Properties

Label 112.10.i.a
Level $112$
Weight $10$
Character orbit 112.i
Analytic conductor $57.684$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2} - 24 \beta_1 - 24) q^{3} + (2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} - \beta_{2} + 364 \beta_1) q^{5} + (\beta_{5} + 18 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} - 1353 \beta_1 + 459) q^{7} + (18 \beta_{5} - 25 \beta_{4} + 27 \beta_{3} - 9 \beta_{2} + 3033 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2} - 24 \beta_1 - 24) q^{3} + (2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} - \beta_{2} + 364 \beta_1) q^{5} + (\beta_{5} + 18 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} - 1353 \beta_1 + 459) q^{7} + (18 \beta_{5} - 25 \beta_{4} + 27 \beta_{3} - 9 \beta_{2} + 3033 \beta_1) q^{9} + ( - 111 \beta_{5} - 127 \beta_{4} - 37 \beta_{3} + 164 \beta_{2} + \cdots - 931) q^{11}+ \cdots + ( - 93411 \beta_{5} + 93411 \beta_{4} + 186822 \beta_{3} + \cdots + 435444723) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 71 q^{3} - 1085 q^{5} + 6796 q^{7} - 9106 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 71 q^{3} - 1085 q^{5} + 6796 q^{7} - 9106 q^{9} - 2555 q^{11} + 36140 q^{13} + 706146 q^{15} + 20759 q^{17} - 1220649 q^{19} + 1951577 q^{21} + 1960903 q^{23} - 863572 q^{25} - 5777318 q^{27} - 5212292 q^{29} + 9377989 q^{31} - 10778103 q^{33} - 20361719 q^{35} - 25814913 q^{37} - 37990822 q^{39} + 418500 q^{41} - 5888616 q^{43} - 37350558 q^{45} - 48391269 q^{47} + 108466086 q^{49} + 75441987 q^{51} + 102186411 q^{53} + 456557402 q^{55} + 86169178 q^{57} - 144220135 q^{59} + 280936871 q^{61} - 33751838 q^{63} - 186819738 q^{65} - 170710399 q^{67} - 1807308342 q^{69} - 939517376 q^{71} + 613838539 q^{73} - 902254676 q^{75} - 729499715 q^{77} - 197445809 q^{79} + 887872901 q^{81} + 2148362872 q^{83} + 822545038 q^{85} - 753428670 q^{87} + 805730427 q^{89} - 17178248 q^{91} - 1721516327 q^{93} - 1799421743 q^{95} - 4525836188 q^{97} + 2607425268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2716901 \nu^{5} + 2705071 \nu^{4} - 10923076698 \nu^{3} + 181829734103 \nu^{2} - 44277546197779 \nu - 3390681949200 ) / 778518660033660 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4027 \nu^{5} + 16261026 \nu^{4} + 146824014 \nu^{3} + 65915341423 \nu^{2} - 1153921789020 \nu + 166009397495697 ) / 690992893521 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 345984829 \nu^{5} - 17803471041 \nu^{4} - 2254218225362 \nu^{3} - 13053923250853 \nu^{2} + \cdots - 39\!\cdots\!70 ) / 389259330016830 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2081085611 \nu^{5} + 53013142539 \nu^{4} + 8366833294278 \nu^{3} - 139277523654233 \nu^{2} + \cdots + 25\!\cdots\!00 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 708061219 \nu^{5} + 18402640069 \nu^{4} - 165226309702 \nu^{3} + 192995085868737 \nu^{2} + \cdots + 21\!\cdots\!20 ) / 778518660033660 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} + 5\beta_{4} + 3\beta_{3} - \beta_{2} - 7\beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -15\beta_{5} - 167\beta_{4} - 5\beta_{3} + 172\beta_{2} - 37744\beta _1 - 37744 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4027\beta_{5} - 4027\beta_{4} - 8054\beta_{3} - 15814\beta_{2} + 1883693 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 214646\beta_{5} + 1723787\beta_{4} + 321969\beta_{3} - 107323\beta_{2} + 302364503\beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25289214 \beta_{5} - 42965987 \beta_{4} - 8429738 \beta_{3} + 51395725 \beta_{2} - 6330121189 \beta _1 - 6330121189 ) / 14 \) Copy content Toggle raw display

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
−35.2598 + 61.0717i
26.1296 45.2579i
9.63015 16.6799i
−35.2598 61.0717i
26.1296 + 45.2579i
9.63015 + 16.6799i
0 −86.7292 150.219i 0 −1172.77 + 2031.30i 0 6347.68 + 246.066i 0 −5202.42 + 9010.85i 0
65.2 0 −38.4345 66.5705i 0 546.130 945.925i 0 −6111.84 + 1731.76i 0 6887.08 11928.8i 0
65.3 0 89.6637 + 155.302i 0 84.1442 145.742i 0 3162.16 5509.48i 0 −6237.66 + 10803.9i 0
81.1 0 −86.7292 + 150.219i 0 −1172.77 2031.30i 0 6347.68 246.066i 0 −5202.42 9010.85i 0
81.2 0 −38.4345 + 66.5705i 0 546.130 + 945.925i 0 −6111.84 1731.76i 0 6887.08 + 11928.8i 0
81.3 0 89.6637 155.302i 0 84.1442 + 145.742i 0 3162.16 + 5509.48i 0 −6237.66 10803.9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.3
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.a 6
4.b odd 2 1 14.10.c.b 6
7.c even 3 1 inner 112.10.i.a 6
12.b even 2 1 126.10.g.e 6
28.d even 2 1 98.10.c.l 6
28.f even 6 1 98.10.a.h 3
28.f even 6 1 98.10.c.l 6
28.g odd 6 1 14.10.c.b 6
28.g odd 6 1 98.10.a.g 3
84.n even 6 1 126.10.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.c.b 6 4.b odd 2 1
14.10.c.b 6 28.g odd 6 1
98.10.a.g 3 28.g odd 6 1
98.10.a.h 3 28.f even 6 1
98.10.c.l 6 28.d even 2 1
98.10.c.l 6 28.f even 6 1
112.10.i.a 6 1.a even 1 1 trivial
112.10.i.a 6 7.c even 3 1 inner
126.10.g.e 6 12.b even 2 1
126.10.g.e 6 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 71T_{3}^{5} + 36598T_{3}^{4} + 2541603T_{3}^{3} + 1165610574T_{3}^{2} + 75455153775T_{3} + 5717239655625 \) acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 71 T^{5} + \cdots + 5717239655625 \) Copy content Toggle raw display
$5$ \( T^{6} + 1085 T^{5} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} - 6796 T^{5} + \cdots + 65\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{6} + 2555 T^{5} + \cdots + 31\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( (T^{3} - 18070 T^{2} + \cdots + 368974841338200)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 20759 T^{5} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{6} + 1220649 T^{5} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{6} - 1960903 T^{5} + \cdots + 97\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2606146 T^{2} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 9377989 T^{5} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{6} + 25814913 T^{5} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( (T^{3} - 209250 T^{2} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 2944308 T^{2} + \cdots - 85\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 48391269 T^{5} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{6} - 102186411 T^{5} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{6} + 144220135 T^{5} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{6} - 280936871 T^{5} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{6} + 170710399 T^{5} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{3} + 469758688 T^{2} + \cdots + 12\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 613838539 T^{5} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{6} + 197445809 T^{5} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{3} - 1074181436 T^{2} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 805730427 T^{5} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{3} + 2262918094 T^{2} + \cdots + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
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