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Properties

Label 112.4.p.g
Level $112$
Weight $4$
Character orbit 112.p
Analytic conductor $6.608$
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] = [112,4,Mod(31,112)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.p (of order \(6\), degree \(2\), 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: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.12258833328.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 2 \beta_{3}) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 9 \beta_{3} - 4) q^{7} + ( - 2 \beta_{4} - 25 \beta_{3} + \cdots - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 2 \beta_{3}) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 9 \beta_{3} - 4) q^{7} + ( - 2 \beta_{4} - 25 \beta_{3} + \cdots - 25) q^{9}+ \cdots + (8 \beta_{5} + 23 \beta_{4} + \cdots + 113) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} - 15 q^{21} + 15 q^{23} + 306 q^{25} - 362 q^{27} - 348 q^{29} - 205 q^{31} - 471 q^{33} + 1185 q^{35} - 249 q^{37} - 288 q^{39} + 2118 q^{45} - 75 q^{47} + 702 q^{49} + 2505 q^{51} - 645 q^{53} - 918 q^{55} - 6 q^{57} - 321 q^{59} - 1707 q^{61} + 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} + 555 q^{77} + 3447 q^{79} + 225 q^{81} - 24 q^{83} + 3786 q^{85} - 3642 q^{87} - 2607 q^{89} + 2448 q^{91} - 2991 q^{93} - 2085 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\nu^{5} - 841\nu^{4} + 1629\nu^{3} - 23432\nu^{2} + 19488\nu - 428592 ) / 45520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{5} + 989\nu^{4} + 5459\nu^{3} + 30793\nu^{2} - 82172\nu + 596328 ) / 68280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -203\nu^{5} + 197\nu^{4} - 5713\nu^{3} - 986\nu^{2} - 159176\nu - 4176 ) / 136560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -133\nu^{5} + 1012\nu^{4} + 4792\nu^{3} + 24959\nu^{2} + 35804\nu + 543504 ) / 68280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 581\nu^{5} + 221\nu^{4} + 16351\nu^{3} + 2822\nu^{2} + 481472\nu + 11952 ) / 45520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 3\beta_{3} - 2\beta_{2} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{5} + 2\beta_{4} - 111\beta_{3} - \beta_{2} - 11\beta _1 - 111 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{4} + 29\beta_{2} + 46\beta _1 - 51 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 116\beta_{5} - 11\beta_{4} + 1029\beta_{3} + 22\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 396\beta_{5} - 1642\beta_{4} + 1851\beta_{3} + 821\beta_{2} - 425\beta _1 + 1851 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.426664 − 0.739004i
2.68858 − 4.65676i
−2.61524 + 4.52973i
0.426664 + 0.739004i
2.68858 + 4.65676i
−2.61524 − 4.52973i
0 −3.53129 + 6.11637i 0 1.05999 − 0.611983i 0 −16.5026 + 8.40621i 0 −11.4400 − 19.8147i 0
31.2 0 2.38417 − 4.12950i 0 14.6315 − 8.44749i 0 8.89981 + 16.2417i 0 2.13148 + 3.69182i 0
31.3 0 4.64712 − 8.04905i 0 −17.1915 + 9.92549i 0 −18.3972 − 2.13127i 0 −29.6915 − 51.4271i 0
47.1 0 −3.53129 − 6.11637i 0 1.05999 + 0.611983i 0 −16.5026 − 8.40621i 0 −11.4400 + 19.8147i 0
47.2 0 2.38417 + 4.12950i 0 14.6315 + 8.44749i 0 8.89981 − 16.2417i 0 2.13148 − 3.69182i 0
47.3 0 4.64712 + 8.04905i 0 −17.1915 − 9.92549i 0 −18.3972 + 2.13127i 0 −29.6915 + 51.4271i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.4.p.g yes 6
4.b odd 2 1 112.4.p.f ✓ 6
7.c even 3 1 784.4.f.g 6
7.d odd 6 1 112.4.p.f ✓ 6
7.d odd 6 1 784.4.f.h 6
8.b even 2 1 448.4.p.f 6
8.d odd 2 1 448.4.p.g 6
28.f even 6 1 inner 112.4.p.g yes 6
28.f even 6 1 784.4.f.g 6
28.g odd 6 1 784.4.f.h 6
56.j odd 6 1 448.4.p.g 6
56.m even 6 1 448.4.p.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f ✓ 6 4.b odd 2 1
112.4.p.f ✓ 6 7.d odd 6 1
112.4.p.g yes 6 1.a even 1 1 trivial
112.4.p.g yes 6 28.f even 6 1 inner
448.4.p.f 6 8.b even 2 1
448.4.p.f 6 56.m even 6 1
448.4.p.g 6 8.d odd 2 1
448.4.p.g 6 56.j odd 6 1
784.4.f.g 6 7.c even 3 1
784.4.f.g 6 28.f even 6 1
784.4.f.h 6 7.d odd 6 1
784.4.f.h 6 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 7T_{3}^{5} + 104T_{3}^{4} - 241T_{3}^{3} + 5216T_{3}^{2} - 17215T_{3} + 97969 \) acting on \(S_{4}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 7 T^{5} + \cdots + 97969 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 168507 \) Copy content Toggle raw display
$7$ \( T^{6} + 52 T^{5} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} + 99 T^{5} + \cdots + 4876875 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 2167603200 \) Copy content Toggle raw display
$17$ \( T^{6} - 9 T^{5} + \cdots + 710833347 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 1106959441 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 8915001507 \) Copy content Toggle raw display
$29$ \( (T^{3} + 174 T^{2} + \cdots - 4622400)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 2349411462841 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 14539922265625 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 38591270836992 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 83474849412288 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 20686781241 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 825904716849 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 912934776095523 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 34\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 48\!\cdots\!03 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 65\!\cdots\!23 \) Copy content Toggle raw display
$83$ \( (T^{3} + 12 T^{2} + \cdots - 308212992)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 60\!\cdots\!03 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 19\!\cdots\!08 \) Copy content Toggle raw display
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