 12\!\cdots\!40 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 29\!\cdots\!18 \beta_{17} + \cdots - 24\!\cdots\!89 ) / 64 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 67\!\cdots\!75 \beta_{17} + \cdots + 19\!\cdots\!39 ) / 32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 47\!\cdots\!81 \beta_{17} + \cdots - 40\!\cdots\!72 ) / 64 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 11\!\cdots\!83 \beta_{17} + \cdots + 30\!\cdots\!46 ) / 32 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 38\!\cdots\!70 \beta_{17} + \cdots - 34\!\cdots\!19 ) / 32 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−126.660 − 0.866025i
−96.8241 − 0.866025i
−62.8682 − 0.866025i
−29.1700 − 0.866025i
20.3320 − 0.866025i
21.2894 − 0.866025i
71.8084 − 0.866025i
76.1752 − 0.866025i
126.418 − 0.866025i
−126.660 + 0.866025i
−96.8241 + 0.866025i
−62.8682 + 0.866025i
−29.1700 + 0.866025i
20.3320 + 0.866025i
21.2894 + 0.866025i
71.8084 + 0.866025i
76.1752 + 0.866025i
126.418 + 0.866025i
0 −121.660 − 210.722i 0 637.834 − 1104.76i 0 3287.44 + 5435.66i 0 −19761.0 + 34227.1i 0
65.2 0 −91.8241 − 159.044i 0 −609.854 + 1056.30i 0 2978.43 − 5610.93i 0 −7021.84 + 12162.2i 0
65.3 0 −57.8682 − 100.231i 0 574.146 − 994.450i 0 −6307.33 − 755.801i 0 3144.05 − 5445.65i 0
65.4 0 −24.1700 − 41.8637i 0 −904.391 + 1566.45i 0 −2996.45 + 5601.33i 0 8673.12 − 15022.3i 0
65.5 0 25.3320 + 43.8764i 0 1186.59 − 2055.24i 0 4635.61 − 4343.36i 0 8558.08 − 14823.0i 0
65.6 0 26.2894 + 45.5345i 0 89.1449 − 154.404i 0 5406.72 + 3334.81i 0 8459.24 − 14651.8i 0
65.7 0 76.8084 + 133.036i 0 −1042.94 + 1806.43i 0 489.665 − 6333.55i 0 −1957.55 + 3390.58i 0
65.8 0 81.1752 + 140.600i 0 416.935 − 722.153i 0 −6076.65 − 1851.48i 0 −3337.32 + 5780.41i 0
65.9 0 131.418 + 227.622i 0 53.0367 − 91.8623i 0 992.571 + 6274.43i 0 −24699.8 + 42781.3i 0
81.1 0 −121.660 + 210.722i 0 637.834 + 1104.76i 0 3287.44 − 5435.66i 0 −19761.0 − 34227.1i 0
81.2 0 −91.8241 + 159.044i 0 −609.854 − 1056.30i 0 2978.43 + 5610.93i 0 −7021.84 − 12162.2i 0
81.3 0 −57.8682 + 100.231i 0 574.146 + 994.450i 0 −6307.33 + 755.801i 0 3144.05 + 5445.65i 0
81.4 0 −24.1700 + 41.8637i 0 −904.391 − 1566.45i 0 −2996.45 − 5601.33i 0 8673.12 + 15022.3i 0
81.5 0 25.3320 − 43.8764i 0 1186.59 + 2055.24i 0 4635.61 + 4343.36i 0 8558.08 + 14823.0i 0
81.6 0 26.2894 − 45.5345i 0 89.1449 + 154.404i 0 5406.72 − 3334.81i 0 8459.24 + 14651.8i 0
81.7 0 76.8084 − 133.036i 0 −1042.94 − 1806.43i 0 489.665 + 6333.55i 0 −1957.55 − 3390.58i 0
81.8 0 81.1752 − 140.600i 0 416.935 + 722.153i 0 −6076.65 + 1851.48i 0 −3337.32 − 5780.41i 0
81.9 0 131.418 − 227.622i 0 53.0367 + 91.8623i 0 992.571 − 6274.43i 0 −24699.8 − 42781.3i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.10.i.f 18
4.b odd 2 1 56.10.i.a ✓ 18
7.c even 3 1 inner 112.10.i.f 18
28.f even 6 1 392.10.a.i 9
28.g odd 6 1 56.10.i.a ✓ 18
28.g odd 6 1 392.10.a.l 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.10.i.a ✓ 18 4.b odd 2 1
56.10.i.a ✓ 18 28.g odd 6 1
112.10.i.f 18 1.a even 1 1 trivial
112.10.i.f 18 7.c even 3 1 inner
392.10.a.i 9 28.f even 6 1
392.10.a.l 9 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 91 T_{3}^{17} + 120657 T_{3}^{16} - 8012536 T_{3}^{15} + 9742637182 T_{3}^{14} + \cdots + 19\!\cdots\!21 \) acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 28\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 72\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 27\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 46\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 39\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 79\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 35\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 23\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 47\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 96\!\cdots\!36)^{2} \) Copy content Toggle raw display
show more
show less